The main characteristic of a plug flow reactor (PFR) is that no mixing occurs in the direction of flow; however, complete mixing is assumed within a cross-sectional area of the reactor. Water and all suspended flocs of bacteria move with the same velocity along the tube reactor. In contrast to the CSTR, the PFR exhibits a continuous decrease in substrate concentration and an increase in bacterial concentration in the direction of flow (Fig. 6.5).
Because of the continuous change of concentrations, the balances for bacteria and substrate must be written as differential balances. They can be obtained by writing first an integral balance for the volume element AxA in a steady state:
with A as cross-sectional area, which is not correct, because of the change of S within Ax; but it can be corrected, by forming the differential:
lim Sl Sj
lim Sl Sj
dx or for the bacterial balance, it yields: dX
Assuming Monod kinetics for bacterial growth, then: dS UmaxX S
result with the conditions x = 0, S = SM and x = 0, X = XM.
If we move along the tube reactor with the flow rate w(1 +nR) and measure S and X at time t, the process could be described with Eqs. (6.3) and (6.4). Therefore, by using:
Equations (6.60) and (6.61) can be transformed into Eqs. (6.3) and (6.4). With the appropriate initial conditions, the solution in Eq. (6.18) is achieved. Batch reactor and plug flow reactor can be described by the same balances if we use the transformation in Eq. (6.62). After consideration of Eq. (6.15) and integration:
With Eq. (6.58) the critical sludge age tRXC can be calculated. The critical mean retention time tRX follows from Eq. (6.67):
Equation (6.67) was published by Fan et al. (1970), as the mathematical limit of a model of CSTRs in series for n with n as the total number of reactors in the system. This must correspond with the plug flow model. It follows from Eq. (6.67) that tRC and tRX increase with decreasing nR; for nR=0 (no recycle) tRC is infinitely large. This result is to be expected because there is neither recycle of bacteria into the inlet feed nor mixing of fluid.
Figure 6.6 compares the results of PFR substrate removal and efficiency with those of a CSTR (S0 = 400 mg L-1 BOD5, pmax=16.7 d-1, KS = 100 mg L-1 BOD5, kd=0.05 d-1, YoX/S = 0.6 g MLSS (g BOD5)-1). These results were published by Benefield and Randall (1980). Equation (6.49) was used for the CSTR and Eq. (6.63) for the PFR. The advantages of a PFR are clear: at tRX = 0.4 d, the effluent of the PFR Se is already very low. By comparison, the CSTR shows an effluent concentration of S = 70 mg L-1 BOD5. Only at tRX = 1.6 d does the CSTR shows an S as low as nearly 10 mg L-1 BOD5.
In Fig. 6.6 tRXC for CSTR and PFR are nearly the same, because of the relative high sludge recycle nR. For lower nR, tRC and tRXC increase in PFR systems.
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