U = function defined in eq 21 u = velocity, ft/sec
IH = length of hypothetical duct, ft LH = dimensionless length of hypothetical duct
V = dimensionless velocity = v/ug0. V* = dimensionless velocity referring to the combination of given and hypothetical duct = u/vgo* z = axial coordinate, ft 2 = dimensionless axial coordinate
--z
3 coo* P g u g o - u 1 0 H T K ( ugo
y2
Greek Symbols /3 = half angle of divergence (see Figure l),deg
rn = mass flow rate, lbm/hr M = liquid to gas mass flow rate ratio = ml/m, p = local pressure, lbf/ft2 PT = local total pressure = p p g u g 2 / 2 g c , lbt /ft2 P = dimensionless pressure loss = gc(po - p)Y% pgugo2) PI = component of PT associated with liquid flow (see eq 14 and 15) PT = dimensionless total pressure loss = -R ~ .( P T O- P T ) /
+
(Y2 P g U g 0 2 )
P, = component of PT associated with wall friction (see eq 14, 16, and 17) P* = dimensionless pressure drop referring to the combination of given and hypothetical duct = gc(po - p ) / ( l h PdkO*Z.)
ro = radius (for circular duct) and half-width (for rectangular duct) at the start of a given duct (see Figure l), f t Ro = dimensionless radius, roll Re = drop Reynolds number = (ug - u i ) d / u S = dimensionless parameter characterizing slope of the given duct
p
= density, lbm/ft'
u
= kinematic viscosity of gas, ft2/hr
Subscripts g = ofthegas H = of the hypothetical duct 1 = oftheliquid 0. = at the start of duct T = total w = wall friction 1,2,3,4,5 = refers to duct numbers in Appendix B corresponding to Figure 9 Superscript * = applying to combination of given and hypothetical ducts
Literature Cited Boll, R. H., 1hd. Eng. Chem., Fundarn., 12, 40 (1973). Calvert, S., A.1.Ch.E. J., 16, 392 (1970). Dickinson, 0. R., Marshall, W. R. Jr., A.I.Ch.E. J., 14, 541 (1968). Gieseke, J. A,, Ph.D. Thesis, University of Washington, Seattle, Wash., 1963 ... Lapple, C. E., Shepherd, C. B.. lnd. Eng. Chern., 32,605 (1940). Nukiyama. S.,Tanasawa. Y.. Trans. SOC. Mech. Eng. (Tokyo), 4, 5, 6 (1938-1940), translated by E. Hope, Defence Research Board, Department of National Defence (Canada), Ottawa, Mar 18. 1950.
S* = S referring to the combination of given and hypothetical duct
Received for review December 10,1973 Accepted September 9,1974 T h i s w o r k was supported f i n a n c i a l l y by a grant a n d fellowship f r o m t h e X a t i o n a l Research C o u n c i l of Canada.
Interparticle Percolation: a Fundamental Solids Mixing Mechanism Andrew M. Scott and John Bridgwater* Department of Engineering Science, Oxford University. Oxford OX7 3 f J , England
One fundamental physical process in mixing cohesionless particulate materials is interparticle percolation, t h e drainage of smaller particles in the mixture through t h e larger ones, normally under t h e influence of gravity and strain. A simple shear cell has been developed to study strain-induced interparticle percolation under closely controlled conditions. T h e uniform deformation in t h e cell allows percolation rates to b e determined readily. T h e amount of percolation d e p e n d s mainly on total strain, t h e relative sizes of large and small particles, and also on rate of strain, even at low strain rates. It is believed that this is t h e first extensive information on this mixing mechanism. T h e results indicate t h e potential significance of interparticle percolation in understanding solids mixing and in those chemical reaction syst e m s in which t h e detailed distribution of particles is significant. Introduction The principal flow pattern in a deforming powder or granular material is that of convection of one block of material over another. In between these is a failure zone, 22
Ind. Eng. Chern., Fundarn., Vol. 14,
No. 1 , 1975
some 10-15 particle diameters wide, in which the strain rate is high. The ideas developed in critical state soil mechanics (Schofield and Wroth, 1968) suggest that a material may be in motion everywhere only if it is very loosely packed. Excluding the effects of interstitial fluid, the
modes of mixing for cohesionless materials may be classified as bulk convection, interparticle percolation, and surface effects. Interparticle percolation is size, shape, or density segregation in a failure zone. In part, percolation is determined by the gravitational force on particles and hence does not occur for cohesive materials. In practical terms, it may be important if particle size exceeds 30 j t ; it is commonplace if it exceeds 100 j t . Surface effects arise as particles may be mobile near a free surface, allowing either mixing or segregation to occur. However, fundamental studies of the mechanics of powder mixing that are both rational and quantitative have proved to be hard to make. Traditionally, many difficulties have beset the evaluation of mixture quality and the understanding of basic mechanisms. Experimental reproducibility has proved difficult to achieve. The spatial distribution of the components of a mix is hard to measure. The quality is usually assessed by withdrawing samples from a mixture using a thief probe and determining the variance of sample compositions. Empirically about 40 are desirable which makes the process laborious and it is often unclear whether the sampling is accurate. Goel (1973) has shown that thief probes can give serious systematic discrepancies. Frequently the sample size has not been reported, workers failing to appreciate its influence on variance. The spatial separation of samples, even if known, normally has not been used. It is known that random mixtures are not usually achieved and that, unless species to be blended are absolutely identical, mixture quality often deteriorates if the mixing time is too long. However, few fundamental generalizations can be made (Campbell and Bauer, 1966). The failure to develop a random structure may be attributed directly to interparticle percolation. The fall in mixture quality may be caused by the development by percolation of strong regions of preferred concentration that partially resist the mixing action (Bridgwater, 19711972). Segregation during pouring onto a free surface is related to percolation and has been investigated; see, for example, Tanaka (1971) and Lawrence and Beddow (19681969). Self-diffusive mixing in rotating horizontal drums can be satisfactorily analyzed (Hogg, et a1 , 1966). but if the particles are not all identical, radial or banded axial classification may occur (Donald and Roseman, 1962). Such behavior may be explained, though it has yet to be proved, in terms of a surface percolation process together with the formation of strong regions (Bridgwater, et al., 1969). Sugimoto (1967) points out that small and large particles may have utterly different residence time distributions in such a drum. While classification near free surfaces is well known, bulk percolation has received only bare recognition. Industrially important cohesionless materials include polymer chips, sodium carbonate, sand, cement, detergents, fertilizers, coal, mineral ores, and foodstuffs. In some processing applications the detailed structure of the material may be unimportant but in others an even one may be of crucial significance. Gas maldistribution is known to affect the performance of catalytic packed beds (Stanek and Szekely, 1972). This may arise from the uneven packing of the solids caused by surface percolation during the charging process (Craven, 1970). In any reactor system involving cohesionless particles, the particulate microstructure is relevant to fluid flow, heat transfer, and mass transfer. It seems reasonable to suppose that percolation may be particularly significant in hot spot or selectivity problems or in processes such as oxide ore roasting with coke in which both components of a mix are under-
Armngcmcnt of hemispheres on base-plate
Figure 1. Simple shear cell Mark I.
going reaction. Percolation may occur unexpectedly. For instance, when the inclination of a conveyor belt to the horizontal is changed, say by passing the belt over a roller, segregation of a well-mixed material may take place. The present work shows that interparticle percolation does indeed occur. In addition, the rate is sufficiently great that any basic approach to the mechanics of cohesionless powders must take account of it. It is shown how the phenomenon can be examined and understood in a test system which does not depend on laborious sampling. Experimental Procedure Campbell and Bridgwater (1973) established that strain-induced percolation took place in a failure zone developed adjacent to a roughened wall in a discharging hopper. Detailed information on the motion of particles at the front wall was found by analysis of cine film. In such equipment the total strain is limited, the velocity profile must be determined empirically, and the analysis of film is tedious. These problems can be avoided by examining percolation in a simple shear cell. In such a device a bed of particles is sheared so that its shape changes from a rectangle to a parallelogram. Every point in the bed undergoes the same shear strain and hence the percolation rate can be established by measuring the strain necessary for a tracer particle to cross the bed. Data collection is thus considerably simplified. Reversing the direction of shear allows unlimited total strain to be applied. The first cell, denoted Mark I, is shown in Figure 1. The lid rested on the cell contents and a normal stress was applied by weights suspended on hangers which were constrained to be perpendicular to the lid. Any rotation of the lid from the horizontal was thus opposed by a restoring couple. The bulk material was 18.6-mm diameter phenolic resin spheres and the cell thus measured approximately 13 x 13 x 13 particle diameters. In the very earliest version all the wall surfaces were planar. On shearing, the packing structure and the percolation rate both altered substantially as hexagonal close-packing developed. This was prevented by glueing 37-mm hemispheres to the top and bottom surfaces in a regular array (Figure l),chosen to disrupt close-packing of the 18.6-mm spheres and yet to prevent jamming in hollows or in the lower corners of the apparatus. In addition, holes cut in the base plate between hemispheres allowed percolating particles up to 11-mm diameter to pass out of the bed. Ind. Eng. Chem., Fundarn., Vol. 1 4 , No. 1, 1975
23
~,.~
Fiaure 3. Tvnical residence strain distribution for snherical oar" ~~
tid e s ; 172 individual readings.
Figure 2. Simple shear cell Mark IIA
".
l s t m .A mnra ti.n hsinhi nf l n m- o h n u n tho ".."mmtal ...""I.nr._""_ ..."-" n-,,-.In ~ n packing structure was thereby obtained immediately above the base plate which reduced the hold-up. With this modification the apparatus was designated Mark IIB. I
Denote the distance between the plane surfaces to which the large hemispheres are stuck by h, the cell material diameter by D, and the percolating particle diameter by d. In a typical experiment the cell was loaded with 18.6mm spheres, about 50 smaller spheres were put near the center of the top surface, and the lid was replaced. The cell was pushed to and fro by hand through a fixed angle until all the smaller particles had left the bed. The total strain when each particle left was noted. The strain per stroke and the cell height . could each be set for a given experiment. In some experiments the spherical percolating particles were replaced bv i r r e d a r l v shaped ones. In others, percolation through beds of irregular particles was examined, although this accentuated wedging in the corners of the bed. Here it was necessary to revert to planar top and bottom surfaces and also to incorporate flaps to exclude particles from the lower comers. In this case the position of percolating- particles after a given strain was found by . stripping the cell. The Mark I was suoerseded bv the Mark IIA (Fiaure 2). A variable-speed motor drove the cell via a slider and variable-throw crank. The lid was free to rise and fall vertically, but was held horizontal by parallel arms. The same hulk packing was used and the apparatus measured some 20 X 20 X 13 particle diameters. The larger area reduced the probability of a percolating particle hitting a side or end wall. The base was identical with that in the Mark I. The residence time of a single particle in the bed was recorded automatically. On entering through a hole in the lid the particle cut a light beam, thereby starting a counter-timer, and on leaving it fell onto a metal plate to which a microphone was attached, thereby stopping the counter-timer. A data-logger punched this time onto paper tape. The percolating particle was pneumatically conveyed back to the top of the cell, where a trap door allowed it to re-enter the bed at a predetermined point in a cycle. Some problems were experienced with the base plate. Careful maintenance was needed to avoid both wedging of 18.6-mm spheres in the comers and the hold-up of percolating spheres. The second problem was alleviated by fitting metal pins which stuck through the center of each hole I
~
24
Ind. Eng. Chem., Fundam., Vol. 14, No. 1,1975
l"
. . . . I1
"I"._
Results and Discussion The experiments fell into two categories: those to ascer. . - n n A : + k n c nnnlrl hr ...vyIu tain whether changes in nnll n---o+:nrationally interpreted and those tn invoctioate nermlat.inn as a function of particle properties The strain to cause a particle to percolate from the top to the bottom of the bed was Ineasured repeatedly. The strain, which is analogous to the residence time of a tracer in flow systems, was expressed a1s the tangent of the total angle, irrespective of direction, t hrough which the cell had passed. The residence strain wa s thus N y . where N was . . the residence strain in strokes, ana ye, the strain per stroke, was equal to 2 tan a,a being the inclination of the end wall to the vertical at the extreme position. A typical residence strain distribution based on 172 individual readings is shown in Figure 3. In general, results were reproducible. The hypothesis that the bed deformed uniformly was tested by measuring the mean and variance of the residence strain as a function of bed height, h. Typical results are shown in Figure 4. The mean residence strain is proportional to h, confirming the hypothesis. Despite the well-known difficulty of estimating variances accurately, these also appear to he proportional to h. The intercepts arise because h includes the height occupied by the hemispheres on the top and bottom surfaces where there is little resistance to percolation. In addition the layers of spheres near the hemispheres may offer a different resistance to percolation. The intercept varied from cell to cell hut was substantially constant for various percolating species in a given cell. The existence of an intercept implies that percolation rate should he estimated from the slope of a graph such as Figure 4 and this was the procedure --. adnnt.nd. For given =..-. . .. ~ ..~~ . onerntine conditions. ~.results from the three cells are seen to be consistent. As the Mark II was I arger than the Mark I and as the base plates of thc? Mark;s IIA and IIB were different, it is deduced that wal 1 .I :(..,,acuca,l, men.+ ...sla Im.ll +hot +ho normlatiny effect,, wClr ybIbv.y..v.~ rates obtained were general for the materials studied. It might be thought that the mean residence strain cell vpcLar.Aag
IYIIYIYIVI.U
.
~
,.
r
ll..y.l
.
~
~~~~~~~
~
lllyl
~
~
~
~
~
~
O
60 ‘dead movement* model
0
I
1
I
I
Figure stroke.
percolatingKey.
I
I
bulk
partfclcs
particles
o spheres
x brwd beans
spheres gravel spheres spheres
A grovel
grovel
spheres
+ gravel
S S C MkIlB
d/D = O 322 h = IL5mm
I
y,=zo
Rate of Slrain
-
(s-‘)
Figure 5. The dependence of mean residence strain on mean rate of strain.
would be independent of the mean rate of strain, $, at low rates of strain, but this was not the case, (Figure 5). As the mean rate of strain rose from 0.061 sec-1 to 0.62 sec-1, the greatest range available in the present cells, the residence strain increased by 55%. The reasons for this change and the remarkable linearity found are not clear. In general the mean residence strain depended on strain per stroke, being least at greatest strain per stroke (Figure 6). Also, slight ordering of the packing could be detected if the strain per stroke were smaller than 0.6. This is a potentially serious objection to the use of a simple shear cell. However, it was noted that the bed consolidated immediately after the direction of shearing was reversed, a minimum bed volume occurring at a strain of about 0.24. With further strain the bed dilated. Therefore it was suggested that after each reversal there was a certain fixed strain, yf,which did not cause any percolation. Thus NV(.~, - .it)
=
r
(1)
where r is the total strain effective in promoting percolation. Hence
r -
= 2 tan
CY
- yf
(2)
L ‘\
Plotting 1/N against tan a gave a straight line with intercept 0.12, whence yf was 0.24 (standard error 0.04). The simple model thus accounts for the observed variation of mean residence strain with strain per stroke, and the consolidating strain and the “dead movement” per stroke yf were approximately equal.
Eqwalent M l u m Diameter Ratio, d/D -+
Figure 7 . Corrected dimensionless percolation velocity u / T D for various percolating and bulk particles as a function of equivalent volume diameter ratio. For spheres-spheres 5 = 0.215 sec-l. For other systems, examined in the Mark I, 4 was approximately the same.
A number of experiments in the Mark I showed that if the normal stress was varied from 3.2 X 10-3 N mm-2 to 5.4 X 10-3 N mm-2, there was little influence on percolation. On varying the percolating particles it was found that by far the most important parameter was the relative size of large and small spheres, expressed as the diameter ratio, d / D . The percolation rate decreased markedly as d / D increased, the mean strain required for a percolating particle to drop a distance equal to one bulk particle diameter being 0.51 when d / D = 0.32 and 19 when d / D = 0.75. From the geometry of close-packed spheres, a small sphere should be able to percolate spontaneously if d / D < 0.155, but it was observed that for d / D < 0.25 the small spheres often dropped a considerable distance once in motion. A dimensionless percolation velocity is defined as the number of bulk particle diameters moved per unit of strain, corrected for “dead movement.” This is plotted against d / D in Figure 7 . For nonspherical particles, equivalent volume diameters are used. In all cases the bulk material was closely sized. The percolation rates for all systems were of the same order at a given volume diameter ratio. The broad beans had principal axes of lengths approximately 1:2:3 and percolated slightly faster than more compact particles of the same volume, but this shape effect is small. The gravel had no preferred orientation. Ind. Eng. Chem., Fundam., Vol. 14, No. 1, 1975
25
t
small percolating particles often dropped spontaneously more than one packing diameter. The dependence of percolation rate on relative size can be modeled by using the diffusion coefficient of the bulk material. Suppose that before a spherical percolating particle can drop relative to its nearest neighbours, one of the bulk balls must undergo a displacement, Ax, in a specified direction of approximately (d - dsD,where ds is the diameter of the largest particle which can just percolate spontaneously. The time, At, for this movement is approximately D/u where u is the mean percolation velocity, corrected for “dead movement.” Substituting in Einstein’s (1926) one-dimensional diffusion equation
9. I_
r
S S C MkIIB i0215 5.’ y,= 2
0
-
0
0 05
-
0 10 ldlD-015512
-
0 15
Figure 8. Test of model predicting percolation rate for spherical particles.
AX‘ = 2 E A t
where E is the diffusion coefficient of bulk balls gives 2ED N
-
Accurate estimates of percolation rate have not yet been made for d/D > 0.5. However, experiments in an annular shear cell showed that percolation caused segregation of closely sized polyethylene chips. After prolonged strain the equivalent volume diameter of material from the bottom of the bed was 0.985 that of material at the top, some ten particles away. Although the main work is restricted to systems where a few small particles percolate through closely sized larger granules, this particular observation suggests that percolation also occurs in materials having a continuous size distribution. Examination of this is now in hand. Density effects were examined using hollow and filled 9.5-mm diameter aluminum spheres as percolating particles. The filling increased the density from 0.74 to 7.66 g cm - 3 . The corresponding increase in measured dimensionless percolation velocity was from 0.516 to 0.647, but this change was not statistically significant. The accuracy of estimation of the mean residence strain was determined mainly by the distribution of residence strains. Errors in individual measurements were very small. Percolation velocities were usually based on 300 readings of residence strain at each of five different bed heights, enabling the rate to be estimated with a standard error of 8%. Differences of less than about 20% in percolation rate are thus not significant at the 95% confidence level. The variance of the residence strain was more difficult to estimate accurately as it was very sensitive to occasional high readings caused by hold-up of the percolating particle on the base plate. If more accurate experiments were performed, effects due to particulate density or normal stress might be detected. Percolation would still be noticed if a tracer particle were identical with the bulk particles, as the former would eventually reach the base by chance. Other work in the simple shear cell established that movement of the bulk particles conformed to a diffusive mechanism and that the apparent dimensionless percolation velocity of such a particle would be 0.012. The percolation velocities found were much higher than this (Figure 7 ) , implying that the smaller particle moved systematically downward. Percolation could be modelled either by a series of equal stirred tanks or by plug flow with axial dispersion. In neither case is the fit to a residence strain distribution sufficiently good to suggest that these were entirely satisfactory. However, if d/D 0.5 the best-fit stirred-tanks model had approximately one tank for each bulk particle diameter descended. When d/D 31 0.3 this dropped to one tank for every three or four bulk particle diameters, which supports the empirical observation that, once in motion, 26
Ind. Eng. Chem.,
Fundam., Vol. 14, No. 1, 1975
(d - d,)‘
U
(4)
For close-packed spheres ds/D = 0.155 and hence (5) where 4is the mean rate of strain. DT/u is plotted as a function of (d/D - 0.155)2 in Figure 8 for d/D < 0.5. The functional dependence on d/D is in accordance with eq 5, but it underestimates residence strain by a factor of about 2. Conclusion It has been shown that when a noncohesive granular material is strained, a smaller particle will move downward. The rate of percolation has been measured by means of a simple shear cell. This develops a uniform strain which enables the rate to be found without recourse to exhaustive sampling or cine photography. Some of the more obvious variables have been examined. For the materials examined so far the amount of percolation depends on the strain of the material, the relative volume of percolating and bulk particles and, to a lesser extent, on the rate of strain, even at low rates of strain. The relative density of percolating and bulk particles appears to be unimportant and at low normal stresses increasing the normal stress by a factor of 2 had little effect on percolation. If the percolation rate is expressed as a function of strain, it is determined primarily by the relative volumes of bulk and percolating particles. If these particles are sufficiently different in size or if the total strain is sufficiently great with particles of very similar sizes, interparticle percolation can produce a significant redistribution of components of a mixture of .cohesionless particles. Any model of a chemical reactor using cohesionless particles should take account of the phenomenon. Nomenclature d = diameter of spherical percolating particle or equivalent volume diameter of nonspherical percolating particle, L ds = diameter of spherical particle which percolates spontaneously, L D = diameter of spherical bulk particles or equivalent volume diameter of nonspherical bulk particles, L E = self-diffusion coefficient of bulk particles, L2T-I h = bed height measured between plane surfaces in simple shear cell, L N = mean residence strain of percolating particle expressed as a number of strokes of the simple shear cell t = time. T
u = mean vertical percolation velocity of a tracer particle which would be observed if there were no dead movement, LT-I x = horizontal coordinate, L
Greek Letters a = inclination of end wall of simple shear cell to vertical - a t extreme position i. = mean rate of strain of material in simple shear cell, T-1 yf = "dead movement," i.e., strain per stroke which does not cause percolation ys = strain of bulk material per stroke of the simple shear cell I' = effective mean residence strain of percolating particles, i.e., measured mean residence strain minus dead movement
Literature Cited Bridgwater, J., Powder Technol., 5, 257 (1971/1972). Bridgwater. J., Sharpe, N. W., Stocker, D. C., Trans. lnst. Chem. Eng., 47, T114 (1969).
Campbell, A . P., Bridgwater, J.. Trans. lnst. Chem. Eng., 51, 72 (1973). Campbell, H., Bauer, W. C., Chem. Eng. (New York), 179 (Sept 12, 1966), Craven, P., Brit. Chem. Eng., 15, 916 (1970) Donald, M. B., Roseman, B., Brit. Chem. Eng., 7, 749, 823, 922 (1962). Einstein, A., "Investigations on the theory of the Brownian Movement," Furth, R . , Ed., Cowper, A. D., Translator, Methuen. London, 1926. Goel, A. K . , Ph.D. Thesis. University of Bradford, Bradford. England, 1973. Hogg, R., Cahn, D. S., Healy, T. W., Fuerstenau, D. W., Chem. Eng. Sci., 21, 1025 (1966). Lawrence, L. R., Beddow, J . K., Powder Technol., 2,253 (1968/1969). Schofield, A. N., Wroth. C. P., "Critical State Soil Mechanics," p 109, McGraw-Hill, London, 1968. Stanek, V . , Szekely, J . , Can. J. Chem. Eng., 50, 9 (1972). Sugimoto, M., Kagaku Kogaku, 31, 145 (1967). Tanaka, T.. lnd. Eng. Chem., Process Des. Develop., 10, 332 (1971).
Received for reuieu, January 28, 1974 Accepted October 1, 1974 The work is supported by an S.R.C. research grant. Thanks are also due to the S.R.C. for a research studentship (A.M.S.) and to the Esso Petroleum Company Limited (UK) for a Senior Kesearch Fellowship in Chemical Engineering a t Hertford College, Oxford (J.B.).
Interfacial Temperatures and Evaporation Coefficients with Jet Tensimetry E. James Davis,* Randolph Chang, and B. Damian Pethica Department of Chemical Engineering, Clarkson College of Technology, Potsdam, New York 73676
The problem of interfacial evaporation from a liquid jet exposed to a vacuum is analyzed to predict interfacial temperatures and evaporation coefficients. The problem is formulated both in terms of integral equations for the general case and for the approximation of small penetration. When the surface temperature does not deviate greatly from the initial temperatu1e of the fluid jet an analytical solution is obtained. In.this case the average evaporation coefficient, €, over the length of exposed surface is obtained explicitly in terms of the average flux of mass from the jet. Experimental data of Maa have been reexamined and evaporation coefficients have been recalculated using this analysis to show that the evaporation coefficient is approximately unity for all of the single-component chemical species studied.
The jet tensimeter, which consists of a carefully designed nozzle, a laminar liquid jet, and a receiver for the jet, has been used by Hickman (1965) and Maa (1967,1970) to measure evaporation coefficients of pure components (water, carbon tetrachloride, toluene, and isopropyl alcohol) and solutions thereby partly resolving controversy over the magnitude of the Coefficient. The advantage of the device is that a fresh liquid surface is exposed to its vapor, thus avoiding problems of surface contamination that occur in static systems. Maa (1969a,1969b) and Maa and Hickman (1972) also used the jet tensimeter to study condensation of vapors on liquids of the same and different species. To use the jet tensimeter as a tool for the quantitative measure of evaporation characteristics it is necessary to determine the interfacial temperature of the jet, for the vapor pressure of the evaporating (condensing) species is a strong function of the interfacial temperature. Maa and Hickman used an approximate iterative technique to estimate the surface temperature of the jet. The technique involved the introduction of an initially unknown heat transfer coefficient to describe the interfacial boundary condition. To solve the governing equations the coefficient was assumed to be constant (independent of
axial position) at an appropriate average value, and iteration was used to determine the heat transfer coefficient. It is the purpose of this paper to show that the interfacial temperature can be predicted in a rigorous manner without recourse to iterative procedures. Furthermore, the analysis leads to an analytical solution for the interfacial temperature as a function of axial position (or time) when the temperature difference between the inlet temperature and the surface temperature is not too large. The evaporation coefficient can also be obtained implicitly and, under some conditions, explicitly in terms of the measured average mass flux. Problem Formulation The system under consideration and its coordinates are shown in Figure 1. A laminar jet of radius R (where R is nearly constant because of the small amount of evaporation or condensation encountered in experiments) is formed by means of a shielded nozzle. The shield isolates the liquid jet from its vapor in the region of the developing flow. In this way a nearly uniform velocity distribution is attained a t the plane x = 0 where the liquid surface first contacts the surrounding vapor. The temperature field in the laminar jet is governed by Ind. Eng. Chern., Fundam., Vol. 14, No. 1, 1975
27
