On the Calculation of Pneumodynamic Speed Regulators
Abstract Generated abstract
The paper addresses the calculation of pneumodynamic speed regulators for carburetor engines when experimental partial engine characteristics and throttle moment curves are unavailable. It proposes a method for constructing the required regulator characteristic by approximating operation along the regulating branch at constant nominal speed, determining the throttle pressure drop from discharge coefficient relations, intake resistance data, and throttle flow area, and then calculating the mixture moment on the throttle axis. The resulting moment versus throttle closure curve is used to select the regulator spring and transmission mechanism so that the spring moment matches the aerodynamic moment. Application to an L-6/3 engine indicated a maximum spring moment error of about 20 percent, reducible to about 15 percent when variation of the flow coefficient is included, which the author considers satisfactory for design purposes.
Full Text
On the Calculation of Pneumodynamic Speed Regulators
V. P. Beltyukov
(Presented at the joint scientific seminar of the Departments of Theoretical and Applied Mechanics, TPI)
The calculation of a pneumodynamic speed regulator for a carburetor engine reduces to determining the characteristic of the spring and to synthesizing the kinematic scheme of the mechanism that transmits the action of the spring to the throttle valve. The spring characteristic and the kinematic scheme of the transmitting device must be chosen so that the turning moment produced by the spring on the axis of the throttle valve, \(M_{pr}\), at any angle of its closing \(\beta\), is equal to the turning moment produced by the flow of the working mixture, \(M_{cm}\), on the same valve when the engine operates along the regulating branch of the speed characteristic. Therefore, the initial curve in calculating a pneumodynamic regulator is the curve \(M_{pr}=f(\beta)\). This curve determines the operation of the pneumodynamic regulator in exactly the same way as the curve \(F_{pr}=f(x)\) determines the operation of an ordinary centrifugal regulator; hence in what follows we shall call it the regulator characteristic.
When experimentally obtained partial characteristics of the engine and curves of the dependence of the turning moment of the mixture on the throttle-valve axis on the engine speed for the same fixed throttle positions are available, the regulator characteristic can be obtained by the method set forth in [1]. Below a method is proposed that makes it possible to determine the characteristic of a pneumodynamic regulator in the absence of partial engine characteristics and curves of the mixture moment—for example, when designing a regulator for an engine under development.
As shown in [2], the magnitude of the turning moment produced by the flow of the working mixture on the axis of the throttle valve is determined by the pressure drop across the valve, \(\Delta p_{13}\), which in turn is a function of the angle of closing of the valve and of the engine speed. However, taking into account the comparatively small change in \(\Delta p_{13}\) when the engine speed varies within the limits determined by the degree of nonuniformity of the regulator, one may take the engine speed during operation along the regulating branch of the speed characteristic to be constant and equal to the nominal engine speed \(n_d\). Such an assumption is usual in calculating speed regulators with a relatively small degree of nonuniformity [3].
The assumption adopted considerably simplifies the problem. Now, in order to construct the characteristic of the regulator, it is necessary to know only the dependence of the pressure drop \(\Delta p_{13}\) at the throttle valve on the angle of its closure at a given constant number of revolutions \(n_k\). This dependence can be established from the expression for the discharge coefficient. The results of our experiments [4] showed that the discharge coefficient determined by the formula
\[ \mu = \frac{1}{ f \sqrt{ \left(\frac{P_3}{P_1}\right)^{\frac{2}{\kappa}} + \frac{P_1}{q_1}\frac{\psi^2}{2g} } }, \]
where \(P_3\) is the absolute pressure at the place of greatest contraction of the jet behind the throttle valve,
\(P_1\) is the absolute pressure ahead of the throttle valve,
\(q_1\) is the velocity head ahead of the throttle valve,
\[ \psi = \sqrt{ 2g \frac{\kappa}{\kappa-1} \left[ \left(\frac{P_3}{P_1}\right)^{\frac{2}{\kappa}} - \left(\frac{P_3}{P_1}\right)^{\frac{\kappa+1}{\kappa}} \right] }, \]
\(g\) is the acceleration of gravity,
\(\kappa\) is the adiabatic exponent, is practically independent of the shape of the throttle valve and changes little with the angle of its closure. By assigning values of the relative flow area of the throttle \(f\) (the angle of closure of the valve \(\beta\)), one can, from previously constructed graphs
\[ Z=\frac{1}{\mu^2 f^2}=f\left(\frac{P_3}{P_1}\right), \]
determine the corresponding values of \(\frac{P_3}{P_1}\), and consequently also \(\Delta p_{13}\).
The construction of the graphs \(Z=f\left(\frac{P_3}{P_1}\right)\) is carried out as follows:
1) We determine \(q_{1k}\), the velocity head ahead of the throttle valve at the initial (setting) angle of closure \(\beta_k\). For this purpose we first construct a graph of the dependence of the engine air consumption per second \(G_{\mathrm{sec}}\) on the velocity head ahead of the valve \(q_1\) (Fig. 1). The graph is constructed point by point for a number of selected values of \(q_1\), using the formula
\[ G_{\mathrm{sec}} = F \sqrt{2g\gamma_1 q_1}, \]
where
\[ F=\frac{\pi d_1^2}{4} \]
is the cross-sectional area of the intake pipe ahead of the throttle valve,
\[ \gamma_1=\gamma_0 \frac{T_0}{P_0}\frac{P_1}{T_1} \]
is the specific weight of the air in this section,
\(\gamma_0\), \(T_0\), and \(P_0\) are the specific weight, temperature, and pressure of the surrounding medium,
\[ T_1 = T_0+\Delta t \]
is the temperature ahead of the throttle valve, taking heating into account.
The absolute pressure ahead of the throttle valve is
\[ P_1=P_0-\Delta p_{01}=P_0-\xi_{\mathrm{in}} q_1. \]
The resistance coefficient of the engine intake system \(\xi_{\mathrm{in}}\) can be determined analytically, for example, by the method proposed by B. A. Kurov [5], or from tests on an experimental installation.
The results of experiments carried out by the author on the L-6/3 engine showed that the value \(\xi_{vs}\), over a wide range of changes in speed, does not depend on the engine speed operating condition and at first remains almost unchanged with the angle of throttle closing (Fig. 2).
Calculating by the known formula
\[ G_{sek\,n}=\frac{V_h n_n \eta_{ion}\gamma_0}{120}. \]
Fig. 1.
Fig. 2.
the air consumption per second by the engine at the rated-speed operating condition, for the initial angle of closure of the throttle valve; from the plotted graph \(G_{\mathrm{sec}}=f(q_1)\) we find the value \(q_{1n}\) (see Fig. 1). The corresponding value of the vacuum ahead of the throttle valve is
\[ \Delta p_{01n}=\Delta p_{01\max}=\xi_{\mathrm{sc}} q_{1n}. \]
2) From the graph
\[ \frac{\Delta p_{01}}{\Delta p_{01\max}}=\varphi(f) \]
(Fig. 3) we determine the values of \(\Delta p_{01}\) for the other adopted values of the relative flow area of the throttle, \(f\). Experiments carried out by the author on the L-6/3 engine
Fig. 3.
give grounds for asserting that, at a constant angle of closure of the valve, the ratio
\[ \frac{\Delta p_{01}}{\Delta p_{01\max}} \]
may be taken as independent of the engine speed.
3) From the found values of \(\Delta p_{01}\) we determine the pressure \(P_1=P_0-\Delta p_{01}\) and the velocity head ahead of the valve
\[ q_1=\frac{\Delta p_{01}}{\xi_{\mathrm{sc}}} \]
for the adopted values of \(f\).
4) From the obtained values of \(P_1\) and \(q_1\) we construct the graphs
\[ Z=f\!\left(\frac{P_3}{P_1}\right) \]
for the adopted values of \(f\), specifying the ratio \(\frac{P_3}{P_1}\). The values of \(Z\) are calculated from the formula
\[ Z=\left(\frac{P_3}{P_1}\right)^{\frac{2}{\kappa}}+\frac{P_1}{q_1}\frac{\psi^2}{2g}. \]
After the graphs
\[ Z=f\!\left(\frac{P_3}{P_1}\right) \]
have been constructed for \(f=\mathrm{const}\), we determine the values
\[ Z=\frac{1}{\mu^2 f^2} \]
for the adopted series of values of \(f\) (the angles of closure of the valve \(\beta\)) and find the corresponding \(\frac{P_3}{P_1}\), and consequently \(P_3\), and the static pressure drop across the valve \(\Delta p_{13}=P_1-P_3\).
The values of the flow coefficient \(\mu\) may, to a first approximation, be taken as constant: equal to \(0.7\) for a four-cylinder engine and \(0.5\) for a two-cylinder engine [4].
To determine the magnitude of the restoring moment of the mixture on the throttle-valve axis for the obtained value of \(\Delta p_{13}\), we shall use the formula [2]:
\[ M_{c.m}=\left(C_0+C_e\frac{e}{B}\right)\Delta p_{13}SBK, \]
where \(C_0\) is the moment coefficient of a centrally located throttle valve;
\(C_e\) is the moment coefficient due to the eccentric location of the throttle valve,
\(e\) is the eccentricity of the throttle valve,
\(S\) is the area of the throttle valve,
\(B\) is the length of the major axis of the throttle-valve ellipse,
\(K\) is the mismatch coefficient.
The curve \(M_{c.m}=f(\beta)\) will represent the desired characteristic of the regulator, since in the steady-state speed regime the moment produced by the flow of the working mixture on the axis of the throttle valve is balanced by the moment produced by the regulator spring.
The problem now consists in selecting, according to the obtained regulator characteristic, a spring (wire diameter, number of turns, winding diameter) and a kinematic scheme of the mechanism transmitting the action of the spring to the regulator throttle valve in such a way as to ensure coincidence of the actual characteristic of the regulator with the desired one. This problem may be solved by trial selection, or by means of the method proposed by Professor V. K. Nechaev [6]. Using this method, one can ensure from three to five points of coincidence of the regulator characteristic with the desired one.
The calculation, by the proposed method, of the throttle-regulator of the L—6/3 engine manufactured by us showed that the maximum error in determining the spring moment is about \(20\%\) (see Fig. 4). Taking into account the variation of the flow coefficient with the angle of closure of the throttle valve, the error can be reduced to \(15\%\). Bearing in mind that, under existing technical conditions, fluctuations in the stiffness of the springs of pneumodynamic regulators are permitted within limits of up to \(16\%\) [7], the accuracy of such a calculation of the regulator of the engine being designed should be regarded as quite satisfactory.
Fig. 4.
References
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Beltyukov V. P. Determination of the discharge coefficient through a slot between a throttle valve and a pipe. Proceedings of TPI, vol. 96, issue 1, Tomsk, 1959.
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Kurov B. A. Investigation and calculation of intake pipelines of carburetor engines. NAMI, Technical Report No. 48, Moscow, 1949.
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Nechaev V. K. On the calculation of mechanisms of certain limiting pneumatic regulators. Proceedings of TPI, vol. 75, Tomsk, 1954.
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