Why do we need constitutive equations in the modeling process?
To complete the system of differential equations and boundary conditions.
To describe the physical behavior and properties of materials.
To link macroscopic quantities to microscopic interactions.
Essential for predicting the transport and transformation of matter and energy in systems.
How can we describe fluxes in homogeneous phases?
Mass Diffusion:
Described by gradients of chemical potential, pressure, force, and temperature.
ji=jDi+jPi+jFi+jTi
Energy Transfer:
Fourier’s law: qF=−λ∇T
For turbulent flows: qF,turb=−λturb∇T
Momentum Transfer:
Described by Newton’s law for viscous stresses in Newtonian fluids.
More complex relationships for non-Newtonian fluids.
Which constitutive equations do you know for molecular diffusion? What are the respective advantages and disadvantages?
Maxwell-Stefan Equations:
Advantages: More accurate for multi-component mixtures, considers interactions between different species.
Disadvantages: Implicit and hard to solve, requires detailed knowledge of diffusion coefficients (Di,k) and activity coefficients (γi).
Fick’s Law:
Advantages: Simpler, easy to implement, useful for binary mixtures.
Disadvantages: Less accurate for multi-component mixtures, does not account for interaction between species explicitly.
What does Fick’s law say, and where do we use it?
Describes the flux of a species due to concentration gradients.
JiD=−Di,effc∇xi
Used in modeling mass transfer in binary mixtures and dilute solutions.
Assumes diffusion coefficient D is constant or depends weakly on concentration.
Where would we use Fourier’s law?
Fourier’s Law:
Describes heat conduction due to temperature gradients.
qF=−λ∇T
Used in modeling thermal conduction in solids, liquids, and gases.
Applies to both laminar and turbulent conditions with appropriate adjustments for turbulence.
In which cases do we need constitutive equations for source terms?
Volumetric sources: When considering internal heat generation, chemical reactions, or other volumetric effects.
Surface sources: When dealing with heterogeneous reactions occurring at interfaces.
Necessary for accurately modeling the generation or consumption of mass, momentum, and energy within a system.
Which interfaces can be considered in a tubular reactor? How do the constitutive equations change in every case?
Interfaces in a Tubular Reactor:
Fluid-Wall Interface: Heat transfer described by Fourier’s law.
Wall-Cooling Medium Interface: Combined conductive and convective heat transfer.
Fluid-Cooling Medium Interface: Lumped parameter models for well-mixed conditions.
Constitutive Equations:
Depend on thermal equilibrium, temperature gradients, and physical properties of materials at each interface.
What are closure equations? Give examples.
Equations that relate the number of unknowns to the number of available equations to ensure a solvable system.
Examples:
Mass Fraction Summation: ∑xi=1
Density Relation: ∑ρi=ρ
Phase Equilibrium: μi+=μi−, p+=p−, T+=T−
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