g = gravitational acceleration, ft/sec2 g, = Newton's law conversion factor, 32.2 f t lb,/lbf sec2 h = heat transfer coefficient, Btu/hr ft2 OF H L = height of stagnant bubble-free liquid above gas distributor, ft
HT = total height of bubble-agitated liquid above gas distributor, f t j factor for heat transfer, equal to NStNpro.6,dimensionless k = thermal conductivity of the liquid, Btu/hr f t O F p = average heat flux, Btu/hr ft2 Q = total heat input, Btu/hr AT = temperature difference between vessel wall and liquid, O F t = gas holdup time, sec U B = average bubble rise velocity relative to the vessel, ft/sec U s = superficial gas velocity through column ft/sec =
jH =
VIA
V = volumetric gas flow rate, ft3/sec VB = volume occupied by gas bubbles, ft3 VL = volume of bubble-free liquid, ft3 VT = total volume = VB VL, ft3 cy,p,y,b = empirical constants, dimensionless K = absolute viscosity of the liquid, lb,/ft sec 4 = fractional gas holdup = VB/VT, dimensionless p = density of the liquid, lb,/ft3 Ap = difference in density between the liquid and gas phases, lb,/ft3
Nw = N N =~ Np, = N R =~ Nst =
Froude number, Us2/Dg, dimensionless Nusselt number, hD/k, dimensionless Prandtl number, c,lk, dimensionless Reynolds number, DUs p / l , dimensionless Stanton number, hlc,Usp, dimensionless
Literature Cited Fair, J. R., Chem. Eng., (July 3 , 1967). Fair, J. R., private communication, Aug. 1966. Fair, J. R., Lambright, A. J.. Anderson, J. W., ind. Eng. Chem., Process Des. Dev., 1 ( l ) , 33 (1962). Hart, W. F., Oklahoma State University, "Heat Transfer in Bubble-Agitated Systems," 1965. Kast, W., lnt. J. Heat Mass Transfer, 5, 329 (1962). Kolbel. H., Borchers, E., Martins, J., Chem. ing. Tech., 32, 84 (1960). Kolbel, H., Borchers, E., Muller, K., Chem. ing. Tech., 30, 729 (1958a). Kolbel. H., Siemes. W.. Mass, R.. Muller, K., Chem. ing. Tech., 30, 400 (1958b). Konsetov, V. V., int. J. Heat Mass Transfer, 9, 1103 (1966). McAdams, W. H., "Heat Transmission," 2nd ed, p 186, McGraw-Hill. New York, N.Y., 1942. Novosad. Z.,Chem. Listy, 48, 946 (1954). Shulman, H. L., Molstad, M. C., hd. Eng. Chem., 40, 1058 (1950). Yoshitome, H. M., Kagaku Kogaku, 29 ( l ) , 19 (1965).
+
Received for review M a r c h 6,1975 Accepted July 16,1975 T h e bulk of t h i s work was done as a Master's degree research p r o g r a m a t Oklahoma State U n i v e r s i t y under t h e advisorship of Dr. K e n n e t h J. Bell.
The Development and Testing of a Mathematical Model for Complex Adsorption Beds James K. Ferrell;
Ronald W. Rousseau, and Marvln R. Branscome
Department of Chemical Engineering, North Carolina State University, Raleigh, North Carolina 27607
A simple general purpose model is developed for the dynamic behavior of fixed-bed adsorption systems. The model is based on an overall coefficient, the form of which results from the first moment of a step response to an adsorption process whose mechanism is controlled by one or more steps: for example, transfer to an adsorbent particle, pore diffusion, and surface reaction. It is shown that the model may be used even when insufficient data are available for the determination of the parameters associated with these resistances. The overall coefficient model is shown to be very useful in correlating the transmission curves of CCGN2 mixtures through beds of carbon impregnated foam material. Furthermore, the overall coefficient is correlated as a function of temperature, vapor concentration, and gas flow rate, giving the model a predictive capability. This approach has considerable potential in the design, scale-up, and control of adsorption processes.
Introduction The mathematical modeling of the dynamics of adsorption processes is well documented with extensive literature on the subject. While a number of significantly different approaches has been used, the most complete and useful for the kind of system discussed here is of the general form described by Schneider and Smith (1968) for a bed of particles in which adsorption is considered to occur as a threestep mechanism: diffusion from the bulk gas phase to the external surface of the particle, diffusion into the particle, and adsorption on the particle's surface. A number of variations of this type of model have also been published. Masamune and Smith (1965) have presented special cases that describe the above mechanisms as controlling resis114
Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 1, 1976
tances which greatly simplifies the mathematics of the model. Models, such as that of Schneider and Smith, assume that the adsorption onto the particle surface is of a mass transfer type; i.e., the rate of removal from the gas phase is proportional to the difference between the adsorbate concentration and an equilibrium adsorbate concentration. In this paper this term is given by
For the above equation KA' is the slope of the linear portion of the adsorption isotherm relating equilibrium adsorbate surface and gas phase concentrations. Accordingly, eq 1 is valid only for a linear isotherm.
A second often used model assumes that the adsorption process is a reaction between the adsorbate and active sites on the solid surface, and in the symbols used here, is given by aCads/at = k C [ N o - Cads]
(la)
This approach has been described by Vogel et al. (1974) and others and has been shown to work well for many cases (most of them involving a chemical reaction with the solid). Most models which use eq l a also assume that the adsorption process is the controlling resistance (external mass transfer and intraparticle diffusion resistance are negligible). Models of this type could not be made to fit the data presented here and it is doubtful if such a model is a good description of the adsorptive process for such cases as the adsorption of organic vapors on activated charcoal. An objective of this paper is to show the utility of the Schneider and Smith type model in a much simplified form which is also applicable when the adsorption bed consists of adsorbent particles of unknown size or size distribution, when the adsorbent particles are mixed with or supported on an inert material, or when the flow passages through the bed are difficult to characterize. In order to accomplish this we will begin with a presentation of the complete model and a development of the simplification. I t will be shown that this simple model, making use of an overall coefficient of mass transfer, can be used to describe the adsorption dynamics of a complex adsorption bed. This model shows promise of being an accurate and useful model for dealing with many adsorption processes of industrial importance. Development of the Model The model for .adsorption processes presented by Schneider and Smith (1968) is given below.
adsorption constant, KA’, must be known and the rate constants, E A , D,, kf, and k & must be determined either from experimental data taken during adsorption runs on the bed or from other independent experimental measurements. For the case where the axial dispersion coefficient (EA) may be neglected, eq 1-8 have been solved by Rosen (1952) using Laplace transforms and the inversion integral to give an expression for C(t), the gas concentration exiting the bed, in the form of an infinite integral. With this solution, it is theoretically possible to determine the three rate constants by curve fitting or by the method of moments from experimental data taken on an adsorption bed. As will be discussed later, this presents several difficulties. The above model can be modified to make it easier to apply to an adsorption bed where the particle size and configuration of the adsorbent bed is not well known and where the adsorbent may be mixed with or supported on an inert material. Begin by defining the following terms. p m = average density of solid material in the bed,
g of bed material/cm3 of bed volume 1 - t = volume fraction of adsorbent material cm3 of
adsorbent material/cm3 of bed volume a = void volume fraction of bed
Af = cross sectional area of bed z = depth of bed
6 = weight fraction adsorbent material to total solids
KA’ = equilibrium adsorption constant of adsorbent material Then the total weight of the bed, Wt, is
Mass balance of the adsorbable component in the gas phase E A ~ ~ CaC a
az2
az
aC at
3Dc 1 - a
R
a
aCi -
i
ar
>,=,
=0
I
(2)
W i = Afzpm
(9)
and the volumetric flow rate through the bed, Q, is:
Mass balance of this component in the particle
Rate of adsorption (assumed to have a linear dependence on driving force)
External diffusion boundary condition =
The apparent bed material density is
k f ( c - ci)
ar r = R Dc (%) Internal diffusion boundary condition d C-i - 0 at r
ar Initial and inlet conditions
= 0 for t
>0
1-t
(4)
(5)
P r n = T P p
Assuming that the particles of adsorbent material are spherical and of radius R , the surface area of the adsorbent material per unit weight of bed, A , is
A = - 36
C =O,atz >Ofort = O
(6)
Ci = 0, a t r I 0 for t = 0
(7)
>0
(8)
C = CO,a t z = 0 for t
A new equilibrium constant KA, can be defined on a weight basis of total bed material as
Conditions 6 and 7 state that the adsorbent does not contain any adsorbed species at the start of the run and condition 8 specifies that a t the bed inlet the concentration of the adsorbable species is maintained constant. The model is derived for a uniform bed of spherical particles and its application requires that the properties of the bed, p p , CY, p, and R be known. In addition the equilibrium
(14)
PPR
Using the above definitions, the only change in the model is in eq 2 which becomes aC - aC - -Dcl-t -u az at R a
(aCi) ar
r=R
=o
(la)
T o work with moments which have finite values the transmission curve is transformed so that the j t h moment is
M~ =
tj(1-
F) dt
With the assumption that p, the internal void fraction of an Ind. Eng. Chem., Process Des. Dev., Vol. 15. No. 1. 1976
115
I d
M
I t,
Y
2 +
2 +
I+
r(
. 2.
a m -c
P
2
r
II
U"
l l
I1
I1
a
a"
R
. 2 t
d
s
I1
a
m
n
a
F 1 8 STQ
rl
I
h
I 0 I1
rc $12 P
Q
I
II
U
116
Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 1, 1976
II
s.
II
a
-&
Table 11. Comparison of Model ResultFVarying UA;Model 1, D,A/R Controlling Model; Model 2, kfA,kads6 or UA Model K A = 15.53
Time, min
Model 1,
c/c,
Model 2,
c/c,
Model 1,
CIC,
Model 2,
c/c,
Time, min
U A = 9.9735 Model 1, Model 2,
UA = 7.9785 Model 2,
UA = 5.9850
U A = 3.9900
Model 1,
c/c,
c/c,
C/C,
c/c,
0.000 0.016
0.000 0.021 0.102 0.292 0.900 0.996
~~
30 40 50 75 100 125
0.000
0.028 0.102 0.474 0.808 0.952
0.000 0.039 0.109 0.464 0.807 0.955
0.000 0.008 0.054 0.449 0.846 0.977
Table 111. Comparison of Model Results-Varying
40 50 60
0.000 0.014 0.061 0.443 0.845 0.980
70 100
125
0.000 0.029 0.128 0.316 0.876 0.989
0.000 0.035 0.131 0.316 0.876 0.991
0.199 0.290 0.899 0.994
K A ;Model 1,D,A/R Controlling Model; Model 2,
kfA,k n d r 6 ,or UA Model U A = 7.9785 K , = 3.88
K A = 11.64
Time, min
Model 1 ,
Model 2,
Time, min
10 15 20 25 30 40
0.000 0.128 0.545 0.876 0.980 1.000
0.000
25 30 40 50 75 100
c/c,
c/c,
0.131 0.536 0.877 0.983 1.000
Model 1,
c/c,
0.000 0.000 0.052 0.246 0.876 0.996
adsorbent particle is small compared to p p K ~ ’the , first two moments are
Higher moments are obtained from eq 15. The general solution and the moment equations contain only total weight of the adsorption bed, volumetric flow rate, the equilibrium adsorption constant, and three groups characteristic of the three rate constants
It should be possible to determine K A from the zeroth moment and the three rate constant groups, as defined above, from the three independent equations for the first, second, and third moments. No further information is needed to evaluate the transmission curve from the general solution of the model. While the approach used here will not give values for the constants kf, D,, and hads without accurate values for A, R, and 6, it is consistent practice in many mass transfer operations. For example, in many applications of interfacial mass transfer where the interfacial area is not known, i t has been found useful to combine the coefficient and the area and report them as a mass transfer coefficient as in kfA above. The determination of four parameters from four moments is asking a great deal from experimental data, and most investigators using the method of moments have found this to be impossible, as it proved to be in this work. An alternate approach is to assume that only one of the three rates is important and constitutes the controlling mechanism for the process. This means that the other two are large compared to the controlling coefficient, and greatly simplifies the mathematics. When it is actually true, it is as exact as the complete model. The adsorption equilibrium constant, K A , and any one of the three rate constants can be determined from eq 16 and 17 from a single adsorp-
Model 2,
c/c,
0.000 0.000
0.058 0.242 0.877 0.997
Time, min 50 60 65 75 100
125
K A = 19.41 Model 1 ,
c/c,
0.000 0.020 0.042 0.128 0.545 0.876
Model 2,
c/c,
0.000
0.026 0.048 0.131
0.535 0.876
tion run. In this case two of the three rate constants are assumed infinite, and the other determined from eq 17. The complete solution of the model for each of the three rate constants controlling has been presented in the literature and the several solutions are shown in Table I using the definitions described above. As can be seen from the table, the solutions for the cases of kfA controlling and for kads6 controlling are mathematically identical when the appropriate rate constant is determined from eq 17 and then used in evaluating the solutions for the transmission curve. This means that it would be impossible to distinguish between the two models even though they appear to be for two entirely different mechanisms. While it is not mathematically identical to the other two, the solution for the case of D c ( A / R )controlling has been shown to be numerically the same over a wide range of conditions. Results of these computations are presented in Tables 11,111,and IV. From the analysis described in the two previous paragraphs, it is clear that for a situation where a single rate constant model fits the experimental data, it is not possible to determine which rate constant represents the controlling mechanism without changing significantly experimental conditions such as temperature, flow rate or concentration or by varying adsorbent particle sizes and properties. Such experimentation is, however, very time consuming and is not always possible especially for a complex adsorption bed of the kind described earlier. The form of the equation for the first moment, eq 17, suggests that the mechanism of adsorption might be described by a single overall coefficient in the following way
This is analogous to the definition of the overall heat transfer coefficient, used so successfully in heat transfer design, for three thermal resistances in series. The three mass transfer resistances for the adsorption process are not exactly in series since pore diffusion contributes a variable resistance that is a function of the spherical coordinate, r , Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 1, 1976
117
Table IV. Comparison of Model Results-Varying .kfA,kads6,or UA Model KA = 3.88 UA = 1.9950
Time, min 5 15 25 30 40 50 75 100 125
K A and U A ;Model 1, D,A/R Controlling Model; Model 2,
KA = 7.76 UA = 3.9900
Model 1
Model2
Model 1
Model2
Model 1
Model 2
0.04 0.331 0.755 0.873 0.972 0.995 1.000
0.018 0.319 0.745 0.872 0.977 0.997
0.000
0.000 0.000 0.001
1.000 1.000
1.000 1.000
0.000 0.009 0.109 0.226 0.541 0.804 0.993 1.000 1.000
0.000 0.000 0.003 0.014 0.090 0.278 0.846 0.991 1.000
c/c,
c/c, -____
1.ooo
c/c,
0.003 0.102 0.229 0.554 0.808 0.99 1.000 1.000
c/c,
and may occur in parallel with surface adsorption. However, one may picture the pore diffusion resistance term, 1/(5D,A/R), as an average resistance, or the pore diffusion resistance at the effective path length that all molecules must travel before being adsorbed on the surface. The numerical work mentioned above shows that this may be a valid approximation. The definition of an overall coefficient provides a means of accounting for the effects of the three resistances as an average or overall resistance. This overall coefficient when evaluated from the first moment would then reflect the influences of each of the three resistances and would simplify the modeling in that higher moments equations would be unnecessary in evaluating UA. The overall coefficient model is: Mass balance in the gas phase
aC
u-
az
aC +-+ at
p*--
1 -taCads CY
at
-0 -
(19)
Rate of adsorption
With the same initial and inlet conditions used before. Note that, even though an overall coefficient is used, eq 20 has the seme form as eq 1 and indicates a mass transfer type mechanism. For the solutions listed in Table I, the differential equations were solved using the procedure outlined by Marshall and Pigford (1947) and were also put into the form presented by Masamune and Smith (1965) for comparison. For the cases of k f controlling or kads controlling (i.e., making the substitutions U A = kfA or UA = kads6, respectively), the overall coefficient model is identical with the two corresponding single resistance models. For the case of pore diffusion controlling, the overall coefficient model gives the same results within the range of numerical error; Le., it is impossible to differentiate between the various models. Experimental Section The experiments described in this section were carried out as a part of a research program sponsored by the U S . Army Natick Laboratory. The work reviewed here is the adsorption of carbon tetrachloride by a carbon-impregnated clothing material from a nitrogen-carbon tetrachloride mixture. The material in sheet form consists of a layer of polyurethane foam bonded to a nylon tricot and impregnated with activated carbon held on the material with a polymer latex binder. Thickness of the material was approximately 0.18 cm. Pictures taken under a scanning electron microscope are shown in Figures 1 and 2. Figure l a is an edge view and shows that approximately six cells of the 118
Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 1, 1976
KA = 19.41 UA = 9.9735
KA = 11.64 UA = 5.9850
c/c,
0.008 0.083 0.284 0.847 0.989 1.000
c/c,
Time, min 40 50 55 60 65 70 75 100
125
Model 1
Model 2
0.000 0.001 0.003 0.010 0.025 0.053 0.099 0.543 0.899
0.000
c/c,
c/c,
0.002 0.006 0.014 0.030 0.059 0.102 0.535 0.900
polyurethane foam must be traversed to penetrate the foam. It also indicates that the gas passage is very tortuous with many changes in the direction of flow. Figure l b shows the foam side of the material and reveals the variation in carbon particle distribution throughout the foam. Figures 2a and 2b show the carbon particles clustered in the foam structure and on the nylon, respectively, and indicate that a wide range of particle sizes is present. A schematic diagram of the apparatus for the study of the dynamics of carbon tetrachloride vapor adsorption by charcoal impregnated foam material is shown in Figure 3 with a list of components in Table V. This apparatus consisted of a flow system of 0.25-in. stainless steel tubing with an adsorption chamber encased in a Plexiglas box for constant temperature control. A pure nitrogen stream from the nitrogen supply cylinder was split a t the inlet and sent through lines 1 and 2. Nitrogen in line 2 flowed through a carbon tetrachloride bubbler (4) immersed in an ice bath to saturate the nitrogen stream with carbon tetrachloride vapor a t OOC. The pure nitrogen stream in line 1 flowed through an orifice ( 5 ) with a hook-gauge manometer (6) and rotameter in series and was mixed with the nitrogencarbon tetrachloride stream for dilution to the desired concentration. The flow control of the pure nitrogen stream in line 1 was critical because of the high dilution factor. This flow rate was set roughly by the rotameter (8) and then adjusted by measuring the pressure drop across the orifice to 0.0001 in. of water with the hook-gauge manometer. The diluted nitrogen-carbon tetrachloride mixture was sent into the constant-temperature box (25) through a temperature equilibrating coil (14) and into a manifold (17). A vapor stream was drawn from the manifold (at 18) and sent through the sample holder (19) a t a flow rate set by rotameter 22. This sample holder consisted of two stainless steel cups between which the foam material was sandwiched, tightened with a clamp, and sealed with wax. The top cup contained a perforated metal sheet to assure uniform gas flow through the sample. The vapor stream leaving the sample holder was sent through a gas collection coil on the chromatograph and the remainder of the vapor mixture entering the manifold was vented through an exhaust hood (23). Analysis of the carbon tetrachloride concentration in the stream exiting the sample cup was made by a Perkin-Elmer gas chromatograph using a column of silicone oil D.C. # Z O O (Perkin-Elmer Column C) with a thermal conductivity detector. A Moseley strip chart recorder monitored the chromatograph output. Reference peaks for the initial concentration were obtained a t the beginning and end of each adsorption run by taking a sample stream from the manifold at (26) and sending it through the chromatograph. Injections were made using the gas sampling valve
a
a
Figure 1. a. Photo edge view X26 (nyb
terial: 1preg-
nated foam materia
b of the chromatograph a t 3-min intervals, and from the recorded output (chromatographic curves), transmission curves of C ( t ) / C ovs. time were generated. The adsorption bed was composed of either single or multiple layers of the carbon impregnated foam material: for this study beds of one and three layers were considered. In this manner the effects of different bed depths could be studied.
Resufts The results of typical runs are shown in Figures 4 and 5. The points shown are experimental data and the solid curve is that from the overall coefficient model. In each case the values of K A and UA were determined from the zeroth and first moments for the run. As was discussed in Llr p'Lrvruur DTLblvI~ the results would he identical for any of the models listed in Table I. The effect of changing the equilibrium adsorption constant, KA, a t constant UA, is shown in Figure 6, and the effect of UA is shown in Figure 7. The effect of K A is mainly on the location of the transmission curve and that of UA is mainly on the shape of the curve. It is clear that the single rate parameter model is ade-
Figure 2. a. Photographs of carbon impregnated foam material: foam side X1000. b. Photographs of carbon impregnated foam ma-
terial: nylon side X300.
quate to fit the data well and that very little improvement could be made by a more complicated mo&:l. Even in the case where there is substantial scatter in the data the model gives a best fit that probably could not be suhstantially improved by a different model. The values of the equilibrium adsorption constant, K A , determined from the moments agreed well with those determined from adsorption isotherm data in the linear portion of the isotherm. At the temperatures used in this study the linear portion extended to concentrations of approximately 8 mgA. Figure 6 shows the effect of changing the equilibrium adsorption constant, KA. as predicted by the model. The characteristic shape of the curve is the same but the area is different. This reflects a change in the total amount adsorbed which is proportional to the area under the 1 C/Co curve and proportional to K A . Figure 7 shows the effect of changing the overall coefficient by 425%. The curve for UA = 7.9785 goes between Ind. Eng. Chem.. ProcessDes. Dev, Vol. 15, No. 1, 1976
119
II
5
17
=-
TO VENT
=-
N2
10 G.C.
INLET
-
Figure 3. Schematic of vapor test apparatus.
I
Table V. Components as Indicated in Figure 3
7
RUN 3-13
--
32.5.C 1.0 I /mm C.. 7.87 m g l l T
P
TIME (min)
Figure 4. Model prediction and experimental data for one layer run. 1.0 RUN 3-17 T
a C .,
--
32.5.C 1.0 I/min 7.87 mg/l
I
I
1
0.2
I
'I20
I
160
1
I
I
200
240
280
I
320
I
360
TIME (min)
Figure run.
5.
Model prediction and experimental data for three layer
the two shown and is left out t o show more effectively the change in the curve's characteristic shape. The transmission curve is not very sensitive t o changes in U A and indicates t h a t small variations in the rate parameter due t o experimental error may be acceptable. The area under the 1 - C/Co curve is the same for both curves drawn and re120
Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 1, 1976
1. Control valve size cv = 0.038 2. Control valve size cv = 0.00145 3. U-tube manometer 4. Carbon tetrachloride bubbler chilled at 0°C 5 . Orifice meter 6. Hook-gauge manometer 7 . Control valve size cy = 0.15 8. Rotameter 9. Rotameter 10. On-off valve 11. Rotameter 12. On-off valve 13. Thermocouple 14. Temperature equilibrating coil 15. Tangential entry of gas into manifold to facilitate mixing 16. Wire mesh obstruction to improve mixing 17. Stainless steel manifold 18. On-off valve 19. Sample holder 20. U-tube manometer 21. Thermocouple 22. Rotameter 23. Control valve size c, = 0.15 24. Control valve size c, = 0.15 25. Plexiglas compartment 26. Exit for reference sampling
flects the fact that K A (and thus total amount adsorbed) was held constant. The three independent variables that influence carbon tetrachloride adsorption are temperature ( T ) , concentration of the carbon tetrachloride in the inlet gas stream (CO) and flow rate through the sample ( Q ) . A central composite statistical design of experiments was set up t o determine quantitatively the effects of these variables on the equilibrium constant, KA, and on the overall mass transfer coefficient, UA. A linear fit of K A with the three run variables showed the equilibrium constant to be a significant function of concentration and temperature given below for bolt 2 carbon impregnated foam material
K A = 17.4 - 0.58Co - 0.12T for 5.2 mg/l. ICOI12.5 mg/l. and 25'C 5 T 5 4OoC, with a standard deviation of 0.72 and a value of K A (mean) = 9.09. As expected, the variation with flow rate, Q, was not
UA * 9.9735
0.6
20
40
60 TIME
80
100
120 TIME ( m i d
(rnin)
Figure 6. Effect of changing K A 10% as predicted by mathematical model.
statistically significant. These values of K A from the dynamic adsorption runs (calculated from the zeroth moment equation) compared very favorably with the values obtained from the adsorption isotherms obtained on a McBain balance. However, a slight deviation was noticed at the higher concentrations where the isotherm became nonlinear. This is due to the assumption of a linear rate of removal made in the adsorption model that is not valid for those concentrations in the nonlinear region of the adsorption isotherms. A statistical fit for the overall coefficient from the statistical design experiments revealed that U A was a function of 1/Co and Q
U A = 2.10 4- 4.80/Co
+ 0.75Q
for 5.2 mg/l. ICOI12.5 mg/l. and 0.5 l./min IQ I1.3 l./ min, with a standard deviation of 0.60 and a value of U A (mean) = 3.060. The variation of U A with temperature, T , was not statistically significant; however, this was probably due to the small temperature range (25OC I T i 4OoC) used. Figure 8 shows the results of the above correlation produced from the statistical analysis of 38 runs, and the data points shown are average values of U A for specific run conditions. The figure shows clearly that U A is an inverse function of inlet concentration and directly proportional to flow through the sample. This further indicates that the overall coefficient represents more than one resistance controlling since D,(A/R) should not be function of Q, flow through the sample, and kfA should not be a function of Co, inlet concentration. The material on which this study was based is a very complex adsorption bed with considerable variability from sample to sample, even where samples are cut from a single sheet. This results in considerable scatter of the results as shown by the statistical correlations for K A and U A . Duplicate runs made on a single sample, regenerated between runs, showed very little scatter. Conclusions The use of an overall coefficient model to describe adsorption dynamics is not limited to complex adsorption systems. Its main advantage, however, is in modeling a system in which the controlling resistances are unknown and constraints on system or variable variations do not allow the determination of a controlling resistance. In fact, under the system operating conditions all resistances to adsorption may be significant. The model was shown to be effective in describing trans-
Figure 7. Effect of changing U A 25% as predicted by mathematical model.
UA
0.7
0.9
1.1
1.3
Q
1.7
1.5
(Ilmin)
Figure 8. Correlation of U A from statistical analysis.
mission curves obtained for NZ-CC14 mixtures flowing through a carbon-impregnated foam material. The ranges on flow rates, temperature, and inlet concentration were not broad enough to isolate individual resistances to adsorption but a correlation for the overall coefficient as a function of these process parameters was obtained. The extension of the overall coefficient model to other systems such as beds of carbon fiber cloth, mixtures of carbon and inert particles, etc. should be possible. By constructing a simple statistical design of experiments, the overall coefficient can be correlated as a function of run conditions, making it possible to characterize mathematically the adsorption dynamics of such systems. The method should be useful for the design, analysis, and scale-up of adsorption systems. Nomenclature
A = surface area of active carbon particles per unit weight of foam material, cm2/g Af = flow area of foam sample, cm2 C = concentration of adsorbable gas in the interparticle space, mg/l. Cads = concentration of adsorbed gas per unit weight of adsorbent, mg of CC14/g of particle ci = concentration of adsorbable gas in the intraparticle space, mg/l. Co = initial concentration of CC14, mg/l. D , = effective intraparticle diffusion coefficient, cm2/sec k = adsorption rate-constant in eq l a k& = adsorption rate constant, g of particles/(mg of CCll min) Ind. Eng. Chem., Process Des. Dev., Vol. 15,
No. 1, 1976
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K A = adsorption equilibrium constant, ml/g of foam material KA' = adsorption equilibrium constant of pure carbon particles, l./mg of particles kf = mass transfer coefficient, cm/sec M o = zeroth moment o f t vs. (1 - C/Co) curve, min M I = first moment, min2 Mz = second moment, min3 M3 = third moment, min4 No = capacity of adsorbent Q = flow through sample, l./min R = radius of spherical particle of adsorbent, cm r = length coordinate in the spherical particle of adsorbent, measured from the center of the particle, cm T = temperature of sample and gas, OC t = time,min 1/UA = overall mass transfer coefficient, cc of CC14/(g of foam min) V = interstitial velocity, cm/min Wt = sample weight,g z = length coordinate of bed of adsorbent, cm
Greek Letters a = cloth porosity (void volume/total volume) = intraparticle void fraction of carbon particle 6 = weight of particles per unit weight of foam material t = void fraction representing the bed volume not occupied by carbon particles divided by the total volume pm = apparent density of foam material, g of solids/cc of total volume pp = apparent particle density, g of carbon/cc carbon Literature Cited Aris, R., "On the Dispersion of Linear Kinematic Waves", Roc. Roy. SOC. London, Ser. A, 245, 268 (1958). Marshall, W. R., Jr., Pigford, R. L., "The Appleation of Differential Equations to Chemical Engineering Problems". University of Delaware, Newark, Del., 1947. Masamune. S., Smith, J. M., AIChE J., 11, 34 (1965). Rosen, J. E., J. Chem. Phys., 20, 387 (1952). Schneider. P.. Smith, J. M., AlChEJ., 14, 762 (1968). Vogel, R. F., Mitchell, 6.R., Massoth. F. E.,Environ. Sci. Techno/., 8, 432
(1974).
Receiued for review M a r c h 10, 1975 Accepted September 29,1975
Startup Solvent Selection for the Liquefaction of Lignite Warren P. Scarrahl and Richard R. Dlllon Project Lignite. Department of Chemical Engineering, University of North Dakota, Grand Forks, North Dakota 5820 1
The liquefaction of lignite involves the reaction of lignite and hydrogen in the presence of an organic solvent which acts as a hydrogen donor. During continuous operation the solvent will be produced from the lignite: however, during the initial startup-a solvent from an outside source must be provided. Batch autoclave liquefactions were run with 15 coal-derived and 10 petroleum-derived solvents to screen potential startup solvents. The criteria for comparing the solvents were divided into (1) pre-liquefaction characteristics and (2) liquefaction performance. Pre-liquefaction characteristics included (a) the percentage of liquefaction solvent contained in the raw solvents and (b) the fluidity of the raw solvents. Liquefaction performance included (a) lignite conversion and product yields, (b) flow and filtration characteristics, (c) product hydrogen:carbon atom ratios and sulfur contents, and (d) recycle solvent properties (aromaticities, sulfur contents, and recoveries). Several coal and petroleum-derived solvents were found to be suitable for startup purposes.
Introduction Solution hydrogenation is a process being extensively investigated for converting coal into liquid, gaseous, and upgraded solid fuel products. Coal and hydrogen gas are reacted in the presence of an organic solvent which acts as a hydrogen donor; the hydrogenation products are then separated from the unconverted coal and mineral matter (Kloepper et al., 1965; Severson et al., 1970, 1973). At the University of North Dakota one goal of Project Lignite, established under the sponsorship of the Office of Coal Research, is to study the solution hydrogenation (liquefaction) of lignite. A continuous process development unit (PDU) is currently under construction; the principal product will be an upgraded solid fuel, solvent refined lignite (SRL). A requirement for a viable process is that it generate its own solvent. However, during the initial startup, there will have been no previous operation of the process and solvent will have to be supplied from an outside source. The experiments to be described were run to screen potential solvents and select one or more suitable for the PDU startup. 122
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Previous tests had established the fraction of the reactor discharge slurry (RDS) which would be separated and recycled as solvent. Light material had little tendency to dissolve the lignite while vacuum distillation of the filtrate fixed the upper temperature a t which liquid could be readily separated from the dissolved solid product. Although some of the solvents investigated were used "as received" from the supplier, the "standard" solvent that will normally be recycled throtigh the process will consist of material boiling between 100 and 2 3 O O C a t an absolute pressure of 1.6 mmHg. While it appears logical that a coal-derived solvent should be used in the initial startup of the PDU, the current demand for coal-derived solvents is so great that their availability for large-scale operation is questionable. Therefore it was decided that potential petroleum-derived solvents should also be screened. Coal-Derived Solvents. The solvents evaluated were creosote oil, anthracene oil, Light Creosote, and MiddleHeavy Creosote. The first two solvents were obtained from a manufacturer of coal tar products while the latter two
