LOYAL- AV-MQRL

JOHN LANGTON

From tKe

ESTATE OF JOHN LANGTON

to tKe
UNIVERSITY OF TORONTO

1920

PRACTICAL LAWS AND DATA

ON THE

CONDENSATION OF STEAM

IN

COVERED AND BARE PIPES

TO WHICH IS ADDED
A '

TRANSLATION OF PECLET'S "THEORY AND EXPERI-
MENTS ON THE TRANSMISSION OF HEAT
THROUGH INSULATING
MATERIALS."

O

CHARLES P. PAULDING, M.E.

The D. Van Nostrand Company

intend this book to be sold to the Public
at the advertised price, and supply it to
the Trade on terms which will not allow
of discount.

Copyright, 1904,

BY
D VAN NOSTRAND COMPANY.

PREFACE

The main object of this book is to bring to the attention of
engineers an accurate and rational method of estimating the loss
of heat from steam pipes and boilers covered with any of the well
known non-conducting materials now on the market. This
method worked out long ago by P6clet seems to have been
generally overooked in this country, perhaps because no trans-
lation of Peclet's work existed.

The principles involved are so general in their application
that it is believed that the fuller explanation of them, to be found
in the appended translation, together with the experiments on
which they are based, will be of some general interest. This
matter will be of use to heating engineers in cases which are not
taken care of by their usual rules, and of great value as an aid
to the cultivation of that broad view of a problem so necessary
to practical success. To the refrigerating engineer and designer
of cold storage warehouses these principles are indispensable.

CONTENTS

Preface

Loss of Heat from Covered Steam Pipes —

Values of K. Method of determining the value of
the co-efficient of conductivity C from an experi-
ment. Barms' T^ests. Jacobus' Tests. Brill's
Tests. 130 Ibs. Steam Pressure Test. Norton's
Tests. Summary of Tests. Applications of the
theory. Table for use with the Formulas on page 7, 1-19

Loss of Heat from Bare Steam Pipes, ..... 20-28

Chapter I — Emission and Transmission of Heat —

Emission of Heat from a surface maintained at a
constant temperature, , . 29-46

Chapter II — Transmission of Heat Through Solid
Bodies —

Conductivity of metals. Tables of values of C.
Solid material. Material in a State of Powder.
Textile materials, 47~7o

Chapter III — Applications of the Formulas —

Discontinuous Walls. Transmission of Heat
through Cylindrical Envelopes. Transmission of
Heat through Spherical Envelopes. Diffusion of
Heat. Influence of the variations of exterior tem-
perature on the quantity of heat transmitted through
walls. Intermittent Heating. Heat lost by walls
during the suspension of Heating. Temporary
Heating of a Room, 71-100

Notes on the Use of the Formulas, 101, 102

THE LOSS OF HEAT FROM COVERED STEAM

PIPES

In the last few years a number of elaborate and careful tests
have been made on various coverings for steam pipes. These
tests have been made for the intending purchaser with the point
of view of ascertaining the most efficient of the pipe coverings
regularly on the market. They have no doubt served this pur-
pose admirably, but it has occurred to the writer that they might
also be made to serve the purpose of guiding the manufacturer as
to the true values of the materials used in the coverings, and to
aid the engineer in computing the losses from pipes already cov-
ered, or the saving to be effected by increasing the covering or
using a more efficient material.

The true value of the different materials used in the tests
has been notably obscured by the different thicknesses, and no
law governing the general subject has been even hinted at in the
published reports of the tests.

As long ago as 1850 the great French physicist Peclet (Pec-
let-Traite de la Chaleur — Paris, 1860) investigated with wonder-
ful skill and patience the laws of the emission of heat from a
surface maintained at constant temperature, and the laws of con-
duction of heat through materials of low conductivity. His
experiments, though on a small scale and strictly laboratory
ones, were so cleverly planned and skillfully executed that the laws
empirically deduced from them could hardly fail to be correct.

The loss of heat from covered pipes is only one of their many
practical applications.

When an iron steam pipe, of customary thickness, covered
with material of low conductivity, is filled with steam at rest or
moving with ordinary velocity, the amount of heat escaping
through the covering is so small compared with what the metal
of the pipe could transmit, that the outer surface of the pipe
attains the same temperature as the steam within it.

Leaving the surface of the pipe, the heat is transmitted

2 Loss of Heat from Covered Steam Pipes

through the cylindrical covering by virtue of the conductivity of
that covering, and when it has reached the surface it is dissipated
to the surrounding objects and the surrounding air by radiation
and contact of air.

We have then two phenomena to deal with, the conduction
of the heat through the covering and its escape from the surface
of the same.

For the latter P£clet gives the following laws : ' ' The quan-
tity of heat emitted by a surface at constant temperature depends
on the radiation and the contact of air. "

"The quantity of heat emitted by radiation, per square foot
of surface, per hour, is independent of the form and size of the
body, provided that it has no reentrant portions. It depends
solely on the nature of the surface of the body, on the excess of
its temperature over that of the objects to which radiation takes
place, and on the absolute value of the temperature of these
objects."

For paper and cloth, P6clet found that color had no influence
on the radiation.

The following table gives the values found by him for radi-
ation from different surfaces :

VALUES OF K.

B. T. U. PER HOUR PER SQUARE FOOT PER ONE DEGREE.

Tin plate 086 Cast iron, new . . . .650

Polished sheet iron . . .092 " " rusty . . . .688

Ordinary " " . . .567 Sheet iron, rusty . . .688

Oil paint " " . . .759 Paper 772

Plaster or wood . . . .737 Calico or canvas . . .747

The coefficients by which these numbers must be multiplied
for any excess of temperature are given in figure 2 . The tem-
perature of the objects radiated to would generally be the same
as that of the surrounding air. It is taken so in all calculations
throughout this article.

The coefficients by which the numbers in the table must be
multiplied for any given temperature of the objects radiated to
are given by figure 3.

For an example take a covering on a hot steam pipe. The
surface of these coverings is invariably formed by canvas.

Loss of Heat from Covered Steam Pipes

Consider a square foot and let its temperature be 125° F.,
and that of the surrounding objects and air 85° F. , then the excess
of temperature will be 40 degrees.

The loss per square foot per hour due to radiation will be,
RY. 40=(-747X i.iyX 1. 1 2) (40) =39.2.

"The loss of heat arising from air contact is independent of

Coeff. for air contact.

o» bo o Vo £* b» oo

•— —

=

— -

1.8
1.15
1.4
LS
1.0

&

.6

==»•

— —

•»—

— —

[— —

^**

*~~

*--

--"

*•

•^

X-

^

Fig. 1
AIR CONTACT

f

X

/

/

t

/

/

20 40 60 80 100 150 200 250 300 350
Diff. of temp, between surface and surrounding air.

Radiation Coeff. Radiation Coeff.

o to '>&. 'o bo o io ',&. CT co o

3.0
2.8
2.6
2.4
2.2
2.0
1.8
1.0
1.4
1.3
1.0

/

Fig. 3
RADIATION

/

^

J

,/

x

/J

X

X1

x

•^

/

^

*

X

s

s

x

2

0

4

0

6

0

8

)

100^

<S

150

^<

K)

^

X

T

emp

.Ots

^llj

)C

s

ra

diati

jd

t<

J

,

X*

^

**

|

.^

'

X

^

f**

1*

^j

^

^

**

**"

^

^

^x-

'

^

^ *"

. —

Fig. 2
RADIATION

^

^>

^-*

^

:

~~-

i —

.-— '

.*- -*

20 40 60 80 100 150 200 250 300 350
Dlff. of Temp, between surface and objects radiated to.

the nature of the surface of the body, and of the absolute temper-
ature of the surrounding air ; it depends solely on the excess
of temperature of the surface of the body over the temperature
of the surrounding air, and on the form and dimensions of the
body."

Figure 4, for horizontal cylinders, and Figure 5 for vertical

Loss of Heat from Covered Steam Pipes

Values ot.K

|«

to co '—

Loss of Heat from Covered Steam Pipes

^ JO S9tlITJA

6 Loss of Heat from Covered Steam Pipes

ones, give the values of the heat loss by air contact per square
foot per hour per one degree.

Figure i gives the coefficients by which these values are to
be multiplied for any given difference of temperature between the
surface of the body and the air.

Assume that the outside diameter of the covering in the ex-
ample used above for radiation loss is six inches, and that the pipe
is horizontal. Then under the same conditions, a difference of
temperature of 40 degrees, the air contact loss is
A X 40= (.52 X 1.13) (40) = 23.5-

The combined loss for this square foot of surface of covering
under the conditions given is therefore,

39.2 + 23.5 = 62.7 B. T. U. per hour.

In a test the temperature of the air should be measured be-
fore the air reaches the heated covering and the thermometer
should be protected from radiation from the covering.

We may now consider the conduction of the heat through the
covering.

The law for a flat plate of insulating material is very simple ;
the quantity of heat transmitted per square foot per hour varies
directly as the conductivity of the material, inversely as its thick-
ness, and directly as the difference of temperature between the
two surfaces of the plate.

Note that it is the temperatures of the surfaces of the plate,
not of the air in contact with the surfaces.

The formula for a flat plate is then,

.

E
where / and ( are the surface temperatures,

C the coefficient of conductivity,

E the thickness in inches, and

M the heat transmitted in B. T. U. per square foot per hour.

For a cylinder, the principle is the same but the expression
changes. Consider a section one foot long.

lyet R and R be the inside and outside radii, in feet, of the
cylinder of insulating material,

/ and f the respective surface temperatures, and

0 the temperature of the surrounding air.

Loss of Heat from Covered Steam Pipes 7

Consider an infinitely thin annular element of the covering
at radius r, its thickness is dr, its area, 2*r, its conductivity for
one foot thickness is C' and the difference between the temper-
atures of its inner and outer surfaces is dt. Then treating it as
a flat plate we have Mt_ 2*r C' dt

1VJ. - ~

dr
Integrating we have M'= — \. — - . . . . . (i)

where N=2. 3 (log R' — log R}.

Now we have already shown that for a square foot of surface
M=(A + R) (/' — 0) and remembering that M'=2itR' M and
calling A -f- R= Q we have

M'=2KR' Q (/'— 0) ....... (2)

The amount of heat passing through the covering must equal
that escaping from the surface, so setting (i) equal to (2) we get

i + QR' N

C

It is more convenient to use C the coefficient of conductivity
for one inch thickness instead of C the coefficient for one foot.
Since C= 1 2 C' we get

QR' N ...... (4)

C

It is really more convenient to keep R' in feet. In working
up the value of N=2.^ (log R — log R' ) from a table of loga-
rithms we can, of course, take R and R' in inches if we wish.

For an example take a lo-inch pipe, covered with a thickness
of i -j^- inches of Keasbey's Magnesia; steam temperature 365.2°
F.; air temp. 66° F.; C=.^.

The difference between the logarithms of the inner and outer
radii is .087 X 2.3=.2=7V.

We cannot yet compute the exact value of Q because we do
not know the temperature of the surface of the covering, but we
will assume it to be i .7.

Then ^=3.44X1.7(365.2-66) _

*

i+/..7X.548X.«\
\ -45 /

45

TABLE FOR USE WITH THE FORMULAS OF

PAGE 7

NOMINAL
PIPE SIZE

THICKNESS
3F COVERING
IN INCHES

RADIUS
IN FEET

«(*§
< ftj

N

to

M

NOMINAL
PIPE SIZE

THICKNESS OF
COVERING IN
INCHES

RADIUS IN
FEET

«H
wx

fc°5=

w*g

<&"

a0'

(A

N

H

R'
"

27TR'

i-> —

R'

2*R'

o

.356

o

.078

|

.097

.609

•567

.660

%

. 121

• 759

•458

.665

^£

.108

.676

.669

.864

H

•132

.828

•521

.823

x/

.118

.-38

7s8

y*

. 142

.890

• 594

I.OOQ

i"

i

.128
.138

.806
.870

.847
.921

1.300
1.530

Ij4"

•'53
.163

• 959
1. 022

.668
•731

• ^^7

I. 22O
I.425

i/4

.149

• 937

•997

1.785

i^s

• 173

1.090

• 796

1.655

I}i

.159

1. 000

1.062

2.028

iji

.183

1.152

• 852

1.875

'ft

.170

1. 068

1.128

2.302

iji

• 194

i. 220

.910

2. 120

I*

.180

1.130

1.185

2.560

us

.204

1.283

•959

2-35°

Q

.62

o

. 120

!i4i

.89

• 351

•593

.162

i. 02

.298

•579

8

.152

• 95

.426

• 775

5^

.172

i. 08

• 358

• 738

y±

.162

1.02

.489

• 950

Ji

.183

I-I5

.419

.918

2"

i

.173
.183
.193

I. 08

1.15
1.22

• 554
.610
.668

I-I45

I:§i

2^"

I

.194
.204
.213

1.22
1.28

1-34

.477
.528
• 574

1. 106
1.290
1.469

i &

.204

1.28

.722

1.767

*}i

.223

1.40

.620

1.651

19»

.244

1-35

.770

1.976

i%

• 234

1.47

.667

1.872

i%

.224

1.41

.814

2.190

i«

• 245

i-54

•713

2.093

o

. 146

2

o

.188

1.18

.188

l!i8

•251

• 565

.229

1-44

.201

•553

8

• 199

1.25

.308

• 734

ft

.240

1.51

.247

.710

Yi

. 208

I.M

.356

.890

%

.250

1-57

.288

.864

3"

i

.219
.229

1.38
1-44

.407
-451

1.070
1.240

4"

i

.260
.271

1.64
1.70

•332
.368

1.038
1. 196

ij£

.240

I-5I

•497

1.430

1/8

.282

1.77

.407

1-374

'If

.250
.261

1-57
1.64

;Jg

1.614
1.815

I|j

.292

.303

1-83
1.90

.442
.478

1-545
1-736

IJi

.271

1.70

!6i8

2.010

I«

• 313

1.96

• 5"

I-9I5

o

212

1.46

O

.276

1-73

• *j"
•273

1.72

.165

•541

" / v

.318

2. OO

. 140

-534

ya

.284

1.79

.203

.692

$£

.328

2.06

• 173

.682

y*

• 294

1.85

.238

.840

54

.338

2.13

.203

.824

S"

i

•SOS
•3'5

1.92
1.98

•275
• 307

I. 006

1. 160

6"

i

• 349
• 359

2. 19
2.26

.235
.263

.984
I.I33

1/4

.325

2.04

• 338

1.318

i/4

• 370

2.32

• 293

1.300

iJi

•336

2. II

• 372

1.500

i&

• 380

2-39

.320

1.460

lf$

•347

2.18

.402

1.671

1^6

•391

2-45

•348

1.630

i«

• 357

2.24

•432

1.847

IJi

.401

2.52

•373

1-795

Q

ISO

2.26

o

. ddQ

2 82

• jjy
.401
• 412

2.52

.no
.137

.529
.678

N

* *T*t7

.490
.501

3.08
3-15

.088

.110

:&

y+

.422

2.65

.161

.815

3i

•5"

3-21

.130

.796

8"

i

• 433
• 443
• 453
• 463
• 474
• 484

2.72
2.78
2.84
2.91
2.98
3-04

.186
.209
.232

•255
.278
.299

.965
i. no
1.260
1.426
1.583
1-738

10"

i

.522
• 532
• 542
.552
.564
• 574

3-28
3-34
3-41

3-47
3-54
3.61

.150
.170
.190
.209
.228
.247

.940
1.085
1.236

1.385

1.700

Loss of Heat from Covered Steam Pipes

Then, by (2)

293 = 3-44 X 1.7 (/ — 66),

t =116. Then 1 16 — 66 = 50.
We may now find Q for 50° difference of temp.
^=.75 X 1.2 X i. 02 = .916
A = .47 X 1.2 =-565

1.48
Then M> = - 3.44X1.5(299) = 28

/i.5X.548X.2\

+ I - J

V -5 /

I2

-45
and t = 121.5

It is evidently unnecessary to make another and closer
approximation to Q.

It will be noticed that a difference of about twelve per cent.
in the value of Q only made a difference of two per cent, in the
results. So it is quite unnecessary to be too particular about the
value of Q, and the smaller the pipe the less effect does an error
in Q have.

The following values of C for different materials were deter-
mined by Peclet :
Plaster ...... 3.44 Hempen Canvas. . . -4i8-

Oak . . . . . . . 1.70 Smooth White Paper . .346

Walnut ....... 86 Cotton Wool ..... 323

Fir ........ 75 Sheep Wool ..... 323

Powdered Charcoal . . '637 Eiderdown ..... 314

Wood Ashes ..... 484 Blotting Paper . . . .274

As far as the theory goes it evidently makes no difference
whether the steam in the pipe is at rest or in motion, for the
inner surface of the covering is at the same temperature in either
case; namely that of the steam. Mr. Barrus made tests at the
Manhattan Railway Co.'s new power house, in which he found
the same rate of condensation from a covered pipe whether the
steam was at rest or moving with a velocity of 18 feet per sec-
ond. He also found this to be true for a bare pipe. (Power ;
Dec., 1901.)

As this theory is of such old origin it seemed best to apply it
to a number of recent pipe covering tests and see how well calcu-
lations made by it would agree with their results.

Loss of Heat from Covered Steam Pipes 9

Of course for any given pipe covering, we have to know the
value of C, the coefficient of conductivity. This we can obtain
by analyzing a test of the particular covering, then we can test
the theory by calculating the loss of heat for the conditions ob-
taining in some other test of the same covering under different
steam pressure, on a different sized pipe and with a different
thickness of covering.

Or we may analyze this second test and determine C. If
this is the same as the C of the previous experiment, we know
that our calculation would have given the same loss of heat as
the experiment.

It is this latter method that has been used here, and in this
way a table of values of C for nearly all the well known cover-
ings now on the market has been obtained.

By the aid of the formulas just given and this table, one
may calculate the loss of heat from a covered pipe or boiler
under any conditions.

METHOD OP DETERMINING THE VAL,UE OP THE COEFFICIENT OP
CONDUCTIVITY C FROM AN EXPERIMENT.

As an example we will take Mr. Barrus' test on Keasbey's
Magnesia on a 2-inch pipe with i inch thickness of covering, and
155 B. T. U. lost per sq. ft. of pipe surface per hour.

Temperature of steam, 365.2° F.

Temperature of air, 64.6° F.

The B. T. U. per foot run of pipe = ^~- = 96.2.

1.61

The surface of covering in sq. ft. per foot of pipe = 1.15.

For a first approximation, take Q = 1.7.

Then 96.2 = 1.15 X 1.7 (/ — 64.6), (Eq. 2) and /, the tem-
perature of the surface of the covering, equals 114°. The differ-
ence between this and the temperature of the air, 64.6°, is 49.4
degrees. We can now make a closer approximation to Q as fol-
lows : For a canvas covering, K = .75, and for a cylinder 4^
inches in diameter, 1C = .56 from Figure 4.

Then for a difference of 50 degrees Fahrenheit.
K = -75 X 1.2 X 1.02 = .92

A = .56 x 1.2 = AT

io Loss of Heat from Covered Steam Pipes

The above coefficients by which we have modified K and A'1
are taken from Figures i , 2 and 3 for a difference of 50 degrees,
and an air temperature of 65 degrees.

Then 96. 2 = 1. 15 X 1.7 (/' — 64.6) and/'=ii7. 117 — 64.6 =
52.4, which is so near 50 that we may take the value of 1.6 for
Q as final.

We now know the temperatures of the inside and outside of
the covering. Transposing our fundamental equation for the
conduction of heat through a cylindrical covering, (Eq. i) we
get

c= M'XN

2 TT X difference of temps.

The temperature of the inside of the covering we know to
be that of the steam, 365.2° F. ; the temperature of the outer
surface we have found to be 117; the difference is therefore
248.2°.

„ 96.2 X .61

C=6.28X 248.2 XI2==-453

All the tests have been analyzed in this way and the value
of C determined.

BARRUS' TESTS.

These tests are on a far larger scale than any previously
made. Every precaution was taken to secure accuracy. The
final results have not yet been published, but a preliminary
description was published in Power of December, 1901, and the
average of the maxima and minima condensations there given
are used here.

Through the kindness of Mr. Barrus, I am able to say that
those of these figures used here stand practically correct and the
temperatures, thickness of coverings and areas of pipe surface
have been given me by him for this paper.

The steam pressure was 150 pounds.

The pipes to which the covering were applied were 2 -inch
about 100 feet long, and lo-inch about 35 feet long.

The length of each test was about nine hours, but the tests
were repeated day after day for a number of days, and the figures
given are the average.

The tests were made in 1901.

B. T. U.

Com-

Thickness
Size of of
Pipe. Covering.

Temp, of
Steam,
Degs.
Fahr.

Temp, of
Air,
Degs.
Fahr.

per sq. ft.
of Pipe
Surface
per hour.

puted
Value
of
C.

2" I#"

364.8

60.7

145

•451

10" !?/%"

364.8

62.8

85

•413

2// j//

365-2

64.6

155

•453

10 " 1^"

365-2

66.0

103

•459

d 2" itf"

365-2

64.6

I76

•570

to" ift"

365-2

66.8

112

•579

Loss of Heat from Covered Steam Pipes 1 1

Name of
Covering.

Asbesto- Sponge
Felted
Do.
Magnesia

Do.

Asbestos Navy Brand -2"
Do.

The covering was measured before being applied to the pipes.
The temperature of the steam is that corresponding to the
pressure observed by a gauge.

In view of the fact that each figure in this table is an aver-
age of a number of tests, and considering the length of the test
pipes and the care with which the experiments were made, we
may confidently assert that having determined the value of C
from an experiment with a given covering we may therefrom
compute the loss of heat from any sized pipe with any thickness
of covering, provided the steam pressure is the same as that
obtaining during the experiment.

JACOBUS' TESTS.

An account of these tests by Professor Jacobus is to be found
in the STEVENS INSTITUTE INDICATOR for July, 1901.

Thickness
Name of of

No.
of

Steam
Pressure,

Temp, of
Steam,
Degs.

Temp, of per. sq. ft.
Air, of Pipe
Degs. Surface

puted
Value,
of

Covering. Covering.

Tests.

I,bs.

Fahr.

Fahr.

per hour.

C.

Hair Felt

.96 "

2

55-4

3O2.8

71,4

89.6

.T.2

I. P. Remanit.'

.88"

6

57-2

304.5

73-3

17

100.3

o

•34

H. P. Remanit

1.3"

7

59-5

306.6

76.1

83-7

•37

Asbestos Sponge

Felted i

• H"

T.

62.0

3OQ.2

7Q.4-

so. 7

.7Q

Magnesia i

.08 "

•j
•I

64.2

*J 7

^lO.Q

/ y*T

81.6

oy* t

60.8

oy

AC

Asbestos Navy

*j

Vfaf. «

o*-^"y

W7

•HO

Brand i

.20"

•I

62.0

1.OQ.2

7Q.4

6Q.Q

.48

A-s-b-e-s-t-o-c-e-l..i

.07"

\j

I

54-2

\j ?
301.8

/ y*r

77-2

7 J

143.0

•*T"

.61

Asbestos 'Air Cell..

.96"

I

55-9

303-3

72-3

165-5

•67

Asbestos Fire Felt

•99"

2

60.2

307-4

72.5

1 80.0

•74

The steam pressure was from 55 to 75 pounds.

The tests pipes were standard 2 -inch pipes, 12 feet long.

The tests were about four hours long, but the figures here

1 2 Loss of Heat from Covered Steam Pipes

given are averages of several tests in most cases. The tests were
made in 1901.

The values of C were for the most part worked up from the
separate tests and averaged.

The "Remanit" coverings are of German origin. They
were encased in canvas.

In the case of the "Hair Felt," a layer of asbestos paper -£%
of an inch thick was first bound around the pipe, over this was
bound the hair felt, then a layer of paper, and outside of all a
canvas covering.

BRILL'S TESTS.

These tests were published in Vol. XVI. of the Transactions of
the American Society of Mechanical Engineers.

The steam pressure was 1 10 to 117 pounds.

The test pipe was a standard 8-inch steam pipe about 60 feet
long.

The tests were about four hours long and the figures given
here are in each case the average of three tests.

The tests were made in 1894 or J895.

B. T U. Corn-
Temp, of Temp, of per sq. ft. puted

Thickness Steam Steam, Air, of Pipe Value
Name of of Pressure, Degs. Degs. Surface of

Covering. Covering. I,bs. Fahr. Fahr. per hour. C.

Magnesia i.25x/ no 344.1 66.3 106.6 .527

Rock Wool i. 60" no 344.1 63.0 72.1 .395

Mineral Wool I-30" no 344.1 58.3 81.3 .381

Champion Mineral Wool.. 1.44" 113.3 346-1 74-3 86.1 .470

Asbestos Fire Felt 1.30" in 344.7 79 133.5 .755

Manville Sectional 1.70" 112.3 345.5 78.3 • 93.4 .600

Hair Felt 82" 117 348.3 69 117.9 -396

Riley Cement 75" 116.3 347-9 74-3 260.8 1.200

Fossil Meal 75" "5 347-1 75-3 238-8 1-050

The ' ' Riley Cement ' ' and ' ' Fossil Meal ' ' were mixed with
water and plastered on the pipe.

The ' ' Hair Felt ' ' was bound very tightly around the pipe
and had no canvas covering. It had a layer of asbestos paper
under it. Its tightness would increase its conductivity, and fur-
thermore it had no canvas covering, which throws a doubt as to
what value to asign to K for the radiation. This would be suf-
ficient to explain the increased value of C over that found in Prof.

Loss of Heat from Covered Steam Pipes 13

Jacobus' tests without assuming a different quality of "Hair
Felt."

The three coverings, "Rock Wool," " Mineral Wool " and
"Champion Mineral Wool," are all mineral wools, and it is in-
teresting to note that the chemical composition of the first two is
almost identical, both having about 18 per cent, of magnesia,
while the " Champion Mineral Wool " has only 3 per cent.

130 POUNDS STEAM PRESSURE TEST.

The results of this test have not previously been published.
The writer was personally connected with it and worked up the
original report.

The test pipes were 2-inch standard steam pipes 80 feet long.

The test was forty -eight hours long, and was made in 1896.

B. T. U. Computed

Thickness per sq. ft. Pipe Value

Name of Covering of Covering. Surface per hr. of C.

Magnesia 1.09" 155.8 .534

Manville 1.31" *57-° -606

Asbestos Fire Felt i.oo" 198.0 .680

The average steam pressure was 128.7; temperature corre-
sponding, 354.7 ; average temperature of air 80. i.

NORTON'S TESTS.

These tests were published in the Transactions A. S. M. E.,
Vol. XIX. They were made in 1896-1897.

They are interesting inasmuch as the pipes under test were
filled with oil, heated by an electric current passing through a
coil. The oil was agitated by two small propellers.

We have already shown that the outside of a covered pipe
carrying steam takes the temperature of the steam, and in these
experiments it would take the temperature of the oil, and for equal
temperatures the loss of heat would be the same. The following
computed values of C bear out this opinion fairly well :

B. T. U. Corn-
Temp, of Temp, of per sq. ft/ puted
Size Thickness Steam, Air, Pipe Value

of of Degs. Degs. Surface of

Name of Covering. Pipe. Covering. Fahr. Fahr. per hour. C.

Magnesia 4" 1.12" 388 72 147 .52

Manville 4" 1.25" 388 72 143 .55

Asbestos Air Cell... 4" 1.12" 388 72 166 .60

The corresponding steam pressure would be 201 pounds.

14 Loss of Heat from Covered Steam Pipes

Two test pipes were employed, one 4 inches in diameter, and
one 10 inches; both were vertical and 36 inches in length.

In the calculations in this case, the value of K' was taken
from Figure 5, instead of from Figure 4, as in the other tests.

Further experiments with the lo-inch pipe with great thick-
nesses of coverings were vitiated by the fact that the ends of the
pipes emit heat, according to a very different law from that obtain-
ing for the cylindrical portion, and further, the ends being covered
by a thickness equal to that on the cylindrical portion the length
of cylindrical surface is increased. This seriously impairs the
value of Mr. Norton's conclusions as to the relative advantage of
increased thickness of covering.

Mr. Norton also made a number of tests on the loss of heat
from his test pipes without any covering.

An analysis of these reveals inconsistencies which are easily
explained by a study of Pellet's experiments on the conductivity
of metals. Mr. Norton's agitating arrangement though sufficient
for the slow loss of heat from a covered pipe was quite inadequate
for the loss from a bare pipe and the results of his experiments
were falsified by the resistance of the oil. We may however draw
certain general conclusions that are useful.

By exposing one of his bare pipes to the draft from an elec-
tric fan the loss of heat was increased by about 50 per cent. The
radiation would be unaffected by the draft, but the air contact
loss, which in still air was about 40% of the total loss, must have
been increased by 120% or to more than double its value in
still air.

Now if we take the case of the magnesia covering in still
air we have by the experiment a loss per sq. ft. of pipe of 147 B.
T. U. per hour. In this case the value of A' (radiation) was .96
and K' (air contact) .73. If we increase K' by 120% we have
K' equal to 1.6, A' remains .96 as before, and the computed value
of the loss per square foot of pipe would become 166, an increase
of 13%.

Mr. Norton stated during the discussion that in no case had
he been able to increase the loss with any of the covered pipes
more than 10% by the draft from an electric fan. This agree-
ment between our calculation and his experiment gives us some

Loss of Heat from Covered Steam Pipes 15

idea of what allowance to make for any covering when in a posi-
tion exposed to strong drafts.

SUMMARY OF TESTS

TABLE OF VALUES OF C

Name of Test.

Barrus

Barrus

Jacobus

Brill

130 Ibs

Norton

Year of Test

1901.

2 ins

100 ft

150 Ibs

•45
•45

•57

1901.
10 ins
Soft
150 Ibs

I

.41
.46

.58

1901.

2 ins

12 ft

55-75 Ibs
^alues oj

•32
•34

•37

•39

•45

.48
.61

.67
•74

I895.
8 ins
60 ft
110-117 Ibs

• c.

.40

.38
.40

•53

•47

.60

.0

.76
1.05

1.20

1896.

2 ins
80 ft
I29lbs

•53
.61

.68

1896.
4 ins

3ft
200 Ibs

•52

•55
.60

Size of Pipe

Length of Pipe, about....
Pressure of Steam

Name of Covering
Hair Felt

Int. Press. Remanit

High Press. Remanit
Mineral Wool

Rock Wool

Asbestos Sponge F'l'd....
Magnesia

Champion Min. Wool
Asbestos, Navy Brand
Manville Sect

A-s-b-e-s-t-o-c-e-l

Cast's Ambler Asbestos
Air Cell

Asbestos Fire Felt

Fossil Meal

Riley Cement

Before analyzing the meaning of the results shown in this
table, it is best to recur to the method by which these figures
were obtained. We have already shown this using as an example
Mr. Barrus' test of magnesia covering on a 2 -inch pipe.

We will now assume an error of 5 per cent, in the condensa-
tion, then instead of 96.2 B. T. U. per foot run we will have 101.
Then 101 = 1.15X1.6 (/— 64.6)
and t' = 120 instead of 117.
101 X .61

Then C =-

12 = .481

6.28 X 245.2

We found before that C = .453. An error of 5% then in
the experiment causes an error of 6T27 per cent, in the computed
value of C.

Our object in examining these experiments was to prove
that Peclet's theory was in agieement with them, or perhaps it is
better to say that we wish to prove that, having determined by

1 6 Loss of Heat from Covered Steam Pipes

an experiment the value of C we were then in a position to com-
pute by Peclet's theory, the loss of heat from any sized steam pipe
under any pressure of steam, with any reasonable thickness of
the given covering, and with any usual temperature of external
air.

L,et us now examine the figures of the table.

Considering the large scale on which Mr. Barms' tests were
made, the number of tests taken to give an average figure, and
the care with which the tests were made, we are justified in hold-
ing that the theory is amply proven for different sized pipes, and
different thicknesses of coverings under the same steam pressure,
by the Barrus tests on "Asbesto- Sponge Felted," "Magnesia"
and "Asbestos Navy Brand." That the theory takes care of
any difference of steam pressure is proven first of all by Barrus'
and Jacobus' " Magnesia " and "Asbesto-Sponge Felted," sec-
ondly by Brill's, 130 Ibs'. and Norton's " Magnesia," by Jacobus'
and Brill's " Fire Felt," and by Brill 'sand 130 Ibs'. " Manville."

Each of these examples are additional proof that different
pipe sizes antl different thicknesses of coverings are perfectly
taken care of by the theory.

The only discrepancy that I think worth noticing is that be-
tween Barms' and Jacobus' "Asbestos Navy Brand."

There is quite a difference between the values of C for
"Magnesia" tested in 1895 and in 1901. I think it is fair to
assume that the material has been improved in that time.

The discrepancy in the case of ' ' Hair Felt ' ' has already
been explained. Prof. Jacobus' figure is no doubt the correct
one.

Pipe coverings are not absolutely homogeneous, and experi-
ments of any kind are seldom in perfect agreement. Taken
altogether the experimental proof is very strong.

APPLICATIONS OF THE THEORY.

Effect of Thickness and Conductivity. — Figure 6 shows clearly
the very great saving of heat that is obtained by even very mod-
erate thicknesses of coverings. It also shows how quickly the
economical limit of thickness of a covering is reached.

In regard to conductivity, we see that halving this for a cov-
ering one inch thick, a usual thickness for this size of pipe,

Loss of Heat from Covered Steam Pipes

reduces the loss from 150 to 85, a reduction of forty-three per
cent, instead of fifty. This is because the surface resistance to
loss is the same in both cases. In all these examples the cover-
ings are supposed to be surrounded by a thin canvas jacket.

It is of interest, though perhaps of no particular practical
value, to note that if we employ a pipe covering having a con-
ductivity greater than 6.00, which is not so very much greater
than that of some plaster, the loss of heat will increase with the
thickness of the covering, and may even be greater than for the
bare pipe. This is because the action of the increased surface
outweighs the feeble resistance of the covering.

B.T.U. per hour per ft. run

,

Fig, 6

I

EFFECT OF THICKNESS AND CONDUCTIVITY,

\\

STANDARD 2 PIPE.

STEAM PRESSURE 125 LBS

\

\

AIR TEMPERATURE 65°

\

\

\

\

\

X

\

^"^

^^

-^--

(

- —

V.

^^

c

w-

—

,

T^-T

-— _

1.5 2. 2.5

Thickness of covering in inches.

3.5

The Effect of Different Steam Pressures. — Figures 7 and 8
should prove of value in the judicious choice of thickness of cov-
ering for very high steam pressure on the one hand and for the
conveyance of hot water on the other.

The Effect of Different Sized Pipes. — A little consideration
will show that if, in our efforts to stop the escape of heat by an
increase of thickness, we at the same time present a larger area
for its passage, our gain will be but slight. This is precisely what
happens with thick coverings on small pipes, as shown in Figure
9. We see that a one-inch pipe requires one and three-quarters
inches of covering to keep the loss per square foot on its surface
the same as for a ten -inch pipe with only eight-tenths of an inch
of covering. This Figure is an excellent example of the danger
of interpreting a theory from a too narrow point of view. It evi-
dently indicates that coverings for large pipes should be thinner

18

Loss of Heat from Covered Steam Pipes

, '

•

— —

— —

— •

— -

r— --

_- — '

,,---

^~

^-^

Fig. 7

x

^-^

^-^"

EFFECT OF DIFFERENT STEAM PRESSURES.

/

STANDARD 2"p|PE. AIR TEMPERATURE 65'

/

/

COVERING lf6"THICK. C=.41

^

I

l<

50

100

150
Steam Pressure.

200

860

than for small ones. The usual practice of the manufacturers of
pipe coverings is directly opposed to this, and there are very good
reasons for their course.

In the first place the cost of a large pipe line warrants the
expenditure of sufficient money on the covering to have it of a

1.6
1.4

1
•gl-2

gl.O
M

r

.6
.4

^^^-*

-^

^

^ — '

^

^

^ — '

^ —

•^

^

^

^

^

f

ESSES
NT LO
1RENT

OT RUI
E. AIR

ig. 8

^

^

THICKN
CONST A
FOR DIFFE

SS PER FO
ARD 2"PIF

DF<

3S F
STE

I-""

JOVERI
ER FO
.AM PR

8 B.T.L
MPERA

NQ FOR
DT RUN
ESSURES.

1. PER HOI
TURE 65.°

/

/

/

LO
STAND

IR.
C=.41

/

/

TE.

/

/

/

/

z_

50 IflO 150 200 260

Steam Pressure.

liberal thickness, and thus to secure a low heat loss. Secondly
the bulk of the small pipes would be very objectionable in many
cases if they were so thickly covered and the expense of cover-
ing would be out of all proportion.

Loss of Heat from Covered Steam Pipes

Thickness of covering In Inches.

i-1 t+ r r r

CD O N *- O> OO

1

\

Fic

F COV
\RE FO
ERENT

PE SUF

-BS.

.9

ERI
OT
SI2

!FA
MR

^Q
OF
ES

CE

THICKNESSES C
LOSS PER SOU
FOR DIFF

=>ER SQ. FT. OF P
K PRESSURE 125 I

-OR
PIP
OF

- 1'

CONSTANT
E SURFACE
PIPES.

>5 B.T.U. PER
RATURE 65.°

\

LOSS
STEAH

HOUR
C=.41

\

TEMPE

\

\

\

S

\.

~~-.

— —.

—

— —

-

•^TT-

34567
Actual outside diain. of pipe.

10

The useful lesson to be drawn from Figure 9 is that the loss
of heat from small pipes will always be large in proportion to
their surface, and it is also to be remembered that the proportion
of surface to weight of steam passed is much greater for small
pipes than for large.

In conclusion it may be mentioned that the very important
quantities of durability and non- inflammability of a covering are
entirely outside of the scope of these notes.

NOTE.— The table at the end of the book makes the solution of practical problems by
the formulas of page 7 much simpler. It obviates the use of logarithms and much calcu-
lation for ordinary thicknesses of coverings on standard wrought iron steam pipes.

THE LOSS OF HEAT FROM BARE STEAM PIPES

In a previous paper it was shown that Peclet's theory of the
loss of heat from a surface maintained at constant temperature
gives results in agreement with extensive practical tests of the
loss of heat from covered steam pipes.

The purpose of the present paper is to show that the theory
is of the same practical value for bare steam pipes.

The subject is chiefly of value to manufacturers of pipe
coverings, since they are not infrequently required to guarantee
a certain saving by the use of their covering over the loss that
would take place with a bare pipe, and it was at the suggestion
of one of these gentlemen that this second paper was under-
taken.

In tests of pipe coverings, one of the test pipes is usually
tried bare, and the result used for determining the saving due to
the various coverings.

Any unreasonable claims on the part of the manufacturers are,
therefore, liable to prompt exposure, and some reliable method
of estimating the saving due to their covering under the particu-
lar circumstances of the test is evidently of value to them.

When a metal steam pipe of customary thickness is filled
with steam at rest or in motion at ordinary velocities there is a
constant escape of heat through the walls of the pipe. This heat
traverses the pipe by virtue of the conductivity of the metal and
on reaching the outer surface a portion is radiated to the sur-
rounding objects and the remainder is carried off by the contact
of the surrounding air.

The conductivity of all the metals is so high that we may
without perceptible error neglect this part of the process and
simply assume the temperature of the outer surface of the pipe
to be the same as that of the steam within . We need only consider
then the manner of the escape of heat from the surface.

Bare Steam Pipes

21

Coeff. for air contact.

OS 00 O IO t^ OS GO

-—

— •

i=*

^r

1.8
1.6
1.4
1.2
1.0
£

,— •

— —

•=~

—

— —

— •

— —

—• -

«~^

.— —

--•

--'

^

—

X

Fig. 1
AIR CONTACT

,,

X

/

/

j

>

I

1

I

20 40 60 80 100 150 200 250 300

Diff. of temp, between surface and surrounding air.

350

1.8

1.0
1.1

1.0
.8
1.6
1.4

1.2
1.0

^

Fig. 3
RADIATION

/

.

^

/

^x

x

'

X

t

X

^

x

'

x

/

x

,x

S

0

i

0

6

0

8

)

K

JOx

•s

150

21

0

4

x

1

eihp

of.

<

).i

1C

a

ni

iiat(

;<1

tc

)

^

X

^

(

^

'

x

'

^

^~

x

^

^

*>

^

•^

^

^*

^

.^

^*

Fig. 2
RADIATION

^

^

. —

— *

^--

^^*

-— -

, — •

—• — •

— -•

20 40 60 80 100 150 200 250 300 350
Dill', of Temp, between surface and objects radiated to.

2.S

2.15

2.1

2.2

2.0

1.8

1.0

22 Loss of Heat from Bare Steam Pipes

The loss of heat by air contact depends on the diameter of
the pipe, on its position, whether vertical or horizontal, and on
the difference of temperature between its outer surface and the
surrounding air. We may express it by the following equation :

Loss by air contact in B. T. U. per square foot per hour =
A" X C X diff. of temp., in which A" is the term affected by the
diameter and position of the pipe and is given by Figs. 4 and 5.
C is a coefficient determined by the amount of the difference of
temperatures and its value is given by Fig. i . The difference of
temperatures is that of the steam and the surrounding air.

The loss of heat due to radiation depends on the nature of
the surface of the pipe, on the difference of temperatures and on
the temperature of the surrounding objects thus :

Loss due radiation in B. T. U. per sq. ft. per hour = A~X C X
C" X diff. of temp., where A' is a number depending on the
nature and condition of the surface of the pipe, C a coefficient
depending on the amount of the difference of temperatures, its
values being given by Fig. 2, and C" a coefficient, given in Fig.
3, depending on the temperature of the surrounding objects.
The temperature of the surrounding objects we must usually
consider to be the same as the temperature of the surrounding
air.

Summarizing we have :

Loss of heat in B. T. U. per sq. ft. per hour
= (A + R) X diff. of temp.
= [(A" X O + (A~ X C X C"] X [diff. of temp.]

A study of this formula will show that all its parts except
A' are rigorously fixed by Peclet's deductions from his experi-
ments. For that matter K is given by his experiments within
narrow limits and might be expected to have the value .64.

There are, however, enough reliable experiments on a large
scale to make it preferable to deduce K directly from them.

The table on the following page gives the results of a num-
ber of careful and 'reliable tests, most of which were on a large
enough scale to give results of assured practical value.

In the previous paper references will be found to the pub-
lished data of these tests.

Prof. Jacobus has kindly furnished additional data in regard

Loss of Heat from Bare Steam Pipes

w

A^ 03

P* G-i ^^ M

P

3

o>
o

•*>

£

W

s

•o

rt

3
rr>

r1

-

14* to 4* Co

1 oo • • •

Approx. Dura-
tion in hours.

-

M M l-l CO ^1 •*-!

No. of tests
averaged.

00

. H-t
to to Cn O to to

Size of pipe.

Co

^J O 00 OOCo Co

Square feet of

ON ON h~( Co vO Cn
Co ON Co Co to "*-4

pipe surface.

M
H

0

M t— ( M M

Cn to Co 4* 4^- 00
Co oo Cn \O \O to

Pressure of

Cn

4— *"4 O OJ O> to

Co

Co Co Co Co Co Co
O 4^- OOCn Cn Cn

Temp, of
Steam

Cn

00-<i O Co 4^ to

Cn

HI O "^J Oo Co ON

Temp, of Air.

Cn

to w O ON to ON

to

ON

to to to to Co to
to *^J vO ^O O ON

vO 4* M "-i to 00

Difference of
Temp's

O

ON ON O ""J to ON

M M M

Lbs. of steam

00
CO

•^J vO O O •"" ^
O vD Cn OOCn i-<
•»»J 4* O Cn O Cn

per sq. ft.
per hour.

to

to Co Co Co Co Co

B. T. U. per

"2

OOCo O OOCn M

sq. ft. per hr.
per i deg.

H

00
Co

00 OOvO O OOvO
to Cn M O ^O O
vO ONCn ONCn 4^

Value of K.

Loss of Heat from Bare Steam Pipes

Values of K

Loss of Heat from Bare Steam Pipes

26 Loss of Heat from Bare Steam Pipes

to his test and some additional data in regard to Mr. Barrus' tests
has also been obtained.

The results of the tests are shown graphically in Fig. 6.
The curves drawn thereon were computed as follows :

Steam pressure 100 Ibs., temperature 338° Fahr., tempera-
ture of air 65° Fahr. Difference of temperatures 273°. Outside
diameter of pipe 2.38 ins.

Taking the value of A" at .87 we have

R=.%-j X 1.04 X 2.03 = 1.83,

the coefficient 1.04 for air temperature having been found from
Fig. 3, and the coefficient for difference of temperatures from
Fig. 2.

The coefficient K' we find from Fig. 4, since the test pipes
were all horizontal, to be .68, choosing from Fig. i the proper
coefficient for difference of temperatures we have,
^4=.68X 1.77 = 1.20.

Then the loss in B. T. U. per square foot per hour per i°
difference of temperatures between the steam and the surround-
ing air,

= ^ + ^4 = 1.83+ 1.201=3.03.

Proceeding thus for several different steam pressures the
curve for the 2 -inch pipe was drawn, and that for the 8 -inch pipe
was constructed in the same way.

The curve for the 2 - inch pipe agrees within less than two
per cent with all the experiments on that size of pipe. The curve
for the 8 -inch pipe agrees with the only experiment on that sized
pipe that we have.

It is, however, to be noted that Mr. Barrus' experiment on
the lo-inch pipe is not at all in agreement with the theory.
This experiment was made on two lo-inch pipes parallel to one
another and only 24 inches distant between centers, certainly not
the very best arrangement for a test of this kind. Although the
radiation would be somewhat decreased the air contact action of
the ascending currents of air might easily have been considerably
increased.

We may say that Fig. 10 demonstrates that for a given size
of pipe the theory takes care of any steam pressure between zero
and 150 pounds. Brill's test would show that the theory also

Loss of Heat from Bare Steam Pipes

27

8.30

3.30

P.

1
H
,-,3.10

o

«H
1
9

Ba

rr

is

>"

w 1

LOSS OF HEAT FROM
BARE STEAM PIPES.
Fig. 1O

x

/•

X

X

f

X

Bt

IT

is

10

'i

SOI

bs.

"5

V

?

X

1

/

X

II

ad

5or

-B

Bill

e

X

'

|£

o.d.

Q

x

I

ai

ru

s2

•

s

^

x

ft

£

o1

02
SJ2.90

P.

d

w
I

TWO

tj

EH

H

2.70
2.60

^

^

•^

9

|

-/

tl

n^

x

^

x

?

>/

7

X

X

B

>

4

X

c<
/

^

/

*

x>«

X

™

/

•

Ja

CO

1U!-

8

'

, i

^;

/

/V

7

yP

^

n

2

^•N

*x

> Bril

18

'

/

a

x

/

x

x

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X

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40

60 80 100 120

SteamJEressure above Atmosphere

140

allows for the different loss of a different size of pipe, but Barrus
test on the lo-inch pipe is equally strong evidence that it does
not. Hudson -Beare's test also agrees with the theory.

Fortunately we are not entirely dependent on these tests for
this point. The previous paper on the "Loss of Heat from
Covered Steam Pipes" showed by a large number of tests that
the theory did in that case most certainly allow correctly for dif-
ferent outside diameters of coverings. Now, the loss of heat
from the surface of a bare pipe must be governed by the same
laws as the loss of heat from the surface of a pipe covering, and
we may therefore conclude that the theory does cover the loss,
from a pipe of any diameter.

28 Loss of Heat from Bare Steam Pipes

We have then a simple means of accurately estimating the

loss of heat from a steam pipe or boiler.

I am at a loss to explain why the experiments require the

value of /fto be .87 when P£clet's experiments point to a value

of .64. It may have been because his tests were made with very

much smaller differences of temperature.

The effect of the condition of the surface of the pipe may be

inferred from the following values found by Peclet :

Ordinary sheet iron , . .567

Rusty sheet iron 688

New cast iron 650

Rusty cast iron 688

The effect of air currents is very marked in the case of bare

pipes. Mr. Norton in his experiments found that the draft from

an electric fan increased the loss by one -half in the case of a

vertical 4 -inch pipe.

Mr. Barrus showed conclusively by several tests that the

loss of heat from a bare pipe is the same whether the steam

therein is at rest or in motion at ordinary velocity.

EMISSION AND TRANSMISSION OF HEAT*
CHAPTER I.

EMISSION OF HEAT FROM A SURFACE MAINTAINED AT A CON-
STANT TEMPERATURE.

775. The case under consideration is that of a pipe heated
within by steam, and with its outer surface exposed to the air ;
that of a vessel full of warm water, and so on. The quantity of
heat emitted by a surface maintained at constant temperature,
and exposed to the air depends on the area of its surface, on its
form, on its temperature, and on that of the air to which it is ex-
posed. It is important to know the quantity of this heat in heat
units, per unit of surface, during unity of time, as a function of
the elements which cause it to vary, at least for the cases which
ordinarily occur in practice.

In order to understand how this quantity of heat emitted
may be determined, consider the case of a metallic vessel full of
warm water ; the metals being very good conductors of heat, the
exterior surface of the vessel will be at the temperature of the
water which it contains, let the weight of water, augmented by
that of the vessel, multiplied by its specific heat, be P kilograms,
.S the area of the surface of the vessel in square meters, O° C the
temperature of the surrounding air, 0 the time in seconds which
is required for the water to cool from T° to T — i degrees.

The quantity of heat units lost during the time 0 is evidently
equal to P, and ought to be sensibly the same as that quantity
which would have escaped from the vase in the same time if the
temperature had been constant and equal to the mean of T and
T — i , that is to say to T — % . According to this the quantity of
heat M which the surface of the vessel would emit per hour per
square meter, if the temperature were maintained at T — y2 would
be

M - *L v 36o° _ J_ p x
S * * o

* From " Trait6 de La Chaleur " by E. PSclet.

29

30 Emission and Transmission of Heat

Thus, by observing the intervals of time 0, 0' , 0", and so on,
which correspond to successive coolings of one degree, one may
therefrom compute the quantities of heat which would be emit-
ted per hour per square meter for the corresponding excesses of
temperature. It then remains to find by trial the law which these
results follow expressed as a function of the excess of temper-
ature.

776. When a vessel filled with warm water cools, we call
the ratio between the infinitely small variation of temperature dt,
and the time do in which this variation takes place, the rate of

cooling thus, we have z>=~ Pdt represents the quantity

of heat emitted in the time do.

If the temperature of the vessel is kept constant, the quan-
tities of heat emitted during equal intervals of time will also be
constant, and that quantity emitted in unity of time will evi-

dently equal P— or Pv.

Since the second represents unity of time we have

Making v = — which is to assume that the rate is constant

during the cooling through one degree, we get the same value of
Mas found in 775.

777. Newton's Law. — Newton's hypothesis was that the
rate of cooling in air was) proportional to the excess of the tem-
perature of the body above that of the air, and his formula was

v = qt

t being the excess |of temperature and q a coefficient varying with
the nature of the body.

This law is, however, inexact, the rate of cooling varying
much more rapidly.

Dulong and Petit 's Laws. — Dulong and Petit have made
numerous experiments on the cooling of the thermometer placed
in a closed vessel, maintained at a constant temperature by a
water bath and filled with different gases under different pressures.
These skilful physicists have established the following facts :

Emission and Transmission of Heat 31

i St. The cooling of a body results from its radiation and
from the contact of the surrounding gas.

2d. The rate of cooling due to radiation is independent of
the substance of which the body is composed ; but its absolute
value varies with the nature of the surface of the body.

It is represented by the formula,

v=maQ (a* — i)

in which m is a number depending on the nature of the surface,
a the number 1.0007, ^ the temperature of the surroundings and
/ the excess of the temperature of the body over that of the sur-
roundings in degrees Centigrade.

3d. The rate of cooling due to the contact of the surround-
ing gas is also independent of the substance of which the body is
composed, but its absolute value is independent of the nature of
the surface ; it depends solely on the form of the body and the
excess of its temperature over that of the surroundings.

This rate for air at 760 mm. pressure is given by the formula

v = nt1-2™

in which n is a number varying with the form and the extent of
the surface of the body, and t the excess of temperature of the
body above that of the surrounding air in degrees Centigrade.

779. New Experiments. — While admitting the exactness of
these laws, the formulas which represent them are of no service
as long as the coefficients m and n are unknown for surfaces of
different natures and for bodies of different forms. I may add
that Laprevotaye and Desains have found, in certain cases, results
that do not at all agree with the above formulas.

I have therefore thought it best to take up the question
again, but limiting it to the study of the cooling of a body in air
under ordinary pressure and in a chamber with dull walls ; for
the cooling of a body in different gases, under different pressures
and in a chamber with gilded walls, is a purely speculative ques-
tion which never occurs in practical applications.

780. There will be found at the end of this book the details
of the apparatus and the methods of calculation employed in the
experiments ; I shall here confine myself to a general description
and to the setting forth of the results arrived at.

781. Experiments, for the purpose of finding the absolute

32 Emission and Transmission of Heat

values of rates of cooling, can not be made simple on thermome-
ters. I have used spheres of thin brass with diameters ranging
from two inches to twelve inches, a number of cylinders with
diameters of one and a quarter inches to twelve inches and
lengths of two to twenty inches, and also several rectangular
vessels of different dimensions.

All of these vessels have been employed successively bare
and covered with different substances. The water which they
contained was continuously agitated. The temperatures were
measured by very sensitive thermometers. Intervals of time
were determined by means of a Breguet counter.

The vessels were placed in an open chamber having a double
wall. The interval between the walls was filled with water, and
the contained air was constantly renewed through passages
which gave to the entering air the temperature of the walls of
the chamber.

782. Figure 149 represents a vertical section of the constant
temperature chamber, figure 150 a plain of the same. ABCDEF
and A' B' CHEF are two cylinders of sheet iron plated with
lead. They are concentric and the intervening space is filled
with water.

This envelope is formed of two halves separated in a vertical
plane and held together by suitable fastenings. The interior cyl-
inder is 39 inches high, and 32 inches in diameter ; the interval
between the two cylinders is i ^ inches and the water contained
in this interval is frequently agitated by horizontal annular
plates, to which are attached vertical iron rods projecting
through the little stuffing boxes GG.

The temperatures of the water in each half of the envelope is
given by thermometers inserted at /and I. JfLMsmd K1 L' M' are
two vertical enclosures fastened on the outside of each half of the
chamber. They are open above and communicate below with the
openings NP (fig. 149) fashioned in the lower part of each of the
two halves of the chamber. The outer wall of these enclosures
is made of wood. They contain throughout their height, strips
of sheet iron soldered perpendicular onto the outer surface of
the chamber, each 4 inches high, and extending to the wooden
outside of the enclosure, and the strips of one horizontal row are

Emission and Transmission of Heat

33

placed midway between those of the two adjacent rows. Similar
strips are placed in the opening A*"and P. QR and Q R' are two
sealed half cylinders of tin full of water at the oridinary tempera-
ture, they serve to close, to the desired extent, the opening AF
of the chamber. ST is an adjustable tripod bearing three glass
tubes terminated by wooden stoppers, into which penetrate cop-
per stems soldered to the vessel of which the rate of cooling is to
be determined.

Fig. 149

783. Figures 151 and 152 show a vertical and horizontal
section of a spherical vessel and its agitator.

The plates, which agitate the water are carried by six pieces
of iron wire in the shape of a semi -circle. These are attached
below to the axial stem and above to a small horizontal ring which
allows of the insertion of the thermometer through the tubular
orifice shown. The cylindrical vessels of large diameter are ar-.
ranged in the same way (fig. 153 and 154).

When the cylinders are of small diameter the agitator is
placed on one side (fig. 155 and 156).

784. I show in figures 157 to 161 the different methods of

34

Emission and Transmission of Heat

supporting the vertical and horizontal cylinders in a frame placed
within the chamber. The object of these arrangements is to ren-
der the cylinders perfectly motionless in spite of the movements
of the agitator. The frames are of iron or brass, the stems a, a, a,

LJ

Fig. 152.

Fig 153 and 154.

Figs. 155 and 156

are very thin and of fir wood ; they project into very small me-
tallic appendices soldered to the vessels. When the cylinders are
placed horizontally, the stem of the agitator turns in a cork which
closes the tubular opening. There is a little play between the
stem and cork, but the water does not escape on account of the
expansion of the small quantity of air remaining in the vessel.
This expansion is due to the contraction of the water by cooling.

Figure 162 is a section of a cylindrical vessel, with hemi-
spherical ends, provided with two agitators.

When the vessels are long and of small diameter, cylinders

Emission and Transmission of Heat

35

of iron filled with mercury are employed, an agitator is no longei
necessary, and a thermometer with a long reservoir inserted in the
vessel, gives its temperature accurately.

785. Figure 163 shows the little apparatus
used in reading the thermometers, ab is a small
plate covered with white paper, and bearing at
the middle of its height two projecting rods, one
on each side of the
stem of the ther-
mometer. These
rods are parallel to
one another and
perpendicular to
the plate ab. They
have fastened upon
them two hairs
which determine a
plane perpendicu-
lar to the stem of
the thermometer
and in which plane
the eye of the ob-
server should be
placed.

c and d are two rings through which the
stem of the thermometer passes. They are lined
with cork which can be more or less compressed
by thumb screws.
This apparatus is necessary on account of the motion caused
by the agitator which renders the use of a cathetometer impos-
sible.

786. Figure 164 shows the apparatus used to refill the ves-
sels at certain stages of their cooling, which is necessary in order
that the cooling surface may remain constant.

An opaque vessel, in which the level of the water has de-
scended, has to be refilled without taking it out of the constant
temperature chamber and without spilling any water.

The apparatus consists of a glass tube AB open at both ends

Fig. '59-

Figs. 157 and 158.

Emission and Transmission of Heat

and provided with a reservoir C, at its side another glass tube
DEF, bent over, and likewise open at both ends. The ends B

Fig. 160.

Fig. 161.

Fig. i6a.

and D are at the same level. Both tubes are inserted at their
lower ends in a stopper which enters freely into the tubular open-
ing of the vessel, this stopper bears at its upper end a plate of
brass of a diameter greater than the opening of the
vessel, which allows the stopper to be inserted al-
ways to the same distance. When the stopper is
put in , the point B is at the height which the liquid
should reach. To fill the vessel the stopper clos-
ing the opening is removed, and is replaced by
the stopper bearing the glass tubes ; the
reservoir C being full of water and the end
A closed by a finger. The end A is then
opened and the operator sucks on the
tube F, and evidently when water ap-
pears in the tube F, the water level in
the vessel will have reached the height
BD ; at this moment the extremity A
is again closed, the apparatus is lifted
out and the ordinary stopper re-
placed. Fig. 164.

The mode of precedure was as follows : the vessel hav-
ing been filled with hot water and placed upon its support the

Emission and Transmission of Heat 37

constant temperature chamber was closed, the opening at the top
for the escape of air was so adjusted that its area was approxi-
mately equal to that of a horizontal section of the vessel, the agi-
tator within the vessel was turned continuously, and those of the
chamber were put in motion from time to time.

The time required at a number of stages, for the thermome-
ter to fall through a few divisions was observed. The temper-
atures indicated by the thermometer were reduced to what they
would have been if the whole stem had been plunged in the
water, on the assumption, proved both by experiment and calcu-
lation, that the stem was exactly at the temperature of the sur-
rounding air.

788. From these experiments the rate of cooling can be
readily deduced.

After obtaining the values of the rates of cooling v, and
therefore (776) the values of M> the quantities of heat emitted,
for excesses of temperatures ranging from 45 to 117 degrees
Fahrenheit, I sought to connect them by some simple law, and
found that they satisfied the following,

M=at(* + bt)

This agrees perfectly with the formulas of Dulong and Petit
within the limits of temperature mentioned above, and it results
that these formulas are very probably exact up to an excess of
temperature of 470° F. as these two celebrated men have indi-
cated.

Since cooling results from simultaneous radiation and con-
tact of air, it is necessary to separate the effects of these separate
causes, in order to determine the coefficients used in the formula.
To do this I have employed the following method. Let us sup-
pose that M represents the quantity of heat lost by a vessel coated
with lamp black, M' that which is lost by the same vessel with
a brilliant surface, A the quantity of heat lost by air contact, and
which is the same for both surfaces, and R and Rf the quantity
of heat lost by radiation from the lamp black, and the metal
respectively. Then

M= A + R; M' = A + R', and it follows that

M— M' = R— R'.
Let R=cR' then the last equation becomes

38 Emission and Transmission of Heat

M—M' = R' (c— i) and R' = M~M'and as

c — i

M=at (i + b() and M' = a't (i + b't} the value of R' would
be

<: — / c — i

Having thus obtained a general expression for the value of
R1 ', that for A is easily deduced from it since A = M' — R '.

I employed the following method in obtaining the ratio c of
the radiations ; it depends on one of Dulong and Petit 's laws. Two
metallic vessels, one side of each being a vertical plane, bare or
covered with different materials, are placed with their plane faces
opposite and parallel, and equally distant from a thermopile con-
nected to a very sensitive galvanometer. One of the surfaces is
maintained at a constant temperature while the temperature of
the other is caused to vary until the effects produced on the ther-
mopile are the same, that is to say until the needle of the galvan-
ometer returns to zero. Designating by m and m' the radiat-
ing power of the two surfaces, by t and /' the excesses of their
temperatures above 0, that of the thermopile, we have for the
quantities of heat radiated according to Dulong and Petit ma®
(a* — i), and m'aQ (a1' — i), and as these quantities are equal, we
deduce that

_. _ R _m av — i

~tf~~~tnr = a''— i

From all these experiments there result the following
formulas.

789. The quantity of heat emitted by radiation to surround-
ings at a temperature differing but little from 12° C., and for
excesses of temperature between 25° C. and 65° C. is given by
the formula

R=Kt(\ + .00560 ...... (a)

A" is a coefficient depending on the form and the dimensions
of the body, / is the excess of temperature in degrees Centi-
grade.

790. The quantity of heat lost by contact of air in the
same circumstances is given by the formula,

A=fCt(i+ .00750 ..... (*)

Emission and Transmission of Heat 39

A' is a coefficient depending on the form and the dimensions of
the body, / is the excess of temperature in degrees Centigrade.

791. When the excess of temperature is but slight, one
may neglect the terms of the second degree and we then have for
the total quantity of heat emitted

M=R + A = (K+l?}t = Qt
which is Newton's L,aw.

The formulas (#) and (£) have only been proven for ex-
cesses of temperature ranging between 45° F. and 117° F. for
greater excesses of temperature we must employ Dulong and
Petit 's formulas.

We will therefore enunciate the formulas in a general form
and give the values of the coefficients K and K' for different sur-
faces and bodies of different form according to the results of our
experiments.

General Formulas Relative to the Emission of Heat in Air.

792. The quantity of heat emitted by a surface maintained
at constant temperature depends on the radiation and the contract
of air. If we designate by Mthe total quantity of heat emitted
in a certain time, by R and A the quantities respectively due to
radiation and air contract we have ;

M=R + A (i)

793. Heat Emitted by Radiation. — The quantity of heat
emitted by radiation, for unit surface and per unit of time, is in-
dependent of the form and size of the body — provided that its
surface has no reentrant portions ; it depends solely on the nature
of the surface, on the excess of its temperature over that of the
objects to which radiation takes place, and on the absolute value
of the temperature of these objects.

794. When a body is surrounded by objects having dul
surfaces, which is nearly always the case except in laboratory
experiments, the quantity of heat R is given by the formula :

^=124.72 A^O' — i) (2)*

Where 0 represents the temperature of the surrounding objects
/ the excess of the temperature of the surface above that of the
surroundings, a a constant having the numerical value 1.0077,

* All the quantities in formula (2) and (3) are in French units.

40 Emission and Transmission of Heat

and K a number depending on the nature of the surface of the
radiating body.

Values of K for different surfaces. In B. T. U. per hour per
square foot per i° F. excess of temperature.

Polished silver . . . .027 Zinc 049

Silvered paper . . . .085 Polished tin 044

Polished brass . . . .053 Tin plate 086

Gilded paper 047 Sheet iron polished . . .092

Polished copper . . . .033 Sheet iron leaded . . .133
Cast iron — new . . . .649 Sheet iron (ordinary ) . .567
Cast iron rusted . . . .688 Sheet iron rusted . . .688

Glass 596 Building stone . . . .737

Powdered chalk . . . .680 Plaster and brick . . .737

Saw dust 723 Wood 737

Powdered charcoal . . .700 Woolen cloth . . . .753

Fine sand 741 Calico or canvas . . .747

Oil paint 759 Silk 759

Paper 772 Water 1.087

Lampblack 820 Oil ....... 1.482

795. For paper and cloth, color has no influence. It
appears from this table that powdered materials have very nearly
the same emissive power. Masson has already recognized that
all substances in a very finely powdered state, obtained by pre-
cipitation and not crystallized, have the same emissive power.

796.*

797. Heat Transmitted by Air Contact. — The loss of heat
arising from air contact is independent of the nature of the surface
of the body, and of the absolute temperature of the surrounding
air ; it depends solely on the excess of the temperature of the
body over that of the surrounding air, and on the form and dimen-
sions of the body.

* The tables given by P6clet in this paragraph are here replaced by the curves of
figures 2 and 3, page 21.

It is more convenient, as will be apparent later on, to use Newton's I,aw for compu-
ting the loss of heat from a given surface and to modify the values of K given in the table
above by coefficients obtained from formula ( 2 ) shown graphically in figures 2 and 3.

Figure 2 gives the coefficients by which K must be multiplied for differeut excesses
Of temperature. The product thus obtained is multiplied by a second coefficient obtained
from figure 3 which corrects for different temperatures of the body radiated to.

Emission and Transmission of Heat 41

This loss of heat per square meter per hour is given by the

formula,

,4=0.552 A"/1'233 ...... (3)

where / represents ( in Centigrade degrees) the constant excess of*
the temperature of the body, over that of the surrounding air,
and K' a number which varies with the form and dimensions of
the bodies.*

798. For spherical bodies we have, in English units,

, 1.048
K = .363 H -- —

where r= radius in inches.

799. For horizontal cylinders of circular section we have,
in English units,

where r equals the radius in inches, f

800. In the case of vertical cylinders, the cooling depends
both on their height and diameter, and

here r = radius in inches and h = height in feet.J

802. For vertical plane surfaces, the value of A" is given
by the formula

where h is the vertical height of the surface in feet.§

804. Figure i, page 21, gives the coefficients by which
the value of A" obtained by the formulas above, must be multi-
plied in order to correct for the differences in temperature between
the body and the surrounding air.

805 . It is apparent from an inspection of Figures i , 2 and
3 that Newton's Law is extremely inaccurate; the coefficients or
the values of R and A, instead of remaining constant vary fof

*It is important to note that by the temperature of the surrounding air is meant its
temperature before it is in any way influenced by the heat emitted from the cooling body

t Figure 4, page 24 gives the values of K' for the horizontal cylinders up to 16 inches,
diameter.

t Figure 5, page 25, gives the values of A"' for cylinders up to 18 inches diameter.

\ See Figure n.

Emission and Transmission of Heat

Emission and Transmission of Heal 43

excesses of temperature between 20° F. and 350° F., the first in
the ratio of i to 2.2, the second in the ratio i to 2. Newton's
Law is approximately true for small excesses of temperature
only.

806. To sum up we have

M=R + A = i2$.>]2Kae (y — 0 + 0.552 A"/1'233*
But we can always in practice calculate the valves of R and
A by Newton's L,aw corrected by the coefficients of figures i, 2
and 3, thus obtaining by simple calculations results that are quite
sufficiently accurate.

807. We will apply this method to a case which frequently
presents itself; that of a horizontal cast iron pipe containing
steam at 212° F. and with a surrounding temperature of 59°.
Forr=2" ;¥=i53X .688 X 1.52 X i + 153 X .58 X 1.56 = 298

= 4"M= " " " + " X-50X "=279-

= 6" M= " " " + " X-47X "=261

The results are in B. T. U. per square foot of surface per hour.
The weights of steam condensed by direct experiment are a little
greater, probably on account of water mechanically entrained by
the steam.

809. For a horizontal pipe of sheet iron 10 inches in dia-
meter containing air at 302° F., the exterior air being at 59°, we
would have M = 243 X .567 X I.88X i + 243 X .48 X 1.73=462
B. T. U. per hour per square foot.f

Emission of Heat from Pipes to Air.

821. ^mission of heat from the surface of a pipe to the air
traversing the pipe. The surface being maintained at a constant
temperature.

Let us consider a metal pipe, the surface of which is main-

* In French units.

t It will be noticed that in these examples the second coefficient by which K is multi-
plied in order to obtain R is unity. This is because the temperature of the surrounding:
objects is 59° F. See figure 3, page 21.

To sum up we have for our working formula,

In which,

M= B. T. U. per square foot per hour.

R= " " due radiation.

A — " " due air contact.

T = temperature of the body emitting heat in F °.

44 Emission and Transmission of Heat

tained at a constant temperature, and through which passes a
current of air ; and let us suppose that all the elementary veins
have sensibly the same velocity or that a thin slice of air taken
perpendicularly to the axis of the pipe at the entrance preserves
its form while it traverses the pipe.

During its passage the circumference of the section will be
at the temperature of the pipe, and the heat will propagate itself
from the circumference towards the centre. After a certain time
the whole section will have attained sensibly the temperature of
the pipe ; if at this moment it has not reached the end of the
pipe, the rest of its passage will evidently be without influence,
if at its exit from the pipe the heat has not had time to reach the
centre of the section the mean temperature of the section will be
the higher the longer it has been in the pipe. The temperature
of the escaping air will depend then on its velocity, and on the
length of the pipe.

We have supposed the pipe to be circular and the slice of air
to be always limited by two plans, but all that we have said is
equally true for a pipe of any form whatever, and in spite of the
difference in velocity of the elementary veins which always takes
place ; except that the time necessary for the centre of the vein
to take the temperature of the circumference augments with the
difference of velocity.

We may add that when the pipe is horizontal or more or less
inclined, the propagation of the heat depends not only on the
transmission through the air, but also on the movement of the
air due to its being heated.

It is easy to see from these considerations how complicated
are the phenomena which take place during the heating of air
while it traverses a pipe maintained at a constant temperature,
we may however deduce certain general principles from the pre-
ceding reasoning, which will be useful under certain conditions.

/ =temperature of objects to which radiation takes place in F°.
/' = " of surrounding air in F°

For value of K see 794.
" " A" see 798-802.

" C see Figure 2, page 21.
" C see Figure 3, "

C' see Figure i, " " •

Emission and Transmission of Heat 45

i st. When air traverses a tube maintained at a constant
temperature greater than that of the air and supposing that the
velocity of the air, at first very small, increases as it progresses
through the tube, the air will emerge at the temperature of the
tube up to a certain limit of velocity, depending on the perimeter
of the tube, on the shape of its section and the inequality of the
velocities of the different elementary veins. This velocity will
increase in proportion as the section of the pipe diminishes. It
is impossible to foresee whether, under the same conditions, this
velocity would be greater in a vertical pipe than in a horizontal
one, because in the first case the increments of velocity resulting
from heating against the walls brings out the air from the centre,
while in the second case the layers of air in contact with the lower
surface are constantly displaced, circumstances which both tend
to distribute the heat.

2nd. When the limit of velocity which I have just mentioned
has been reached, the air escapes at a decreasing temperature be-
cause in each section the temperature is decreasing from the cir-
cumference to the centre, and the greater the velocity of the air
the more is this the case. But the quantity of heat carried off
by the air increases with its velocity, this fact is thoroughly
proved by experiment and is easily explained by admitting that
the sum of the quantities of heat diffused through each section
increases very rapidly with time, for the number of sections pass-
ing in unity of time being proportional to the velocity, and the
time of passage of each section being inversely proportional to
the velocity, it follows that the quantity of heat carried off by the
air will increase with the velocity provided that the quantity of
heat which diffuses itself through a section increases very rapidly
with the time.

822. In practice we may admit as a sufficiently close ap-
proximation that the quantity of heat emitted by the pipe is sen-
sibly equal to that which it would emit in the open air by air
contact to the surrounding air at a temperature the mean of the
observed temperatures of the air at entering the pipe and at leav-
ing it.

I have verified this principle by means of a cylindrical vessel
1 6 inches high, and 8 inches in diameter, pierced at the centre by

46 Emission and Transmission of Heat

a tube 4 inches in diameter with its surfaces entirely covered by
paper. By observing the cooling when the orifice of the tube
was open and then when closed, I have found that in the last case
the loss of heat was very nearly equal to half of that of the central
tube when surrounded by free air.

823. Emission of heat to air traversing a conduit enclosing a
pipe maintained at a constant temperature.

This case is very similar to the preceding one, except that
the diffusion of the heat takes place more rapidly, because the
concentric layers of air undergo a continual increase of surface
as they recede from the surface of the pipe, and at the same
time the interior surface of the conduit, heated by radiation,
heats the layers of air from the opposite direction. Here, as in
the preceding case, there is a limiting velocity, below which the
heating of the air is complete, and beyond which the temperature
of the air diminishes, although the quantity of heat carried away
increases with the velocity.

824. For this case we may estimate the quantity of heat
emitted to be approximately equal to that which the pipe would
emit in the open air by radiation and air contact, the temperature
of the surroundings being taken as the mean of the observed
temperatures of the air at entering the conduit and at leaving it.

CHAPTER II
Transmission of Heat through Solid Bodies.

When a solid body is limited by two parallel surfaces, main-
tained at temperatures constant but different, it is traversed by a
constant flow of heat proportional to the distance between these
two surfaces. This law may be deduced from the very nature it-
self of the movement of heat. Consider a homogeneous plate,
of thickness <?, of unit superficial area, and of which the surfaces
are maintained at the constant temperatures / and /, imagine the
thickness of the plate to be divided into a very great number of
extremely thin layers. The entire thickness of the plate being
obviously traversed simultaneously by a quantity of heat equal to
that which traverses any part whatsoever of its thickness, an
equal quantity of heat must pass simultaneously through each of
the elementary layers in question, then, since for any particular
body the quantity of heat transmitted can only depend on the
thickness, and on the difference of temperature, and since the
thicknesses are the same, it follows necessarily that the differences
of temperature of the surfaces of the elementary layers are the
same, and in consequence, that throughout the thickness of the
plate the temperature varies uniformly from / to /. As the quan-
tity of heat which traverses any layer is the same as that which
traverses any number whatsoever of layers, and since the differ-
ence of temperature of the outer surfaces is proportional to the
number of layers, and as the thickness is also proportional to the
number of layers, the equality in question can only exist as long
as the flow of heat is proportional to the total thickness, and we
have

M=

In this expression M represents the quantity of heat trans-
mitted by unit surface in unit time, /and/' the temperatures of
the two surfaces, e the thickness of the plate and Cthe conductiv-

47

48 Emission and Transmission of Heat

ity of the material, that is to say the value of M for /— /'=i and

This formula has been verified by experiment.

We will first examine the conductivity of bodies which trans-
mit heat well, that is to say the metals, and then the conductivity
of bodies which transmit heat poorly.

CONDUCTIVITY OF METALS

827. When one end of a bar of metal is maintained at a con-
stant temperature, heat is transmitted along the bar and is dissi-
pated from its surface. If we suppose the bar to be so long that
the heat does not reach its farther extremity, and its section to be
so small that all the points of the same section have sensibly the
same temperature, we may readily find by calculation, for the
excess of temperature^ of any section above that of the air, at a
distance x from the extremity, the relation,

\~Ph~'

y=Ae

in which A represents the constant excess of temperature of the
heated extremity over that of the air, 5" the section of the bar, P
its circumference, h its coefficient of cooling, and C the conduc-
tivity of the bar.

828. In i8i6M. Biot verified the truth of this formula by
numerous experiments (Traite de Physique, 1816). Later in
1836 M. Despretz made further experiments which confirmed those
of M. Biot and which permitted him to determine the relative con-
ductivity of the metals, given in the following table:*

Silver . . . 1000 Iron 119

Copper . . . 736 Steel 116

Gold . . . 532 Lead 85

Tin .... 145 Plantinum ... 84

829. These numbers cannot however be considered very ex-
act since to begin with, Newton's Law on which the formula
(827) depends, departs too far from the truth when the excesses of
temperature are considerable, and further because in measuring
the temperatures of the various sections of the bar holes were

* The table given here is from more recent experiments of Frantz and Wiedermann
inwhich the errors due to boring holes in the bars were avoided by the use of thermo-
piles.

Emission and Transmission of Heat 49

bored for the insertion of the thermometers, thus producing points
of reduced sectional area, which must have had some influence
on the results.

830. Since the numbers found by M. Despretz were relative
only, they could not serve for any applied calculation so long as
the absolute conductivity of no one of the metals was known.
In 1841 I took up the question and in order to determine the con-
ductivity in heat units I made a great number of experiments of
which a detailed description will be found at the end of this work;
I shall limit myself here to a brief description of the apparatus
employed and to the setting forth of the results.

A cylindrical vessel full of water and surrounded by non-
conducting material was provided with a bottom formed by a
disc of metal of which the circumference was separated from the
walls of the vessel by a ring of cork. The lower surface of the
disc was heated by steam, and the heating of the water contained
in the vessel was observed. The water was kept in constant
motion by an agitator. If we admit that the amount of heat
traversing the disc is proportional to the difference of temperature
of the two surfaces, and remembering that this amount of heat is
proportional to the rate of cooling (776) we have dT=aTdt;
and m log. T=C—a t, m being the modulus of the table of loga-
rithms , 2 . 3025 and T the difference of temperature after the time /.

If we designate the temperature corresponding to t=o by A
we find C=m log. A, since in this case T=A, and the formula

m.
gives a=— (log. A— log.T)

The logarithms are those of the tables and a is the cooling
which would take place in one second for a difference of tempera-
ture of one degree. Two observations gave a value of a, and the
identity of these values deduced from different pairs of observa-
tions proved the truth of the hypothesis on which the formula
was based. These experiments repeated with discs of lead, zinc,
tin, sheet iron, and cast iron, gave the same results. But what
at first astonished me very greatly was the fact that the values of
a were sensibly the same for these different metals, no matter
what their thickness was, although it varied from one to twenty
millimeters. In all the experiments, I noticed that the speed of

5O Emission and Transmission of Heat

rotation of the agitator had a very perceptible influence; the value
of a increased or diminished notably with this speed. When we
consider that the steam in condensing must cover the lower -sur-
face of the disc with a layer of almost motionless water it is very
probable that in these experiments this surface was not at the
temperature of the steam, and likewise that the upper surface was
not at the temperature indicated by the thermometer. The heat
was really traversing a sheet of metal, comprised between two
sheets of water, one of which was practically motionless and the
other was only slowly changed; and since the conductivity of
water is very low compared to that of the metals, the influence of
the conductivity of the metals disappeared.

831. To verify this conjecture I abandoned the heating by
steam. I filled the upper vessel with water at O°Cand I immersed
the disc which formed its bottom to a depth of several millimeters
in a vessel filled with water at ordinary temperature. The interior
agitator was provided with brushes which grazed the surface of
the disc and the water in contact with the other surface was re-
newed by cords stretched in a frame to which a rapid reciprocat-
ing motion was imparted.

With this arrangement the heating of the water in the vessel
was very slow and the water in contact with the disc could be
changed with great rapidity.

With the discs of lead from one to twenty-five millimeters
thick, the values of a varied from .00060 to .00025. The other
metals gave similar results.

From this I concluded that, by greatly increasing the rate of
renewal of the waters bathing the two faces of the disc, and by
employing thick discs of the metals with the lower conductivities,
coefficients would be obtained that would be inversely proportional
to the thicknesses.

832. I devised for this purpose a new apparatus, in which
the interior agitator was driven by gearing; the external agitator
was also geared and consisted of a horizontal wheel, mounted ec-
centrically to the vessel and having spokes formed by tightly
stretched cords, which in their motion rubbed against the outer
surface of the disc.

Figure 165 is a vertical section of this last apparatus, figure

Emission and Transmission of Heat

1 66 a horizontal projection, and figure 167 a section on a largei
scale of the lower part of the interior vessel, A B C D is a vessel
of tin closed at its lower end by the disc of metal E, of which the
manner of attachment is shown by figure 167. This vessel

has within it a copper tube
bearing paddles at different
heights and having at its
lower end a horse hair brush.
This tube is guided by two
rings secured to it by the
rods //and /'/' and it bears
at its upper end a pinion
driven by means of the gear
and crank M' .

The vessel is closed by a
tight cover through which
the central copper tube pro-
jects. This cover supports
a ring O, one centimeter
above the top of the tube, in
K which is placed a pierced
stopper through which pass-
es the stem of the thermom-
eter, whose reservoir is
nearly at the center of the vessel, and which remains fixed regardless
of the motion of the agitators. A' B' C D' is a second vessel
surrounding the first to
which it is fastened by
three glass rods F F F,
it is filled with cotton
wool and its three feet
are provided with screws
engaging in the supports
NN soldered to the low-
er vessel G G. The ves-
sel A' B' C D' supports
the crank and gear that
drives the pinion of the
central tube. Finally the

V Fig. 166

Fig. 165

M

52 Emission and Transmission of Heal

vessel G G contains a horizontal
wheel R R having spokes formed of
cords which rub as the wheel re-
volves against the lower surface of
the disc E. This wheel is driven by
Fig. xey the pinion P, the gear U and the

crank M of which the shaft passes
through the stuffing box H.

By means of this apparatus I was able to renew 1600 times
per minute the water in contact with the surfaces of the metallic
disc.

With water at very nearly 24° Cin the open vessel, and at
the atmospheric temperature in the inner vessel, and employing
discs of lead of twenty and fifteen millimeters in thickness, the
duration of equal amounts of heating of the water in the interior
vase was 500 seconds for the first disc and 380 for the second,
this last figure only differs by five seconds from three-quarters of
the first.

We may therefore consider the law of thicknesses as having
been proven by direct experiment.

In these experiments the mean temperature of the outer bath
was 24.04° Cand only differed from the extreme temperatures
by a small fraction of a degree. The excesses of temperature at
the beginning and the end of the experiment were 8.91° Cand

9-55° C.

Then the quantity of heat transmitted through the 20 mm.
disc was .000294 per minute; the weight of water enclosed in the
vessel increased by the weight of the vessel multiplied by its spe-
cific heat being 3.287 kilograms, the quantity of heat which
would be transmitted through the disc for a difference of i° C
would equal, .000294X3. 287=. 000966 and as the surface of the
disc was .005026 square meters the quantity which tinder the
same circumstances would be transmitted per square meter would
equal, .OOO966X .mr^rsr =0.192 and for a disc one meter thick,
and during one hour would equal .192 X .02 X 3600=13.83* and
this figure is the conductivity of the metal.

833. Then if we admit the correctness of the relative con -

*H3 B. T. U. per hour per square foot per i° F. per I inch thickness.

Emission and Transmission of Heat 53

following figures for the quantity of heat in British thermal units,
which would be transmitted in an hour through plates with a
superficial area of one square foot, a thickness of one inch, and
of which the surfaces are maintained at constant temperatures dif-
fering by one degree Fahrenheit.

Silver .... 1330 Iron .... 158
Copper .... 980 Steel . . . . 154
Gold .... 706 Lead . . . . 113
Tin 193 Platinum . . 112

But these conductivities have been obtained under special
circumstances which are never met with in practice.

According to some old experiments of Clement, a sheet of
copper 10.8 square feet in area, and one-tenth of an inch in thick-
ness, bathed by steam on one side and water at 82° on the other,
condensed 220.5 pounds per hour for a temperature difference of
130 degrees, which is equivalent to 1.7 pounds for a difference of
one degree Fahrenheit.

According to the more recent experiments of Thomas & Lau-
rens, with a single copper tube of small diameter, there was
evaporated 81.8 pounds of water per hour, per square foot, for a
temperature difference of 81°, which makes one pound for a dif-
ference of i° F.

The figure obtained by Thomas & L,aurens is much greater
then Clements, for the surface of transmission being a tube of
small diameter, the air was completely expelled, a circumstance
which greatly augments the amount of steam condensed.

It is evident that even under the most favorable circumstan-
ces, the figure obtained for the transmission of heat through cop-
per, when the liquid wetting the surfaces is not renewed, is much
less than that resulting from the experiments we have been con-
sidering. This is on account of the sensibly motionless layer of
water which covers at least one of the surfaces.

834. Thus, although the laws admitted by physicists for the
transmission of heat through plates may be exact, these laws are
not applicable to the transmission of heat from one liquid to an-
ductivities of the metals found by M. Despretzf we obtain the

tSee foot note to paragraph 829. The figures given here are based on Pdclet's experi-
ment with lead and the relative values of Franz and Wiedermann.

54 Emission and Transmission of Heat

other through a sheet of metal; and we may admit that, within
the limit of thicknesses generally employed, the material and thick-
ness of the metal are without sensible effect; but if in certain cases
there should be a great advantage in increasing the rate of this
transmission, even at the expense of doing work for this purpose,
it could be effected by a stirring which would very rapidly renew
the layers of liquid in contact with the plate.

All of the preceding remarks suppose both surfaces of the
plates to be bathed by liquids, and should therefore be only ap-
plied to heating of liquids by liquids, or by steam; for steam in
condensing on the surfaces of the plates wets them and all takes
place as if the heating were done by a liquid. But when liquids
are heated by gases, and when gases are heated by other gases is
it still the same ? It is this which we will examine next.

835 . Consider first the heating of liquids by gases, this is for
example, the case with steam boilers, at least for that part of a
boiler which does not receive the radiation from the grate. I
have made no direct experiments on this subject, but the results
of practice leave no room for doubt that if the nature and thick-
ness of the metal have any influence it is only very small; for it
has been recognized for a long time that boilers of cast iron, of
copper, and of wrought iron, of the same dimensions but of widely
differing thicknesses, give sensibly the same results under the
same circumstances. It is a fact upon which all engineers agree.
One may easily account for it: when the thickness of metal in-
creases or its conductivity decreases, the temperature of its outer
surface rises; a well proven fact since in cast iron boilers the ex-
terior often becomes red, and as to wrought iron boilers, the
change of shape which they experience from the heat increases
with their thickness. Now since the quantity of heat transmitted
increases as the exterior temperature increases, it is easy to see
that the effect of the nature and thickness of the metal is but
slight.

This has been proven further by recent experiments of M.
Boutigny; who evaporated water in silver capsules of the same
form and exterior dimensions but of very different thicknesses;
the quantities evaporated under the same conditions and in the
same time were precisely the same.

Emission and Transmission of Heat 55

836. In regard to the transmission of heat from one gas to
another through a plate of metal, as for equal volume gases have
much less capacity for heat than liquids and as their conductivity
is very feeble, we may regard the influence of the nature and thick-
ness of the plate as absolutely nonexistent, since the quantity of
heat which the plate can transmit, even under the most unfavor-
able conditions, is incomparably greater than that which actually
traverses it, and consequently in no case can the thickness of the
metal delay the transmission . The quantity of heat which tra-
verses the plate is determined solely by the difference of tempera-
tures of the two gases, the absorbing and radiating powers of the
two surfaces of the plate, and above all by the movements of the
sheets of gas in contact with these surfaces.

Thus we see that in every case the rapid renewal of the layers
of gas or liquid, which bathe the surfaces of the metal plate, has
a very great influence on the transmission of heat, but that this
circumstance is of much greater importance with gases than with
liquids.

837. One should then in designs seek the disposition
most favorable to this renewal, taking advantage of the motions
which the fluids must have in order to pass through the appara-
tus, and those resulting from heating and cooling. But with gases,
one may also artificially produce motions throughout their masses
which would cause rapid renewals of the layers in contact with
the metallic surfaces, either in some direct way requiring only a
small amount of power, or by employing a part of the force re-
sulting from the passage of the gas.

838. Consider, for example, air heated to an extremely high
temperature, escaping through a horizontal cylindrical conduit
surrounded by water which it is intended to heat. The layers of
air in contact with the metal cool very quickly; but as all the
little elementary veins possess motion only in the direction of the
axis of the conduit, the layers will change place but slowly, for
the sole cause of change lies in the increase of density resulting
from cooling ; it only exists for the upper half of the conduit,
and it only operates slowly. This same state of affairs will con-
tinue no matter what the direction of the conduit. We may
imagine according to this, that if the area of the conduit were

56 Emission and Transmission of Heat

very great, and the velocity of the air at its exit very high, that
the greater part of the central veins would not come into contact
with the wall of the conduit, and would preserve their original
temperature. But if one provided the conduit with fan wheels
mounted on a shaft driven by power, the central veins would be
thrown out against the metal of the conduit and one would thus
obtain much greater cooling of the gas. The movement of the
gas from the centre to the circumference could even be produced
by the passage of the gas itself. It would suffice to place in the
conduit a number of wheels with vanes inclined as are the vanes
of a windmill ; these wheels on which the movement of transla-
tion of the gas would impress a certain velocity of rotation, would
evidently produce the same effect as wheels with blades set in the
plane of the axis of rotation.

839. The transmission of heat may be increased by still
another proceeding which is in some circumstances very efficaci-
ous.

We have seen that in the transmission of heat through a
metal plate, it was necessary to distinguish between the absorp-
tion by one face, the emission by the other and the transmission
through the metal ; and we know that under ordinary circum-
stances, the quantity of heat that the plate can transmit is much
greater than that which it can absorb or emit. It results from
this that if we use plates provided with projecting rods that dip
into the two liquids or gases, the one of which is to heat the
other, we increase the extent of the surfaces in contact with the
two fluids, and in consequence the effect produced ; so much the
more since the veins of fluid in contact with the rods will be con-
stantly renewed by the motion of translation of the fluids.

We may take for an illustration a horizontal conduit trav-
ersed by highly heated air intended to heat the air, moving in
an opposite direction, in a surrounding conduit. If the surface
of the inner conduit is provided with rods projecting inside and
out, and if they are not arranged in parallel rows, the inner por-
tion of the bars will absorb heat throughout their length and this
heat will be dissipated by the exterior portions; the current of
hot air will be cooled uniformly throughout its section and the
heat will be transmitted to all points of the section of the exterior

Emission and Transmission of Heat 57

current of air. This arrangement may be useful, especially
when it is important to effectuate the transfer of heat in a small
space; but it often has drawbacks, either on account of difficulty
in construction, or in cleaning the surfaces of absorption and
transmission.

840. Most important of all, whenever it is a question of
transferring the heat of one body in motion to another, is it that
the latter body moves in a direction opposite to that of the
first, for as the hot body progresses it encounters the body, to
which it is to transfer its heat, at a lower temperature, and the
transmission continues, which would not be the case if the two
bodies were moving in the same direction. Consider, for ex-
ample, two concentric tubes, the inner tube conveying hot air or
water, and the interval between the two tubes, which we will
assume equal in sectional area to the inner tube, conveying cold
air or water in the opposite direction. If the tubes are suffi-
ciently long and the motion of the fluid slow enough, it is evi-
dent that there will be a complete exchange of temperature be-
fore the exit of the air, or water, whilst if the two fluids trav-
elled in the same direction, as one cooled and the other heated,
their temperatures would constantly approach and would finally
become the same, and from this point on all transmission of
heat would cease.

CONDUCTIVITY OF POOR CONDUCTORS OF HEAT.

841. The case of materials which are poor conductors of
heat is different from that of the metals, in nearly every case they
propagate all the heat which they are really able to transmit and it
is very important to know their conductive power.

M. Despretz is the single physicist who has occupied himself
with the conductivity of a few of these bodies. Experiments on
the propagation of heat through bars of marble, porcelain and
brick gave 23.6, 12.2, and 11.4 as the relative values of the con-
ductivities of the three materials. But the manner of making
these experiments presents not only the causes of error already
explained (829) in speaking of the conductivity of metals, but also
that resulting from the hypothesis of uniformity of temperature
at all points of the same section, an hypothesis the more untrue
the less the conductivity of the material under experiment. Even

58 Emission and Transmission of Heat

supposing these experiments to be perfectly exact, the results are
only relative and therefore cannot be used in practical applica-
tions.

842. In the second edition of this work I have given the re-
sults of some experiments on the conductivities of the materials
under consideration, but the methods then employed were not
exact enough to give in every case sufficiently accurate results.
I have taken up these researches again using varied and more
precise methods; the details of which will be found in an appen-
dix to this edition, I shall limit myself here to a review of the
methods used, and a statement of the results obtained.

843. In the first method which I have employed, the
material under experiment was enclosed between two spheres of
thin copper; the inner sphere was filled with warm water con-
stantly stirred, and the outer sphere was submerged in a bath, of
considerable size, of water at the ordinary temperature; this latter
was constantly stirred in the immediate vicinity of the sphere. It
can be shown by calculation that the cooling of the warm water
follows Newton's law, provided that certain relations between
the temperatures of different points of the spherical envelope hold
true at the beginning of the experiment, and these relations are
satisfied by bodies which are not too poor conductors of heat.
The coefficient of conductive power may then easily be deduced
from the rate of cooling. For quartz sand, mahogany sawdust
and starch paste, the rate of cooling followed exactly the law in-
dicated above, and I was able to obtain the conductivity of these
materials. But for cotton and powdered charcoal the cooling fol-
lowed a more rapid law and no definite conclusions could be
drawn from the experiments except that in the case of cotton the
rate of cooling under the same circumstances was sensibly the
same for densities varying from .0077 to .076, and I arrived at the
conclusion that the conductivity of this material was independent
of its density and in consequence that the conductivity of the fibre
composing it must be equal to that of motionless air.

844 . Figure 1 68 represents a vertical section of the apparatus,
A B CD is a copper frame placed in a leadplated iron vessel filled
with water at ordinary temperature. The lower member B C of the
frame is provided with three toothed wheels E fand G (fig 169).

Emission and Transmission of Heat

59

The wheel E is driven by the vertical shaft HI provided with a
steadiment at L and a crank K at its upper end. The wheel G is

loose on the fixed shaft
M, in which is sup-
ported the stem A7" sol-
dered to the surface of
the outer sphere,
which it is intended
to support.

The wheel G carries
a horizontal ring with
eight spokes , each pro-
vided with a copper
plate inclined at 45°;
at its circumference are
eight vertical stems
Q Q, bound together
at the top by another
horizontal ring; each
of these rods holds
arcs of circles R R,
R' R', provided with a large number of small inclined copper
plates, with edges almost touching the outer sphere,
copper cylinder fastened
at the upper part of the
outer sphere. It covers
three stuffing boxes
communicating with the
i n n er sphere ; through
one of these passes a ther-
mometer, through anoth-
er the stem of an agita-
tor, and the third, ordi-
narily closed, may re- V
ceive the stopper carrying the tubes designed to refill the inner
sphere when the level of the water has sunk.

The outer surface of the small sphere and the inner surface
of the large sphere are covered with lamp black. The outer

Fig. 168

V©

Fig. 169

6o

Emission and Transmission of Heat

sphere is in halves which fit into
one another, the joint being made
water tight with wax. Thermome-
ters, not shown in the figures, give
the temperature of the water in the
outer vessel. Throughout an ex-
periment the water in the inner
sphere was agitated by an apparatus
similar to that employed in the ex-
Fig- 17° periments on cooling in air (782)

and the water in the outer vessel was stirred by turning the

crank K.

The rate of cooling was measured in the same way as in the
experiments on cooling in air, and with the same corrections in
regard to the water introduced into the vessel before each obser-
vation in order to maintain it full.

845. In the second method I have used a hollow cylinder of
the material of which I wished to determine the conductivity; its
interior was heated by steam and its exterior exposed to the air in
the constant temperature chamber used in the experiments on
cooling in air. When the r6gime is established, the quantity of
heat passing through the walls of the cylinder is evidently equal
to that escaping from its surface. Now there exists a very simple
relation between the inner and outer radii of the cylinder, the tem-
peratures of the steam, of the outer surface of the cylinder and of
the surrounding air, the rate of cooling of^ the outer surface, and
the conductivity of the material forming the cylinder. All these
quantities can be easily measured, except the temperature of the
surface of the cylinder. We shall show presently how this has
been determined.

846. Figure 171 shows the general arrangement of the ap-
paratus. For powdered substances I used a cylinder of tin a b cd
eight inches in height, and of different diameters, covered with
paper; it was surmounted by a tube ef closed by a stopper at its
upper end through which passed the stem of a thermometer with its
212° mark just above the top of the stopper; this tube was pro-
vided with an outlet /£• communicating with a vessel producing
steam. Attached to the bottom of the cylinder was a tube h i k,

Emission and Transmission of Heat

61

bent at i, serving as a drain for the water of condensation and any
excess of steam. Three quarters of an inch below the cylinder a
bed, the tube h i was provided with three projecting pieces of iron

Fig. 171

three quarters of an inch wide, serving to sustain the wooden disc
/ m, which was put in place by removing the horizontal tube i k
and passing the iron projections through corresponding openings
in the disc; by a slight twist the disc was then held in position. Be-
fore introducing the disc there had been fixed upon its upper side
a cylinder n p q r of very thin glass, open at both ends, and of the
same diameter as the disc, and maintained in position by means of
a band of paper glued to both the glass and wood. The whole
surface of the glass cylinder was covered with paper and the sheet
of paper projected at both ends for nearly four inches. The mater-
ial to be experimented on was placed between the two cylinders
abed and n p q r, and the paper cylinders forming prolongations
of the glass cylinder were filled with carded cotton. The appa-
ratus which we have just described was suspended within the con-
stant temperature chamber, used in the experiments on cooling, by
means of the supports s t. Steam was allowed to flow into the tin
cylinder a b cd, often for several hours, so as to be well assured that
a constant regime had been established.

847. The temperature of the outer surface of the cylinder

62 Emission and Transmission of Heat

was obtained in the following manner ; a very thin band of iron
four-tenths of an inch wide, and six and a half feet long, was
soldered at each end to bands of copper of the same dimensions ;
one of the joints and the adjacent portions of the bands were
bound to the cylinder of glass, the other joint was secured in like
manner to a cylindrical vessel M containing water, an agitator
and thermometer and provided with a lamp by which the water
could be readily heated, finally the two free ends of the copper
bands were connected through a very sensitive galvanometer N.
When it was desired to measure the temperature of the surface of
the cylinder, the circuit was closed, and the water in the vessel
M heated until the needle of the galvanometer returned to zero ;
at this instant both joints between the iron and copper bands
were at the same temperature and consequently that of the sur-
face of the cylinder was the same as that of the water in the ves-
sel M. To fasten the joints against the surfaces of the two cyl-
inders I used a band of cotton, three quarters of an inch wide,
which enveloped the metal band and pressed it against nearly the
whole circumference of the cylinder ; the band was held clasped
by a buckle, and the free ends of the metal emerged from slits in
the cotton. The thermometer serving to indicate the tempera-
ture of the water in the vessel whose surface was in contact with
the second junction of the bands, was placed horizontally that it
might be the more easily read ; its stem being protected by a ver-
tical board beneath it ; one could easily estimate one-hundreth of
a degree. The vessel was heated by an alcohol lamp. The gal-
vanometer made by M. Ruhmkorff was sensitive enough to in-
dicate one-fortieth of a Fahrenheit degree difference in the tem-
peratures of the two junctions.

848. The cylinder of glass had no influence on the results,
on account of its thinness and the high conductivity of the ves-
sel. I also used a paper envelope glued to three tin cyl-
inders four-tenths of an inch wide of the same diameter
and with the same axis, held in position by two narrow
strips of tin (Fig. 172). The strips of tin made it possi-

ble to fasten closely the metallic band serving to measure Fig. 173
the temperature of the surface of the cylinder. This meth-
thod gave the same results as the glass cylinder.

Emission and Transmission of Heat 63

849. For solid bodies such as stone and marble I used hol-
low cylinders painted inside with oil, and covered on the outside
with paper; more often to be the better assured that water could
not penetrate the material the outer surface was covered with
tin foil. The cylinders were closed at each end by a sheet
of rubber and then by a disc of wood. In the case of the wooden
cylinders, the steam entered at top, and escaped at bottom as
with the tin ones. They were supported from the lower end,
and insulated at both ends by means of paper cylinders filled
with cotton. (Fig. 173). For wood I used the same arrange-
ment when determining the conductivity perpen-
dicular to the fibres ; but to measure it parallel

with them I employed portions of cylinders hav-
ing surfaces perpendicular to the fibres, these were
strongly compressed against a tin cylinder covered
with paper, having an outer radius equal to the
inner radius of the wooden cylinder.

850. Papers and cloths were rolled around a

tin cylinder, always covered with paper, and they Flg" 173 Flg- 1?4
were held in position by a cylinder of strong paper having a height
double that of the tin cylinder. This paper cylinder was held by
the metallic band (Fig. 174).

85 1 . The formula which gives the conductivity is as fol-
lows:

Q R'm(t"-t'") (log. JT-lQg. R)

t'-t"
/

In this formula C is the conductivity of the material, R, R'
the radii of the inner and outer cylinders; #2 = 2.3025; /,' /," /,'"
the temperature of the steam, of the outer surface of the envelope
and of the air; Q represents the coefficient of emission of the heat
from the surface, and equals K-\-K' (791), the values of A' and
K' having been corrected by the coefficients taken from the
curves of figure i, 2 and 3.

852. Finally in the last method I employed vertical rectang-
ular plates, insulated around their perimeter; one of the surfaces
was heated by steam, and the other exposed to the free air of a
constant temperature chamber. A formula even more simple
than that of the cylinders permitted the calculation of the con-

64

Emission and Transmission of Heat

ductivity from the same elements; but I have employed a differ-
ent and much more exact method of measuring the temper-
ature of the exposed surface. Opposite the heated plate was
placed a vessel of the same form filled with water; the opposing
surfaces of the two vessels were covered with paper, and the
poles of a very sensitive thermopile, connected with a galva-
nometer, were placed exactly equidistant from both; it is evident
that when the needle of the galvanometer was at zero, the tem-
perature of the exposed service of the plate was the same as that
of the water in the vessel.

853. Figure 175 is a plan of the apparatus, figures 176 and
177 elevations of the two longer sides, figure 178 a section per-
pendicular to the eleva-
tions . In all these figures
the same letters indicate
the same objects. ABC
D figure 175 is a rectang-
ular closed tank of sheet
lead full of water; it is
surrounded by a wooden
box X X X X; the bot-
toms and ends of these
two vessels are separated by spaces of two and a half inches.
The long sides of the tank are only partly covered with wood.
The spaces A X X C
and B X XD between
the ends of the tank
and the box are pro-
vided with a great num-
ber of metal strips sol-
dered vertically on the
tank, the object of this
is to impart to the air
passingdownthespac-

esAXXCand£XX Plg< I?6

D the temperature of

the water in the tank. G G and H H are two vertical rectangular
conduits passing entirely through the tank and communicating

X

L

P

r

L

X

5fe

» X

s

,

IT

L

L

X

X

Emission and Transmission of Heat

Fig. 177

Fig. 178

by the space beneath it with the openings A XX C and B XX D.
II is an opening in the tank but it does not penetrate completely
through it. K K (fig 178) are two horizontal tubes with the
same axis and of the same diameter, open at both ends, LLLL
are openings eight inches square formed in the long sides of the
tank, which pierce from the outside into the conduits G G, H H.
These conduits, cylinders and openings are surrounded by water,
MMMMo.ro. funnels surmounting openings through which pass
agitators. A7" TV are openings for thermometers. O is a thermo-
pile in the enclosure //, opposite the common axis of the two
tubes K K and equidistant from their inner ends; its poles are con-
nected with a very sensitive galvanometer not shown in the figures.
/'/'are substantial double screens, capable of movement about
the axis Q Q, and intended to intercept the rays of heat reaching
the thermopile through the tubes K K. The openings LLLL
receive the bodies whose surfaces act upon the thermopile; but as
these surfaces must always be at the same distance from the ther-
mopile, the inner edges of the openings LLLL are provided
with four stops RRRR against which the bodies are supported.
Figure 179, representing one side of the tank when the opening
LLLL is empty, shows the arrangement of these stops.

854 . One of the openings L L L L is always occupied by
a copper vessel 6* T (fig 178) full of water, and provided with two
agitators and a very sensitive thermometer; a part of its lower sur-
face is so arranged as to be easily heated by an alcohol lamp. This
vessel is surrounded by a frame of fir wood, which separates it

66

Emission and Transmission of Heat

from the perimeter of the opening, so as to avoid heating the wa-
ter in the tank, it is held in place by small wooden wedges.

M

X

X

f

1- • =

x

Bfl

L

X X

Fig. 179

Fig. 180

855. In the other opening are placed the plates of which the
the conductivity is to be measured ; these rest against the stops R
R R R and their outer surface is heated by steam. Their con-
struction is suitably varied for solid or powdered material. In
figure 178 we have assumed the material under test to be solid;
figure 1 80 shows the construction on a larger scale; a b cd\s a rec-
tangular box of copper having one face provided with a large
opening shown in figure 181 and 182, the first of which is an eleva-

Fig. 181

Fig. 183

tion of the face a b and the second a section perpendicular to this
face. Steam is admitted by the tube e f, and the water of con-
densation and along with it any excess steam escapes by the tube
g h\ a thin narrow rubber gasket is laid around the opening in the
face a b, and against this the plate is held by copper rods, hooked
at one end and provided with thumb screws at the other. The
vessel is fitted with a thermometer for ascertaining the tempera-
ture of the steam, and is surrounded by a wooden frame like that

Emission and Transmission of Heat

Fig. 183

of the water vessel in the opposite opening, it is secured in the same
way by wooden wedges.

Figure 183 shows the construction for pow-
dered material, the steam box is solid and sur-
rounded by a wooden frame to insulate it from
the tank, and the front of the frame is closed
by a thin sheet, either of glass or tin plate, cov-
ered on both sides with paper, the outer paper
serving to fasten the sheet in place. The pow-
dered matter is placed between the steam box
and the thin sheet.

856. Figure 184 shows the thermopile and
the means of adjusting its position, a b cd is the
pile, e and /are two terminals in contact with the poles and to
which are connected the wires from the galvanometer. The ther-
mopile is held by two rods g h, fixed
to a frame ik, carrying a pinion,
turned by a key, engaging in the
rack Im. By this means the pile
can be adjusted so that the needle of
the galvanometer is at zero when
the two surfaces, radiating to the op-
posite faces of the pile, are the same
and at the same temperature. The rack is supported by the two
rods / n, terminating in the threaded rings q r, into which are
screwed cylinders which pass through the openings KK in the
tank (fig 178) and are fastened therein by slips of wood.

857. Figure 185 represents the
arrangement used to determine the re-
lative radiating powers of different sur-
faces as needed for the purpose ex-
plained in paragraph 788. In the
openings L L L L, of the apparatus
just described, are placed two copper
vessels, one heated by steam, the other
containing water and so arranged that
the water can be heated to a suitable
degree; the inner faces of the vessels
are covered with the substances of which we wish to determine the

p

Fig. 184

68 Emission and Transmission of Heat

relative radiating powers, that which radiates the least being on
the vessel heated by steam. For glass I used the arrangement
shown in figure 178, the opening in the vessel being closed by a
thin sheet of glass held in the same way as already described for
thick plates of the same material.

858. As in the former methods, we may write an equation
between the quantity of heat traversing the plate and that emitted
from the exposed surface, and we thus arrive at the very simple
formula ;

e Q (?-?>)
/-/'

In which Cis the conductivity of the material, e the thickness of
the plate ; Q the value of K-\-K' modified by the coefficients cor-
responding to the temperatures (figures i , 2 and 3), /, / and /,"
the temperatures of the steam, of the outer surface of the plate
and of the surrounding air.

859. I have deduced from these experiments the following
tables of values of C. These numbers give the quantity of heat
in B. T. U. which would pass in one hour through a plate of the
given material one inch thick, one square foot in area, and of
which the two surfaces were at temperatures differing by one de-
gree Fahrenheit.

TABLE OF VALUES OF C.

B. T. U. per hour, per square foot, per inch, per one degree.
SOLID MATERIALS.

Density. C.

Marble, gray, fine grained .... 2.68 '28.1

Marble, white, coarse grained . . . 2.77 22.4

Limestone fine grained . . . . 2.34 16.8

Limestone do do . . . . .2.27 13.6

Limestone do do . .... 2.17 13.7

Limestone coarse grained . . . .2.24 10.6

Limestone do do .... 2.22 10.2

Plaster, ordinary . ..... 2.22 2.67

Plaster do very fine . . . . 1.25 4.20

Brick 1.98 5.56

Brick 1.85 4.11

Fir, (wood) transmission perpendicular to fibres .48 .75

Emission and Transmission of Heat 69

Density. C.

Fir, do do parallel do do .48 1.37

Walnut, do perpendicular do do .48 .86

Walnut, do parallel do do 1.40

Oak, do perpendicular do do 1.70

Cork . . . , . . . . .22 1.15

India rubber . ...... .22 1.37

Gutta-percha . . . . . . .22 1.39

Starch paste ...... 1.02 3.43

Glass ....... 2.44 6.05

Glass ........ 2.55 7.10

MATERIALS IN A STATE OF POWDER

Density. C-

Quartz sand ....... 1.47 2.18

Brick dust — large grains ..... i.oo 1.12

Brick dust — passed through a sieve of silk . . 1.76 1.33

Brick dust — fine powder obtained by decantation 1.55 1.13

Chalk, in powder, slightly damp ... .92 .897

Chalk, in powder, washed and dried . . . .85 .694

Chalk, in powder, washed, dried and compressed 1.02 .855

Flour of potatoes . . . . . . .71 .790

Wood ashes • . . .45 .484

Mahogany sawdust . . . . . . .31 .524

Charcoal, powdered . . . . . . .49 .637

Charcoal, powdered and passed through silk . .41 .653
Coke, in powder ...... .77 1.290

TEXTILE MATERIALS

Density. C.

Canvas of Hemp, new ..... .54 .420

Canvas, do do, old ..... .58 .347

White writing paper 85 .347

Gray blotting paper ...... .48 .274

Calico, new, of any density .... .403

Cotton wool, do do . . . . . .323

Sheep's wool, do do . . . . . .323

Eider down, do do . . . . . .315

860. It is important to notice that as the conductivity of
textile materials is independent of their density, it follows that

7° Emission and Transmission of Heat

their conductivity is the same as that of motionless air. The
conductivity of starch-paste may also be regarded as that of mo-
tionless water. I have also noticed, in the case of poor conduc-
tors of heat, that dampness greatly increases their conductivity.*

•TRANSLATOR'S NOTE. Although the experiments described above were made more
than fifty years ago there is but little to add due to more recent research. Pficlets figures
for the conductivity of the metals have been shown to be too low, due probably to the
fact the liquids in contact with the surfaces of the metal plate were not frequently enough
renewed. This defect would not disturb the accuracy of the experiments on the materials
of low conductivity, "and it is only these that we use in practical applications of this sub-
ject. Peclets figures for these substances have been repeatedly confirmed by more recent
experimenters. Peclet evidently made no experiment on the conductivity of motionless
air, simply reasoning that it must be the same as that of the textile materials. It has quite
recently been proven by direct experiment to be 0.152 which is about one-half of Peclet's
value, but in practice we never obtain perfectly motionless air, the process of convection
aiding to at least some extent that of conduction. In 879 we find experimental evidence
of the value of Peclets figure for the kind of practical application in which we most want
to use it.

For further notes on the reliability of this portion of Peclets work the reader is refer-
red to a very interesting paper by Mr. A. B. Reck of Copenhagen, presented in December,
1901, before the American Society of Heating and Ventilating Engineers.
The following table is from Jude and Gossln,— Physics.— 1899.

VALUES OF c

Silver 4440 Lead 334

Copper 3192 Ice 17.42

Gold 2100 Snow 2.03

Zinc 888 Water 4.41

Tin 572 Air 0.16

Iron 476

The ratios between these conductivities for the metals is almost the same as Frantz
& Wiedermann's and the value for lead is about three times as large as that found by
Peclet.

CHAPTER III
APPLICATIONS OF THE FORMULAS

861. We have already seen (826) that in designating by M,
the quantity of heat that traverses in one hour a plate with paral-
lel surfaces, of an area of one square foot, and with its surfaces
maintained at the constant temperatures / and /', we have

M=C-S<±> ....... (a)

In this expression e represents the thickness of the plate in
inches and C the conductivity, that is to say the value of Mfor
t-t'=i° F and e=i inch.

862. If the body were formed of two superimposed plates,
of thicknesses e and e' and conductivities C and C', designating
by 0 the common temperature of the surfaces in contact, we have
evidently when the regime is established:

n. C(t-e) n, C'(e-t')
M=—- - -and M= — -L, — -
e d

Eliminating 6 we have:

And for any number of plates:

863. By means of the tables (859) and the preceding formu-
las we may easily compute the quantities of heat which will be
transmitted through plates when the temperatures of their sur-
faces are known . But these temperatures never are known exactly
and can only be measured by very delicate experiments, impos-
sible in practical work. Furthermore, in making estimates, it is
necessary to have at least an approximate value of the quantity
transmitted, in terms of the temperatures of the air inside and out-
side of the surfaces.

864. Consider first a room enclosed by walls of which one
only is exposed to the outside air. Let the temperature of the

71

72 Applications of the Formulas

air within the room be T and that of the outside air be 0. The
r6gime once established the quantity of heat which would traverse
the wall exposed to the air would evidently be equal to that which,
in the same time, would penetrate its inner surface, and to that
which would escape during the same time from its outer surface.
The interior surface would be at a temperature / lower than T and
the exterior surface would be at a temperature /' higher than 0,
We may admit that the heating of the inner surface and the cool-
ing of the outer take place according to the same laws. Thus in
designating by M the quantity of heat transmitted per square foot
per hour we would have three expressions for M, one in terms of
of the conductivity C of the material of the wall, the two others
in terms of the coefficients K and K' of cooling by radiation and
contact of air, from which equations we may deduce the values of
/ and /' in terms of known quantities. . But if we employ Dulong's
formulas for cooling (794-797)^6 calculation would be impossible,
and even admitting the simpler formulas of 789 and 790 we would
be led to an equation of the second degree, rather complicated
and extremely difficult to handle. It is better to admit for the
cooling and the heating Newton's law (791) which is sufficiently
exact for small differences of temperature, and besides in all cal-
culations relative to the transmission of heat one can never expect
more than a rather rough approximation because there are cir-
cumstances which are impossible to take into account, such as
the increase of temperature of the outer surface according to its
height above the ground, the action of the wind, of the sun, and
so forth. According to this we have:

equations which give,

.T(C+Qe)+C9 0(C+Qe)+TC

865. From this formula follow several important conse-
quences. If Qe was very small relatively to 2C and could be

neglected, the formula reduces to M—-- (T-0) and the value of

CQ(T-e)

Applications of the Formulas 73

M would be independent of C and e, that is to say of the material
of the body and of its thickness. This case may occur when the
value of e is very small relatively to that of C.

Consider for example lead, the poorest conductor among the
metals, and for which C= 113; supposing the surfaces of the plate
to be dull, Q will equal approximately 1.25 and for thicknesses of
•395"> -79" and 1.18" the values of zC-\-Qe would be 226+ .494,
226+ .988, 226+1.48, numbers which differ very slightly. The
differences would be even less for the other metals. If we imagine
a plate of woolen material, for which the lowest value of C is .323,
to have a thickness of .004", which is about that of a sheet of
paper, the value of zC-\-Qe would be equal to .646 + .005; the
second term being again small relative to the first may be neglected
and the value of M would be the same as in the preceding case.
Thus a sheet of paper transmits the same amount of heat as a
metal plate, of which the thickness may be varied through quite
a wide extent.

The case of thin sheets of glass is similar. Since C for glass
equals 6.05; <2=.6o+.45 = i.o5 and 2C-\-Qe=i2.i-}-i.o5e and
for thicknessesof .04", .08" and .12", this last expression becomes,

I2.I-j-.042, I2.I+.O84, I2.I + .I26.

866. If we suppose C very small and the thickness e so large
that 2Ccan be neglected in comparison with Qe, the value of M

C

reduces to — (T—0), consequently it would be independent of the

nature of the surface, and inversely proportional to the thickness
e; but it is necessary, even for the poorest conductors, that the
thickness be very great. For example for woolen cloth with^=
19.7" we would have 2C-\-Qe=. 646+ 1.23X19.7=. 646+ 24. 2.

867. If we had two walls in immediate contact with one
another, admitting that there is no sudden change of temperature
in the passage of heat from the first to the second, which is con-
firmed by experiment, and designating by x the temperature of
the junction of the two walls, by e and e' their thicknesses, by C
and C their conductivities, we would have after the regime was
established:

** M=C <x7t:) M=Q(T-t)

74 Applications of the Formulas

M=Q (t'—O), equations which give,
M= -

For any number of walls, of thicknesses e, e' y e*', and con-
ductivities C, C, C":

868. To illustrate the method of using these formulas we
will apply the first (a) (864), to a particular case. Imagine a
wall built of limestone 32.8 feet high with a conductivity (7=13.71 ,
the coefficients K and K' being .737 (794) and .40 (figure 7) we
have Q =1.1 37; we will suppose 7~=59° and #=.42.8°, the value
of T is the ordinary temperature of dwellings and that of 0 about
the average value of the exterior temperature at Paris during the
seven months of heating.

The curves of figure 12 show the effect on m, t and /', of the
increase in thickness of the wall.

869. The preceding formulas apply to the transmission of
heat through a wall exposed to the open air, only when all the
other walls of the room may be considered as having approxi-
mately the same temperature as the air within the room, a con-
dition which can hardly exist unless the first wall is the only one
exposed to the outer air. When all the walls of the room are
exposed to the outer air, all the inner surfaces are at temperatures
differing very little one from another, and lower than that of the
air in the room, and consequently for the same temperature of the
inside air, the quantity of heat transmitted under the same cir-
cumstances, per square foot per hour is smaller than in the case
previously considered. This is the case with isolated buildings
with only one room on a story, and also with churches.

870. In the case where all the walls of a room are exposed
to the open air, the heating of the inside wall surfaces is effected
entirely through the movement of the air, for since these inside
surfaces have the same temperature their mutual radiation is with-
out effect. Then, keeping to the preceding notation, we have:

Applications of the Formulas 75

i '^ M=Q(t'—0) M=K' (T—t)

From these equations;

Q(eICT+Ce)+CK' T

Q(eK' Q+CQ)+CK' T

K'CQ(T-e)

871. If the wall was made up of two walls, in immediate
contact, of thicknesses e and e' and conductivities C and C' we
would have:

e e'

M= Q ( t'— 0) M=K[ (T— t)

and consequently:

K'Q(T-V)

If there were any number of walls the general formula
would be:

M= ^Q(T-o)

Q + K'+K' i

Applying this case to a wall as in 868 with C= 13.71,
^=.737 and A"=.4o, Q=\.\y] 7^=59° and 0=42. 8° we get
the results shown by the dotted lines of figure 1 2 .

It is useful to notice that if the walls had a height of 66 feet
the value of A"' would be .39 instead of .40 and we would obtain
practically the same numerical results.

872. The values of ^/obtained in this last case are much
smaller than in the first; this is due, as already explained, to the
much lower temperature of the interior surface of the walls.

873. I would remark that, in the two cases just examined,
it is necessary to take some precautions in measuring the temper-
ature ; if the thermometer were exposed freely to the air, the tern-

76 Applications of the Formulas

perature which it would indicate would be that of the air modi-
fied by the mutual radiation between its bulb and the surroundings,
and in consequence it would show a temperature lower than that
of the air. It is necessary to prevent radiation from the bulb to
the walls of the room by surrounding the bulb with several con-
centric envelopes open at top and bottom, and the air between
the envelopes should be rapidly renewed. The effect on a ther-
mometer of the fall of temperature of the inner surface of the walls
would evidently be felt by a person in the room, and consequently
in order that the sensation of heat should remain the same, the
temperature of the air must be increased as the temperature of the
inner surface of the walls decreases.

874. The two cases which we have been considering are
never precisely fulfilled in practice ; in the first case the inner sur-
faces of the unexposed walls are never exactly at the temperature
of the air, on account of their radiation to the other walls and the
windows ; in the second case there are always portions of the inner
surface of the rooms, such as the ceilings and floors which are not
exposed to exterior cooling, and often there exist interior walls,
such as those separating the naves of churches. These walls are
heated by air and radiate to the inner surfaces of the outer walls.
Finally in both cases, if there is heating by radiating surfaces, by
stoves, radiators or pipes, the rays of heat reach the inner surfaces
of the walls and raise the temperature of these surfaces. But, as
we shall see further on, we may consider the effects produced by
these two cases as the extreme limits of those which are generally
met with in practice.

DISCONTINUOUS WALLS

875. So far we have supposed the walls to be continuous,
but if they were made up of walls with parallel surfaces, separated
by intervals occupied by air, the quantity of heat transmitted
might be considerably smaller. If we suppose the intervals to be
sufficiently large to permit free movement of air, we may admit
without fear of departing very far from the truth, that the quantity
of heat transmitted across the air spaces is represesented by

Q (x — x' ), x and x' representing the temperatures of the surfaces
forming the air space. If the air space was filled up with a mate-
rial of thickness e and conductivity C the heat transmitted would

Applications of the Formulas

equal — (x — x' ); thus we may obtain the value of M, in the two
cases that we have considered by replacing in the general formu-
las -~ by -= - .

Thus the general formula relative to the first case, assuming
one and then two air spaces, becomes:

M=_ Q(T-«)

M=_ Q(T-o)

f e i .e' i e"
\C + Q ^C ^~Q^C"

If the walls were all of the same material and of the same
thickness e, and n in number, we would have

Assuming a continuous wall of the same total thickness, the
quantity of heat transmitted would be:

M'= Q (T~ 0) Q(T-O)

and we would have

M'

876. If we assume the air spaces and the walls each to be
.79" thick, and the walls to be of brick, we would have Q- =1.14
(7=4.84 and the ratio:

M_ 2-f.i85 (in — ij
W'~2-\-.iS5 n+ (n— i )
making n successively equal to

i 2 3 4 5 10

we find for M^-M'

i -75 -64 -57 -53 -43

78 Applications of the Formulas

877. It would seem at first sight, that it would be advan-
tageous to diminish the thickness of the layers of air in order to
render it motionless; but then there would be a direct transmission
of heat through the air and if the thickness were too small, the
transmission would be greater than if the air could move freely.
In such an air space the quantity of heat transmitted is given by

(C \* C

A"-| -- ; 1 and if we suppose e=.jg" — would equal

which is practically the value of A" and for a smaller

value of e, the factor \K-\- - - j would be greater than Q.\

In order that air spaces may diminish the transmission of

€ I

heat, it is necessary that — ^- be always smaller than . . , — .\

L K + .32

e
Evidently then the air spaces will be especially advantageous

when the surfaces of the walls forming them have a very feeble

radiating power.

It results from the preceding that hollow bricks should trans-

mit much less heat than solid ones of the same thickness, and

this is perfectly confirmed by experiment.

878. Similar results will be obtained in the second case that
we have examined (870), by making the same modifications in
the general formula relative to superimposed walls of different
materials; there will still be, as in the case which we have just

g
studied, a decrease in the transmission whenever -^ would be

larger than the transmission through a sheet of air, augmented by
the radiation between the opposite surfaces of the air space.

879. It is now easy to find the loss of heat from a vessel
surrounded by envelopes separated by extremely small air spaces.

*Radiation = K (x—x1)

Conduction = (x— x')

Then M=(x-x')( K+- )
Note that in 875 we assumed M=Q (x—x')=(K+K') (x—x')

flf we take the value of Cfor air given in the foot note to page — , — does not
equal .4 until e has been diminished to .38".

t This may be seen by observing the general formulas of 875.

Applications of the Formulas 79

Thus assuming a single envelope and designating by / and /' the
temperatures of the surfaces of the vase and the envelope, and by
/" the temperature of the exterior air, we have:

, M=(t—t') K+-~
from which :

Assuming two envelopes and designating by 0 the tempera-
ture of the second surface, we have:

M=(t—0')

\ c /

M=(6-t') '-' CN ^rom which

(t-t")

Increasing successively by one the number of envelopes we
find that the coefficient of ( K^ K' ) in the denominator of the
fraction increases successively by one, and we are led to the gen-
eral formula:

c

K+
(t-t")

in which K is the radiation from the interior surfaces, Kv that of
the exterior surface, K' the quantity of heat abstracted from the
exterior surface by contact of the external air, C the conductivity
of the air, e the thickness of the air spaces, and J/the number of
envelopes.

If we take ^=^ = .772, A"=.788 C=.$2 and ^=.04" for
the following values of m,

01234
we find that the relative values of M are,

i .87 .77 .69 .62

8o Applications of the Formulas

Direct experiments have given,

i .90 .75 .67 .60

If we create a vacuum between the envelopes, the transmis-
sion of heat will only take place by radiation, and the quantity
transmitted may evidently be deduced from the preceding for-
mula, by making C therein equal to zero. It then becomes,

Assuming the envelopes to be of zinc, we would have K=
^=.492; supposing their number to be ten, and taking A"=.82,
/=2i2, /"=59 we find M=.jS. To compare this transmission
with that that would take place if the interval between the vessel
and the outermost envelope were filled with eiderdown, one of
the poorest of conductors, assume the interval to be .4". The
formula for the transmission of heat through a plate is
C Q ( t — t" }* y in which C is the conductivity of the material,

C-\-Qe which for eiderdown is .29, Q the loss of heat
from the outer surface which is here .87, and e the thickness
which we have taken as .4". For these numerical values we find
the quantity of heat transmitted to be equal to 68.2 nearly ninety
times greater than with the envelopes and vacuum.

The means that I have just indicated for diminishing the
transmission of heat is the most efficacious that I know. To
apply it to two concentric metal cylinders it is necessary to close
both ends of the interval separating them with poorly conducting
material, then to solder near one end a very small lead pipe,
which serves to form the vacuum and is then closed by squeezing
it together and melting off the end in the flame of a blow pipe.

This arrangement would be especially advantageous in appa-
ratus destined to make ice.

TRANSMISSION OF HEAT THROUGH WINDOWS

880. We will examine two cases of the transmission of heat
through windows ; the first will be the same as the case first con-
sidered of transmission through a wall, and, the second the case of
an enclosure formed entirely of glass.

*Note that in this case we know the temperature of the inner surface of the wall
and of the outer air, and thus M= (t—f) and M=Q (t'—t"). Combining these we get
the equation given above.

Applications of the Formulas 8 1

88 1. Consider first that the windows are placed in the
exposed wall of a room having one only of its walls exposed to
the outer air, the inner surfaces of the other walls will be sensibly
at the temperature of the interior air. As the rays of obscure
heat do not pass through glass, the windows will be heated on
one side by radiation from the inner surfaces of the walls and by
the contact of the warm air, and will cool on the other side from
analogous causes. Admitting that the heating and cooling take
place in the same way, for the same excesses of temperature, and
remarking that for the small thicknesses of window glass, the
quantities of heat transmitted are independent of the thickness,
as we have already seen (865), we will have, preserving the pre-
vious notation, M= (T—x) Q M= (x — 0) Q; from which

"p Q 'p 0

x=- — andJ/= Q

2 2

For heights in feet of:

3.28 6.56 9.84 13.12 16.40

the values of K' (802) being equal to

.492 .453 .437 .427 .420

and the radiation of glass being equal to .596, we find for these
different heights and for a difference of temperature of i° Fahren-
heit between T and 0, the following values of M.

.542 .524 .516 .511 .508

The greatest of these numbers is smaller than that I formerly
found by direct experiment, because I used a window of smaller
height, and did not take all the precautions which I have since
recognized as necessary.

If the inside temperature was 59° and the otside 42.8°, that
of the glass would be 50.9°, and the quantities of heat emitted in
B T U per hour per square foot would be, for the heights we
have been considering :

8.79 8.50 8.36 8.28 8.24

882 . Let us consider now an entirely glazed enclosure heated
by warm air, and let us disregard the effect produced by the floor
or ground. The glass will only be heated by the air, since all
the surfaces are at the same temperature, and their mutual radi-
ation would therefore have no effect. We would then have, pre-

82 Applications of the Formulas

serving the previous notation: M=(T — x)K' M=Q(x — 0)
equations which give: ,

K'T+QO .. QK'(T-e)
;r= — — and M= —

We.find, as in the preceding case, that for heights in feet of:

3.28 6.56 9.84 13-12 16.40

the quantities of heat transmitted in B T U per hour per square
foot per i° Fahrenheit difference, are:

.338 .816 .305 .301 .297

For an interior temperature of 59° and an exterior tempera-
ture of 42.8°, the quantities of heat transmitted are:

5.48 5.11 4.94 4.87 4.80

These numbers are smaller than those we found in the preceding
case (88 1 ) because the glass is at a lower temperature.*

883. The two cases which we have just examined, like those
relative to the transmission of heat through walls, never actually
occur. In the first case the walls opposite the windows are always
at a lower temperature than the air, and in the second case there is
always a part of the enclosure which is not glazed, and when the
heating is effected in part by the radiation from hot surfaces, the
rays of heat which fall on the glass increase the quantity of heat
which it transmits. But in any case, the quantity of heat actu-
ally transmitted is comprised between those which we have com-
puted for the two extreme cases. We will return to this question
in speaking of the heating of dwellings.

*The following note from London Engineering, Nov. I, 1902, is of interest here:
"In cold countries double glazing is sometimes resorted to, in order to reduce the
heat lost from a room to the exterior through the windows. Some experiments made by
H. Schoentjes, of Ghent, show that there is a certain distance of separation between the
glasses, at which the heat lost is a minimum. The glass used in his experiment was .08"
thick and the loss was least when the distance between the opposing sheets was some-
where between 2.6" to 4.6". The loss in calm air through one thickness of the glass was
at the rate of about .42 B T U per square foot per hour for each degree Fahr. of the dif-
ference of the temperature on the opposite side of the sheet. The experiments were made
over a range of 12.6° to 40° Fahr. and the rate of loss was somewhat greater as the differ-
ence of temperatures increased but the mean was as stated. . With double walls at the
best distance apart the rate of loss was about halved. Wetting the outer surface of the
glass increased the loss about 39 per cent; whilst if at the same time, a current of air was
directed over the outer surface the rate of loss was still further increased up to about .93
B T C/per square foot of glass per hour. The utility of the second layer of glass which
can be kept dry and in still air would in such a case as this, be very marked.'

Note that the rate of loss .42 B T U is about a mean between Peclet's calculations
for his two extreme cases. The relative value of the double window appears somewhat
greater however.

Applications of the Formulas 83

MULTIPLE WINDOWS

884. In the case of several parallel windows, separated by
intervals sufficiently large to allow the air to move freely, the two
surfaces of each window are sensibly at the same temperature,
and we may obtain the value of M, in the first case that we have
considered, by supposing the thickness e, e* , e" equal to zero in
the general formula (875). Then for 2, 3, 4 . . n windows,
the values of M would be
Q(T—e) Q(T—0) Q(T—Q) Q (T—0)

2 + 1 2 + 2 2 + 3 2 + » — I

and the relative values ot these quantities to that of a single
window are

212 2

3; 2; 5 i+»

Curtains, more or less thick, would produce sensibly the
same effect. If the distance between the windows was less than
.79", the transmission would be increased, because the transmis-
sion through the air ;f-f is equal to the transmission by contact
of air and its renewal, and at smaller distances the transmission
through the motionless air would be greater.

In the second case, that we considered in regard to walls,
the calculation relative to the transmission of heat through win-
dows, would be extremely complicated, because it is necessary
to take into account the surfaces of the windows, and we could
arrive at no exact conclusions, on account of the influence of
floors and ceilings, the effect of which could not be evaluated.

The case first examined evidently gives the maximum trans-
mission ; it is the only one in which we can calculate with suffi-
cient exactitude the quantity of heat transmitted by the windows,
and the figures obtained suffice in any case to give a sufficient
appreciation of the quantity of, heat lost.

TRANSMISSION OP HEAT THROUGH CYLINDRICAL ENVELOPES

885. The case that we will now consider is, for example,
that of a metal pipe traversed by steam, and surrounded by some
material of low conductivity, in order to diminish the loss of
heat from the steam during its passage.

Let M represent the quantity of heat in B T U transmitted

84 Applications of the Formulas

per foot of length, in one hour, R and R' the internal and exter-
nal radii of the cylindrical covering in feet, / and /' the tempera-
tures of the internal and external surfaces, and 0 the temperature
of the surroundings. When the regime is established the quantity
of heat which passes through the covering is equal to that which
at the same time traverses an infinitely thin annular element of
radius r, this quantity being equal to the product of the surface
2 *r of this element, the conductivity C' of a thickness of one
foot of the material, and the temperature difference dt of the two
surfaces, divided by the distance apart dr of these surfaces. We
will have then :

M= ; / from which C1 d t=

dr 2 iir

The minus sign shows that the variations of the tempera-
ture and of the radius of the covering take place in opposite
directions. Integrating the last equation between the limits /
and /' for d t, and R and R' for d r it becomes :

C (t — f ) = M-m (logR'—logR)

27T C (t—f)

and M= j^-

where m is the modulus of the table of logarithms and equals
2.3025 and N represents m (log R' — log R).

But we also know that M=2 - R' Q ( t' — t) ): eliminating /'
from these equations we have :

_ 27T R' Q (t—0)

>Q*'N ' (a)

C'

If there are two contiguous coverings, designating by x the
temperature of the common surface, we have:

M .. ~ , , M ,.

C (t—x)= — N; C,' ( x—t' ) = — N'

27T 27T

and M=z * R" Q ( t' — 0 ) equations which give, after eliminat-
ing*.-

M= 2-^£ *" " *}

Applications of the Formulas 85

Repeating the calculations for 3, 4 and so on, coverings, we
are lead to the general formula :

2T.Q R<*) (t—0)
M=

886. Returning to the formula relative to a single covering:

z*R'QC(t-0)
C' + QR' m (log R' — log R)

if we suppose C' to be very small relatively to Q R' N, the for-
mula reduces to

2 TT C (t—Q)
m (log R— log R' )

an expression independent of Q and decreasing as R' increases;
thus in this case the transmission does not change with the nature
of the surface. If, on the contrary, the value of C' was very
large relatively to the following term we would have M= 2 K Q
R' ( t — 0J, an expression independent of (T'and increasing in
proportion to the increase of R' .

The first supposition is realized with a covering of wool ; the
second if we suppose the covering to have almost the conductiv-
ity of the metals.

887. The relation of this value of Mto the quantity of heat
which would be transmitted under the same circumstances by a
bare pipe is evidently equal to

C_ R_

R C' + QR' m (log R'—log R)

An inspection of this formula will show that it is not always
advantageous, as regards the loss of heat, to apply a covering,
even one of low conductivity, for the value of this expression is
not necessarily less than unity; and for a given value of C' it
varies with R and R' . There are certain values of C', belonging
to bodies reputed to be poor conductors, which give for ^/values
greater than those for a bare pipe, and consequently for these
bodies the increase of surface has more effect than the resistance
to the transmission of heat through their thickness.

888. I<et us take for an example a horizontal cast iron
pipe, 2.25" radius and one foot long, filled with steam at 212°
F.( and covered successively with different thicknesses of hair

86 Applications of the Formulas

felt. If we assume the temperature of the surrounding air to be
59° F., it results from what we have said in (807) that the quan-
tity of heat emitted per square foot per hour when the pipe is
bare is 298 B T £7, and for one foot of length this becomes 352.
To find the quantities of heat emitted per lineal foot per
hour, by the pipe when it is covered with a layer of hair felt of

.5" 1.0" 1.5" 2.0" 2.5" 5-0"

in thickness, we must, in the formula (885), make C'=-:ff =
.03 R= .188 feet and give to R' the successive values, in feet:
.229 .271 .313 .354 .396 .604

To obtain the values of Q, we must remember that Q = K
-f- K' , K and K' being the coefficients of cooling by radiation
and air contact ; if we suppose the covering material to be
wrapped with canvas, A' will equal .75; and the values of K'
we may take from figure 4.

These values of K' are as follows :

•53 -5i -50 -49 -49 -46

then the values of Q are :

1.28 1.26 1.25 1.24 1.24 i. 21

The values of N are :

.200 .367 .510 .636 .747 1.170

Substituting the constants in formula b (886) it becomes :

28.84 QR'
W+QR'N

and we obtain the following results :

Thickness of covering

in inches. Value of M.

= ^9

9-86

1.0 -=63.6

.155

11.30

1.5 - = 49.2

.230

12.65

2.0 -=41.0

•309
14.12

=35'7

21.08

5-0

Applications of the Formulas

If the coverings were enveloped with a sheet of tin we
would have A"=.O9, and consequently the values of Q would
become :

.62 .60 .59 .58 .58 .55

and we would find for the values of M :

70.7 52.2 43.0 37.0 32.9 22.4

The influence of the feeble radiation from the tin surface
diminishes as the thickness of the covering increases, because the
value of C' relatively to the term Q R' N is diminishing and
because if C' could be neglected, the value of M would be.
entirely independent of Q.

889. In the preceding calculations we have taken C= .36
which gave .03 for C' ; if we suppose a conductivity two, four;
eight. . . . N times as large, it would suffice, to obtain
the corresponding values of M, to multiply the numerators o/
the fractions in the preceding table of values of M, by 2, 4, 8

. N, and to add to the denominators :
•03, .09, .21, 03 f«— i)

It is in this way that we have obtained the numbers in the
following table. These correspond to the same values of R' anrt
to the same temperatures. The quantity of heat emitted frou
the bare pipe would of course remain 352 B T U.

Quantities of Heat in B T £/ transmitted per foot run per hour by a
horizontal pipe 4. 5" outside diameter, heated to 212° and with
external air at 59° and covered with materials of different conduc-
tivities and different thicknesses :

Value of

.36

5.76

Value of
C'

.03

.48

Thickness of covering in inches.

•5

I.O

1-5

2.0

2-5

5-

B. T. U.

94-9

63.6

49.2

4I.O

35-7

23-7

251-

261.

265.

266.

267.

252.

266.

292.

312.

325-

34i.

372.

11.52 .96

890. We see from this table that the values of ^/decrease
very rapidly with increase of thickness when the conductivity is
very low ; that the variations are very small for values of C in the
neighborhood of 5.76, and that for greater values of Cthe values
of M increases with the thickness of the covering. For pipes of

88 Applications of the Formulas

other diameters, the same phenomena recur, but at other thick-
nesses.

891. *.

892. In all that precedes in regard to cylindrical coverings,
I have supposed the value of Q to be constant, that is to say that
Newton's law of cooling held true for any excess of temperature ;
but this is not the case as we have seen in 805, and the value of
Q increases quite rapidly with the excess of temperature ; thus the
different values of M calculated by the formula of 885 , can only
be considered approximations, the more exact the smaller they
are, since the excess of temperature decreases with M. But it is
easy in any particular case to obtain a value of the quantity of
heat which is very close to the truth. Suppose that we have first
calculated the value of M by the formula (885); dividing this
result by 2 K R' Q we will have the temperature /' of the surface,
and by means of the curves of figures i, 2 and 3, we may deduce
a new value Ql of Q, then new values M^ and /1} and we may
repeat the process, until two consecutive values of M are the
same.

I will take for an example a horizontal cast iron pipe 4.5"
outside diameter covered with one -half inch of hair felt, filled
with steam at 212° and surrounded by air at 59°. We have
already found that for this case ^/equals 94.9. Since the surface
per foot run is 1.18 sq. feet and Q was equal to 1.28 we have

//'_0i=94^ =62.8°

' 1.18X1.28

We know that ^=.75 and A"' = .53 then from figures i, 2
and 3,

Qi= -75Xi. 22X1 + .53X1. 25 = i. 58

Then ^ = 102 and (t' — O) 1=54.7° then £2 = i -55 and
M2 = 101.5

Thus the value of Afis 101.5 instead of 94.9, not a large
difference, and it would have been much smaller if the covering
had been thicker.

893. In general when the pipes are of small diameter, the
second term of the denominator of the expression for M is very
large in relation to the first term, at least when the covering is a

*The table given in 891 is not reproduced as it has but little value for modern condi-
tions.

Applications of the Formulas 89

poor conductor of heat. In this case we see that the denomina-
tor, like the numerator is nearly proportional to Q, and therefore
the value of Q has only a small influence. Thus the numerical
results that we have given may be considered sufficiently accurate
for many practical purposes, but this would no longer be true if,
for the same difference between the inside and outside radii, the
value of the inside radius was very large. In this case we find a
notable difference between the direct calculation and the method
of approximation which I have just indicated.

894. I will take a steam boiler for an example. The sur-
face is generally covered with a poor conductor of heat, and it is
important to recognize the influence on the loss of heat of the dif-
ferent materials which may be employed, I will assume that the
cylinder forming the boiler has a diameter of 3.28 feet; the quan-
tity of heat emitted per hour per square foot is given by the fol-
lowing formula :

M= Q ° (l-e) ' • <*>

C+,Q R' m (log R' — log R)

Assume as before £?— -43+-75 = i-i8 and that the covering
material is sawdust mixed with a little clay and cows hair to ren-
der it plastic ; we may take C= .80 and therefore C' = .067.

We will suppose the boiler to be filled with steam at 212° and
the surrounding air to be at 59° Fahrenheit.

With the following thicknesses of the covering
.4" .8" 1.2" 1.6" 2.0"

the formula (a) gives directly for M

113.0 81.8 63.7 52.6 44.7
and by the method of approximation indicated

132.0 89.5 67.4 54.5 45.8

If the surface of the covering is finished by a sheet of Russia
iron, we have by the formula

63.8 52.7 45.0 38.8 34.2
and by the method of approximation,

82.7 63.0 51.0 42.8 36.5 (4}

It follows from these numbers and from the fact that the
emission from the bare surface is 260, that the coverings reduce
the transmission to,

.509 .344 .259 .210 .176

90 Applications of the Formulas

and the same coverings covered with Russia iron :

.318 .243 .196 .165 .141 (6)

The influence of the Russia iron is very great, but it dimin-
ishes with the thickness, for the relations of the corresponding
numbers of series 5 and 6 are :

.63 .71 .76 '.79 .80 (-])

As the steam pressure increases the percentage of saving due
to a covering somewhat increases.*

i

TRANSMISSION OF HEAT THROUGH SPHERICAL ENVELOPES

895. Preserving the same notation as before, we find

4 TT r * C d t x*» j .. *jl r

M=—- -- -j - ; or 4* C' dt= — M-~
dr r2

and integrating between the limits / and /' for t and R and R' for
r we find

whence M=—

K — K

But since M=<\ n R'2 Q ( t — 0 ), we may eliminate / and obtain;

**CQRR'(t-e)

CR+QR' (R'—R)

an equation in which M represents the quantity of heat in B. T.
U. per hour emitted by the total surface of the sphere. To
obtain the emission per square foot, we must evidently divide M
by 4 *,£'.*.

DIFFUSION OF HEAT

896. In all that precedes we have only considered the trans-
mission of heat through a body after the establishment of a per-
mament r6gime of temperature. In this case the laws of the trans-
mission are very simple, and the formulas that we have given

*For a more modern example we may take a Scotch boiler twelve feet in diameter,
under liio pounds steam pressure, with the surrounding air at 100°, and covered with two
inches of magnesia. The values of C', K, and K', will be .04, .74 and .42

By the formula ./>/=- 53. 7 but by successive approximations thus becomes 55.7.

If we jacket the covering with Russia iron the formula gives 44.7 and successive
approximations brings this to 48.2.

The heat loss from the bare boiler would be (by 807) 685 B. T. TJ. per hour per square
foot. The plain covering has a loss of only eight per cent of this amount and when
jacketed with Russia iron only seven per cent.

Applications of the Formulas 91

allow the computation of the quantities of heat transmitted in the
different cases which ordinarily present themselves. But before
the establishment of the regime, in bodies limited by two surfaces
of which one receives the heat and the other emits it, and during
the entire duration of heating, for bodies unlimited in one direc-
tion, the temperatures of different points vary with their position
and with time, according to very complicated laws, which depend
at the same time on the form of the body, on the conductivity of
the material of which it is constituted, on its specific heat and on
its density ; thus bodies formed of materials which are the best
conductors of heat are not always those which disperse it most
rapidly, because the dispersion depends on the relation between
the conductivity of the material and its specific heat.

897. It results from mathematical calculations, too compli-
cated to give here, that if we consider a plane surface of unlimited
extent, maintained at a temperature 7, and beneath this surface
a homogeneous body, of very great thickness, at the temperature
o° Centigrade, after one minute the temperatures at distances of

.04" .4" 4-" 40."

will be for sand 0.98870. 88y7o. 15470.000 7

limestone 0.99770.97370.7457 o.ooi7

iron 0.99970.99270.92470.3397

coarse grained marble 0.9997*0. 99370. 938?" 0.4397

plaster 0.99670.96270.62470.0007

" motionless water 0.99370.93970.39970.0007

The formula by which these numbers have been com-
puted is a rigorous deduction from the fundamental principle of
the transmission of heat and this principle has been proven by
too great a number of experiments to allow any doubt of its
exactitude ; but in establishing this formula the effects of expan-
sion and of variation of specific heat with temperature have been
neglected; however, since for solid bodies the expansion, varia-
tion of specific heat and variation of conductivity are small we
may regard the formula as very approximately representing the
facts. Thus the numbers which we have given show with what
rapidity heat diffuses itself through bodies even when of low con-
ductivity, after the establishment of a permanent regime of tem-
perature.

92 Applications of the Formulas

899. If we apply the formula to air supposedly motionless,
as it would be if heated from above, we would certainly obtain
but a rather vague approximation, on account of the large expan-
sion that it would experience and the unknown variations of its
conductivity with increase of temperature. However, as the
results of the calculation may at least give an idea of the rapidity
with which heat disseminates itself in air we will indicate them.
After one minute and at distances :

.04" 4." 4" 40"

the temperatures given by the formula are:

0.9997" 0.9967" 0.9607" O.62O7"

If the formula employed was truly applicable to air it would
result that the dispersion of heat in air would be grater than in
any other substance ; but we may certainly conclude from these
figures that the diffusion of heat in air takes place with great
rapidity. Furthermore this fact explains many phenomena which
appear very strange.

900. In churches heated by warm air escaping from a num-
ber of openings in the floor, the temperatures of the air at heights
of 6^ feet and 66 feet differ by less than two degrees as has been
proven at L,a Madeleine and Saint -Roch.

In the cooling of bodies by contact of air, the quantity of
heat emitted diminishes very slowly with increase of height of
the body, which can only be explained by the rapid diffusion of
the heat in the surrounding air. Another result of this fact is
that in the heating of rooms by open fire places, not only a part
of the radiation is utilized but also that portion of the heat pro-
duced which is transmitted by diffusion to the surrounding air.

901 . M. Darcy , chief engineer of the ponts et chausstes made
very interesting experiments on the cooling of warm water pass-
ing through pipes buried in tke ground. The total length of the
cast iron pipe was 7610 feet. Its diameter varied between 6.4"
and 9.84". The weight of water passing in one second was 8.12
pounds, the cooling was from 80.16° to 69.64° that is to say
10.52°, the loss of heat per second in B. T. U. was then 8.12 X
10.52 = 85.5 and per hour 85.5 X 3600=307800; and as the pipes
had a total surface of 16450 square feet, the quantity of heat in
B. T. U. per square foot per hour was 1 8. 68 for a mean temper-

Applications of the Formulas 93

ature of 74.8°. The time employed by a particle of the liquid in
traversing the entire length of the pipe was eight and one-half
hours. The liquid when motionless cooled 9.9° in seven hours.
It is probable that the transmission is proportional to the excess
of the temperature of the pipe above 32°, and for steam would
be between 75 and 100 B. T. U.

INFLUENCE OF THE VARIATIONS OF EXTERIOR TEMPERATURE
ON THE QUANTITY OF HEAT TRANSMITTED THROUGH WALLS

902. In what we have said regarding the transmission of
heat through bodies of low conductivity, we have supposed the
regime to have been established, and consequently that the inter-
ior and exterior temperatures were constant ; ordinarily the heat-
ing is regulated so that the interior temperature does not change,
but the transmission is always affected by the variations of exter-
ior temperature. These variations are of two sorts ; the general
decrease and increase of the mean exterior temperature during
the season of heating, and the accidental variations which mani-
fest themselves almost every day. We will examine the influence
of these two in turn.

In our climate, * heating generally is necessary from the first
of October to the end of April, and during these seven months,
the mean exterior temperatures, deduced from the records of ten
years, are
Oct. Nov. Dec Jan. Feb. Mar. Apr.

5i-8 45-i 37-4 36-1 39-7 43-9 5°.9

If we assume that the interior temperature is maintained at
59°, that the walls are all exposed to the outer air, and that their
thickness is 39.37", the total quantity of heat transmitted per
square foot during the total duration of the heating, admitting
that the regime is constantly established would be (871) ;

2.59X210X24=13050 #. 7". U. f
and the total heat above 32° contained in the wall at 59°;

3.28X62 4 X 2. 2X0.2 X 27 = 2430.

This last quantity being less than two-tenths of the first, and
the cooling of the wall being never complete, it is easy to see that
if the variations of temperature took place gradually, without

*Paris, France.

fThe average exterior temperature for the seven months is here taken as 42.8°

94 Applications of the Formulas

sudden oscillations, that whatever the law according to which the
wall cooled during the first part of the winter, and heated during
the second part , the quantities of heat emitted and absorbed by
the wall, could have but a slight influence on the transmission,
according to the hypothesis of a constantly maintained transmis-
sion . We see furthermore that during the decrease of the exterior
temperature the cooling of the walls diminishes by a small amount
the quantity of heat which must be furnished in order to main-
tain the interior temperature, and that during the increase of the
exterior temperature, there will be more heat to be furnished to.
reestablish the original regime throughout the wall.

903. According to what we have just said, the curve of
mean monthly temperatures of the heating months presents but
one minimum; but each day there are several variations in oppo-
site directions so that the actual curve of temperatures shows a
great number of sinuosities around the curve of mean temperature.
These variations act directly on a heated room through the win-
dows, because the windows take almost instantaneously a tem-
perature which is a mean between the interior and exterior tem-
peratures. It is otherwise with the walls; they furnish, when
the exterior temperature falls, a certain quantity of heat, and
when the exterior temperature rises again to the original point,
they absorb the same quantity of heat, so that the quantity of
heat to be furnished to maintain a constant interior temperature,
varies much less rapidly than the exterior temperature. Since
these variations are equal and of opposite signs in regard to the
curve of mean temperature, in whatever way the partial cooling
and heating of the wall takes place, the losses and gains end by
compensating one another, and the total expenditure of heat dur-
ing the heating season remains the same as if the exterior tem-
perature had followed the curve of mean monthly temperatures,
or as if the exterior temperature had remained constant at the
mean temperature for the whole season. This is confirmed by
experience.

904. The phenomena produced in walls by sudden varia-
tions of the exterior temperature are very complicated. If a fall
of temperature takes place there is an increase of loss from the
exterior surface, and a decrease in its temperature which spreads

Applications of the Formulas 95

little by little even to the inner surface, and if the new tempera-
ture of the exterior air lasts sufficiently long, a new regime is
established in the wall. During this interval the temperatures of
the different points of the walls would undergo changes which it
would be actually impossible to calculate, since the calculations
would be even more complicated than those regarding the trans-
mission of heat in an unlimited medium at constant temperature
(897) . But, since walls are rarely more than twenty inches thick,
and as the dissemination of heat through bodies, even of the rather
low conductivities of materials of construction, takes place with
great rapidity, and as the difference of temperature of the two sur-
faces are not generally more than a smallnumber of degrees, we may
assume that, during all the changes of temperature which precede
the establishment of the new regime, the temperatures of the dif-
ferent points of the wall increase uniformly from the exterior to
the interior. This supposition is never realized, but it allows us
to follow approximately the phenomena which accompany a fall
of the exterior air temperature.

Consider a wall belonging to a room of which none of the
other walls are exposed to the exterior air; assume as in para-
graph 868, 7^=59°, 0=42.8, C=i3.7i and e=ig.jo, we will
have /=54.62°, /'48.i9° and M=$.o8, If the exterior temper-
ature became 32°, the formula (a) (864) would give ^=51. 56,
/'=39-42, and .$/=8.45, the quantity of heat lost by the wall
per square foot in passing from the former regime to the latter
would be :

and as this cooling takes place while the temperature of the outer
surface falls from 48.19 to 39.42, the cooling is decreasing;
admitting the hypothesis of a uniform variation of temperature,
this cooling would take place in the same time as if the excess of

temperature of the outer surface was equal to— or 11.81;

and since for an excess of temperature of 16.19, the loss of heat

*This figure is just twice that given in P6clet although the equations are other-
wise identical. This error appears in both the later editions.

96 Applications of the Formulas

per hour is 5.98, * the cooling in question would take place in
=6 t

5.98

This supposes the interior temperature to be maintained at
59° , and the cooling of the wall to take place in the way that we
have assumed; but in reality the cooling would be much less
rapid, because the temperatures of the outer surface would be
much lower than we have admitted and because the temperatures
of the different layers of the wall would succeed each other
according to a different law which would also aid in retarding
the cooling.

905. We see from this that if the cooling of a room took
place through the walls alone, the variations of the exterior tem-
perature would manifest themselves within very slowly, and
feebly. But rooms have always glazed windows, and as the glass
almost instantaneously assumes the mean of the inner and outer
temperatures, we must in order to maintain a constant tempera-
ture, supply an increased quantity of heat which will vary with
the exterior temperature and is in general very much greater than
that which would result from the transmission of heat through
the walls.

*When the loss was 5.98 B. T. U. the exterior temperature was 42.8° and the temper-
ature of the outer surface was 48.19° ; this is an excess of temperature of 5.39° not 16.19°.

fThis figure is just twice that given in the original text for the reason explained
above.

The reasoning inlthis paragraph is certainly difficult to follow. We may consider
the problem in another way. P6clet appears to admit that during the cooling of the wall
the mean temperature of the outer surface is a mean between its temperatures at the
beginning and end of the process. Then we may assume the same to be true for the
mean temperature of the inner surface. Then the difference between the emission from
the outer surface and the absorption by the inner surface per hour will be the mean
amount of the hourly cooling of the wall. Then the total amount of cooling, which we
have already calculated, divided by this quantity, will give the time of cooling.

The mean temperature of the outer surface is- ' ——=43.81°

and its excess over the outer temperature is 43.81°— 32°= 11.81°.

Taking Q as in 858 =1.14 we have ; B. T. U. emitted per hour per square foot = 1.14
X 11.81 = 13.47.

The mean temperature of the inner surface is— — — 53-O9

The excess is 59 — 53.09 = 5.91 and; B. T. U. absorbed per hour per square foot = 1.14
X 5-91 = 6-74-

Then the mean hourly loss of heat by wall = 13.47—6.74=6.73 and the duration of

cooling= =42 hours.

Applications of the Formulas 97

Consider for example, a room with but one wall exposed to
the outer air, with 4.305 square feet of glazed windows and 64.6
square feet of walls of 19.7 inches thich ; the interior temperature
being 59° and the outer temperature 42.8° the total quantity of
heat transmitted will be ; (S68 and 88 ij.

43.05X8.5 + 64.6X5-05 = 366 + 326 = 692.

If we suppose the exterior temperature to fall to 32 °, the
quantity of heat transmitted by the windows will rise immediately
from 366 to 6 1 1 , whilst the transmission of heat through the walls
will rise very slowly in 32 hours, * from 326 to 546, and as a
matter of fact the rise will be much slower. Thus the windows
have a much greater influence than the walls on the variation of
the interior temperature or on the quantity of heat which must
be supplied to maintain this temperature, at least unless the walls
are very thin and their area very large relative to that of the win-
dows.

906. As it is important to have a clear idea of the variations
of temperature which take place at the surfaces of walls during
the heating season, as well as the quantities of heat transmitted
and the quantities of heat contained in the walls, I have computed
these different elements for walls of 19.70, 39.37 and 59.07 inches
in thickness according to the formulas (870) which assume all
the walls exposed to the outer air. I have taken C=i3.jilF=
.737 and A^ = .4o whence 15=1.137; and have assumed the
specific gravity of the stone to be 2.2 and its specific heat to be .2.
Then the quantity of heat contained per square foot of wall at
the temperature v will be

i X — X62.4X 2.2X2 z>=2.288 e v
12

When the temperatures vary uniformily from /to /', between the
two surfaces, the quantity of heat contained in the wall, reckoned

from O° would be /+ /' we will designate this quantity

2.288^— — ;
2

by A and we will have, calling the interior temperature of the
room T and the exterior temperature 0 :
for ^=19.7 inches.

*See foot-notes to 904

98 Applications of the Formulas

,820, M=.2i (r-Oj,A = 45 (.33

.67*)

for <? = 39-37 inches.

t=.6oT+.4oO, /' = .i4r+.86^, M=.i6(T— 0), A = 90 (.37

for ^=59.07 inches.

/= 68r+.330, /' = .i2r+.890, M=.iT,(T— OJ,A = i35 (.40

7H-.600)

INTERMITTENT HEATING

907 . We have hitherto supposed the heating to be continu-
ous ; but it is often suspended during the night and at other times
it-only takes place during a very limited time : we have then two
cases to consider, the loss of heat due to the suspension of the
heating at night, and the quantity of heat which must be furnish-
ed in order to maintain a room at a certain temperature during a
certain length of time.

HEAT LOST BY WALLS DURING THE SUSPENSION OF HEATING

908. During this suspension which generally takes place at
night, the heat emitted from the outer surface of the walls is the
cause of a certain amount of cooling throughout their mass and
consequently a certain cooling of the interior which is added to that
arising from the windows. This cooling of rooms during the inter-
vals between heating is a very important question, unfortunately
very complicated, but in regard to which theoretical considera-
tions may nevertheless lead to some useful practical conclusions.

909 . Considering the simplest case , that in which al 1 the walls
of a room are exposed to the outer air ; all the interior surfaces
will be sensibly at the same temperature, and the heat emitted
by the walls will be solely that contained within them. I have
attempted to calculate the temperatures of different points of the
wall at different periods of its cooling by employing the principles
and formulas of Fourier (TraitS analytique de la Chaleur);& very
simple equation giving the temperature of any point as a function
of all the given quantities of the question is in this way easily
arrived at. This equation contains arbitrary constants of which
the values are easily determined on the assumption that at the
beginning of cooling, the temperatures at the different points of

Applications of the Formulas 99

the wall follow one another according to the regime of cooling ;
but, to determine them on the assumption that at the beginning
of cooling, the temperatures follow one another according to the
regime of transmission, we would be lead to very long calcula-
tions, to formulas composed of an infinity of terms and which
would not really be of any practical value. However, we may
admit the following facts to have been well demonstrated both by
experience and calculation.

i St. When a wall is left to its own cooling, the temperature
of its outer surface falls- rapidly, the more so the smaller its con-
ductivity and its specific heat. The line of temperatures of the
points within the wall changes from the straight line of regular
transmission, to a curve intersecting the straight line at distances
increasing with the time. The curve approaches more and more,
as time goes on, to that curve which calculation would give for
the case of regular cooling on the assumption that at the begin-
ning of cooling the temperature of the points within the wall
follow one another according to the regime of cooling.

2nd. The quantity of heat transmitted by the wall, result-
ing solely from its own cooling, is never more than a very small
fraction of that which it would allow to pass in a regular regime
of transmission .

3rd. The quantity of heat lost by a wall in ten hours is gen-
erally but a small part of the heat contained in the wall at the
beginning of cooling, at least for walls as thick as 19.7 inches;
for examplefor T=5g°, 0=42.8°, ^=50.7°, t' =45. 8°, J/=3.36
(871), the quantity of heat above 32° contained in the wall
per square foot of surface would be 1.64 X 62.4 X 2.2 X .2 (49 — 32)
=764. Whilst the quantity of heat that would be lost in ten
hours by regular transmission would be only 3.36X10=33.6,
and the quantity lost by cooling would be much smaller.

4th. Cooling, during suspension of heating, results princi-
pally from the transmission through the glass of the windows
and the infiltration of air through the cracks of doors and win-
dows when the room is provided with a chimney.

TEMPORARY HEATING OF A ROOM

910. When a room is only to be occupied for a very short

ioo Applications oj the Formulas

time, we may heat it by warm air alone or by warm air and radi-
ation ; in both cases the loss of heat is made up of that trans-
mitted by the windows and that absorbed by the walls.

The heat received by the walls through radiation or through
the currents of air cooled by contact with their surfaces, is propa-
gated step by step throughout their thickness according to
extremely complicated laws. When the air of the room has been
brought to a certain temperature and the heating ceases, the cool-
ing of the air takes place with rapidity, because the heat is
absorbed by the walls and windows, not alone by the air contact
action but also by the diffusion of the heat through the mass of
air; thus, to maintain the air at a sensibly constant temperature,
the heating must be continuous.

NOTES ON THE USE OF THE FORMULAS

911. None of these calculations which we have just been
making in regard to the transmission of heat, can be considered
rigorously exact. Those relating to elementary transmission
rest on two hypotheses which are only true within certain limits ;
Newton's Law of cooling, which is only approximate even for
small differences of temperature, and the supposition that all
points of a body exposed to the air are at the same tempera-
ture; which is not exact, for the lower portions are always at a
lower temperature than those above. In regard to the formulas
representing the transmission of heat through walls the two
assumptions that we have made (864 and 870) are actually only
extreme cases, between which any actual case will be found to
lie. But we must not deceive ourselves in regard to the practical
importance of rigorous exactness in the formulas or precision in
the calculations ; the lightest movement of the air has a great
influence on the quantity of heat which it carries away and for
bodies exposed to the open air the accidental and periodical
variations which it experiences never permit the existence of a
permanent regime in the interior temperatures. In the prelimi-
nary calculations we are obliged to use figures which are not per-
fectly exact for any of the materials except textile materials, for
they vary with the density, for stones with the state of their crys-
tallization, for wood with the direction of the fibres. Thus we
can only regard the results of the calculations as sufficient
approximations for the guidance of engineers. But heating and
ventilating apparatus has always under ordinary circumstances
an excess of power, because it should be calculated for the most
unfavorable conditions, and this excess of power is destroyed,
according to the indications of thermometers and anemometers,
by the adjustment of registers or the supply of fuel ; thus the
uncertainty of the calculation, which is always within sufficiently
narrow limits, is taken care of by the excess of power of the
apparatus, an excess of power which is only necessary in cases
of rare occurrence and of short duration. This is not at all

IO2 Notes on the Use of the Formulas

peculiar to heating apparatus ; in all practical applications of
theory, the calculations always depend on certain given quanti-
ties or conditions which are never more than approximately
known, and every apparatus or machine, for whatever purpose it
may be destined, should possess an excess of power or of strength
designed to cover the uncertainties of calculation, or to satisfy
exceptional circumstances.

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New York, 1897. '$0.50

Redwood. Theoretical and Practical Ammonia Refrigeration. A Prac-
tical Handbook for the use of those in charge of Refrigerating Plants.
Illustrated with numerous Tables. i2mo. New York, 1896. $1.00

Wallis-Tayler. Refrigeration and Cold Storage; being a complete
Practical Treatise on the Art and Science of Refrigeration. 8vo,
cloth. Illustrated. net, $4.50

LIST OF BOOKS.

INDICATORS.

Bacon. Treatise on the Richards Steam Engine Indicator. With a
Supplement, describing the latest Improvements in the Instruments for
Taking, Measuring, and Computing Diagrams. Also an Appendix, con-
taining Useful Formulas and Rules for Engineers. 23 diagrams. 4th
edition. i6mo, flex. New York, 1883. $1.00

Ellison. Practical Applications of the Indicator. With reference to the
Adjustment of Valve Gear on all Styles of Engines. 2d edition. 8vo.
100 engravings. Chicago, 1897. $2.00

Hemenway. Indicator Practice and Steam Engine Economy. With
Plain Directions for Attaching the Indicator, Taking Diagrams, Comput-
ing the Horse-Power, Drawing the Theoretical Curve, Calculating Steam
Consumption, Determining Economy, Locating Derangement of Valves,
and making all desired deductions ; also, Tables required in making the
necessary computations, and an Outline of Current Practice in Testing
Steam Engines and Boilers. 6th edition. i2mo. New York, 1899.

#2.00

Le Van. The Steam Engine Indicator and its Use. A Guide to Practi-
cal Working Engineers for greater economy, and the better Working of
Steam Engines. 1 8 mo, boards. New York, 1900. $0.50

The Steam Engine and the Indicator : Their Origin and Progressive

Development, including the most recent examples of Steam and Gas
Motors, together with the Indicator, its Principles, its Utility, and its Ap-
plication. Illustrated by 205 engravings, chiefly of Indicator-cards. 8vo.
Philadelphia, 1890. #4.00

Porter. A Treatise on the Richards Steam Engine Indicator, and the
Development and Application of Force in the Steam Engine. 5th edi-
tion, revised and enlarged. 8vo. London, 1894. $3-oo

Pray. Twenty Years with the Indicator. Being a Practical Text-book
for the Engineer or the Student, with no Complex Formulae. With
many illustrations and rules as to the best way to run any Steam Engine
to 'get the most economical results. How to Adjust Valves and Valve
Motions Correctly. Full directions for working out Horse-Power, the
Amount of Steam or Water per Horse-Power, Economy and Fuel. Ex-
tended directions for Attaching the Indicator, what Motions to use and
those not to use. Full directions for Computation of Power by Planim-
eter and other methods, with many tables and hints. 8vo. New York,
1896. 12.50

D. VAN NOSTRAND COMPANY.

INJECTORS.

Kneass. Practice and Theory of the Injector. 8vo. New York, 1898.

#1.50
Nissenson. Practical Treatise on Injectors as Feeders of Steam Boilers.

Illustrated. 8vo, paper. New York, 1890. $0.50

Pochet. Steam Injectors : Their Theory and Use. i6mo, boards. New

York, 1890. $0.50

INSTRUCTIONS TO ENGINEERS, FIREMEN,
AND BOILER ATTENDANTS.

Bale. A Hand-Book for Steam Users, being Rules for Engine Drivers
and Boiler Attendants, with Notes on Steam Engine and Boiler Manage-
ment and Steam Boiler Explosions. I2mo. London, 1890. $0.80

Edwards. 900 Examination Questions and Answers for Engineers and
Firemen (Stationary and Marine), who desire to obtain a U. S. Govern-
ment or State License. A new, revised, and enlarged edition. 32mo,
mor. Philadelphia, 1901. $1.50

Grimshaw. Steam Engine Catechism. A Series of Direct Practical
Answers to Direct Practical Questions. Mainly intended for Young En-
gineers. i8mo. New York, 1897. $2.00

Grimshaw. The Engine Runner's Catechism. Telling how to Erect,
Adjust, and Run the principal Steam Engines in use in the United States.
Illustrated. i8mo. New York, 1898. $2.00

Hawkins. Maxims and Instructions for the Boiler Room. Useful to
Engineers, Firemen, and Mechanics, relating to Steam Generators, Pumps,
Appliances, Steam Heating, Practical Plumbing, etc. 184 illustrations.
8vo. New York, 1901. $2.00

— Aids to Engineers' Examinations. Prepared for Applicants of all
Grades with Questions and Answers. A Summary of the Principles and
Practice of Steam Engineering. I2mo, leather, gilt edge. New York,
1901. $2.00

Reynolds. The Engineman's Pocket Companion and Practical Educator
for Engineman, Boiler Attendants, and Mechanics. Illustrated. i6mo,
4th edition. London, iqoo. $1.40

Roper. Instructions and Suggestions for Engineers and Firemen who
wish to Procure a License, Certificate, or Permit to take charge of any
class of Steam Engines or Boilers, Stationary, Locomotive, and Marine.
i8mo, mor. Philadelphia, 1894. $2.00

LIST OF BOOKS.

Rose. Key to Engines and Engine-running. A Practical Treatise
upon the Management of Steam Engines and Boilers for the use of
those who desire to pass an examination to take charge of an engine
or boiler. I2mo, cloth. New York, 1899. $2.50

Questions and Answers for Engineers. This little book contains all

the questions that Engineers will be asked when undergoing an exami-
nation for the purpose of procuring licenses, and they are so plain that
any Engineer or Fireman of ordinary intelligence may commit them to
memory in a short time. 6th edition. i8mo, mor. Philadelphia. $2.00

Stephenson. Illustrated Practical Test Examination and Ready Refer-
ence Book for Stationary, Locomotive, and Marine Engineers, Firemen,
Electricians, and Machinists, to procure Steam Engineer's license. i6mo,
Chicago, 1892. $1.00

Stromberg. Steam User's Guide and Instructor. Plain and Correct Ex-
planations in regard to Engines, Pumps, Dynamos, and Electricity. Prac-
tically, so that Engineers, Machinists, Firemen, and Electricians of Lim-
ited Education can understand and become expert practical engineers.
i6mo. St. Louis, l$th edition. 1900. $2.50

Watson. How to Run Engines and Boilers. Practical Instruction for
Young Engineers and Steam Users. 2d edition. Illustrated. i6mo.
New York, 1896. $1.00

Zwicker. Practical Instructor in questions and answers for Machinists,
Firemen, Electricians, and Steam Engineers. 241110. St. Louis, Mo.,
1898. $0-75

LOCOMOTIVE ENGINEERING.

Grimshaw. Locomotive Catechism. Containing nearly 1,300 Questions
and Answers Concerning Designing and Constructing, Repairing and
Running Various Kinds of Locomotive Engines. Intended as Exami-
nation Questions and to Post and Remind the Engine Runner, Fireman,
or Learner. 1 76 illustrations. I2tno. New York. 1898. $2.00

Hill. Progressive Examinations of Locomotive Engineers and Firemen.
i6mo. Terre Haute, Ind., 1899. $0.50

Meyer. Modern Locomotive Construction. 1,030 illustrations. 410. New
York, 1899. $10.00

Phelan. Air Brake Practice, being a description of the construction, ob-
jects sought, and results obtained, by the Westinghouse automatic air
brake, as well as complete directions for operating it under the many
diverse conditions in daily practice. 3 large folding plates. I2mo. New
York. 4th edition. $1.00

D. VAN NOSTRAND COMPANY.

Reagan. Locomotive Mechanism and Engineering. i2mo, with 145 il-
lustrations. New York, 1898. $2.00

Reynolds. Locomotive Engine Driving. A Practical Manual for Engi-
neers in charge of Locomotive Engines. 8th edition, enlarged. i2mo.
London, 1892. $1.40

The Model Locomotive Engineer, Fireman, and Engine Boy : Com-
prising a Historical Notice of the Pioneer Locomotive Engines and their
Inventors. I2mo. London, 1895. $1.80

Continuous Railway Brakes. A Practical Treatise on the several

Systems in use in the United Kingdom ; their Construction and Perform-
ance. Numerous illustrations and tables. 8vo. London, 1882. $3.60

Engine Driving Life : Stirring Adventures and Incidents in the Lives

of Locomotive Engine Drivers. 2d edition, with additional chapters.
I2mo. London, 1894. $0.80

Rogers. Pocket Primer or Air Brake Instruction. Stiff paper cover. $0.50

Roper. Hand-Book of the Locomotive; including the construction of
engines and boilers and running of locomotives. I5th edition, revised.
i2mo, mor. tucks. Philadelphia, 1897. $2.50

Sinclair. Locomotive-Engine Running and Management. A Practical
Treatise on Locomotive Engines, showing their performance in running
different kinds of trains with economy and Despatch. Also, directions
regarding the care, management, and repairs of Locomotives and all their
connections. Illustrated by numerous engravings. 2ist edition, revised.
I2mo. New York, 1901. $2.00

Stretton. The Locomotive Engine and its Development. A Popular
Treatise on the Gradual Improvements made in Railway Engines be-
tween the years 1803 and 1892. Illustrated. i2mo. 5th edition. Lon-
don, 1896. $i-5o

Synnestvedt. Diseases of the Air Brake System. Their Causes, Symp-
toms, and Cure. Illustrated. I2mo. 1894. $1.00

Woods. Compound Locomotives. 2d edition, revised and enlarged by
D. L. Barnes. 8vo. Illustrated. Chicago, 1894. $3-oo

MACHINE TOOLS AND APPLIANCES.

Harrison. The Mechanic's Tool Book, with Practical Rules and Sugges-
tions for Machinists, Iron Workers, and others. I2mo. New York,
1882. $1.50

LIST OF BOOKS.

Hasluck. The Mechanics' Work-shop Handy Book. A Practical Man-
ual on Mechanical Manipulation. Embracing Information on Various
Handicraf . Processes, with Useful Notes and Miscellaneous Memoranda.
1 21110. London, 1895. $0.50

Knight. Mechanician. A Treatise on the Construction and Manipulation
of Tools, for the Use and Instruction of Young Engineers and Scientific
Amateurs. 4th edition. 410. London, 1888. #7.25

Lukin. Young Mechanic. Containing directions for the use of all kinds
of Tools and for construction of Steam Engines and Mechanical Models,
including the Art of Turning in Wood and Metal. Illustrated. I2mo.
New York. #i-75

Rose. Complete Practical Machinist. Embracing Lathe Work, Vise
Work, Drills and Drilling, Taps and Dies, Hardening and Tempering,
the Making and Use of Tools, Tool Grinding, Marking out Work, etc.
Illustrated by 356 engravings, igth edition, greatly enlarged. i2mo.
Philadelphia, 1901. $2.50

Shelley. Work-shop Appliances. Including descriptions of some of the
Gauging and Measuring Instruments, Hand Cutting Tools, Lathes, Drill-
ing, Planing, and other Machine Tools used by Engineers. loth edition,
with an additional chapter on Milling, by R. R. Lister. Illustrated.
I2mo. London, 1897. $1.50

Smith. Cutting Tools worked by Hand and Machine. 14 plates and 51
illustrations. 2d edition. I2mo. London, 1884. $1.50

Usher. Modern Machinist. A Practical Treatise on Modern Machine
Shop Methods, describing in a comprehensive manner the most Approved
Methods, Processes, and Appliances Employed in Present Practice, etc.
257 illustrations-. i2mo. New York, 1895. $2.50

Watson. Modern Practice of American Machinists and Engineers. 1 21110.
Illustrated. Philadelphia, 1897. $2.50

MECHANICAL DRAWING AND MACHINE
DESIGN.

Andre. Draughtsman's Hand-Book of Plan and Map Drawing; including
Instructions for the preparation of Engineering, Architectural and Me-
chanical Drawings, with numerous illustrations, and colored examples.
8vo. London, 1891. $3-75

Appleton's Cyclopaedia of Technical Drawing. Embracing the Principles
of construction as applied to Practical Design. With numerous illustra-
tions of Topographical, Mechanical, Engineering, Architectural, Perspec-
tive, and Free-hand Drawing. 8vo, leather. New York, 1896. $9,00

D. VAN NOSTRAND COMPANY

Barber. Engineers' Sketch Book of Mechanical Movements, Devices,
Appliances, Contrivances, Details employed in the Design and Con-
struction of Machinery for every Purpose. Collected from numer-
ous sources and from actual work. Classified and arranged for
reference. Nearly 2000 Illustrations. 4th edition. 8vo. London,
1902. $4.00

Building and Machine Draughtsman. A Practical Guide to the Pro-
jection and Delineation of Subjects met with in the practice of the
engineer, machinist, and building constructor, etc.; by practical
draughtsmen. I2mo. London, 1891. $2.00

Burns. Illustrated Architectural Engineering and Mechanical Draw-
ing Book. For the use of Schools, Students, and Artisans, roth
edition, revised and corrected, with additional sections on impor-
tant departments of the art. 8vo. 284 illustrations. New York,
1893. $1,00

Cathcart, Prof. Win. L. Machine Design: Fastenings, with Tables,
Diagrams and Engravings. 8vo, cloth. Illustrated. In Press.

Davidson. Drawing for Machinists and Engineers. Comprising a com-
plete course of Drawing adapted to the requirements of Millwrights and
Engineers ; also, course of practical instruction in the coloring of me-
chanical drawings. 4th edition. i6mo. London. $1-75

Donaldson. Drawing and Rough Sketching for Marine Engineers, with
Proportions, Instructions, Explanations, and Examples ; also How to De-
sign Engines, Boilers, Propellers, Paddle Wheels, Shafts, Rods, Valves,
etc. 6th edition. Illustrated. London, 1899. $3.00

Faunce. Mechanical Drawing, prepared for the use of the students of
the Mass. Institute of Technology. 2d edition, revised and enlarged.
Illustrated and 8 plates. I2mo. Boston, 1900. $1.25

Fox, Wm., and C. W. Thomas, M. E. A Practical Course in Mechan-
ical Drawing. 2d edition, revised. i2mo, cloth, with plates. $1.25

Halliday. First Course in Mechanical Drawing (Tracing). Folio,
paper. London, 1889. $0.75

Mechanical Graphics. A Second Course In Mechanical Draw-
ing, with Preface by Professor Perry. 8vo. London, 1889. $2.00

Hulme. Mathematical Drawing Instruments and How to Use Them.
4th edition. I2mo. New York, 1890. $1.50

Klein. Elements of Machine Design. Notes and Plates, 8vo. Beth-
lehem, Pa., 1892. $6.00

LIST OF BOOKS.

Low and Bevis. Manual of Machine Drawing and Design. 36 edition,
753 illustrations. 8vo. London, 1901 $2.50

MacCord. Practical Hints for Draughtsmen. Illustrated with 68 dia-
grams and full page plates. 3d edition, 4to. New York, 1890. $2.50

Kinematics, or Practical Mechanics. A Treatise on the Transmis-
sion and Modification of Motion and the Construction of Mechanical
Movements. For the use of Draughtsmen, Machinists, and Students of
Mechanical Engineering, in which the laws governing the motions and
various parts of Mechanics, as affected by their forms and modes of con-
nection, are deduced by simple geometrical reasoning, and their applica-
tion is illustrated by accurately constructed diagrams of the different
mechanical combinations discussed. 4th edition. 8vo. New York,
1899. $5.00

Minifie. Mechanical Drawing. A Text-Book of Geometrical Drawing,
for the use of Mechanics and Schools, in which the Definitions and Rule?
of Geometry are familiarly explained : the Practical Problems are ar-
ranged from the most simple to the more complex, and in their descrip.
tion technicalities are avoided as much as possible. With illustrations
for Drawing Plans, Sections, and Elevations of Buildings and Machin-
ery; an Introduction to Isometrical Drawing, and an Essay on Linear
Perspective and Shadows. Illustrated by over 200 diagrams, engraved
on steel. With an Appendix on the Theory and Application of Colors.
8vo. New York, 1893. $4.00

Geometrical Drawing. Abridged from the octavo edition, for the

use of Schools. Illustrated with 48 steel plates, gth edition. Revised
and enlarged. I2mo. New York, 1890. $2.00

Palmer. Mechanical Drawing, Projection Drawing, Geometric and Oblique

Drawing, Working Drawings. A Condensed Text for Class Room use.

8vo. Columbus, O. 1894. #i.ot

Reinhardt, Chas. W. Lettering for Draftsmen, Engineers and Students.

A Practical System of Free-hand Lettering for Working Drawings.

nth thousand. Oblong, boards. $1.00

The Technic of Mechanical Drafting. A Practical Guide to

neat, correct and legible Drawing, containing many Illustrations,
Diagrams and full-page Plates. 410, cloth. Illustrated. $1.00

D. VAN NOSTRAND COMPANY.

Ripper. Machine Drawing and Design for Technical Schools and Engi-
neer Students. Being a complete course of Instruction in Engineering
Drawing, with Notes and Exercises on the Application of Principles to
Engine and Machine Design, and on the Preparation of Finished Col-
ored Drawings. Illustrated by 52 plates and numerous explanatory
drawings. 8vo. London, 1897. $6.00

Roberts. Drawing and Designing for Marine Engineers. 21 large fold-
ing plates and many other illustrations throughout the text. 8vo. Lon-
don, 1898. $3.00

Rose. Mechanical Drawing Self-Taught. Comprising Instructions in
the Selection and Preparation of Drawing Instruments, Elementary In-
struction in Practical Mechanical Drawing, together with Examples in
Simple Geometry and Elementary Mechanism, including Screw Threads,
Gear Wheels, Mechanical Motions, Engines and Boilers. Illustrated by
330 engravings. 4th edition, revised. 8vo. Philadelphia, 1902. $4.00

Shaw. Mechanical Integrators. Including the various Forms of Pla-
nimeters. i8mo, boards. Illustrated. New York, 1886. $0.50

Smith. Graphics, or the Art of Calculation by Drawing Lines, applied
especially to Mechanical Engineering. Part I. Text, with Separate Atlas
of Plates — Arithmetic, Algebra, Trigonometry, Vector and Lecor Addi-
tion, Machine Kinematics, and Statics of Flat and Solid Structures. 8vo.
London, 1888. $5.00

Stanley. Descriptive Treatise on Mathematical Drawing Instruments,
their Construction, Uses, Qualities, Selection, Preservation, and Sugges-
tions for Improvements, with Hints upon Drawing and Coloring. 5th
edition. I2mo. London, 1878. $2.00

Tomkins. Principles of Machine Construction ; being an application of
Geometrical Drawing for the Representation of Machinery. Text ismo,
Plates 4to. New York. $3. 50

Unwin. Elements of Machine Design. Part I. General Principles, Fas-
tenings, and Transmissive Machinery. i6th edition. I2mo. London,

- Part II. Chiefly on Engine Details. I2mo. I3th edition, revised
and enlarged. London, 1901. $1.50

Warren. Elements of Machine Construction and Drawing : or, Machine
Drawing, with some elements of descriptive and rational kinematics.
2 vols. Text and plates. 8va New York. $7-$Q

LIST OF BOOKS.

MECHANICAL ENGINEERS' HAND-BOOKS.

Adams. Hand-Book for Mechanical Engineers. 2d edition. Revised
and enlarged, ismo. London, 1897. $2.50

Appleton's Cyclopaedia of Applied Mechanics : a Dictionary of Mechani-
cal Engineering and the Mechanical Arts. Edited by Park Benjamin.
Nearly 7,000 illustrations. Revised and improved edition. 2 vols. 8vo,
leather. New York, 1896. $15.00

Bale. Steam and Machinery Management : A Guide to the Arrangement
and Economical Management of Machinery, with Hints on Construction
and Selection. Illustrated. 2d edition. i2mo. London, 1890. (Weale's
Series.) $1.00

Benjamin. Wrinkles and Recipes. Compiled from the Scientific Ameri-
can. A collection of Practical Suggestions, Processes, and Directions,
for the Mechanic, Engineer, Farmer, and Housekeeper. With a Color
Tempering Scale and numerous Wood Engravings. 5th edition.
I2mo. New York, 1901. $2.00

Byrne. Hand-Book for the Artisan, Mechanic, and Engineer. Compris
ing the Grinding and Sharpening of Cutting Tools, Abrasive Processes,
Lapidary Work, Gem and Glass Engraving, Varnishing and Lackering
Apparatus, Materials and Processes for Grinding and Polishing, etc. 8vo.
Illustrated. Philadelphia, 1887. $5.00

Carpenter. Text-Book of Experimental Engineering. For Engineers and
for Students in Engineering Laboratories. 249 illustrations. 5th revised
edition. 8vo. New York, 1898. $6.00

Chordal. Extracts from Chordal's Letters. Comprising the choicest
selections from the Series of Articles which have been appearing for the
past two years in the columns of the American Machinist. With over 50
illustrations. I2mo. gth edition. New York, 1901. $2.00

Clark. Manual of Rules, Tables and Data for Mechanical Engineers,
based on the most recent investigations. With numerous Diagrams.
7th edition. 1,012 pages. London, 1897. $5.00

Half morocco. $7.50

— Mechanical Engineers' Pocket-Book of Tables, Formulae, Rules,

and Data. A Handy-Book of Reference for Daily Use in Engineer-
ing Practice. i6mo, mor. 4th edition. London, 1899. $3.00

Dixon. The Machinists' and Steam Engineers' Practical Calculator.
A Compilation of useful Rules and Problems, arithmetically solved,
together with general information applicable to Shop Tools, Mill
Gearing, Pulleys and Shafts, Steam Boilers and Engines. Embracing
valuable Tables and Instructions in Screw Cutting, Valve and Link
Motion. 3d edition. 16010, mor., pocket form. New York, 1901. $1.25

D. VAN NOSTRAND COMPANY,

Engineering Estimates, Costs, and Accounts. A Guide to Commercial
Engineering. With numerous Examples of Estimates and Costs of Mill-
wright Work, Miscellaneous Productions, Steam Engines and Steam
Boilers, and a Section on the Preparation of Costs Accounts. By a Gen-
eral Manager. 2d edition. 8vo. London, 1896. $4.80

General Machinist, Being a Practical Introduction to the Leading Depart-
ments of Mechanism and Machinery, the Communication of Motion or
the Transmission of Force by Belt, Rope, Wire Rope, and Pulley Gearing
— Toothed- Wheel and Frictional Gearing ; together with the details of
the component and essential parts of mechanism — Shafts, Pedestals,
Hanger, Clutches, etc., and of the methods of fitting up Machines, Screw
Bolts, Riveting, etc. By various practical writers and machinists. 75
illustrations and 4 folding plates. 8vo. London, 1891. $2.00

Grimshaw. Hints to Power Users. Plain, Practical Pointers, free from
high Science, and intended for the man who pays the bills. i2mo. New
York, 1891. $1.00

Hasluck. Mechanic's Workshop Handy-Book. A Practical Manual on
Mechanical Manipulation. Embracing Information on Various Handi-
craft Processes, with Useful Notes, and Miscellaneous Memoranda.
I2mo. London, 1888. $0.50

Haswell. Engineers' and Mechanics' Pocket Book, Containing Weights
and Measures, Rules of Arithmetic, Weights and Materials, Latitude and
Longitude, Cables and Anchors, Specific Gravities, Squares, Cubes, and
Roots, etc. ; Mensuration of Surfaces and Solids, Trigonometry, Me-
chanics, Friction, Aerostatics, Hydraulics and Hydrodynamics, Dynamics,
Gravitation, Animal Strength, Windmills, Strength of Materials, Limes,
Mortars, Cements, etc. ; Wheels, Heat, Water, Gunnery, Sewers, Com-
bustion, Steam and the Steam Engine, Construction of Vessels, Miscel-
laneous Illustrations, Dimensions of Steamers, Mills, etc.; Orthography
of Technical Words and Terms, etc. 6?th edition. Revised and
enlarged. I2mo, mor. tuck. New York, 1902. $4.00

Hawkins. Hand-Book of Calculations, for Engineers and Firemen ;
relating to the Steam Engine, the Steam Boiler, Pumps, Shafting,
etc. Illustrated. 8vo. New York, 1902. $2.00

Button. Works Manager's Hand-Book of Modern Rules, Tables, and
Data for Civil and Mechanical Engineers, Millwrights, and Boiler Makers,
Tool Makers, Machinists, and Metal Workers, Iron and Brass Founders,
etc. 5th edition, revised, with additions. 8vo, half-bound. London,
1895. $6.00

LIST OF BOOKS.

Button. Practical Engineer's Hand-Book. Comprising a Treatise on
Modern Engines and Boilers, Marine, Locomotive, and Stationary, and
containing a large Collection of Rules and Practical Data Relating to
Recent Practice in Designing and Constructing all kinds of Engines,
Boilers, and other Engineering Work. 5th edition, carefully revised, with
additions. 370 illustrations. 8vo. London, 1896. . $7-oo

Kent. Mechanical Engineers' Pocket-Book. A Reference Book of Rules,
Tables, Data, and Formulae, for the Use of Engineers, Mechanics, and
Students. 1,087 pages. 5th edition. I2mo. New York, 1902. $5.00

Knight. American Mechanical Dictionary. A Descriptive Word Book
of Tools, Instruments, Chemical and Mechanical Processes ; Civil, Me-
chanical, Railroad, Hydraulic, and Military Engineering. A History of
Inventions. General Technological Vocabulary, and Digest of Mechani-
cal Appliances in Science and the Industrial and Fine Arts. 3 vols.
Illustrated, 8vo. Boston, 1884. $24.00

Supplement to the above, $9.00

The 4 vols., complete, $27.50

Lockwood's Dictionary of Terms used in the Practice of Mechanical En-
gineering. Embracing those current in the Drawing Office, Pattern Shop,
Foundry, Fitting, Turning, Smiths' and Boiler Shops, etc., comprising
upwards of 6,000 definitions. Edited by a Foreman Pattern Maker.
3d edition. i2mo. London, 1902. $3.00

Molesworth. Pocket-Book of Useful Formulae and Memoranda for
Civil and Mechanical Engineers. 24th edition, revised and enlarged.
Pocket-book form. London, 1901. $2.00

Moore. Universal Assistant and Complete Mechanic : Containing over
One Million Industrial Facts, Calculations, Receipts, Processes,
Trade Secrets, Rules, Business Forms, Legal Items, etc. Illustrated.
i2mo. New York. $2.50

Rankine. Useful Rules and Tables relating to Mensuration, Engineering
Structures, and Machines. 7th edition, thoroughly revised by W. J.
Millar. With Electrical Engineering Tables, Tests, and Formulae for the
use of Engineers, by Prof. A. Jamieson. I2mo. London, 1889. $4.00

Roper. Engineers' Handy-Book. Containing a full explanation of the
Steam Engine Indicator, and the Use and Advantage to Engineers and
Steam Users. With Formulae for estimating the Power of all Classes

D. VAN NOSTRAND COMPANY.

or Steam Engines ; also Facts, Figures, Questions, and Tables, for Engi-
neers who wish to qualify themselves for the United States Navy, the
Revenue Service, the Mercantile Marine, or to take charge of the better
class of stationary Steam Engines. Illustrated. I5th edition.
i6mo, mor. tucks. Philadelphia, 1901. $3.50

Scribner. Engineers' and Mechanics' Companion. Comprising United
States Weights and Measures, Mensuration of Superfices and Solids;
Tables of Squares and Cubes ; Square and Cube Roots ; Circumference
and Areas of Circles ; the Mechanical Powers ; Centres of Gravity ; Gravi-
tation of Bodies ; Pendulums ; Specific Gravity of Bodies ; Strength,
Weight, and Crush of Materials ; Water-wheels, Hydrostatics, Hydraulics,
Statics, Centres of Percussion and Gyration ; Friction Heat ; Tables of
the Weight of Metals, Scantling, etc. ; Steam and Steam Engine.
2ist edition, revised. i6mo, full mor. New York, 1902. $1.50

Spons' Tables and Memoranda for Engineers, and convenient refer-
ence for the pocket. I2th edition. 64mo, roan, gilt edges. London,
1901. In cloth case. $0.50

Mechanics' Own Book. A Manual for Handicraftsmen and

Amateurs. Complete in one large vol., 8vo, containing 700 pages
and 1,420 Illustrations. 6th edition. London, 1901. $2.50

• Dictionary of Engineering. Civil, Mechanical, Military, and Naval,

with Technical Terms in French, German, Italian, and Spanish. 8 vols.
8vo, cl. London, 1874. Each, #5.00

— Supplement to above. 3 vols., cl. London, 1881. Each, $5.00

Templeton. Practical Mechanics' Workshop Companion. Completing
a great variety of the most useful Rules and Formulas in Mechanical
Science, with numerous Tables of Practical Data and Calculated Results
for Facilitating Mechanical Operations. i8th edition, revised, mod-
ernized, and considerably enlarged, by Walter S. Hutton. i6mo,
leather. London, 1902. $2.00

Engineers', Millwrights', and Mechanics' Pocket Companion.

Comprising Decimal Arithmetic, Tables of Square and Cube Roots, Prac-
tical Geometry, Mensuration, Strength of Materials, Mechanical Powers,
Water Wheels, Pumps and Pumping Engines, Steam Engines, Tables of
Specific Gravity, etc. Revised, corrected, and enlarged from the 8th Eng-
lish edition, and adapted to American Practice, with the addition of much
new matter. Illustrated by J. W. Adams. lamo, mor. tucks. New York,
1893. $2.00

LIST OF BOOKS.

Van Cleve. English and American Mechanic. An every-day Hand-Book
for the Workshop and the Factory. Containing Several Thousand Re-
ceipts, Rules, and Tables indispensable to the Mechanic, the Artisan, and
the Manufacturer. A new, revised, enlarged, and improved edition.
Edited by Emory Edwards, M.E. I2mo. Philadelphia, 1893. $2.00

MECHANICS (ELEMENTARY AND APPLIED).

Church. Notes and Examples in Mechanics ; with an Appendix on the
Graphical Statics of Mechanism. 128 illustrations and 6 plates. 8vo.

2d edition, New York, 1900. $2.00

Cotterill. Applied Mechanics, ajn Elementary General Introduction to
the Theory of Structures and Machines. Illustrated. 3d edition. 8vo.
London, 1895. $5.00

and Slade. Lessons in Applied Mechanics. i2mo. London, 1894.

Net $1.25

Dana. A Text-Book of Elementary Mechanics for the use of Colleges

and Schools. I2th edition. I2mo. New York, 1901. $ 1.50

DuBois. Elementary Principles of Mechanics. Designed as a Text-Book
for technical schools. 3 vols. 8vo. New York.

Vol. I. Kinematics. $3-S°

Vol. II. Statics. $4-00

Vol. III. Kinetics. $3-5o

Garnett. Treatise on Elementary Dynamics. For the use of Colleges
and Schools. 5th edition. 8vo. London, 1889. Net $1.50

Geldard. Statics and Dynamics. Illus. i2mo. London, 1893. $1.50

Goodeve. Principles of Mechanics. New edition, rewritten and enlarged.
I2mo. London, 1889. $2.50

Manual of Mechanics. An Elementary Text-Book for Students of

Applied Mechanics. Illustrated. i2mo. London, 1881. $0.80

Hancock. Text-Book of Mechanics and Hydrostatics. With over 500
diagrams. 8vo. New York, 1894. #I-75

Hughes. Condensed Mechanics : a selection of Formulae, Rules, Tables,
and Data for the Use of Engineering Students, Science Classes, etc., in
accordance with the requirements of the Science and Art Department.
I2mo. London, 1891. $1.00

Jamieson. Elementary Manual of Applied Mechanics. Specially ar-
ranged for the use of First Year Science and Art, City and Guilds of
London Institute, and other Elementary Engineering Students.
4th edition, revised and enlarged. I2mo. London, 1900. $1.25

D. VAN NOSTRAND COMPANY.

Kennedy. Mechanics of Machinery. With numerous illustrations, izmo.

London, 1886. $3-5<J
Kinematics of Machinery; or, The Elements of Mechanism. i6mo,

boards. New York, 1881. $0.50

Nystrom. New Treatise on Elements of Mechanics. 8vo. Philadelphia,

1875. $2.00

Perry. Applied Mechanics. Illustrated. i2mo. London, 1901.

$2.50

Practical Mechanics. Being the Fourth Volume of " Amateur Work Il-
lustrated." Plates and illustrations. 4to. London. $3-oo

Rankine. Applied Mechanics, comprising Principles of Statics, Cinemat-
ics, and Dynamics, and Theory of Structures, Mechanism, and Machines.
I2mo. i6th edition, thoroughly revised, by W. J. Millard. Lon-
don, 1901. $5.00

and Bamber. Mechanical Text-Book ; or, Introduction to the

Study of Mechanics and Engineering. With numerous Diagrams.
4th edition, revised. 8vo. London, 1890. $3«5o

Stahl and Woods. Elementary Mechanism. A Text-Book for Stu-
dents of Mechanical Engineering, nth edition, revised and en-
larged. Illustrated. I2mo. New York, 1901. $2.00

Weisbach. Theoretical Mechanics, with an Introduction to the Cal-
culus. Translated from the 4th German edition by E. B. Coxe.
gth edition, revised. 8vo, cloth. New York, 1899. $6.00

Sheep. $7-oo

Vol. II., Part i. Hydraulics and Hydraulic Motors. $5-oo

Vol. II., Part 2. Heat, Steam, and Steam Engines. $5-oo

Vol. III., Part i. Kinematics and Machinery of Transmission. $5.00
Vol. III., Part 2. Machinery of Transmission and Governors. $5.00

Wood. Elements of Analytical Mechanics. With numerous examples
and illustrations. For use in Scientific Schools and Colleges. 7th edi-
tion, revised and enlarged, comprising Mechanics of Solids and Mechanics
of Fluids, of which Mechanics of Thirds is entirely new. 8vo. New
York, 1900. $3-°°

Principles of Elementary Mechanics. Fully illustrated. Qth edition.

I2mo. New York, 1894. $1.25

Wright. Text-Book of Mechanics. With numerous examples. 3d edi-
tion. I2mo. New York. $2.50

LIST OF BOOKS.

MISCELLANEOUS.

Amateur Mechanic's Workshop. A Treatise containing plain and concise
directions for the manipulation of Wood and Metals, including Casting,
Forging, Brazing, Soldering, and Carpentry. By the author of " The
Lathe and its Uses." 7th ed. Illustrated. 8vo. London, 1888. $3,00

Hiscox, Gardner D. Compressed Air ; Its Production, Uses, and Appli-
cations. Comprising the Physical Properties of Air from a Vacuum
to its Liquid State, its Thermodynamics, Compression, Transmis-
sion, and Uses as a Motive Power. With 40 Air Tables and 545
Illustrations. 8vo, cloth. New York, 1901. $5.00

Half morocco. $6.50

Plympton, Prof. Geo. W. How to become an Engineer ; or, the Theo-
retical and Practical Training necessary in fitting for the Duties of
the Civil Engineer. (Van Nostrand's Science Series). $0.50

STEAM AND STEAM ENGINES.

Alexander. Model Engine Construction. With Practical Instructions to
Artificers and Amateurs. Containing numerous illustrations and twenty-
one Working Drawings, from Original Drawings by the Author. i2mo.
London, 1895. $3-oo

Baker. Treatise on the Mathematical Theory of the Steam Engine.
With Rules at length and Examples worked out, for the use of practical
men, with numerous diagrams. 8th edition. London, 1890. $0.60

Bale. How to Manage a Steam Engine ; a Handbook for all who use-
Steam. Illustrated, with examples of different Types of Engines and
Boilers; with Hints on their Construction, Working, Fixing, Economy
of Fuel, etc. 7th edition. i2mo. London, 1890. $0.80

Barrus, Geo. H. , S.B. Engine Tests; embracing the results of over
100 Feed-Water Tests and other investigations of various kinds of
Steam Engines, conducted by the author. Illustrated. 8vo, cloth.
New York, 1900. $4.00

Bourne. Catechism of the Steam Engine in its various Applications to
Mines, Mills, etc. New edition, enlarged. Illustrated. I2mo. New"
York, 1897. $2.00

Hand-Book of the Steam Engine, containing all the Rules required

for the right Construction and Management of Engines of every Class
with the easy Arithmetical Solution of those Rules. Illustrated. I2mor
New York, 1892. $i-75.

Burn. Steam Engine, its History and Mechanism. 3d edition. 8vo»
Illustrated. London. i«K7. $J QQ

D. VAN NOSTRAND COMPANY.

Clark. Steam and the Steam Engine, Stationary and Portable. (Being
an Extension of the Elementary Treatise on the Steam Engine, of Mr.
John Sewell.) 4th edition. London, 1892. $1.40

The Steam Engine. A Treatise on Steam Engines and Boilers ;

comprising the Principles and Practice of the Combustion of Fuel, the
Economical Generation of Steam, the Construction of Steam Boilers,
and the principles, construction, and performance of Steam Engines,
Stationary, Portable, Locomotive, and Marine, exemplified in Engines
and Boilers of recent date. Illustrated by above 1 ,300 figures in the text,
and a series of folding plates drawn to scale. 2 vols. 8vo. London,
1895. #15.00

Colyer. Treatise on Modern Steam Engines and Boilers, including Land,
Locomotive, and Marine Engines and Boilers. For the use of Students.
With 46 plates. 4to. London, 1886. $5.00

Cotterill. Steam Engine considered as a Thermodynamic Machine. A
Treatise on the Thermodynamic Efficiency of Steam Engines. Illus-
trated by tables, diagrams, and examples from practice. 3d edition, re-
vised and enlarged. 8vo. London. 1896. net $4.50

Diesel. Theory and Construction of a Rational Heat Motor. Translated
from the German by Bryan Donkin. With eleven figures in the text and
three plates. 8vo. London, 1894. $2.50

Edwards. American Steam Engineer, Theoretical and Practical. With
Examples of the latest and most approved American Practice on the De-
sign and Construction of Steam Engines and Boilers of every description.
For the use of Engineers, Machinists, Boiler Makers, etc. Illustrated by
77 engravings. I2mo. Philadelphia, 1897. $2.50.

Practical Steam Engineers' Guide in the Design, Construction, and

Management of American Stationary, Portable, and Steam Fire Engines,
Steam Pumps, Boilers, Injectors, Governors, Indicators, Pistons, and
Rings, Safety Valves and Steam Gauges. For the use of Engineers,
Firemen, and Steam Users. Illustrated. 3d edition, revised and cor
reeled. i2mo. Philadelphia, 1900. $2.50

Evers. Steam and other Prime Movers. A Text-Book both Theoretical
and Practical. Illustrated. I2mo. London, 1890. $1-50

Steam and the Steam Engine ; Land, Marine, and Locomotive II

histrated. I2mo. New York. $1.00

Ewing. Steam Engine and other Heating Engines. Illustrated. 8vo.
Cambridge, 1899. #3-75

LIST OF BOOKS.

Goodeve. Text-Book on the Steam Engine. With a Supplement on Gas
Engines and on Heat Engines. I3th edition, enlarged. i2mo. 143
illustrations. London, 1896. $2.00

Gould. Arithmetic of the Steam Engine. i2mo. N. Y. 1898. $1.00

Grimshaw. Steam Engine Catechism. A series of direct practical an
swers to direct practical questions, mainly intended for young engineers
and for examination questions, nth edition, enlarged and improved.
iSiuo. New York, 1897. $2.00

Haeder. Hand-Book on the Steam Engine with especial Reference to
Small and Medium sized Engines. For the Use of Engine Makers, Me-
chanical Draughtsmen, Engineering Students, and Users of Steam
Power, i, i oo illustrations, ismo. London, 1896. $3.00

Henthorn. Corliss Engine and its Management. Edited by E. P. Watson.
3d edition, enlarged with an appendix, by Emil Herter. Illustrated.
i8mo. New York, 1897. $1.00

Holmes. Steam Engine. 212 illustrations. loth edition. I2mo. London,
1900. $2.00

This is a complete practical and theoretical treatise on the steam-engine, written in
very clear and beautiful style, rendering the more abstruse principles of the subject as
plain and simple as it is probably possible to make them. It is one of the best, if not the
best, combinations of theoretical investigation and practical applications in the whole lite-
rature of the subject, and forms an admirable companion to Ripper's smaller and more
exclusively practical treatise.

Jamieson. Text-Book of Steam and Steam Engines, isth edition,
with numerous Diagrams, four folding Plates, and Examination
Questions. I2mo. London, 1901. $3.00

— Elementary Manual on Steam and the Steam Engine. With
numerous Diagrams, Arithmetical Examples, and Examination
Questions. 8th edition. I2mo. London, 1900. $1.40

Lardner. Treatise on the Steam Engine, for the Use of Beginners.
i6th edition. Illustrated. London, 1893. $0.60

Le Van. Steam Engine and the Indicator; their Origin and Progres-
sive Development, including the most recent examples of Steam
and Gas Motors, together with the Indicator, its Principles, its
Utility, and its Application. Illustrated by 205 Engravings, chiefly
of Indicator Cards. 8vo. Philadelphia, 1900. $4.00

Mallet. Compound Engines. i6mo, boards. New York, 1884. $0.50

Marks. Relative Proportions of the Steam Engine. Illustrated. 3d edi-
tion. I2mo. Philadelphia, 1896. $3.00

Peabody. Table of the Properties of Saturated Steam and other
Vapors. 6th edition. 8vo. New York, 1901. $1.00

D. VAN NOSTRAND COMPANY.

Pray. Steam Tables and Engine Constants. For facilitating all calcu-
lations upon Indicator Diagrams or Various Problems connected with
the operation of the Steam Engine, from reliable data and with precision
compiled from Regnault, Rankine, and Dixon directly, making use of the
exact records. 8vo. New York, 1894. $2.00

Rankine. Manual of the Steam Engine and other Prime Movers, with
numerous tables and illustrations. i2mo. I4th edition. London, 1897.

$5.00

Rigg. Practical Treatise on the Steam Engine, containing Plans and
Arrangements of Details for Fixed Steam Engines, with Essays on the
Principles involved in Design and Construction. Copiously illustrated
with woodcuts and 96 plates. 4to. 2d edition. New York, 1894.

$10.00

Ripper. Steam. Illustrated. i2mo. London, 1889. $1.00

This work is based upon a course of lectures given to an evening class of young me-
chanical engineers on steam, steam-engines, and boilers. It is remarkably clear, concise,
and practical ; no superfluous matter is introduced, and every page goes directly to the
point. It is the best book for beginners, and also for those who wish to have a manual
embracing the practical features of the subjects in small compass.

Roper. Hand-Book of Modern Steam Fire Engines ; including the run-
ning, care, and management of Steam Fire Engines and Fire Pumps.
2d edition, revised and corrected by H. L. Stellwagen. Illustrated.
I2mo, mor. tucks. Philadelphia, 1897. $3-5O

• Hand-Book of Land and Marine Engines, including the Modelling,

Construction, Running, and Management of Land and Marine Engines
and Boilers, gth edition, revised, enlarged, and improved. i2mo, mor.
tucks. Philadelphia, 1897. $3- S°

Catechism of High Pressure or Non-Condensing Steam Engines,

including the Modelling, Constructing, Running, and Management of
Steam Engines and Steam Boilers. 2Oth edition, revised and enlarged.
Illustrated. i2mo, mor. tucks. Philadelphia, 1897. $2.00

Young Engineer's Own Book. Containing an Explanation of the

Principle and Theories on which the Steam Engine as a Prime Mover is
based, with a description of different kinds of Steam Engines, Condens-
ing and Non-Condensing, Marine, Stationary, Locomotive, Fire, Trac-
tion, and Portable. 106 illustrations. 3d edition, revised. 16 mo, mor.
tucks. Philadelphia, 1897. #3-oo

Rose. Modern Steam Engines. An Elementary Treatise upon the
Steam Engine, written in Plain Language ; for use in the Workshop as
well as in the Drawing Office. Giving Full Explanations of the Con-

LIST OF BOOKS.

struction of Modern .Steam Engines ; including Diagrams showing their
Actual Operation ; together with Complete but Simple Explanation of
the Operations of various kinds of Valves, Valve Motions, and Link
Motions, etc., thereby enabling the ordinary engineer to clearly under-
stand the Principles involved in their Construction and use, and to Plot
out their movements upon the Drawing Board. New edition, revised
and improved. 453 illustrations. 410. Philadelphia, 1900. $6.00

- Key to Engines and Engine Running. A Practical Treatise upon
the Management of Steam Engines and Boilers, for the use of those
who desire to pass an Examination to take Charge of an Engine or
Boiler. With numerous Illustrations and Instructions upon Engineers'
Calculations, Indicator Diagrams, Engine Adjustments, and other Valu-
able Information necessary for Engineers and Firemen. I2mo. N. Y.
1899. $2.50

Thurston. History of the Growth of the Steam Engine. 4th revised

edition. Illustrated. I2mo. New York, 1897. $2.50

— Manual of the Steam Engine. For Engineers and Technical

Schools. Part I, Structure and Theory. Illustrated. 5th edition,

revised. 8vo. New York, 1900. $6.00

Part II. Design, Construction, and Operation. Illustrated. 4th edi-
tion, revised. 8vo. New York, 1900. $6.00
Or in sets. $10.00

• Hand-Book of Engine and Boiler Trials, and of the Indicator

and Prony Brake, for Engineers and Technical Schools. Illustrated.
4th and revised edition. 8vo. New York, 1897. $5.00

— Stationary Steam Engines, Simple and Compound, especially

as adapted to Electric Lighting Purposes. 5th edition, revised, with
additions. Illustrated, i2mo. New York, 1893. $2.50

Turnbull. Treatise on the Compound Engine. 2d edition, revised
and enlarged by Prof. S. W, Robinson. i6mo, boards. New York,
1884. $0.50

Watson, E. P. Small Engines and Boilers. A manual of Concise
and Specific Directions for the Construction of Small Steam Engines
and Boilers of Modern Types from five Horse-power down to model
sizes. Illustrated. I2tno, cloth. $1.25

Weisbach. Heat, Steam, and Steam Engine. Translated from the
4th edition of Vol. II. of Weisbach's Mechanics. Containing Notes
giving practical examples of Stationary, Marine, and Locomotive
Engines, showing American practice, -by R. H. Buel. Numerous
illustrations. 8vo. New York, 1891. $5-oo

D. VAN NOSTRAND COMPANY

Whitham. Steam Engine Design. For the use of Mechanical Engi-
neers, Students, and Draughtsmen. 3d edition, revised. With 210
illustrations. 8vo. New York, 1902. $5.00

TRANSMISSION OF POWER, BELTING, ETC.

Kerr, E. W,, M.E. Power, and Power Transmission. A 'Series of
Lectures delivered to Students, on the Elementary Principles of
Engineering. Containing numerous full-page Diagrams, Figures
and Tables. Illustrated. 8vo, cloth. New York, 1902. $2.00

Toothed Gearing. A Practical Hand-Book for Offices and Workshops.
By a Foreman Pattern Maker. 184 illustrations, tamo. London,
1892. $2.50

Unwin. On the Development and Transmission of Power from Cen-
tral Stations. Being the Howard Lectures delivered at the Society
of Arts in 1893. Illustrated. 8vo, New York, 1894, $3.50

VALVES AND VALVE GEARS.

Auchincloss. Practical Application of the Slide-Valve and Link-
Motion to Stationary, Portable, Locomotive, and Marine Engines,
with new and simple methods for proportioning the parts. Illus-
trated. I4th edition, revised and enlarged. 8vo. New York,
1901. $2.00

Bankson. Slide Valve Diagrams. A French Method of Obtaining
Slide Valve Diagrams. 8 Plates. i6mo. New York, 1892. $0.50

Begtrup, Julius. Slide Valve and its Functions. With special refer-
ence to modern Practice in the United States. With many Dia-
grams. Illustrated. 8vo, cloth. $2.00

Buel. Safety Valves. 3d edition. i6mo, boards. New York,
1898. $0.50

Halsey. Slide Valve Gears ; an explanation of the Action and Con-
struction of plain and cut-off Slide Valves. Analysis by the Bilgram
Diagram. 79 illustrations. 7th edition, revised and enlarged,
ismo. New York, 1901. $1.50

Worm and Spiral Gearing. With Illustrations and Diagrams.

Van Nostrand's Science Series, No. 116.) $0.50

LIST OF BOOKS.

Le Van. Safety Valves ; Their History, Antecedents, Invention, and
Calculation. 69 Illustrations. I2mo. New York, 1892. $2.00

MacCord. Treatise on the Movement of the Eccentric upon the Slide
Valve, and explaining the Practical Process of Laying out the Move-
ments, adapting the Valve for its various duties in the Steam Engine,
for the Use of Engineers, Draughtsmen, Machinists, and Students of
Valve Motion in general. 2d edition. 4to. Illustrated. New York,
1883. #2.50

Peabody. Valve Gears and Steam Engines. 33 Plates. 8vo. New
York, 1900. $2.50

Rose. Slide Valve Practically Explained. Embracing Simple and
Complete Practical Demonstrations of the Operations of each Element
in a Slide-Valve Movement, and illustrating the effects of variations in
their proportions, by examples carefully selected from the most Decent
and successful practice. Illustrated. I2mo. Philadelphia, 1895. $1.00

Spangler. Valve Gears. 2d edition, revised and enlarged. 8vo. New
York, 1900. $2.50

Welch. Treatise on a Practical Method of Designing Slide Valve Gear-
ing by Simple Geometrical Construction, based upon the principles enun-
ciated in Euclid's Elements, and comprising the various forms of Plain
Slide Valve and Expansion Gearing; together with Stephenson's, Gooch's,
and Allen's Link Motions, as applied either to reversing or to variable
expansion combinations. i2mo. London, 1875. $I-5°

Zeuner. Treatise on Valve Gears, with Special consideration of the link
motions of locomotive engines. 4th edition. Translated by Prof. J. F.
Klein. 8vo. London, 1884. $S-°°

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