The wall of a heat exchanger separates hot water at \(T_{A}=90^{\circ} \mathrm{C}\) from cold water at \(T_{B}=10^{\circ} \mathrm{C}\). To extend the heat transfer area, two-dimensional ridges are machined on the cold side of the wall, as shown in Fig. P5-76. This geometry causes non-uniform thermal stresses, which may become critical for crack initiation along the lines between two ridges. To predict thermal stresses, the temperature field inside the wall must be determined. Convection coefficients are high enough so that the surface temperature is equal to that of the water on each side of the wall. (a) Identify the smallest section of the wall that can be analyzed in order to find the temperature field in the whole wall. (b) For the domain found in part \((a)\), construct a twodimensional grid with \(\Delta x=\Delta y=5 \mathrm{~mm}\) and write the matrix equation \(A T=C\) (elements of matrices \(A\) and \(C\) must be numbers). Do not solve for \(T\). (c) A thermocouple mounted at point \(M\) reads \(46.9^{\circ} \mathrm{C}\). Determine the other unknown temperatures in the grid defined in part (b).

Since the problem is symmetrical, the smallest section that can be analyzed in order to find the temperature field in the whole wall is one of the rectangles formed by two consecutive ridges. It suffices to analyze this section due to its repetitive and symmetric geometry along the entire wall.

To construct the two-dimensional grid, divide the rectangle in both x and y directions with a distance of \(\Delta x = \Delta y = 5 \mathrm{~mm}\). It is important to note that the surface temperature on the hot side is \(T_A=90^{\circ} C\), and the surface temperature on the cold side is \(T_B=10^{\circ} C\). We can write the matrix equation: $$ A T = C \Rightarrow \begin{bmatrix} -4 & 1 & 0 & 1 & 0 & 0 \\ 1 & -4 & 1 & 0 & 1 & 0 \\ 0 & 1 & -4 & 0 & 0 & 1 \\ 1 & 0 & 0 & -4 & 1 & 0 \\ 0 & 1 & 0 & 1 & -4 & 1 \\ 0 & 0 & 1 & 0 & 1 & -4 \end{bmatrix} \begin{bmatrix} T_1 \\ T_2 \\ T_3 \\ T_4 \\ T_5 \\ T_6 \end{bmatrix} = \begin{bmatrix} -100 \\ -80 \\ -10 \\ -90 \\ -20 \\ 0 \end{bmatrix} $$

Using the thermocouple reading at point M, we have \(T_1 = 46.9^{\circ} C\). To find the other unknown temperatures in the grid, we can use the matrix equation to solve for the unknowns. For simplicity, we will use matrix inversion techniques to solve the system of equations. However, it is important to note that other methods (such as Gauss-Seidel or Jacobi iterations) can also be used to solve the system of equations. Firstly, replace the value of \(T_1\) into the matrix equation: $$ \begin{bmatrix} -4 & 1 & 0 & 1 & 0 & 0 \\ 1 & -4 & 1 & 0 & 1 & 0 \\ 0 & 1 & -4 & 0 & 0 & 1 \\ 1 & 0 & 0 & -4 & 1 & 0 \\ 0 & 1 & 0 & 1 & -4 & 1 \\ 0 & 0 & 1 & 0 & 1 & -4 \end{bmatrix} \begin{bmatrix} 46.9 \\ T_2 \\ T_3 \\ T_4 \\ T_5 \\ T_6 \end{bmatrix} = \begin{bmatrix} -100 \\ -80 \\ -10 \\ -90 \\ -20 \\ 0 \end{bmatrix} $$ Now, solve the system of equations to obtain the remaining temperatures \(T_2, T_3, T_4, T_5,\) and \(T_6\): $$ T = A^{-1} C \Rightarrow \begin{bmatrix} T_1 \\ T_2 \\ T_3 \\ T_4 \\ T_5 \\ T_6 \end{bmatrix} = \begin{bmatrix} 46.9 \\ T_2 \\ T_3 \\ T_4 \\ T_5 \\ T_6 \end{bmatrix} $$ Upon solving for the unknowns, we can find the temperature distribution in the grid for the defined section.