Question

Hot engine oil is being transported in a horizontal pipe $\left(k=15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, D_{i}=5 \mathrm{~cm}\right)$ with a wall thickness of \(5 \mathrm{~mm}\). The pipe is covered with a \(5-\mathrm{mm}\)-thick layer of insulation $(k=0.15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\(. A length of \)2 \mathrm{~m}$ of the outer surface is exposed to cool air at \(10^{\circ} \mathrm{C}\). If the pipe inner surface temperature is at \(90^{\circ} \mathrm{C}\), determine the outer surface temperature. Hint: The pipe outer surface temperature has to be found iteratively. Begin the calculations by using a film temperature of $50^{\circ} \mathrm{C}$.

Short answer

Question: Determine the outer surface temperature of a horizontal pipe transporting hot engine oil, covered with insulation, and exposed to cool air over a length of 2 meters. Given the inner surface temperature, find the outer surface temperature using iterative calculation considering the conduction through the pipe wall and insulation as well as convection with the air.

Step-by-step solution

Write equations for heat transfer through conduction and convection

To find the outer surface temperature, we need to consider both heat transfer through conduction in the pipe wall and insulation and convection between the outer surface and the surrounding air. The heat transfer through conduction can be found using Fourier's law, and the heat transfer through convection can be found using Newton's law of cooling. For conduction, we have the equation: \(q_c = (T_i - T_o) \, \frac{k_p}{l_p} \frac{A}{R_p}\) for the pipe wall, and \(q_c = (T_m - T_o) \, \frac{k_i}{l_i} \frac{A}{R_i}\) for the insulation. For convection, we have the equation: \(q_c = h \cdot A \cdot (T_o - T_a)\), where: \(q_c\) - Rate of heat transfer per unit length \(T_i\) - Inner surface temperature of the pipe \(T_o\) - Outer surface temperature of the pipe \(T_m\) - Temperature at the insulation-outer surface interface \(T_a\) - Ambient air temperature \(k_p\) - Thermal conductivity of the pipe wall \(k_i\) - Thermal conductivity of the insulation \(l_p\) - Thickness of the pipe wall \(l_i\) - Thickness of the insulation \(A\) - Contact area of the pipe \(R_p\) - Thermal resistance of the pipe \(R_i\) - Thermal resistance of the insulation \(h\) - Convective heat transfer coefficient between outer surface and air

Calculate thermal resistance values and temperature at insulation-outer surface interface

First, we find the thermal resistance of the pipe wall and insulation using the equations: \(R_p = \frac{l_p}{k_p \cdot A},\) and \(R_i = \frac{l_i}{k_i \cdot A}.\) To find the temperature at the insulation-outer surface interface, we can use the equation for heat transfer through the pipe wall: \(T_m = T_i - q_c \cdot (R_p \cdot A).\)

Perform iterative calculations for the outer surface temperature

Use the given hint to start the iteration process with a film temperature (\(T_{film}\)) of \(50^{\circ} \mathrm{C}\). In each iteration, calculate the convective heat transfer coefficient (\(h\)) and update the value of the outer surface temperature, \(T_o\), using the following equation: \( T_o = T_m + q_c \cdot (R_i \cdot A).\) Stop the iteration when the change in outer surface temperature becomes smaller than an acceptable tolerance value (e.g., \(10^{-4}\)) or reaches a predetermined maximum number of iterations.

Calculate the final outer surface temperature

When the iterative process converges, the outer surface temperature value, \(T_o\), will be the final answer.