Question

Consider the flow of oil at \(10^{\circ} \mathrm{C}\) in a \(40-\mathrm{cm}\)-diameter pipeline at an average velocity of $0.5 \mathrm{~m} / \mathrm{s}\(. A \)1500-\mathrm{m}$-long section of the pipeline passes through icy waters of a lake at \(0^{\circ} \mathrm{C}\). Measurements indicate that the surface temperature of the pipe is very nearly \(0^{\circ} \mathrm{C}\). Disregarding the thermal resistance of the pipe material, determine \((a)\) the temperature of the oil when the pipe leaves the lake, \((b)\) the rate of heat transfer from the oil, and \((c)\) the pumping power required to overcome the pressure losses and to maintain the flow of oil in the pipe.

Short answer

Question: Determine (a) the temperature of the oil when the pipe leaves the lake, (b) the rate of heat transfer from the oil, and (c) the pumping power required to overcome the pressure losses and to maintain the flow of oil in the pipe.

Step-by-step solution

Calculate the temperature of the oil after leaving the lake

For this step, we will use the logarithmic mean temperature difference (LMTD) method to calculate the temperature difference between the inlet and outlet temperatures. The LMTD formula is: \[ \Delta T_\mathrm{lm} = \frac{\Delta T_\mathrm{in} - \Delta T_\mathrm{out}}{\ln(\Delta T_\mathrm{in} / \Delta T_\mathrm{out})} \] where - \(\Delta T_\mathrm{in}\) is the initial temperature difference between the oil and the lake (10°C - 0°C = 10°C), - \(\Delta T_\mathrm{out}\) is the final temperature difference between the oil and the lake (unknown), - \(\Delta T_\mathrm{lm}\) is the logarithmic mean temperature difference. We also have a heat transfer equation: \[ q = U \cdot A \cdot \Delta T_\mathrm{lm} \] where - \(q\) is the rate of heat transfer, - \(U\) is the overall heat transfer coefficient, - \(A\) is the surface area of the pipe. We will use these two equations to determine the temperature of the oil after leaving the lake.

Calculate the rate of heat transfer from the oil

We can calculate the rate of heat transfer from the oil using the following equation: \[ q = m \cdot C_p \cdot \Delta T \] where - \(m\) is the mass flow rate of the oil, - \(C_p\) is the specific heat capacity of the oil, - \(\Delta T\) is the temperature change. To calculate the mass flow rate of the oil: \[ m = \rho \cdot A_\mathrm{c} \cdot v \] where - \(\rho\) is the density of the oil, - \(A_\mathrm{c}\) is the cross-sectional area of the pipeline, - \(v\) is the average velocity of the oil. We now have all the equations needed to calculate the rate of heat transfer from the oil.

Calculate the pumping power required

To calculate the pumping power required to overcome the pressure losses and maintain the flow of oil in the pipe, we can use the following equation: \[ W_\mathrm{p} = \Delta P \cdot \frac{m}{\rho} = m \cdot g \cdot h_\mathrm{loss} = m \cdot (h_\mathrm{p} - h_\mathrm{loss}) \] where - \(W_\mathrm{p}\) is the pumping power, - \(\Delta P\) is the pressure loss, - \(g\) is the acceleration due to gravity, - \(h_\mathrm{loss}\) is the head loss due to friction, - \(h_\mathrm{p}\) is the pump head. We can calculate the head loss using the Darcy-Weisbach equation: \[ h_\mathrm{loss} = f \cdot \frac{L}{D} \cdot \frac{v^2}{2g} \] where - \(f\) is the friction factor, - \(L\) is the length of the pipe, - \(D\) is the diameter of the pipe. Now we have all the equations needed to calculate the pumping power required. Using the information provided and solving these equations, you will be able to determine \((a)\) the temperature of the oil when the pipe leaves the lake, \((b)\) the rate of heat transfer from the oil, and \((c)\) the pumping power required to overcome the pressure losses and to maintain the flow of oil in the pipe.