"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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[jay165]

Pete,

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.

 

[MikeClay]

See

http://www.maintenanceworld.com/Articles/mcnally/theaffinity.html

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

--Mike--

 

[electricpete]

Jay - what I mean by "the DP of pump and DP of system must match" is that the operating point is the intersection of the pump curve (let's assume a vertical line) and the system curve. The DP of the pump equals the DP of the system.

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. Now my 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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[imok2]

In the positive displacement pump,if we will double the speed, the capacity or amount of fluid you are pumping varies directly with this number.

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

 

[electricpete]

Thx imok. It looks like you have copied from Mike's link. As Mike points out the assumption that hp doubles means we are pumping into a system with constant dp. (BHP~Q*DP).

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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[25362]

Some thoughts:

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 Q² in a centrifugal pump with the system's friction drop as function of Q² 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 Qⁿ, where n is 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's Handbook 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.

 

[electricpete]

thx 25362.

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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[smckennz]

This is getting a bit technical.
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

 

[imok2]

electricpete:
"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.

 

[25362]

To electricpete, again some thoughts.

Contrary to your basic statement, the pump laws do not describe 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 Q². 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 N² 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_r², and Q_r=N_r, where r means ratio.

Thus H2 at the new speed would be a function f(Q1,Q2²), not just Q2². Does this satisfy your query ?

 

[tonyh]

Or to put it another way, the affinity laws for a centrifugal pump do not and are not supposed to be dependent on duty. They are merely a way of predicting the charcteristic curve of the pump when you have a known characteristic curve and wish to make a variation in speed of the pump (or diameter of the impeller).
When you have your new pump characteristic curve you can superimpose the system curve and find out where it will now run.

 

[smckennz]

Thank you Tony

engineering has often been criticised as the "science of coefficients"
How apt.
Keep your head up.

Cheers

Steve

 

[electricpete]

Steve Mck - You wrote: "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."

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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[25362]

To electricpete, "Eppur si muove": there is no relation with the system characteristics. To such a point that if one modifies the discharge system, or connect the same pump alternatively to two or more totally different parallel systems where H is proport. to Qⁿ, 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 not a sole function of Q2², but of Q1 and Q2², meaning curves are parallel to each other at different speeds. This is because the affinity law is not H prop to N² 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 inside the pump, and not 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.

 

[electricpete]

25632 - Thanks for responding. I leave open the possibility that I have misunderstood your next to last post. You say flat pump curve... meaning pd pump? If so, is it your intent to express that "H1=100, Q1=20 => H2=400, Q2=40" would be a "typical" response for pd pump (typical = implicit unstated assmption about system curve)? I would like to hear you say yes.

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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[25362]

A flat curve is a characteristic curve of low specific speed pumps where the differential head remains constant at various flow rates from zero to almost the BEP. These type of pumps may be needed when the head pressure must stay constant as for spray systems working in parallel.

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.

 

[quark]

Electricpete!

You forgot one point that the system curve is a parabola with equation of the kind y = cx² 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,

 

[electricpete]

25362 - So the gist of your comment is that centrifugal pump laws are not exact. OK. At this point my main focus is to get someone to admit that a square-law system characteristic is inherent in the pump laws.

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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[abeltio]

for PD pumps as well as for centrifugal pumps the viscosity parameter creates a set of curves... which operate on the pump curves in opposite ways...
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.