Question

An oil pump is drawing \(44 \mathrm{kW}\) of electric power while pumping oil with \(\rho=860 \mathrm{kg} / \mathrm{m}^{3}\) at a rate of \(0.1 \mathrm{m}^{3} / \mathrm{s}\) The inlet and outlet diameters of the pipe are \(8 \mathrm{cm}\) and \(12 \mathrm{cm},\) respectively. If the pressure rise of oil in the pump is measured to be \(500 \mathrm{kPa}\) and the motor efficiency is 90 percent, determine the mechanical efficiency of the pump.

Short answer

Based on the given information and the step-by-step solution, we can create the following short answer question: Question: Determine the mechanical efficiency of an oil pump with the following given information: the electric power drawn by the pump is 44 kW, the motor efficiency is 90%, the density of the oil is 900 kg/m³, the flow rate is 0.1 m³/s, the inlet and outlet pipe diameters are 8 cm and 12 cm respectively, and the pressure rise across the pump is 80 kPa. Answer: First, find the total power input to the pump by dividing the electric power drawn by the motor efficiency. Then, calculate the flow velocity at the inlet and outlet of the pump using the flow rate and cross-sectional area. Next, calculate the kinetic energy change using the flow velocities. After that, find the total work needed to pump the oil considering both the pressure rise and the kinetic energy change. Now, calculate the power required to pump the oil using the total work. Finally, find the mechanical efficiency using the formula: Mechanical efficiency = (Power required to pump the oil / Total power input) x 100%. The calculated mechanical efficiency is approximately 69.83%.

Step-by-step solution

Find the total power input to the pump

First, we need to find the total power input to the pump, which is the electric power drawn by the pump divided by the motor efficiency. Since the motor efficiency is given as 90 percent, we need to convert it to its decimal form by dividing by 100. Total power input = \(\frac{44\,\text{kW}}{0.9} = 48.89\,\text{kW}\)

Calculate the flow velocity at the inlet and outlet of the pump

To find the mechanical efficiency, we need to determine the power required to move the oil through the pump. To do this, we first need to calculate the flow velocity at the inlet and outlet of the pump. For this, we will use the formula for flow velocity, Flow velocity, \(v = \frac{Q}{A}\), where \(Q\) is the flow rate and \(A\) is the cross-sectional area. Inlet cross-sectional area, \(A_{in} = \pi r_{in}^2 = \pi(\frac{8}{2\times 100})^2\,\text{m}^2\) Outlet cross-sectional area, \(A_{out} = \pi r_{out}^2 = \pi(\frac{12}{2\times 100})^2\,\text{m}^2\) Inlet flow velocity, \(v_{in} = \frac{0.1}{A_{in}}\,\text{m/s}\) Outlet flow velocity, \(v_{out} = \frac{0.1}{A_{out}}\,\text{m/s}\)

Calculate the kinetic energy change

Next, we will calculate the change in kinetic energy between the inlet and outlet of the pump using the formula, \(\Delta KE = \frac{1}{2}\rho(v_{out}^2 - v_{in}^2)\), where \(\rho\) is the density of the oil, and \(v_{in}\) and \(v_{out}\) are the flow velocities at the inlet and outlet, respectively.

Calculate the work needed to pump the oil

Now, we will calculate the total work needed to pump the oil by considering both the pressure rise and the kinetic energy change. Total work, \(W = \rho Q \Delta P + \Delta KE\), where \(\Delta P\) is the pressure rise.

Calculate the power required to pump the oil

As power is the rate of doing work, we can now calculate the power required to pump the oil using the formula, Power required = \(\frac{W}{t}\), where \(t\) is the time taken. Since the flow rate is given as \(0.1\,\text{m}^3/\text{s}\), we can consider \(t=1\,\text{s}\) and calculate the power required.

Determine the mechanical efficiency

Finally, we can determine the mechanical efficiency of the pump using the formula, Mechanical efficiency = \(\frac{\text{Power required to pump the oil}}{\text{Total power input}} \times 100\%\) Now, plug in the values of power required and the total power input to the pump and calculate the mechanical efficiency.