A meter for continuous indication of dissolved air in water

867

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NAVY DEPARTMENT
THE DAVID W. TAYLOR MODEL BASIN

WASHINGTON 7, D.C.

A METER FOR CONTINUOUS INDICATION OF
DISSOLVED AIR IN WATER _

by

H.M. Fitzpatrick
Gnd:
M.F Harkleroad

October 1954 Report 867

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A METER FOR CONTINUOUS INDICATION OF DISSOLVED AIR IN WATER

by

H.M. Fitzpatrick
and
M.F. Harkleroad

October 1954 Report 867
NS715-102

TABLE OF CONTENTS

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18

ABSTRACT

An instrument is described which continuously measures the concentra-
tion of dissolved gases in liquids. The operation of the instrument is based on
the establishment of equilibrium between the continuously flowing liquid sample
and a gas space of constant volume. The equilibrium pressure in the.gas space
indicates the concentration of dissolved gases. Experimental confirmation of the
validity of the method is presented for air dissolved in water. The design of a

practical instrument and criteria for its improvement are given.

INTRODUCTION

Methods of determining the concentration of dissolved gases in liquids find applica-
tions in such fields as medical and biological research, chemical industries, operation of
steam turbines, and hydrodynamic research. The necessity for control of the air content in
water used in cavitation experiments has long been recognized. Procedures for testing model
ship propellers in variable-pressure water tunnels have included the standardization of the
air content of the water in the tunnel.!»2 Whereas, in general, higher air content appears to

3 experiments and observations on

make for inception of cavitation at higher local pressures,
this and various other effects of dissolved air are neither complete nor in total agreement.
Continued interest in the role of air, both dissolved and undissolved, in cavitation phenomena
is directed not only toward the standardization of water-tunnel test procedures but also to-
ward fundamental studies of inception, growth, and collapse of cavities and of the noise

and erosion produced.

Methods employed at the David Taylor Model Basin for the determination of the air
content of water used in cavitation experiments have included the Winkler chemical test for
dissolved oxygen,* the Cambridge Instrument Company’s meter which measures the thermal
conductivity of the extracted gases,” and the method of Van Slyke.> In this last, the dis-
solved gases in a measured sample of the solution are extracted under a Torricellian vacuum
and the pressure of the extracted gases is measured upon recompression to a specified volume.
The various methods of measurement do not, in general, determine directly the same physical
quantity, especially if the solution is of a mixture of gases having different solubilities.

This report describes a method for obtaining a continuous indication of gas content,
discusses some of the principles involved in the cesign and operation of an instrument employ-
ing the method, presents experimental confirmation of its feasibility, and provides design in-

formation necessary for construction of a practical instrument

I peferences are listed on page 18.

to

The method was developed and the instrument constructed for use with the Taylor
Model Basin Flow Facility in the study of cavitation phenomena under the Bureau of Ships
Fundamental Hydrodynamics Research Program (NS715-102).

Operation of the instrument is based on certain well-known principles concerning the
solution of gases in liquids. The first of these principles relates to the condition for equili-
brium between the concentration of dissolved gas in solution and the pressure exerted by the
gas on the exposed surface of the liquid. At a given temperature, the concentration of dis-
solved gas will be, at equilibrium, a function of the pressure exerted by that gas on the ex-
posed surface of the liquid. If the pressure of the gas on the exposed surface is made higher
than the equilibrium pressure corresponding to the existing concentration of dissolved gas,
more gas will go into solution; if the pressure is reduced below the equilibrium value, gas will
be evolved from the liquid.

For most gases and liquids at moderate pressures, the relation between the concentra-
tion of dissolved gas and the equilibrium pressure is a simple proportionality. This generali-
zation, known as Henry’s Law,® while not essential to the operation of the instrument des-
cribed here, permits a simplified description of its behavior and will be assumed in the subse-
quent discussion. It will likewise be assumed that the perfect gas law applies. Henry’s Law
states, then, that at equilibrium, the mass of gas dissolved in a given volume of the liquid is
proportional to the pressure exerted by that gas on the surface of the liquid. Since the mass
of free gas contained in a fixed volume at a given temperature is, by the simple gas law, like-
wise proportional to the pressure, the solubility of a gas in a liquid may be expressed as a
solubility coefficient 3 defined as the ratio, at equilibrium, of the amounts (i.e., the mass) of
gas contained in equal volumes below and above the surface. The ratio G for each combina-
tion of a gas and a liquid depends upon the temperature. Table 1 gives values of the solubi-
lity coefficients for nitrogen and for oxygen in water.*

The rapidity of establishment of equilibrium between the concentration of a gas in solu-
tion and the pressure exerted by the gas on the surface of the liquid is limited by the rates of
diffusion obtaining on both sides of the gas-liquid boundary. For oxygen and nitrogen in the
usual atmospheric proportions in contact with water, the imposed limitation ordinarily arises
principally in the rates of diffusion of the dissolved gases within the liquid. Since the mole-
cular diffusivity of oxygen and nitrogen in water is very low for most practical purposes,**
aeration and de-aeration of water are ordinarily effected by some combination of spraying,
bubbling, agitation, and other means for increasing the area of contact and for accelerating

mixing within the liquid.®

*The values shown in the table were derived from data in the ‘‘Handbook of Chemistry and Physics??? which
gives values of an absorption coefficient @ whose definition is such that B/@ =T/ To: Here J is the absolute

temperature at which the values of @ and B apply and T> is the absolute temperature corresponding to 0°C.
5

**The observed values of the molecular diffusivity for oxygen and nitrogen in water are of the order of 2°10

cm2/sec.

TABLE 1

Solubility Coefficients for Oxygen and Nitrogen in Water

Temperature e

A second principle relevant to the operation of the instrument is known as Dalton’s
Law.® Dalton’s Law states that in a mixture of two or more gases, each gas behaves inde-
pendently of the other gases present, the equilibrium concentration of each gas in solution
being determined by the partial pressure exerted only by that gas on the exposed surface of
the liquid and by its own solubility in that liquid. A consequence of this behavior is that the
relative proportions of the dissolved gases may be different from the corresponding relative
proportions in the gas mixture with which the solution is in equilibrium. A pertinent example
is the case of a solution consisting of oxygen and nitrogen in water, in equilibrium with the
ordinary atmosphere. The solubility coefficient of nitrogen is only about one-half that of oxy-
gen. Consequently, the dissolved gas, when extracted, proves to be a mixture of the two
gases in the proportion of two volumes of nitrogen to one volume of oxygen though the atmos-
phere is a mixture having the proportions of four volumes of nitrogen to one volume of oxygen.

The instrument described in this report makes use of the principles outlined above and
in the following section to establish an equilibrium condition between the gas pressure in a
fixed volume in a closed chamber and a continuously flowing sample of liquid passing through
the chamber. In the construction whose evaluation is discussed, water is sprayed into the
chamber through one or more ‘‘atomizing’’ nozzles. The necessary surface exposure and
mixing take place not only in the spray itself but also in a film of liquid on the inside walls
of the closed chamber, especially in the upper part of the film which is supported and agitated
by the impinging spray. This arrangement was selected for tests on the basis of apparent
simplicity rather than for maximum effectiveness in mixing. Various other types of scrubbing
and absorption devices are known and used in industry.® It is recognized that either analy-
sis or experiment may prove some of these more effective than that chosen. Results obtained
indicate that the method used is a valid and practical one and that it provides a continuous
reading with acceptable accuracy. Further development may be aimed at increasing the rapi-
dity of response which, at present, does not allow significant indications of changes in air

content of the inflowing sample within a period of less than about 10 minutes.

THEORY
PRINCIPLE OF OPERATION

The basic components illustrating the principle of operation are sketched in Figure 1.
A continuously flowing sample of the liquid is sprayed into the upper part of a closed cham-
ber and subsequently falls to the bottom or runs down the inside walls of the chamber. Liquid
is continuously withdrawn from the bottom of the chamber; the inflow and outflow rates are
controlled so that the volume of the gas space in the upper portion of the chamber remains
constant. The outflowing sample liquid, once having passed through the meter chamber is
either returned to the reservoir from which it was taken or disposed of in whatever manner is
convenient. Thus, the sample liquid being exposed to the gas space within the chamber is
renewed continually. If, with respect to its dissolved gas content, the incoming liquid is in
equilibrium with the partial pressure of each gas trapped in the gas space in the upper portion
of the chamber, no net exchange of gas will take place between the flowing liquid and the gas
space. If, however, the partial pressure of any gas in the gas space is different from the
pressure corresponding to equilibrium with the concentration of that gas in solution in the in-
coming liquid, an exchange will take place and gas will be evolved or absorbed by the liquid
until the quantity of that gas in the gas space is just sufficient to exert the pressure corres-
ponding to equilibrium with the concentration of that gas in the incoming liquid. The result
is that the pressure in the chamber approaches an equilibrium value indicative of the concen-

; tration of dissolved gas in the sample liquid. The pressure is
Inflow

measured by any ordinary means connected to the chamber.

Proper account must be taken of the pressure due to the vapor
of the solvent.

It is manifest that an instrument operating on the princi-
ple just described will give the correct indication if the flow
of liquid through the chamber is slow enough so that actual
equilibrium is maintained between the outgoing liquid and the
gas trapped in the gas space. This is true since, in the limit

Water of extremely slow flow and long exposure, the pressure of each
gas in the gas space will, in fact, reach equilibrium with the
concentration of dissolved gas in the liquid. It is necessary
to determine by experiment, however, whether the instrument
can be made to operate under practical requirements, such as
workable physical size and flow rate, sufficiently rapid ap-
proach to equilibrium, and freedom from perturbation of the

eae equilibrium pressure caused by the methods employed to pro-

}

Outflow

duce the required amount of mixing and exposure of the liquid.

Figure 1 - Basic Form of These requirements are closely interdependent. An under-

Meter Chamber standing of the relations involved will be facilitated by a

discussion of the manner in which the gas pressure in the chamber approaches the equilibrium

value.

RESPONSE TO FLUCTUATIONS IN INPUT

Considering first the case of a single gas dissolved in the inflowing sample, let the
instantaneous pressure of that gas in the gas space be denoted by p and let the equilibrium
pressure corresponding to the concentration of dissolved gas in the inflowing sample be P.
Denote by Q the volume flow rate of the liquid sample. The temperature will be considered
to be constant. The inflowing sample carries dissolved gas into the chamber at a rate such
that if it were completely extracted and added to the gas space of volume J, the rate of in-
crease in pressure would be PQB/V. This follows from the definitions given for the flow rate
Q, solubility coefficient 8, and equilibrium pressure P. Similarly, if, in passing through the
chamber, the sample liquid is brought into complete equilibrium with the existing pressure p
in the gas space, there will be carried out from the chamber an amount of dissolved gas suffi-
cient to reduce the pressure in the gas space at the ratepQB/V. The net rate of change of
pressure in the gas space will then be (QB/V)(P-p).

In general, however, the exchange of gas between the sample liquid and the gas space
will not be complete so that the actual rate of change of pressure will be given by an equa-
tion of the form

ae

= -1(P-p) (1)

where 7 has been written for V/QBn. The factor 7, which will be called the mixing efficiency,
has some yalue between zero and unity. Thus the mixing efficiency 7 represents the actually
utilized fraction of the capacity of the flowing sample for carrying dissolved gas into or out
from the chamber. Its value depends upon the area of exposure, the intimacy of mixing, and

the degree of agitation as well as upon the molecular diffusivity in both the gaseous and
liquid phases.

It is instructive to consider the case in which, at time ¢ = 0, the pressure p in the gas
space has some arbitrary value p, and the equilibrium pressure P remains constant for sub-
sequent time, corresponding to some constant concentration of dissolved gas in the inflowing
sample. This condition may be recognized as corresponding to a sudden change (at time
t = 0) in the concentration of dissolved gas in the inflowing sample after previous establish-
ment of equlibrium. The time function describing the response to such a “‘step’’ change in the
input is sometimes called the indicial response® and is a useful characterization of the be-
tavior of an instrument with respect to fluctuations in the quantity being measured. In the

case being considered, the integral of Equation [1] gives the response:

p=P)+(P-p)(1-e '/7) [2]

The significance of the indicial response function (1 - e~*/7) is readily apprehended. Equa-
tion [2] states that the response of the instrument to a suddenly occuring step change in the
concentration of dissolved gas in the incoming sample is not an immediate corresponding
step change in the pressure p but rather an asymptotic exponential approach to the new read-
ing with “‘response time’? (i.e., time required for the reduction by a factor 1/e of the depar-
ture of the reading from the new equilibrium) equal to7.

It is, of course, essential to the usefulness of the instrument that the response time
T (= V/QBn), be made reasonably short. It is desirable, in fact, to make the rapidity of re-
sponse as great as possible. This can be accomplished by making the gas-space volume V
small and the flow rate Q and mixing efficiency y high. Since the mixing efficiency cannot
be made greater than unity, the practical problem becomes one of maximizing the ratio of the
flow rate to the volume of the gas space while maintaining adequate exposure and mixing.
The desirable choices of gas-space volume, flow rate, and mixing efficiency militate against
one another in various ways so that it is difficult to determine the most suitable compromise
in design. Thus, an extremely small gas volume would make for rapid response, but a small
volume is somewhat incompatible with the combination of a high flow rate and a large area of
exposure.

Equation [2] describes the response of the instrument for a case in which the change in
input involves only one gas. Where the response to a mixture of gases is required, a slightly
more general expression applies. Dalton’s Law states that in a mixture of gases, each be-
haves independently of the others. In the experiments to be described, a step change in the
concentration of nitrogen and oxygen dissolved in water is simulated; the changes in equili-
brium pressures are approximately in the atmospheric proportions of 80 percent nitrogen to 20
percent oxygen. The expected indicial response for this case consists of a linear super-
position of the responses due to each gas separately, if it is assumed that the mixing effi-
ciency for one gas is not affected by the presence of the other. Accordingly, instead of Equa-
tion [2], the equation describing the approach to equilibrium for a mixture of oxygen and

nitrogen in atmospheric proportions after a sudden displacement from equilibrium is:

P= Py + (P- pod(t — ge (Y/ QBymt_ 26-0 n*)
—t/T hy —t/T Ee
= fy PP = Dy) (: — .6e — .2e °)
Here p is the instantaneous value of the total gas pressure in the gas space, P is the total
equilibrium pressure corresponding to the new concentrations of nitrogen and oxygen in the
inflow, and By and 8 are the solubility coefficients for nitrogen and oxygen respectively.

For simplicity, the mixing efficiencies for the two gases are assumed to be equal and

independent of the various pressures.* Since practically, Bq = 2B y> the response time Ta
for oxygen may be taken to be one-half that for nitrogen. Consequently, fitting a response
function of the prescribed form to experimental data involves the determination of a single
parameter only. In the discussion which follows, the observed response will be characterized
by giving the response time 7), for nitrogen thus inferred.

It would be incorrect to interpret the preceding discussion as implying that the instru-
ment can give a Significant indication only when the equilibrium condition exists. Consider
first the behavior where the inflowing sample contains only one dissolved gas whose concen-
tration is changing with time. Rearrangement of terms in Equation [1] gives the instantaneous
relation between the pressure P, corresponding to the instantaneous gas content of the in-
flowing sample, and the indicated pressure p corresponding to the instantaneous gas pressure

in the chamber:
P 2 [1a]
=p+rT— a
se ite

Ideally, then, the instantaneous value of the gas content of the inflowing sample may be de-
termined without delay even though its value may be fluctuating in an arbitrary manner. In
practice, however, there is a limitation: Equation [1] assumes that the gas-space volume is
constant. If fluctuations in gas-space volume occur, corresponding fluctuations in the gas
pressure p will accompany them. Whether the extrapolation procedure indicated by Equation
[la] is made by the observer or by an automatic device incorporated in the indicating or record-
ing instrument, spurious fluctuations proportional to the time derivative of the volume fluc-
tuations will appear in the indication of the meter. The extent to which the lag inherent in
the response may be compensated depends, therefore, upon the accuracy with which the
volume of the gas space can be maintained constant. A reduction by a factor of 2 or 3 in the
response time is probably the most that can be obtained with the simple type of level control

contemplated for the Model Basin installation.

*Neither of these assumptions is strictly correct; the evolution or absorption of the two gases at rates result-
ing in unequal net velocities of flow into or out from the liquid surface results in concentration gradients ina
thin gas film adjacent to the liquid with a consequent interaction between the rates of absorption or evolution.
However, because of the low solubility of oxygen and nitrogen, the gradients in the gas film may be considered
negligible and the rates of absorption or evolution are determined principally by the liquid film. In the liquid,
rates of diffusion of the two gases are determined near the surface by molecular diffusion and in the interior of
the liquid by a combination of molecular diffusion and convection. The molecular diffusivity of oxygen in water
is about 10 percent greater than that of nitrogen. The convective diffusivity, if such obtains, is of course the
same for both dissolved substances. The assumed relation 8Moxygen = 2(81) Nitrogen represents, then, the
limiting condition in which molecular diffusion plays a negligible role in the ‘‘mixing’’ as contrasted with ‘‘tur-
bulent’’ diffusion occurring in the agitated upper portion of the film. The opposite extreme condition in which
molecular diffusivity accounts for the major portion of the distribution of the dissolved substance within the
liquid would accordingly result in the relation (Moxygen = 2.2 (BN) Nitrogen’ While the former condition is be-
lieved to correspond more ¢losely to that actually realized in the tests (except Run 1), Equation {3] should be

considered as a convenient empirical form rather than as one having a strict theoretical basis.

Where the dissolved gas consists of more than one kind, a compromise value of T
should be used in performing the extrapolation indicated by Equation [la]. Thus for air, a

suitable value is nine-tenths of the response time Tj.

EXPERIMENT
APPARATUS AND PROCEDURE

The earliest experimental tests of the principle outlined in the Introduction were per-
formed with laboratory equipment arranged in the fashion shown in Figure 1. These experi-
ments indicated the feasibility of the method and led to the version of the instrument shown
in Figures 2 and 8, The glass dome enclosing the nozzle and gas space is an inverted
Pyrex test tube. The mouth of the tube is sealed with litharge-and-glycerine compound into
a flanged receptacle. The latter is fastened with cap screws to the lower part of the cham-
ber, and the joint is sealed by means of an ‘‘O’’-ring. Except for the glass dome, all the
parts of the chamber are made of brass. The purpose of the bulbous lower portion of the
chamber is to reduce the velocity of the outgoing liquid so that gas will not be carried out
in the form of entrained bubbles. It is shown in the Appendix that the presence of undissolv-
ed gas in the outflow introduces an error in the indication of the instrument. A glass sight
tube sealed into the dome at the top and at the side is intended as an aid in determining the
level of the liquid in the chamber under conditions of operation, where the main surface is

disturbed by the falling liquid.

Meter

| trol
Level Contro Chamber

Electrodes

ae nee ose tre irc]

Pressure
Cell

Motor
Control

To Reservoir

Figure 2 - Schematic Diagram Showing

Installation of Air Content Meter aa
The automatic control was not installed at the time Figure 3 - Meter Chamber, Pressure Gage,

of the experiments described in the text. and *lowmeter Installed

The installation of the meter is shown schematically in Figure 2. The reservoir for
the test water is the 5000-gallon head tank of the Flow Facility. The head tank can be pres-
surized or evacuated. The water can be circulated through spray nozzles mounted inside the
tank, and thus any desired air content can be obtained. Once established, a given air con-
tent can be maintained for hours or even days because of the large volume of water and the
low diffusivity of dissolved gases. The pressure in the meter chamber was measured by
means of a telemetering pressure recorder (Automatic Temperature Control Co.). The pumps
are constant displacement gear pumps (Eastern Industries model GW-1) driven at reduced
speed by d-c shunt motors (gearmotors).

In order to determine more definitely the feasibility of the method and the validity of
the analysis presented above, some experimental tests of the meter were conducted. These
tests consisted of runs in which an initial displacement of the pressure in the gas-space
was effected and the reapproach to equilibrium was monitored and recorded. Before a test
run, the meter was allowed to operate until the pressure in the gas space appeared to have
reached an equilibrium value. The inflow and outflow rates were carefully regulated manually
so that the water level in the sight tube(and hence the volume of the gas space) remained con-
stant. After equilibrium had been thus established, the volume of the gas space was allowed
to change quickly to a new value either larger or smaller than that which had previously been
maintained. The effect of this last action is to simulate a ‘‘step’’ change in the concentra-
tion of dissolved gas in the incoming sample since, as a result of the change in volume, the
new value of the pressure in the gas space is lower or higher than the equilibrium pressure.
The operation of the meter was then continued with the volume of the gas space carefully
maintained at the new value and readings were taken at intervals in order to determine the
manner in which the pressure returned to the equilibrium values. The flow rate was maintained
as nearly constant as possible and its exact value noted and recorded with each pressure
reading taken. A thermometer suspended inside the meter chamber indicated the temperature.

As testing of the meter proceeded, successive modifications were made to improve its
performance. Figure 4 shows details of the arrangement of the nozzles and other components
in the upper part of the chamber. For the first two runs, the gas space was unnecessarily
large (Figure 4a) and the response time was excessive. A revised version is shown in the
sketch of Figure 4b. A micarta core whose volume occupies the greater portion of the glass
tube reduces the gas-space volume but not the surface available for exposure of the liquid.
The single nozzle was replaced by a triad of nozzles to allow a higher flow rate and thus
further decrease the response time. A ramp consisting of flexible plastic tubing wound in a
helix about the central core was added to minimize the formation of bubbles. A later addition
was the spray baffle. Its purpose was to prevent the spray from clogging the opening of the
sight tube. In some of the test runs made before the installation of the baffle, erratic results
were obtained because a small slug of water occupied a portion of the vertical section of the

sight tube just above the dome, thus introducing errors in the adjustment of the volume of the

gas space.

10

Sight
Tube
Baffle
(Level Control )
aa
\
\
\ Water __
\ Level
/
250 cc_
Inflow
Inflow
a |

Figure 4 - Details of Three Arrangements of Meter Chamber

The inside diameter of the inverted tést tube forming the glass dome is 48 millimeters.

All nozzles used were ‘‘atomizing’’ nozzles designed to produce a hollow cone spray.*

RESULTS OF EXPERIMENTS

Figure 5 shows typical results of the tests of the response characteristics of the
meter. The circles represent the measured values of gas pressure in the chamber while the
broken line indicates the simulated step change in the input. The solid line was obtained by
fitting a curve of the form given by Equation [3] to the observed pressure readings; the re-
sponse time 7, was chosen to give the best fit. This procedure determines the value of the
mixing efficiency 7 since V, Q, and By are known.

Table 2 summarizes the results of eleven test runs with four different combinations of

nozzle size and internal arrangement. For each run, the arrangement, flow rate, gas-space

*Spray Systems Company type %4LN1, %4LN2, Y%4LN3 and 4LN4. The smaller numbers designate nozzles producing
a finer spray. The flow rate obtained is roughly proportional to the designating number and to the square root of

pressure drop across the nozzle; it is about 120 cc/min. for a single No. 2 nozzle at 20 psi.

11

1.4 Tea + T
| | | |
ee | | |
13 + ut
o Run 8 : |
S | Volume: 178cc
4 | Flowrate: 511 cc/min
ee ee - - +—
|
° |
€ | |
_ |
< Gos Pressure in Chamber |
Sa i |
© |
= LEquation [3]
wo
wo i
io} Nee }
ae Equilibrium tel
Loot. = = Lt SSS Se == +—t,5
y= s L ee ees
(0) 20 40 60 80 100 120
Time in Minutes
Igaa T Sasa ale Waal
Equilibrium Pressure |
@ 1.0 -
= |
2 i
tS |
[ou
3
©)
E | quation [3]
< |
c Gas Pressure in Chamber
eo] :
5 Run 10
a Volume: 162 cc
= Flowrate: 524 cc/min
OF 27} ——

salt
fo) 20 40 60 80 100 —«:120
Time in Minutes

Figure 5 - Response of Meter to Sudden Change in the Concentration of Dissolved Air
in the Incoming Sample

The broken line indicates the simulated step change in the air content of the inflowing sample. The circles
indicate the observed gas pressure inthe chamber. The solid line represents a response of the form given by

Equation [3].

volume, temperature, and the solubility coefficient for nitrogen are given in the first six col-
umns. The next two columns give the observed response time and the computed mixing effi-
ciency. The response time given is that for nitrogen; the corresponding response time for
oxygen is one-half that for nitrogen inasmuch as the solubility coefficient for oxygen is twice
that of nitrogen. The initial pressure, i.e., the pressure in the gas space immediately after
the simulated step in the concentration of the inflowing sample, and the equilibrium pressure

are also shown in the table.

COMPARISON WITH VAN SLYKE DETERMINATIONS

In the preliminary tests of the method with the apparatus represented in Figure 1, com-
parisons were made with the indications given by the Van Slyke apparatus and substantial
agreement between the two methods was obtained. During the tests summarized in Table 2,
comparisons were made for the first two runs and for the last two. The last two columns in

Table 2 show the pressure approached by the meter and the equilibrium pressure corresponding

12

TABLE 2

Summary of Test Runs

Le Equilibrium Pressure P
Coefficient of Mixing , Initial atmospheres
Run Gas Volume V | Temperature Solubility for Efficiency Response Time Ty| Pressure p| Indicated ] Van Slyke
Nitrogen By min atm by Meter | Determination
Fig. 4b 0.92 A 0.72
One nozzle, No.3
2 | Fig 4b 110 0.0162 0.95 146 0.95 1.25 1.24, 1.26
One nozzle, No. 1
3*| Fig. 4b r 453 0.0166 0.68 29 1.33 1.14
4 | Three nozzlesNo.2} 340 191 0.0172 0.72 46 0.75 0.96
5 | No spray baffle 333 155 0.0172 0.70 39 1.18 0.96
6 | Fig. 4b 720 Alay 152 0.0166 0.77 16 1.28 1.13
7 | Three nozzlesNo.4) 594 147 0.0166 0.83 18 1.18 1.01
8 | No spray baffle | 511 ‘178 0.0167 0.80 26 1.34 0.92
9 | Fig. 4b 517 170 0.0165 0.64 31 0.76 0.90
10 | Three nozzlesNo.4} 524 162 0.0162 0.72 27 0.70 1.00 1.00, 1.03
11 | Spray baffle 527

0.0162 0.71 | 22 1.245 1.01 1.00, 1.04
—— |
*Measurements affected by clogging of sight tube.

to the air content indicated by the Van Slyke determination.*

DISCUSSION

The data furnished by the test runs and summarized in Table 2 are not as complete as:
might be desired. Nevertheless, a number of conclusions deducible from the tabulated data
or from observations made during the experiments permit the design of a meter chamber ade-
quate for measurement of the air content of water employed for cavitation studies in water
tunnels and similar facilities.

A pertinent question affecting the choice of the internal arrangement of the meter
chamber is the relative effectiveness of the various means employed for providing Eontact be-
tween liquid and gas phase. Thus, if it were determined that the ‘‘mixing’’ takes place pre-
dominantly in the spray itself, emphasis would be directed toward the employment of finer
nozzles and away from the provision of a large area of the inside walls of the chamber. Simi-
larly, if the falling film or the helical ramp were determined as the principal area of effective
mixing, optimum design would emphasize that feature. In each case the minimum gas-space
volume consistent with proper function would be sought.

In all of the test runs except that indicated as Run 1, a mode of mixing of the air and
water was observed which is almost certainly more effective than the combination of all those

mentioned above. At and above the level at which the spray impinges upon the inside wall of

*In converting the indication given by the Van Slyke apparatus to equilibrium pressures, it is, of course, neces-
sary to know the relative proportions of the various gases as well as their separate coefficients of solubility.
For this purpose, the relative proportions of nitrogen, oxygen, and carbon dioxide in ordinary atmospheric air and

their solubility coefficients as given in Reference 5 were used.

13

the chamber, a cap of violently agitated foam is formed and supported by the spray. The inti-
mate mixture and vigorous agitation maintained in this film of emulsified air and water appear
to provide an extremely effective process of exposure. This conjecture is supported by the
experimental results in conjunction with estimates of the extent of the exposure effected out-
side of the emulsified area.

The extent of the exposure of the sample water which occurs as a result of diffusion of
the dissolved gas out of or into the vertical film formed as the water runs down the inside
walls of the chamber can be estimated. The thickness 6 of the film and the flow velocity u
are given by the well-known Nusselt relations obtained by equating viscous and gravitational

forces on each lamina:

= ea)“

bs EA
_— 8Qloy yy?

v =a PE -Sl el

in which vp is the kinematic viscosity of the liquid, y is distance from the wall, and C is the
width of the film. (For the experiments being considered, C, the inside circumference of the
glass dome, is 15 cm.)

Let the coordinate z represent distance measured in the flow direction, i.e., downward,
from the top of the film. Then if p(x, y) represents the excess concentration of dissolved gas

in the film, the equation describing the steady-state distribution resulting from convection in

the z-direction and diffusion in the y-direction is!®
a2
u ep pa =0 [6]
Ox 2

Here D is the diffusivity of the dissolved gas.

The mixing efficiency 7“ which would result if the exposure of the sample were restric-
ted to that which results from diffusion out of such a laminar* film of vertical height 2 is the
fraction of excess (or deficit) of dissolved gas which is removed as the film falls a distance
z. If the dissolved gas is assumed to be uniformly distributed at the top of the film, e.g.,

p(0, ¥) = po; the fraction which has been removed in vertical distance « is then given by

, Cuye
sy ghisatgn Ol 7
7 A he ee dy [7]

since the total rate at which the dissolved gas enters the top of the film is p,@, and the rate

*Experience has shown!! that laminar flow obtains if a Reynolds number 4Q/Cy is smaller than 1000. This

condition is well satisfied even for the highest flow rates employed in the present experiments.

14

1.0 T

O Run 2
0.8 O Run 7 _ |

b Run | Film alone
0.6 IL | +

X, Film Height
D, Diffusivity of Dissolved Gas |
g, Acceleration of Gravity
v, Kinematic Viscosity
L—G, Film Width

Q, Flowrate

Mixing Efficiency

| | | l | |
On ©2 Os ©4 ©S ©O8 Or Of

ss o(2) 1/3 (s)""

Figure 6 - Comparison of Observed Mixing
Efficiency with Mixing Efficiency Estimated
for Vertical Film Alone

The solid line indicates the estimated mixing efficien-
cy for an arrangement in which the exposure of the sam-
ple water is confined to that provided by the vertical
film which runs down the inside wall of the meter cham-
ber. The circles indicate observed mixing efficiencies.
This comparison indicates that the exposure provided by

the vertical film is insignificant in the design tested.

at which dissolved gas is conveyed past the
section at 2 is the width C multiplied by the
integral in Equation [7].

Equation [6] may be integrated by
methods too detailed to be included here. With
the aid of Equations [4], [5], and [7], the
essential result is shown in the form of the
graph of Figure 6 which allows the estimated
mixing efficiency to be determined directly
from the flow rate, chamber dimensions, and
properties of the fluids. Conversely, the film
height which would be necessary to result in
a given mixing efficiency by exposure in the
film alone may be determined. The observed
mixing efficiencies for three representative
conditions are plotted and may be compared
with the mixing efficiencies which would have
been obtained for the same flow rate and film

height if the exposure provided by the film had

been the only means of mixing. It is apparent that an insignificant part of the observed ex-

posure is attributable to the film.

A significant difference in mixing efficiency appears between comparable tests with a

relatively fine nozzle (Runs 3, 4, and 5) and with a coarse nozzle (Runs 6, 7, and 8). The

difference is in the opposite direction from that

which is to be expected if the mixing process

depends primarily upon the fineness of the spray. It is believed that the difference in favor

of the coarser nozzle is due to the fact that, for

a given flow rate, the vertex angle of the

conical spray pattern is smaller for the coarser nozzle and that, consequently, a more exten-

sive area of emulsion is formed.
The results of Runs 9, 10, and 11, for wh

ich the spray baffle had been installed, show

a significant decrease in the mixing efficiency from the values obtained for the three runs

immediately previous. The reason for this was apparent: a considerable portion of the flow

drained from the top of the chamber down the support of the spray baffle, thus reducing the

volume of water supported and agitated by the spray in the form of an emulsion.

DESCRIPTION OF F

INAL DESIGN

In accordance with observations made during the experiments and with conclusions

indicated above, the version of the meter chamber illustrated in Figure 4c was designed.

As a measure for reducing the response time, provision of a large wall area was aban-

doned in favor of minimizing the gas-space volume. This choice follows the conclusion

15

indicated above that the extent of exposure provided by the vertical falling film is relatively
unimportant.

The connection of the sight tube to the side of the chamber instead of to the top
eliminates the need for the spray baffle.

The internal core tapers to a smaller diameter immediately below the intended level
of the liquid surface. The resulting lowering of the velocity of outflow facilitates the rapid
return of entrained bubbles to the surface. The lower bulbous portion is retained as an
additional precaution.

A stopcock connected to the upper part of the chamber is useful for venting the gas
space for the purpose of establishing nearly the equilibrium pressure before starting. This
is desirable because, even with the method of compensation described above, the instrument
begins to give accurate indications only after the equilibrium condition has been approximated
and the time required for this is small if the starting condition roughly approximates the
equilibrium condition. An additional reason for the provision of such a means for venting is
the possibility that some relatively insoluble gas might be carried into the gas space in un-
dissolved state and accumulate there, thus producing an erroneously high reading for a con-
siderable period thereafter. This, of course, is hardly possible where the body of liquid is
exposed only to atmospheric air, as in the present application.

From the observed performance of the arrangements tested, e.g., Runs 7 and 8, the
following operating characteristics may be expected for the meter chamber designed accord-
ing to the sketch, Figure 4c:

Flow rate: 600 cc/min with 3 No. 4 nozzles

Air-space volume: 110 cc

Mixing efficiency: 0.75

Response time (Nitrogen, 20°C): 15 min. (uncompensated)

These characteristics are suitable for indicating the air content of the water used in
a hydrodynamic facility such as the flow facility or a large water tunnel. If a shorter response
time is required, or if the presence of undissolved air in the inflow requires a higher value
of the mixing efficiency, a different compromise in design is desirable. Thus, an increase in
the flow rate will reduce the response time, though at some detriment to mixing efficiency,
whereas the use of finer spray nozzles might improve the mixing efficiency but require a
higher nozzle pressure. Similarly, a reduction in the dimensions of the meter chamber would
reduce the response time but with attendant difficulties associated with control of the flow
rate and gas-space volume. It appears that no absolute limit exists with regard to an approach
to a mixing efficiency of unity or increase in'the rapidity of response but that practical con-
siderations such as those indicated will dictate the optimum compromise for a given applica-
tion.

For practical use, it is desirable that the meter provide a continuous indication while

unattended. For this purpose a servomechanism arranged to maintain constant gas-space

16

volume is necessary. A simple control system designed for the Model Basin installation pro-
vides two-point control of the water level in the sight tube. Electrical contacts made between
the water and metal electrodes fused into the sight tube actuate, through relays, a motor-
driven rheostat differentially controlling the fields of the two motors driving the inlet and out-
let pumps. The servo system tas not been tested at this writing so that no experimental data
concerning the adjustments required for satisfactory operation are available, but no extreme

difficulties in this regard are anticipated.

SUMMARY

The air-content meter described in this report determines the concentration of dissolved
air in water by measuring the air pressure which is in equilibrium with the water. In this way
the meter can be used to indicate directly the degree of saturation of the water, relative to
any arbitrary air pressure. If it is desired to express the air content in terms of volume or
weight concentration, these quantities can be calculated from the measured equilibrium pres-
sure by using well-known tables.

The meter operates continuously and can be used to provide a continuous indication of
air content. However, the meter is limited in the rapidity with which it can follow a change
in air content. For the present design, the response time is about 15 minutes but, by a meth-
od of compensation discussed in the text, it may be reduced effectually to about 5 minutes.
This response is obtained with the sample water flowing at 600 cubic centimeters per minute.
For applications where faster response is necessary, modifications in design have been sug-
gested.

Comparison of the indications given by the continuously operating meter with those of
the more conventional Van Slyke apparatus show good agreement over the restricted range of

air content considered.

17

APPENDIX
EFFECT OF UNDISSOLVED GAS

The presence of undissolved gas bubbles in either the inflow or outflow can be shown
to be detrimental to the accuracy of indication. It is convenient to express the concentration
of undissolved gas in terms of the equilibrium pressure which would correspond to its concen-
tration if it were dissolved. Let the equilibrium pressure corresponding to the undissolved
gas in the inflow be denoted by P, and that corresponding to the ¢o¢al gas in the inflow be P,
so that (P — P,) is the equilibrium pressure corresponding to the dissolved gas in the inflow.
Also let P, be the equilibrium pressure corresponding to the undissolved gas in the outflow.
Then the rate of change of the pressure p in the gas space may be determined as follows:

It may be assumed that the undissolved gas in the inflow is wholly released into the
gas space regardless of the mixing efficiency and the existing pressure, whereas the dissolved
portion is brought toward equilibrium withthe existing pressure p at a rate dependent upon the
mixing efficiency.* The rate of change of the pressure p ascribable to the difference between
the concentration of the dissolved gas in the inflow and in the outflow is then given by an

expression similar to that appearing in Equation [1], namely,

SOUP 8 = 2)

whereas the contribution to the rate at which p changes due to the undissolved gas is

Adding these two contributions gives
dp _ 98
op = 28n(p - P, -p) + BP, -P,) [6]

The equilibrium condition dp/d¢ = 0 requires that 7(P -F - p) + (P, — P,) =0 or

pePel tad oe [7]

Since the total gas content corresponds to a pressure P, the terms on the right side of
[7] involving P, and P, represent the errors arising from undissolved gas in the inflow and in
the outflow respectively. It may be observed that while the error due to undissolved gas in
the inflow vanishes as the mixing efficiency approaches unity, that due to the same condition
in the outflow does not. It is desirable, therefore, to make the mixing efficiency as large as

practical and, in addition, to eliminate the convection of significant amounts of undissolved

*This statement is really a definition of ‘undissolved gas’’ combined with the assumption that the gas in the
inflow, however, actually dispersed, may be considered to consist of two phases only, dissolved gas and undis-

solved gas, each of which behaves as indicated in the above treatment.

18

gas in the outflow. In the instrument described in the text, a bulbous enlargement of the
lower portion of the meter chamber is provided in order to reduce the velocity of outflow and
so minimize the downward convection of any minute bubbles which may have persisted in the
outflow.

The property of the instrument indicated above suggests the possibility of discrimina-
ting between dissolved and undissolved gas in the inflow if this were desirable. This could
be effected, for example, by simultaneous operation of two meter chambers in one of which

the mixing efficiency is maintained purposely at some known low value.

REFERENCES

1. Bowers, W.H., ‘‘The 12-Inch Variable Pressure Water Tunnel Propeller Testing Pro-
cedure,’’ David Taylor Model Basin Report 505 (November 1943).

2. Borden, A., ‘‘Design, Operation, and Maintenance of a Meter for Recording the Air Con-
tent of Water in the David Taylor Model Basin Water Tunnels,’’ David Taylor Model Basin
Report 549 (December 1946).

3. Crump, S.F., ‘‘Determination of Critical Pressures for the Inception of Cavitation in
Fresh and Sea Water as Influenced by Air Content of the Water,’’ David Taylor Model Basin
Report 575 (October 1949).

4. Griffin, R.C., ‘‘Technical Methods of Analysis, ‘‘McGraw-Hill (1927), p. 703.

5. Van Slyke, D.D., and Neil, J.M. ‘‘The Determination of Gases in Blood and Other
Solutions,’’ Journal of Biological Chemistry, Vol. 61 (1924), p. 523.

6. Glasstone, Samuel, ‘‘Textbook of Physical Chemistry,’’ Second Edition, D. Van Nos-
trand Co., New York (1946), pp. 693-703.

7. ‘‘Handbook of Chemistry and Physics,’’ Twenty-Seventh Edition, Chemical Rubber
Publishing Co., Cleveland, Ohio, pp. 1328-29.

8. Sherwood, Thomas K. and Pigford, Robert L., ‘Absorption and Extraction,’’ Second
Edition, McGraw-Hill (1952).

9. Bush, Vannevar, ‘‘Operational Circuit Analysis,’’ J. Wiley and Sons (1937).

10. Prandtl, Ludwig, ‘‘Essentials of Fluid Dynamics,”’ Hafner Publishing Co. New York
(1952), p 400.

11. Dukler, A.E., and Bergelin, O.P., ‘“‘Characteristics of Flow in Falling Liquid Films,”
Chemical Engineering Progress 48 (November 1952).

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ay) Sounsvew Ajsnonuuos yoy paqliosep si UeuNIQSUT Uy

‘sjor “say ‘sopquy ‘jour “d 6T ‘i “FEET 1eqo}00

“peowapyVH “AW pus youjedziia “WH Aq “YALVM NI HIV
GHA TOSSIGC AO NOLLVOIGNI SQOANLLNOD NOU YALAW V

198 qdoy *ulseg Jopow Jo[Avy, *M plaeq

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POA[OSSIp Ite JO} poyueseid st poyjoul ayy Jo AqIptywa eyy Jo uoljeU
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ueeMjoq WNIIgI{Inbo jo yWeWYsI{qe}se ey) Uo paseq si JUSUNASUT
oyj Jo uoysiedo oy, “spinbi] ul sesed peajossip jo uoljyeiqueou0s
oy} Soinsvow A[snonunuos YoIyM peqiosep SI yUeWNIYSUT Uy

*sjor “sayy ‘seyqey ‘jour *d GT “i “FEET 4040390

*peoiep By “AW pus yorgedzqta "WH Aq “YALVM NI UIV
QA TOSSIC AO NOLLVOIGNI SQOANILNOD HOA YALAW V
198 ‘ydoy ‘utsug [opow 10[AvT, *M plavq

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SI JO} BIJoJTID puv yUOUINQSUL [vojousd BJO USISep oy, “JOVBM Ut
PEATOSSIp div JO) pajuesoid st poyjew oy jo AyIptpBA a1 jo uote
-IyUOd [BjUeULIedxg *sesvd peayOssip Jo UONRIQUeDUOD ay) SeyEo
-Iput eovds sed oy) ul oinssosd wintaqi{inbe ayy, ‘awnyoA URS
-uoo jo vovds sed v pux efduies pinbiy Surmoyy AJSnonuyuoo ayy
veenjog wntiqitinbs jo yuewWYst{qe}se ey} UO peseq SI quaWNASUT
ay} JO uotjwiedo ayy, *spinbi] ul sosed peajossip jo uonvaqusouo
ey} SoMsvoew ATSNONUUOS YOIYM pequoasep si yuewWNaysul uy

*sjor say ‘soyqe) Jour d GI ‘It “FEET 1eqo}00

*PBolOT IVY “AW pus youqedzqiy “WH Aq ‘MALVM NI UIV
GHA TOSSIG AO NOILVOIGNI SQOANILNOO YON UALAN V¥
‘uIseg [OpoW 1o[ABy, *\\ plAud

*198 “doy

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POATOSSIp IIe Jo} pajueseid st poyjou ay) jo AjIpt{BA ayy Jo uOTVBU
+IJ1juOod [vJUeWIIedxy “Sess PeATOSSIp JO UOBAyUBdUOD ay) Seaivo
-{put aovds sed ay} ul ounssead Wintiqi{Inbe ayy, *eun,oA quBIS
-uoo jo eouds sud v pus ojdues pinbi] surmoqy Ajsnonuuoo ayy
ueeMjog WNligi{Inbe jo JU@eWYSI]qu}Se ey] UO pase SI qUeLINASUE
dy} JO uoIqvsedo ayy, ‘spinbi] ul sosvsd peajossip Jo uoivs}Uue.UCD
oy Soansvew Ajsnonuuos YOIyYM pequosop st quoWNASUI Uy

*sjod “say ‘soyquy "jour sd GT ‘I “peg, 10qo}00

*peo1o[yIBy “A"\N pus yorgedzqty “WH Aq ‘NALVM
GHA TOSSIG AO NOLLVOIGNI SQOANILNOOD YOU UALAN V

198 dey *utseg Jepoy so[Avy, *\\ plavd

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