1. Environ. Eng. Course Note 10 (Reactor II)
Review of Ideal Reactor Models
- CMBR
- CM(C)FR
- PFR
Advanced Ideal Reactor Problems
Non-Ideal Reactor: Advection-Dispersion-Reaction Equation
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2. Ideal Reactors Q-In Q-out Q-In Q-out
Completely Mixed Batch Reactor (CMBR)
Completely Mixed Continuous Flow Reactor (CMCFR)
Q-In
Q-out
Plug-Flow Reactor (PFR)
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3. Unbalanced Flow in a CMFR
Consider a CMFR containing water. Initially, the reactor is only partially filled with water. For some period thereafter (0 < t < ∞), the inlet and outlet flows are steady but unequal (Figure 5.A.20). A contaminant species enters the reactor with the inlet flow and decays by a first-order process. Derive a material balance that describes the rate of change of the contaminant concentration in the reactor.
Control volume
Qin
Qout
V(t)
C(t)
Cin C
Figure 5.A.20 Schematic of a CMFR with unbalanced fluid flow

4. Dissolved Oxygen Consumption by BOD in a CMFR
Water containing biochemical oxygen demand (BOD) and dissolved oxygen (DO) flows into a CMFR. Within the reactor the BOD undergoes first-order decay, consuming DO in the process (cf. §3.D.5). What is the steady-state concentration of DO in the reactor?
Control volume
[BOD]in Q
[Do]in V
[BOD]
[DO]
[BOD],[DO] Q
Figure5.A.21 Schematic of biochemical oxygen demand and dissolved
oxygen in a CMFR.

5. Coupled Reactors
Consider the experimental apparatus depicted in Figure 5.A.22. It consists of a reactor that is partially filled with water. The air and water independently well mixed. The reactor has air supply and discharge lines and is operated with balanced flow. There is no water flow through the reactor. Benzene undergoes interfacial mass transfer across the interface area A between air and water according to the two-resistance model. Initially, the reactor contains pure water and benzene-free air. Humidified (RH = 100 percent) air flows through the reactor. At t = 0, the benzene content of the air flowing into the reactor is suddenly increased from 0 to a partial pressure Pin, which is maintained indefinitely. Describe the time-dependent concentration of benzene in air and water.
Control volume Qa Qa J Vw A
Figure5.A.22 Schematic of an experimental apparatus in which benzene
is transferred from air to water.

6. Unbalanced Flow in a CMFR
A common goal in environmental engineering research and practice is to measure the total amount of a contaminant that is released into an environment through some process or activity. This example demonstrates an experimental approach for evaluating emissions using a CMFR model. The specific data come from a study of the episodic release of air pollutants from dishwashing (potentially significant, but demonstrated to be minor) (Wooley et al., 1990).
A simulated dishwashing activity was conducted in a room-sized test chamber (V = 20 m3). The concentration of ethanol (a constituent of dishwashing detergent that is of concern as a potential contributor to photochemical smog) was measured during and after 20 minutes of dishwashing, with the results shown in Figure 5.A.23. The chamber was ventilated with ethanol-free air at a rate of 0.7 m3 min-1. Assuming that ethanol is nonreactive, show how the total mass of ethanol emitted by the dishwashing can be determined from these data.
1.4
1.2
1.0
Control volume
Area = ∫C(t)dt
0.8
C in = 0 V
C(t)
Concentration (mg m-3) Q
0.6
0.4
0.2
C(t)
E(t) Q
Time (min)
Figure5.A.23 Ethanol concentration in chamber air resulting from a dishwashing activity conducted during the period 0 to 20 minutes

7. Plug-Flow Reactor with Recycle
Dingbat Engineering has a contract to design a new wastewater treatment plant. They studied Example 5.A.10 and know that a PFR works better than a CMFR for treatment processes involving first-order decay.
The clever engineers at Dingbat reason that they can do even better by recycling a fraction of the outlet flow of the PFR back to the inlet. They think that this will allow the contaminant to react for a longer period and so yield even better conversion efficiency for a fixed reactor volume. Figure 5.A.24 shows a schematic of their design.
Consider the use of this reactor to remove BOD that decays according to
r = -kC
Where C represents the aqueous BOD concentration. In answering the following question, assume that the flow rates, Q and αQ, and the influent BOD concentration, Cin, are constant and that steady-state conditions prevail. A design goal is to minimize the reactor volume such that a fixed fractional of BOD is achieved (e.g., 90 percent destruction so Cout/Cin = 0.1).
(a) Derive an expression for BOD in the effluent (Cout ) in terms of the system parameters (Q, α, A, L, k, and Cin)for the limiting case of no recycle(i.e., α → 0).
(b)Derive an expression for Cout that is value of α.
(c)Do the Dingbat engineers have a good idea? In other words, for α > 0 , does their configuration perform better, worse, or the same relative to a conventional PFR?
Control
volume2
Area =A L
Cin
Cin *
Cout
Control
Volume 1 Q
Q+αQ Q x Δx αQ
Cout
Figure5.A.24 Schematic of a PFR with recycle.

8. Advection-Dispersion-Reaction (ADR) EquationCX
CX+ΔX
CIN
V (water velocity) is varying! A X ΔX

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1-D ADR Equation
- (δNx/δx) + r = δC/ δt
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10. 1-D Solute ADR Equation
In a porous medium, q = nV (here n = porosity)

11. Dispersion Number
Dispersion Number

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Example
Waste Water
Treatment
Dispersion Coeff, Dd = 10 km2/day
A bathing beach
L = 10 km
C at L
(beach)
Velocity of River, V = 10 km/day
Instantaneous Increase &
Continuous Input
of E. coli
CIN = 400 cells/100 mL
The 1st order die-off rate of E. coli in river, k=0.5 1/day (reaction rate = -kC)
Question: Will the bathing standard be violated at the beach?
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Stepped Input
Analytical solution is available 1- and 2-D
for homogeneous systems with uniform velocity
Initial and Boundary Conditions
C=Co
at x=0
t > 0
x =>∞
t>0
C=0 at t=0 0 ≤ x ≤ ∞
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Assumptions
The first order decay rate: r = - k * C (valid for decay of bacterial cells when food is limited.)
2. Steady State Assumption: If the river flow and bacteria count at the discharge point are reasonably constant in time (i.e., V=constant, CIN=constant at t>0).
Governing Equation and Solving the Problem
ADR Equation can be applied in this analysis.
2. Initial and Boundary Conditions?
3. Use the analytical solution.
C=400*0.619 = 248 cells/100mL > standard
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15. Effect of Dispersion Number on C/CIN at Effluent
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16. Break Through Curve at EffluentCo V
3000 m
Break Through Curve at Effluent

17. A+D: Advection, dispersion; A+10xD: Advection + 10X Dispersion
A+D+R: Advection+Dispersion+1st Order Decay