Two cylinders of identical material having a constant thermal conductivity k, and lengths L and L...

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Two cylinders of identical material having a constant thermal conductivity k, and lengths L₁ and L₂ are joined together as shown in the figure below (Fig. 1). In order to maintain the 1-D heat conduction, the lateral surface of the cylinder is covered with the insulating material. One of the two open boundaries is maintained at a constant temperature To and the other boundary is exposed to a surrounding cold fluid with a temperature T₁ and the convective heat transfer coefficient h. If the steady state heat transfer occurs, determine the temperature distribution of the cylinder with a constant cross-sectional area A. For this problem, it is assumed that the rate of internal volumetric heat generation (q") in both the cylinders which is as a linear function of temperature, q" =qo (1+aT), where go and a are constant. Due to the thermal contact resistance at the joining section, a temperature drop exists by Ta. Determine also the heat transfer rate through the cylinders. If the heat transfer is a function of time, derive the governing energy equation. To Ta>To Insulation (A, k) L₁ L₂ Ta, h Fig. 1 A schematic diagram of two attached cylinders Two cylinders of identical material having a constant thermal conductivity k, and lengths L₁ and L₂ are joined together as shown in the figure below (Fig. 1). In order to maintain the 1-D heat conduction, the lateral surface of the cylinder is covered with the insulating material. One of the two open boundaries is maintained at a constant temperature To and the other boundary is exposed to a surrounding cold fluid with a temperature T₁ and the convective heat transfer coefficient h. If the steady state heat transfer occurs, determine the temperature distribution of the cylinder with a constant cross-sectional area A. For this problem, it is assumed that the rate of internal volumetric heat generation (q") in both the cylinders which is as a linear function of temperature, q" =qo (1+aT), where go and a are constant. Due to the thermal contact resistance at the joining section, a temperature drop exists by Ta. Determine also the heat transfer rate through the cylinders. If the heat transfer is a function of time, derive the governing energy equation. To Ta>To Insulation (A, k) L₁ L₂ Ta, h Fig. 1 A schematic diagram of two attached cylinders

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