M A S S A N D H E A T TRANSFER R A T E S FOR L A R G E G R A D I E N T S OF C O N C E N T R A T I O N AND TEMPERATURE W I L L I A M
E.
RAN2 AND P H I L I P F. D I C K S O N ’
Lkpnrtnieni of C‘hmicul Erigineerinc
CiiiLers2ty of M i n n e s o t a . i2linneapolzs, Minn
Data on the evaporation of water, benzene, carbon tetrachloride, and heptane into hot air, at temperatures as high as 545’ F., and dissolution of 2-butanone, methyl formate, and cyclopentanone into water at room temperatures were obtained and compared with the predictions of an integral boundary layer analysis. Stagnation regions and a turbulent boundary layer on a flat surface were chosen as general flow configurations. Attention was focused on simultaneous transfer (with measurement of wet-bulb temperature and heat transfer rate), large gradients in concentration and temperature, large concentrations of diffusing cornponent (with mass fractions as high as 0.8), convection caused b y diffusion, and boundary conditions.
methods of predicting inlerTfacial. ismassneedandforheatgeneralized transfer rates in fluids \\.hen gradients HERE
of concentration and temperature are large. Moreover, there is a paucity of experimental data accurate enough to check theory. This paper concerns a project in which the problem of large-gradient transfer was reviewed. experimental data were obtained in well controlled systems, and correlation was based on an “approximate” (integral) analysis. Stagnation regions and a completely turbulent boundary layer on a flat plate were chosen as general flo\v configurations worthy of study. At1:ention \vas focused on simultaneous transfer, variations in transport coefficients. large gradients in concentration and temperature, large concentrations of diffusing component, convection caused by diffusion, and boundary conditions. In addition to empirical correlations, which can be based on dimensional analysis of the equations of change, and the simple concept of a ”stagnant film,” Prandtl’s boundary layer theory (27) has been extended to yield so-called “exact” and “nearly exact’‘ solutions for heat and mass transfer rates in boundary layers. Most of these solutions are for vanishingly small gradients-that is, for constant or nearly constant physical properties. Recent publications by Merk (76. 77), Spalding and Evans (72-74, 22-27), Steivart (79, 28). and Acrivos (7-3) are cited here. Many of these results include effects of large mass transfer rates! and some account for varying properties. Some of Spalding’s solutions can be characterized as “approximately exact’‘ ; Acrivos‘ solutions are ”asymptotically exact” for very large or ’TPr,for very small .\’sc or Npr,or for very large mamss transfer rates. Although it lacks mathematical sanction, the Von KarmanPohlhausen “approximate” (integral) method (27) for boundary layer solution is a general and practical way of attacking new problems. I t is also easy to use on old problems having new complications. Justification for integral methods lies primarily in results. Problems, Lvhose “exact’‘ solutions appear beyond the reach of present mathematics, submit to integral analysis with only algebraic complexities to be overI
Present address, Colorado School of Mines, Golden, Colo
come. Moreover. there is expectation that theoretical objections \vi11 eventually be blunted by estimation theorems of the type developed by Nickle (78). These theorems permit close bracketing of the unknoxzn profile with trial profiles of integral type, giving a criterion for estimating error without access to the exact solution. For present purposes approximate solutions of the boundary layer equations are used as the basis of data correlation. This choice \vas made because approximate methods emphasize physical happenings on and near the boundaries of f l o ~ v ,can incorporate the complications under study. give solutions which match rather \vel1 the form of available exact solutions? and are successful in correlating the data. Results of Approximate Solutions and Bases of Data Correlation
Table I. shoxzing transfer numbers for vanishingly small mass transfer rates, and Tables I1 and 111: shokving correction factors to be applied for significant mass transfer rates: summarize approximate solutions of the boundary layer equations applied to the region near a stagnation point. T o obtain these results linear gradients of mass flux, velocity, mass fraction of component i, and enthalpy \\.ere assumed to hold over boundary layers of different thicknesses. Furthermore, i t \vas assumed that a two-component diffusivity could be applied to each component ( 4 ) and that concentrations and temperatures at the surface and far from the surface \$.ere uniform in value. Transfer numbers for vanishingly small mass transfer ratesthat is, for *\-+ 0-are given explicitly in Table I. Equations 1 through 6 are convenient base equations to xvhich can be applied a correction factor for &Y. Equations 1 and 2 are of familiar form. It should be noted that 5 and 9 2, as critical values of .Ysc and ,Ypr. are much larger than the values near unity found for boundary layers on flat plates. Equation 3 for mass transfer u4th .YSc belo\v the critical value and Equation 6 for heat transfer ivith .\-PI below the critical value are also of familiar form. Equation 4 sholvs a curious additional effect of simultaneous diffusion on heat conduction. In a similar \vay. Equation 5 sho\vs a curious additional effect of VOL. 4
NO. 3 A U G U S T 1 9 6 5 345
~~~
~
Transfer Numbers for Small Mass Transfer Rates (N
Table 1. Bilateral Symmetry
Eq. No.
Nsc > 5 or no diffusion and Np, conduction-that is, 6 ~ < ' 6'
Case A.
> 5 or no heat > aio:
-P
0)
Axial Symmetry
Ns,> 9/2 or
no diffusion and iVp, > 9/2 or no heat conduction-that is, 6 ~ '< 6 " > 6 , ' :
Case A .
= ( A I / ~ ) ~ ~ ~ ( ~ ~ ~ ~ ~ ' ~ , ' ' ~ ) N ( 2R) , ~ ' ' ( &/ ?pm)Avxu = ( A 1/6)' [( 9 /2)1'6.vp,' '1 -\'R,"~ Aise < 5 with diffusion and Np, > N S , or no heat Case B. N s , < 9 / 2 with diffusion and iVpr > ,\'sC or no conduction-that is, 6' < 6,' > 6 ~ ' : heat conduction-that is, 6" < 6,' > 6 ~ ' :
(?p,/?p,)~~,,o
Case B.
A
A
( c p ~ / c ~ ~ )=i \( 'A~116)' ~ "( ~ i : 9 0 " 6 ~ ~ p , 1 1 3 ) N R e 1 ' 2 Case C. .VsC > S p , or no diffusion and ,Vp, < 5 with heat conduction-that is, 6,' < a H 0 > 6 ' :
Table II.
A
.
.
( C p o / C p ~ ) .=V ~( Au1o/ 6 ) " ~ ( N 5 ~ " 6 S p , l'),YR,"'
(4)
> 'Vp, or no diffusion and -Vpr < 9/2 with hea conduction-that is, 6,' < aH0 > 6 ' :
Case C.
2Vsc
Correction Factors to Be Applied to Mass Transfer Numbers in Table I
Eq. N o .
Bilateral Symmetry 1. For N s c < N p , or no heat transfer-that that is. 6 > 6,
is, 6,
> 6H-and
for Y, from Eq. 7 > Y , from Eq. 8-
For ,%'so < N p , or no heat transfer-that that is, 6, > 6
is, 6,
> 6H-and
for Yj from Eq. 8
2.
For N B , > NP, and with heat transfer-that that is, 6 > 6~ Use Eq. 7 4. For Ns, > N p , and with heat transfer-that that is, 6~ > 6
is, AH
3.
Axial Symmetry Substitute 9/2 for 5 in Eq. for bilateral symmetry
>
Y , from Eq. 7-
Same as Eq. 8 for bilateral symmetry
> +-and for Y, from Eq. 7 >
Y , from Eq. 9-
Use Eq. 7 above (7)
is, 6 8
> 6,-and
for Yj from Eq. 9
>
Yj from Eq. 7-
Same as Eq. 9 for bilateral symmetry (9)
Table 111.
1.
For N p , < .Vs, or no diffusion-that that is, 6 > 6~
2. For iVpr < that is. 6~
3. 4.
346
or no diffusion-that
>6
is, 6~
> 6,-and
is, 6 8
> &-and for
1
Axial Symmetry Substitute 912 for 5 in Eq. 10 for bilateral symmetry
>
YH from Eq. 11-
>
YH from Eq. 10-
Same as Eq. 11 for bilateral symmetry
>
YH from Eq. 1 2
Use Eq. 10 above
> 6H-and for Y H from Eq. 12 >
Y Hfrom Eq. 10
Same as Eq. 12 for bilateral symmetry
for YH from Eq. 10
YH from Eq. 11
ds-4 sp,2
1 LVH'XH + 2 N p , For 'Vp, > Nso and with diffusion-that is, 6, -that is. 6 > 6. Use Eq. 10 For LVpr> .Yg, and with diffusion-that is, 6, -that is, 6i > 6
1 =
YH
Correction Factors to Be Applied to Heat Transfer Numbers in Table I
AvH4xH2
,Vpr +
I&EC FUNDAMENTALS
> 6H-and for Y Hfrom Eq.
10
simultaneous heat conduction on diffusion. In Table I: .41 is the dimensionless ncrmal gradient of the normal velocity of the potential flo\v. A very close similarity between bilateral and axial symmetry is apparent, the only difference being an unimportant difference in critical values of .Vsc and .Ypr. Tables I1 and I11 give correction factors to be applied to the mass and heat transfer numbers for the various conditions specified and when there are significant mass transfer rates. Because mass transfer alters the values of .Ysc and A\-pr at which the largest bou ndary layer thickness changes among 6. 6,. and 6 H . the corrections appear complex. However, for each case the correction factor, Y;reduces to a function of dimensionless mass transfer X: .\ae. and -VPr. If exact solutions \vere possible. there would result continuous rather than segmented functions. Figure 1 illustrates these correction factors for axial svmmetry and for component diffusion \Then .YSc < A Y ~orr there is no heat conduction or for heat conduction \Then -YPr < or there is no componmt diffusion. Ynder these conditions I’ is a function of X and .Yet or AYp,, T h e corrections are slightly smaller than those for constant property, laminar boundary layers on sharp-edged flat plates (5). (If special interest ,are the limiting conditions ivhen 6 is ali\-ayc rhe largest boundary layer thickness and ivhen or J H is alivays the largest boundary layer thickness. For the + m ) former condition (-\Jc or A\-s,
and for the latter (.Vsoor *YPr+ 0) 1 -
Y2
I .6
=
-\
1
+ (XlY)
(1 4)
I
I.4
1.2
I‘ = X / ( e X - 1)
(15)
also closely approximates Equation 14 in the range - 1 < X < 1. Since Equations 14 and 15 lie between corrections for stagnation flows and flows on flat plates, either represents a good engineering approximation for all types of transfer at a complex surface submerged in a flow. When component i is the only component being transferred ( n , , ~ = niz,o), Equation 13 for mass transfer reduces further to
and Equation 14 to
”hen
also
azo? =
0: Yican be combined with -\Bh(l - at,)
to give relatively simple correlations of experimental data in
is: lVSh(1 -
terms of a modified Sherwood number-that ai,)2i3 in the case of Equalion 16 and nSh(1
-
wio)112 in the
case of Equation 17. The quantity (1 - at,)is related to the so-called ”pressure factor” of mass transfer correlations for gas systems. Results here indicate that for single-component transfcr and aim = 0 an empirical correlation \vi11 probably result when the pressure factor is included in the definition of Sherwood number and raised to some power between unity and one half. In this \ l a y one accidentally makes rough corrections for concentration of diffusing component and significant mass transfer rates. Integral solutions of the type given above can be extended and modified to analyze a variety of situations. For example, surface concentrations and temperatures can become furctions of 3: as in the cases of an electrode with certain areas active for electrolysis and a heater with certain areas heated. In particular, a mobile interface, such as that encountered on the front of a liquid drop or gas bubble, can be included in the analysis. T h e limiting case of a completely mobile interface is of special interest, since i t defines a maximum variation from the rigid surface case. Here 6 = 0, and Equation 14 is applicable. Experimental Method
I .o
0.a
0.6
-I
-0.5
0.5
0
I
TRANSFER FACTOR X
Figure 1. Correction factor for large transfer rates and axial symmetry When Nsc = Nno When Np, = NP,
All of the curves shown in Figure 1 lie between the curves defined by Equations 13 and 14. I’ is primarily a function of X and either Equation 13 or 14 represents an engineering approximation for all the cases presented. The correction factor for stagnant film theory ( 5 )
< Np,
or there i s no heat transfer:
< Nsc or there Is no dlffuslon:
Y = Y,; X = X,; X
Y = YH; X = XH;
X
Transfer rates near a stagnation point were measured by studying the evaporation of volatile liquids into a hot air stream. Temperature difference, concentration difference, and concentration of diffusing component could be varied over a wide range of values. T h e air \vas directed dobvnward from rectangular and circular jets onto the liquid surface, ivhich was confined to a small area near the stagnation point. The liquid \vas made to seek its “wet-bulb temperature”-that is, all of the heat for evaporation came to the surface from the gas stream above. In this way heat transfer, as \vel1 as mass transfer, data \\.ere obtained. Mass transfer rates \vere also studied at higher values of ,\,, and .VRe by measuring the dissolution of one liquid into another (8)in a flow Lvhich \vas geometrically similar to that for evaporation studies. Apparatus
T h e experimental apparatus for evaporation studies consisted of an air flow system and a thermostated test section. ComVOL. 4
NO. 3
AUGUST
1965
347
pressed air, from a laboratory source a t 68 p.s.i.g., was reduced to operating pressure by a Cash-Acme pressure reducing and regulating valve, passed through a knock-out trap and a KMaster air filter to six Brooks rotameters, Model 110, mounted in parallel for air metering. Rotameter delivery capacity at 14.7 p.s.i.a., and 70” F. ranged from 90 to 33 cu. feet per minute. The air, after passing through a Chromalox circulation heater, Type GCH-6, 12 k w . , 240 volts, three-phase, entered the approach section through a 2-inch 0.d. brass tube. This approach section consisted of a 5-inch 0.d. brass tube, 3 feet long, with a straightening vane section of 0.5-inch 0.d. brass tubes 1 foot long. From this approach, air issued from a 1.500inch diameter smoothly converging circular nozzle onto an evaporating liquid surface 0.531 inch away (shown in Figure 2) and from a 1.525- X 0.666-inch smoothly converging rectangular nozzle onto a surface 0.375 inch away. Air flow rate and temperature were varied independently. T h e liquid pool, shown in Figure 2, was nearly filled with a 0.5-inch diameter porous Teflon plug in a 0.505-inch brass collar mounted in the test section. Thermostat fluid from the controlled thermostat, circulated through the test section by a gear pump, flowed against the underside of the evaporating liquid and returned to the thermostat. So that the evaporating surface would receive heat only from the hot air impinging on it, the thermostat fluid temperature was made to seek the temperature of the evaporating surface ( 2 9 ) by opposing two thermocouples, one in the evaporating surface (No. 40 copper-constantan) and the other in the thermostat (No. 28 copper-constantan). For more accurate control the e.m.f. difference of the opposed thermocouples was amplified by a Leeds Northrup d.c. microvolt amplifier (No. 9835). T h e amplified signal was fed to an L & N Speedomax Type H strip chart indicating recorder-controller which, through a control unit (L & N M.E.C. control unit, Catalog No. 10762) and valve motor (L & S Electric Control, Catalog No. 10224), actuated a General Radio V-10 Variac (10 amperes). Nozzle air temperature and temperature profiles between nozzle and surface were measured by means of a platinumplatinum, 1Oyorhodium thermocouple probe \vith 1- or 3mil diameter wires mounted on a two-directional traversing device. Both air and surface temperatures were obtained from millivolt readings indicated on an L & N precision portable potentiometer. Liquid level of the evaporating surface was maintained by manual control of a Gilmont micropipet-buret, 10-ml. capacity, connected to the evaporating surface by a ‘,!*-inch diameter copper feed line. Data collected during steady-state operation, assumed when a constant nozzle air temperature was obtained for the particular flow rate under study and when the temperature difference between evaporating surface and thermostat fluid was less than 1 F., were air flow rate, evaporation rate (amount of liquid necessary to maintain surface level for a measured time), nozzle air temperature, liquid temperature, inlet air temperature, line pressure, barometric pressure, and temperature difference between liquid surface and thermostat fluid. @
Experimental Results
Figures 3. 4, 5. and 6 record experimental results for mass and heat transfer rates obtained in evaporation studies. Four liquids-distilled water, L.S.P. grade benzene, carbon tetrachloride. and heptane-\+ere employed. Air temperatures as high as 640@F. are represented. Concentration of diffusing component varied in the range 0.05 < w z o < 0.8; Schmidt number in the range 0.5 < .Va, < 2; Prandtl number in the range 0 6 < .Vpr < 0.8; Reynolds number in the range 1500 < ’$-Re < 26,000. T o correlate data as dimensionless quantities, physical and transport properties were calculated by the methods listed below :
cpo,ED=.
A Tabulated experimental values for individual components averaged on the basis of mass fraction. B D t U , Vapor-air diffusivity calculated according to 348
l&EC FUNDAMENTALS
ObT
IN
ObT
CIRCULATED THERMOSTAT FLUID
Figure 2.
Test section for evaporating liquids
Bird, Stewart, and Lightfoot (5) in the case of nonpolar vapors and according to Hirschfelder, C u r t i s , and Bird (75) in the case of polar vapors. C. k,. Tabulated experimental values for individual components averaged according to (5). D. po. Values for individual components calculated and averaged according to (5) in the case of nonpolar vapors and (75) in the case of polar vapors. E. p,,, pm. Calculated for ideal gas mixtures of vapor and air. F. pia. Calculated from tabulated experimental values of vapor pressure, assuming ideal gas mixtures and local thermodynamic equilibrium between gas and liquid at the surface. G. X z q . Tabulated experimental values of heats of vaporization. In Figures 3 and 5 mass transfer data are displayed in terms of an approximate correlation, ‘VSh(I - ~ i , , ) ’ / us. ~ iVS,.VR,. T h e approximation is that LVS, is never much larger than iVp, and always less than the critical value. I n Figures 4 and 6 heat transfer data are displayed in terms of a correlation [l suggested directly by the analysis .VS~(&,, :Vpr/.l’~?iu(Cpo,!Cp,) I 1 i 2 us. AVPr.V~e. Here experimental values of iVxu were obtained by applying calculated corrections for radiant heat transfer. All data are recorded by Dickson ( 9 ) . If all heat transfer to the surface was by conduction and radiation from above, -nizE0 times the latent heat of vaporization of the liquid represents the sum of conductive and radiant heat flux. Radiant heat flux was calculated as net radiation between a “black” orifice at T , to a “gray” evaporating surface at To,using appropriate values for emissivity and view factor. The conductive heat flux was then calculated to be the observed flux corrected for radiant flux. A
I
epm)+
liquid Dissolution Experiments
T o obtain mass transfer data at large values of iVB, and at appreciable values of uio.studies were also made of the dissolution of organic liquids highly soluble in water ( 8 ) .
T h e test section and axisymmetrical flow configuration are shown in Figure 7. T h e diameter of the distilled water stream was 0.375 inch; the diameter of the transfer sector was 0.156 inch; the stagnation point was located approximately 0.375 inch from the water nozzle opening. As shown in Figure 7, the transfer surface was nearly that of a thin spherical sector confined by a ground-glass rim not wetted by organic liquid. T h e height of the sector was such that the diameter of the water jet \vas less than the diameter of the spherical surface and, being smaller, became the characteristic flow dimension. Dissolution rates were measured by mercury displacement of organic liquid from a 1-ml. micropipet when a leveling bulb was raised by a variable-speed motor drive. A cathetometer was used for viewing the interface and measuring the height of the spherical sector. I n test runs: water rate was varied to maintain a constant sector height at constant dissolution rate. Measured quantities during steady state were dissolution rate, water rate: and water temperature. Equilibrium concentrations, densities, and Saybolt viscosities of water and saturated aqueous solutions of the organic liquids were measured a t 25' and 30' C . in separate tests. Organic liquids employed were U.S.P. grade 2-butanone, methyl formate, and cyclopentanone. Since there was no appreciable heat of solution, the interface temperature was assumed to be equal to the measured water temperature. For correlating purposes values of Dt, were determined according to the recommended procedure (20). Concentration of diffusing component varied in the range 0.25 < ull0< 0.30; Schmidt number in the range 540 < .VS, < 1100; Reynolds number in the range 300 < < 2500.
r-
0
3-
4
2
lo3
Figure 3. data
6
8
IO4
Mass transfer rates from evaporation
