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A hollow insulating sphere of a material with thermal conductivity kA has internal diameter D1 and external diameter D2. An electrical heater is located in the hollow, such that there is heat flow by conduction through the wall of the sphere. (a) Perform an energy balance over a thin spherical shell of inner radius r and outer radius r + dr, when the system is at steady state. [2 marks] (b) Solve the differential energy equation from part (a) and derive an expression for the rate of heat transfer by conduction through the wall, with the temperature at the internal surface T1, and temperature at the external surface T2. [5 marks] (c) Consider the case of internal diameter D1 = 8 cm and external diameter D2 = 15 cm. The thermal conductivity of the insulating material is kA = 0.2 W m-1 K-1. An electrical heater of power 20 W is switched on, and eventually steady-state is achieved. The external surface of the sphere is in contact with ambient air at temperature Ta = 20 ºC and winds at velocity va = 30 km h -1. For the properties of air, assume density ρ = 1.2 kg m−3 , dynamic viscosity µ = 1.8×10−5 Pa s, thermal conductivity ka = 0.026 W m−1 K−1 , and heat capacity Cp = 1000 J kg−1 K−1 . To estimate the convective heat transfer coefficient h for a sphere, use the correlation: 𝑁𝑢 = 2 + (0.4𝑅𝑒1/2 + 0.06𝑅𝑒2/3)𝑃𝑟0.4 . (c-i) Calculate the value of the heat transfer coefficient (h) for forced convection from the external surface to the ambient air. [8 marks] (c-ii) Calculate the temperature at the external surface (T2). [2 marks] (c-iii) Calculate the temperature at the internal surface (T1). [4 marks] (c-iv) If the heater power is doubled to 40 W, estimate the new external surface temperature (T2,new) and internal surface temperature (T1,new).
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![A hollow insulating sphere of a material with thermal conductivity kA has internal diameter D1 and external diameter D2. An electrical heater is located in the hollow, such that there is heat flow by conduction through the wall of the sphere. (a) Perform an energy balance over a thin spherical shell of inner radius r and outer radius r + dr, when the system is at steady state. [2 marks] (b) Solve the differential energy equation from part (a) and derive an expression for the rate of heat transfer by conduction through the wall, with the temperature at the internal surface T1, and temperature at the external surface T2. [5 marks] (c) Consider the case of internal diameter D1 = 8 cm and external diameter D2 = 15 cm. The thermal conductivity of the insulating material is kA = 0.2 W m-1 K-1. An electrical heater of power 20 W is switched on, and eventually steady-state is achieved. The external surface of the sphere is in contact with ambient air at temperature Ta = 20 ºC and winds at velocity va = 30 km h -1. For the properties of air, assume density ρ = 1.2 kg m−3 , dynamic viscosity µ = 1.8×10−5 Pa s, thermal conductivity ka = 0.026 W m−1 K−1 , and heat capacity Cp = 1000 J kg−1 K−1 . To estimate the convective heat transfer coefficient h for a sphere, use the correlation: 𝑁𝑢 = 2 + (0.4𝑅𝑒1/2 + 0.06𝑅𝑒2/3)𝑃𝑟0.4 . (c-i) Calculate the value of the heat transfer coefficient (h) for forced convection from the external surface to the ambient air. [8 marks] (c-ii) Calculate the temperature at the external surface (T2). [2 marks] (c-iii) Calculate the temperature at the internal surface (T1). [4 marks] (c-iv) If the heater power is doubled to 40 W, estimate the new external surface temperature (T2,new) and internal surface temperature (T1,new).](https://numengineering.com/wp-content/uploads/2024/12/Screenshot-2024-12-10-195436.png)
Heat transfer
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