Problem 90 - Fundamentals of Physics

Question

The sewage 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 needed is approximately 43500 Pa.

Step 1: Identify the Given Information

We know that the outlet is located 6.59 meters below street level, and the sewer is 2.16 meters below street level. The density of the waste is given as 1000 kg/m³. We need to find the pressure difference needed to pump the waste.

Step 2: Calculate the Height Difference

Determine the height difference between the outlet and the sewer. Since the outlet is at 6.59 m below street level and the sewer is at 2.16 m below street level, the height difference is:\[ \Delta h = 6.59 - 2.16 = 4.43 \text{ meters} \]

Step 3: Use the Hydrostatic Pressure Formula

The pressure difference required to move the waste vertically is due to the height difference, which can be calculated using the hydrostatic pressure formula:\[ \Delta P = \rho \cdot g \cdot \Delta h \]where \( \rho = 1000 \ kg/m^3 \) (density of the waste) and \( g = 9.81 \ m/s^2 \) (acceleration due to gravity).

Step 4: Calculate Pressure Difference

Substitute the values into the hydrostatic pressure formula:\[ \Delta P = 1000 \times 9.81 \times 4.43 \]Calculate to find:\[ \Delta P = 43542.3 \ Pa \]

Step 5: Round Off the Result

Generally, results are rounded to reflect the precision of the original data. Since our input data has three significant figures, we should round to:\[ \Delta P \approx 43500 \ Pa \]

Key Concepts

Pressure Difference

Understanding the pressure difference is crucial when dealing with fluids, such as sewage, in hydrostatics. Pressure difference refers to the variance in pressure between two points in a fluid. This difference is a driving force that can cause fluid flow from one location to another. In cases like the sewage outlet, the pressure difference helps in moving waste from a lower elevation to a higher one.
The pressure difference can be determined using the hydrostatic pressure formula:

• \( \Delta P = \rho \cdot g \cdot \Delta h \)
• Here, \(\rho\) is the fluid's density, \(g\) is the gravitational acceleration, and \(\Delta h\) is the height difference.

This formula tells us that the pressure difference depends directly on the fluid's density and the vertical height it needs to travel. A higher density or a larger height difference increases the pressure difference needed to move the fluid.

Density

Density plays a significant role in hydrostatics, especially when calculating pressure differences. It is defined as the mass of a substance per unit volume. In mathematical terms, density \(\rho\) is expressed as:

• \( \rho = \frac{m}{V} \)
• Where \(m\) is the mass and \(V\) is the volume.

In the context of the sewage system, the waste's density is given as 1000 kg/m³. This is a typical density for water, and many waste fluids have similar densities. The higher the density, the more force is required to move the fluid across a height difference.
Understanding density helps us predict how substances will behave under gravitational forces, particularly in closed systems like pipes.

Height Difference

Height difference is another critical aspect in understanding fluid movement in hydrostatic scenarios. It refers to the vertical distance between two points in a fluid system. In the sewage pumping problem, the height difference \(\Delta h\) between the sewage outlet and the sewer was calculated as 4.43 meters.
This measurement is vital because it influences the pressure needed to move fluid. With a larger height difference, more pressure is required. Calculating this difference accurately ensures that the correct amount of force is applied for effective fluid transport. The principle behind this is due to gravitational pull, which necessitates higher pressure for ascending fluids over greater heights.

Fluid Mechanics

Fluid mechanics is the broader scientific discipline that covers the behavior of fluids, both liquids and gases, and the forces acting on them. This field combines various principles, such as pressure differences, density, and height differences, to understand and solve real-world problems.
In the given exercise, fluid mechanics principles help determine the pressure difference needed for the upward movement of sewage against gravity. It explains how different factors, including waste density, gravitational forces, and vertical distances, interact to necessitate a specific pressure level. The practical application of fluid mechanics ensures efficient and safe design and operation of systems like sewage pumps.
For students and engineers, mastering fluid mechanics is essential to designing systems that efficiently manage fluid flow under various conditions.