# Full text of "Handbook of Formula and Physical constants"

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```Table of Contents

TOPIC PAGE

SI Multiples 1

Basic Units (distance, area, volume, mass, density) 2

Mathematical Formulae 5

Applied Mechanics 10

Thermodynamics 21

Fluid Mechanics 28

Electricity 30

Periodic Table 34

Names in the Metric System

VALUE

EXPONENT

SYMBOL

PREFIX

1 000 000 000 000

10 12

T

tera

1 000 000 000

10 9

G

giga

1 000 000

10 6

M

mega

1 000

10 3

k

kilo

100

10 2

h

hecto

10

10'

da

deca

0.1

10" 1

d

deci

0.01

10" 2

c

centi

0.001

10" 3

m

m i Mi

0.000 001

10" 6

H

micro

0.000 000 001

10" 9

n

nano

0.000 000 000 001

10 12

P

pico

Conversion Chart for Metric Units

CD

>

O

O

To
Milli-

To
Centi-

To
Deci-

To

Metre,
Gram,
Litre

To
Deca-

To
Hecto-

To
Kilo-

Kilo-

x10 6

x10 5

x10 4

x10 3

x10 2

x10 1

Hecto-

x10 5

x10 4

x10 3

x10 2

x10 1

x10" 1

Deca-

x10 4

x 10 3

x10 2

x10 1

x10" 1

x10" 2

Metre,
Gram,
Litre

x 10 3

x 10 2

x10 1

x10" 1

x10" 2

x10" 3

Deci-

x 10 2

x 10 1

x10" 1

x10" 2

x10" 3

x10 4

Centi-

x 10 1

x10" 1

x10" 2

x10" 3

x10 4

x10" 5

Milli-

x 10" 1

x10" 2

x10" 3

x10 4

x10" 5

x10" 6

Page 1

BASIC UNITS

SI

IMPERIAL

DISTANCE

1 metre (1 m) = 10 decimetres (10 dm)

12 in. = 1 ft

= 100 centimetres (100 cm)

3ft = 1yd

= 1000 millimetres (1000 mm)

5280 ft = 1 mile

1760 yd = 1 mile

1 decametre (1 dam) = 10 m

1 hectometre (1 hm) = 100 m

1 kilometre (1 km) = 1000 m

Conversions:

lin.

= 25.4 mm

1ft

= 30.48 cm

mile

= 1.61km

1yd

= 0.914 m

lm

= 3.28 ft

Area

1 sq metre (lm 2 ) = 10 000 cm 2

= 1 000 000 mm 2

1 sq hectometre (1 hm 2 ) = 10 000 m 2

= 1 hectare (1 ha)

1 sq km (1 km 2 ) = 1 000 000 m 2

lft 2

lyd 2

1 sq mile

144 in. 2
9 ft 2
640 acre

1 section

Conversions:

lin. 2

lm 2

1 acre

1 sq mile

6.45 cm 2
10.8 ft 2
0.405 ha
2.59 km 2

645 mm

Page 2

SI

IMPERIAL

Volume

lm'

= 1 000 000 cm'
= lxl0 9 mm'

1ft'
1yd'

= 1728 in.'
= 27 ft 3

1dm'

1 litre

1 mL

lm'

= 1 litre
= 1000 cm'
= 1 cm'
= 1000 litres

1 (liquid) U.S. gallon

1 U.S. barrel (bbl)
1 imperial gallon

= 231 in.'
= 4 (liquid) quarts
= 42 U.S. gal.
= 1.2 U.S. gal.

Conversions:

lin.' =

16.4 cm'

lm' =

35.3 ft'

1 litre =

61 in.'

lU.S.gal =

3.78 litres

1 U.S. bbl =

159 litres

1 litre/s =

15.9 U.S. gal/min

Mass and Weight

1 kilogram (1 kg)
1000 kg

1000 grams
1 tonne

2000 lb
1 long ton

1 ton (short)
2240 lb

Density

Conversions:

1 kg (on Earth) results in a weight of 2.2 lb

mass density :

m fkg

V m

mass
volume

weight density

w

V

jb
ft 3

weight
volume

Conversions:

(on Earth) a mass density of 1 — 3 results in a weight density of 0.0623 —

Page 3

SI

RELATIVE DENSITY

In SI R.D. is a comparison of mass density
to a standard. For solids and liquids the
standard is fresh water,
water.

Imperial

In Imperial the corresponding quantity is
specific gravity; for solids and liquids a
comparison of weight density to that of

Conversions:

In both systems the same numbers
hold for R.D. as for S.G. since
these are equivalent ratios.

RELATIVE DENSITY (SPECIFIC GRAVITY) OF VARIOUS SUBSTANCES

Water (fresh) 1.00

Water (sea average) .... 1.03

Aluminum 2.56

Antimony 6.70

Bismuth 9.80

Brass 8.40

Brick 2.1

Calcium 1.58

Carbon (diamond) 3.4

Carbon (graphite) 2.3

Carbon (charcoal) 1.8

Chromium 6.5

Clay 1.9

Coal 1.36-1.4

Cobalt 8.6

Copper 8.77

Cork 0.24

Glass (crown) 2.5

Glass (flint) 3.5

Gold 19.3

Iron(cast) 7.21

Iron (wrought) 7.78

Magnesium 1.74

Manganese 8.0

Mercury 13.6

Mica 2.9

Nickel 8.6

Oil (linseed) 0.94

Oil (olive) 0.92

Oil (petroleum) 0.76-0.86

Oil (turpentine) 0.87

Paraffin 0.86

Platinum 21.5

Sand (dry) 1.42

Silicon 2.6

Silver 10.57

Slate 2.1-2.8

Sodium 0.97

Steel (mild) 7.87

Sulphur 2.07

Tin 7.3

Tungsten 19.1

Wood (ash) 0.75

Wood (beech) 0.7-0.8

Wood (ebony) 1.1-1.2

Wood (elm) 0.66

Wood (lignum-vitae) ..1.3

Wood(oak) 0.7-1.0

Wood (pine) 0.56

Wood (teak) 0.8

Zinc 7.0

Page 4

Greek Alphabet

Alpha

a

Beta

P

Gamma

Y

Delta

A

Epsilon

s

Zeta

c

Eta

Tl

Theta

e

Iota

i

Kappa

K

Lambda

X

Mu

n

Nu

V

Xi

S

Omicron

Pi

71

Rho

P

Sigma

I, a

Tau

X

Upsilon

u

Phi

®A

Kai

i

Psi

V

Omega

Q, co

MATHEMATICAL FORMULAE

Algebra

1. Expansion Formulae

(x + y) = x + 2xy + y

(x - y) = x - 2xy + y
x 2 -y 2 = (x-y) (x + y)

(x + y) = x + 3x y + 3xy + y
x + y = (x + y) (x - xy + y )

(x-y) = x" - 3x y + 3xy - y
x 3 -y 3 = (x - y) (x 2 + xy + y 2 )

Ifax 2 + bx + c = 0,

Then x

■b±Vb 2 -4ac

2a

Page 5

Trigonometry

1. Basic Ratios

Sin A = — , cos A = —

tan A:

y

2. Pythagoras' Law

2,2 i 2

x +y = n

3. Trigonometric Function Values

Sin is positive from 0° to 90° and positive from 90° to 180°
Cos is positive from 0° to 90° and negative from 90° to 180°
Tan is positive from 0° to 90° and negative from 90° to 180°

4. Solution of Triangles
a. Sine Law

Sin A Sin B Sin C

b. Cosine Law

a 2 + b 2 - 2 ab Cos C

b 2 + c 2 - 2 be Cos A

a 2 + c 2 - 2 ac Cos B

Page 6

Geometry

1. Areas of Triangles

a. All Triangles

base x perpendicular height

Area:

Area

be Sin A ab Sin C ac Sin B

and,

Area = ^J s (s - a) (s - b) (s - c)

where, s is half the sum of the sides, or s

a + b + c

b. Equilateral Triangles

Area = 0.433 x side 2

2. Circumference of a Circle

C = 7id

3. Area of a Circle

A = TOf

circumference x r %

d 2 =0.7854d 2

4. Area of a Sector of a Circle

arcxr

A = — — x 71 r 2 (9 = angle in degrees)

A :

360

9°r 2

Page 7

5. Area of a Segment of a Circle

A = area of sector - area of triangle

Also approximate area = — h 2 J— - 0.608

6. Ellipse

A=*Dd

Approx. circumference = it

(D + d)

t
d

Triangle

Segment

7. Area of Trapezoid

H a — H

A :

a + b'

-H

8. Area of Hexagon

A = 2.6s 2 where s is the length of one side

9. Area of Octagon

A = 4.83s 2 where s is the length of one side

10. Sphere

Total surface area A =4tuc

Surface area of segment A s = 7tdh

4 3
Volume V= — 7ir
3

Volume of segment
V s = ^(3r-h)

V. = -^-(h 2 + 3a 2 ) where a = radius of segment base

Page 8

11. Volume of a Cylinder

V = — d 2 L where L is cylinder length

12. Pyramid

Volume

V = - base area x perpendicular height
Volume of frustum

Frustum

V F = — (A + a + v Aa) where h is the perpendicular height, A and a are areas as shown

13. Cone

Area of curved surface of cone:

A :

7i DL

Area of curved surface of frustum

A F

7c (D + d)L

Volume of cone:

_ base area x perpendicular height

Volume of frustum:

_ perpendicular height x tt (R 2 + r 2 + Rr)

Page 9

APPLIED MECHANICS

Scalar - a property described by a magnitude only

Vector - a property described by a magnitude and a direction

, T , mjL , A displacement
Velocity - vector property equal to — —.

The magnitude of velocity may be referred to as speed
In SI the basic unit is ™, in Imperial ^

Other common units are ^H, ^

h h

^ . , m . _„ ft

Conversions: 1 — = 3.28 —

, km „ ^„, mi

1 = 0.621 —

h h

Speed of sound in dry air is 33 1 ^ at 0°C and increases by about 0.61 ^ for each °C

s
rise

Speed of light in vacuum equals 3 x 10 8 m

Acceleration - vector property equal to

s

change in velocity
time

In SI the basic unit is — -, in Imperial — -

s s

-. . , m _ _„ ft

Conversion: 1— = 3.28 —

s s

Acceleration due to gravity, symbol "g", is 9.81 — or 32.2 — -

s s

Page 10

LINEAR VELOCITY AND ACCELERATION

u initial velocity

v final velocity

t elapsed time

s displacement

a acceleration

v = u + at
'v + u

t

v

1 9

s = ut + y at

2 - u 2 + 2 as

Angular Velocity and Acceleration

co angular velocity (radians/s); coi = initial, co 2 = final

a angular acceleration (radians/s 2 )

co 2 = coi + at

9 = coi + co 2 x t

2

9 = coit + V^ar

co 2 2 = coi 2 + 2a9

linear displacement, s = r 9

linear velocity, v = r co

linear, or tangential acceleration, a T = r a

Page 1 1

Tangential, Centripetal and Total Acceleration

Tangential acceleration ax is due to angular acceleration a

a T = roc

Centripetal (Centrifugal) acceleration a c is due to change in direction only

a c = v 2 /r = r co 2

Total acceleration, a, of a rotating point experiencing angular acceleration is the vector sum
of a T and a c

a = ax + a c
FORCE

Vector quantity, a push or pull which changes the shape and/or motion of an object

In SI the unit of force is the newton, N, defined as a ,

s

In Imperial the unit of force is the pound lb

Conversion: 9.81 N = 2.2 lb
Weight

The gravitational force of attraction between a mass, m, and the mass of the Earth
In SI weight can be calculated from

Weight = F = mg , where g = 9.81 m/s 2

In Imperial, the mass of an object (rarely used), in slugs, can be calculated from the known
weight in pounds

Weight ft

m = — g — § = 32 V

Page 12

Newton's Second Law of Motion

An unbalanced force F will cause an object of mass m to accelerate a, according to:
F = ma (Imperial F = ^ a, where w is weight)

Torque Equation

T = I a where T is the acceleration torque in Nm, I is the moment of inertia in kg m 2
and a is the angular acceleration in radians/s 2

Momentum

Vector quantity, symbol p,

p = mv (Imperial p = ^ v, where w is weight)

. „ x . . kg m
in SI unit is — ^ —

Work

Scalar quantity, equal to the (vector) product of a force and the displacement of an object. In
simple systems, where W is work, F force and s distance

W = Fs

In SI the unit of work is the joule, J, or kilojoule, kJ

1 J = 1 Nm

In Imperial the unit of work is the ft-lb

Energy

Energy is the ability to do work, the units are the same as for work; J, kJ, and ft-lb

Page 13

Kinetic Energy

Energy due to motion
E v = imv 2

In Imperial this is usually expressed as E k = ^v 2 where w is weight
Kinetic Energy of Rotation

E R = — mk 2 (D 2 where k is radius of gyration, co is angular velocity in rad/s

or

Ep = — Icd 2 where I = mk 2 is the moment of inertia

R 2

CENTRIPETAL (CENTRIFUGAL) FORCE

F c = where r is the radius

or

F c = m co 2 r where co is angular velocity in rad/s
Potential Energy

Energy due to position in a force field, such as gravity

E p = m g h

In Imperial this is usually expressed E p = w h where w is weight, and h is height above some
specified datum

Page 14

Thermal Energy

In SI the common units of thermal energy are J, and kJ, (and kJ/kg for specific quantities)

In Imperial, the units of thermal energy are British Thermal Units (Btu)

Conversions: 1 Btu = 1055 J

1 Btu = 778 ft-lb

Electrical Energy

In SI the units of electrical energy are J, kJ and kilowatt hours kWh. In Imperial, the unit of
electrical energy is the kWh

Conversions: 1 kWh = 3600 kJ

1 kWh = 3412 Btu = 2.66 x 10 6 ft-lb

Power

A scalar quantity, equal to the rate of doing work
In SI the unit is the Watt W (or kW)

lW=l|
In Imperial, the units are:

Mechanical Power - ~ — , horsepower h.p.

Thermal Power - =^±

Electrical Power - W, kW, or h.p.

Conversions: 746 W = 1 h.p.

1 h.p. = 550 &^

1 kW = 0.948 =^=

Page 15

Pressure

A vector quantity, force per unit area

In SI the basic units of pressure are pascals Pa and kPa

lPa=l4
m

In Imperial, the basic unit is the pound per square inch, psi
Atmospheric Pressure

At sea level atmospheric pressure equals 101.3 kPa or 14.7 psi
Pressure Conversions

1 psi = 6.895 kPa

Pressure may be expressed in standard units, or in units of static fluid head, in both SI and
Imperial systems

Common equivalencies are:

1 kPa = 0.294 in. mercury = 7.5 mm mercury
1 kPa = 4.02 in. water = 102 mm water
1 psi = 2.03 in. mercury = 51.7 mm mercury
1 psi = 27.7 in. water = 703 mm water
1 mH 2 = 9.81 kPa

Other pressure unit conversions:

1 bar= 14.5 psi = 100 kPa

.2

1 kg/cm =98.1 kPa= 14.2 psi = 0.981 bar
1 atmosphere (atm) = 101.3 kPa = 14.7 psi

Page 16

Simple Harmonic Motion

Velocity of P = oo a/R 2 -x :

Acceleration of P = (D xm/s

m

s

2

2%

The period or time of a complete oscillation = — seconds

CO

General formula for the period of S.H.M.

T = 2jc.

(displacement
acceleration

Simple Pendulum

T = 2ti / — T = period or time in seconds for a double swing

V §

L = length in metres

The Conical Pendulum

(elevation)

force diagram

R/H = tan 9= F c /W = co 2 R/g

Page 17

Lifting Machines

F = force applied

MA.

effort

W

F

V.R. (velocity ratio)

n

efficiency

effort distance

M.A.

V.R.

1. Lifting Blocks

V.R. = number of rope strands supporting the load block

2. Wheel & Differential Axle

Velocity ratio

2tiR

27i(r - r, )

2

2R

r-r,

2R

„ . . 2D

Velocity ratio

(d-d,)

3. Inclined Plane

V.R

length
height

4. Screw Jack

V.R

circumference of leverage

Gearing

S=

n

Snatch block

Pitch

A

*

Effort

Page 18

Indicated Power

LP. = P m A L N where LP. is power in W, P m is mean or "average" effective pressure in
Pa, A is piston area in m , L is length of stroke in m and N is number of
power strokes per second

Brake Power

B.P. = Lcd where B.P. is brake power in W, L is torque in Nm and co is angular

STRESS, STRAIN and MODULUS OF ELASTICITY

Direct stress

Direct strain

area A

extension A£
original length L

Modulus of elasticity

„ direct stress P/A

PL

direct strain AiVL AAl

Shear stress x

Shear strain

force

area under shear

L

Modulus of rigidity

_ shear stress

(j

shear strain

Force

Force

x Force

Page 19

General Torsion Equation (Shafts of circular cross-section)

T
J

G0

1. For Solid Shaft

T % 4 7id 4
J = — r =

2 32

2. For Hollow Shaft

J = f(r: 4 -r 2 4 )

32

(d 4 -d 4 )

T = torque or twisting moment in newton metres

J = polar second moment of area of cross-section

x = shear stress at outer fibres in pascals

r = radius of shaft in metres

G = modulus of rigidity in pascals

= angle of twist in radians

L = length of shaft in metres

d = diameter of shaft in metres

Relationship Between Bending Stress and External Bending Moment

M
I

(7

y

E
R

1. For Rectangle

D

-

BD 3

M
I

c

y

E
R

external bending moment in newton metres

second moment of area in m

bending stress at outer fibres in pascals

distance from centroid to outer fibres in metres

modulus of elasticity in pascals

12

2. For Solid Shaft

7iD 4

64

Page 20

THERMODYNAMICS

Temperature Scales

°C = -(°F-32) °F=-°C + 32

9 5

°R = °F + 460 (R Rankine) K = °C + 273 (K Kelvin)
Sensible Heat Equation

Q = mcAT

m is mass

c is specific heat

AT is temperature change

Latent Heat

Latent heat of fusion of ice = 335 kJ/kg

Latent heat of steam from and at 100°C = 2257 kJ/kg

1 tonne of refrigeration = 335 000 kJ/day

= 233 kJ/min

Gas Laws

1. Boyle's Law

When gas temperature is constant

PV = constant or

PiV, = P 2 V 2

where P is absolute pressure and V is volume

2. Charles' Law

V
When gas pressure is constant, — = constant

or -=i = -=!■ , where V is volume and T is absolute temperature

Page 21

3. Gay-Lussac's Law

When gas volume is constant, — = constant

T

P P

Or — = — , where P is absolute pressure and T is absolute temperature
lj l 2

4. General Gas Law

PiV, _ P 2 V 2 _
T, T 2

= constant

P V=mRT

where P

V

T

m

R

Also

PV = nPvoT

where P

V

T

N

Ro

absolute pressure (kPa)

volume (nr )

absolute temp (K)

mass (kg)

characteristic constant (kJ/kgK)

absolute pressure (kPa)

volume (m 3 )

absolute temperature K

the number of kmoles of gas

the universal gas constant 8.314 kJ/kmol/K

SPECIFIC HEATS OF GASES

Specific Heat at

Specific Heat at

Ratio of Specific

Constant Pressure

Constant Volume

Heats

kJ/kgK

kJ/kgK

T = C p /C v

GAS

or

or

kJ/kg °C

kJ/kg °C

Air

1.005

0.718

1.40

Ammonia

2.060

1.561

1.32

Carbon Dioxide

0.825

0.630

1.31

Carbon Monoxide

1.051

0.751

1.40

Helium

5.234

3.153

1.66

Hydrogen

14.235

10.096

1.41

Hydrogen Sulphide

1.105

0.85

1.30

Methane

2.177

1.675

1.30

Nitrogen

1.043

0.745

1.40

Oxygen

0.913

0.652

1.40

Sulphur Dioxide

0.632

0.451

1.40

Page 22

Efficiency of Heat Engines

T - T

Carnot Cycle r\ = —^-= — - where Ti and T 2 are absolute temperatures of heat source and
lj

sink

Air Standard Efficiencies

1. Spark Ignition Gas and Oil Engines (Constant Volume Cycle or Otto Cycle)

1 . . cylinder volume

T| = 1 - — — — where r v = compression ratio =

r; Y " } v clearance volume

specific heat (constant pressure)

y ~

specific heat (constant volume)

2. Diesel Cycle

(R Y — V)

n = 1 — ^ — where r = ratio of compression

r v T y(R-l)

R = ratio of cut-off volume to clearance volume

3. High Speed Diesel (Dual-Combustion) Cycle

kp y -l

n = l-

where r v

k

rr[(k-l) + yk(P-l)]
cylinder volume

P

clearance volume

absolute pressue at end of constant V heating (combustion)
absolute pressue at beginning of constant V combustion

volume at end of constant P heating (combustion)

clearance volume
4. Gas Turbines (Constant Pressure or Brayton Cycle)

1

r, = l-

Page 23

where r p = pressure ratio

compressor discharge pressure
compressor intake pressure

W

o
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Internal
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Page 24

Heat Transfer by Conduction

n= AAtAT
V d

where Q = heat transferred in joules

A, = thermal conductivity or coeficient of heat

transfer in

J x m

or

W

m 2 x s x °C m x °C

A = area in m 2
t = time in seconds
AT = temperature difference between surfaces in °C
d = thickness of layer in m

COEFFICIENTS OF THERMAL CONDUCTIVITY

Material

Coefficient of

Thermal Conductivity

W/m °C

Air

0.025

Aluminum

206

Brass

104

Brick

0.6

Concrete

0.85

Copper

380

Cork

0.043

Felt

0.038

Glass

1.0

Glass, fibre

0.04

Iron, cast

70

Plastic, cellular

0.04

Steel

60

Wood

0.15

Wallboard, paper

0.076

Page 25

Thermal Expansion of Solids

Increase in length

where L

a

(T 2 - Ti )

L a (T 2 - Ti )

original length

coefficient of linear expansion

rise in temperature

Increase in volume
Where V

P
(T 2 - T, )

V P (T 2 - Ti )

original volume

coefficient of volumetric expansion

rise in temperature

coefficient of volumetric expansion

P

coefficient of linear expansion x 3

3a

SPECIFIC HEAT and LINEAR EXPANSION OF SOLIDS

Mean

Coefficient

Mean

Coefficient

Specific Heat

of

Specific Heat

of

between 0°C

Linear Expansion

between 0°C

Linear Expansion

Solid

and 100°C

between 0°C and

Solid

and 100°C

between 0°C and

kJ/kgK or

100°C
(Multiply by 10" 6 )

kJ/kgK or

100°C
(Multiply by 10" 6 )

kJ/kg°C

kJ/kg°C

Aluminum

0.909

23.8

Iron (cast)

0.544

10.4

Antimony

0.209

17.5

Iron (wrought)

0.465

12.0

Bismuth

0.125

12.4

0.131

29.0

Brass

0.383

18.4

Nickel

0.452

13.0

Carbon

0.795

7.9

Platinum

0.134

8.6

Cobalt

0.402

12.3

Silicon

0.741

7.8

Copper

0.388

16.5

Silver

0.235

19.5

Glass

0.896

9.0

Steel (mild)

0.494

12.0

Gold.

0.130

14.2

Tin

0.230

26.7

Ice

2.135

50.4

Zinc

0.389

16.5

(between -20°C

and 0°C)

SPECIFIC HEAT and VOLUME EXPANSION FOR LIQUIDS

Liquid

Specific Heat

(at20°C)

kJ/kgKorkJ/kg°C

Coefficient of
Volume Expansion
(Multiply by 10" 4 )

Liquid

Specific Heat

(at 20°)

kJ/kgKorkJ/kg°C

Coefficient of
Volume Expansion
(Multiply by 10-4)

Alcohol (ethyl)

2.470

11.0

Olive Oil

1.633

Ammonia

0.473

Petroleum

2.135

Benzine

1.738

12.4

Gasoline

2.093

12.0

Carbon Dioxide

3.643

1.82

Turpentine

1.800

9.4

Mercury

0.139

1.80

Water

4.183

3.7

Page 26

Chemical Heating Value of a Fuel

Chemical Heating Value MJ per kg of fuel = 33.7 C + 144 (h 2 - ^) + 9.3 S

8

C is the mass of carbon per kg of fuel

H 2 is the mass of hydrogen per kg of fuel

O2 is the mass of oxygen per kg of fuel

S is the mass of sulphur per kg of fuel

Theoretical Air Required to Burn Fuel

Air (kg per kg of fuel) = [-C + 8 (h 2 - ^_) + s]

100

23

Air Supplied from Analysis of Flue Gases

Air in kg per kg of fuel = 33 (C( ^ 2 + CQ) x C

C is the percentage of carbon in fuel by mass

N 2 is the percentage of nitrogen in flue gas by volume

C0 2 is the percentage of carbon dioxide in flue gas by volume

CO is the percentage of carbon monoxide in flue gas by volume

Boiler Formulae

m,(h!-h 2 )

Equivalent evaporation
Factor of evaporation =
Boiler efficiency

2257 kJ/kg

2257 kJ/kg
m s( h i- h 2 )

m f x calorific value of fuel

where rh s = mass flow rate of steam

hi = enthalpy of steam produced in boiler

h 2 = enthalpy of feedwater to boiler

rh, = mass flow rate of fuel

Page 27

FLUID MECHANICS

Discharge from an Orifice

Let A

and A c
then A c

or C c

cross-sectional area of the orifice = (7t/4)d

cross-sectional area of the jet at the vena conrtacta = ((tt/4) d ^

C C A

A

where C c is the coefficient of contraction

Vena contracta

At the vena contracta, the volumetric flow rate Q of the fluid is given by

Q = area of the jet at the vena contracta x actual velocity
= A c v
or Q = C C AC V s/^gli

The coefficients of contraction and velocity are combined to give the coefficient of discharge,

C d

i.e. (^j — ^A^„

and Q = C d A ^/^gh
Typically, values for Cd vary between 0.6 and 0.65
Circular orifice: Q = 0.62 A y/2gh
Where Q = flow (m 3 /s) A = area (m 2 ) h = head (m)
Rectangular notch: Q = 0.62 (B x H) ^sfl\$\

Triangular Right Angled Notch: Q = 2.635 H 5/2
Where H = head (m above sill)

Page 28

Bernoulli's Theory

H

H

h

P

w 2g
height above datum level (metres)
pressure (N/m 2 or Pa)

w

V

force of gravity on 1 m of fluid (N)
velocity of water (metres per second)

Loss of Head in Pipes Due to Friction

Loss of head in metres = f-

?L v 2

L

d
pipes

length in metres
diameter in metres

d 2g

v =
f =

velocity of flow in metres per second

constant value of 0.01 in large pipes to 0.02 in small

Note: This equation is expressed in some textbooks as

T v 2
Loss = 4f=^ ^— where the f values range from 0.0025 to 0.005
d 2g °

Actual Pipe Dimensions

Scheduled (SI Units)

Nominal

Pipe Size

(in)

Outside
Diameter

(mm)

Inside

Diameter

(mm)

Wall

Thickness

(mm)

Flow
Area
(m 1 )

l
s

10.3

6.8

1.73

3.660 x 10 ~*

i

4

13.7

9.2

2.24

6.717 x 10" 5

1
8

17.1

12.5

2.31

1.236 x 10"*

1
2

21.3

15.8

2.77

1.960 x KT 4

1

4

26.7

20.9

2.87

3.437 x 10"*

1

33.4

26.6

3.38

5.574 x 10 " 4

li

42.2

35.1

3.56

9.653 x 10 ~*

ii

48.3

40.9

3.68

1.314 x 10" 3

2

60.3

52.5

3.91

2.168 x 10" 3

2i

73.0

62.7

5.16

3.090 x 10" 3

3

88.9

77.9

5.49

4.768 x 10" 3

8}

101.6

90.1

5.74

6.381 x 10 " 3

4

114.3

102.3

6.02

8.213 x 10 -3

5

141.3

128.2

6.55

1.291 x 10 -2

6

168.3

154.1

7.11

1.864 x KT 2

8

219.1

202.7

8.18

3.226 x 10 -2

10

273.1

254.5

9.27

5.090 x 10 " 2

12

323.9

303.2

10.31

7.219 x 10" 2

14

355.6

333.4

11.10

8.729 x 10 -2

16

406.4

381.0

12.70

0.1140

18

457.2

428.7

14.27

0.1443

20

508.0

477.9

15.06

0.1794

24

609.6

574.7

17.45

0.2594

Page 29

ELECTRICITY

Ohm's Law

or h

where I
E
R

E
R

IR

current (amperes)
electromotive force (volts)
resistance (ohms)

Conductor Resistivity

a
where p = specific resistance (or resistivity) (ohm metres, Q-m)
L = length (metres)
a = area of cross-section (square metres)

Temperature correction

R t = Ro (1 + at)

where R, = resistance at 0°C (Q)
R t = resistance at t°C (Q)
a = temperature coefficient which has an average value for copper of 0.004 28

(Q/Q°C)

R 2 = Rl (l±^il
(1 + atJ

where Ri = resistance at ti (Q)
R2 = resistance at t2 (Q)

a Values

Q/Q°C

copper

platinum

nickel

tungsten

aluminum

0.00428

0.00385

0.00672

0.0045

0.0040

Page 30

Dynamo Formulae

Average e.m.f. generated in each conductor = — rr r

° 60c

where Z = total number of armature conductors

c = number of parallel paths through winding between positive and negative brushes
where c = 2 (wave winding), c = 2p (lap winding)

O = useful flux per pole (webers), entering or leaving the armature
p = number of pairs of poles
N = speed (revolutions per minute)

Generator Terminal volts = E G - I a R a

Motor Terminal volts = E B + I a R a

where E G = generated e.m.f.

E B = generated back e.m.f.

I a = armature current

R a = armature resistance

Alternating Current

R.M.S. value of sine curve = 0.707 maximum value

Mean value of sine curve = 0.637 maximum value

., , R.M.S. value 0.707 1 „

Form factor of sinusoidal = = = 1.11

Mean value 0.637

pN
Frequency of alternator = cycles per second

Where p = number of pairs of poles
N = rotational speed in r/min

Page 31

Slip of Induction Motor

Slip speed of field - speed of rotor , nn

— - — xlOO

Speed of field

Inductive Reactance

Reactance of AC circuit (X) = 27ifL ohms
where L = inductance of circuit (henries)

Inductance of an iron cored solenoid = — ; — henries

LxlO 8

where T = turns on coil

[i = magnetic permeablility of core

A = area of core (square centimetres)

L = length (centimetres)

Capacitance Reactance

Capacitance reactance of AC circuit = ohms

2jrfC

r

Total reactance

27ifL ohms

2ti fC J

Impedence (Z) = -^/(resistance) 2 + (reactance) 2

R 2 + (2tc fL - — *— ) 2 ohms
I 27tfC

Current in AC Circuit

impressed volts

Current ■

impedance

Page 32

Power Factor

p.f.

true watts
volts x amperes

also p.f. = cos O, where O is the angle of lag or lead
Three Phase Alternators

Star connected
Line voltage
Line current

Delta connected
Line voltage

Line current

V3 x phase voltage
phase current

phase voltage

V3 x phase current

Three phase power

P = V3 E l I l cosO
E l = line voltage
II = line current
cos O = power factor

\ E 2

P

\ R

E

E \

/ E I

R \

/ I 2 R

1 P

I

\ R

E

\ I

\ R

E

/ VPR /

\y^ E

2 \^^

I R ^ /

\ P

/ P

p

/ I 2

i

Page 33

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Page 34

ION NAMES AND FORMULAE

MONATOMIC

POLYATOMIC

Ag +

silver ion

B0 3 3 "

borate ion

Al 3+

aluminum ion

C2H3O2

acetate ion

Au + and Au 2+

gold ion

CIO"

hypochlorite ion

Be 2+

beryllium ion

C10 2 "

chlorite ion

Ca 2+

calcium ion

CIO3-

chlorate ion

Co 2+ and Co 3+

cobalt ion

CIO4"

perchlorate ion

Cr 2+ and Cr 3+

chromium ion

CN"

cyanide ion

Cu + and Cu 2+

copper ion

CO3 2 "

carbonate ion

Fe 2+ and Fe 3+

iron ion

C2O4 2 "

oxalate ion

K +

potassium ion

Cr0 4 2 "

chromate ion

Li +

lithium ion

Cr 2 7 2 "

dichromate ion

Mg 2+

magnesium ion

HCO3"

hydrogen carbonate or bicarbonate ion

Na +

sodium ion

H 3 +

hydronium ion

Zn 2+

zinc ion

HPO4 2 "

hydrogen phosphate ion

H2PO4"

dihydrogen phosphate ion

HSO3"

hydrogen sulphite or bisulphite ion

HSO4"

hydrogen sulphate or bisulphate ion

Mn0 4 "

permanganate ion

N 3 "

azide ion

NH 4 +

ammonium ion

N0 2 "

nitrite ion

N0 3 "

nitrate ion

2 2 -

peroxide ion

OCN"

cyanate ion

OH"

hydroxide ion

P0 3 3 "

phosphite ion

P0 4 3 "

phosphate ion

SCN"

thiocyanate ion

SO3 2 "

sulphite ion

S0 4 2 "

sulphate ion

S2O3 2 "

thiosulphate ion

Page 35

power engineering

TRAINING SYSTEMS

This material is owned by Power Engineering Training Systems and may not be modified from its original form.
Duplication of this material for student use in-class or for examination purposes is permitted without written approval.

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