The onedimensional wall in the figure is modeled using one element with three nodes. There is a uniform heat source inside the wall generating \(Q=200 \mathrm{~W} / \mathrm{m}^{3}\). The thermal conductivity of the wall is \(k=2 \mathrm{~W} /\left(\mathrm{m} \cdot{ }^{\circ} \mathrm{C}ight)\). Assume that \(A=1 \mathrm{~m}^{2}\) and \(l=1 \mathrm{~m}\). The left end has a fixed temperature of \(T=20^{\circ} \mathrm{C}\), while the right end has zero heat flux.

a. Calculate the shape function \([\mathbf{N}]=\left[N_{1}, N_{2}, N_{3}ight]\) as a function of \(x\).

b. Solve for the nodal temperature \(\{\mathbf{T}\}=\left\{T_{1}, T_{2}, T_{3}ight\}^{\mathrm{T}}\) using boundary conditions. Plot the temperature distribution in an \(x T\) graph.

c. Calculate the heat fluxes at \(x=0\) and \(x=1\) when the solution is \(\{\mathbf{T}\}=\{20,58.25,71\}\).

Transcribed Image Text:

No heat flow 1 20C T T 2 13 7 -8 1 16 -8 1 -8 7 x [K]T= [K]T=-8