Distribution of Residence Times for Chemical Reactors

Topics

1. 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 X MM and Xseg

6. RTD and Multiple Reactions

1. 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)) 

= 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

 

13.113.1 Mean Residence Time

13.113.1 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) : 0 1 2 3 4 5 6 C(mg/m3) : 0 0 0.

1 0.2 0.

3 0.1 0

We can obtain                                                                           

 

->  ->   ->  ->

13.213.2 Calculate E(t), t and s2

13.213.2 Using the E(t) curves

 

2. 2. RTD for Ideal Reactors top

for Ideal Reactors

PFR- Inject a pulse at t=0

  Dirac Delta Function

     

 

CSTR

Laminar (LFR)

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

13.413.4 Matching Reactors with Tracer Step Inputs

13.513.5 Matching Reactor Models with E(t)

  3. 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

The CSTR

  Concentration  

       

  RTD Function  

       

  Cumulative Function  

       

  Space Time  

a. Perfect Operation

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)

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. 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 Model Maximum Mixedness Model                        

One parameter models

Tanks-in-Series Model Dispersion Model

Two parameter models

Bypassing Dead Space Recycle

 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 real reactor 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.

 

Solve for X(t) for a first order reaction in a batch reactor.

For the batch reactor the conversion-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 integral and 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.

4A.3 Segregation Model Applied to a CSTR

4A.4 Mean Concentration for Multiple Reactions

Solutions Using Software Packages

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.

13.613.6 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.

           

                                                 

 

                           

 

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,

                        

                                                   

 

13.713.7 Maximum Mixedness Model

13.313.3 Calculate Xmm and Xseg   5. 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. 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:

 

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.

Ideal Reactor Design EquationsResidence Time

theoretical mean residence time

reactor volume

reactor flow rate

Step input for completely mixed or continuously stirred tank reactor CSTR

concentration

initial concentration

time

theoretical mean detention time

Pulse Input for completely mixed or continuously stirred tank reactor CSTR

concentration

initial concentration

time

theoretical mean detention time

concentration at n tank in series

Plug Flor Reactor PFR

step input:when t < tR, C = 0when t >= tR, C = C0

pulse input:when t = tR, C = C0

when t < tR, C = 0when t > tR, C = 0

Where tR  = theoretical mean residence timeV  = reactor volume Q  = reactor flow rate C  = concentration at time C0 = initial concentrationCn = concentration at tank nt  = timen  = number of tanks

Models for Non ideal Reactors

Topics

1. One Parameter Models

2. Two Parameter Models

1. 1. One Parameter Models top

A.  Tanks-In-Series    

A real reactor will be modeled as a number of equally sized tanks-in-series. Each tank behaves as an ideal CSTR.  The number of tanks necessary, n (our one parameter), is determined from the E(t) curve.

For n tanks in series, E(t) is

It can be shown that

In dimensionless form

Carrying out the integration for the n tanks-in-series E(t)

For a first order reaction

For reactions other than first order and for multiple reactions the sequential equations must be solved

 

B.  Dispersion Model

The one parameter to be determined in the dispersion model is the dispersion coefficient, Da. The dispersion model is used most often for non-ideal tubular reactors.  The dispersion coefficient can be found by a pulse tracer experiment.

 

 

After a very, very narrow pulse of tracer is injected, molecular diffusion (and eddy diffusion in turbulent flow) cause to pulse to widen as the tracer molecules diffuse randomly in all directions.  The convective transport equation is:

Finding the Dispersion Coefficient

1)     For laminar flow Taylor-Aris Dispersion, the molecules diffuse across radial 

        streamlines as well as axially to disperse the fluid.

2)     To find Da for pipes in turbulent flow see Figure 14-6

3)     To find Da for packed bed reactors see Figure 14-7

    

Using the RTD to find the Dispersion Coefficient.

       Results of the tracer test can be used to determine Per from E(t).  We need to

       consider two sets of boundary conditions.

1. Closed-Closed Vessels

       2. Open-Open Vessels

            

 

For a Closed-Closed Vessel

We note there is a discontinuity at the entering boundary in the tracer concentration.

 

Due to the discontinuity at boundary due to forward diffusion,  Ct(O-)=Ct0>Ct(O+)

However, at the end of the reactor,  Ct(L-)=Ct(L+)

 

Returning to the unsteady tracer balance

(1)

Let λ = z/L, and

Then in dimensionless form

For the closed-closed boundary condition the solution to the tracer balance at the exit (λ = 1) , at any time Θ, i.e., Ψ(1,Θ) gives

where

(2)

From page 530 of Froment and Bischoff Chemical Reactor Analysis, 2 nd Edition, John Wiley & Sons, 1966.      Eigen Values αi are found from the equation

For Pe = 10           α1=1.5           α2=2.4           α3=4.8           α4=5.1           α5=7.9           α6=8.1

F

0 ≤ λ ≤ 1

Integrating Equation (3) using Equation (2) gives

(3)

We see from the equation for E(theta) that the exponential term dies out as the values of alpha i become large.

then tm and Per is found from the relationship:                                                            

 

Use RTD data to calculate tm, 2, and then Per (i.e. Da) and then use Da in calculating conversion.

For the solution to with the above boundary conditions we find

    

For an Open-Open Vessel

Dispersion occurs upstream, downstream and within the reactor.

Per>100, for long tubes, the solution at the exit is 

 

Because of the dispersion the mean residence time is greater than the space time.  The molecules can flow out of the reactor and then diffuse back in.

Use RTD data to calculate tm, 2, and then Per (i.e. Da) and then use Da in calculating conversion. It can be shown that at steady state, the open-open boundary conditions reduce to the Dankwerts Boundary Conditions.

 

14.114.1 Sketch of F curves

 

Dispersion with Reaction

For a First Order Reaction

        

Let Ψ = CA/CA0, λ = z/L , Pe = UL/Da , and Da = kτ. The dimensionless balance on the concentration of A in the reaction zone is

Danckwerts Boundary conditions

At λ = 0 then

At λ = 1, then

the solution is [See John B. Butt, Reaction Kinetics and Reactor Design, 2 nd Edition, page 378, Marcel Dekker, 2000.]

where

The Polymath program used to plot Ψ versus λ is given below

Polymath Program

Sketches of the dimensionless concentration profiles for different values of Peclet and Damköhler numbers are shown below

Note how Ψ(0+) changes as Pe and Da change.

     The exit conversion is

The following figures given the Polymath solutions for y versus l for different values of Pe and Da

Open Open System

Upstream of the reaction zone the balance on A in dimensionless form is

For these boundary conditions

the solution is

Rearranging

A typical profile is

 

The following profiles were obtained from the Polymath Program given above. Here Ψ1 is the dimensionless concentration of A upstream of the reaction section where Da is greater than zero and Ψ is the dimensionless concentration of A in the reaction section.

Use RTD data to calculate tm, 2, and then Per (i.e. Da) and then use Da in calculating conversion.    

2. 2. Two Parameter Models top

The goal is to model the real reactor with combinations of ideal reactors.

CSTR with Bypass and Dead Volume

Two parameters and , the fraction of volume that is well-mixed (alpha), and the fraction of the stream that is bypassed (beta).

   

Reactor Balance

We now find the parameters a and b from a tracer experiment. We will choose a step tracer input

The balance equations are:

 

14.114.1 Finding a two parameter model

OTHER SCHEME INCLUDED

          Real Reactor                      Model  of Reactor

Two parameters, and :

1. Use the tracer data to find and . 2. Then use the mole balances and the rate law to solve for CA1 and CA2.

Reactor 1:

(Eqn. A)

Rate Law:

for a first order reaction or

for a second order reaction

Reactor 2:

(Eqn. B)

Rate Law:

Solve Equations (A) and (B) to obtain CA1 as a function of , , k, , and CA0.

Overall Conversion