pkladisios
Mechanical
Greetings. I am currently trying to model a photovoltaic module, mounted on a roof, using an implicit scheme of FDM (finite difference method). My primary concern is the boundary conditions. To be more precise, assuming a one dimensional heat flow, the heat diffusion equation is ∂T/∂t=(k/ρCp)∂[sup]2[/sup]T/∂x[sup]2[/sup].
Internal nodes
Applying the implicit scheme to the heat diffusion equation leads us to the following equation, valid for internal nodes:
(-kdt/ρCpdx[sup]2[/sup])T[sub]i-1[/sub][sup]p+1[/sup]+(1+2kdt/ρCpdx[sup]2[/sup])T[sub]i[/sub][sup]p+1[/sup]+(-kdt/ρCpdx[sup]2[/sup])T[sub]i+1[/sub][sup]p+1[/sup]=T[sub]i[/sub][sup]p[/sup]
Boundary conditions
Upper surface (i=0)
Q[sub]sol[/sub]-Q[sub]conv[/sub]-Q[sub]rad[/sub]=Q[sub]cond[/sub]→-k(T[sub]1[/sub][sup]p+1[/sup]-T[sub]0[/sub][sup]p+1[/sup])/dx=Q[sub]sol[/sub][sup]p+1[/sup]-Q[sub]conv[/sub][sup]p+1[/sup]-Q[sub]rad[/sub][sup]p+1[/sup]→-k(T[sub]1[/sub][sup]p+1[/sup]-T[sub]0[/sub][sup]p+1[/sup])/dx=Q[sub]sol[/sub][sup]p+1[/sup]-h[sub]c[/sub](T[sub]0[/sub][sup]p+1[/sup]-T[sub]air[/sub][sup]p+1[/sup])-εσ([sup]Τ[sub]0[/sub][sup]p+1[/sup]4[/sup]-[sup]T[sub]air[/sub][sup]p+1[/sup]4[/sup])
Lower surface (i=n-1)
Q[sub]cond[/sub]=Q[sub]conv[/sub]+Q[sub]rad[/sub]→-k(T[sub]n-1[/sub][sup]p+1[/sup]-T[sub]n-2[/sub][sup]p+1[/sup])/dx=Q[sub]conv[/sub][sup]p+1[/sup]-Q[sub]rad[/sub][sup]p+1[/sup]→-k(T[sub]n-1[/sub][sup]p+1[/sup]-T[sub]n-2[/sub][sup]p+1[/sup])/dx=h[sub]c[/sub](T[sub]n-1[/sub][sup]p+1[/sup]-T[sub]air[/sub][sup]p+1[/sup])-εσ([sup]Τ[sub]n-1[/sub][sup]p+1[/sup]4[/sup]-[sup]T[sub]air[/sub][sup]p+1[/sup]4[/sup])
where:
Q[sub]sol[/sub]: insolation
Q[sub]conv[/sub]: convective heat losses
Q[sub]rad[/sub]: radiative heat losses
k: thermal conductivity
Cp: specific heat capacity
ρ: density
ε: emissivity
σ: Stefan-Boltzmann constant
T: temperature
Subscripts:
Spatial i=0,1,...,n-1
Temporal p=0,1,...,m-1
My questions are:
Am i right so far? If i am, what happens when unknown temperatures to the power of four are inserted into the system of equations? Do i assume an overall/combined coefficient h=h[sup]conv[/sup]+h[sup]rad[/sup] to linearize the radiation factor (Q[sub]rad[/sub]+Q[sub]conv[/sub])=h(T[sub]surface[/sub]-T[sub]air[/sub])? I really need help on this one.
Thank you in advance and i apologise for any unclarities.
Internal nodes
Applying the implicit scheme to the heat diffusion equation leads us to the following equation, valid for internal nodes:
(-kdt/ρCpdx[sup]2[/sup])T[sub]i-1[/sub][sup]p+1[/sup]+(1+2kdt/ρCpdx[sup]2[/sup])T[sub]i[/sub][sup]p+1[/sup]+(-kdt/ρCpdx[sup]2[/sup])T[sub]i+1[/sub][sup]p+1[/sup]=T[sub]i[/sub][sup]p[/sup]
Boundary conditions
Upper surface (i=0)
Q[sub]sol[/sub]-Q[sub]conv[/sub]-Q[sub]rad[/sub]=Q[sub]cond[/sub]→-k(T[sub]1[/sub][sup]p+1[/sup]-T[sub]0[/sub][sup]p+1[/sup])/dx=Q[sub]sol[/sub][sup]p+1[/sup]-Q[sub]conv[/sub][sup]p+1[/sup]-Q[sub]rad[/sub][sup]p+1[/sup]→-k(T[sub]1[/sub][sup]p+1[/sup]-T[sub]0[/sub][sup]p+1[/sup])/dx=Q[sub]sol[/sub][sup]p+1[/sup]-h[sub]c[/sub](T[sub]0[/sub][sup]p+1[/sup]-T[sub]air[/sub][sup]p+1[/sup])-εσ([sup]Τ[sub]0[/sub][sup]p+1[/sup]4[/sup]-[sup]T[sub]air[/sub][sup]p+1[/sup]4[/sup])
Lower surface (i=n-1)
Q[sub]cond[/sub]=Q[sub]conv[/sub]+Q[sub]rad[/sub]→-k(T[sub]n-1[/sub][sup]p+1[/sup]-T[sub]n-2[/sub][sup]p+1[/sup])/dx=Q[sub]conv[/sub][sup]p+1[/sup]-Q[sub]rad[/sub][sup]p+1[/sup]→-k(T[sub]n-1[/sub][sup]p+1[/sup]-T[sub]n-2[/sub][sup]p+1[/sup])/dx=h[sub]c[/sub](T[sub]n-1[/sub][sup]p+1[/sup]-T[sub]air[/sub][sup]p+1[/sup])-εσ([sup]Τ[sub]n-1[/sub][sup]p+1[/sup]4[/sup]-[sup]T[sub]air[/sub][sup]p+1[/sup]4[/sup])
where:
Q[sub]sol[/sub]: insolation
Q[sub]conv[/sub]: convective heat losses
Q[sub]rad[/sub]: radiative heat losses
k: thermal conductivity
Cp: specific heat capacity
ρ: density
ε: emissivity
σ: Stefan-Boltzmann constant
T: temperature
Subscripts:
Spatial i=0,1,...,n-1
Temporal p=0,1,...,m-1
My questions are:
Am i right so far? If i am, what happens when unknown temperatures to the power of four are inserted into the system of equations? Do i assume an overall/combined coefficient h=h[sup]conv[/sup]+h[sup]rad[/sup] to linearize the radiation factor (Q[sub]rad[/sub]+Q[sub]conv[/sub])=h(T[sub]surface[/sub]-T[sub]air[/sub])? I really need help on this one.
Thank you in advance and i apologise for any unclarities.