[Princess38]

Hello,

My question is, how can i concert the time step from the finite difference explicit method to the fourier number / dimensionless number? My time step is delta_t = t_Max / (t_Nodes - 1)
The equation for Fourier number is Fo = alpha*t/L. Is the t every time step in my simulation? Where is the definition that my timestep has the unit seconds?

I need the dimensionless time to validate my results.

My Boundary is:

rho * c * dT/dt = k * (d2T/dr2 + 2/r * dT/dr)

After approximation (explicit method):

rho * c * (T(n+1,m) - T(n,m)/delta_t = k * {[T(n,m-1) - 2T(n,m) + T(n,m+1)]/delta_r^2 + 2/[(m-1)*delta_r] * [T(n,m+1) - T(n,m-1)]/(2*delta_r)}


The equation for the temperature at next step is:

T(n+1,m) = (k*delta_t)/(rho*c*delta_r^2)*{T(n,m-1) - 2T(n,m) + T(n,m+1) + (T(n,m+1) - T(n,m-1))/(m-1)} + T(n,m)

One boundary condition on the surface is:

rho*c*dT/dt = k*dT/dr + h*(T∞ - T(n,m))

-> T(n+1,m) = (delta_t)/(rho*c) * {k*(T(n,m+1) - T(n,m)/delta_r + h*(T∞ - T(n,m))} + T(n,m)

[FMDenaro]

First, Fourier number is Fo = alpha*t/L^2.
I suggesto to rewrite your equations in the non-dimensional form to get it explicitly.

[mprinkey]

http://www.ewp.rpi.edu/hartford/~ern...Notes/ch03.pdf

[Princess38]

FMDenaro thanks for your answer!

Here are the non dimensional equations:

With mesh Fourier number Fo=k*delta_t/(rho*c*delta_r^2) the equation for heat transfer is:

rho * c * dT/dt = k * (d2T/dr2 + 2/r * dT/dr)

T(n+1,m) = Fo*[T(n,m-1)-2T(n,m)+T(n,m+1) + (T(n,m+1)-T(n,m-1))/(m-1)]

With Bi = h*r_delta/k the equation for boundary condition:

rho*c*dT/dt = k*dT/dr + h*(T∞ - T(n,m))

T(n+1,m) = 3*Fo*(T(n,m+1)-T(n,m-1)) + 3*Fo*Bi*(T∞-T(n,m)) + T(n,m)

Am i on the right way with this equations?

[FMDenaro]

The type of discretization of the heat equation can be done in several ways, implicit/explicit time integration, high order discretization and so on.
I suggesto to follow some classical method for computational heat transfer, your problem is somehow a homework.