5950
Ind. Eng. Chem. Res. 2004, 43, 5950-5956
GENERAL RESEARCH Computerized Solution of the Dynamic Sorption Process for a Binary System in a Heterophase Medium Olaosebikan A. Olafadehan* and Alfred A. Susu Department of Chemical Engineering, University of Lagos, Lagos, Nigeria
A mathematical model equation for the binary adsorption-reaction process is developed and illustrated for the catalytic dehydrogenation of cyclohexane to benzene on platinum-rhenium/ alumina catalyst with unadsorbed hydrogen in the inert (argon, helium) and active (hydrogen) carrier gases using pulse and continuous flow techniques. The optimization routine of the NelderMead simplex algorithm is developed with a view toward estimating the surface reaction rate and adsorption equilibrium constants at different temperatures, which, in turn, are used to determine the activation energies and adsorption equilibrium energies for cyclohexane dehydrogenation in inert and active carrier gases using pulse and continuous flow techniques. Introduction and Literature Review Many investigators have worked successfully on the modeling of binary and multicomponent adsorption, without reaction, onto porous media in fixed beds,1-9 but none of these models consider the adsorption-reaction process. Thus, relatively little or no model equation development has been done for multicomponent adsorption and reaction in porous media. In this study, the subject of adsorption-reaction is considered with application to the dehydrogenation of cyclohexane to benzene on platinum-rhenium/alumina catalyst in inert and active diluents using pulse microcatalytic and continuous flow reactors. The catalytic dehydrogenation of cyclohexane to benzene is an important reforming reaction. The study of this reaction lends itself as a model for catalytic studies using pulse and continuous flow techniques. Therefore, the objective of this study is to develop a comprehensive mathematical model for binary adsorption and reaction in a porous medium and the necessary numerical solution method using the Nelder-Mead modified simplex method, orthogonal collocation, and Michelsen’s third-order semi-implicit Runge-Kutta method combined with a step-size adjustment strategy. The data set of Susu et al.10 is used for the kinetics of cyclohexane dehydrogenation on platinum-rhenium/alumina catalyst. The numerical solution developed permits determination of kinetic and adsorption equilibrium constants with a view toward estimating the activation energy and heat of adsorption for cyclohexane dehydrogenation on Pt-Re/Al2O3 catalyst in active and inert diluents environment using pulse and continuous flow techniques. Mathematical Model The model describes the adsorption-reaction process on the catalyst surface where diffusive and convective mass transfers play predominant roles. A material * To whom correspondence should be addressed. E-mail:
[email protected].
balance applied to the adsorbate carried by the flowing fluid stream (the macroscopic system) gives the following equation11
( )( )
∂Ci ∂2Ci ∂Ci 1 - b 3 ) DLi 2 - Uf K (C - Xi)|r)R ∂t ∂z b R fi i ∂z (1) The initial and boundary conditions needed to complete the definition of the macroscopic system are
(i) Ci ) C0i
at t e 0, zT g z g 0
(ii) UfC0i(t) - UfCi|z)0 + DLi
(iii)
|
∂Ci ∂z
z)zT
)0
(2)
∂Ci )0 ∂z at z ) 0, t > 0 (3)
at z ) zT, t > 0
(4)
A mass balance applied to the adsorbate in the pore fluid, with the reaction term, gives11
p ∂
(
r Dpi
r2 ∂r
)
∂Xi
2
∂r
+ hf ri ) p
∂Xi ∂t
n
+
( )( ) ∂q/i ∂Xj
∑ j)1 ∂X
j
(5)
∂t
subject to the initial and boundary conditions
(i) Xi ) 0
t ) 0, 0 e z e zT
( )|
(ii) pDpi
(iii)
|
∂Xi ∂r
∂Xi ∂r
r)R
r)R
(6)
) Kfi(Ci - Xi) 0 e z e zT, t > 0
) 0 0 e r e R, t > 0
(7) (8)
Equation 1 in dimensionless form for a binary system gives
10.1021/ie030877h CCC: $27.50 © 2004 American Chemical Society Published on Web 08/04/2004
Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004 5951
∂C hi ∂C hi ∂2C hi - ψi3(C hi - X h i)|σ)1 ) -ψi1 + ψi2 ∂τ ∂Z ∂Z2 i ) 1, 2 (9) where
(i) X hi ) 0 (ii)
UfA ψi1 ) zTDp1 ψi2 ) ψi3 )
i ) 1, 2
ψi1 Pei
(10)
i ) 1, 2
( )( )
1 - b 3 KfiA b R Dp1
(11)
i ) 1, 2
(12)
subject to the transformed initial and boundary conditions
(i) C hi ) 1
i ) 1, 2; τ ) 0; 1 g Z g 0
(ii) C h i(0,τ) ) 1 + (iii)
Equations 19 and 20 are subject to the transformed initial and boundary conditions
|
∂C hi ∂Z
Z)1
)0
( )|
hi 1 ∂C Pei ∂Z
σ)0
) (C hi - X h i)|σ)1 σ)1
)0
i ) 1, 2; 0 e Z e 1; τ > 0 (25) i ) 1, 2; 0 e Z e 1; τ > 0
Development of Mechanistic Reaction Model for Cyclohexane Dehydrogenation with Unadsorbed Hydrogen The scheme of the dehydrogenation reaction of cyclohexane has been considered as a consecutive scheme through the intermediate cyclohexene12,13
i ) 1, 2; τ > 0 (14)
i ) 1, 2; Z ) 1; τ > 0
[ (
(15)
)
)
[ (
]
)
∂2X ∂X h2 2R1 h2 ∂X h2 1 4σ +6 ) 2 ∂τ R1R4 - R2R3 Sh2 ∂σ ∂σ pAR3 ∂2X ∂X h1 h1 4σ + 6 + QR1 - PR3 (17) ∂σ R2 ∂σ2
)
]
However, the dehydrogenation of cyclohexane on PtRe/Al2O3 catalyst can be considered as occurring through the following reaction network
In the development of the rate expressions for various rate-determining steps that follows, note that A represents cyclohexane, D represents cyclohexene, B represents benzene, and S represents a vacant catalyst site. On the assumption that the hydrogen participating in the dehydrogenation process is unadsorbed, the following reaction scheme was considered kA
A• + S y\ z A‚S k′
where
A
R1 ) p + R2 )
(26)
Z)0
∂2X ∂X h1 pAR4 h1 ∂X h1 1 4σ +6 ) 2 ∂τ R1R4 - R2R3 R2 ∂σ ∂σ 2R2 ∂2X ∂X h2 h2 4σ +6 + PR4 - QR2 (16) 2 Sh2 ∂σ ∂σ
(
|
∂X hi (iii) ∂σ
(24)
(13)
The mass balance of the adsorbate in the pore fluid, with chemical reaction, reduces to the following expressions for a binary system, in dimensionless form
(
( )|
hi 4 ∂X Shi ∂σ
i ) 1, 2; τ ) 0; 1 g σ g 0
q/01
∂Q/1
(18)
C01 ∂X h1
q/01 ∂Q/1 C01 ∂X h 2
s1
(19)
ks2
z B‚S + 2H2 D‚S y\ k′ s2
( )
2q/02R2Kf1 ∂Q/2 R3 ) ApSh1C02Kf2 ∂X h1
(
ks1
A‚S y\ z D‚S + H2 k′
kD1
(20)
)
2R2Kf1 ∂Q/2 R4 ) pC02 + q/02 ApSh1C02Kf2 ∂X h 2
(21)
P)
hf r1A Dp1C01
(22)
Q)
hf r2R Kf2C02
(23)
B‚S y\ zB+S k′ D1
(27) (28) (29) (30)
where kA and k′A are the chemisorption rate coefficients for the forward and backward reactions, respectively; ks1, k′s1, ks2, and k′s2 are the surface reaction rate coefficients; kD1 and k′D1 are the rate constants for the desorption steps; KA ) kA/k′A is the adsorption equilibrium constant; Ks1 ) ks1/k′s1 and Ks2 ) ks2/k′s2 are the surface reaction equilibrium constants; and KD1 ) kD1/k′D1 ) 1/KR1 are the adsorption equilibrium constants. The thermodynamic equilibrium relation for the overall reaction is given by K ) KAKs1Ks2/KR1.
5952 Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004
Site balance requires
Hence, for each component in the binary system, the reaction rate expression, hfri, is given by
CS + CAS + CDS + CBS ) 1
(31)
(
kA CA -
r1 ) 1+
(
K
KR1CBCH22 CH2 Ks2
) )
CBCH2
Ks1
(32)
+ 1 + KR1CB
(
r2 )
1 + KACA +
CBCH2
KR1CBCH22 Ks2
)
r3 )
)
r4 ) 1 + KACA +
- CB
)
KAKs1CA KKR1CA + CH2 C 3
(
CH22 Ks2
(
CH22 Ks2
)
(38)
)
(39)
+ 1 KR1CB
+ 1 KR1CB
Hence, the equilibrium functions are given by eqs 38 and 39. The partial derivatives contained in eqs 18-21 are thus obtained, and consequently, R1, ..., R4 are evaluated through the following expression
(34)
∂Q/i C0i ∂q/i ) / ∂X hi q ∂Xi
(40)
0i
4. If the desorption of adsorbed benzene is ratedetermining, then
CH23
)
(37)
+ 1 KR1CB
KR1X2
q/2 ) CBS )
2
+ 1 KR1CB
KAX1
q/1 ) CAS )
(33)
KAKs1CA 1 + KACA + + KR1CB CH2
(
Ks2
1 + KAX1 +
(
kD1KR1
(
)
(36)
The adsorption equilibrium relationships for cyclohexane (component 1) and benzene (component 2) were obtained by considering their respective adsorptions on the catalyst surface using eqs 27 and 30. Therefore, the respective concentrations of the adsorbed species are given by
+ KR1CB
CBCH2 CA CH2 K
KCA
1 + KAX1 +
CH22
1 + KAX1 +
3. If the conversion of adsorbed cyclohexene to adsorbed benzene is rate-determining, then
ks2KAKs1
Ks2
ks1KAX2
hf r2 )
3
K
(
CH22
1 + KAX1 +
3
2. If the surface conversion of adsorbed cyclohexane to adsorbed cyclohexene is rate-determining, then
ks1KA CA -
ks1KAX1
hf r1 ) -
The mechanistic rate equations obtained for the dehydrogenation of cyclohexane to benzene with unadsorbed hydrogen, assuming that any of the elementary reaction steps outlined above is rate-determining, are presented below. 1. If the adsorption of cyclohexane is rate-determining, then
(35)
H2
Mathematical Model for the Dehydrogenation of Cyclohexane with Unadsorbed Hydrogen and Discretization Routine The model equations for the binary adsorption and reaction are given by eq 9 for the fluid phase and eqs 16 and 17 for the pore-fluid phase with the reaction term included. These equations are illustrated for the case of the binary adsorption-reaction system of the dehydrogenation of cyclohexane to benzene on a supported bimetallic catalyst (Pt-Re/Al2O3), as the adsorption of hydrogen atoms on the catalyst surface is ignored. The reaction rate expression is given by eq 33 as step 2 in the reaction mechanism has been shown to be the slowest step by investigations at lower pressures;13 at high pressures,14 cyclohexane could be dehydrogenated much more slowly than cyclohexene, so step 2 is again rate-determining.
Method of Numerical Solution The method of numerical solution for the resulting model equations for the binary adsorption-reaction case is the method of orthogonal or optimal collocation.5,15-24 When the method of orthogonal collocation is applied to the space variables Z and σ, eq 9 becomes
dC h i,j
N+2
) -ψi1(
N+2
Aj,lC h i,l) + ψi2( ∑ Bj,lC h i,l) ∑ l)1 l)1
dτ ψi3(C h i,j - X h i,N+1)
i ) 1, 2; j ) 2, 3, ..., N + 1 (41)
The initial and boundary conditions are transformed as follows
C h i,j ) 1 C h i,1 ) 1 +
i ) 1, 2; j ) 1, 2, ..., N + 2; τ ) 0 (42) 1 Pei
N+2
(
A1,lC h i,l) ∑ l)1
i ) 1, 2; Z ) 0; τ > 0 (43)
N+2
AN+2,lC h i,l ) 0 ∑ l)1
i ) 1, 2; Z ) 1; τ > 0
(44)
Also, applying orthogonal collocation to the system
Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004 5953
model equations for the spherical particle radial direction, we have
dX h 1,j
[
dτ
NR+1 pAR4 (4σj Bj,lX h 1,l + ) dτ R1R4 - R2R3 R2 l)1 NR+1 NR+1 NR+1 2R2 Aj,lX h 1,l) (4σj Bj,lX h 2,l + 6 Aj,lX h 2,l) + 6 Sh2 l)1 l)1 l)1
1
∑
∑
∑
∑
]
PR4 - QR2 dX h 2,j
)
dτ
[
1
2R1
j ) 1, 2, ..., NR (45)
∑
(4σj
R1R4 - R2R3 Sh2 l)1 NR+1 NR+1 pAR3 6 Aj,lX h 2,l) (4σj Bj,lX h 1,l + l)1 l)1 R2
NR+1
Aj,lX h 1,l) + QR1 - PR3 ∑ l)1
dC h 2,j dτ
]
dτ
j ) 1, 2, ..., NR (46)
N+1
h 2,j + )P h e2f(t) + ψ′6C
4
( Shi
N+1
C h i,j -
C h i,l )
N+1
+
A1,N+1
∑ l)2
Pei
1+
AN+2,l
+
A1,N+1 A1,1
-
A1,N+2
AN+2,N+2
N+1
A1,1 - Pei
+
∑ l)2
(
AN+2,1
i ) 1, 2
A1,l
-
-
AN+2,l AN+2,1
)
(49)
C h i,l
A1,N+2 A1,1 - Pei
i ) 1, 2 (50)
Use of the four expressions (eqs 49 and 50 with i ) 1, 2) to eliminate the concentrations at Z ) 0 and Z ) 1 (C h 1,1, C h 2,1, C h 1,N+2, C h 2,N+2) from eq 41 results in a reduction of the number of terms in the summations. After the above substitutions have been made, the summations in eq 41 run from l ) 2 to l ) N + 1. The resulting forms of these equations for the fluid phase using the pulse technique are
(54)
NR
4
∑AN+1,lXh i,l Sh l)1 4 Shi
i ) 1, 2
(55)
AN+1,N+1
Use of eq 55 to eliminate X h 1,N+1 and X h 2,N+1 from eqs 45 and 46 yields two expressions having summations ranging from l ) 1 to l ) NR. The resulting forms of these equations for the pore-fluid phase using the pulse and continuous flow techniques are
dX h 1,j
)
dτ dX h 2,j dτ
AN+2,N+2
A1,1 - Pei
AN+2,N+2
)
C h i,l
AN+2,1
-
A1,N+2
Pei C h i,N+2 )
(
A1,l
(53)
NR
Fj,lC h 2,l + ∑A h ′′N+1,lX h 2,l ∑ l)2 l)1
i
X h i,N+1 )
AN+1,lX h i,l) ) C h i,j - X h i,N+1 ∑ l)1
Pei
NR
Equation 48 was used to reduce the sums in eqs 45 and 46 such that they are taken from l ) 1 to l ) NR to yield the following expression
NR+1
where NR is the number of collocation points in the radial direction inside the catalyst particle. It should be noted that the boundary points Z ) 0 and Z ) 1 are taken as external collocation points in eq 9, whereas only the boundary point σ ) 1 is taken as an external collocation point. Equation 43 was used to reduce the number of terms in the summations in eq 41 such that it is taken from l ) 2 to l ) N + 1 by solving for C h 1,1, C h 2,1, C h 1,N+2, C h 2,N+2, to give the following expressions
A h ′′N+1,lX h 2,l ∑ l)1
Tj,lC h 1,l + ∑A h ′′N+1,lX h 1,l ∑ l)2 l)1
h 2,j + )P h e2 + ψ′6C
i ) 1, 2, 3; j )1, 2, ..., NR + 1; τ e 0 (47)
i ) 1, 2, 3; j ) 2, 3, ..., NR + 1; τ > 0 (48)
∑ l)2
NR
Fj,lC h 2,l +
N+1
h 1,j + )P h e1 + ψ′3C
subject to
X h i,j ) 0
A h ′N+1,lX h 1,l ∑ l)1
whereas for the continuous flow technique, we have
dC h 2,j
∑
∑ l)2
NR
Tj,lC h 1,l +
(52)
dτ
Bj,lX h 2,l +
N+1
h 1,j + )P h e1f(t) + ψ′3C
(51)
dC h 1,j
NR+1
∑
6
dC h 1,j
)
1 R1R4 - R2R3
1 R1R4 - R2R3
NR
(Vj,lX h 1,l - Wj,lX h 2,l) + ψ h ′7C h 1,j ∑ l)1
[
h 2,j + PR4 - QR2] (56) ψ h ′8C NR
(Zj,lX h 2,l - Uj,lX h 1,l) + ψ h ′′8C h 2,j ∑ l)1
[
h 1,j + QR1 - PR3] (57) ψ h ′′7C
In these equations, the concentrations corresponding to σ ) 1 have been eliminated. Thus, the problem is now to solve eqs 51, 52, 56, and 57 simultaneously using the pulse technique and eqs 53, 54, 56, and 57 using the continuous flow technique and the initial conditions given by eqs 42 and 47, with the aim of determining the kinetic rate constant and the adsorption isotherm parameters for the dehydrogenation of cyclohexane with unadsorbed hydrogen. This set of nonlinear coupled ordinary differential equations constitutes 4N simultaneous equations in 4N unknown concentrations (exclusive of the initial concentrations). The development of eqs 51-54, 56, and 57 for the binary adsorption-reaction system is given by Olafadehan.25 The {Aj,l} and {Bj,l} were evaluated on the basis of power series in the concentrations. The orthogonal polynomials were taken to be the Jacobi polynomials P(0,0) N (Z) of order N ) 8 for the equations of the moving fluid, because this value proved to be sufficient to obtain differences in only the fourth digit as compared to lower and higher approximations.26 The roots of these poly-
5954 Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004
nomials were taken as the interior collocation points. Also, for the diffusion equations, the collocation was performed by use of the roots of the orthogonal polynomials P(0,0) NR (σ) of order NR ) 8, and the number of the interior points of the orthogonal process was taken to be 8 for the reason given previously. Simulation Results and Discussion The resulting stiff, coupled ordinary differential equations were solved using the semi-implicit Runge-Kutta method combined with a step-size adjustment strategy. The Dirac δ pulse input was reflected in the function FT that took the following values
FT ) 1.0 at τ ) 0
(58)
FT ) 0.0 at τ * 0
(59)
Table 1. Kinetic Rate and Equilibrium Constants for Cyclohexane Dehydrogenation with Unadsorbed Hydrogen on Pt-Re/Al2O3 Catalyst in Argon Carrier Gas at Different Temperatures Using the Pulse Technique adsorption isotherm parameters temp ks1a K KA Ks1 Ks2 KR1 (K) (kg/m3s) [(kg/m3)3] (m3/kg) [(kg/m3)2] (kg/m3) (m3/kg) 429 433 457 480 a
6.85 7.78 15.87 29.42
1.09 0.90 0.30 0.11
0.064 0.058 0.029 0.019
0.052 0.046 0.024 0.015
200.13 168.08 70.41 25.33
0.609 0.502 0.165 0.063
Surface reaction rate constant.
Table 2. Kinetic Rate and Equilibrium Constants for Cyclohexane Dehydrogenation with Unadsorbed Hydrogen on Pt-Re/Al2O3 Catalyst in Helium Carrier Gas at Different Temperatures Using the Pulse Technique adsorption isotherm parameters
The exit reactor concentrations obtained were then integrated over the time period to obtain the area under the chromatogram. The integration was evaluated from the exit concentration predictions using the LegendreJacobi quadrature method.27 Theoretical prediction of conversion in the reactor is thus given by
∫0 CEXIT dτ
(60)
∫01CEXIT dτ ) area under the chromatogram
(61)
calculated conversion ) 1.0 -
1
where
The Nelder-Mead modified simplex method was then used to determine the surface reaction rate constant and adsorption equilibrium constant parameters for the cyclohexane dehydrogenation. This method is quite robust, efficient, effective, and easily implementable on a digital computer, and it avoids the computation of derivatives in the complex solutions of the exit concentration wave solutions. Derivative methods, although more efficient, are not recommended because of the difficulty of performing both analytical and numerical differentiation.28 A six-dimensional search procedure was employed as there were six constant parameters (ks1, K, KA, Ks1, Ks2 and KR1) to be determined. In the binary adsorptionreaction mathematical model equations, the kinetic rate and equilibrium constants for each of the models in eqs 32-35 were obtained using the scheme discussed earlier. Discrimination among the models was based on the positiveness of the rate and equilibrium constants, the goodness of fit as determined by the values of the objective function, and the increase and decrease with temperature of the kinetic rate and equilibrium constants, respectively. Before the ordinary differential equations for the binary adsorption-reaction study using pulse and continuous flow techniques are solved, the required physical constants must first be determined. The physical model parameterssthe pore diffusivities, axial dispersion coefficients, and mass-transfer coefficientss were obtained independently using correlations available in the literature.29-32 Here, cyclohexane and benzene were taken as components 1 and 2, respectively, and the adsorption of hydrogen was ignored. The parameter values used for this binary adsorption-reaction case were as follows: b ) 0.360, p ) 0.940, Uf ) 0.0015 m/s,
temp ks1a K KA Ks1 Ks2 KR1 (K) (kg/m3s) [(kg/m3)3] (m3/kg) [(kg/m3)2] (kg/m3) (m3/kg) 442 457 474 a
2.05 3.35 5.65
3.81 1.82 0.83
0.080 0.049 0.052
0.064 0.039 0.042
1603.54 981.09 175.75
2.13 1.02 0.46
Surface reaction rate constant.
Table 3. Kinetic Rate and Equilibrium Constants for Cyclohexane Dehydrogenation with Unadsorbed Hydrogen on Pt-Re/Al2O3 Catalyst in Hydrogen Carrier Gas at Different Temperatures Using the Continuous Flow Technique adsorption isotherm parameters temp ks1a K KA Ks1 Ks2 KR1 (K) (kg/m3s) [(kg/m3)3] (m3/kg) [(kg/m3)2] (kg/m3) (m3/kg) 553 583 603 623 a
530.74 1664.77 3348.84 6440.84
0.124 0.029 0.012 0.005
0.033 0.011 0.007 0.005
0.026 0.008 0.006 0.004
9.82 5.25 2.13 0.69
0.069 0.016 0.007 0.003
Surface reaction rate constant.
zT ) 0.5 m, R ) 0.025 m, DL1 ) DL2 ) 7.38 × 10-3 m2/s, Dp1 ) Dp2 ) 1.62 × 10-6 m2/s, and Kf1 ) Kf2 ) 0.348 m/s. The values of zT, Uf, and b were taken from the work of Harwell et al.,32 and the p and R values were taken from the works of Liapis and Rippin6 and Susu et al.,10 respectively. The objective of this study was achieved by estimating the surface reaction rate constant, ks1, and adsorption equilibrium constants, Ki, using the Nelder-Mead modified simplex algorithm28 to minimize the sum of squares of all errors between the experimental and predicted conversions, as given by ni)tb
e)
(XA,cal - XA,obs)2 ∑ n )t i
(62)
a
where e is the error. The number of experimental and simulated data points used in eq 62 was 6. These experimental data were measured in a microcatalytic reactor where the mass of catalyst was very small (592 mg).10 The optimization routine employed initial guesses for all constants until it found no other values that produced a smaller error. All four mechanistic reaction kinetic models in eqs 32-35 were tried in the Nelder-Mead search method, and discrimination among rival rate expressions was
Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004 5955 Table 4. Kinetic Rate and Equilibrium Constants for Cyclohexane Dehydrogenation with Unadsorbed Hydrogen on Pt-Re/Al2O3 Catalyst in Helium Carrier Gas at Different Temperatures Using the Continuous Flow Technique adsorption isotherm parameters
a
temp (K)
ks1a (kg/m3s)
K [(kg/m3)3]
KA (m3/kg)
Ks1 [(kg/m3)2]
Ks2 (kg/m3)
KR1 (m3/kg)
523 543 583
287.15 412.73 791.33
0.005 0.003 0.001
1.55 × 10-3 1.23 × 10-3 9.00 × 10-4
1.24 × 10-3 9.81 × 10-4 7.19 × 10-4
8.27 4.01 0.88
2.97 × 10-3 1.64 × 10-3 5.63 × 10-4
Surface reaction rate constant.
Table 5. Activation Energies and Heats of Adsorption for Cyclohexane Dehydrogenation with Unadsorbed Hydrogen on Pt-Re/Al2O3 Catalyst in Different Carriers carrier gas
reaction mode
temp range (K)
∆E (kJ/mol)
ks1o (s-1)
∆H (kJ/mol)
Ko (s-1)
argon helium hydrogen helium
pulse pulse continuous continuous
429-480 442-474 553-623 523-583
48.65 56.57 102.14 44.44
5.76 × 106 9.52 × 107 2.36 × 1012 7.74 × 106
-76.25 -82.86 -129.47 -70.16
5.68 × 10-10 6.17 × 10-10 7.30 × 10-14 5.24 × 10-10
based on the positiveness of the rate and adsorption equilibrium constants, on the goodness of fit, and on the increase and decrease with increasing temperature of the kinetic rate and adsorption equilibrium constants of the reaction, respectively. The best fit of the experimental conversion against W/F data with positive rate and equilibrium constants was obtained for model 2 (eq 33), in conformity with what was obtained in the literature,10,12 In this model, the conversion of adsorbed cyclohexane to adsorbed cyclohexene was the ratedetermining step, thus providing justification for its use in the mathematical model equation for the adsorptionreaction of cyclohexane dehydrogenation. Particular difficulties associated with the optimization problem arise because nonlinear systems can have more than one optimum. The optimization method of Nelder and Mead is capable of finding one or more of the optimum points depending on the initial guesses and step sizes used. It was observed that at initial guesses and step sizes of 0.5, 1.0, and 5, there was no effect of change in step size on the values of the optima. The sets {Ki} of the required constant parameters that minimize the error as given in eq 62 are the required values of the constant parameters. Tables 1 and 2 report the predicted results in argon and helium carrier gases, respectively, in a pulse reactor, whereas Tables 3 and 4 report the results in hydrogen and helium carrier gases, respectively, in a continuous flow reactor. Tables 2 and 4 are for the same carrier gas in different flow regimes. Thus, it expected that the kinetic parameters would be different as revealed in these tables. The data used to generate the results in Table 2 were collected in a pulse reactor with 1 µL of reactant being injected at any given time, so that the catalyst surface was relatively free of carbonaceous material. This was not the case for the data set used to generate the results in Table 4, for which cyclohexane was continuously fed through a continuous reactor system. In the latter case, it was shown10 that conversion probably occurred on a coked surface, hence the difference in the kinetic constants reported in Tables 2 and 4. From the kinetic constants, the activation and adsorption energies with corresponding preexponential factors were evaluated using the least-squares technique via the application of the Arrhenius law and plots. The calculated values of ∆E and ∆H for the binary adsorption-reaction case are reported in Table 5. The activation energies are fractions of the adsorption energies, that is, ∆E ) R∆H, and this fraction is
between 0.63 and 0.68 for the inert carriers (argon and helium) regardless of whether the reaction is carried out in the pulse or continuous flow reactor. The activation energy is not only higher in hydrogen (102.14 kJ/mol) but it represents a higher fraction of the adsorption energy (0.79). It, of course, follows that a higher adsorption energy will necessitate a higher activation energy for bond scission. Nomenclature A ) cross-sectional area of the column, m2 Aj,l ) constant generated in the orthogonal collocation method Bj,l ) constant generated in the orthogonal collocation method Ci ) concentration of solute i in the fluid phase of the column, kg/m3 C h i ) dimensionless concentration of solute i in the fluid phase of the column (Ci/C0i) C0i ) inlet concentration of solute i in the column, kg/m3 DLi ) axial diffusivity for component i in the fluid phase, m2/s Dpi ) diffusivity of component i in the fluid phase within the pore of an adsorbent, m2/s F ) volumetric flow rate of carrier gas, m3/s hfri ) reaction rate, kg/m3s Kfi ) film mass-transfer coefficient of solute i, m/s Ki ) adsorption equilibrium constant, m3/kg ks1 ) surface reaction rate constant, kg of catalyst/m3s ko ) preexponential frequency factor, s-1 N ) number of interior collocation points in the axial direction NR ) number of collocation points in the radial direction Pei ) Peclet number of component i / Qi ) dimensionless concentration of solute i in the solid / / (or adsorbed) phase (qi /q0i ) / qi ) concentration of solute i in the solid (or adsorbed) phase, kg/m3 / q0i ) concentration of solute i in the solid (or adsorbed) phase, kg/m3 R ) external radius of the adsorbent, m r ) radial distance in the particle (radius of the adsorbent pellet as measured from the center of the pellet), m Rei ) Reynolds number of component i Shi ) Sherwood number of component i t ) time from the start of the sorption process, s Uf ) linear velocity in the positive z direction, m/s W ) catalyst weight, kg W/F ) contact time, kg s/m3
5956 Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004 XA,cal ) calculated conversion of cyclohexane XA,obs ) observed conversion of cyclohexane Xi ) concentration of solute i in the pore-fluid phase, kg/ m3 X h i ) dimensionless concentration of solute i in the porefluid phase (Xi/C0i) z ) axial distance in the column, m zT ) length of the column, m Z ) dimensionless distance (z/zT) Greek Letters b ) void fraction in the bed (i.e., volume of voids per unit volume of bed) p ) void fraction in the particles (i.e., volume of pores in the pellet per unit volume of pellet) σ ) dimensionless radius (r2/R2) τ ) dimensionless time (tDpi/A) Superscript * ) equilibrium value Subscripts i, j ) indexes L ) liquid phase p ) pore phase s ) solid phase
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Received for review December 19, 2003 Revised manuscript received May 12, 2004 Accepted May 17, 2004 IE030877H
