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Chapter 7: Problem 36

The following dimensionless groups are often used to present data on centrifugal pumps: flow coefficient \(\varphi=\frac{Q}{\omega D^{3}},\) head coefficient \(\psi=\frac{g H}{\omega^{2} D^{2}},\) power coefficient \(\xi=\frac{\dot{W}}{\omega^{3} D^{5}},\) efficiency \(r_{i}=\) \frac{\rho g Q H}{\dot{W}}, \text { specific speed } N_{s}=\frac{\omega \sqrt{Q}}{(g h)^{3 / 4}}, \text { specific diameter } D_{s}=\frac{\omega \sqrt{Q}}{(g h)^{1 / 4}} Show that the last three groups can be formed from combinations of the first three groups.

Short Answer

Expert verified
The efficiency \( r_{i} \) can be expressed as \( \frac{\varphi \psi}{\xi} \). The specific speed \( N_{s} \) can be shown to be \( \frac{\varphi^{1/2}}{\psi^{3/4}} \) and the specific diameter \( D_{s} \) can be shown to be \( \frac{\varphi^{1/2}}{\psi^{1/4}} \). Hence, the last three groups can indeed be formed from combinations of the first three groups.

Step by step solution

01

Express the Efficiency

Begin by expressing the efficiency (\( r_{i} \)) in terms of the flow coefficient (\( \varphi \)), head coefficient (\( \psi \)), and power coefficient (\( \xi \)) as follows: \( r_{i} = \frac{\varphi \psi}{\xi} \). The result is obtained from the relation \( r_{i} = \frac{\rho g Q H}{\dot{W}} = \frac{\varphi \psi}{\xi} \).
02

Express the Specific Speed

Next, express the Specific Speed (\( N_{s} \)) in terms of the flow coefficient, head coefficient, and power coefficient. Following the given relation \( N_{s} = \frac{\omega \sqrt{Q}}{(g H)^{3 / 4}} \), we find that \( N_{s} = \frac{\varphi^{1/2}}{\psi^{3/4}} \). The result is \( N_{s} = \frac{\varphi^{1/2}}{\psi^{3/4}} \).
03

Express the Specific Diameter

Finally, express the Specific Diameter (\( D_{s} \)) in terms of the flow coefficient, head coefficient, and power coefficient. Taking a look at the given equation \( D_{s} = \frac{\omega \sqrt{Q}}{(g H)^{1 / 4}} \), we derive the relation \( D_{s} = \frac{\varphi^{1/2}}{\psi^{1/4}} \). The result is \( D_{s} = \frac{\varphi^{1/2}}{\psi^{1/4}} \).

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Most popular questions from this chapter

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