Calculation of Axial-Flow Machines with Counter-Rotating Impellers
Abstract Generated abstract
This paper develops a unified calculation methodology for axial machines with counter-rotating impellers, including fans, compressors, pumps, and hydraulic turbines. It bases the design and verification procedures on Zhukovsky’s lift-force method, using airfoil and cascade characteristics to construct velocity triangles, determine head distribution, profile blade rows, and analyze off-design operation for paired counter-rotating wheels. The treatment derives relations for design conditions with axial inlet and outlet, extends them to unequal swirl and flow-rate regimes, discusses blade-number selection and cavitation considerations for pumps, and indicates how the same framework applies to engine machines. Experimental comparisons cited in the text are used to support the predicted relation between equal wheel power and maximum efficiency.
Full Text
CALCULATION OF AXIAL MACHINES WITH COUNTER-ROTATING IMPELLERS
Yu. N. Sokolov
(Presented by Honored Worker of Science and Technology Prof. I. N. Butakov)
The use of counter-rotation of impellers under certain conditions may be expedient in axial machines of various types: fans, blowers and compressors, pumps and hydraulic turbines. To design such machines with counter-rotation, it is necessary to have a well-founded methodology whose application would ensure reliable results confirmed by experiment.
In the existing literature there are no specific recommendations on such a methodology, either as applied to any particular type of axial machine or, still less, for all of the above-mentioned types. The task of the present work is to fill this gap.
The methodology for calculating hydraulic and blower machines, if in the latter separate stages with a small pressure rise at moderate (subsonic) velocities are considered, is based on identical concepts of the flow around the flow passages of these machines by an incompressible fluid and can therefore be completely of one type. Nevertheless, in calculating axial fans, axial compressors, pumps, and hydraulic turbines, different practical procedures are usually used, associated with the application of particular experimental materials or theoretical constructions that differ in each individual case.
Considering here the calculation of axial machines of all the above-mentioned types with counter-rotation of their impellers, we deem it expedient to apply to them a unified methodology. The basis of such a methodology is taken to be N. E. Zhukovskii’s lift-force method, using the aerodynamic characteristics of isolated profiles and experimental material that makes it possible to pass to the characteristics of these profiles in cascades of specified parameters, or else the results of blowing tests of plane cascades of profiles.
In application to machines with counter-rotating impellers, this method is all the more expedient because the flow around a pair of profile cascades during their counter-motion is associated with such a complex pattern of phenomena that investigating them by the methods of hydrodynamic theory of cascades would lead to practically unsolvable equations, or would require such simplifying assumptions as would devalue the calculation results.
The indisputable advantage of using the aerodynamic characteristics of isolated airfoils in all cases is the possibility of drawing upon very extensive material on the testing of airfoils of various parameters.
In what follows, we shall consider the application of the lift-force method to the calculation of axial machines with counter-rotating impellers, regardless of the type of machine; moreover, within the pair of wheels under consideration the fluid will be regarded as incompressible at its mean specific weight (for gas machines)
\[ \gamma = \frac{1}{2}(\gamma_1+\gamma_2). \]
Speaking first of machines that transfer energy to the flow (pumps, fans, compressors), by their total useful head we shall in all cases understand the quantity
\[ H = H_p + H_k = \frac{p_2-p_1}{\gamma}+\frac{c_2^2-c_1^2}{2g}, \]
where \(p_2\) and \(p_1\) are the final and initial pressures, and \(c_2\) and \(c_1\) are the corresponding absolute velocities. For compressor machines, when \(\gamma \approx \mathrm{const}\), the quantity \(H_p\) is identified with the work of pressure increase (allowing for suction and discharge),
\[ L_p=\int_{p_1}^{p_2}\frac{dp}{\gamma}. \]
Design operating condition
The most expedient organization of the working process in a pair of counter-rotating axial wheels transferring energy to the flow is taken to be that [5] in which, at equal peripheral velocities of these wheels, \(|u'|=|u''|\), axial exit from the second wheel is ensured with axial entry into the first. With the same theoretical head of each of the wheels,
\[ H_t' = H_t'' = \frac{1}{g}\,u c_u \tag{1} \]
in this case the changes in the circumferential components of the absolute velocities must also be identical,
\[ \Delta c_u'=\Delta c_u''=c_u, \]
Fig. 1. Velocity triangles of a pair of counter-rotating axial wheels at the design operating condition.
and the superposed velocity triangles for the given annular element of the pair of wheels will correspond to the diagram of Fig. 1, where, as below, \(c\), \(w\), and \(u\) are the absolute, relative, and peripheral velocities; the lower subscripts 1 or 2 denote inlet or outlet; the primes \('\) and \(''\) denote the first or second wheel; and the subscript \(m\) refers to the vectors of the mean geometric velocity. This operating condition will be taken as the design condition.
The portion of the total useful heads of the first and second impellers used to increase the pressure under such a design operating condition must be determined as
\[ H_p'=\frac{1}{g}\,c_u\left(u-\frac{c_u}{2}\right)\eta_p'; \tag{2} \]
\[ H_p''=\frac{1}{g}\,c_u\left(u+\frac{c_u}{2}\right)\eta_p''. \tag{3} \]
The magnitude and direction of the mean geometric velocities are then determined from the relations:
\[ w_m'=\sqrt{c_a^2+\left(u-\frac{c_u}{2}\right)^2}; \tag{4} \]
\[ w_m''=\sqrt{c_a^2+\left(u+\frac{c_u}{2}\right)^2}; \tag{5} \]
\[ \operatorname{tg}\beta_m'=\frac{c_a}{u-\frac{c_u}{2}}; \tag{6} \]
\[ \operatorname{tg}\beta_m''=\frac{c_a}{u+\frac{c_u}{2}}, \tag{7} \]
where \(c_a\) is the axial component of the absolute velocity, assumed constant within the limits of the given stage.
Method of Design Calculation
Under the design operating condition considered above, the total useful head produced by a pair of counter-rotating axial impellers is determined by the corresponding pressure rise
\[ H=H_p\frac{p_2-p_1}{\gamma}. \]
Assuming that this quantity is specified in the calculation, in order to determine the design theoretical head of the pair of impellers it is necessary to estimate the efficiency of this pair. For a preliminary estimate of the latter one may take \(\operatorname{tg}\beta_m'\simeq \operatorname{tg}\beta_m''\simeq \frac{c_a}{u}\), which leads to determination of the efficiency of the annular element of the pair of impellers by the approximate formula
\[ \eta_{bb}^{\mathrm{el}}=\eta_p= \frac{1-\psi_p\frac{c_a}{u}} {1+\psi_p\frac{u}{c_a}}, \tag{8} \]
which follows from equality, in this case, of the efficiencies of the blade rows of the first and second impellers and of the efficiency of the entire stage. Such an assumption, usually
accepted in the calculation of axial machines of the usual type is even more permissible here, since, as \(\frac{c_u}{u}\) increases under counter-rotation, the efficiency of the blade row of the first wheel increases, while that of the second decreases, as a result of which \(\eta_{bb}^{\mathrm{el}}\) changes little. Calculations show that the error under ordinary conditions is \(0.1 \div 0.2\%\).
According to generally accepted standards for machines of this type, the ratio \(\frac{c_a}{u}\) is also evaluated at the outer radius. On other annular elements this ratio will be determined by the law, adopted as the basis of the calculation, for the variation of axial velocities along the radius (or by their constancy).
Having specified the assumed value of the inverse quality of the profiles in the blade row, \(\psi_p\), one can thus calculate \(\eta_{bb}^{\mathrm{el}}\) and determine the design head of the pair of wheels on the given annular element:
\[ H_t = H_{tp} = H_p \cdot \eta_{bb}^{\mathrm{el}} . \tag{9} \]
Using equation (1), from the value \(H_t' = H_t'' = \frac{H_t}{2}\) for each annular element one can establish the necessary values of the quantities \(u\) and \(c_u\), without going beyond the permissible limits of \(u_R\) at the outer radius, and the values of the ratios \(\frac{c_u}{u}\) that are normal for machines of this type. Since under the normal operating condition of counter-rotating wheels \(c_u = c_{u_2}' = c_{u_1}''\), while \(c_{u_1}' = 0\) and \(c_{u_2}'' = 0\), \(u\) and \(c_u\), for a known \(c_a\), completely determine the design velocity triangles on each annular element.
The subsequent calculations associated with profiling the blades for the first of the counter-rotating axial wheels, whose inlet is axial and whose outlet has swirl \(c_u\), are carried out according to the usual relations used in calculating the impeller of an axial machine of the normal type. When profiling the blades of the second wheel, however, the magnitude and direction of the mean geometric velocity must be determined from equations (5) and (7).
Having determined, from the usual relations, the necessary values \(C_{yp}\) and \(\frac{b}{t}\) for each of the counter-rotating axial wheels on each of its annular elements, one can select the corresponding profile cascades for them, based on experimental data.
For reasons related to the undesirability of all the blades of the second wheel simultaneously entering the aerodynamic wakes from the blades of the first—which would lead to nonuniformity of the torque transmitted to the second wheel—the numbers of blades of the first and second wheels should be taken unequal and not multiples of one another.
Operation of a Pair of Counter-Rotating Axial Wheels at Off-Design Conditions
In the verification calculation of a pair of counter-rotating axial wheels, it is necessary to take into account the possibility of their operation also at such (off-design) conditions when the inlet to the first wheel is axial, natural
under ordinary conditions at any operating condition, no longer ensures an axial exit from the second wheel. The equality of \(\Delta c'_u\) and \(\Delta c''_u\), and consequently also of the energy transferred at each of the wheels, is thereby violated.
The pressure rise produced in the flow as it passes around a rotating annular cascade of profiles must be determined, on the one hand, by L. Euler’s equation, which relates this quantity to \(u\) and \(\Delta c_u\), and, on the other hand, by N. E. Zhukovsky’s equation, which determines the same quantity through the axial components \((P_u)\) of the forces acting on the profiles being flowed around. For an elementary annular section of the first of the counter-rotating axial wheels, the inlet to which remains axial under all operating conditions, the corresponding equations will be:
\[ \Delta p'=\rho \Delta c'_u\left(u-\frac{\Delta c'_u}{2}\right)\eta'_p \]
\[ \Delta p'=\rho\,\frac{b'}{t'}\,w_m^{\prime 2}K'_{ap} =\rho\,\frac{b'}{t'}\left(\frac{u-\dfrac{\Delta c_u}{2}}{\cos\beta'_m}\right)^2 K'_{ap}, \]
where, in addition to the notation used above,
\[ \frac{b'}{t'} \text{ is the solidity of the profile cascade under consideration,} \]
\[
K'_{ap}=\frac{dP_u}{\rho\cdot ds\cdot w_m^{\prime 2}}
= C_{yp}\cos\beta'_m - C_{xp}\sin\beta'_m
\]
is the head coefficient of the cascade, determining the axial projection of the forces acting on the elementary blade area \((ds=b\cdot dl)\), in the same way as the coefficients \(C_{yp}\) and \(C_{xp}\) determine, respectively, the lift force and profile drag.
Equating the right-hand sides of these equations, we obtain the condition determining the direction of the mean geometric velocity of flow around the profile cascade of the first of the counter-rotating axial wheels, i.e., the relation between the magnitude of the angle \(\beta'_m\) and the relative swirl of the flow \(\dfrac{\Delta c'_u}{u}\):
\[ \frac{\Delta c'_u}{u}\eta'_p = \frac{b'}{t'}\cdot \frac{1-\dfrac{1}{2}\cdot\dfrac{\Delta c'_u}{u}} {\cos^2\beta'_m} \cdot K'_{ap}. \tag{10} \]
In doing so it should be taken into account that \(\eta'_p\) and \(K'_{ap}\), in turn, depend on the angle \(\beta'_m\), which determines the angle of attack and the numerical values of the coefficients \(C_{yp}\) and \(C_{xp}\).
The angle \(\beta'_m\) also determines the relative axial velocity \(\left(\dfrac{c_a}{u}\right)\), which, for axial inlet, corresponds to the equality
\[ \frac{c_a}{u} = \left(1-\frac{1}{2}\frac{\Delta c'_u}{u}\right)\tan\beta'_m . \]
Thus, by equation (10), there is ultimately established a dependence between the axial velocity \((c_a)\) and the swirl of the flow \((\Delta c'_u)\) on the annular element of the first wheel.
The problem of determining the operating conditions of the second of the counter-rotating axial wheels under off-design conditions is complicated by the fact that the exit direction here may in general be arbitrary, while the inlet direction is determined by the exit from the first wheel. Depending on the operating condition, the velocity triangles determining the flow around the blade rows of the first and second wheels when the axial velocity \((c_a)\) changes may take the forms \(a\), \(b\), and \(c\) (Fig. 2). In all these cases, however, the following equalities will hold:
Fig. 2. Velocity triangles of a pair of counter-rotating axial wheels under different operating conditions.
\[ \tg \beta''_m = \frac{c_a}{u + \Delta c'_u - \dfrac{\Delta c''_u}{2}} ; \]
\[ w''_m = \frac{ u + \Delta c'_u - \dfrac{\Delta c''_u}{2} }{ \cos \beta''_m }. \]
Accordingly, the pressure rise on the annular element of the second of the counter-rotating axial wheels, under any operating condition, must be determined by the equations:
\[ \Delta p''=\rho\,\Delta c_u''\left(u+\Delta c_u'-\frac{\Delta c_u''}{2}\right)\eta_r' , \]
\[ \Delta p''=\rho\,\frac{b''}{t''}\left(\frac{u+\Delta c_u'-\dfrac{\Delta c_u''}{2}}{\cos \beta_m''}\right)^2 K_{ap}'' . \]
Thus, the general condition determining the direction of the mean geometric velocity in the flow around the blade cascade of the second of the counter-rotating axial wheels will be:
\[ \frac{\Delta c_u''}{u}\cdot \eta_r' = \frac{b''}{t''}\cdot \frac{1+\dfrac{\Delta c_u'}{u}-\dfrac{\Delta c_u''}{2u}} {\cos^2\beta_m''} \cdot K_{ap}'' . \tag{11} \]
Analysis of this equation shows that, for each specific pair of blade cascades, with decreasing \(\dfrac{c_a}{u}\), the ratio \(\dfrac{\Delta c_u''}{u}\) increases faster than \(\dfrac{\Delta c_u'}{u}\). Therefore, within the limits of separation-free flow around such a pair of cascades, at higher axial velocities \(\Delta c_u''<\Delta c_u'\), and at low velocities \(\Delta c_u''>\Delta c_u'\). Under certain conditions \(\Delta c_u''=\Delta c_u'\), which corresponds to the design operating condition adopted by us for the pair of wheels.
Fig. 3. Example of calculated relationships \(\dfrac{\Delta c_u'}{u}\left(\dfrac{c_a}{u}\right)\) and \(\dfrac{\Delta c_u''}{u}\left(\dfrac{c_a}{u}\right)\) for an annular element of a pair of counter-rotating axial wheels.
Fig. 4. Powers I and II of counter-rotating axial wheels at various operating conditions according to experimental data.
In Fig. 3 an example is given of the variation of the ratios \(\dfrac{\Delta c_u'}{u}\) and \(\dfrac{\Delta c_u''}{u}\) as a function of \(\dfrac{c_a}{u}\) (or of the flow rate \(Q\), proportional to this quantity), according to the results of calculation, and in Fig. 4 curves are presented for the variation of pow-
velocities consumed by the first and second counter-rotating wheels, according to experimental data. The analogous character of these curves in the region of continuous flow and the close coincidence of the regimes \(N''=N'\) and \(\eta_{\max}\) according to experimental data testify to the experimental confirmation of the propositions set forth above.
Method of Verification Calculation
Equations (10) and (11), valid for any operating regime of a pair of counter-rotating axial wheels (with equal peripheral speeds in magnitude), may be used in their verification calculation. The calculation itself has to be carried out by a selection method for the individual annular elements of the pair of wheels under consideration, the geometry of the blade cascades and the peripheral speed of which are specified.
The calculation for an annular element of the first of the wheels, the inlet to which in all cases is assumed to be axial, can be carried out analogously to that recommended by the TsAGI method [2] for the verification calculation of axial fans. Specifying a number of values \(\Delta c'_u\), one selects for each of them such an angle of attack as would correspond to \(\eta'_p\) and \(K'_{ap}=C'_{yp}\cos\beta'_m-C'_{xp}\sin\beta'_m\), satisfying equation (10). This determines the pressure rise \(\Delta p'\) and the corresponding axial velocity on the given annular element
\[ c_a=\left(u-\frac{1}{2}\Delta c'_u\right)\operatorname{tg}\beta'_m. \]
In calculating the corresponding annular element of the second of the counter-rotating wheels, for each of the regimes considered in the calculation of the first wheel it is necessary to select such a value \(\Delta c''_u\) that would provide values of \(\beta''_m\) (and of the corresponding angle of attack), \(\eta''_p\), and \(K''_{ap}=C''_{yp}\cos\beta''_m-C''_{xp}\sin\beta''_m\), satisfying equation (11). This determines the pressure rise on the annular element of the second wheel \((\Delta p'')\) for each value of the already known axial velocity \(c''_a=c'_a=c_a\).
For each of the annular elements considered in the course of the calculation, it is thus possible to establish the relation
\[ \Delta p=\Delta p' + \Delta p'' = f(c_a), \]
and, from the dimensions of the element, for each \(c_a\) one can also compute the corresponding elementary flow rate \(\delta Q=c_a\cdot \delta F\).
Assuming that in the sections swept by the blades the law of constancy (with respect to radius) of the circulation of the central vortex is observed, one may consider that the pressure rise created by the calculated axial wheels as a whole is determined by that on the outermost elements of their blades. Therefore, in order to determine the pressure rise \(\Delta p=\Delta p' + \Delta p''\) corresponding to each design regime, it is sufficient to restrict oneself to carrying out the verification calculation described above only for the outermost annular elements of the pair of counter-rotating axial wheels, with the introduction of the corresponding corrections for the finite blade length and the influence of the radial clearance.
As for determining the flow rates corresponding to each design regime (and to the design value of \(\Delta p\)), for this it is necessary to know the axial velocity in each of the design annular sections \(\delta F\). In this connection, for each of the cal-
considered regimes may also be carried out for other (besides the outermost) annular elements, but these calculations may be limited to only the first impeller.
Example of a checking calculation and comparison of its results with experimental data
As an example, we carry out a checking calculation for a pair of axial impellers of the experimental model “VDV-E,” the results of which can be compared with the characteristics of this model obtained from experiments performed on it.
1. Basic dimensions and parameters of the profiles and cascades
Inner diameter of the casing bore . . . . . . . . . . . . . . . \(D_k = 400\ \mathrm{mm}\).
Outer diameter of each of the impellers . . . . . . . \(D = 398\ \mathrm{mm}\).
Hub diameter . . . . . . . . . . . . . . . . . . . . . . \(d = 240\ \mathrm{mm}\).
Swept area . . . . . . . . . . . . . . . . . . . . . .
\[
F=\frac{\pi}{4}\left(D^2-d^2\right)=0.0792\ \mathrm{m}^2.
\]
Radial length of the blades . . . . . . . . . . . . .
\[
l=\frac{1}{2}(D-d)=79\ \mathrm{mm}.
\]
Radial clearance . . . . . . . . . . . . . . . . . .
\[
s=\frac{1}{2}(D_k-D)=1\ \mathrm{mm},
\]
\[
\bar{s}=\frac{s}{l}\,100=1.27\%.
\]
Operating rotational speed of each of the impellers . . .
\[
n'=n''=3700\ \mathrm{rpm}.
\]
Peripheral speed at the outer elements . . .
\[
u_R=\frac{\pi D n}{60}=77.1\ \mathrm{m/sec}.
\]
In all blade sections an English screw profile is adopted, with relative thickness
\[ \bar{\delta}=\frac{\delta}{b}=0.12. \]
The number of blades on the first impeller is \(z_1=8\), and on the second \(z_2=5\). The blades are twisted. Their geometry is determined as follows:
| Parameter | 240 (at the hub) |
330 (mean) |
398 (outer) |
|---|---|---|---|
| Section diameter (mm) | 240 | 330 | 398 |
| Blade width, I impeller, \(b'\) (mm) | 68.5 | 57.5 | 51.8 |
| Setting angle, \(\theta'\) (deg., min.) | \(33^\circ 00'\) | \(23^\circ 00'\) | \(18^\circ 20'\) |
| Blade width, II impeller, \(b''\) (mm) | 85.8 | 79.4 | 75.6 |
| Setting angle, \(\theta''\) (deg., min.) | \(26^\circ 00'\) | \(20^\circ 00'\) | \(16^\circ 40'\) |
2. Aerodynamic characteristics of the profiles and cascades
To estimate the pressure coefficients of isolated profiles,
\[ K_u=C_y\cos\beta_m-C_x\sin\beta_m \]
and the profile drag coefficients, we use the corresponding curves \(K_{ap}(\alpha,\beta)\) and \(\eta_{\mathrm{prof}}(\alpha,\beta)\), given in the work of K. A. Ushakov [2].
Using the experimental material of V. I. Bogdanovskii [3] and extrapolating the curves obtained by him where necessary, we estimate the parameters
\[ \chi=\frac{dC_{yp}}{d\alpha}:\frac{dC_y}{d\alpha} \]
and \(\Delta a_0=z_{op}-a_0\) for the considered-
selected subsequently for the profile cascades. Thus, for example, the following was obtained.
For the end elements of the blades \((D = 398\ \text{mm})\):
1) first impeller . . . \(\chi = 1.19;\ \Delta \alpha_0 = 0^\circ 35'\);
2) second ” ” . . . \(\chi = 1.18;\ \Delta \alpha_0 = 0^\circ 42'\).
For the sections of the first impeller:
3) at diameter \(366.5\ \text{mm}\) . . \(\chi = 1.20;\ \Delta \alpha_0 = 0^\circ 30'\);
4) ” ” \(288.5\ \text{mm}\) . . \(\chi = 1.22;\ \Delta \alpha_0 = 0^\circ 26'\).
Fig. 5. Example of the aerodynamic characteristics of a single profile and of a profile cascade.
Having the dependence \(C_y(\alpha)\) for a single profile and knowing \(\chi\) and \(\Delta \alpha_0\), for each of the cascades under consideration we construct the curves \(C_{yp}(\alpha)\) and \(\kappa_p(\alpha)\), similarly to what is shown in Fig. 5. For each of the profile cascades considered in the calculation process, it is now possible, from the angle of attack \(\alpha = \Theta - \beta_m\), to determine \(\eta_{\text{prf}}\) and \(K_a\), as well as
\[ \kappa_p = \frac{C_{yp}}{C_y} \simeq \frac{K_{ap}}{K_a} \]
and \(K_{ap} = K_a \cdot \kappa_p\), which is used subsequently.
3. Check calculation of impellers I and II at the blade end elements
For seven calculated operating conditions we prescribe changes in the circumferential components of the absolute velocity at the end elements of the blades of the first impeller, \(\Delta c'_{uR} = 5,\ 6,\ 7,\ 8,\ 9\) and \(10\ \text{m/sec}\), and for each case we carry out the calculation, first selecting such a value \(a'\) for which equation (10) is satisfied, and then such a \(\Delta c''_u\) for which condition (11) is satisfied1. In view of the small changes in pressure, the air density in all cases was taken as constant and equal to
\[ \rho = 0.122\ \frac{\text{kg}\,\text{sec}^2}{\text{m}^4}. \]
The results of the calculation are summarized in a table.
4. Calculation of the flow rate and corrections to the calculated pressure-rise value
To establish the probable law of variation of the axial component of the absolute velocity over the radius, we carry out a check calculation, analogous to that for the blade end elements, for other sections as well, but for the same calculated operating conditions. As was indicated, it is sufficient to do this only for the first impeller.
Omitting the details of such a calculation, we give only its final result in the form of the curves \(c_a(r)\) for each calculated operating condition, presented in Fig. 6.
Table
| Regime | I | II | III | IV | V | VI | VII |
|---|---|---|---|---|---|---|---|
| $\Delta c_u' = \Delta c_{uR}'$ (m/sec) | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| $\alpha'$ — adopted (deg) | $-2^\circ10'$ | $-1^\circ30'$ | $-0^\circ50'$ | $-0^\circ20'$ | $+0^\circ20'$ | $+1^\circ10'$ | $+2^\circ00'$ |
| $\beta_m' = \theta' - \beta_m'$ (deg) | $20^\circ30'$ | $19^\circ50'$ | $19^\circ10'$ | $18^\circ40'$ | $18^\circ00'$ | $17^\circ10'$ | $16^\circ20'$ |
| $\eta_{\mathrm{prof}}'$ | 0,84 | 0,85 | 0,87 | 0,885 | 0,90 | 0,915 | 0,93 |
| $K_a'$ | 0,137 | 0,170 | 0,205 | 0,228 | 0,260 | 0,304 | 0,345 |
| $\kappa_p$ | 0,854 | 0,924 | 0,951 | 0,996 | 1,02 | 1,03 | 1,05 |
| $K_{ap}' = K_a' \cdot \kappa_p$ | 0,117 | 0,157 | 0,195 | 0,225 | 0,265 | 0,314 | 0,363 |
| $\Delta p_{c_u}' = \rho\,\Delta c_u'\left(u-\dfrac{\Delta c_u'}{2}\right)\eta_{\mathrm{prof}}'$ | 30,7 | 39,1 | 47,3 | 55,7 | 64,2 | 73,2 | 81,8 |
| $\Delta p_{ka}' = \rho\,\dfrac{b'}{t'}\left(\dfrac{u-\dfrac{\Delta c_u'}{2}}{\cos\beta_m'}\right)^2 K_{ap}'$ | 30,4 | 39,9 | 48,5 | 54,9 | 63,2 | 73,4 | 82,8 |
| $\Delta p' = \dfrac{1}{2}\left(\Delta p_{c_u}' + \Delta p_{ka}'\right)$ | 30,6 | 39,5 | 47,9 | 55,3 | 63,7 | 73,3 | 82,3 |
| $c_a = \left(u-\dfrac{\Delta c_u'}{2}\right)\operatorname{tg}\beta_m'$ | 28,1 | 26,9 | 25,8 | 24,9 | 23,7 | 22,4 | 21,1 |
| $\Delta c_u''$ — adopted (m/sec) | 3,1 | 4,6 | 6,0 | 7,3 | 8,5 | 9,8 | 11,3 |
| $\beta_m'' = \operatorname{arctg}\dfrac{c_a}{u+\Delta c_u' - \dfrac{\Delta c_u''}{2}}$ | $19^\circ27'$ | $18^\circ37'$ | $17^\circ52'$ | $17^\circ10'$ | $16^\circ20'$ | $15^\circ25'$ | $14^\circ30'$ |
| $\alpha'' = \theta'' - \beta_m''$ | $-2^\circ47'$ | $-1^\circ57'$ | $-1^\circ12'$ | $-0^\circ30'$ | $+0^\circ20'$ | $+1^\circ15'$ | $+2^\circ10'$ |
| $\eta_{\mathrm{prof}}''$ | 0,810 | 0,836 | 0,855 | 0,870 | 0,890 | 0,910 | 0,925 |
| $K_a''$ | 0,110 | 0,152 | 0,190 | 0,225 | 0,265 | 0,312 | [[unclear]] |
| $\kappa_p$ | 0,864 | 0,954 | 1,00 | 1,04 | 1,075 | 1,09 | 1,10 |
| $K_{ap}'' = K_a'' \cdot \kappa_p$ | 0,095 | 0,145 | 0,190 | 0,234 | 0,285 | 0,340 | 0,395 |
| $\Delta p_{c_u}'' = \rho\,\Delta c_u''\left(u+\Delta c_u' - \dfrac{\Delta c_u''}{2}\right)\eta_{\mathrm{prof}}''$ | 24,4 | 37,5 | 50,2 | 62,4 | 74,7 | 88,3 | 104,0 |
| $\Delta p_{ka}'' = \rho\,\dfrac{b''}{t''}\left(\dfrac{u+\Delta c_u' - \dfrac{\Delta c_u''}{2}}{\cos\beta_m''}\right)^2 K_{ap}''$ | 24,9 | 38,0 | 49,7 | 61,3 | 74,7 | 89,0 | 103,0 |
| $\Delta p'' = \dfrac{1}{2}\left(\Delta p_{c_u}'' + \Delta p_{ka}''\right)$ | 24,6 | 37,8 | 50,0 | 61,8 | 74,7 | 88,6 | 103,5 |
| $\Delta p = \Delta p' + \Delta p''$ (kg/m²) | 55,2 | 77,3 | 97,9 | 117,1 | 138,4 | 162,3 | 185,8 |
Dividing the entire section swept by the blades into four rings of approximately equal areas, with mean diameters of 382, 348, 309, and 265 mm, for each of them, at each of the calculated operating conditions, using the curves of Fig. 6 we find \(c_{ai}\) and compute \(\partial Q_i = c_{ai}\cdot \partial F_i\), where \(\partial F_i\) is the section of the given ring. For each of the calculated operating conditions, the flow rate of the machine as a whole is thus determined as \(Q = \sum \partial Q_i\).
Fig. 6. Variation of the axial velocity over the radius of the first rotor for the calculated operating conditions (for the example of a rotational calculation).
Fig. 7. Comparison of the results of the rotational calculation with experimental data for a pair of counter-rotating axial rotors (VDV-E model).
To the pressure rises \(\Delta p_{s=0}\) calculated for the same operating conditions we introduce corrections for the finite blade length \((k_l)\) and for the influence of the radial clearance \((k_s)\). Using the corresponding empirical formulas and experimental material, for the case under consideration \((l = 79\ \text{mm},\ s = 1.27\%)\) we shall have:
\[ \Delta p = \Delta p_{s=0}\cdot k_l \cdot k_s = 0.859\,\Delta p_{s=0}. \]
As a result of these calculations and the introduction of corrections, we finally obtain:
| Operating condition | I | II | III | IV | V | VI | VII |
|---|---|---|---|---|---|---|---|
| \(Q\), m³/sec | 2.29 | 2.20 | 2.10 | 2.01 | 1.91 | 1.82 | 1.72 |
| \(\Delta p\), kg/m² | 47.5 | 66.5 | 84.0 | 100.5 | 119.0 | 139.5 | 159.5 |
This determines the calculated characteristic of the considered pair of counter-rotating axial rotors (within the limits covered by the calculated operating conditions) at the given number of revolutions \((n' = n'' = 3700\ \text{rpm})\).
In Fig. 7 the results of the calculation performed are compared with material obtained by the author in tests of the subjected to rotational—
for the point-by-point calculation of the pair of wheels. The calculated points I—VI fit well near the main experimental characteristic in its working part. The calculated dependence $\Delta c'_{u2}(Q)$ agrees sufficiently closely with the curve $c'_{u2}(Q)$ obtained on the basis of measurements of the flow direction behind the first impeller (with the corresponding processing of the experimental data under the assumption that the axial velocity of flow rate is constant along the radius).
A certain upward deviation from the experimental curves of the last calculated points (at large $\Delta c_u$) is apparently explained by the incomplete correspondence of the experimental dependences of V. I. Bogdanovskii used in the calculation process to the cascades of profiles employed (V. I. Bogdanovskii’s experimental material had to be extrapolated), and also by the circumstance that the law $c_a=\text{const}$, adopted in processing the experimental data on the direction of the velocities, is not actually fulfilled, especially noticeably precisely at regimes of large $\Delta c_u$ (as is seen from Fig. 6).
The correctness of the assumptions underlying the calculations concerning the essence of the flow process for a pair of counter-rotating axial wheels at off-design regimes is also confirmed by the fairly close agreement (in flow rate) of the calculated regime $\Delta c''_u=\Delta c'_u$, which provides discharge without swirl losses, with the experimentally established regime of maximum efficiency.
The quite satisfactory overall agreement of the results of the verification calculation with the experimental data confirms the applicability of the basic (design) calculation method for counter-rotating axial wheels set forth above, since the latter is based on the same theoretical foundations, calculation relations, and experimental material used.
The following circumstances also confirm the applicability of the proposed method for the design calculation and profiling of blades of counter-rotating axial wheels. The impellers of the experimental model VDV-E fully correspond in design to those of the VPD-4 turbofan unit, the aerodynamic calculation of the fan part of which was carried out by the author at one time essentially according to the same design-calculation method described above. The design parameters of the VPD-4 fan were: $Q=140\ \text{m}^3/\text{min}=2.33\ \text{m}^3/\text{s}$ and $\Delta p=240\ \text{kg}/\text{m}^2$ at a rotational speed of each of the counter-rotating wheels $n=4750\ \text{rpm}$. At the working rotational speed of the VDV-E model, $n=3750\ \text{rpm}$, according to the laws of similarity these correspond to $Q=1.815\ \text{m}^3/\text{s}$ and $\Delta p=146\ \text{kg}/\text{m}^2$ (see the corresponding point in Fig. 7); according to the experimental characteristic, at $Q=1.815\ \text{m}^3/\text{s}$, $\Delta p=140\ \text{kg}/\text{m}^2$ was obtained. The slight discrepancy here is explained by an insufficiently accurate allowance for the mutual influence of the blades in the initial calculation of the fan part of the VPD-4 unit.
Features of the Calculation of Hydraulic Machines
The calculation method recommended here, using the results of profile blowing in the form of the dependences $C_y(\alpha)$ and $C_x(\alpha)$, does not make it possible to determine the pressure distribution over the blade surface. In the calculation of hydraulic machines, however, the latter is important for establishing their cavitation properties. Therefore, for hydraulic machines it is necessary to carry out an additional check for cavitation of the impeller blades profiled according to the above-described method.
Of the two counter-rotating axial impellers of the pump, the first will obviously be in the dangerous conditions (with respect to cavitation), since the pressure in the flow entering the second impeller will already be increased. Therefore only the first impeller of the counter-rotating axial pump is subject to a cavitation check; under the scheme we have adopted (axial inlet), its operating conditions do not differ in any way from those for the runner of a conventional axial pump.
Here the method proposed by V. S. Kvyatkovskii may be used for the cavitation calculation of propeller-turbine blades when they are profiled by the lift method, refined by introducing corrections for mutual influence [4]. This method is based on the use of the relation
\[ \frac{p_{\min}-p_{\text{boil}}}{\gamma}=H(\sigma_y-\sigma_p), \tag{16} \]
in which
\[ \sigma_y=\frac{H_a-H_{\text{boil}}-H_s}{H} \]
is the cavitation coefficient of the installation, and
\(\sigma_p\) is the cavitation coefficient of the profile cascade, determined through the cascade parameters and the coefficient of the minimum pressure of the profile in the cascade \(\chi_{p\min}\);
\(p_{\min}\) is the minimum pressure on the blade surface in the cascade;
\[ p_{\text{boil}} \quad \text{and} \quad H_{\text{boil}}=\frac{p_{\text{boil}}}{\gamma} \]
are the boiling pressure and the corresponding head;
\[ p_a \quad \text{and} \quad H_a=\frac{p_a}{\gamma} \]
are the pressure and head of atmospheric air;
\(H\) is the total head of the installation;
\(H_s\) is the suction height.
These relations, introduced by V. S. Kvyatkovskii for a hydraulic-turbine installation, will be fully applicable also to a pump installation, if by \(H\) one understands the total useful head of the pump.
The condition for cavitation-free flow around the blades will obviously be
\[ p_{\min}>p_{\text{boil}} \]
or, according to (16),
\[ \sigma_p<\sigma_y. \]
For the cavitation coefficient of the profile cascade of a vertical axial hydraulic turbine, V. S. Kvyatkovskii derives the formula
\[ \sigma_p=\chi_{p\min}\frac{w_2^{\prime\,2}}{2gH}+\eta_{\text{out}}\frac{c_2^2}{2gH}, \]
where \(w'_2\) and \(c_2\) are the relative and absolute velocities at the exit from the runner cascade, and \(\eta_{\text{out}}\) is the draft-tube coefficient, accounting for all energy losses as the water passes from the turbine runner into the lower pool.
It is not difficult to show that, for an axial pump with a horizontal shaft (Fig. 8), the cavitation coefficient of the cascade at zero approach velocity \((c_a^0 = 0)\) must be expressed as
\[ \zeta_p = \chi_{p\min}\cdot \frac{w_1^2}{2gH} + \left(1-\xi_{\mathrm{vs}}\right)\frac{c_1^2}{2gH} + \frac{r}{H}, \tag{17} \]
where \(w_1\) and \(c_1\) are the corresponding velocities at the inlet to the cascade of profiles;
\(\xi_{\mathrm{vs}}\) is the resistance coefficient of the suction system;
\(r\) is the radius of the annular section of the blades defining the cascade under consideration.
Fig. 8. For determining the cavitation coefficient of a pumping installation.
V. S. Kvyatkovskii recommends determining the coefficient of minimum pressure of the profiles in cascade from the coefficient of minimum pressure of a single profile
\[ \chi_{\min} = \frac{w_{\max}^2 - w^2}{w^2} \]
(\(w_{\max}\) is the maximum velocity on the surface of the profile, and \(w\) is the velocity of the oncoming flow at a considerable distance) and from the ratio of these coefficients
\[ \frac{\chi_{p\min}}{\chi_{\min}}. \]
The latter is established by means of spe-
...of special graphs constructed on the basis of experimental investigations of hydraulic-turbine cascades.
Numerical values of the minimum-pressure coefficients of single profiles may be established on the basis of special investigations carried out at VITM. To estimate the ratio
\[ \frac{x_{p\min}}{z_{\min}} \]
for diffuser cascades of pumps, it is necessary to have the corresponding experimental material, which, apparently, will be obtained in the near future at scientific institutions of the USSR.
On the Calculation of Counter-Rotating Axial Wheels of Engine Machines
For a pair of counter-rotating axial wheels of an engine machine, in principle the same calculation methods and procedures may be applied as those considered above for machines transmitting energy to the flow. In doing so, however, one must take into account certain changes in the operating conditions of the pair of wheels under consideration and in the calculation relations applied to it, arising from the specifics of the case considered.
Fig. 9. Flow scheme and velocity triangles for a pair of counter-rotating axial wheels of an engine machine with axial inlet and axial outlet.
Without repeating the particulars and details of the calculation, we shall dwell only on these specific features. We shall assume here that, at the design operating condition, the pair of wheels of the engine machine also has an axial approach to the first wheel and an axial discharge from the second (Fig. 9). Operation according to such a scheme, with equal rotational speeds of each of the wheels, leads to equality of the heads utilized on these wheels.
Since in the present case \(\beta'_m < \beta''_m\), the efficiency of the cascade of the first wheel will be less than that of the second \((\eta'_p < \eta''_p)\), while the hydraulic efficiency \((\eta_{\Gamma}^{\mathrm{el}})\) of the corresponding annular element of the pair of wheels must be greater than \(\eta'_p\) and less than \(\eta''_p\).
Adopting in the course of the calculation the simplifying substitution \(\operatorname{tg}\beta_m \simeq \dfrac{c_a}{u}\),
and in this case it may be assumed that \(\eta'_p \simeq \eta''_p \simeq \eta_{\Gamma}^{\mathrm{el}}\).
Then, having estimated \(\eta_{\Gamma}^{\mathrm{el}}\) for the pair of corresponding elementary cascades, from the total available head we find the design head of each of these cascades:
\[ H'_{\mathrm{calc}} = H''_{\mathrm{calc}} = \frac{1}{2} H \eta_{\Gamma}^{\mathrm{el}} . \tag{18} \]
Dividing the entire flow passing through the turbine into separate elementary turbines with annular cross section, we establish the mean radius of each of these sections and the corresponding circumferential velocity. From the equation \(gH'_{\mathrm{calc}} = u c_u\) we find \(c_u\), which makes it possible to construct the velocity triangles for the cascades of the first and second counter-rotating wheels of the given elementary turbine according to the diagram in Fig. 9.
The considerations examined here make it possible to carry out both design and checking calculations of counter-rotating axial wheels of machines of all types of interest to us.
REFERENCES
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Ushakov K. A. Aerodynamic calculation of an axial fan. Transactions of TsAGI, issue No. 277, 1936.
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Ushakov K. A., Polikovskii V. I., Nevelson M. I., Tatarinov M. P. Mine Ventilation Installations. GONTI, 1938.
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Bogdanovskii V. I. Experimental investigation of the influence of a hydrodynamic cascade in an oblique axial pump. Report on the VIGM topic (manuscript), 1951.
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Kvyatkovskii V. S. The working process of an axial hydraulic turbine. Transactions of VIGM, issue XIV, 1951.
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Sokolov Yu. N. On axial machines with counter-rotation of impellers. Proceedings of the Tomsk Polytechnic Institute, vol. 80, 1955.
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With some experience, such a selection is carried out quite quickly—after two or three attempts. ↩