Question

A hot fluid $\left(k_{\text {fluid }}=0.72 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right)$ is flowing as a laminar fully developed flow inside a pipe with an inner diameter of \(35 \mathrm{~mm}\) and a wall thickness of $5 \mathrm{~mm}\(. The pipe is \)10 \mathrm{~m}$ long, and the outer surface is exposed to air at \(10^{\circ} \mathrm{C}\). The average temperature difference between the hot fluid and the pipe inner surface is $\Delta T_{\text {avg }}=10^{\circ} \mathrm{C}$, and the inner and outer surface temperatures are constant. Determine the outer surface temperature of the pipe. Evaluate the air properties at \(50^{\circ} \mathrm{C}\). Is this a good assumption?

Short answer

Question: Determine the outer surface temperature of a pipe with hot fluid flowing inside, and evaluate the air properties assumption at \(50^{\circ} \mathrm{C}\). Answer: The outer surface temperature of the pipe is approximately \(66.75^{\circ} \mathrm{C}\). The air properties assumption at \(50^{\circ} \mathrm{C}\) is reasonable for this specific case.

Step-by-step solution

Calculate the pipe's inner and outer radii

In order to analyze the heat transfer through the pipe wall, we first need to determine the radii of the inner and outer pipe surfaces. Given the inner diameter of the pipe (35 mm) and the wall thickness (5 mm), we can calculate the inner and outer radii as follows: $$R_{in} = \frac{d_{in} }{2}= \frac{35}{2} \mathrm{mm}= 17.5 \mathrm{mm}$$ $$R_{out}= R_{in} + t = 17.5 + 5 \mathrm{mm} = 22.5 \mathrm{mm}$$

Calculate the heat transfer through the pipe wall

To find the transfer of heat through the pipe wall, we will use the formula: $$q = \frac{2 \pi L \Delta T}{\ln{\frac{R_{out}}{R_{in}}} } \cdot k_{\text{fluid}} $$ Where \(q\) is the heat transfer, \(L\) is the length of the pipe, \(k_{\text{fluid}}\) is the thermal conductivity of the fluid, \(R_{in}\) and \(R_{out}\) are the inner and outer radii of the pipe, and \(\Delta T\) is the temperature difference between the hot fluid and the pipe inner surface. Substitute the known values for the variables and calculate the heat transfer: $$q = \frac{2 \pi (10 m) (10^{\circ} \mathrm{C})}{\ln{\frac{22.5}{17.5}} } \cdot 0.72 \frac{\mathrm{W}}{\mathrm{m} \cdot \mathrm{K} }$$ $$q \approx 75.751 W$$

Calculate the convection heat transfer coefficient on the outer surface

Now we need to calculate the convection heat transfer coefficient, \(h\), for the outer surface exposed to air. This coefficient depends on the specific air properties at a certain temperature (which, according to our assumption, will be evaluated at \(50^{\circ} C\)). Assuming natural convection, we can estimate \(h\) from correlations: $$h \approx 9 \, \mathrm{W/m^{2} \cdot K}$$

Calculate the outer surface temperature

Finally, we can calculate the outer surface temperature of the pipe by applying the following energy balance equation: $$q = h A (T_{out} - T_{\infty})$$ Where \(A\) is the outer surface area of the pipe and \(T_{\infty}\) is the temperature of the air. To calculate \(A\), use the formula: $$A = 2\pi R_{out} L = 2\pi \times 0.0225 m \times 10 m$$ $$A \approx 1.41 m^2$$ Now, rearrange the energy balance equation to solve for \(T_{out}\): $$T_{out} = \frac{q}{hA} + T_{\infty}$$ Substitute the known values and calculate the outer surface temperature: $$T_{out} = \frac{75.751 W}{9 \frac{\mathrm{W}}{\mathrm{m^{2} \cdot \mathrm{K} }} \times 1.41 \mathrm{m^{2}}} + 10^{\circ} \mathrm{C} $$ $$T_{out} \approx 66.75^{\circ} \mathrm{C}$$

Evaluate the air properties assumption

We assumed air properties at \(50^{\circ} \mathrm{C}\), and we found the outer surface temperature to be \(66.75^{\circ} \mathrm{C}\). Since this value is close enough to our assumption of air temperature, it can be considered as a reasonable assumption for this specific case.