Mass Peclet number

The model discretization or the number of collocation points necessary for accurate representation of the profiles within the reactor bed has a major effect on the dimensionality and thus the solution time of the resulting model. As previously discussed, radial collocation with one interior collocation point generally adequately accounts for radial thermal gradients without increasing the dimensionality of the system. However, multipoint radial collocation may be necessary to describe radial concentration profiles. The analysis of Section VI,E shows that, even with very high radial mass Peclet numbers, the radial concentration is nearly uniform and that the axial bulk concentration and radial and axial temperatures are nearly unaffected by assuming uniform radial concentration. Thus model dimensionality can be kept to a minimum by also performing the radial concentration collocation with one interior collocation point. 

Similarly for the mass Peclet number PeM, the mass balance design equation for this case is... 

Number of particles Ny, total number N , number per unit mass Peclet number for axial mixing, UL/E, dimensionless Number of increments or screens number of particles in spot sample also, exponent in equation for flow from bins... 

Here the numerical constants employed are consistent with in BTU/h —°R, T is in °R, h is the solid-gas heat transfer coelScient [c.f. equation (7-83)], Kg the thermal conductivity of the solid phase, Npe the mass Peclet number, G the mass velocity, Cp the heat capacity, dp the particle diameter, and e is the bed porosity. The second term in brackets represents the radiative heat transfer contribution to the apparent thermal conductivity this is normally negligible for operation at less than 300-350 °C. The quantity / is a surface emissivity factor that normally can be estimated as 0 < / <0.1 (see Argo and Smith). Keep in mind that equation (7-153) is for gas flow in packed beds with Npe < 40. Other situations are discussed by both Beck, and Argo and Smith. Note also that, with the possible exception of /, there are no adjustable parameters in equation (7-153). Since the radiative contribution is normally quite small except at very elevated temperatures, a typical working form of equation (7-153) is... 

If one were to attempt to determine any communality in the discussion of models given in this chapter, about the best would be to say that the parameters invoked are derivatives of the model, as would be inferred from the titles of the previous sections. For example, there is the overall heat-transfer coefficient, h, that appears in the nonisothermal, one-dimensional axial dispersion model, which is not to be confused with the wall heat transfer coefficient, a y, that belongs to the radial dispersion model. Similarly, would the bed thermal conductivity be the same in an axial dispersion model as in a radial dispersion model What is the difference between a mass Peclet number and a thermal Peclet number and so on. In fact, let us take a moment... 

Thermal axial dispersion must be treated with care. Even if axial dispersion of mass is negligible, the same may not be true for heat transport. The dispersion coefficient that appears in the thermal Peclet number is very different from the dispersion coefficient of the mass Peclet number. The combination of a plug-flow model for the mass balance and a dispersion... 

Stanton Number = a /puCp) mass Peclet number = dpu/D) axial heat dispersion Peclet number = (dpu/Xa) radial heat dispersion Peclet number = dpu/A,.)... 

Fig.5 illustrates the PO selectivity (S) as a function of mass Peclet number. The selectivity was increased with increasing the total reaction gas flow rate (F) passing through the membrane pores, such as S= 18-41% according to F= 70-130 cmVmin at 483K, indicating an effective role of the convection flow in the membrane pores for the enhancement of S compared to intraparticle diffusion in spherical catalyst solid supports used for conventional packed bed reactors. This result experimentally proves the validity of our previous work based on the mathematical analysis [7]. 

The dimensionless velocity profiles are given in Section 8.2.1 and the shape factor (S) takes the values of 0 or 1, for parallel plates or circular channels, respectively. The two dimensionless parameters in Equation 8.16 are the transverse mass Peclet number... 

The Peclet number for the process and the various aspect ratios of the fixed-bed reactor determine the impact of the dispersion terms. The mass Peclet number quantifies the ratio of bulk mass transport to diffusive mass transport. We define the mass Peclet number as... 

Peclet number, (Re)tf grmai = uL/E, should be identical to the mass Peclet number, Pe = uLjD. At lower Reynolds numbers, one would expect iiL/E to depend on a thermal Schmidt number, (Sc) ,a] = p, pap = pCpj k, which is more commonly called the Prandtl number. The inside heat transfer coefficient, h, can be estimated from standard correlations such as Equation (5.38). 

For a large mass Peclet number, the above equation can be rewritten as ... 

Pe)m Peclet number for molecular diffusion given in Eq. 14.40 (Pe)r radial, mass Peclet number given in Eq. 14.41... 

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