1. CHEM-E7130 Process Modeling Lecture 3

2. Lecutre 3 outline Physical models vs. mathematical similarity
Numerical methods and tools to solve initial value ordinary differential equations
Programming and modeling with Matlab
Stiff ordinary differential equations

3. One model from the previous lecture
What did this describe?
How molar flux along the reactor was modeled?
What happens at steady state?
What happens if Peclet number is very high?
High Pe, negligible
axial dispersion:
Steady state

4. A dynamic stirred tank reactor
Initially, a batch is set
Reactor is stirred vigorously so that it is uniform in composition
No feed or heat transfer during the batch, let’s assume isothermal operation
Write a material balance for this kind of reactor

5. A dynamic stirred tank reactor
Integral form
Differential form
Negligible volume change

6. Comparison Dynamic stirred tank Constant volume Plug-flow reactor
Steady state
Constant velocity
No axial dispersion

7. Comparison
Residence time in the plug flow reactor
Plug-flow reactor

8. Comparison Dynamic stirred tank Plug-flow reactor
Mathematically the problem is the same
 The same numerical solution!

9. Some solutions
Model
Initial condition
Closure:
first order reaction

10. Separable form, analytical solution possible

11. How would you solve this numerically with e.g. Excel?
Euler’s method
Explicit Euler if
is calculated at tn
Implicit Euler if
is calculated at tn+1

12. Numerical solution: Euler’s method
For the test problem:
Explicit Euler
Implicit Euler

13. Numerical solution: Euler’s method
Implicit Euler
For this linear test problem we have an explicit formulation
Usually implicit methods require solution of a nonlinear set of equations at each time step
For example,
pth order reaction

14. Example
c0=1
k=0.1
t=1

15. Example
c0=1
k=0.5
t=1

16. Example
c0=1
k=1
t=1

17. Example
c0=1
k=1.5
t=1

18. Example
c0=1
k=2
t=1

19. Example
c0=1
k=2.5
t=1

20. Shorter time steps
c0=1
k=2.5
t=0.2

21. Other options
Calculate
at average of tn and tn+1

22. Crank-Nicolson method

23. Crank-Nicolson method
Stable,
but
oscillates!

24. ODE solvers available in Matlab
Solves These Kinds of Problems
Method
ode45
Nonstiff differential equations
Runge-Kutta
ode23
ode113
Adams
ode15s
Stiff differential equations and DAEs
NDFs (BDFs)
ode23s
Stiff differential equations
Rosenbrock
ode23t
Moderately stiff differential equations and DAEs
Trapezoidal rule
ode23tb
TR-BDF2
ode15i
Fully implicit differential equations
BDFs

25. A catalyst particle model
We want to study dynamical behavior of a single catalyst particle. Some assumptions:
Spherical particle
One phase outside the particle (vapor or liquid)
Uniform and constant (time invariant) conditions (concentrations, temperature, pressure) around the particle
Uniform conditions inside the particle (but different from those outside)
Catalytic reaction inside the particle
Heat and mass transfer resistance outside the particle

26. A sketch
Particle is assumed to be spherical (this assumption needs to be stated since we need the surface area
Outside region (”box”) shape is irrelevant

27. Model equations Material balance Spherical porous particle
Particle properties time invariant

28. Model equations Enthalpy balance Catalyst particle enthalpy
Guess all these
If we want to reduce the number of guessed parameters, average heat capacity
Guess only these

29. Model equations Review: What assumptions so far? One fluid phase
Constant heat capacity
Apparent density and heat capacity
Be careful with the heat of reaction convention!
For multiple phases, it is advisable to use formation enthalpies

30. Model equations
Material balance
Enthalpy balance
Reaction rate

31. General programming scheme for these models
Main program, initialization
User model giving source terms or errors in the balances
Solver
Colors indicate how much a user needs to do/know about various blocks. Solver is typically coded already, but user model needs to be given.
Initialization means initial values or initial guesses for the variables, initialization of paramters etc.
Source terms typically for differential equations, errors in the balances for algebraic set of equations (e.g. CSTR)
In the previous example, parameters were given in the user model, but very often they are given in the initialization and then transferred to the user model
Post-processing results (graphs etc.)

32. Matlab code Explain what happens here to a person sitting next to you!
function catparticle(tint)
% initial variables
c0=100; % concentration (mol/m3)
T0=600; % temperature (K)
% tighten integration tolerances
options = odeset('RelTol',1e-10,'AbsTol',1e-10);
[t,st] = [0,tint], [c0,T0], options);
figure(1)
plot(t,st(:,1))
title('concentration (mol/m3)','FontSize',12)
xlabel('time (s)','FontSize',12)
figure(2)
plot(t,st(:,2))
title('temperature (K)','FontSize',12)

33. Matlab code A person sitting next to you will explain this to you!
function dx=dsdt(t,x)
c=x(1); % first variable is concentration (mol/m3)
T=x(2); % second is temperature (K)
eps=0.5; % void fraction
kr=2e9; % kinetic pre-exponential factor
E=100000; % activation energy (J/mol)
R=8.3144; % gas constant (J/Kmol)
DHr= ; % heat of reaction (J/mol)
cp=1000; % average heat capacity (J/kgK)
rho=2000; % average density (kg/m3)
k=0.02; % mass transfer coefficient (m/s)
h=600; % heat transfer coefficient (W/m2K)
d=0.001; % particle diameter (m)
cout=100; % outside fluid concentration same than initial
Tout=600; % outside temperature
rate=kr*exp(-E/(R*T))*c;
dcdt=1/eps*
(-rate+6*k/d*(cout-c));
dTdt=1/(cp*rho)*
(-rate*DHr+6*h/d*(Tout-T));
dx(1,1)=dcdt;
dx(2,1)=dTdt;

34. Some results
kr=1e9

35. Some results
kr=2e9

36. Some results
kr= e9

37. Some results
one hour
kr= e9

38. Some results
one day
kr= e9

39. Some results
one week
kr= e9

40. Some results a ten second period at
6 days, 13 h, 1 min and 10 s after start
kr= e9

41. Some results
kr=2.4e9

42. Stiff differential equations
Formally ratio of eigenvalues
In practice, if there are simultaneously very fast phenomena dictating step sizes, and very slow phenomena dictating simulation time, system is stiff. E.g. reaction:
slow reaction between molecular species, perhaps mass transfer limitations...
very rapid reaction approaching equilibrium, e.g. ion reactions

43. kf kr
in batch reactor
kf determines step size in numerical solution but kr the total simulation (reaction batch etc.) time.
This case leads to ”pseudo-steady state” for cA, but sometimes this kind of simplification is not so easy.

44. kf kr
in batch reactor
How many variables you would need to solve this in a) isothermal case, b) adiabatic case?
List some other assumptions that are needed to be done / checked
Is this linear model? Can it be solved analytically?
Note! Nothing is said if there are other than these three components!
How many phases, if mixing is ideal etc...
Yes it is, yes it can! d(c)/dt=[A](c)

45. Summary Dimensionless numbers can be obtained from
traditional dimensional analysis
ratios of two dimensionally similar objects
by non-dimensionalizing model equations
Models can be classified in mathematical or physical point of view. There are numerous ways to classify models.
Model classification helps to choose the best numerical methods for solution
Simulation, design, and optimization problems are different ways of looking at the same physical process