Consider 1-D, transient, uniform heat generation problem. a) \( L=0.1 \mathrm{~m} \) b) \( k_{\text {wall }}=30 \frac{\mathrm{W}}{\mathrm{m}^{\circ} \mathrm{C}}, c_{p}=400 \frac{\mathrm{j}}{\mathrm{kg}^{\circ} \mathrm{C}} \), \( ho=2000 \mathrm{~kg} / \mathrm{m}^{2} \) c) \( T_{i}=70^{\circ} \mathrm{C}, T_{s 1}=20^{\circ} \mathrm{C}, T_{s 2}=70^{\circ} \mathrm{C} \) d) \( \dot{e}_{\text {gen }}=10^{6} \mathrm{~W} / \mathrm{m}^{3} \) e) Derive formulations via energy balance

method. You may choose explicit or implicit approach. f) Solve the equations with a suitable method and obtain the time-dependent temperature distribution. Use a minimum of 10 grids. Move forward in time by 60 seconds. g) Plot the temperature distributions at 30 and 60 seconds. h) Compare the results with the analytical solution: \[ \begin{array}{l} A=2 \frac{1-(-1)^{n}}{n \pi}\left(T_{i}-T_{s 1} ight)+\frac{2(-1)^{n}}{n \pi}\left[\left(T_{s 2}-T_{s 1} ight)+\left(\dot{e}_{g e n} L^{2} ight) /(2 k) ight]- \\ \frac{\left[\left(\pi^{2} n^{2}-2 ight)(-1)^{n}+2 ight]}{(n \pi)^{3}} \frac{\dot{e}_{g e n} L^{2}}{k} \\ \frac{T(x, t)-T_{s 1}}{T_{s 2}-T_{s 1}}=\frac{x}{L}+\frac{\dot{e}_{g e n} L^{2}}{2 k\left(T_{s 2}-T_{s 1} ight)}\left[1-(x / L)^{2} ight](x / L)+ \\ \sum_{n=1}^{\infty} \frac{A}{T_{s 2}-T_{s 1}}\left[\exp \left(\frac{-\alpha n^{2} \pi^{2} t}{L^{2}} ight) ight] \sin (n \pi x / L) \end{array} \]

Obtain the temperatures at the following points using the analytical solution \[ \begin{array}{lllll} T_{(x=0.01, t=30)} & T_{(x=0.025, t=30)} & T_{(x=0.05, t=30)} & T_{(x=0.075, t=30)} & T_{(x=0.09, t=30)} \\

T_{(x=0.01, t=60)} & T_{(x=0.025, t=60)} & T_{(x=0.05, t=60)} & T_{(x=0.075, t=60)} & T_{(x=0.09, t=60)} \end{array} \]

For each time \( (30,60) \), connect the \( 5(+2 \) boundary temperatures) points with a spline and add them to the previous figure.

Course Code / Title: AEE331 / HEAT TRANSFER Instructor: Asst. Prof. M. Mollamahmutoglu Pr. 4 Group person limit: 6, Group limit: only one group can choose this topic Consider 1-D, transient, uniform heat generation problem. a) L=0.1m b) kwall=30mCW,cp=400kgCj, =2000kg/m2 c) Ti=70C,Ts1=20C,Ts2=70C d) egen=106W/m3 e) Derive formulations via energy balance method. You may choose explicit or implicit approach. f) Solve the equations with a suitable method and obtain the time-dependent temperature distribution. Use a minimum of 10 grids. Move forward in time by 60 seconds. g) Plot the temperature distributions at 30 and 60 seconds. h) Compare the results with the analytical solution: A=2n1(1)n(TiTs1)+n2(1)n[(Ts2Ts1)+(egenL2)/(2k)](n)3[(2n22)(1)n+2]kegenL2TS2TS1T(x,t)TS1=Lx+2k(Ts2Ts1)egenL2[1(x/L)2](x/L)+n=1Ts2Ts1A[exp(L2n22t)]sin(nx/L) Obtain the temperatures at the following points using the analytical solution T(x=0.01,t=30)T(x=0.01,t=60)T(x=0.025,t=30)T(x=0.025,t=60)T(x=0.05,t=30)T(x=0.05,t=60)T(x=0.075,t=30)T(x=0.075,t=60)T(x=0.09,t=30)T(x=0.09,t=60) For each time (30,60), connect the 5(+2 boundary temperatures) points with a spline and add them to the previous figure