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Consider one-dimensional conduction in a plane composite
Chapter , Problem 3.84(choose chapter or problem)
Consider one-dimensional conduction in a plane composite wall. The outer surfaces are exposed to a fluid at \(25^{\circ} \mathrm{C}\) and a convection heat transfer coefficient of 1000 \(W/m^2.K\). The middle wall B experiences uniform heat generation \(\dot{q}_{\mathrm{B}}\), while there is no generation in walls A and C. The temperatures at the interfaces are \(T_1\) = 261°C and \(T_2\) = 211°C.
(a) Assuming negligible contact resistance at the interfaces, determine the volumetric heat generation \(\dot{q}_{\mathrm{B}}\) and the thermal conductivity \(k_B\).
(b) Plot the temperature distribution, showing its important features.
(c) Consider conditions corresponding to a loss of coolant at the exposed surface of material A (h = 0). Determine \(T_1\) and \(T_2\) and plot the temperature distribution throughout the system.
Questions & Answers
QUESTION:
Consider one-dimensional conduction in a plane composite wall. The outer surfaces are exposed to a fluid at \(25^{\circ} \mathrm{C}\) and a convection heat transfer coefficient of 1000 \(W/m^2.K\). The middle wall B experiences uniform heat generation \(\dot{q}_{\mathrm{B}}\), while there is no generation in walls A and C. The temperatures at the interfaces are \(T_1\) = 261°C and \(T_2\) = 211°C.
(a) Assuming negligible contact resistance at the interfaces, determine the volumetric heat generation \(\dot{q}_{\mathrm{B}}\) and the thermal conductivity \(k_B\).
(b) Plot the temperature distribution, showing its important features.
(c) Consider conditions corresponding to a loss of coolant at the exposed surface of material A (h = 0). Determine \(T_1\) and \(T_2\) and plot the temperature distribution throughout the system.
ANSWER:Step 1 of 4
The thermal contact resistance for wall A is calculated as,
\({R_A} = \dfrac{1}{h} + \dfrac{{{L_A}}}{{{k_A}}}\)
For \(h = 1000\;{\rm{W/}}{{\rm{m}}^{\rm{2}}} \cdot {\rm{K,}}\,{L_A} = 30\;{\rm{mm}} = 0.03\;{\rm{m}}\), and \({k_A} = 25\;{\rm{W/m}} \cdot {\rm{K}}\),
\({R_A} = \dfrac{1}{{\left( {1000\;{\rm{W/}}{{\rm{m}}^{\rm{2}}} \cdot {\rm{K}}} \right)}} + \left( {\dfrac{{0.03\;{\rm{m}}}}{{25\;{\rm{W/m}} \cdot {\rm{K}}}}} \right)\)
\({R_A} = 2.2 \times {10^{ - 3}}\;{{\rm{m}}^{\rm{2}}} \cdot {\rm{K/W}}\)
The thermal contact resistance for wall C is calculated as,
\({R_C} = \dfrac{1}{h} + \dfrac{{{L_C}}}{{{k_C}}}\)
For \(h = 1000\;{\rm{W/}}{{\rm{m}}^{\rm{2}}} \cdot {\rm{K,}}\,{L_c} = 20\;{\rm{mm}} = 0.02\;{\rm{m}}\), and \({k_A} = 50\;{\rm{W/m}} \cdot {\rm{K}}\),
\({R_C} = \dfrac{1}{{\left( {1000\;{\rm{W/}}{{\rm{m}}^{\rm{2}}} \cdot {\rm{K}}} \right)}} + \left( {\dfrac{{0.02\;{\rm{m}}}}{{50\;{\rm{W/m}} \cdot {\rm{K}}}}} \right)\)
\({R_C} = 1.4 \times {10^{ - 3}}\;{{\rm{m}}^{\rm{2}}} \cdot {\rm{K/W}}\)
The heat flux on the side A of the wall is calculated as,
\({q_A} = \dfrac{{{T_\infty } - {T_1}}}{{{R_A}}}\)
For \({T_\infty } = \left( {25 + 273} \right)\;{\rm{K}} = 298\;{\rm{K,}}\,{T_1} = \left( {261 + 273} \right)\;{\rm{K}} = 534\;{\rm{K}}\), and \({R_A} = 2.2 \times {10^{ - 3}}\;{{\rm{m}}^{\rm{2}}} \cdot {\rm{K/W}}\),
\({q_A} = \dfrac{{298\;{\rm{K}} - 534\;{\rm{K}}}}{{2.2 \times {{10}^{ - 3}}\;{{\rm{m}}^{\rm{2}}} \cdot {\rm{K/W}}}}\)
\({q_A} = - 107272.73\;{\rm{W/}}{{\rm{m}}^{\rm{2}}}\)
The heat flux on the side C of the wall is calculated as,
\({q_C} = \dfrac{{{T_2} - {T_\infty }}}{{{R_C}}}\)
For \({T_\infty } = \left( {25 + 273} \right)\;{\rm{K}} = 298\;{\rm{K,}}\,{T_2} = \left( {211 + 273} \right)\;{\rm{K}} = 484\;{\rm{K}}\), and \({R_C} = 1.4 \times {10^{ - 3}}\;{{\rm{m}}^{\rm{2}}} \cdot {\rm{K/W}}\),
\({q_C} = \dfrac{{484\;{\rm{K}} - 298\;{\rm{K}}}}{{1.4 \times {{10}^{ - 3}}\;{{\rm{m}}^{\rm{2}}} \cdot {\rm{K/W}}}}\)
\({q_C} = 132857.14\;{\rm{W/}}{{\rm{m}}^{\rm{2}}}\)