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

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

(OP)

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.

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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## RE: "affinity laws"... for pd pumps ?!?

Not sure what you mean by "the DP of pump and DP of system must match".

A PD pump puts out the same fixed volume (assuming constant speed) no matter what the "system" pressure is. Horsepower will go up, of course. The volume a centrifugal will put out will depend on the discharge pressure of the system - the volume is not fixed.

## RE: "affinity laws"... for pd pumps ?!?

But I must admit, the horsepower vs speed discussion does not include the increase in frictional head.

--Mike--

## RE: "affinity laws"... for pd pumps ?!?

And yes, I am aware of the curves for centrifugal pumps and pd pumps.

Let me re-phase the question:

If you change the speed of a pd pump (without changing anything else in the system), we all agree that the volumetric flow rate changes approximatley in proportion to speed. Nowmy question is what happens to the dp?Your answer will likely include: it is determined by the system. My response: yes it does, and that is also true for the centrifugal pump. The centrifugal pump will respond accoring to Q~N and DP~N^2 if and only if the system characteristic is DP~Q^2. After all, we cannot simultaneously satisfy Q~N and DP~N^2 unless Q~DP^2 (right?).

So if you follow my logic, the assumption underlying centrifugal pump affinity laws is that the system characteristic follows DP~Q^2. More questions about this assumption:

1 - Is this a realistic assumption for most systems? (I think it is as long as we don't transition between laminar or turbulent flow?).

2 - If we apply this same assumption (DP~Q^2 system) to pd pumps, along with Q~N, isn't it perfectly logical that the new operating point will satisfy DP~N^2? And assuming BHP~Q*DP, then BHP~N^3?

If the answer to 1 and 2 is yes, why don't we just say the affinity laws apply to pd pumps?

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## RE: "affinity laws"... for pd pumps ?!?

Example: 100 Gallons per minute x 2 = 200 Gallons per minute

There is no direct change in head with a change in speed. The pump generates whatever head or pressure that is necessary to pump the capacity.

The horsepower required changes by the number

Example : A 9 Horsepower motor was required to drive the pump at 1750 rpm.. How much is required now that you are going to 3500 rpm?

We would get: 9 x 2 = 18 Horse power is now required.

The NPSH required varies by the square of the speed

Example 9 feet x (2)2 = 36 feet

Rotary pumps are often used with high viscosity fluids. There is a set of Affinity Laws for changes in viscosity, but unlike changes in speed the change in viscosity does not give you a direct change in capacity, NPSH required, or horsepower. As an example: an increase in viscosity will increase the capacity because of less slippage, but twice the viscosity does not give you twice the gpm.

Since there are a variety of Rotary Pump designs operating over a wide range of viscosities, simple statements about changes in operating performance are hard to make, but the following relationships are generally true.

Here are the Viscosity Affinity Laws for Rotary(PD) Pumps:

Viscosity 1>Viscosity 2 = gpm 1 > gpm 2

Viscosity 1>Viscosity 2 = BHP 1 > BHP 2

Viscosity 1>Viscosity 2 = NPSHR 1 > NPSHR 2

Viscosity 1>Viscosity 2 = No direct affect on differential pressure.

So they do not follow the affinity laws of centrifiguls

## RE: "affinity laws"... for pd pumps ?!?

If I were to use that constant-dp system on a centrifugal pump, I would not get dp~N^2 (since I would get dp constant). Agreed?

Let us forget the constant dp system. I am interested most in a single-pump closed loop system with an constant-in-time system characteristic. I believe the reasonable assumption for system is system DP~ system Q^2, and this is the assumption inherent in the centrifugla pump laws.

Please see my questions above.

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## RE: "affinity laws"... for pd pumps ?!?

1. The affinity laws for centrifugal pumps hold, more or less, for equal efficiencies, and, in particular at the BEP.

2. Speed changes of up to 10% don't change the efficiencies. Large speed changes, on the other hand, involve changes of efficiency and the affinity ratios at the pump/system meeting point don't hold anymore.

3. Equating the relation of H as prop. to Q

^{2}in a centrifugal pump with the system's friction drop as function of Q^{2}may be in error on two counts:a. the head developed by a centrifugal pump is independent -and not a function- of the friction drop in the discharge-side of the system;

b. the friction drop in the system is proportional to Q

^{n}, wherenis generally but not necessarily = 2. In fact, it may be quite different depending on the system's configuration (i.e., not just round straight pipes) and the pumped fluid (suspensions, non-Newtonian fluids, going through heaters and vaporizing creating two-phase flow situations, etc). See, for example, Chopey'sHandbook of Chemical Engineering Calculations, fig. 6-4, on the friction drop of paper stock in a 4" pipe. Even for clear liquids the Hazen-Williams formula calls for a dependence on Q^{1.85}in fully developed turbulent flow.Electripete and others, please comment.

## RE: "affinity laws"... for pd pumps ?!?

Items 1 and 2 identify limitations of affinity laws. Agreed.

Item 3a/3b seems to be at the center of my issue.

3a. "the head developed by a centrifugal pump is independent -and not a function- of the friction drop in the discharge-side of the system;"

3a question 1 - So if I reposition a throttle valve on the output there is no change in the operating point, including head? I disagree. The operating point is a function of both the pump and the system.

3a question 2 - More important than question 3.a.1 - Don't you agree I cannot make any conclusion about how much the dp will change upon change in speed (the pump laws) unless I know something about the system. I can draw two complete pump curves (one for each speed). But I can't pick two points to say dp~speed^2 unless I connect them with a system curve. (shutoff head is a special case).

3b. - Agreed. It is a simplification to say DPsystem~Q^2system. But it is a necessary simplification in order to satisfy centrifugal pump laws. Let's say I put this pump onto your system with system characteristic DP=K*Q^1.85. Record initial conditions Q0, DP0=K*Q0^1.85, N0. Now change speed to N1. By pump laws the final conditions are Q1=Q0*(N1/N0), DP1 = K*DP0*(N1/N0)^2 = K*Q0^1.85*(N1/N0)^2

Does these predicted final conditions satisfy the system curve? No. To satisfy the system curve we would need

DP1=K*Q1^1.85

Substitute in for DP1 and Q1 predicted by pump law:

K*Q0^1.85*(N1/N0)^2 = K* {Q0*(N1/N0)}^1.85

K*Q0^1.85*(N1/N0)^2 = K* Q0^1.85*(N1/N0)^1.85

divide both sides by K*Q0^1.85

(N1/N0)^2 = (N1/N0)^1.85

It is not a consisten conclusion because the pump laws describe the behavior of a pump which is connected to a system with DP=k*Q^2 (which includes as a special case shutoff head, limit k->infinity forces Q to zero). They are not consistent with any other system curve. Notice that if you change 1.85 to 2 the conclusion would be true.

Let me ask you the same question in another way. Let's say I give you a centrifugal pump operating in a system at 100% speed and tell you to predict the conditions as a function of speed.

Find initial Q100,DP100

Calculate Q90=Q100*90%, DP90=DP100*(90%)^2 and plot

Calculate Q80=Q100*80%, DP80=DP100*(80%)^2 and plot

Calculate Q70=Q100*70%, DP70=DP100*(70%)^2 and plot

Calculate Q60=Q100*60%, DP60=DP100*(60%)^2 and plot

etc (you get the idea).

What is the shape of these curves we are plotting? It is a system curve Q~DP^2.

Sorry if this is getting more complicated than seems necessary. I think the most important issue to me is to define the conditions under which you'all think the pump laws apply if it is not within any of the contexts I have identified above (fixed system). Is there anyone who thinks we can have centrifugal pump Q~N and DP~N^2 and NOT have DP~Q^2?

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## RE: "affinity laws"... for pd pumps ?!?

The thrust of the discussion appears to be chasing a single duty intersection point up the speed curve. As pump discharge head must equal system resistance, the conclusions to date cannot be considered unexpected.

Perhaps it is worth going back to basics and considering what the simplest pump laws (Q vs N and Q vs H) are normally used for. I use them for generating a centrifugal pump head/flow curve at a different speed from the one in my published data. This is completely independent of any system resistance, and merely tells me the approximate head/flow characteristic at another speed.

With a PD pump,no such curve is necessary (ignoring secondary effects such as leakage); the volumetric flow is proportional to speed and independent of head/pressure. Therefore a family of speed curves is not necessary; straight vertical lines will suffice. The characteristics of an ideal PD pump are too simple to require a set of laws for performance prediction.

Sorry to sound negative but there appeared to be a lot of effort going into reinventing the wheel.

Cheers

Steve McKenzie

## RE: "affinity laws"... for pd pumps ?!?

"If I were to use that constant-dp system on a centrifugal pump, I would not get dp~N^2 (since I would get dp constant)". Agreed?

Yes, and I got my info from McNally"s web site,I never read Mikes..cogito ergo sum.

## RE: "affinity laws"... for pd pumps ?!?

Contrary to your basic statement, the pump laws

do notdescribe the performance of a pump connected to a system in which P2/P1=(Q2/Q1)^{2}.The characteristic pump curves depend on the internal design of the pumps and their specific speed. Low SS pumps have quite flat curves, and the location of the meeting point with the system curve depends solely on the system.

Let's make an exercise, assuming a flat type of pump curve and that the speed is doubled:

H1=100, Q1=20 => H2=400, Q2=40

H1=100, Q1=40 => H2=400, Q2=80

H1=100, Q1=60 => H2=400, Q2=120

Namely, the change in speed doesn't oblige making H prop. to Q

^{2}. What it actually does is move the pump curve to a different position keeping approximately its original form; flat, drooping, inverted parabolic or rising, as it was before the change of rotating speed.I pressume that the error in your interpretation is in making H prop to N

^{2}and Q prop to N, and the real presentation would be (H2/H1)=(N2/N1)^{2}, and (Q2/Q1)=N2/N1, namely the proportionality of N is with the ratios.In other words, H

_{r}=N_{r}^{2}, and Q_{r}=N_{r}, where r means ratio.Thus H2 at the new speed would be a function f(Q1,Q2

^{2}), not just Q2^{2}. Does this satisfy your query ?## RE: "affinity laws"... for pd pumps ?!?

When you have your new pump characteristic curve you can superimpose the system curve and find out where it will now run.

## RE: "affinity laws"... for pd pumps ?!?

engineering has often been criticised as the "science of coefficients"

How apt.

Keep your head up.

Cheers

Steve

## RE: "affinity laws"... for pd pumps ?!?

Yes. If we have 100% speed curve and want to generate 90% speed curve.

Pick point A100 off 100% curve and read off values DP100A, Q100A. Now compute the corresponding 90% speed point A90 as DP90A=DP100A*(0.9)^2 and Q90A=Q100A*(0.9).

Repeat with point B... Pick point B100 off 100% curve and read off values DP100B, Q100B. Now compute the corresponding 90% speed point as DP90B=DP100B*(0.9)^2 and Q90B=Q100B*(0.9).

Continue with points C, D, E etc until you have a curve. Very good excercize.

Now, under what conditions does the affinity laws Q~N and DP~N^2 apply? It will apply when we map point A100 to point A90. Or when we map point B100 to point B90.

What allows us to map A100 to A90? The assumption that DP~Q^2. If we discard DP~Q^2 we have no physical reason to associate the particular point A100 with the particular point A90. There are many points on both curves. Under what conditions do pump laws apply? When we map points between the two speed curves using DP~Q^2. If you pick any other curve to map between the speed curves you will not obtain points which obey the pump laws. Agreed?

25632 - You show me an excercize.

"H1=100, Q1=20 => H2=400, Q2=40" etc.

QUESTION: Under what conditions can your reliationship be true?

ANSWER: Only when the pump is connected to a system where DP~Q^2. Agreed? [hint: 400/100~(40/20)^2]. Thank you for proving my point.

Tony - I disagree that the laws have any truth without considering duty. Please see my response to 25632.

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## RE: "affinity laws"... for pd pumps ?!?

^{n}, for different n values, the pump's performance curve will still be the same. How would you explain that in pumps having flat curves as needed for spray nozzle systems working in parallel, differential H is constant and doesn't depend on Q ?Not forgetting that system heads are the result of static and flowing components...

Again for a given pump, H2

is nota sole function of Q2^{2}, but ofQ1 and Q2, meaning curves are parallel to each other at different speeds. This is because the affinity law^{2}is notH prop to N^{2}and Q prop to N, but(H2/H1)=(N2/N1).^{2}and (Q2/Q1)=(N2/N1)The squared exponent in regard to rpm is a function of Bernoulli's law applied

insidethe pump, andnot necessarily outside, on the system. Consider the case of paper stock flowing in 4" pipes at 8-10 fps, where H is prop to Q^{0.35}. The pump affinity laws still apply even when the system behaves quite differently. QED.## RE: "affinity laws"... for pd pumps ?!?

You bring up as others bring up that I have have on a piece of paper two curves for centrifugal pump at speeds N1 and N2. They exist and the pump curves describe the operation of the pump for ANY system. 100% true. The PUMP CURVES are independent of the system. Did I ever say they weren't?

(I said the PUMP LAWS are dependent on the system.)

So let's study these two centrifugal pump curves atspeed N1 and N2 plotted as dp vs Q that we are all familiar with. Why is it that you say they are described by the pump laws? Under what conditions can I pick a set of points of the two curves and conclude Q~N? It certainly does not hold if we read horizontal for all dp's (it only works for dp=0...special case of DP=kQ^2 and k=0). Under what conditions can I pick a set of points off the two curves and conclude DP~N^2? It certainly does not hold for all Q's. (It only holds for Q=0 special case of DP=kQ^2 and k approach infinity).

So far we have only two points where we say the pump laws hold (the axis intercepts where DP=0 or Q=0).

So please tell me why it is that you think the PUMP LAWS apply to these PUMP CURVES.My answer is that if we draw any system curve following the relationship DP=K*Q^2 onto my paper, I can read off a sets of two operating points (one per pump curve) which obey the pump law. Any other set of system curves will not produce a set of points obeying the pump laws.

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## RE: "affinity laws"... for pd pumps ?!?

The affinity (or similarity) laws are an approximation.

For small changes in speeds, such as 5-10%, the efficiencies don't change much, and the new curve could be based on the affinity laws with a greater degree of accuracy.

For a large change in rotating speeds, say 2 to 1, typically the best curve fitting for any type of curve is taking two points: at shut-off -as you say- and at the BEPs then draw a parallel. The farther away from the BEP the larger the deviation from the correlations set by the affinity laws.

## RE: "affinity laws"... for pd pumps ?!?

You forgot one point that the system curve is a parabola with equation of the kind y = cx

^{2}where abscissa relates to flow rate and ordinate to the head. So when ever you change the speed of the pump, the operating point just shifts along the system curve and this is the principle used in variable speed pumping systems. So for all Q1 and Q2 your H1 and H2 will match. The tangents to the two pump performance curves at various speeds are almost parallel. (in layman's words the two pump curves are almost parallel)For PD pumps Steve already explained it. PD pumps operate on back pressure principle. They will develop any pressure at constant flow rate. If you reduce the system resistance the pumps discharge pressure reduces and if you increase the system resistance the pump pressure increases, yet they maintain constant flow rate. There is no need to change the speeds here.

Regards,

## RE: "affinity laws"... for pd pumps ?!?

quark - I don't think I forgot to mention it. My frist post "I believe there inherent in the above an assumption that the system flow characteristic curve is unchanged and follows a relationship DP~Q^2"

It sounds like you are the first one to agree with this statement. I was beginning to think I was talking to a wall. Thanks.

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## RE: "affinity laws"... for pd pumps ?!?

for centrifugal pumps the higher the viscosity the lower the head developed at each flow... while the effect is just the opposite with PD pumps (the higher the viscosity of the fluid the flow gets nearer a vertical line - i.e. the ideal flow at given rpm´s).

affinity laws are nothing more than scaling. i.e. the laws that govern the modelling.

affinity laws are another way of expressing the specific speed of a pump... which is defined for centrifugal pumps but not for PD pumps (as far as i remember without my karassik, messina, fraser at hand).

another very important relationship is the suction specific speed for centrifugal pumps (again never found it defined for PD pumps) this relationship was found to be quite important in the failure rate of centrifugal pumps... where values below 7000 are recommended and often specified because experience shows that pumps with such values have a much lower failure rate.

HTH

saludos.

a.

## RE: "affinity laws"... for pd pumps ?!?

System curves follow an equation such as H=a+b*Q

^{n}, 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 ?

## RE: "affinity laws"... for pd pumps ?!?

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*Q

^{2}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,

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## RE: "affinity laws"... for pd pumps ?!?

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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## RE: "affinity laws"... for pd pumps ?!?

Can we ever have A~C and B~C^2 when we don't have B~A^2?

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## RE: "affinity laws"... for pd pumps ?!?

_{2}/A_{1}=C_{2}/C_{1}, B_{2}/B_{1}=(C_{2}/C_{1})^{2}, then(A_{2}/A_{1})^{2}=B_{2}/B_{1}We are actually after A

_{2}, and B_{2}, given C_{2}/C_{1}.A

_{2}=A_{1}(C_{2}/C_{1})=A_{1}(B_{2}/B_{1})^{0.5}thusA

_{2}^{2}=A_{1}(B_{2}/B_{1}), and since A_{1}=f(B_{1}),A

_{2}^{2}=B_{2}*f'(B_{1}) orB_{2}=A_{2}^{2}/ f'(B_{1})It is the factor

1/f'(Bthat is missing from your otherwise correct presentation._{1})## RE: "affinity laws"... for pd pumps ?!?

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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## RE: "affinity laws"... for pd pumps ?!?

^{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.

## RE: "affinity laws"... for pd pumps ?!?

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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## RE: "affinity laws"... for pd pumps ?!?

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(Q

^{n}) equation applies for the system.## RE: "affinity laws"... for pd pumps ?!?

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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## RE: "affinity laws"... for pd pumps ?!?

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.

## RE: "affinity laws"... for pd pumps ?!?

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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## RE: "affinity laws"... for pd pumps ?!?

^{n}, 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=kN^{n}.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.

## RE: "affinity laws"... for pd pumps ?!?

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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## RE: "affinity laws"... for pd pumps ?!?

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' equationA typical equation for a conventional "steep" pump characteristic curve is H=A.N

^{2}-B.N.Q-C.Q^{2}, where A, B, C are constants. H, head; N, speed of rotation; Q, flow rate.If B=>0, we are left with H=A.N

^{2}-C.Q^{2}, i.e., your example B. If C=>0, the equation becomes H=A.N^{2}-B.N.Q (similar to your example A).2.

The (square) similarity ruleI 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 NQfor a particular pump, and as a result of^{0.5}/H^{0.75}Q being proportional to N, independently of whether the system follows, or not, a quadratic rule.In other words:

^{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 pointsThe 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.

## RE: "affinity laws"... for pd pumps ?!?

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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## RE: "affinity laws"... for pd pumps ?!?

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.

## RE: "affinity laws"... for pd pumps ?!?

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

## RE: "affinity laws"... for pd pumps ?!?

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

## RE: "affinity laws"... for pd pumps ?!?

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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## RE: "affinity laws"... for pd pumps ?!?

1) We are already in agreement that DP~Q^2 for the purposes of preparing a second pump curve. From what you have previously said, you are aware that this relationship does not necessarily hold for a pump/system combination, where Q is the actual flow. This is the classic: " I reduced the pump speed by 20% and now nothing comes out the end of the pipe. According the the pump laws the flow should have reduced by only 20%" I thought this was the concept you are having difficulty with. Obviously not.

2) DP ~ Q^2 is not an assumption, it is a derivation from the pump laws as you have already shown. The pump laws can be proven but are obviously an idealised case. So your question reduces for a request to provide an example in conflict with the pump laws.

You final statement has me a little mystified inasmuch as your DP~Q^2 was derived from the pump laws, and your curves will be derived from the same laws. So the statement appears to answer itself.

The above assumes that all the "Q"s are on pump curves and do not relate to fixed system flows, in accordance with your recommendations.

I have however, found the squiggle key.

Cheers

Steve

## RE: "affinity laws"... for pd pumps ?!?

^{2}, I"ll refer to the true/false questions of your message of February 16th.Since the affinity rules were arrived at by dimensional analysis assuming internal pump hydraulics independent of the Re number, and the pump/impeller set being defined by the constancy of the specific speed, I'd rephrase your first question as follows:

We cannot have (H2/H1)=(Q2/Q1)as a result ofg the constancy of the specific speed for a particular pump. BTW, the last equality, is also valid (with reserves) for rotary PD pumps.^{2}or (H2/H1)=(N2/N1)^{2}without (N2/N1)=(Q2/Q1)As for the second question, in which it is stated that CP laws, as above, are based on the assumption that the

systemcomplies with (H2/H1)=(Q2/Q1)^{2}, I beg to disagree on three counts:First, historically the pump laws were derived from dimensional analyses, in total disregard of the system characterisitcs.

Second, the law applies for centrifugal pumps, no matter what the characteristic pump curves appear to be, mostly at equal efficiencies (id est, small speed changes), and optimally at the BEP locus.

Third, although the system's friction losses are indeed proportional to f.Q

^{2}, it happens that 'f' is also a function of Q, thus (H2/H1)=(Q2/Q1)^{m}where m=2 only applies to very special cases in piping systems.For example, water flowing in sch 40 steel pipe with typical rugosities (taken from tables prepared by the Hydraulic Institute):

gpm nom. size, in Friction, ft/100 ft of pipe m

100 3 2.39

200 3 8.90 1.93

------------------------------------------------------------

100 4 0.624

200 4 2.27 1.91

------------------------------------------------------------

200 6 0.299

400 6 1.09 1.91

------------------------------------------------------------

For the square law (m=2) to apply in piping, the values of Re numbers should be greater than 2,500,000 for 3" pipes, 3,000,000, for 4" pipes, and 4,500,000 for 6" pipes, and so forth. For smooth tubes in which the rugosity ratios are of the order of 0.000001, Re should be larger than 100,000,000 for m=2 to apply!

For flows across staggered tubes m=1.8; if these tubes are finned, m=1.68; for tubular helices, m=1.75. All this is beside various lower 'm' values for slurries such as paper stock.

Therefore, it appears to me that, in the majority of cases, the intersections of the system curve with the pump curves (i.e., operating points),

would not correspondto the "square" rule.I have other comments regarding the exercises we both did with pump curves answering to certain types of equations, but I feel it is time I should rest my case.

## RE: "affinity laws"... for pd pumps ?!?

One thing I believe we have reached agreement on: The earlier contention that pump laws can exist in a vacuum without contemplating a system characteristic relationship I believe has been proven false. (Pump curves can exist in a vacuum, but we cannot relate them by the pump laws unless we consider a system characterstic form).

Steve you said "DP ~ Q^2 is not an assumption, it is a derivation from the pump laws as you have already shown"

Now I have to delve into the semantics. Saying DP~Q^2 is derived from pump laws (rather than pump laws are based upon assumption that DP~Q^2) seems like backward logic to me.

Do we agree on the following equivalent statements:

1 - Pump laws cannot be true unless we consider / apply / contemplate a system characteristic curve with DP~Q^2.

2 - Pump laws are true ONLY IF system DP~Q^2 is true.

If the previous statements 1 and 2 agreed on, then I'm not sure why anyone would disagree with the what I consider another equivalent statement:

3 - System DP~Q^2 is an assumption of the pump laws.

25352

You show that not all systems obey DP~Q^2. That is agreed. Therefore we cannot predict DP and Q as function of speed in these systems using pump laws.

You mention a derivation based on specific speed. I am not that familiar with the subject. It seems irrelevant to my question if we accept statements 1, 2, 3 (do you?).

If 1,2,3 are not true, then Steve or 25 please provide an example of application of pump laws which does not in some way rely on an assumption that system DP~Q^2

Interesting discussion by the way. I know it seems like we are going around in circles. But still interesting

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## RE: "affinity laws"... for pd pumps ?!?

Take, for example, a pump with a drooping curve (regularly found with low-specific-head power-efficient pumps) as its characteristic having

two flow rates for one head.This would be a curve of the type h=a+bQ+cQ

^{2}, where a (=KN^{2}), b and c are constants for a given speed, describing a parabola having its axis parallel to the "h" axis, and its apex at Q=-b/2c. If this apex is way to the right of shutoff, the curve will show a detectable droop. Upon applying the square law, increasing speeds would move the apex to the right, farther away from shutoff.Pump affinity rules still hold for such a pump, but how can we speak of an effective dependence or correspondence with a given system ?

Of course, a system is needed for a pump to work on, but the pump laws apply w/o the need of assuming the presence of a particular system.

In a similar manner available NPSH for a given system can be estimated without a pump being even installed, but, of course, to physically verify it, one has to have the pump.

All these seems rather philosophical. Electricpete, I'm looking forward to reading your comments.

## RE: "affinity laws"... for pd pumps ?!?

Of course, a system is needed for a pump to work on, but the pump laws apply w/o the need of assuming the presence of a particular system."

I say we need BOTH of two 2 things for affinity laws to apply

1 - Pump with the proper characteristics. This may be positive displacement pump or example B above (DP=k1*N^2 – k2*Q^2). Perhaps there are other forms that will work but certainly not every conceivable pump curve will be able to behave according to affinity laws (example A and example C did not).

AND2 - Assumed quadratic form of system curve.

I think your recent e-mail focus on the first aspect. Yes, there are certain characteristics of the pump required for pump laws to apply.

I think you cannot deny that #2 is also a firm prerequisite for the pump laws to apply. You have said it yourself about 10 messages ago:

"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."

So, we are in full agreement that pump laws doen't work when pump is used in a non-quadratic system, but we can't agree that quadratic system is a prerequisite for pump laws. It is somewhat philosophical and maybe I am being semantic, but I still don't have any comprehension of how anyone believes the pump laws are not based on assumption of quadratic system.

I go back to a repeating question which will silence my objection. Can you give me one example application of the pump laws which does NOT in some way assume a quadratic-form system?

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## RE: "affinity laws"... for pd pumps ?!?

^{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'm really saying the pump quadratic law applies to pump curves, but not necessarily to system curves and "operating points".The fact that

all kind of pump curvescan be drawn following the square rule, whilst not all system curves need to answer to a square "H/Q" relation, is -to my grasping- a proof of the independence of the pump affinity laws from any system characteristic.## RE: "affinity laws"... for pd pumps ?!?

I thought it had already been well established.A

PumpQ = System Q

Pump DP = System DP

Let's use non-quadratic system:

SystemDP = K*Q^M, m not equal 2.

Change the speed. If pump laws were true:

Q2 = Q1*N2/N1

DP2 = DP1*(N2/N1)^2 [Equation 1]

Substitute into equation 1 values for DP1 and DP2 based on system relationships (DP1= K*Q1^M, DP2=K*Q2^M_:

K*Q2^M = K*Q1^M*(N2/N1)^2 [Equation 2]

Substitute into equation 2 Q2 = Q1*N2/N1

K*(Q1*N2/N1)^M = K*Q1^M*(N2/N1)^2

Cancel out the K's

(Q1*N2/N1)^M = Q1^M*(N2/N1)^2

Distribute the power of M over the items on LHS

Q1^M*(N2/N1)^M = Q1^M*(N2/N1)^2

Cancel out Q1^M

(N2/N1)^M = (N2/N1)^2

But we started out by saying M (the exponent of our system characteristic) was not 2!

We see that the pump-laws are self-consistent ONLY IF M=2. (Only if we consider a quadratic system)

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## RE: "affinity laws"... for pd pumps ?!?

This is OK:

Pump Q1,Q2 = System Q1,Q2: Operating points.

Pump DP = System DP

Let's use non-quadratic system:

SystemDP = K*Q^M, m not equal 2.

The following is not OK:

As I see it, the conceptual error appears in assuming "a priori" that both points, [DP1,Q1] and [DP2,Q2] follow the pump affinity laws. The new point for the pump law to apply should be [DP3,Q3]. Thus your Equation 1 changes as follows:

Change the speed. If pump laws were true:

Q3=Q1*N2/N1DP3=DP1*(N2/N1)^2 [Equation 1]One has to show that Q3=Q2 and DP3=DP2And this could be done only when the system's

DP2/DP1=(Q2/Q1)

^{m}, and m=2.Since our pre-condition was that m is not 2, the point defined by [DP3,Q3], is not equal to the point [DP2,Q2].

Do you follow my thinking ?

## RE: "affinity laws"... for pd pumps ?!?

The following equations are meaningless to me:

Q3=Q1*N2/N1

DP3=DP1*(N2/N1)^2

What is the speed at which Q3 and DP3 are measured?

We have a speed N2 but no associated DP or Q... then what does speed N2 have to do with anything? Can you formulate this as a complete problem statement.

In general one will intepret Q~N to mean

Q2/Q1 = N2/N1.

In general one will intepret DP~N^2 to mean

DP2/DP1 = (N2/N1)^2

Have I misunderstood the meaning of proportionality? Is there another definition for proportionality that I should be aware of?

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## RE: "affinity laws"... for pd pumps ?!?

thirdpoint (H2',Q2') -if you don't like (H3,Q3)- on curve N2, resulting from applying the curve affinity laws to (H1,Q1).The "operating points", belong to, and are indeed related by, the system's curve. But it is wrong to assume -a priori- that they are also related by the pump's affinity law square equation.

The "correct" third point on pump curve (N2), called (H2',Q2'), follows H2'/H1=(Q2'/Q1)

^{2}=(N2/N1)^{2}by the pump laws.The object of the exercise is to show that "operating" point (H2,Q2) coincides with "pump" point (H2',Q2'). This can only be shown to be the case when H2/H1=(Q2/Q1)

^{2}on the system curve. Otherwise they'll not coincide.Did I make myself clear ?

## RE: "affinity laws"... for pd pumps ?!?

Start with one system curve and two pump curves (at speed N1 and N2). Creates operating points 1 and 2.

Apply the affinity laws to point 1 and we come up with [ficticious] point 3.

Point 3 is equal to point 2 IF AND ONLY IF the system curve follows quadratic relationship.

Repeat. Point 3 has no physical significance if the system does not follow quadratic relationship.

So what you’re telling me is the purpose of the pump laws is to predict operating points which have NOT PHYSICAL SIGNFICANCE unless the system is quadratic (in which case they will be the operating point). But in spite of this you don’t think that the pump laws assume the system is quadratic?

Tell me why on earth do we need a law to predict a ficticious point with no physical significance. Give me an example of a real-world problem that needs to be solved where we are concerned about this ficticious point.

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## RE: "affinity laws"... for pd pumps ?!?

exists on pump curve No. 2. Dismissing it as fictitious, would be equivalent to dismissing the whole curve No. 2 as irrelevant and unreal.An example: whenever the system needs a different flow rate as in point 3 -as when commissioning a flow control valve- while using pump curve No. 2, after a change in speed, it will encounter point 3 defining the new [H3,Q3] conditions.

## RE: "affinity laws"... for pd pumps ?!?

In your example, it sounds like you are controlling flow by a control valve. Yes, if flow #3 happens to be the target flow, the valve will drive it there. But again, there is no special reason to prefer point #3 over any other point.

The only reason we ever conceieved of point #3 is that it is predicted by pump laws for change in speed N1->N2 given previous point #1. It has no other significance. Sure you can attach a magical significance by inventing a demand for exactly this flow. But you could also invent a demand for any other flow on the curve.

Step back and take a deep breath and re-read our last few messages. Are you really serious or just yanking my chain?

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## RE: "affinity laws"... for pd pumps ?!?

The same principles govern both, the idealized affinity laws for centrifugal pumps (CP), and the friction drops for systems on Newtonian fluids. As for the square law for pumps and the relation between H and Q on flow systems one sees that both can be arrived at by dimensional analyses.

The important difference I found is that for CP affinity laws, head is normally

independentof the Re number, meaning we have rough and fully developed turbulent flow (where the Darcy friction factor doesn't change any more as a function of Re), while "system heads" are normally dependent on Re, from laminar to turbulent flow, and the H=Q^{m}relationship can thus change from m=1 to m=2."New" characteristic pump curves of various geometrical contours can be constructed from points on "existing" curves at differing speeds, and this is a practical, "real" not imaginary, procedure.

[H,Q] parabolic curves can vary, and are in fact being varied, by factors such as control valves. I sincerely thought the example you asked from us could include the FCVs.

The main conclusion I see is that the pump affinity laws and the system curves, while stemming from the same hydraulic concepts, are, however, independent from each other.

Please don't be disappointed. As for myself I feel honoured from getting your considered attention. The governing premise is that all of us are seeking the truth in this discussion. And I still think my viewpoints on this subject are not wrong.

## RE: "affinity laws"... for pd pumps ?!?

“New characteristic pump curves of various geometrical contours can be constructed from points on existing curves at differing speeds, and this is a practical, real not imaginary, procedure.”

I agree we can construct pump curves without any assumption on the system.

I agree during the process of generating pump curves, we will generate points on curve 2 from point on curve 1 using (Q2,DP2) = (Q1*N2/N1, DP1*N2^2/N1^2) (ie the pump laws). When we are done with the curves there is no particular reason to assume those two points remain associated with each other (unless we are making an assumption that the pump will be connected to a square-law system). There is no reason to suggest that the two pump curves are related to each other by the pump laws unless we are assuming a square-law system.

“[H,Q] parabolic curves can vary, and are in fact being varied, by factors such as control valves. I sincerely thought the example you asked from us could include the FCVs.”

I asked for an example where the pump laws could be applied without regard to requirement for system to be quadratic (prerequisite for applying the pump laws). You gave me an example where we have a non-square law piping system, but when we change speed we want to adjust the FCV to make it move to the exact target operating point (3) that it would move to IF connected to a square-law system. It is a very artificial scenario in which the pump laws are only relevant because the target point that you defined is that which would result if the same change in speed were applied to a square-law system. (It is like saying y’’=-m*g is not relevant to falling-body motion because I can create the same y’’ if I apply a total force unrelated to gravity which is numerically equal to m*g - let’s say using rocket propulsion in space). You are changing two parameters (speed and valve position) when the intent of the pump laws clearly is to describe effect of change in speed only. Change in system characteristics is not incorporated into the pump laws as can plainly be seen by the form of the pump laws (Q~N and DP~N^2)

“The main conclusion I see is that the pump affinity laws and the system curves, while stemming from the same hydraulic concepts, are, however, independent from each other. “

Pump curves of varying speeds are independent of the system curves.

Pump laws (Q~N and DP~N^2) can never be mathematically satisfied without having DP~Q^2.

Can you give me a sentence description of what the pump laws mean. We have written the equations but where do we apply them?

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## RE: "affinity laws"... for pd pumps ?!?

One must remember that the affinity laws as described until now by us, are not complete.

The impeller radius r, or diameter D, should be included.

Using (a.p.t.) for "about proportional to" the laws would include:

H (a.p.t.) r

^{2}N^{2}Q (a.p.t.) r

^{3}N, and as a resultP (a.p.t.) r

^{5}N^{3}T (a.p.t.) r

^{5}N^{2}Where P and T are power and torque, respectively.

It may well be to repeat that the affinity laws in this form neglect any effects of Reynolds number and that they are restricted to incompressible flow.

Applications of the affinity laws are common for pump users and makers. Changes of rotating speeds and of impeller diameters are frequently done to modify the pump's output at varying operating conditions.

The affinity laws enable also attaining a variety of specific speeds N

_{s}=NQ^{0.5}/H^{0.75}and specific radiuses, r_{s}=rH^{0.25}/Q^{0.5}meaning differing pump designs, to approach optimum performance, i.e., the BEP's.Can I rephrase your last sentence by saying pump laws

(DP (a.p.t.) Q

^{2}and DP (a.p.t.) N^{2}) can never be mathematically satisfied without having Q (a.p.t.) N ?It was not I that tried to tie up the system's and pump's characteristics by the common "square law". I brought in the FCV to show that systems change to adapt to differing process conditions. Only when these changes aren't feasible or economical any more, pump curve modifications are sought after, for example by varying the impeller diameter or the rotating speed.

## RE: "affinity laws"... for pd pumps ?!?

Thanks for all the help.

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## RE: "affinity laws"... for pd pumps ?!?

perhaps you could close by summarising what you have learned through the effort put into this discussion.

My own view of a simplified centrifugal pump is a "G" machine (rotation) in series with an orifice plate(flow path resistance). The "G" portion of the machine is cabable of producing a zero flow head according to Eulers V^2/g. As flow increases, the available (outlet)head varies as the G head less the head loss across an orifice; commonly considered to vary as the square of the flow. Perhaps this is another way of looking at your H~Q^2 proposition.

Cheers

Steve

## RE: "affinity laws"... for pd pumps ?!?

However, I don't really feel like I've learned anything. I have unsuccessfully tried to communicate the basis of my question, which I still believe to be correct. At this point the conversation is going in circles and no useful end in sight.

Thanks again to all who replied.

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