Chapter 31, Problem 11P

Find the temperature distribution in a rod (Fig. P31.11) with internal heat generation using the finite-element method. Derive the element nodal equations using Fourier heat conduction

$q_k=-kA\frac{dT}{\partial x}$

and heat conservation relationships

${\displaystyle \sum \left[q_k+f\left(x\right)\right]=0}$

where $q_k=$ heat flow (W), $k=$ thermal conductivity $\left(\text{W/}\left(\text{m}\cdot °\text{C}\right)\right)$, $A=$ cross-sectional area $\left({\text{m}}^2\right)$, and $f\left(x\right)=$ heat source $\left(\text{W/cm}\right)$. The rod has a value of $kA=100\text{ Wm/}°\text{C}$. The rod is 50 cm long, the x-coordinate is zero at the left end, and positive to the right. Divide the rod into five elements (six nodes, each 10 cm long). The left end of the rod has a fixed temperature gradient and the temperature is a variable. The right end has a fixed temperature and the gradient is a variable. The heat source $f\left(x\right)$ has a constant value. Thus, the conditions are

${\frac{dT}{\partial x}|}_{x=0}=0.25°\text{C/m}{T|}_{x=50}=100°\text{C}f\left(x\right)=30\text{ W/cm}$

Develop the nodal equations that must be solved for the temperatures and temperature gradients at each of the six nodes. Assemble the equations, insert the boundary conditions, and solve the resulting set for the unknowns.