17
Ind. Eng. Chem. Fundam. 1980, 19, 17-21 Sladek, K. J., Gilliland, E. R., Baddour, R. F., I d . Eng. Chem. Fundam., 13, 100 (1974). Suzuki, M., Smith, J. M., Chem. €ng. Sci., 26, 221 (1971). Tamaru, K., Adv. Catal., 15, 65 (1964). Villermaux, J., Matras, D., Can. J . Chem. €ng., 51, 636 (1973). Youngquist, G. R., Allen, J. L., Elsenberg, J., rnd. ~ n gchem. . prod. Res. BV., 10, 308 (1971).
Received f o r review October 27, 1978 Accepted September 13, 1979 The authors gratefully acknowledge the support of the National Research Council of Canada and the J. W. McConnell Foundation.
Methods of Interpreting Transient Response Curves from Dynamic Sorption Experiments Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on September 11, 2015 | http://pubs.acs.org Publication Date: February 1, 1980 | doi: 10.1021/i160073a003
Agyapal S. Oberol, 0. Maynard Fuller,’ and James F. Kelly Department of Chemical Engineering, McGill University, Montreal, Quebec H3A 2 A 7, Canada
Two new methods for obtaining sorption and diffusion parameters are based on an integrodifferential form of the conservation equation for the sorbate in the bulk gas phase of a “gradient-free’’ sorption vessel. Both methods and the method of moments are evaluated with respect to the agreement of theoretical and experimental response curves, the variability within sets of parameter values, linearizationerrors, and computing effort. They are about equal in providing agreement between theoretical and experimental curves, but all three suffer from significant linearizationerrors when used for sorption isotherms that include a pronounced “knee”. When programming and computing effort are important, the method of moments is preferred: if not, direct fitting of the response curve is preferred.
Introduction Several methods of interpreting the response curves that are obtained from transient sorption experiments have recently been reviewed in this journal by Ramachandran and Smith (1978). This paper is a report on two new methods and a slight variant of an old one. We are particularly concerned with the reproducibility of the original transient response curves, the variability of the results, the handling of linearization errors, and computing effort. All of the methods discussed here will be illustrated using Kelly’s (1975) experimental data and the mathematical model of Kelly and Fuller (1980). The Representation of Response Curves by Empirical Equations We found that Kelly’s experimental curves could be represented with reasonable accuracy by functions of the form 7 = -aiexp(-bit) ti-1
5t
e
ti; i = 1, 2,
(1)
...
where 1 + 7 is the dimensionless response to a step input. The constants ai and bi for each segment were found by linear regression of In (-7) on t. Most of the curves could be represented by two line segments and none required more than three. Examples of the approximations that were combined for typical experimental response curves are presented in Figure 1. An empirical representation such as eq 1 removes the noise in an experimental curve. This permits the curve to be differentiated without amplifying the noise and it permits the curve to be extrapolated to very large values of time, for which the signal-to-noise ratio is poor. Both of these features help in interpreting the experimental curves. 0019-7874/80/1019-0017$01 .OO/O
Methods DTF and ITF. For method DTF, one of the two differential equations in the model of Kelly and Fuller is solved analytically (see Appendix) to give
-d?l + - -*+ +7 dt (2)
and 7(0+) = -1. This integro-differential equation is then solved numerically to give a function 7=
f(4;D , J,)
(3)
that is made to agree closely with an experimental response curve by varying the parameters D and J,. When L 2 / Dis small, the solution is insensitive to changes in the value of D and the integral in eq 2 may be approximated to permit an analytical solution 7 = -exp[-t/[~+
+ 7*($ + tm)]]
(4)
in which J, is varied to fit the experimental response data. For method ITF, eq 2 is first solved for J, + t,
By differentiating with respect to time, we obtain
7’7’’) = 0 (6) 0 1980 American Chemical Society
18
Ind. Eng. Chern. Fundam., Vol. 19, No. 1, 1980
Table I. Moment Eauations for pairs of moments
il,
and
D
equations
+ e m = (p1 - 7+)/‘r* (loa) D = (2Lz/3)(b, - 7+)/(pZ - 2 ~ ~ ’ ) ( l o b ) p 1 and p 3 & + Em = (wl - T + ) / T * (Ila) D is found from the positive root of ( p 3 - 6 p 1 3 ) 0 - 4Lzpl(p1- 7+)D(llb) (4L4/5)(p1 - 7’) = 0 u. and u. b and D are found bv trial from p1 and p 2
\ ------
EXPERIMENTAL RESPONSE CURVE
I
FITTED CURVE (EO. 1)
+
~ * ( ie m ) l
R U N IO T = 108.5’ C
-
Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on September 11, 2015 | http://pubs.acs.org Publication Date: February 1, 1980 | doi: 10.1021/i160073a003
= 0.897
0.011
100 t,
SECONDS
200
300
I
Figure 1. Comparisons of experimental response curves with empirical curves from eq 1. Semilogarithmic coordinates.
This equation does not contain $, so it may be solved to obtain values of the parameter D, provided that q, q‘, and q” are available. If q” is not available, we can eliminate $ from eq 5 to give
where tl and t 2 are two different values of time. After D is found from eq 6 or 7, $ is obtained from eq 5. Both eq 5 and 6 (or 7) must be solved for a specific value oft. For best results this values of t should be chosen so that 0.1 < -q(t) < 0.5. When L2/D is less than or approximately equal to 7+, a value of D cannot be obtained by method ITF. In this case, $ may still be determined from eq 4 or from q $+€m=--
+ r+q‘ T*q’
(8)
Although the equations that are used for methods DTF and ITF are essentially the same, the procedures are quite different. In the latter, the two parameters are not determined together but sequentially, and the theoretical response curve is not directly compared to the entire experimental curve. However, the effect is the same so that method ITF is essentially an indirect means of fitting an experimental response curve in the time domain. Methods M a n d MS. The method of moments, originated by Kubin (1965) and Kucera (1965), gives the relations shown in Table I between the moments of the response to a pulse input and the sorption and diffusion parameters. Using Jeffreson’s (1970) relations between the moments of responses to a pulse and a step input, the moments of the response to a pulse input are also given by pn =
Lt‘tn(n+
l)(-q(t))dt
Theoretically, tf should be infinite in this equation, but the integration must be stopped at finite values of tf if an experimental response curve is used. Sater and Levenspiel
+ 6w + 7 * ( $ + 5 D 2 ) ( 1+
Em)
(1966) avoided this problem by fitting an exponential decay function to the tail of the response to a pulse input. The contributions of the tail of the response curve to the moments were then determined analytically. A simple extension of this idea is to obtain the moments from an empirical representation of the response curve, such as eq 1. The method of moments will be called method M when an experimental curve is used and method MS when eq 1 is used. According to Anderssen and White (1971), Wakao and Tanaka (1973), and Rony and Funk (1971), the factor tn in eq 9 and the low signal-to-noise ratio for large t combine to make the higher moments inaccurate and dependent on the finite upper limit of integration in method M. The avoidance of this problem by method MS raises the question whether the values of $ and D calculated from one pair of moments are consistent with the values calculated by another pair. We compared the values of $ and D calculated from the first and second moments to those calculated from the first and third moments by the equations shown in Table I. The values of $ are the same because eq 10a and l l a are the same. For Kelly’s experiments, the root mean square of the percentage difference in the values of D was 2.6%, and the largest difference was 10.7%. Spot checks of the values of J/ and D obtained from the second and third moments indicate that these are also within a few percent of the values obtained from the other two pairs of moments. The consistency in the sets of values of and D shows that the errors in fitting eq 1 to the experimental response curves were reasonably small. Agreement with Experimental Response Curves A useful criterion for evaluating methods of interpreting response curves is that the theoretical curves generated from values of $ and D should agree with the experimental curves from which they were derived. Examples of typical response curves generated using eq 2 are shown in Figure 2. Method DTF satisfies the criterion BS a matter of course, because eq 2 is used in this method. The computed and experimental curves match quite well for all of Kelly’s experiments. If they did not, this would show that the mathematical model was not adequate. For example, the assumption of negligible diffusional resistance leads to eq 4, but the response curves obtained from this equation only match the experimental ones for very thin pellets of sorbent. The root-mean-square differences between an experimental response curve and computed curves are shown for several experiments in Table 11. The rms differences for methods DTF, ITF, and MS are not very different over wide ranges of $ and D. The rms difference for method M is sometimes about the same as the others, as in experiment 166, but more often it is much larger, as in experiment 14.
+
Ind. Eng. Chem. Fundam., Vol. 19, No. 1, 1980 I
-
RUN 14 T = 1565°C
X
EQUATION 4
-
METHOD M--.-----. METHOD ,TF ...............
-?
METHOD DTF & MS--EXPERIMENTAL-
+ E,,,)/D,~~~
Table 111. Variability of %‘(I)
method
n
mean
std dev
95% conf int of the mean
M MS,
36 49 49 48 49
1.105 0.986 0.980 1.045 1.018
0.441 0.234 0.216 0.200 0.124
t0.149 i0.067 i0.062 t0.058 i0.036
MS,
= 0.306
ITF DTF
10
a F o r all cases h2 = 4. Values of could not be calculated for all experiments by methods M and ITF. The results for method M are from Fuller and Kelly (1980). The first and second moments were used for MS, and the first and third moments were used for MS,.
Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on September 11, 2015 | http://pubs.acs.org Publication Date: February 1, 1980 | doi: 10.1021/i160073a003
Table IV. Estimates of Linearization Errors
1 METHOD ITF--
\
’\
1 , SECONDS
Figure 2. Comparisons of experimental response curves with theoretical response curves from several methods. The agreements for the curves from method DTF are typical of the results for this method. Table 11. Comparisons of Response Curves
expt no.
no. of points method
9
7
24
8
30
7
14
10
166
10
DTF ITF MS DTF ITF MS DTF ITF MS DTF ITF MS M DTF ITF MS M
b,cmz/ $
s x IO3
5.0 4.9 5.2 28.6 28.4 27.5 8.8 8.8 9.1 40.0 40.8 39.9 38.6 12.6 13.0 12.7 12.8
4.2 2.8 5.2 0.92 1.13 0.99 3.8 2.1 2.7 0.50 0.61 0.58 0.90 2.5 2.5 2.3 2.3
rms diff from exptl curve X 10’ 1.3 1.2 1.5 0.37 0.33 0.31 1.1 1.5 1.2 0.69 0.82 0.38 2.6 0.95 0.79 0.98 0.98
Variability For macropore diffusion and sorption of hydrocarbons in zeolites, Youngquist _et al. (1971) and Ruthven and Derrah (1972) show that D is related to as well as gaseous diffusivity and pore structure. In the notation of Kelly and Fuller (1979)
mean
maximum
expt no.
temp, “C
ITF
MS
ITF
MS
3-11 13-21 22-33 160-168 170-179
108.5 156.5 204.5 233.5 252.5
0.13 0.06 0.02 0.05 0.03
0.26 0.16 0.10 0.06 0.06
0.34 0.09 0.05 0.07 0.04
0.41 0.26 0.21 0.10 0.08
Linearization Errors The combination of truncation errors made in the linearized eq A2d and noninfinitesimal step inputs leads to systematic errors in $ and D. Ruthven and Derrah (1972) have shown how the apparent diffusivity changes with increasing step size for fitting in the time domain, and Kelly and Fuller (1979) have given an error bound on rC, for method M. The size of the relative error in 1c/ is a third criterion for evaluation. Let the error in $ be represented by
p
=
p-p
(14)
and the truncation error of eq A2d be
where
Then, estimates of the relative error in $ are given by
+
where t, and X2 are constants that depend on pore structure. This equation provides another criterion: the variability of R within sets of $ and D values obtained by one method of interpret,ing response curves. The standard deviation and confidence limits of R obtained from Kelly’s response curves using the four methods are shown in Table 111. Method DTF gives the smallest standard deviation and method M the largest, while methods MS and ITF give standard deviations that are about equal and between the other two.
for method ITF and
for methods M and MS. In eq 16, the same value of t used to calculate p by eq 5 must be used. The means and maxima in the relative errors for several sets of Kelly’s data are given in Table IV. This comparison shows that method ITF is better than method MS by this criterion. A comparison between the sets of experiments a t lower and higher temperatures shows that the step inputs must be made smaller as the “knee” of the isotherm becomes more pronounced at lower temperatures.
20
Ind. Eng. Chem. Fundam., Vol. 19, No. 1, 1980
Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on September 11, 2015 | http://pubs.acs.org Publication Date: February 1, 1980 | doi: 10.1021/i160073a003
Although we have not presented an error analysis of method DTF, we would expect its performance to be similar to that of method ITF because both are based on eq 2. If detector performance forces step inputs to be larger than desired for accuracy, the effect of truncation errors may be reduced by averaging the values of $ from a pair of experiments at each steady-state concentration. A positive step input would be used in one experiment of each pair and a negative step input would be used in the other. In the estimates of the error of the average $ the terms involving $(I) have the opposite sign and partially cancel one another. This means of reducing the systematic error due to linearization should work equally well for both methods MS and ITF.
Computing Effort The determination of parameters by curve fitting in the time domain may have the disadvantage of requiring an excessive amount of programming and computing time. For example, differential equations Ala and A2a are linked through the time-dependent boundary condition A2d. A completely numerical solution requires the repetitive solution of eq A2a at each step in the solution of eq Ala. Then, the solutions of eq Ala must be varied in-a twodimensional search for the best values of $ and D. One of the virtues of method DTF is that it avoids the repetitive numerical solution of eq A2a. Methods MS, DTF, and ITF all require the determination of the constants for eq 1. After this, the programming and computing efforts increase in the order MS, ITF, and DTF, but all three are relatively small. The moments for method MS were calculated using a 14-statement program. For method ITF, eq 6 was solved for D (by bisection) and eq 5 was solved for $ + t, using two 56statement programs. For method DTF, eq 2 was solved for 7 as a function o f t using a 40-statement program. All programs were written in BASIC and were run on a Wang 2200 microcomputer. The computer time for method DTF was considerably larger than the other two, but only because of the two-dimensional search for the best parameter estimates. Conclusions Methods DTF, ITF, and MS are about equal in providing agreement between experimental and theoretical response curves and all of them suffer from significant linerization errors when used for sorption isotherms that include a well defined "knee". The linearization error may be reduced by reducing the size of a step input or by averaging the results from responses to positive and negative steps at the same steady-state concentration. If computing effort is important, method MS should be preferred among the methods considered here, and, if not, method DTF should be preferred, because it gives a smaller variability of R within sets of experiments. The combinations of the first and third moments and the first and second moments in method MS gave results that were nearly the same. Appendix Equations 2,5,6, and 7 of methods DTF and ITF are derived from the mathematical model of Kelly and Fuller (1980). It has been rewritten below in terms of deviations from the final steady state because this leads to smaller linearization errors in method ITF.
aw - (t,O) = 0 az
where
and M = the difference between the final and initial steady-state values of Co. We transform eq A2 to obtain tanh p 2=1
where p 2 = sL2/D. An approximation to the inverse for small L2/D is dw -(t,l) dz
dt
L2
$
az
(Ala)
$L2 dY0 =-
D
dt
The inverse of eq A5 without approximation is
where 64(S;q) is the fourth Jacobian 6 function, q = exp(-L2/Dt), and a change of variables has been made to eliminate the factor, t-'J2, from the expression under the integral. For all of the equations of this paper, { = 0, so we suppress the first argument and only write 64(q). The infinite product form of the function 64(q) (Rainville, 1960) should be used for calculation because of the large round-off errors and slow convergence of the infinite series form for L2/Dt C 1. The substitution of eq A7 into eq Ala and rearrangement gives eq 2.
Nomenclature ai,bi = empirical constants in eq 1 C, = concentration of sorbate in,the gas of the macropores of the porous sorbent, g-mol/cm3 macropore volume C, = concentration in the micropores of zeolite crystals, gmol/cm3 micropore volume D = effective diffusivity defined by eq A4, cm2/s D, = gaseous diffusivity of propylene in macropores, cm2/s D,= surface diffusivit of propylene on the boundary surfaces of macropores, cmP/s e = the truncation error made in linearizing the sorption isotherm G(C,) = the representation of C, as a function of C,; sorption isotherm L = pellet depth, cm M =-the maximum value of yo D = (L2S/D)1/2 q = expi-Li/Dt) R = D X 2 ( + + cm!/Dgt, s = Laplace variable, s-l t = time, s tf = upper limit of integration in eq 9 L~
rdY +o =Dr* - rei + ern a w yo(t) - - -- (t, 1);t > 0
(-45)
Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on September 11, 2015 | http://pubs.acs.org Publication Date: February 1, 1980 | doi: 10.1021/i160073a003
Ind. Eng. Chem. Fundam. 1900, 19, 21-26
ti = empirical constants in eq 1 tl,t2 = values of time used in eq 7 u = deviation of C, from its final steady-state value, g-mol/cm3 micropore volume w = E,,u, g-mol/cm3 pellet volume x = mole fraction corresponding to y o yi, yo = deviations from their final steady states of the concentrations of sorbate in the gas streams flowing into and out of the sorption vessel, g-mol/cm3 Yo' = dY,/dt z = dimensionless distance from the closed end of the zeolite pellet Greek Letters em = macropore porosity of a pellet, cm3macropore volume/ cm3 pellet volume E,, = micropore porosity of a pellet, cm3micropore volume/cm3 pellet volume { = first argument of the Jacobian 0 function, e4 t) = Yo/M t)',~'' = dv/dt and d2q/dt2,respectively 0 ({,q) = the fourth Jacobian 0 function (see Rainville, 1960) Ai = correction factor for the tortuoisty and diameter variations of macropores kn = the nth moment of the response to an ideal pulse input 5 = dummy variable in eq A7, s1j2 T * , '7 = time constants for the sorption vessel (see Kelly and Fuller, 1980) I,L = c,(aG/aC,)~ (g-mol sorbed/pellet volume)/(g-mol in gas/gas volume) Subscripts
21
i = inflow = outflow
0
Superscripts a = approximate t = true t = error - = the final steady-state value of a variable Literature Cited Anderssen, A. S., White, E. T., Chem. Eng. Sci., 25, 1015 (1970). Jeffreson, C. P., Chem. Eng. Sci., 25, 1319 (1970). Kelly, J. F., Ph.D. Thesis, McGill University, Montreal, 1975,copies available from the National Library of Canada, Ottawa. Kelly. J. F., Fuller, 0. M.. Ind. €no. Chem. Fundam.. Drecedina article in this issue, 1980. Kubin, M., Collect. Czech. Chem. Commun ., 30, 1 104 (1965). Kucera, E., J. Cbromafogr., 19,237 (1965). Ralnville, E. D., "Special Functions", Macmillan, New York, 1960. Ramachandran, P. A., Smith, J. M., Ind. Eng. Cbem. Fundam., 11, 148 (1978). Rony, P. R., Funk, J. F., J . Cbromafogr. Sci.. 9, 215 (1971). Ruthven, D. M., Derrah, R. I., Can. J . Chem. Eng., 50, 743 (1972). Sater, V. E., Levenspiel, O., Ind. Eng. Chem. Fundam., 5, 86 (1966). Wakao, N., Tanaka, K., J . Chem. Eng. Jpn., 8, 338 (1973). Youngquist, G.R., Allen, J. L., Eisenberg, J.. Ind. Eng. Chem. Prod. Res. Dev., 10, 308 (1971).
Received for review October 27, 1978 Accepted September 13, 1979 The authors gratefully acknowledge the support of the National Research Council of Canada, the J. W. McConnell Foundation, and Mrs. R. B. Fuller. We acknowledge with pleasure the suggestions and criticisms of Professors J. Grace and H. de Lasa and Mr. N. Nguyen-Dinh.
Gas Holdup and Bubble Diameters in Pressurized Gas-Liquid Stirred Vessels 1.Srldhar and Owen E. Potter" Depaltment of Chemical Engineering, Monash University, Clayton, Victoria, 3 168 Australia
The effect of system pressure on gas holdup has been experimentallydetermined in a stirred gas-liquid reactor. The effect of increased gas density is to increase gas holdup. The contribution of the power supplied through the gas stream to the total power dissipated in the stirred vessel increases as the pressure increases, other things being equal. Bubble diameters decrease as pressure is increased. Measurements at varying pressures (and hence varying gas kinetic energies) but at constant gas volumetric flow rate and temperature were used to study the effect of kinetic energy on the characteristics of the gas-liquid dispersion. The data were correlated by a simple modification of Calderbank's equations. The equations presented reduce to Calderbank's equation at atmospheric pressure. The fact that the same correction factors enter in the equations for interfacial area, gas hoklup, and bubble diameters lends confidence to the recommended equations. Since data were obtained with one small vessel only, caution is required in scaling up to larger vessels.
Introduction Gas-liquid mass transfer and reacting systems constitute an important processing tool. Knowledge of the surface area of gas-liquid dispersions of holdup of gas in the dispersion and of mean bubble diameter and mass transfer rates are of fundamental importance. Such systems may comprise bubble-towers without agitation other than by the bubble-stream or agitated vessels where the liquid is circulated throughout the vessel by the agitator with or without turbulence depending on the degree of agitation. 00 19-7874/80/10 19-0021$0 1 .OO/O
Holdup, bubble diameter and surface area are related a=-
6H
DBM While there has been ample experimentation at atmospheric pressure, determinations of the above properties in pressurized agitated systems have not been made, so that the available models all involve extrapolation from atmospheric pressure operation to pressurized operation. The most notable feature of pressurized operation is that 0 1980 American
Chemical Society
