A 10 -m-long and 10 -mm-inner-diameter pipe made of commercial steel is used to heat a liquid in an industrial process. The liquid enters the pipe with \(T_{i}=25^{\circ} \mathrm{C}, V=0.8 \mathrm{~m} / \mathrm{s}\). A uniform heat flux is maintained by an electric resistance heater wrapped around the outer surface of the pipe, so that the fluid exits at \(75^{\circ} \mathrm{C}\). Assuming fully developed flow and taking the average fluid properties to be \(\rho=1000 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=\) \(4000 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, \mu=2 \times 10^{-3} \mathrm{~kg} / \mathrm{m} \cdot \mathrm{s}, k=0.48 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), and \(\operatorname{Pr}=10\), determine: (a) The required surface heat flux \(\dot{q}_{s}\), produced by the heater (b) The surface temperature at the exit, \(T_{s}\) (c) The pressure loss through the pipe and the minimum power required to overcome the resistance to flow.

Question: Determine the required surface heat flux, the surface temperature at the exit, the pressure loss through the pipe, and the minimum power required to overcome the resistance to flow in a 10-meter long and 10-mm-inner diameter pipe with an inlet temperature of 25°C and an outlet temperature of 75°C. Answer: 1. Calculate the mass flow rate and Reynolds number: Mass flow rate, \(\dot{m}= \rho AV\) Reynolds number, \(\text{Re}= \frac{\rho V d}{\mu}\) 2. Calculate the Nusselt number and heat transfer coefficient: Nusselt number, \(\text{Nu}= 0.023\; \text{Re}^{0.8}\; \operatorname{Pr}^{n}\) Heat transfer coefficient, \(h= \frac{\text{Nu}\cdot k}{d}\) 3. Calculate the required surface heat flux: Required surface heat flux, \(\dot{q}_{s}= \frac{\dot{m}c_{p}(T_{o}-T_{i})}{L}\) 4. Calculate the surface temperature at the exit: Surface temperature at the exit, \(T_{s}= T_{o}+ \frac{\dot{q}_{s}}{h A_{s}}\) 5. Determine the pressure loss through the pipe and the minimum power required to overcome the resistance to flow: Pressure loss, \(\Delta P= f \frac{L \rho V^{2}}{2 d}\) Friction factor, \(f= 0.079\; \text{Re}^{-0.25}\) Minimum power required, \(P_{\text{min}}= \frac{\Delta P \dot{V}}{\rho}\)