Korean J. Chem. Eng., 9(, 4-48 (0 DOI: 0.007/s84-0-00-7 INVITED REVIEW PPER High-order approximations for unsteady-state diffusion and reaction in sla, cylinder and sphere catalyst Dong Hyun Kim and Jitae Lee Department of Chemical Engineering, Kyungpook National University, Daegu 70-70, Korea (Received 6 Feruary 0 accepted May 0 stract High-order approximations for unsteady-state diffusion, a linear adsorption and a first-order reaction in a sla, cylinder and sphere catalyst are developed. The approximations are ased on a first-, a second-, a third- and a fifth-order approximation of the Laplace domain solutions of the exact model for the catalyst of three geometries. The coefficients in the approximations are functions of Thiele molus of the respective geometry and easy to determine. The accuracy of the approximation is shown to increase markedly with the approximation order. Key words: Mathematical Modeling, Simulation, pproximation, LDF Formula, Transient Response INTRODUCTION In modeling of unsteady-state packed-ed reactors or adsorers, the asic element is the model for catalyst particles or adsorent particles. The model for the particle is then coupled with the model for the flowing phase in the reactor or adsorer. The exact model for the particles is given in the form of a partial differential equation (PDE to descrie the transient concentration profile in the particle. The profile is then used to determine the mass exchange rate etween the particle and the flowing fluid phase. The exchange rate can also e expressed in terms of the average concentration in the particle. In this case, the reactor model does not involve the space variale in the particle and the resulting model is easier to solve, since the difficulty in the solution of the model is often commensurate with the numer of independent variales in the model. The exact average concentration, however, can only e otained from the exact concentration profile in the catalyst. This requires again the exact model in the form of PDE. But for approximate average concentrations, there have een developed many formulas without the space variale in the particle. For transient adsorption and diffusion in an adsorent, Glueckauf [] first proposed a formula that the mass exchange rate etween the particle and its surrounding is proportional to the difference etween the average concentration and the outside concentration of the particle. This type of equation has een called the LDF (linear driving force equation. Later Kim [] proposed LDF equations for transient diffusion, adsorption and reaction in catalyst particles. For cyclic adsorption processes such as pressure swing adsorption, approximation formulas are separately developed for adsorents with monodisperse pore structures [,4] and idisperse pore structures [5]. More accurate high-order approximations have also een proposed [6-9]. ll these approximations are for spherical adsorents or spherical catalysts. For other geometries such as an infinite sla and an infinite cylinder, approximations have een proposed recently. Patton et al. [0] To whom correspondence should e addressed. E-mail: dhkim@knu.ac.kr developed LDF approximations for sla and cylinder adsorents. Szukiewicz [] and Kim [] otained LDF approximations for adsorption and reaction in sla and cylinder catalysts. These LDF approximations for sla and cylinder are of first-order in the order of approximation and hence have a limited accuracy, eing applicale to very slowly changing conditions. It has een shown that the accuracy of approximation increases significantly with the order of approximation [9]. To complement the first-order approximations, high-order approximations for adsorption and reaction in sla and cylinder catalysts are developed in this study. The new high-order approximations are shown to e highly accurate and as good as the exact model for pore diffusion in most practical applications. THEORY The dimensionless unsteady-state mass alance for a linear adsorption and a first-order reaction in a catalyst is [] c c S ---- = -- τ x + -- c φ c x x In the equation, φ is the Thiele molus. In the asence of reaction, φ =0, Eq. ( ecomes the pore diffusion equation. S depends on the shape of the catalyst (S=0 for an infinite sla, S= for an infinite cylinder and S= for a sphere. The oundary conditions at the external surface and at the center are cτ (, = f( τ, c x x=0 = 0 f(τ is an aritrary function of time. When Eq. ( is coupled with the alance equation for the phase flowing through the reactor, f(τ ecomes the concentration in the flowing phase. With a zero initial condition, c(x, 0=0, Eq. ( with Eq. ( can e easily solved y the method of Laplace transform. In reactor simulations, the mass exchange rate etween the catalyst particle and its surrounding is more important than the concentration profile in the catalyst. The exchange rate can e expressed in terms of d c/ +φ c, where c is the volume-average concentration, determined y ( ( 4
High-order approximations for unsteady-state diffusion and reaction in sla, cylinder and sphere catalyst 4 Tale. Sla catalyst (S=0. Transfer function and coefficients tanh φ + s G( s =, φ = L S k φ + s D e Z 0 = -- φ Z Z = - -- φ φ Z( Z ( Z Z = ---- ---- + -- 4φ 8φ 4 8φ 5 ( Z ( Z Z( Z 5( Z 5Z = - 4φ 4 + ---- + ---- 6φ 6 6φ 7 4φ 5 ( Z ( Z Z 5( Z ( Z 5Z( Z 4 = + ---- - 48φ 5 96φ 6 64φ 7 5( Z 5Z -- + --- 8φ 8 8φ 9 ( Z 5 = - 5Z 4 5Z ( Z ( + + - ( Z Z 480φ 6 φ 7 7( Z + ---- Z 7Z( Z 6( Z 6Z ( + --- + -- ---- 8φ 8 φ 9 56φ 0 56φ Z=tanh(φ Tale. Sphere catalyst (S=. Transfer function and coefficients coth φ + s k G( s = - φ + s φ, φ = R S ---- + s D e 0 = Y --- φ ---- φ = -- Y -- φ -- Y --- φ + --- φ 4 = -- Y ( Y 9( - 4 φ -- Y 9 - + -- Y --- 8 φ 4 8φ 5 --- φ 6 ( = -- Y ( Y -- + -- Y ( Y 5( - + Y 5-8 φ 4 4 φ 5 6 φ 6 Y --- 6φ 7 + --- φ 8 ( 4 = Y ( Y Y 5 ( -- + Y ( Y 45 -- 6 φ 5 φ 6 Y Y - ( 64 φ 7 05( --- Y 05-8 φ 8 + --- Y ---- 8φ 9 φ 0 ( Y 5 = - 5Y 4 5Y ( Y 60φ 6 ( + + - 8Y φ 7 ( 6Y ( Y + - 6Y Y( Y 8φ 8 ( + -- φ 9 89( Y 89Y + 56φ 0 ---- 56φ + φ c( τ = ( S + cx (, τx S dx 0 ( Y=coth(φ The solution in the Laplace domain can e integrated with respect to x as in Eq. ( to have the Laplace domain solution for c(τ. The solution for c(τ can e expressed as G(sF(s, where F(s is the Laplace transform of f(t. G(s is usually called the transfer function. G(s for the three geometries is listed in Tale (sla, (cylinder and (sphere. To develop approximations of G(s, we need the Taylor expansion of G(s. G(s= 0 + s+ s + s + 4 s 4 + 5 s 5 (4 The coefficients 0 5 are all functions of φ and listed in Tales - for the three geometries. The approximations of G(s are in the form of rational functions which correspond to simple expressions in the time domain. The approximations developed in the present Tale. Cylinder catalyst (S=. Transfer function and coefficients I ( φ + s k G( s = ---, φ = R C ( φ + si 0 ( φ + s D e φ I ( 0 = --- φi 0 ( φ I ( φ I ( φ = --- φ -- I 0 ( φ φ I 0 ( φ φ I ( φ + φi 0 ( φi ( φ ( φ 4I 0 ( φ I ( φ φi 0 ( φ = --- φ 5 I 0 ( φ = --- φ I ( φ 4 +φ I 0 ( φi ( φ + ( φ 4φ I 0 ( φ I ( φ φ 7 I 0 ( φ 4 + ( 4 0φ I 0 ( φ I ( φ + ( φ φi 0 ( φ 4 6φ 4 I ( φ 5 + 0φ I 0 ( φi ( φ 4 + ( 70φ 0φ 4 I 0 ( φ I ( φ 4 = --- 48φ 9 I 0 ( φ 5 + ( 00φ 5φ I 0 ( φ I ( φ + ( 96 5φ + 4φ 4 I 0 ( φ 4 I ( φ + ( 7φ 48φI 0 ( φ 5 5φ 5 I ( φ 6 90φ 4 I 0 ( φi ( φ 5 + ( 0φ 5 55φ I 0 ( φ I ( φ 4 5 = -- 40φ I 0 ( φ 6 + ( 5φ 4 450φ I 0 ( φ I ( φ + ( 7φ 5 + 7φ 548φI 0 ( φ 4 I ( φ + ( 47φ 4 + 08φ 480I 0 ( φ 5 I ( φ + ( φ 5 47φ + 40φI 0 ( φ 6 Korean J. Chem. Eng.(Vol. 9, No.
44 D. H. Kim and J. Lee study are G G 5 ( s 0 = --, G s 0 + s ( = ----, G s 0 + s ( = -- and a 0 + s a 0 + s a 0 + a s + s ( s 50 + 5 s + 5 s = --- a 50 + a 5 s + a 5 s + s Tale 4. The first-order approximation G 0 ( s = -- a 0 + s 0 0 a 0 = 0 = pproximation formula: ---- = a 0 c + 0 f Tale 5. The second-order approximation G ( s = ---- + s 0 a 0 + s a 0 = 0 = 0 - = 0 pproximation formula: df ---- = a 0 c + 0 f + ---- Tale 6. The third-order approximation G ( s = 0 + s -- a 0 + a s + s a 0 = 0 -- a = 0 + -- 0 = 0 a 0 = 0 0 --- pproximation formula: ---- = a c + f + u = a 0 c + 0 f (5 The coefficients in G, G 5 are determined y matching the series expansions of the approximations and Eq. (4. The numer of matching terms in the expansions depends on the numer of coefficients in the approximations. For example, G By matching the first two terms in Eq. (4 and Eq. (6, a 0 and 0 of G (s can e determined in terms of 0 and, ie. s G is accurate to the first-order term, s, in the series expansion, it can e called a first-order approximation of G. Similarly, since G 5 (s is made to e exact up to the sixth term, from 0 to 5 s 5, it is a fifth-order approximation. s the numer of matching terms increases, the accuracy of the approximate functions of G(s increases Tale 7. The fifth-order approximation 50 + 5 s + 5 s -- a 50 + a 5 s + a 5 s + s a 50 =( 4 0 4 + + 0 /Γ a 5 =( 5 + 4 + 0 4 + 0 5 /Γ a 5 =[ 0 ( 4 5 + ( 5 4 + ( 4 ]/Γ 50 = 0 a 50 5 =[ 0 ( 5 4 + ( 4 0 5 5 =+ ( 0 + 0 ]/Γ 5 =[ 0 ( 4 5 + ( 4 5 + ( 0 4 5 =+ ( + 0 + 0 ( 5 4 ]/Γ Where Γ = 4 + 5 + 4 5 pproximation formula: ---- = a 5 c + 5 f + u = a 5 c + 5 f + v = a 50 c + 50 f Tale 8. Taylor series coefficients of the transfer function G(s for φ =0 Sla, S=0 Cylinder, S= Sphere, S= 0 -- -- 8 5 4 5 0 ( s = -- 0 = a 0 + s s 0 + 0 a 0 0 a 0 =, 0 = a 0 5 7 --- 5 6-85 8 -- 5595 48-07 9 ---- 070 47 44680 --- 5-575 ---- 85 8-8765 (6 (7 January, 0
High-order approximations for unsteady-state diffusion and reaction in sla, cylinder and sphere catalyst 45 markedly. This type of approximation is called the Pade approximation [6]. The time domain approximation formulas and the coefficients of G, G, G and G 5 are listed in Tales 4-7, respectively. DISCUSSION. pproximations in the sence of Reaction (φ =0 The coefficients in Eq. (4 ecome constants, which are listed in Tale 8 for the three geometries. The resulting approximations are listed in Tale 9. Previous approximations have een developed mainly for spherical geometry. Eq. (T9 in Tale 9 is the well-known linear driving force (LDF formula, first proposed y Glueckauf []. Eq. (T0 is a second-order approximation, proposed y Kim []. The third-order and the fifth-order approximation, Eqs. (T and T(, were developed y Lee and Kim [6]. Recently, Patton et al. [0] and Kim [] developed Eq. (T and Eq. (T5 for sla Tale 9. pproximations of G(s and the corresponding approximation formulas for φ =0 Catalyst shape pproximation of G(s pproximation formula ---- s + ---- = ( f( τ c (T s +5 ---- 6s +5 5 ---- = -- ( f( τ c + -- - df( τ 6 (T Sla (S=0 0s +05 ---- s + 45s +05 ---- = 45c + u +0f( τ =05( f( τ u (T s +60s +095 s + 0s + 475s +095 ---- = 0c + u + f( τ = 475c + v +60f( τ =095( f( τ c (T4 8 ---- s + 8 ---- = 8( f( τ c (T5 s + 4 4s + 4 ---- = 6( f( τ c + -- - df( τ 4 (T6 Cylinder (S= 4s + 84 s + 7s + 84 ---- = 7c + u + 4f( τ = 84( f( τ u (T7 48s + 840s + 46080 s + 88s + 9600s + 46080 ---- = 88c + u + 48f( τ = 9600c + v + 840f( τ = 46080( f( τ c (T8 5 - s +5 ---- =5( f( τ c (T9 s +05 0s +05 ---- = ( f( τ c + df - ( τ 0 (T0 Sphere (S= 4s + 945 -- s +05s + 945 ---- = 05c + u + 4f( τ = 945( f( τ u (T 8s + 86s +55 s + 78s +75s +55 ---- = 78c + u + 8f( τ = 75c + v + 86f( τ =55( f( τ c (T Korean J. Chem. Eng.(Vol. 9, No.
46 D. H. Kim and J. Lee Fig.. (a Step responses of the approximations for a cylinder adsorent. ( Redrawing of the step responses in the time interval (0, 0.. and cylinder adsorents, respectively. The high-order approximations, Eqs. (T-(T4 and Eqs. (T6-(T8, are new approximations developed in this study. Fig. (a compares the approximations for the case of unit step f(τ= and cylinder geometry (S=. The exact response is otained y numerical inversion of the Laplace transform G(s/s. ll the approximations approach the steady-state value c= as τ increases. The first-order approximation, Eq. (T5, shows some deviation from the exact response, ut the third-order and the fifth-order approximations, Eqs. (T7 and (T9, virtually coincide with the exact step response. It is seen that the accuracy of approximation consideraly increases with the order of approximation. Fig. ( is a magnification of Fig. (a at small τ values. Here the fifth-order approximation is seen to e as good as the exact model. s the approximations are ased on the Taylor series expansion around s=0, the approximations are valid for slowly varying f(τ. This is examined for a sinusoidal change in the surface concentration (f(τ=sin(ωτ. For f(τ=sin(τ, the responses of all the approximations including the first-order approximation coincide with the exact model response (not shown. s the frequency increases, however, the approximation starts to ecome less accurate. Fig. (a and Fig. ( show the sinusoidal responses of the approximations and the exact model for f(τ=sin(0τ and f(τ=sin(00τ, respectively. Fig.. Sinusoidal responses of the approximations for a cylinder adsorent. (a f(τ=sin(0τ, ( f(τ=sin(00τ. In Fig. (a, it is seen that the first-order approximation deviates consideraly from the exact response, while the third-order and the fifth-order approximations are indistinguishale with the exact response. For f(τ=sin(00τ, shown in Fig. (, only the fifth-order approximation is shown to e valid. With increasing order of approximation, the applicale range of the approximation is extended to higher frequencies.. pproximations in the Presence of Reaction (φ>0 Recently, simple first-order approximations for the three geometries were developed []: = {( S + + φ } 0 f τ ( ( c where S is the shape factor. The same equation can e derived from the present first-order approximation for each geometry y considering the asymptotes of 0 and at small and large values of φ. Eq. (8 gives an approximate time constant of the catalyst particle, which is τ 0 = - ( S + + φ The time constant characterizes the response to a time-varying input. Physically, the constant represents the time it takes to reach 6.% of its steady-state value after it receives a step input. smaller time (8 (9 January, 0
High-order approximations for unsteady-state diffusion and reaction in sla, cylinder and sphere catalyst 47 φ=0 is around 0 times faster than the response for φ=. When plotted in the time range of 0-5τ 0 (0-0.5 for φ= and 0-0.05 for φ=0, the two figures look similar: the first-order approximation eing least accurate and the high-order approximations eing indistinguishale with the response of the exact model. If, however, the figures are plotted on the same time scale (0-0.5 as shown in Fig. 4, it is seen that the first-order approximation is as good as the exact model when φ=0. This is ecause the error in the small time span is not apparent if it is seen on the time scale much greater than the time constant of the particle, and also the particle responds very quickly to the external disturances. CONCLUSION For the three typical geometries of catalysts (sphere, cylinder and sla, high-order approximation formulas have een developed for unsteady-state diffusion, a linear adsorption and a first-order reaction in catalysts. In reactor modeling, the approximations would e useful y removing the space variale in catalysts and making the resulting model more tractale. CKNOWLEDGEMENT This work was funded y Priority Research Centers Program through the National Research Foundation of Korea (NRF. NOMENCLTURE Fig.. Step responses of the exact model and the first-order approximation for a cylinder catalyst. (a φ =, ( φ =0. constant means a faster dynamic response. ccording to Eq. (9, the time constant is a function of φ and decreases with increasing φ. Figs. (a and ( show step responses of a cylinder catalyst (S= for φ= (τ 0 =0. and φ=0 (τ 0 =0.0048. The step response for i : i-th order coefficient in the series expansion of G(s C 0 : reference concentration [mol m ] C p : concentration in catalyst [mol m ] c : dimensionless concentration in catalyst (=C P /C 0 c : average concentration in catalyst, dimensionless D e : effective diffusivity in catalyst [m s ] f : concentration at the outer surface of catalyst, dimensionless G(s : transfer function of catalyst G i : i-th order approximation of G(s I 0, I : the zeroth- and first-order modified Bessel functions of the first kind k : reaction rate constant [s ] L S : half of sla thickness [m] R C : radius of cylindrical catalyst [m] R S : radius of spherical catalyst [m] S : shape factor of catalyst, dimensionless s : Laplace-domain variale x : space variale in catalyst, dimensionless Greek Letters φ : Thiele molus, defined in Tale, and τ : time, dimensionless REFERENCES Fig. 4. Step responses of the exact model and the first-order approximation for φ = and φ =0.. E. Glueckauf, Trans. Farad. Soc., 5, 540 (955.. D. H. Kim, IChE J., 5, 4 (989.. D. H. Kim, Chem. Eng. Sci., 5, 47 (996. 4. H.-K. Hsuen, Chem. Eng. Sci., 55, 475 (000. Korean J. Chem. Eng.(Vol. 9, No.
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