Could you tell me how changing the number of stages in a pump affects the system, both theoretically and mathematically? And can you tell me what equations to use when there is a multi stage pump? And how those formulas relate to efficiency?
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PK Bot’s answer:
Below is a discussion of how adding stages to a pump affects the system, along with the key ideas and equations that you can use.
• In a single-stage pump the impeller (or rotor) produces a change in total pressure (or head). For an ideal, homogeneous design, if one stage produces a head H_1 then in an ideal multi‐stage pump (with little or no interstage losses) the total head is approximately
H_{\mathrm{total}} = m\,H_1\ ,
where m is the number of stages in series. Here we are assuming implicitly that the stages are in series. If you connect n turbomachines in parallel, you don’t call it a multistage pump, you call it an array of pumps or a bank of pumps or something like that.
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The volumetric flow rate Q, assuming that each stage is designed identically, is the same by conservation of mass as long as the pumped fluid is roughly incompressible e.g., a liquid. If the multistage pump is pumping gas, then the mass flow \dot{m} is unchanged from stage to stage but the volumetric flow rate Q will change.
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In practice, each extra stage introduces additional losses but I won’t consider that here.
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the key thing that happens when the pump is multistage is that because the head or total-pressure change is divided amongst m stages, each stage has an effective specific speed that is higher (or equivalently, an effective specific diameter that is lower). So each stage is faster than the whole, or equivalently you can make a multistage pump that is slower than you could with a single-stage pump.
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for gases, multistage axial flow pumps are the norm because they are much easier to build. for liquids, if you are making a multistage pump it is almost certainly a radial flow pump geometry, because you are trying to get to a low specific speed and radial flow geometries are the way to do that.
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putting multiple stages moves the system to a lower-speed range, or equivalently says that each stage can operate at a higher specific speed than a single-stage pump would to get the the same system specific speed. I fyou are pumping an approximately incompressible fluid (i.e., liquid) you can write an expression for the specific speed of the system as
\omega_p=\frac{\omega (nQ)^{1/2} \rho ^{3/4}}{(m\Delta p_0)^{3/4}}
in contrast to the specific speed of each stage as
\omega_p=\frac{\omega Q^{1/2} \rho ^{3/4}}{\Delta p_0^{3/4}}
So if you have a centrifugal pump that works well at a specific speed of 1 you could put two in series, and the multistage system would operate at similar efficiency with a system output of 2^{-3/4}\simeq 0.6. By connecting stages in series, you have made the pumping system better at generating higher total dynamic head at the same flowrate, and you quantify that through specific speed.
In crafting this response, I dedicated some time to reviewing the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2), and I found these sections to be particularly helpful in formulating my answer. I encourage you to explore the pertinent ones directly:
Textbook sections:
- Chapter 6.3: Pump types (RELEVANT)
- Chapter 2.1: Turbomachines Part 2 (RELEVANT)
- Chapter 2.3: Selecting a Turbomachine Class (RELEVANT)
- Chapter 2.5: Combining and Resizing Turbomachines (RELEVANT)
- Chapter 6.1: Pumps (RELEVANT)
- Chapter 6 Fundamentals: Combining the language of fluid mechanics with the language of hydraulics en… (RELEVANT)
- Chapter 2.2: Nondimensional Turbomachine Parameters (MARGINALLY RELEVANT)
- Chapter 2.7: Limitations (MARGINALLY RELEVANT)
- Chapter 9.6: Centrifugal Pump Design Part 1 (MARGINALLY RELEVANT)
- Chapter 6.6: Models (MARGINALLY RELEVANT)
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