I
S. S. KWONG' and J. M. SMITH2 Purdue University, Lafayette, Ind.
Radial Heat Transfer in Packed Beds A new method of evaluating point thermal conductivities in a packed bed has been used to obtain data for different gases and solid particles
RADIAL
heat transfer rates in packed beds are important in the design of solid catalyst reactors and packed-bed heat exchangers. In 1931 Colburn (4) measured the heat transferred to air flowing through a tube packed with solid particles. His results were reported in terms of heat transfer coefficients between the air flowing through the bed and the tube surface. As interest developed in the mechanism of heat transfer within the bed, the concept of an effective thermal conductivity, k,, was used to represent radial heat transfer rates. This quantity has been calculated mathematically by supposing the transfer of energy to be entirely by molecular conduction. Recognition of the real situation is then introduced by considering the effective thermal conductivity to be made up of contributions from each of the actual mechanisms by which heat is transferred. Singer and Wilhelm (72) have suggested a method for evaluating the contributions of several of the mechanisms. Argo and Smith (2) have divided the total process into a set of mechanisms operating in series and parallel and proposed a theory for combining the individual contributions. The resulting equations permit calculation of the effective thermal conductivity from a knowledge of the fundamental characteristics of the system-that is, the void fraction, certain physical properties of the fluid and the solid particles, the diameters of the tube and the particle, and the mass velocity of the fluid in the bed. The application of the theory requires 1 Present address, C. F. Braun Co., Alhambra, Calif. 2 Present address, University of New Hampshire, Durham, N. H.
894
data on the heat transfer coefficient between particle and fluid and on the mass transfer rate in the fluid phase. If point values of k, are to be obtained, these data and the void fraction and mass velocity must be known at all radial positions in the bed. In the past the problem of obtaining temperature profiles across the packed bed and interpreting the measurements in terms of point values of the effective thermal conductivity to compare with the theory has been laborious. Extensive graphical differentiation was required, introducing uncertainties in the results. One objective of the present work was to develop a new method of obtaining k, which required less experimental data and substituted numerical integration techniques for graphical differentiation. The labor of the integrations can be reduced by employing computer facilities. Argo and Smith tested the theory with data for two particle sizes in a single pipe. A second objective of this work was to obtain additional k, data by the new calculational procedures for different gases and solid particles for comparison with the predicted results. Range of Experimental Data
Temperature measurements were made in the 2-inch tube with air and with ammonia with 0.25-inch steel packing and 0.25- and 3/s-inch alumina packing plus a run with air with 5/32inch steel packing; in the 4-inch tube measurements were made in air with 5 / 3 1 and 0.25-inch steel packing and 0.25inch alumina packing. In a11 cases the particles were spherical in shape. The thermal conductivity could be computed
INDUSTRIAL AND ENGINEERING CHEMISTRY
for each indicated combination of gas, packing material, packing size, and pipe size. The average mass velocity of gas was varied from 150 to 1200 lb./(hr.) (sq. ft.) for the air runs and from 150 to 650 lb./(hr.) (sq. ft.) for ammonia. Procedure
The apparatus and experimental procedure used were similar to those employed by Argo (7). Temperatures were measured across the diameter of the bed at two depths of packing, 0 and 4 inches, by inserting thermocouples in the axial direction into the bed. In this way, bare thermocouple junctions were exposed in the gas stream at a series of nine radial positions across the entire diameter of the bed. A schematic diagram of the entire apparatus is shown in Figure I, and a more detailed illustration of the thermocouples in the packed bed is provided in Figure 2. Operation and construction of this type of equipment are described by Argo and Smith (2) and Schuler and others ( 9 ) . Only the more important experimental problems are discussed here. It is important to reduce to a minimum the disturbances to the flow caused by the entrance and exit geometrical arrangements. At the entrance the zero depth reading was taken at an actual depth of approximately 7 inches of packing. The second temperature profile was measured 4 inches higher, and beyond this point the bed was packed for an additional 3 inches. The thermocouple junction extended approximately 0.25 inch beyond the I/s-inch insulated tube containing the thermocouple wires.
It was necessary to dismantle the bed after the zero-bed-depth measurements and replace the thermocouples at the 4inch level; otherwise the thermocouple tubes for the zero bed depth measurements would remain in the 4-inch section and affect the flow and temperature distributions. By using the two control thermocouples indicated in Figure 2 it was possible to reproduce the zero-bed-depth temperature profile after reassembling the bed for the 4-inch measurements. Readings of these couples and the lead bath temperature were adjusted to the same values for the zero and 4-inch bed depth measurements. The inlet gas temperature at the center of the tube was fixed at 200' C. for the zero-bed-depth condition. The pipe wall was maintained at about looo C. by boiling water in the jacket surrounding the pipe. Actual wall temperatures were measured with copper-Constantan couples peened into the metal of the wall. The thermocouples in the bed were Chromel-Alumel. All were calibrated by comparison with a platinumplatinum-rhodium thermocouple supplied by the National Bureau of Standards. Temperatures were recorded to 0.1 'C. However, the greatest error in the combined temperature-position measurements was in locating the radial distance of the thermocouple from the tube wall. The measurement was made by making impressions of the thermocouple junctions on a circular piece of modeling clay mounted on a special holder that could be inserted into the unpacked tube. The assembly of thermocouples was held together as a unit by spacers in the upper, unpacked part of the tube. The resulting temperature traverses (8) formed parabolic-type curves. As the new computation method employed to compute the effective thermal conductivity did not require differentiating the data, it was not necessary to prepare temperature-profile graphs.
Calculation of Effective Thermal Conductivity
Employing the concept that all the heat is transferred radially by conduction, the equation defining k. is as follows :
where T = t - t , = gas temperature in wall temperature. bed The equation is based upon the gas temperature in the bed. Small differences are known to exist between the
-
temperatures of the gas and solid particles. However, it is convenient to base the thermal conductivity upon the gas temperature, as it is easy to measure. The term for axial transfer of heat by diffusion has been neglected because of the small temperature gradient in the direction of flow. Equation 1 does take into account the variation of k, with radial position and permits the utilization of a variable mass velocity-two phenomena known to exist. The derivation of the equation is given in detail by Schuler (9), Stallings, and Smith. I n previous work Equation 1 has been solved for k, as a function of radial position by graphically differentiating the temperature data to obtain the partial derivatives. For the second derivative this is a hazardous procedure. Furthermore, it is necessary to measure temperature profiles for at least four bed depths to determine satisfactory values of bt/bz. Two new schemes, which require data at but two bed depths, are based upon integrating the differential equation by numerical methods. The boundary conditions of Equation 1, when the tube wall is maintained at a constant temperature (the experimental condition), are :
To illustrate the application of Equation 3, the right-hand side of the equation did not exceed 0.5 in this study. Then if k, values are desired at five radial positions, Ar/ro = 0.2 and Az/L must not exceed 0.02. In other words, 50 axial increments were required in order to integrate over the total bed depth of 4 inches. Numerical Method A. Another approach to Equation 1 is to postulate that the solution can be represented as the product of two functions, R and 2, where R is a function of radial distance only and Z depends only upon bed depth z. I n terms of the eigen values Xi, the solution may be written:
(4) The differential equation expressing the variation of R with r may be written in the form of a Sturm-Louiville problem : (rk.
$) + Xc,GrR
= 0
(5)
If the bed depth is large, only the first term in Equation 4 is significant. This makes it possible to evaluate XI by numerically integrating the expression
T = f ( r ) : e = 0 (zero-beddepth profile)
Numerical Method B. This method is simple in principle but requires extensive calculations because of its trial and error feature. A radial distribution of k, values is assumed. Using the three boundary conditions, Equation 1 can be integrated numerically to yield the temperature profile at the bed depth where the data are available. I n the present study this bed depth was 4 inches. Comparison of the calculated and experimental temperatures at this depth suggests modifications in the assumed distribution of thermal conductivities. Continuing the process leads to a set of k, values consistent with the measured temperatures and Equation 1. In such stepwise integration procedures attention must be directed to the propagation of errors from one step (bed-depth increment) to the next. To avoid this effect it is necessary to operate with increments no larger than the critical value. For the present application the critical values of increments Az and Ar must be such that the following restriction is satisfied :
where TO= temperature at any radius at zero bed depth and T , = temperature at any radius at bed depth L. T o apply Equation 6 it is only necessary to know the temperature profile across the radius of the tube at two bed depths. Once XI is available, k, at any radial position is obtainable from Equations 1 and 4. If Equation 4 is differentiated with respect to bed depth at a constant radial position, and if again only the first term of the series is important,
At bed depth L, this expression becomes
If the solution for T given by Equation 4, is substituted in the original differential equation (Equation 1) there results G,,
Gr az =
(rk.
g)
(9)
Combining Equations 8 and 9 and integrating with respect to r at bed depth L yields rks
(g)
VOL.49,
= -k~Lc,Gr TLdr
NO. 5
MAY 1957
(10)
895
I
"3
CYAINDERS
lowing terms in Equation 4 wiil be serious. If L is too large, the temperature profile is so flat that accurate values of the temperature differences used in the difference equations substituted for Equations 6 and 11 cannot be obtained. After some study, a depth of 4 inches was decided upon as optimum, and this determined the bed depth for the second set of temperature measurements. The total error involved in Method A can be judged by applying the Bessel test. If the mass velocity and effective thermal conductivity are both supposed t o be constant with respect to r, the solution of Equation 1 is the summation of products of Bessel functions (first order, first kind) and e--xiz, The Bessel test may be applied in the following way:
I
AIR FILTER
AIR
Figure 1 .
Apparatus for measuring radial temperature profiles
Equation 31 may be employed to calculate k , a t any radius r by numerical integration to that radius. There is an optimum choice for bed depth L in applying Equation 11. If L is too small, the error involved in neglecting the second, third, and fol-
This equation may be solved for k, as follows: -
c p G r T L dr
(11)
k, =
HEAD
iI
11
1. Using the experimental temperature profile a t zero bed depth, and an assumed value of k,, the temperature profile at depth L can be evaluated from the analytical solution of Equation 1 in the form of Bessei functions. 2. Using these temperatures at L computed in step 1, Method A may be employed to calculate the fhermal conductivity at each radial position. 'The values of k, so determined should be the same and equal to those assumed in the first step.
Figure 3 shows the results of this test. The conductivities obtained by Method A are in excellent agreement with the assumed result except a t radial position 0.1. Analysis of the method indicates that the largest errors are to be expected a t the center of the bed. Comparison of Methods. T o compare the methods, the temperature data for 0.25-inch alumina spheres in a 2inch pipe with air flow at G = 300 Ib./ (hr.) (sq. ft,) were used to compute the effective thermal conductivity (Figure
4). The agreement is good for this type of computation, w-ith the greatest deviation near the center of the tube. It is
DEPTH SPACER
4
Figure 2.
Integral reactor Figure 3. Bessel's equation test of numerical Method A
Assumed CONSTANTA COUPLES
l
El
FLANGE INDUSTRIAL AND ENGINEERING CHEMISTRY
JACKET
3
THERMOCOUPLE^ 896
+. v-
0.38
k e = 0.36Btu/ hr. ft. "F
HEAT TRANSFER IN PACKED B ~ D S
Stallings' data ( 4 5 ) P e l l e t : 1/4" cylinders
0.71
0.7
0
0.6
0
Numerical method A Graphical method
0.5 LL'
Pellet : 1/4" diameter alumina sphere P i p e size: 2 " diameter
0.4 i
c \
Gas: A i r
3 0.3
m
.
y" 0.3
300 Ib: ./hr. ft.'
G
a,
Y
0.2
Method A Method B
0
* 0.1
Run No. : 2A-C-1/4-300 1
C
t
0.2
1
I
I
0.4
I
0.6
1
t
0.8
1
1 I
t- /ro Figure 4.
Comparison of numerical Methods A and B
believed that Method A is in error at this point, because both the numerator and denominator of Equation 11 are approaching zero; hence the equation is very sensitive to errors in temperature measurements near the center of the bed. This same effect was illustrated by the Bessel test (Figure 3). Aside from this disadvantage, Method A is preferred. It involves no trial and error procedure and hence the programing for the computer is easier and simpler. In this work an IBM card-programed calculator was used, and the time required to obtain one set of thermal conductivity values, such as shown in Figure 4, was 3 minutes. In contrast, Method B necessitated about 25 minutes of computer time per trial. It is also of interest to compare Method A with the graphical procedure. Figure 5 illustrates the results obtained by the two methods for air flowing through a bed packed with 0.25-inch cylindrical packing in a 2-inch pipe. These conditions were chosen because Schuler and coworkers (9) had measured temperatures for this bed of cylindrical particles at a sufficient number of bed depths to apply the graphical scheme. While the curves are in reasonably good agreement, Method A results are judged to be most accurate because of the differentiation errors inherent in the graphical method.
0
0.2
0.4
r/ro
0.8
0.6
I 3
Figure 5. Comparison of graphical method with numerical Method A
The values at T / Y O = 0.1 for Method A are out of line with those for other bed depths, even though they are in better agreement with the results of the graphical solution. This is consistent with the conclusions reached earlier in comparing Methods A and B.
A and C, k, values were also computed by numerical Method B. These results are shown at 0.2 increments of T / Y O in the last column.
Results
The effect of packing depth on transport properties in packed beds has long been an uncertainty. After a depth is reached such that the flow and temperature profiles are well established, presumably there should be no further effect. However, for shallow beds heat and mass transfer rates would be expected to be a function of the depth of packing. This problem was not studied thoroughly in this work, but temperature measurements were obtained for 0.25inch cylindrical pellets at a prepacking of 1 inch-that is, the nominal zero bed depth was located after the gas had passed through 1 inch of packing. The results for these conditions are compared in Figure 6 with those for the same packing at a prepacking of 7 inches as obtained by Stallings (73). The agreement is of the same degree as the accuracy of the experimental method and calculational procedures (see Figures 4 and 5 ) . Hence, it is concluded that for this arrangement of packing, pipe, and entrance temperature conditions, 1 inch
Using the experimental temperature measurements, k, values as a function of radial position were computed for each of the conditions given in Table I by Method A. The computations required a knowledge of the mass velocity and void fraction profiles for each particle and tube size. This information was obtained from the velocity measurements of Schwartz and Smith (70) at isothermal conditions. As the effect of temperature on the velocity profiles was not known, it was assumed that the mass velocity was independent of temperature. The results of the calculations are shown in Table I. I n addition to the effective thermal conductivity, values of the temperature read from smoothed curves and the modified Peclet number
are included.
In two cases, Table I,
Effects of Prepacking and Temperature level
VOL. 49, NO. 5
e
MAY 1957
897
Table 1. T/ro
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Point Effective Thermal Conductivity and Modified Peclet Number for Heat Transfer
t(0. n) t(L, n) G ks Pe; kea A. Air ; 6/sz-Inch Steel Spheres; 2-Inch Pipe G = 150 Ib./&r.) (sq. it.) 198.3 195.3 190.4 184.0 176.2 167.3 157.9 147.5 134.0
122.6 122.1 121.4 120.6 119.8 118.6 117.3 114.2 109.0
124.5 125.4 128.7 138.8 154.7 168.3 179.1 186.8 160.4
0.282 0.324 0.392 0.484 0.509 0.502 0.344 0.204 0.125
1.40 1.22 1.04 0.91 0.96 1.06 1.63 2.91 4.01
0.314 0.474
0.194
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
198.9 195.8 191.0 184.5 176.6 167.9 158.4 148.4 135.3
135.6 134.4 133.1 131.2 129.4 127.2 124.4 119.8 112.1
249.0 250.8 257.4 277.5 309.3 336.6 358.2 373.5 320.7
0.271 0.320 0.382 0.458 0.538 0.527 0.417 0.282 0.183
155.4 140.7
G = 750 lb./(hr.) (sq. ft.) 622.5 0.476 157.5 627.0 0.503 155.8 643.5 0.545 153.5 693.8 0.597 150.3 146.6 773.3 0.678 841.5 0.772 142.7 895.5 0.682 138.5 933.8 0.438 132.0 801.8 0.268 119.8
199.2 197.4 194.6 190.5 185.8 179.0 170.2 159.7 145.6
G = 1200 lb./(hr.) 996.0 166.8 1,003.2 164.9 162.2 1,029.6 1,110.0 158.4 153.9 1,237.2 149.2 1,346.4 144.0 1,432.8 1,494.0 137.0 1,282.8 123.8
199.2 197.0 193.2 188.3 182.3 175.0
166.0
(sq. ft.) 0.605 0.648 0.701 0.770 0.864 0.967 0.914 0.600 0.358
2.90 2.48 2.13 1.92 1.82 2.02 2.72 4.19 5.53 4.16 3.96 3.75 3.69 3.60 3.44 4.15 6.74 9.41
5.22 4.91 4.66 4.58 4.54 4.40 4.95 7.87 11.27
B.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Air ; 0.25-Inch Steel Spheres ; 2-Inch Pipe G = 150 lb./(hr.) (sq. it.) 2.05 129.6 0.320 198.9 121.9 1.85 132.0 0.362 196.6 121.4 1.65 137.1 0.421 192.5 120.8 0.493 1.50 145.7 187.2 119.9 0.635 1.28 160.7 180.6 119.1 1.32 172.1 0.660 173.3 118.2 2.84 179.9 0.320 164.5 117.1 0.179 5.11 181.8 154.7 113.2 5.62 153.0 0.137 143.0 107.2
G = 300 lb./(hr.) (sq. it.) 0.1 0.2 0.3 0.4 0.5 0.6
0.7 0.8 0.9
198.9 196.8 193.0 188.0 182.0 175.1 167.5 158.6 146.9
135.8 135.1 134.0 132.4 130.7 128.8 126.7 121.8 112.5
259.2 264.0 274.2 291.3 321.3 344.1 359.7 363.6 306.0
0.463 0.494 0.534 0.584 0.679 0.746 0.501 0.279 0.187
2.83 2.70 2.60 2.52 2.39 2.33 3.63 6.58 8.24
G = 750 lb./(hr.) (sq. ft.) 0. I 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
199.0 197.0 193.5 189.1 183.5 177.4
171.0 164.0 150.7
156.5 155.4 153.4 151.0 148.0 144.7 140.9 133.9 119.9
648.0 660.0 685.5 728.3 803.3 860.3 899.3 909.0 765.0
0.630 0.684 0.751 0.827 0.901 0.971 0.748 0.431 0.277
5.22 4.90 4.63 4.48 4.51 4.48 6.08 10.65 13.90
G = 1200 lb./(hr.) (sq. ft.) 0.1 0.2
898
199.0 197.0
164.0 162.8
1,036.8 1,056.0
0.902 0.967
5.84 5.52
INDUSTRIAL AND ENGINEERING CHEMISTRY
0.320 0.458
0.530 0.587 0.762
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
197.7 195.5 192.1 187.3 181.2 174.6 167.9 159.6 145.1
0.755 0.945
1.050 1.148 1.260 1.251 0.971 0.631 0.348
5.30 5.16 5.18 5.56 7.49 11.65 17.79
125.2 124.8 124.2 123.4 122.4 121.4 120.3 117.4 110.0
132.8 136.5 147.8 150.3 164.7 175.5 180.5 174.0 145.2
0.413 0.439 0.472 0.517 0.590 0.686 0.440 0.188 0.114
1.96 1.89 1.90 1.77 1.70 1.56 2.49 5.60 7.71
0.377 0.453 0.483 0.189
G = 300 lb./(hr.) (sq. ft.)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
199.2 197.4 194.2 189.6 184.5 179.0 172.7 164.5 148.5
139.8 138.9 137.8 136.1 134.2 132.0 129.5 125.2 115.7
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
198.4 196.1 192.9 188.9 184.4 179.9 174.1 166.9 153.0
156.1 155.0 153.5 151.6 149.0 146.1 142.5 136.7 124.4
265.5 273.0 295.5 300.6 329.4 351.0 360.9 348.0 290.4
0.456 0.489 0.534 0.585 0.641 0.678 0.544 0.301 0.171
3.54 3.32 3.36 3.12 3.12 3.15 4.11 7.02 10.28
0.490 0.557 0.641 0.275
G = 750 lb./(hr.) (sq. ft.)
0.421
0.699
1,096.8 1,165.2 1,285.2 1,376.4 1,438.8 1,454.4 1,224.0
G = 150 lb./(hr.) (sq. ft.)
0.517 0.275
160.8 158.3 155.2 151.8 147.2 139.7 126.3
C. Air; 0.25-Inch Alumina Spheres; 2-Inch Pipe
0.492
G = 300 lb./(hr.) (sq. ft.) 0. I 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
194.0 189.8 184.8 179.0 173.1 166.2 154.5
0.3 0.4 0.5 0.6 0.7 0.8 0.9
G
0.518
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
199.1 197.9 195,4 192.0 188.0 183.4 178.0 170.9 157.4
D.
663.8 682.5 738.8 751.5 823.5 877.5 902.3 870.0 726.0
= 1200 lb./(hr,)
164.6 163.5 161.8 159.3 156.5 153.3 149.6 143.6 129.9
1,062.0 1,092.0 1,182.0 1,202.4 1,317.6 1,404.0 1,443.6 1,392.0 1,161.6
0.770 0.839 0.920 0.997 1.059 1.087 0.879 0.510 0.262
5.26 4.96 4.90 4.60 4.73 4.80. 6.24 10.35 16.75
0.918 0.998 0.440
(sq. ft.) 1.087 1.138 1.204 1.311 1.445 1.531 1.280 0.700 0.335
5.96 5.86 5.99 5.60 5.53 5.57 6.85 12.09 21.07
Air; s/,-Inch Alumina Spheres; 2-Inch Pipe
G = 150 lb./(hr.) (sq. ft.) 198.0 195.8 192.3 187.5 182.1 175.7 169.0 161.9 150.0
125.4 124.9 124.1 123.1 122.0 120.7 119.1 116.3 110.1
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
197.5 195.2 191.7 186.8 180.9 174.1 167.0 159.6 148.6
138.5 137.6 136.4 134.8 132.9 130.8 128.5 125.0 117.2
0.1 0.2 0.3
198.8 196.9 193.5
153.4 152.5 151.0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.819
144.6 148.2 153.9 163.5 177.5 192.0 188.9 169.7 126.5
0.350 0.377 0.422 0.490 0.542 0.544 0.423 0.227 0.128
3.01 2.87 2.66 2.43 2.39 2.57 3.26 5.45 7.20
G = 300 lb./(hr.) (sq. ft.) 289.2 296.4 307.8 327.0 354.9 384.0 377.7 339.3 252.9
0.463 0.496 0.542 0.597 0.667 0.733 0.639 0.367
0.170
4.56 4.36 4.14 4.00 3.88 3.82 4.31 6.75 10.80
G = 750 lb./(hr.) (sq. ft.) 723.0 741.0 769.5
0.893 0,949 1,030
5.93 5.72 5.47
1.093 1.264 1.465 0.596
Table 1. 0.4 0.5 0.6 0.7 0.8 0.9
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Point Effective Thermal Conductivity and Modified Peclet Number for Heat Transfer (Continued)
189.2 183.8 178.0 171.6 164.8 154.4
149.1 146.8 144.5 141.8 137.6 124.8
817.5 887.3 960.0 944.3 848.3 632.3
1.150 1.318 1.491 1.269 0.573 0.267
198.4 196.5 193.6 189.7 184.9 179.2 173.9 167.2 157.4
G = 12001b./(hr..) 164.1 1,156.8 162.8 1,185.6 160.7 1,231.2 158.2 1,308.0 154.9 1,419.6 151.5 1,536.0 147.8 1,510.8 142.9 1,357.2 131.5 1,011.6
(sq. ft.) 0.979 1.025 1.104 1.200 1.320 1.514 1.445 0.840 0.334
5.20 4.91 4.73 5.42 10.80 17.26
0.5 0.6 0.7 0.8 0.9
186.4 181.6 176.5 170.1 157.2
G. 8.74 8.47 8.17 7.98 7.87 7.40 7.63 11.78 22.10
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0.9
198.2 196.0 192.9 188.5 183.5 177.3 170.4 161.4 146.7
198.5 196.6 193.6 189.6 184.7 179.2 173.0 165.5 152.8
G = 300 155.0 153.9 152.2 149.9 147.1 144.1 140.2 132.7 119.5
lb./(hr.) (sq. it.) 259.2 0.690 264.0 0.727 274.2 0.773 291.3 0.831 0.925 321.3 0.948 344.1 0.668 359.7 0.271 363.6 0.208 306.0
3.16 2.88 2.67 2.53 2.47 2.26 3.85 6.91 8.41
713.7 760.5 782.0 754.0 629.2
2.070 2.375 1.736 0.832 0.390
4.82 4.47 6.26 12.55 22.30
Ammonia; '/a-Inch Alumina Spheres; 2-Inch Pipe G: = 150 lb./(hr.) (sq. it.) 0.522 4.64 197.2 140.0 144.6 4.51 139.2 148.2 0.553 195.3 4.32 153.9 0.596 192.2 137.9 4.18 136.1 163.5 0.652 188.1 3.96 134.2 177.5 0.746 183.1 3.51 132.1 192.0 0.908 177.5 3.47 188.9 0.901 130.1 171.8 0.437 5.83 127.3 169.7 165.8 11.27 119.2 126.5 0.184 150.0
E. Ammonia; 0.25-Inch Steel Spheres; 2-Inch Pipe G = 150 lb./(hr.) (sq. it.) 137.4 129.6 0.489 136.6 132.0 0.532 137.1 0.595 135.4 145.7 0.666 133.9 132.1 160.7 0.750 172.1 0.875 130.2 0.536 128.2 179.9 181.8 122.7 0.300 113.0 153.0 0 . 2 0 6
156.2 153.7 151.0 145.3 131.2
G = 300 lb./(hr.) (sq. ft.) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
197.2 195.3 192.2 188.1 183.1 177.5 171.8 165.8 153.2
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
198.0 196.4 193.5 189.7 185.1 180.1 174.6 169.0 156.3
153.0 152.1 150.8 149.1 147.1 144.8 142.5 138.2 125.9
289.2 296.4 307.8 327.0 354.9 384.0 377.7 339.3 252.9
0.845 0.915 1.013 1.130 1.260 1.435 1.177 0.520 0.234
5.76 5.44 5.09 4.85 4.70 4.46 5.34 10.81 17.72
1.392 1.480 1.594 1.750 1.995 2.320 1.790 0.795 0.335
7.60 7.32 7.06 6.80 6.46 6.00 7.68 15.34 26.95
G = 650Ib.l '(hr.) (sq. it.)
4.39 4.23 4.14 4.07 4.03 4.19 6.20 10.09 12-83
165.2 164.3 162.8 160.8 158.5 156.0 153.4 148.2 133.8
626.6 642.2 666.9 708.5 769.0 832 .O 818.4 735.2 548.0
H. Air; 0.25-Inch Alumina Spheres, 4-Inch Pipe 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ?./To
G = 650 lb./(hr.) (sq. ft.) 197.2 162.2 561.6 1.285 195.8 161.0 572.0 1.340 193.2 1.401 159.3 594.1 1.510 189.9 156.7 631.2 1.700 185.6 153.8 696.2 1.910 180.9 150.5 745.6 1.420 175.1 147.1 779.4 168.0 140.0 787.8 0.790 125.6 663.0 0.446 155.8 t(0, n) t(L, n) G
0 = 150 lb./(hr.) (sq. ft.) 5.11 4.99 4.95 4.87 4.76 4.52 6.34 11.48 16.95
ks
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
G = 150 lb./(hr.) (sq. it.) 132.8 141.6 199.1 140.8 136.5 197.1 147.8 139.7 194.0 150.3 189.6 138.1 136.4 164.7 184.2 134.5 175.5 178.4 132.3 180.5 172.3 127.7 174.0 165.0 117.2 145.2 148.5
0.512 0.559 0.618 0.686 0.797 0.853 0.604 0.305 0.182
3.63 3.41 3.33 3.05 2.87 2.85 4.12 7.84 10.93
0.750 0.805 0.881 1.968 1.093 1.254 0.852 0.417 0.220
4.97 4.76 4.70 4.34 4.20 3.89 5.88 11.52 15.03
G = 300 lb./(hr.) (sq. ft.) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
198.7 196.8 193.9 189.7 184.8 179.2 173.3 165.6 151.3
0.1 0.2 0.3 0.4
198.4 196.8 194.2 191.5
155.9 155.0 153.6 151.7 149.5 147.1 144.6 138.6 125.2
265.5 273.0 295.5 300.6 329.4 351.0 360.9 348.0 290.4
G = 650 lb./(hr.) (sq. it.) 1.515 575.3 162.7 1.607 161.9 591.5 1.724 160.4 640.3 1.864'1 158.6 651.3
5.34 5.17 5.21 4.89
1.28 1.27 1.23 1.15 1.14 1.29 1.83 2.68 4.56
G = 300 lb./(hr.) (sq. it.) 0.550 213.6 170.5 0.561 216.3 168.0 0.585 218.7 165.4 0.621 223.8 162.0 0.657 235.5 157.0 0.672 151.0 262.8 0.590 145.0 331.5 0.450 137.8 400.2 0.272 124.0 393.6
2.37 2.35 2.28 2.20 2.18 2.39 3.42 5.41 8.81
199.2 197.0 193.0 187.2 180.5 173.2 165.0 155.0 137.8
0.1 0.2 0.3 0.4
0.8 0.9
199.1 197.1 194.3 190.8 186.3 179.9 171.7 161.4 141.4
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
200.2 199.7 197.6 194.7 190.1 185.0 178.4 168.8 147.0
= 750 Ib./(hr.) (sq. ft.) 534.0 0.850 185.0 540.8 0.854 183.0 546.8 0.862 180.5 559.0 0.910 176.4 588.8 1.080 170.4 657.0 1.190 163.4 828.8 1.120 155.5 1,000.5 0.840 145.9 0.430 129.4 984.0
3.83 3.86 3.88 3.74 3.32 3.36 4.50 7.25 13.82
0.1 0.2 0.3 0.4 0.5 0.6 0.7
200.8 200.1 198.9 196.8 193.9 188.9 182.5
G = 1200 lb./(hr.) (sq. ft.) 0.970 191.5 854.4 865.2 0.975 189.0 0.980 186.8 874.8 1.030 183.0 895.2 1.270 177.0 942.0 1,051.2 1.500 169.8 1,326.0 1.280 161.5
5.36 5.42 5.44 5.30 4.52 4.26 6.30
Pe
F. Ammonia; 0.25-Inch Alumina Spheres; 2-Inch Pipe
0.510 0.520 0.544 0.595 0.630 0.620 0.550 0.455 0.261
0.5 0.6 0.7
144.3 142.5 140.2 137.8 134.6 130.9 127.0 122.7 115.0
106.8 108.2 109.4 111.9 117.8 131.4 165.8 200.1 196.8
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
.G
VOL. 49, NO. 5
MAY 1957
899
~~
Table 1. 0.8 0.9
173.3 152.3
Poiht Effective Thermal Conductivity and Modified Peclet Number for Heat Transfer (Continued) 151.1 132.5
1,600.8 1,574.4
0.960 0.560
10.15 17.00
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
198.3 195.9 192.5 188.1 181.5 172.0 160.9 148.7 129.0
141.9 140.0 138.0 136.0 132.5 128.3 123.9 118.7 111.0
129.9 131.4 132.6 134.4 139.5 149.6 177.0 206.1 202.8
0.605 0.608 0.610 0.615 0.627 0.640 0.605 0.475 0.340
0.5 0.6 0.7 0.8 0.9
198.7 197.1 194.7 190.8 184.0 175.1 164.3 150.2 131.0
164.7 162.2 149.7 154.9 148.9 142.2 135.5 128.0 116.0
259.8 262.8 265.2 268.8 279.0 299.2 354.0 412.2 405.6
0.672 0.675 0.677 0.680 0.696 0.725 0.660 0.520 0.422
1.96 1.97 2.00 2.00 2.02 2.09 2.72 4.02 4.85
G = 750 lb./(hr.) (sq. ft.) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
200.1 199.0 196.9 194.2 190.5 184.5 173.9 159.1 136.7
185.9 183.4 180.0 175.1 168.4 161.5 152.0 139.1 121.5
649.5 657.0 663.0 672.0 697.5 747.8 885.0 1,030.5 1,014.0
0.930 0.930 0.931 0.934 0.938 0.955 0.865 0.730 0.545
3.54 3.58 3.61 3.65 3.76 3.96 5.18 7.15 9.40
G = 1200 lb,/(hr.) (sq. ft.) 0.1 0.2 0.3 0.4 0.5 0.6 0.7
200.1 199.4 198.4 196.9 194.1 189.0 181.2
193.8 191.5 188.7 184.3 177.9 169.1 158.5
1,039.2 1,051.2 1,060.8 1,075.2 1,116.0 1,196.4 1,416.0
of prepacking is sufficient to establish constant values of the thermal conductivity. Theoretical predictions suggest that the contribution to k , due to radiation from one particle to its neighbor in the bed should be small a t the temperature levels encountered in this study. T o obtain experimental justification for this conclusion, a special run was made with the air entering the pipe a t 300' C. instead of 200" C. as shown in Table I. The results, plotted in Figure 7 , show no significant difference and hence reaffirm the unimportance of radiation a t these relatively low temperature levels. Theoretical Prediction of Thermal Conductivity Argo proposed that the radial transfer of heat in a packed bed occurred by the following mechanisms: 1. Molecular conduction in fluid phase. 2. Radiation from solid particle to adjacent partide. 3. Turbulent mixing in fluid phase. 4. A series process by which heat is
900
0.961 0.970 0.976 0.994 1.050 1.120 1.040
5.48 5.50 5.51 5.49 5.37 5.40 6.88
1,648.8 1,622.4
0.880 0.759
0.6 0.7 0.8 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
199.5 197.2 192.5 185.4 176.4 166.1 155.5 142.7 123.0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
200.9 199.8 197.0 192.2 184.3 174.8 163.5 150.4 132.0
0.2 0.3 0.4
0.5
G = 300 lb./(hr.) (sq. ft.) 155.2 284.1 0.868 153.1 285.0 0.870 150.0 286.8 0.871 146.7 289.2 0.872 293.4 0.880 141.4 135.0 297.0 0.892 311.7 0.845 128.4 338.4 0.705 120.9 330.9 0.540 111.5
9.46 10.79
0.55 0.55 0.55 0.55 0.55 0.55 0.63 0.83 1.11 1.04 1.04 1.04 1.05 1.05 1.05 1.17 1.52 1.93
G = 750 lb./(hr.) (sq ft.)
0.931 0.934 0.937 0.941 0.945 0.951 0.910 0.795 0.575
2.42 2.42 2.42 2.43 2.46 2.47 2.70 3.36 4.52
G = 1200 lb ./(hr.) (sq. it.) 1,136.4 1.370 198.9 186.2 1,140.2 1.383 198.0 183.7 1,147.2 1.428 196.5 180.0 174.5 1,156.8 1.498 193.5 188.0 167.5 1,173.6 1.593 159.8 1,188.0 1.675 180.0 149.8 1,246.8 1.490 169.0 137.8 1,353.6 1.181 155.5 122.0 1,323.6 0.830 137.0 Calculated by Method B.
2.62 2.61 2.54 2.44 2.33 2.24 2.64 3.60 5.00
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
transferred from one solid particle to its neighbor via the intervening fluid. 5. Conduction from particle to particle by direct solid contact. These mechanisms were combined into an over-all expression for the thermal conductivity by considering heat transfer a t any plane to be the sum of that through the solid and fluid phases. For the purposes of this investigation the contribution of radiation has been shown to be negligible. Then the resultant equation is
Equation 12 includes the effect of the fundamental variables of the bed on k,-that is, the effect of void fraction 6, gas conductivity k,, particle size and conductivity d, and k,, and gas mass velocity G. The influence of tube size is given by its effect upon G and Pe', and 6. The expression was developed by making simplifying assumptions concerning the geometry of a packed bed. I t contains no arbitrary constants but
INDUSTRIAL AND ENGINEERING CHEMISTRY
145.7 125.3
Air; 6/az-Inch Steel Spheres ; 4-Inch Pipe G = 150 lb./(hr.) (sq. ft.) 0.812 198.7 133.5 142.1 0.817 131.5 142.5 196.6 0.821 192.1 129.4 143.4 0.830 127.0 144.6 184.9 0.840 124.9 146.7 176.2 0.850 168.1 120.7 148.5 0.784 158.1 117.3 155.9 0.645 144.9 112.8 169.2 0.472 125.8 107.2 165.5
0.1 1.09 1.09 1.10 1.11 1.13 1.18 1.48 2.20 3.00
G = 300 lb./(hr.) (sq. it.) 0.1 0.2 0.3 0.4
169.9 145.9
K.
J. Air; 0.25-Inch Steel Spheres; 4-Inch Pipe G = 150 lb./(hr.) (sq. ft.) 0.1
0.8 0.9
183 7 180.8 176.9 172.0 164.5 155.0 145.0 133.8 118.2 I
710.3 712.5 717.0 723.0 733.5 742.5 779.0 846.0 827.3
does require for application auxiliary heat transfer data. Thus the average coefficient of heat transfer between fluid and particle, h,, and the equivalent conductivity for transfer from particle to particle, k,, must be known. Equation 12 can be used to predict average values for k, for the entire bed by using average values of the void fraction, mass velocity, and Peclet number, and to estimate point values of k, at any radial position, provided corresponding information for 6, G, and Per, is known. Applied in the latter manner curves of effective conductivity us. radial position r/ro can be computed for comparison with the experimental data in Table I. Comparison of Predicted and Experimental Results Average Effective Thermal Conductivities, Equation 12 was employed to compute average values of k , for the conditions of Table I, using the average value of the mass velocity and the over-all void fraction of the bed as reported by Schwartz and Smith (70). The experimental results for air given in
H E A T T R A N S F E R IN PACKED BEDS
0.4
Pellet: 1/4" alumina cylinder Pipe size : 2 " diameter
, 0.81
Gas: A i r Pellet: 1/4" alumina spheres G = 1 5 0 Ib. /hr. ft.'
o,7-
A
200°C total temp. change
0
02
04
06
0.8
0.1
I
0
r/ro Figure 6.
Effect of prepacking height on k,
Table I were averaged for the several radial positions, and these averages were compared with the predicted results. Comparisons for 0.25-inch spheres are shown in Figure 8. The agreement is good at higher Reynolds numbers, but the predicted values are low at low Reynolds numbers. On the same chart are shown predicted and experimental results for S/16-inch cylindrical particles as determined by Argo ( 2 ) . Cobedy and Marshall (3) measured k, values for 0.25-inch Celite cylinders, assuming a constant conductivity and a wall resistance. These are also indicated in Figure 8. Because k, is an increasing function of particle size, the results for 0.25- and 3/16-inch particles should not form a single line on Figure 8. In fact, comparison of the set of dotted lines and the set of solid ones shows the predicted and experimental effects of particle size on the average thermal conductivity, and indicates that the theory predicts the effect of d,, in a satisfactory manner. Comparisons for other particle sizes show similar results. Effect of Radial Position on Point Values of k,. Because of the change in void fraction in a packed bed with radial position, the mass velocity, and
r/ ro
Figure 7. Effect of total radial temperature change on k,
dPG/A Figure 8.
Average k, vs. modified Reynolds number 0.
V. A.
X a / l ~inch alumina cylinder ( I ) 0.25 X 0.25 inch Celite cylinder (3) 0.25-inch steel spheres (this work)
"18
VOL. 49, NO. 5
MAY 1957
901
2 .c 0
,B--.8
x
\
/ // B
a
\ * \
G.150 G=300 G=750 G =I500
I .E
I.€
I.4
1.2
I .c
0.E
0.E
I
I
I
I
I
0.4
0.2
1
I
0.4
I
I
0.8
0.6
r/r, 0.2
Figure 9. Comparison of predicted and experimental k, for air runs with 0.25-inch steel spheres
0
Figure 10. Comparison of predicted and experimental k, for ammonia runs with 0.25-inch steel spheres hence the thermal conductivity, should vary across the diameter, decreasing as the tube wall is approached. The theory, through Equation 12, should predict this effect. Figures 9 and 10 show predicted and experimental results for 0.25-inch steel spheres and air and ammonia in a 2-inch pipe. The predicted values decrease with radial position as the wall is approached and are in 12 IO
08
PREDICTED
~
AIR
x
CALCDFROM T E M P DATA
04 c--
o
0.2 I
0
I 200
902
CALC FROM TEMP DATA 1 ,
I
400
1000 Ib /(hr.)(sq ft.) 800
600
G a s mass v e l o c i t y
Figure 1 1.
PREDICTED
‘ I
-
I200
Effect of particle conductivity on effective thermal conductivity
INDUSTRIAL AND ENGINEERING CHEMISTRY
1
0.2
1
1
1
0.4
1
0.6
r/ro
good agreement with the experimental data in this region. However, the peak in k, values observed experimentally is not reproduced. A major reason for this lies in the inadequacy of the velocity and heat transfer data that are available for applying the theory. Thus Schwartz isothermal velocity data (70) must be used for the nonisothermal conditions in the bed. The other separate corre-
06 0.e
1
1
1
0.8
1
I 3
lations of experimental data needed in applying Equation 12 are: The void fraction profile; for this the measurements of Shaffer (77) reported by Schwartz (70) were employed. The convection heat transfer coefficient between particle and fluid; the correlations of Hougen and Wilkie (7) and Gamson, Thodos, and Hougen ( 6 ) were used for estimating values of h,. Finally Fahien’s work ( 5 ) on mass transfer under isothermal conditions was used to predict values of Peh. Figures 9 and 10 also show that the agreement between predicted and experimental results is generally better at high mass velocities than at low. This was also observed in Figure 8 in the comparison of average values of k,. Effect of Particle Conductivity. The thermal conductivity of the alumina particles was only about 2% of that for the steel balls, 0.52 us. 26 B.t.u./ (hr.) (ft.) ( O F.). Because only a small part of the total resistance to radial heat transfer in a packed bed is due to the particles themselves, the resultant effect on k, of this difference in k,, is small. This means that the heat transfer characteristics of a packed bed cannot be greatly improved by changing the packing material.
H E A T T R A N S F E R IN P A C K E D BEDS Nomenclature cp d?,
De G
h, x
\
k,
yr)
I
k
k,
Y
k,
0
200 400 600 Gos moss velocity- lb /(hr.)(sq.ft.)
Figure 12.
Effect of fluid properties on effective thermal conductivity
Equation 12 was used to predict the ratio of k, for alumina and steel particles; results are shown in Figure 11 for the 0.25-inch size. Average void fraction values and mass velocities were employed in the calculations, so that average conductivities were obtained. Curves are shown for both air and ammonia as the fluid flowing through the bed. Experimental values of k,A,umins/ke8tea,for this packing were obtained by averaging the data in Table I, B, C, E, and F, and are plotted on Figure 11 for comparison with the predicted curves. The agreement is a measure of the validity of the theory and the method of estimating the magnitude of conduction from particle to particle by direct contact. The only available data for estimating the latter quantity, k,, is the empirical correlation of Wilhelm and coworkers (75) expressed by the equation: log
kp
=
- 1.76 f
k 0.0129 f
(13)
This result was based upon an analysis of data reported by Schuhmann and Voss (74) for heat transfer with and without flow of gas through a single packed bed. The uncertain magnitude of heat transfer by convection in the gas phase under no-flow conditions makes interpretation of such data uncertain. Therefore? the accurate value of the contribution of particle to particle contact is in doubt. This uncertainty is probably responsible for some of the disagreement between the predicted and experimental ratios shown in Figure 10. The experimental results at high mass velocities are apparently in error, as the ratio cannot exceed unity. At high velocities the turbulent diffusion contribution becomes the controlling factor in the value of k,, and this is the same for different kinds of packing ma-
terials; hence the ratio should approach but not surpass unity as G increases. In this respect the predicted results are better than the experimental values. Both predicted and experimental ratios indicate the small effect of k, on k,. Effect of Fluid Properties. There is less uncertainty in determining the value of the theory for predicting the effect of fluid properties on k,. The specific heat, viscosity, and thermal conductivity of air and ammonia are accurately known and can be used in Equation 12 to predict values of the ratio kCNH8/ kdsir. The results are shown as solid curves in Figure 12 for 0.25-inch particles. Experimental ratios prepared from Table I, By C , E, and F, are indicated as points. The agreement is reasonably good, suggesting that Equation 12 does account for the effect of physical properties of the fluid on the effective thermal conductivity. The higher specific heat of ammonia results in a sizable increase in effective thermal conductivity. Conclusion
The theoretical method of determining k, values previously developed is, in general, satisfactory for predicting the effect of gas and solid packing properties. However, the large variation in effective thermal conductivity with radial position, observed experimentally, is not reproduced. This failure is probably due to the inadequate data available for the heat transfer coefficients between fluid and particle and for mass velocity profiles. With respect to average conductivities for the entire bed, theoretical and experimental values are in good agreement at high Reynolds numbers, but the predicted results are lower at low Reynolds numbers.
k, L Pe;
specific heat of gas at constant pressure = diameter of particle = effective diffusivity = superficial mass velocity of gas flowing through bed = convection heat transfer coefficient between particle and Auid = molecular thermal conductivity of fluid = effective thermal conductivity = equivalent thermal conductivitv for heat transfer from particle to particle given by Equation 13, B.t.u./(hr.) (ft.) (" F.) = molecujar ihermal conductivity of solid particle = total beddepth = Peclet number for heat transfer, dPCPG kt! = Peclet number for mass transfer, dPP/D* = function of r for radial solution of Equation 1 = Reynolds number, dPG/p = radial distance from center of bed = radiusofbed = gas temperature at radius r in bed = wall temperature =t-tzo = bed depth measured in direction of flow = eigenvalue in solution of Equation 5 = voidfraction =
-
Pe;
R Re' r 70
t
t,
T
z
h
6
literature Cited (1) Argo,. W.. B., Ph.D. thesis, Purdue Universitv. June 1952. (2) Argo, W. B., Smith, J. M . , Chem. Eng. Progr. 49, 443 (1953). (3) Coberly, C. A., Marshall, W. R., Jr. Ibid., 47, 141 (1951). (4) . . Colburn, A. P., IND.ENG.CHEM.23, 910 (1931). (5) Fahien,R. W., Smith, J. M., A.1.Ch.E. Journal 1,28 (1955). (6) Gamson, B., Thodos, G., Hougen, 0. A., Trans. Am. Inst. Chem. Engrs. 39, l(1943). ( 7 ) Hougen, 0.A., Wilkie, C. R., Zbid., 41.445 (1945). (8) Kwong, S. S.,'Ph.D. thesis, Purdue University, February 1955 (Purdue University Library). (9) Schuler, R. W., Stallings, V. P., Smith, J. M., Chem. Eng. Progr., Symposium Ser. No. 4, 48, 19 (1952). (10) Schwartz, C. E., Smith, J. M., IND. END. CHEM.45, 1209 (1953). (11) Shaffer, M.R.,M. S. thesis, Purdue University, August 1952. (12) Sin er, E., Wilhelm, R. H., Chem. j n g . Progr. 46, 343 (1950). Stallings, V. P., Ph.D. thesis, Purdue University, June 1951. Schuhmann, T. E. W., Voss, V., Fuel 13, 249 (1934). Wilhelm, R. H., Johnson, W. C., Wynkoop, R., Collier, D. W., Chem. Eng. Progr. 44, 105 (1948).
RECEIVED for review May 24, 1956 ACCEPTEDOctober 15, 1956 VOL. 49,
NO. 5
MAY 1957
903
