FUNDAMENTALS OF A METHODOLOGY FOR APPROXIMATE CALCULATION OF HIGH-SPEED FLOWMETERS WITH TRANSDUCERS OF THE “PROPELLER” TYPE*

S. D. Smiragin, V. P. Shuk

(Presented by Prof. Yu. N. Sokolov)

At the present time, high-speed flowmeters with transducers of the “propeller” type have become widely used for measuring the velocity (flow rate) of various liquids under a variety of conditions. In this connection, it becomes necessary to develop a methodology for calculating the design parameters that determine the rational dimensions and the prescribed operating regime of the transducers.

The relationship between the flow velocity \(v\), the angular velocity of rotation of the transducer \(\omega\), its design parameters, the flow parameters, and friction in the bearings is determined approximately by the equation of the static characteristic [1]

\[ v = \omega \left( \sqrt{\frac{a_1 + \alpha b_1 + \alpha^2 c_1}{a_2 + \alpha a_3 + \alpha^2 a_4}} + \frac{\alpha f_2 - f_1}{f_3 - \alpha f_4} \right), \tag{1} \]

where \(\alpha\) is a coefficient due to the manifestation of the viscosity of the liquid; the remaining coefficients are determined by the functions

\[ F(L, z, d_{\text{st}}, d_{\text{n}}, H_{\text{g}}, f); \tag{2} \]

here \(L\) is the blade length, \(z\) is the number of blades, \(d_{\text{st}}\) is the hub diameter, \(d_{\text{n}}\) is the outside diameter, \(H_{\text{g}}\) is the geometric pitch, and \(f\) is the coefficient of frictional resistance in the transducer bearings.

Equation (1) indicates that the operation of the transducer depends on the physical properties of the liquid being measured. However, the use of this equation for an exact calculation of transducers is impossible because of the absence of numerical data on the resistance coefficient \(\alpha\) for various Reynolds numbers and on the friction coefficient \(f\) for liquids of different viscosity. Nevertheless, equation (1) points to the possibility of developing a methodology for the approximate calculation of transducers.

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

Let us determine under what conditions equation (1) will be numerically identical for different transducers. This is possible when

\[ \sqrt{\frac{a_1+\alpha b_1+\alpha^2 c_1}{a_2+\alpha d_3+\alpha^2 a_4}} + \frac{\alpha f_2-f_1}{f_3-\alpha f_4} = \mathrm{idem}. \tag{3} \]

For equality (3) to be satisfied, it is necessary that

\[ F\left(L, z, d'_{\mathrm{st}}, d'_{\mathrm{n}}, H_{\mathrm{r}}, f\right)=\mathrm{idem}. \tag{4} \]

It is obvious that equation (4) will hold when the arguments are equal. Since \(L, z, d'_{\mathrm{st}}, d'_{\mathrm{n}}, H_{\mathrm{r}}\) are geometric parameters, in relative units they will be numerically identical for all geometrically similar transducers.

The coefficient \(f\) depends on the quality of the contacting surfaces, the type of lubricant, etc.; and, if certain conditions are fulfilled, its constancy can be ensured.

Consequently, for geometrically similar transducers, equality (4) will be valid.

In (3), the coefficient \(\alpha\) is a dimensionless resistance coefficient depending on the nature of the flow and on the design of the transducer. It is an obvious aim to design the transducer so that the head losses in it are small. In this case, under steady flow of the transducer blades by the liquid stream, the total resistance is practically determined by frictional resistance. In [2], for similar conditions, the function

\[ \alpha = F_1(\mathrm{Re}, N_{ji}), \tag{6} \]

was obtained, indicating that the resistance coefficient depends on the Reynolds number \(\mathrm{Re}\) and on the turbulence numbers \(N_{ji}\).

It follows from (6) that when \(\mathrm{Re}=\mathrm{idem}\), \(N_{ji}=\mathrm{idem}\), then also \(\alpha=\mathrm{idem}\). The number \(\mathrm{Re}\) entering into (6) is determined by the known relation. Difficulties arise in determining the numbers \(N_{ji}\). However, experiments show that, as \(\mathrm{Re}\) increases, the influence of \(N_{ji}\) on \(\alpha\) decreases. Therefore, for turbulent flow one may approximately take

\[ \alpha = F'_1(\mathrm{Re}). \tag{7} \]

In similarity theory it is proved that, in order to obtain \(\mathrm{Re}=\mathrm{idem}\), it is necessary and sufficient to observe mechanical similarity, which is determined by the geometric, kinematic, and dynamic similarity of the flows. Numerous experiments show [3] that in turbulent flows, careful observance of the geometric similarity of the design as a whole and of the relative roughness of the blade surfaces in particular automatically ensures, with acceptable error, both kinematic and dynamic similarity of the flows.

Consequently, the resistance coefficient \(\alpha\) will be numerically identical for different transducers operating under diverse conditions, provided that the following are observed:

1) geometric similarity of the transducers,
2) geometric similarity of the relative roughnesses of the transducer blades,
3) Reynolds numbers.

If these conditions are fulfilled, equality (3) will hold; consequently, equation (1) in relative units will be numerically the same for different transducers.

It follows from (1) that, when \(\mathrm{Re}=\mathrm{idem}\) is observed, for all geometrically similar transducers the static characteristics in absolute units will differ by the value of the constant of geometric similarity. If there are two transducers, 1 and 2, with sensitivities \(a_{01}\) and \(a_{02}\), differing in that the linear dimensions of transducer 2 are \(\kappa\) times smaller than the same dimensions of transducer 1, then

\[ a_{02}=\frac{a_{01}}{\kappa}. \tag{8} \]

This makes it possible, from the static characteristics obtained as a result of an experimental study of transducers of one standard size on a given liquid, to calculate the characteristics of all geometrically similar transducers intended for measuring the velocity of various media in pipelines of different diameter.

For transducers operating in the automodel region, the lower limit of validity of the recalculation results is determined from the relation

\[ \mathrm{Re}^{p}_{\min}=\frac{v^{p}_{\min}L^{p}}{\nu^{p}} \geq \mathrm{Re}^{op}_{\text{pred.}}= \frac{v^{op}_{\min}L^{op}}{\nu^{op}}, \tag{9} \]

where \(v^{op}_{\min}\) is the minimum flow velocity at which the automodel regime of flow around the blades of the experimental transducer begins;

\(L^{op}\) is the characteristic geometric parameter of the experimental transducer;

\(\nu^{op}\) is the viscosity of the liquid on which the static characteristic of the experimental transducer was taken;

\(v^{p}_{\min}, L^{p}, \nu^{p}\) are the same quantities for the calculated transducer.

The construction of the methodology consists of the following:

1) recalculation, according to (8), of the experimental dependences of the sensitivity of the transducers on the corresponding parameters \((L, z, d_{\mathrm{n}}, H_{\mathrm{g}})\) to transducers geometrically similar to those tested;

2) indication, from (9), of the lower limit of validity of the recalculated characteristics with respect to the number \(\mathrm{Re}\). As the characteristic dimension entering into (9), it is expedient to take the internal diameter of the pipeline \(D_{\mathrm{tr}}\).

The results of such calculations for four standard pipeline diameters are presented in the form of nomograms:

Fig. 1—for transducers differing in \(L\);

Fig. 2—for transducers differing in \(d_{\mathrm{n}}\) (in this nomogram, \(\Delta\) is the clearance between the transducer and the pipeline; consequently, \(d_{\mathrm{n}}=D_{\mathrm{tr}}-2\Delta\));

Fig. 3—for transducers differing in \(z\);

Fig. 4—for transducers differing in \(H_{\mathrm{g}}\).

On the nomograms, the dependence corresponding to \(D_{\mathrm{tr}}=93\) mm was obtained experimentally.

The design parameters of the calculated and experimental (corresponding to \(D_{\mathrm{tr}}=93\) mm) transducers are summarized in Table 1.

Use of the nomogram is as follows. From the measurement specification, the required sensitivity of the transducer \(a_0^p\) and the number \(\operatorname{Re}_{\min}^p\), corresponding to the lower measurement limit \(v_{\min}\), are determined. From \(a_0^p\) and \(D_{\mathrm{tr}}\), in which the measurement is assumed to be made, the corresponding nomogram determines

Fig. 1. Nomogram for determining blade length

\[ \operatorname{Re}_{\mathrm{pred.}}^{\mathrm{op}} \]

and one of the design parameters \(\Delta\), \(z\), \(H_r\), or \(L\) (depending on which nomogram is used). When using the nomogram in Fig. 2,

Fig. 2. Nomogram for determining outside diameter

\[ \operatorname{Re}_{\mathrm{pred.}}^{\mathrm{op}} = 18000, \]

whereas in Figs. 1, 3, and 4 the determination of \(\operatorname{Re}_{\mathrm{pred.}}^{\mathrm{op}}\) is indicated by arrows. After \(\operatorname{Re}_{\mathrm{pred.}}\) has been determined, the relation

\[ \operatorname{Re}_{\mathrm{pred.}}^{\mathrm{op}} \leq \operatorname{Re}_{\min}^{p} \]

is checked. If this condition is not satisfied, it is necessary to determine, from the corresponding nomogram, another of the parameters

Fig. 3. Nomogram for determining the number of blades

Fig. 4. Nomogram for determining the geometric pitch

\(\Delta, z, H_{\mathrm{r}}, L\) until the inequality is satisfied. After this, from Table 1, for the given \(D_{\mathrm{tr}}\), the remaining design parameters of the transducer are taken.

Table 1

We shall explain the application and accuracy of the proposed method by an example.

Example.

a. Technical conditions for the measurement.

In a pipeline with \(D_{\mathrm{tr}} = 72\ \mathrm{mm}\), it is necessary to measure the water velocity at temperature \(t_{\mathrm{w}} = 15^\circ\mathrm{C}\). The range of measured velocities is
\(v = 0.3\text{–}3\ \mathrm{m/sec}\). It is necessary to provide, at the input of the measuring instrument, an electrical-signal frequency \(f = 6 \div 66\ \mathrm{Hz}\).

b. Calculation of the transducer.

Assume that, for one revolution of the transducer, the tachometric device gives two pulses. Then the number of revolutions of the transducer will be \(n = 3 \div 33\ \mathrm{rev/sec}\).

The required sensitivity of the transducer is

\[ a_0^{\mathrm{p}}=\frac{\Delta v}{\Delta n}=\frac{2.7}{30}=0.09\ \mathrm{m/rev}. \]

The Reynolds number is

\[ \operatorname{Re}^{\mathrm{p}}_{\min} = \frac{0.3\cdot 0.072}{1.01\cdot 10^{-6}} = 21400. \]

From \(a_0^{\mathrm{p}} = 0.09\ \mathrm{m/rev}\) and \(D_{\mathrm{tr}} = 72\ \mathrm{mm}\), from the nomogram of Fig. 2 (in this case any other nomogram may be used), we determine \(\Delta = 1.2\ \mathrm{mm}\). We take \(\operatorname{Re}^{\mathrm{op}}_{\mathrm{pred.}} = 18000\).

The condition \(\operatorname{Re}^{\mathrm{op}}_{\mathrm{pred.}} \leq \operatorname{Re}^{\mathrm{p}}_{\min}\) is satisfied. In accordance with \(\Delta = 1.2\ \mathrm{mm}\), the outer diameter of the transducer is equal to \(d_{\mathrm{n}} = 69.6\ \mathrm{mm}\). From Table 1, for \(D_{\mathrm{tr}} = 72\ \mathrm{mm}\), we write out the design parameters of the transducer:

\[ d_{\mathrm{st}} = 20.2\ \mathrm{mm},\quad d = 11.2\ \mathrm{mm},\quad t = 2.3\ \mathrm{mm},\quad L = 78.5\ \mathrm{mm}, \]

\[ H_{\mathrm{r}} = 107.4\ \mathrm{mm},\quad z = 2. \]

Experimental verification of the calculated transducer showed that the discrepancy between the calculated and experimental data does not exceed \(2.5\%\) and is due to the error of the nomogram graphs, ne-

the manufacturing accuracy of the transducer and the uncontrollability of the quality of the bearings. Reducing these causes to a minimum will undoubtedly increase the accuracy of calculating transducers by the method set forth.

Conclusions

Thus, the fundamentals of an approximate methodology for calculating transducers of the “propeller” type have been set forth above. For practical use, the methodology requires only that the measurement conditions be specified and does not require special experimental determination of any coefficients. The calculation error does not exceed 2.5%. To increase the accuracy of the calculation, quantitative allowance for friction in the transducer bearings is necessary.

References

1. S. D. Smiryagin, V. P. Shuk. In: Proceedings of the Scientific and Technical Conference on the Results of Research Work for 1964–1965, Thermal-Power Engineering Section, Subsection on Automation of Production Processes. MEI, Moscow, 1965.

2. A. N. Patrashov. Hydromechanics. Voenno-Morskoe Izd., Moscow, 1953.

3. M. V. Kirpichev. Theory of Similarity. Publishing House of the Academy of Sciences of the USSR, 1953.