Conversion According to the Dispersion Model

In Sec. 6-5 a non-steady-state mass balance for a tubular-flow reactor (plug flow except for axial dispersion) was used to evaluate an effective diffusivity. Now we consider the problem of calculating the conversion when a reaction occurs in a dispersion-model reactor operated at steady-state conditions. Again a mass balance is written, this time for steady state and including reaction and axial-dispersion terms. It is considered now that the axial diffusivity is known. 

Note the similarities and differences between this steady-state equation and the transient one, Eq. (6-23). The boundary conditions for the two expressions are the same, that is, Eqs. (6-25) and (6-26). For first-order kinetics the solution is straightforward. Introducing r = k C and using a dimensionless concentration C — C/Cq and reactor length z — z/L, we obtain for the differential equation and boundary conditions 

If DJuL, and of course kd, are known, Eq. (6-45) gives the conversion predicted by the dispersion model, provided the reaction is first order. For most other kinetics numerical solution of the differential equation is necessary.  

As mentioned earlier, the dispersion model, like other models, is subject to two errors inadequate representation of the RTD and improper allowance for the extent of micromixing. We can evaluate the first error for a specific case by using the dispersion model to obtain an alternate solution for Example 6-5. The second error does not exist for first-order kinetics. However, the maximum value of this error for second- and halforder kinetics was indicated in Sec. 6-8. 

Example 6-6 In Example 6-2, DJuL = 0.117 was found to provide the best fit of the dispersion model to the reported RTD. Use this result to evaluate the conversion for a first-order reaction, with = 0.1 sec and B = 10 sec. 

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