91Ó°ÊÓ

A housing development is served by a water tower with the water level maintained between 20 and 30 meters above the ground, depending on demand and water availability. Responding to a resident's complaint about the low flow rate of water at his kitchen sink, a representative of the developer measured the water pressure at the tap above the kitchen sink and at the junction between the water main (a pipe connected to the bottom of the water tower) and the feed pipe to the house. The junction is \(5 \mathrm{m}\) below the level of the kitchen tap. All water valves in the house were turned off. (a) If the water level in the tower was 25 m above tap level, what should be the gauge pressures (kPa) at the tap and junction? (b) Suppose the pressure measurement at the tap was lower than your estimate in Part (a), but the measurement at the junction was as predicted. State a possible explanation. (c) If pressure measurements corresponded to the predictions in Part (a), what else could be responsible for the low water flow to the sink?

Step by step solution

01

Calculation of Water Pressure at the Tap

First, we determine the pressure at the tap in kilopascals (KPa) using the water density (\( \rho = 1000 \, \mathrm{kg/m^3} \)), the acceleration due to gravity (\( g = 9.81 \, \mathrm{m/s^2} \)), and the height difference between the water level and the tap (\( h = 25 \, \mathrm{m} \)). The formula to use here is \( P = \rho g h \). After calculations, the pressure can be found in pascals (Pa), so to convert that into KPa, the answer should be divided by 1000.

02

Calculation of Water Pressure at the Junction

The same equation will be used to calculate the pressure at the junction. However, in this case, the height difference is \( h = 25 + 5 = 30 \, \mathrm{m} \), because the junction is 5m below the tap. The obtained result should be converted from Pa to KPa.

03

Explanation for Discrepancy in Pressure Measurements

If the pressure measurement at the tap is lower than calculated but the measurement at the junction is as predicted, it could be due to a blockage or leak in the pipe connecting the junction to the tap. This is because if the flow path is obstructed, the pressure at the tap will be less than what it should be.

04

Identifying Potential Causes for Low Water Flow

If the pressure measurements match the predictions, yet there is low water flow to the sink, the problem could be caused by a partial blockage in the pipes, such as build-up of sediment or scale, or there could be an issue with the faucet itself, such as a clogged aerator or cartridge.

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Key Concepts

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

Water Pressure Calculations

Calculating water pressure is a fundamental concept in fluid mechanics and is crucial to understanding how water systems work. In this exercise, we explore the calculation of gauge pressure at two separate points: the tap and the junction.
When calculating pressure with a known height difference, we use the formula \( P = \rho g h \), where:

• \( P \) is pressure in pascals,
• \( \rho \) is the density of water, which is \( 1000 \, \mathrm{kg/m^3} \),
• \( g \) is the acceleration due to gravity, \( 9.81 \, \mathrm{m/s^2} \),
• \( h \) is the height difference.

The exercise involves calculating the pressure at two heights, which requires modifying \( h \) accordingly:

• At the tap, the height is given as \( 25 \, \mathrm{m} \), leading to the calculation of the pressure at this point.
• For the junction, we add the additional \( 5 \, \mathrm{m} \) below the tap, giving a total \( h = 30 \, \mathrm{m} \).

Once you've found the pressure in pascals, converting it to kilopascals (KPa) can be done by dividing by 1000. Understanding these calculations is vital for applying fluid mechanics principles to real-world water pressure issues.

Hydrostatics

Hydrostatics deals with fluids at rest and is a crucial part of analyzing and solving water pressure problems. In this exercise, we assume that the water within the system is at rest as all water valves are turned off.
The key hydrostatic principle used is that pressure within a fluid increases with depth due to the weight of the fluid above. Since the water in a tower is elevated, this potential energy translates into pressure at any point below the water level.
This principle helps explain how the system works:

• The height of water in the tower provides potential energy that converts to pressure, driving water flow.
• The difference in heights—between the water in the tower and the tap or junction—affects the pressure difference at these points.

Understanding hydrostatics allows players to predict behavior of the water system under static conditions and evaluate discrepancies like those addressed in this problem.
Such analysis is essential for ensuring adequate water supply in development and troubleshooting any issues that may arise from pressure variations.

Problem Solving in Engineering

Problem solving in engineering is about applying fundamental principles to diagnose and remedy issues in practical scenarios. In this exercise, multiple steps involve predictions and analysis based on established metrics to address a homeowner's concern.
Some key points engineers consider when solving similar problems include:

• Initial calculations to set a baseline expectation, like estimating water pressure from the known height in the tower.
• Comparing actual measurements against expectations to identify potential issues, such as a pressure drop pointing towards a leak or blockage.
• Considering alternative explanations if initial tests do not resolve the issue, such as a blockage causing low water flow despite normal pressure readings.

Effective problem solving often involves diagnosing the problem with a clear understanding of theory but adapting to practical challenges. By applying basic principles of fluid mechanics and iterative testing, engineers efficiently address real-world engineering problems in a structured and effective manner.