Thiele modulus second-order reaction

Example 5 Application of Effectiveness For a second-order reaction in a plug flow reactor the Thiele modulus is ( ) = SVQ, and inlet concentration is C50 = 1.0. The equation will he integrated for 80 percent conversion with Simpsons rule. Values of T) are... 

A second order reaction takes place in a flow reactor with catalyst particles in the shape of lamellae. At the inlet the concentration is 2 lbmol/cuft and the Thiele modulus is

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A catalyst mass is made up of a mixture of porous spheres with a range of diameters. The average diameters of 10% cuts are shown. A second order reaction with k = 2.5 and C0 = 1.2 is performed in a PFR. The Thiele modulus... 

A second order reaction is catalyzed in a PFR by a porous catalyst made up of small spheres. At the inlet the Thiele modulus is 0O = 10. Integrate the rate equation up to 90% conversion. 

A great deal of attention has been devoted to this topic because of the interesting and often solvable mathematical problems that it presents. Results of such calculations for isothermal zero-, first-, and second-order reactions in uniform cylindrical pores are summarized in Figure 17.6. The abscissa is a modified Thiele modulus whose basic definition is... 

Effectiveness factor versus Thiele, modulus for different values of the Biot number second-order reaction in a cylindrical pellet. 429... 

Exercise 7.4 Thiele modulus and a second-order reaction... 

For a second-order reaction, the basic equation is similar to Eq. (4.21) but with the right-hand term replaced with k2C /D. The Thiele modulus is based on kiC, which has the units of a first-order rate eonstant ... 

For this second-order reaction, the effectiveness factor changes substantially from the inlet of the catalyst bed to the outlet, because the Thiele modulus depends on the concentration of A at the external surface of the catalyst particle Ca,s-... 

Figures 3a and 3b show the variation in dimensionless concentration in the catalyst slab as a function of Thiele modulus for second order and half order reactions respectively. In the case of these reaction orders there is no analytical solution to the...

From Equation 7.13 it then follows that the error <5 i0 does not depend on the catalyst geometry or a Thiele modulus like <5 J it only depends on the value of and the reaction kinetics. This is illustrated in Figure 7.3 where <5W is plotted versus for zeroth-, first- and second-order kinetics and for exothermic reactions ( > 0). The curves were obtained from the formulae given in Table 7.2, which were calculated with the aid of Equations 7.13 and 7.26. 

The rate processes of diffusion and catalytic reaction in simple square stochastic pore networks have also been subject to analysis. The usual second-order diffusion and reaction equation within individual pore segments (as in Fig. 2) is combined with a balance for each node in the network, to yield a square matrix of individual node concentrations. Inversion of this 2A matrix gives (subject to the limitation of equimolar counterdiffusion) the concentration profiles throughout the entire network [14]. Figure 8 shows an illustrative result for a 20 X 20 network at an intermediate value of the Thiele modulus. The same approach has been applied to diffusion (without reaction) in a Wicke-Kallenbach configuration. As a result of large and small pores being randomly juxtaposed inside a network, there is a 2-D distribution of the frequency of pore fluxes with pore diameter. 

A plol of the effectiveness factor as a function of the Thiele modulus is shown in Figure 12-5. Figure 12-5a shows "q as a function of (j> for a spherical catalyst pellet for reactions of zero-, first-, and second-order. Figiue 12-5b corresponds to a first-order reaction occurring in three differently shaped pellets of voliune Vp and external siuface area A,. When volume change accompanies a reaction, the corrections shown in Figure 12-6 apply to the effectiveness factor for a first-order reaction. 

Recognizing the second tenii is just the ratio of a reaction rale to a diffusion rate for a zero order reaction, we call this ratio the Thiele modulus, We divide and multiply by two to facilitate the integration ... 

This is a system of stiff, second-order partial differential equations which can be solved numerically to yield both transient, and steady state concentration profiles within the layer. Comparison of the experimental calibration curves and of the time response curves with the calculated ones provides the verification of the proposed model from which it is possible to determine the optimum thickness of the enzyme layer. Because the Thiele modulus is the controlling parameter in the diffusion-reaction equation it is obvious from Eq.6 that the optimum thickness L will depend on the other constants and functions included in the Thiele modulus. Because of this the optimum thickness will vary from one kinetic scheme to another. 

Here the dimensionless variables are concentration c, radius r, and (f) = R kjDg is the Thiele modulus, a dimensionless munber used to describe the relation between the reaction and diffusion rates. In this ODE-BVP, the first boundary condition specifies the concentration at the catalyst surface, and the second boundary condition specifies no flux at the center of the particle. In order to use the shooting method, the BVP is rewritten using dCjdr = w, which gives a system of first-order ODEs,... 

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