Results of a Study of Velocity Flowmeters with Propeller-Type Transducers*

S. D. Smiryagin, V. P. Shuk

(Presented by Prof. Yu. N. Sokolov)

The theoretical equation of the static characteristic of “propeller”-type transducers, obtained with the aid of the hydrodynamic theory of cascades, has the form [1]

\[ v=\omega\left( \sqrt{ \frac{ a_1 + a b_1 + a^2 c_1 + \dfrac{d_1 - a d_2}{z \rho \omega} + \dfrac{e_1 + a e_2}{z \rho \omega^2} }{ a_2 + a a_3 + a^2 a_4 } } + \frac{a f_2 - f_1}{f_3 - a f_4} \right), \tag{1} \]

where \(v\) is the fluid velocity, \(\omega\) is the angular velocity of rotation of the transducer, \(a\) is a coefficient determined by the manifestation of the fluid viscosity, \(\rho\) is the density of the fluid; the coefficients \(a_1, a_2, a_3, a_4, b_1, c_1, f_1, f_2, f_3, f_4\) are determined by functions of the form \(F_1(L, z, d_{\mathrm{st}}, d_{\mathrm{n}}, H_{\mathrm{g}}, f)\), and \(d_1, d_2, e_1, e_2\) by functions \(F_2(L, z, d_{\mathrm{st}}, d_{\mathrm{n}}, H_{\mathrm{g}}, f, M_0, M_y)\); here \(L\) is the blade length, \(z\) is the number of blades, \(d_{\mathrm{st}}\) is the hub diameter, \(d_{\mathrm{n}}\) is the outside diameter, \(H_{\mathrm{g}}\) is the geometric pitch, \(f\) is the coefficient of friction in the bearings, \(M_0\) is the initial friction moment, and \(M_y\) is the reaction of the tachometric device.

The design cascade of profiles of the transducer under study, and the velocity and force triangles for the profile in the cascade, for which equation (1) was obtained, are shown in Fig. 1.

The following requirements are imposed on the static characteristic (1):

1) the characteristic must correspond to the maximum sensitivity of the transducer;

2) the characteristic must deviate as little as possible from the straight line passing through the origin of the coordinates;

3) the characteristic must be reproducible and stable in time.

Let us consider ways of satisfying these requirements.

A violation of the proportionality between \(v\) and \(\omega\) may occur for two reasons. First, when the resisting moment due to friction in the bearings and the reaction of the tachometric devi-

* The study was carried out at the Department of Thermal Control and Automation of the Moscow Power Engineering Institute.

become commensurate with the torque. In this case, in (1) the terms \(\dfrac{d_1-\alpha d_2}{z\rho\omega}\), \(\dfrac{e_1-\alpha e_2}{z\rho\omega^2}\) increase. Secondly, when operation of the transducer goes beyond the region auto-

Fig. 1. Cascade of profiles, velocity and force triangles of a profile in the cascade

model with respect to the Reynolds number. In this case the rectilinearity of the characteristic is disturbed because \(\alpha\) is not constant.

A reduction in $\dfrac{d_1-\alpha d_2}{z\rho\omega}$ and $\dfrac{e_1-\alpha e_2}{z\rho\omega^2}$ is promoted by increasing the number of blades. This is confirmed experimentally**. When $z$ is increased from four to ten, the minimum flow velocity at which proportionality between $v$ and $\omega$ is preserved decreases from $0.3\ \text{m/sec}$ to $0.1\ \text{m/sec}$.

Therefore, to ensure the rectilinearity of the static characteristic in the lower measuring range, the use of multiblade transducers with $z=6 \div 10$ may be recommended. This is valid when measuring the velocity of a liquid with insignificant viscosity. When measuring a viscous liquid, it is necessary to use a smaller number of blades, $z=3 \div 5$, in order to avoid premature violation of proportionality due to a change in $\alpha$.

The friction coefficient in the supports of the transducer, $f$, has a substantial effect on the static characteristic, since in equation (1) all coefficients, except $a$, depend on $f$. It is shown in [1] that the influence of $f$ decreases as the diameter of the bearings is reduced. At the same time, the terms $\dfrac{d_1-\alpha d_2}{z\rho\omega}$ and $\dfrac{e_1-\alpha e_2}{z\rho\omega^2}$ also decrease. It has been found experimentally that, when the bearing diameters are reduced by a factor of 1.5, the lower measuring limit, while maintaining rectilinearity of the characteristics, decreases from $0.1\ \text{m/sec}$ for a ten-blade transducer to $0.054\ \text{m/sec}$ for a two-blade transducer, i.e., by 45%.

Therefore, to extend the measuring range by reducing the initial velocities, the maximum possible reduction of the bearing diameter may be recommended.

Since the coefficients of equation (1) are in a complex dependence on the friction coefficient [1], the reproducibility and stability of the calibration characteristics depend to a considerable extent on the quality of the supports. From published materials [2] it is known that, with velocity flowmeters, measurements can be made with an accuracy of $0.3 \div 0.5\%$. To ensure this accuracy in repeated measurements after dismantling the sensor, it is first of all necessary to establish standards for the quality of the supports and to develop methods for monitoring it.

It follows from equation (1) that, to eliminate the influence of the viscosity of the liquid on the operation of the transducers, the coefficients at $\alpha$ must be set equal to zero. However, physically this is impossible, and only their minimization is possible through an appropriate choice of design parameters.

Calculations performed on a digital computer showed that, in order to minimize the dependence of transducer operation on the viscosity of the liquid, it is necessary to use two-blade transducers with a minimum blade length. The influence on this dependence of the outside diameter, the hub diameter, and the geometric pitch is insignificant. Therefore, to obtain the required sensitivity of the transducer, these parameters may be varied over wider limits.

As noted, in addition to the rectilinearity of the static characteristic, it is important that it provide the greatest sensitivity—

**) Here and below, experimental data are presented that were obtained by the authors of the present article as a result of a specially conducted experimental study of transducers.

...ness of the transducer. It follows from Fig. 1 that the force \(P\) acting on an element of the transducer blade is resolved into a circumferential component \(P_u\), producing a torque, and an axial component \(P_0\), producing a resistance moment in the supports. The rotational speed of the transducer is determined by the action of both components. Calculations show that, with changes in the design parameters of the transducer, the vector \(P\) changes in magnitude and direction.

Therefore, in order to obtain the greatest rotating force and, consequently, sensitivity, the optimum design will be that transducer which provides the greatest value of the vector \(P\) in modulus at the smallest \(\Theta\) (Fig. 1). This corresponds to maximum \(P_u\) and minimum \(P_0\). Let us determine the conditions that ensure \(P_u=\max\) and \(P_0=\min\).

As the principal parameter determining the operation of the transducer, we shall take \(\dfrac{Lz}{2\pi r}\) and investigate the function \(P_u-P_0=F\!\left(\dfrac{Lz}{2\pi r}\right)\).

The results of calculating this function by the formulas of [1], without taking into account the viscosity of the liquid, are presented graphically in Fig. 2. The graphs characterize the distribution of \(P_u-P_0\) over the height of the blade as a function of the blade length \(L\) and the number of blades \(z\).

Fig. 2. Distribution of \(P_u-P_0\) over the height of the transducer blade as a function of \(\dfrac{Lz}{2\pi r}\)

In Fig. 2 three regions can be distinguished:

1) for \(\dfrac{Lz}{2\pi r}\leq 0.1\), all the curves have a tendency to increase, and, on the whole for the blade, \(P_u=\max\), \(P_0=\min\), which corresponds to the greatest sensitivity of the transducer;

2) at \(0.1 < \dfrac{Lz}{2\pi r} \leqslant 1.0 - 1.2\), the periphery of the blade causes an increase in \(P_0\) (the curves \(r = 0.3183;\ 0.2865;\ 0.2546;\ 0.2223\) drop sharply), which increases the braking torque in the supports and worsens the sensitivity of the transducer;

3) at \(\dfrac{Lz}{2\pi r} > 1.2\), all curves become parallel to the abscissa axis, \(P_u\) and \(P_0\) assume constant values, and the sensitivity remains unchanged.

On the basis of the calculated data we arrive at the conclusion that short-bladed transducers have greater sensitivity than long-bladed ones. The graphs in Fig. 2 indicate ways of achieving the greatest sensitivity of the transducer. For this it is necessary to profile the blades so that the ratio \(\dfrac{Lz}{2\pi r}\) for each \(r\) falls at the maximum of the corresponding curve, i.e.,

\[ \dfrac{Lz}{2\pi r_{0.3183}} \approx 0.1,\quad \dfrac{Lz}{2\pi r_{0.2865}} \approx 0.15, \]

and so on. In practice, this complicated profiling can, with a sufficient degree of accuracy, be replaced by satisfying the conditions

\[ \dfrac{Lz}{2\pi r_{\mathrm{n}}} = 0.1;\quad \dfrac{Lz}{2\pi r_{\mathrm{st}}} = 0.8 \div 1.2. \]

The theoretical results obtained are confirmed by experimental data, presented in Fig. 3 in the form of the dependence

Fig. 3. Dependence of the sensitivity of transducers on blade length

of the sensitivity of the transducers \(a_0\) on the blade length \(L\). It follows from Fig. 3 that the greatest sensitivity is possessed by the transducer with \(L = 13\ \mathrm{mm}\), which corresponds to \(\dfrac{Lz}{2\pi r_{\mathrm{n}}} = 0.09\). With an increase in

increasing the blade length to \(L = 140 \div 160\) mm, which corresponds to

\[ \frac{Lz}{2\pi r_{\mathrm{n}}} = 0.97 \text{–} 1.11, \]

the sensitivity of the transducers deteriorates and, with a further increase in blade length, remains unchanged.

Theoretical and experimental results make it possible to recommend, in order to ensure the greatest sensitivity of the transducers, that the following conditions be satisfied:

\[ \frac{Lz}{2\pi r_{\mathrm{n}}} \approx 0.1; \qquad \frac{Lz}{2\pi r_{\mathrm{st}}} \approx 0.8 \div 1.2. \]

The blade length has a substantial effect not only on sensitivity, but also on the lower limit of measurement, i.e., on the value of the initial velocity of the liquid at which the transducer begins to rotate. An experimental study of three transducers having blade lengths of, respectively, 217.2 mm, 27 mm, and 13 mm, and identical with respect to the other geometrical parameters, showed that the transducer with \(L = 217.2\) mm had the lowest initial velocity. This circumstance is explained by the difference in the flow conditions around long and short blades. Calculations show that blades with \(L = 13\) mm have a hydrodynamic angle of attack \(\mu\) more than twice as large as blades with \(L = 217.2\) mm. As \(\mu\) increases, the liquid velocity at which turbulization of the boundary layer occurs at the blade–liquid interface increases. As a result, the initial velocity at which the transducer begins to rotate increases.

Therefore, in order to ensure the smallest lower limit of measurement, it is possible to recommend satisfying the conditions

\[ \frac{Lz}{2\pi r_{\mathrm{n}}} \approx 0.8 \div 1.0; \qquad \frac{Lz}{2\pi r_{\mathrm{st}}} \approx 1.5 \div 2.0. \]

To reduce the initial velocity of measurement by short-bladed transducers, which are more sensitive, the use of artificial turbulizers appears expedient. However, their effectiveness can be determined only experimentally.

In conclusion, we formulate the principal results. To ensure linearity of the static characteristic of the transducers and to reduce the lower limit of measurement, the following conditions must be satisfied:

1) one should choose:

a) \(z = 6 \div 10\) — for a low-viscosity liquid,

b) \(z = 3 \div 5\) — for a viscous liquid,

c)

\[ \frac{Lz}{2\pi r_{\mathrm{n}}} \approx 0.8 \div 1.0 \quad \text{and} \quad \frac{Lz}{2\pi r_{\mathrm{st}}} \approx 1.5 \div 2.0. \]

2) the dimensions of the supports should be reduced as much as possible.

To ensure the greatest sensitivity of the transducer, one should choose multi-bladed transducers (\(z\) up to 10) and take

\[ \frac{Lz}{2\pi r_{\mathrm{st}}} \approx 0.8 \div 1.2; \qquad \frac{Lz}{2\pi r_{\mathrm{n}}} \approx 0.1. \]

To minimize the effect of the viscous properties of the liquid being measured on the operation of the transducers, it is necessary to use two-bladed transducers with short blades.

Conclusions

The article presents practical recommendations for selecting the number of blades, their length, and the dimensions of supports, applicable to various standard sizes of transducers. With the aid of a theoretical equation for the statistical characteristic, the mechanism by which each of the listed parameters affects the operation of the transducers has been clarified. This makes it possible to gain a deeper understanding of the basic regularities governing the operation of transducers of the “propeller” type.

References

1. S. D. Smirnagin, V. P. Shuk. Proceedings of the Scientific and Technical Conference on the Results of Research Work for 1964–1965, thermal-power section, subsection on automation of production processes. MEI, Moscow, 1965.

2. L. L. Boshnyak. Study of transducers for impeller-tachometric flowmeters. Candidate dissertation. LMI, Leningrad, 1962.