Langmuir 2005, 21, 3503-3510
3503
Modeling of Temperature-Programmed Desorption Thermograms for the Determination of Adsorption Heat Considering Pore and Surface Diffusion Jose´ A. Delgado* and Jose´ M. Go´mez Department of Chemical Engineering, Universidad Complutense Madrid, 28040 Madrid, Spain Received November 23, 2004. In Final Form: January 27, 2005 In this work, two partial differential equation-based models have been proposed for the quantitative analysis of temperature-programmed desorption (TPD) thermograms when the adsorption cell can be modeled as a well-mixed reactor, using the Langmuir equation as the adsorption isotherm and including the effect of diffusional resistance. One model considers pore diffusion, and the other considers surface diffusion. For both models, the rate of adsorption is proportional to the gas pressure. By nondimensionalizing these models, the range of design parameters for which the accumulation in the gas cell, diffusional resistance, and readsorption have an important effect on the TPD signal is proposed. An important conclusion is that the dimensionless numbers accounting for the diffusional resistance and the corresponding range of parameters are quite different for both mechanisms. The models have been validated with two systems where surface and pore diffusion are the relevant mechanisms: (i) CO2-Na-mordenite and (ii) CO2Na-mordenite pellets.
Introduction Temperature-programmed desorption (TPD) is an experimental technique which is often used to characterize the adsorption properties of porous catalysts and sorbents, for example, the acidity and basicity of zeolites. Commonly, the amount and strength of adsorption sites are evaluated from the desorbed amount of the sorbate and peak maximum temperature, respectively. However, because the shape and position of the TPD thermograms can be affected by phenomena such as diffusional resistance and readsorption and by the system parameters, such as the weight of catalyst to flow of carrier ratio,1,2 the interpretation of the TPD data is often difficult. The adequate parameter to estimate the adsorption strength is then the enthalpy of adsorption (∆H) extracted from the TPD data, instead of the peak maximum temperature.1,3 For this purpose, a physical model considering the effect of the system parameters on the TPD thermogram must be used. This problem has already been addressed in the literature, and several models based on different assumptions have been proposed. These models are normally derived from a mass balance on the adsorption cell and can be divided into two types: (i) the ordinary differential equation (ODE)-based models and (ii) the partial differential equation (PDE)-based models. In the first type, the model consists of a system of ODEs. It is commonly assumed that no diffusional resistances are present and the accumulation of sorbate in the adsorption cell is negligible, whereas in the second kind, consisting of a system of PDEs, these effects are considered, taking into account the accumulation terms in the mass balance on the adsorption cell. A well-known example of the first kind is the model proposed by Cvetanovic and Amenomiya,4 where the adsorption enthalpy is estimated using a * Corresponding author. Phone: +34 91 3944119. Fax: +34 91 3944114. E-mail:
[email protected]. (1) Kapustin, G. I.; Brueva, T. R.; Klyachko, A. L.; Beran, S.; Wichterlova, B. Appl. Catal. 1988, 42, 239. (2) Sawa, M.; Niwa, M.; Murakami, Y. Zeolites 1990, 10, 307. (3) Joly, J. P.; Perrard, A. Langmuir 2001, 17, 1538. (4) Cvetanovic, R. J.; Amenomiya, Y. Adv. Catal. 1967, 17, 103.
linearized equation, considering the Langmuir’s equation for the adsorption isotherm. The experimental data needed to estimate the adsorption enthalpy are the peak maximum temperatures obtained for different heating rates. In a later work, Sawa et al.2 revised the model of Cvetanovic and Amenomiya and included the effect of the saturation capacity of the sorbent, proposing a new linearized expression considering the effect of this parameter. More recently, Joly and Perrard3 have proposed a general linearized equation valid for TPD experiments performed under a vacuum and under an inert stream, evaluating the Langmuir constant from statistical thermodynamics. In all these models, several TPD experiments are necessary to estimate the adsorption enthalpy of the sorbate-sorbent system. With respect to the PDE-based models, the one proposed by Gorte5 has widely been cited in the literature. This work is intended for the case when the adsorption cell can be modeled as a well-stirred tank. Later works6,7 addressed the effect of diffusion and readsorption when the sample is placed in the adsorption cell in a packed-bed configuration. In the work of Gorte,5 several dimensionless groups of parameters were proposed, allowing a priori determination of the limit values that may be used to obtain reliable TPD data. Although this model gave good insight into the effect of the design parameters on the TPD signal, it is based on several assumptions that must be commented: (i) The adsorption isotherm is linear. In many systems, this assumption is not valid, because the adsorbed concentration used in the TPD experiments is outside the linear range of the isotherm (ii) The adsorption rate is proportional to the gas concentration. Most adsorption models in the literature for gas systems consider that this rate is proportional to gas pressure.3 (iii) The pore size of the sorbent is large enough so that there exists a gaseous phase inside the sorbent pores. It (5) Gorte, R. J. J. Catal. 1982, 75, 164. (6) Rieck, J. S.; Bell, A. T. J. Catal. 1984, 85, 143. (7) Demmin, R. A.; Gorte, R. J. J. Catal. 1984, 90, 32.
10.1021/la0471238 CCC: $30.25 © 2005 American Chemical Society Published on Web 03/01/2005
3504
Langmuir, Vol. 21, No. 8, 2005
Delgado and Go´ mez
Figure 2. Pore size distribution of the mordenite pellets used in this work estimated with Hg porosimetry.
Figure 1. Schematic representation of the diffusion mechanism in porous materials. (a) Pore diffusion and (b) surface diffusion in small pore zeolites.
is assumed that mass transport inside the pores is dominated by diffusion in the gas phase. Among all these assumptions, the last one deserves a deeper analysis, especially for zeolites, which are widely studied with the TPD technique. Diffusion in these materials has many features in common with surface diffusion, and the same type of activated diffusion mechanism can often be used to model the process.8 The mathematical modeling of flux and the boundary conditions is different for the pore and surface diffusion mechanisms, because of the different physical description of the system. This description is represented in Figure 1. For pore diffusion, the flux is proportional to the gradient of concentration in the gas phase,
N)
Dp* ∂p RTrp ∂x
(1)
where N is the flux, p is the partial pressure of the adsorbing gas (considering an ideal gas and neglecting the total pressure gradient), R is the gas constant, T is the temperature, Dp* is the effective pore diffusivity (based on the total particle cross-sectional area), rp is the particle radius (for spherical particles), and x is the dimensionless spatial coordinate. For surface diffusion, the flux is proportional to the gradient of concentration in the adsorbed phase, q (in mol kg-1)
N ) FpDs
∂q ∂x
(2)
where Fp is the particle density and Ds is the diffusivity in the adsorbed phase. Diffusivities in zeolitic materials are commonly estimated using eq 2.8 Previous models in the literature for TPD systems have apparently not considered this mechanism. In this work, two PDE-based models are proposed for the quantitative analysis of TPD thermograms when the (8) Ka¨rger, J.; Ruthven, D. M. Diffusion in Zeolites; John Wiley and Sons: New York, 1992.
adsorption cell can be modeled as a well-mixed reactor, one considering pore diffusion and the other considering surface diffusion. For the pore diffusion model, similar to the one proposed by Gorte,5 nonlinearity of the adsorption isotherm has been included considering the Langmuir equation. For both models, the rate of adsorption is proportional to the gas pressure. By non-dimensionalizing these models, the range of design parameters for which the accumulation in the gas cell, diffusional resistance, and readsorption have an important effect on the TPD signal is proposed. An important conclusion is that the dimensionless numbers accounting for the diffusional resistance and the corresponding range of parameters are quite different for both mechanisms. The models have been validated with two systems where surface and pore diffusion are the relevant mechanisms: (i) CO2-Namordenite and (ii) CO2-Na-mordenite pellets. Experimental Section A commercial pelletized Na-mordenite zeolite was used in this work, supplied by CU Chemie Uetikon AG (Si/Al ) 6). The BET surface of this material was of 350 m2 g-1, and the pore volume estimated with nitrogen adsorption was 0.13 cm3 g-1 (Micromeritics ASAP 2000). The particle density of the pellets was 1.26 g cm-3, and the pore volume estimated with Hg porosimetry was 0.276 cm3 g-1 (ThermoFinnigan Pascal 140440). The pore size distribution obtained with this technique is shown in Figure 2. One of the experiments was performed with pieces of the pellets directly (particle diameter of 3 mm), and the rest was performed with the pellets crushed previously. TPD experiments were carried out with a commercial apparatus (Micromeritics TPD/TPR 2900). Helium was used as the carrier gas (35 cm3 min-1). In a typical TPD experiment, 169 mg of crushed sample was put into the adsorption cell, dispersed in quartz wool (free volume ) 1.8 cm3). The sample previously outgassed overnight at 250 °C was saturated with pure CO2 at 1 atm at room temperature for 1 h, purged with helium for 1/2 h, and then heated under a helium flow at 10 K min-1.
Model Description The models used to predict the TPD signal are based on the assumption that the sample cell is a well-stirred tank of volume V, that is, there are no concentration gradients in the fluid phase, being pumped at a volumetric flow rate Q. The mass balance of sorbate in this volume is
net desorption rate
dc ) Qc + V (mol s ) dt
(3)
where c is the concentration in the adsorption cell. For the pore diffusion model, the net desorption rate is
Modeling of TPD Thermograms
net desorption rate )
Langmuir, Vol. 21, No. 8, 2005 3505
-3W Dp* ∂p Fprp2 RT ∂x
( )
x)1
(4)
The initial conditions for the differential equations are p ) 0, pcl ) 0, and θ ) θ0. The normalized signal predicted by both models is given by
where W is the weight of sorbent. For the surface diffusion one,
net desorption rate )
-3WDs ∂q rp2 ∂x
( )
x)1
(5)
N.S. )
p p
( )
pcl QT p + β VT0 cl T
x)1
dpcl -3WRTDsqsat ∂θ ) dt ∂x r 2V
pcl QT pcl + β VT0 T
(6)
(13)
To obtain the model in dimensionless form, the following dimensionless variables were defined
Taking into account the ideal law of gases and the variation of temperature with time in both mass balances,
dpcl -3WDp* ∂p ) dt F r 2V ∂x
Q(pcl/RT0) Wqsatθ0
p p0
v)
T T0
Y)
(14)
where p0 is the standard pressure (1 bar). The diffusivity and the kinetic constants were considered as dependent variables, their temperature dependence being given by
( (
) ) ) )
(7)
ka ) ka0 exp
-Ea,ads RT
(15)
where pcl denotes the sorbate pressure in the adsorption cell, whereas p is the pressure inside the pore, qsat is the maximum adsorption capacity of the sorbent, θ is the surface coverage (q/qsat), T0 is the initial temperature, and β is the heating rate (T ) T0 + βt). The differential mass balances in the gas and the adsorbed phase for the pore diffusion mechanism,
kd ) kd0 exp
-Ea,des RT
(16)
( )
p
-
x)1
( )
FpqsatRT p kdθ + β (8) T dθ ) kap(1 - θ) - kdθ dt
(
(9)
x)0 x)1
The differential mass balance in the adsorbed phase for the surface diffusion model is given by
∂θ Ds 1 ∂ 2∂θ ) x ∂t r 2 x2 ∂x ∂x p
( )
(11)
For simplicity, the concentration dependence of surface diffusivity8 is not considered. Although the intracrystalline diffusivity in zeolites is concentration-dependent, it must be mentioned that, in many cases, only order-of-magnitude information on the diffusivity is available,9 so the proposed assumption seems reasonable. The boundary conditions for eq 11 are given by
x)0 x)1
∂θ )0 ∂x 3 ∂θ Ds ) kap(1 - θ) - kdθ rp2 ∂x
(9) Yang, R. T. Gas Separation by Adsorption Processes; Imperial College Press: Singapore, 1997.
-Ea,diff RT
(18)
x8RT πM
(19)
T Tref
(20)
( )
1.75
where rav is an average pore radius for a sorbent with pores of different sizes, M is the molecular weight of the diffusing molecule, and the subscript ref indicates a reference temperature. The corresponding dimensionless variables were defined as the variable divided into an average value, which was estimated at Tav ) 0.5(T0 + 723 K). This parameter has been used with initial temperatures near the ambient one, and it was proposed to avoid the dependence of the dimensionless numbers on the final temperature so that they do not change for the same experiment performed with a different final temperature. The following expressions are obtained:
( )
ka ) kav
ka kav
( )
kd ) kdav
( )
Dpav* ) Dpav* (12)
(17)
τ
2 DK ) rav 3
Dm ) Dm,ref (10)
-1
where the subscript 0 indicates the preexponential factor and Ea is the activation energy of the corresponding process. For the effective pore diffusivity, is the particle porosity and τ is the tortuosity. The temperature dependences of DK (Knudsen diffusivity) and Dm (molecular diffusivity) are given by10
where ka (Pa-1 s-1) and kd (s-1) are the adsorption and desorption rate constants, respectively. The boundary conditions for eq 8 are
∂p )0 ∂x p ) pcl
1 1 + DK(T) Dm(T)
Ds ) Ds0 exp
FpqsatRT ∂p Dp* 1 ∂ 2∂p ) kap(1 - θ) + x ∂t r 2 x2 ∂x ∂x p
(
Dp* )
Dp* Dpav*
kd kdav
( )
Ds ) Dsav
Ds (21) Dsav
Substituting these variables into eqs 6-12 along with the heating schedule (T ) T0 + βt) gives for the pore diffusion model (10) Do, H. D.; Do, D. D. Chem. Eng. Sci. 1998, 53, 1239.
3506
Langmuir, Vol. 21, No. 8, 2005
( ) (
)( )( )
3WDpav* Dp* -∂v Vβ dvcl ) QT0 dY Fprp2Q Dpav* ∂x
Delgado and Go´ mez Table 1. Parameters Used in the Sensitivity Analysis
- Yvcl +
x)1
pore diffusion
( )
Vβ vcl (22) QT0 Y
( ) ( )[ ( )] ( )( ) ( )( ) ( ) βrp2 dv Dp* 1 ∂ 2∂v ) x T0Dpav* dY Dpav* x2 ∂x ∂x 2
rp FpqsatRT0kav ka v(1 - θ) + Dpav* kav
rp2FpqsatRT0kdav
(
Dpav*p0
2
) (
2
βrp ∂θ rp kavp ) T0Dpav* ∂Y Dpav*
kd βrp2 v (23) Yθ + kdav T0Dpav* Y
)( )
0
ka v(1 - θ) kav
(
)( )
rp2kdav kd θ (24) Dpav* kdav
The following equations are obtained for the surface diffusion model:
( ) (
) ( )( ) ( ) ( ) ( )[ ( )] ( )( )( ) ( )( ) ( )
3DsavWRT0qsat Ds -∂θ Vβ dvcl ) Y 2 0 QT0 dY Dsav ∂x rp p Q
Vβ vcl (25) QT0 Y
βrp2 dθ Ds 1 ∂ 2∂θ ) x T0Dsav dY Dsav x2 ∂x ∂x
x)1
- Yvcl +
x)1
3Dsav 2
rp kdav
(26)
Ds ∂θ ) Dsav ∂x
kavp0 ka kd vcl(1 - θ) θ (27) kdav kav kdav
It is now possible to study the effect of the different dimensionless groups on the TPD signal. In this work, the importance of the accumulation of gas in the adsorption cell, diffusional resistance, and readsorption will be addressed. The models will also be validated later by comparison with experimental TPD data. The complete models were solved numerically using the PDECOL program,11 which uses orthogonal collocation on finite elements technique. Further information about the calculation and fitting procedures used in this work can be found elsewhere.12 Results and Discussion Sensitivity Analysis. To study the effect of the different dimensionless parameters, the calculations were performed starting from typical values of the model parameters and changing the pertinent parameters so that the effect of the desired phenomenon is observed. The starting values for both models are shown in Table 1. These parameters simulate the desorption of ammonia from zeolitic materials. The criterion used to the determine (11) Madsen, N. K.; Sincovec, R. F. ACM Transactions of Mathematical Software 1979, 5, 326. (12) Delgado, J. A.; Nijhuis, T. A.; Kapteijn, F.; Moulijn, J. A. Chem. Eng. Sci. 2002, 57, 1835.
parameter Q (m3 s-1) V (m3) W (kg) Fp (kg m-3) T0 (K) β (K s-1) rp (m) Dm,ref (m2 s-1) Tref (K) rav (m) M (kg mol-1) τ θ0 ka0a (Pa-1 s-1) Ea,ads/Ra (K) kd0a (s-1) Ea,des/Ra (K) Ds0b (m2 s-1) Ea,diff/Rb (K) qsatc (mol kg-1)
surface diffusion
value 5.83 × 10-7 1.8 × 10-6 10-4 1300 0.35 298 0.166 10-4 5 × 10-5 293 2 × 10-7 1.7 × 10-2 4 0.8 0.8 0 1013 16 365 1
value 5.83 × 10-7 1.8 × 10-6 10-4 298 0.166 10-6
1.7 × 10-2 0.8 0.8 0 1013 16 365 0.1134 5800 1
a
Taken from ref 3 for ammonia in HY. b Estimated values, considering the diffusivity of ammonia in NaX at 298 K as 4 × 10-10 m2 s-1 (ref 8), and taking the activation energy of diffusion of ammonia in analcime.13 c Taken from ref 2 for ammonia in HMOR and HZSM5.
if an effect is important was to observe a displacement of the peak maximum temperature or the peak maximum height within 2 and 2.5% with respect to the signal obtained in the absence of the effect studied. The values of the dimensionless groups estimated in this case are denoted limit values. Pore Diffusion Model. According to eq 22, if the accumulation of gas in the adsorption cell is not important in the pore diffusion model, the dimensionless group Vβ/ (QT0) must be a small number, which represents the average residence time for the adsorption cell.5 In this case, the mass flow rate passing through the TPD detector is equal to the net desorption rate, which is proportional to the partial derivative ∂v/∂x at the particle surface. By changing the value of this group, it was observed that this assumption is valid when Vβ/QT0 < 0.02. For higher values, the TPD peak is broadened and slightly shifted to higher temperatures and a long tail is observed in the high-temperature part of the peak (Figure 3a). The limit observed is similar to the one proposed by Gorte5 (0.01), although the dimensionless number proposed by this author is slightly different {Vβ/[Q(Tfinal - T0)]}, indicating that the nonlinearity of the adsorption isotherm in the present model and the temperature effect in the gas concentration do not change the effect of average residence time much. The effect of the diffusional resistance is governed by the group multiplying ∂v/∂x at the particle surface, that is, 3WDpav*/(Fprp2Q). This number represents the ratio between the net desorption rate and the carrier gas flow rate, and the smaller this number is, the lower the sorbate pressure at the particle surface, and the steeper the pore concentration gradient. The diffusional resistance can be quantified by the inverse of this group, being negligible for Fprp2Q/(3WDpav*) < 1. For larger values, the TPD peak is shifted to higher temperatures (Figure 3b). The equivalent limit in the Gorte’s model is lower (0.1), which can be attributed to the softening effect of the Langmuir (13) Barrer, R. M. Zeolites and Clay Minerals as Sorbents and Molecular Sieves; Academic Press: London, 1978.
Modeling of TPD Thermograms
Langmuir, Vol. 21, No. 8, 2005 3507
isotherm on the adsorbed concentration gradient with respect to a linear isotherm, for the same pressure gradient in the pore and the same Henry’s constant. The dimensionless group multiplying the variation with time of v and θ [βrp2/(T0Dpav*)] indicates the effect of accumulation of sorbate in the gas phase inside the pores. It was observed that the effect of this phenomenon is negligible when realistic values of the parameters are used in the model, in agreement with previous results.5 The dependence of the readsorption effect on the model dimensionless groups is more difficult to determine, because this effect depends on two factors, the adsorption rate constant and the partial pressure of sorbate in the gas phase. This effect was studied by comparing the TPD signal obtained when the adsorption constant is equal to 0 with the one obtained increasing this parameter. For simplicity, the simulations were performed in the absence of diffusional resistance [Fprp2Q/(3WDpav*) < 1] and neglecting the accumulation of sorbate in the adsorption cell (Vβ/QT0 < 0.02), because it is not practical to perform TPD experiments with the interference of these phenomena. In this case, the net desorption rate is equal to the flow rate of sorbate purged by the carrier gas, and the adsorbed concentration gradient is negligible so that the following dimensionless equation is obtained:
1)
(
)( )
kavWqsatRT0 ka (1 - θ) Q kav kdavWqsatRT0
(
0
Qp
)( )
kd θ (28) kdav v
The effect of readsorption is shown in Figure 3c, which is important when the dimensionless group kavWqsatRT0/Q is larger than 0.1. When readsorption is important, broader peaks are obtained and peak maxima are shifted to higher temperatures. The so-called free readsorption case, that is, the attainment of quasi-steady-state equilibrium at the sorbent surface, was studied by reducing both the adsorption and desorption constants by the same factor, starting from the free readsorption case, so that the group kavWqsatRT0/Q is reduced. It was observed that free readsorption can be assumed for kavWqsatRT0/Q > 20. Surface Diffusion Model. The effect of accumulation of gas in the adsorption cell is independent of the diffusion mechanism, so the same criterion must be valid for the pore and surface diffusion models (Vβ/QT0 < 0.02), which was verified by simulation. The same applies to the effect of readsorption in the absence of diffusional resistance. It was checked that identical TPD thermograms were obtained with both models when the diffusional resistance is negligible. As it was commented before, the effect of accumulation in the pore gas phase was negligible for the pore diffusion model. The limits for diffusional control were studied first by changing the value of the dimensionless group equivalent to the one proposed for the pore diffusion model, that is, rp2p0Q/(3DsavWRT0qsat; eq 25). It was observed that a constant limit could not be defined, because the limit changed very much when different design parameters were used. From the previous study with the pore diffusion model, it was deduced that the diffusional resistance is very much related to the external partial pressure. This is so because this pressure affects directly the boundary condition for the pore mass balance (eq 10). For the surface diffusion model, the effect of the external pressure is less direct, being combined with the impacts of the adsorption and desorption constants and the diffusional time (eq 27). Therefore, it seems clear that the different effect of the dimensionless groups on the diffu-
Figure 3. Effect of dimensionless groups for the pore diffusion model. (a) Effect of average residence time: Vβ/(QT0) ∼ 0 (solid line), 0.02 (dashed line), and 0.2 (dotted line). (b) Effect of diffusional resistance: Fprp2Q/(3WDpav*) ∼ 0 (solid line), 1 (dashed line), and 10 (dotted line). (c) Effect of readsorption: kavWqsatRT0/Q ∼ 0 (solid line), 0.1 (dashed line), and 20 (dotted line).
sional resistance effect is due to the different boundary condition. In this case, it was observed that the diffusional resistance depended on the value of the dimensionless groups appearing in the pore mass balance [βrp2/(T0Dsav)] and in the boundary conditions 3Dsav/(rp2kdav) and kavp0/ kdav, so that the heating rate and the adsorptiondesorption constants have an important effect on this phenomenon. The limits of the values of the four groups involved in the diffusional resistance [i.e., rp2p0Q/ (3DsavWRT0qsat) ) G1, 3Dsav/(rp2kdav) ) G2, kavp0/kdav ) G3, and βrp2/(T0Dsav) ) G4] were estimated for different values of the design parameters, reducing Dsav until the influence of diffusional resistance is observed. To simplify the study, the calculation was performed for the freereadsorption and no-readsorption cases, so the group indicating the importance of readsorption was also included (kavWqsatRT0/Q ) G5). The changed parameters were Q, β, kdav, and kav, except for the no readsorption case, where the effect of kav was not considered. Results are shown in Table 2, where the subscript T1 indicates the value given in Table 1. The data obtained for the noreadsorption case were obtained with QT1 × 103, (kav)T1 ×
3508
Langmuir, Vol. 21, No. 8, 2005
Delgado and Go´ mez
Table 2. Limit Values of the Dimensionless Groups Involved in the Diffusional Resistance Effect for the Surface Diffusion Model parameter changed
G1
G2
G3
G4
nonea (kav)T1 × 10-2 (kdav)T1 × 102 QT1 × 103 QT1 × 103, (kdav)T1 × 10-2 βT1‚0.1
6.448 × 102 1.013 × 102 1.013 × 102 3.941 × 104 2.533 × 105 2.533 × 103
6.924 × 10-4 4.406 × 10-3 4.406 × 10-5 1.130 × 10-2 1.762 × 10-1 1.762 × 10-4
Free Readsorption 1.516 × 105 4.534 1.516 × 103 7.125 × 10-1 1.516 × 103 7.125 × 10-1 1.516 × 105 2.771 × 10-1 1.516 × 107 1.781 1.516 × 105 1.781
none (kdav)T1 × 10-2 βT1‚0.1, (kdav)T1‚0.1 βT1‚0.1 QT1 × 104
1.436 × 105 8.866 × 105 1.419 × 106 5.456 × 105 1.419 × 106
3.110 × 10-1 5.036 3.147 × 10-1 8.183 × 10-2 3.147 × 10-1
No Readsorption 1.516 × 102 1.000 × 10-2 1.516 × 104 6.234 × 10-2 1.516 × 103 9.975 × 10-3 1.516 × 102 3.837 × 10-3 1.516 × 102 9.975 × 10-3
G5
G4/(G2‚G5)
G4/G2
3.396 × 105 3.396 × 103 3.396 × 105 3.396 × 102 3.396 × 102 3.396 × 105
2 × 10-2b 5 × 10-2 5 × 10-2 7 × 10-2b 3 × 10-2 3 × 10-2
6.5 × 103 1.6 × 102 1.6 × 104 2.4 × 101 1.0 × 101 1.0 × 104
3.396 × 10-3 3.396 × 10-3 3.396 × 10-3 3.396 × 10-3 3.396 × 10-4
9.6 3.6 9.3 14 93
3.2 × 10-2 1.2 × 10-2b 3.2 × 10-2 4.7 × 10-2b 3.2 × 10-2
b a For all the calculations, V ) 0.1‚V T1 was used to ensure no effect of the average residence time. In these cases, setting the limit to the average value, the effect of resistance is clearly visible, although the displacement is then within 1.5-2 (for the high values) or within 2.5-3% (for the low values).
For the free readsorption case, a criterion was found using a combination of three groups (Figure 4):
rp2β/T0Dsav [3Dsav/(rp2kdav)](kavWqsatRT0/Q)
Figure 4. Effect of diffusional resistance for the surface diffusion model. (a) Free readsorption: G4/(G2‚G5) ∼ 0 (solid line), 0.05 (dashed line), and 0.5 (dotted line). (b) No readsorption: G4/G2 ∼ 0 (solid line), 0.05 (dashed line), and 0.5 (dotted line). The definitions of the G groups are given in the text.
10-3, and WT1 × 10-2, if nothing else was indicated. From Table 2, it is clear that the importance of diffusion limitations cannot be predicted considering only one dimensionless group among those appearing explicitly in the model differential equations. Combinations of two groups were only successful for the no-readsorption case, resulting in the following criterion to neglect the diffusional resistance (Figure 4):
rp2β/(T0Dsav) 3Dsav/(rp2kdav)
< 0.05
(29)
This result is expected considering that when no readsorption occurs, the only dimensionless groups having an effect on the model equations (eqs 25-27) are those included in the criterion, because the adsorption rate is negligible and the external partial pressure is not relevant.
< 0.05
(30)
This criterion includes the effect of the adsorption equilibrium constant, given by kav/kdav. After analyzing both criteria, it is remarkable to observe that the effect of diffusional time on the internal resistance is quadratic, stronger than for the pore diffusion model. The desorption process favors the internal resistance, because the surface coverage at the particle surface decreases faster, whereas the adsorption one has the opposite effect. The carriergas flow rate has the same effect as the desorption constant in the free readsorption case. From these results, it is deduced that very low diffusivities are necessary to have diffusional limitations for the typical values of the model parameters in Table 1 (about 10-16 m2 s-1), much lower than those predicted with the pore diffusion model (10-8 m2 s-1). Model Validation. The proposed models were tested in the analysis of the TPD of CO2 on Na-mordenite. This system gives two different peaks (Figure 5), indicating that two kinds of adsorption sites are present. First, the surface diffusion model was fitted to a TPD thermogram obtained with crushed pellets (Figure 5a, run 1). The surface diffusion model was used because it gives a more realistic representation of this system than the pore diffusion model. The two peaks observed were deconvoluted using a Gaussian fitting routine (Origin 5.1), and the model was fitted to each of the two resulting peaks, previously normalized. It was assumed that the free readsorption case was applicable, because, as it is commented below, it was observed that the peak maximum temperature changed when the weight of the sample did, which indicates that free readsorption is applicable.14 It was checked by simulation that this parameter is insensitive to the weight of sample in the no-readsorption case. Model parameters are given in Table 3. The sorbent saturation capacity qsat was estimated in our laboratory from adsorption equilibrium data of the CO2-Namordenite system. The adjustable parameters were the ratio ka0/kd0 and -∆H ) Ea,des - Ea,ads, resulting in ka0/kd0 ) 7.7 × 10-12 Pa-1 and -∆H ) 46.3 kJ mol-1 for the first peak and ka0/kd0 ) 2.1 × 10-9 Pa-1 and -∆H ) 47 kJ mol-1 (14) Kapustin, G. I.; Brueva, T. R.; Klyachko, A. L.; Beran, S.; Wichterlova, B. Appl. Catal. 1988, 42, 239.
Modeling of TPD Thermograms
Langmuir, Vol. 21, No. 8, 2005 3509 Table 3. Parameters Used for Fitting and Predicting the Experimental TPD Data parameter Q (m3 s-1) V (m3) W (kg) Fp (kg m-3) T0 (K) β (K s-1) rp (m) Dm,ref (m2 s-1) Tref (K) rav (m) M (kg mol-1) τ ka0/kd0 (Pa-1), peak 1 -∆H (kJ mol-1), peak 1 ka0/kd0 (Pa-1), peak 2 -∆H (kJ mol-1), peak 2 Ds0 (m2 s-1) Ea,diff (kJ mol-1) qsat (mol kg-1)
Figure 5. TPD thermograms of CO2 on Na-mordenite. (a) Effect of weight and particle size. Run 1, solid line, W ) 169 mg, q0 ) 0.26, and 0.68 mol kg-1. Run 2, dashed line, W ) 268 mg, q0 ) 0.34, and 0.70 mol kg-1 for each peak. Run 3, dotted line, rp ) 1.5 × 10-3 m, q0 ) 0.33, and 0.65 mol kg-1. Other parameters are listed in Table 3. (b-d) Comparison between experimental and fitted curves (dashed lines) for run 1 and between experimental and predicted curves (dashed lines) for runs 2 and 3.
for the second one. A comparison between experimental and calculated data is given in Figure 5b, where a good reproduction is observed. The adsorption heats for both peaks compare well with the adsorption heats given in the literature for this system at a similar surface coverage13 (50 kJ mol-1 for θ ) 0.1). The different preexponential factors of the adsorption equilibrium constants indicate different entropies of adsorption on the sites associated to each peak. As the preexponential factor is related to the partition functions of rotation and vibration of the adsorbed phase,15,16 the different entropies can indicate a different vibration frequency in the adsorbed phase for both peaks and/or a difference in the rotational degrees of freedom in the adsorbed phase. In this calculus, it was also assumed that no diffusional limitations were present. Although diffusivity data of CO2 in Na-mordenite could not be found in the literature, the diffusivity of CO2 in 4A,8 a zeolite with a smaller pore size (Ds0 ) 9 × 10-10 m2 s-1, Ea,diff ) 23 kJ mol-1), suggests that no diffusional resistance is present. The group defined previously G4/ (G2‚G5) was of the order of 10-5 in the worst case using these diffusion parameters. This assumption was also checked by simulation. To see the prediction capability of the proposed models, two additional experiments were performed, changing the sample weight and the particle size (Figure 5, runs 2 and 3). Run 2 was performed with the crushed sample, and run 3 was performed with large pieces of the original pellets. The TPD thermogram for run 2 was predicted with the surface diffusion model, using the adsorption equilibrium parameters estimated for run 1 (Figure 5c). The model parameters used are given in Table 3. It can be observed that the model predicts reasonably the shift of the peaks caused by the change in the sample weight. (15) Rudzinski, W.; Borowiecki, T.; Panczyk, T.; Dominko, A. J. Phys. Chem. B 2000, 104, 1984. (16) Derrouiche, S.; Bianchi, D. Langmuir 2004, 20, 4489.
run 1
run 2
run 3
5.83 × 10-7 5.83 × 10-7 5.83 × 10-7 1.8 × 10-6 1.8 × 10-6 1.8 × 10-6 1.69 × 10-4 2.68 × 10-4 1.73 × 10-4 1260 0.35 296 303 299 0.166 0.166 0.166 0.5 × 10-6 0.5 × 10-6 1.5 × 10-3 5.65 × 10-5 293 3 × 10-7 4.4 × 10-2 4 7.7 × 10-12 7.7 × 10-12 7.7 × 10-12 46.3 46.3 46.3 2.1 × 10-9 2.1 × 10-9 2.1 × 10-9 47.0 47.0 47.0 9 × 10-10 23 5.47 5.47 5.47
The TPD thermogram for run 3 was predicted with the pore diffusion model, because this model is more realistic in this case, considering the pore size distribution of the pellets (Figure 2). The average pore radius in eq 19 was estimated as the most frequent pore radius of the pellets, 300 nm. The molecular diffusivity of CO2 in helium was estimated with the Chapman-Enskog equation,17 resulting in 5.65 × 10-5 m2 s-1. The tortuosity was set to 4, a reasonable value for this parameter,8 and the values of the particle density and porosity were the ones estimated experimentally (Table 3). The comparison between the experimental data is given in Figure 3d, where it is again observed that the model predicts reasonably the shift of the peaks caused by the diffusional resistance. Therefore, it may be concluded that the proposed models are valid to describe the effect of design parameters on the TPD thermograms. Conclusions Two PDE-based models have been proposed for the quantitative analysis of TPD thermograms when the adsorption cell can be modeled as a well-mixed reactor, using the Langmuir equation as the adsorption isotherm, and including the effect of diffusional resistance. One model considers pore diffusion, and the other surface diffusion. For the pore diffusion model, the following criteria have been found to estimate the conditions in which diffusional resistance and readsorption are negliglible and free readsorption is a valid assumption: (1) negligible diffusional resistance, Fprp2Q/(3WDpav*) < 1; (2) no readsorption, kavWqsatRT0/Q < 0.1; and (3) free readsorption is valid, kavWqsatRT0/Q > 20. In the readsorption criteria, it is assumed that no diffusional resistance is present. For the surface diffusion model, the same criteria are applicable to determine the effect of readsorption. When free readsorption is applicable, diffusional control is negligible when [rp2β/ (T0Dsav)]/{[3Dsav/(rp2kdav)](kavWqsatRT0/Q)} < 0.05. When readsorption is negligible, the criterion is [rp2β/(T0Dsav)]/ [3Dsav/(rp2kdav)] < 0.05. (17) Bird, R.; Stewart, W.; Lightfoot, E. Transport Phenomena; Wiley: New York, 1960.
3510
Langmuir, Vol. 21, No. 8, 2005
The models have been validated with two systems where surface and pore diffusion are the relevant mechanisms: (i) CO2-Na-mordenite and (ii) CO2-Na-mordenite pellets. Nomenclature β c D Dpav* Ea G ka kd M N N.S. p q Q q R rp Fp
heating rate, K s-1 concentration, mol m-3 diffusivity, m2 s-1 effective pore diffusivity, m2 s-1 activation energy, J mol-1 particle porosity dimensionless group adsorption rate constant, Pa-1 s-1 desorption rate constant, s-1 molecular weight, kg mol-1 flux, mol m-2 s-1 normalized signal partial pressure, Pa adsorbed concentration, mol kg-1 volumetric flow rate, m3 s-1 surface coverage gas constant, 8.31 J mol-1 K-1 particle radius, m particle density, kg m-3
Delgado and Go´ mez T t τ V v W x Y -∆H
temperature, K time, s tortuosity adsorption cell volume, m3 dimensionless pressure sample weight, kg dimensionless spatial coordinate dimensionless temperature adsorption heat, J mol-1
Subcripts s K m cl sat av 0 ads des diff
surface Knudsen molecular adsorption cell maximum adsorption capacity average initial, preexponential adsorption desorption diffusion
Superscripts 0
standard
LA0471238
