# Modeling Nonideal Reactors

The use of RTDs for the prediction of reactor performance is a complex subject and the reader is encouraged to consult other sources for more complete coverage. "v" The techniques commonly used in environmental engineering are relatively straightforward, however, and thus we will look at them briefly.

4.4.1 Continuous Stirred Tank Reactors in Series Model

The simplest way to model a reactor with a nonideal flow pattern is as a series of CSTRs and this technique will be used extensively in this book. The basis for doing this may be seen by considering the response of such a system to a step input of tracer. Consider a chain of N CSTRs, each with volume V, receiving a flow F, giving each CSTR a mean hydraulic residence time t (Figure 4.5). At time zero the feed to the first tank is switched to one with a tracer concentration SI(1. The response

Figure 4.5 CSTRs in series.

from the first tank is the feed to the second tank, the response of the second is the feed for the third, etc. If we write and solve the mass balance equations for each (with no reaction term since the tracer is assumed to be inert), we obtain the following expression:

Tracer concentration profiles, i.e., F(t) curves, for various numbers of tanks in series are shown in Figure 4.6 while the corresponding £(t) curves are shown in Figure 4.7. The curve for N = 1 is the classical response of a single CSTR. More significantly, however, the curve for N = ^ is the classical response for a PFR, i.e., a step change in effluent concentration after one HRT. This suggests that the step response for N CSTRs in series will lie somewhere between that of a single CSTR and a PFR, with the pattern depending on the number of tanks in the chain. Furthermore, this implies that a real reactor which has the response of neither a CSTR nor a PFR can be simulated as N CSTRs in series. The easiest way of determining the appropriate value for N, which is sufficiently accurate in many cases, is to plot either the F(t) or the £(t) curve for the reactor in question and compare it to the curves in Figures 4.6 and 4.7, thereby selecting the value of N which corresponds most closely.

The tanks in series model has been used frequently in environmental engineering practice. For example, Murphy and Boyko" found that many conventional activated sludge systems had RTDs that were equivalent to three to five CSTRs in series.

Figure 4.6 Responses of N CSTRs in series to a step tracer input, t in the abscissa is the total HRT for N CSTRs in series.

Figure 4.7 Responses of N CSTRs in series to an impulse tracer input, t in the abscissa is the total HRT for N CSTRs in series.

Time, t

Figure 4.7 Responses of N CSTRs in series to an impulse tracer input, t in the abscissa is the total HRT for N CSTRs in series.

### 4.4.2 Axial Dispersion Model

An alternative approach is to superimpose some degree of backmixing upon a plug flow of fluid. The magnitude of the backmixing is assumed to be independent of the position in the reactor and is expressed by the axial dispersion coefficient, D|, which is analogous to the coefficient of molecular diffusion in Fick's law of diffusion. Modeling the superimposed backmixing by axial dispersion requires adding another transport term to the mass balance equation for the differential element of the PFR in Figure 4.2. In addition to the advective transport term used in Eq. 4.7, a term for transport by axial dispersion must be included, thereby increasing the number of terms in the resulting partial differential equation:

dSA d2SA F 3Sa

The equation is usually rewritten to include the term D,/vL (the dispersion number):

in which v is the longitudinal velocity through the basin (F/AJ, L is the basin length, z is dimensionless distance along the basin (x/L) and 9 is dimensionless lime (t/-r).

When the dispersion number is zero, there is no axial dispersion and therefore plug flow, whereas when it is infinitely large, complete backmixing exists and the reactor behaves as a CSTR.

The effect of the value of the dispersion number on the RTD may be seen by solving Eq. 4.23 with the appropriate initial and boundary conditions for a step input of an inert tracer.8 The solution, expressed in the form of an F(t) curve, is shown in Figure 4.8. To characterize the mixing pattern in a reactor by this technique, we need a way to select the appropriate value for the dispersion number. One way is to evaluate the derivative of the F(t) function at one mean residence time: