Ordinary Differential Equation Models

1
Ordinary Differential Equation Models

• Classification of ODE models
• Liquid storage tanks
• Chemical reactors
• Cellular metabolic networks

2
Mathematical Modeling

• Motivation
• Yields improved understanding of physicochemistry
• Provides rigorous framework for data analysis
• Allows prediction without additional
  experimentation
• Core competency of any chemical engineer
• UMass ChE curriculum
• ChE 120 mass energy balance modeling
• ChE 226, 325 thermodynamic modeling
• ChE 230 fluid transport modeling
• ChE 320 chemical reaction modeling
• ChE 333 heat mass transfer modeling
• ChE 338 separation unit modeling
• ChE 444 plant-wide modeling
• ChE 446 - control system modeling

3
Classification of ODE Models

• Order
• Linearity
• Explicit versus implicit

4
Classification of ODE Models cont.

• Dimension
• Current focus systems of first-order, explicit
  ODEs

5
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

6
Mass Balance Models

• Mass balance on tank
• Steady-state operation
• Valve characteristics
• Linear ODE model
• Nonlinear ODE model

7
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

8
Plug-Flow Chemical Reactor

• Assumptions
• Pure reactant A in feed stream
• Perfect plug-flow
• Steady-state operation
• Isothermal operation
• Constant physical properties (r, k)

9
Plug-Flow Chemical Reactor cont.

• Component balance

• Overall mass balance

10
Cellular Metabolic Networks
11
Yeast Glycolysis
12
Model Formulation

• Intracellular concentrations
• Intermediates S1, S2, S3, S4
• Reducing capacity (NADH) N2
• Energy capacity (ATP) A3
• Reaction scheme

13
Stoichiometric Model

• Assumptions
• Intracellular extracellular steady state
• Measure glucose influx J0 acetaldehyde/pyruvate
  efflux J
• Steady-state mass balances

14
Kinetic Model

• Intracellular concentrations
• Intermediates S1, S2, S3, S4
• Reducing capacity (NADH) N2
• Energy capacity (ATP) A3
• Mass action kinetics for v2-v6
• Mass action kinetics ATP inhibition for v1

15
Kinetic Model cont.

• Intracellular mass balances
• Extracellular mass balance
• j cell density
• k cell membrane permeability
• ODE model