Effectiveness factor isothermal model

The temperature for methane and butane calculated with the isothermal model is a factor 1.4 times greater than the average temperature measured by Lihou and Maund (1982) in their small-scale tests, although higher local maximum temperatures were measured. In this model, combustion is stoichiometric, thus leading to very high fireball temperatures which, in turn, lead to high radiation emissions. Effective surface emissions measured experimentally were one-half the value calculated from this model, because combustion is not stoichiometric and emissivity is less than unity. 

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

Suppose that catalyst pellets in the shape of right-circular cylinders have a measured effectiveness factor of r] when used in a packed-bed reactor for a first-order reaction. In an effort to increase catalyst activity, it is proposed to use a pellet with a central hole of radius i /, < Rp. Determine the best value for RhjRp based on an effective diffusivity model similar to Equation (10.33). Assume isothermal operation ignore any diffusion limitations in the central hole, and assume that the ends of the cylinder are sealed to diffusion. You may assume that k, Rp, and eff are known. 

Figure 2.1 Dependence of the effectiveness factor on the Thiele modulus for a first-order irreversible reaction. Steady-state diffusion and reaction, slab model, and isothermal conditions are assumed.

This section is concerned with analyses of simultaneous reaction and mass transfer within porous catalysts under isothermal conditions. Several factors that influence the final equation for the catalyst effectiveness factor are discussed in the various subsections. The factors considered include different mathematical models of the catalyst pore structure, the gross catalyst geometry (i.e., its apparent shape), and the rate expression for the surface reaction. 

The equations for effectiveness factors that we have developed in this subsection are strictly applicable only to reactions that are first-order in the fluid phase concentration of a reactant whose stoichiometric coefficient is unity. They further require that no change in the number of moles take place on reaction and that the pellet be isothermal. The following illustration indicates how this idealized cylindrical pore model is used to obtain catalyst effectiveness factors. 

The Effectiveness Factor for a Straight Cylindrical Pore Second- and Zero-Order Reactions. This section indicates the predictions of the straight cylindrical pore model for isothermal reactions that are zero- and second-... 

The activity calculated from (7) comprises both film and pore diffusion resistance, but also the positive effect of increased temperature of the catalyst particle due to the exothermic reaction. From the observed reaction rates and mass- and heat transfer coefficients, it is found that the effect of external transport restrictions on the reaction rate is less than 5% in both laboratory and industrial plants. Thus, Table 2 shows that smaller catalyst particles are more active due to less diffusion restriction in the porous particle. For the dilute S02 gas, this effect can be analyzed by an approximate model assuming 1st order reversible and isothermal reaction. In this case, the surface effectiveness factor is calculated from... 

The equations describing the concentration and temperature within the catalyst particles and the reactor are usually non-linear coupled ordinary differential equations and have to be solved numerically. However, it is unusual for experimental data to be of sufficient precision and extent to justify the application of such sophisticated reactor models. Uncertainties in the knowledge of effective thermal conductivities and heat transfer between gas and solid make the calculation of temperature distribution in the catalyst bed susceptible to inaccuracies, particularly in view of the pronounced effect of temperature on reaction rate. A useful approach to the preliminary design of a non-isothermal fixed bed catalytic reactor is to assume that all the resistance to heat transfer is in a thin layer of gas near the tube wall. This is a fair approximation because radial temperature profiles in packed beds are parabolic with most of the resistance to heat transfer near the tube wall. With this assumption, a one-dimensional model, which becomes quite accurate for small diameter tubes, is satisfactory for the preliminary design of reactors. Provided the ratio of the catlayst particle radius to tube length is small, dispersion of mass in the longitudinal direction may also be neglected. Finally, if heat transfer between solid cmd gas phases is accounted for implicitly by the catalyst effectiveness factor, the mass and heat conservation equations for the reactor reduce to [eqn. (62)]... 

The catalyst intraparticle reaction-diffusion process of parallel, equilibrium-restrained reactions for the methanation system was studied. The non-isothermal one-dimensional and two-dimensional reaction-diffusion models for the key components have been established, and solved using an orthogonal collocation method. The simulation values of the effectiveness factors for methanation reaction Ch4 and shift reaction Co2 are fairly in agreement with the experimental values. Ch4 is large, while Co2 is very small. The shift reaction takes place as direct and reverse reaction inside the catalyst pellet because of the interaction of methanation and shift reaction. For parallel, equilibrium-restrained reactions, effectiveness factors are not able to predict the catalyst internal-surface utilization accurately. Therefore, the intraparticle distributions of the temperature, the concentrations of species and so on should be taken into account. 

The experimentally-determined effectiveness factor is determined as the ratio of the experimental macro reaction rate to the intrinsic reaction rate under the same interface (bulk) composition and temperature. Based on the experimental conditions of the macrokinetics, the predicted effectiveness factors of the methanation reaction and the WGSR are obtained by solving the above non-isothermal one-dimensional and two-dimensional reaction-diffusion models for the key components. Table 1 shows the calculated effectiveness factors and the experimental values. By... 

The effectiveness factor for a wide range of reaction kinetic models differs little from that of the first-order case. For an isothermal particle, the first-order reaction effectiveness factor is... 

A summary of reactor models used by various authors to interpret trickle-bed reactor data mainly from liquid-limiting petroleum hydrodesulfurization reactions (19-21) is given in Table I of reference (37). These models are based upon i) plug-flow of the liquid-phase, ii) the apparent rate of reaction is controlled by either internal diffusion or intrinsic kinetics, iii) the reactor operates isothermally, and iv) the intrinsic reaction rate is first-order with respect to the nonvolatile liquid-limiting reactant. Model 4 in this table accounts for both incomplete external and internal catalyst wetting by introduction of the effectiveness factor r)Tg developed especially for this situation (37 ). 

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... 

The simple isothermal pseudo-homogeneous model The use of a very simple model to investigate the implications of this nonmonotonic kinetics will give the opportunity to study its effect away from the combined effect of other phenomena which will interfere if a more sophisticated model is used. Xu and Froment s (1989a) rate equations were used in a one dimensional isothermal model for the catalyst tube, with a constant effectiveness factor, taken from Soliman et al. s (1988) work, to bring the actual reaction rates down to reasonable values. 

The molecular weight differences between lignin and its model compounds also complicate the use of model compound kinetics in a predictive simulation. The mobility of a high-molecular weight polymer would be much less than that of smaller model substrates (14). As for catalyst decay, a simple model was used to probe transport issues. For a first order, irreversible reaction in an isothermal, spherical catalyst pellet with equimolar counterdiffusion, the catalyst effectiveness factor and Thiele modulus provide the relevant information as... 

Another crucial factor is an adequate inclusion of these diffuse-layer and discreteness-of-charge effects into the model, which play a marked role if the overall electrolyte concentration is well below 1 M. It is why the adsorption parameters found by the interpretation of experimental data with the use of various isotherms derived mostly for uncharged species (Flory-Huggins, Frumldn, Temldn, etc) may strongly deviate from their proper values. 

The blockage of active sites by impurities can be related to the impurity concentration in solution through the Langmuir adsorption isotherm, and a number of models utilizing this concept have been proposed (Davey and Mullin, 1974 Black et al., 1986 King, 1993). A recent refinement, which incorporates the concept of an impurity effectiveness factor (Kubota and Mullin, 1995), offers an opportunity to explain several hitherto anomalous patterns of behaviour and may be summarized as follows. 

In order to compare the model predictions, the sum of the square difference of experimental and predicted values of CO conversion was performed and Model 2 led to the highest sum being the worst model. The sum of the squares of Model 2 was used as normalization factor and the normalized sum of all models are presented in Figure 4.5.2. It can be concluded that two hypotheses played a key role in the reactor modeling firstly the isothermal behavior secondly, the effectiveness factor. Further changes were import to improve, but not considerably, the data fit. It must also be noted that a few number of parameters were estimated, indicating an important likelihood feature of the reactor model. Finally, it is worth stressing that the correlation coefficient between experimental data and model predictions of Model 5 was equal to 0.8. 

Precise comparisons between packed beds and ceramic foam structures are complex since many factors - activity, effectiveness factors, mass and heat transfer, and pressure drop - are all interdependent. We have simulated the performance of a conventional steam reformer and compared it to one containing a ceramic foam cartridge loaded to achieve equivalent intrinsic activity per gram of catalyst. A 1-D model developed and tested previously for heat-pipe reformers with isothermal walls was used [57]. Pressure drop, mass transfer and heat transfer correlations for the packed bed were known to be accurate for commercial catalysts those used for the foam were determined in the studies described above. Process conditions and results are given in Table 7. 

A commercial cumene cracking catalyst is in the form of pellets with a diameter of 0.35 cm which have a surface area. Am, of 420 m g and a void volume, Vm, of 0.42cm g. The pellet density is 1.14g cm. The measured l -order rate constant for this reaction at 685K was 1.49cm s g . Assume that Knudsen diffusion dominates and the path length is determined by the pore diameter, dp. An average pore radius can be estimated from the relationship fp = 2Vm/Am if the pores are modeled as noninterconnected cylinders (see equation 4.94). Assuming isothermal operation, calculate the Thiele modulus and determine the effectiveness factor, tti, vmder these conditions. 

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