double-Wiebe Function Research Articles

Construction and verification of dual-fuel engine combustion model

In order to achieve real-time mapping and online optimization of the combustion process of a dual-fuel engine that is in prolonged operation, this paper is the first to combine the Wiebe function with a deep learning neural network to propose a zero-dimensional (0-D) combustion prediction model for a biodiesel-diesel dual-fuel engine. First, the parameters of the double Wiebe functions are calculated by the Pelican Optimization Algorithm (POA), and the operating parameters and Wiebe parameters are used as input and output parameters of neural networks, respectively, to construct parameter identification models. Then, the combustion process is simplified and reconstructed by combining the Wiebe function with the deep learning neural network, and a 0-D prediction model based on the hybrid model-driven and data-driven method is established, which can further obtain combustion results such as cylinder pressure curve and indicated mean effective pressure (IMEP). The results show that the coefficient of determination (R2) value of the dual-fuel engine 0-D prediction model based on the double Wiebe function combined with POA–CNN–Bi-LSTM is 0.9827, and the model has good prediction accuracy and generalization. The development of the combustion model provides reliable numerical model support for the online evaluation and optimization of dual-fuel engine performance.

Estimation of Wiebe Function Parameters for Syngas and Anode Off-Gas Combustion in Spark-Ignition Engines

Abstract Wiebe functions, analytical equations that estimate the fuel mass fraction burned (MFB) during combustion, have been effective at describing spark-ignition (SI) engine combustion using gasoline fuels. This study explores if the same methodology can be extended for SI combustion with syngas, a gaseous fuel mixture composed of H2, CO, and CO2, and anode-off gas; the latter is an exhaust gas mixture emitted from the anode of a Solid Oxide Fuel Cell, containing H2, CO, H2O, and CO2. For this study, anode off-gas is treated as a syngas fuel diluted with CO2 and vaporized water. Combustion experiments were run on a single-cylinder, research engine using syngas and anode-off gas as fuels. One single Wiebe function and three double Wiebe functions were fitted and compared with the MFB profile calculated from the experimental data. It was determined that the SI combustion process of both the syngas and the anode-off gas could be estimated using a governing Wiebe function. While the detailed double Wiebe function had the highest accuracy, a reduced double Wiebe function is capable of achieving comparable accuracy. On the other hand, a single Wiebe function is not able to fully capture the combustion process of a SI engine using syngas and anode off-gas.

A Double-Wiebe Function for Reactivity Controlled Compression Ignition Combustion Using Reformate Diesel

AbstractReactivity controlled compression ignition (RCCI) combustion has previously been proposed as a method to achieve high fuel conversion efficiency and reduce engine emissions. A single-fuel RCCI combustion strategy can have decreased fuel system complexity by using a reformate fuel for port fuel injection and the parent fuel (diesel) for direct injection. This paper presents a one-dimensional computational model of a compression ignition engine with single-fuel RCCI. A Wiebe function is used to predict the combustion process by representing the mass fraction burned (MFB) on a crank angle resolved basis. One single-Wiebe function (SWF) and two double-Wiebe functions(DWFs) were fitted to experimentally derive MFB data using the least-square method. The fitted results were compared with MFBs calculated from experimental data to verify the accuracy. The SWF did not fully capture the MFB curve with high fidelity while the detailed DWF captured the MFB curve within a root mean square error of 1.4%. The reduced double-Wiebe function (RDWF) also resulted in a predicted combustion profile with similar accuracy. Hence, the RDWF was used in a GT-power thermodynamic study to understand the effects of the low-temperature heat release (LTHR) fraction and combustion phasing on combustion characteristics. At optimum phasing of 5–10 crank angle degree after the top dead center, increasing the LTHR fraction from 20% to 60% resulted in the fuel conversion efficiency increasing from 39.5% to 41.1%, thus suggesting that the reformate fuel-based RCCI strategy is viable to unlock improved combustion performance.

Theoretical Investigation of Combustion and Performance Analysis of Diesel Engine under Low Load Conditions

Diesel engine is using prominently in islands and remote areas due to its reliability and stability for power generation. In recent years, most of the isolated power systems (e.g., islands and remote areas) have integrated renewable energies to reduce both the cost and pollution in diesel power generating system. However, due to intermittent and stochastic behaviour of renewable sources (e.g., solar and wind), it is unable to eliminate diesel generation entirely. In that case, low-load diesel operation (operation < 30% of maximum rated load) is particularly relevant for its ability to support higher levels of renewable penetration. In this paper, a thermodynamic model was developed using MATLAB for diesel engine combustion and performance. This model includes sub models such as heat release rate, heat transfer, double-Wiebe function, and ignition delay correlation. Engine thermal efficiency (TE), brake power (BP), indicated mean effective pressure (IMEP) and brake specific fuel consumption (BSFC) has been taken into consideration for performance analysis. The simulation results show that at 25% load, in-cylinder pressure and temperature are 168 bar and 2300 K which are the cause of lower heat release rate (74 J/deg) and longer ignition delay (0.25 ∼0.5 ms higher than that of conventional mode) and significantly responsible for lower efficiency (18%), brake power (4kW) and higher brake specific fuel consumption (1.2 g/kWh).

Machine Learning Integration With Combustion Physics to Develop a Composite Predictive Model for Reactivity Controlled Compression Ignition Engine

Abstract Phasing of combustion metrics close to the optimum values across operation range is necessary to avail benefits of reactivity controlled compression ignition (RCCI) engines. Parameters like start of combustion occurrence crank angle (CA) (θsoc), occurrence of burn rate fraction reaching 50% (θ50), mean effective pressure from indicator diagram (IMEP), etc. are described as combustion metrics. These metrics act as markers for the macroscopic state of combustion. Control of these metrics in RCCI engine is relatively complex due to the nature of ignition. As direct combustion control is challenging, alternative methods like combustion physics-derived models are a subject of research interest. In this work, a composite predictive model was proposed by integrating trained random forest (RF) machine learning and artificial neural networks (ANNs) to combustion physics-derived modified Livengood–Wu integral, parametrized double-Wiebe function, autoignition front propagation speed-based correlations, and residual gas fraction model. The RF machine learning established a correlative relationship between physics-based model coefficients and engine operating condition. The ANN developed a similar correlation between residual gas fraction parameters and engine operating condition. The composite model was deployed for the predictions of θsoc, θ50, and IMEP as RCCI engine combustion metrics. Experimental validation showed an error standard deviation (σ68.3,err) of 0.67°CA, 1.19°CA, 0.223 bar and symmetric mean absolute percentage error of 6.92%, 7.87%, and 4.01% for the predictions of θsoc, θ50, and IMEP, respectively, on cycle to cycle basis. Wide range applicability, lesser experiments for model calibration, low computational costs, and utility for control applications were the benefits of the proposed predictive model.

Design, development and testing a hybrid control model for RCCI engine using double Wiebe function and random forest machine learning

Reactivity controlled compression ignition (RCCI) engine technology significantly reduces emissions of NOx and soot while improving thermal efficiency. RCCI engine development requires optimum in-cylinder combustion metrics as desired objective across different operating conditions. Physics-inspired control models are a cost-competitive option to achieve these objectives in engine control systems. Combining these control models with machine learning can offer computational efficiency, higher accuracy and lower number of experiments needed for control system calibration. In this context, the present paper proposed a novel hybrid control model by combining physics-inspired parametrized double Wiebe function (D-W) and Random Forest Machine Learning (RFML) to predict both the average and cyclic variation trends of combustion metrics in an RCCI engine. In this work, firstly, an optimized D-W parameters matrix was formulated that can stochastically reconstruct pressure cycles in agreement with 99.7% of the experimental confidence interval. RFML was then employed to develop correlative learning models between experimental control variables such as premix ratio, engine load, injection timing etc. with corresponding D-W parameters matrix. The hybrid control model was used to predict peak pressure (PP), indicated mean effective pressure (IMEP), crank angle for 50% of fuel mass fraction burnt (θ50) and maximum pressure rise rate (MPRR) across 60 test cases and 36000 cycles. Results showed an error mean and standard deviation (μerr±σerr) of 0.068 ± 1.840 bar, 0.007 ± 0.264 bar, 0.15 ± 0.98° CA and 0.018 ± 0.221 bar/° CA between predictions and experiment for PP, IMEP, θ50 and MPRR, respectively in the tenable RCCI operation regime. Proposed hybrid control model exhibited good accuracy and was able to capture the local cyclic variation trends.

Stochastic Cycle to Cycle Prediction in a Reactivity Controlled Compression Ignition Engine Using Double Wiebe Function

&lt;div class="section abstract"&gt;&lt;div class="htmlview paragraph"&gt;The present paper proposed a cycle to cycle prediction of in-cylinder pressure profile using double Wiebe function in a reactivity controlled compression ignition (RCCI) engine. RCCI engines lack direct control over combustion by means of any explicit hardware such as fuel injection or spark timing. Therefore, cylinder pressure sensor based control or model driven control is necessary for RCCI engines. In this work, an iterative algorithm to generate double Wiebe function parameters, were designed to model cycle average measured cylinder pressure. The model and measured data were in good agreement. However, when, this model was compared with measured cycles, the error and regression exhibited a near normal distribution. The quality of error and regression was found to deteriorate with premix energy share (ES) in RCCI mode due to higher cycle to cycle variations. To address this challenge, a range of double Wiebe parameters were generated to mimic the inner and outer most pressure cycle at three times the standard deviation from the cycle average cylinder pressure (99.7% confidence interval). This parameter range was randomized using Gaussian distribution and model cycles were generated stochastically. Leveraging this modelling approach for realistic sensing of selected combustion control parameters, indicated robustness and reliability. The stochastic approach was found to be an innovative strategy for RCCI engine model driven control.&lt;/div&gt;&lt;/div&gt;

Characterizing Two-Stage Combustion Process in a Natural Gas Spark Ignition Engine Based on Multi-Wiebe Function Model

Abstract The Wiebe function is a simple and cost-effective analytical approach to approximate the burn rates in internal combustion (IC) engines. Previous studies indicated that a double-Wiebe function model can better describe the two-stage combustion process inside diesel engines retrofitted to natural gas (NG) spark ignition (SI) compared with a single-Wiebe function. Specifically, the two Wiebe functions are associated with the bowl burn and the squish burn. However, the long tail in the energy release at the end of combustion produces some differences between experiment and model, which can be attributed to the complexity of the late oxidation process inside the post-flame zone and the incomplete combustion of the unburned mixture flowing out from engine crevices. To improve the matching between the model and experimental data, this paper investigated the effect of adding a third Wiebe function just to describe the long tail in the energy release at the end of combustion. The results indicated that such a methodology greatly improved the fitting accuracy in terms of phasing and magnitude of the heat release rate in each combustion stage.

Partially premixed combustion optimization using double injection strategy in transient operation

Partially Premixed Combustion (PPC) has been proved a high efficiency combustion concept, along with ultra-low soot and NOx emissions. However, the nature of high pressure rise rate prevents PPC from operating range extension and further practical use. This paper aims to optimize mixture preparation to achieve the benefits of PPC with the constraints of engine states through injection strategy during transient operation. A control-oriented model (COM) is developed based on double-Wiebe function to predict combustion process, where a new linear algorithm is proposed to identify the model parameters. The root mean square error (RMSE) of the predicted cylinder pressure is less than 2.58 bar and peak error is less than 5% against experimental measurements of steady states. A constrained model predictive controller (MPC) is designed and implemented in a PPC engine. Simulation and experiment results show that the proposed controller manipulates injection events to optimize premixing period and fuel distribution towards more benefits of PPC concept. In the testing scenario, soot, NOx and pressure rise rate are regulated within 0.15 mg/m3, 400 ppm and 8 bar/deg, respectively. Consequently, cumulative soot and NOx emissions are reduced by 43.2% and 6.8% in the whole transient cycle.

Improving combustion chamber heat transfer correlation of an HCCI engine in order to achieve a more accurate thermodynamic model

An appropriate heat transfer rate prediction for an internal combustion engine is important from many perspectives. This study investigates the appropriate coefficients of heat transfer correlation and provides a new method for calculating the convective heat transfer coefficient of combustion chamber walls of an HCCI engine. Therefore, a thermodynamic code including the combustion and heat transfer processes, which are coupled with an optimization algorithm, is used to estimate P-θ behavior of an HCCI engine. Double-Wiebe function and Woschni correlations are applied for combustion and heat transfer respectively in a single-zone model. The results show that the use of heat transfer correlation without improvement in coefficients and exponents for different engines, leads to significant errors in the prediction of in-cylinder pressure. Comparison of computational results with the experimental data shows that original Woschni correlation in the single-zone model cannot predict in-cylinder pressure appropriately. Modifications made in this paper resulted in a change of more than 40% in some coefficients of the existing heat transfer correlation, to provide accurate prediction of the engine thermodynamic behavior of an HCCI engine.

Single and double Wiebe function combustion model for a heavy-duty diesel engine retrofitted to natural-gas spark-ignition

The conversion of existing diesel engines to natural-gas spark-ignition operation would reduce the dependence on oil imports, reduce the burden on refining capacity, and increase U.S. energy security. The use of simple combustion models such as the Wiebe function to analyze or predict the combustion process in such retrofitted engines can accelerate and optimize the engine conversion. This study compared the standard single Wiebe function to a double-Wiebe function to investigate if the latter will improve the predicted mass fraction burned and, if yes, which of formats of the duel-Wiebe function in the literature described the best the mass fraction burned in such converted engine. The results showed that while the standard Wiebe function could not predict the mass fraction, a double-Wiebe function (one for the fast burn inside the piston bowl and a second for the slower burning process inside the squish region) predicted with good accuracy the mass fraction burned, heat release rate, in-cylinder pressure, and combustion phasing in such converted engines (∼90% reduction in error) for the conditions investigated here, especially if the start of the second Wiebe was delayed. Moreover, such a condition-dependent model as the double-Wiebe model is limited to the operating conditions used for determining the model parameters. However, the model can be developed into a condition-independent model if enough experimental data or CFD simulations are available to find the unique set (or sets) of parameters that minimizes the error at most conditions.

Improving the Thermal Efficiency of the Homogeneous Charge Compression Ignition Engine by Using Various Combustion Patterns

The efficiency of an internal combustion engine (ICE) is essential for automobiles and motorcycles. Several studies have demonstrated that the homogeneous charge compression ignition (HCCI) is a promising technology for realizing engines with high efficiency and low emissions. This study investigated the combustion characteristics of the HCCI using a 125 cc motorcycle engine with n-heptane fuel. The engine performance, combustion characteristics, and thermal efficiency were analyzed from experimental data. The results revealed that a leaner air–fuel mixture led to higher engine efficiency and output. The improvement of engine output is contradictory to the general trend. Energy balance analysis revealed that lower heat loss, due to the low cylinder gas temperature of lean combustion, contributed to higher efficiency. A double-Wiebe function provided excellent simulation of the mass fraction burned (MFB) of the HCCI. Air cycle simulation with the MFB, provided by the double-Wiebe function, was executed to investigate this phenomenon. The results indicated that a better combustion pattern led to higher thermal efficiency, and thus the engine output and thermal efficiency do not require a fast combustion rate in an HCCI engine. A better combustion pattern can be achieved by adjusting air–fuel ratio (AFR) and the rates of dual fuel and exhaust gas recirculation (EGR).

Predictions of the heat release rate model of hydraulic-free piston engines

Hydraulic-free piston engine are associated with the challenges of large cyclic variations and poor running stability due to the piston free states. It is therefore necessary to investigate the cyclic stability of this engines. For this purpose, the heat release rate was fitted with the double Wiebe function. The error transfer calculation is carried out for the Wiebe characteristic parameters. Then, some cylinder pressures and scavenging duration affecting the engine cycle are selected as detection parameters after cyclic volatility analysis. Finally, the relationship between the characteristic parameters and the detection parameters was then subjected to training on the basis of the training samples that were used to predict the heat release rate by the method of support vector machine. This method will help establish the interrelationship between the parameters of the cycle and establish a foundation for investigating the cyclic stability of HFPE and formulating relevant instability control strategies.

Combustion Model for a Homogeneous Turbocharged Gasoline Direct-Injection Engine

Homogeneous charge is a preferred operation mode of gasoline direct-injection (GDI) engines. However, a limited amount of work exists in the literature for combustion models of this mode of engine operation. Current work describes a model developed to study combustion in a homogeneous charge GDI engine. The model was validated using experimental data from a 1.6 L Ford EcoBoost® engine, tested at the U.S. EPA. The combustion heat release was approximated using a double-Wiebe function, to account for the rapid initial premixed combustion followed by a gradual diffusion-like state of combustion, as observed in this GDI engine. Variables of Wiebe correlations were adjusted into a semipredictive combustion model. The effectiveness of semipredictive combustion model was tested in prediction of in-cylinder pressures. The root-mean-square (RMS) errors between experiments and numerical results were within 2.5% of in-cylinder peak pressures during combustion. The semipredictive combustion model was further studied to develop a predictive combustion model. The performance of predictive combustion model was examined by regenerating the experimental cumulative heat release. The heat release analysis developed for the GDI engine was further applied to a dual mode, turbulent jet ignition (DM-TJI) engine. DM-TJI is a distributed combustion technology with the potential to provide diesel-like efficiencies and minimal engine-out emissions for spark-ignition engines. The DM-TJI engine was observed to offer a faster burn rate and lower in-cylinder heat transfer compared to the GDI engine.

Optimization of the working cycle of a transport diesel engine by simulating the combustion process with a double Wiebe function

For modeling the working cycle of transport diesels, models with two laws of combustion according to the Wiebe, mathematically related to each other and corresponding to the initial and main periods of the process, are increasingly being used. However, known mathematical models give different results in estimating the distribution of fuel shares between these periods. These models also do not allow to simulate a smooth start of the combustion process. Therefore, a new mathematical model was developed for calculating the working cycle of a diesel engine with two independent laws of combustion of fuel, described by the functions of Wiebe. Communication between them is carried out through a new parameter, which is the proportion of fuel, burning in the initial period, from the total cyclic feed. The novelty of the model is also the ability to set, within a wide range, independently of each other, the angles of the start and duration of the first and second combustion laws for the initial and base periods, respectively. A program for calculating the operating cycle with simulation of the combustion process using a new mathematical model is developed. Calculations have been made for a transport diesel engine equipped with a fuel-supply system of a battery type. The results of the calculation showed that an increase in the fraction of the fuel burned according to the first law from 0.1 to ~ 0.3 with other parameters being constant leads to a significant increase in the maximum pressure in the diesel cylinder and some improvement in the indicator indexes. With a further increase in the fuel fraction to 0.5, the maximum gas pressure continues to increase, and the indicator indexes slightly deteriorate. At the same time, in the initial period, the value of the index of the combustion character has practically no effect on the indicator index of the diesel engine and the maximum pressure in the cylinder. In the main period, the value of the indicator of the nature of combustion, on the contrary, significantly affects the indicator indexes of the diesel engine and the maximum pressure in the cylinder. The calculation determines the optimal heat release law in the diesel cylinder of ChN15/16 type, which provides the best indicator values when the maximum pressure in the cylinder is limited.

Determining Parameters of Double-Wiebe Function for Simulation of Combustion Process in Overload Diesel Engine with Common Rail Fuel Feed System

To analyze the peculiarities of the combustion process in an overload diesel engine with the system of Common Rail type with one-stage injection, the indicator diagram was registered. The parameters of the combustion process simulated by the double-Wiebe function were calculated as satisfactorily reconstructing the law of burning rate variation. The main parameters of the operating cycle obtained through the indicator diagram processing and the double-Wiebe function calculation differed insignificantly. And the calculated curve of the cylinder pressure differed notably only in the end of the expansion stroke. To improve the performance of the diesel engine, a two-stage fuel injection was recommended.

Heat release rate and performance simulation of DME fuelled diesel engine using oxygenate correction factor and load correction factor in double Wiebe function

A computer simulation scheme with a rapid thermodynamic model is developed to predict the diesel engine HRR (heat release rate) and performance characteristics with DME as a fuel. The bmeps (MPa) are simulated as 0.5 (optimum power output) and 0.3, 0.4, 0.6 (other power outputs) with diesel and with DME in four steps sequentially. Initially, HRR is predicted by modifying four double Wiebe function parameters; the equations of heat release and combustion duration of diffusion phase and the efficiency factors of premixed and diffusion phase as 2.25 and 3.25 respectively (obtained by qualitative and approximate fitting by trial and error method). Obviously 99% combustion efficiency assumption with efficiency factor as 6.9 is excluded. Secondly, all Wiebe parameters are computed by using LCF (load correction factor which is the ratio between mass flow rates of diesel of optimum and other power output) alone and by using LCF and OCF (oxygenate correction factor which is the ratio between mass flow rates of DME and diesel of identical power output) in other steps. Premixed and diffusion phase peaks (J/degree) with DME are 13.73 and 31.47 (0.3 MPa); 23.18 and 47.25 (0.6 MPa). Performance predicted with 1.2% accuracy is validated by SAE and USA Army literature.

Predictive model of the diesel engine operating in dual-fuel mode fuelled with different gaseous fuels

In this work, a predictive model has been developed in order to investigate the dual-fuel diesel engine operating on different gaseous fuels. A thermodynamic single zone modeling of the engine in dual-fuel (DF) mode was carried out to understand the characteristics of combustion and to predict the cylinder pressure. Tow primary fuels were tested, natural gas (NG) and biogas. The biogas fuel is composed of 60% methane (CH4) and 40% carbon dioxide (CO2). They are introduced into the intake manifold in order to be mixed with the air. Diesel fuel is the pilot fuel. A single zone model was coupled with a double–Wiebe function and was adopted to simulate the heat release rate. The double-Wiebe function includes six parameters, determined by using the least squares method, based on the experimental values obtained on a single cylinder test bench engine. Also, this predictive model is based on sub-models such as the ignition delay, the heat losses through walls and the flow rate through the intake and exhaust valves. The simulated results were compared with experimental results under different engine loads. The present model reproduces the cylinder pressure with high precision. Moreover, the calculation of other engine parameters such as indicated thermal efficiency and indicated power shows a satisfying and good agreement between both experimental and simulated results with a relative error lower than 5%.

Automatic calibration algorithm of 0-D combustion model applied to DICI diesel engine

To automatically acquire high accuracy 0-D combustion model based on Wiebe function, in this work, an automatic calibration algorithm is developed to determine and then calibrate standard and double Wiebe function, which combines advantages of the algebra and Levenberg-Marquardt (LM) algorithm and avoids the two methods’ disadvantages. Up to two combustion phases models have been analyzed. An innovative method is proposed to confirm the transition angle between premixed and diffusive combustion phases. The algebra method is used to provide the initial value of Wiebe parameters, then the LM algorithm is applied to obtain the further refined Wiebe parameters. The automatic calibration algorithm is applied to a turbocharging medium-speed diesel engine for a wide range of engine speeds and loads. The effectiveness of this algorithm is quantified by the accuracy for describing the burnt fraction (xb). The fitting results indicated that the algebra method is an effective measure to provide the initial values of Wiebe parameters. All the fitting results are convergent and the iterations of LM algorithm are all less than 8. The R2 of xb are all at a high level (larger than 0.995), and RMSE are all at a low level (smaller than 0.015). The proposed automatic calibration algorithm presents high reliability and precision, moreover, its computation time is less than 70 ms. Therefore it is a promising algorithm to realize online fitting, diagnosing and controlling combustion process.

Development and validation of a marine sequential turbocharging diesel engine combustion model based on double Wiebe function and partial least squares method

Developing an accurate combustion model for a Sequential Turbocharging diesel engine is an efficient way to reduce the time consumption and the expensive experiment costs in the process of determining the control strategy of a Sequential Turbocharging system. In this paper, 142 operations of a marine diesel engine with two Sequential Turbocharging systems are used for this research. A novel method is proposed to calculate the initial values and limit bounds of the Wiebe parameters, and the use of this method could effectively avoid the occurrence of impossible solutions or multiple solutions caused by only using nonlinear least square fitting. The calculated Wiebe parameters could achieve a high level of accuracy that the mean value of the determination coefficient between the simulated and experimental heat release rate profiles is 0.98 for all the operations, and they are accurate enough for developing the combustion model. The partial least square regression is first imposed to relate the Wiebe parameters to the operating conditions. For the ability to solve the multiple correlations among the operating conditions, the prediction performance of the combustion model developed by partial least square regression is much higher than that by least square regression. The mean absolute percentage errors of the brake specific fuel consumption and maximum in-cylinder pressure simulated by the model are 0.284% and 1.39%, respectively. The model could accurately predict the engine performance and is sufficient to make the reasonable control strategies for the different Sequential Turbocharging systems.