Theoretical Process Modeling

1. Theoretical Process Modeling Background Liquid holding tank Continuous stirred tank heater Continuous chemical reactor Continuous biochemical reactor Metabolic reaction network

2. Background Basic modeling approaches Theoretical � based on conservation principles Empirical � based on fitting of experimental data Goals of theoretical modeling Quantify and improve process understanding Operator training Control system design and analysis Process optimization Important considerations Ultimate use of model determines appropriate level of detail Detailed modeling typically require considerable time and effort Model always an approximation to the real process

3. Liquid Holding Tank Assumptions Constant liquid density r Constant cross-sectional area A Other possible assumptions Steady-state operation Outlet flow rate qo known function of liquid level h

4. Derivation of Model Equations Dynamic mass balance on tank Steady-state operation: Valve characteristics Linear ODE model Nonlinear ODE model

5. Continuous Stirred Tank Heater Assumptions Constant volume Perfect mixing Negligible heat losses Constant physical properties (r, Cp)

6. Derivation of Model Equations Mass balance Energy balance Initial value problem Given physical properties (r, Cp), operating conditions (V, w, Ti, Q) and initial condition T(0) Integrate model equation to find T(t)

7. Continuous Chemical Reactor Assumptions Constant volume Pure A in feed Perfect mixing Negligible heat losses Constant properties (r, Cp, DH, U) Constant cooling jacket temperature Constitutive relations Reaction rate/volume: r = kcA = k0exp(-E/RT)cA Heat transfer rate: Q = UA(Tc-T)

8. Derivation of Model Equations Mass balance Component balance Energy balance

9. Model Structure Properties 2 ordinary differential equations (ODEs) Time is the independent variable ? dynamic model Nonlinear and coupled Initial value problem requires numerical solution Degrees of freedom 6 unknowns 2 equations Must specify 4 variables Nomenclature cA(t) and T(t) are state variables q(t), cAi(t), Ti(t) and Tc(t) are possible input variables System order = number of state variables

10. Continuous Biochemical Reactor Assumptions Sterile feed Constant volume Perfect mixing Constant temperature & pH Single rate limiting nutrient Constant yields Negligible cell death

11. Derivation of Model Equations Cell mass VR = reactor volume F = volumetric flow rate D = F/VR = dilution rate Non-trivial steady state: Washout: Product

12. Model Derivation cont. Substrate S0 = feed concentration of rate limiting substrate Steady-state: Model structure State variables: x = [X S P]T Third-order system Input variables: u = [D S0]T Vector form:

13. Yeast Energy Metabolism

14. Model Formulation Intracellular concentrations Intermediates: S1, S2, S3, S4 Reducing capacity (NADH): N2 Energy capacity (ATP): A3 Mass action kinetics for r2-r6 Mass action kinetics and ATP inhibition for r1

15. Dynamic Model Equations Mass balances Conserved metabolites Matrix notation