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# (Solved):   Question 6 (14 marks) Figure 1: The cross-section of the pipe in qu ...

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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.

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