I
JOHN E. POWERS School of Chemical Engineering, University of Oklahoma, Norman, Okla.
Transient Behavior of Thermal Diffusion Columns Numerical analysis of batch thermal diffusion columns should be useful for planning experiments and interpret ing ex per iment a I data
yielded a value of the thermal diffusion constant in agreement with results of other investigations. A similar analysis qualitatively supported the predicted influence of temperature difference and plate spacing on column performance.
Theoretical
A
thermogravitational thermal diffusion columns currently used for separating materials and testing phenomenological theories of column behavior are of the batch type without reservoirs. In spite of this fact, the theoretical analyses of transient behavior of such columns reported in the literature apply only over limited concentration ranges. [Since this article was first submitted, an analytical solution of an analogous equation has been reported ( 8 ) , which can be applied in analyzing transient behavior of batch columns without reservoirs over the entire concentration range. ] None of the analytical solutions are general enough to aid in interpretating much of the data currently appearing in the literature. Therefore, it was decided to solve the general nonlinear partial differential equation describing such column behavior by numerical means using a medium speed digital computer. Experimental results obtained in three laboratory columns could then be compared with results of the numerical computations to check predicted transient behavLor as well as effects of important operating variables. In carrying out this combined theoretical and experimental investigation, a general computer program was developed to predict transient behavior of batch thermogravitational columns without reservoirs for any size separation with an initial charge of any composition. Calculations were made for an initial charge of an equicomposition mixture over the entire range of interest and the results presented graphically. Comparison of thermal diffusion data and theory is complicated in that results of mathematical treatment of the theory are not simple in form and agreement is not quantitative. Over the range of variables studied, both steady state and transient solutions of the equation describing column behavior were entirely adequate for correlating experimental data using two empirical constants for each set of data. Comparison of these empirical constants with theoretically predicted values MAJORITY
O F THE
The “transport equation” of Furry and others ( I ) relates the net amount of one component in a binary mixture which passes through a cross section normal to the walls of a thermogravitational thermal diffusion column, 71, to the fractions of the two components a t that cross section, C1 and CZ, to the gradient of the concentration in the direction parallel to the walls, and to two parameters, H and K , which are dependent on the column dimensions, operating conditions, and properties of the mixture being separated: 71
= HClCz
- K(dCl/ay)
(1)
The transport equation can be combined with continuity conditions to analyze the transient behavior of a thermogravitational column, yielding :
Boundary conditions corresponding to reasonable experimental arrangements are : at t = 0, CI = COf9r a l l y
(3)
at y = 0, C1 = CQfor all t
(4)
at
y = L, dCl/dy = H C I C J K
(5)
The bouLiclary condition at y = 0 implies a degree of symmetry and was used to reduce the number of calculations; it defines the pointy = 0. This point is located a distance from the top of the column, L,, equal to a fraction of the total column length, LT,
calculated using the relation:
For C, = 0.5, y = 0 is located at the vertical midpoint of the column. Equation 2 and the boundary conditions were adapted for numerical solution, and a general program was written for the IBM 650 computer with indexing accumulators and floating decimal unit. The program is quite flexible and yields as output the dimensionless time, 5, and the fraction of component 1 a t five equally spaced positions in the column. These positions correspond to the location of sample taps in the laboratory columns when y = 0 is at the vertical midpoint of the column (CO= 0.5). No analytical tests for convergence or stability were applied to the nonlinear difference equations used in the computations, but sufficient calculations were made to establish proper limits on the increments’ size and values of moduli appearing in the difference equations. T h e solution agrees with results of a restricted analytical solution ( 4 ) as A .--$ 0. I t was convenient to present the results of numerical computations in terms of five dimensionless parameters corresponding to the concentration of charge material (Co), position in the column (A’), separation (A/Am)x, time ( E ) , and the column and system parameters ( A ) . All calculations were restricted to C, = 0.5. Individual graphs for X = 0.2, 0.4, 0.6, 0.8, and for X = 1.0 (Figure 1) corresponding to the locations of five sample points were prepared. Calculations were made for values of A from .O to 1500, and the results were presented as two sets of graphs, one set for ‘4 < 20 and the other set for 20 < A 6 1500.
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I I I I I I I I I 1 I I I I I
the complete manuscript, including all equations and derivations, figures, tables, and additional text, by Powers. After one year this material can be obtained from the AD1 Auxiliary Publications Project, Library of Congress, Washington 25, D. C., as Document No. 6668. The price will be $3.50 for microfilm and $10.00 for photostat copies.
Clip and mail coupon on reverse side. VOL. 53, NO. 7
JULY 1961
577
that graphical results for C, = 0.5 can be applied in conjunction with Equation 6 to estimate column performance for CO# 0.5. Experimental Equipment. Three concentric-cylinder columns, each 6 feet long, were used. Basic construction has been described (3). Columns were modified by adding eight additional thermocouples and rewrapping, using alternate layers of insulating material and aluminum foil. 'These modifications permitted greater accuracy in determining average column temperature, T , and temperature difference A T and provided for more even distribution of heat in the column. Cooling was provided by circulating water from a large constant temperature bath. The 10 sample taps were modified by replacing sample probes with hypodermic tubing attached to a syringe fitting and stoppered with a Teflon plug to facilitate sampling and eliminate loss of solution from sample taps. The annular spacing (2w) is the mosl influential variable, and care was taken to obtain accurate measurements and correct for variation with changes in A T . Values of 2w for the three columns at 38' C. and A T = 0 are 0.0309, 0.0750, and 0.1142 cm. System. The system investigated was 50 mole yo a-heptane-benzene. The n-heptane was Phillips "pure grade,'' and the benzene was Baker Analyzed reagent grade! 0.4 C. boiling range. Ne?Lher showed any appreciable difference in index of refraction when treated individually in a thermal diffusion column. Physical properties for the system were from the literature ( 4 ) . Procedure. The column was clean, dry, and as nearly isolhermal as possiblc before feed was charged to the column to initiate the run. After filling the column, flow of cooling water \vas started and heating initiated at a higher than normal rate to minimize the time required to attain thermal steady-state (approximately 1 minute was required). A much longer time was required for the column to attain a steady-state concentration profile. During this transient period, samples were withdrawn periodically from two sample taps, each equidistant from the center of the column. at as nearly the same time as possible. TZ'hen the concentration distribution reached steady state: samples were taken at each of the 10 sample taps. Samples, maintained as small as possible at 0.1 ml., were analyzed using an Abbt refractometer at 25' C. Calibrated a t that temperature using experimental reagents. Temperature measurements were made using a Brown potentiometer Model No. 156x15~-x-C. Mean temperature was maintained at 100 O F. for all experiments. Results. Transient and steady state
-
f
DIMENSIONLESS
TIME
Figure 1. Typical graph shows results of numerical solution of Equation 2 in dimensionless form Some interesting observations concerning the predicted effect of several important operating variables on the time required to approach equilibrium were made, based on study of the graphical results. These observations (Tablr I) are based on the assumption that mass transfer by convective flow within the column is much greater than mass transfer by diffusion ( K , >> K d ) . This assumption is suspect for A > 20. Example Calculation. Graphs representing results of numerical computation can be used to predict transient behavior of batch columns wirhout
Table 1. Effect of Column Variables on Time to Attain 95% Equilibrium Separation Depends on Value of A Effect on t,, of Plate
A 0-1 %20 >20 >20
Sample Taps All All X 6 0.8
X
= 1.0
Length
L2 L1 L' Lo
Temp. diff. (AT)--l (AT)-* (AT)-z (AT)-z
reservoirs which are initially filled with an equiromposition binary mixture. Thus, the length of time required to attain steady staie conditior's in a column can be estimated, the concentration profile in a column at any time calculated. and other estimates made as required. The importance of such estimates is illustrated by considering the time required to approach equilibrium under typical operating conditions, as influenced by the value of the plate spacing ( 2 w ) (Table 11). Although no calculations were made for values of C, # 0.5, the program is general enough to make such computations. If 0.3 < C, < 0.5, it is believed
Table II. Time Required to Attain 95% Equilibrium Separation Is Strongly Influenced by Plate Spacing
spaciiig 2wP6 ZW-~ 2w-2 2w"
2 w , Cm.
tm
0.01 0.02
5 . 4 mo.
0.04
0.07 0.10
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578
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INDUSTRIAL AND ENGINEERING
CHEMISTRY
5 . 5 wk. 13 days 41 hr. 4.8 hr.
THERMAL DIFFUSION COLUMNS data were obtained over a range of temperature differences ( A T ) from 2 " to 35' C. Over 50 runs were made with over 1000 samples (Figure 2 ) .
In addition to the tables and figures contained in this condensed version, the unabridged manuscript (see coupon) contains those listed below. Supplementary material (transient data and concentration ptofiles at steady state) i s available now from the American Documentation Institute (see footnote a t end of article). TABLES
Interpretation of Results To draw conclusions from the large mass of data, data reduction was based on results of the theoretical analysis. Steady State. At steady state (&',/at) = 0 and the analytical solution of Equation 2 is:
[*)Im=
in[?]
X K HL- X A
Illustration o f the Use o f Figures 2-1
3
Typical Experimental Results (Run No.
37)
To Predict the Time Required to Achieve Steady State in Columns with Various Plate Spacings To Predict the Concentration Profile in a Column at Any Time
Summary of Experimental Conditions Summary of Analysis of Data Obtained a t X = k0.8 (Sample Taps 2 and 10)in Run No. 37 Complete Listing of Empirical Parameters Obtained by Correlating Data within Theoretical Framework
Estimate o f the Concentration Profile In the Enriching Section In the Stripping Section
Column Dimensions a t
38" C.
(7)
FIGURES Therefore, a plot of In us. X should yield a straight line with slope
equal to A E HL/K a n d intercept o f In [ ( I CO)/COl. In most experiments, three steadystate traverses were obtained. These data yielded a straight line relation and values of A and CO,calculated in most cases by fitting the data to a straight line by the method of least squares. Transient Data. T h e values of A determined from measurements made after the column reached a steady state separation were used to compare data obtained under transient conditions with results of the numerical solution of Equation 1. Values of (Am)x corresponding to a particular pair of sample taps (X)were calculated using Equation 7 and used in combination with experimentally determined values of A, to calculate (A/Am)x. These values, together with the previously determined value of A , were used to determine values of dimensionless time, $, from results of the numerical calculation. For every value of [ thus determined, a corresponding value of actual time, t, had been recorded during the experimental run. From the definition of $, the ratio t / [ should be a constant, independent of time and position in the column for any particular set of operating conditions :
-
t / E = 4pL2/?raK
~~~~
X f0.2 f0.4 f0.6 f0.8 11.0
Weighted av.
Dimensionless Separation Factor, A, as a Function o f Plate Spacing ( 2 w ) and Temperature Difference (AT) for the System Benzene a t 31 1 O K. (LT = 6 Feet) Ratio o f Actual Time, t, t o Dimensionless Time, E, as a Function o f Plate Spacing ( 2 w ) and Temperature Difference (AT) for the System n-Heptane-Benzene a t 31 1 K. ( I T = 6 Feet)
t/&
Min.
1417 1318 1150 1176 1447 1192
(24 On On
H K
for comparison with data plots. I n these calculations, the value of A (as determined from the steady-state data) and a weighted average value of t / { were used in conjunction with Equation 7 and results of the numerical solution of Equation 1 to calculate theoretical curves for all five pairs of sample taps (Figure 2). Other data showed similar degrees of correlation.
to represent such data by using an average value for the set. T h e variation in the average values of t/E from data obtained a t the five sets of sample taps were likewise random (Table 111). Other results show similar variation with sample tap location (Table IV). T h e degree to which empirical A and t / [ values correlate experimental data is shown by calculating theoretical curves __
Correlation of Data with Results o f Numerical Computation Indicates Good Agreement between Theory and Experiment
Table IV.
Kexptl.3
RW3
t/$,
No.
X
A
28
zt0.6
0.121
32
11.0 10.8 f0.6 f0.4 10.2
0.629 0.629 0.629 0.629 0.629
34
11.0 f0.8 &0.6 f0.4 f0.2
7.03(?) 7.03(?) 7.03(?) 7.03(?) 7.03(?)
43
11.0 10.6
0.121 0.121
~
Table 111. Average Values of t / [ for Five Sample Taps Show Random Variation
-
For l o w Values of A For A >20
(8)
The variation in t / [ calculated for any pair of samples taps was random and strongly indicated that the form of the numerical solution was adequate
Plot of Steady State Data to Determine A and CO Illustration o f Use o f Results o f NUmerical Solution t o Determine Values of Dimensionless Time, E, from EXperimental Values of (A/Am)x 0.8 (Run No. 37,A = 0.60) Comparison of Experimental Data with Values Calculated by Computer Program and Plotted on IBM 407 Printer Plot of adH VS. d K to Determine a and Relation Between d H and 4K Experimental Check on Theoretically Predicted Effect of Plate Spacing
Half-Element at the End o f a Batch Thermogravitational Thermal Diffusion Column Unsteady-State Separation in a Batch Thermogravitational Column a t X = 1.0, 0.8, 0.6, 0.4, and 0.2
Hemt~. , Gram/Sec.
Min.
X lo4
17.62
17.39
586 501 526 503 611
1,743 2.039 1.941 3.031 1.672
3608 4603 5172 6215 6807
1.418 1.112 0.989 0.823 0.752
125.8 82.9
2.445 3.710
GramCm./Sec.
x
103
1310 25.33 29.63 28.20 29.51 24.30 1.842 1.444 1.285 1.069 0.977 184.3 279.7
VOL. 53, NO.
Y
4 H
4K
0.6855
0.6145
0.9489 1.110 1.0565 1.106 0.9104
0.8521 0.9966 0,9487 0.9927 0.8175
1.103 0.8646 0.7696 0.640 0.585
2.925 2.293 2.041 1.698 1.550
0.8506 1.291
0.7407 1.124
JULY 1961
579
030
I
I EXPERIMENTAL DATA
- THEORETICAL
- RUNX37
I
I
I
= thermal diffusion constant
--
=
-
CURVES
A * 0.60
=
Figure 2. Results of numerical cornputations provide accurate representation of experimental data
= =
= =
= =
TIME
-
4"""
MINUTES
92
Therefore, it was concluded that the lations predict that both H a n d K should two empirical constants A and t / [ adebe proportional to ( A T ) 2 ,and A should quately represent the experimental data be independent of AT. For the column (69 points including measurements made with the largest annular spacing, the at steady state for data shown in Figure data indicate that H is proportional to 2). Such data reduction permitted (AT)*J, K to (A73I.96, and A to (AT)OJ4. meaningful investigation of theoretically For the column intermediate in annular predicted effects of column parameters. spacing, both H and K are found to be T h e parameters H a n d K in Equation proportional to (A27l.85; A shows only 1 are more fundamental to thermal difslight variation with A T . Data on the fusion theory than either A or t / [ , third (smallest annular spacing) column Therefore, values of Hexpt1. and were obtained a t only one A T value. were determined from empirical values of Plate Spacing. According 10 the A and ti.$ using Equations 8 and the theory for the range of this investigation, definition A = H L / K . These experiH is proportional to ( 2 ~ and ) ~ K to mental values were compared with ( 2 ~ ) 7 . Data at A T = 31.4' C . for the and Ktheor,caltheoretical values? Htheor. three columns indicate that H is indeed culated from relations presented. Values proportional to ( 2 ~ ) ~but : K may be of cy are, in general, not known acproportional to (2w)C.3. Because data curately, and, therefore, the ratio obtained in the column with the smallest ( H / c Y ) ~ which ~ ~ ~ ~is. ,independent of a, plate spacing arc questionable, it is prewas calculated instead of H. sumptive to draw definite conclusions, and additional experiments are planned. Values of # K = Koxntl./~tt,eor. and = Hexpt~,/(H/~)~~,,r. as defined Nomenclature previously (5) were also calculated (Table IV). T h e fact that values of A = HL/K A, = HLT/2K # K are approximately equal to 1.0 in B = width of column (mean cirmost cases gives stong support to theoreticumference of annular space), cal developments. cm. Estimation of Thermal Diffusion C ( x ) = fraction of component 1 at Constant. I t has been proposed position X (5) that the empirical correction terms CI, Cz = fraction of components 1,2 in $ H and q!IK are related by: a binary solution = fraction of material charged '$" 4; (9) into the column = ordinary diffusion coefficient, at least for the limited range of column sq. cm./sec. operation used to obtain data with = acceleration of eravitv liquid systems. From a plot of cy4x us. GK on log-log graph paper, cy was determined using the two columns with the larger annular spacings. The value of a = 1.1 determined for this system is gram-cm./sec. in good agreement with results presented = one half total column length, by other investigators (2, 6, 7). The cm. data obtained from the column with = distance from top of column to the smallest annular spacing are well y = 0, cm. = total column length, cm. represented by Equation 9 and yield = exponent in Equation 9 essentially the same value of the ex= absolute temperature, ' K. 1.0) as data from the other ponent (11 = time, sec. columns. The value of cy determined = time required to attain 957, from these data is about 0.37. I t is equilibrium separation, sec. believed that the sampling rate was too = 9,'L high for this column. = axis parallel to plates in direcTemperature Difference. For all tion of normal convective cases investigated the theoreical reflow, cm.
=
580
INDUSTRIAL AND ENGINEERINS CHEMISTRY
-dp/bT, g~ams/ml./' K.
= difference in concentration in
column measured a t = X difference in temperature of hot and cold plate, ' K. coefficient of viscosity, cp. amount of solution per unit cclumn length = 2wBp, grams/cm. dimensionless time = n2K;4pL2 numerical constant densit)., grams,'ml. amount of component 1 passing through cross section of column normal to plates, grams/ sec.
PK
= =
w
= one half annular spacing of
m
=
IL!,tl ifftheor. K e x u t , . iKci,eor.
thermogravitational column, cm. subscript to denote conditions a t steady state
Acknowledgment
The suggestions and criticisms of C.
M.Sliepcevich and W. J. \'iavant were very helpful. John Stevens and Sohio Oil Co., Oklahoma City, Okla., ivere generous in donating personal and computing machine time in initial phases of this project, and computations were completed in the University of Oklahoma Computing Laboratory by hfargaret Crawley. Carl Crownover obtained most of the experimental data, S. S. Dhillonn programmed the data analysis for the computer, and F. C. Lforris prepared the figures. literature C i k d
(1) Furry, W. H., Jones, R. C., Onsager, L., Phys. Reo. 5 5 , 1083-93 (1939). (2) Heines, T. S.,Larson, 0. A., Martin, J. J., IND.EXG.CHEY.49, 1911 (1957). (3) Jones: A. L., Milberger, E. C.: Zbid., 43, 2689 (1953). (4) Powers, J. E., "Proc. Conf. Thermodynamic and Transport Properties Fluids? London, 1957;" Inst. Mechanical Engineers, London, 1958. ( 5 ) Powers, J . E., Univ. Calif. Radiation Lab., UCRL-1G18, 1954. ( 6 ) Powers, J. E., Wilke, C. R., -4.I.Ch.E. Journal 3, 213-22 (1957). (7) Trevoy, D. J.. Drickamer, H. G.; J . Chem. Phys. 17, 582-83 (1949). (8) Von Halle, E. H.. Y. S. .4t. Energy Comm. Rept. K-1420, 1959. RECEIVED for review November 13, I959 ACCEPTED March 28: i961 Division of Industrial and Enpineering Chemistry, 138th Meeting, XCS, New York, September 1960. Research initiated with funds from American Petroleum Institute ; continued with grant from National Science Foundation. Transient data and concentration profiles at steady state have been deposited as Document No. 6669 with the AD1 Auxiliary Publications Project, Photoduplication Service, Library of Congress, Washinqton 25, D. C. 4 copy may be secured by citing the document number and by remitting $2.50 for photoprints or $1.75 for 35-mm. microfilm. Advancr payment is required. Make checks or money order4 payable t o Chief, Photoduplication Service, Library of Congress.
