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# Pump Power Question 5

## Pump Power Question

(OP)

I have a question that I am stuck, something that's confusing me.

Looking at pump power needed per the first law, it is directly related to GPM so if the flow is reduced by 50% then brake HP is reduced by 50%, but per the second law , it is to the power of 3 . Not sure what I am missing. Any thoughts? Thank you for taking a look

1. Pump Power Law
Brake HP= ((GPM) x (Delta-H) x SG)) / (( 3960 X eff))

2. Pump Affinity Laws
bhp2= bhp1(Q2/Q1) ^3

### RE: Pump Power Question

If the flow is reduced by 50% then brake HP is reduced by 50% if the head is constant

Reducing flow for the same/similar pump (affinity laws) will increase the head.

### RE: Pump Power Question

Two separate ideas there.
Power is what power is.

The other (affinity law calculation)is actually comparing the power used by one pump running at speed 1, or speed 2, which will also result in two different corresponding flow rates, Q1 and Q2, and two different heads H1 and H2.

Flow is directly proportional to rpm speeds, but head is proportional to the rpm speed ^2, thus power at two different speeds, multiplying Flow by head then becomes a cube of speed.

--Einstein gave the same test to students every year. When asked why he would do something like that, "Because the answers had changed."

### RE: Pump Power Question

(OP)
Thanks guys for the input..

### RE: Pump Power Question

Beware that the devil is in the detail and BHP in both circumstances isn't as easy as that. Efficiency has a big input into the BHP and that can change if you change something like speed or flow by substantial amounts

Remember - More details = better answers
Also: If you get a response it's polite to respond to it.

### RE: Pump Power Question

2
Why are you looking at these inaccurate "rules of thumb"?

What happened to the performance curves for your particular pump and impeller ?

.... and, if you answer "they were lost" or "client has discarded all plant records" ... then, explain why you cannot find the original pump vendor, guess at the impeller diameter (based on your actual flow rate) and determine what you need from a similar sized pump !!!

Centrifugal pumps are not precision equipment .... how close do you ave to be with your estimate anyway ??

MJCronin
Sr. Process Engineer

### RE: Pump Power Question

(OP)

MJCronin, this was a question in a sample PE exam. I did it by the first equation to arrive at pump power when I looked at answer they have used affinity laws. So I got confused. But yes in reality we can do the approach what you are suggesting

### RE: Pump Power Question

I'll chime in to review my understanding of the Affinity laws:

#### Quote:

2. Pump Affinity Laws
bhp2= bhp1(Q2/Q1) ^3

I didn't recognize it until MJCronin pointed it out, but even if we substitute FHP, that is not necessarily one of the affinity laws (depends on context)
https://en.wikipedia.org/wiki/Affinity_laws

Roughly speaking to my understanding the affinity laws can help us map one curve to a new curve if the speed changes on a given impeller, or the diameter changes on geometrically similar impellers

For speed change of a given impeller: Q2/Q1 = N2/N1; DP2/DP1 = (N2/N1)^2; FHP2/FHP1 = (N2/N1)^3
They are a self-consistent set of relations only if we use FHP (since FHP=Q*DP). If you use BHP then efficiency has to show up and that will undoubtedly be harder to analyse.

IF we were building a curve at speed 2 from a curve at speed 1 then we could map point (q1,dp1) to point (q2,dp2) using dp2 = dp1*(N2/N1)^2. And we'd have to repeat for many points to build the new curve. (I am curious if that was the context of the question).

Next question I ask myself: how accurate are the affinity laws when used as above?

I'll quote myself only because that's where I know how to find it (not because I'm a pump expert.... I am clearly not that). As shown in my post within thread thread407-477916: Affinity law on open type impeller , the affinity laws can be derived from dimensional analysis with only 2 assumptions:

#### Quote (electricpete)

Assumption 1 - Compare only geometrically similar pump designs so L1/L2, L1/L3 are constant (and of course dimensionless) => these can dimensionless constants can be dropped from the RHS as independent variables (they will be absorbed into the function definition).
Assumption 2 - Assume turblent flow. The viscosity term would end up non-dimensionalizing to something like a Reynold’s number which will be a constant friction factor IF highly turbulent flow. As long as we have only turbulent with inertia forces much larger than viscuous forces, then we can neglect viscous forces and drop viscosity from the RHS as a variable (the dimensionless constant friction factor will be absorbed into the function definition)
... accordingly I presume the extent to which affinity laws are accurate depends upon the extent to which those two assumptions are met.

To my thinking assumption 2 is more likely to be violated when mapping points at lower flow rate (at some point very far to the left of curve, as flow rate approaches zero eventually everything becomes laminar and assumption 2 is violated). Also recirculation flow through small channels around wear rings is more likely to be laminar so I think pump stages that are low flow, high dp (per stage) likely deviate from assumption 2 more. Also an oil pump deviates more than a water pump (everything else being equal) due to the higher viscosity of the pumped fluid. Of everything I've said, the stuff in this last paragraph is what I'm least sure of. (I'd welcome any corrections or clarifications if I've said something wrong).

EDIT - I notice the relationships for varying diameter is different in wiki than what I derived. First I want to outline there should imo be two different relationships corresopnding to two different contexts:

context 1: for point-by-point mapping of curves for geometrically similar impellers at constant density and speed
relationship 1: from my derivation: (thread407-477916: Affinity law on open type impeller)
• Q~D^3
• DP~D^2
• i.e. the point [Q1,DP1] maps to [Q2,DP2]=[(D2/D1)^3*Q1,(D2/D1)^2*DP1]
context 2: for point-by-point mapping of curves for minor trimming of a given impeller at constant density and speed (not geometric scaling of similar impellers).
relationship 2: from wiki: (https://en.wikipedia.org/wiki/Affinity_laws)
• Q~D
• DP~D^2
• i.e. the point [Q1,DP1] maps to [Q2,DP2]=[(D2/D1)*Q1,(D2/D1)^2*DP1]
I claim that relationship 1 belongs with context 1 and relationship 2 belongs with context 2. But Wiki seems to imply relationship 2 applies to both contexts. Specifically wiki lists relationship 2 twice and the first time refers to it as if it was derived from Buckingham Pi theorem for dimensional analysis(which it is not). Invoking that theorem would only be appropriate (imo) when comparing geometrically similar impellers (which is not what's going on in context 2). So I think wiki is misleading on this. Then again I may be totally missing something. I'm open to comment.

### RE: Pump Power Question

Ive always heard that impeller variations should not exceed 10% diameter.
With all the square and cubed exponents floating around, things can guickly run off the track.

--Einstein gave the same test to students every year. When asked why he would do something like that, "Because the answers had changed."

### RE: Pump Power Question

It is a little baffling to me that wiki presents relationship 2 as if it is derived from dimensional analysis. Indeed I see a lot of references that list relationship 1 and call it an "affinity law" which at one time I assumed was equivalent to geometric similarity scaling, but there's that ugly word again (assume) and indeed by the end of the post I'll confess that is an incorrect assumption.

"Centrifugal Pumps: Design and Application" by Lobanoff and Ross has a lot of related info that I need to digest:
• First they have a section on "affinity laws" which match the relationship 2 but they don't exactly clarify the context.
• Second they have a section "correction for impeller trim". It gives a 2-step process for figuring out how much to trim an impeller: First step is to calculate impeller size ratio from affinity laws (relationship 2). Second step is to apply a correction from Figure 2-6. The correction factor is close to 1 when very little is trimmed and drops farther below 1 as you trim more and more (which matches what you said that the relationship becomes less accurate with more trimming). It was also mentioned pumps become inefficient if too much is trimmed.
• Third they have a section on "model law". Seems to be a limited subset of theory of comparison of geometrically similar pumps for an oddly-specific purpose (comparing geometrically similar pumps at same dp... select speed ratio to make velocities the same in corresponding sections).
• Fourth (finally!) they give what they call "Factoring law" and it matches relationship 1 for context 1 (geometric scaling). They do caution specific speed generally shouldn't change more than 10%, which makes sense since a given geometric design is optimized toward a given specific speed. They don't mention any limit on the scaling factor but I think (as long as those two assumptions are met) it can be applied to a large scaling factor because I think I remember hearing about 25% scale model of pumps being constructed back in the old days before computational fluid dynamics was a thing.
So one lesson for me is to be careful of exactly what is meant by the phrase "affinity law". Apparently (according to Lobanoff above) it is not the same as scaling of geometrically similar impellers (which he calls "factoring law"). I'm still not 100% clear on the purpose of the affinity law with a diameter ratio (relationship 2) but I think it's primarily for making predictions about trimming (because that's the way Lobanoff uses it, albeit with an additional correction factor). To me, it seems illogical that they lump together the dimensionally-derived speed relationships together with the more empirical/fuzzy trimming relationships (relationship 2) under the common name "affinity laws". (and if you're talking about that relationship 2 you shouldn't imply it comes from dimensional analysis like wiki did). It would make more sense to me to lump together the speed relationships together with the relationship 1 for scaling geometrically-similar impellers, since both of those fall out of the same dimensional analysis together.

### RE: Pump Power Question

KSB uses the phrase "affinity laws" to describe relationship 1 above. https://www.ksb.com/en-global/centrifugal-pump-lex...

Wiki uses the phrase "affinity laws" to describe relationship 2 above https://en.wikipedia.org/wiki/Affinity_laws

I can find a couple more examples of usage in both ways. So it seems to be an ambiguous phrase.

Based on the dictionary definition of the English word "affinity", I would assume the phrase was originally intended to describe context 1...

But my assumption doesn't mean a whole lot. I'm going to steer clear of that phrase and look for further clarification of either the context or the intended relationship whenever I see that phrase.

### RE: Pump Power Question

2
My opinion is that diameter change should NOT be mentioned in the same sentence with affinity relationships. The concepts behind each one is entirely different.

--Einstein gave the same test to students every year. When asked why he would do something like that, "Because the answers had changed."

### RE: Pump Power Question

Thanks 1503. It's good to know I'm on roughly the same page as you on that aspect.

### RE: Pump Power Question

"Centrifugal and Axial Flow Pumps" (Stephanoff) 2nd edition 1957.

#### Quote ("Centrifugal and Axial Flow Pumps" (Stephanoff) 2nd edition 1957.)

Specific Head, hs = gH/(n^2 D^2)
Input energy per unit mass per revolution and with an impeller diameter of 1 ft.
It remains constant for all similar impellers.

Affinity laws follow this property of the specific head: for a given diameter D, head varies directly as speed^2. If n is kept constant, then head varies directly as the Diameter^2.

is Ψ = -H/(u^2/g) = gH/π^2

For the flow to be dynamically similar while speed, size and viscosity are varied, it is necessary that all three criteria remain constant. In a practical sense it is impossible to to comply to that requirement. If the viscosity is kept constant, and speed and size are varied, Reynolds number will vary.

The Rn in various parts of the pump is not well understood, thus all affinity relationships will have deviations, but Rn affects similarity of flow only so far as skin friction and velocity distribution are concerned, which is small in good pumps.
Accuracy is sufficient for practical purposes.

For varying impeller diameters, it comes down to specific speed, which should remain the same.
That takes me away from the pure affinity laws. You can ascribe an affinity relationship to diameter variation, but just because you can do it, doesn't make D part and parcel of the law itself. As Long as specific speed doesn't vary too much, you can get away with it.

--Einstein gave the same test to students every year. When asked why he would do something like that, "Because the answers had changed."

### RE: Pump Power Question

Pump Handbook (Mcgraw Hill) 2nd edition

Diameter reduction.
Exit velocity triangles must be similar.
Apply only to an altered impeller, not to geometricly similar impellers.
Diameter reductions of more than 10 to 20% are rarely made in practice. (There is little experience of larger reductions).
N x D should remain constant, as not to severely affect efficiency.
For Mixed flow impeller, Stephanoff recommended calculations based on average of inside and outside diameter.

--Einstein gave the same test to students every year. When asked why he would do something like that, "Because the answers had changed."

### RE: Pump Power Question

All good stuff above. I think there are way too many assumptions made when using the Affinity Law, and we all know what happens when we assume. Lol! I think the biggest use for the Affinity Law is to make people think they are saving energy by purchasing something that varies the pumps speed, which is not true. No one ever seems to add back in the loss of efficiency for the pump/motor at reduced speeds or the actual energy used by the VFD itself. Most people do not even understand that head is lost by the square of the speed. They just see that third part about reducing energy used by the cube of the speed and nothing else seems to matter.

### RE: Pump Power Question

"Before we begin, understand that the definition of the word law is pretty strong. In its simplest form, the term law means a principle that has been proven to be true for all cases. As we will learn, the affinity laws have limitations. Maybe a better term would be Affinity Guides or Affinity Postulates;

In the real world, physical systems operate with inherent losses. What goes in does not necessarily come out. Efficiency is a measure or indication of the amount of loss. The term entropy is used to define unavailable or lost energy; entropy is ever increasing.

Lines of constant efficiency are plotted on manufacturer’s pump performance curves as superimposed incomplete parabolic arcs generally opening in the positive direction. An rough estimate of the efficiency value using these plots is sometimes the best that can be accomplished from the pump performance curve."

### RE: Pump Power Question

I'll just mention that 1503-44 is using the phrase affinity laws to refer to geometric scaling (as we agreed was "logical") and Pierre's link is using the same phrase to refer to impeller trimming. Myself I'm going to try avoid that, maybe I'll just call them the geometric scaling laws and the impeller trimming laws. (*)

1503 points out it is misleading to refer to geometric scaling laws in a form Q~D^3, DP~D^2 since there usually would be speed change involved. So a better form would be Q~D^3*N, DP~D^2*N^2 (in both cases assuming constant rho).

Realistically I guess most users won't be involved in geometric scaling laws unless they are selecting from an offerred family of pumps, in which case hopefully there is some manufacturer's data to look at. It seems more the domain of interest for pump manufacturers (at least for order of magnitude estimates,.. I know they have a lot more sophisticated ways to analyze things)

Pump trimming is probably a more common thing that users would be involved with. So I can see why the pump trimming laws are more commonly presented to users along with the speed dependence laws for a given impeller since those are the 2 things users might change in the field.

The speed dependence laws are derived alongside the geometric scaling laws (they're all one relationship as mentioned). The pump trimming laws derivation is a little more mysterious to me but I now gather from 1503 that assumptions about pump velocity triangle play a role in that estimate. There are no doubt a whole lot more variables involved in pump trimming than just diameter, like filing to adjust the angle, and the type of pump.

> N x D should remain constant, as not to severely affect efficiency

In our plant most of the large pumps are fixed speed. I know we have done some limited impeller trimming on some of those fixed speed pumps where the pump capacity was too high and destaging was not an option. I guess the devil is in the details though, I don't know many of the details.

The document from Pierre presents mostly what we know but reinforces one particular usage of the term affinity laws to include the pump trimming relationships. Revisiting my comments regarding Lobanoff I had tentatively concluded that what he calls "affinity laws" (relationship 2) was applicable to pump trimming and the correction factor was a refinement of that. The subsequent discussion and the attachment help convince me that is the case (I don't have to be tentative about that any more).

(*) Valvecrazy - Just saw your last post. Sorry for (ab)using that word law. I agree law brings to mind things like Newton's laws, first law of thermodynamics. It conveys more exactness and certainty than might be appropriate. Heck, maybe even Murphy's law carries a higher degree of certainty then some of this stuff!

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