Note on the flow of vapor between liquid surfaces

Reprinted from THE JOURNAL OF CHEMICAL PHYSICS, Vol. 20, No. 5, 790-793, May, 1952 Printed in U. S. A.

Note on the Flow of Vapor Between Liquid Surfaces*

MILTON S. PLESSET

California Institute of Technology, Pasadena, California (Received January 30, 1952)

The mass flow of vapor from a liquid surface at temperature To to another surface of the same liquid at temperature Ti(Ti< To) may be very readily determined in the case of one-dimensional flow from the con-servation relations for mass and momentum. These relations involve both the coefficient of evaporation and the coefficient of condensation. It is therefore possible to determine the condensation coefficient which has heretofore not been accessible to measurement. A new method for determining the evaporation coefficient is also made available. The temperature of the vapor between the liquid surfaces may also be found when viscous effects in the vapor flow are neglected.

I. STATEMENT OF THE PROBLEM

THE

surface at x= 0 is maintained at a temperaturefollowing problem is considered : A liquid

To, and a second surface of the same liquid composition

at x= a is maintained at the temperature T1 with

T,<T 0. It will be supposed that the intervening space contains only the vapor of this liquid and that the flow

of the vapor from x= 0 to x= a is steady and

one-dimensional, i.e., that the vapor density p, the vapor

pressure p, the vapor temperature T, and flow velocity

u are functions of x only. It will also be assumed that

the flow velocity u is small compared with the velocity

of sound in the vapor and further that any deviations

of the vapor from the perfect gas law are negligible.

H. ALGEBRAIC RESULTS FROM CONSERVATION OF MASS AND MOMENTUM

By assumption p= p(x), p= p(x),

u=u(x), and

T=T(x), and the conservation of mass is expressed by

pu= pouo ₍₁₎

and the conservation of momentum by

p+pu2=po+ Polio% (2)

where po= P(0) Po p(0), and uo= u(0). Since it is

sup-* This study was supported by the ONR.

posed that the vapor obeys the perfect gas law

p pRT,

Eq. (2) becomes

pRT-1-- pie= poRTo+ Pou02, (2')

where T o= T(0). From Eq. (1) p= pouo/u so that Eq.

(2') may be written

(RT / u)-E u= (RT uo)-F uo. ₍₃₎

The speed of sound in the vapor is (yRT)i, where 7 is the ratio of the specific heat of the vapor at constant pressure to the specific heat sat constant volume. The

case of present interest is one in which the,flow velocity is very small compared with the speed of sound so that

RT/uu; RT uo>>uo.

(4)

Equation (3) is a quadratic equation in u, and in view of the relations (4) the root of concern is given

ap-proximately by

u= uoT/To. (5)

If the term of next higher order is included, this root

of Eq. (3) is

T u02 (T

u=

---)1.

T02-791

FLOW OF VAPOR BETWEEN LIQUID SURFACES

1.10 1.00 .9 .8 .7 .4 3 " .2 .1 0

FIG. 1. The density of water vapor pi, adjacent to a water surface at temperature 7'1, is shown by the solid curves as a function of T,, the temperature of the cooler surface, for various values of To where To is the temperature of the warmer surface. The dotted extensions of the curves give the density of water vapor pi adz jacent to a water surface as a function ()fits temperature Ti where now these values apply to the warmer surface. pi° is the density of saturated water vapor in equilibrium with its liquid at tem-perature T1. The evaporation coefficient a has been taken to have the same value as the accommodation coefficient 0.

The net current density .1 involves the evaporation

coefficient a so that measurement of J leads to a value for a. This coefficient has been determined by previous

investigators from the rate of evaporation of a liquid

into a vacuum, but it may be useful to have a different

experimental procedure available. To the author's

knowledge, it has not been possible to measure the

ac-commodation coefficient for condensation #. It is ap-parent that a measurement of pa would determine #

when a is known, or alternatively, if a measurement of the velocity uo can be made, ft may be found directly. Clearly a and have the same value under equilibrium

conditions. It has often been assumed that they have

the same value under non--equilibrium conditions; this assumption may not be justified.

Computations using Eqs. (12), (13), and (14) have

been made for water vapor. Figure 1 gives a comparison of p1 and po with the equilibrium values pi° and po° for

various values of To and Ti. Figures 2 and 3 show the,.

vapor flow velocities u0 and ul, respectively, and Fig. 4 shows the variation of the current density J with To and Ti. In the evaluation of these quantities the evaporation

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The first correction term is proportional to the square

of the ratio of the vapor velocity to the velocity of sound and will be omitted in the analysis to follow. If one uses

the approximation of Eq. (5) for u, then from Eq. (1)

one has pT= constant so that is also constant.

There-fore,

pT= poTo (6)

P=

(7)

in the present approximation.'

The mass flow of vapor per unit area per unit time Or the current density at the liquid surface x=0 is the

difference between the rate per unit area at which

molecules evaporate from the liquid and the rate per

unit area at which they condense so that

pouo= J+(o)J_(0). (8)

From kinetic theory one has

4(0) = a(RT0/27rM)Ip00, (9)

where a is the evaporation coefficient for vaporization from the liquid surface, M is the molecular weight of the

vapor, and poe is the equilibrium or saturation vapor

density at the temperature To. The vapor density

adjacent to the liquid surface at x=0 is Po(Po<Poe) and the temperature is To so that

J_(0)= (RT0/ 27rM)ipo,

where 13 is the accommodation coefficient for

condensa-tion on the liquid surface. Equacondensa-tion (8) therefore

be-comes

Pouo= (RTo/27rM)(aPoe-13)90). (10)

The current densities at the liquid surface x= a may be considered in the same way; thus,

J_(a)= a (RT1/277-M)4pi°, J.f.(a)= Li(RT1/21-31)i pi,

where pie is the equilibrium vapor density for the tem-perature Ti(Pi> pie). Then

Pgii= (RTI/ 2rM)1(opi «pie). (11)

The coefficients a and are taken to be independent of temperature. From Eqs. (5) and (6)

ul/Ti= uo/To, PoTo=

and these relations together with Eqs. (10) and (11)

readily give the following:

Po=

a

( Ti)

(Ple/P08)(2-VT0)1

# To 1+ (T1/7'0)1

u0=13(_27310 _{(7;0 .)i 11: ((pP:ve /pP:ee))((TTilicroo))i.}

RT0 )i 1- (Pie /Poe)(T I/ To)

= Pah= a Po

e-27rM 1+ (TilT0)1

1 Equation (1) is :an approximation to the relation p=po +7P1(142/co2) (1 .u/ uo, where the sound velocity Co= (7P,/ PO.

50 60 70 80 .90 100 110

T, (°C)

FIG. 2. The value of the flow velocity for water vapor uo at the warmer liquid surface To is given as a function of the temperature of the cooler liquid surface of temperature T1. The accommodation coefficient for water vapor has been taken to be fl=0.04. The dotted portions of the curves correspond to interchange of warmer and cooler liquid surfaces (T1> To).

coefficient a for water vapor has been taken to be 0.04,2 and the accommodation coefficient j3 has been given the same value.

III SOLUTION OF THE ENERGY EQUATION

The temperature field T(x) for 0<x<a may be de-termined from the energy equation; the velocity u(x) and the density p(x) may then be found from Eqs. (5)

and (6), respectively. When effects of viscosity or

turbulence are neglected, the energy equation' becomes for steady one-dimensional flow

dT d2T pu dp

(15) dx dx2 p dx

where G is the specific heat of the vapor at constant

volume and k is its thermal conductivity. Equation (15)

neglects any variation of k. From Eq. (6) p= poTo/T so that

dp poTo dT

dx T2 dx

2 G. Wyllie, Proc. Roy. Soc. (London) A197, 383 (1949).

8S. Goldstein, Modern Developments in Fluid Dynamics (Oxford

University Press, New York, 1938), Vol. II, §260.

0.

MILTON S. PLESSET

792 and consequently pu dp dT

= I Rpu--.

p dx dx

Thus, Eq. (15) may be written as

( R\=7

dT dT

k d2T

dx dx dx2

where 7 is the ratio of specific heats. Further, from Eqs. (5) and (6) up= uopo so that the equation to be

solved becomes

dT Do d'T

7_7=

777.,

dx uo dx2

where the thermal diffusivity Do= (k/ Pocv). The bound-ary conditions for Eq. (16) are T(0)= To and T(a)= T1.

The appropriate solution of Eq. (16) is readily found

to be

T(x)= BeTuoxiDo, (17)

where the constants of integration A and B have the

values Toe72ó"0 T1 (18) ru00/D0 1 To--Ti

B=

(19) ortroa/Do__1 (16) 60 70 80 (°C)

FIG. 3. The value of flow velocity u1 at the cooler liquid surface of temperature T1 is shown for various temperatures To of the warmer liquid surface. The.accommodation coefficient 0=0.04.

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793

FLOW OF VAPOR BETWEEN ,LIQUID SURFACES

It should be remarked that under ordinary

circum-stances e"°41D° is a very large number so that to a good approximation

A70, B (To ne-7144/Do,

and the temperature field may be written

To (To T i)e-(7uoiD0)(a-x). (20)

It is apparent from 'Eq. (20) that T(x) remains very

nearly equal to To from x=0 to within a distance of the order of Dohuo from _x= a; the temperature then drops rapidly to T1 at x= a. This behavior may be illustrated by considering an example of water vapor flow. Then Do is approximately 0.2 cm2/sec, and -y is roughly 1.3;

for a vapor flow velocity of 100 cm/sec, one has Do/ Tito

1.5X 10-2 cm. This distance is still large compared

with the mean free path so that the use of ordinary kinetic theory relations is permissible. If with these

same numerical values, a is 1 cm, then uoa /Do e630,.

SO that A is very closely equal to To, B is very near

° zero, and the second term in Eq. (17) or (20) becomes

significant only when x is nearly equal to a. The be-havior of u(x) and p(x) will, of course, be similar to that of T(x).

As a final remark it may be pointed out that the

results of Eqs. (5),

(6), and (7) apply to the

one-dimensional flow of any gas (obeying the perfect gas

law) at low velocity. Likewise the solution for the energy

equation given by Eq. (17) has general validity for a

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perfect gas with constant thermal conductivity. For

example, the thickness of a one-dimensional flame front as determined by thermal conductivity would be given by Eq. (17). ...Z.P 120°C

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---...._... To6 0° C--?...*°,..,.. '...

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120 60 70 80 90 100 To (°G)

FIG. 4. The inais floN'v, J= pu, of water vapor per unit area per unit time from a warmer water surface with temperature To to a cooler surface with temperature T1 is shown as a function of T1 for various values of To. The dotted portions of the curve refer to reversed flow where the warmer and cooler liquid surfaces are interchanged (Ti> To). The evaporation coefficient a= 0.04.