r t V”,m
ve, vd
07,
radial position coordinate radius of time or conduit time covariant derivative of vi with respect to x m coordinate = velocity components (contravariant) in directions of r , 0, and $I coordinates = mean velocity of flow through porous medium or conduits = superficial mean velocity based on total crosssectional area = coordinate labels, spherical coordinate system = Deborah number, Equations 1 and 19-19c = Reynolds number, Equation 3 = isotropic pressure (arbitrary for incompressible fluids) = deformation rate or shear rate = fractional void volume = density = relaxation time of fluid = characteristic process time = viscosity of fluid = stress tensors; T denotes total stresses and Q’ stresses relative to isotropic pressure, cy = second invariant of deformation rate tensor, Equation 12 = = = =
R
V V, xr, xo, xd
hiDeb NR~ cy
r E
P
*rz 0, I I.1 T, T’ nd
literature Cited Ackerberg, R. C., Ph.D. thesis, Johns Hopkins University, Baltimore, Md., 1962. Astarita, G., IND.ENG.CHEY.FUNDAMENTALS 4, 354 (1965). 6. 257 (1967). Astarita. G.. IND.ENG.CHEM.FUNDAMENTALS Astarita; G.; Metzner, A. B., Atti. Accad. Lincei 40, 606 (1966). Bird, R. B., Stewart, \Y. E., Lightfoot, E. N., “Transport Phenomena,’’ pp. 196-207, Wiley, New York, 1960. Christopher, R. H., Middleman, Stanley, IND. ENG. CHEM. FUNDAMENTALS 4, 422 (1965). Dauben, D. L., Ph.D. thesis, University of Oklahoma, Norman, Okla.. 1966. Ergun, Sabri, Chem. Eng. Progr. 48, 89 (1952). Etter, Irwin, Schowalter, W. R., Trans. SOC.Rheol. 9 (2), 351 (1965). Feig, J. L., M.Ch.E. thesis, University of Delaware, Newark, Del., 1966. Fredrickson, .4.G., “Principles and Applications of Rheology,” Prentice-Hall, Englewood Cliffs, N. J., 1964. Gaitonde, N. Y., M.S. thesis, University of Rochester, Rochester, N. Y., 1966. Gaitonde, N. Y., Middleman, Stanley, IND.ENG.CHEM.FUNDAMENTALS 6, 147 (1967). Ginn, R. F., M.Ch.E. thesis, University of Delaware, Newark, Del., 1963; Ph.D. thesis in preparation, 1967. Ginn, R. F., Metzner, A . B., Proceedings of 4th International Congress on Rheology, p. 583, Interscience, New York, 1965. Hawkins, G. A , , “Multilinear Analysis for Students in Engineering and Science,” Lt’iley, New York, 1963.
Hermes, R. A., Meeting, Society of Petroleum Engineers, December 1966. Jones, W.M., Maddock, J. L., Paper 1686, Society of Petroleum Engineers, December 1966. Kapoor, N. N., Kalb, J. W., Brumm, E. A., Fredrickson, A. G., IND.ENG.CHEM.FUNDAMENTALS 4,186 (1965). Langlois, \Y. E., “Slow Viscous Flow,” Macmillan, New York, 1964. LaNieve, H. L., Ph.D. thesis, University of Tennessee, Knoxville, Tenn., 1966. Lodge, A. S., “Elastic Liquids,” Academic Press, New York, 1-O,h4, .
McConnell, A. J., “Applications of Tensor Analysis,’’ Dover, New York, 1957. McKinlev. R. M.. Jahns. H. 0.. Harris. W. W.. Greenkorn. R. A,. A.I.Ch.E. J . 12,’17 (1966). ’ Metzner, A. B., A.I.Ch.E. J . 13, 316 (1967). Metzner, A . B., Astarita, G., A.I.Ch.E. J., in press. Metzner, A. B., Houghton, W. T., Hurd, R. E., Wolfe, C. C., Proceedings of International Symposium on Second-Order Effects in Elasticity, Plasticity, and Fluid Dynamics, p. 650, Pergamon Press, Oxford, 1964. Metzner, A. B., Johnson, M., annual meeting, Society of Rheology, 1966. Metzner, A. B., White, J. L., A.I.Ch.E. J. 11, 989 (1965). Metzner, A. B., \Yhite, J. L., Denn, M. M., A.I.Ch.E. J . 12, 863 Il966ai. Metzner,’A. B., White, J. L., Denn, M. M., Chem. Eng. Progr. 62 (12), 81 (1966b). Oliver, D. R., Can. J . Chem. Eng. 44, 100 (1966). Pipkin, A. C., Quart. J . Appl. Math. 23, 297 (1966). Pruitt. G. T.. Crawford. H. R.. Report to David Taylor Model B a s k Cloniract Nonr-4306(00) Cl$65). Pyi, D. J.: J . Petrol. Technol. 1 6 , 911 (1964). Reiner, M., Physics Today 17, 62 (January 1964). Rosenhead, L., “Laminar Boundary Layers,” Oxford University Press, Oxford, 1963. Sadowski, T. J., Trans. SOC. Rheol. 9 (2), 251 (1965). Sadowski, T . J., Bird, R. B., Trans. Soc. Rheol. 9 (2), 243 (1965). Savins, J. G., private communication, 1966. Savins, J. G., SOC.Petrol. Eng. J . 4, 203 (1964). Schlichting, Hermann, “Boundary Layer Theory,” 4th ed., McGraw-Hill, New York, 1960. Seyer, F. A,, Metzner, A. B., Can. J . ChPm. Eng., in press. Shertzer, C. R., Ph.D. thesis, University of Delaware, Newark, Del., 1965. Slatterv, J. C., Paper 1684, Society of Petroleum Engineers, December 1966. Tokita, N., White, J. L., J . Appl. Polymer Sci.10, 1011 (1966). Uebler. E. A,. M.Ch.E. thesis. University of Delaware, 1964. Uebler; E. A,; Ph.D. thesis, University of Delaware, 1966. Vela, Saul, Kalb, 3. LY., Fredrickson, A. G., A.I.Ch.E. J . 11, 288 (1965). White, J. L., Metzner, A. B., J . Appl. Polymer Sci. 7 , 1867 (1963). RECEIVED for review November 1, 1966 ACCEPTED April 25, 1967 Work supported by the Water Resources Center, University of Delaware.
HOLDUP IN IRRIGATED RINGIPACKED
TOWERS BELOW T H E LOADING POINT J.
E. B U C H A N A N
University of hrew South Wales, Kensington, N.S. W., Australia
holdup may well be considered as the basic liquidside dependent variable in packed tower operation. Holdup has been shown to have a direct influence on liquidphase mass transfer (2),on loading behavior (72), on gasphase pressure gradient (72), and on mass transfer (9). Ir. itself it is important only in the consideration of unsteadystate behavior of a tower-e.g., in batch distillation ( 8 ) . IQIJID
400
l&EC FUNDAMENTALS
Many workers (3, 4, 7, 70-72) have measured holdup, with or without gas flow, and have produced empirical descriptions of their results. Only the correlation of Otake and Okada (7) is in dimensionless form and can claim any generality. This correlation fits the available experimental data very well but it is derived by essentially empirical methods. I t is desirable therefore to justify this form of relation theo-
Two limiting dynamic regimes for liquid flow on an irrigated packing may be distinguished: the gravityviscosity regime at low Reynolds numbers and the gravity-inertia regime at high values. Examination of simple models of the two modes suggests the form of the holdup relation for each case and gives order of magnitude estimates of the coefficients. Experimental results lie between the limits but are satisfactorily correlated by an interpolation formula.
retically or to find a n expression with a firmer theoretical basis.
form of expression used is considerably simpler if the holdup approaches zero when liquid rate approaches zero. This is the case for operating holdup but not for total holdup.
Modes of Holdup
loading
Three modes of liquiid holdup have been discussed in the literature (77), all expressed as cubic feet of liquid per cubic foot of packed volume, a dimensionless unit.
Because most towers are operated below the loading point and because the problem is considerably simplified thereby, this treatment deals only with operating holdup in the preloading range of flow rates. T o show the significance of this specification and to justify the methods of data selection it is necessary to discuss briefly the meaning of loading and the relation between holdup and loading. Following White (73), the loading point is usually defined in terms of the gas-phase pressure gradient by reference to a plot of log (pressure gradient) us. log (gas flow rate) at constant liquid rate, such as Figure l a . When experimental data are plotted in this way, most of the points usually fall convincingly on three straight lines, the points not on the straight lines showing smooth transitions between them. T h e lowest line has a slope 1.8 to 2 and the next about 4 ; the last line is practically vertical. T h e point of intersection of extrapolations of the two lower lines defines the loading point. Visual observation of a tower shows that as gas velocity increases from zero the liquid flow pattern is unchanged until the loading region is approached. Then the pattern begins
TOTAL HOLDUP, the total amount of liquid on the packing a t a given operating condition. STATICHOLDUP, the amount of liquid remaining on the packing after it has been fully wetted and drained for a long time. OPERATING HOLDUP, the difference between total and static holdups. The static holdup is clearly a function of static properties only. In a similar way it is usually taken that flow rates and dynamic properties affect only the operating holdup. This assumption implies that the static holdup remains stagnant and in place under all operating conditions. A partial justification for this view can be suggested, but in general it must be taken as only a n approximation. At high liquid rates the static holdup makes only a small contribution to the total and little error is occasioned by accepting the assumption. Largely, of course, the holdup is divided into two types as a matter of convenience in producting correlations. T h e
I
Water
- Ai r 1.0
n
*.u. .4-
* l c. . b
v n 3 U d
0
I
0.1 1
I
I
I
I
1
0.1 Superficial Figure 1.
Gas
VeIoci ty
(ft./Soc.)
Holdup and pressure gradient VOL. 6 NO. 3
AUGUST 1967
401
Liquid flow rate Liquid flow properties Density Viscosity Surface tension Local gravitational acceleration Shape of the bed Scale
to change and a buildup of liquid on the packing may be observed. This phenomenon was the origin of the name “loading point” and was one early way of defining it. Measurements of holdup and pressure gradient over the same range confirm the close connection, implied above, between holdup and pressure gradient near the loading point. This also is shown in Figure 1, using some experimental data of Elgin and Weiss ( 3 ) . At low gas rates the holdup increases very slowly, if at all, about linearly with gas rate. Near the loading velocity the holdup increases sharply and a t a n increasing rate. T h e region in which holdup begins to increase corresponds closely to that in which the Slope of the pressure gradient line increases. T h e loading point could well be defined in this way. At high liquid rates holdup is indeed a more sensitive indicator of loading than is pressure gradient. In such cases the pressure gradient curves change their form. The experimental points now clearly define a continuous curve, the slope in the lower part being considerably less than the usual value of about 1.9. The position of the loading point is effectively obscured. This is the natural result of plotting on log-log coordinates a line that has nonzero intercepts. The holdup curves, however, follow their usual course, loading naturally, occurring at lower gas rates (Figure lb). At the highest liquid rate there is a change in the holdup behavior, as indicated in Figure 2. I t is evident that at the highest rates used in this series of experiments (72) the loading point occurs a t gas rates velocities near zero or negative. This fact too is concealed by the usual log-log plot. Use of a logarithmic plot implies some absolute significance of a zero value of the quantity plotted and no significance of negative values. When, as in the two cases cited, this assumption is unsound, the resulting picture is likely to be misleading.
Three items of the list-surface tension, shape, and scalerequire further discussion. I t is assumed that all of the packing surface is wet, though not necessarily active ; thus the only relevant surface tension is that between gas and liquid phases. As is established at greater length below, the experiments of Shulman et al. (77), in which surface tension was a n experimental variable, show that its effect is small; it is neglected in the following discussion. When applied to a packed bed, “shape” has two aspects: the shape of the individual packing pieces and that of the assembly. The following treatment, while general in its application to all packing shapes of the film type, is applied only to Raschig rings. For these the height and diameter are equal for all sizes of rings. For mechanical reasons, however, small rings tend to have relatively thicker walls than large ones and so geometric similarity may not be exactly maintained through the full range of sizes. All of the data used are for dumped, random packings. It is not certain that such packings can be considered fully random in the statistical sense. Indeed, some recent work suggests that they are definitely not. Yet, so long as the bed diameter is sufficiently greater than the ring diameter and if the voidages are about the same, all beds may be taken to be equivalent and to have virtually the same shape. T h e usual requirement that D 3 8d has been met by almost all of the data which I have used. I n this work the relevant linear dimension is taken to be the ring diameter, d. Commonly, in treatments of packed beds the dimension is taken to be the diameter of a sphere of the same surface area or some other such equivalent diameter, which, with the voidage, is considered a sufficient description
Independent Variables
Holdup is almost independent of gas rate below the loading point. With little loss of precision it can be taken to be completely independent, and may be treated as a function of the liquid flow variables only:
1 Water- A i r
I
I
1
I
1r 0.028 I
I
I
I
I
I
I
I
---
--
u, =
135rnrn.Raschig R i n g s
0.015 ft./sec.
.
0,02 SuperficiaL Figure 2. 402
I&EC FUNDAMENTALS
Gas
Velocity
(ftlsec.)
Holdup at high liquid flows
I
of scale and shape. But even in the pressure drop studies for which it was derived this assumption has proved to be of limited value ( 5 ) . Where shape is virtually constant, as in the case being considered, there is no need for such a n elaboration and any relevant linear dimension will do. Exact geometric similarity requires that all the packing pieces have the same shape and that beds be formed by stacking these pieces in the same way. For such assemblies the voidage, e, and dimensionless packing density, Nd3, should be the same. Where these quantities are available they are listed in the summary (of data used (Table I). The data are not perfectly homogeneous, which no doubt accounts for some of the scatter in the results. Data Selection and Methods of Measurement
Holdup data have been taken from several sources, discussed separately below. The best experimental methods were those of Shulman, Ullrich, and LVells (7t3), who used a tower mounted on a weighing scale so that water holdups could be measured directly by weight. For each liquid rate, measurements were taken over a ranye of gas flow rates, so that it can be established that operalion was definitely below the loading point. These data have been accepted unreservedly. Uchida and Fujita (72) used an arrangement where both the liquid inlet and the outlet to the tower could be cut off simultaneously. After stable operation had been established, the valves were closed and the volume of liquid which drained from the packing was measured. This was essentially a measurement of operating holdup. Again the holdup was measured over a range of positive gas flow rates for each liquid rate. Some of the results for very high liquid rates have been rejected because no region where holdup was independent of gas flow could be found in the results. I n these cases the loading point was evidently near zero or in the negative gas flow rrgion, as in cocurrent flow. Shulman, Ullrich, LYells, and Proulx (77) used the apparatus described by Shulman, Ullrich, and Wells (70) with a large variety of liquids. All measurements were taken at zero gas rate, but the absence of loading could be checked from the Uchida and I’ujita data and all of their results were accepted. Otake and Okada (7) used a method similar to that of Uchida and Fujita but in this case air flowed freely through the toiver, the flow being restricted only partly at a measuring orifice. It I S here assumed that this arrangement would avoid loading and that the measured holdup would be the value independent of gas velocity.
Table 1.
I n all of the other cases the holdup value used was either that at zero gas flow (77) or extrapolated to zero gas flow (70, 72). Operating Holdup
Limiting Flow Regimes. Forces acting on fluid particles are gravity, viscous drag, and inertia. While gravity is always the driving force, either viscous or inertial forces may predominate as the resistance. Thus, two limiting flow regimes may be distinguished-the gravityviscosity and gravity-inertia regimes. I n general, these will occur at low and high values of Reynolds number, respectively-that is, a t low velocity or high viscosity losses will be mainly due to viscous drag in a n essentially laminar flow. I n the converse case the losses will be caused mainly by turbulence arising at sudden changes of flow path. Gravity-Viscosity Control. Behavior of this regime has been described by several workers, notably Nusselt (6) and Davidson ( 2 ) . T h e basic assumption is that the liquid is everywhere at its terminal velocity; no accelerations need be considered. The model used is a n assembly of flat surfaces inclined at angle, 8, to the horizontal and having a liquid film of uniform thickness, A, flowing down the surface of total area, a, sq. feet per cu. foot. The liquid loading is L lb./(sec.) (sq. ft.) of horizontal cross section. At any cross section of the assembly the width of the surface will be a sin 8 and the liquid loading may be expressed in.
L
= L’ lb./(sec.) (ft. width of surface) another way as 7a sin 0 Then by the well known derivation of Nusselt ( 6 )
A =
[-]
H=
[ X I
3pL’ p2g sin 8
Now H = a A and 1/3
p2g sin2 8 For standard packings the packing size, d, is a more convenient quantity than the area, a. For packings of constant shape ad = k, a dimensionless constant. Noting also that L/p = U , the superficial liquid velocity, we may write:
Summary of Data V,
Workers
Ref.
Shulman et al.
( 70)
Shulman et ai. Otake and Okada Uchida and Fujita
a
Assumed.
(77) (7)
( 72)
Diameter, d 0 . 5 in. 1 . O in.
1 . 5 in. 1 . O in. 1 . 0 2 cm. 1.60 cm. 2.55 cm. 1 . 5 cm. 1 . 6 cm. 2 . 6 cm. 3 . 5 cm.
e
0.61 0.73 0.72 0.72 0.59 0.64 0.74 0.73 0.73 0.76
113
Nd3 0.79 0.79 0.84
Dld
...
20 10 7 10
0.82
10
0.82 0.83
16
...
... ... ...
10
17 16
10-14 7
Centistokes
0.74-125 1 1 1
1 20-330 1-280 1-310
Surface Tension, DyneslCm.
73a 735 73a 38-86 730 73a 73a 735 30 30-730 30-73a
Symbol
w A X
0
The liquid was water.
VOL. 6 N O . 3
AUGUST 1967
403
0
8 x 0
7m
H 6 -
Fi
5 4 -
3 21.
01
1
-2
-1
I
l
0
1
l
3
2
T h e first term is a coefficient depending on the inclination of the surfaces. The second is equal to the quotient of the Froude and Reynolds numbers, referred to as the film number, Fi. The third term may be combined with the first as a factor shape for a given packing. The equation may be written briefly as
H = S(Fi)i/3
-
Figure 4. Gravity-inertia model
(1)
For dumped Raschig rings ad 5; and for 0 taken to be from 60' to 80°, S is calculated to be in the range 4.2 to 4.5. Figure 3 is a plot of H/Fi1j3 us. Re. At low Reynolds numbers H/FiIi3 approaches a constant value in the range S = 2 to 3. At high Re the holdup is higher than Equation 1 would predict. Gravity-Inertia Control. The assumption made here is that energy losses owing to continuous viscous drag are negligible. The only losses occur when the natural liquid flow path is impeded from time to time and energy is dissipated in turbulence. This dissipation is an effect of liquid viscosity, but the amount of energy dissipated is not strongly dependent on the magnitude of the viscosity. The model for this situation is similar to the previous one, except that the flow is interrupted, at intervals 1 by steps, as shown in Figure 4; at each interruption the liquid loses a fraction, F, of its kinetic energy before proceeding down the next slope. Other losses are negligible. If the initial velocity is Vi, we may write
At distance r down the plate
V,Z
+ 2 gr sin 0
=
Vi2
=
2g sin
e
V , = (2g sin Now
1
[(l/n) 0)1/~
+ r]
[(l/n)
+ ,I1/'
A(r) = L ' / ( P ~ ~ )
I
and Amean = 1 / 1
A(r) dr
-
n V t = 2gl sin 0 where
n = F/(1 404
l&EC FUNDAMENTALS
- F)
z/zL p a sin e (91 sin e)llz
=-
(n
+ l)1I2 - 1 n1/2
A
01
Figure
L/p
But and, as before,
H =
=
Lr
H
= aLean
[:I1’’ . [2-1 . sin3 B
Log ( R e ) 5. Data plotted according to Equation 2 2.4
(n
+
I
- 1
flow rate, the equation reduces to: = S’ (Fr)”*
I
I
nl12
If it is assumed that I , F, and 6’ are independent of liquid
H
I
S 2 .2
(2)
Xow the value of 1, in a packing must be related to the packing
size and may be expected to be of about the same magnitude. Taking I , to be equal to d.
For values of F ranging from 0.5 to 0.9 and 6‘ from 60’ to 80” calculated values of S’ are in the range 0.61 to 1.26. Holdup in falling drops or streams would follow a similar law, 6’ now being 90’. Figure 4 is a plot of iY/Fr”* us. Re. In this case it may be seen that at high Reynolds number H/Fr112 approaches a constant value of about 3. At low R e the holdup is greater than can be accounted for by Equation 2. Correlation
From Figures 3 and 5 it is evident that except for some of the points at very low Reynolds numbers the experimental results cannot be described by either Equation 1 or 2 but appear to be in a transition region. An interpolating expression is required to cover this i.ransition. A simple expression is the sum of the terms for the liniiting conditions.
fJ =
S Fill3
+ S’
Fr”2
(3)
For convenience in developing a correlation this equation is transformed to the straight-line expression H/Fi1/3 = S + S‘ Fr1’2iFill3
(4)
The experimental data have been examined using this equation and coefficients S and 5” found to give the line of best fit. A joint 95% confidence ellipse ( 7 ) for S and 5” is shown in Figure 6. The indicated best values of 2.2 and 1.8 compare well with the order of magnitude estimates of 4 and 1 given above. VOL. 6
NO. 3
AUGUST 1967
405
The experimental points are shown in Figure 7 plotted according to Equation 4; the line of best fit is the final proposed equation,
H
=
2.2 Fi1'3
+ 1.8 F
P
For comparison the Otake and Okada relationship is shown in the same way in Figure 9. The equations are:
(5)
H = 8.1 FrO.44Re-0.37 for 0.01
As may be seen from Figure 8, this equation, with two arbitrary coefficients, satisfactorily correlates all the data over a range of almost five orders of magnitude in Reynolds number.
and
H
=
6.3 Fr0.44 Re-0.20 for 10
1u
/
/
/
0
1
0
2
3
4
k
Fr'/Fi
Data plotted according to Equation 4
Figure 7.
0.3 H = 2.2 F i k + 1 . 8 Fr
0.2 n
Q
g
0.1
I
"0 d
e
r -0.1
Y
ul
e
0
a -0.2
e
e
-0.3
-2
-1
0
1
2
Log (Re) Figure 8. 406
l&EC FUNDAMENTALS
< R e < 10
This correlation compared with experiment
e
< Re < 200
0.3
H = 8.1 Fr’44R6”
0.2
I
6.28 F
~R ~ -*. * o ~
~
n
n. x al 0.1 3=
‘44 0 0 0
I Y
-0.1
cn
0
0
-I
I’
”
I
-0.2
I
-0.3 Figure 9.
Otake and Okada correlation compared with experiment
and the agreement is only slightly better. However, the range of Reynolds numbers had to be separated into two parts, leading to a total of six empirical constants in the two equations. Effect of Surface Tension. This general correlation has been developed without taking any account of surface tension. I t is desirable to re-examine the data to see whether any surface tension effect can be found. For this purpose the data of Shulman et al. (77) are relevant. I n these experiments a wide range of surface tensions was examined, using both low surface tension organic liquids and aqueous solutions of a surface-active agent. T h e relevant data are shown in Figure 10 plotted according to Equation 5. The points cover a wide range but in a n apparently random fashion. No residual surface tension effect is discernible. Scope of Correlation. T h e experimental data used in developing the correlation were taken from experiments with ceramic Raschig rings only and the resulting equation applies strictly only to such packings.
m
4
2
1
3
Log (Re) Figure 10.
Effect of variable surface tension
liquid Water Calcium chloride sohtion Petrowet solution Petrowet solution Petrowet solution Methanol Benzene
Surface Tension, DyneslCm. 73 a6 58 43 3a 23 29
Symbol 0
m
+ A
v X
+
An equation of the same form should be applicable, however, to any packing of the film type, but the shape factors, S and S’, must be expected to take on different values when the packing shape is changed. Nomenclature Fi = film number = Fr/Re, dimensionless = Froude number, U2/gd, dimensionless Fr = Reynolds number Ud/v, dimensionless Re U = interfacial area of packing, sq. ft./cu. ft. D = tower diameter, ft. d = packing size(ring diameter), ft. = local gravitational acceleration, ft./sec2 g = operating holdup, cu. ft./cu. ft., dimensionless. H L = liquid rate, lb./hr., sq. ft. L’ = liquid rate, lb./hr. ft. width of surface N = number of packing pieces per cubic foot, ft.-3 S, S’ = shape factors, dimensionless. U = superficial liquid velocity, ft./sec. V = liquid velocity, ft./sec. n, F, k = dimensionless constants GREEKLETTERS A = film thickness, ft. E = void fraction, dimensionless = angle of inclination of surface to horizontal 8 = liquid dynamic viscosity, lb./(ft.) (sec.) fi = liquid kinematic viscosity, sq. ft./sec. V = liquid density, lb./cu. ft. P literature Cited ( I ) Acton, F. S.,“Analysis of Straight-Line Data,” Wiley, New York, 1959. (2) Davidson, J. F., Trans. Znst. Chem. Engrs. 37, 131 (1959). (3) Elgin, J. C . , Weiss, F. B., Ind. Eng. Chem. 31, 435 (1939). (4) Jesser, B. W., Elgin, J. C., Trans. A m . Znst. Chem. Engrs. 39, 277 (1943). (5) Liang-Tseng Fan, Can. J . Chem. Eng. 38, 138 (1960). (6) Nusselt, W., Z . Ver. Deut. Zng. 60, 541 (1916). (7) Otake, T., Okada, K., Kagaku Kogaku 17, 176 (1953). (8) Rose, 24.,LVelshans, L. M., Ind. Eng. Chem. 32, 673 (1940). (9) Shulman, H. L., Savini, C. G., Edwin, R . V., A.I.Ch.E. J . 9, 479 (1963). (10) Shulman, H. L., Ullrich, C. F., W’ells, N., Ibid., 1, 247 (1955). (11) Shulman, H. L., Ullrich, C. F., Wells, N., Proulx, A. Z., Ibid., 1, 259 (1955). (12) Lchida, S., Fujita, S., J . SOC.Chem. Ind. ( J a p a n ) 39, 876, 432B (1936); 40, 538, 238B (1937); 41, 563, 275B (1938). (13) White, A . M . , Trans. A m . Inst. Chem. Engrs. 31, 390 (1935). RECEIVED for review October 10, 1966 ACCEPTEDMarch 27, 1967 VOL. 6
NO. 3 A U G U S T 1 9 6 7
407
