DETERMINATION OF THE DISCHARGE COEFFICIENT THROUGH A GAP BETWEEN A THROTTLE PLATE AND A PIPE

V. P. BELTYUKOV

(Presented by Professor, Doctor G. I. Fuks)

Knowledge of the magnitude of the discharge coefficient through the gap between a throttle plate and a pipe proved necessary in calculating pneumodynamic speed governors for carburetor internal-combustion engines, in order to establish the dependence of the turning moment on the plate axis on the angle of its opening and on the engine speed.

To determine the discharge coefficient, one often (e.g., [1]) uses the formula

\[ \mu = \frac{G}{F_4 \sqrt{2g \cdot \Delta p_{12}\gamma_1}}, \tag{1} \]

where

\[ \begin{aligned} G &\text{— mass flow rate of air,}\\ F_4 &\text{— area of the pipe opening at section } 4\text{—}4 \text{ (Fig. 1),}\\ \gamma_1 &\text{— specific weight of the air before the plate,}\\ \Delta p_{12} &= P_1 - P_2 \text{ — pressure drop across the plate.} \end{aligned} \]

Fig. 1

Formula (1) is obtained from the discharge equation for an incompressible fluid; therefore, its application in studying the question of outflow ...

… cannot be justified for a strongly covered throttle valve. In addition, experiments carried out by the author showed that the value of the discharge coefficient determined by formula (1) varies greatly depending on the angle of closure and the shape of the throttle valve (Fig. 2).

Fig. 2

It also seems incorrect to us to use the expression [3, 4]

\[ \mu=\frac{G}{f F_1 \sqrt{P_0\gamma_0}\,\psi}, \tag{2} \]

where \(F_1\) is the cross-sectional area of the pipe before the valve,

\[ f=\frac{F_4}{F_1} \]

is the relative flow area of the throttle.

First, in formula (2) the pressure \(P_1\) and the specific weight of the air \(\gamma_1\) before the valve are replaced by the atmospheric pressure \(P_0\) and the specific weight of the intake air \(\gamma_0\). Such a replacement is permissible only if the resistance upstream of the valve is very small, which cannot be said of the resistances in the intake system of internal-combustion engines.

Second, formula (2) does not take into account the velocity \(V_1\) in section \(1—1\) before the throttle valve. Our calculations have shown that the influence of the velocity \(V_1\) on the value of the discharge coefficient can be neglected only when the throttle is strongly covered. With the throttle fully open, the error in determining \(\mu\) may amount to \(10—15\%\).

Third, the quantity \(\psi\) entering formula (2) is determined as a function of

\[ \frac{P_2}{P_1}, \]

i.e., as a function of the difference \(\Delta p_{12}\). But the quantity of air passing through the pipe in which the throttle valve is installed is determined not by the difference \(P_1-P_2\), but by the pressure drop \(\Delta p_{13}\) between the section before the valve \(P_1\) and the place of greatest contraction of the jet near it, \(P_3\) (Fig. 1). The pressure difference \(\Delta p_{12}\) is not proportional to the difference \(\Delta p_{13}\) and, moreover, depends strongly on the point at which the pressure \(P_2\) is taken.

A formula for determining the discharge coefficient as a function of the difference \(P_1-P_3\) was derived by us on the basis of the following considerations.

The energy balance, referred to \(1\ \mathrm{kg}\) of air, for sections \(1—1\) and \(3—3\) is written in the form:

\[ \frac{V_1^2}{2g}+\frac{P_1}{\gamma_1}+z_1+\frac{U_1}{A} = \frac{V_3^2}{2g}+\frac{P_3}{\gamma_3}+z_3+\frac{U_3}{A} +\xi\,\frac{V_3^2}{2g}. \tag{3} \]

Neglecting the change in elevation head over the section \(1—3\) and considering the outflow process adiabatic, after transformations we obtain

\[ \frac{V_1^2}{2g} = \frac{V_3^2}{2g} - \frac{k}{k-1}\, \frac{P_1}{\gamma_1} \left[ 1-\left(\frac{P_3}{P_1}\right)^{\frac{k-1}{k}} \right] +\xi\,\frac{V_3^2}{2g}. \tag{4} \]

From this expression

\[ V_3=\sqrt{\frac{1}{1+\xi}}\cdot \sqrt{V_1^2+2g\,\frac{k}{k-1}\,\frac{P_1}{\gamma_1} \left[1-\left(\frac{P_3}{P_1}\right)^{\frac{k-1}{k}}\right]} =\varphi V_{3\text{theor}}. \tag{5} \]

Here \(\varphi=\sqrt{\dfrac{1}{1+\xi}}\) is the velocity coefficient,

\[ V_{3\text{theor}.}= \sqrt{V_1^2+2g\,\frac{k}{k-1}\,\frac{P_1}{\gamma_1} \left[1-\left(\frac{P_3}{P_1}\right)^{\frac{k-1}{k}}\right]} \]

is the velocity in section 3—3 in the absence of friction losses (theoretical case \(\xi=0\)).

According to the continuity equation, the air flow rate through section 3—3 is

\[ G=V_3 a F_4 \gamma_3, \tag{6} \]

where \(a=\dfrac{F_3}{F_4}\) is the jet contraction coefficient.

For the adiabatic outflow process adopted by us,

\[ \gamma_3=\gamma_1\left(\frac{P_3}{P_1}\right)^{\frac{1}{k}}. \tag{7} \]

Substituting expressions (5) and (7) into equation (6), we obtain

\[ G=\varphi \sqrt{V_1^2+2g\,\frac{k}{k-1}\cdot\frac{P_1}{\gamma_1} \left[1-\left(\frac{P_3}{P_1}\right)^{\frac{k-1}{k}}\right]}\cdot a f F_1 \gamma_1 \left(\frac{P_3}{P_1}\right)^{\frac{1}{k}} = \]

\[ =\mu f \sqrt{ G^2\left(\frac{P_3}{P_1}\right)^{\frac{2}{k}} + F_1^2\gamma_1 P_1\psi^2 }, \tag{8} \]

where \(\mu=a\varphi\) is the discharge coefficient,

\[ \text{and}\quad \psi= \sqrt{ 2g\,\frac{k}{k-1} \left[ \left(\frac{P_3}{P_1}\right)^{\frac{2}{k}} - \left(\frac{P_3}{P_1}\right)^{\frac{k+1}{k}} \right] }. \tag{10} \]

From equation (8) we determine the discharge coefficient

\[ \mu= \frac{1}{ f\sqrt{ \left(\dfrac{P_3}{P_1}\right)^{\frac{2}{k}} + \dfrac{P_1}{\gamma_1 V_1^2}\cdot\psi^2 } }. \]

It is more convenient to present the latter expression in the form

\[ \mu= \frac{1}{ f\sqrt{ \left(\dfrac{P_3}{P_1}\right)^{\frac{2}{k}} + \dfrac{P_1}{q_1}\cdot\frac{\psi^2}{2g} } }, \tag{11} \]

where \(q_1=\dfrac{\gamma_1 V_1^2}{2g}\) is the velocity head in section 1—1 before the damper.

In comparison with the formulas ordinarily used [5, 2], formula (11) differs in that, first, the discharge coefficient determined by it takes into account only the contraction of the jet and the friction losses when the gas flows through the slot between the throttle valve and the pipe, whereas ordinarily \(\mu\) also takes into account the magnitude of the relative flow area of the throttle, the flow velocity ahead of the valve, the velocity distribution over the flow cross section, and the influence of the locations of pressure tapping; second, formula (11) takes into account the air velocity in section \(1—1\) before the valve; and, third, the discharge coefficient is determined as a function of the pressure drop before the valve and at the place of greatest contraction of the jet near it.

The latter circumstance is in complete agreement with the theory of efflux. The basic equations of this theory are a consequence of applying Bernoulli’s equation to sections \(1—1\) and \(3—3\), in which the streamlines are parallel to the axis of the pipeline and, consequently, the pressure is constant over the entire flow cross section. True, the calculation formulas usually include pressures before and after the throttling device, which are more conveniently measured, while the deviation caused by changing the pressure-tapping locations is taken into account by introducing an additional coefficient [5].

Experimental determination of the pressure at the place of greatest contraction of the jet presents considerable difficulty; therefore the dependence of \(P_3\) on the pressure \(P_2\) in some section behind the valve, which can be readily determined experimentally, was derived.

Usually [7] the process of gas throttling is calculated as an isothermal process (the gas being taken as ideal). To calculate the throttling process of a real gas, it is necessary to specify the equation of state of the real gas and the dependence \(c_p=f(P)\). We do not have these data for the gasoline-air mixture, and experiments on the L6/3 engine showed that the temperature of the mixture during throttling does not remain constant; therefore the pressure \(P_3\) was determined by us in the following manner.

Neglecting the change in internal energy and the friction losses between the third and second sections, we obtain:

\[ P_3=P_2\frac{\gamma_3}{\gamma_2} -\frac{\gamma_3 V_2^2}{2g} \left[ \frac{1}{f^2\left(\dfrac{\gamma_3}{\gamma_2}\right)^2}-1 \right]. \tag{12} \]

In this expression the quantity \(\gamma_3\)—the specific weight of the air in section \(3—3\)—is unknown to us. Taking, as a first approximation, \(\gamma_3=\gamma_2\), we obtain the approximate expression for \(P_3\)

\[ P'_3=P_2-\frac{\gamma_2 V_2^2}{2g} \left(\frac{1}{f^2}-1\right). \tag{13} \]

The third and second sections are located close to one another; therefore it may be approximately assumed that the efflux process in the segment \(3—2\) is adiabatic. Then the value of \(\gamma_3\) can be determined from the formula

\[ \gamma_3=\gamma_2\left(\frac{P'_3}{P_2}\right)^{\frac{1}{k}}. \tag{14} \]

Substituting the value found into expression (12), we obtain the refined value \(P_3\). If necessary, subsequent approximations may be made.

Formula (11) was also used by us to determine the coefficient of flow through the gap between the throttle valve and the pipe, both under conditions of a steady air flow (during blowing on the installation with an air blower) and under conditions of a pulsating flow of the working mixture (when processing the results of experiments on a running engine).

The installation for blowing consisted of an air blower driven in rotation by an electric motor, a system of pipelines, and a receiver. On a straight section of the pipeline a normal diaphragm for measuring the air flow rate was installed, made in full accordance with “Rules No. 169”... [5]. A special device for blowing valves was attached to the receiver by means of a short branch pipe with a flange. The layout of the installation is shown in Fig. 3.

Fig. 3

The experiments were carried out with a flat centrally located valve, the same valve located eccentrically, and the throttle valve of a production K-49 carburetor. The parameters of the valves being blown are given in Table 1.

Blowing three throttle valves differing in shape on an installation with an air blower makes it possible to assert that, under conditions

Table 1

of a constant air flow, the value of the discharge coefficient \(\mu\), determined by formula (11), does not depend on the shape of the throttle flap and the angle of its covering, and is on average \(0.70 \pm 0.04\) (Fig. 4 and Table 2).

Table 2

Value of the discharge coefficient \(\mu\) for a flat centrally located flap

A certain decrease of \(\mu\) with an increase in the angle of covering of the flap up to

\[ \frac{\beta}{\beta_{\max}} \simeq 0.6 \]

(Fig. 4) can be explained by a decrease in both components of its value—the jet-contraction coefficient \(\alpha\) and the velocity coefficient \(\varphi\). As the angle of covering increases, the velocity of the air motion, referred to the clear section, increases, and this leads to an increase in losses and a decrease in the area of the effective section \(F_3\).

The steep rise of the curve of the discharge coefficient in the region of the beginning—

Fig. 4

Fig. 5

...of complete closing of the throttle is explained, in our opinion, by the presence of an unaccounted-for clearance between the pipe and the throttle when the latter is fully closed. Indeed, if there is absolutely no clearance, then with the throttle fully closed \((f = 0)\) the flow rate will be zero and \(\mu\) will be indeterminate. In the presence of an unaccounted-for clearance with the throttle fully closed, the flow rate will no longer be zero and \(\mu = \infty\).

To determine the value of the discharge coefficient under conditions of a pulsating flow of the working mixture, the author carried out experiments on an operating L-6/3 engine¹.

The values of \(\mu\) were calculated by formula (11). The results of experiments to determine the discharge coefficient through the slot between the throttle valve and the walls of the mixing chamber, according to scavenging data on the L-6/3 engine, are presented in Table 3 and Fig. 5.

Table 3

The nature of the change in the curve \(\mu = f(\beta)\) is entirely analogous to the behavior of the discharge-coefficient curve when the throttles are scavenged on the installation with air blowing and, in our view, is explained by the same causes.

Up to \(\beta = 45^\circ\) \((f = 0.076)\), with accuracy sufficient for practical purposes, it may be considered that the discharge coefficient under engine operating conditions also depends neither on the angle of throttle closure nor on the flow rate (engine speed) and is \(0.52 \pm 0.03\). For angles of closure greater than \(45^\circ\), the values of \(\mu\) increase with increasing \(n\) and much more sharply with increasing \(\beta\) (Figs. 5 and 6).

Experiments on the L-6/3 engine showed that the discharge coefficient depends strongly on the pulsation frequency of the flow, determined, at constant speed, by the number of engine cylinders. If under steady-flow conditions (experiments on the installation with air blowing) \(\mu \simeq 0.7\), then on the two-cylinder L-6/3 engine \(\mu \simeq 0.5\). Such a decrease in the discharge coefficient with a decrease in the number of engine cylinders can be confirmed theoretically.

Indeed, during operation of a piston engine, the pressure in the intake pipeline is not constant. For example, for the GAZ-51 engine [6], the pressure variation in the intake system is represented by the curves shown in Fig. 7. Pressure oscillations in the absence of resonance phenomena in the intake pipeline—

in water will be greater the smaller the number of cylinders of the engine and the more slow-running it is. Owing to the inertia of the liquid, ordinary U-shaped differential manometers show not the actual

Fig. 6

Fig. 7

pressure in the pulsating flow, but rather a certain pressure averaged over the operating cycle of the engine.

The air consumption by the engine can be calculated from the value of the mean air velocity in the intake pipeline \(V_{cp}\)

\[ G = V_{cp} F_1 \gamma_1 T, \tag{15} \]

where \(T\) is the time between two intake strokes.

At the same time, assuming in the first approximation [6] that the air velocity during the suction process varies according to a sine law (Fig. 8), we obtain

\[ G = F_1 \gamma_1 \int_{0}^{T_{bc}} V_{\max}\sin \frac{\pi t}{2T_{bc}} = V_{\max} F_1 \gamma_1 \frac{2T_{bc}}{\pi}. \tag{16} \]

If compressibility is neglected, then the gas velocity can be determined from the formula

\[ V = \mu f \sqrt{\Delta p}. \tag{17} \]

Fig. 8

Taking this into account, equation (15) can be written in the form

\[ G = \mu_{cp} f F_1 \gamma_1 \sqrt{\Delta p_{cp}}\, T, \]

and equation (16) in the form

\[ G = \mu_{\max} f F_1 \gamma_1 \sqrt{\Delta p_{\max}}\, \frac{2T_{bc}}{\pi}. \]

Equating these expressions, we find

\[ \mu_{cp} = \mu_{\max}\sqrt{\frac{\Delta p_{\max}}{\Delta p_{cp}}}\cdot \frac{2T_{bc}}{\pi T}. \tag{18} \]

From Fig. 8 it is seen that

\[ \Delta p_{cp} T = \int_{0}^{T_{bc}} \Delta p \cdot dt. \]

After simple transformations we obtain

\[ \Delta p_{cp} T=\Delta p_{\max}\cdot \frac{T_{bc}}{2}. \]

Whence

\[ \sqrt{\frac{\Delta p_{\max}}{\Delta p_{cp}}}=\sqrt{\frac{2T}{T_{bc}}}. \tag{19} \]

Substituting the expression obtained into formula (18), we find

\[ \mu_{cp}=\mu_{\max}\sqrt{\frac{2T}{T_{bc}}}\cdot \frac{2T_{bc}}{\pi T} =\frac{\mu_{\max}}{\pi}\,2\sqrt{\frac{2T_{bc}}{T}}, \]

or (see Fig. 8)

\[ \mu_{cp}=\frac{\mu_{\max}}{\pi}\,2\sqrt{\frac{2\varphi_{bc}}{\varphi}}. \tag{20} \]

For a four-stroke engine, the angle of rotation between two intake strokes is \(\varphi=\dfrac{720^\circ}{i}\), where \(i\) is the number of cylinders.

Substituting the value of \(\varphi\) into expression (20), we finally obtain:

\[ \mu_{cp}=\frac{\mu_{\max}}{\pi}\sqrt{\frac{i\varphi_{bc}}{90}}. \tag{21} \]

If we take \(\mu_{\max}=0.704\) (the value of the discharge coefficient obtained when blowing through a flat eccentrically located throttle valve on a setup with an air blower), \(\varphi_{bc}=213^\circ\) (the duration of the intake stroke of the L-6/3 engine), then for a single-cylinder engine \(\mu_{cp}\) is equal to 0.345, for a two-cylinder engine—0.491, and for a four-cylinder engine—0.691. Thus, already for a four-cylinder engine the influence of pulsation on the magnitude of the discharge coefficient may be neglected.

The values of \(\mu\) calculated for a two-cylinder four-stroke engine are close to the values of the discharge coefficient determined by blowing on the L-6/3 engine (\(\mu \simeq 0.5\)). This makes it possible to hope that formula (21) correctly reflects the dependence of the discharge coefficient on the number of engine cylinders.

Literature

1. Krivolutskii V. A. Tests of the throttle valve of the supercharger of the VK-105 engine, Transactions of TsIAM, issue 12, 1947.

2. Makarov A. N. and Sherman M. Ya. Calculation of measuring and regulating throttle devices, Metallurgizdat, Moscow, 1953.

3. Akopyan S. I. Internal-combustion engines with fuel injection and electric ignition, GONTI, Moscow, 1945.

4. Orlin A. S. (ed.). Internal-combustion engines, vol. I, Mashgiz, Moscow, 1951.

5. Rules No. 169 for measuring the flow rate of liquids, gases, and steam by means of nozzles and orifice plates, and guidelines to Rules No. 169, Katalogizdat NKM, 1938.

6. Matveev E. A. Hydraulic calculation and investigation of the working process of an emulsion carburetor (dissertation), Moscow, 1949.

7. Litvin A. M. Technical Thermodynamics, GEI, Moscow-Leningrad, 1947.

1. Two-cylinder four-stroke carburetor engine. Power 6 hp at 2200 rpm. Cylinder displacement 597 cm³. The throttle is flat, eccentrically located, \(d_1 = 32.2\) mm, \(t_1 = 0.95\) mm, \(\beta_{\max} = 50.1^\circ\), \(\dfrac{e}{B} = 0.03\). ↩