ON THE EFFICIENCY OF A PAIR OF COUNTER-ROTATING AXIAL ROTORS

Yu. N. Sokolov

(Presented by Honored Worker of Science and Technology Prof. I. N. Butakov)

The use of counter-rotation of the rotors of axial-flow machines can make it possible to increase their economy. It is important here to establish within what limits and under what conditions this is possible.

Below we consider the question of the efficiency (e.f.f.) of a pair of counter-rotating axial rotors of a machine transmitting energy to a flow of liquid or gas (pumps, fans, compressors) under the normal mode of its operation. The possible value of this efficiency is compared with the efficiency of axial-flow machines of one-directional rotation.

By normal mode we shall mean the operation of a pair of counter-rotating axial rotors with axial inlet to the first and axial outlet from the second, which ensures full use of the possibilities of counter-rotation, eliminating the need to have guide or straightening apparatuses [1]. In this case the pair of rotors may also be considered as a stage of a multistage axial-flow machine.

The efficiencies of individual annular elements of an axial rotor of any type may, as is known, be unequal, and the efficiency of the rotor as a whole must, strictly speaking, be determined as

\[ \eta_i = \frac{ \displaystyle \int_{r_0}^{R} \eta_{\mathrm{el}} c_a r\,dr }{ \displaystyle \int_{r_0}^{R} c_a r\,dr }. \]

where \(\eta_{\mathrm{el}}\) is the efficiency of an annular element between radii \(r\) and \(r+dr\), \(r_0\) is the hub radius, \(R\) is the outer radius of the rotor, and \(c_a\) is the axial velocity.

In doing so, one must also take into account the losses associated with end effects and the finite blade length, as well as losses in the clearance.

For practical calculations, however, it is customary to define the internal (hydraulic—in the terminology of pumps and fans) efficiency of an axial-machine stage as

\[ \eta_{\mathrm{st}} = \eta_{\mathrm{sr}} \cdot k_l \cdot k_s, \]

where \(\eta_{\mathrm{sr}}\) is the efficiency of the annular element of the stage at the mean diameter, \(k_l\) is the correction for the finite blade length, and \(k_s\) is the correction for the influence of the radial clearance.

Assuming that the profiling of the blades of the stages being compared in axial-flow machines, both with one-directional and with counter-rotation, is carried out under identical conditions, the correction coefficients \(k_l\) and \(k_s\) for them may be regarded as identical, and only the efficiencies of the mean annular elements may be compared. In view of this, in what follows we shall consider only the latter.

Under the above-noted normal operating condition of a pair of counter-rotating axial rotors with equal rotational speeds, the circumferential components of the velocities at the exit from the first rotor and at the inlet to the second must be equal [2]:

\[ c'_{u2} = c'_{u1} = c_u . \]

The total energy (the theoretical head) imparted at each of the rotors will also be the same:

\[ L'_t = L''_t = \frac{1}{g} u c_u . \]

This energy is expended on the work of pressure increase (taking into account suction and discharge)—\(L_p\)—and on losses \((W_p)\) in the blade rows. For the first and second counter-rotating axial rotors we obtain, respectively:

\[ L'_t = L'_p + W'_p; \]

\[ L''_t = L''_p + W''_p. \]

The efficiencies of the blade rows of these rotors may then be determined from the generally known approximate formulas:

\[ \eta'_p = \frac{L'_p}{L'_p + W'_p} = \frac{1 - \mu'_p \tg \beta'_m}{1 + \mu'_p \ctg \beta'_m}; \tag{1} \]

\[ \eta''_p = \frac{L''_p}{L''_p + W''_p} = \frac{1 - \mu''_p \tg \beta''_m}{1 + \mu''_p \ctg \beta''_m}, \tag{2} \]

where \(\mu'_p\) and \(\mu''_p\) are the inverse qualities of the profiles in these blade rows, and \(\beta'_m\) and \(\beta''_m\) are the angles determining the direction of the mean geometric velocities of the relative flow.

From comparison of the velocity triangles of a pair of counter-rotating axial rotors operating at the above-mentioned normal condition [2], it is evident that \(\beta''_m < \beta'_m\), and therefore, for the same profile quality in the corresponding blade rows, the efficiency of the blade row of the second rotor must be less than that of the first:

\[ \eta''_p < \eta'_p . \]

With axial inlet to the first of these rotors and axial exit from the second, the kinetic energy of the flow passing through the pair of rotors may be regarded as unchanged, and the useful energy of this pair will be determined by the work of pressure increase [2]:

on the first rotor—

\[ L'_p = \frac{1}{g} c_u \left( u - \frac{c_u}{2} \right) \eta'_p, \]

on the second—

\[ L_p''=\frac{1}{g}\,c_u\left(u+\frac{c_u}{2}\right)\eta_p''. \]

The efficiency of a pair of counter-rotating axial rotors (the stage efficiency constituted by this pair in a multistage compressor machine, or the hydraulic efficiency in pump and fan terminology) on the given annular element will then be

\[ \eta_{vv} = \frac{L_p'-L_p''}{L_t'+L_t''} = \frac{ \frac{1}{g}c_u\left(u-\frac{c_u}{2}\right)\eta_p' + \frac{1}{g}c_u\left(u+\frac{c_u}{2}\right)\eta_p'' }{ 2\frac{1}{g}u c_u } \]

\[ = \left(\frac{1}{2}-\frac{1}{4}\frac{c_u}{u}\right)\eta_p' + \left(\frac{1}{2}+\frac{1}{4}\frac{c_u}{u}\right)\eta_p''. \tag{3} \]

It is evident that \(\eta_p' > \eta_{vv} > \eta_p''\).

To estimate the decrease in the efficiency of a pair of counter-rotating axial rotors due to the reduction of the cascade efficiency on the second rotor, it is convenient to express the difference \(\eta_p' - \eta_{vv}\) through \(\Delta\eta_p=\eta_p'-\eta_p''\):

\[ \eta_p' - \eta_{vv} = \left(\frac{1}{2}+\frac{1}{4}\frac{c_u}{u}\right)\Delta\eta_p. \tag{4} \]

When the swirl velocity \(c_u\) changes, the angles \(\beta_m'\) and \(\beta_m''\) change. In this case the quality of the profiles in the cascades may also change. However, if one considers the operation of different machines at design conditions and assumes that in all cases the selection of profiles was carried out on the basis of the condition of optimum flow around the corresponding cascades, then, for comparison of the efficiencies of these machines, the profile qualities may be regarded as identical and close to their optimum values.

The results of calculating the efficiencies of the cascades of the first and second rotors (\(\eta_p'\) and \(\eta_p''\)) and the efficiency of their pair in counter-rotation (\(\eta_{vv}\)) by formulas (1), (2), and (3), with \(\mu_p'=\mu_p''=0.02\), for various design conditions determined by the relative swirl velocity \(\frac{c_u}{u}\), and for two limiting values \(\frac{c_a}{u}=0.2\) and \(\frac{c_a}{u}=0.4\), are represented by the corresponding curves in Fig. 1. From comparison of these curves it follows that, while with increasing \(\frac{c_u}{u}\) \(\eta_p'\) increases and \(\eta_p''\) decreases, as a result \(\eta_{vv}\) is practically independent of the choice of the design value \(\frac{c_u}{u}\) in the usually used interval (the plot in Fig. 1 is drawn on a large scale, so that the noticeable deviation of the \(\eta_{vv}\) curve from the horizontal in fact corresponds to a change of this quantity by only 0.5%).

With an increase in the relative axial velocity \(\frac{c_a}{u}\), the efficiency of the pair of counter-rotating rotors increases noticeably, which corresponds to the usual increase in cascade efficiency with increasing angle \(\beta_m\).

In order to establish the relative economy of axial machines with counter-rotating impellers, let us compare the efficiency calculated by formula (3) with the corresponding efficiencies of axial machines of the usual type.

Figure 1. Efficiency of a pair of counter-rotating axial wheels and its comparison with the efficiency of a stage of the usual type.

As applied to fans, such a comparison should be made, first of all, with the efficiency of a single-stage axial fan without guide or straightening vanes. Passing to the terminology and notation adopted in the theory of axial fans, let us note that the efficiencies of the cascades, \(\eta_p\), given above correspond to the concept of the profile \((\eta_{prf})\) efficiency of a fan, while the efficiency of a pair of counter-rotating wheels as a whole \((\eta_{vv})\) corresponds to the hydraulic efficiency of a fan with such a pair \((\eta_g^{vv})\). As is known, the hydraulic efficiency of a fan is

\[ \eta_g = \eta_{prf}\cdot \eta_{kr}, \]

where

\[ \eta_{kr}=1-\frac{\rho \dfrac{c_u^2}{2}}{H_t} =1-\frac{1}{2}\cdot\frac{c_u}{u} \]

is the so-called swirl efficiency, which determines the nonutilization of the swirl energy of the outgoing flow,

\[ \rho \frac{c_u^2}{2}; \]

\(\rho\) is the density of air, and \(H_t=\rho u c_u\) is the theoretical fan head (in \(\mathrm{kg}/\mathrm{m}^2\)).

The natural condition for comparison in the case under consideration will obviously be equality of the relative swirl velocities on the given annular element of the single wheel of the single-stage axial fan and of each of the counter-rotating axial wheels:

\[ \left(\frac{c_u}{u}\right)_1 = \left(\frac{c_u}{u}\right)_{vv} = \frac{c_u}{u}. \]

The theoretical head of the pair of counter-rotating axial wheels will then be twice as large as that of the single-stage machine whose impeller has the same peripheral...

the velocity, as is each of the counter-rotating ones. For axial entry onto the impeller of a single-stage fan, the velocity triangles of the annular element under consideration must coincide completely with those for the first of the counter-rotating wheels and, consequently, their profile efficiencies will be the same: \(\eta'_{\mathrm{prf}}=\eta_{\mathrm{prf}}\).

The ratio of the hydraulic efficiencies of the annular element of a fan with a pair of counter-rotating axial wheels and of a single-stage fan

\[ \frac{\eta_g^{\mathrm{vv}}}{\eta_g^1} = \frac{ \left(\dfrac{1}{2}-\dfrac{1}{4}\dfrac{c_u}{u}\right)\eta'_{\mathrm{prf}} + \left(\dfrac{1}{2}+\dfrac{1}{4}\dfrac{c_u}{u}\right)\eta''_{\mathrm{prf}} }{ \eta'_{\mathrm{prf}}\cdot \eta_{\mathrm{kp}} } = \]

\[ = \frac{1}{2} + \frac{1}{2}\cdot \frac{ \left(1+\dfrac{1}{2}\cdot\dfrac{c_u}{u}\right)\eta''_{\mathrm{prf}} }{ \left(1-\dfrac{1}{2}\dfrac{c_u}{u}\right)\eta'_{\mathrm{prf}} } \tag{5} \]

will therefore depend on the quality of the profiles in the cascades and on the ratios adopted in the design conditions, \(\dfrac{c_u}{u}\) and \(\dfrac{\dot c_u}{u}\). The ratio of the total efficiencies of the fans being compared may be considered in the same way.

Fig. 2. Comparison of the hydraulic efficiencies of a single-stage axial fan and of a pair of counter-rotating axial wheels.

Figure 2 gives the results of the corresponding calculations for the same profile quality in all cascades, \(\mu_p=0.02\). The relative increase in efficiency when changing from a single-stage axial fan without guide and straightening apparatuses to a pair of counter-rotating axial wheels in high-pressure machines (large \(\dfrac{c_u}{u}\)) reaches 20–25%. This is explained by the reduction to zero of the swirl loss under counter-rotation of the impellers.

The use of guide or straightening apparatuses in a single-stage or two-stage axial fan converts it respectively into one or two stages of an axial compressor machine of the usual type (with one-sided rotation). Therefore, the further comparison of the efficiency of a pair of counter-rotating axial wheels is carried out with such a stage, consisting of an impeller and a straightening apparatus, returning again to the terminology and notation adopted for compressor machines.

The efficiency of a stage for an annular element of an axial compressor machine of the usual type can be determined as

\[ \eta_{\mathrm{stp}} = \frac{L_{p}^{\mathrm{r.w.}}+L_{p}^{\mathrm{g.v.}}}{L_t} = \frac{ \dfrac{c_u}{g}\left(u-\dfrac{c_u}{2}\right)\eta_p + \dfrac{c_u^2}{2g}\eta_n }{ \dfrac{1}{g}c_u u } = \left(1-\frac{1}{2}\frac{c_u}{u}\right)\eta_p + \frac{1}{2}\frac{c_u}{u}\eta_n, \tag{6} \]

where \(\eta_p\) is the efficiency of the rotor cascade, and \(\eta_n\) is the efficiency of the guide-vane apparatus.

For identical values of the ratio \(\dfrac{c_u}{u}\) on the annular element under consideration, a single rotor of a normal stage and each of the counter-rotating rotors, the latter, at identical peripheral speeds, transmit twice as much energy, \(L_{\mathrm{cr}}=2L_{\mathrm{stp}}\). Comparison of expressions (3) and (6) in this case leads to the conclusion that the efficiency of a pair of counter-rotating rotors will be higher than the efficiency of a normal stage if

\[ \left(\frac{1}{2}+\frac{1}{4}\frac{c_u}{u}\right) \left(\eta_p' - \eta_p''\right) < \frac{1}{2}\frac{c_u}{u}\left(\eta_p-\eta_n\right). \tag{7} \]

With the same quality of profiles in the cascades being compared and with axial inlet to the rotor of the normal stage, \(\eta_p=\eta_p'\). The comparison of interest to us is therefore determined by the relative magnitude of the efficiency of the second of the counter-rotating rotors \((\eta_p'')\) and the guide-vane apparatus \((\eta_n)\) of the normal stage. The latter can be calculated from the formula

\[ \eta_n= \frac{1-\xi_p^n \tan \alpha_m} {1+\xi_p^n \cot \alpha_m}, \]

where

\[ \cot \alpha_m = -\frac{c_u}{2c_a} \]

determines the direction of the mean geometric velocity of flow past the guide-vane apparatus for a stage with axial inlet and axial outlet.

The results of comparing the economy of stages with counter-rotation and with one-sided rotation of the rotors will thus depend on the values \(\dfrac{c_a}{u}\), \(\dfrac{c_u}{u}\), and on the quality of the profiles in the cascades of the rotors and of the guide-vane apparatus.

As an example of such a comparison, Fig. 1 shows the corresponding curves, from consideration of which it is evident that in the given case \(\left(\text{for } L_{\mathrm{stp}}=\dfrac{1}{2}L_{\mathrm{cr}}\right)\) the relative economy of counter-rotation increases with increasing \(\dfrac{c_a}{u}\) and with decreasing \(\dfrac{c_u}{u}\). The difference between \(\eta_{\mathrm{cr}}\) and \(\eta_{\mathrm{stp}}\) is then insignificant and, under ordinary conditions, does not exceed 1–2%.

If, however, the quality of the profiles in the cascade of the guide-vane apparatus is lower than the quality of the profiles of the rotor cascades (which, for example,

characteristic of axial fans), counter-rotation at sufficiently large values of \(\frac{c_a}{u}\) will provide higher efficiencies. This is also confirmed by the results of the corresponding experimental investigations, according to which axial fans with counter-rotating impellers give higher efficiencies than two-stage axial fans of the usual type [3], [4].

It may be of definite interest to compare the efficiency of stages with counter-rotation and with single-direction rotation in the particular case when, for identical diametral dimensions and ratios \(\frac{c_a}{u}\), a pair of counter-rotating wheels transmits the same energy as a normal stage consisting of an impeller and a guide apparatus \((L_{cr}=L_{st})\). In this case the circumferential velocity of each of the counter-rotating wheels \((u_{cr})\) must be less than that of the wheel of the normal stage \((u_{st})\), and, obviously,

\[ u_{cr}=\frac{u_{st}}{\sqrt{2}}. \]

To ensure equal capacity of the stages being compared, equality of the axial velocities is necessary and, consequently,

\[ \left(\frac{c_a}{u}\right)_{st} = \left(\frac{c_a}{u}\right)_{cr}\cdot\frac{1}{\sqrt{2}}. \]

This changes the conditions for comparing efficiency as compared with those considered above, since the values of \(\frac{c_a}{u}\) in the left- and right-hand sides of inequality (7) will now be unequal.

As an example of comparing efficiencies under these new conditions, the corresponding curves are plotted in Fig. 3, with the ratio \(\frac{c_a}{u}\) for counter-rotation taken in all cases as equal to 0.4. Under these conditions, counter-rotation provides higher efficiencies over the entire practically used range of ratios \(\frac{c_a}{u}\), even with identical profile quality in the blade rows of the impellers and guide apparatus.

Fig. 3. Comparison of the efficiency of a stage with counter-rotation and with single-direction rotation for equal energy transmission.

The relationships considered here for fans and compressor machines (assuming, in the latter, the air density within the stage to be unchanged) can also be applied to hydraulic machines, with the corresponding change in notation and terminology. The results of comparing the efficiency of machines with counter-rotation and with single-direction rotation will in this case, obviously, be the same.

The theoretical determination of efficiency carried out in the present work shows that the use of counter-rotation of the impellers of axial-flow machines should substantially improve economy only in comparison with an axial impeller without guide or straightening vanes. In comparison with a normal stage consisting of an impeller and a straightening apparatus, a pair of counter-rotating axial impellers should give higher efficiencies in those cases when the quality of the profiles in the cascades of the straightening (or guide) apparatuses of machines of the usual type is appreciably lower than on the impellers, or when, with the same profile quality, larger values of the ratio \(\dfrac{c_a}{u}\) are used in counter-rotation. With the same profile quality and the same \(\dfrac{c_a}{u}\), the economy of stages with counter-rotation and with one-direction rotation is, within the usual ranges of the ratio \(\dfrac{c_u}{u}\), practically the same.

An indisputable advantage of counter-rotation of axial impellers in all cases remains, however, the absence of the need to use guide or straightening apparatuses.

LITERATURE

1. Sokolov Yu. N. On axial-flow machines with counter-rotation of the impellers. Izv. TPI, vol. 80, 1955.

2. Sokolov Yu. N. Calculation of axial-flow machines with counter-rotation of the impellers. Izv. TPI, vol. 109, 1960.

3. Arshakov A. P. Operation of a two-stage axial fan with counter-motion of the impellers. Proceedings of the Leningrad Shipbuilding Institute, issue XVII, 1955.

4. Sokolov Yu. N. On the application of counter-rotation of impellers in axial compressor machines. Proceedings of the 1st Interuniversity Conference on Aircraft Vane Machines, Oborongiz, 1958.