Modeling pv module using FDM (implicit scheme) 1

[pkladisios]

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.

 

[BigInch]

Simply the math and the model. By far the greatest heat loss will usually be convective, unless you have attached some thermal jacketing. I'd leave out the conduction and radiation losses for now. See how well that matches actual performance then decide about adding them back in, if and when you need to.

 

[corus]

If you have radiation in the boundary condition then you'd need to iterate until convergence was achieved. In such schemes it's normal to use the first step of Newton's method to reduce the quartic down and improve convergence. Look at this site

https://www.math.ohiou.edu/courses/math3600/lecture34.pdf

as an example