One Parameter Models  top 
A real reactor will be modeled as a number of equally sized tanksinseries. 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 tanksinseries E(t)
For a first order reaction
For reactions other than first order and for multiple reactions the sequential equations must be solved
The one parameter to be determined in the dispersion model is the dispersion coefficient, D_{a}. The dispersion model is used most often for nonideal 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 TaylorAris Dispersion, the molecules diffuse across radial
streamlines as well as axially to disperse the fluid.
2) To find D_{a} for pipes in turbulent flow see Figure 146
3) To find D_{a} for packed bed reactors see Figure 147
Using the RTD to find the Dispersion Coefficient.
Results of the tracer test can be used to determine Pe_{r} from E(t). We need to
consider two sets of boundary conditions.
1. ClosedClosed Vessels
2. OpenOpen Vessels
For a ClosedClosed 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, C_{t}(O^{})=C_{t0}>C_{t}(O^{+})
However, at the end of the reactor, C_{t}(L^{})=C_{t}(L^{+})
Returning to the unsteady tracer balance
(1) 
Let λ = z/L, and
Then in dimensionless form
For the closedclosed 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 α_{i} are found from the equation

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 t_{m} and Pe_{r} is found from the relationship:
Use RTD data to calculate t_{m}, ^{2}, and then P_{er} (i.e. D_{a}) and then use D_{a} in calculating conversion.
For the solution to with the above boundary conditions we find
For an OpenOpen Vessel
Dispersion occurs upstream, downstream and within the reactor.
Pe_{r}>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 t_{m}, ^{2}, and then P_{er} (i.e. D_{a}) and then use D_{a} in calculating conversion. It can be shown that at steady state, the openopen boundary conditions reduce to the Dankwerts Boundary Conditions.
For a First Order Reaction
Let Ψ = C_{A}/C_{A0}, λ = z/L , Pe = UL/D_{a} , and D_{a} = 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 D_{a} 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 Ψ_{1} is the dimensionless concentration of A upstream of the reaction section where D_{a} is greater than zero and Ψ is the dimensionless concentration of A in the reaction section.
Use RTD data to calculate t_{m}, ^{2}, and then Pe_{r} (i.e. D_{a})
and then use D_{a} in calculating conversion.
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 wellmixed (alpha), and the fraction of the stream that is bypassed (beta).
We now find the parameters a and b from a tracer experiment. We will choose a step tracer input
The balance equations are:
OTHER SCHEME INCLUDED
Real Reactor Model of Reactor
Two parameters, and :
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 C_{A1} as a function of , , k, , and C_{A0}. 
Overall Conversion
^{*} All chapter references are for the 4th Edition of the text Elements of Chemical Reaction Engineering .