Rapid thermodynamic simulation model for optimum performance of a four-stroke, direct-injection, and variable-compression-ratio diesel engine

A thermodynamic simulation model for the performance of a four-stroke, direct-injection diesel engine is developed. The simulation model includes detailed sub-models for fuel burning rate, combustion products, thermodynamic properties of working fluid, heat transfer, fluid flow, and both soot and oxides of nitrogen (NO_x) formation mechanisms. To validate the model, comparisons between experimental and predicted results for different engines, operating under different conditions, were conducted. The comparisons show that there is a good concurrence between measured and predicted values. An optimization analysis is conducted for seeking an optimum variation of compression ratio to achieve pre-set objective targets such as constant minimum brake-specific fuel consumption and constant maximum torque. The optimization analysis is performed under the constraint that the maximum pressure and temperature inside the cylinder do not exceed the maximum allowable pressure and temperature of the conventional engine (constant compression ratio).The varying compression ratio is optimized with each of the previous conditions separately. Results indicated that varying the compression ratio to achieve previous targets leads to saving fuel consumption, higher brake efficiency and power, and reduction in soot emission from the engine. Also, an increase in NO_x is noticed at low speed. This drawback is considerable and can be overcome by reducing the operation speed range.

Thermodynamic models (zero-dimensional) and turbulent models (multi-dimensional) are the two types of models that have been used in internal combustion engine simulation modeling. Nowadays, trends in combustion engine simulations are towards the development of comprehensive multi-dimensional models that accurately describe the performance of engines at a very high level of details. However, these models need a precise experimental input and substantial computational power, which make the process significantly complicated and time-consuming [1-6]. On the other hand, zero-dimensional models, which are mainly based on energy conservation (first law of thermodynamics) with a limited set of experimental data, are preferred due to their simplicity, of being less time-consuming, and their relatively accurate results [7-10].

Requirements to provide environmentally friendly vehicles (i.e., low fuel consumption with less emission) have resulted in using the most complicated and intellectual means of internal combustion engine development. Nowadays, new ideas, which could not be discussed two decades ago, are being considered by automotive manufacturers. In particular, many leading automotive companies have approached practically the very complicated design ideas with different aspects of diesel/petrol engine design. These aspects have been under extensive theoretical and experimental investigations. The most important aspect of design is aimed to vary the engine compression ratio (r_c) depending on load, speed, or both. Several trials have been done in that respect with extensive design, experimentation, and measurements [11-19]. All attempts to change the r_c are achieved by one or more of the following concepts [11]: (1) moving the cylinder head, (2) variation of combustion chamber volume, (3) variation of piston deck height, (4) modification of connecting rod geometry (usually by means of some intermediate member), (5) moving the crankpin within the crankshaft (effectively varying the stroke), and (6) moving the crankshaft axis. Benefits and challenges of variable-compression-ratio (VCR) spark-ignition engines had been illustrated, examined, and critically reviewed by Roberts [11]. Also, the implications for volume manufacture and the strategy for VCR implementation in order to produce the maximum benefit had been discussed.

Ladommatos and Balian [12] studied the effects of an increased clearance volume on the performance of a direct-injection diesel engine. This study was conducted to overcome problems with combustion at standard r_c for high-speed, direct-injection (DI) diesel engine. Modifications were done in the injection system and operating conditions to improve the power output and fuel consumption. Sobotowski et al. [13] modified a prototype DI diesel engine to incorporate BP Oil's VCR engine concept. Their objective was achieved by altering the phase relation between two pistons linked to separate crankshafts and operating in two cylinders arranged in an opposed engine or interconnected through a transfer port. According to their concept, a novel crankshaft phasing mechanism was employed to achieve r_c variation with a number of significant advantages over other VCR engine designs. These advantages include (a) mechanical simplicity and compact design, (b) capability to externally control r_c during engine operation, (c) automatic variation of valve timing, and (d) compatibility with current production technology.

Rychter et al. [14] evaluated the concept of varying r_c to give different expansion and compression ratios by means of a theoretical simulation of a turbocharged diesel engine. They varied the ratio of connecting rod length to crank throw to study the effect of r_c on engine performance at a fixed load. The principal benefits, at 3/4 load and r_c of 20, were a reduction in fuel consumption by about 2% and a reduction in ignition delay that leads to an estimated 6-dB reduction in combustion noise. Variable-stroke engine mechanism patented by Freudenstein and Maki [15] was used by Yamin and Dado [16] as it gives availability to change the stroke length and the corresponding r_c in the range of 6.82 to 10. Their results showed a significant improvement in the engine's power and reduction in the fuel consumption, in addition to an increase in carbon monoxide and nitric oxides compared with the actual engine. Finally, they concluded that the engine performance can be optimized for a full range of driving conditions, such as acceleration, speed, and load. Such optimized performance was obtained by incorporating some means to suppress the increase in emission rate to be applicable in the automotive field due to the strict anti-pollution laws.

The effect of varying compression ratio and other parameters such as injection timing, engine load, and fuel on the performance of diesel engine was recently studied experimentally [17-20]. Raheman and Ghadge [17] studied the performance of a Ricardo EG engine using biofuel and its blend with high-speed diesel fuel at varying compression ratio, injection timing, and engine load. They concluded that the performance of the engine reduced with the increase in the percentage of biofuel in the blend above 20% compared to that of conventional engine. This reduction in engine performance can be overcome by increasing the compression ratio and injection timing. Jindal et al. [18] investigated the effect of compression ratio and injection pressure on the performance of DI diesel engine running on Jatropha methyl ester. They demonstrated that an increase in compression ratio associated with an increase in injection pressure improves the performance of the engine in terms of brake-specific fuel consumption, brake thermal efficiency, and emissions (i.e., hydrocarbon, NO_x, and smoke opacity). They concluded that for fuelling the engine with bio-diesel, one should go for higher compression ratio associated with higher injection pressure. Muralidharan et al. [19] estimated the performance, emission, and combustion characteristics of a single-cylinder, four-stroke, VCR, multi-fuel engine fuelled with waste cooking oil methyl ester and its blends with standard diesel. The performance parameters include brake thermal efficiency, brake-specific fuel consumption, brake power, indicated mean effective pressure, mechanical efficiency, and exhaust gas temperature. They compared and analyzed their results with standard diesel, and a considerable improvement in the performance parameters as well as exhaust emissions was observed especially at full load. Gandure and Ketlogetswe [20] carried out their work to compare the performance of variable compression engine fuelled with native marula seed oil and petrodiesel fuels. The results indicated that engine performance when powered with marula oil as fuel was very close to that when powered with petrodiesel and at an 80% load, the compression ratio of 16:1 yields optimum performance in terms of engine torque and brake power for both petrodiesel and marula oil fuels.

The purpose of this study is to develop a simple, rapid, and accurate simulation model based on thermodynamic analysis without the need for a great deal of computational power or knowledge of precise engine geometrical data to predict the range of compression ratio corresponding to the optimum engine performance (minimum brake-specific fuel consumption (bsfc) and maximum brake torque) without an increase in the maximum cylinder pressure and temperature of the conventional engine. The developed model is based on mass and energy conservation laws, equation of state, heat transfer between the working fluid and cylinder walls, fuel energy release rate, and engine friction losses. Single-zone combustion is assumed, and thermodynamic properties are calculated from chemical equilibrium principles for six and/or eleven species. A stepwise solution is carried out for the varying compression ratio and the engine speed over its entire range (i.e., 1,500 to 2,800 rpm). In calculations, the temperature and pressure cannot exceed the maximum allowable temperature and pressure of the conventional engine, i.e., constant compression ratio, which are 2,100 K and 10.7 MPa, respectively.

Mathematical model

A thermodynamic analysis of a four-stroke, direct-injection diesel engine is conducted. It is based on a single-zone combustion modeling with a rapid premixed burning phase followed by a slower mixing-controlled burning phase. Although a single-zone model is an old one, it was selected in this study for the sake of its simplicity and of being less time-consuming for the program execution. Works with two- and multi-zone models are in progress. The analysis includes detailed sub-models representing different engine aspects integrated in the main engine simulation model. These sub-models are engine geometry, combustion products, thermodynamic properties, engine friction and heat transfer, mass exchange, combustion rate, temperature-pressure, and emission mechanisms. The analysis presented in this article is based on the following assumptions:

1. The content of the cylinder is a homogeneous gas mixture of air (79% N₂ and 21% O₂ by volume), where fuel vapor, combustion products, and its thermodynamic properties are calculated using ideal gas law with temperature-dependent specific heats.

2. Pressure and temperature inside the cylinder are uniform and vary with the crank angle.

3. The gas motion inside the cylinder (created by pressure gradients due to piston motion) is neglected.

4. A single-zone combustion process starts with a rapid premixed burning phase, followed by a slower mixing-controlled burning phase.

5. Temperatures of the cylinder head, cylinder walls, and piston crown are assigned constant values.

In the present work, the detailed sub-models are formulated as follows:

Engine geometry

The rate of change of the volume with respect to time is given as

Combustion product model

For temperature <1,600 K, the equilibrium composition of gases produced by the combustion of a general hydrocarbon fuel with air (having a general fuel-air molecule in the form C_wH_xO_yN_z) is calculated based on the assumption that only gaseous phases are considered. The general combustion equation of the fuel in air can be written as

For temperature ≥1,600 K, 11 combustion products are assumed. The chemical reaction can be written as

Obtaining the moles of the 11 combustion products, 11 equations are needed (i.e., four equations from the mass balance of C, H, O, and N in Equation 3, and seven equations from the chemical equilibrium). These equations have been solved by the procedure mentioned in [21].

Thermodynamic properties of cylinder content model

During each thermal cycle, the cylinder contains, in general, air, fuel vapor, and combustion products with different proportions depending on the engine conditions under consideration. The specific heat under constant pressure, enthalpy, and entropy of the cylinder content are given by

where , and zz (number of different gases) is equal to 6 and 11 for T < 1,600 K, and T ≥ 1,600 K, respectively.

Engine friction model

Empirical correlations of losses are in terms of compression ratio, engine speed, bore, stroke, intake and exhaust manifold pressures, and valve geometry [22] are used in this work. These correlations were concluded to the following friction categories as follows:

1. Piston losses:

2. Blowby losses:

3. Exhaust and inlet system throttling losses:

4. Crankcase mechanical losses:

5. Combustion chamber and valve pumping losses:

The intake and exhaust manifold gage pressures (p_im, p_em) in Equation 7 are taken as reported by Eichelberg [23] as .

The overall engine frictional power losses are obtained by

Engine heat transfer model

Eichelberg [23] derived a simple equation for calculating the instantaneous heat flux out of the engine (in Watts per square meter) in terms of cylinder temperature, pressure, and piston mean velocity. Rakopoulos and Hountalas [24] modified this equation into the form

Radiation heat transfer was neglected in some studies (e.g., [25-28]), but it had been considered in others (e.g., [29-31]). In the present simulation, the Eichelberg model [23] as modified by Rakopoulos and Hountalas [24] is used with assumed specified values for piston crown, cylinder head, and cylinder wall temperatures.

Mass exchange model

Air intake and exhaust exchange. The mass flow rates through the valves are usually described by the equation for compressible, one-dimensional, isentropic flow. The real flow effects are included by means of an experimentally determined discharge coefficient. In the present simulation, the discharge coefficient is taken as reported by Campbell [29] and is equal to . The gas flow rate is related to the upstream stagnation pressure, stagnation temperature, and static pressure just downstream of the flow restriction. For the flow into the cylinder through intake valve, the upstream stagnation pressure and temperature are the intake manifold pressure and temperature (p_im, T_im), respectively, and the downstream static pressure is the cylinder pressure (p). For the flow out of the cylinder through exhaust valve, the upstream stagnation pressure and temperature are the cylinder pressure and temperature (p, T), respectively, and the downstream static pressure is the exhaust manifold pressure (p_em) [29,32]. The mass flow rates entering and leaving the cylinder are defined as follows [33]:

For subsonic inflow where

For sonic and supersonic inflow where

For subsonic outflow where

For sonic and supersonic outflow where

Mass of injected fuel. The mass of fuel injected into the cylinder m_f (relative to total injected mass m_ft) is written (according to Zelenik [34] and Gu and Ball [35]) in terms of crank angle (θ) as follows:

Differentiation of the above equation with respect to time leads to the following equation:

Combustion rate model

Watson et al. [36] developed equations for fuel energy release appropriate for diesel engine simulations. In their development, the combustion process starts from a rapid premixed burning phase (represented by function f₁), followed by a slower mixing-controlled burning phase (represented by function f₂), with both functions empirically linked to the duration of ignition delay () and the duration of combustion (Δθ_comb). According to their model, the burned mass of fuel at any angle (θ) related to total injected mass (m_ft) is

where , , and

The ignition delay () is calculated in degrees of crank angle by empirical correlations mentioned by Heywood [32] and Mehta et al. [37]. These correlations are in terms of intake manifold temperature (T_im, in Kelvin) and pressure (p_im, in bars), compression ratio (r_c), fuel cetane number (CN), and mean piston velocity (S_p, in meters per second) as follows:

The ignition delay period can be expressed in milliseconds according to . Values of the empirical coefficients k₁, k₂, k₃, and k₄ are reported by Heywood [32] and Rossini et al. [38] as follows: , , , and . The fuel burning rate with respect to the crank angle (θ) can be obtained by differentiating Equation 18 as follows:

Temperature-pressure relationships

The changes of pressure and temperature with respect to time are defined as follows:

where

Soot mechanism

Hiroyasu and Kadota [39] developed a simple soot model to predict the production rate of soot mass () using a single-step competition between the soot mass formation rate and the soot mass oxidation rate according to . Morel and Keribar [40] and Wahiduzzaman et al. [41] suggested models for soot formation and their oxidation in diesel engines. In their models, the time rates of soot formation and oxidation, respectively, are given by

where m_fb is the mass of fuel burned, T_af is the adiabatic flame temperature at an equivalence ratio of 1.1, d_s (the soot particle diameter) = 0.022 × 10⁻⁶ m, ρ_s (soot density) = 1,800 kg/m³[42].

NO_x mechanism

The extended Zeldovich mechanism for NO_x formation rate is formed as follows [29]:

where . The constants above are taken as follows: , , , and .

Solution procedure

The governing equations (Equations 1 to 25) of the comprehensive model subjected to the assumptions previously mentioned in the mathematical model are solved for different processes of the engine cycle. The fourth-order Runge-Kutta method is used to simulate the comprehensive model equations at the prescribed initial conditions. The engine cycle starts from the moment the exhaust valve closes. The solution is carried out with end results of each process taken as the starting conditions for the following process and end values of the completed thermal cycle taken as the starting values for the subsequent cycle. The solution is iterated and considered satisfactory when the end values of the temperature and pressure of the cycle are within ±0.001% from the starting values of the temperature and pressure of the same cycle. Inputs to the code are the engine specifications, operating conditions, and fuel data listed in Table 1. The numerical output results are the instantaneous pressure, temperature, volume, heat transfer, work, mass exchange, fuel heat release, and engine performance variables such as bsfc, indicated mean effective pressure, brake mean effective pressure, indicated power, brake power, and volumetric, indicated, brake, and mechanical efficiencies.

Simulation results and its validation

The numerical simulation results are performed for a conventional DI diesel engine with constant compression ratio (specifications and operating conditions are given in Table 1). Samples of these results represent the engine thermodynamic cycle, the temperature and pressure variation throughout the cycle, the rates of mass entering and leaving the cylinder, and the in-cylinder NO_x and soot history along the cycle which are displayed in Figures 1, 2, and 3, respectively.

In order to validate the present developed model, comparisons between the numerical results and the available measurements are conducted. Two sets of experimental data reported by Han and Reitz [3] and Wahiduzzaman et al. [41] are used to validate the developed model as shown in Figure 4. Figure 4 shows the comparison between the predicted and measured pressures and heat release versus the crank angle. Based on this figure, one can observe that there is a good concurrence between measured and predicted results for different engines under different operating conditions. More details about the simulation results and validation can be found in a previous work of the authors [43].

Variation of the compression ratio

The numerical simulation results are used to determine the range of r_c corresponding to the optimum engine performance for two cases: (1) minimum bsfc and (2) maximum brake torque. This was computed using optimization techniques under the restriction of maximum allowable pressure and temperature. The numerical results of the optimization process, for engine specifications previously mentioned as shown in Table 1, are presented in Figures 5,6,7,8,9, Figures 10,11,12,13,14. It is worth mentioning that for all figures, the solid lines represent the performance of optimum variable r_c, while dashed lines represent the performance of conventional constant r_c engine.

Case 1: minimum brake-specific fuel consumption

Figures 5, 6, 7, 8, and 9 show the results at minimum bsfc. Figure 5 shows the comparison between the operation under constant compression ratio (r_c = 16.4) and the hypothetical operation under varying compression ratio to achieve working with minimum bsfc (230 g/kW.h) as possible. The figure shows that the bsfc for the constant-compression-ratio engine changes between 238.5 and 261 g/kW.h passing through a minimum value of 230 g/kW.h at an engine speed of about 1,900 rpm. In addition, for the variable-compression-ratio engine, the bsfc is kept constant at its minimum value of 230 g/kW.h by varying the compression ratio between 16.4 and 17.8. The shown variation trend in compression ratio emphasizes that the variation in compression ratio should be considered to decrease the bsfc to its minimum possible value. Since the increase in compression ratio is accompanied by an increase in both cylinder pressure and brake power, consequently, a decrease in bsfc is achieved. The percentage reduction in bsfc ranges between 3.7% at 1,500 rpm and 4.4% at 2,460 rpm passing through 0% at 1,900 and 2,800 rpm. Changing the compression ratio for an operating engine is desirable and could be attained. Its effect on maximum cycle pressure and temperature must be studied as they represent important parameters in the design of engine's different parts.

Figure 6 shows the increase in maximum pressure and temperature when operating under variable compression ratio compared to the operation under constant compression ratio. Compared to the case of a constant compression ratio, the increase in maximum cycle pressure is about 0.53 MPa at 1,500 rpm and 0.55 MPa at 2,460 rpm passing through 0 MPa at both 1,900 and 2,800 rpm with variable compression ratio. The noticeable increase in maximum temperature is about 72 K at 1,500 rpm and 40 K at 2,460 rpm passing through 0 K at both 1,900 and 2,800 rpm. These increases still do not exceed the engine design's maximum pressure and temperature of the conventional engine. Changes in brake power and torque with speed for the variable compression ratio at constant minimum bsfc are shown in Figure 7. It shows an increase in brake power by about 1.67 kW at 1,500 rpm and 1.6 kW at 2,460 rpm. Moreover, the brake torque increases by 9 N.m at 1,500 rpm and 8 N.m at 2,460 rpm.

Figure 8 shows that varying the compression ratio results in a decrease in the delay period, volumetric efficiency, and residual gas fraction (RGF) over the engine speed range (1,500 to 2,460 rpm). It is seen that this decrease changes between 0.2° of crank angle at 1,500 rpm and 0.2° of crank angle at 2,460 rpm. The decrease occurs due to the fact that the increase in compression ratio results in an increase of the air pressure and temperature at the end of the compression process which are inversely proportional to ignition delay. The decrease in the ignition delay is desirable as it causes ignition to occur before greater amounts of fuel is injected in the cylinder. Consequently, the rates of heat release and pressure rise are then controlled primarily by the rate of injection and air-fuel mixing, resulting in a smoother engine operation. Long ignition delay allows for more fuel to be injected before auto-ignition occurs, which results in a very rapid pressure rise as the combustion starts and reaches high-peak pressures. Under extreme conditions, when auto-ignition of most of the injected fuel occurs, an audible knocking sound is noticed, often referred to as a ‘diesel knock.’ Also, if long ignition delay occurs sufficiently late and reaches the expansion process, the burning process may be quenched. This leads to incomplete combustion, reduced power output, and poor fuel-power conversion efficiency. Also, the figure shows that the volumetric efficiency and RGF are reduced by about 4.3% and 0.3%, respectively, at a speed of 1,500 rpm. These reductions continue to decrease until reaching 0% at 1,900 rpm. These reductions, then, increase again until reaching about 4.3% and 0.1% at 2,460 rpm. This variation was attributed to the increase in compression ratio. It means that a decrease in clearance volume leads to a decrease in charging-up the cylinder. The general decrease in residual gas fraction is desirable because it decreases the dilution of the charge in the cylinder and increases the scavenging efficiency of the engine. However, these reductions are small and can be neglected.

The effect of the optimal variation of the compression ratio corresponding to the minimum bsfc on soot and NO_x emissions is shown in Figure 9. For all reported speed range, the soot emission from the engine with variable compression is less than that of the conventional engine. The decrease in soot emission is found to be about 21% at 1,500 rpm and decreases until it reaches 0% at 1,900 rpm. Then, it increases again to 9% at 2,460 rpm and returns to 0% at 2,800 rpm. On the other hand, the formed NO_x inside the variable-compression-ratio engine is much higher compared to that inside the constant-compression-ratio engine. The maximum increase reaches about 11.2 g/kg of fuel instead of 6.4 g/kg of fuel (r_c = constant) at 1,500 rpm. Beyond 1,900 rpm, the NO_x emission from the engine with variable compression is also greater than that of conventional engine and is about 0.97 g/kg of fuel at 2,460 rpm. The increase of NO_x is a drawback to the advantages mentioned when using variable-compression-ratio engines to achieve minimum bsfc (maximum efficiency). This drawback can be overcome by reducing the engine speed range from 1,500 to 2,800 rpm to 1,665 to 2,800 rpm as illustrated on the figure.

To conclude based on Figures 6, 7, 8, and 9, varying the compression ratio, within the entire range of engine speed of 1,500 to 2,800 rpm, to attain a minimum bsfc results in significant variations in the predicted performance variables compared to those predicted at constant compression ratio (r_c = 16.4). These variations can be summarized as follows: (1) at an engine speed of 2,460 and 2,800 rpm, there is no variation in all investigated performance variables; (2) for the other values of engine speed, the maximum pressure, maximum temperature, brake torque, brake power, and unfavorable NO_x are increasing. On the other hand, the delay period, volumetric efficiency, RGF, and soot formation are decreasing. Table 2 summarizes the predicted results of performance variables at different values of engine speed.

Case 2: maximum brake torque

The predicted performance variables corresponding to maximum brake torque compared to those predicted at a conventional constant compression ratio (r_c = 16.4) are displayed in Figures 10, 11, 12, 13, and 14. Figure 10 shows that the brake torque, in the case of a constant compression ratio, changes between 281 and 252 N.m passing through a maximum of 295 N.m at about 1,900 rpm. For the variable compression ratio, the brake torque is kept constant at its maximum value of 295 N.m over a speed ranging from 1,500 to 2,460 rpm by varying the compression ratio between 16.4 and 19.3. This means that the brake torque increases by 5.4% at 1,500 rpm and 4.6% at 2,460 rpm. The shown variation trend in compression ratio urges that actual variation in compression ratio should be considered to increase the brake torque to its constant maximum value.

Figure 11 shows the increase in p_max and T_max when operating under variable compression ratio compared to the operation under constant compression ratio. The increase in p_max is about 0.9 MPa at 1,500 rpm and 0.6 MPa at 2,460 rpm passing through 0 MPa both at about 1,900 and 2,800 rpm. The noticeable increase in T_max is about 60 K at 1,500 rpm and 40 K at 2,460 rpm passing through 0 K both at 1,900 and 2,800 rpm. These increases in p_max and T_max are tolerable, as they are still below the allowable pressure and temperature of the conventional engine. Similar observations for the changes of brake efficiency and bsfc versus engine speed for operation under constant and variable compression ratios are shown in Figure 12. If the compression ratio is constant, the brake thermal efficiency varies between 36% and 32.9% passing through a maximum value of 37.3% at an engine speed of 1,900 rpm. In case of varying compression ratio (between 16.4 and 19.3 for maximum torque), the efficiency changes between 37.6% at 1,500 rpm and 36.5% at 2,460 rpm. In addition to gains in brake torque and efficiency, the figure shows a decrease in the bsfc over an engine speed ranging from 4.6% at 1,500 rpm to 2.7% at 2,460 rpm passing through 0% both at 1,900 and 2,800 rpm.

Figure 13 shows a significant decrease in delay period, volumetric efficiency, and RGF over an engine speed ranging from 1,500 to 2,460 rpm with values between 15.3% (0.45° of crank angle) at 1,500 rpm and 6.8% (0.3° of crank angle) at 2,460 rpm. Also, the figure shows that the volumetric efficiency is reduced by about 0.48% at a speed of 1,500 rpm and 0.25% at 2,460 rpm passing through 0% both at 1,900 and 2,800 rpm. Also, this figure shows that variation of the compression ratio for optimum brake torque results in a general decrease in RGF by about 6.5% at 1,500 rpm and 5% at 2,460 rpm passing through 0% both at 1,900 and 2,800 rpm.

The effect of the optimal variation of compression ratio for maximum brake torque on soot and NO_x emission is shown in Figure 14. The soot emission from the engine with variable compression ratio is less than that of the conventional engine by about 9.7% at 1,500 rpm, 0% both at 1,900 and 2,800 rpm, and 12% at 2,460 rpm. The formed NO_x inside the variable-compression-ratio engine is very high compared to that inside the constant-compression-ratio engine by about 21.7% (2.65 g/kg of fuel) at 1,500 rpm, 90.7% (0.461 g/kg of fuel) at 2,460 rpm, and 0% at both 1,800 and 2,800 rpm. This increase is due to the increase of cylinder temperature with increasing compression ratio. The increase in NO_x at low speed, i.e., 1,500 rpm, can be avoided by avoiding the engine to run in the speed range of 1,500 to 1,620 rpm as illustrated in Figure 14.

The conclusion based on Figures 10, 11, 12, 13, and 14 is that varying the compression ratio of the diesel engine, within the entire range of engine speed of 1,500 to 2,800 rpm, to achieve a maximum engine torque leads to variations in the predicted performance variables compared to those predicted at constant compression ratio (r_c = 16.4). These variations can be summarized as follows: (1) at an engine speed of 1,500 and 2,460 rpm, there is no variation in all studied variables; (2) for the other values of engine speed, the maximum pressure, maximum temperature, brake torque, thermal efficiency, and unfavorable NO_x are increasing. On the contrary, the delay period, volumetric efficiency, residual gas fraction, and soot formation are decreasing. Table 3 summarizes the predicted results of performance variables at different values of engine speed.

The present rapid thermodynamic simulation for the direct-injection diesel engine is based on a single-zone combustion modeling with a rapid premixed burning phase followed by a slower mixing-controlled burning phase. Comparison with other experimental results (under constant compression ratio) verifies very well the simulation predictions of cycle performance. The main conclusions of this study indicate that varying the compression ratio to achieve minimum bsfc or maximum brake torque for moderate diesel engine under the restriction that the maximum cylinder pressure and temperature do not exceed those of the conventional engine results in saving fuel consumption, higher brake efficiency and power, and reduction in soot emission from the engine. Also, an increase in NO_x is noticed at low speed. This drawback is considerable and can be overcome by reducing the operation speed range from 1,500 to 2,800 rpm to 1,665 to 2,800 rpm in the case of minimum bsfc and from 1,500 to 1,800 rpm to 1,620 to 2,800 rpm in the case of maximum brake torque.

CN: fuel cetane number; c_p, c_v: constant pressure and volume specific heats; D_iv: intake valve head diameter; D: cylinder bore; H, h: enthalpy and specific enthalpy; K: specific heat ratio; M: mass; N: crank speed; N: number of moles; n_c: number of cylinders per engine; n_iv, n_r: number of intake valves and rings per cylinder; p: pressure; Q, q: heat transfer and heat flux; r_c: compression ratio; R: gas constant; S_p: mean piston velocity; T: cylinder temperature; V, V_c, V_d: swept, clearance, and maximum displacement volumes; X: mole fraction; Θ: crank position angle; Λ: connecting rod length to crank radius ratio; θ_evo, θ_ivo: exhaust and intake valve opening angles; Δθ_eopen: opening duration angle of exhaust valve; Δθ_iopen: opening duration angle of intake valve; ρ: density; ϕ: equivalence ratio; τ, τ: ignition delay period and ignition delay angle; B: backward; comb: combustion; e: exit, equilibrium; em: exhaust manifold; ex: exhaust; f: fuel, forward; fb: fuel burning; ft: total fuel; i: indicated, inlet, injection start; ig: ignition; im: intake manifold; in: intake; o: ambient condition; p: piston; s: stoichiometry, soot; sf: soot formation; so: soot oxidation; w: wall, atoms of carbon in fuel-air molecule; x: hydrogen atoms in fuel-air molecule; y: oxygen atoms in fuel-air molecule; z: nitrogen atoms in fuel-air molecule.

The authors declare that they have no competing interests.

MM conceived the concept and procedures of the present work, developed the model, and carried out the computations and analysis. MA checked the equations and analysis, and reviewed the revised manuscript. YM reviewed the graphs and results. All authors read and approved the final manuscript.

MM earned his Ph.D. degree in Mechanical Engineering from the University of Manitoba, Canada, in 2007. He is currently an assistant professor in the Department of Mechanical Engineering at the Assiut University, Egypt. He has been interested in heat and mass transfer relating to droplets and spray modeling, turbulence, combustion, and modeling and simulation of internal combustion engines. MA received his Ph.D. degree in Mechanical Engineering from Mississippi State University, USA, in 1994. He is currently a professor in the Department of Mechanical Engineering at the Assiut University, Egypt. His current research interests focus on sprays and liquid atomization, fuel cells, solar hydrogen generation, and two-phase flows. YM received his Ph.D. degree in Mechanical Engineering from Kansas State University, USA, in 1984. Now, he is an emeritus professor in the Department of Mechanical Engineering at the Assiut University, Egypt. His research interests focus on combustion and management and optimization of thermal systems.

This work has been fully supported by Assiut University and the Mechanical Engineering Department.

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