Selected Differential System Examples from Lectures

1. Selected Differential System Examples from Lectures

2. wi V = Ah wo Liquid Storage Tank • 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. MassBalance • 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. qi, CAi qo, CAo CA(z) Dz z Plug-Flow Chemical Reactor • Assumptions • Pure reactant A in feed stream • Perfect plug flow • Steady-state operation • Isothermal operation • Constant physical properties (r, k)

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

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

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. Assumptions Sterile feed Constant volume Perfect mixing Constant temperature and pH Single rate limiting nutrient Constant yields Negligible cell death Continuous Bioreactor Model • 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%