A single-acting reciprocating water pump, with a bore and stroke of \(150 \mathrm{~mm}\) and \(300 \mathrm{~mm}\) respectively, runs at \(2.51 \mathrm{rad} \cdot \mathrm{s}^{-1}(0.4 \mathrm{rev} / \mathrm{s}) .\) Suction and delivery pipes are each \(75 \mathrm{~mm}\) diameter. The former is \(7.5 \mathrm{~m}\) long and the suction lift is \(3 \mathrm{~m}\). There is no air vessel on the suction side. The delivery, pipe is \(300 \mathrm{~m}\) long, the outlet (at atmospheric pressure) being \(13.5 \mathrm{~m}\) above the level of the pump, and a large air vessel is connected to the delivery pipe at a point \(15 \mathrm{~m}\) from the pump. Calculate the absolute pressure head in the cylinder at beginning, middle and end of each stroke. Assume that the motion of the piston is simple harmonic, that losses at inlet and outlet of each pipe are negligible, that the slip is \(2 \%\), and that \(f\). for both pipes is constant at 0.01. (Atmospheric pressure \(10.33 \mathrm{~m}\) water head.)

Step by step solution

01

Calculate the volumetric flow rate

First, calculate the volumetric flow rate of the water pump. Use the formula for the flow rate in a reciprocating pump:\[ Q = A \times L \times N \]Where:- \(A\) is the cross-sectional area of the bore.- \(L\) is the stroke length.- \(N\) is the rotational speed (in cycles per second).Given data:- Bore diameter (\(D\)) = 150 mm = 0.15 m- Stroke length (\(L\)) = 300 mm = 0.3 m- Rotational speed (\(N\)) = 0.4 rev/sFirst, calculate the cross-sectional area (\(A\)):\[ A = \frac{\pi\times D^2}{4} = \frac{\pi \times (0.15)^2}{4} = 0.0177 \text{ m}^2 \]Now, calculate the flow rate (\(Q\)):\[ Q = 0.0177 \text{ m}^2 \times 0.3 \text{ m} \times 0.4 \text{ rev/s} = 0.00213 \text{ m}^3/s \]Take into account the slip (2%):\[ Q_{useful} = 0.00213 \times (1 - 0.02) = 0.00209 \text{ m}^3/s \]

02

Determine the velocity of water in pipes

Calculate the velocity of water in both suction and delivery pipes using flow rate and cross-sectional area:Suction pipe diameter = 75 mm = 0.075 mCross-sectional area (suction) = \[ A_s = \frac{\pi D_s^2}{4} = \frac{\pi (0.075)^2}{4} = 0.00442 \text{ m}^2 \]Velocity in suction pipe:\[ v_s = \frac{Q_{useful}}{A_s} = \frac{0.00209 \text{ m}^3/s}{0.00442 \text{ m}^2} = 0.473 \text{ m/s} \]Delivery pipe diameter = 75 mm = 0.075 mCross-sectional area (delivery) = \[ A_d = \frac{\pi D_d^2}{4} = \frac{\pi (0.075)^2}{4} = 0.00442 \text{ m}^2 \]Velocity in delivery pipe:\[ v_d = \frac{Q_{useful}}{A_d} = \frac{0.00209 \text{ m}^3/s}{0.00442 \text{ m}^2} = 0.473 \text{ m/s} \]

03

Calculate head loss due to friction

Use Darcy-Weisbach equation to calculate head loss due to friction in both pipes:\[ h_f = f \times \frac{L}{D} \times \frac{v^2}{2g} \]Where:- \( f \) = 0.01 (friction factor)- \( L \) = length of the pipe- \( D \) = diameter of the pipe- \( v \) = velocity- \( g \) = acceleration due to gravity (9.81 \text{ m/s}^2)For the suction pipe:\[ h_{fs} = 0.01 \times \frac{7.5}{0.075} \times \frac{(0.473)^2}{2 \times 9.81} = 0.0114 \text{ m} \]For the delivery pipe (up to air vessel):\[ h_{fd} = 0.01 \times \frac{15}{0.075} \times \frac{(0.473)^2}{2 \times 9.81} = 0.0227 \text{ m} \]

04

Calculate pressure heads

Determine the pressure head at the beginning, middle, and end of the stroke.At the beginning of the stroke (0 degrees):- To account for suction lift and friction losses:\[ H_{beginning} = H_{atm} - H_s - h_{fs} = 13.33 \text{ m} - 3 \text{ m} - 0.0114 \text{ m} = 10.3186 \text{ m} \]At the middle of the stroke (90 degrees):- Simple harmonic motion assumption implies velocity at max:\[ H_{middle} = H_{atm} - H_s - h_{fs} - \frac{v_{max}^2}{2g} = 10.3186 \text{ m} - 0.2255 \text{ m} = 10.0931 \text{ m} \]At the end of the stroke (180 degrees):- Velocity decreases to zero again:\[ H_{end} = H_{atm} - H_s - h_{fs} = 10.3186 \text{ m} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Volumetric Flow Rate

In fluid dynamics, the volumetric flow rate (\(Q\)) represents the volume of fluid flowing per unit time through a cross-sectional area. It is a vital metric for understanding the capacity of pumps and pipes.
For a reciprocating pump, the flow rate can be calculated using the formula:
\( Q = A \times L \times N \)
where:

• \(A\) is the cross-sectional area of the bore.
• \(L\) is the stroke length.
• \(N\) is the rotational speed.

Using this formula helps quantify how much fluid moves through the system. The pump's bore and stroke dimensions and its operational speed directly influence this rate.
Once calculated, adjustments for inefficiencies, like slip, must be considered. Slip accounts for the percentage of fluid not contributing to actual flow, offering a more accurate measure of useful flow rate.

Velocity Calculation

To determine how fast fluid moves through a pipe, you need to calculate the velocity (\(v\)). This is essential for analyzing kinetic energy and pressure changes.
Using the volumetric flow rate (\(Q\)) and the pipe’s cross-sectional area (\(A\)), the velocity is determined by:
\( v = \frac{Q}{A} \)
For pipes, accurate diameters are crucial because the area (\(A\)) is based on diameter (\(D\)):
\( A = \frac{\pi D^2}{4} \)
A clear understanding of these calculations assists in designing systems with optimal flow, minimizing losses and ensuring sufficient capacity for the required application. When comparing different pipe sections like suction and delivery, ensure diameters are accurately measured for precise velocity computation.

Darcy-Weisbach Equation

Frictional losses in pipes, affecting the efficiency of fluid systems, are calculated using the Darcy-Weisbach equation:
\( h_f = f \times \frac{L}{D} \times \frac{v^2}{2g} \)
This formula allows us to compute the head loss (\(h_f\)) due to pipe friction, where:

• \(f\) is the friction factor,
• \(L\) is the length of the pipe,
• \(D\) is the pipe diameter,
• \(v\) is the velocity of the fluid,
• \(g\) is the gravitational acceleration (\(9.81 \, \text{m/s}^2\)).

Using this equation indicates where energy losses occur, helping in the design and optimization of piping systems. Understanding frictional losses signifies better management of pump power, reducing operational costs, and enhancing overall system performance.

Pressure Head

The pressure head is an expression of the fluid’s potential energy, representing the height of the fluid column due to pressure. It's vital for determining the required pump pressure to achieve desired flow conditions.
Calculated using the simplified Bernoulli's principle, which states the sum of pressure, kinetic, and potential energy remains constant along a streamline:

• Beginning of stroke (minimal kinetic energy):
  \( H_{beginning} = H_{atm} - H_s - h_{fs} \)
• Middle of stroke (maximum kinetic energy):
  \( H_{middle} = H_{atm} - H_s - h_{fs} - \frac{v_{max}^2}{2g} \)
• End of stroke (velocity decreases to zero):
  \( H_{end} = H_{atm} - H_s - h_{fs} \)

Understanding pressure head ensures the system is adequately designed to manage fluid transport, allowing for precise pump specifications to meet operational demands.

Simple Harmonic Motion

Simple Harmonic Motion (SHM) describes oscillatory motion where the restoring force is proportional to the displacement. For pumps, the piston’s motion is often assumed to follow SHM, simplifying velocity and acceleration calculations.
During the pump's cycle:

• At the beginning and end of the stroke, velocity is zero.
• At mid-stroke, velocity reaches a maximum.

Understanding SHM facilitates accurate timing of fluid dynamics through the system, leading to efficient design and function of reciprocating pumps. It provides a basis to analyze transient fluctuations in flow and pressure within the pump cycle.