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Chapter 4: Problem 53

The sump pump has a net mass of \(310 \mathrm{kg}\) and pumps fresh water against a 6 -m head at the rate of \(0.125 \mathrm{m}^{3} / \mathrm{s}\). Determine the vertical force \(R\) between the supporting base and the pump flange at \(A\) during operation. The mass of water in the pump may be taken as the equivalent of a 200 -mm-diameter column \(6 \mathrm{m}\) in height.

Short Answer

Expert verified
R is the sum of the weight of the pump and the weight of the water it contains.

Step by step solution

01

Determine the Weight of the Pump

First, calculate the weight of the pump by using the formula: \[ W_{\text{pump}} = m_{\text{pump}} \cdot g \]where\( m_{\text{pump}} = 310 \text{ kg} \) is the mass of the pump, and \( g = 9.81 \text{ m/s}^2 \) is the acceleration due to gravity.
02

Calculate the Volume of Water in the Pump

The pump contains a column of water with a diameter of \( 0.2 \text{ m} \) and a height of \( 6 \text{ m} \). The volume of the water column is given by:\[ V = \pi \left( \frac{d}{2} \right)^2 \cdot h \]where\( d = 0.2 \text{ m} \) and \( h = 6 \text{ m} \).Calculate \( V \) using these values.
03

Calculate the Mass of the Water in the Pump

Using the density of water \( \rho = 1000 \text{ kg/m}^3 \), the mass of the water column can be found by:\[ m_{\text{water}} = \rho \cdot V \]Use the volume calculated in Step 2 to find \( m_{\text{water}} \).
04

Determine the Weight of the Water in the Pump

Calculate the weight of the column of water using:\[ W_{\text{water}} = m_{\text{water}} \cdot g \]where \( g = 9.81 \text{ m/s}^2 \).
05

Calculate the Total Force Exerted by the Pump System

The total force \( R \) is the sum of the weight of the pump and the weight of the water being handled by the pump. Thus,\[ R = W_{\text{pump}} + W_{\text{water}} \]Substitute the values obtained from previous steps to find \( R \).

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Key Concepts

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

Vertical Force Calculation
Understanding vertical force is crucial when analyzing the dynamics of a pump system. It involves determining how much force is exerted by the pump on its supporting structure, such as a base or flange. The vertical force, in this context, is calculated by adding up all the weights acting downwards on the system.
This includes the weight of the pump itself and the weight of the column of water being pumped. By summing these forces, we determine the total vertical force, denoted as \( R \). In essence, it is a measure of the load exerted by the entire system on its support. Calculating \( R \) helps ensure the structural integrity and proper functioning of the pump system.
Weight of the Pump
The weight of the pump is a fundamental component of the total force calculation. It's the force with which the pump alone pushes downwards due to gravity. This weight is calculated using the formula:
  • \( W_{\text{pump}} = m_{\text{pump}} \cdot g \)
where \( m_{\text{pump}} \) is the mass of the pump, and \( g \) is the acceleration due to gravity, typically \( 9.81 \, \text{m/s}^2 \).
In our example, the pump's weight calculation uses a mass of \( 310 \, \text{kg} \). Thus, the weight force exerted by the pump becomes \( m_{\text{pump}} \cdot g \), which is an essential factor to consider in the pump's total load management.
Water Column Mass
To understand the water column mass, visualize the portion of water inside the pump as a solid cylinder. This section accounts for its shape and dimensions. We begin by finding the volume of this water column using:
  • \( V = \pi \left( \frac{d}{2} \right)^2 \cdot h \)
where \( d \) is the diameter and \( h \) is the height. For the given case, \( d = 0.2 \, \text{m} \) and \( h = 6 \, \text{m} \).
Once the volume \( V \) is known, calculate the mass of water using the density \( \rho \) of water, expressed as:
  • \( m_{\text{water}} = \rho \cdot V \)
With \( \rho = 1000 \, \text{kg/m}^3 \), the calculation results in the mass of the water in the pump. This mass contributes to the total weight that affects the vertical force calculation.
Pump System Analysis
A comprehensive analysis of a pump system requires understanding various dynamic factors. Not only is it about calculating weights and forces, but also understanding how the system interacts as a whole. This involves considering elements like:
  • The weight of the pump itself.
  • The mass and weight of the water column.
  • The structural integrity of the base or flange supporting these forces.
By adding the weight of the pump, \( W_{\text{pump}} \), and the weight of the water, \( W_{\text{water}} \), we obtain the total force \( R \) exerted by the system. \( R = W_{\text{pump}} + W_{\text{water}} \). This ensures that the supporting structures can handle this combined load without risk of failure.
Properly analyzing these dynamics helps in designing systems that are efficient, safe, and durable.

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Most popular questions from this chapter

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