I n d . E n g . C h e m . Res. 1994, 33, 1585-1592
1585
Non-Isothermal Differential Adsorption Kinetics for Binary Gas Mixture Shivaji Sircar Air Products and Chemicals, Inc., 7201 Hamilton Boulevard, Allentown, Pennsylvania 18195
Analytical solutions are developed for non-isothermal adsorption kinetics of a binary gas mixture in a differential adsorption test (DAT). Linear driving force models are used t o describe the adsorption kinetics of individual components. It is demonstrated that a very small change in the adsorbent temperature can introduce a substantial difference between isothermal and non-isothermal kinetic behaviors of the components of the mixture. The fractional uptake of a component of the mixture can exceed unity and go through a maximum value during the DAT due to the non-isothermal effects. An isothermal kinetic model for binary mixture adsorption using both straight and cross transport coefficients can also describe such uptake behavior, but the model parameters will be artificial due to the ignorance of adsorbent non-isothermality.
Introduction Heat is evolved (consumed) during the measurement of gas ad(de)sorption kinetics in a conventional gravimetric or volumetric apparatus. It is generally not possible to instantaneously remove (supply) the heat from (to) the adsorbent mass used in the experiment. As a result, ad(de)sorption kinetic measurements are carried out under non-isothermal conditions except for the special case where the rate of ad(de)sorption is so much slower than the heat transfer rate from (to) the adsorbent that the adsorbent mass remains isothermal. Consequently, a non-isothermal kinetic process model must be used for estimation of adsorbate diffusivity or mass transport coefficient from the measured data. These transport properties, however, can be strong functions of temperature and amounts adsorbed. Therefore, the kinetic model must account for the variation in the transport properties caused by the change of the adsorbent temperature and adsorbate loading during the kinetic experiment. This imposes a major problem in the analysis of kinetic data when the experiment is carried out using a relatively large change in the adsorbate loading (accompanied by a large change in the adsorbent temperature) because the functional dependence of the transport properties on the temperature and adsorbate loading is generally not known a priori. In fact, the objective of the kinetic test is to measure these dependencies. Use of preassigned functionalities to describe these dependencies in the kinetic model creates the problem of estimating too many model parameters from the kinetic data, and that introduces ambiguity in the data analysis. The problem can be simplified by using a differential adsorption test (DAT) where the adsorbent is subjected to a very small change in the adsorbate loading (accompanied by a very small change in adsorbent temperature) during the experiment. The DAT does not eliminate the need for non-isothermal data analysis (Chihara et al., 1976; Sircar, 1981)but it allows the use of a constant value of the transport property (at the base temperature and adsorbate loading of the test) in the kinetic model. Furthermore, the adsorption equilibria can be linearized as functions of temperature and gas-phase adsorbate concentration within the bounds of a DAT which significantly simplifies the mathematics of the kinetic model. This last advantage has produced several analytical solutions of simultaneous mass and heat transfer equations describing the DAT for ad(de)sorption of a pure gas using different transport mechanisms (Armstrong and Stannet, 0888-588519412633-1585$04.50/0
1966;Chihara et al., 1976;Ruthvenet al., 1980,1981;Sircar et al., 1983, 1984). These studies have provided much insight into the role of adsorbent non-isothermalityduring the kinetic process. The purpose of this work is (a) to theoretically investigate the effect of adsorbent non-isothermality on the ad(de)sorption kinetics of a binary gas mixture during a DAT and (b) to estimate the error involved in calculating the transport properties of the components by the assumption of isothermal adsorption process.
Binary Gas DAT The DAT for binary gas adsorption consists of equilibrating the adsorbent mass with a binary gas mixture having a gas-phase partial pressure of pio for component i (i = 1 , 2 ) at temperature To and then introducing a very small step change in the gas-phase partial pressure of component i to pi" at the start of the kinetic test (time t = 0). The adsorbate loading of component i changes from nio (equilibrium capacity atpi' and T o )to ni" (equilibrium capacity at pi" and T o )during the test. The adsorbent temperature (T)increases (decreases) from TO, reaches a maximum (minimum), and then decreases (increases) to Towhen thermal equilibrium between the adsorbent mass and the surroundings is reached. The direction and magnitude of adsorbent temperature change is governed by the net thermal change (heat evolved or consumed) during the process. Component i is adsorbed when ni" > nio, and it is desorbed when ni" < nio. Mathematical Model and Solution It is assumed that the overall rate of transport of each component of the gas mixture into the adsorbent mass is described by the linear driving force (LDF) model:
where ni(t) is the adsorbate loading of component i at time t after the kinetic test started. ki is the overall adsorptive mass transfer coefficient of component i. ni* is the equilibrium adsorption capacity of component i at T and pi". T is the temperature of the adsorbent mass at time t. Equation 1 assumes that there is no cross transport coefficient for adsorption of component i and that the entire adsorbent mass is at the same temperature. The LDF model has been successfully used to describe pure 0 1994 American Chemical Society
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Ind. Eng. Chem. Res., Vol. 33, No. 6, 1994
gas ad(de)sorption kinetics by DAT (Sircar, 1983). It also provides an adequate model framework for design of adsorptive separation processes (Sircar, 1991). The overall heat balance for the adsorbent mass in a DAT is given by d0 dni C p - = cqi-- (ha)O (i = 1,2) (2) dt i dt where 8(=(T - TO)) is the differential change in the temperature of the adsorbent mass a t time t. C p is the heat capacity of the adsorbent mass. qi is the isosteric heat of adsorption of component i a t the adsorbate loading of ni". h is the external film heat transfer coefficient per unit area for heating or cooling the adsorbent mass. a is the external surface area per unit weight of the adsorbent mass. The boundary conditions for the DAT are
-
of T m, qio is the constant isosteric heat of adsorption of pure component i. R is the gas constant. Equations 6-8 can be,combined to get
A , = m0," where dim (=ni"/m) is the fractional coverage of the adsorbent surface by component i at Toand pi". Equations 9a and 9b show that Ai can be easily calculated for a binary Langmuirian system. Use of adsorption thermodynamics in a DAT also gives
ni"(To,pi")- nio(To,pio)= c a i j ( p j m- p i o )
0' = 1,2)
I
dni -- ki(ni"- nio) at dt
t
-
aij =
0
(10)
(z)
(11)
PPi+j
It can be easily shown for a Langmuirian binary system that
-
ni(t) = nim, O(t) = O a t t m (3d) The equilibrium amount adsorbed of component i (ni*) a t any given partial pressure @ j ) of the components of the mixture and temperature T may be functionally written as
ni* = ni*[pj,TI 0' = 1, 2) (4) Thus, using adsorption thermodynamics, one may write for a binary gas mixture: ni*[T,piml- ni"[To,pi"l= -AiO
(5)
n," - nIo= mO,"[(l- 8,")Al - 0,"A21
(12a)
n2- - n20 = mO,"[(l- 02")A2 - 8,"A,l
(12b)
where Ai = (pirn- pio)/pi"
(13) Thus, Ai represents the fractional change in partial pressure of component i used in the DAT. Ai is positive or negative depending on whether component i is adsorbing or desorbing during the kinetic test. Equations 1,2, and 5 can be simultaneously solved for a binary adsorption system using the boundary conditions of eq 3 to obtain
+
+
\k1(7) = CleY1' CZeYz' C3eys'
Equations 5 and 6 are valid for a DAT only when the differences between pio and pi" as well as those between nio and nim are kept small so that the change in the temperature of the adsorbent mass during the test is small. This also permits the calculation of Ai by obtaining the slopes of the binary adsorption isotherms a t To and pi" and by obtaining the isosteric heat of adsorption of component i at nim. For example, the mixed binary Langmuir absorption isotherm is given by (Young and Crowell, 1962)
mbg, constant T
0' = 1,2)
CAZeYzr + C3L3eYS'l(15) 8(7) =
[
n i m ~ l n i o[CISleyl' ]
+ C2S2eyz'+ C3S3eY3'I(16)
where y 1 = -1
(7) 72
bi = bio exp[qio/RTl (8) ni* is the equilibrium amount adsorbed of component i (=l,2) a t temperature T when the gas-phase partial pressure of component j (=l,2) is pj. bj is the gas-solid interaction parameter for adsorption of pure gas j . m is the saturation adsorption capacity for both components. Equation 8 describes the temperature coefficient of bi for the Langmuir isotherm. bi" is the value of bi a t the limit
(14)
=
-(F - (Y) + [ ( F2ff
(17) -~(H/(Y)]~'~
(18)
Ind. Eng. Chem. Res., Vol. 33, No. 6, 1994 1587
(k = 1,2,3) (25)
Thus, the isothermal kinetics of adsorption of both components can be described independent of each other when there are no cross transport coefficients. Adiabatic Adsorption. For the case where the adsorbent mass is insulated (h = o), so that all heat evolved (consusmed) during the ad(de)sorption process is contained within the adsorbent, it follows from eqs 18, 22, and 24 that
H = O ;7,=0
(38)
Consequently, modified forms of eqs 14-16 where the second terms on the right hand sides of these equations become CZ,C277,and C2,respectively, describe the \ki and B functions. The values of \ki and 0 at the limit o f t are
-
\k2(t--)
O(t--)
[
w = a 2 1 + p 1 + p -k2q2 n2" - n2O] kiqi nlm- nlo
(32)
The variables Jli in eqs 14 and 15 are defined by \ki(7)
fiC.1
=
= 1- f i ( 7 ) (i = 1,2) ni(7) - nio
ni" - nio
(i = 1,2)
(33) (34)
is the fractional uptake (loss) of component i during the ad(de)sorption kinetic process in the DAT at dimensionless time 7 which is defined by
= at
A , nl" - nl0
c,Ai n2- - n20
=
c,n,"A1- nl0
Heat Transfer Control. For the case where the adsorbate mass transfer coefficients are infinitely large, local equilibrium prevails between the gas and adsorbed phases at all times during the ad(de)sorptionprocess, and the change in adsorbate loading is controlled by the rate of change of adsorbent temperature which, in turn, is governed by the rate of heat loss (gain) by the adsorbent mass. The rate of adsorbate transport, in this case, is given by dni _ dt - - A i @ dt
fi(7)
7
=
(i = 1,2)
(42)
Equations 2 and 42 can be combined and integrated to get
(35)
(43)
a is the positive real root (having a value between k1 and
k2) of the following cubic equation: a3- Fa2
+ GCY- H = 0
(36)
For the special case where the mass transfer coefficients of the adsorbates are equal ( k l = kp = k), a is equal to k. The variables 71,72,and 7 3 , given by eqs 17-19, are negative quantities. Thus, \ki and 0 approach zero when t approaches infinity. Equations 14-36 provide a complete analytical solution of the binary non-isothermal DAT. They can be used to calculate f i ( t ) , the fractional uptake (loss) of both components as a function of time as well as the corresponding adsorbent temperature change during the kinetic process. The independent variables consist of ki,Ai, qi, C,, ha and (nim- nio). Special Cases Equations 14-16 can be simplified for several special cases. They are given below: Isothermal Adsorption. For isothermal adsorption (qi = 0; 0 = 01, eq 1 can be directly integrated for each component to obtain the following trivial solution: \ki(t)= e-kit (i = I, 2)
Equation 43 describes the adsorbent temperature as a function of time for this case. O* is the temperature reached by the adsorbent mass instantaneously after the gas phase partial pressures are changed from pio to pi". It can be easily shown that the changes in adsorbate loadings during this process are given by \k,(t)=
O(t)
(45)
- -
Both \ki and 0 are exponential functions of time for this case, and they approach zero (fi 1, 0 0) at infinite time.
Interesting Feature of Non-Isothermal DAT A very interesting characteristic of non-isothermalDAT can be seen by rewriting eq 34 in view of eq 5 fi(t)
(37)
"Ai
ni - nio
+
AiO ni(t)- ni"(T,pi") = 1- nim- nio nim- nio
(46)
1588 Ind. Eng. Chem. Res., Vol. 33, No. 6, 1994
-
The third term on the right-hand side of eq 46 vanishes when local mass transfer equilibrium [ni(t) ni"(T(t),pi"jl is reached during the DAT. It follows that f i ( t )can then be greater than unity when 6 > 0 and (n," - yo)< 0, or when 6 < 0 and (ni" - ni") > 0. Otherwise, f i ( t )will be less than unity. In other words, the fractional uptake (loss) of a component can be larger than unity during the DAT if that component is desorbing during the process while the adsorbent mass temperature is larger than Todue to adsorption of the other component. f i for a component can also be larger than unity even if that component is adsorbing if the adsorbent temperature is less than To due to desorption of the other component. The relative values of (ni"- ni")and qi will decide whether f i will exceed unity during the DAT. Clearly, f i ( t )cannot exceed unity if the ad(de)sorption process is isothermal (eq 37) and the cross transport coefficients are absent. It can only be caused by the effect of adsorbent non-isothermality in that case. This phenomenon of the transitional fractional uptake of a component of a binary system exceeding the equilibrium limit has also been observed for selective absorption of H2S and C02 into aqueous solutions of methyldiethanolamine (Savage et al., 1986). I t is caused by the differences in the rates of simultaneous dissolution and reactions of H2S and C02 with the amine.
Isothermal Binary Gas Ad(de)sorption Kinetics with Cross Transport Coefficients The excursion of fractional uptake (loss) of a component above unity during the adsorption kinetics measurement has been experimentallyobserved (Maroutsky and Bulow, 1987). It is traditionally explained by the presence of cross transport coefficients for adsorption of a gas mixture (Bulow, 1991). According to this concept, the rate of adsorption of component i of a gas mixture (containing j components) in an isothermal process can be written as dni(t) -= kii[ni"- ni(t)l + z k i j [ n j "- nj(t)l dt
(47)
j#i
The adsorbent mass is subjected to a step change in partial pressure of component i from pio to pi" at time t = 0. ni" and ni" are, respectively, the equilibrium adsorption capacities of component i at temperature To (constant during the process) and partial pressures pi" and pi". ni(t) and n,(t) are instantaneous adsorbate loadings of components i and j of the mixture at time t. kii is the straight transport coefficient for component i. kij are the cross transport coefficients of component i with respect to component j ( # i). For a binary system (i = 1,j = 2), eq 47 can be integrated using the boundary conditions given by eqs 3a and 3b and the same definition of \ki as described by eq 33 and 34 to
c, =-
- Yz
(52)
Y1-72
w - Y1 c, = '
w=
-[
k,,
\ - - I
Y2-Y1
+ k,, nZm- n2" n l m- nl0
1
L , = - - Y1+ h,, n," - n1O 4 2 n z m -n2"
-
\ki(t)approaches zero cf; 1)when 72 are negative quantities.
t
-
(54)
(55)
because y1 and
I t can be shown from eq (48) that f ~ ( tvaries ) between zero ( t = 0) and unity ( t a), and it can go through a maximum > 1)at ( t = t m m ) given below:
I t is, however, required that a combination of kii and ki, values exist so that tmaxis real for a given set of (ni" - ni"). It will be shown in the next section that the isothermal binary gas adsorption model with cross transport coefficients can be used to describe the non-isothermality effect in a DAT where the adsorbent temperature change is very small and isothermal behavior can be wrongly assumed. That, however, will generate misleading(evenmeaningless) values of transport coefficients.
Model Examples of Non-Isothermal Binary Gas DAT The adsorption of binary nitrogen (component 1) + oxygen (component 2) gas mixture on Na-mordenite is considered as a model example of DAT. The equilibrium adsorption isotherms of both pure components and their mixtures can be describedvery well by the Langmuir model (Kumar and Sircar, 1986). The model parameters are m = 1.65 mol/kg, b1" = 1.56 X atm-l, bzO = 6.98 X atm-l, 91" = 6.25 kcal/mol, and qzo = 4.53 kcal/mol. N2 has a larger isosteric heat of adsorption than 0 2 . It is assumed that the DAT is carried out at a base temperature (To) of 300 K and the initial-gas phase partial pressures of both components (pi")are 0.5 atm. Thus, the fractional adsorbate loadings of component 1 (elo)and component 2 ( 8 2 " ) at the start of the experiment are equal to 0.207 and 0.0515, respectively. The equilibrium selectivity of adof N2 over 02 on the zeolite sorption ( S = p1"n2~/p2"n1") at 300 K is 4.0. It is further assumed that a *5.0% step change (=Ai X 100) in the partial pressures of the components is made in the DAT. Figures 1-6 show the fractional uptake (loss) curves for N2 and 0 2 calculated by using the non-isothermal DAT model presented in this work. They also show the corresponding adsorbent temperature changes with time. The time scales in the plots represent a dimensionless time (=kit). The real time for component i can be obtained by dividing the dimensionless time with the appropriate value of ki (s-l). The dimensionless times for 6-t plots are given by (kzt). The dashed lines in these figures represent the fractional uptakes (loss)for an isothermal system. They
Ind. Eng. Chem. Res., Vol. 33, No. 6,1994 1589 1.0
1.1 1
0.9 0.8 0.8
0.7
t-
c
- NON-ISOTHERMAL
0.6
0.5
t
---
I
0.4
- NON-ISOTHERMAL
kl = 0.6 S-1 k, = 0.2 S-l
t
-L
ISOTHERMAL A1
=
A2=
---
0.05
0.4
0.3
1.3
0.2
1.2
0.3
-t
0.2
Y a
0.1
1.1
0.1 0
1
0 0
kit-'
Figure 1. Calculated binary uptake and adsorbent temperature profiles during a DAT: kl = 0.6 s-l; k2 = 0.2 a-l; A1 = A2 = 0.05.
ISOTHERMAL
A i = 0.05, A 2
= -0.05
L 0.3 0.2
0.1
z
0
0
20
10
k, = 0.6 S-' k, = 0.2 S-'
0.5
10
20
kit-'
30
Figure 2. Calculated binary uptake and adsorbent temperature profiles during a DAT kl = 0.6s-l;k:! = 0.2 s-l; A1 = 0.05;A:! = -0.05. 1.0
coincide for both components according to eq 37. The ki and Ai values for the DAT are given in the figures. The model parameters governing the thermal effects in the DAT were chosen to be C, = 0.25 cal/(gK) and X = 0.05 s-l. These values are typical for a DAT (Sircar, 1983). Therefore, the other dimensionless model parameters were calculated to be A1 = 9.02 X lo4, A2 = 1.43 X lo4, p1 = 0.2255, and PZ = 0.0258. Figure 1represents the case where the more selectively adsorbed component with the larger isosteric heat of adsorption (component 1)has also a larger mass transfer coefficient (It1 = 0.6, KZ = 0.2 s-l) than component 2. Both components are in the adsorption mode (A1 = A2 = 0.05). The adsorbent temperature rises rapidly to a maximum 8 value of 0.24 in about 4 s and then it slowly goes down to To as the adsorbent cools. The fractional uptakes of both components are initially rapid, and they are comparable to those for isothermal uptakes. Then they appear to slow down significantly. Component 1 takes much longer time than component 2 to reach 98 % of equilibrium as compared to their respective isothermal equilibration times. This apparent slower uptakes of the components is caused by the adsorbent non-isothermality during the process, albeit small. It will be shown later that both components are close to local equilibrium at the appropriate adsorbent temperature in the later parts of the uptake curves. The deviation from the isothermal uptake behavior is caused by the difference between the quantities ni*(T,pi") and nim(To,pim) created by the adsorbent nonisothermality. This shift in equilibrium amount adsorbed for component 1is much larger than that for component 2 a t any T because A1 is much larger than A2 (eq 5). As a result, non-isothermal uptake of component 1deviates more than that for component 2 as compared to those for isothermal uptakes. Figure 2 shows the uptakes for the components when their mass transfer coefficients are the same as those in Figure 1but when component 1is in the adsorption mode (A1 = 0.05) and component 2 is in the desorption mode (A2 = -0.05). The temperature of the adsorbent again rises fast to its maximum value of 8 = 0.22 in about 4 s and then it decreases to To.However, the net heat evolved during this process is less than that for the case of Figure 1because component 2 is desorbing by consuming a part of the heat
0.9
0.8 0.7
-t-
- NON-ISOTHERMAL
0.6
0.5 0.4
0.3
---
i
t
ISOTHERMAL Al
=
I-
A2=
0.05 - 0.3 0.2
t
Q v
W
0.1 0
0
20
10
30
kit+
Figure 3. Calculated binary uptake and adsorbent temperature profiles during a DAT: kl = 0.2 s-l; kz = 0.6 s-l; A1 = A2 = 0.05.
generated bythe adsorption of component 1. This explains the lower maximum adsorbent temperature and, in general, a lower adsorbent temperature at any time than that for the case of Figure 1. The fractional uptake of component 1 has a pattern very similar to that for Figure 1. The fractional loss of component 2, on the other hand, is close to that of the isothermal process a t the start of the test. Thenf2 at any given time becomes larger than that for the isothermal process, and goes above unity and through a maximum value of f2 = 1.025, before approaching unity a t t m. The reason for this is explained earlier (eq 46). Figure 3 represents the case where both components are adsorbing (A1 = A2 = 0.05) except that component 1 has a lower mass transfer coefficient (kl = 0.2 s-l) than component 2 (k2 = 0.6 s-l). The adsorbent temperature, in this case, rises more slowly than the previous cases and the maximum temperature (8 = 0.207) is reached after about 8 s. The subsequent rate of decrease of the adsorbent temperature is also slower. This is caused by the lower mass transfer coefficient of component 1which slows down
-
1590
Ind. Eng. Chem. Res., Vol. 33, No. 6, 1594 1.1
1.0
1.0
0.9
0.9
0.8
0.8
0.7
0.7
-t-
0.6 0.5 0.4
0.3
- NON-ISOTHERMAL
I 7I
-t-
kl = 0.2 S 1 k, = 0.6 S 1
+ I t
--A1
0.6 0.5
ISOTHERMAL
= 0.05,
A,
= -0.05
i
0.3
L
0.2
0.4 0.3
t
0.2 I
0.1
0.1
0
10
20
kl = 0.4 S-l k, = 0.4 S 1
---
I
ISOTHERMAL EQUATIONS (48,491 al = A , = 0.05
0.3 0.2
-gt
yst 0
0.1 0
0
0
I
NON-ISOTHERMAL
0
30
20
10
30
kit+
kit-'
Figure 4. Calculated binary uptake and adsorbent temperature profilesduring a DAT: k l = 0.2 s-l; kz = 0.6 s-l; A1 = 0.05;A2 = -0.05.
Figure 5. Calculated binary uptake and adsorbent temperature profiles during a DAT: k l = kz = 0.4 s-l; A1 = AZ = 0.05.
the rate of generation of heat in the adsorbent mass and prolongs the heat generation process throughout the DAT. The fractional uptake plots of component 1 and 2 in the dimensionless time coordinate cross each other, indicating that f 2 is initially closer to isothermal uptake behavior and then f 2 deviates from it much more than fl for component 1. However, f 2 is always larger than f l in the real time scale. Component 2 is essentially at local equilibrium with the adsorbent in the later part of the process. Hence, its uptake is controlled by the adsorbent temperature which decreases very slowly in this case. Consequently, a very slow rate of change of f 2 with the dimensionless time is exhibited. A similar effect of low k l on the adsorbent temperature change and the fractional uptake of component 2 is observed when the component 1 is in the adsorption mode (A1 = 0.05) and component 2 is in the desorption mode (A2 = -0.05) as shown by Figure 4. The initial rate of adsorbent temperature rise is much slower in this case (about 12 s reach the maximum at 0 = 0.15) which is followed by a slow cooling. f 2 again rises above unity and reaches a maximum value of f 2 = 1.045, followed by a very slow decrease to unity. Figures 5 and 6 show another example of binary fractional uptake (loss) curves and adsorbent temperature changes. In this case, both components have the same mass transfer coefficients (kl = k2 = 0.4 s-l). Both components are in the adsorption mode (A1 = A2 = 0.05) in the case of Figure 5 while component 1 is in the adsorption mode and component 2 is in the desorption mode ( A I = 0.05, A2 = -0.05) in the case of Figure 6. The maximum adsorbent temperature changes for these cases are 0.24 and 0.175 K, respectively, for Figures 5 and 6, and they occur at about 4 s after the start of the kinetic test. The fractional uptake (loss)curves exhibit similar patterns as before. It should be mentioned here that the fractional uptake (loss) profiles of components 1 and 2 given by Figures 2, 4, and 6 also represent their kinetic behavior when component 1 is desorbing (A1 = -0.05) and component 2 is adsorbing (A2 = 0.05). The only difference in that case is that the adsorbent cools down during the process (negative O ) , reaches a minimum, and then heats up to T O .
1.1 1.0 0.9
0.8
- NOM-ISOTHERMAL
0.7
-t-
k, = 0.4 S-1 k, = 0.4 S-1
0.6 0.5 0.4
I
1
Ai=
EQUATIONS (48,491 0.05, A 2 = -0.05
0.3 0.2
0.1 0 20
10
30
kit-
Figure 6. Calculated binary uptake and adsorbent temperature profiles during a DAT k l = 122 = 0.4 s-l; A1 = 0.05; A2 = -0.05.
The absolute value of B a t any time is the same as that given by these figures. It is clear from the above described examples that adsorbent non-isothermality plays a key role in determining the fractional uptake (loss) behavior of the components of a gas mixture. A very small change in the adsorbent temperature may have a significant effect on the apparent kinetic behavior of the components. The departure from the isothermal kinetic behavior can be substantial even for a very small adsorbent temperature change. It is primarily caused by the small but critical shift in the equilibrium adsorption capacities of the components at the prevailing gas phase partial pressures due to the change in the adsorbent temperature. The relative magnitudes of adsorbate mass transfer coefficients, the isosteric heats of adsorption of the components of the gas mixture, the heat capacity and external heat transfer coefficients of the adsorbent mass, and the size and direction of the step
Ind. Eng. Chem. Res., Vol. 33, No. 6, 1994 1591 change in the adsorbate partial pressure in the DAT determine the final shape of the fractional uptake (loss) curves. The maximum adsorbent temperature changes for the examples of Figures 1-6 were less than *0.3 K from a base temperature of 300 K. It may not always be possible to detect such a small change in the adsorbent temperature during the kinetic process. Consequently, the assumption of adsorbent isothermality during the DAT in order to extract adsorbate mass transfer coefficients from the fractional uptake (loss) profiles can be very misleading. For example, the fractional uptake (loss) profiles shown by Figures 5 and 6 can be very well described by the isothermal binary adsorption kinetic model with cross transport coefficients (eqs 48-56). The closed circles in these figures show the best fit of the non-isothermal uptake (loss) curves by the isothermal model. The isothermal model parameters are, respectively, (kll = -0.584, k 2 2 = 1.113, k12 = 3.862, k21 = -0.173 s-l) and (kll = 0.135, k22 = 0.304, k l z = -0.611, k2l = -0.042 s-l for Figures 5 and 6. This exercise shows that the adsorbent non-isothermality can be easily ignored and an isothermal kinetic model with added complexity (cross transport coefficients in this case) can be used to describe the effects of nonisothermal adsorption in a DAT. Cross transport coefficients have been used in past to describe experimental fractional uptake curves from mixtures where fi for a component went above unity and exhibited a maximum value (Bulow, 1991). The present study shows that the measurement of adsorbent temperature during the DAT is critical to resolve the possible ambiguity in interpreting the data. The straight and cross transport coefficients required to describe the data of Figures 5 and 6 have both positive and negative values. A similar characteristic has been observed for describing experimental binary uptake of n-pentane + n-heptane gas mixture on 5A zeolite (Maroutsky and Bulow, 1987). Although the existence of negative cross transport coefficients in a binary LDF model may not be physically inconsistent, the present study shows that they may be artifacts of ignoring adsorbent nonisothermality in data analysis. On the other hand, a negative straight transport coefficient which is required to describe the data of Figure 5 by the isothermal binary LDF model may be physically meaningless. Furthermore, two substantially different sets of straight and cross transport Coefficientswere required to describe the uptakes of the same adsorbates under the same base loadings by the isothermal model in Figures 5 and 6. The only difference between these two cases was the direction of differential step changes in the adsorbate loadings. That proves the incompatibility of the isothermal model in the present case. A clear signature of adsorbent non-isothermality, even in a DAT, is the rapid rise of fractional uptake with time a t the initial portion of the kinetic process followed by a very slow change in the fractional uptake with time as shown by the uptakes of component 1in Figures 1-6. This behavior can be easily mistaken as adsorbate mass transfer in two different pore structures in the adsorbent (Ruckenstein et al., 1971) or as the existence of some complex loading dependence of mass transfer resistances.
Limiting Mass Transfer Rates The two limiting cases of non-isothermal adsorption in a DAT are the adiabatic adsorption and adsorption under heat transfer control described earlier. Figures 7 and 8 show the uptakes of the components and the corresponding
1.0
"
2
0.9
2 0.8
--- ADIABATIC - HEAT TRANSFER
0.7 0.6
t A-
0.5
CONTROL NON-ISOTHERMAL
I
f kl = 0.2
S-l
fl
0.4 4 k,
= 0.6 S-l al = 0.05, = 0.05
1
f2
O
e
A
--------
-
0.3
0.2
-
0.2
0.1
-
0.1
0.3
b
f
01
I
0
10
kit+
t 8 a
10 30
I 20
Figure 7. Limiting binary uptake and adsorbent temperature profiles during a D A T k l = 0.2 s-l; k2 = 0.6 s-l; A1 = A2 = 0.05. 1
1.1 1 02
1.0 0.9
---
0.8
0.7
t-
0.6
u-
0.5 0.4
ADIABATIC
- HEAT TRANSFER -I I
CONTROL NON-ISOTHERMAL
f kl = 4 I
0.2 S-' k, = 0.6 S-1 = 0.05, A*= -0.05
' O
fl f2
0
"'k;; A
-
0.3
0.3
t
A
0.1
0.1
0
0
0
10
kit-.
8 m
30
20
Figure 8. Limiting binary uptake andadsorbent temperature profiles during a DAT: k l = 0.2 s-l; k2 = 0.6 s-l; A1 = 0.05; A2 = -0.05.
adsorbent temperature changes under these limiting conditions for the exemplary cases of Figures 3 and 4, respectively. The dashed and solid lines in Figures 7 and 8 represent the adiabatic and heat transfer control limits, respectively. For heat transfer control, fi and 0 instantaneously reach a finite value at t = 0. The fractional uptakes (losses) then slowly increase with time as the adsorbent cools down. The closed circles vi),the open circles and the closed triangles (8) in Figures 7 and 8 are the fractional uptakes and adsorbent temperature profiles replotted from Figures 3 and 4, respectively. It may be seen that these profiles approximately follow the adiabatic profiles at the start of the DAT and then they approach the heat transfer control profiles a t larger times. Thus these two limiting cases approximately represent the short- and long-time behavior of the profiles in a DAT. In particular, component 2, for this case, closely traces the heat transfer control profiles a t larger times because of its higher mass transfer
vz),
1592 Ind. Eng. Chem. Res., Vol. 33, No. 6, 1994
coefficient. This kind of behavior has been experimentally verified for adsorption of pure gases in a DAT (Sircar, 1981).
Consequently, the heat transfer coefficients for the adsorbent mass in a DAT can be evaluated using the later part of the uptake curves while the mass transfer coefficients must be evaluated from the initial portion of the uptake data.
Conclusions A very small change in the adsorbent temperature during the measurement of ad(de)sorption kinetics of a binary gas mixture may introduce a substantial difference in the uptake behavior of the components as compared with the corresponding isothermal adsorption kinetics. It is primarily caused by the small but critical shift in the equilibrium adsorption characteristics of the adsorbates during the non-isothermal kinetic process. Under certain conditions, the adsorbent non-isothermality can cause the fractional uptake of a component of the mixture to go above unity, reach a maximum, and then approach unity when mass and thermal equilibrium is reached. The overall effects of adsorbent non-isothermality on the uptake behavior of the components depend on the relative adsorbate mass transfer rates, the magnitudes of isosteric heats of adsorption of the components, and the heat capacity and external heat transfer coefficients of the adsorbent mass used in the test. The size and direction of the adsorbate loading changes during the test are also critical in determining the uptake behavior. A nonisothermal kinetic model must be used to account for the adsorbent temperature changes during the process in order to estimate the component mass transfer coefficients. The use of a complex isothermal kinetic model which includes straight and cross mass transport coefficients or which incorporates the dependence of mass transfer coefficients on adsorbate loadings may adequately describe the effect of adsorbent non-isothermality, but they are liable to produce physically inconsistent values of these coefficients. Measurement of adsorbent temperature change during the kinetic process may be necessary for correct interpretation of the experimental data. Nomenclature a = external heat transfer surface area per unit weight of adsorbent mass aij = thermodynamic parameter defined by eq 11 A = thermodynamic parameter defined by eq 6 b = Langmuirian gas-solid interaction parameter for pure gas bo = limiting value of b at T m C, = specific heat capacity of adsorbent mass Ck = constants defined by eqs 29-31,52, and 53 f i = fractional uptake (loss) of component i in DAT F = parameter defined by eq 20 G = parameter defined by eq 21 h = external heat transfer coefficient per unit surface area H = parameter defined by eq 22 ki = mass transfer coefficient for component i kii = straight-mass transfer coefficient for component i ki, = cross-mass transfer coefficients for component i Lk = parameters defined by eqs 25, 55, and 56 m = Langmuirian specific saturation adsorption capacity n, = specific amount adsorbed of component i p = gas-phase partial pressure q = isosteric heat of adsorption q o = isosteric heat of adsorption for Langmuir isotherm Yk = parameters defined by eqs 17-19, 50, and 51 R = gas constant
-
Sk = parameter defined by eq 26 S = selectivity of adsorption t = time T = temperature w = parameter defined by eqs 32 and 54 Greek L e t t e r s a = roots of eq 36
Pi = AiqiICp
Ai = fractional change in partial pressure of component i in DAT 8 = kl/a q = parameter defined by eq 28 A = (ha)/C, 0 = (2'- To) 0i = fractional surface coverage of component i 0* = instantaneousadsorbenttemperature change in adiabatic DATatt = O 1 = at \ki = 1 - f i
--
Superscripts a n d Subscripts O
= start of DAT ( t
m
= end of DAT ( t
0) m)
* = equilibrium condition Literature Cited Armstrong, A. A,; Stannet, V. Temperature Effects During the Sorption and Desorption of Water Vapor in High Polymers. Makromol. Chem. 1966,90,145-160. Bulow, M.Sorption Kinetics of Gas Mixtures on Molecular Sieves. Proceedings of Seiken Symposium on Adsorptive Separation; Suzuki, M., Ed.; Institute of Industrial Science: Tokyo, Japan, 1991;Vol. 7,pp 209-226. Chihara,K.; Suzuki, M.; Kawazoe, K. Effect of Heat Generation on Measurement of Adsorption Rate by Gravimetric Method. Chem. Eng. Sci. 1976,31, 505-507. Kumar, R.; Sircar,S. Skin Resistance for Adsorbate Mass Transfer intoExtruded Adsorbent Pellets. Chem.Eng. Sci. 1986,41,22152223. Marutovsky,R. M.; Bulow,M. SorptionKineticsof Multicomponent Gaseous and Liquid Mixtures on Porous Sorbents. Gas Sep. Purif. 1987,I , 66-76. Ruckenstein, E.; Vaidyanathan,A. S.;Youngquist, G. R. Sorptionby Solids with Bidisperse Pore Structure. Chem. Eng. Sei. 1971,26, 1305-1318. Ruthven, D. M.; Lee, L. K. Kinetics of Non-Isothermal Sorption Systems with Bed Diffusion Control. AZChE J. 1981,27,654663. Ruthven, D. M.; Lee, L. K.; Yucel, J. Kinetics of Non-Isothermal Sorption in Molecular Sieve Crystals. AZChE J. 1980,26,16-23. Savage, D. W.; Funk, E. W.; Yu, W. C.; Astarita, G. Selective Absorption of H2S and COz into Aqueous Solutions of Methyldiethanolamine. Znd. Eng. Chem. Fundam. 1986,25,326-330. Sircar, S. On the Measurementof Sorption Kinetics by Differential Test: Effect of the Heat of Sorption. Carbon 1981,19,285-288. Sircar, S . Linear Driving Force Model for Non-Isothermal Gas Adsorption Kinetics. J. Chem. SOC.,Faraday Trans. I 1983,79, 785-796. Sircar,S. Pressure Swing Adsorption: Research Needs by Industry. Proceedings of Third International Conference on Fundamentals of Adsorption, Sonthofen, Germany; Mersmann, A,, Ed.; Engineering Foundation: New York, 1991;pp 815-843. Sircar, S.; Kumar, R. Non-Isothermal Surface Barrier Model for Gas Sorption Kinetics on Porous Adsorbents. J.Chem. Soc., Faraday Trans. I 1984,80,2489-2507. Young, D. M.; Crowell, A. D. Physical Adsorption of Gases; Butterworth London, 1962.
Received for review October 14, 1993 Revised manuscript received January 26, 1994 Accepted February 9, 1994'
* Abstract published in Advance A C S Abstracts, April 1,1994.
