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Non-isothermal adsorption in a pellet
Chemical Engineering Science. Vol. 46, No.
1. pp. 69-74. 1991.
ooo!-2509/91 $3.00 + 0.00 0 1990 Pergamon Press plc
Printed in Great Britain.
NON-ISOTHERMAL
ADSORPTION
IN A PELLET
J. H. HILLS Department of Chemical
Engineering,
University
Park,
Nottingham
NG7
2RD,
U.K.
(First received 25 August 1989; accepted in revised form 26 February 1990) Abstract-An analytical solution is presented for the temperature rise in an adsorbent pellet in the presence of both internal and external mass transfer resistance, and external heat transfer resistance. A linear, temnerature-indemndent isotherm and constant diffusion coefficient are assumed. Predictions are comp&d with literat& data.
lNTROUCTlON
Such a complex system would require a very long numerical procedure to solve it, and the general case would involve so many parameters that only particular solutions could be given. Marcussen (1982) included most of the elements, and his paper illustrates the problems involved. Most other workers introduced considerable simplification in order to make the problem tractable. Kondis and Dranoff (1971) used a relatively complete version of the system in their study of the adsorption of ethane on a 4 A molecular sieve. Their numerical solution considered a concentration- and temperature-dependent diffusion coefficient in the micropores, and a Langmuir isotherm with temperature-dependent equilibrium constant. However, they assumed micropore diffusion control and constant temperature throughout the pellet. Their predicted temperature variations were much greater than those measured, but they point out that this could have been due to experimental difficulties. Brunovska et al. (1978) gave numerical solutions for the case of negligible external heat and mass transfer resistance, internal mass transfer by pore diffusion with a coefficient independent of temperature and concentration, and instantaneous gas-solid equilibrium at all points within the pellet according to a Langmuir isotherm with temperature-dependent coefficients. Later, the same workers gave a very restricted set of numerical solutions which included an external heat transfer resistance (Brunovska et al., 1980), and a simplified version in which internal heat transfer resistance was neglected (Brunovska et al., 1981). Chihara ef al. (1976) assumed the isotherm to be linear in both concentration and temperature, and represented the overall mass transfer resistance by the linear driving force model with a coefficient independent of temperature. With these assumptions, they were able to obtain a simple analytical solution to the problem which they used to propose a criterion for the validity of the isothermal adsorption model. However, the linear driving force model is only exact when external mass transfer controls the rate, or when the concentration profile within the particle is parabolic; the latter condition holds approximately at long contact times (Rice, 1982), but in most practical cases internal mass transfer resistance is important, and
In the design of packed bed adsorbers both equilibrium data (adsorption isotherms) and rate data (effect of internal and external heat and mass transfer resistances) are needed to predict breakthrough curves. While there are a great deal of data on the former, it is much harder to obtain reliable mass transfer data, which are consequently less common in the literature. These data may be obtained from experimental breakthrough curves, but a number of simplifying assumptions must be made whose validity is hard to check. A safer method is to subject a small quantity of adsorbent to a large excess of fluid containing various fixed concentrations of adsorbate, and measure the rate of uptake as a function of time. Mass transfer resistance can arise from a number of factors: diffusion through the fluid film surrounding the adsorbent pellet, transport along the macropores leading from the pellet surface to the interior, and transport along the micropores of individual crystals. A popular simple method of analysing data is in terms of an overall effective diffusion coefficient, obtained from the well-known equation for unsteady-state diffusion in a sphere (Crank, 1975a) or an overall mass transfer coefficient obtained from the simpler linear driving force approximation (Glueckauf and Coates, 1947). These methods are only valid when the effective intraparticle diffusion coefficient is independent of concentration, and when there is no temperature rise in the pellet. If either of these constraints is violated, a time-consuming numerical integration will be needed, which is particularly onerous in the case’of temperature effects, since the heat and mass balances are strongly coupled. While a temperature rise in the fluid is easily detected, it is the solid temperature which is important; this is hard to measure, and must normally be inferred on theoretical grounds. The most general solution of this problem would allow for both concentration and temperature gradients within the adsorbent pellet and across the fluid film, for temperature- and concentration-dependent diffusion coefficient within the pellet, and for a general, non-linear adsorption isotherm, q* = f(c, T). The heat of adsorption and other physical properties may also be temperatureand concentrationdependent. 69
J. H. HILLS
70
non-isothermality is most likely at short contact times. Ruthven et al. (1980) studied the case of internal mass transfer control with a diffusion coefficient independent of temperature and concentration, and an isotherm varying linearly with temperature. Their analytical solution is in terms of the sum of an infinite series, and while it was developed for the micropore adsorption in molecular sieves, it diffusion-controlled can also be applied, with minor modifications, to macropore diffusion control, as shown later. Bowen and Rimmer (1973) assumed the isotherm to be temperature-independent when expressed in terms of relative humidity. They used the quadratic driving force approximation for mass transfer, with external heat transfer resistance. The equations can be solved numerically apart from the very early stages, when the rate of adsorption is infinite; in this region, Bowen and Rimmer proposed using external mass transfer control, which leads to a simple analytical solution. In the present paper the effects of simultaneous internal and external mass transfer resistance are combined with external heat transfer resistance to give a simplified analytical solution. Both the diffusion coefficient and the linear adsorption isotherm are assumed independent of temperature, which can be justified if the calculated temperature rise is, in fact, small. The solution is presented for short adsorption times as a simple graphical correlation which can be used to estimate the maximum solid temperature rise, and hence to decide whether the data can fairly be analysed by an isothermal model.
external mass transfer resistance: F
6vz exp ( - fi.2~)
4 - 40 = -4m - 40 =
ka
*E1 /.?;12[tl;f +
l-
v-
Dt
r =
-,
KD
4--o _=-
3
4m -%
v(
exp (v2r) - erfc (vJr)
‘y, Jr . 7 1
--I+
The relative error involved in using eq. (2) rather than (1) is less than 1% for vz < 20. The heat balance for a single pellet can be written:
= 4na2h( T -
TB)
4
DERIVATION
OF THE ESMhlATE
assumptions
are made.
(a) Uniform, spherical pellets of adsorbent. (b) Mass transfer resistance can be represented externally by a fluid-to-particle mass transfer coefficient, k, and internally by a constant effective diffusion coefficient, D. (c) The isotherm is linear with distribution coefficient K. (d) There is no internal heat transfer resistance, so the pellet is at uniform temperature. Externally, fluidto-particle heat transfer is governed by a heat transfer coefficient, h. (e) Bulk fluid concentration cg and temperature Tn are constant. (f) K, D, k and h are constants, independent of the pellet temperature. This assumption, which will only be true for very small temperature rises, effectively uncouples the heat and mass balances, and means that a standard isothermal mass transfer equation can be used, with the temperature rise calculated from the rate of adsorption. Crank (1975b) gives the analytical solution for isothermal adsorption in a sphere with both internal and
(l)
and p,, are the roots of a2 B. cot 8. + v - 1 = 0. qm, the equilibrium solid phase concentration of adsorbate, is given by q_ = Kc,. Equation (1) is exact, but difficult to use at short exposure times (r + 0) because an excessive number of terms are needed for convergence. In this case, an asymptotically correct solution can be derived from the solution for penetration into a semi-infinite solid, also given by Crank (197%). For small times, adsorbate only penetrates the surface layers of the sphere, so the effects of curvature can be neglected. Expressed in terms of v and T, the result is: where
+ 3 2za3psCp, The following
1)]
v(v -
dT -
[ dt
TB = 0 and replacing
Now, writing T mensionless 2:
1 .
t by the di-
dt3 dr
0
where a = 3ah/Dp,C,. Differentiating eq. (2) and substituting for dq/dr in (3) gives:
de
0z
+ a0 = 3vGl,
- 40)
exp (v2r). erfc (vJT).
AC, (4)
To integrate eq. (4), use exp (CX) as integrating factor, and integrate the right hand side by parts. The result is:
8. exp (ax) =
3v4q.m - 40) PaC& x
1
+ v2)
erfc(vJ*)-exp
[(a +
v2)r] - 1
(5)
Non-isothermal adsorption in a pellet
71
The remaining integral in eq. (5) can be transformed by writing c& = 4’. The result is: 8
3v4q.n - 40) = p.C,(c!
exp (v’r).erfc
+ v2) (
- exp ( - 2as) +
,::a,
(vJr)
F(Jaz)
s
1
Y
where F(y)
= exp( - y2)
exp (r$2). d4 is known as
Dawson’s integral, and c& be found in standard tables and computer libraries. A final transformation of eq. (6) can be made by the introduction of the dimensionless groups: x
=
$7
V2
=
-
K2D
h
Fig. 2. Dimensionlesspeak temperature,B,,&, x,, as a functionof p.
t
The corresponding given by:
- qo)k Kh
.
E
-
1
exp (x) - erfc ( Jx)
- exp
(
- z
$@,,,,,,,.
> (7)
Figure 1 shows a graph of f?/f3,as a function of x for various values of the parameter F. The curves pass through a maximum at a value of dimensionless time x given by the implicit equation: fi. exp (x,)
. erfc (Jx,)
= 2
maximum
temperature
8,/8, = exp (n,) . erfc (Jx,).
The resulting expression is:
--
I rem
10
I 1000
c
and time,
3K2Dh
4q,
=
I
I 10
ak’p, C,,
‘=;= e
k2
I cl,
J@FuM
rise is (9)
x, and 8,/8, are functions of p alone, and are plotted in Fig. 2. Thus the effects of internal and external mass transfer resistance on particle temperature can be combined in a single parameter I(. and Fig. 2 can be used directly to estimate the magnitude of any temperature rise in the adsorbent. All the parameters involved in 0, and p can be determined experimentally: a, p. and C,, can be measured for the solid particles, k and h estimated from standard correlations, and K and D calculated from the adsorption experiments themselves by assuming a negligible temperature rise. DISCUSSION
03)
Fig. 1. Dimensionless temperatureprofiles. Lines: theory [eq. (7)] for valuesof p as given. Points: data of Bowen and Rimmer (1973) (JJ= 0.84).
Because of experimental difficulties, very few workers have actually measured the temperature of the adsorbent during an adsorption experiment, and the majority of those who have measured it have done so in conditions where there is no external mass transfer resistance (pure adsorbent gas). Marcussen (1970) worked with water vapour in air adsorbing onto alumina pellets and made some temperature measurements, but does not record the results in enough detail to check the present theory. Bowen and Rimmer (1973) measured the rates of adsorption and the temperature rise when 2.83 mm pellets of activated alumina were used to adsorb water vapour from a stream of nitrogen at 30°C. As explained above, they compared their experimental data with the results of a numerical integration using the quadratic driving force approximation, but for short exposure times they replaced this with external mass transfer control, leading to a very simple analytical result which, expressed in our nomenclature, is: Xm’P
(10)
8, = 0,.
(11)
J. H. HILLS
72
Comparison with Fig. 2 shows that eq. (10) underestimates x, for p -E 20, while eq. (11) grossly overestimates 8, except at very low values of p. Both the simple equations (10) and (11) and the more complex (8) and (9) predict x, to be independent of gas-phase concentration of adsorbent and 0, to be proportional to this concentration when K and D are concentration-independent. The experimental data presented in Table 1 of Bowen and Rimmer’s paper can be used to estimate K and D at different humidities. Since the temperature measurements (although not the mass transfer ones) were made on a single pellet in the flowing gas stream, equation k and h can be estimated from the F&sling as given by Rowe et al. (1965): Nu = 2 + 0.69 Re112 Pr113
(12)
Sh = 2 + 0.69 R~“‘SC”~
RH
(93
Fig. 3. Peak temperature as a function of relative humidity. (0) Data of Bowen and Rimmer (1973); (0) predicted in Table 1. Line: eq. (11).
(13)
which gives k = 0.071 ms-’ and h = 0.076kW m- * K- 1 at a Reynolds number of 48.9. (The authors quote Re = 40, but this is inconsistent with their other data.) From the literature, pS = 1154 kg m- 3 and C,, = 0.84 kJ kg-’ K-i, and in another paper (Rimmer and Bowen, 1972) the authors give a value of 1= 2720 kJ kg-’ for this experiment. Table 1 presents the results of the calculations for the time, f,, at which the peak temperature occurs, and the maximum temperature rise, B,,,. Since the isotherm is not completely linear, both K and D vary somewhat with the relative humidity of the inlet gas, which leads to a small predicted variation of t, as shown in the table. Figure 3 compares the measured temperature rise at different relative humidities with the predictions of Table 1, and of Bowen and Rimmer’s simple theory [eq. (1 l)]. It can be seen that eq. (11) gives a large over-prediction in all cases, whereas eq. (9) (Table 1) gives good predictions for relative humidities up to about 50%. For higher RI-I, the temperature rise exceeds 10°C. which almost certainly affects both K and D significantly, so the whole basis of the present theory breaks down.
The actual values of v and z, for these experiments are such that the use of eq. (2) in place of eq. (1) introduces an error of about 1%. The complete experimental temperature profile for a 29.9% RH feed (estimated p = 0.84) is plotted in Fig. 1 in dimensionless coordinates (e/f?, against x) for comparison with the predictions of eq. (7). The peak height and the shape of the curve are seen to be well predicted, although the experimental peak time (10.6 s) occurs later than predicted (8 s). However, the time constant of the thermistors used (1 s) is itself almost enough to explain the difference in the observed and predicted peak times. Bowen and Rimmer also investigated the effect of flow velocity on the temperature profile. They present profiles measured with 96% RH at velocities of 0.045, 0.093 and 0.361 m s- t, corresponding to Reynolds numbers of 6.1, 12.6 and 48.9. Only k and h are affected by a change in Re, and, from eqs (12) and (13), both are affected to a similar extent, so that 0, (proportional to k/h) will remain almost constant, while p (proportional to k’/h) will increase with Re, reducing 0,. Table 2 presents the calculated results, and compares them with Bowen and Rimmer’s data. It can be seen that the predicted variation of t, is qualitatively
Table 1. Predicted maximum temperature rise on adsorption (data of Bowen and Rimmer, 1973) RH (%) 9.3 17.0 32.1 44.1 49.7 57.5 43.6 49.5 74.5 83.1 88.1 96.7
K2D (m’ s- ‘)
P
0, (R)
x,
L/e,
r, (s)
0,1”C)
0.0340 0.0407 0.0261 0.0213 0.241 0.274 0.0326 0.0368 0.0446 0.0535 0.0600 0.0451
0.687 0.574 0.895 1.096 0.969 0.852 0.716 0.634 0.524 0.437 0.390 0.518
7.16 13.09 24.72 33.95 38.27 44.27 48.97 53.51 57.36 63.98 67.83 74.45
1.22 1.05 1.52 1.80 1.63 1.46 1.27 1.14 0.98 0.84 0.77 0.97
0.401
0.420 0.371 0.350 0.363 0.377 0.396 0.409 0.431 0.451 0.464 0.432
8.3 8.5 7.9 7.6 7.8 8.0
2.9 5.5 9.2 11.9 13.9 16.7 19.4 21.9 24.7 28.9 31.5 32.2
::4’ 8.6 8.9 9.1 8.7
Non-isothermal
adsorption in a pellet
73
Table 2. Effect of Reynolds number on temperature rise (data of Bowen and Rimmer, 1973) tin (s)
e, WI
RI?
@kP-3
P
%n
pred.
meas.
pred.
meas.
6.1 12.6 48.9
74.4 74.2 73.9
0.269 0.318 0.477
0.563 0.647 0.904
17.4 14.2 8.8
27.5 21.0 10.5
37.7 36.2 32.6
48.8 48.2 50.4
justify the isothermal assumption. Direct measurement of this temperature rise, which is experimentally difficult, is thus avoided. Acknowledgements~I should like to thank Dr. J. H. Bowen
and Professor L. Marcussen for their help in attempting to find enough experimental data to test the proposed model. NOTATION
I
Fig. 4. Temperature profile. Points: data of Bowen and ) This work, p = 0.84. (- - -) Rimmer (1973). (Ruthven et al. (1980), 01= 1440, B = 9.26.
correct, but that there is actually a small increase in 61, instead of the predicted decrease. This is presumably connected with the fact that the high values of 8, invalidate assumption (f) above. It is worth comparing the predictions of the present model with those of Ruthven et al. (1980), who neglected external mass transfer resistance, but allowed for the temperature effect on the isotherm. The experimental data presented by Bowen and Rimmer can also yield values of the parameters in this model, which for the run at 29.9% RH are cx= 1440 and /_I= 9.26. (For macropore diffusion control, a is the same for both models.) Figure 4 compares the theoretical curves for the two models with the experimental data of Bowen and Rimmer already shown in Fig. 1. The present model is seen to be a better representation of the experimental data than that of Ruthven et al., which suggests that the result of neglecting the effects of temperature on the isotherm is less serious than that of neglecting external mass transfer resistance. CONCLUSIONS
The theory summarized in the curves of Fig. 1 gives a good prediction of the maximum temperature rise of the adsorbent in conditions where both internal and external maSs transfer are important, provided that this temperature rise does not exceed about 10°C. In experiments where the rate of uptake of adsorbate by a small batch of adsorbent is measured and used to estimate the effective diffusion coefficient, D, from an isothermal model, Fig. 2 can be used to check whether the temperature rise of the solid is small enough to
a
pellet radius
CP
specific heat capacity of solid concentration of adsorbate in the gas phase effective diffusion coefficient of adsorbate in the solid fractional saturation of adsorbent pellet heat transfer coefficient, gas/solid mass transfer coefficient, gas/solid gradient of isotherm Nusselt number for gas film heat transfer Prandtl number for gas concentration of adsorbate in solid phase initial value of q final, steady state value of q Reynolds number for gas flow Schmidt number for gas time temperature of solid bulk temperature of gas dimensionless time, k2t/K ‘D dimensionless time to peak temperature
C D
F h k K Nu Pr 4 40 4m Re SC 2
T TB X X,
Greek letters Fj” Ok Rn 1 p Y
dimensionless group, Sah/Dp, C, roots of 8. cot 8. + v - 1 = 0 [eq. (l)] temperature difference, solid/gas reference temperature rise, L(q, - qo) k/Kh maximum temperature attained by solid heat of adsorption dimensionless variable, equal to v’/a ratio of external to internal mass transfer,
ka/KD PS
T
fP
density of solid dimensionless time, Dt/az dummy variable of integration, equal to ,/(a=)
REFERENCES Bowen, J. H. and Rimmer, P. G., 1973, Prediction of nonisothermal sorption in single pellets using the quadratic driving force equation. Chem. Engng .I. 6, 145-156.
74
J. H. HILLS
Brunovska, A., Hlavacek, V., Ilavsky, J. and Valtyni, J., 1978, An analysis of a nonisothermal one-component sorption Sci. 33, in a single adsorbent particle. Chem. Engng 1385-1391. Brunovska, A., Hlavacek, V., Ilavsky, J. and Valtyni, J., 1980, Nonisothermal one-component sorption in a single adsorbent oarticle. Effect of external heat transfer. Chem. Engng SEi. 35, 757-759. Brunovska. A.. Ilavskv. J. and Hlavacek, V., 1981, An analysis of a nonisoth&mal one-componcni sorption in a single adsorbent particle-a simplified model. Chem. Engng Sci. 34, 123-128. Chihara, K., Suzuki, M. and Kawaxoe, K., 1976, Effect of heat generation on measurement of adsorption rate by eravimetric method. Chem. Encna Sci. 31, 505. Crank, J., 1975a, The Mathem&& of Difision, 2nd edn, D. 90. Oxford University Press, Oxford. Crank, J., 1975b, The Mathematics of Di@iuion, 2nd ecln, p. 96. Oxford University Press, Oxford. Crank, J., 1975c, The Mathematics of Difision, 2nd edn, p. 36. Oxford University Press, Oxford. Glueckauf, E. and Coates, J. I., 1947, Theory of chromatography. Part IV. The influence of incomplete equilibrium
on the front boundary of chromatograms and on the effectiveness of separation. J. them. Sot. 1315. Kondis, E. F. and Dranoff, J. S., 1971. Nonisothermal sorption of ethane by 4 A molecular sieves. C.E.P. Symp. Ser. 67 (117). 25-34. Marcussen, L., 1970, Comparison of experimental and predieted breakthrough curves for adiabatic adsorption in a fixed bed. Chem. &ng Sci. 25, 1487. Marcusscn, L., 1982, Comparison of experimental and predieted breakthrough c&es for adiabatic adsorption in fixed bed. Chem. Engng Sci. 37, 299-309. Rice, R. G., 1982, Approximate solutions for batch, packed tube and radial flow adsorbers
’
1. pp. 69-74. 1991.
ooo!-2509/91 $3.00 + 0.00 0 1990 Pergamon Press plc
Printed in Great Britain.
NON-ISOTHERMAL
ADSORPTION
IN A PELLET
J. H. HILLS Department of Chemical
Engineering,
University
Park,
Nottingham
NG7
2RD,
U.K.
(First received 25 August 1989; accepted in revised form 26 February 1990) Abstract-An analytical solution is presented for the temperature rise in an adsorbent pellet in the presence of both internal and external mass transfer resistance, and external heat transfer resistance. A linear, temnerature-indemndent isotherm and constant diffusion coefficient are assumed. Predictions are comp&d with literat& data.
lNTROUCTlON
Such a complex system would require a very long numerical procedure to solve it, and the general case would involve so many parameters that only particular solutions could be given. Marcussen (1982) included most of the elements, and his paper illustrates the problems involved. Most other workers introduced considerable simplification in order to make the problem tractable. Kondis and Dranoff (1971) used a relatively complete version of the system in their study of the adsorption of ethane on a 4 A molecular sieve. Their numerical solution considered a concentration- and temperature-dependent diffusion coefficient in the micropores, and a Langmuir isotherm with temperature-dependent equilibrium constant. However, they assumed micropore diffusion control and constant temperature throughout the pellet. Their predicted temperature variations were much greater than those measured, but they point out that this could have been due to experimental difficulties. Brunovska et al. (1978) gave numerical solutions for the case of negligible external heat and mass transfer resistance, internal mass transfer by pore diffusion with a coefficient independent of temperature and concentration, and instantaneous gas-solid equilibrium at all points within the pellet according to a Langmuir isotherm with temperature-dependent coefficients. Later, the same workers gave a very restricted set of numerical solutions which included an external heat transfer resistance (Brunovska et al., 1980), and a simplified version in which internal heat transfer resistance was neglected (Brunovska et al., 1981). Chihara ef al. (1976) assumed the isotherm to be linear in both concentration and temperature, and represented the overall mass transfer resistance by the linear driving force model with a coefficient independent of temperature. With these assumptions, they were able to obtain a simple analytical solution to the problem which they used to propose a criterion for the validity of the isothermal adsorption model. However, the linear driving force model is only exact when external mass transfer controls the rate, or when the concentration profile within the particle is parabolic; the latter condition holds approximately at long contact times (Rice, 1982), but in most practical cases internal mass transfer resistance is important, and
In the design of packed bed adsorbers both equilibrium data (adsorption isotherms) and rate data (effect of internal and external heat and mass transfer resistances) are needed to predict breakthrough curves. While there are a great deal of data on the former, it is much harder to obtain reliable mass transfer data, which are consequently less common in the literature. These data may be obtained from experimental breakthrough curves, but a number of simplifying assumptions must be made whose validity is hard to check. A safer method is to subject a small quantity of adsorbent to a large excess of fluid containing various fixed concentrations of adsorbate, and measure the rate of uptake as a function of time. Mass transfer resistance can arise from a number of factors: diffusion through the fluid film surrounding the adsorbent pellet, transport along the macropores leading from the pellet surface to the interior, and transport along the micropores of individual crystals. A popular simple method of analysing data is in terms of an overall effective diffusion coefficient, obtained from the well-known equation for unsteady-state diffusion in a sphere (Crank, 1975a) or an overall mass transfer coefficient obtained from the simpler linear driving force approximation (Glueckauf and Coates, 1947). These methods are only valid when the effective intraparticle diffusion coefficient is independent of concentration, and when there is no temperature rise in the pellet. If either of these constraints is violated, a time-consuming numerical integration will be needed, which is particularly onerous in the case’of temperature effects, since the heat and mass balances are strongly coupled. While a temperature rise in the fluid is easily detected, it is the solid temperature which is important; this is hard to measure, and must normally be inferred on theoretical grounds. The most general solution of this problem would allow for both concentration and temperature gradients within the adsorbent pellet and across the fluid film, for temperature- and concentration-dependent diffusion coefficient within the pellet, and for a general, non-linear adsorption isotherm, q* = f(c, T). The heat of adsorption and other physical properties may also be temperatureand concentrationdependent. 69
J. H. HILLS
70
non-isothermality is most likely at short contact times. Ruthven et al. (1980) studied the case of internal mass transfer control with a diffusion coefficient independent of temperature and concentration, and an isotherm varying linearly with temperature. Their analytical solution is in terms of the sum of an infinite series, and while it was developed for the micropore adsorption in molecular sieves, it diffusion-controlled can also be applied, with minor modifications, to macropore diffusion control, as shown later. Bowen and Rimmer (1973) assumed the isotherm to be temperature-independent when expressed in terms of relative humidity. They used the quadratic driving force approximation for mass transfer, with external heat transfer resistance. The equations can be solved numerically apart from the very early stages, when the rate of adsorption is infinite; in this region, Bowen and Rimmer proposed using external mass transfer control, which leads to a simple analytical solution. In the present paper the effects of simultaneous internal and external mass transfer resistance are combined with external heat transfer resistance to give a simplified analytical solution. Both the diffusion coefficient and the linear adsorption isotherm are assumed independent of temperature, which can be justified if the calculated temperature rise is, in fact, small. The solution is presented for short adsorption times as a simple graphical correlation which can be used to estimate the maximum solid temperature rise, and hence to decide whether the data can fairly be analysed by an isothermal model.
external mass transfer resistance: F
6vz exp ( - fi.2~)
4 - 40 = -4m - 40 =
ka
*E1 /.?;12[tl;f +
l-
v-
Dt
r =
-,
KD
4--o _=-
3
4m -%
v(
exp (v2r) - erfc (vJr)
‘y, Jr . 7 1
--I+
The relative error involved in using eq. (2) rather than (1) is less than 1% for vz < 20. The heat balance for a single pellet can be written:
= 4na2h( T -
TB)
4
DERIVATION
OF THE ESMhlATE
assumptions
are made.
(a) Uniform, spherical pellets of adsorbent. (b) Mass transfer resistance can be represented externally by a fluid-to-particle mass transfer coefficient, k, and internally by a constant effective diffusion coefficient, D. (c) The isotherm is linear with distribution coefficient K. (d) There is no internal heat transfer resistance, so the pellet is at uniform temperature. Externally, fluidto-particle heat transfer is governed by a heat transfer coefficient, h. (e) Bulk fluid concentration cg and temperature Tn are constant. (f) K, D, k and h are constants, independent of the pellet temperature. This assumption, which will only be true for very small temperature rises, effectively uncouples the heat and mass balances, and means that a standard isothermal mass transfer equation can be used, with the temperature rise calculated from the rate of adsorption. Crank (1975b) gives the analytical solution for isothermal adsorption in a sphere with both internal and
(l)
and p,, are the roots of a2 B. cot 8. + v - 1 = 0. qm, the equilibrium solid phase concentration of adsorbate, is given by q_ = Kc,. Equation (1) is exact, but difficult to use at short exposure times (r + 0) because an excessive number of terms are needed for convergence. In this case, an asymptotically correct solution can be derived from the solution for penetration into a semi-infinite solid, also given by Crank (197%). For small times, adsorbate only penetrates the surface layers of the sphere, so the effects of curvature can be neglected. Expressed in terms of v and T, the result is: where
+ 3 2za3psCp, The following
1)]
v(v -
dT -
[ dt
TB = 0 and replacing
Now, writing T mensionless 2:
1 .
t by the di-
dt3 dr
0
where a = 3ah/Dp,C,. Differentiating eq. (2) and substituting for dq/dr in (3) gives:
de
0z
+ a0 = 3vGl,
- 40)
exp (v2r). erfc (vJT).
AC, (4)
To integrate eq. (4), use exp (CX) as integrating factor, and integrate the right hand side by parts. The result is:
8. exp (ax) =
3v4q.m - 40) PaC& x
1
+ v2)
erfc(vJ*)-exp
[(a +
v2)r] - 1
(5)
Non-isothermal adsorption in a pellet
71
The remaining integral in eq. (5) can be transformed by writing c& = 4’. The result is: 8
3v4q.n - 40) = p.C,(c!
exp (v’r).erfc
+ v2) (
- exp ( - 2as) +
,::a,
(vJr)
F(Jaz)
s
1
Y
where F(y)
= exp( - y2)
exp (r$2). d4 is known as
Dawson’s integral, and c& be found in standard tables and computer libraries. A final transformation of eq. (6) can be made by the introduction of the dimensionless groups: x
=
$7
V2
=
-
K2D
h
Fig. 2. Dimensionlesspeak temperature,B,,&, x,, as a functionof p.
t
The corresponding given by:
- qo)k Kh
.
E
-
1
exp (x) - erfc ( Jx)
- exp
(
- z
$@,,,,,,,.
> (7)
Figure 1 shows a graph of f?/f3,as a function of x for various values of the parameter F. The curves pass through a maximum at a value of dimensionless time x given by the implicit equation: fi. exp (x,)
. erfc (Jx,)
= 2
maximum
temperature
8,/8, = exp (n,) . erfc (Jx,).
The resulting expression is:
--
I rem
10
I 1000
c
and time,
3K2Dh
4q,
=
I
I 10
ak’p, C,,
‘=;= e
k2
I cl,
J@FuM
rise is (9)
x, and 8,/8, are functions of p alone, and are plotted in Fig. 2. Thus the effects of internal and external mass transfer resistance on particle temperature can be combined in a single parameter I(. and Fig. 2 can be used directly to estimate the magnitude of any temperature rise in the adsorbent. All the parameters involved in 0, and p can be determined experimentally: a, p. and C,, can be measured for the solid particles, k and h estimated from standard correlations, and K and D calculated from the adsorption experiments themselves by assuming a negligible temperature rise. DISCUSSION
03)
Fig. 1. Dimensionless temperatureprofiles. Lines: theory [eq. (7)] for valuesof p as given. Points: data of Bowen and Rimmer (1973) (JJ= 0.84).
Because of experimental difficulties, very few workers have actually measured the temperature of the adsorbent during an adsorption experiment, and the majority of those who have measured it have done so in conditions where there is no external mass transfer resistance (pure adsorbent gas). Marcussen (1970) worked with water vapour in air adsorbing onto alumina pellets and made some temperature measurements, but does not record the results in enough detail to check the present theory. Bowen and Rimmer (1973) measured the rates of adsorption and the temperature rise when 2.83 mm pellets of activated alumina were used to adsorb water vapour from a stream of nitrogen at 30°C. As explained above, they compared their experimental data with the results of a numerical integration using the quadratic driving force approximation, but for short exposure times they replaced this with external mass transfer control, leading to a very simple analytical result which, expressed in our nomenclature, is: Xm’P
(10)
8, = 0,.
(11)
J. H. HILLS
72
Comparison with Fig. 2 shows that eq. (10) underestimates x, for p -E 20, while eq. (11) grossly overestimates 8, except at very low values of p. Both the simple equations (10) and (11) and the more complex (8) and (9) predict x, to be independent of gas-phase concentration of adsorbent and 0, to be proportional to this concentration when K and D are concentration-independent. The experimental data presented in Table 1 of Bowen and Rimmer’s paper can be used to estimate K and D at different humidities. Since the temperature measurements (although not the mass transfer ones) were made on a single pellet in the flowing gas stream, equation k and h can be estimated from the F&sling as given by Rowe et al. (1965): Nu = 2 + 0.69 Re112 Pr113
(12)
Sh = 2 + 0.69 R~“‘SC”~
RH
(93
Fig. 3. Peak temperature as a function of relative humidity. (0) Data of Bowen and Rimmer (1973); (0) predicted in Table 1. Line: eq. (11).
(13)
which gives k = 0.071 ms-’ and h = 0.076kW m- * K- 1 at a Reynolds number of 48.9. (The authors quote Re = 40, but this is inconsistent with their other data.) From the literature, pS = 1154 kg m- 3 and C,, = 0.84 kJ kg-’ K-i, and in another paper (Rimmer and Bowen, 1972) the authors give a value of 1= 2720 kJ kg-’ for this experiment. Table 1 presents the results of the calculations for the time, f,, at which the peak temperature occurs, and the maximum temperature rise, B,,,. Since the isotherm is not completely linear, both K and D vary somewhat with the relative humidity of the inlet gas, which leads to a small predicted variation of t, as shown in the table. Figure 3 compares the measured temperature rise at different relative humidities with the predictions of Table 1, and of Bowen and Rimmer’s simple theory [eq. (1 l)]. It can be seen that eq. (11) gives a large over-prediction in all cases, whereas eq. (9) (Table 1) gives good predictions for relative humidities up to about 50%. For higher RI-I, the temperature rise exceeds 10°C. which almost certainly affects both K and D significantly, so the whole basis of the present theory breaks down.
The actual values of v and z, for these experiments are such that the use of eq. (2) in place of eq. (1) introduces an error of about 1%. The complete experimental temperature profile for a 29.9% RH feed (estimated p = 0.84) is plotted in Fig. 1 in dimensionless coordinates (e/f?, against x) for comparison with the predictions of eq. (7). The peak height and the shape of the curve are seen to be well predicted, although the experimental peak time (10.6 s) occurs later than predicted (8 s). However, the time constant of the thermistors used (1 s) is itself almost enough to explain the difference in the observed and predicted peak times. Bowen and Rimmer also investigated the effect of flow velocity on the temperature profile. They present profiles measured with 96% RH at velocities of 0.045, 0.093 and 0.361 m s- t, corresponding to Reynolds numbers of 6.1, 12.6 and 48.9. Only k and h are affected by a change in Re, and, from eqs (12) and (13), both are affected to a similar extent, so that 0, (proportional to k/h) will remain almost constant, while p (proportional to k’/h) will increase with Re, reducing 0,. Table 2 presents the calculated results, and compares them with Bowen and Rimmer’s data. It can be seen that the predicted variation of t, is qualitatively
Table 1. Predicted maximum temperature rise on adsorption (data of Bowen and Rimmer, 1973) RH (%) 9.3 17.0 32.1 44.1 49.7 57.5 43.6 49.5 74.5 83.1 88.1 96.7
K2D (m’ s- ‘)
P
0, (R)
x,
L/e,
r, (s)
0,1”C)
0.0340 0.0407 0.0261 0.0213 0.241 0.274 0.0326 0.0368 0.0446 0.0535 0.0600 0.0451
0.687 0.574 0.895 1.096 0.969 0.852 0.716 0.634 0.524 0.437 0.390 0.518
7.16 13.09 24.72 33.95 38.27 44.27 48.97 53.51 57.36 63.98 67.83 74.45
1.22 1.05 1.52 1.80 1.63 1.46 1.27 1.14 0.98 0.84 0.77 0.97
0.401
0.420 0.371 0.350 0.363 0.377 0.396 0.409 0.431 0.451 0.464 0.432
8.3 8.5 7.9 7.6 7.8 8.0
2.9 5.5 9.2 11.9 13.9 16.7 19.4 21.9 24.7 28.9 31.5 32.2
::4’ 8.6 8.9 9.1 8.7
Non-isothermal
adsorption in a pellet
73
Table 2. Effect of Reynolds number on temperature rise (data of Bowen and Rimmer, 1973) tin (s)
e, WI
RI?
@kP-3
P
%n
pred.
meas.
pred.
meas.
6.1 12.6 48.9
74.4 74.2 73.9
0.269 0.318 0.477
0.563 0.647 0.904
17.4 14.2 8.8
27.5 21.0 10.5
37.7 36.2 32.6
48.8 48.2 50.4
justify the isothermal assumption. Direct measurement of this temperature rise, which is experimentally difficult, is thus avoided. Acknowledgements~I should like to thank Dr. J. H. Bowen
and Professor L. Marcussen for their help in attempting to find enough experimental data to test the proposed model. NOTATION
I
Fig. 4. Temperature profile. Points: data of Bowen and ) This work, p = 0.84. (- - -) Rimmer (1973). (Ruthven et al. (1980), 01= 1440, B = 9.26.
correct, but that there is actually a small increase in 61, instead of the predicted decrease. This is presumably connected with the fact that the high values of 8, invalidate assumption (f) above. It is worth comparing the predictions of the present model with those of Ruthven et al. (1980), who neglected external mass transfer resistance, but allowed for the temperature effect on the isotherm. The experimental data presented by Bowen and Rimmer can also yield values of the parameters in this model, which for the run at 29.9% RH are cx= 1440 and /_I= 9.26. (For macropore diffusion control, a is the same for both models.) Figure 4 compares the theoretical curves for the two models with the experimental data of Bowen and Rimmer already shown in Fig. 1. The present model is seen to be a better representation of the experimental data than that of Ruthven et al., which suggests that the result of neglecting the effects of temperature on the isotherm is less serious than that of neglecting external mass transfer resistance. CONCLUSIONS
The theory summarized in the curves of Fig. 1 gives a good prediction of the maximum temperature rise of the adsorbent in conditions where both internal and external maSs transfer are important, provided that this temperature rise does not exceed about 10°C. In experiments where the rate of uptake of adsorbate by a small batch of adsorbent is measured and used to estimate the effective diffusion coefficient, D, from an isothermal model, Fig. 2 can be used to check whether the temperature rise of the solid is small enough to
a
pellet radius
CP
specific heat capacity of solid concentration of adsorbate in the gas phase effective diffusion coefficient of adsorbate in the solid fractional saturation of adsorbent pellet heat transfer coefficient, gas/solid mass transfer coefficient, gas/solid gradient of isotherm Nusselt number for gas film heat transfer Prandtl number for gas concentration of adsorbate in solid phase initial value of q final, steady state value of q Reynolds number for gas flow Schmidt number for gas time temperature of solid bulk temperature of gas dimensionless time, k2t/K ‘D dimensionless time to peak temperature
C D
F h k K Nu Pr 4 40 4m Re SC 2
T TB X X,
Greek letters Fj” Ok Rn 1 p Y
dimensionless group, Sah/Dp, C, roots of 8. cot 8. + v - 1 = 0 [eq. (l)] temperature difference, solid/gas reference temperature rise, L(q, - qo) k/Kh maximum temperature attained by solid heat of adsorption dimensionless variable, equal to v’/a ratio of external to internal mass transfer,
ka/KD PS
T
fP
density of solid dimensionless time, Dt/az dummy variable of integration, equal to ,/(a=)
REFERENCES Bowen, J. H. and Rimmer, P. G., 1973, Prediction of nonisothermal sorption in single pellets using the quadratic driving force equation. Chem. Engng .I. 6, 145-156.
74
J. H. HILLS
Brunovska, A., Hlavacek, V., Ilavsky, J. and Valtyni, J., 1978, An analysis of a nonisothermal one-component sorption Sci. 33, in a single adsorbent particle. Chem. Engng 1385-1391. Brunovska, A., Hlavacek, V., Ilavsky, J. and Valtyni, J., 1980, Nonisothermal one-component sorption in a single adsorbent oarticle. Effect of external heat transfer. Chem. Engng SEi. 35, 757-759. Brunovska. A.. Ilavskv. J. and Hlavacek, V., 1981, An analysis of a nonisoth&mal one-componcni sorption in a single adsorbent particle-a simplified model. Chem. Engng Sci. 34, 123-128. Chihara, K., Suzuki, M. and Kawaxoe, K., 1976, Effect of heat generation on measurement of adsorption rate by eravimetric method. Chem. Encna Sci. 31, 505. Crank, J., 1975a, The Mathem&& of Difision, 2nd edn, D. 90. Oxford University Press, Oxford. Crank, J., 1975b, The Mathematics of Di@iuion, 2nd ecln, p. 96. Oxford University Press, Oxford. Crank, J., 1975c, The Mathematics of Difision, 2nd edn, p. 36. Oxford University Press, Oxford. Glueckauf, E. and Coates, J. I., 1947, Theory of chromatography. Part IV. The influence of incomplete equilibrium
on the front boundary of chromatograms and on the effectiveness of separation. J. them. Sot. 1315. Kondis, E. F. and Dranoff, J. S., 1971. Nonisothermal sorption of ethane by 4 A molecular sieves. C.E.P. Symp. Ser. 67 (117). 25-34. Marcussen, L., 1970, Comparison of experimental and predieted breakthrough curves for adiabatic adsorption in a fixed bed. Chem. &ng Sci. 25, 1487. Marcusscn, L., 1982, Comparison of experimental and predieted breakthrough c&es for adiabatic adsorption in fixed bed. Chem. Engng Sci. 37, 299-309. Rice, R. G., 1982, Approximate solutions for batch, packed tube and radial flow adsorbers
’