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Chapter 4: Problem 54

A water pump operating at steady state has \(76 \mathrm{~mm}\) diameter inlet and exit pipes, each at the same elevation. The water can be modeled as incompressible and its temperature remains constant at \(20^{\circ} \mathrm{C}\). For a power input of \(1.5 \mathrm{~kW}\), plot the pressure rise from inlet to exit, in \(\mathrm{kPa}\), versus the volumetric flow rate ranging from \(0.015\) to \(0.02 \mathrm{~m}^{3} / \mathrm{s}\).

Short Answer

Expert verified
Plot pressure rise from inlet to exit versus volumetric flow rate using \[ \triangle p = \frac{1500 \times 1000 \times 9.81}{Q} \].

Step by step solution

01

- Understand the Problem

Determine what is asked: to find the pressure rise from inlet to exit for a given range of volumetric flow rates. The water pump operates at steady state and water is incompressible with constant temperature. A power input of 1.5 kW is given.
02

- Use the Power Equation

The relationship between power input, pressure rise, and volumetric flow rate can be given by \[ P = \frac{\triangle p \times Q}{\rho \times g} \] where \(P = 1.5 \text{kW} = 1500 \text{W}\), \(Q\) is the volumetric flow rate, \( \triangle p \) is the pressure rise in Pascal (Pa), \( \rho \) is the density of water (approximately \( 1000 \text{kg/m}^3 \)), and \( g = 9.81 \text{m/s}^2 \).
03

- Solve for Pressure Rise

Rearrange the power equation to solve for pressure rise: \[ \triangle p = \frac{P \times \rho \times g}{Q} \].
04

- Calculate Pressure Rise for Given Flow Rates

Substitute the given values into the equation. Calculate \[ \triangle p \] for \(Q = 0.015 \text{m}^3/\text{s}\) and \(Q = 0.02 \text{m}^3/\text{s}\).
05

- Convert to KPa

Convert the pressure rise from Pascal (Pa) to kilopascal (kPa) by dividing the result by 1000.
06

- Plot the Results

Plot \( \triangle p \) in kPa versus \( Q \) in \( \text{m}^3/\text{s} \) for the specified range of volumetric flow rates.

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Key Concepts

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

steady state operation
In a steady state operation, the system's properties do not change over time. Imagine a pump that's always running the same way without fluctuations. This means the flow rate of water, pressure, and other properties remain constant. It simplifies our calculations because we don't need to account for changes with time.
When problems state 'steady state,' it tells us to assume conditions are constant. This makes it easier to use average values in our equations.
For the water pump, knowing it's in steady state means the pressure rise and volumetric flow rate can consistently be related to the power input.
incompressible fluid
An incompressible fluid has a constant density. Water is a common example.
This term means if we squeeze or stretch the water in the pump, its volume remains nearly the same.
For calculations, this simplifies equations because we assume no significant change in density.
So, for the given problem, knowing water is incompressible helps in using a consistent density value, usually around 1000 kg/m鲁. This consistency is crucial for accurately solving the pressure rise.
volumetric flow rate
Volumetric flow rate, denoted as Q, is the volume of fluid passing through a section of the pipe per unit time. It鈥檚 measured in cubic meters per second (m鲁/s).
Understanding Q helps us analyze how much water the pump moves over time.
In this exercise, we have a range of 0.015 to 0.02 m鲁/s. Different Q values will affect the pressure rise.
Remembering that Q's changes directly impact how we calculate the pressure rise is important when plotting the results.
power input
Power input to the pump is the energy provided, measured in watts (W) or kilowatts (kW).
In our exercise, the pump's power input is 1.5 kW.
This power is used to increase the pressure of the water as it moves through the pump.The given power helps in determining the pressure rise by using the power equation. Up next is to understand how to use the power input in solving for pressure rise.
pressure rise calculation
Pressure rise, denoted as 螖p, is the increase in pressure from the pump鈥檚 inlet to its outlet. We calculate it as:
\[ \triangle p = \frac{1500 \times 1000 \times 9.81}{Q} \text{Pa}\]Plug in different values for Q (volumetric flow rate) to find the corresponding pressure rise. Then, convert the pressure rise from Pascal (Pa) to kilopascal (kPa) by dividing by 1000.Plot these values to visualize how pressure rise changes with different flow rates. This method helps in understanding how efficiently the pump operates under various conditions.

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Most popular questions from this chapter

Generating power by harnessing ocean tides and waves is being studied across the globe. Underwater turbines develop power from tidal currents. Wave-power devices develop power from the undulating motion of ocean waves. Although tides and waves long have been used to meet modest power generation needs, investigators today are aiming at large-scale power generation systems. Some see the oceans as providing a potentially unlimited renewable source of power. Critically evaluate the viability of tidaland wave power, considering both technical and economic issues. Write a report, including at least three references.

Ammonia enters a refrigeration system compressor operating at steady state at \(-18^{\circ} \mathrm{C}, 138 \mathrm{kPa}\), and exits at \(150^{\circ} \mathrm{C}, 1724 \mathrm{kPa}\). The magnitude of the power input to the compressor is \(7.5 \mathrm{~kW}\), and there is heat transfer from the compressor to the surroundings at a rate of \(0.15 \mathrm{~kW}\). Kinetic and potential energy effects are negligible. Determine the inlet volumetric flow rate, in \(\mathrm{m}^{3} / \mathrm{s}\), first using data from Table A-15, and then assuming ideal gas behavior for the ammonia. Discuss.

At steady state, a well-insulated compressor takes in air at \(15^{\circ} \mathrm{C}, 98 \mathrm{kPa}\), with a volumetric flow rate of \(0.57 \mathrm{~m}^{3} / \mathrm{s}\), and compresses it to \(260^{\circ} \mathrm{C}, 827 \mathrm{kPa}\). Kinetic and potential energy changes from inlet to exit can be neglected. Determine the compressor power, in \(\mathrm{kW}\), and the volumetric flow rate at the exit, in \(\mathrm{m}^{3} / \mathrm{s}\).

The electronic components of a computer consume \(0.1 \mathrm{~kW}\). of electrical power. To prevent overheating, cooling air is supplied by a 25-W fan mounted at the inlet of the electronics enclosure. At steady state, air enters the fan at \(20^{\circ} \mathrm{C}, 1\) bar, and exits the electronics enclosure at \(35^{\circ} \mathrm{C}\). There is no significant energy transfer by heat from the outer surface of the enclosure to the surroundings, and the effects of kinetic and potential energy can be ignored. Determine the volumetric flow rate of the entering air, in \(\mathrm{m}^{3} / \mathrm{s}\).

A rigid tank of volume \(0.75 \mathrm{~m}^{3}\) is initially evacuated. A hole develops in the wall, and air from the surroundings at 1 bar, \(25^{\circ} \mathrm{C}\) flows in until the pressure in the tank reaches 1 bar. Heat transfer between the contents of the tank and the surroundings is negligible. Determine the final temperature in the tank, in \({ }^{\circ} \mathrm{C}\).
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