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# Minor comment about the way the centrifugal pump affinity laws are sometimes discussed

## Minor comment about the way the centrifugal pump affinity laws are sometimes discussed

(OP)
I typically see discussions like this:

#### Quote:

The Affinity Law for speed of a centrifugal machine: Power required by the load (pump or fan) varies by the cube of the speed change.
This principle (power ~ N^3) is then used to estimate how much the motor load might change if the speed is changed (neglecting changes in motor and pump/fan efficiency).

I think it’s a very reasonable approach for most purposes (more later). But it’s not completely accurate to say that we only need the centrifugal pump affinity laws to get there.

The affinity laws predict change in centrifugal pump/fan CURVES as speed changes. A pump curve would be any curve involving any relationship among the variables flow, dp and horsepower. In order to map a curve from one speed to another speed, we apply Q~N, DP~N^2, HP~N^3 to BOTH coordinates of a point on a curve at old speed to find a point on the curve at the new speed… repeat for many points to generate the new curve.

The affinity laws do not in general predict the change in operating point as speed changes. The new operating point is of course the intersection of the new pump/fan curve and the system curve (which is unchanged for simple systems which don’t have something like auto-flow control valve or other feedback mechanism).

There is a special case where the affinity laws WILL predict change in operating point in the simple manner (Q~N, DP~N^2, HP~N^3). That happens IF AND ONLY IF the pump/fan is connected to a system which obeys DP~Q^2. That is a system characteric curve which is dominated by turbulent flow resistance. That is the case in closed loop piping systems with reasonably high flow velocities.

But if you happen to have an open fluid system, there may be pressure changes in the system that are independent of flow. Example a pump sends water from atmospheric pressure reservoir through a pipe into a pressurized tank at the same elevation– the pump must overcome both pipe pressure drop and the DP between tank and reservoir – the latter component does not vary with flow. The DP vs Q curve does not pass through (Q,DP)=(0,0) as a DP~Q^2 curve would. For DP=0 (no pump DP), you would have negative (reverse) flow (neglecting any check valves). If you tried to force Q=0 by installing a blank flange at the pump you would have a positive DP across the blank flange.

Likewise laminar flow would follow DP~Q instead of DP~Q^2, although I think most pumped fluid systems operate more in turbulent flow.

(I realize there are some circumstances which it makes more sense to deal in pump head rather than pump dp, but it doesn’t matter if we are not contemplating change in density.)

I’m not trying to pick on anyone. I saw something like this from a very respected member in another thread that spurred this comment. I decided not to clutter up that thread with my picky comment. It is something I was confused about myself awhile back before the guys on the pump forum straightened me out, so I thought others might be confused the same way I used to be (well… I’m still plenty confused, just not about this particular thing). I apologize if I’m telling you guys something you already know.

=====================================
(2B)+(2B)' ?

### RE: Minor comment about the way the centrifugal pump affinity laws are sometimes discussed

it's a good reminder, because there are several Affinity Laws and we should be careful not to mix the messages. You are referring to the one for FLOW, I was referring only to the one for SPEED in that other thread, but I made no mention of the affect of speed change on flow or head. The context was the affect of speed change on the MOTOR load. Flow is an entirely different matter, as is head.

" We are all here on earth to help others; what on earth the others are here for I don't know." -- W. H. Auden

### RE: Minor comment about the way the centrifugal pump affinity laws are sometimes discussed

(OP)
My comment is addressing the discussion of what happens with change in speed.

The change in speed affects the pump CURVEs in a predictable way.

We simply cannot predict the change in operating point (which means flow, dp, and power) without knowing something about (or assuming something about) the fluid system characteristics.

=====================================
(2B)+(2B)' ?

### RE: Minor comment about the way the centrifugal pump affinity laws are sometimes discussed

(OP)

#### Quote:

The context was the affect of speed change on the MOTOR load. Flow is an entirely different matter, as is head.
they are of course all related:

fluid horsepower = volume flow rate * dp

shp = bhp = fluid horsepower / pump efficiency

motor input power = bhp / motor efficiency

I'm sure I'm not telling you something you don't already know.

=====================================
(2B)+(2B)' ?

### RE: Minor comment about the way the centrifugal pump affinity laws are sometimes discussed

(OP)
My starting point was to consider the three laws together:
Q~N
DP~N^2
P ~ Q*DP ~ N^3
In general, these laws can only describe operating points (at intersection of system curve and multiple scaled pump curves) if the system curve also obeys DP~ Q^2 (barring contrived cases).

========

But there are a lot of ways to look at it.

The affinity laws map a pump curve at one speed to a pump curve at another speed.

Examine effect of change of speed on two systems with same initial operating point, same pump curve, but different system curves. The final operating points will obviously be different, but they both lie on the same final pump curve. We could only claim that all possible system curves give the same final power if all the points along the final pump curve share the same power. Do all points along that pump curve share the same power? No (*). So the effect on power of a change in speed for a given pump cannot be the same for all systems sharing the same initial operating point. So in this sense we cannot predict the final operating power without knowing the system.

* A pump curve is neither constant fluid horsepower FHP nor constant brake horsepower. But, I think BHP is more constant along a range of operating points than FHP.

If you’d prefer a different assumption to justify projecting power proportional to speed cubed, you could instead assume that BHP is approximately constant as you travel along a fixed speed pump curve within a narrow range of interest, and limit yourself to evaluating small speed changes. Then since we are mapping small speed change, the new operating point won't be very far along the pump curve from where we would land if we followed a DP~Q^2 system curve from the initial operating point to it's intersection with the final pump curve; and since these two points are close the BHP are approximately the same by assumption. That seems like another valid way to state the required assumption. With that assumption, you could say the system characteristic is irrelevant for purposes of estimating BHP over that small range. That's probably more along the lines of the assumption that most people have in mind, and would be reasonable for the recent thread where only a small speed change was discussed.

It’s a little bit in the weeds. But it brings out that there are some subtleties in the pump laws that aren’t often mentioned.

=====================================
(2B)+(2B)' ?

### RE: Minor comment about the way the centrifugal pump affinity laws are sometimes discussed

LOL, that's why I don't mention it. My approach is almost always on only the electrical aspects of people wanting to play fast and loose with pumps and VFDs, usually resulting in disasters for the very reasons you pointed out; its far far more complex than it might seem to the uninitiated.

Case in point; I once put a VFD on a 100HP pump for a customer per a bid spec and tied it to an ultrasonic level sensor through a PID loop to maintain a minimum tank level in a collection system outflow pump so as to avoid sucking in air and losing prime on the pump. Relatively simple task. But the system I was working on was ONLY the pump-out end, I had no control of the INFLOW and the inflow totally overwhelmed my pump. I was blamed for not having my PID loop tuned correctly, to which I responded that the VFD was at 100% speed, the PID loop was totally irrelevant. Everyone was confused, so I looked closer at their tank and saw an old plugged 8" outlet pipe. I asked what that was, it was the pipe that went to the old pump. I found the old pump, it was 400HP! Turns out that the PUMP SALESMAN had determined that a 100HP pump + a VFD was going to be able to handle it, because "the VFD can run the pump at 7.37x the rated speed and get 400HP from that motor."!!! Uh huh... Pump salesman who did not understand the Affinity laws...

In case you are wondering (I was), I determined later that he took the cube root of 400 (7.368) as the multiple of the based speed of a 100HP motor to get 400HP of pumping capacity. Don't ask me why he thought that was right...

" We are all here on earth to help others; what on earth the others are here for I don't know." -- W. H. Auden

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