Thank you in advance for any help with this assignment. I promise, I have spent hours trying various sources to understand these terms and operators, but I just cannot make the fundamental leap necessary to know what's going on.
The problem statement is quite vague, in my opinion. It is for an engineering systems & modeling class, so it is abstract and has been very frustrating. I have had lots of trouble finding a consistent resource to read for better understanding.
The goal is to model heat flux through a two-dimensional plane experiencing steady-state conduction. The plane is a rectangle, with one edge held at constant temperature, $T_\text{ref}$. The heat flux abides Fourier's Law: $q = -k\nabla T$ A source term applies to the domain: $f = aT + bx + cy^2$ The requirement for this part is to derive the governing equation, and make 'stencils' in the most general form.
I understand that $\nabla T$ represents the gradient of $T$, which is $[\frac{\partial T}{\partial x}, \frac{\partial T}{\partial y}]$ I know that $\kappa$, is the thermal conductivity of the material, which is a constant physical property.
The system is in steady-state operation, so $q'=0$. Looking through my old notes from class on 2-D finite differences, I see $-\nabla \cdot q + f =0$. And, later on, I see $\nabla \cdot q = \kappa(\frac{\partial^2 T}{\partial x^2}) + \kappa(\frac{\partial^2 T}{\partial y^2}) + f = 0$. This makes some sense to me, as the $\nabla \cdot q$ implies the gradient of $q$, which would be the partial differentials of $q$, which would be second-order differentials overall.
I also have in my notes and reaffirmed in a heat transfer textbook the finite differences using nearby nodes.
$\frac{\partial T}{\partial x} \bigg|_{m+(1/2),n} = \frac{T_{m+1,n} - T_{m,n}}{\partial x}$
$\frac{\partial T}{\partial x}\bigg|_{m-(1/2),n} = \frac{T_{m,n} - T_{m-1,n}}{\partial x}$
$\frac{\partial^2 T}{\partial^2 x}\bigg|_{m,n} = \frac{T_{m+1,n} + T_{m-1,n} - 2T_{m,n}}{\partial x^2}$
I cannot seem to put all the parts together and end up going in circles. Things I am confused on:
- Should I apply partial derivatives to the source term, $f$? If so, it would be :$f(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y})= b*i* + 2cy*j*$
- what it means when the source term is a function of $T$ This seems circular and unsolvable in my opinion.
The heat source is coming from the right side held at constant $T$, correct? Does the 'source term' also imply that each node in the plane is generating heat? Or is this simply a weird trick, in that the $aT$ term disappears when taking the partials, as done above.
Again, I'm not making sense of it and making myself more confused the more I try to research. Thank you for your help.
