Question 4 As shown in Figure Q4, the heat conduction in a thin alloy rod is...

 

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Question 4 As shown in Figure Q4, the heat conduction in a thin alloy rod is governed by the equation: 8²T 4 1JT + əx² a at 2cm Figure Q4 Thermal diffusivity of the rod is a = 1.5 cm²/sec, thermal conductivity is k = 5x104 W/(cm °C), the rod is 8cm in length and subjected to the uniform internal heating à = 1x105 W/cm³. At initial time, t=0 sec, the temperature of the rod is 10°C except 0°C at the left end and 90°C at the right end. The boundary conditions at the left end of the rod, T(x=0cm) = 0°C and at right end of the rod, T(x=8cm) = 90°C, are applied when t >=0 sec. Simultaneously, the uniform internal heating, 4 = 1x105 W/cm³, is also applied when 0<t<2 sec; and it cannot change the boundary conditions at both ends of the rod. a. Calculate the temperature distribution of the rod when the time t = 1 sec by explicit finite difference equations, of the above one-dimensional transient PDE with Ax = 2 cm, and At = 0.5 sec b. Check the stability of the above derived parabolic PDE solution. c. Derive the finite difference form of the simple implicit method for the above problem (do NOT solve the equations). = 90°C Question 4 As shown in Figure Q4, the heat conduction in a thin alloy rod is governed by the equation: 8²T 4 1JT + əx² a at 2cm Figure Q4 Thermal diffusivity of the rod is a = 1.5 cm²/sec, thermal conductivity is k = 5x104 W/(cm °C), the rod is 8cm in length and subjected to the uniform internal heating à = 1x105 W/cm³. At initial time, t=0 sec, the temperature of the rod is 10°C except 0°C at the left end and 90°C at the right end. The boundary conditions at the left end of the rod, T(x=0cm) = 0°C and at right end of the rod, T(x=8cm) = 90°C, are applied when t >=0 sec. Simultaneously, the uniform internal heating, 4 = 1x105 W/cm³, is also applied when 0<t<2 sec; and it cannot change the boundary conditions at both ends of the rod. a. Calculate the temperature distribution of the rod when the time t = 1 sec by explicit finite difference equations, of the above one-dimensional transient PDE with Ax = 2 cm, and At = 0.5 sec b. Check the stability of the above derived parabolic PDE solution. c. Derive the finite difference form of the simple implicit method for the above problem (do NOT solve the equations). = 90°C

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