Mass Transfer with Chemical Reactions in Porous Catalysts: A Discussion on the Criteria for the Internal and External Diffusion Limitations

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1 Defect and Diffusion Forum Online: ISSN: , Vols , pp doi:0.408/ 03 Trans Tech Publications, Switzerland Mass Transfer with Chemical Reactions in Porous Catalysts: A Discussion on the Criteria for the Internal and External Diffusion Limitations S. Gültekin,a University Malaysia Sarawak (UNIMAS), Kota Samarahan-Malaysia Uskudar University, Istanbul-Turkey (on leave) a sgultekin@feng.unimas.my Abstract. In this study, criteria for internal mass transfer given in the literature were investigated by considering tortuosity (τ) in the porous catalysts. Uncertainties in τ, which may have values between to 7, have a big impact on the effective diffusivity (D eff ) which also affects the Thiele Modulus (Φ). Since effectiveness factor (η) is function of Φ, then the criteria given for limitations are questionable. The value of D eff, and in turn the value of Φ calculated for the τ= to 7. At low Φ the effects are very small, but when the Φ increases the effect becomes more pronounced. As a result, when using internal mass transfer limitations, one has to be very careful not to get trapped by the disguised kinetics, results of which may end up with disaster. Introduction In heterogeneous catalytic reactions with porous catalysts, there are seven steps involved: External and internal mass transfer of reactants, adsorption, surface reaction, and internal and external of diffusion of the products to the bulk phase []. The magnitude of the diffusivity and its dependence on temperature and pressure for different phases are given in Table. Table - The magnitude of the diffusivity and its dependence on temperature and pressure for variety of cases []. In solid catalyzed reactions, internal (intraphase) diffusion resistances may be eliminated by reduction of catalyst particle size at the expense of high pressure drop in fixed bed reactors.. Similarly, external diffusion resistances may be eliminated by increasing velocity of feed stream through the catalyst bed and reducing the particle size. Chemical engineers would, mostly, like to work in a diffusion free region, be it mass diffusion or heat diffusion. All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans Tech Publications, (# , Pennsylvania State University, University Park, USA-7/09/6,06:08:7)

2 80 Diffusion in Solids and Liquids VIII Although, in the literature, theoretical considerations (like pore diffusivity, thermal conductivity, axial dispersion, heat and mass transfer coefficients, nature of kinetics and so on) give some guidelines for limitations criteria, the best criteria are determined experimentally. Effectiveness factor (η), which is defined as Diffusion affected rate Observed rate η = = () Diffusion unaffected rate Rate would be observed ifallsurfacewereat C s should be close to unity (η>0.95 is preferable) for isothermal systems. For exothermic nonisothermal systems η could be much greater than (T bulk <T catalyst for H<0) as can be seen in Figure []. Figure - Intraphase non-isothermal effectiveness factor versus Thiele modulus []. Some of the useful criteria for laboratory and pilot-plant scale are given below: Thiele modulus (Φ) is defined as follows [3] Surface reaction rate Φ= () Diffusion rate for small value of Φ, closer the effectiveness factor to unity. Whenever η>0.95, it is assumed the internal diffusion resistance is eliminated. Further, for an irreversible power-law kinetics, η>0.95 if Φ s <6 for zero- order, Φ s < for first-order, and Φ s <0.3 for second order reactions [4]. For non-isothermal systems two new groups are defined: Cs( H)Deff β= (Heat generation group) (3) kts and E γ= (Arrhenius group) (4) RT s In fact, the effectiveness factor is, in general, a function of some dimensionless numbers as below ( Φ, γγβ, Bi, Bi ) η= f (5) s m h where, Bi m and Bi h are Biot numbers for mass and for heat, respectively and defined as follows Interphase masstransfer Bi m = Intraphase masstransfer (6) Interphase heat transfer Bi h = Intraphase heat transfer (7)

3 Defect and Diffusion Forum Vols If both temperature and concentration gradients occur simultaneously, the criterion for 0.95<η<.05 or diffusion free region the following criteria must be satisfied φ< (8) n γβ This criterion applies to endothermic as well as exothermic reactions. As for interphase heat transfer limitations, if ( H).(-Rx)Rp RTb < 0.5 (9) ht E observed rate not to deviate more than 5% due to the temperature difference between bulk fluid and the particle [5]. Under isothermal conditions no external mass transfer resistance available if (-Rx)Rp/C b k c <0.5/n. The above criteria assume each can be treated separately (whereas the gradients may interact with each other in a very complex manner). In order to get the usage of effectiveness factor (η) as a function of Thiele modulus (φ) a steadystate mole balance on a spherical shell for a first- order reaction, one gets the following differential equation [6] d ψ dψ + φ ψ = 0 (0) d λ λ dλ with the boundary conditions: B.C. : ψ= at λ = (a) B.C. : ψisfinite at λ = 0 (b) where Ψ and λ are dimensionless concentration and dimensionless distance, respectively. Φ n knrp Cs = = Deff Surface reaction rate Diffusion rate and remembering that Φ is called Thiele modulus. The relationship between effectiveness factor and Thiele modulus for a first-order reaction can be shown that [6] η = 3/ ϕ.[ ϕ.coth( ϕ) -] (3) This relationship is plotted with MATLAB package program [7,8] and depicted in Figure. () Plot of Effectiveness Factor versus Theile Modulus for a First-order Reaction on a Spherica 0.9 Reaction Rate-limiting Region UNIMAS 0.8 Effectiveness Factor Strong Internal Diffusion-limiting Region Thiele Modulus Figure - Effectiveness Factor vs. Thiele modulus for a first-order reaction.

4 8 Diffusion in Solids and Liquids VIII Results and Discussions The relationship between tortuosity and effective diffusivity is given by Deff = DABθ/τ (4) where D AB = bulk diffusion, θ=porosity of catalyst and τ= tortuosity [9] The values of τ can change between and 7 (if not known, τ=4 is assumed). This uncertainty will be reflected to effectiveness factor in the following way: Φ is inversely proportional to the square root of D eff. If we assume τ= as a basis, then for τ=3, Φ 3 = 3/. Φ (5) and for τ =i (where i 7) Thiele modulus becomes Φi = i/. Φ (6) A plot of η versus Φ i with τ as the parameter is plotted in Figure 3 and the relevant MATLAB program is given below: MATLAB Program (Notice the element-by-element operations) % UNIMAS, Chemical Engineering and Energy Sustainability Department % Determining the Effect of Tortuosity on the Effectiveness Factor for i=:7 for Φ=0.:0.:4 η (i)=3./((i/)* Φ.^)(sqrt(i/).* Φ.*coth (sqrt(i/).* Φ)-) end f=0.:0.:4 plot(φ, η (i)) end Internal Effectývenes Factor Effect of Tortuosity on the Effectiveness factor Tortuosity decreases UNIMAS t= t=3 t=4 t=5 t=6 t= Thiele Modulus Figure 3 - Effect of tortuosity on the effectiveness factor. As can be realized from the plot that as τ increases, η decreases which is consistent with physical understanding. Again, the effect of τ on η is more pronounced as Φ increases. When there are strong diffusion limitations it can be shown that + n E n App = and E App = (7)

5 Defect and Diffusion Forum Vols where n App and E App are the apparent (observed) reaction order and apparent activation energy, respectively. Similarly, n and E are the true reaction order and true activation energy, respectively. These are referred to disguised or falsified kinetics [6]. Serious consequences could occur if the laboratory data were taken in disguised regime or criteria were taken into consideration in disguised regime and the reactor was operated in different regime. For example, what if particle size were reduced so that internal diffusion limitations became negligible? The higher activation energy, E, would cause reaction to be much more temperature sensitive [0] and there is possibility for runaway conditions to occur. As a result, when using internal mass transfer limitations, one has to be very careful not to get trapped by the disguised kinetics, result of which may be disaster. Conclusions We may itemize the results of this study with the following points: - When the tortuosity increases the effectiveness factor decreases; - The effect of tortuosity is more pronounced as Thiele modulus gets higher values; - When there are strong diffusion limitations, one may get into the disguised kinetic regime with false kinetic parameters. Therefore, serious consequences could occur if the laboratory data were taken in disguised regime or criteria were taken into consideration in disguised regime and the reactor was operated in different regime. Therefore, when using internal mass transfer limitations, due to uncertainties in D eff, one has to be very careful not to get trapped by the disguised kinetics, result of which may be extremely dangerous. Nomenclature µ: Viscosity; Bi h : Heat Biot number; Bi m : Mass Biot number; C b : Bulk concentration; C s : Surface concentration; D AB : Bulk diffusivity; D eff : Effective diffusivity; D i : Diffusion coefficient of component i; E: True activation energy; E App : Apparent activation energy h: Heat transfer coefficient; k= Heat conductivity; k c : Mass transfer coefficient; k g : Mass transfer coefficient of gas; k n : Reaction rate constant; n: True reaction order; n App : Apparent reaction order; P=Pressure; R: Universal gas constant; R p : Catalyst particle radius; -Rx: Rate of reaction of recant; T: Observed temperature; T b : Bulk temperature; T s =Surface temperature Greek Letters β: Heat generation group; γ: Arrhenius group; H: Heat of reaction; η: Effectiveness factor; θ: Porosity λ=dimensionless distance; τ: Tortousity; Φ: Thiele Modulus Acknowledgement The author would like to express his thanks to University Malaysia Sarawak (UNIMAS) for its financial support. References [] H.S. Fogler: Elements of Chemical Reaction Engineering (Prentice-Hall, 006). [] J.J. Carberry: Chemical and Catalytic Reaction Engineering (McGraw-Hill, 976). [3] C. Wagner: Z. Phys. Chem Vol. A93 (943), p. [4] C.N. Satterfield: Heterogeneous Catalysis in Practice (McGraw-Hill, 980). [6] H. S. Fogler: Essentials of Chemical Reaction Engineering (Prentice-Hall, 0). [6] D. Mears: Ind. Eng. Chem. Process Des. Dev. Vol. 0 (97), p. 54 [7] Matworks Co.: MATLAB Guide, 00. [8] S. Gültekin: Numerical Analysis with MATLAB and its Applications to Science and Engineering (Papatya Yayıncılık, Istanbul, 0). (in Turkish) [9] C.N. Satterfield: Heterogeneous Catalysis in Industrial Practice ( nd Ed., McGraw-Hill, 990). [0] O. Levenspiel: Chemical Reaction Engineering (3 rd Edition, Wiley, 999).