ON METHODS OF APPLYING THE LAWS OF PROPORTIONALITY IN THE OPERATION OF TURBOMACHINES

B. M. TITOV

In the operation of mine turbomachines (fans, pumps, and turbocompressors), it is sometimes necessary to determine how the capacity \(Q\), head \(H\), and shaft power \(N\) of a turbomachine will change if, with the same characteristic of the network (pipeline), the number of revolutions of the turbomachine shaft is changed. This problem is usually solved by means of the well-known laws of proportionality:

\[ \frac{Q_2}{Q_1}=\frac{n_2}{n_1}, \tag{1} \]

\[ \frac{H_2}{H_1}=\left(\frac{n_2}{n_1}\right)^2, \tag{2} \]

\[ \frac{N_2}{N_1}\simeq\left(\frac{n_2}{n_1}\right)^3, \tag{3} \]

\[ \eta_2\simeq \eta_1 . \tag{4} \]

The laws of proportionality are often incorrectly called the laws of the French professor A. Rateau, who formulated them in 1892 on the basis of laboratory experiments on fans and pumps [7] and asserted:

1) the action of the laws of proportionality extends to all turbomachines when they operate at any constant equivalent orifice;

2) the laws of proportionality can be proved only on the basis of experimental results.

A. Rateau arrived at these erroneous conclusions because, in laboratory tests of pumps, it is impossible to create a significant geodetic head; moreover, he did not set himself the task of verifying the operation of turbomachines in the presence of static resistance in the pipeline, for example, geodetic head.

The analytical proof of the laws of proportionality belongs to Academician A. P. German, who first published his conclusions in 1925 [1].

The equations of the characteristics of turbomachines obtained by A. P. German enabled him to prove that the laws of proportionality characterize not the properties of turbomachines when they operate at any constant equivalent orifice, as A. Rateau imagined, but only the nature of their behavior when operating in similar regimes\(^1\), i.e., in networks without static resistance.

A. P. German was the first to draw attention to the fact that the laws of proportionality can be used directly only in the case when the characteristic of the pipeline is represented by a quadratic parabola—

\(^1\) Similar regimes are those operating regimes of a turbomachine in which the similarity of the velocity triangles is preserved.

passing through the origin, since in this case the proportionality curve coincides with the network characteristic:

\[ H = RQ^2, \tag{5} \]

where \(R\) is the pipeline constant.

As an example of such a network one may cite the ventilation network of a mine in the absence of natural draft, the pipeline of the circulation pump of a heating system, etc.

In mine drainage, compressor, and sometimes also fan installations, however, in addition to dynamic resistance there is also static resistance: \(H_g\)—the geodetic height of water lifting for pumps, \(H_{\text{back}}\)—the air pressure before the consumers for turbocompressors, \(H_{\text{nat}}\)—the magnitude of the natural draft for fans, \(H_0\)—the pressure difference in the common section of an air duct in flanged ventilation, created by other fans operating in parallel; therefore, in most practical cases the network characteristics do not pass through the origin and are expressed by an equation of the form:

\[ H = H_{\text{st}} + RQ^2. \tag{6} \]

In these cases the operating regime of a turbomachine when the speed of its shaft is changed must be determined not directly from the laws of proportionality, but by the graphical-analytical method. In applying the graphical-analytical method two cases may occur:

1) it is required to determine the operating regime of the turbomachine for a given network characteristic and a previously known new speed \(n_2\);

2) it is required to determine the speed \(n_2\) at which the turbomachine, in the given pipeline, will produce the necessary capacity and head.

The graphical-analytical method is described in general terms in the literature [2], [3]; here, to explain its application, we shall give only specific examples for each case.

Example 1. A drainage installation equipped with a three-stage fifty-cubic-meter pump of the “Komsomolets” type delivers water to a height \(H_g = 30\) m. The hourly inflow of water is \(Q_{\text{in}} = 20\ \text{m}^3/\text{h}\). The pump characteristic and its operating regime at \(n_1 = 1450\) rpm are known (see Fig. 1, point \(K\)): \(Q_1 = 61\ \text{m}^3/\text{h}\), \(H_1 = 71\) m water column, \(\eta_1 = 0.57\), \(N_1 = 20.7\) kW. It is required to determine the operating regime of this pumping installation at \(n_2 = 960\) rpm and to check whether it will cope with the inflow.

Solution.

1. We recalculate the pump characteristic from \(n_1 = 1450\) rpm to \(n_2 = 960\) rpm, using the laws of proportionality (1), (2), and (4). The results of the calculation are given in Table 1, from which the pump characteristic at \(n_2 = 960\) rpm is plotted in Fig. 1 (dashed line).

2. We determine the pipeline constant, substituting \(Q\) and \(H\) in the dimensions in which the pump characteristic has been plotted:

\[ R = \frac{H_1 - H_g}{Q_1^2} = \frac{71 - 30}{61^2} = 0.011, \]

and, assigning different values of \(Q\), construct the network characteristic from equation (6)

\[ H = 30 + 0.011Q^2. \]

The results of the calculation for plotting the network characteristic are given in Table 2, and its construction is made in Fig. 1.

1. From the point of intersection of the characteristics \(K'\) we determine the operating regime of the pump at \(n_2 = 960\) rpm:

\[ Q_2 = 25\ \text{m}^3/\text{h}; \qquad H_2 = 37\ \text{m water column}; \qquad \eta_2 = 0.57. \]

Table 1

Fig. 1. Characteristics of the “Komsomolets” pump. \(Q = 50\) m\(^3\)/h.
1—at \(n = 1450\) rpm,
2—at \(n = 960\) rpm,
3—proportionality curve,
\(i\)—number of impellers.

Table 2

In our example it turned out that \(\eta_2=\eta_1\), but this is only a special case, since \(\eta_1\) lies on the descending branch, and \(\eta_2\) on the ascending branch of the efficiency curve; see Fig. 1.

\[ N_2=\frac{25\cdot37\cdot1000}{3600\cdot102\cdot0.57}=4.4\ \text{kW}. \]

Since \(Q_{\mathrm{pr}}=20\ \text{m}^3/\text{h}<Q_2=25\ \text{m}^3/\text{h}\), the pump with the existing drive is quite adequate.

In this example the pump delivery could have been reduced not by changing the rotational speed, but by reducing the number of impellers; however, in that case (see Fig. 1), at \(i=2\) the pump delivery decreases only slightly and is equal to \(46\ \text{m}^3/\text{h}\), while at \(i=1\) the pump delivers no water at all; i.e., the most successful solution of the problem is to change the rotational speed of the pump shaft.

If the same problem is solved by a simple recalculation according to the laws of proportionality, the following result is obtained:

\[ Q_2'=Q_1\frac{n_2}{n_1}=61\cdot\frac{960}{1450}=40\ \text{m}^3/\text{h}, \]

\[ H_2'=H_1\left(\frac{n_2}{n_1}\right)^2 =71\cdot\left(\frac{960}{1450}\right)^2 =31\ \text{m water column}, \]

\[ N_2'=N_1\left(\frac{n_2}{n_1}\right)^3 =20.7\cdot\left(\frac{960}{1450}\right)^3 =6\ \text{kW}. \]

This occurs because the proportionality curve intersects the pump characteristic at \(n_2=960\ \text{rpm}\) at point \(K''\), and not at point \(K'\); see Fig. 1.

Consequently, with direct recalculation according to the laws of proportionality, the following errors are obtained in this example:

\[ \Delta Q\%=\frac{Q_2'-Q_2}{Q_2}\cdot100 =\frac{40-25}{25}\cdot100=+60\% \]

\[ \Delta H\%=\frac{H_2'-H_2}{H_2}\cdot100 =\frac{31-37}{37}\cdot100=-16.2\% \]

\[ \Delta N\%=\frac{N_2'-N_2}{N_2}\cdot100 =\frac{6-4.4}{4.4}\cdot100=+36.5\% \]

Example 2. For the water-drainage installation of the preceding example, determine what rotational speed must be given to the pump shaft in order to obtain a delivery \(Q_2=61\ \text{m}^3/\text{h}\), if the characteristic and operating condition of the pump at \(n_1=960\ \text{rpm}\) are known (see Fig. 1, point \(K'\)): \(Q_1=25\ \text{m}^3/\text{h}\), \(H_1=37\ \text{m water column}\), \(N_1=4.4\ \text{kW}\), \(\eta_1=0.57\).

Solution.

1. We determine the pipeline constant

\[ R=\frac{H_1-H_2}{Q_1^2}=\frac{37-30}{25^2}=0.011 \]

and, analogously to the preceding example, construct the pipeline characteristic in Fig. 1.

1. From the constructed network characteristic and the specified delivery \(Q_2=61\ \text{m}^3/\text{h}\), we determine (see Fig. 1, point \(K\)) the required pump head \(H_2=71\ \text{m water column}\).

1. We determine the constant \(C\) of the proportionality curve intersecting the pipeline characteristic at the specified capacity \(Q_2 = 61\ \text{m}^3/\text{h}\)

\[ C=\frac{H_2}{Q_2^2}=\frac{71}{61^2}=0.0191, \]

and, assigning various values to \(Q\), we construct the proportionality curve from the equation:

\[ H=CQ^2=0.0191Q^2. \]

The calculations required for constructing the proportionality curve are collected in Table 3, and the construction itself is shown in Fig. 1 by the dashed line.

Table 3

1. From the intersection of the proportionality curve with the pump characteristic at \(n_1=960\ \text{rpm}\) (point \(K''\)), we find \(Q''=40\ \text{m}^3/\text{h}\) and determine the number of revolutions that must be imparted to the pump shaft in order to obtain the specified capacity \(Q_2=61\ \text{m}^3/\text{h}\):

\[ n_2=n_1\frac{Q_2}{Q''}=960\frac{61}{40}=1450\ \text{rpm}. \]

When solving this same problem by simple recalculation according to the laws of proportionality, one obtains:

\[ n_2'=n_1\frac{Q_2}{Q_1}=960\frac{61}{25}=2340\ \text{rpm}. \]

Consequently, the error in determining the number of revolutions is:

\[ \Delta n\%=\frac{n_2'-n_2}{n_2}\cdot100=\frac{2340-1450}{1450}\cdot100=+61\%. \]

The examples given prove that, when there is a geodetic head in the pipeline, the operating condition of the pump at a new speed of its shaft must necessarily be determined by the graphoanalytical method.

However, it is necessary to note and emphasize that the graphoanalytical method, which follows from the instructions of Academician A. P. German, is not mentioned at all in the pump manuals published by Ugletekhizdat, for example: [4], [5], [6]; moreover, some of them \([4,40]\), \([6,27]\) mislead production workers. For example, in the brochure Mine Centrifugal Pump “GIM”, Ugletekhizdat, 1952, p. 27, it is written: “The indicated values \(Q\), \(H\), and \(N\) change depending on the change in the number of revolutions \(n\) according to the laws of proportionality, and deviation from these laws indicates abnormality in operation …”

pump, under which its continued operation is impermissible”¹).

In reality, however, for geodetic heads corresponding to the actual operating conditions of mine pumps, this deviation is inevitable, but it indicates not abnormal operation of the pump, but only an incorrect use of the laws of proportionality.

The formulation of the laws of proportionality in the indicated instructions (as well as in [4], [5]) is imprecise; for example, there, on p. 27, we read: “The quantity of water delivered by a pump is directly proportional to its number of revolutions,” i.e., it is not indicated at all under what conditions this proportionality is observed.

It would be more correct to formulate the laws of proportionality as follows: the capacity of a turbomachine when operating at similar regimes varies in direct proportion to the first power of the ratio of the numbers of revolutions of its shaft. Or: the capacity of a turbomachine when operating on an unchanged quadratic characteristic of a network without static resistance varies in direct proportion to the first power of the ratio of the numbers of revolutions of its shaft.

The same applies to the formulation of the remaining laws of proportionality, since the laws of proportionality do not have universal significance, but are valid only in the particular case of operation of a turbomachine on a network without static resistance.

Conclusion

1. The laws of proportionality reflect not the properties of turbomachines when operating on any constant equivalent orifice, as the French professor A. Rateau imagined, but only the character of their behavior when operating at similar regimes.

2. The analytical derivation and correct explanation of the laws of proportionality were first given by Academician A. P. German.

3. The operating regime of a turbomachine when the shaft speed is changed and the network characteristic remains the same should be determined:

a) if the resistance of the pipeline is purely dynamic (\(H = RQ^2\)), then by direct recalculation according to the laws of proportionality;

b) if there is static resistance in the pipeline (\(H = H_{\mathrm{st}} + RQ^2\)), as is the case in mine drainage, compressor, and sometimes also fan installations, then necessarily by a graphical-analytical method, using the laws of proportionality only for recalculating the characteristic of the turbomachine;

c) if there is a static resistance in the pipeline that is small, in percentage terms, relative to the total head, and there is no graphical characteristic of the turbomachine, it is possible to use direct recalculation according to the laws of proportionality, taking into account, however, that the result obtained in this case will contain an error proportional to the static resistance of the pipeline and to the change in the shaft speed of the turbomachine.

1. In the future, when preparing instructions for pumps, the Bureau of Technical Information must explain the graphical-analytical method for determining the operating regime of a pump when the speed of its shaft is changed and indicate the range of application of the laws of proportionality.

¹ B. Titov.

1. The examples presented show what impermissible errors result from the simplified application of the laws of proportionality.

REFERENCES

1. A. P. German. Turbomachines. KUBUCH Publishing House, 1925.
2. V. I. Polikovskii. Fans, Blowers, Compressors. Energoizdat, 1938.
3. V. S. Pak and V. G. Geier. Mine Ventilating and Dewatering Installations. Ugletekhizdat, 1950.
4. The “Komsomolets” Mine Centrifugal Pump. Ugletekhizdat, 1948.
5. N. F. Gorokhov. The GMS-2 Mine Centrifugal Pump. Ugletekhizdat, 1950.
6. The “TIM” Mine Centrifugal Pump. Ugletekhizdat, 1952.
7. A. Rato. High-Pressure Centrifugal Pumps and Fans. St. Petersburg, 1907, supplement 1.