Question

The sewer outlets of a house constructed on a slope are \(8.16 \mathrm{~m}\) below street level. If the sewer is \(2.08 \mathrm{~m}\) below street level, find the minimum pressure differential that must be created by the sewage pump to transfer waste of average density \(926 \mathrm{~kg} / \mathrm{m}^{3}\)

Short answer

The minimum pressure differential required is found to be approximately \(55,039.97 \mathrm{Pa}\) or \(55.04 \mathrm{kPa}\)

Step-by-step solution

Identify the Required Variables

First, the given values need to be identified. The height difference between the sewage outlet and the street level is \(8.16 \mathrm{m} - 2.08 \mathrm{m}\). This difference is \(6.08 \mathrm{m}\). The average density of waste is given as \(926 \mathrm{kg} / \mathrm{m}^{3}\). The constant \(g\), acceleration due to gravity, is approximately \(9.81 \mathrm{m/s}^{2}\).

Use the Pressure Formula.

Now, we can use the formula for pressure differential, which is \(P = \rho \cdot g \cdot h\). Substituting the above values into this formula, we obtain \(P = 926 \mathrm{kg/m^{3}} \cdot 9.81 \mathrm{m/s^{2}} \cdot 6.08 \mathrm{m}\).

Compute the pressure differential.

Finally, calculate the pressure differential by carrying out the multiplication. The result is the minimum pressure differential that must be created by the sewage pump to transfer waste.

Key concepts

Pressure Differential

Understanding the concept of pressure differential is crucial in fluid dynamics, especially when dealing with systems involving fluid movement, like a sewage pump. Pressure differential refers to the difference in pressure between two points in a fluid system. This difference is what drives the movement of the fluid from areas of high pressure to low pressure.
- In our example, the sewage pump needs to create enough pressure differential to move the waste from a lower elevation (sewer outlets) to a higher one (street sewer level).
- Mathematically, pressure differential can be calculated by the formula: \[ P = \rho \cdot g \cdot h \] Where, \( P \) is the pressure differential, \( \rho \) is the density of the fluid, \( g \) is the acceleration due to gravity, and \( h \) is the height difference between the two points.
This formula helps us understand and compute the energy needed by the pump to ensure the efficient movement of waste.

Sewage Pump

Sewage pumps play a critical role in waste management systems. They are designed to move waste material from lower to higher elevations, often necessary in buildings situated below street level.
- These pumps need to generate a specific pressure to ensure smooth and effective fluid transfer.
- In our scenario, the sewage system is below street level, requiring the pump to exert extra force to lift the waste up.
- It is important that the pump can create at least the minimum pressure differential calculated, ensuring that the waste is moved efficiently against gravitational forces.
Proper understanding and application of these pressure systems prevent backflow and potential leakages in the sewage system.

Density of Fluids

The density of a fluid is a fundamental property that is crucial in calculating the pressure differential in a fluid system. Defined as mass per unit volume, it helps in determining how much mass is contained in a given volume of fluid.
- For liquid waste, the average density is provided as 926 kg/m³, a key factor in our calculations.
- Higher density fluids will result in higher pressure at a given depth compared to lower density fluids.
Understanding fluid density helps in predicting how the fluid will behave under different pressure conditions and is essential to ensure that the sewage pump is designed to handle the mass of the fluid correctly.

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at equilibrium due to the force of gravity. It increases with depth in a fluid because of the increasing weight of the fluid exerting downward force from above.
- In the sewage system case, the hydrostatic pressure is a component that the pump has to work against to move the waste uphill.
- It is directly proportional to the density of the fluid and the height of the fluid column above the point of measurement, which is why in the formula for pressure differential, all these factors are crucial.
By understanding hydrostatic pressure, we gain insight into the pump's requirements to overcome gravitational forces and maintain effective waste management systems.