No of reactors in series
Reactor volume V 5 K(Q/k) Removal efficiency, %
85 90 95 98
1 5.67 9.00 19.00 49.00
2 3.16 4.32 6.94 12.14
4 2.43 3.11 4.46 6.64
6 2.23 2.81 3.89 5.52
8 2.14 2.67 3.63 5.05
10 2.09 2.59 3.49 4.79
Plug flow 1.90 2.30 3.00 3.91
a Volume of individual reactors equals value in table divided by the number of reactors in series.
Figure 1–12
Definition sketch for the total volume required versus the number of complete-mix reactors in series for various removal efficiencies. The K value on the vertical axis is multiplied by the flowrate and divided by the reaction coefficient to obtain the total volume required. The volume of an individual reactor is equal to the total volume divided by the number of reactors in series.
Constant K
0 5
0 10 15 20 25 30
2 4
98%
6 8 10
Number of complete-mix reactors in series, n 95%
90%
85%
Total reactor volume V = K (Q/k) Volume of individual reactor = V/n
EXAMPLE 1–4
Solution
Comparison of Required Reactor Volumes for Second-order Kinetics
Assuming that second-order kinetics apply (rc52kC 2), compare the required volume of a complete-mix reactor to the volume of a plug-flow reactor to achieve a 90 percent reduc- tion in the concentration (Co5 1 and Ce5 0.1).
1. Compute the required volume for a complete-mix reactor in terms of Q/k.
a. At steady state, a mass balance for a complete-mix reactor yields 05QCo2QCe2kC2eV
b. Simplify and substitute the given data.
V5Q kaCo2Ce
C2e
b 5Q k
120.1 (0.1)2 590Q
k
2. Compute the required volume for a plug-flow reactor in terms of Q/k.
a. At steady state, a mass balance for a plug-flow reactor yields
05 2Q dC
dxdx1Adx(2kC2) b. The integrated form of the steady-state equation is
V5 Q k#CCoe
dC C2 5Q
k
1
C`
ce
co
5 Q ka 1
Ce
2 1
Co
b c. Substituting the given concentration values yields
V5Q ka 1
0.121 1b 59Q
k
3. Determine the volume ratio.
VCMR
VPFR
5 (90Q/k)
(9Q/k) 510
1–11 Introduction to Treatment Process Modeling 49
Plug-Flow Reactor with Axial Dispersion and Reaction
In most full-scale plug-flow reactors, the flow usually is non-ideal because of entrance and exit flow disturbances, axial dispersion, and dispersion caused by advection (see Appendix I for an expanded discussion of dispersion and advection). Depending on the magnitude of these effects, the ideal effluent-tracer curves may look like the curves shown on Fig. 1–13.
Using first order removal kinetics, Wehner and Wilhelm (1958) have developed a solution for a plug-flow reactor with dispersion numbers varying from complete-mix (d 5 q) to ideal plug-flow (d 5 0). The equation developed by Wehner and Wilhelm is as follows:
C
Co5 4a exp(1/2d)
(11a)2exp(a/2d)2(12a)2exp(2a/2d) (1–85)
where C 5 effluent concentration, ML23
Co5 influent concentration, ML23
a5 "114ktd
d 5 dispersion factor 5 D/yL [see Eq. (I–9), Appendix I]
k 5 first order reaction constant, T21, (1/h) t5 hydraulic detention time, V/Q, T, (h)
To facilitate the use of Eq. (1–85) for the design of treatment processes such as stabiliza- tion ponds and natural systems, Thirumurthi (1969) developed Fig. 1–14, in which the
Figure 1–13
Theoretical and generalized nonideal response curves for a plug-flow reactor with axial
dispersion. C/Co
0 0.5 1.0
0
0.5 1.0 2.0
u (a)
C/Co
0 0.5 1.0
0
0.5 1.0 1.5 2.0
u (b)
Ideal plug flow Ideal plug flow
Nonideal response curves
Nonideal response curves
1.5
Figure 1–14
Values of kt in the Wehner and Wilhelm equation (Eq. 1–85) versus percent remaining for various dispersion factors and first order kinetics for a plug flow reactor. (Adapted from Thirumurthi, 1969.)
d = 0.0625
Value of kt
2 2 6 8
4
3 6 8 10 20
Percent remaining, C/Co
40 60
4
3
d = 4 d = 2 d = 1
d = 0.5 d = 0.25 d = 0.1
Plug flow d = 0
Completely mixed flow
d = x 7
5
EXAMPLE 1–5
Solution Comment EXAMPLE 1–5
Solution
Comment
term kt is plotted against C/Co for dispersion factors varying from zero for an ideal plug- flow reactor to infinity for a complete-mix reactor. The application of Fig. 1–14 is illus- trated in the following example.
Comparison of the Performance of a Treatment Process Occurring in a Plug-flow Reactor Without and with Axial Dispersion A treatment process reactor was designed assuming ideal plug-flow with a first order BOD removal rate constant of 0.5/d at 20°C and a detention time of 5 d. Once in operation, a considerable amount of axial dispersion was observed in the reactor. What effect will the observed axial dispersion have on the performance of the treatment process? The dispersion factor for the reactor, d, has been estimated to be about 0.5. Determine how much longer the detention time must be for a reactor with a dispersion factor of 0.5 to achieve the same degree of treatment as expected initially with the ideal plug-flow reactor.
1. Estimate the percentage removal for an ideal plug-flow reactor using Eq. (1–84).
a. The BOD remaining is:
C Co
5e2kt
C Co
5e20.53550.08258.2%
b. The percentage removal is
Percentage removal 100 2 8.2 5 91.8%
2. Determine the percentage removal for the reactor using Fig. 1–14.
a. The value of kt equals
kt5 (0.5/d 3 5 d) 5 2.5 b. The percent remaining from Fig 1–14 is equal to
C/Co5 0.20 5 20%
Percentage removal 100 2 20 5 80.0%
3. Determine the required detention time to achieve 91.8 percent removal a. The value of kt from Fig. 1–14 for a C/Co value of 8.2% is 4.6.
b. The required detention time is
kt5 4.6 t5 4.6/0.5 5 9.2 d
Clearly, axial dispersion can affect the predicted performance of a treatment process designed to function as an ideal plug-flow reactor. Because of axial dispersion and temperature effects, the actual performance of the treatment process will generally be less than expected.
Other Reactor Flow Regimes and Reactor Combinations
In the previous discussions of complete-mix and plug-flow reactors, a single-pass straight- through flow pattern has been used for the purpose of analyses. In practice, other flow regimes and reactor combinations are also used. Some of the more common alternative
1–11 Introduction to Treatment Process Modeling 51
flow regimes are shown schematically on Fig. 1–15. The flow regime shown on Fig. 1–15(a) is used to achieve intermediate levels of treatment by blending various amounts of treated and untreated wastewater. The flow regime used on Fig. 1–15(b) is often adopted to achieve greater process control and will be considered specifically in Chaps. 9 and 10 which deal with biological wastewater treatment. The flow regime shown on Fig. 1–15(c) is used to reduce the loading rate applied to the process. On Fig. 1–15(d), the return flow is not mixed with the influent, but is introduced at the entrance of the reac- tor to achieve greater initial dilution of the wastewater to be treated. Each of these hydrau- lic regimes is considered further in the following chapters.
Among the numerous types of reactor combinations that are possible and that have been used, two combinations using a plug-flow reactor and a complete-mix reactor are shown on Fig. 1–16. In the arrangement shown on Fig. 1–16(a), complete-mixing takes place second; in the arrangement shown on Fig. 1–16(b), it occurs first. If no reaction takes place and the reactors are used only to equalize temperature, for example, the result will be identical. If a reaction is occurring, however, the product yields of the two reactor systems can be different. The use of such hybrid reactor systems will depend on the spe- cific product requirements. Additional details on the analysis of such processes may be found in Denbigh and Turner (1965), Kramer and Westererp (1963), and Levenspiel (1972).
Figure 1–15
Flow regimes commonly used in the treatment of wastewater: (a) direct input with bypass flow (plug-flow or complete-mix reactor), (b) direct input with recycle flow (plug-flow or complete-mix reactor), (c) stepped input with recycle (recycle flow mixed with influent, recycle type 1), and (d) stepped input with recycle (recycle flow introduced at influent end of reactor, recycle type 2).
Reactor Effluent t
n e u l f n I
Bypass (a)
Recycle Reactor Influent
Effluent
(c)
Reactor Effluent t
n e u l f n I
Recycle (b)
Recycle Reactor Influent
Effluent
(d)
Recycle may occur before or after another treatment process Recycle may
occur before or after another treatment process
Figure 1–16
Hybrid reactor systems: (a) plug- flow reactor followed by complete-mix reactor and (b) complete-mix reactor followed by
plug-flow reactor. (a)
(b)