Abstract Generated abstract
This paper derives an expression for the indicated efficiency of an internal combustion engine by replacing the actual indicator cycle with an equivalent thermodynamic cycle having the same thermal efficiency. The analysis introduces a constant heat capacity working gas approximation, estimates the resulting error in heat supply as small, and relates efficiency to the heat release law, compression ratio, and properties of the combustion products. It defines a reduced compression ratio and a coefficient of active heat utilization, showing that indicated efficiency is governed chiefly by the fraction of active heat released to the working medium and the effectiveness with which that heat is converted into work. The resulting formulas are proposed for evaluating existing engine indicator diagrams and for estimating efficiency in engine design when empirical values of the governing coefficients are available.
Full Text
UDC 621.43
HEAT ENGINEERING
Academician B. S. STECHKIN
ON THE INDICATED EFFICIENCY
OF AN INTERNAL COMBUSTION ENGINE
In the present work we shall find an expression for the indicated efficiency of an engine.
Figure 1 gives the indicator diagram of an engine over two working strokes with a constant amount of working substance in the cylinder. The expansion and compression lines from the beginning of exhaust and from the end of intake are extended to the lower dead point (L.D.P.) in logarithmic coordinates as straight lines. As the working substance we take the combustion products, to which heat is supplied from outside in such a way that the working substance describes the actual cycle \(1 — a — c — 2\) (Fig. 1). In this case the cycle can be closed by cooling the working substance from point 2 to 1 with rejection of heat \(Q_2\) to the cold source. Let along the path \(1 — a — c — 2\) heat \(Q_1\) (active heat) have been supplied to the working substance, and let the indicated work, equivalent to the area of the diagram, be equal to \(L_i\); then
Fig. 1. Cycle No. 1—the actual cycle \(1 — a — c — 2\); cycle No. 2 (dashed)—the cycle \(1 — 3 — d — 4 — 2\), equivalent to cycle No. 1
\[ \eta_t = 1 - Q_2 / Q_1 = AL_i / Q_1 = AL_i / (AL_i + Q_2); \tag{1} \]
\[ \eta_i = AL_i / H_u = \xi \eta_t,\qquad Q_1 = \xi H_u, \tag{2} \]
where \(\eta_t\) is the thermal efficiency; \(\eta_i\) is the indicated efficiency; \(H_u\) is the heating value of the working mixture; \(\xi\) is the coefficient of release of active heat.
If another working substance with heat capacity \(c_v'\) described the cycle \(1 — a — c — 2 — 1\), then, in order to preserve \(\eta_t\), it would be sufficient to have
\[ Q_2 = \int_1^2 c_v dT = \int_1^2 c_v' dT,\qquad R = R', \]
where \(R\) is the gas constant.
For the case \(c_v' = \mathrm{const}\) and \(c_v = a + bT\), we find
\[ c_v' = a + 0.5b\,(T_2 + T_1). \tag{3} \]
The law of heat supply to the working substance along the path \(1 — a — c — 2\) for \(c_v'=\mathrm{const}\) and \(c_v=\mathrm{var}\) will be somewhat different. The discrepancy in the amount of active heat supplied to the working substance is found from the equation
\[ d(Q_{\mathrm{ak}}' - Q_{\mathrm{ak}})/d\varphi = (c_v' - c_v)\,dT/d\varphi, \]
where \(\varphi\) is the angle of rotation of the crankshaft. The maximum discrepancy occurs at \(T_{\mathrm{avg}}=0.5(T_2+T_1)\), when \(c_v=c'_v\). In this case we obtain
\[ \frac{|Q'_{\mathrm{ak}}-Q_{\mathrm{ak}}|_{\max}}{Q_1} = \frac{T_2-T_1}{9[a/\theta+T_{\mathrm{avg}}]}\cdot\frac{Q_2}{Q_1}, \]
which amounts to no more than \(2\%\) of \(Q_1\). Taking hereafter as the working body a gas with constant heat capacity \(c_v'\), we shall preserve the value \(\eta_t\) and almost not change the law of heat supply. Let our new working body describe (Fig. 1) the cycle \(1—3—d—4—2—1\) (cycle No. 2), consisting of the combustion line \(3—d—4\) and two adiabats \(1—3\) and \(4—2\). Let, on the combustion line in cycle No. 2, heat continue to be supplied in the amount \(Q_1\), and let the rate of heat supply with respect to the angle of rotation of the crankshaft be \(dQ_2/d\varphi=\mathrm{const}\). The thermal efficiency of cycle No. 2 will obviously be equal to the thermal efficiency of the cycle \(1—a—c—2\) (cycle No. 1). At the same time, each position of point 3 must correspond to its own constant value of the rate of heat release, so that, in passing from the adiabat \(1—3\) to the adiabat \(4—2\), exactly the heat \(Q_1\) would be supplied. We choose the position of point 3 so that the rate of heat release is minimal. We note that in the actual cycle No. 1 the supply of the greater part of the heat occurs at an almost constant rate, while the beginning of heat release is regulated in order to obtain the maximum \(\eta_i\), which, as we shall see below, is equivalent for cycle No. 2 to the minimum of the rate of heat release.
Thermodynamic cycles Nos. 1 and 2 are, as having the same \(\eta_t\), equivalent, and we shall seek the thermal efficiency of cycle No. 2, estimating the law of heat supply in cycle No. 1 by the value of the minimum rate of heat release of cycle No. 2. We take the first law of thermodynamics in the form
\[ (k-1)\frac{dQ}{A}v^{k-1}=d(pv^k) \tag{4} \]
and, applying it (clockwise) to cycle No. 1 or 2, find:
\[ \oint \frac{k-1}{A}\,dQ\cdot v^{k-1}=0 = \int_1^2 \frac{k-1}{A}v^{k-1}dQ - (k-1)\frac{Q_2}{A}v_1^{k-1}, \]
whence, for \(\varepsilon=v_1/v_0\), we have
\[ (1-\eta_t)\varepsilon^{k-1} = \int_1^2 \frac{dQ}{Q_1}\left(\frac{v}{v_0}\right)^{k-1}; \tag{5} \]
for cycle No. 2 we obtain
\[ (1-\eta_t)\varepsilon^{k-1} = \int_3^4 \frac{dQ}{Q_1}\left(\frac{v}{v_0}\right)^{k-1}. \tag{6} \]
Introduce a new variable \(x\) (for \(\varphi\leq 40^\circ\) from TDC)
\[ x=\pm\sqrt{\frac{v}{v_0}-1}\simeq \frac{\sqrt{\varepsilon-1}}{2}\varphi, \tag{7} \]
where the plus sign corresponds to cylinder volumes after TDC. By the rate of heat release we shall understand the quantity
\[ W=\frac{dQ}{dx}\simeq \frac{2}{\sqrt{\varepsilon-1}}\frac{dQ}{d\varphi}. \]
Noting that \(W\) is constant in the interval \(x_4-x_3\), we find from equation (6)
\[ (1-\eta_t)\varepsilon^{k-1} = \frac{W}{Q_1}\int_3^4\left(\frac{v}{v_0}\right)^{k-1}dx \tag{8} \]
and also
\[ Q_1 = W(x_4 - x_3). \tag{9} \]
Whether we now, on the basis of equations (8) and (9), seek \(W_{\min}\) for \(\eta_t=\text{const}\), or, for a given \(W\), seek \(\eta_{t\max}\), in both cases we find \(v_4=v_3\) and \(x_4=-x_3\) (with \(Q_1=\text{const}\)).
For \(v_4/v_0 \leqslant 2.6\) one may, with sufficient accuracy, put
\[ \frac{W}{Q_1}\int_{-x_4}^{+x_4}\left(\frac{v}{v_0}\right)^{k-1} dx = \frac{W}{Q_1}\,2x_4(1+x_4^2)^{0.4(k-1)} = \left(\frac{v_4}{v_0}\right)^{0.4(k-1)}, \]
and then from equation (8) we find
\[ \eta_t = 1 - \frac{1}{\varepsilon_1^{k-1}}, \tag{10} \]
where
\[ \varepsilon_1 = \varepsilon (v_0/v_4)^{0.4} = \psi \varepsilon . \]
We shall call \(\varepsilon_1\) the reduced compression ratio, and \(\psi\) the coefficient of utilization of active heat.
From equation (2) we shall have
\[ \eta_i = \xi\left[1-\frac{1}{(\psi\varepsilon)^{k-1}}\right]. \tag{11} \]
Noting that
\[ (k-1)=AR/c_v' = AR/[a+0.5b(T_2+T_1)], \tag{12} \]
\[ (1-\eta_t)=1/\varepsilon_1^{k-1}=Q_2/Q_1=(T_2-T_1)[a+0.5b(T_2+T_1)]/\xi H_u, \tag{13} \]
from equations (12)—(13) we find
\[ \varepsilon_1^{k-1}= \frac{\xi H_u\mu (k-1)} {4[2/\mu b\,(k-1)-a/b-T_1]}; \tag{14} \]
where \(\mu\) is the molecular weight of the combustion products, and \(T_1\) is their temperature at point 1, differing from the temperature of the fresh working mixture \(T_{1c}\), according to the equation \(T_{1c}=T_1R/R_c\). Equation (14) relates \(\varepsilon_1\), \(\xi\), and \((k-1)\).
According to equation (11), the indicated efficiency in the most general case is determined by two parameters: 1) the coefficient of liberation of active heat \(\xi\), showing what fraction of the available heat is imparted in the cycle to the working medium, and 2) the reduced compression ratio \(\varepsilon_1\), determining the degree of utilization of active heat for work. In this case the heat-utilization coefficient \(\psi\) reaches its maximum value \(\psi_{\max}=1\) only in the ideal cycle of rapid combustion (the Otto cycle), provided that the real working medium has a heat capacity independent of temperature; i.e., \(\psi\) takes into account (estimates) both the law of heat addition to the working medium and the heat capacity of the working medium.
The coefficients \(\xi\) and \(\psi\) (when compared with unity) may serve to evaluate the perfection of the working process of an internal-combustion engine. Having taken an indicator diagram from the engine and having found on it \(T_1\), \(T_2\), \(L_i\), we find, from equations (12), (15), and (16), \((k-1)\), \(\xi\), and \(\varepsilon_1=\psi\varepsilon\):
\[ \xi H_u = (T_2-T_1)[a+0.5b(T_2+T_1)] + AL_i, \tag{15} \]
\[ \varepsilon_1^{k-1}=1+\frac{AL_i}{(T_2-T_1)[a+0.5b(T_2+T_1)]}. \tag{16} \]
When designing a new engine, the expected indicated efficiency can be found from equations (11) and (14), if experimental material is available that permits \(\xi\) and \(\psi\) to be estimated in advance.
Received
4 XI 1967