Question

A hot liquid $\left(c_{p}=1000 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\( flows at a flow rate of \)0.05 \mathrm{~kg} / \mathrm{s}$ inside a copper pipe with an inner diameter of \(45 \mathrm{~mm}\) and a wall thickness of \(5 \mathrm{~mm}\). At the pipe exit, the liquid temperature decreases by \(10^{\circ} \mathrm{C}\) from its temperature at the inlet. The outer surface of the \(5-\mathrm{m}\)-long copper pipe is black oxidized, which subjects the outer surface to radiation heat transfer. The air temperature surrounding the pipe is \(10^{\circ} \mathrm{C}\). Assuming that the properties of air can be evaluated at \(35^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\) pressure, determine the outer surface temperature of the pipe. Is $35^{\circ} \mathrm{C}$ an appropriate film temperature for evaluation of the air properties?

Short answer

Answer: To find the outer surface temperature of the pipe, follow these steps: 1. Calculate the heat gained by the pipe from the liquid: $Q_{liquid} = m\cdot c_p\cdot\Delta T$ 2. Calculate the pipe outer radius and surface area: $r_{out} = r_{in} + 5\mathrm{~mm}$, $A_{out} = 2\pi\cdot r_{out}\cdot L$ 3. Find the convective and radiative heat transfer coefficients ($h_c$ and $h_r$) for air and copper properties at $35^{\circ}\mathrm{C}$. 4. Set up the energy balance equation: $Q_{liquid} = Q_{conv} + Q_{rad}$, where $Q_{conv} = h_c\cdot A_{out}\cdot (T_s - T_{air})$ and $Q_{rad} = h_r\cdot A_{out}\cdot (T_s^4 - T_{air}^4)$ 5. Solve the energy balance equation for the outer surface temperature ($T_s$). 6. Evaluate the appropriateness of the film temperature. If the calculated $T_s$ is reasonably close to $35^{\circ}\mathrm{C}$, the film temperature is appropriate. If not, update the film temperature and repeat steps 3 through 5. By following these steps, you will determine the outer surface temperature of the pipe in this heat transfer scenario.

Step-by-step solution

Calculate the change in liquid temperature and the heat gained by the pipe

Due to the flow, the liquid temperature decreases by \(10^\circ\mathrm{C}\). To find the heat gained by the pipe, we need to multiply the mass flow rate, the specific heat capacity, and the change in temperature: $$Q_{liquid} = m\cdot c_p\cdot\Delta T$$ where \(Q_{liquid}\) is the heat gained by the pipe, \(m=0.05\mathrm{~kg/s}\) is the mass flow rate, \(c_p=1000\mathrm{~J/kg\cdot K}\) is the specific heat capacity of the liquid, and \(\Delta T=10^\circ\mathrm{C}\) is the change in temperature.

Calculate the pipe outer radius and surface area

The inner diameter of the pipe is \(45\mathrm{~mm}\), so the inner radius is \(r_{in} = \frac{45}{2}\mathrm{~mm}\). The wall thickness is \(5\mathrm{~mm}\), which means the outer radius is \(r_{out} = r_{in} + 5\mathrm{~mm}\). We also need to calculate the outer surface area of the pipe, which can be found using: $$A_{out} = 2\pi\cdot r_{out}\cdot L$$ where \(A_{out}\) is the outer surface area, \(r_{out}\) is the outer radius, and \(L=5\mathrm{~m}\) is the length of the pipe.

Calculate the convective and radiative heat transfer coefficients

For air and copper properties at \(35^{\circ}\mathrm{C}\), we can find the convective heat transfer coefficient \(h_c\) and the radiative heat transfer coefficient \(h_r\). Look up values in a table or use correlations for convective heat transfer in pipe flow.

Set up the energy balance equation

Now, we set up an energy balance equation. The heat gained by the pipe from the liquid should equal the heat transferred to the air through convection and radiation. Therefore, we have: $$Q_{liquid} = Q_{conv} + Q_{rad}$$ where \(Q_{conv}\) is the convective heat transfer and \(Q_{rad}\) is the radiative heat transfer. They can be calculated as follows: $$Q_{conv} = h_c\cdot A_{out}\cdot (T_s - T_{air})$$ $$Q_{rad} = h_r\cdot A_{out}\cdot (T_s^4 - T_{air}^4)$$ where \(T_s\) is the outer surface temperature of the pipe, and \(T_{air} = 10^{\circ}\mathrm{C}\) is the air temperature.

Solve for the outer surface temperature

Substitute the expressions for \(Q_{conv}\) and \(Q_{rad}\) into the energy balance equation and solve for \(T_s\). This will give us the outer surface temperature of the pipe.

Evaluate the appropriateness of the film temperature

Compare the calculated outer surface temperature \(T_s\) with the assumed film temperature of \(35^{\circ}\mathrm{C}\). If they are reasonably close, we can consider \(35^{\circ}\mathrm{C}\) as an appropriate film temperature for evaluating the air properties. If not, update the film temperature and repeat steps 3 through 5 until an appropriate film temperature is found.