I’m looking to model the steady-state flow of coolant through a porous, heat-generating medium. I have a heat equation to solve for the solid, and a heat equation to solve for the fluid. The two temperature profiles are linked by convection between the solid and the fluid.
For the solid:
Q_{fuel-generation} = Q_{fuel-conduction} + Q_{convection}
Q_{fuel-generation} = q''' V
Q_{fuel-conduction} = kA\nabla u_{fuel}
Q_{convection} = hA_{bubbles}(u_{fuel}-u_{coolant})
For the fluid:
Q_{convection} = Q_{advection}
Q_{advection} = C_{p}\dot{m}\frac{du_{coolant}}{dx}
Where:
Q_{fuel-generation} is the rate at which heat is generated by the fuel (W).
q''' is the volumetric heat generation rate (W/m^3).
V is the volume of a differential shell (m^3).
Q_{convection} is the rate at which heat is transferred from the fuel to the coolant via convection (W).
h is the convective heat transfer coefficient (W/m^2).
A_{bubbles} is the area of coolant exposed to the fuel within a differential shell (m^2).
A_{shell} is the surface area of a differential shell of fuel (varying with x) (m^2).
Q_{coolant-adv} is the rate at which the coolant within a differential shell loses heat due to advection (W).
C_p is the specific heat of the coolant (J/kg K).
\dot{m} is flow rate of coolant (kg/s).
u_{fuel} is a function describing the temperature of the fuel as it varies with position x (K).
u_{coolant} is a function describing the temperature of the coolant as it varies with position x (K).
How do I translate this into a set of variational formulae? How do I use the temperature profile of the coolant (u_{coolant}) as a variable in the differential equation for the fuel, and vice versa?
Any help at all would be tremendously appreciated.
