how can I solve the problem Bellow with a three coupled PDEs , knowing that thoses equations represent the energy heat balance of a thermal energy storage system?

I work in problem of thermal storage energy where the fluide is the air, and I want to realise a modelisation to have the profil of temperature of the fluid, the solid and the wall . Theses tree equations describe the energy heat balance where the fluid, the solid and the wall are identified by "f", "s", "w" respectively as follows:

eq 1 : ρf*cpf ∂Tf/∂t + ρf*cpf*v*∂Tf/∂x = λf*∂²Tf/∂x² + h ( Ts_in− Tf).

eq 2: ρs*cps ∂Ts/∂t = λs*∂²Ts/∂x² + ∂²Ts/∂r² + (1/r)*∂Ts/∂r + h*( Tf − Ts_in) + hw*Aw,f *Vf ( Tw − Tf ).

eq 3 : ρw*cpw ∂Tw/∂t = hw Aw,f Vw ( Tf − Tw ).

with : rho_f = 1.27 # Densité du fluide (kg/m³)

cp_f = 1006.0 # Capacité thermique du fluide (J/kg.K)

lambda_f = 0.025 # Conductivité thermique du fluide (W/m.K)

v_f = 1.24 # Vitesse du fluide (m/s)

h = 1000.0 # Coefficient d'échange thermique entre le solide et le fluide (W/m².K)

rho_s = 3.91 # Densité du solide (kg/m³)

cp_s = 800.0 # Capacité thermique du solide (J/kg.K)

lambda_s = 1.61 # Conductivité thermique du solide (W/m.K)

h_w = 500.0 # Coefficient d'échange thermique entre le solide et le mur (W/m².K)

rho_w = 900.0 # Densité du mur (kg/m³)

cp_w = 1000.0 # Capacité thermique du mur (J/kg.K)

h_ext = 10.0 # Coefficient d'échange thermique entre le mur et l'extérieur (W/m².K)

A_w = 0.01 # Surface du mur (m²)

V_f = 0.001 # Volume du fluide dans le lit (m³)

V_w = 0.0001 # Volume du mur (m³)

Tf_in = 800.0 # Température d'entrée du fluide (°C)

Ts_in = 800.0 # Température initiale du solide (°C)

Tw_in = 60.0 # Température initiale du mur (°C)

the results shoud appear like the photo.

Answer 1

[Torsten]

Choose a grid in x and r direction. Replace the spatial derivatives by finite difference approximations in the grid points. Incorporate the boundary conditions. Set the initial conditions and solve the resulting system of ordinary differential equations in the grid points using ODE15S. Look up "method-of-lines" for more details.

There is no MATLAB program you could use to do this except the time integrator - all will depend on your skills to numerically discretize parabolic partial differential equations and incorporate the boundary conditions.