23年 4月 21日 CEE3330-01 Joonhong Park Copy Right

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

23年 4月 21日 CEE3330-01 Joonhong Park Copy Right

Ideal Reactors

Completely Mixed Batch Reactor (CMBR)

Completely Mixed Continuous Flow Reactor

(CMCFR)

Q-In

Q-out

Plug-Flow Reactor (PFR)

Q-In Q-out

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.

V(t)

C(t)

Control volume

Qin

Cin

Qout

C

Figure 5.A.20 Schematic of a CMFR with unbalanced fluid flow

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?

V

[BOD]

[DO]

Control volume

Q

Q

[BOD],[DO]

[BOD]in

[Do]in

Figure5.A.21 Schematic of biochemical oxygen demand and dissolved oxygen in a CMFR.

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

Figure5.A.22 Schematic of an experimental apparatus in which benzene is transferred from air to water.

Qa QaJ

Vw A

Unbalanced Flow in a CMFRA 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.

V

C(t)

E(t)

C in = 0

Q

Q

C(t)

Control volume

1.4

1.2

1.0 0.8

0.6

0.4

0.2

0

0 50 100

Conce

ntr

ati

on (

mg

m-3)

Time (min)

Area = ∫C(t)dt

Figure5.A.23 Ethanol concentration in chamber air resulting from a dishwashing activity conducted during the period 0 to 20 minutes

Plug-Flow Reactor with RecycleDingbat 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?

Controlvolume2

Cin

Q

Cin*

Area =A L

Q+αQ

x Δx αQ

Cout

Q

CoutControlVolume 1

Figure5.A.24 Schematic of a PFR with recycle.

Advection-Dispersion-Reaction (ADR) Equation

CIN

V (water velocity) is varying!

XΔX

CX CX+ΔX

A

t

CrN i

23年 4月 21日 CEE3330-01 Joonhong Park Copy Right

1-D ADR Equation

t

CkC

dx

CdD

x

CV

xAt

CxrA

x

CAD

x

CADAVCAVC

dd

xxddxddxxx

2

2

)][]([

• - (δNx/δx) + r = δC/ δt

x

CDN iddxi

,VCN xi ,

IDDD lddd kCr

t

Cnr

x

CnD

xx

Cq h

'

1-D Solute ADR Equation

t

Cr

x

CD

xx

CV dd

In a porous medium, q = nV (here n = porosity)

Dispersion Number

Lv

D

Advection

Dispersion

x

d

Dispersion Number

CEE3330-01 May 29, 2006 Joonhong Park Copy Right

Example

A bathing beach

Instantaneous Increase &

Continuous Input

of E. coli

CIN = 400 cells/100 mL

Velocity of River, V = 10 km/day

Waste Water

Treatment

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?

L = 10 km

C at L

(beach)

Dispersion Coeff, Dd = 10 km2/day

CEE3330-01 May 29, 2006 Joonhong Park Copy Right

Stepped Input

t

CkC

x

CV

X

CD

x h

Analytical solution is available 1- and 2-D

for homogeneous systems with uniform velocity

C=0 at t=0 0 ≤ x ≤ ∞

C=Coat x=0t > 0

0

X

C

x =>∞

t>0

Initial and Boundary Conditions

CEE3330-01 May 29, 2006 Joonhong Park Copy Right

0

rX

CD

xx

CV d

1. ADR Equation can be applied in this analysis.2. Initial and Boundary Conditions?3. Use the analytical solution.

Assumptions

1. 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

Lv

D

v

kL

DLv

ExpDLv

Exp

DLv

Exp

C

C

x

d

xD

d

xDD

d

xDD

d

xD

IN

41

)5.0()1()5.0()1(

)5.0(4

22

C=400*0.619 = 248 cells/100mL > standard

CEE3330-01 May 29, 2006 Joonhong Park Copy Right

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.001 0.01 0.1 1 10 100 1000 10000 100000

Dispersion Number

C/C

IN

k=0.5 1/d

k=1.0 1/d

Effect of Dispersion Number on C/CIN at Effluent

3000 mCo

V

Break Through Curve at Effluent

0

0 . 2

0 . 4

0 . 6

0 . 8

1

1 . 2

0 5 0 0 1 0 0 0 1 5 0 0

D i s t a n c e (m e t e r )

C/C

o

C / C o @ 0 . 0 2 h

C / C o @ 0 . 2 h

C/Co Profile at X=3000 m

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5

Time (hours)

C/C

o

t=0

C/ Co Profile at 100 cm

0

0.2

0.4

0.6

0.8

1

1.2

0 100 200 300 400

Time (min)

C/Co

A+D A+D+R A+10xD

A+D: Advection, dispersion; A+10xD: Advection + 10X Dispersion

A+D+R: Advection+Dispersion+1st Order Decay