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# Affinity Laws or Similarity laws for Impeller Dia Changes3

## Affinity Laws or Similarity laws for Impeller Dia Changes

(OP)
Hi,
I have come accross an interesting problem when looking to change impeller diameter to increase the flow output of a centrifugal pump.

Some books quote using the affinity laws:
Q1/Q2 = (D1/D2)
H1/H2 = (D1/D2)^2
P1/P2 = (D1/D2)^3
to redraw a new pump curve for the changed impeller.  The new flow can be taken from the intersection of the exisitng system curve on the new pump curve.

Other books quote the similarity laws:
Q1/Q2 = (D1/D2)^3
H1/H2 = (D1/D2)^2
P1/P2 = (D1/D2)^5

I have 4 questions:

1. Why the different schools of thought?
2. Which is correct?
3. Where does each set of laws apply?
4. Which set should be used in which circumstances?

Many Thanks

### RE: Affinity Laws or Similarity laws for Impeller Dia Changes

Actually the affinity and similarity laws are the same thing, and are correctly expressed as the formats you've shown as Similarity laws.  The requisite for their application is that the impellers be geometrically similar, radial flow impellers with a specific speed generally below about 4000 rpm, hence the name, under similar dynamic conditions.  They can be used to compare different impellers with the same specific speed.

The first set of relations, which you listed as affinity laws, are actually a derivative of the affinity laws and apply only to a situation where you're trimming the diameter of a known impeller - once trimmed, the specific speed changes, so the affinity laws no longer accurately apply.  They're also not widely applied as trimming the diameter of an impeller changes it's geometric similiarity and efficiencies become who knows what, outside of more than a 15% change in impeller diameter any resemblance between your estimate and the actual performance you get will be purely coincidental, and even under 15% you're flipping a coin.  I'm under the impression this is due to different clearance between volute tips and diffuser or cutwater, and different vane tip geometry associated with trimming.  That's why if you look at a pump curve the BEP at maximum trim usually has a higher efficiency than at minimum trim. You can use it to ballpark your final performance, but once in the ballpark your best bet is to have the impeller manufacturer run the numbers, most either have historic test data or software than can predict the efficiency losses.

Hope that helps

### RE: Affinity Laws or Similarity laws for Impeller Dia Changes

(OP)
Thanks for that,

So in essence what you are saying is that the first set of equations are only applied for trimming (or increasing)the impeller diameter within the same sized volute and the second set should be applied when the volute size is changed within a family of like pumps?

### RE: Affinity Laws or Similarity laws for Impeller Dia Changes

Scipio is right on. The first set of equations are used for pumps which have same casing size but with different diameter impellers. The second set of equations relate to geometrically similar pumps of different sizes. All these equations can be used just to get an approximation. I believe this topic was discussed in the past and you may get actual exponentials (emperical) by doing a search.

Regards,

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

### RE: Affinity Laws or Similarity laws for Impeller Dia Changes

(OP)
OK cheers for that,
Can anyone tell me why the second set of equations are not just used in all circumstances.

I plotted the pump curve using both sets of equations and there is very little difference, until the flows are higher, for small impeller diameter changes since D1/D2 is close to 1, so why not just use the single second set of equations?

Why do many technical handbooks say use the first set whereas text books seem to tell you to use the second?

### RE: Affinity Laws or Similarity laws for Impeller Dia Changes

To Bygeorge, the logic behind the first set of equations is that when the diameter is changed, the peripheral velocity changes. This suggests that a change in diameter could be translated into a change in rotating speed, N, as long as the efficiency doesn't change much. And since the affinity laws for a change in speed, are:

Q1/Q2=N1/N2; H1/H2=(Q1/Q2)2=(N1/N2)2

it is reasonable to assume that:

Q1/Q2=D1/D2; H1/H2=(D1/D2)2

Deviations from the rules stem from the fact that large diameter reductions involve changes in the geometry of the blades (i.e., outlet width, blade angle and blade length) increasing the mismatch with the casing volute.

On the other hand, and to confirm your statement on the possible use of the "model" affinity equations, Sulzer (Centrifugal Pump Handbook-Elsevier) tells us that when changing diameters of radial-flow impellers the relations would approximately be:

Q1/Q2=(D1/D2)m; H1/H2=(D1/D2)m

where m=2 for changes of greater than 6%, and m=3 for corrections of less than 1%.

Radial-flow impellers are those having specific speeds Ns<4200 for single suction, and <6,000 for double suction (in USCS units).

### RE: Affinity Laws or Similarity laws for Impeller Dia Changes

Bygeorge

"Why do many technical handbooks say use the first set whereas text books seem to tell you to use the second?"

Technical handbooks are used by services techs (mechanics) who have a pump in their hands, and if they want to change the performance of a specific pump this is the result. The only item they change easily is the impellor trim so the lower order laws apply.

Text books are used by engineers who have many more things the engineer can change, if an engineer wishes to design a pump for performance 'X1' which is some factor of an existing pump that performs 'X2', apply the laws to get a first draft, build, test, and adjust from there.

The first set of equations is based upon the impellor diameter. The second set of equations are based upon a characteristic length, which applies to all the dimensions,  (the impellor is used as a convenient measuring stick)
An example, a pure centrifugal pump, when the mechanic trims the impellor the thickness does not change but the diameter does, the eye remains the same, the volute remains the same, this results moderate flow changes and significant pressure changes,  pressure is directly related to the speed squared just as in the Bernoulli equation.  Now power is flow times pressure giving the cube function.
But when the engineer designs a new pump, every dimension is changed, the flow goes up (or down) by the cube because all the dimensions increased including the eye, impellor diameter, and thickness, and the volume of the volute increased by the cube,  this causes the flow to increase by the cube, the pressure still increases by the velocity squared which is proportional to impellor tip speed, and when you multiply the flow and pressure the power increased to the 5th.

Hydrae

### RE: Affinity Laws or Similarity laws for Impeller Dia Changes

(OP)
This is interesting - more so that the first set of equations hold for speed but not necessarily for dia changes yet so many publications (including some text books i have trawled up) seem to refer to them and only a few refer to the second set

further comment?

### RE: Affinity Laws or Similarity laws for Impeller Dia Changes

For the second set of laws, which, BTW, also applies to fans, the conditions for similarity of two pumps (model and prototype) are geometric, kinematic and dynamic.

The three resulting constants of proportionality are:

k1=DN/H0.5;   k2=Q/D2H0.5;   k3=P/D2H1.5

Where the constants have "absorbed" variables such as fluid density, Earth's gravity, and efficiency, assumed constant.

### RE: Affinity Laws or Similarity laws for Impeller Dia Changes

I do not want to confuse you but you mentioned speed: when you put a pump on a drive the laws are as follows.

Q1/Q2 = (n1/n2)
H1/H2 = (n1/n2)^2
P1/P2 = (n1/n2)^3

Notice it is an 'n' not a 'd'.
Hydrae

### RE: Affinity Laws or Similarity laws for Impeller Dia Changes

The two variables generally considered when applying the similarity equations for different pumps are: D (impeller diameter) and N (rotating speed). From the equations I brought above:

Q1/Q2=(N1/N2)(D1/D2)3
H1/H2=(N1D1/N2D2)2
P1/P2=(N1/N2)3/(D1/D2)5

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