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Comprehensive Differential Systems in Chemical Reactor Models and Their Applications

This document explores selected differential system examples from lectures on chemical reactors, including liquid storage tanks, stirred tanks, and plug-flow reactors. It covers essential concepts such as mass balance, steady-state operation, and model characteristics (linear and nonlinear). The text includes critical assumptions, overall mass balance, and equations for continuous biochemical reactors, along with growth rate modeling of substrates and products. The analysis highlights bioprocess dynamics, including stability and washout states, plus Gaussian quadrature in analytical solutions.

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Comprehensive Differential Systems in Chemical Reactor Models and Their Applications

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  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%

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