I've been reading a paper which models heat transfer in solids using a finite element model. Elements are connected to each other through a thermal resistance. The heat equation given for conduction in the model is:
[itex]\frac{dT_{m}(t)}{dt}[/itex]= [itex]\sum_{x+,x-,y+...} (\frac{T_{n}(t)-T_{m}(t)}{C_{m}(R_{m}+R_{n})})[/itex] (1)
Where m is the cell in question and n is the neighbor.
[itex]\Sigma[/itex] sums up the heat additions/subtractions in each direction
C is the thermal capacitance ([itex]C=C_{specific} \rho V)[/itex]
R is the thermal resistance of each element (R=[itex]\frac{x}{Ak})[/itex]
Is it possible to show this equation is equivalent to the conduction term of a 'standard' heat equation where the heat source term Q is zero?:
[itex]\frac{\partial T}{\partial t}=\frac{1}{\rho C}\nabla (k\nabla T)[/itex] (2)
EDIT: I've tried showing that the two equations are compatible in my attachment but I'm not sure if my working is correct (especially as I'm left with a 'dt' that should not be there). Any help in proving (or disproving) the equations are describing the same process would be appreciated.
[itex]\frac{dT_{m}(t)}{dt}[/itex]= [itex]\sum_{x+,x-,y+...} (\frac{T_{n}(t)-T_{m}(t)}{C_{m}(R_{m}+R_{n})})[/itex] (1)
Where m is the cell in question and n is the neighbor.
[itex]\Sigma[/itex] sums up the heat additions/subtractions in each direction
C is the thermal capacitance ([itex]C=C_{specific} \rho V)[/itex]
R is the thermal resistance of each element (R=[itex]\frac{x}{Ak})[/itex]
Is it possible to show this equation is equivalent to the conduction term of a 'standard' heat equation where the heat source term Q is zero?:
[itex]\frac{\partial T}{\partial t}=\frac{1}{\rho C}\nabla (k\nabla T)[/itex] (2)
EDIT: I've tried showing that the two equations are compatible in my attachment but I'm not sure if my working is correct (especially as I'm left with a 'dt' that should not be there). Any help in proving (or disproving) the equations are describing the same process would be appreciated.