On the Analytical Expression of the External Characteristic of an Internal Combustion Engine

V. P. Beltyukov

(Presented by Prof. V. K. Nechaev, Doctor of Technical Sciences)

In solving the equation of motion of an automobile, the problem arises of integrating expressions containing the engine torque \(M\). The characteristics of internal combustion engines are usually represented graphically—in the form of curves obtained during testing. In this connection there arises the need for an analytical expression of these characteristics. It is desirable that this expression be as simple as possible, be easily integrable, and provide good agreement of the curve it describes with the experimentally obtained characteristic.

The formulas known to us for the analytical expression of the external characteristic of an internal combustion engine may be divided into two categories.

The coefficients in formulas of the first type are constant for all engines of a given method of mixture formation. These formulas [1, 2] are simple; to determine the coefficients entering into them, it is usually necessary to know only one point of the experimental curve, but the deviation of the value of the torque calculated from them from the actual value may reach 20%. Such a degree of accuracy cannot be considered satisfactory.

The coefficients in formulas of the second type are constant only for a given engine. These formulas [3] are sufficiently accurate, but in order to construct from them the external characteristic of an engine it is necessary to know two points on this curve, while the calculation of the cumbersome coefficients entering into them requires much time.

The author proposes¹ using, for the analytical expression of the external characteristic of an internal combustion engine, a formula that is sufficiently simple, accurate, and that reflects to a certain extent the physical essence of the processes occurring in the engine. The coefficients entering into it are easily calculated from two experimentally determined points of the external characteristic.

As is known, the torque developed by an engine is proportional to the mean effective pressure \(p_e\). The decrease of \(p_e\) (and consequently also of \(M\)) on the descending portion of the characteristic is explained by two circumstances: a decrease in the mean indicated pressure \(p_i\) and an increase in the mean friction pressure \(p_{mp}\). Thus, at any number of revolutions \(n\) greater than \(n_m\),

\[ p_i = p_{i m} - \Delta p_i, \]

\[ p_{mp} = p_{mp m} + \Delta p_{mp}, \]

and

\[ p_e = p_i - p_{mp} = p_{i m} - \Delta p_i - p_{mp m} - \Delta p_{mp} = p_{e m} - \Delta p_i - \Delta p_{mp} \tag{1} \]

(in these expressions \(p_{i.m}\), \(p_{\mathrm{tr}.m}\), and \(p_{e.m}\) are, respectively: the indicated pressure, the friction pressure, and the effective pressure when the engine is operating in the maximum-torque regime (Fig. 1).

The change in the mean indicated pressure is determined by a whole series of circumstances. The most important of these are: the amount of mixture entering the engine cylinder (characterized by the filling coefficient \(\eta_v\)), and the quality of the working process (characterized by the indicated efficiency \(\eta_i\)).

Fig. 1

efficiency. When operating according to the external characteristic, \(\eta_i\) may, with sufficient accuracy, be regarded as a constant quantity. The amount of mixture entering the engine in one cycle, however, decreases as the speed increases, owing to the increase in the aerodynamic resistances of the intake system. At speeds corresponding to the usual velocity of motion of the working mixture in the intake pipe (70–150 m/sec), the aerodynamic resistances increase approximately in proportion to the square of this velocity. Therefore, when the engine speed increases by the amount \(\Delta n = n - n_m\) (Fig. 1), the intake losses will increase according to the law

\[ \Delta p_i = A \Delta n + B \Delta n^2, \tag{2} \]

where \(A\) and \(B\) are certain constant quantities.

The friction losses increase directly in proportion to the increase in engine speed,

\[ \Delta p_{\mathrm{tr}} = D \Delta n, \tag{3} \]

where \(D\) is the coefficient of proportionality.

The adopted laws governing the variation of intake and friction losses are confirmed by experimental investigations [4, 5].

Substituting expressions (2 and 3) into equation (1), we obtain

\[ p_e = p_{e.m} - A \Delta n - B \Delta n^2 - D \Delta n = p_{e.m} - B \Delta n^2 - C \Delta n. \tag{4} \]

Passing to the expression for the torque, we shall have

\[ M=M_m-B\Delta n^2-C\Delta n . \tag{5} \]

After substituting for \(\Delta n\) its value \(n-n_m\) and performing the simplest transformations, we obtain

\[ M=M_N\left(a+b\frac{n}{n_N}-c\frac{n^2}{n_N^2}\right). \tag{6} \]

The coefficients \(a\), \(b\), and \(c\) in this equation will be determined by the method of selected points [6], which is simpler than the usually recommended [7] least-squares method and gives sufficient accuracy for practical purposes.

We shall require that the following conditions be satisfied:

\[ \begin{array}{llll} 1) & M=M_m & \text{at} & n=n_m,\\[6pt] 2) & \dfrac{dM}{dn}=0 & \text{at} & n=n_m,\\[10pt] 3) & M=M_N^{\,1)} & \text{at} & n=n_N. \end{array} \]

Here \(M\) is the value of the torque at an engine speed equal to \(n\);

\(M_N\) and \(n_N\) are the torque and engine speed when operating at the maximum-power condition;

\(M_m\) and \(n_m\) are the same quantities at the maximum-torque condition.

Solving the resulting system of three equations, we determine

\[ c=\frac{\beta-1}{(\alpha-1)^2}; \qquad a=\beta-\alpha^2 c; \qquad b=2\alpha c, \tag{7} \]

where

\[ \alpha=\frac{n_m}{n_N}\quad \text{and}\quad \beta=\frac{M_m}{M_N}. \]

The check that we carried out showed that the proposed formula (6) is sufficiently universal and, provided the coefficients entering it are determined by formulas (7), gives good agreement both for the descending and for the ascending branch of the torque curve. Curves of the torques of carburetor engines and diesels with various mixture-formation methods, both two-stroke and four-stroke, constructed from it give, over the entire speed range, deviations from the actual characteristics not exceeding 3%. Such accuracy should be regarded as quite sufficient, since the errors in the experimental determination of characteristics reach approximately the same magnitude.

Formula (6) can be used not only for the analytical expression of an experimental torque curve, but also for constructing the torque curve of an engine being designed, for which \(M_N\) and \(n_N\), the adaptability coefficient \(\beta\), and the speed ratio \(\dfrac{n_m}{n_N}\) are known.

Using the known relation

\[ N=\frac{Mn}{716.2}, \]

the resulting curve can be transformed into the power characteristic of the engine.

\(^{1)}\) It can be shown that the second known point of the external characteristic may be arbitrary, and need not necessarily correspond to the maximum power.

References

1. Leiderman S. R. The characteristic of automobile engines, Automotive Industry, No. 9, 1948.

2. Lenin I. M. Weight filling and the speed characteristic of an automobile carburetor engine. Problems of mechanical engineering. Collection of articles dedicated to the 60th anniversary of Academician Chudakov, Moscow, 1950.

3. Khlystov F. L. Traction calculations of tracked and wheeled motor vehicles, ONTI, 1937.

4. Kurov B. A. Investigation of intake pipelines of carburetor engines, Transactions of NAMI, issue 59, 1951.

5. Maslennikov M. M. Determination of the indicated efficiency and friction power of an aircraft engine, Transactions of TsIAM, No. 132, 1947.

6. Uorsing A. and Geffner D. Methods for processing experimental data, Foreign Literature Publishing House, Moscow, 1949.

7. Soroko-Novitskii. Engine tests, Moscow, 1954.

1. The work was first reported at the scientific seminar of the departments “Internal Combustion Engines” and “Automobiles and Tractors” of TPI, October 26, 1950. ↩