An axisymmetric annular region in the r-z plane is 1≤ r≤2, 0≤ z≤1

An axisymmetric annular region in the \(r\)-\(z\) plane is \(1\le r\le2\), \(0\le z\le1\). Temperature satisfies \(-\nabla^2T=0\) with constant conductivity \(k=10\,\mathrm{W/(m\,K)}\). The boundary \(r=1\) is held at \(100^\circ\mathrm{C}\); the boundaries \(z=0\) and \(z=1\) are insulated. On \(r=2\), convection to \(T_\infty=20^\circ\mathrm{C}\) occurs with \(h=5\,\mathrm{W/(m^2\,K)}\). Use meridional nodes \(1=(1,0)\), \(2=(2,0)\), \(3=(2,1)\), \(4=(1,1)\) and linear triangular elements \((1,2,3)\) and \((1,3,4)\). Starting from the axisymmetric weak form with measure \(2\pi r\,dr\,dz\), integrate \(r\) exactly in each element and on the convecting edge. Assemble the conduction and Robin contributions, impose \(T_1=T_4=100^\circ\mathrm{C}\), solve for \(T_2\) and \(T_3\), and determine both the total convective heat loss at \(r=2\) and the total reaction heat input at \(r=1\).

For each triangle, use \(2\pi k(\int_{\Omega_e}r\,dA)\nabla N_i\cdot\nabla N_j\).