Calculation of Effectiveness Factor

Here it is assumed that it is possible to use the concept of an effective diffusion coefficient without making too large an error. Hence the effect of micro properties will not be studied here and it is assumed the value of De is known. The discussion is restricted to the impact of the macro properties and reaction properties on the effectiveness factor. Furthermore only simple reactions are discussed. Generalized formulae are provided that enable calculation of effectiveness factor for varying properties of the catalyst or the reacting system. 

The subroutine for the calculation of effectiveness factor is used at each step of the solution to obtain the values for //, which are substituted in the material balance equation. 

After the input has been read and sorted, heat transfer coefficients and other thermodunamic data are calculated at the beginning of each catalyst zone. Temperature and conversion profile in the catalyst bed is then calculated by an axial integration. The mathematical model used in the integrations is described in. This model allows in principle the determination of diffusion restrictions and calculation of effectiveness factors for each reaction in cases where several reactions take place simultaneously. In such cases the concept of effectiveness factor may become rather dubious as shown below for the methanol synthesis, and this may be reflected in difficulties in the calculations. 

Calculate the effectiveness factor of a single catalyst pore and of a catalyst slab for zeroth-order... 

An enzyme which hydrolyzes the cellobiose to glucose, /3-glucosidase is immobilized in a sodium alginate gel sphere (2.5 mm in diameter). Assume that the zero-order reaction occurs at every point within the sphere with k0 = 0.0795 mol/sm3, and cellobiose moves through the sphere by molecular diffusion with Ds = 0.6 x 10 5 cm2 /s (cellobiose in gel). Calculate the effectiveness factor of the immobilized enzyme when the cellobiose concentration in bulk solution is 10 mol/m3. 

Derive the material and energy-balance design equations, including the equations of the catalyst pellets to calculate the effectiveness factor rj. 

Here we consider a spherical catalyst pellet with negligible intraparticle mass- and negligible heat-transfer resistances. Such a pellet is nonporous with a high thermal conductivity and with external mass and heat transfer resistances only between the surface of the pellet and the bulk fluid. Thus only the external heat- and mass-transfer resistances are considered in developing the pellet equations that calculate the effectiveness factor rj at every point along the length of the reactor. 

In this section we check our models against two industrial adiabatic fixed-bed methanators. Along the length of the methanators we calculate the effectiveness factor from the dusty gas model (DGM) and our simplified models (A) and (B). 

The main objective of the pellet model is to calculate the effectiveness factors for the six reactions that take place inside the reactor. These factors are defined as the ratios of the actual rates occurring inside the pellet and the rates occurring in the bulk phase, i.e., inside the pellet when diffusional resistances are negligible. 

Roberts and Satterfield [87, 88] analyzed this type of reaction. On the basis of numerical calculations for a flat plate, these authors presented a solution in the form of effectiveness factor diagrams, from which the effectiveness factor can be determined as a function of the Weisz modulus as well as an additional parameter Kp s which considers the influence of the different adsorption constants and effective diffusivities of the various species [91], The constant K involved in this parameter is defined as follows ... 

Other attempts have been made to arrive at modified Thiele moduli for different forms of reaction kinetics. For example, Valdman and Hughes [11] have proposed a simple approximate expression for calculating the effectiveness factor for Langmuir-Hin-shelwood kinetics of the type... 

In summary, it can be said that calculating the effectiveness factor for bimolecular reactions does not pose any new problems as compared to simple reactions. Criteria for isothermal operation and criteria which determine whether or not CBj may be substituted for CB can easily be determined. From the last criteria it can be seen that the conclusion of Doraiswamy and Sharma [4], that can be substituted for CB if the key reactant... 

Therefore, Equations 8.48 and 8.49 can be combined into one equation for concentration only. The effectiveness factor for the case considered can be calculated with the technique described in Chapter 7. It is important to stress that the effectiveness factor changes along the reactor because parameters of the reaction rate expression, Equation 6.18, e and a depend on the surface concentration and temperature. The calculated modified effectiveness factors for nonisothermal first-order reaction at different conversions = (1- CAJCa) are shown in Figure 8.9 versus the ratio of inner and outer diameters of the hollow cylinder. The parameters chosen for the calculation are ... 

Hence, since > 0, the reaction is exothermic. The geometry of the catalyst is given by X = 1 and i = 14 (Equations 6.46 and 6.47). Then from Figure 6.10 the geometry factor T equals 1.11. The problem is to calculate the effectiveness factor. 

Since An < 0, approximation 6.59 cannot be used. To calculate the effectiveness factor exactly involves solving partial differential equations, which is very time consuming. The effectiveness factor is therefore estimated as follows construct an infinite slab in such a way, that for an exothermic zeroth-order reaction, it has the same Aris numbers as given above. Since the Aris numbers are generalized the hollow cylinder under consideration and the constructed slab will have almost the same effectiveness factor. Calculation of the effectiveness factor for a slab is relatively easy. Hence an estimate for the effectiveness factor for the hollow cylinder is obtained relatively easily. 

Since theoretical calculation of effectiveness is based on a hardly realistic model of a system of equal-sized cylindrical pores and a shaky assumption for the tortuosity factor, in some industrially important cases the effectiveness has been measured directly. For ammonia synthesis by Dyson and Simon (Ind. Eng. Chem. Fundam., 7, 605 [1968]) and for SO2 oxidation by Kadlec et al. Coll. Czech. Chem. Commun., 33, 2388, 2526 [1968]). 

Mathematical models [216] for calculating these effectiveness factors involve simultaneous differential equations, which on account of the complex kinetics of ammonia synthesis cannot be solved analytically. Exact numerical integration procedures, as adopted by various research groups [157], [217]-[219], are rather troublesome and time consuming even for a fast computer. A simplification [220] can be used which can be integrated analytically when the ammonia kinetics are approximated by a pseudo-first-order reaction [214], [215], [221], according to the Equation (21) ... 

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