Prof. V. N. Pinegin.

The Banki Turbine and Its Theoretical Efficiency

The end of the 19th and the beginning of the 20th century, approximately up to 1918, may be noted as a period of lull in the appearance of new forms of water turbines. Both theory and practice, apparently, were occupied exclusively with questions of expanding the fields of application and improving the qualities of two types of turbines that had already succeeded in establishing themselves—Pelton’s and Francis’s. Indeed, the successes achieved¹ in this direction surpassed all expectations: it became possible to raise the efficiency of Francis turbines to 0.937, of Pelton turbines to 0.90; the power of individual units rose, for Francis turbines, to 55,000 h.p., and for Pelton turbines to 30,000 h.p.; on the other hand, the very same turbines are manufactured with capacities down to tenths of a horsepower. Thanks to such technical possibilities and to the considerably expanded limits of speed, it became fully possible, in all cases of utilizing water energy, to manage with only these two types of turbines; moreover, the choice of one type or the other, according to local conditions, presented no difficulties. It seemed that science and technology in the field of turbine construction had reached such perfection that there was nowhere further to go. But beginning in 1918, new forms of turbines unexpectedly began to appear—and, moreover, turbines which, in certain of their qualities, yielded little to the former forms.

Leaving aside such turbines as the Hydrauccon–Francis turbine² and the Moody turbine³, of which the first differs essentially from the Francis turbine only by the special form of its draft tube, while the second is a transitional form from Francis turbines to the Kaplan turbine indicated below, we shall dwell somewhat more fully on the following three types of turbines, which indeed present greater peculiarities: the Kaplan, Latwaczeck, and Banki turbines.

The first two turbines belong to the group of so-called turbines with overpressure, or, according to the old nomenclature, reaction turbines; the last belongs to the group of turbines without pressure, or impulse turbines.

The Kaplan turbine, which at one time bore the name of the Nagler turbine⁴, may, by the form of its runner, be called a propeller turbine, since its blades, the number of which in it is reduced to a minimum—4—do indeed have the form of propeller blades. Thanks to such a limited num-

in view of the number of blades, the introduction of water between them according to the generally accepted theory of turbines is entirely inadmissible: until recently it was considered not only incorrect, but also impossible, to reduce the number and width of the blades to such an extent that, when the turbine is at rest, water could pass between the blades without even touching them. However, in Kaplan’s turbine such passage of water, which is indeed possible when it is at rest, is completely eliminated during its operation, owing to the extremely high number of its revolutions and the mutual interaction of the water particles upon one another; the extremely high number of revolutions constitutes the merit of this turbine. The specific-speed coefficient of Kaplan turbines has at present been brought up to 2000 (reduced number of revolutions—550), although, to be sure, official tests are known only for turbines with a specific speed of 700–1200 (reduced number of revolutions 190–330), and it turned out that at specific speeds of 700–800 the efficiency had a value of 0.839, while at specific speeds of 1200 the efficiency did not exceed 0.775¹).

Quite recently Kaplan has considerably improved his turbine by making the blades of the runner adjustable, whereby constancy of the turbine efficiency under variable operating conditions has been attained. The field of application of Kaplan turbines is low-head installations.

Lawaczeck’s turbine likewise has a very limited number of blades, and moreover blades of a very distinctive form: their surface is a helicoidal surface whose generating curve is a parabola. Owing to the considerably greater length of the blade surfaces, the proper guidance of water between the blades is here fully ensured, and free passage of water, as between the blades in Kaplan’s turbine, no longer exists. The specific-speed coefficient of Lawaczeck turbines ranges between 450–800 (reduced number of revolutions 120–220). The efficiency of such a turbine, when tested at Charlottenburg, proved to be 0.87²). The field of application of Lawaczeck’s turbine is for such small heads for which even the highest-speed Francis turbines already become inapplicable.

The Banki turbine is nothing other than a water wheel to which the water is supplied along a wide, but very small in height, rectangular nozzle from the outside; after passing over the curved blades of the wheel and having given up the last part of its energy, the spent water enters the interior of the wheel and is again directed, now from within, toward the blades, to which, after giving up the next portion of energy, it finally exits outward. Thus, in the Banki turbine the water works as if according to the principle of double action, whereby, on the one hand, the loss at the outlet of water from the wheel can be considerably reduced, and on the other hand the runner, thanks to the possibility of considerably widening the wheel, is made independent in its diameter of the quantity of water; this latter circumstance makes it possible to impart extraordinarily high numbers of revolutions to the wheel and considerably to reduce its weight (down to 1 kg per 1 h.p.³).

In this possibility of reducing the weight of the turbine, as well as in the high efficiency (reaching up to 0.92), lies the advantage of the Banki turbine over the Pelton turbine. In its applicability the Banki turbine embraces to a considerable extent the fields of application of both Pelton and Francis turbines. In view of the extraordinary interest presented by the Banki turbine, by virtue of the properties indicated, I, without pausing for the present over the other turbines enumerated above, in the present article make an attempt at a theore—

¹) Mitteilungen über Bremsungen des Kaplan-Turbinen-Konzernes in Brünn. Die Wasserkraft. 1922. H. No. 4, p. 73.
²) Budau. Neuere Turbinen Konstruktionen. Die Wasserkraft. 1922. H. No. 3, p. 46 ff.
³) Ibid.

...of a theoretical investigation of this turbine, chiefly of its efficiency coefficient, all the more so since, as far as I know, this has not been done up to the present time. First of all, it is not difficult to show that the Banki turbine also satisfies that fundamental relation between the quantity of working water (\(Q\) in m\(^3\)/sec.), the head (\(H\) in m), the number of revolutions per minute (\(n\)), and the reduced number of revolutions (\(n_0\)), to which the Francis and Pelton turbines are subject¹:

\[ n=n_0\frac{\sqrt[4]{H^3}}{\sqrt{Q}} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (1) \]

Indeed, it is obvious that for the given wheel, as for other wheels, the peripheral velocity (\(u_1\)) may be represented by the relation:

\[ u_1=\frac{\pi D_1 n}{60}=\lambda_0\sqrt{2gH}, \ldots \ldots \ldots \ldots \ldots \ldots \ldots (2) \]

where, in addition to the notations already indicated, \(D_1\) denotes the outside diameter of the wheel in meters, and \(\lambda_0\) is a coefficient usually taken equal to \(0.44\)—\(0.46\); from the latter relation we have

\[ n=\frac{60\lambda_0\sqrt{2gH}}{\pi D_1}. \ldots \ldots \ldots \ldots \ldots \ldots \ldots (3) \]

Since, further, denoting the width of the nozzle opening by \(b\), and its height by \(a\), we may always, on the one hand, put

\[ \frac{a}{b}=m,\quad \frac{b}{D_1}=k, \ldots \ldots \ldots \ldots \ldots \ldots \ldots (4) \]

and on the other hand, obviously, denoting the velocity of outflow of water from the nozzle by \(c_0\), with \(c_0=\varphi\sqrt{2gH}\), where \(\varphi \sim 0.96\)—\(0.98\), we have

\[ Q=ab\,c_0, \]

then, substituting in the latter relation the values of \(a\) and \(b\) from (4) and the value of \(c_0\), we obtain

\[ Q=mb^2\varphi\sqrt{2gH}=mk^2D_1^2\varphi\sqrt{2gH}; \ldots \ldots \ldots (5) \]

substituting now from the latter relation the value of \(D_1\):

\[ D_1=\frac{\sqrt{Q}}{k\sqrt{m\varphi}\sqrt[4]{2gH}} \ldots \ldots \ldots \ldots \ldots (6) \]

in expression (3) for \(n\), we have

\[ n=\frac{60\lambda_0 k\sqrt{m\varphi}\,(2gH)^{3/4}}{\pi\sqrt{Q}} \ldots \ldots \ldots \ldots (7) \]

Putting in expression (7) \(Q=1\) and \(H=1\), we obtain the expression for the reduced number of revolutions

\[ n_0=\frac{60\lambda_0\sqrt{m\varphi}\,k(2g)^{3/4}}{\pi} \ldots \ldots \ldots \ldots (8) \]

and, dividing relation (7) by (8), we have

\[ n=n_0\frac{\sqrt[4]{H^3}}{\sqrt{Q}}, \ldots \ldots \ldots \ldots \ldots \ldots (9) \]

which is what was required to prove.

Let us now see what relation must exist between the velocities of the water flow along the blades of the wheel, the peripheral velocities, and the elements of the wheel—

Banki turbine wheel. In Fig. No. 1 the runner of the turbine under consideration is shown schematically in section; the designations are indicated there for all velocities of the water flow, circumferential velocities, and the angles made by the blades with the inlet and outlet surfaces. According to this drawing, first of all we have

\[ w_1^2=u_1^2+c_0^2-2u_1c_0\cos\alpha_0 =\lambda_0^2(2gH)+\varphi^2(2gH)-4\lambda_0\varphi gH\cos\alpha_0, \]

whence

\[ w_1=\sqrt{2gH}\cdot \sqrt{\lambda_0^2+\varphi^2-2\lambda_0\varphi\cos\alpha_0} \tag{10} \]

If we further assume that the absolute exit velocity \((c_2)\) of the water into the interior of the wheel is normal to the inner circumference of the wheel, then evidently there must exist the relation

\[ w_2^2=u_2^2+c_2^2, \]

where

\[ u_2=u_1\frac{R_2}{R_1}=\lambda_0\mu u_1, \]

denoting by \(\mu\) the ratio of the inner radius \((R_2)\) of the wheel to the outer radius \((R_1)\). With velocity \(c_2\) the water leaves the blades, passes through its inner space, and again approaches the blades, but now from the inside; for normal entry, evidently, there must exist the relation

\[ w_3^2=u_2^2+c_2^2, \]

whence it follows

\[ w_2=w_3, \]

which is quite probable. Since, further, the velocity \(c_2\) plays on entry into the wheel from the inside the same role as \(c_0\) on entry from the outside, we must put

\[ u_2=\frac{\lambda_0}{\varphi}\sqrt{2g\frac{c_2^2}{2g}} =\frac{\lambda_0}{\varphi}c_2, \]

or we have

\[ c_2=\frac{\varphi u_2}{\lambda_0} =\frac{\varphi\mu u_1}{\lambda_0} =\varphi\mu\sqrt{2gH} \]

\[ u_2=\lambda_0\mu\sqrt{2gH} \]

The latter relations make it possible to determine \(w_2=w_3\):

\[ w_3=w_2=\mu\sqrt{2gH}\sqrt{\varphi^2+\lambda_0^2}. \tag{11} \]

Putting further

\[ c_3=\sqrt{2gH\Delta}, \]

and assuming that this velocity is normal to the direction of the relative exit velocity \(w_4\), we have

\[ w_4^2=u_1^2-c_3^2, \]

whence, after the appropriate substitutions,

\[ w_4=\sqrt{2gH}\sqrt{\lambda_0^2-\Delta}. \tag{12} \]

Let us now assume that the ratio of \(w_4\) to \(w_3\) is the same as that of \(w_2\) to \(w_1\), i.e. let us assume that

\[ \frac{w_2}{w_1} = \frac{\mu\sqrt{\varphi^2+\lambda_0^2}} {\sqrt{\lambda_0^2+\varphi^2-2\lambda_0\varphi\cos\alpha_0}} = \frac{w_4}{w_3} = \frac{\sqrt{\lambda_0^2-\Delta}} {\mu\sqrt{\varphi^2+\lambda_0^2}}, \]

whence

\[ \mu= \sqrt{ \frac{\sqrt{\lambda_0^2-\Delta}} {\varphi^2+\lambda_0^2} \cdot \sqrt{\lambda_0^2+\varphi^2-2\lambda_0\varphi\cos\alpha_0} }; \tag{13} \]

for any other ratios \(\frac{w_4}{w_3}\) to \(\frac{w_2}{w_1}\) we shall, obviously, obtain another value of \(\mu\).

Let us now try to determine the hydraulic losses in the Banki turbine and its efficiency. These losses, evidently, like the losses in

in a Pelton turbine, are divided into losses in the runner and losses in the supply organs; the former, in turn, are divided into losses:

1) at entry into the runner;
2) due to friction as the water flows over the blades;
3) due to curvature of the path as the water flows over the blades;
4) at the exit of the water from the runner. Losses in the supply organs are divided into losses in
5) the nozzle
6) the supply pipe.

The loss at entry into the runner is determined from the following considerations: although the blades of the runner have a small thickness, 3–5 mm, nevertheless the angle \((\beta)\) of their sharpening at the entrance is hardly made less than \(8—10^\circ\); assuming\(^1\) even that the inlet angle \(\alpha_1\) is formed by the inlet surface and a straight line bisecting the sharpening angle \((\beta)\), we find that the velocity lost to impact, which the jet experiences upon entering the blade, is determined by the quantity \(w_1 \sin \gamma = w_1 \sin \beta/2\).

Thus, the head lost to impact will be represented by the relation

\[ h_1=\frac{w_1^2\sin^2\gamma}{2g}, \]

and, substituting the value of \(w_1\), we have

\[ h_1=\frac{2gH(\lambda_0^2+\varphi^2-2\lambda_0\varphi\operatorname{Cos}\alpha_0)\operatorname{Sin}^2\gamma}{2g}. \]

The work lost to impact will obviously be

\[ L_1=\delta Qh_1=\delta H(\lambda_0^2+\varphi^2-2\lambda_0\varphi\operatorname{Cos}\alpha_0)\operatorname{Sin}^2\gamma\, mk^2D_1^2\varphi\sqrt{2gH}, \]

denoting the weight of one cubic meter of water by \(\delta\), and substituting the value of \(Q\) from (5); entering into the latter relation the value \(D_1\) from (3), we have

\[ L_1=\delta(\lambda_0^2+\varphi^2-2\lambda\varphi\operatorname{Cos}\alpha_0)\operatorname{Sin}^2\gamma\, mk^2\varphi(2g)^{3/2}60^2\lambda_0^2\frac{H^{5/2}}{\pi^2 n^2}. \]

The ratio of the inlet loss in work to the total work \((\delta QH)\) of the turbine will be

\[ \frac{L_1}{L}=(\lambda_0^2+\varphi^2-2\lambda\varphi\operatorname{Cos}\alpha_0)\operatorname{Sin}^2\gamma\, mk^2\varphi(2g)^{3/2}\frac{60^2\lambda_0^2}{\pi^2}\frac{H^{3/2}}{Qn^2}=\frac{A_1}{n_0^2}, \tag{14} \]

denoting the whole factor multiplying \(\dfrac{H^{3/2}}{Qn^2}\) by \(A_1\) and having in view relation (9).

The loss due to friction as the water flows over the blades of the runner is determined by the known relation

\[ h_2=\xi_2\frac{l}{R}\frac{w^2}{2g}, \]

where \(\xi_2\) is an experimental coefficient, \(R\) is the hydraulic radius, \(l\) is the length of the water path over the blade, and \(w\) is the velocity of the water flow. In view of the fact that the velocity \(w\) changes all the time and the exact law of its variation with the length of the path is unknown, we shall divide the entire path into 4 parts and for each of them adopt velocities respectively equal to \(w_1, w_2, w_3\), and \(w_4\); then the preceding relation will take the form, if by \(l\) we shall still understand the full length of the water path on the blades of the wheel:

\[ h_2=\frac{\xi_2}{R}\frac{l}{8g}\{w_1^2+w_2^2+w_3^2+w_4^2\} =\frac{\xi_2}{R}\frac{l}{8g}\{w_1^2+2w_2^2+w_4^2\}. \]

\(^1\) To reduce the sum of all resistances due to curvature it would be more advantageous to incline the straight line bisecting the blade sharpening angle slightly from the direction of the jet inward; we do not do this, in view of the comparative smallness of the sharpening angle \((\beta)\); see on this point R. Thomann. Die Wasserturbinen. 1908, p. 67.

The work lost in impact is determined as follows:

\[ L_2=\delta Q h_2=\frac{\delta \xi_2}{8g}\frac{1}{R}\{w_1^2+2w_2^2+w_4^2\}\,mk^2D_1^2\varphi\sqrt{2gH}; \]

and, substituting the values of \(w_1,\ w_2,\ w_4\) and \(D_1\), we have

\[ L_2=\delta \xi_2\frac{1}{4R}mk^2\varphi(2g)^{3/2} \frac{60^2\lambda_0^2H^{5/2}}{\pi^2n^2} \{2\lambda_0^2+\varphi^2-2\lambda_0\varphi\operatorname{Cos}\alpha_0 +2\mu^2(\varphi^2+\lambda_0^2)-\Delta\}. \]

The ratio of the work lost in friction to the total work of the turbine will be

\[ \frac{L_2}{L} = \xi_2\frac{1}{4R}mk^2\varphi(2g)^{3/2} \frac{60^2\lambda_0^2}{\pi^2} \{2\lambda_0^2+\varphi^2-2\lambda_0\varphi\operatorname{Cos}\alpha_0 +2\mu^2(\varphi^2+\lambda_0^2)-\Delta\} \times \]

\[ \times \frac{H^{3/2}}{n^2Q} = \frac{A_2}{n_0^2}, \qquad \ldots\ldots\ldots\ldots\ldots (15) \]

bearing in mind again relation (9) and denoting the entire expression preceding \(\frac{1}{n_0^2}\) by \(A_2\).

The head loss due to curvature of the path of the water jets as they flow along the blades of the runner may also be determined from the known relation\(^1\):

\[ h_3=\frac{\delta_0}{90}\xi_3\frac{w^2}{2g}, \]

where \(\delta_0\) is the angle of deflection of the jet from the initial direction, and the coefficient

\[ \xi_3=0.124+3.104\left(\frac{e}{\rho}\right)^{3.5} \]

is for the case of a rectangular jet, where \(2e\) is the thickness of the jet and \(\rho\) is the radius of curvature of the curved path of the jet. In view of the variable value of \(w\), here too the whole path has to be divided into 4 parts, taking the velocities for each of them respectively equal to \(w_1,\ w_2,\ w_3\), and \(w_4\); therefore we shall have

\[ h_3=\frac{\delta_0}{360}\frac{\xi_3}{2g}\{w_1^2+2w_2^2+w_4^2\}. \]

The work lost through curvature is represented as

\[ L_3=\delta Qh_3 = \frac{\delta\delta_0}{360}\frac{\xi_3}{2g} \{w_1^2+2w_2^2+w_4^2\} mk^2D_1^2\varphi\sqrt{2gH}, \]

and, substituting the values of \(w_1,\ w_2,\ w_4\) and \(D_1\), we shall have

\[ L_3= \frac{\delta\delta_0}{360}\xi_3mk^2\varphi(2g)^{3/2} \frac{60^2\lambda_0^2H^{5/2}}{\pi^2n^2} \{2\lambda_0^2+\varphi^2-2\lambda\varphi\operatorname{Cos}\alpha_0 +2\mu^2(1+\lambda_0^2)-\Delta\}. \]

For the ratio of the work lost through curvature to the total work we obtain the expression

\[ \frac{L_3}{L} = \frac{\delta_0}{360}\xi_3mk^2\varphi(2g)^{3/2} \frac{60^2\lambda_0^2}{\pi^2} \{2\lambda_0^2+\varphi^2-2\lambda\varphi\operatorname{Cos}\alpha_0 +2\mu^2(1+\lambda_0^2)-\Delta\} \times \]

\[ \times \frac{H^{3/2}}{Qn^2} = \sigma\frac{A_3}{n_0^2}, \qquad \ldots\ldots\ldots\ldots\ldots (16) \]

introducing again the reduced number of revolutions and denoting the entire expression preceding \(\frac{1}{n_0^2}\) by \(A_3\).

The loss at the exit of the water from the runner is found in the following way: the absolute exit velocity of the water from the wheel is expressed as

\[ c_3=\sqrt{2gH\Delta}, \]

\(^1\) Ph. Forchheimer. Hydraulik. 1914, p. 243.

whence we have the loss in head

\[ h_4=\frac{c_3^2}{2g}=\Delta H. \]

The work lost at the outlet will be

\[ L_4=\delta Q h_4=\delta \Delta H\, m k^2 \varphi (2g)^{3/2} \frac{60^2\lambda_0^2 H^{3/2}}{n^2\pi^2}, \]

and for the ratio of the work lost at the exit of the water from the wheel to the total work of the turbine we shall have the expression

\[ \frac{L_4}{L} = \Delta m k^2 \varphi (2g)^{3/2} \frac{60^2\lambda_0^2}{\pi^2} \frac{H^{3/2}}{Q n^2} = \frac{A_4}{n_0^2}, \qquad \ldots\ldots (17) \]

introducing, as before, the reduced number of revolutions and denoting the entire expression preceding \(\dfrac{1}{n_0^2}\) by \(A_4\).

Thus, the total relative loss in the runner is represented in the form:

\[ S_1=\sum_{i=1}^{i=4}\frac{L_i}{L} = \frac{A_1+A_2+A_3+A_4}{n_0^2} \sim \frac{A}{n_0^2}, \qquad \ldots\ldots (18) \]

if the sum \(A_1+A_2+A_3+A_4\) is denoted by \(A\).

Passing now to the determination of the losses in the supply organs, we shall first of all consider the loss in the nozzle; this loss, obviously, is easily determined from the relation

\[ h_5=\xi_5\frac{c_0^2}{2g}. \]

In fact, since \(Q=ab\,c_0\), then

\[ c_0=\frac{Q}{ab}=\frac{Q}{mb^2}; \]

and therefore

\[ h_5=\frac{\xi_5}{2g}\frac{Q^2}{b^4m^2}, \]

or, substituting the value of \(b\) through \(D_1\) from (4), and the value of \(D_1\) from (6), we have

\[ h_5=\frac{\xi_5}{2g}\, \frac{Q^2\pi^4 n^4}{m^2 k^4 60^4 \lambda_0^4 (2gH)^2}. \]

The work lost in the nozzle is represented as

\[ L_5=\delta Q h_5 = \frac{\delta \xi_5}{2g}\, \frac{Q^3\pi^4 n^4}{m^2 k^4 60^4 \lambda_0^4 (2gH)^2}, \]

and the ratio of the work lost in the nozzle to the total work of the turbine will be

\[ \frac{L_5}{L} = \frac{\xi_5\pi^4}{m^2 k^4 60^4\lambda_0^4(2g)^3} \cdot \frac{n^4 Q^2}{H^3} = B_1 n_0^4, \qquad \ldots\ldots (19) \]

introducing here also the reduced number of revolutions and denoting the entire expression preceding \(n_0^4\) by \(B_1\).

Finally, the loss in head during the flow of water in the supply pipe is found in exactly the same way as we did this¹ for the pipe supplying water to the Pelton turbine. Assuming that the mean diameter of the supply pipe is \(D_0\), we have, according to Dupuit’s formula, for the lost head

\[ h_6=\xi_6\frac{Q^2}{D_0^5}\,L, \]

where \(\xi_6\) is a coefficient taken equal to \(0.0025\), and \(L\) is the length of the pipe in meters; this expression may be rewritten as

\[ h_6=\xi_6\,\frac{Q^2}{D_0^4}\,\frac{L}{D_0}; \]

we can always put \(D_0=D_1\frac{D_0}{D_1}=\nu D_1\), and also \(\frac{L}{D_0}=\sigma\), and then

\[ h_6=\xi_6\,\frac{Q^2}{\nu^4D_1^4}\,\sigma \]

and now, substituting the value of \(D_1\), we have

\[ h_6=\xi_6\,\frac{Q^2\sigma\pi^4 n^4}{\nu^4 60^4\lambda_0^4(2gH)^2}; \]

the work lost in the pipe will be expressed as

\[ L_6=\delta Qh_6= \frac{\delta\xi_6 Q^3\sigma\pi^4 n^4}{\nu^4 60^4\lambda_0^4(2gH)^2}; \]

for the ratio of the lost work to the total work of the turbine we find

\[ \frac{L_6}{L}= \frac{\xi_6\sigma\pi^4}{\nu^4 60^4\lambda_0^4(2g)^2}\cdot \frac{Q^2 n^4}{H^3}. \]

The total relative loss in work in the nozzle and pipe will be

\[ S_2=\sum_{i=1}^{i=2} B_i\cdot n_0^4=B n_0^4, \tag{20} \]

denoting the sum \(B_1+B_2\) by \(B\).

Finally, the total relative loss in the work of the runner, in the nozzle and the supply pipe, will be represented as

\[ S=S_1+S_2=\frac{A}{n_0^2}+B n_0^4, \tag{21} \]

whence the condition for a minimum of the losses will be

\[ \frac{dS}{dn_0}=-\frac{2A}{n_0^3}+4Bn_0^3=0 \]

and consequently we obtain the most advantageous reduced number of revolutions in the form

\[ n_0=\sqrt[6]{\frac{A}{2B}}. \tag{22} \]

Substituting this value of \(n_0\) into relation (21), we find the minimum magnitude of the losses

\[ S_{\min}=1.89\sqrt[3]{A^2B}; \tag{23} \]

if only the runner and the nozzle are considered, then, obviously, the relative minimum losses will be represented in the form

\[ S'_{\min}=1.89\sqrt[3]{A^2B_1}. \tag{24} \]

Let us apply the derived relations to the following example: it is required to find the principal dimensions and the efficiency for a Banki turbine with the following data: \(Q=0.0447\ \mathrm{m^3/sec.}\), \(H=20\ \mathrm{m.}\), \(n=300\).

We take: \(\varphi=0.97,\ \lambda_0=0.46,\ \alpha_0=20^\circ,\ a=0.01,\ \gamma=5^\circ,\ \xi_2=0.0025,\ \Delta=0.01,\ \delta_0=160^\circ,\ D_0=0.2,\ L=40\ \mathrm{m.}\)

We determine:

\[ n_0=n\,\frac{\sqrt{Q}}{\sqrt[4]{H^3}} = 300\,\frac{\sqrt{0.0447}}{\sqrt[4]{20^3}} =21.2. \]

\[ c_0=\varphi\sqrt{2gH} = 0.97\cdot\sqrt{2\cdot9.81\cdot20} = 19.2\ \mathrm{m/sec.} \]

To V. N. Pinegin’s article, “The Banki Turbine and Its Theoretical Coefficient of Efficiency.”

Drawing 1.

Drawing 2.

\[ f\;(\text{nozzle area})=ab=\frac{0.0447}{19.2}=0.00233\ \text{m}^2. \]

\[ b=\frac{f}{a}=\frac{0.00233}{0.01}=0.2333\ \text{m}. \]

\[ m=\frac{a}{b}=\frac{0.01}{0.2333}=0.0429. \]

\[ D_1=\frac{60\cdot\lambda_0\sqrt{2gH}}{\pi\cdot n}=0.580\ \text{m}. \]

\[ k=\frac{b}{D_1}=\frac{0.2333}{0.580}=0.402. \]

\[ \mu'=\sqrt{\frac{\sqrt{\lambda_0^2-\Delta}}{\varphi^2+\lambda_0^2}\cdot \sqrt{\lambda_0^2+\varphi^2-2\lambda_0\varphi\cos\alpha_0}}=0.468. \]

\[ D_2=\mu D_1=0.272\ \text{m}. \]

\[ R=0.01. \]

\[ \xi_3=0.06. \]

After all these preliminary assumptions and calculations, it is no longer difficult to compute the values \(A_1,\ A_2,\ A_3,\ A_4,\ B_1\) and \(B_2\); they will be: \(A_1=0.1072\), \(A_2=1.1286\), \(A_3=1.2167\), \(A_4=0.4460\), \(B_1=0.0000282\), \(B_2=0.0000154\), with \(l=0.396\).

Thus, relation (21) takes the form:

\[ S=\frac{2.99}{n_0^2}+0.0000436\,n_0^4. \]

The most advantageous reduced number of revolutions will be

\[ n_0=\sqrt[6]{\frac{A}{2B}} =\sqrt[6]{\frac{2.99}{0.0000872}}\simeq 21.2, \]

i.e. exactly the same as what we found from relation (9).

The smallest magnitude of the losses for the entire installation is found from relation (23):

\[ S_{\min}=1.89\sqrt[3]{A^2B} =1.89\sqrt[3]{2.99^2\cdot 0.0000436}\simeq 0.13, \]

and for the runner with nozzle (the turbine), from relation (24):

\[ S'_{\min}=1.89\sqrt[3]{A^2B_1} =1.89\sqrt[3]{2.99^2\cdot 0.0000282}\simeq 0.11. \]

Whence it follows that the efficiency coefficient of the entire installation will be

\[ \eta'=0.87, \]

and the efficiency coefficient of the turbine

\[ \eta=0.89. \]

With more careful construction, with a smaller curvature of the blades, and with a smaller radial length of the blades, it is evidently possible to increase the efficiency coefficient still further and bring it to the value \(0.92\), which was found in the test, as indicated above.

1. V. N. Pinyagin. The reduced number of revolutions in Pelton turbines. Proceedings of Siberian Engineers. 1923, No. 2. ↩↩↩

2. Zeit. des Vereines d. Ingenieure. Festnummer zur Kasseler Hauptversammlung 1921. ↩

3. Zeit. des Vereines d. Ingenieure. 1922, p. 852, and also Zeit. des Vereines d. Ing. 1923, p. 136. ↩

4. In my article “The Utilization of Water Energy in Its Past and Present” I called this turbine Nagler’s turbine; at the present time the priority of the invention has been established in favor of Kaplan; see on this matter Zeit. d. Ver. d. Ing. 1921, No. 8, p. 190, and Zeit. d. Ver. d. Ing. 1921, No. 22, p. 587. ↩