In analyzing the thermodynamic performance of an internal combustion engine, only the compression, combustion, and expansion processes affect thermodynamic performance. During the adiabatic compression process, mechanical work is transformed into internal energy of the working fluid. During the adiabatic combustion process, chemical energy in fuel is converted into heat energy which in turn is transformed into internal energy of working fluid. During the adiabatic expansion process, a portion of the internal energy of working fluid is transformed into mechanical work. The remaining internal energy of working fluid is lost through the exhaust gas. The increment in the internal energy dE of working fluid is equal to the difference between the increment of heat dQ accumulated by the working fluid and the increment of work done dW by working fluid.

dE = dQ – dW (1)

The compression work W done on working fluid from V0 to V is equal to the definite integral of pdV from V0 to V with p equal to p0(V0/V)k (Boyle’s law) the average value of pressure distribution throughout the volume. The work done W0 is zero at V0.

W = p0(V0)k(1/V)k-1/(k-1) (2)
E = p0(V0)k/(k-1)(1/V)k-1/J (3)
E2/E1 = (V1/V2)k-1 (4)
T2/T1 = cv2T2/ cv1T1 = E2/E1 = (V1/V2)k-1. (5)

Equation (2) is the mechanical work done on cylinder gas by a piston moving from V0 to V. Equation (3) is the internal energy E transformed from the mechanical work of Equation (2). The difficult question is what are the values of p0 and V0? This difficulty is overcome by taking the ratio of two internal energies at two different volumes to obtain Equation (4). Equation (4) is law of conservation of energy and can be applied to internal combustion engines, while Equation (5) is true only for ideal gas with cv2 = cv1.

For an adiabatic constant-volume air cycles, at the beginning of a compression process 1-2, E1 is known. At point 2, E2 is computed by Equation (4). There is an endless possibility of heat release curves for the same Q. Because E is a state variable, it depends only on the state of gas and not on any process that produces that state. In other words, E3 depends on V3 alone. Traditionally,  of an air cycle is computed based on work balance. Work balance requires Wc = cv(T1-T2) and We = cv(T3-T4). These two equations are true only for ideal cycles (having constant cv). For an engine cycle, E3 of working fluid consists of two parts. The first part is transformed from work done during a compression process, and the second part from Q is transformed from fuel chemical energy during a combustion process. E4 is the part of internal energy not transformed into work. By definition  = (E3 – E4)/E3. Utilizing internal energy balance not only meets the law of conservation of energy, but it is also much simpler to calculate.

It is clear that indicated efficiency is a function of Rc,  and Re. These three parameters are independent from one another. The compression ratio Rc controls the required compression temperature. Equivalence ratio controls the combustion temperature and thus controls the NOx formation and heat loss to coolant. The expansion ratio Re controls indicated efficiency . The higher the compression temperature, more completely the fuel burns, but the temperature should not be high enough to cause pre-ignition. The higher the , the higher is the power density, but it should not be high enough to produce NOx or too much heat loss to coolant. The higher the expansion ratio, the larger is the indicated efficiency  but it should not be too high to reduce geometrical efficiency excessively. Based on these considerations, engine designers can choose appropriate combinations of these three parameters for developing new reciprocating engines to achieve the maximum possible fuel efficiency with minimum emissions.

As an example, three values 14.5, 0.334, and 16 are chosen for Rc,, and Re respectively for this discussion. Figure 1 below shows the diagram E versus V. At the beginning of a compression process 1-2, E1 = 95.73 (average absolute temperature of ambient air times cv). At point 2, E2 = 290.0 computed by Equation (4) T2 = E2/cv =942.8, and p2 = 713.0. At point 3, E3 = E2 + 400 = 690, T3 = E3/cv = 2242.0 and p3 = p2(T3/T2) = 1696.0. At point 4 for V4 = V1, E4 = E3(V3/V1)k-1 = 227.6. The indicated efficiency  is equal to (E3 – E4)/E3 = (690 – 227.6)/690.0 = 0.670. By keeping one-third of exhaust gas in the cylinder, the internal energy of 0.67(1.0 - 0.67)/3 can be added to Q in the ensuing cycle to increase indicated efficiency from 0.67 to 0.744. For the expansion process of a GDI engine at k value of 1.3 has been taken. For new cycle at k is 0.38 (0.889 x 1.4 + 0.111 x 1.3).E4 = E3(V3/V4)0.38 = 240.6, and  = 0.651, a 34.9% (1- 0.561) of fuel chemical energy is lost in the exhaust gas.

Figure 1

The indicated efficiency of a constant-volume adiabatic air cycle 1-2-3-4-1 having a compression ratio of 10.0 is computed for comparison. At point 2, E2 = 240.4, T2 = 781.2, and p2 = 369.2. At point 3, E3 = E2 + 1200 = 1440.4, T3 = E3/cv = 4680.0, p3 = 2212.0. At point 4, E4 = E3(V3/V4)k-1 = 573.4 nt = (1440.4 – 573.4)/1440.4 = 0.602. For GDI engine cycle at k is 1.3 and E4 = E3(V1/V4)0.3 = 721.9 and  = (1440.4 – 721.9)/1440.4 = 0.50; 50% of fuel chemical energy is lost in the exhaust gas. By increasing the expansion ratio from 10.0 to 16.0 and reducing the fuel equivalence ratio from 1.0 to 0.334, the fuel chemical energy lost in the exhaust gas is reduced from 50% to 34.9%. This reduction of 15.1 percentage points is the result of adiabatic air cycle computation based on conservation of energy law and is achievable.

Traditionally, brake efficiency is defined as the product of indicated efficiency and mechanical efficiency without taking account of heat loss to coolant during combustion. To meet the conservation of energy law, brake efficiency is defined as the difference between the indicated efficiency and the sum of exhaust gas internal energy, direct heat loss, and friction loss. The direct heat loss and friction loss are very difficult to compute. They are estimated from the combustion temperature and pressure. The internal energy of exhaust gas of GDI engine is 0.50. For an assumed brake efficiency of 0.25, the sum of direct heat loss and friction loss is 0.25 (0.50 – 0.25). Operating at T3 and p3 are 4680.0 and 2212.0, respectively. For the new engine operating at T3 and p3 are 2242.0 and 1361.0, respectively. The combustion chamber wall temperature is about 360. Heat loss to coolant is proportional to the difference between combustion temperature and combustion chamber wall temperature. The ratio of these differences is (2242 – 360)/(4680 – 360) = 0.44. The friction loss is roughly proportional to a pressure ratio of 0.62 (1361.0/2212). About one-third of 0.25 is due to friction loss. The estimated sum of direct heat loss and friction loss for the new engine is equal to 0.25x0.44x0.666 + 0.25x0.62x0.334 = 0.125. Then for the new engine, the brake efficiency is 0.526 (0.651 – 0.125). The brake efficiency ratio between the GDI engine and new engine is 47.5% (0.25/0.526). The specific fuel consumption and GHG of the new engine are both reduced to less than one half of that of the GDI engine. Therefore, there is no urgent need for electrification to replace the conventional internal combustion engines.