"affinity laws"... for pd pumps ?!? 2

[electricpete]

We always hear the affinity laws for centrifugal pumps:
Given two speeds N1, we have:
Q1/Q2~N1/N2, DP1/DP2 ~(N1/N2)^2, (Power1/Power2)~(N1/N2)^3

I believe there inherent in the above an assumption that the system flow characteristic curve is unchanged and follows a relationship DP~Q^2. Without assumed system there is no basis for drawing any conclusion. (if we are not moving along that curve, then please tell me how we find the two operating points at which the relationships hold).

Now let's look at a positive displacement pump. Assume piston type or gear type: a fixed volume is trapped and moved for every revolution of the shaft. It seems to me very likely that Q~N.

Now if I hook up that pump to that same system with a fixed characteristic curve DP~Q^2, the DP of pump and DP of system must match, I must have that DP~Q^2 ~ N^2.
Now look at fluid power Power~Q*DP~N*N^2~N^3.

Hmmm, looks very familiar. It looks to me like positive displacement pump also follows the affinity laws.

If I'm right, why are the affinity laws taught as applicable to centrifugal pumps, without mention of pd pumps?

If I'm wrong, what was my error?
Thx in advance.


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[25362]

To electricpete.
System curves follow an equation such as H=a+b*Qⁿ, where a, b are constants. Indeed, the most common case is n=2.
Why do you insist in repeating this fact ?
What is the relation between system curves and centrifugal pumps affinity laws that you intend to show or find ?

 

[quark]

Electricpete!

I am sorry, you did say that(The wall is yet to be broken). You said, "the system characteristic is inherent in the pump curves". True if the system resistance is only dynamic. This is not applicable if you have a system with high static discharge head(where system resistance will not change as per the square law). Still, centrifugal pumps behave as per affinity laws. That is why the flow control in recirculated piping system is done by the differential pressure across supply and return headers(here comes the system curve equation H = a+b*Q² as suggested by 25362).

PS: Anyhow I will take a printout of this thread and read it leisurely at home. I would be delighted if you require further clarification.

Regards,


Eng-Tips.com : Solving your problems before you get them.

 

[electricpete]

25632 - My original question related to affinity laws/pd pumps also seeks to gain agreement that the centrifugal pump laws (Q~N, DP~N^2,P~N^3) are based on a premise that the connected system follows the square law. I have been repeatedly challenged on this premise throughout the thread. I feel there is no productive discussion on pd pumps until I can gain a modest agreement on this basic premise regarding centrifugal pump laws.

Do you agree that the centrifugal pump laws (Q~N, DP~N^2,P~N^3) are based on an assumption that the connected system follows the square law?

Quark - you stated "This is not applicable if you have a system with high static discharge head(where system resistance will not change as per the square law). Still, centrifugal pumps behave as per affinity laws."

I believe this is also in conflict with my basic assumption. Centrifugal pump laws assume the pump is within a system following DP~Q^2. If assumption is not met the laws do not apply.

I have utterly failed in communicating what seems to be a basic point. I would like to suggest to forget the physical variables for the moment and consider only that math.
If we assume three variables in our world A, B, C.
Can we ever have A~C and B~C^2 when we don't have A~B^2?

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[electricpete]

Dyslexic correction to last line:
Can we ever have A~C and B~C^2 when we don't have B~A^2?

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[25362]

Mathematically correct only when remembering that A, B, and C are themselves ratios. A₂/A₁=C₂/C₁, B₂/B₁=(C₂/C₁)², then

(A₂/A₁)²=B₂/B₁​

We are actually after A₂, and B₂, given C₂/C₁.

A₂=A₁(C₂/C₁)=A₁(B₂/B₁)^0.5 thus
A₂²=A₁(B₂/B₁), and since A₁=f(B₁),
A₂²=B₂*f'(B₁) or

B₂=A₂²/ f'(B₁)​

It is the factor 1/f'(B₁) that is missing from your otherwise correct presentation.

 

[electricpete]

Hi 25632. Thanks for entertaining my question.

I see you invented a function f. It appears to be a function that maps B (DP) into A (flow) at speed 1.

Then you invent function f'. Since it operates on B1 it can also be presumed to be speed 1? Or is it a derivative. Doesn't make any sense to me. Perhaps you can illuminate.

In the meantime, you seem to take exception to the contention that A~C and B~C^2 implies B~A^2? OK, let's use the ratios:

Given 2 equations:
equation 1 A2=A1(C2/C1) => C2/C1 = A2/A1
equation 2 B2 = B1 * (C2/C1)^2

Substitute the expression for C2/C1 from equation 1 into equation 2:
B2 = B1 * (A2/A1)^2
That sounds like B~A^2 to me. Do you agree?

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[25362]

You aim at B2=K*(A2)² which is right.
All I wanted to say is that K is not a constant and depends on whatever the relation between A1 and B1 for a particular pump.

 

[electricpete]

Thanks, it sounds like we have agreed on a mathematical relationship.

Now let's talk about pumps. Do you agree that the centrifugal pump laws in their ideal form (Q~N, DP~N^2) are applicable only if the pump is connected to a system with characteristic DP~Q^2?

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[25362]

Sometimes on testing a pump, when a calibrated motor of suitable size is unavailable, or there are restrictions in power supply, or limitations in the testing equipment, the test is carried out at a different speed than specified.

In such a case affinity laws -although not foolproof- are accepted and usually produce correct results.

Although the tests are carried out with water on systems where the square law applies, I still think that affinity laws are an intrinsic property of the pumps, whatever the "n" value in the H=f(Qⁿ) equation applies for the system.

 

[electricpete]

"I still think that affinity laws are an intrinsic property of the pumps, whatever the "n" value in the H=f(Qn) equation applies for the system"

I have proved in my 2/4/04 message that the pump laws do NOT predict the correct change in speed when the pump is connected to a non-square law system specifically DP~Q^1.85. Do you disagree with what I have proven?

"Sometimes on testing a pump, when a calibrated motor of suitable size is unavailable, or there are restrictions in power supply, or limitations in the testing equipment, the test is carried out at a different speed than specified."

I agree there is a process for generating alternate pump curve from given pump curve using known change in speed change. I have described it in my comments to Steve on 2/5/04. If we say there is a 1:1 mapping between points on our two curves, that 1:1 mapping falls along system characteristic Q=k*DP^2. In this case on the basis of that mapping (assumption that we are connected to a square-law system), the pump laws relate the two curves. If we do not provide any 1:1 mapping between the curves, there is no mathematical basis to make the claim that Q~N and DP^N^2. It certainly does not apply if we pick a random fixed value of Q and examine the behavior of DP. It certainly does not apply if we pick a random fixed value of DP and examin the value of Q. It certainly does not apply if we examine the behavior along some system charactersistci which does not follow Q~DP^2. It ONLY applies if we pick two points related by Q=k*DP^2. (I consider that the axis intercept Q=0 is a special case where k=0 and the axis intercept DP=0 is a special case where k-> infinity).

So I don't understand what basis you apparently continue to disagree with my simple statement that the centrifugal pump laws in their ideal form (Q~N, DP~N^2) are applicable only if the pump is connected to a system with characteristic DP~Q^2 (square-law system is an assumption inherent in the pump laws).

However, I would like to for the moment imagine that someone out there agrees that a square-law system is an inherent assumption of the pump law.

Now please hook up a positive displacement pump to that same system. Measure flow, dp, power, speed. Now vary speed. We have agreed we can predict the change in speed Q~N. Can we agree (with assumed square-law system) that we can predict the change in DP by DP~N^2?

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[25362]

I vaguely remember that the affinity laws were obtained as a result of dimensional analysis on pump hydraulics unrelated to system characteristics.

I have -of late- seen a graph for a pump of N_s=1250 showing the BEP curve at different speeds of rotation, generated by using the specific speed formula, given the BEP at one speed on a H-Q conventional diagram.

The specific head of the impeller/pump set N_s=N(Q)^0.5/H^0.75, which characterizes a particular geometry and performance of a pump, served to construct the BEP curve.

The H-Q pump curves at different speeds were drawn parallel to themselves cutting the BEP curve at points where H and Q relate as per the affinity laws. Again without referring to a particular system friction curve.

Let's summarize by saying that I am definitely not a pump expert. Experts' opinion on this issue should be heard to confirm or deny the assumptions aired in this thread.

 

[electricpete]

Let me put a different twist on the subject

In order to have operating points vary in accordance with pump laws upon change in speed we need BOTH of the following two prerequisites:
Prerequisite 1 – A specific type of pump curve and it’s dependence on speed
Prerequisite 2 – A system where DP~Q^2.

So far we focused very much on #2. But now if we step back and focus on #1 there are many hypothetical pump curves we could invent (not centrifugal or pd) which would not satisfy the pump laws even when connected to our system DP~Q^2.

For example A assume pump curves were straight lines on theP/Q graph given by DP=N^2*(k2 – k3*Q)
Find operating point when connectecd to our square law system by Substituting DP = k4*Q^2
K4*Q^2=N^2*(k2 – k3*Q)
The solution of the quadratic equation in Q will not in general be proportional to N. Hence our operating point also will not follow Q~N and DP~N^2.
Pump laws do not apply.


But now for example B let’s say we use a form: DP=k1*N^2 – k2*Q^2
(I think this is similar to centrifugal pump)
Find operating point when connectecd to our square law system by Substituting DP = k4*Q^2
K4*Q^2=k1*N^2 – k2*Q^2
Q^2 (K4+k2) = K1*N^2
Q^2 = N^2* K1/(K4+k2)
Q = N*sqrt(K1/(K4+k2))
DP = k4*Q^2 = N^2*k4*K1/(K4+k2))
Pump laws do apply.

So the idealized centrifugal pump curve has unique characteristics which allow it to satisfy Prerequisite #1 (these characteristics are met in example B but not example A). That means when we put centrifugal pumps into system which satisfies prerequisite #2, the operating pont will follow the pump laws.. There is no doubt in my mind that prerequisite #2 (system square law) also is a prerequisite for pump laws to apply. I think it has been proven beyond a shadow of a doubt.

Let me go one more time to a simple question. If we put an idealized pd pump (Q~N) into our DP~Q^2 system, do we get DP~N^2?

If there is anyone who agrees with any of the above questions, please let me know. Maybe they are obvious but I am feeling the acknowledgement of these facts is approaching a vacuum.

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[25362]

I concur in what refers to rotary pd pumps: if the system's dP=kQⁿ, then a idealized pump discharge (in the pump: no slippage, no losing suction, no abnormal rheology effects, no non-condensables, no change of phase, no erosion, no corrosion, etc.) would show dP=kNⁿ.

In respect to your conclusions on centrifugal pumps, let me think a bit more about the issue . In the meantime, a pump expert may give you his considered answer for us to learn.

 

[electricpete]

Thx 25362 for your patience in attempting to answer my question. It seems we have gone through a lot and yet there remains lack of response to my original question.

With the considerable pump expertise available, will someone tackle the following questions:

True/False: We cannot have DP~N and Q~N^2 without DP~Q^2.

True/False: The centrifugal pump laws DP~N and Q^N^2 are based on the assumption that the system DP~Q^2.

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[25362]

Sorry for taking so long in coming back. I see that pump experts somehow refrain from giving their opinion.

There is a typo error in the formulas: H2/H1=(N2/N1)², and Q2/Q1=N2/N1. Not the other way round.

My thoughts:

1. Pumps' equation

A typical equation for a conventional "steep" pump characteristic curve is H=A.N²-B.N.Q-C.Q², where A, B, C are constants. H, head; N, speed of rotation; Q, flow rate.

If B=>0, we are left with H=A.N²-C.Q², i.e., your example B. If C=>0, the equation becomes H=A.N²-B.N.Q (similar to your example A).

2. The (square) similarity rule

I still think, at the risk of repeating myself, that the square rule, for c.p. operating at different speeds, more or less applies on points of constant pump efficiencies, more exactly at the locus of the BEP, as a result of the constancy of the specific pump speed NQ^0.5/H^0.75 for a particular pump, and as a result of Q being proportional to N, independently of whether the system follows, or not, a quadratic rule.

In other words:

(NQ^0.5/H^0.75)₁=(NQ^0.5/H^0.75)₂​

or (H2/H1)=(N2/N1)^4/3(Q2/Q1)^2/3

Only when N2/N1=Q2/Q1, this being a basic assumption, we obtain: (H2/H1)=(N2/N1)²=(Q2/Q1)²

3. Operating points

The operating points at different values of N, are the intersections of the system's curve with the pump (quasi-parallel) curves. If the system's curve isn't quadratic, for example H2/H1=(Q2/Q1)^m, with m not equal 2, it will intercept the pump (quasi-parallel) curves at points (one on each pump curve) not interconnected by the similarity rule.

I still hope I'm right.

 

[electricpete]

Thanks. Your are right about the obvious typo.

Your item 1
You propose a form of the Head (I call it DP) as follows:
DP=AxN^2-BxNxQ-CxQ^2

Let’s use the same logic as before and call this example C
Assume DP=AxN^2-BxNxQ-CxQ^2
(25362’s form for to centrifugal pump)
Find operating point when connectecd to our square law system by Substituting DP = D*Q^2
D*Q^2=AxN^2-BxNxQ-CxQ^2
Q^2(C+D) + Q(BN) – AN^2 = 0
Quadratic equation:
Q =[-0.5/(AN^2)] * [ - BN + / - sqrt(B^2N^2 + 4(C+D)AN^2 ]
I don’t see this as being proportional to N in general under assumption of square-law system.
I think the form assumed in example B supports pump-law behavior when connected to a system DP~Q^2.


Your item 2
The relevance of this discussion to my question escapes me. Are we saying that the purpose of the pump laws is to descripe the locus of BEP points? That is the first I have heard of it.

Your item 3
"If the system's curve isn't quadratic, for example H2/H1=(Q2/Q1)m, with m not equal 2, it will intercept the pump (quasi-parallel) curves at points (one on each pump curve) not interconnected by the similarity rule."

I agree. I interpret that to mean my two true/false questions in my Feb 16 post above are true.

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[25362]

On item 1, the equation for conventional pump curves is not my creation, but has been taken from the technical literature.

On item 2, although it is not always admitted, it appears that pump experts refer to the specific speed as that estimated at the BEP, and call it sometimes "optimum" specific speed. It is at the locus of the BEP that the square rule best applies.

On item 3, apparently the square affinity rule was deducted for Re numbers greater than, say, 500,000. So the equation of head being proportional to the square of flow rate holds, becoming independent of the Re numbers, as in fully rough turbulent pipe flow.
The affinity laws neglect any effect from the Re numbers and are restricted to incompressible flow.

 

[smckennz]

With reference to electricpetes two questions on 16 Feb

Q1: your corrected meaning Q = f(N)
and H = f(N)^2 (cant find the squiggle button)
I use H as head instead of your DP through laziness.
so H = f(Q)^2 true.
But what does it mean?
It means that if the speed is changed, the new pump curve head point is proportional to the square of the new flow point. It means nothing that we didnt already know except we lost knowing what caused the flow point to change.

Q2: Typos as for Q1 apply.
False. A pump curve is independent of any system resistance curve. It is important that you understand this (upper case). A pump curve would exist in a world where there were no system curves. If the pump speed could be changed then there would be an infinite family of pump curves with no system curve to produce a line solution of duty points.
There is no assumption of system curve shape.

Nothing.

The pump "laws" are based on how a pump is likely to respond to changes in speed and impeller diameter in terms of its head flow characteristic. Nothing to do with the system.
Draw your pump curve, with extra speed curves according to the pump laws, if required. Then overlay your system resistance curve or curves if you are dealing with a real world scenario. The pump curves existed before you overlaid, therefore they are independent of system interaction.
It is the interstection of the pump curve and system curve that tells you whether or not the pump (or system) will do what you want.

It is important that you understand this point (which I had some considerable difficulty with) or things will appear much more difficult than they really are.

A subsequent discussion referred to the validity of the pump laws. It is highly unlikely that they are absolutely accurate in practice, but the difference over normal interpolation ranges is likely to be negligible. The pragmatic validation of this is that most, if not all pump test codes permit pump test results to be scaled according to the pump laws (within limits) if the on-test pump speed is not the same as the specified in-service pump speed. I am quite sure that all hell would have broken loose by now if the pump laws didnt work. I can do some math to support all this if you like, by there appears to have been quite enough of that sort of thing done already.

The purpose of this post is to help you understand, not to disagree with you or, for that matter, for you to disagree with me.

Think:
Pump curve.
System curve.
Two separate things.
If they cross we have a duty point.
Synchronicity.

Cheers

Steve

 

[smckennz]

Just after I sent the previous drivel I realised a question that might help the cause:

Q: I know the head, flow and pump of my system but nothing else. Can I predict the head and flow of my system if I change the pump speed?

A: In nearly all real world situations, the answer is "no".
The only time it is "yes" is when the system resistance passes through the pump "zero" and follows the square law. We normally need to know the shape of the system curve to estimate the outcome of a change in pump speed or diameter.

Cheers

Steve

 

[electricpete]

25632 - thanks.

smckennz:

"so H = f(Q)^2 true.
But what does it mean?
It means that if the speed is changed, the new pump curve head point is proportional to the square of the new flow point."

You are picking two points from the two pump curve and comparing them. What allows you to pick and associate those particular two points on the pump curve? An assumed system characterstic curve (of course). And that system curve must follow DP~Q^2, correct? If not please tell me the basis on which you picked two points for comparison.

"False. A pump curve is independent of any system resistance curve"
I am well aware of that fact. What statement of mine leads you to believe I am not?

"There is no assumption of system curve shape."
Agree there is no assumption on system characteristic for the pump CURVES. I am talking about the PUMP LAWS.

"It is the interstection of the pump curve and system curve that tells you ..[the operating point]...It is important that you understand this point"
Agree. What statatement of mine leads you to believe I do not understand this point?

"I can do some math to support all this if you like"
1 - Please do me a math proof that Q can vary proportional to N and DP can vary proportional to N^2 without DP varying proportional to Q^2.
2 - Give me an example of application of the pump affinity law to predict change in flow and dp as function of change in speed which does not have inherent assumption that we are picking two points off the two pump curves using assumption DP~Q^2

"Think:
Pump curve.
System curve.
Two separate things.
If they cross we have a duty point."
Agreed. What statement of mine leads you to believe this is not understood?

My recommendation to you.
Think:
Pump curve.
Pump law (the subject of my question)
Two separate things.
The pump law describes the relationship between two pump curves of different speeds IF AND ONLY IF we are comparing two points on the two pump curves related by DP~Q^2.

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