analogy between heat transfer and mass transfer

Analogy between Heat Transfer and Mass Transfer A Psychrometric Study

CHARLES HOSMER BEDINGFIELD, JR.’, AND THOMAS BRADFORD DREW

COLUMBIA UNIVERSITY. NEW YORK $?7, N. Y .

A careful theoretical study shows that the analogy between heat transfer and mass transfer may be justified in the sense that

where functions fare identical in form and p is a correction factor. New psychrometric data obtained with volatile solids as “wet bulbs” are presented. They are consistent with other data in showing no effect of Reynolds number or temperature and in satisfying the relation

which appears consistent with the analogy in so far as can be determined a t present.

HE utility of the analogy between heat transfer and mass T transfer ( 4 ) lends importance to a study of the circumstances in which it may be formally justified. Because these conditions seemed inadequately developed even by such excellent treatments as that of Klinkenberg and Mooy (19) or hy the very numerous experimental studies, the investigation here reported was under- taken. The particular comparison used as an example is heat transfer from a cylinder of uniform wall temperature exposed to a stream of fluid versus a cylinder of volatile solid evaporating into a stream of fluid.

Diffusion of matter is flow of the constituents of a mixture relative to each other. Largely because composition of relative velocities is thus necessarily involved, the use of vector methods in its analysissweeps away much niathematical fog. Such methods are therefore used in the theoretical portions of this paper. Readers unfamiliar with vector analysis should observe that all the equations in the discussion of velocities of diffusing matter and nearly all in the discussion of the formal basis of the analogy (exceptions are Equations 18 and 19) will be correct and per- fectly intelligible if read as ordinary algebra and calculus, provided that all flows of heat and matter are assumed to parallel the same straight line, say the z-axis. In making this interpretation V should be read as the partial derivative symbol, b/bx8 Although this restricted interpretation will leave the argument incomplete, the essential features should be clear.

VELOCITIES OF DIFFUSING M A l T E R

No inconsiderable confusion in the study of diffusional phenom- ena is avoided if it is recognized at the outset that linear velocity is a concept of multiple meanings for matter in which diffusion is occurring. The everyday notion that the local linear velocity of a fluid is the “velocity of a point moving with the fluid” be-

ton, Del. I Present address, E. I. du Pont ds Nemours & Company, Ino., Wilming-

comes ambiguous when tho different constituents of the fluid are moving in different directions at various speeds.

Imagine a amall plane element, 88, so located within a body of matter that it contains a point, P, fixed relative to the observer. Let aW denote the net rate of flow of mass-e.@;., pounds per hour-across 68, considering matter of all kinds prasent taken together. Now, provided 6s is so small that for each of ita possible orientations about P the conditions of flow are sensibly uniform over its surface, SW will be proportional to the projection of SS upon the plane in which it lies when 6W is a maximum. The limiting value of the constant of proportionality, as SS is taken smaller and smaller, is the magnitude of a vector,G, the local mms velocity at P. The direction of G is normal to 6s when positioned for the maximum 6W and flies with the matter crossing the element. Analogous considerations applied to the mass rate of flow, We, across 88 of the qth constituent of a mixture lead to the concept of Ga, the local mass velocity of the qth constituent. It is easily shown that, vectorinlly,

G = ZG, (1) where the summation includes all constituents.

Molal Mass Velocities. If matter is measured in mol- instead of in mass units the same in size for all substances, the foregoing analysis leads to vector molal mass velocities which will be de- noted by w for the mixture aa a whole and by w* for the qth constituent. Evidently a t any given point

Ga wdfe (2 ) where M e is the molecular weight of q. Thus, for individual con- stituente the two types of mass velocity have the same direction. The mass velocity, G, and the molal mass velocity, w, are, however, respectlvely

G = C G q = ZW&, and

(3)

( 4 )

and, unless the M,‘s are all equal, these vector sums will usually have different directions because of the different weightings of their components; they may even have opposite directions. It is important to note that w i s not in general obtained by dividing G by the mean molecular weight corresponding to the local composition of the mixture. , Local Linear Velocity. Seemingly, a reasonable characteristic

of a point ‘‘moving with a body” should be that there is no transport of matter across a small plane element containing the point, whatever the orientation of the element. If we restrict ouraelvea to the consideration of the qbh constituent, a point can then he said to be moving with that constituent if, relative to an observer riding on the point, the mass velocity and molal mass velocity of q are iero. If V, is the linear velocity of such a point in the fixed coordinates of an observer in the laboratory, V, may be called the loral linear velocity of substance q. In t e r m of G, and I, aa mewured relative to the laboratory coordinates,

Vo Gg/~q * W J f q f Y / P A (5)

1164

lune 19% I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y 116s

where yo ia the local mass density of q in the mixture and p r is the local molal density of q.

To derive Equation 5, consider the flow across a plane element, 68, oriented in any manner at a fixed point P. Let n be a unit vector normal to 68 and considered posibve when it points generally downstream with respect to q, Now the maas of p which will crom 68 in the interval 68 is G,.n 68 68 and all this material must be contained in the cylinder which would be swept out were 68 allowed to move with velocity V9 during the interval. Thus

G,m SS 60 = yqV,.n SS 6% (6) because the right-hand side of Equation 6 is simply the product of the local mms density of q and the volume of the cylinder. From equation 6

which can be true for all orientations of n only if the first form of Equation 5 is true.

G,.n = y9V,.n (7)

In extending the same ideas to the mixture m a whole, there are two possibilities. The element 68 may reasonably be supposed to be moving with the mixture when there is no net transport acrosa it as measured (a) in equal mass units like pounds or ( b ) in moles. The first possibility leads to

where y is the total local maea density and q,, the local mass fraction of Q; the second, to

rl

where p is the total local molal density and ye the local mole fraction of q. The proof of these equations is analogoua to that of Equation 5.

Linear Velocity of Mixture. A volume element bounded by surfaces moving with the fluid in the sense of V has constant maa8 and the momentum of such an element is correctly found by the usual formula of mechanics if V is taken as the linear velocity of mixture. [Some writers (3) in kinetic theory call V the “mass velocity,” a usage inconsistent with current engineering terminology in the United States; the dimensions of V are those of linear velocity.] To distinguish U, which frequently differs from V both in magnitude and direction, the name “molal linear velocity” is suggested.

FORMAL BASIS OF THE ANALOGY

The Fourier equation (13) for heat conduction in a stationary isotropic medium,

- is simply a definition: the thermal conductivity, k, is the ne@- tive of the ratio of the heat current density, q, to the temperature gradient a t the point considered. For ordinary measure- menta the point is, of course, supposed to be fixed relative to the medium 89 well as relative to the observer. In a fluid moving without diffusion, Equation 10, with k determined for the stationary fluid, will correctly give the heat flux relative to a point moving with the fluid in the sense that no fluid crosses an element of surface affixed to the point. Thus, if k is constant, a heat balance on an element of volume bounded by surfaces moving with the fluid in the sense of V gives the Fourier-Poisson equation (I)

(11)

if (a) there are no internal heat sources, (b) the dissipation of mechanical energy by viscosity makes a negligible contribution, ( c ) there is no change in phase, ( d ) cp is, 88 is usually to be expected, a su5cient approximation to the generalized specific heat which properly should appear in the denominator on the right (91, and (e) there is no diffusion.

The fundamental equation of debition of D,, the molal diffu- sivity, aa used by Walker, Lewis, McAdams, and Giliiland (96) for an isotropic binary mixture, is

-DmVVA m PA(VA - u) WA - YAW (12)

(12a) PVA (1 - YA)(VA - VB)

where M = M B + YA ( M A - M E ) is the mean molecular weight. I t is supposed that no other causes of diffusion are present--e.g., temperature or prassure gradients. In words, Dm is the negative of the ratio between the molal mass velocity of substance A relative to a surface element moving with velocity U and the gradient of the mole fraction of A . The choice of U as the basis relative t o which diffusional flows are measured for the purpose of defining “diffusivities” is usual for binary gaseous systems; there seems to be no well standardized convention in the c u e of solid and liquid systems. A primary reason for this choice for gases is that both theory and experiment (9,B) indicate small variation of D , with composition in the binary case a t ordinary temperatures and pressures.

Equation 12 is true whether U is zero or not. When it is used to compute a material balance with respect to substance A for an element of volume bounded by surfaces moving with the fluid in the sense of U, the result, if D , is constant, is

provided substance A is neither created nor destroyed within the element and conditions are nearly enough isopiestic and isothermal to make pressure and thermal diffusion negligible. This is the general differential equation for diffusion as ordinarily given for a binary system.

In Equations 11 and 13 it must be remembered that while the “mobile operators’’ on the left, D( )/DO, and D( )/De,, are the same with respect to hest conduction because U and V are then identical, they differ when diffusion is occurring. In fact, then

the latter form following from Equation 12b. The physical significance of the distinction can be seen when it is realized that the turbulent fluctuations of the linear velocity, V, are among the quantities determined by the Reynolds number. When diffusion is occurring, the turbulent fluctuations of U in some regions may differ from those of V in she, direction, and timing in a way governed by the concentration gradients and molecular weights. The true analog in binary gaseous diffusion of Equation 11 i n heat transfer is not Equation 13 unless M A = ME; but instead i t is

For

Hnd

D In M MA - MB DYA D v M D v

-c---

eo that Equation 16 reduces to the result of substituting 15 in 13. The principles of dimensional similarity are most readily applied

when the equations have been expressed in terms of reduced vari-

1166 I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y Vol. 42, No. 6

ables such that the limits of integration are purely numerical and the reduced dependent variables have identical numerical values a t corresponding limits. For the problem here to be considered experimentally-that of a gas flowing past a cylinder with which it exchanges matter or heat-the following substitutions, appro- priately selected, reduce Equation 11 or 16 as the case may be to the same nondimensional form:

Table I. Substitutions for Equation 17 In Equation 11 or 16

Do = diameter o fcylinder; V = magnitude of mean linear velocity at a distance

8’ VOID0 V i = DoV R’ = R/Do where R is the position vector with respect to which V

is the derivative dR’ V’ =de, = v/v

For hest-transfer, Equation 11 x = L x . k

t, - t, k K = -

DO Vcgr For diffusion, Equation 16

In M - In M e In M , - In M . x =

For each problem the boundary conditions are:

At a point far removed from the cylinder-Le., at R’ = m ;

At the surface of the cylinder-i.e., if T(R’) = 0; X x = o

and, subject to certain limitations discussed below, the reduced velocity, V’, which appears in the mobile operator is given by the same function of the Reynolds number, time, and position

1

V’ = T [( D Q V ~ T), O ’ , R ’ ]

if geometrically similar situations are considered and the charac- ters of the initial streams are dynamically similar (14).

Equation 18 follows from the laws of fluid dynamics if fi and y are constant or if they always vary with relative time and position in the same way. When this is not the case, some additional dimensionless groups must appear in the equation. Because 1.1 and y do not vary with t in the same manner as with In M, no reason immediately appears for supposing that the correction can be made for heat transfer and for diffusion in an analogous way. Clearly, however, for sufficiently small temperature variations, on the one hand, and sufficiently small concentration variations, on the other, the effect of nonuniformity of 1.1 and y will be negligible.

Another reason why Equation 18 will fail to be the same for heat transfer and for diffusion is that in the first case it is ordinarily fitted to the specification that V’ = 0 at all points of the heating surface, a condition which obtains in the latter case only if there is zero net mass flow across the boundar? (19,27) . In the diffusional case V’ usually has a finite component normal to the wall, but again, it is true that if this component is small enough its influence will be negligible.

Equation 18 miKht conceivably also fail to he identical for the two processeR if in diffusion the relative flowsof the constituents result, even for streamline flow, in a transport of momontum analogous to that due to the “Reynolds stresses” in turbulent flow. This would result in an apparent viscosity depending on the diffusional stream at each point, rather than solely upon the local composition, and the equations of hydrodynamics would involve

those of diffusion. As has been pointed out by others (%), no real investigation of this matter seems to have been made, but again for sufficiently small gradients of composition its effect must be negligible. If the circumstances are such that Equation 18 is the same for heat transfer and for diffusion, the integral of Equation 17

X = n [ K , (DoVy/1.1), e’, R’l (19)

whether found experimentally or otherwise, is equally valid whether X and K are interpreted for heat transfer or diffusion. In the light of present knowledge, only experience can determine how far conditions may be allowed to deviate from those imposed above and yet admit of this conclusion. Throughout it must be remembered that the cases compared are heat transfer without diffusion and diffusion without heat transfer.

In the case of hest transfer, the usual coefficient h is related to the temperature gradient a t the wall by the formula

-k Do = - [VIXI,

because all the heat leaving the wall enters the fluid. The second form is simply the expression of the first in reduced variables. In the diffusional case, the Drew-Colburn mass transfer coefficient, F, is by definition such that

where z = W A / W , both molal mass velocities being the normal components at the wall. It, must, analogously, satisfy

where

(22%)’

- Yw M E M E - M A Y M =

Y. - Yw z - Y w Y , = - Yv - Yw

For certain special cases p has special forms which may be deduced from the corresponding form of Equation 22. In particular, if there is no net mass flow at the wall, I = M E / ( M B - M A ) and p reduces to unity. For equimolal oounter flow at the wall, the last two factors of the denominator of Equation 23 are replaced by the difference of the y’s. Estimation of p in the general cllse is facilitated by Figure 1, in which

is plotted as ordinates versus Y as absoissas. To find p: (1) Read BM, found when the abscissa is taken to be Yy; (2) read B,, found when the abscissa is taken as Yz; (3) then = BM/B,.

Normally, quantities h and F are measured for a Rteady state. Presuming this, one finds [V ,XI, in either case by differentiating Equation 19 vectorially with respect to R’ and evaluating the

June 1950 I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y 1167

P * m m

12 I 3 14 I5 16 18 2 0 25 30 3S 4 6 6 7 9 II

Y

A8

t o s p 0,e

0 4

0.3 0 2 03 04 O S 0 6 O B I O I S 20 2s ? 4 6 8 8 IO

(-VI FiOure 1. Chart fer Estimation of B

Uw up r dlapram when Y is positive- lower diagram when Y la nogo- tive. x t h e upper dlagram the equadion of the ordlnata aoala Is y mi{Y in IY / (Y - l ) ] - l I a n d t h a t o f t h e a b ~ ~ I ~ ~ ~ a i e , x =mt{ ln(Y- l ) ) . Slmpie logarithmio scales are uwd in the lower diapram. ,9 - BM/BI

(mq text)

time average for the point on the surface where the values of h and F are wanted. For mean values over the whole surface, a surface average must be taken. In either case, for geometrically identical arrangements in the two processes the result for either will be

[ v I X I ~ - -f[K,(DoV~/p)l (24) although, if point values are considered, function f will vary in form from point to point. The substitution of Equation 24 in Equations 208 and 22s with appropriate evaluation of K gives

or alternatively

The functions f are identical for the two processes and, if found for one, may be used to describe the other; the same is true of 4. As a prttctical matter, these formulas suffor from the defects

usual in those typically deduced by the aid of dimensional analysis. Heat tranRfer cannot, in fact, be experienced in normal fluids without significant variation of p, even though the variation of the other physical properties may be relatively small. Gas phase diffusional transport cannot generally occur in an essentially isopiestic and isothermal field, as here necessary to be supposed, without variation in the mean moleculw weight and so, in ‘yo How far into the domain of reality the formal analogy of Equations

25a and 26b may be extended by judiciously selected empirical rules for evaluating the physical properties is a question for experiment. The formal justification of the analogy given here for the gas phase has only limited validity for the liquid phase. It seems futile to discuss the liquid case in detail until more information is available concerning the variation of D, for that case with composition. Only then can i t be discovered whether there exists a class of interesting cases for which liquid phase diffusion may be adequately described by an equation like 16; if such exist, the dependent variable may well be other than In M and the constant other than DJp.

EXPERIMENTAL WORK

The psy6hrometric method has a great advantage in a study of the analogy, in that the ratio F/h is very easily calculable from accurate wet and dry bulb data. Indeed, for the authors’ data, the simple formula

sufficed; refinements such as that due to Ackerman (1 ) were found to give insignificantly different results. Although several appropriate sets of psychrometric data ( d , 10, ,86,33) have been collected for organic liquids and water, i t seemed desirable to add data for “wet bulbs” of volatile solids. For such materials a wick is unnecessary, and moreover the Schmidt number range for conveniently obtainable volatile solids evaporating in air ie somewhat above that for most of the liquids that have been studied. The errors to which the usual variations of the wet wick

ta

0 I

E 6 ‘

P

0

I O V T ~ K

Figure 2. Comparison of Vapor Pressures

I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y Vel. 42. No. 6

~taiice as far as the psychrometric experiments were concerned. The measured vapor p r w ~ u w ~ vera used in all calculations. Listed also in Table I1 a x the relative latent heats of sublimation iound from these vapor pressures by using the Clausius-Clapeyroo equation in perfect gas form to interpret the slope of B plot of the logarithm- of tho vapor presvures versus those for a reference sub- stnnce, in this case mter .

The wct, bulb8 for the nuvchromet& measurements ere made

Figure 3. Slide wl th Cylinder Mounted, Ready for Insertion into Air Stream

method are susceptible liiive lireii icwgnired iliid diswsaed by many (8, ?, 8, 10, I f , ZO, 88, $0, SI. $3, $4) .

The following d d e were used :

.'iaphtllakne, crysthl Wagmt, Baker & Adamson Camphor, U.S.P., synthetio powder, Du Pont p-Dichlorabenzcne, technical, Amend Dru and Chemical p-Dibromobenzene, c.P., Amend Drug m i Chemical

Their vapor pres~ures were determind by parsing aL a t B mtc. alwsys low tlisn 4 cubic feet per hour through a thermostateil 1 . la ioch sluminurn tube packed to a depth of 8 inches with irregular '/Isinch pellets of the desired substanre. From tlw datu of Hurt (IJ) 8 irrches is estimated to be 2'/* Cimes tlie depth required ior 99% saturation of the air; tests indicated that the aii leaving WGS indeed saturated and was at tho temperature 01 the lhermostat. From the loss in wight of the tube during tho pas- sage of a measured volume of &?, the vapor pressures given in Table I1 were found by aid of the ideal grw laws. The diffrreol iuus could be reproduced within 1%. The results are eompurexl with valuesof the Internstlonal Criticirl Tubles (16) in Figure 2. The pehlorobeneene was undoubtedly impure and almnst certainly contained the ortho isomer. Even in this case, however, in the subsequent measurements no trend with time wili. observed, so the material appeared to bshilvc like B pure 6~11-

Table 11. Vapor Pressures and Latent Heats

4 5 . 5 0 3 1 4 50.0 0761 ,500 1.230 20.2 i d ? i o 1 0.580 55.0 1.142 55.0 1.872 35.2 2.32 54.0 0.838 50.0 1.092 MI.0 2.240 I I . O 8 6 6 50.0 1.288 88.0 2 855 88.0 3.240 45.1 I !IS ~ 5 . 0 5 1.840

not d l sieoCwerc used with every solid. The eylitidors w& mounted singly, with itsia vertienl, on LL slide (Figure 3) which could be itiserted in i i I x I foot air duct as shown ~n Figure 4. The Iomtion of the slide was receded bv B 7.5-foot run of

tion, B I I ~ an impsit tube for traveruing'ttie duct.' The position ai the test cvlirider in the cross section was BO selected that it WYBS

by I-inch 85% magnesia block hished with asbestos cement and canvas. The upper return duet contains steam coils for regulsting the temperature oi the air.

Comer-constants" thermmounles were used throuehhout. after

88GZ portable' potentiometer. This instrument had B limit i f error eauivalent to &bout 0.4' F. The wet bulb dcoresaion wlls dr-

Figure 4. Cylinder In Position, Ready for Run

lune 1950 I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y 1169

R u n NO.

44 45 56 57 68 59 62 79 80 81 82 83

65 66 67 70

104 129 130 131 132 133

16 18 20 21 26 27 28 29 75 76 77 78

126

85 8G 87

100 101 102 103

120 121 122 128

88 89 90 91 93 94 95 96 97 98

112 113 114 118

4. 143.7 156.2 139.7 140.8 136.8 136.7 140.5 183.3 176. 8 176.6 177.6 178.4

137.5 134.8 176.9 120.7

154.1 157.8 161.2 153.4 153.4 164.4

69.7 69.2 70.8 70.8 67.0 67.8 67.8 69.0

117.5 114.1 114.1 114.5

116.5

149.3 144.7 157.7

178.2 180.5 174.4

179.a

146.9 146.7 146.9 163.9

154.1 158.0 152.0 156.2 127.6 123.0 134.4 133.3 135.0 165.0

139.7 141.7 135.8 166.2

f . - tt., F.

4.26 6.90 3.95 4.20 3.39 3.55 4.11

14.77 13.10 12.60 13.18 11,68

3.50 3.30

13.30 1.77

6.70 7.42 8.15 6.40 6.37 9.10

1.62 1.51 1.62 1.74 1.35 1 ,54 1.51 1.50

10.13 9.05 9.35 8.93

9.73

4.82 4.49 6.27

11.34 10.55 10.80 9.75

4.92 4.92 5.00 8.06

5.16 6.08 5.05 5.57 1.88 1.60 2.60 2.44 2.68 7.55

3.40 3.40 2.76 a.16

Table I I I. Experimental Data and Results V

La - fr, I, Fe&/ ooa (PB),V., ' W / h ) F. Mm.Hg Sea. Mm%g ry Mm. Hg 1.1)6 X 10'

Nnphthalene, 3.5 Inches by 0.375 Inch in Diameter 1.76 2.15 0.37 1.77 1.73 2.23 1.36 6.06 7.54 8.33 7.94 8.36

1.90 1.86

-1.59 1.43

2.40 2.49 4.65 4.78 5.02 5.10

a. 18 1.70 1.74 1.74 1.35 1.75 1.57 1.50 9.00 9.25 8.25

11.50

10.40

2.29 2.90 2.30 4.51 5.34 5.93 4.86

1.88 2.90 3.32 5.45

0.81 2.31 3.13 2.36 0.63 1.05 0.95 0.37 0.00 8.21

0.00 1.89 1.86 4.21

762 762 755 755 755 755 755 757 757 757 757 756

765 766 765 765

757 772 772 772 772 772

755 755 755 755 769 769 769 709 757 757 757 757

756

764 764 764 756 755 755 755

761 761 761 756

758 757 758 758 758 758 758 758 758 758

766 756 756 766

63 5 1.67 9SOO 761 18,280 652 45 5 2.51 6:780 762 18,180 705 45 6 1.45 7030 755 18320 679 57 0 1.50 8:780 755 18:310 708 66 4 1.30 10,300 755 18350 659 76 3 1 28 11,850 755 13:350 704 46 2 1 49 7,100 755 18315 696 49 9 5 60 6900 754 17:970 682 59 4 4 50 8:320 755 18,020 764 71 0 4 60 0,910 755 18,020 710 79 2 4.70 ll.060 755 18,015 724 87 0 4 20 12,220 754 18,040 718

Nnphthalene, 6.75 Inches by 0.375 Inch in Diameter 67 0 1.34 10,500 765 18,340 670 81 5 1.21 13.000 766 13,370 698 31 0 4.55 4,360 763 18,020 739 44.0 0.695 7,270 765 18,500 657

Naphthalene, 4.5 Inobes by 0.72 Inch in Diameter 46 4 2.35 13250 756 18,200 731 44 6 2.68 12:830 771 18,160 725 57 4 2 94 16,400 771 18,140 735 77 6 2.30 22,800 771 18,200 736 87 1 2.30 22,500 771 18,200 732 46 5 3.28 13,200 770 18,110 739

p-Dichlorobenscns, 3.5 Inches by 0.375 Ihch in Diameter A

1.45 -

755 755 756 755 769 769

742 690 684 741 687 779

77 0 0.582 15,100 769 19,040 746 83 6 0.617 16,300 769 19,025 697 63.6 3.90 10550 756 18,620 756 79.0 3.50 133280 755 18,640 750 53 0 3.47 8,920 755 18,645 785 71.4 3.60 12,000 755 18,640 726

pDichlorobenaene, 4.5 Inches by 0.72 Inch in Diameter 44.1 3.80 14,100 754 18,625 7G4

Camphor, 8.5 Inches by 0.375 Inoh in Diameter x 1.29

600 641 624 656 628 607 640

Camphor. 4.5 Inohss by 0.72 Inch in Diameter

46.3 2.47 13,560 760 18,255 600 67.6 2.45 I9800 760 18,260 670 86.2 2.4(1 25:200 760 18,260 676 78.7 3.88 21,900 754 18,105 702

p-Dibromobenrcne 3.5 Inches by 0.375 Inch in Diameter x

1.72 -

46.0 I 00 6 , ~ s 757 i8,iao 664 69.3 2.10 10,150 756 18,150 692 86.2 1.74 12,900 757 18,200 719 56.8 2.04 8,400 757 18,160 657 74 0 0 .08 11,900 758 18,430 668 89 5 0.56 14.600 758 18,475 695 82.8 0.91 13,000 758 18365 695 65 9 0.87 IO 400 758 18:375 G77 45 3 0.92 73130 758 18,360 699 47.4 2.70 6,840 757 18,090 698

pDibromobenrcnc, 4.6 Inches by 0.72 Inch in Diameter 46.1 1.10 13,700 755 18,320 740 67.0 1.20 19900 755 18,295 700

47.6 2.79 13,150 755 18,080 735 14.0 0.95 26:zoo 756 18,350 717

B Mf

1.0050 29.10 1.0075 29.14 1.0040 29.10 1.0045 29.10 1.0035 29.10 1.0035 29.10 1.0045 29.10 1.0185 29.35 1.0130 29.28 1.0138 29.28 1.0140 29.30 1.0120 29.23

1.0035 29.10 1.0035 29.10

1.0020 29.03 i .oiao 29.28

1.0070 29.14 1.0080 29.14 1.0080 29.15 1.0068 29.12 1.0068 29.12 1.0095 29.20

1.002 29.02 1.002 29.02 1.002 29.02 1.002 29.02 1.002 29.02 1.002 29.02 1.002 29.02 1.002 29.02 1.013 29.25 1.013 29.25 1.013 29.25 1.013 29.25

1.013 29.25

1.0100 29.16 1.0085 29.16 1.0120 29.22 1.0200 29.40 1.0200 29.40 1.0200 29.40 1.0175 29.35

1.0095 29.10 1.0006 29.16 1.0095 29.16 1.0140 29.25

1.0105 29.23 1.0105 29.23 1.0105 29.23 1.0105 29.23 1.0040 29.10 1.0040 29.10 1.0060 29.10 1.0060 29.10 1.0060 29.10 1.0150 29.30

1.0065 29.10 1.0070 29.10 1.0060 29.10 1.0150 29.30

FM. h e

1.89 2.04 1.97 2.05 1.91 2.04 2.02 1.97 2.17 2.06 2.09 2.07

-

2.02 2.02 2.13 1.90

2.12 2.10 2.13 2.13 2.12 2.14

2.15 2.00 1.98 2.14 1.99 2.26 2.16 2.02 2.18 2.16 2.27 2. 10

2.20

1.73 1.86 1.80 1.89 1.81 1.75 1.86

1.91 1.94 1.95 2.02

1.92 2.00 2 .08 1.95 1.94 2.01 2.01 1.96 2.03 2.01

2.12 2.08

2.12 2.08

1'1 J d X 101

2.56 2.56 2.56 2.56 2.56 2.56 2.56 2.64 2.54 2.54 2.54 2.55

2.56 2.56 2.54 2.57

2.56 2.56 3.66 2.56 2.56 2.65

2.23 2.23 2.23 2.23 2.23 2.23 2.23 2.23 2.21 2.21 2.21 2.21

8.21

2.59 2.59 2.58 2.56 2.56 2.5R 9.57

z.59 2.5.9 2.59 2.58

2.35 2.35 2.35 2.35 2.86 2.36 2.36 2.36

2.34 2.36

2.36

2.34

2 . a ~ 2.36

678 848 805 765 668 668 815 805 830 725 695 665

704 640

1038 765

675 G78 630 548 520 676

885 700 630 660 575 645 580 540 695 625 770 630

020

G I 0 605 68.5 785 710 615 595

600 520 47s 520

750 670 635 710 605 580 610 660 7ao 785

025

480 640

aao

1170 I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y Vol. 42, No. 6

Kinetic theory leads uiie to expect theSchmidt number to vary little with the temperature for B given mixture. Actual experimental datn on the variationof &with temperatureare few and show rnther different rcsults for different g&m, 80 that for an arbitrary mixture there w i a ~ e l y a possibility of reliably knoxing the wtwal vsriation of i l /MD. with temperaturc; it appears to be s m d for small percentage variations in absolute temperature. For the ronditions of psychrometry, the miut,urez involved are usually so dilute that ii may iessonnbly be oonsidered that of pure air. I n the cireunujtsnces i t seems adequate to take *,ID* ss constant a t its value for 0" C . The molecular weight i n the Schmidt number is, however, given its actual vnlue for the mixture. Table I V aives the values of tlre diffusivitv

Figure 5. Interior Of Dust Immediately Upstream of Slide, Showing adopted. mental v~luwwere twvailnbletheg havebeenused:

Heemin&' Shielded Therm-upks and Impact Tube

WIU shout 1 microvolt (0.04" P.). The Type K-2 potentiometer was similarlv used in mearuiinp. the trinmruture differenoe between the suiisce o i tho duct andthe wet i;ulb.

The shielded air temporat,uie junctions were made by soldering the wires to opposite ends of a 1.5-inch longth of cop r tubing, the fomwd wire being p-ed thruugh the tube. Ee forward end of the tube was then mashed c l o d , and the tuba w m filled with sealing wax. This provided B sturdy junction of con- sidernble heat capacity. The copper tube WBS centered in R thin- walled aluminum tube, *I8 inch in diameter and 3.5 inclres long. This tube was in t w n centered in tl thin-walled aluminum tube, 1.25 inches in diameter and 10 inolios long, with the front, end slotted and the projections bent in toward the center line. The concentric tu im were supported in place by laminsted-plartic 8pnocrs. The whole msembly WB^Y 80 h e e d within the duet &s to ensure high air velocity through tbe tuboa. These shielded thermocouples are shorn in Figure 5.

The surface tempernture junction wsq partially embedded i n the Transite floor of the duet, cemented in with B mixture of litherge and glyc~rol, and filed flush with the surface.

The differential pressure between the impact tube and the statio opening w w measured by an inolined gage, and was determined seeu+4y to within 0.5 inch of water, whioli corresponds to a maximum velocit,v error of shout 1%.

necessary readin@ were made and thereilftcr the wet bulb depression w89 rechecked. The baro- metric pressure was recorded for exlr run.

The experimental data we recorded in Table 111 together with various pertinent quantities computed from the datn.

In computing the Reynolds numbers and the Schmidt numbers in Table 111, tlre viscosity w m taken a t a temperature midway between t, and L; the molecular weight M , wss ~ndog~us ly taken for mole irlrction midway between yo snd llu. Actually, for most of the present rune there would have been negligible change in the results, had the mein stresm values of p snd M been used. The mass density, 7 , was evaluated for the main s t r s m .

Ratio P l h was found by Equation 27; h, was evaluated for an sgsumed emkivi ty 0.9 from a large oh& similar to thst of MoAdams (e$), and h in the correction term, by MeAdam' ap- proximate equation for air (U). The vnlue of F/h deduced ia relatively i-ndtive to errom in evaluating the oamwtion term

in seversl cases distinctly inierior correlation of the data rBulted when valuez onlculated from

the Clilliland equation were substituted ior experimental values.

Although it could be sntioipated that # fur the authors' cir- uumstanees would be nearly unity, it was computed for each IUD

88 B matter of interest. l'be highest Y B I U ~ vma 1.0% in rum 100, 101, and 102; in most ease8 the value exceeded unity by legs than 1 %. Such insignificant deviation of 6 from unity is by no mems the rdc; v~lues as high 89 1.23 were found for data from the literature utilized below.

The velus of the Prandtl group throughout this paper i s taken at its d u e for pure air, 0.70 (18); it is e o n s t d over the ranges involved.

A principal purpose of teating cylinders differing in length WM

to verify the prediction of theoretical eslculations that end effects and errors due to conduction along the axial thermocouple Wire would be negligible. No effect of length is apparent in the ie- sults.

Figurn 6. Afr Duot

June 1950 I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y 1171

A

Table IV. Value of Diffurlvity and Sohmidt Number a t 0' C. for Various Vapors in Air

P - Vapor 106Dm MDm

Naphthalene" p-Dichlorobenzene b Cam horb

Digromobenzeneb iLnzene5 Chlorobenzene C Toluene C Carbon tetrachlorideb Ethyl acetateC Tetrachloroethylene b m-Xylene" Watera

Calculated from I.C.T. values of Dtl (17). b Calculated from Gilliland equation (fa) for Dv using molecular volumes

C Calculated k o m Gilliland experimental DV (1.9). from LeBas ($1

CORRELATION WITH THE ANALOGY

In addition to the authors' data, the investigations of Mark (869, Arnold (a), Dropkin (IO), Vint (86), Lohrisch (28), Powell (?IO), and Clapp (6) have been been studied for purposes of comparison with the analogy. In each case the data were recalculated to allow for radiation when necessary, and the results were expressed as values of the quantities appropriate to this investigation.

Work of Mark (26). Psychrometric measurement.? were made, employing water, chlorobenzene, benzene, carbon tetrachloride, tetrachloroethylene ethyl acetate, and toluene to wet the wick of the wet bulb. Oi the 64 runs made, 52 were recalculated for comparison, eliminating only those runs which were obviously in error.

Work of Arnold (2). Psychrometric measurements were made with toluene, chlorobenzene, and m-xylene, in a tunnel 4 inches square. There was ne ligible variation of velocity over a central area of about 4 cm. in jiameter, but the length of the wick on the wet bulb thermometer was 4.3 cm. Arnold's results indicate a variation of F/h with the Reynolds number, although no variation would be expected. The variation can be expressed as a function of the Reynolds number to the 0.07 power. Calculations of the heat conducted by the thermometer stem indicate a possible error, with such a short wick, which varies inversely with velocity from 8 to 14%, and when allowed for tends to reduce the indicated variation of F/h with Reynolds number.

Work of Dropkin (IO). Using water and an adiabatic chamber, Dropkin investigated the deviation o! the wet bulb temperature from the temperature of adiabatic saturation. He did not re ort the humidity of the air passing the wet bulb, but it was possibg to calculate it from the adiabatic saturation temperatures he reported. He covered his wet bulb thermometers with wicks 8 inches in length to make negligible the error due to heat conduction along the thermometer stems. Of 37 runs reported, 32 were recalculated, the other 5 being discarded as in error due to lack of attainment of adiabatic saturation. The results are the most consistent of those studied, the average deviation from the mean value of F/h being 0.6%, and the maximum deviation being 1.7%. Absolute1 no variation of F/h with either temperature or Reyn- olds numler is indicated. The results are so consistent that it is possible to represent by average values the data for each of the six velocities at which he made runs.

Mass transfer measurements were made for the vaporization of)'n-butyl alcohol, toluene, and water, from a cylinder 17.3 inches long and 2.0 inches in diameter. The cylinder was covered with a cloth and the liquid was circulated over it. The rate of transfer was determined from the rate of volume loss of the circulating liquid. The experimental pionts spatter con- siderably, and the results for n-butyl alcohol are inconsistent with the results for all other substances investigated. Vint questioned the results for n-butyl alcohol, and attributed the error to lack of purity of the material. A vapor pressure error so introduced would displace the results in the manner observed. Vint did not report the total pressure for his runs; it was assumed to be 1 at- mosphere in his calculations. For the reasons stated, this work is not included in the general correlation.

Work of Lohrisch (22). In the Lohrisch investigation of mass transfer of water vapor in air, air was assed over c linders of solid sodium hydroxide, 0.394 inch in gameter and i 2 3 inches long, which absorbed moisture from the air stream. The rate of

Work of Vint (35

mass transfer was determined from the increase in weight of the cylinders. The results of these experiments were used b Chilton and Colburn in their original paper on the analogy (,$rand the points were found to lie a little hi h. However, Lohrisch did not measure the temperature of the c$inder; he assumed it to be the same as the air tem erature. It appears to have been generally overlooked that the {eat of reaction of the water with the sodium hydroxide will raise the temperature of the saturated solution, and thus increase the va or pressure of water over the saturated solution. Assuming aJabaticity, the calculated temperature rise amounts to several degrees centigrade, and shifts the points about 40% further above the line representing the analogy in the Chil- ton-Colburn fmm.

For his ammonia experiments, glass tubes of various sizes were wrapped with blotting paper saturated with phosphoric acid, and the amount of ammonia absorbed from the passing air stream was determined by titration. Despite the fact that Lohrisch did not report sufficient data to permit calculation of the partial pressure of the ammonia in the air, it is ossible to calculate the quantities necessary for the present s tu ty from the data reported. The data are found to lie about 40% above the predictions of the analo y when the diffusivit is calculated from the data of Winter- gerst f3'7). The Lohrisch i a t a seem likely to be in error because the ammonia was introduced into the air stream only a short distance before the cylinder, and it i s probable that the stream wm not homogeneous by the t8ime it reached the cylinder.

For the reasons stated, Lohrisch's work is not included in the correlation.

Work otPowell(30). Mass transfer rates werc determined for water va orizing from several different sizes and shapes of ob- jects, inckding c linders from 0.16 to 37.2 cm. in diameter. The water was circuited over a wick covering the surface of the cylinder, and the rate of transfer was determined from loss of weight of the circulating water. The original data are not reported, but it is possible to calculate representative values of the quantities desired, from the curves representing the correlation made by Powell, The results are in excellent agreement with the psychrometric data for water of Mark and Dropkin, for values of the Reynolds number up to 10,000. Beyond this value, Powell's results tend to be high.

Work of Clapp (5). This investigation was concerned primarily with the effect of turbulence upon the heat transfer coefficient. A few mass transfer runs were made, in which water WM circulated over a 3-inch length of co er pipe 1.25-inches in diameter, covered with blotting paper. 8 i p p stated that the accuracy of his mass transfer determinations was low, and as a result he used average values. Owing to the admitted low accuracy, and because the results do not correlate well with those of other investigations, the work of Clapp is not included in the correlation.

When the values of F/h of the above investigations are plotted

Figure 7. (FM,/ht9),v. as a Function of (a,/M,Dm).v. Theory raquirer line to part through P Symbols same as in Figure 8

11'12 Vol. 42, No. 6 I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y

2 3 5 7 IO4 2 3 5 7

This W o r k : o Nophtholene u Camphor V Par a - di c h I o f 0 bent ene D Para- dibromobenttne

Data o f Arnold: J Toluene > Chlorobenzene c Mota-xylene

Data D O 0

Pf e -

Data A V 0 3 A L

n

ot Dropkin: Water of Mark: Water Benzene Chlorobenzene Toluene Carbon Tetrachloride Tet r a ch I o r o e t h y I en e Ethyl Acetate

Figure 8. I d Computed from Experimental Values of FM,/L,9 h taken from McAdams (24). Line represents either Equation 32 or 32a

against either the Reynolds number or temperature there is generally considerable spatter of the points, but there is no clear indicationof variation with either except in the caseof Arnold, who found a small effect of ne. The data of Table I11 show no defi- nite trend with t or Kith Re. The evidence is preponderantly in favor of the view that h and F each vary in such a way with DoV that this quantity either cancels out in F / h or so nearly does so that its influence on F / h is negligible.

In the psychrometric case, it is certainly true that the physical situation in any one run is geometrically identical, whether looked a t from the point of view of heat transfer or mass transfer. I t is also dynamically identical. If, then, we suppose that the F and h in F / h are either negligibly different from the values appropriate to circumstances in which mass transfer and heat transfer, respectively, are proceeding alone or, in computing the ratio suit- able corrections have been applied, one may apply Equations 25a and b or equivalently 26a and 1) . To choose the former pair,

in whichfis the same function of its argument8 in nunicrrrtor and denominator.

?;ow from hcat transfer data we know that over modcrate ranges f can be closely approximated as a power of the Reynolds number multiplied by a constsmt depending on c,,u/k. If this is true exactly, the form of the arguments o f f are such that the vonstant must be proportional to a power of c p p / k . In this caac Equation 28 would reduce to

which would be consistent with the observed invariance of F / h with DpV. If we take as an experimental fact that F / h is invari- ant with DoV, we are forced to the conclusion that Equation 29 is valid because:

1.

2.

Eitherfis a simple power function of Re as supposed above, or it is a more complicated function.

If it is a more complicated function and Dol' cancels out of Equation 28, then a power of the first argumcnt off must factor out off , in which case Equation 29 results.

If the Schmidt nunher and the Prandtl number are equal

so that if the carrier gas is air, a plot of F h l / h p versus the Schmidt number mwt be a curve passing through the point (0.70, 4.16).

Figure 7 is a plot of the data discussed above in the form aug- gested by Equation 29. The molecular weight is taken as MI because, first, it gives slightly less spread of the points than does the choice of M , or M , and, secondly, because such evaluation is eon- sistent with the usual practice relative to c,p/k. The best straight line through the data and the point (0.70, 4.16) i R

The slope -0.56 is inconsistent with the power to which c,M/k apprars in typical heat transfer equations for flow around cylin-

lune 19% I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY 1173

ders. The slope -4 /a is clearly too steep. Actually, the experimental data for heat transfer in the range of interest come almost exclusively from work in which c,p/k varied little from its value for normal air and can therefore be adequately represented by using any power of c&/k (6, $4, $6). It is therefore possible that -0.56 is not really inconsistent with the heat transfer data if the analogy is accepted; obviously, further heat transfer studies we required to settle the point.

The paired masa transfer and heat transfer equations to which the slope of -0.56 in Figure 7 would correspond if existing heat transfer data for air are accepted are:

Figure 8 ahows all the data here discussed plotted in this form. It is, of course, simply a demonstration of self-consistency of the interpretation of the conclusions from Figure 7.

NOMENCLATURE

Bold-face symbols are vectors; the same letter in italics is the

= heat capacity at constant pressure per unit mass, B.t.u./

magnitude of the vector, unless otherwise stated.

r p

D, = molal &ffusivity, Ib. mole/(foot)(hour)(unit Ay) DO c. diameter, foot F = mam transfer coefficient, Ib. mole/(hour)(sq. foot)(unit

G = maas velocity of fluid as a whole, Ib./(hour)(sq. foot) G, - m m velocit of qth constituent, Ib./(hour)(sq. foot) h = coeEcient o f heut trunsfer by convection B.t.u./(hour)

(sq. foot)(’ F.) h , = coefficient of heat transfer by radiation B.t.u./(hour)

(s foot)(’ F.) Jh, ~ d , see % y t i o m 32 and 32a K , see Table k = thermal conductivity, B.t.u./(hour)(foot)( a F.) M , Mq = local molecular weight of mixture and of qbh con-

n = unit vector drawn perpendicular to specified surface p = artial pressure, mm. mercury

= Reat current density, B.t.u./(hour)(sq. foot) b = gsit ion vector with respect to arbitrary origin, foot l i e = eynolds number S = area of cross seztion, sq. foot t - temperature, F. U = molal linear velocity of mixture, foot/hour V, V, = linear velocity of mixture and of qtb constituent, foot/

hour 1’ = magnitude of linear velocity of main stream, foot/hour or

foot /second W = mass rate of flow, Ib./hour w = molal mass velocity of fluid as a whole, Ib. mole/(hour)

(8 I foot) wv = m o d mass velocity of q*b constituent, Ib. mole/(hour)

(sq. foot) X = generalized dependent variable, see Equation 17 and Table

I !/,yv - mole fraction of Q in mixture, subscript used only wlicn

more than two constituents are present

L, zq = !!!! mole fraction of Q in diffusion stream

B A = difference 6 = small increment y, yp = local mass density of mixture and of qtb constituent,

qq fl = time, hours 11 p, p p = local molal density of mixture and of 9th Constituent, Ib.

P = viscosity, Ib./(hour)(foot) X

F. 1

All)

stituent, Ib./lb. mole

W - defined by Equation 23

Ib./cu. foot = masa fraction of Q in mixture

= total pressure, mm. mercury

mole/cu. foot

= molal latent heat of sublimation, B.t.u./lh, mole

Subscripts A , B, 8 , v,w,gdesignatevalues of the uantit a t the surface of a duct, iri

the main stream, at the su4ace o?heat transfer or evapora- tion, and in the “film,” respectively.

refer to substance A , B, and Q, respectively.

Vector notation v p = gradient of p , a vector with the direction of the most rapid

increase of p with change in position and a magnitude equaling the space rate of change in that direction

U * V = product of the magnitude of the vectors U and V multiplied by the cosine of the angle between them

LITERATURE CITED

(1) Ackerman, G., Forschungsheft, No. 382, 1-16 (1937). (2) Arnold, J. H., Physics, 4, 255, 334 (1933). (3) Chapman, S., and Cowling, T. G., “Mathematical Theory of

Non-Uniform Cases,” Chap. 14, London, Cambridge Universi ty Press, 1939.

(4) Chilton, T. H., and Colbuiii, A. P., IND. ENQ. CHEM., 26, 1183 (1 934).

(5) Clapp, J. T., Jr., Ph.D. thesis in chemistiy, University of Illinois, 1942.

(6) Colburn, A. P., personal communication, 1948. (7) Dorfell, K., Verhandl. geophya. Inst. Unw. Leipzig. 6, 4 (1935) (8) Dorfell, K., and Lettau, H., Ann. Hydrographie maritirnen

(9) Drew, T. B., Trans. Am. Inst. Chem. Engrs., 26, 26 (1931). Meteorol., 64, 342 (1936).

(10) Dropkin, D., Cornel1 Univ., Eng. Expt. Sta., Bull. 23 (July

(11) Ekholm, Arkiu Mat. A s t r a . Fuaik, 4, No. 15 (1908). (12) Fourier, J. B., ”Oeuvres,” Vol. 11, p. 601, Paris. Cauthior-

(13) Gilliland, E. R., IND. ENQ. CHEM., 26, 681 (1934). (14) Goldstein, S., ed., “Modern Developments in Fluid Mechanics,”

Vol. I, p. 101, London, Oxford University Press, 1938. (15) Hurt, D. M., IND. ENQ. CHEM., 35, 522 (1943). (16) “International Critical Tables,” Vol. 111, pp. 199-246, New

York, McGraw-Hill Rook Co., 1929. (17) Ibid., Vol. V, p. 62. (18) Keenan, J. H., and Kaye, J., “Thermodynamic Properties o f

(19) Klinkenberg, A., and Mooy, H. H., Chem. Eng. Progress, 44, 17

(20) Kuns, R. F. J., “Air Drying of Cellulosic Solids,” Ph.D. d~suer-

(21) LeBas, G., “Molecular Volumes of Liquid Chemical (‘on,-

(22) Lohrisch, W., Mitt. Forschungsarb., 322, 46 (1929). (23) McAdams, W. H.. ”Heat Tiansmission,” 2nd ed.. p 63, Sc\r

(24) Ibid., p. 222. (25) McAdams, W. H., personal communication, 1948. (26) Mark, J, G., Trans. Am. Inst. Chem. Engrs., 28, 107 (1932). (27) Nusselt, W., 2. angew. Math. Mech., 10, 105 (1930). (28) Onsager, L., Ann. N . Y . Acad. Sn‘., 46, Art. 5, 241-65 (1945). (29) Poisson, 5. D., “ThBorie MathBmatique de la Chaleur,” Paris

1936).

Villars, 1890.

Air,” p. 36, New York, John Wiley & Sons, 1945.

(1948).

tation in chemical engineering, Columbia University, 1942.

pounds,” New York, Longmana, Green & Co., 1915.

York, McCraw-Hill Book Co., 1942.

Bachelier, 1835. (80) Powell, R. W., Trans. Inst. Chem. Engrs., 18, 36 (1940). (31) Powell, R. W., and Griffiths, E., Ibid., 13, 175 (1935). (32) Sherwood, T. K., “Absoi ption and E;xti action,” New York.

(33) Sherwood, T. K., and Comings, E. W., Trana. A m Insf . Ckrm

(34) Svensson, Meteorol. Z., 43, 140 (1926). (35) Vint, A. W.. thesis in chemical engineering, hlamchusetts Insti-

tute of Technology, 1932. (36) Walker, W. H., Lewis, W. K., McAdams, W. H., and Gilliland

E. R.. “Principles of Chemical Engineering,” 3rd ed., p. 444, New York, McGraw-Hill Book Co., 1937.

(37) Wintergeret, E., Ann. Phyaik, 4, 323 (1930).

McGraw-Hill Book Co., 1937.

Engra., 28, 88 (1932).

R E C ~ I V E D Dbcernber 29, 1949. Subnutted by Charlea Hosmer Bedingfield Jr., in partlal fulfillment of the requirements for the degree of doctor of [ihilonophy in the Graduate Faculty of Pure Science, Columbia University, 1848. Contribution No. 1 from the Chemical Engineering LabnratorieP, Engineering Centor. Columbia liniveraity. New York, N I’

analogy between heat transfer and mass transfer

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