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Chapter 8: Problem 34

A pump is to be selected such that it operates at or near its most efficient state when delivering \(100 \mathrm{~L} / \mathrm{s}\) with an added head of \(50 \mathrm{~m}\). A manufacturer has five models of homologous pumps, with specific speeds of \(0.95,0.81,0.61,0.36,\) and \(0.25 .\) Motors with any of the standard-rated speeds can be provided with these pumps. Identify the best choice of specific speed and the rotational speed of the motor that should be used to drive the pump.

Short Answer

Expert verified
Specific speed of 0.61 and rotation speed of 1835 RPM is optimal.

Step by step solution

01

Identify Key Parameters

For the given problem, the two key parameters are the discharge rate of the pump, \(Q = 100 \ \mathrm{L/s} = 0.1 \ \mathrm{m^3/s}\), and the head, \(H = 50 \ \mathrm{m} \). These will be used in the specific speed formula to determine the right pump model.
02

Calculate Specific Speed Formula

The formula to calculate the specific speed \(N_s\) of a pump is given by: \(N_s = \frac{N \sqrt{Q} }{H^{3/4}} \), where \(N\) is the rotational speed in RPM (revolutions per minute), \(Q\) is flow rate in cubic meters per second, and \(H\) is total head in meters.
03

Rearrange Formula for Rotational Speed

Given the specific speed formula and values, we rearrange to solve for \(N\): \[ N = N_s \frac{H^{3/4}}{\sqrt{Q}} \]. Since \(N_s\) are provided and constant for a particular model, calculate \(N\) for each.
04

Calculate Rotational Speed for Each Pump Model

For each specific speed: - For \(N_s = 0.95\): \(N = 0.95 \frac{50^{3/4}}{\sqrt{0.1}}\)- For \(N_s = 0.81\): \(N = 0.81 \frac{50^{3/4}}{\sqrt{0.1}}\)- For \(N_s = 0.61\): \(N = 0.61 \frac{50^{3/4}}{\sqrt{0.1}}\)- For \(N_s = 0.36\): \(N = 0.36 \frac{50^{3/4}}{\sqrt{0.1}}\)- For \(N_s = 0.25\): \(N = 0.25 \frac{50^{3/4}}{\sqrt{0.1}}\)
05

Evaluate Each Result

Calculating each expression:- \(N_s = 0.95\): \(N = 2865\,\mathrm{RPM}\)- \(N_s = 0.81\): \(N = 2440\,\mathrm{RPM}\)- \(N_s = 0.61\): \(N = 1835\,\mathrm{RPM}\)- \(N_s = 0.36\): \(N = 1083\,\mathrm{RPM}\)- \(N_s = 0.25\): \(N = 752\,\mathrm{RPM}\)
06

Select the Most Efficient Choice

The most efficient pump operates closest to medium to high specific speed with reasonable RPM for stability and efficiency. Thus, \(N_s = 0.61\) with \(N = 1835 \ \mathrm{RPM}\) offers a balanced speed between efficiency and mechanical safety.

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Key Concepts

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

Specific Speed
Specific speed is a dimensionless parameter that helps in selecting the best pump for a certain application. It relates the rotational speed, flow rate, and head of the pump. By using specific speed, engineers can compare the efficiency of different pump models irrespective of their size and type.
Specific speed can be calculated using the formula:
  • \(N_s = \frac{N \sqrt{Q} }{H^{3/4}}\)
  • Where \(N\) is the rotational speed in RPM, \(Q\) is the flow rate in cubic meters per second, and \(H\) is the total head in meters.
By comparing specific speeds of available pumps, we can select the pump that will operate most efficiently for the given application's conditions. Specific speed allows us to handle both dynamic behavior and hydraulic efficiency effectively.
Rotational Speed
Rotational speed, often measured in revolutions per minute (RPM), is key in determining the performance of a pump. The speed at which a pump shaft rotates can affect the pump's ability to manage different flow rates and heads effectively. For each specific pump model, the required RPM will vary according to the set specific speed.
For example, using the formula:
  • \(N = N_s \frac{H^{3/4}}{\sqrt{Q}}\)
  • Where \(N_s\) is a given specific speed,
  • \(H\) is the head in meters, and
  • \(Q\) is the discharge rate in cubic meters per second.
This formula helps determine the necessary RPM for any given pump model. Mechanical stability, energy consumption, and wear and tear are influenced by maintaining an appropriate rotational speed.
Discharge Rate
Discharge rate is the measure of the volume of fluid a pump can move in a specific amount of time, often expressed in liters per second or cubic meters per second. This parameter is crucial for pump selection as it dictates the pump's capacity to handle the expected flow demand of the system.
  • A high discharge rate is necessary in systems demanding rapid fluid movement.
  • Conversely, lower discharge rates are suitable for applications requiring precise fluid control.
When selecting a pump, ensure that the discharge rate meets the system's requirement to avoid inefficiencies and potential mechanical stress.
Pump Efficiency
Pump efficiency is a fundamental factor in choosing the right pump. It is defined as the ratio of the useful hydraulic power delivered by the pump to the power supplied to the pump shaft. Efficiency affects how effectively a pump converts input energy into useful work, impacting operating costs and energy consumption.
Factors influencing pump efficiency include:
  • Design and type of pump: Different designs yield varying efficiency levels based on the flow and head characteristics.
  • Operating conditions: Pumps running under optimal conditions generally achieve better efficiency.
  • Mechanical losses: These occur due to friction and other resistances within the pump structure.
For best results, select a pump model that maintains high efficiency within the operating range needed for your application. This ensures less energy is wasted and promotes sustainable operation.

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Most popular questions from this chapter

Water is pumped from a lower reservoir to a higher reservoir. The water surface in the higher reservoir is \(10 \mathrm{~m}\) above the water surface in the lower reservoir. The piping system consists of 200 -mm-diameter ductile iron pipe with a length of \(2 \mathrm{~km}\) and minor losses equal to 6.2 times the velocity head. The pump characteristics are shown in Table 8.3. (a) Determine the expected flow rate through the system, assuming fully turbulent flow. (b) What is the power of the motor required to drive the pump? $$ \begin{array}{l|c|c|c|c|c|c} \text { Discharge (L/s) } & 0 & 10 & 20 & 30 & 40 & 50 \\ \hline \text { Total head (m) } & 25 & 23.2 & 20.8 & 17.0 & 12.4 & 7.3 \\ \hline \text { Efficiency (\%) } & -1 & 45 & 65 & 71 & 65 & 45 \end{array} $$

A Pelton wheel has a diameter of \(5 \mathrm{~m}\) and a bucket angle of \(165^{\circ} .\) The effective head at the nozzle is \(550 \mathrm{~m}\), and the nozzle is set such that the velocity coefficient is 0.94 and the diameter of the jet is \(130 \mathrm{~mm}\). Estimate the optimal flow rate, rotational speed of the wheel, and power output. Assume water at \(20^{\circ} \mathrm{C}\).

A single-jet Pelton wheel hydropower plant is to be operated such that the shaft power developed by the Pelton wheel is equal to \(15 \mathrm{MW}\) when the head just upstream of the nozzle is equal to \(1600 \mathrm{~m}\). The Pelton wheel is constructed such that is has a diameter of \(4 \mathrm{~m}\), has a deflection angle of \(170^{\circ},\) and rotates at a controlled speed of 600 rpm. Determine the appropriate nozzle diameter to be used in the project. Assume water at \(20^{\circ} \mathrm{C}\) and a nozzle loss coefficient of 0.03 .

A pump has an impeller diameter of \(450 \mathrm{~mm}\), and at its most efficient operating point, it delivers water at a flow rate of \(650 \mathrm{~L} / \mathrm{s}\) with an added head of \(9.5 \mathrm{~m}\). The specific speed of the pump is \(1.5,\) and the shaft power delivered by the motor is \(80 \mathrm{~kW}\). (a) Estimate the shutoff head of the pump. (b) Estimate the efficiency of the pump at its best operating point. Assume water at \(20^{\circ} \mathrm{C}\).

The performance of a turbine is being studied using a \(\frac{1}{5}\) -scale model. The prototype (full-scale) turbine operates at a design head of \(35 \mathrm{~m}\) when the flow rate through the turbine is \(64.1 \mathrm{~m}^{3} / \mathrm{s}\) and the angular speed of the runner is \(600 \mathrm{rpm}\). The model is to be tested at a head of \(12 \mathrm{~m}\). (a) What should be the angular speed and flow rate in the model to achieve similarity with the prototype? (b) If the shaft power generated in the model is measured as \(110 \mathrm{~kW}\) and the efficiency in the prototype is assumed to be \(5 \%\) better than the efficiency in the model, estimate the power that is generated in the prototype under design conditions. (c) What is the specific speed of the turbine, and what should be its type? Assume water at \(20^{\circ} \mathrm{C}\).
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