A one-dimensional object (you can treat it as a line) has one end immersed in boiling...

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A one-dimensional object (you can treat it as a line) has one end immersed in boiling water of 100°C and the other end immersed in liquid nitrogen of -196°C. The temperature of the object at different positions can be modelled by the heat equation below: where (a) (f) (b) (c) (d) (e) (g) 5Txx = Tt, 0<x<8, t>0 T(0, t) = 100, T(8, t) = -196, t> 0 T(x,0) = f(x), 0<x<8 f(x)= 37 8 -x²74x + 100. Write down the steady-state heat equation for the above problem. (Denote T, (x) as the solution) Hence, show that the steady-state solution is Ts(x) = -37x + 100 Write down the transient heat equation for the above problem. (Denote Q(x, t) as the solution) Compute the Fourier sine series of f(x) - Ts(x). (You do not have to simplify the Fourier coefficients) Hence, find the solution to the heat equation T(x, t). (5 marks) Explain what will happen to the temperature of the object after a long time. (5 marks) (15 marks) Hence, solve for the transient temperature function Q(x, t), the solution to the transient heat equation. (5 marks) (5 marks) (2 marks) (3 marks) | A one-dimensional object (you can treat it as a line) has one end immersed in boiling water of 100°C and the other end immersed in liquid nitrogen of -196°C. The temperature of the object at different positions can be modelled by the heat equation below: where (a) (f) (b) (c) (d) (e) (g) 5Txx = Tt, 0<x<8, t>0 T(0, t) = 100, T(8, t) = -196, t> 0 T(x,0) = f(x), 0<x<8 f(x)= 37 8 -x²74x + 100. Write down the steady-state heat equation for the above problem. (Denote T, (x) as the solution) Hence, show that the steady-state solution is Ts(x) = -37x + 100 Write down the transient heat equation for the above problem. (Denote Q(x, t) as the solution) Compute the Fourier sine series of f(x) - Ts(x). (You do not have to simplify the Fourier coefficients) Hence, find the solution to the heat equation T(x, t). (5 marks) Explain what will happen to the temperature of the object after a long time. (5 marks) (15 marks) Hence, solve for the transient temperature function Q(x, t), the solution to the transient heat equation. (5 marks) (5 marks) (2 marks) (3 marks) |

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a The steadystate heat equation for the given problem is 5Tssxx 0 0Tss0 100 Tss8 196 b The steadysta... View the full answer