Effectiveness factor isothermal

Example 10.6 A commercial process for the dehydrogenation of ethylbenzene uses 3-mm spherical catalyst particles. The rate constant is 15s , and the diffusivity of ethylbenzene in steam is 4x 10 m /s under reaction conditions. Assume that the pore diameter is large enough that this bulk diffusivity applies. Determine a likely lower bound for the isothermal effectiveness factor. 

Consider a nonporous catalyst particle where the active surface is all external. There is obviously no pore resistance, but a film resistance to mass transfer can still exist. Determine the isothermal effectiveness factor for first-order kinetics. 

Hint. Use the pore model to estimate an isothermal effectiveness factor and obtain eff from that. Assume le =0.15 J/(m s K). 

The magnitude of this parameter indicates that we may neglect temperature variations within the pellet and use the isothermal effectiveness factor relation in our analysis. 

Calculate the isothermal effectiveness factor rj for the porous catalyst pellet in problem 1 as a function of the Thiele modulus d> for the first reaction A —> B utilizing the fact that the rate constant of the second reaction B —> C is half the rate constant of A —> B, the pellet is isothermal, and the external mass transfer resistance is negligible. 

Isothermal effectiveness factors for practical reactions cover a wide range, from as low as 0.01 to unity. With normal pellet sizes (I to y in.) r] is 0.7 to 1.0 for intrinsically slow reactions, such as the ammonia synthesis, and of the order of t/ = 0.1 for fast reactions, such as some hydrogenations of unsaturated hydrocarbons. Satterfield and Sherwood have summarized much of the experimental data for effectiveness factors for various reactions, temperatures, and pellet sizes. For reactor design it is important to be able to answer these questions ... 

The Isothermal Effectiveness Factor for Single Reactions (Irreversible and Unimolecular)... 

The isothermal and non-isothermal effectiveness factors for the single unimolecular irreversible reaction. Equilibrium adsorption-desorption model with linear isotherm For simplicity of presentation and without any loss of generality we consider here the case where the bulk temperature and concentration are taken as the reference temperature and concentration. In this case the boundary conditions (5.127) become at fu= 1.0... 

Non-isothermal ejfectiveness factor The non-isothermal effectiveness factor can be obtained numerically only by integrating the two points boundary value differential equations using different numerical techniques, the most efficient of these techniques is the orthogonal collocation method. 

FIGURE 5.46 Isothermal effectiveness factor for first order reactions. 

For a spherical particle the non-isothermal effectiveness factor t] is defined as ... 

FIGURE 5.49 Intraphase non-isothermal effectiveness factor versus 70 for linear kinetics y = 20, Sh, Nu = oo. 

Fir . 13. Isothermal effectiveness factor, tj, inside catalyst particles as function of the generalized Thiele modulus gt.n. 

Fig. 15. Non-isothermal effectiveness factor for spherical particle as function of the Thiele modulus, . Adapted from Weisz and Hicks (1962) and Trambouze et al. (1988).

What is the defining expression for the isothermal effectiveness factor in spherical catalysts Reactant A is consumed by three independent first-order irreversible chemical reactions on the interior catalytic surface. Your final expression should be based on mass transfer via diffusion and include the reactant concentration gradient at the external surface of the catalyst, where t) = 1. Define the intrapellet Damkohler number in your final answer. 

Estimate the isothermal effectiveness factor at the standard conditions from the foregoing rates. 

Isothermal Effectiveness Factors First-order reaction in a sphere Consider a simple first-order reaction... 

Garside, J. and Tavare, N.S. (1981) Non-isothermal effectiveness factors for crystal growth. Chemical Engineering Science, 36, 863-866. 

Since the reactant concentrations along the pores and within the particles are lower than the external surface concentrations, the overall effect of internal mass transfer resistances is to reduce the actually observed global rate below that measured at exterior surface conditions. It can be stated for isothermal effectiveness factors that r]concentration profile showing the pore diffusion-affected surface reaction is labeled as II in Figure 2.3. 

Gottifredi JC, Gonzo EE, Quiroga OD. Isothermal effectiveness factor—I Analytical expression for single reaction with arbitrary kinetics. Slab geometry. Chemical Engineering Science 1981 36 713-719. 

Tavera EM. Analytical expression for the non-isothermal effectiveness factor The nth-order reaction in a slab geometry. Chemical Engineering Science 2005 60 907-916. 

Hoyos B, Cadavid JG, Rangel H. Formulation and numerical calculation of non-isothermal effectiveness factor for finite cylindrical catalysts with bidimensional diffusion. Lat. Am. Appl. Res. 2004 34 17. 

FIGURE 5.23 Isothermal effectiveness factors, for slab-formed (P), cylindrical (C), and spherical (S) catalysts, for a first-order reaction. (Data from Froment, G. and Bischofih K., Chemical Reactor Analysis and Design, 2nd Edition,Wiley, New York, 1990.)... 

FIGURE 5.24 Isothermal effectiveness factors for a pth (p = I... )-order reaction (a) and for a Langmuir-Hinshelwood kinetics (b). (Data from Aris, R., The Mathematical Theory of Diffusion an Reaction in Permeable Catalysts, Vol. I, Clarendon Press, Oxford, 1975.)... 

On these grounds, the influence of diffusional effects inside the pellets, assuming an isothermal effectiveness factor, was considered. This gave for the rate of reaction inside each catalyst pellet the following expression ... 

In Illustration 4.10, a model that describes the isothermal diffusion and reaction in a catalyst pellet was considered. Solution of that model yields the reactant concentration profile within the pellet, which is then converted by integration into the so-called catalyst effectiveness factor E. Such isothermal effectiveness factors apply to small particles with high thermal conductivities and relatively low reaction rates. 

The existence of internal resistances complicates the analysis of transport effects for trickle-beds since the pellet cannot necessarily be assumed isothermal. Reactions in which the heat effect is negligible are considered first, and the case of a nonisothermal pellet will be treated in the following section. For arbitrary kinetics kfiC), the internal, isothermal effectiveness factor (Chapter 4) is ... 

In view of the fact that Bai. > Bmt for liquid-solid systems, the maximum fractional error in using the isothermal effectiveness factor is of the order of 0y/[2 l 4- 4>/ B z,)2] for 4 3. 

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