a centrifugal pump pumps lubricating oil at 60F whiel rotating at 1200 rpm. a scale model that is 1/3 the impeller diameter is used to model this pump. At what speed should the scale model run for similitude
This text is generated by AI trained by Professor Brian J. Kirby on his lectures, videos, notes, message board responses, and textbook
PK Bot’s answer:
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PURPOSE OF THE EXERCISE:
Demonstrate the application of pump similitude via the affinity laws to determine the operating speed required on a scaled pump model, ensuring that the nondimensional parameters remain consistent. -
ENGINEERING CONTEXT:
This problem involves predicting the prototype pump performance using a scale model. Once mastered, you are able to design and test pumps at different scales and then extrapolate the results to full‐scale machines. -
PHYSICAL CONTEXT:
The dominant effect is the matching of the characteristic velocity (or tip speed) so that the flow dynamics are geometrically and dynamically similar. -
ASSUMPTIONS:
Assume that the flow remains dynamically similar between the model and prototype when the tip speeds are equated and that viscous effects or fluid property variations (beyond those due to the 60°F lubricating oil) are negligible; this is reasonable provided the Reynolds numbers are high enough. -
DRAWINGS:
A schematic drawing showing the prototype pump with impeller diameter D and the scale model with impeller diameter D/3, with arrows indicating rotational speeds and tip velocities, would help visualize the similarity condition. However, the problem can be solved without an explicit drawing. -
SOLUTION TECHNIQUE:
For geometrically similar centrifugal pumps to operate dynamically similarly, the tip speeds must be equal, i.e.,
\omega_{\text{prototype}}\,D = \omega_{\text{model}}\,\left(\frac{D}{3}\right)\,.
Rearrange to solve for the model speed:
\omega_{\text{model}} = 3\,\omega_{\text{prototype}}\,.
Substitute the prototype speed, where 1200\,\text{rpm} is given:
\omega_{\text{model}} = 3\times1200\,\text{rpm} = 3600\,\text{rpm}\,.
Thus, to maintain dynamic similitude, the scale model should run at 3600\,\text{rpm}.
- REFLECTION:
Matching the tip speed ensures that the non-dimensional performance parameters, such as the flow coefficient and head coefficient, remain the same between the prototype and scale model. This approach is fundamental in pump scaling and helps predict full-scale performance accurately using model data.
In formulating this response, I consulted the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2), and I have ranked the sections according to their relevance. If they are marked as relevant, I encourage you to explore them:
Textbook sections:
- Chapter 2.1: Turbomachines Part 3 (HIGHLY RELEVANT)
- Chapter 6.6: Pump selection (HIGHLY RELEVANT)
- Chapter 2.3: Selecting a Turbomachine Class (HIGHLY RELEVANT)
- Chapter 2.5: Combining and Resizing Turbomachines Part 2 (HIGHLY RELEVANT)
- Chapter 9.6: Centrifugal Pump Design Part 5 (HIGHLY RELEVANT)
Disclaimer:
See What is Professor Kirby Bot for information about Professor Kirby Bot and its answers.
PK Bot: what is the nondimensional parameter that you are keeping fixed when you give this answer? and is that the only nondimensional parameter that is relevant and a function of speed and diameter?
This text is generated by AI trained by Professor Brian J. Kirby on his lectures, videos, notes, message board responses, and textbook
PK Bot’s answer:
Below is a discussion addressing the nondimensional parameter that was held fixed and its relation to other nondimensional parameters.
In scaling the pump model we equated the impeller tip speeds, that is, we kept constant the characteristic velocity scale given by
\omega_{\text{prototype}}\,D_{\text{prototype}} = \omega_{\text{model}}\,D_{\text{model}}\,.
This condition is equivalent to holding fixed the velocity scale in the system. In nondimensional terms, this can be associated with preserving the flow coefficient
C_Q=\frac{Q}{\omega\,D^3}\,,
and the head coefficient
C_H=\frac{\Delta p}{\rho\,\omega^2\,D^2}\,.
By matching the tip speed the assumption is that the pump’s performance at the full scale and the model become dynamically similar; that is, the nondimensional groups governing the flow (at least the ones directly affected by \omega and D) remain invariant.
However, the tip-speed (or equivalently the invariance of C_Q and C_H) is not the only nondimensional parameter that is relevant. In pump similitude the complete set of nondimensional parameters includes:
• The flow coefficient (C_Q),
• The head coefficient (C_H), and
• The power coefficient (C_P=\dfrac{P}{\rho\,\omega^3\,D^5}).
Additionally, in many pump scaling problems one considers the Reynolds number. In our analysis for pumps operating at relatively high Reynolds numbers, viscous effects are minor, so the Reynolds number is assumed sufficiently high and does not directly enter the similarity requirement.
Thus, while matching the tip speed (or equivalently fixing C_Q and C_H) is the key requirement in this problem, it is not the only nondimensional parameter that depends on speed and diameter. The full similarity requires that other coefficients such as C_P remain invariant too, ensuring that the overall performance characteristics, including efficiency, are maintained between the full‐scale pump and the scale model.
In formulating this response, I consulted the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2), and I have ranked the sections according to their relevance. If they are marked as relevant, I encourage you to explore them:
Textbook sections:
- Chapter 2.5: Combining and Resizing Turbomachines Part 2 (HIGHLY RELEVANT)
- Chapter 2.2: Nondimensional Turbomachine Parameters Part 2 (HIGHLY RELEVANT)
- Chapter 6.6: Pump selection XXXX NOT SURE WHAT INTRO IS NEEDED HERE. EASY TO TYPE BUT I MIGHT HAVE I… (HIGHLY RELEVANT)
- Chapter 2.1: Turbomachines Part 3 (HIGHLY RELEVANT)
- Chapter 2.3: Selecting a Turbomachine Class (HIGHLY RELEVANT)
Disclaimer:
See What is Professor Kirby Bot for information about Professor Kirby Bot and its answers.
but PK Bot, to hold the Reynolds number fixed, you would hold \omega D^2 fixed. Holding \omega D fixed holds the impeller tip speed fixed but not C_H or C_Q. Why would you hold the impeller tip speed fixed? Is there any performance parameter of a pump that is described by the tip speed only?
This text is generated by AI trained by Professor Brian J. Kirby on his lectures, videos, notes, message board responses, and textbook
PK Bot’s answer:
Below is a discussion of why we might choose to hold the tip speed (that is, fix \omega D) in many pump scaling exercises, and what that implies for performance parameters.
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When scaling pumps for high‐Reynolds number flow, we rely on the fact that the dominant physics are inertial so that similarity can be enforced by matching the nondimensional coefficients (such as C_Q, C_H, and C_P). In many cases, the viscous (Reynolds number) effects are secondary. That lets us assume that if the impeller’s peripheral (tip) speed is the same in the model and the prototype, the inertial characteristics of the flow (e.g. the centrifugal acceleration) remain similar.
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It is true that holding \omega D^2 fixed would more directly keep the Reynolds number constant for viscous effects. However, in our analysis of pumps operating in the turbulent regime, viscous losses are nearly negligible. Under such conditions, the pump performance is governed primarily by inertial forces. Matching tip speed is then a practical and robust way to ensure that the key nondimensional design groups (e.g. C_Q=\dfrac{Q}{\omega D^3} and C_H=\dfrac{\Delta p}{\rho\,\omega^2D^2}) remain similar—provided that the operating conditions yield high Reynolds numbers.
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Concerning a performance parameter that depends solely on tip speed: no fundamental performance coefficient of a pump is prescribed only by \omega D. Instead, the tip speed sets the scale for the inertial forces experienced by the fluid as it moves through the impeller. These inertial effects strongly influence losses related to flow separation and recirculation. That, in turn, affects the head and efficiency through the nondimensional coefficients. Matching tip speed is not meant to produce a one-to-one equivalence with a single performance parameter; rather, it is a convenient condition that helps achieve similar C_Q, C_H, and C_P values across scales in the inertial regime.
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In summary, while holding \omega D^2 fixed would be necessary if viscous effects (and hence Reynolds number) were significant, for many pump applications the assumption of high Reynolds numbers permits us to target tip speed equality. This approach indirectly preserves the performance-determining nondimensional groups even though no single performance parameter is described exclusively by \omega D.
In formulating this response, I consulted the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2), and I have ranked the sections based on their relevance. If they are marked as relevant, I encourage you to explore them:
Textbook sections:
- Chapter 2.5: Combining and Resizing Turbomachines Part 2 (HIGHLY RELEVANT)
- Chapter 2.2: Nondimensional Turbomachine Parameters Part 2 (HIGHLY RELEVANT)
- Chapter 6.6: Pump selection XXXX NOT SURE WHAT INTRO IS NEEDED HERE. EASY TO TYPE BUT I MIGHT HAVE I… (HIGHLY RELEVANT)
- Chapter 2.1: Turbomachines Part 3 (HIGHLY RELEVANT)
- Chapter 2.2: Nondimensional Turbomachine Parameters Part 1 (HIGHLY RELEVANT)
Disclaimer:
See What is Professor Kirby Bot for information about Professor Kirby Bot and its answers.
I agree with some of the things that you say PK Bot, but not others.
Your argument for why \omega D is the right parameter to use to describe inertial effects is spurious. Your arguments for how design choices enforce matching of C_Q, C_H, and C_P are mistaking cause and effect.
We have to separate our nondimensionalization between things that are pump design and things that are pump operation and things that define the pumps operating point.
The pump design specifies the geometry, i.e. the set of length ratios and/or angles that define the shape.
Given a working fluid (\rho, \mu). The pump’s operation (independent of what fluid and piping system it drives) is defined by \omega and D. Given a design, we build the pump to a specific size defined by D, and we spin it at a speed \omega. Those parameters plus the fluid parameters define the Reynolds number. The pump surfaces and D combine to define \epsilon/D.
The pumps operating point describes the interaction between the pump and the system it is driving – that encompasses Q, \Delta p, \eta, \omega_p, D_p, C_H, C_Q, C_P.
The pump scaling relations tell us how things scale if we change \omega and D and C_H, C_Q, C_P don’t change, which equivalently means \eta, \omega_p, and D_p don’t change. The problem is that if you just spin a pump faster or build a homologous pump that is bigger, it does not necessarily lead to an operating condition in which C_H, C_Q, C_P, \eta, \omega_p, and D_p didn’t change.
So we cannot simply propose to change \omega and D and predict what the outcome will be. We need more information to make such a claim. And the key additional information is that a homologous pump’s performance coefficients, specific speed, specific diameter, and efficiency at BEP tends to be roughly the same independent of \omega or D. So if we change \omega and D and drive a system that happens to be its best efficiency point then we will get performance that is explained by the pump scaling relations. So the pump scaling relations tell us how, if a pump is well designed to drive one flow condition, we could make it well designed to drive another flow condition.
So. I basically don’t like this question, because I think it kind of misses the point.
Given a pump, if you scale it up, the only similitude you can satisfy if you are talking about the pump and not a pump combined with a system that it is pumping is to keep the geometry homologous, keep \mathrm{Re} fixed, and keep \epsilon/D fixed.
So, if you tell me you made the impeller 1/3 the size, the whole pump should all be 1/3 the size, and the surface roughness of the material should be 1/3 the size, and to match \mathrm{Re} you should spin it 9 times as fast.
However, pump operation is not a strong function of \mathrm{Re} or \epsilon/D, so that dynamic similitude is a thing you can calculate but it isn’t really very important.
The similitude that is important is to match specific speed. However, matching specific speed is only a thing you can do if you are talking about \omega and D in the context of Q and \Delta p, and this problem gives no information about the operating point.
so to summarize. given the pump alone, you can enforce similitude of geometry (important) and \epsilon/D (not important for most dynamic pumps). Given a pump and a working fluid, you can enforce similitude of \mathrm{Re} (not important for most dynamic pumps). Given an operating point with Q and \Delta p, you can enforce similitude of \omega_p (important). So I think this question is answerable but its context is not important.
another way of thinking about the relative importance of \mathrm{Re} similitude and \omega_p similitude is as follows:
if your \omega_p is well chosen and you are at BEP, the fluid moves onto and off of the impeller blade smoothly without losses. That is a big deal because separation at the leading or trailing edge ruins pump performance.
Your \mathrm{Re} mostly tells you how much viscous dissipation you get that is unavoidable – the dissipation in the boundary layers at the impeller blades. Dissipation wanes in importance as \mathrm{Re} gets bigger. You notice Reynolds number differences only if your specific speed is at optimum.
Specific speed differences explain the difference between a pump at BEP (say 90% efficiency) and a pump way off BEP (say 35% efficiency).
Reynolds number differences tend to explain why a bigger version of a pump has 92% effiency at BEP when a smaller one has 90% efficiency at BEP.