V. G. Narpetsky

Calculation of the Timing Mechanisms of Two-Stroke and Four-Stroke Machines

The scavenging process and the exhaust process involve very complex changes in the state of gases with a variable mass quantity. It does not appear possible to connect all these changes by equations without major assumptions and conventions. All modern literature on this question gives only approximate results of calculation, while indicating in parallel the significance of one or another factor and of the assumption adopted in the calculation. Nevertheless, the results of calculating timing mechanisms lead to dimensions that coincide with the design data of existing machines.

Scavenging of two-stroke machines is carried out during a short period of the working process, at the end of the stroke, when the piston is near its dead-center position. By the beginning of scavenging, the gases in the cylinder must reduce their pressure almost to atmospheric pressure \((p = 1.05—1.10\ \mathrm{kg/cm^2})\), and then, under the pressure of the scavenging air or mixture, must be removed from the cylinder. However short in time the scavenging period may be (from \(1/5\) to \(1/40\) sec.), a whole series of processes can nevertheless be distinguished in it, differing in their nature and therefore requiring separate investigation and calculation.

At the beginning, when the ports open, the gas begins to flow out at the critical velocity, i.e., at the velocity of sound in the given medium. This continues until the pressure ratio in the cylinder and in the exhaust pipe becomes equal to \(p_0:p_1 = 0.528\), since at any greater ratio \((p_0:p_1 > 0.528)\) the flow rate and the velocity of the gas remain constant.

With further lowering of the pressure in the cylinder, the velocity of outflow gradually decreases, depending on the pressure difference in the cylinder and the exhaust pipe.

This second period continues from the pressure \(p_1 = 1.9 p_0\) to a pressure that is selected depending on the scavenging air or mixture in the receiver or in the channel between the working cylinder and the scavenging pump. This pressure must be lower than the pressure in the receiver, so that when the scavenging ports or valve open the products of combustion would not rush into the receiver.

The third period is the scavenging of the cylinder proper, which in liquid-fuel engines is carried out with air, and in gas engines with a mixture; the order of supplying the air and gas is arranged so that air enters the cylinder first, and then the mixture. Such a complex movement of the combustion products, air, and mixture through the cylinder is possible only with calm motion of all gases, and this can be achieved only on condition that each layer does not differ from another by a large pressure difference upon entering the cylinder. From this follows the requirement that the scavenging air or mixture should not have a high pressure.

If the machine is four-stroke, then the scavenging process is absent. The calculation has to be carried out in order to determine the advance of the exhaust of gases, on the condition that at the dead-center position of the piston the pressure in the cylinder should fall to 1.10–1.15 kg.

The calculation of all periods of the exhaust process and of scavenging is reduced to determining the live cross-section of the valve and the time of its lift, expressed by the equation

\[ \int_{t_1}^{t_2} f\,dt=\frac{v}{w}, \]

where \(f\) is the live cross-section of the valve, \(t\) is the time, \(v\) is the volume of the gases, and \(w\) is their velocity.

The latter has to be calculated from the volume of the cylinder; the velocity of the gases is obtained from three equations—

\[ w_1=\varphi\,18.3\sqrt{T}, \]

\[ w_2=\varphi\,44.8\sqrt{T_2\left[1-\left(\frac{p_{\mathrm{tr}}}{p_2}\right)^{\frac{k-1}{k}}\right]}, \]

\[ w_3=\varphi\,24\sqrt{T_3\left(1-\frac{p_{\mathrm{tr}}}{p_2}\right)}, \]

where \(\varphi\) is the velocity coefficient*), \(p_{\mathrm{tr}}\) is the pressure in the pipe, and \(p_2\) is the pressure in the cylinder. The first of these equations is used for calculating the critical velocity, the second for calculating the velocity when the difference between the pressures \(p_{\mathrm{tr}}\) and \(p_2\) is large, and the third when the difference between these same pressures is small.

Thus the question of determining the time and the cross-section is solved as though approximately; but if one takes into account that the quantities \(T\), \(p\), and \(v\) are variable, then the determination of time and cross-section must be carried out according to certain mean values of temperature and pressure for each of the three periods described above. As for the volume of gas which must flow out during one or another period from the cylinder, it will be equal to

\[ v=v_2-v_1, \]

where \(v_1\) is the volume of gas at the beginning of the period at pressure and temperature \(p_1\) and \(T_1\); \(v_2\) is the volume of gas at the end of the period at pressure \(p_2\) and \(T_2\). But this equation is valid only on the condition that the volume \(v\) flows out without residue, i.e., on the condition that the piston remains without motion, which in reality is not the case. The piston moves, and part of the gas remains in this increased volume of the cylinder, and the assumption that the piston does not move leads to an error of about 15%. To correct this error it is necessary to carry out a repeated calculation, introducing a correction for the magnitude of this error.

Not having the possibility, in this brief report, to set forth all the details of the calculations, the author considers it necessary only to note that, unlike his predecessors Balog, Borth, Förpl, Gutmann, and Kreglewski, he carried out his calculations according to the arithmetic mean values of \(p\) and \(T\) of two extreme points. Such a simplification of the problem, in view of the smallness of the periods considered, gave excellent results in the sense of agreement of the valve dimensions by calculation with the dimensions given by practice. But not only this. Calculations by this simplified method give values of the time-section \(\int fdt\) that differ little—

*) Hütte 8, Russian ed., pp. 382 and 386.

depend on the same quantities, obtained by other authors through a more complicated calculation.

The time-section, \(\int f\,dt\), of the exhaust ports is represented by the area \(OAB\) (Fig. 1); the time-section necessary for the preliminary opening of the exhaust ports, corresponding to the first two periods of the exhaust process, is represented by the area \(AMN\). Thus the length of the exhaust ports \(OK\) is determined; it will be greater than the length of the scavenging ports \(OL\), and therefore their covering by the piston during the reverse stroke will occur later. But this is permissible only in crankcase-scavenged two-stroke engines, where it is unavoidable; in all other engines the scavenging ports or valves close either simultaneously with the exhaust ports or somewhat later. In order to determine the opening time of the scavenging ports, the scavenging area is shifted downward on the diagram so that the end of scavenging coincides with the end of exhaust: \(O^{1}A^{1}B^{1}\). The area \(AMN\) will shift to the position \(amn\), as a result of which the length of the exhaust ports will decrease from \(OK\) to \(OK^{1}\). The plotting and shifting of the diagram areas can be carried out only by successive trial with the aid of a planimeter.

For four-stroke machines the calculation is reduced to the calculation of the first two periods of the exhaust process. Since exhaust is effected through a valve, the time-section refers to the opening of this valve. In constructing the valve-lift curve it is assumed approximately that the valve is opened by the piston \(T\), whose diameter \(d\) is taken as \(1/4\) of the cylinder diameter \(D\); the motion of the valve may then be considered as the motion of a piston in a cylinder of smaller diameter \(d\). The path traversed by this piston by the time the working piston reaches dead center will be the valve lift height \(h\). The time-section found must be completed by the dead-center position of the piston, and therefore in diagram 2 it is expressed by the area \(OAB\). The quantity \(OB\) will be proportional or equal to the valve lift.

When plotting the valve-lift curve on the cam washer (Fig. 3), the values \(h\) may be taken from diagram 2, laying them off, for simplicity, along the radii of the washer. If it is desired that the cam nose be obtained as smoothly as possible, then, for smoothness of motion, one must draw a smooth curve through them, making the area \(rsa\) approximately equal to the area \([[unclear: letters]]\).

Everything set forth above becomes especially clear when the question is presented with confirmation by calculations of existing machines; owing to brevity, the author cannot do this in this short abstract.