7/29/2019 Chapter 13 Summary Notes
1/14
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))
http://www.umich.edu/~elements/course/lectures/thirteen/audio/audio31_2.mp3http://openaudioblockwin%28%27audio/audio31_2.mp3',%202)http://openaudioblockwin%28%27audio/audio31_2.mp3',%201)http://www.umich.edu/~elements/course/lectures/thirteen/audio/audio31_1.mp3http://openaudioblockwin%28%27audio/audio31_1.mp3',%202)http://openaudioblockwin%28%27audio/audio31_1.mp3',%201)http://-/?-http://%20openindexwin%28%27mcecc%27%29/http://-/?-http://%20openindexwin%28%27res_td%27%29/http://-/?-7/29/2019 Chapter 13 Summary Notes
2/14
= 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
http://www.umich.edu/~elements/course/lectures/thirteen/audio/audio31_3.mp3http://openaudioblockwin%28%27audio/audio31_3.mp3',%202)http://openaudioblockwin%28%27audio/audio31_3.mp3',%201)7/29/2019 Chapter 13 Summary Notes
3/14
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
http://www.umich.edu/~elements/course/lectures/thirteen/selftest1.htmhttp://www.umich.edu/~elements/course/lectures/thirteen/example2.htmhttp://www.umich.edu/~elements/course/lectures/thirteen/audio/audio31_4.mp3http://openaudioblockwin%28%27audio/audio31_4.mp3',%202)http://openaudioblockwin%28%27audio/audio31_4.mp3',%201)http://www.umich.edu/~elements/course/lectures/thirteen/Femlab_example.pdfhttp://www.umich.edu/~elements/course/lectures/thirteen/test1.htm7/29/2019 Chapter 13 Summary Notes
4/14
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
http://-/?-http://www.umich.edu/~elements/course/lectures/thirteen/self3.htmhttp://www.umich.edu/~elements/course/lectures/thirteen/self2.htmhttp://www.umich.edu/~elements/course/lectures/thirteen/test2.htmhttp://-/?-7/29/2019 Chapter 13 Summary Notes
5/14
The CSTR
Concentration
RTD Function
Cumulative Function
Space Time
a. Perfect Operation
7/29/2019 Chapter 13 Summary Notes
6/14
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)
7/29/2019 Chapter 13 Summary Notes
7/14
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
http://www.umich.edu/~elements/course/lectures/thirteen/audio/audio31_5.mp3http://openaudioblockwin%28%27audio/audio31_5.mp3',%202)http://openaudioblockwin%28%27audio/audio31_5.mp3',%201)http://www.umich.edu/~elements/05chap/html/polymath_tutorial/index.htmhttp://-/?-http://www.umich.edu/~elements/course/lectures/thirteen/sidenote13.1.htm7/29/2019 Chapter 13 Summary Notes
8/14
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.
http://www.umich.edu/~elements/course/lectures/thirteen/audio/audio31_7.mp3http://openaudioblockwin%28%27audio/audio31_7.mp3',%202)http://openaudioblockwin%28%27audio/audio31_7.mp3',%201)http://www.umich.edu/~elements/course/lectures/thirteen/audio/audio31_6.mp3http://openaudioblockwin%28%27audio/audio31_6.mp3',%202)http://openaudioblockwin%28%27audio/audio31_6.mp3',%201)7/29/2019 Chapter 13 Summary Notes
9/14
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.
http://www.umich.edu/~elements/course/lectures/thirteen/segregation.pdf7/29/2019 Chapter 13 Summary Notes
10/14
4A.3 Segregation Model Applied to a CSTR
4A.4 Mean Concentration for Multiple Reactions
Solutions Using Software Packages
7/29/2019 Chapter 13 Summary Notes
11/14
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.
http://www.umich.edu/~elements/course/lectures/thirteen/audio/audio31-8.mp3http://openaudioblockwin%28%27audio/audio31-8.mp3',%202)http://openaudioblockwin%28%27audio/audio31-8.mp3',%201)http://www.umich.edu/~elements/course/lectures/thirteen/test4.htm7/29/2019 Chapter 13 Summary Notes
12/14
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,
http://www.umich.edu/~elements/course/lectures/thirteen/audio/audio31-9.mp3http://openaudioblockwin%28%27audio/audio31-9.mp3',%202)http://openaudioblockwin%28%27audio/audio31-9.mp3',%201)7/29/2019 Chapter 13 Summary Notes
13/14
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:
http://-/?-http://-/?-http://www.umich.edu/~elements/course/lectures/thirteen/exampl1.htmhttp://www.umich.edu/~elements/course/lectures/thirteen/test3.htm7/29/2019 Chapter 13 Summary Notes
14/14
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 .
top
Fogler & Gurmen
2008 University of Michigan
http://-/?-http://www.umich.edu/~elements/13chap/html/obj.htmTop Related