Problem 90 - Fundamentals of Physics

Question

The scwage outlet of a house constructed on a slope is \(6.59 \mathrm{~m}\) below street level. If the sewer is \(2.16 \mathrm{~m}\) below street level, find the minimum pressure difference that must be created by the sewage pump to transfer waste of average density \(1000.00 \mathrm{~kg} / \mathrm{m}^{3}\) from outlet to sewer.

Step-by-Step Solution

Answer

The minimum pressure difference is 43.52 kPa.

Step 1: Identify the Height Difference

Calculate the height difference between the sewer outlet and the sewer, which is the vertical distance that the sewage pump must overcome. The outlet is located 6.59 m below street level and the sewer is 2.16 m below street level. The formula to calculate this difference is:\[\text{height difference} = 6.59\, \text{m} - 2.16\, \text{m} = 4.43\, \text{m}\]

Step 2: Use the Hydrostatic Pressure Formula

The hydrostatic pressure difference required to transfer the sewage is related to the height difference of the fluid column. The formula to determine this pressure difference is:\[\Delta P = \rho \cdot g \cdot h\]where \(\Delta P\) is the pressure difference, \(\rho = 1000\, \text{kg/m}^3\) is the fluid density for water, \(g = 9.81\, \text{m/s}^2\) is the acceleration due to gravity, and \(h = 4.43\, \text{m}\) is the height difference calculated in step 1.

Step 3: Calculate the Minimum Pressure Difference

Substitute the known values into the hydrostatic pressure formula:\[\Delta P = (1000\, \text{kg/m}^3) \times (9.81\, \text{m/s}^2) \times (4.43\, \text{m})\]Solving this gives:\[\Delta P = 43518.3\, \text{Pa} = 43.52\, \text{kPa}\]

Step 4: Interpret the Result

The calculated pressure difference of 43.52 kPa is what the sewage pump must overcome to transfer the waste against gravity from the outlet to the sewer. This is the minimum pressure that the pump should provide.

Key Concepts

Understanding Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at rest due to the gravitational force acting on it. This is a fundamental concept in fluid mechanics. The pressure at a specific depth depends on the density of the fluid, the gravitational acceleration, and the depth itself. It's important to understand that hydrostatic pressure increases linearly with the depth of the fluid.

The formula for calculating hydrostatic pressure is given by:\[ P = \rho \cdot g \cdot h \]Where:

• \( P \) is the hydrostatic pressure.
• \( \rho \) is the fluid density (in kg/m³).
• \( g \) is the acceleration due to gravity (approximately 9.81 m/s² on Earth).
• \( h \) is the height or depth of the fluid column in meters.

By using this formula, one can determine the pressure exerted by a fluid column at any given depth. In practical applications, such as determining the pressure requirements of sewage systems, hydrostatic pressure provides crucial insights on the forces that must be counteracted.

The Role of Sewage Pump Pressure

Sewage pumps are essential in transporting waste against the natural force of gravity from lower levels to higher sewage systems, especially in areas where buildings are situated below street level.

The pump pressure required to move the sewage is dictated by the need to overcome the hydrostatic pressure created by the elevation difference. This is crucial for ensuring that waste is efficiently moved and doesn't settle or cause blockages in the system.

• The pump must generate a pressure difference that equals or exceeds the hydrostatic pressure.
• This means the sewage pump should have a capacity to generate at least 43.52 kPa, as calculated in the original exercise.
• Choosing the right pump ensures not only effective waste management but also the longevity of the sewage infrastructure.

Understanding how sewage pump pressure works can be vital for anyone involved in plumbing, civil engineering, or maintenance of sewage systems.

Calculating Height Difference

Height difference calculation is a critical step in determining how much pressure is required from a sewage pump. This calculation is straightforward, requiring only the subtraction of the lower point from the higher reference point.

In the given problem, the height difference is calculated as follows:

• The sewage outlet is 6.59 meters below street level.
• The sewer itself is 2.16 meters below street level.
• The resulting height difference is \( 6.59 \, \text{m} - 2.16 \, \text{m} = 4.43 \, \text{m} \).

This height difference indicates how far vertically the sewage needs to be pumped. Once you know this difference, you can calculate the hydrostatic pressure that the sewage pump needs to overcome. Such calculations are essential not only for efficient fluid transport but also for ensuring that the system is designed to handle pressure loads appropriately.