1. Selected Differential System Examples from Lectures

2. Liquid Storage Tank Standing assumptions Other possible assumptions
V = Ah wi wo
Standing assumptions
Constant liquid density r
Constant cross-sectional area A
Other possible assumptions
Steady-state operation
Outlet flow rate w0 known function of liquid level h

3. Mass Balance Mass balance on tank Steady-state operation:
Valve characteristics
Linear ODE model
Nonlinear ODE model

4. Stirred Tank Chemical Reactor
Overall mass balance
Component balance
Assumptions
Pure reactant A in feed stream
Perfect mixing
Constant liquid volume
Constant physical properties (r, k)
Isothermal operation

5. Plug-Flow Chemical Reactorz
qi, CAi
qo, CAo
CA(z) Dz
Assumptions
Pure reactant A in feed stream
Perfect plug flow
Steady-state operation
Isothermal operation
Constant physical properties (r, k)

6. Plug-Flow Chemical Reactor cont.z
qi, CAi
qo, CAo
CA(z) Dz
Component balance
Overall mass balance

7. Continuous Biochemical Reactor
Fresh Media Feed
(substrates)
Exit Gas Flow
Agitator
Exit Liquid Flow
(cells & products)

8. Cell Growth Modeling Specific growth rate Yield coefficients
Biomass/substrate: YX/S = -DX/DS
Product/substrate: YP/S = -DP/DS
Product/biomass: YP/X = DP/DX
Assumed to be constant
Substrate limited growth
S = concentration of rate limiting substrate
Ks = saturation constant
mm = maximum specific growth rate (achieved when S >> Ks)

9. Continuous Bioreactor Model
Assumptions
Sterile feed
Constant volume
Perfect mixing
Constant temperature and pH
Single rate limiting nutrient
Constant yields
Negligible cell death
Product formation rates
Empirically related to specific growth rate
Growth associated products: q = YP/Xm
Nongrowth associated products: q = b
Mixed growth associated products: q = YP/Xm+b

10. Mass Balance Equations
Cell mass
VR = reactor volume
F = volumetric flow rate
D = F/VR = dilution rate
Product
Substrate
S0 = feed concentration of rate limiting substrate

11. Exothermic CSTR
Scalar representation
Vector representation

12. Isothermal Batch Reactor
CSTR model: A  B  C
Eigenvalue analysis: k1 = 1, k2 = 2
Linear ODE solution:

13. Isothermal Batch Reactor cont.
Linear ODE solution:
Apply initial conditions:
Formulate matrix problem:
Solution:

14. Isothermal CSTR Nonlinear ODE model
Find steady-state point (q = 2, V = 2, Caf = 2, k = 0.5)

15. Isothermal CSTR cont. Linearize about steady-state point:
This linear ODE is an approximation to the original nonlinear ODE

16. Continuous Bioreactor
Cell mass balance
Product mass balance
Substrate mass balance

17. Steady-State Solutions
Simplified model equations
Steady-state equations
Two steady-state points

18. Model Linearization Biomass concentration equation
Substrate concentration equation
Linear model structure:

19. Non-Trivial Steady State
Parameter values
KS = 1.2 g/L, mm = 0.48 h-1, YX/S = 0.4 g/g
D = 0.15 h-1, S0 = 20 g/L
Steady-state concentrations
Linear model coefficients (units h-1)

20. Stability Analysis Matrix representation Eigenvalues (units h-1)
Conclusion
Non-trivial steady state is asymptotically stable
Result holds locally near the steady state

21. Washout Steady State Steady state:
Linear model coefficients (units h-1)
Eigenvalues (units h)
Conclusion
Washout steady state is unstable
Suggests that non-trivial steady state is globally stable

22. Gaussian Quadrature Example
Analytical solution
Variable transformation
Approximate solution
Approximation error = 4x10-3%

23. Plug-Flow Reactor Examplez
qi, CAi
qo, CAo
CA(z) Dz L

24. Plug-Flow Reactor Example cont.
Analytical solution
Numerical solution
Convergence formula
Convergence of numerical solution

25. Matlab Example Isothermal CSTR model
Model parameters: q = 2, V = 2, Caf = 2, k = 0.5
Initial condition: CA(0) = 2
Backward Euler formula
Algorithm parameters: h = 0.01, N = 200

26. Matlab Implementation: iso_cstr_euler.m
h = 0.01;
N = 200;
Cao = 2;
q = 2;
V = 2;
Caf = 2;
k = 0.5;
t(1) = 0;
Ca(1) = Cao;
for i=1:N t(i+1) = t(i)+h; f = q/V*(Caf-Ca(i))-2*k*Ca(i)^2; Ca(i+1)= Ca(i)+h*f; end plot(t,Ca) ylabel('Ca (g/L)') xlabel('Time (min)') axis([0,2,0.75,2.25])

27. Euler Solution
>> iso_cstr_euler

28. Solution with Matlab Function
function f = iso_cstr(x)
Cao = 2;
q = 2;
V = 2;
Caf = 2;
k = 0.5;
Ca = x(1);
f(1) = q/V*(Caf-Ca)-2*k*Ca^2;
>> xss = xss = >> df iso_cstr(x); >> [t,x] = ode23(df,[0,2],2); >> plot(t,x) >> ylabel('Ca (g/L)') >> xlabel('Time (min)') >> axis([0,2,0.75,2.25])

29. Matlab Function Solution

30. CSTR Example
Van de Vusse reaction
CSTR model
Forward Euler

31. Stiff System Example CSTR model: A  B  C Homogeneous system:
Eigenvalue analysis: q/V = 1, k1 = 1, k2 = 200

32. Explicit Solution Forward Euler First iterative equation
Second iterative equation

33. Implicit Solution Backward Euler First iterative equation
Second iterative equation

34. Matlab Solution function f = stiff_cstr(x) Cai = 2; qV = 1; k1 = 1;
Ca = x(1);
Cb = x(2);
f(1) = qV*(Cai-Ca)-k1*Ca;
f(2) = -qV*Cb+k1*Ca-k2*Cb;
f = f';
>> xo = 1]) xo = >> df stiff_cstr(x); >> [t,x] = ode23(df,[0,2],[2 0]); >> [ts,xs] = ode23s(df,[0,2],[2 0]); >> size(t) ans = >> size(ts) ans = 30 1

35. Matlab Solution cont.
>> subplot(2,1,1) >> plot(t,x(:,1)) >> hold Current plot held >> plot(ts,xs(:,1),'r') >> ylabel('Ca (g/L)') >> xlabel('Time (min)') >> legend('ode23','ode23s') >> subplot(2,1,2) >> plot(t,x(:,2)) >> plot(ts,xs(:,2),'r') >> ylabel('Cb (g/L)')