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Question 6 (14 marks) Figure 1: The cross-section of the pipe in question 6 Consider a circular pipe of length \( L \) with thermal conductivity \( \kappa \), and internal and external radii \( r_{1} \) and \( r_{2} \) respectively as shown in Figure 1. There is some fluid passing through the cavity inside the pipe. The temperature of the fluid inside the cavity of the pipe is \( T_{i} \) and the ambient temperature (outside the pipe) is \( T_{o} \) with \( T_{i}>T_{o} \). The convective heat transfer coefficients of the inside the cavity and outside the pipe are \( h_{i} \) and \( h_{o} \) respectively. The temperatures on the inner and outer surfaces of the pipe are \( T_{1} \) and \( T_{2} \) respectively. Suppose that the heat transfer is in a steady state. (a) By using the Fourier law for conduction, show that the rate of heat transfer \( q \) from the inner surface to the outer surface of the pipe satisfies \[ T_{1}-T_{2}=\frac{q}{2 \pi \kappa L} \ln \left(\frac{r_{2}}{r_{1}}\right) . \] (3 marks)
(b) By using the steady-state heat transfer by convection, show that the rate of heat transfer \( q \) from the outer surface of the pipe to the air in the surrounding due to convection in the steady state is \[ T_{2}-T_{o}=\frac{q}{2 \pi r_{2} L h_{o}} . \] (2 marks) (c) By using the steady-state heat transfer by convection again, show that the rate of heat transfer \( q \) from the cavity to the inner surface of the pipe due to convection in the steady state is \[ T_{i}-T_{1}=\frac{q}{2 \pi r_{1} L h_{i}} . \] (2 marks) (d) Using the previous results, or otherwise, show that \[ \frac{q}{L}=2 \pi\left(T_{i}-T_{o}\right)\left\{\frac{1}{r_{1} h_{i}}+\frac{1}{\kappa} \ln \left(\frac{r_{2}}{r_{1}}\right)+\frac{1}{r_{2} h_{o}}\right\}^{-1} \] (2 marks) (e) Given the data shown in Table 2 , find the rate of heat loss per unit length of the pipe \( \frac{q}{L} \), and the temperatures at the inner and outer surfaces \( T_{1} \) and \( T_{2} .(5 \) marks) Table 2: Data for the pipe, the surrounding and the fluid inside the pipe.

Given : The cross-section of the pipe is shown below T0,h0 Ti,hi T1 T2 r1 r2 ? (a) By using the Fourier law for conduction, show that the rate of he