please solve it, thanks!
Question 7 (14 marks)
T2
To,ho
Figure 3: The cross-section of the pipe in question 6
Consider a circular pipe of length L with thermal conductivity , and internal and external radii ri and r2 respectively as shown in Figure 3. There is some fluid passing through the cavity inside the pipe. The temperature of the fluid inside the cavity of the pipe is T and the ambient temperature (outside the pipe) is T with T > To. The convective heat transfer coefficients of the inside the cavity and outside the pipe are h; and ho respectively. The temperatures on the inner and outer surfaces of the pipe are T and T 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
(21)
(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 surroundings due to convection in the steady state is
a
(22)
(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 9 (23)
(2 marks)
(d) Using the previous results, or otherwise, show that
=2T-T{++
(24)
(2 marks)
(e Given the data shown in Table 2, find the rate of heat loss per unit length of the pipe , and the temperatures at the inner and outer surfaces Ti and T2. (5 marks)
Parameters Numerical value K 50 Wm=1K1 hi 2000 Wm=2K1 ho 600 Wm2K 0.05 m r2 0.06 m To 20C T 110C L 100 m
Table 2: Data for the pipe, the surroundings, and the fluid inside the pipe.
