Selection of Injection Parameters in Hydrogen SI Engines Using a Comprehensive Criterion-Based Approach

Abstract

Direct injection in hydrogen engines enables flexible combustion control, improves engine efficiency, and reduces the risk of abnormal combustion. However, implementing this injection strategy is challenging due to the need to provide a relatively high volumetric fuel flow rate, achieve a specified degree of mixture stratification, and account for the functional and technological limitations of the injection system. These challenges highlight the relevance and objectives of the present study. The mathematical model of a turbocharged engine cycle has been refined to account for the influence of injection parameters on combustion kinetics. On the basis of mathematical modeling, the injection pressure and injector area were determined to ensure the specified injection conditions. For the late injection strategy, a method was proposed to select the start of injection based on a specified value of the “relative ignition timing” criterion. Engine operation was simulated across the full range of operating modes for both early and late injection strategies. The results show that the late injection strategy increases the maximum indicated thermal efficiency by approximately 2%, reduces peak in-cylinder pressure by about 1 MPa, lowers maximum nitrogen oxide emissions by a factor of 1.4, and ensures knock-free operation across all modes compared to early injection.

1. Introduction

The transition to hydrogen-based energy systems necessitates the development and implementation of technical solutions for the efficient utilization of hydrogen in power generation. Hydrogen piston engines represent a particularly attractive option, offering high efficiency comparable to that of conventional diesel engines, negligible CO₂ and hydrocarbon emissions, no reliance on scarce rare-earth materials, and relative tolerance to variations in fuel quality [1,2,3,4,5]. Moreover, the deployment of hydrogen engines facilitates the ongoing advancement of existing technologies, supports workforce retention, and sustains related industrial sectors.

Hydrogen possesses unique fuel properties, including a high gravimetric lower heating value (119.9 MJ/kg), wide flammability limits in air (4–76% by volume), relatively high knock resistance (octane number > 130), low ignition energy (0.02 mJ), and others [1,2,3]. These properties enable the implementation of multiple strategies for mixture formation, injection, ignition, and boosting in engines [2,4]. Among the most important are hydrogen injection strategies.

There are two main types of hydrogen injection: port fuel injection (PFI) and direct injection (DI). Direct injection provides the most flexible control over mixture formation and combustion, reduces fuel compression losses, allows for mixture stratification, and decreases the likelihood of abnormal combustion [1,2,3,4,5]. Therefore, the primary focus of this study is on direct injection strategies.

The implementation of direct injection in a hydrogen engine is accompanied by a number of technological and design challenges. Due to the relatively low volumetric lower heating value of hydrogen (10.7 MJ/m³), approximately three times more fuel must be supplied compared to methane to achieve the same power output [4,5]. This requires either an increase in injection pressure and/or an enlargement of the injector orifice area. Increasing the pressure, however, reduces the efficiency of utilizing the storage gas volume. For example, when the injection pressure is increased from 3 MPa to 20 MPa, the fraction of gas that cannot be utilized from 70 MPa storage tanks rises from 4% to 29%. Enlarging the injector area complicates its integration into the engine. At the same time, it significantly shortens the injection duration at idle and low-load conditions, which may challenge the achievement of the required injector response time. Increasing the number of DI injectors introduces additional packaging challenges, while combining PFI and DI injectors considerably complicates the injection system and reduces its overall efficiency.

Ensuring a balance between injection pressure and injector area necessitates a compromise solution. In the available studies on hydrogen direct injection, experiments were generally conducted at relatively low engine speeds (n, 1000–2000 rpm) [6,7,8,9,10]. In studies investigating a wider speed range of 600–8000 rpm, either gasoline injectors with increased injection pressure were used [11,12], or a combination of gaseous PFI injectors and gasoline DI injectors was applied [13,14]. The issues of justifying the choice of injector orifice area and/or upstream injection pressure have been scarcely addressed.

There are known studies on heavy-duty engines in which the injection parameters have been justified [15,16]. However, the specific power and engine speed range in those cases are relatively low, which leaves open questions regarding the choice of injector parameters across a wider range of fuel flow rates, particularly for automotive engines.

A challenging task is the selection of the start of injection (SOI). Late injection during the second half of the compression stroke leads to significant mixture stratification [9,10,17,18]. In this case, part of the fuel burns in locally rich zones with a high combustion rate, while lean zones are formed, where combustion is prolonged. Late injection, up to a certain point, can increase engine efficiency by 2.0–2.6%, reduce nitrogen oxide emissions, and decrease knock tendency [8,9,18]. However, with further SOI retardation, combustion stability decreases, incomplete combustion increases, efficiency drops sharply, and exhaust gas temperature rises. At the same time, the rationale for SOI selection across a wide range of engine speeds and loads in hydrogen engines has been scarcely addressed in the literature.

The degree of mixture stratification is determined not only by the start of injection but also by other parameters, specifically the duration of injection (DOI) and the ignition timing (θ_ign). Together, these parameters define the time interval between the end of injection (EOI) and ignition, during which mixture preparation occurs. The combined influence of injection parameters on mixture stratification necessitates the search for a criterion that can account for this effect. A review of the literature revealed few examples of comprehensive criteria for determining direct injection parameters in hydrogen engines.

Beyer et al. [8] proposed an integral parameter called “relative ignition timing” (RIT), which accounts for the interrelation between SOI, DOI, and ignition timing and affects mixture stratification. In [8], RIT was used to assess the combined effect of injection parameters on engine performance. Due to its criterion-like nature, RIT can also be used to solve the inverse problem of determining appropriate injection parameters from a specified RIT value.

The literature review shows that selecting parameters for late direct hydrogen injection constitutes a multivariable compromise problem. At the same time, comprehensive solutions covering the full range of engine operating conditions are insufficiently represented in the literature.

The problem of selecting injection parameters can be addressed using mathematical modeling. The authors developed a zero-dimensional (0D) hydrogen combustion model based on the Wiebe function [19]. This model accounts for the characteristics of homogeneous, stratified, and diffusion combustion of hydrogen and can be used to assess the influence of injection parameters on mixture formation and combustion. In a subsequent study [20], the effectiveness of various operational strategies was evaluated for a turbocharged hydrogen automotive engine. However, constant injection parameters were used without justification. Further refinements to the mathematical model [21] allowed the influence of injection parameters on fuel combustion kinetics and engine performance to be taken into account.

The objective of this study is to determine the parameters of direct hydrogen injection in a turbocharged automotive engine across a wide range of operating conditions using a criterion-based approach.

2. Materials and Methods

2.1. Base Mathematical Model

This study was conducted using a Ford 1.6 L EcoBoost engine adapted for hydrogen direct injection. In the mathematical modeling, the engine’s design parameters—including the compression ratio, combustion chamber geometry, valve timing, turbocharging system, and others—were kept unchanged (

Table 1. Main parameters of the Ford 1.6 L EcoBoost engine.

Parameter	Value
Number of cylinders	4, in-line
Bore	75 mm
Stroke	88 mm
Compression ratio	10:1
Power output	132 kW at 5700 rpm
Torque	250 Nm at 2500–4500 rpm
Valvetrain	16-valve DOHC
Injection system	Direct injection
Turbocharger	BorgWarner KP39

).

The mathematical model of the hydrogen engine cycle was developed by the authors and implemented in MATLAB^® R2023a. A detailed description is provided in [19,21]. This model is a quasi-steady thermodynamic model of a spark-ignition turbocharged engine. The model describes the processes in the cylinder, intake, and exhaust manifolds based on the solution of the system of energy and mass conservation equations and the equation of state, written in the form proposed by Dyachenko [22].

The engine cycle mathematical model includes submodels for calculating the thermodynamic parameters of the gaseous fuel, air, and combustion products, the equilibrium composition of the combustion products, turbine and compressor parameters, NO_x emissions, heat transfer to the walls, mechanical losses, and several other submodels.

The hydrogen combustion mathematical model developed by the authors, based on the Wiebe function [19,21], accounts for the characteristics of injection, mixture formation, and a number of other factors. According to this model, the combustion rate is

$${\displaystyle \frac{d\mathrm{x}}{d\theta }}=r_I\left[r_{I1}{\left({\displaystyle \frac{d\mathrm{x}}{d\theta }}\right)}_{I1}+r_{I2}{\left({\displaystyle \frac{d\mathrm{x}}{d\theta }}\right)}_{I2}\right]+r_{II}{\left({\displaystyle \frac{d\mathrm{x}}{d\theta }}\right)}_{II},$$

(1)

where ${\left({\displaystyle \frac{d\mathrm{x}}{d\theta }}\right)}_{I1}$ is the premixed burn rate, ${\left({\displaystyle \frac{d\mathrm{x}}{d\theta }}\right)}_{I2}$ is the diffusion burn rate, and ${\left({\displaystyle \frac{d\mathrm{x}}{d\theta }}\right)}_{\mathrm{I}\mathrm{I}}$ is the burn rate in the lean mixture zones. The parameters r_I1 and r_I2 represent the mass shares of the fuel burned in the premixed and diffusion combustion processes, respectively, while r_I and r_II are the shares of the fuel burned in the rich and lean mixture zones, respectively.

Each burn rate in Formula (1) is calculated using the Wiebe function:

$${\displaystyle \frac{dx}{d\theta }}=-a{\displaystyle \frac{m+1}{{\theta }_z}}{\left({\displaystyle \frac{\overline{\theta }}{{\theta }_z}}\right)}^{m}exp\left[a{\left({\displaystyle \frac{\overline{\theta }}{{\theta }_z}}\right)}^{m+1}\right],$$

(2)

where x and dx/dθ are the fuel fraction burned and the heat release rate, respectively, $\overline{\theta }=\theta -{\theta }_0$ is the crank angle from the start of combustion, θ₀ is the crank angle at the start of combustion, θ is the current crank angle, θ_z is the combustion duration angle, a is a constant, and m is the combustion characteristic exponent.

The coefficients in Formula (2), primarily θ_z and m, are determined either from empirical correlations or set within a specific range, depending on the engine parameters and operating conditions.

In this study, early and late injection strategies are considered, with injection ending before the start of ignition, so that diffusion combustion does not occur. Therefore, the diffusion burn rate ${\left({\displaystyle \frac{d\mathrm{x}}{d\theta }}\right)}_{I2}$ and its fraction r_I2 in Formula (1) can be assumed to be zero. Accordingly, the fraction of premixed combustion is taken to be r_I1 = 1.

The duration of premixed combustion is determined by the formula [19]:

$${\theta }_{Z_{I1}}={\theta }_{Z_p}{\lambda }^{k_1}{\left({\displaystyle \frac{n}{n_p}}\right)}^{k_2},$$

(3)

where ${\theta }_{z_p}$ is the combustion duration in the rated (nominal) mode, λ is the air–fuel ratio, n and n_p are the engine speeds at the specified and rated (nominal) modes, respectively, and k₁ and k₂ are empirical constants.

The specifics of selecting other coefficients in the hydrogen combustion model are discussed in detail in [18,20]. The verification of the mathematical model for the gasoline version of the Ford 1.6 L EcoBoost engine is presented in [20]. The same work also provides a comparison of the simulation results for the hydrogen variant of the engine under early hydrogen injection.

It can be seen that the basic model accounts for the influence of the mixture composition and engine speed—which largely determines the turbulence level in the cylinder—on the duration of hydrogen combustion. The effect of direct injection parameters is accounted for in a simplified manner through the coefficient k₁ in Formula (3), which is assumed constant. For a more accurate consideration of the injection parameters, corresponding refinements were introduced into the combustion model (see Section 3.1).

2.2. Methodology for Selecting Injection Pressure and Injector Area

The injection pressure was selected to ensure choked flow of the fuel through the injector orifices at all engine operating conditions. In this case, the fuel flow through the injector does not depend on the gas parameters in the engine cylinder, which allows accurate fuel metering. Choked fuel flow occurs under the condition [15,17,23]:

$${\displaystyle \frac{p_{cyl}}{p_{inj}}}\le {\left({\displaystyle \frac{2}{\gamma -1}}\right)}^{{\displaystyle \frac{\gamma }{\gamma -1}}}\approx 0.52,$$

(4)

where p_cyl is the cylinder pressure, p_inj is the injection pressure, and γ is the specific heat ratio (for hydrogen, it varies from 1.406 at 20 °C to 1.399 at 100 °C).

Direct injection is performed during the compression stroke, when the pressure in the cylinder rises. The end of injection, in accordance with condition (4), is set before the cylinder pressure reaches 0.52 of the injection pressure (Figure 1). Increasing the injection pressure allows for expanding the time window for the injection process and, accordingly, reducing the injector orifice area and overall injector dimensions. On the other hand, it is desirable to minimize the injection pressure to maximize the effective utilization of the storage tank capacity. In this study, the injection pressure was selected based on the analysis of literature data presented in Section 3.2.

Once the injection pressure is chosen, the injector area is selected. Using the engine cycle mathematical model, the maximum fuel flow rate and the corresponding maximum injection duration are determined at the rated power mode (point 1 in Figure 1b). The start of injection in the simulation is set to coincide with the intake valve closing timing (θ_IVC) to avoid backfire in the intake manifold. The injector area is selected so that, at the given pressure upstream of the injector, the end of injection coincides with the point where the cylinder pressure reaches p_cyl = 0.52 p_inj.

After selecting the injector area, injection is simulated at the minimum stable idle speed (point 2 in Figure 1), when the injection duration is shortest. The injection pressure and injector area are constrained by the minimum injector pulse width. If the injection duration becomes too short, options such as variable injection pressure or the use of multiple injectors with the same total flow area should be considered. The specifics of injector parameter selection for the base engine are discussed in Section 3.2.

2.3. Criterion for Selecting the Start of Injection

The literature analysis has shown that mixture stratification is largely determined by the start of injection, the injection duration, and the ignition timing [6,7,8,9,10,11,18]. These parameters determine the period between the end of injection and the spark timing, during which the fuel and air mix. The shorter this period, the stronger the stratification achieved.

The end of injection can occur either before or after the ignition timing. When injection ends before ignition, the mixture in the cylinder has time to partially mix with the air, forming zones of rich and lean mixtures. When injection ends after the onset of combustion, part of the fuel is injected into the flame front, where it burns via a diffusion process. This process is referred to as jet-guided operation [8,24].

Beyer et al. [8] proposed that ignition timing, the start of injection, and the end of injection under mixture stratification and jet-guided operation can be jointly characterized by the parameter relative ignition timing (RIT) (Figure 2):

$$\mathrm{R}\mathrm{I}\mathrm{T}={\displaystyle \frac{\mathrm{S}\mathrm{O}\mathrm{I}{-\theta }_{ign}}{\mathrm{D}\mathrm{O}\mathrm{I}}}100\%,$$

(5)

where SOI is the start of injection, DOI is the injection duration, and θ_ign is the ignition timing.

For RIT < 0%, combustion cannot occur, since the start of injection is delayed relative to the ignition timing. When RIT is between 0% and 100%, injection begins at or before ignition and ends after ignition. In this case, part of the fuel burns via a diffusion process. For RIT > 100%, injection ends before ignition. By the time of ignition, the mixture has had time to partially or fully mix. The larger the RIT value, the greater the degree of mixture homogenization achieved. Thus, RIT accounts for the time available for mixture formation in the cylinder and can be used as a criterion for the degree of mixture stratification.

The effect of RIT variation on BTE/ITE, determined from [8,10,18,25,26], is shown in Figure 3. It can be seen that under jet-guided operation, when injection ends after ignition (RIT < 100%), engine efficiency is reduced compared to earlier and/or shorter injection. Increasing the degree of homogenization at RIT > 200% also leads to a slight decrease in BTE/ITE. The highest efficiency values are observed in the RIT range from 120% to 200%. For the engines considered in [8,10,18], with a cylinder displacement similar to that of the base engine, the maximum efficiency occurs within an even narrower RIT range of 117% to 128%. Therefore, when intentionally organizing mixture stratification for the base engine, an RIT value of 120% is targeted.

In the present study, with the specified RIT value and the calculated injection duration and ignition timing according to Formula (5), the start of injection is determined. If the end of injection occurs after the cylinder pressure reaches the level of 0.52 p_inj, the start of injection is shifted earlier by the amount of this delay.

2.4. Computational Study Methodology

This study was carried out in three stages. In the first stage, refinements were introduced into the combustion mathematical model to account for the influence of injection parameters across the entire range of engine operating modes. The injection model was validated using experimental data from several engines.

In the second stage, the maximum injection pressure and injector area were selected according to the methodology described above, taking into account the analysis of literature data.

In the third stage, the engine cycle was simulated using the selected injection parameters. Two approaches were considered depending on the choice of the start of injection. According to the first approach, the SOI was selected to maximize mixture homogenization, coinciding with the intake valve closing timing. In the second approach, the SOI was chosen according to the methodology described above, ensuring a specified value of the relative ignition timing and, consequently, controlling mixture stratification.

Calculations were performed across the full range of engine loads and speeds at 348 operating points. A mixture control strategy was employed, in which the air–fuel ratio varied from 4 at idle to 1 at maximum load conditions. The maximum λ value was limited to 4, since further increases lead to a sharp rise in hydrogen combustion duration and incompleteness, unstable operation, and a reduction in engine efficiency [4,18].

The ignition timing is determined by the location of the combustion center, or the MFB50 parameter, which corresponds to the crank angle at which 50% of the injected fuel has burned. According to studies [8,11,17], the highest engine efficiency is achieved when this parameter is approximately 8° crank angle after top dead center (°CA ATDC). Accordingly, in this study, the ignition timing was set to ensure MFB50 = 8 °CA ATDC. If the cylinder pressure exceeds the maximum allowable value for the base engine of 7.5 MPa, the ignition timing is retarded up to 15 °CA ATDC.

Thus, the study proposes a comprehensive methodology, which includes a mathematical model of the hydrogen engine cycle, a method for selecting the injection pressure and injector area, a criterion for determining the start of injection, injection and air–fuel ratio strategies, and a set of constraints. This methodology enables the selection of injection parameters across a wide range of engine operating conditions and modes.

3. Results and Discussion

3.1. Refinements of the Mathematical Models

The coefficient k₁ in Formula (3) can be modified to account for the influence of injection parameters on hydrogen combustion. This coefficient was originally introduced to represent the effect of the air–fuel ratio on the combustion duration. An analysis of experimental data for several PFI engines in [21,27] showed that the influence of mixture composition on the combustion duration decreases with increasing engine speed. The value of this coefficient ranges from approximately 1.2–1.3 at 1000 rpm to about 0.8 at 6000 rpm. In this case, the dependence of k₁ on engine speed is described in [21] by the following expression (curve 1 in Figure 4):

$$k_1= 1.07\times {10}^{-8}{n}^2 - 1.751\times {10}^{-4}n +1.469.$$

(6)

The analysis of experimental data for several DI engines presented in [21] showed that, for early injection shortly after intake valve closing, the influence of λ on combustion duration is the same as, or only slightly smaller than, that for PFI engines. For example, based on the experimental data for a DI engine with early injection reported by Beyer et al. [8], the coefficient k₁ was determined to be 1.2 (point A in Figure 4), which is close to the value calculated using Formula (6) for PFI engines (curve 1 in Figure 4).

However, for late injection, the influence of the air–fuel ratio on combustion duration is significantly lower [8,21]. Figure 5 shows the combined influence of the start of injection and the air–fuel ratio on the combustion characteristics of the hydrogen engine in [8]. Here, the cases of early injection (Figure 5a) and late injection (Figure 5b) are considered. In the case of late injection, the end of injection coincides with the ignition timing, i.e., the degree of mixture stratification is maximal. It can be seen that, for early injection, when λ increases from 1 to 2.5, the combustion duration increases from 10 °CA (curve 1) to 38 °CA (curve 2), whereas for late injection this parameter increases only from about 7 °CA (curve 3) to 18 °CA (curve 4). Accordingly, for late injection, the coefficient k₁ in Formula (3), which accounts for the influence of λ, takes smaller values than in the case of early injection.

Based on the analysis of experimental data for the engine in [8], the coefficient k₁ for the limiting case of late injection was found to be 0.6 (point B in Figure 4). Thus, for the engine in [8], the coefficient k₁ ranges from 1.2 for early injection to 0.6 for late injection at an engine speed of 1500 rpm. The results of combustion characteristic modeling for this engine are shown in Figure 5a for early injection with k₁ = 1.2, and in Figure 5b for late injection with k₁ = 0.6. It can be seen that the appropriate choice of the coefficient k₁ for the cases of early and late injection provides a satisfactory agreement between experimental and calculated data.

Figure 6a,b present the results of combustion characteristic modeling for the engines from Tanno et al. [7] and Eichlseder et al. [28], respectively, under early injection immediately after intake valve closing (curves 2 and 4) and late injection with the end of injection occurring close to ignition timing (curves 1 and 3). The engines operated at 2000 rpm with λ = 2.5–2.6. It can be seen that the influence of the injection timing on the combustion characteristics under early and late injection is similar to that observed for the engine in [8] at the corresponding λ (see Figure 5a). In the modeling of combustion characteristics, a coefficient k₁ = 1.05–1.2 was applied for early injection, while for late injection, k₁ = 0.5–0.6 was used. It is evident that the selected values of coefficient k₁ for early and late injection are close to those obtained for the engine in [8] under comparable conditions. This indicates the possibility of generalizing the obtained data regarding the choice of coefficient k₁ to a wider range of hydrogen engines.

To account for the influence of mixture stratification on hydrogen combustion in the mathematical model, the following assumptions are made:

• The effect of the end of injection on the combustion kinetics of the base engine is the same as that of the engine in [8];
• The degree of mixture stratification is proportional to the time interval between the end of injection and the ignition timing;
• The effect of in-cylinder turbulence on the combustion of a stratified mixture with varying engine speed is the same as for a homogeneous mixture;
• Provided that [θ_ign − EOI] > 150 °CA, the end of injection does not affect the degree of mixture stratification.

The last assumption is indirectly supported by the data in [17], which show that varying the EOI from approximately −300 °CA ATDC to −150 °CA ATDC has virtually no effect on the performance of the hydrogen engine.

Considering the adopted assumptions, the formula for the coefficient k₁ can be written as:

$$\begin{array}{c}\mathrm{for} \left[{\theta }_{ign}-\mathrm{E}\mathrm{O}\mathrm{I}\right]\le 150°\mathrm{CA}: \hfill \\ k_1= \left(0.0033\left[{\theta }_{ign}-\mathrm{E}\mathrm{O}\mathrm{I}\right]+0.49\right)(1.07·{10}^{-8}{n}^2 - 1.751·{10}^{-4}n +1.469);\hfill \\ \mathrm{f}\mathrm{o}\mathrm{r} \left[{\theta }_{ign}-\mathrm{E}\mathrm{O}\mathrm{I}\right]>150°\mathrm{C}\mathrm{A}:\hfill \\ k_1= (0.985)(1.07·{10}^{-8}{n}^2 - 1.751·{10}^{-4}n + 1.469).\hfill \end{array}$$

(7)

Figure 4 shows examples of k₁ values calculated using Formula (7) for the cases of homogeneous and maximally stratified mixtures (curves 2 and 3, respectively) over a wide range of engine speeds. It can be seen that increasing engine speed and mixture stratification both lead to a decrease in the coefficient k₁, and consequently to a reduction in the influence of the air–fuel ratio on the burn rate and combustion duration.

Thus, the refinements introduced into the model make it possible to account for the combined effects of injection timing, injection duration, and ignition timing on the burn rate and combustion duration.

An important component of the engine cycle model is hydrogen injection submodel, since this process significantly affects mixture stratification in the cylinder and, consequently, combustion. Therefore, considerable attention in this research was devoted to selecting appropriate coefficients for the injection model.

In the injection simulation, a key parameter is the effective flow rate, which represents the ratio of the injected fuel mass to the injection duration. Under choked flow conditions, this parameter can be determined using the formula:

$$G_{eff}={\mathrm{C}}_d{G}_{ideal}={\mathrm{C}}_d{\displaystyle \frac{A{p}_{inj}}{\sqrt{{RT}_{inj}}}}\sqrt{\gamma {\left({\displaystyle \frac{2}{\gamma +1}}\right)}^{{\displaystyle \frac{\gamma +1}{\gamma -1}}}},$$

(8)

where C_d is the discharge coefficient, G_ideal—ideal choked mass flow rate, A is the area of the nozzle orifices, R is the specific gas constant for hydrogen, p_inj and T_inj are the injection pressure and temperature, respectively, and γ is the specific heat ratio.

The discharge coefficient accounts for hydrodynamic losses in the injector, as well as the dynamics of the needle lift and closure. This parameter represents the ratio of effective mass flow to ideal isentropic choked flow. Peters et al. [23] show that C_d for several injector designs under consideration ranges from 0.44 to 0.81. Using the experimental data from Hu et al. [10] and Gerke [18] as input for our injection simulations, we obtained C_d values of 0.81 and 0.61, respectively.

The calculated effective flow rate obtained using Formula (8), along with the processed experimental data for the engines reported in [9,10], is presented in Figure 7. For the calculations, the specific heat ratio of hydrogen was assumed to be 1.41, and the fuel temperature upstream of the injector was set to 293 K. The experimental values of the effective mass flow rate were derived from the reported BTE, BMEP, cylinder volume, engine speed, and injection duration in [9,10]. Since [9] does not provide injector orifice area data, an effective injector area of 0.31 mm² was assumed and kept constant for all operating conditions. It can be observed that the use of Formula (8) yields satisfactory agreement between the calculated and experimental data across different injection parameters, engine speeds, BMEP values, and air–fuel ratios.

The injection duration in the mathematical model is determined using the formula:

$$\mathrm{D}\mathrm{O}\mathrm{I}={\displaystyle \frac{b_c}{G_{eff}}}6n$$

(9)

where b_c is the hydrogen cyclic supply, n is the engine speed.

Thus, the refinements introduced into the mathematical model allow determination of injection parameters based on the injector design and injection system settings, as well as accounting for the influence of injection parameters on combustion kinetics.

3.2. Selection of Maximum Injection Pressure and Injector Area

Figure 8 shows the relationship between injector area and maximum injection pressure for DI engines, based on data from [7,10,12,15,17,18,25,29,30,31,32]. Note that in [12,15,17,32], only static mass flow rates and injection pressures were reported; the corresponding injector areas were calculated in the present study. Maximum injection pressures ranged from 2.5 MPa to 22 MPa, while injector areas varied from 0.28 mm² to 8.6 mm². As shown in Figure 8, a clear correlation can be observed between injection pressure and injector area: small injector areas correspond to high injection pressures, and vice versa.

In several studies [10,12,17,18,30], engines with cylinder displacements close to that of the base engine (0.3–0.5 L) were equipped with injectors having relatively small areas, ranging from 0.28 mm² to 1.48 mm². For example, in [12,17], a Bosch HDEV4 gasoline injector was used, which is estimated to have an area of approximately 0.46 mm². Accordingly, to provide an increased fuel flow rate, the maximum injection pressure was raised to 15–22 MPa. However, even at elevated injection pressures, to ensure engine operation under high-load and high-speed conditions, the start of injection was carried out with the intake valves open. In [18], where an injector with a 1.0 mm² area operating at 15 MPa was used, the engine ran at speeds ranging from approximately 800 rpm to 6000 rpm, although the IMEP did not exceed 1.5 MPa. In Wallner et al. [30], an injector with a 1.48 mm² area operating at 10 MPa was employed; however, the engine operated at relatively low speeds, up to 2000 rpm, and an IMEP of up to 0.6 MPa.

The literature review shows that, for direct injection after intake valve closing, operating the base engine across the full speed range from 1000 to 6000 rpm and BMEP from 0 to 1.94 MPa can only be achieved by increasing the maximum injection pressure and/or injector area compared to the engines considered in the aforementioned studies, which had similar cylinder volumes.

The choice of injection pressure is constrained by several technological and functional limitations. Injection systems with maximum pressures above 15 MPa are generally used in studies of late single- and double-injection strategies [7,10,12,17,18] or in racing engines, where vehicle range is not a critical factor (Tafel et al. [13]). In contrast, for the development of engines intended for series production, lower-pressure systems (2–6 MPa) are usually preferred, as they allow more efficient utilization of hydrogen storage and improve injection system reliability [15,16]. To achieve a balance between the injection window width, the feasibility of late injection, and the efficient utilization of gas storage capacity for the base engine, an injection pressure of 6 MPa was adopted (Point A in Figure 8). At a storage pressure of 70 MPa, the residual mass fraction of hydrogen in the tanks would be about 9%, which is considered acceptable.

Section 2.2 shows that the ratio between the injection pressure and the cylinder pressure during the compression stroke determines the time window over which the injection must occur. The maximum fuel flow rate is achieved at the rated power mode. For the base engine, this corresponds to n = 6000 rpm and BMEP = 1.42 MPa. The cylinder pressure during the compression stroke at this mode is shown in Figure 9. The start of injection is chosen to coincide with the intake valve closing timing (θ_IVC = −120 °CA). At the selected maximum injection pressure of 6 MPa, the cylinder pressure at the end of injection should not exceed 3.12 MPa, which corresponds to a crank angle of −20 °CA. Thus, with the chosen start of injection, the available injection window at this operating point is approximately 100 °CA.

Calculations for the rated power mode show that, at an injection pressure of 6 MPa, the specified injection duration of 100 °CA can be achieved with an injector area of 3.08 mm². In the calculations, the discharge coefficient C_d in Formula (8) was assumed to be 0.65.

After selecting the maximum injection pressure and injector flow area, it is necessary to verify the technological feasibility of the chosen injection parameters. For this purpose, the minimum injection duration under idle conditions must be determined. Figure 10 presents the injection duration at idle operation for two hydrogen engines with cylinder volumes similar to that of the investigated engine [18,33]. In the first case, fuel was injected through an injector with an effective flow area of approximately 3.5 mm² at a pressure of 0.3–0.4 MPa [33] (curve 1), while in the second case, the injection was carried out through an injector with a 1 mm² flow area at 15 MPa [18] (curve 2). It can be seen that the minimum injection duration falls within the range of 0.5 to 1.5 ms. For the investigated engine, this parameter is assumed to be 1 ms.

Calculations for the idle mode of the base engine showed that, with an injection pressure of 6 MPa and an injector area of 3.08 mm², the injection duration would be 0.14 ms, which is significantly lower than the adopted minimum value of 1.0 ms. Two possible solutions exist in this case: either using two or more injectors with a combined area of 3.08 mm², operating at the same or different constant pressures, or using a single injector with an area of 3.08 mm² and variable injection pressure.

Both of these solutions have been successfully applied in practice. For example, Tafel et al. [13] report that a racing car with a cylinder volume of 0.5 L used two PFI injectors operating at 1.5 MPa and one DI injector operating at 15 MPa. At low-load and idle conditions, injection is performed predominantly by the DI injector, which ensures precise fuel dosing at these operating points. As engine power and speed increase, the fuel flow through the two PFI injectors, operating in parallel with the DI injector, also increases. This configuration is characterized by a complex layout and reduced reliability of the injection system, as it requires sealing multiple high-pressure cavities and involves a relatively large number of components.

In our view, it is more appropriate to use an injection system with a single injector operating under variable pressure. In this case, the low-pressure circuit contains an electronically controlled gas-metering valve, which is integrated either into the pressure regulator [16,32] or into the fuel rail [15]. According to signals from the electronic control unit (ECU), this valve regulates the amount of fuel entering the fuel rail, thereby controlling the rail pressure. As noted in [15], when the engine load decreases from maximum to idle, the pressure in the rail drops from 3 MPa to 1 MPa.

Calculations for the base engine at 800 rpm indicate that, with the selected injector area, the injection pressure must be reduced to 0.85 MPa to achieve an injection duration of 1.0 ms. Assuming a minimum injection duration of 1.0 ms across the entire idle speed range, the injection pressure would need to be varied from 0.85 MPa at 800 rpm to 1.7 MPa at 6000 rpm. The increase in injection pressure with engine speed at idle is due to the need to raise the cyclic fuel delivery to compensate for the increasing mechanical losses in the engine.

When setting the injection pressure across the engine’s operating range, the following approach was adopted. To achieve the shortest possible injection duration and to maximize the time available for mixture formation, the injection pressure was maintained constant at 6 MPa for most operating conditions. At low-load operating points, the pressure was varied to ensure a prescribed injection duration of 1 ms.

The selected parameters of the hydrogen direct injection system for the base engine are summarized in

Table 2. Selected hydrogen injection system parameters for the base engine.

Parameter	Value
Max. quantity	22.5 mg/cycle
Min. quantity	1 mg/cycle
Min. injection duration	1.0 ms
Max. injection duration at 110 kW and 6000 rpm	3 ms
Max. injection pressure	6 MPa
Min. injection pressure	0.85 MPa
Injector area	3.08 mm2

.

3.3. Early and Late Injection Strategies

In this work, two injection strategies are considered. In the first strategy, the objective is to achieve maximum mixture homogenization. The start of injection is selected to correspond to the intake valve closing timing at all operating points. In the second strategy, a certain level of mixture stratification is intentionally created in the cylinder. The criterion for selecting the start of injection in this case is the relative ignition timing (see Section 2.3). This criterion is set to 120% and is adjusted in cases where the end of injection occurs at an in-cylinder pressure exceeding 0.52 p_inj.

Figure 11a presents the simulation results of injection parameters for early injection at the intake valve closing timing. The θ_IVC is set identical to that of the base gasoline engine. It can be observed that, under this strategy, the end of injection varies from −150 °CA ATDC to −20 °CA ATDC. At all operating points, the ratio of injection pressure to cylinder pressure satisfies the condition for choked fuel flow through the injector orifices. At low-load operating points, the injection duration is 1.0 ms. Therefore, the injection parameters meet the requirements across the full engine operating range.

Under this injection strategy, RIT varies between 134% and 2242%. As the load and engine speed decrease, this parameter increases, indicating that the degree of mixture homogenization differs significantly across operating points. On the other hand, the difference between the end of injection and the ignition timing ranges from 157 °CA to 34 °CA, allowing broad flexibility to adjust the start of injection and, consequently, control potential mixture stratification.

The simulation results of injection parameters for the stratified mixture strategy, based on a prescribed relative ignition timing, are shown in Figure 11b. It can be observed that, under this strategy, the injection duration and injection pressure practically do not differ from those in the early injection strategy. The start of injection varies widely from –123 °CA ATDC to −6 °CA ATDC. As engine speed and load decrease, the start of injection is set later, which corresponds to a longer available injection duration and a reduced fuel shot per cycle.

It can be seen from Figure 11b that achieving the target relative ignition timing value of 120% is possible only within a limited range of engine operating conditions. This is due to the selected injection parameters, which define a specific injection duration and the ratio between the injection pressure and the in-cylinder pressure at the end of injection. Setting the start of injection based on an RIT of 120% leads to the end of injection typically occurring after the in-cylinder pressure exceeds 0.52 p_inj. This necessitates an earlier start of injection and results in an increase in the RIT value up to 468%. Achieving the specified RIT = 120% across the entire operating range would require a significant increase in injection pressure; however, this would conflict with the goal of maximizing the utilization of the compressed gas storage volume. Nevertheless, it should be noted that, for most operating conditions, the RIT lies within the range of 120–200%, where the highest BTE and ITE values for different engines, as discussed in Section 2.3, are observed.

3.4. Engine Performance Under Early and Late Injection Strategies

Figure 12a shows the engine performance under the early injection strategy. It can be seen that the brake thermal efficiency reaches 38.8% in the low-speed and medium-load operating region, which is approximately 4% higher than the maximum BTE of the gasoline engine variant. The decrease in efficiency at BMEP above 1 MPa is caused by delayed ignition timing and suboptimal combustion phasing relative to the cylinder volume change.

The ignition timing at high-load operating points was set later to reduce the peak cylinder pressure and decrease the likelihood of knocking events. Even with ignition timing delayed up to 15 °CA ATDC, the peak cylinder pressure reaches nearly 11 MPa, which is significantly higher than the cylinder pressure of the gasoline engine at corresponding BMEP levels (up to 7.5 MPa).

The knock criterion at low engine speeds and high loads reaches a value of 1.5, indicating a high likelihood of knock and requiring additional measures to reduce it.

Over most operating points, the engine runs on a lean mixture, with enrichment applied only at high-load points to achieve the target power. The excess air at most operating points leads to lower exhaust gas temperatures upstream of the turbine compared to the gasoline engine. For BMEP up to 1 MPa, NO_x emissions generally do not exceed 50 ppm. However, at higher loads, in the range of air–fuel ratios from 1.1 to 1.3, NO_x emissions increase sharply to nearly 8400 ppm, necessitating measures for NO_x emission control.

Thus, when the direct early hydrogen injection strategy is implemented, the engine achieves a relatively high BTE compared to gasoline operation; however, this is accompanied by increased mechanical stress and a higher likelihood of knock.

Figure 12b shows the engine parameters when the late hydrogen injection strategy is implemented. It can be seen that mixture stratification is an effective means of increasing engine efficiency compared to homogeneous combustion. The maximum BTE increases to 40.9%, while the knock criterion does not exceed 0.86, and the peak in-cylinder pressure decreases by approximately 1 MPa. The main reason for the improved engine efficiency with mixture stratification is the intensification and shortening of the combustion at low and medium loads. The maximum NOx emissions at high-load operating points reach 6167 ppm, which is approximately 1.36 times lower than under the homogeneous mixture strategy. The effect of mixture stratification on the exhaust gas temperature upstream of the turbine is approximately the same.

It should be noted that the method used to calculate nitrogen oxide emissions considers the effect of mixture stratification only indirectly, via heat release dynamics and resulting cylinder temperature changes. The distribution of oxygen across cylinder zones is not accounted for, resulting in significant uncertainty in NO_x emission predictions for engine operation on lean mixtures with λ > 1.6 (see Figure 12b).

Thus, it can be concluded that the proposed method for determining the start of injection based on a specified relative ignition timing is an effective way to improve engine efficiency while reducing both the likelihood of knock and the mechanical stress on the engine.

4. Conclusions

In this work, the direct injection parameters for a hydrogen engine were justified for achieving both maximum mixture homogenization and mixture stratification. The main findings are as follows:

• The hydrogen combustion model has been refined to account for the effect of injection parameters on heat release kinetics. Empirical relationships were proposed to consider the influence of the interval between the end of injection and the ignition timing on the duration and shape of the combustion curve. The discharge coefficient values were calibrated for the effective flow rate in the injection model.
• Using the developed engine cycle model, the injection system parameters were determined to ensure the specified minimum injection duration at idle, the maximum fuel shot at rated power, as well as choked flow conditions across all operating points.
• A method is proposed for selecting the start of injection to achieve a specified degree of mixture stratification in the cylinder. According to this method, the start of injection is chosen to maintain a specified constant relative ignition timing, taking into account the limitations imposed by choked flow through the injector orifices.
• The engine operation was simulated using early and late injection strategies across the entire range of operating conditions. It is shown that, compared to early injection, implementing the late injection strategy allows an increase in the maximum brake thermal efficiency by approximately 2%, a reduction of the maximum cylinder pressure by about 1 MPa, a 1.4-fold decrease in peak nitrogen oxide emissions, and enables knock-free operation across all engine operating points.