1. MEP 4120 – Hydraulic Machines (A)
Lecture 6 Centrifugal Pumps: Characteristic Curves for Similar Pumps and Specific Speed

2. Centrifugal Pumps Performance
Again, as we learned previously that the main parameters involved in all hydraulic turbomachines and that influence their performance are: the fluid quantities (represented in the flowrate (Q) and the head (H)) and the mechanical (associated with the machine itself) quantities (represented in the power (P), speed (N), size (D) and efficiency (h)).
Although they are all of equal importance, the emphasis placed on certain of these quantities is different for pumps and turbines.
We concluded also that every machine has its own set of the performance characteristics when it is running at a constant given speed. These characteristics can be derived from the theoretical performance and mainly conducted as three curves (up to now) having the discharge (Q) as an independent variable versus the head (H), the input shaft power (P) and the overall efficiency (h) as dependent variables. These curves are normally plotted together on the same figure in what is so-called “the fundamental performance characteristic curves diagram”.

3. Centrifugal Pumps Performance
In real practice the procedures to obtain these real characteristics curves from the theoretical are headache and inconvenience. Thus, we simply employ the experimental method to determine such curves for each different turbomachine by carrying out a series of experiments on a particular test-rig especially prepared for this purpose in turbomachines production companies.
A typical test-rig is shown in the figure. The experiments are carried out for a given constant speed value in the same order as followed in the accompanying table to fill the columns. This is accomplished by varying the discharge in steps and then registering the corresponding readings of the other variables.

4. Centrifugal Pumps Performance
(All these results are conducted at a given measured value of (N) using a tachometer)
Exp. 1
Exp. 2 ……
Exp. x
Discharge Q (measured) √
Hm, s (measured)
Hm, d (measured)
Hm=Hm, d - Hm, s (calculated)
Psh (measured) using a dynamometer
Output fluid power(calculated) P=rgQHm
Overall efficiency(calculated) h=P/Psh= rgQHm/ Psh

5. Centrifugal Pumps Performance
A typical complete set of experimental results for the performance characteristics of a centrifugal pump, for example, is shown in the following figures.

6. Centrifugal Pumps Performance

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We would like to emphasis here on the idea that every set of these performance characteristics curves are given for a specific machine of certain size (diameter) and running at a specific constant speed. If we have another different machine or the same machine but running at another value of speed, then we will have another completely different set of characteristics curves.
Now, machines belonging to the same family, i.e., being of the same design but manufactured in different sizes and even maybe running at different speeds within practical limits. Thus, they constitute a series of geometrically similar machines. In fact, each size and speed combination will produce a unique set of characteristics, so that for one family of machines, the number of characteristics needed to be determined is impossibly large. Also, too large machines need expensive fixed and running costs for constructing and operating large test-rigs enough to test such machines .

8. Centrifugal Pumps Performance
Here, the importance of the dimensional analysis and similarity theory appears to solve this problem. As we reviewed before, the variables affecting the performance of the machines are replaced by a dimensionless groups which are valid for similar machines that operate under dynamically similar conditions. As we mentioned before, similar machines have the same figure of performance characteristics curves when this figure is plotted between the dimensionless groups of Φ versus each of ψ and Kp. One can note the disappearing of the efficiency curve where similar machines have the same efficiency corresponding to each Φ value.
At the end of the analysis, the dimensionless groups provided the similarity laws governing the relationships between the variables within one family of geometrically similar machines. Thus, if we apply these similarity laws between two different (for example, one prototype and one model) but similar machines and even if they are running at different speeds, one can be able to derive a set of characteristics curves for a prototype machine from a corresponding known set of test data for a geometrically similar model.

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In the following, the similarity laws will be given as:
Or in another form as:
Remembering that:

10. Centrifugal Pumps Performance
The following example is useful in , the similarity laws will be given as:
A centrifugal pump, impeller diameter of 0.5 m, when running at 750 rpm gave on test the following performance characteristics:
Predict the performance of a geometrically similar pump of 0.35 m diameter and running at 1450 rpm. Plot both sets of characteristics.
Performance characteristics of 0.5 m Pump at 750 rpm.
Q (m3/min) 7 14 21 28 35 42 49 56
H (m)
40.0
40.6
40.4
39.3
38.0
33.6
25.6
14.5
Efficiency (%) 41 60 74 83 51

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Let for simplicity the given pump has the suffix 1 instead of p (refer to 0.5 m pump) and the suffix 2 instead of m (refer to 0.35 m pump). Thus, from the equation:
The values of Q1 and H1 are given by the table above. Therefore, by multiplying them by the multipliers calculated above, the values of Q2 and H2 may be tabulated. These, together with the same values of efficiency constitute the predicted characteristic of pump 2 as follows:

12. Centrifugal Pumps Performance
Performance characteristics of 0.35 m Pump ② at 1450 rpm.
Q (m3/min)
4.64
9.28
13.92
18.56
23.2
27.8
32.5
37.0
H (m)
73.2
74.3
74.0
71.9
69.5
61.5
46.8
26.5
Efficiency (%) 41 60 74 83 51
Remembering the characteristic of pump 1 is as follows:
Performance characteristics of 0.5 m Pump ① at 750 rpm.
Q (m3/min) 7 14 21 28 35 42 49 56
H (m)
40.0
40.6
40.4
39.3
38.0
33.6
25.6
14.5
Efficiency (%) 41 60 74 83 51

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The characteristics of both pumps are plotted in the Figure.

14. Centrifugal Pumps Performance
Furthermore, the characteristics curves for the same machine i.e. self-similar , but, running at different speeds can be predicted using the similarity laws by simply putting (D1=D2) to obtain the simple form of the similarity laws (called also the affinity laws) as:

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SCALE EFFECTS:
In the application of the similarity laws it was assumed that all criteria of dynamical similarity are satisfied. i.e. all the dimensionless groups remain the same. In real case, this is not true with regards to the dimensionless groups representing the Reynolds number, the mach number and the relative roughness.
Consider at first the Reynolds number, which is a function of both the speed and the diameter. Thus in real practice, Re is not constant. However, for water and air this effect is usually small because the values of Re are usually very high, the flow being fully turbulent.
A similar consideration of Mach number indicates that an increase of tip speed (through increasing either N or D) will make the Mach number higher. This not only means that one of the conditions of dynamical similarity is not satisfied but, in addition, may also mean that the compressibility effect may now be of considerable importance. The second point must be watched carefully in the application of similarity laws to fans and compressors.

18. Centrifugal Pumps Performance
Consider now the effect of relative roughness. Absolute roughness (e) is the mean height of surface perturbances, which, therefore, remains the same for a given material and process used in the manufacture of the machine, irrespective of its size. Thus, any change of machine size involves a change of relative roughness (e/D). On the whole, the larger the machine, the smaller the relative roughness will be. This tend to make frictional losses relatively less important in larger machines.
In practice, it is also difficult to maintain geometrical similarity in clearances and some material thicknesses. The same gauge of sheet metal, for example, may be used for a range of sizes of fabricated impeller blades. Such deviations from geometrical similarity must obviously cause some departures from the idealized predictions based on the aforementioned similarity laws.
All such departures, which do occur in practice and which due to the Re, Mach number, relative roughness or lack of strict geometrical similarity, are usually referred to as the scale effect. In general, the scale effect tends to improve the performance of larger machines.

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SPECIFIC SPEED:
The performance of geometrically similar machines, i.e. machines belonging to one family, is governed by similarity laws and may be represented for the whole family by a single plot of dimensionless characteristics as have been discussed before.
Thus, the performance of machines belonging to different families may be compared by plotting their dimensionless characteristics on the same graph. In the Figure, for example, three different types of characteristic curves belonging to three different families of pumps are shown in the Figure.

20. Centrifugal Pumps Performance
Detailed comparison may then be achieved by analyzing the various aspects of the sets of curves. This method of comparison is satisfactory and often needed, but it lacks the brevity required in machine classification.
In fact to classify the machines families we need to define a new term called the specific speed or known also as type number.
We all know that every machine is designed to meet a specific duty, usually referred to as the design point. For a pump, for example, this would be stated in terms of the discharge and the head developed and, thus, represents a particular point on its basic performance characteristic. The design point is normally associated with the maximum efficiency of the machine.
It is useful to compare machines by quoting the values of Φ, ψ, and KP corresponding to their design points. However, since for pumps Φ and ψ are the two most important parameters, their ratio would indicate the suitability of a particular pump for large or small volumes relative to the head developed.

21. Centrifugal Pumps Performance
Furthermore, if the ratio is obtained in such a way that the impeller diameter is eliminated from it, then the comparison becomes independent of the machine size.
This can be achieved by raising Φ to power (½) and ψ to power (¾). This process results the specific speed parameter as:
It must be realized that a value of specific speed can be calculated for any point from the infinitely number of points on the characteristic curve. However, all of these points of no practical interest except one point only which lies at the design point or (BEP). Thus, this point is used for define the specific speed, i.e. for machine classification as well as comparison and design purposes.

22. Centrifugal Pumps Performance
The specific speed, since it is obtained from dimensionless coefficients, is also a dimensionless quantity provided a constant system of units, such as SI, is used. Thus, if we use the discharge in m3/s, the head in m and the speed in rev/s or rad/s, then we will obtain the dimensionless specific speed in as(rev) or (rad) as:
Unfortunately, in real practice there are many companies around the world which produce pumps. Usually, they use their own local system of units. Therefore, the specific speed becomes inconsistent and depends essentially on the units used for the definition of each relevant quantity within it. Hence, a conversion factor should be applied when we use different systems of units.

23. Centrifugal Pumps Performance
The conversion factors of specific speed for some different units into SI units are given in the Table:
With the aid of specific speed the various types of pumps may be classified and compared as will be described in the following figures. Also, since the specific speed refers to the design point it is used as the most important design parameter in the field of turbomachinery design.

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