Chapter 13 Summary Notes

7/29/2019 Chapter 13 Summary Notes

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13. DISTRIBUTIONOF RESIDENCE TIMESFORCHEMICAL

REACTORS*

Topics1. Residence Time Distribution

2. RTD for Ideal Reactors

3. RTD to Diagnose Faulty Operation

4. Models to Calculate Exit Concentrations and Conversions

A. Segregation Model

1. Segregation Model Applied to an Ideal PFR

2. Segregation Model Applied to an LFR

3. Segregation Model Applied to a CSTR

4. Mean Concentration for Multiple Reactions

B. Maximum Mixedness Model

5. Comparing XMM

and Xseg6. RTD and Multiple Reactions

1. Residence Time Distribution top

We shall use the RTD to characterize existing (i.e. real) reactors and then use it to predict exit

conversions and concentrations when reactions occur in these reactors.

Inject a tracer and measure exit concentration, CT(t).

From the exit tracer concentration we can determine the following information:

A. RTD (Residence Time Distribution) Function (E(t))
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= Fraction of molecules exiting the reactor that have spent a time between (t) and (t + dt) in

the reactor.

B. The Cumulative Distribution Function F(t)

= Fraction of molecules exiting the reactor that have spent a time t or less in the

reactor.

= Fraction of molecules that have spent a time t or greater in the reactor.

C. Definitions

1. Mean Residence Time
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13.1 Mean Residence Time

13.1 Residence Time Distribution Analysis using COMSOL Multiphysics

2. Variance

3. Space Time - For no dispersion/diffusion and v = v0, the space time equals the mean residence

time.

4. Internal Age Distribution, = Fraction of molecules inside the reactor that have been

inside the reactor between a time and .

5. Life Expectancy = Fraction of molecules inside the reactor with age that are expected to

leave the reactor in a time to .

From our experimental data of the exit tracer concentration from pulse trace test

t(min) :01 2 3 4 5 6

C(mg/m3):000.10.20.30.10

We can obtain

-> -> -> ->

13.2 Calculate E(t), t and s2

13.2 Using the E(t) curves
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2. RTD for Ideal Reactors top

for Ideal Reactors

PFR- Inject a pulse

at t=0

Dirac Delta Function

CSTR

Laminar

(LFR)

13.3 Drawing the F(theta) curves for the above ideal reactors

13.4 Matching Reactors with Tracer Step Inputs

13.5 Matching Reactor Models with E(t)

3. RTD to Diagnose Faulty Operation top

Experimentally injecting and measureing the tracer in a laminar flow reactor can be a difficult task, if

not a nightmare. For example, if one uses tracer chemicals that are photo-activated as they enter

the reactor, the analysis and interpretation of E(t) from the data becomes much more involved.

Diagnostics and Troubleshooting
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The CSTR

Concentration

RTD Function

Cumulative Function

Space Time

a. Perfect Operation


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b. Passing (BP)

c. Dead Volume

A summary for ideal CSTR mixing volume is shown in Figure 13-14

Tubular Reactor

A similar analysis to that for a CSTR can be carried out on a tubular reactor.

a. Perfect Operation of PFR (P)

b. PFR with Channeling (Bypassing, BP)


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c. PFR with Dead Volume (DV)

A summary for PRF is shown in Figure 13-18

In addition to its use in diagnosis, the RTD can be used to predict conversion in existing reactors

when a new reaction is tried in an old reactor. However, the RTD is not unique for a given system,

and we need to develop models for the RTD to predict conversion.

Medicinal Uses of RTD

4. Models to Calculate the Exit Concentrations and Conversions top

If using mathematical software to apply the models described below, you may need to fit C(t) and

E(t) to a polynomial. The procedure for fitting C(t) and E(t) to a polynomial is identical to the

techniques use to fitting concentration as a function of time described in Chapter 5.

Polymath regression analysis tutorial

Use combinations of ideal reactors to model real reactors that could also include: Zero parameter

models

Segregation ModelMaximum Mixedness Model

One parameter models

Tanks-in-Series Model

Dispersion Model

Two parameter models

Bypassing

Dead Space

Recycle
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4A. Segregation Model

Models the real reactor as a number of small batch reactors, each spending a different time in the

reactor. All molecules that spend the same length of time in the reactor (i.e., that are of the same

age) remain together in the same globule (i.e., batch reactor). Mixing of the different age groups

occurs at the last possible moment at the reactor exit.

Mixing of the globules of different ages occurs here.

Little batch reactors (globules) inside a CSTR.

X3>X2>X1

Mixing occurs at the latest possible moment.Each little batch reactor (globule) exiting the realreactor at different times will have a different conversion. (X1,X2,X3...)

But, the mean conversion for the segregation model is

4A.1 Segregation Model Applied To An Ideal PFR

Lets apply the segregation model to an ideal PFR and see if we get the same result for conversion as

we did in Chapter 4.
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Solve for X(t) for a first order reaction in a batch reactor.

For the batch reactor the c onversion-time relationship is

Calculate the mean conversion

which is the same conversion one finds from a mole balance (Chapter 4)

Further Explanation of Mean Conversion in Segregation Model

4A.2 Segregation Model Applied to an LFR

For a Laminar flow reactor the RTD function is

The mean conversion is

The last integral is the exponential integraland can be evaluated from tabulated values. Fortunately,

Hilder developed an approximate formula ( =Da).

Hilder, M.H. Trans. IchemE 59 p143(1979)

For large values of the Damkohler number then there is complete conversion along the streamlines off

the center streamline so that the conversion is determined along the pipe axis.
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4A.3 Segregation Model Applied to a CSTR

4A.4 Mean Concentration for Multiple Reactions

Solutions Using Software Packages


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For multiple reactions use an ODE solver to couple the mole balance equations, dCi/dt=ri, with the

segregation model equations: d /dt=Ci(t)*E(t), where C i is the concentration of i in the batch

reactor at time t and is the concentration of i after mixing the batch reactors at the exit.

13.6 Batch, PFR, CSTR, Segregation

4B Maximum Mixedness Model

Mixing occurs at the earliest possible moment.

Note E(l)=E(t)

E(l)dl =Fraction of molecules that have a life expectancy between l+dl and l.

Modeling maximum mixedness as a plug flow reactor with side entrances.
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Dividing byDland taking the limit asDlgoes to zero. Substitute ,

Differentiating the first term

and recalling we obtain.

We need to integrate backwards from (the entrance) to = 0 (the exit). In real systems we

have some maximum value of (say = 200 minutes) rather than minutes. Consequently we

integrate backward from = 200. However, because most ODE packages will not integrate backwards,

we have to use the transfer

z = T - to integrate forward

Thus

In terms of conversion,
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13.7 Maximum Mixedness Model

13.3 Calculate Xmm and Xseg

5. Comparing Segregation and Maximum Mixedness Predictions top

For example, if the rate law is a power law model

From the product [(n)(n-1)], we see

If n > 1, then > 0 and Xseg > Xmm

If n < 0, then > 0 and Xseg > Xmm

If 0 > n < 1, then < 0 and Xseg < Xmm

6. Multiple Reactions and RTD Data top

For multiple reactions use an ODE solver to couple the mole balance equations, dCi/dt=ri (where ri is

the net rate of reaction), with the segregation model equations: dCi/dt=Ci(t)*E(t) as previously

shown. For maximum mixedness:
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To obtain solutions with an ODE solver, first fit E(t) to a polynomial or several polynomials. Then let z

= T - where T is the largest time in which E(t) is recorded. Proceed to solve the resulting equations.

Object Assessment of Chapter 13

* All chapter references are for the 4th Edition of the text Elements of Chemical Reaction

Engineering .

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Fogler & Gurmen

2008 University of Michigan
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Chapter 13 Summary Notes