Regular Article
Realtime capable virtual NO_{x} sensor for diesel engines based on a twoZone thermodynamic model
Faculty of Mechanical Engineering, University of Ljubljana,
Ljubljana  Slovenia
^{*} Corresponding author: rok.vihar@fs.unilj.si
Received:
25
August
2017
Accepted:
26
February
2018
This paper presents a controloriented thermodynamic model capable of predicting nitrogen oxides (NO_{x}) emissions in diesel engines. It is derived from zerodimensional combustion model using incylinder pressure as the input. The methodology is based on a twozone thermodynamic model which divides the combustion chamber into a burned and unburned gas zone. The original contribution of proposed method arises from: (1) application of a detailed twozone modeling framework, developed in a way that the thermodynamic equations could be solved in a closed form without iterative procedure, which provides the basis for achieving high level of predictiveness, on the level of realtime capable models and (2) introduction of relative airfuel ratio during combustion as a main and physically motivated calibration parameter of the NO_{x} model. The model was calibrated and validated using data sets recorded in two different direct injection diesel engines, i.e. a light and a heavyduty engine. The model is suitable for realtime applications since it takes less than a cycle to complete the entire closed cycle thermodynamic calculation including NO_{x} prediction, which opens the possibility of integration in the engine control unit for closedloop or feedforward control.
© R. Vihar et al., published by IFP Energies nouvelles, 2018
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
The increasing share of diesel engine powered automotive fleet in recent years, being largely due to high thermal efficiency of diesel engines, has become an environmental issue since it represents a major source of NO_{x} emissions [1]. One of the ways to comply with forthcoming increasingly stringent diesel emission limits for NO_{x}, which will comprise also real driving emission, is adoption of engine control strategies on the realtime (RT) basis. Accurate estimation of incylinder NO_{x} formation on the cycletocycle basis can namely be used as an input parameter for active control methods (like Closed Loop Combustion Control) and also for the control of exhaust aftertreatment systems.
Some modern diesel engines are equipped with incylinder pressure transducers, which can be used as a source of combustion related data [2, 3]. Currently incylinder pressure data is mainly used for obtaining information related to aging of engine components as well as for adaptation of the injection strategy with respect to the variation in the fuel quality [4]. Information on the incylinder pressure opens many other possibilities for more precise control of the engine especially in transient operating conditions, as it can be used for calculation of the thermodynamic parameters inside the cylinder being an enabler for the virtual NO_{x} sensor with higher level of predictiveness.
In general, methods for simulations of pollutants in the RT are usually divided into semiempirical and phenomenological approaches [5]. In the semiempirical models where NO_{x} formation process is based on a singleequation correlation which takes into account measured engine variables, the accuracy largely depends on the tuning of the model. Therefore their prediction quality outside the calibration area is very poor [6]. Hence, phenomenological models, e.g. [5, 7], can be considered as a preferred option in terms of general applicability of the virtual NO_{x} sensors. Thus the advanced virtual NO_{x} sensor presented in this paper is also based on the phenomenological modeling approach.
Main thermodynamic parameters that are required for prediction of the NO_{x} formation are rate of heat release (ROHR) and consequently the temperatures and concentrations in the combustion chamber or selected zones of the combustion chamber. These properties can be assessed with thermodynamic models whose accuracies depend on the physical depth, which is directly reflected in the computational complexity of the models [8]. Since the whole control loop of the engine (signal processing, thermodynamic model and potential control of actuators) should be in the automotive application carried out by the engine control unit (ECU), design of a computationally efficient thermodynamic model is of utmost importance.
Combustion of nonhomogeneous mixture of fuel, air and recirculated exhaust gases is characterized by a high heterogeneity of the temperature and concentration field, which have high impact on NO_{x} generation. The most accurate methodology to simulate such phenomena is based on the use of 3D CFD models [9]. Due to their high spatial resolution these models feature a high computational complexity and thus long computing times. On the system level, 0D combustion models, e.g. [10, 11], are often used as models that effectively combine the requirements on multizone spatial resolution and much shorter computational times compared to the 3D CFD models. However, multi zone 0D models featuring many zones to resolve spray propagation are still too complex and thus computationally demanding for RT applications [12]. On the other hand, in the simplest models the entire combustion area is treated as a single zone (singlezone models) [13] and the temperature could be treated as a global variable that is uniform throughout the combustion chamber. As formation of thermal NO_{x} is very sensitive on temperature, the global incylinder temperature provided by the single zone models is not a suitable input for virtual NO_{x} sensors [14].
Twozone models can, on the system level, efficiently comply with the requirements on heterogeneity of the temperature and concentration field as well as short computational times that are in the range of single zone models. Twozone models thus separate the combustion chamber into burned and unburned zone while considering zone specific temperatures and concentrations as well as homogeneity of the pressure field [15, 16]. Threezone models are further variants of twozone models, where by adding complexity, computational complexity is also increased [17]. The model proposed in this paper was developed with the objective on very low computational demand and thus the twozone combustion modeling approach offered a proper basis for achieving good tradeoff between accuracy and computational complexity at low computational demand.
The twozone models mostly differ in terms of complexity of the gas property treatment. Besides the inputs of gas properties this also influences the complexity of governing equations through the inclusion of the partial derivatives of internal energy and gas constant. Some of the RT twozone models are using constant gas properties in the calculation of the thermodynamic parameters [18, 19]. This modeling depth might not provide sufficiently profound basis for achieving high accuracy of the results as will be shown in the Result section of this paper. Therefore, variable gas properties need to be considered to comply with the requirement on sufficient accuracy of the results. This calls for an efficient approach in terms of complexity of the gas property database and of the underlying governing equations, where in addition the computationally efficient treatment of gas property database is preferred. The modeling approach proposed in this paper thus considers dependency of the gas properties on temperature and relative airfuel ratio. In addition, the gas properties are not given as polynomial functions, as for example in [17], which are computationally demanding and might not allow for achieving high accuracy especially if lower order polynomials are applied. In the proposed model, main thermodynamic properties, their derivatives and species equilibrium fractions are therefore stored in the equidistant lookup tables being a computation efficient approach since commercial engine ECUs mostly use the lookup tables for modeling operating point dependent parameters [20]. Moreover, the thermodynamic modeling framework was developed in a way that the thermodynamic equations could be solved in a closed form, without requiring an iterative procedure. This means that governing equations are explicit in mass of burned fuel, while considering twozone combustion model and variable gas properties as well as using an incylinder pressure as an input. This thermodynamic modeling framework is crucial to effectively combine computational efficiency and sufficient modeling depth to achieve high level of predictiveness and it is more detailed compared to other models that were applied for realtime virtual NO_{x} sensors [18, 21].
The twozone models also differ in the approaches for determining the evolution of relative airfuel ratio during combustion (RAFC), which influences the mass, species and enthalpy transport from the unburned to the burned zone. The first group of the models is based on the assumption that at the end of combustion process the entire unburned zone is consumed and thus transferred to the burned zone [22], while the second group of models is based on the assumption that fuel and air are burning at stoichiometric conditions [8, 15, 23]. Both assumptions are not necessarily relevant for the entire operating range of the diesel engine. This can, besides omitting contribution of the prompt NO mechanism whose relative contribution is more pronounced at low engine loads [6, 21, 24, 25], be related to the inability of predicting NO_{x} emissions using extended Zeldovich mechanism with sufficient accuracy without using operating point specific NO_{x} multipliers [8]. As these NO_{x} multipliers to a large extend correct also for the temperature trace in the burned zone − being significantly influenced by the previously mentioned approaches to determine the RAFC − this paper introduces a physically motivated degree of freedom in the calibration of combustion models being the RAFC. It is also important to mention that in some studies other thermodynamic calibration parameters, e.g. heat transfer multipliers, were used to calibrate the NO_{x} mass [21, 24]. Heat transfer multiplier inherently influences the burned zone temperature and thus also NO_{x} mass, however it also has a direct impact on the energy balance and thus a significant impact on other thermodynamic results as MFB. This might lead to inconsistencies, if heat transfer multiplier that best suits for calibration of the NO_{x} mass violates the energy balance. Therefore, the proposed modeling framework is derived in a way that NO_{x} calibration does not violate the energy balance. This is made possible by the application of the RAFC as the main calibration parameter as it influences the transport from the unburned to the burned zone and thus the burned zone temperature and NO_{x} formation. It also has a nearly negligible influence on the overall internal energy of the burned and the unburned zone and thus on the energy balance. Later is true for lean mixtures as overall internal energy of the burned and the unburned zone is only very insignificantly influenced by the composition [26]. RAFC is thus the only independent calibration parameter of the NO_{x} submodel which has minor impact on energy balance. In addition, unlike the approach proposed in [22], the approach proposed in this paper does not require that entire unburned zone is converted to the burned zone at the end of combustion. This is very important to obtain plausible results of NO_{x} especially at low loads, where indeed only a part of the incylinder charged is mixed with the fuel and thus being subjected to combustion.
The proposed twozone thermodynamic modeling framework thus considers an additional thermodynamic NO_{x} calibration parameter, which differs from the two currently used approaches for determining the RAFC that either rely on constant RAFC or lead to full conversion of the unburned zone. This allows applying extended Zeldovich mechanism with fixed model parameters over the entire engine operating range. Moreover, it was shown that it is also possible to obtain plausible NO_{x} results using only a single RAFC input for the entire engine operating regime thus omitting any operating point specific tuning of the airfuel ratio evolution during combustion.
The entire modeling framework, i.e. thermodynamic and NO_{x} model, was validated in multiple operating points of two significantly different diesel engines demonstrating its general applicability. In addition, it was validated using pressure signals from a laboratory pressure sensor and a large series lowcost glow plug pressure sensor. The model was run both in offline analyses on PC and in online analyses utilizing field programmable gate array (FPGA). The proposed modeling framework can thus be considered as an approach offering low parametrization effort and optimized tradeoff between accuracy and calculation time thus opening ways towards its use in the next generation large series commercial hardware.
2 Experimental setup
The experiments were carried out on a two different diesel engines covering light duty (LD) and heavy duty (HD) segment. Both engines were coupled with a Zöllner B350AC eddycurrent dynamometer controlled by Kristel, Seibt & Co control system KS ADAC. A Kistler CAM UNIT Type 2613B shaft encoder provided an external trigger and an external clock (0.1–6 deg CA) for data acquisition system. Incylinder pressure was measured with two piezoelectric pressure transducers: calibrated AVL GH12D transducer in combination with charge amplifier AVL MICROIFEM and preseries Hidria glow plug pressure transducer (PSG) with integrated charge amplifier [27], both connected to 16 bit, 4 channel National Instruments dataacquisition system with maximum sampling frequency 1 MS/s/ch. Top dead center (TDC) was determined by capacitive sensor COM Type 2653. An AVL 730 gravimetric balance fuel meter was employed to measure fuel consumption. Intake air flow was measured with Meriam laminar flow meter 50MC26F. Emissions of NO_{x} were measured with Horiba OBS2200 with the use of CLD analyzer. Sampling attachments were welded on both engines in the exhaust system after the turbine and before aftertreatment devices. In the development phase of the model the laboratory equipment was used for evolution of the proposed modeling approach, while validation was done only with the use of pressure signal from PSG in combination with data provided by the ECU which demonstrates the applicability of the proposed methods for RT use.
NO_{x} emissions were measured continuously while incylinder pressure was recorded over 100 successive cycles at sampling resolution of 0.2 CA for the HD and 0.1 CA for the LD engine. For offline application, a representative pressure trace for the model development was generated by averaging 100 cycles of the individual operating point measured with AVL pressure transducer in order to partially suppress measurement noise from the measured pressure trace. Furthermore, additional filtering of the pressure trace was applied, as suppression of oscillations of the pressure signal is very important in this particular study due to the high impact of pressure oscillations on the burn rate, which strongly influences temperature in the burned zone and consequently NO_{x} concentration. Therefore, pressure traces were further processed by applying low pass FIR filter [28], whereby filter parameters (e.g. filter order, cutoff frequency) were carefully selected in a way to suppress unwanted contributions of vibrational eigenmodes in the combustion chamber and of the measurement noise while preserving contributions of piston kinematics and combustion [29, 30]. For validation of online application, 1 cycle was measured with PSG pressure transducer to validate applicability of the method in most severe realtime applications. In online application the same filter settings as in offline application were applied.
Besides different engine application areas, i.e. light and heavy duty, analyzed engines feature also different emission standards and different fuel injection systems, which confirms validity of the proposed incylinder NO_{x} formation modeling approach. The LD engine was a 1.6 L, 4cylinder, PSA turbocharged diesel engine equipped with a common rail fuel system in combination with solenoid injectors with injection pressure up to 1600 bar. The main characteristics of the LD engine are given in Table 1. The HD engine was a 6cylinder, 4 stroke, turbocharged, 6.87 L MAN diesel engine. The main characteristics of the second engine are given in Table 2.
The proposed modeling approach was implemented in the LabVIEW environment. The model was designed in a modular way and allows for variations in modeling complexity to support analyses of the tradeoff between computational speed and accuracy of the model. Offline validation of the proposed modeling approach was performed on a standard PC, whereas online validation was performed on a Xilinx Kintex7 325T FPGA integrated on a NI cRIO Controller. The matrix of operating points that was used for model validation is shown in Tables 3 and 4 for LD and HD engine, respectively.
LD engine specifications.
HD engine specifications.
Operating points for LD engine.
Operating points for HD engine.
3 Description of the modeling framework
Until the start of combustion (SOC), the incylinder gas mixture is modeled as a single zone model described in Ref. [31] and mean incylinder temperature is determined considering ideal and nonperfect gas. After the SOC, combustion chamber is virtually divided into two zones assuming homogeneous incylinder pressure distribution. SOC was identified by comparing the motored and the measured incylinder pressure after the start of injection (SOI) [6]. The crank angle resolution is 1 deg CA with the exception of first 30 deg CA after the SOC, where it is set to 0.1 deg CA resolution in order to prevent potential instability of the model in the initial phase of the twozone calculation.
3.1 Twozone combustion modeling framework
The twozone modeling framework is derived based on the mass and energy balances applied to each of the zones. The burned zone is denoted with subscript B and the unburned zone with subscript U. In general, nzone modeling frameworks are derived in a way to predict pressure derivative based on the known ROHR, e.g. [32, 33], whereas in the RT modeling ROHR is usually predicted based on the know pressure trace [8, 17, 23]. To avoid iterations, nzone modeling framework, as for example given in [32, 33], was rearranged in a way to be an explicit function of derivative of the mass of burned fuel while considering only two zones. Governing equations are derived for a high pressure phase only while considering mass and enthalpy variation due to the injected fuel and enthalpy variation due to the heat transfer. In the analyzed case, the blowby was not considered, whereas modeling framework can easily be extended to consider this effect. Conservation of mass in the particular zone is expressed as : (1) (2) where the equation for mass transport from the unburned to the burned zone is derived on the basis of equation reported in Ref. [16], considering change of fuel mass (dm_{fuel}) from Equation (8) and relative airfuel ratio of the unburned zone (λ_{U}) from the previous calculation step as: (3)
The conservation of total incylinder mass is expressed as: (4)
Conservation of energy in the particular zone is expressed as: (5) (6) where dH_{B}, dH_{U} represent change of enthalpy and dV_{B}, dV_{U} volume change for burned and unburned zone. Equations were derived under the assumption that combustion only takes place in the burned zone yielding the following expression for the ROHR: (7)where H_{LHV} represents lower heating value of the fuel. After rearranging the equations for mass (1–4) and energy (5–6) the equation for the derivative of the mass of burned fuel reads:
and equations for temperature derivative of the burned and unburned zone read (9) and (10)
m_{UBZ} represents the sum of the mass of the trapped air, recirculated exhaust gases and residual gases and thus determines the oxygen concentration of the unburned zone. From these three contributions it is most demanding to determine the amount of residual gases. The values of the residual gases were determined in the offline analyses and stored in the lookup tables for online applications, which resembles calibration process of control functionalities. Both tested engines were running without the exhaust gas recirculation (EGR) and thus the EGR rate was set to zero. Otherwise this parameter is available in the ECU, while, if required, modeling framework proposed in this section can also be used to determine the EGR rate. As the evaporation of fuel, which occurs only in the unburned zone (Eq. 8), has minor impact on the burned zone temperature, it was assumed that there is no delay between injection rate (dm_{inj}/dφ) and the rate of fuel burned. (EGR)
3.2 Heat transfer submodel
The total heat transfer between the incylinder charge and the combustion chamber walls is estimated by means of convective formulation [34]: (11) where A_{p}, A_{l} and A_{h} are surface areas (indexes represent piston, liner and cylinder head respectively), T_{p}, T_{l} and T_{h} are temperatures and α_{HT} is heat transfer coefficient. The mean temperature in the cylinder (T_{cyl}) is for the purpose of heat transfer submodel obtained from the ideal gas law. In the analyzed case, the Woschni correlation [35, 36] was applied to determine the heat transfer coefficient: (12)where D represents cylinder bore, p is incylinder pressure, V represents actual cylinder volume, v_{u} is circumferential velocity and v_{m} is mean piston speed. T_{IVC}, p_{IVC} and V_{IVC} are temperature, pressure and volume in the cylinder at intake valve closing (IVC).
3.3 Thermodynamic data
Thermodynamic properties such as R_{B}, R_{U}, u_{B}, u_{U} and their partial derivatives ∂R_{B}/∂ T_{B}, ∂R_{U}/∂ T_{U}, ∂R_{B}/∂ λ_{B}, ∂R_{U}/∂ λ_{U}, ∂u_{B}/∂ T_{B}, ∂u_{U}/∂ T_{U}, ∂u_{B}/∂ λ_{B}, ∂u_{U}/∂ λ_{U} were precalculated and stored in the lookup tables as a function of temperature (300–3500 K) and relative airfuel ratio (1–1000). The specific enthalpies of both zones must also be evaluated in order to solve the Equation (8) and are expressed as: (13)
3.4 Determination of RAFC
The evolution of the RAFC gives the amount of air that is consumed while burning a unit of fuel. It thus determines the mass, species and enthalpy transport from the unburned to the burned zone. Therefore, it significantly influences temperature and composition of the burned zone and thus formation of NO_{x}. In contrast to other published approaches, where it is considered that combustion takes place at stoichiometric conditions or that the airfuel ratio is determined in a way that the whole unburned zone is consumed during combustion [22], in the proposed method the RAFC is a physically motivated calibration parameter. As the amount of the burned fuel is not known during the combustion, the variation of the RAFC is not given as function of the relative amount of the burned fuel. Instead, it is given as a function of the crank angle degree to ensure general applicability and robustness of the model.
At the SOC point (Fig. 1), the RAFC is initialized and it is afterwards kept constant until the end of injection (EOI). In the analyzed cases, RAFC was initialized with the values between 1.0 and 1.1. In contrast to some approaches where airfuel ratio in the flame front is assumed to be <1 [37], in this model RAFC was kept above 1.0 during the initial combustion phase. This is aligned with the main focus of the model being NO_{x} formation, as at least some surplus of oxygen in the burned zone is required to allow for NO_{x} formation during the initial and thus also premixed combustion phase, which is indeed characterized by NO_{x} formation [38]. After EOI, RAFC is linearly increasing thereby leaning burned gases. This trend is motivated by the fact that during that period spray detaches from the nozzle, which provokes more pronounced mixing with air as fuel occupies larger volume. The RAFC variation as presented in Figure 1 is given by Equation (14), (14) where k_{lam} and n_{lam} are constants of the linear function and λ_{com_init} is initial value of RAFC.
Fig. 1
Variation of representative RAFC function. 
3.5 NO_{x} submodel
The calculation of species equilibrium fractions is usually performed by the timeconsuming minimization of Gibbs free energy [39]. For minimization of computation effort of complex equilibrium calculations in the present work all species equilibrium fractions were given in precalculated lookup tables determined on the basis of JANAF data tables [26]. The equilibrium calculation is performed for a set of a six relevant species (N, N_{2}, O, O_{2}, NO, OH) as a function of temperature (1000–3500 K) and RAFC (1–4), which covers the relevant range for the burned zone. The formation of NO_{x} was modeled according to the extended Zeldovich mechanism described by the following six reactions [40]: (15)
Even though the NO_{2} formation in compression ignition engines cannot be neglected [41], in the described model approximation on NO_{x} formation was made only on calculation of NO formation. Similar as in, e.g. [8, 42], NO formation and NO_{x} emissions are assumed to be correlated and one constant calibrating factor was applied on the final cycle NO_{x} mass to match the experimental tailpipe NO_{x} emissions. The rate of formation of NO in the burned zone is therefore derived by the following equation [11, 16, 40]: (16) where α_{NO} is the ratio of actual NO concentration and its equilibrium concentration calculated according to Equation (17): (17)
_{n} is engine speed and C_{NO,multi} is multiplicative parameter that also takes into account difference between predicted NO formation and actual NO_{x} formation. Furthermore, c_{e} represents equilibrium species concentration read from the lookup table. K_{2} and K_{4} are help variables calculated according to Equation (18): (18)
Variables denoted with k are determined by the following equations [16]: (19)
The adjustment of the model to the measurements was done with the scaling parameter C_{NO,multi}, which was calibrated in the model calibration phase and was then kept constant for the entire operating range of an engine.
4 Results
4.1 Impact of code execution source
The applied modeling framework of NO_{x} model was developed and calibrated in Lab View environment on a PC, while execution of the code was performed on the FPGA integrated circuit. Due to the different architecture of the chips, the code implemented on the FPGA integrated circuit was restructured in a way to avoid all nonlinear dependencies. Furthermore, on the FPGA integrated circuit it is necessary to specify word length and integer bites for every mathematical operator in order to improve computational times and reduce memory consumption. Therefore, it is important to verify consistency of the results calculated on a PC and on a FPGA integrated circuit. This was done by using the same pressure trace with all the related thermodynamic data as an input for models on both hardware platforms and by comparing the result of calculated temperature in the burned zone and the total NO_{x} mass. As discernable from the results shown in Figure 2, it can be concluded that the difference in the result is within the line thickness and the error between the execution sources is less than 1% regarding total NO_{x} mass. These results thus confirm consistency of the models on both hardware platforms.
Fig. 2
Comparison of NO_{x} mass calculated on a PC and on a FPGA in operating point at 3000 min^{−1} and 160 Nm. 
4.2 Impact of the gas property treatment
Before analyzing NO_{x} emissions, it is meaningful to assess plausibility of the thermodynamic modeling framework and gas property treatment presented in section 3. As some of the RT twozone NO_{x} models are using constant gas properties in the calculation of the thermodynamic parameters [18, 19], in this analysis importance to consider variable gas properties to establish a sound basis for modeling NO_{x} emissions is demonstrated. To demonstrate this, two different cases were analyzed (Fig. 3): 1. the modeling depth as considered in Equations (8)–(10), i.e. variable gas properties of R, u and their derivatives (denoted with: T_base) and 2. a simplified approach with constant values of R, u (denoted with T_u const_R const). The impact of gas property treatment is demonstrated on the case of temperature in the burned zone (Fig. 3) being one of the most influential parameters of NO_{x} kinetics. It can be seen in Figure 3 that the simplified gas property treatment results in unrealistically high temperatures. This can mainly be attributed to neglecting the dissociation. Consequently, simplified gas property treatment yields too low values of specific heats at high temperatures, whereas neglecting associated partial derivatives additionally increases the error. These results indicate that variable gas properties need to be considered to establish a sound basis for modeling NO_{x} emissions.
Fig. 3
Temperature trace in the burned zone in operating point at 3000 min^{−1} and 160 Nm for different modeling depths. 
4.3 Thermodynamic model results
In this section, the traces of thermodynamic parameters and NO_{x} mass are shown in Figures 4 and 5 for the two operating points of the LD engine at 3000 min^{−1} and at 20 Nm and 160 Nm to provide an insight into underlying phenomena and to form the basis for further analyses where only data at exhaust valve opening (EVO) will be shown. Figure 4 shows measured incylinder pressure trace being an input to the model (Fig. 4a)), ROHR (Fig. 4b) calculated by Equations (7) and (8), masses of the burned and unburned zone (Fig. 4c) calculated by Equations (1)–(3), the RAFC (Fig. 4d) given by Equation (14), temperatures of the burned zone, unburned zone and mean incylinder temperature (Fig. 4e) calculated by Equations (9) and (10) as well as actual NO_{x} mass calculated by Equation (16) compared to the equilibrium NO mass at instant temperature and composition in burned zone (Fig. 4f) at 3000 min^{−1} and 20 Nm.
Figure 4b clearly indicates combustion of fuel injected in the pilot and the main injection, which is a direct consequence of the shape of the pressure trace in Figure 4a. ROHR (Fig. 4b) is also reflected in mass trace of the unburned to burned zone (Fig. 4c), where there is first a mass transfer due to combustion of a pilot injection followed by a short period of no mass transfer and then a more pronounced mass transfer due to combustion of the fuel injected during the main injection. As the RAFC (Fig. 4d) increases after the EOI, the rate of mass transfer from the unburned to the burned zone per mass of burned fuel increases towards the EOC. It can be seen in Figure 4c that a large amount of mass is not transported from the unburned to the burned zone despite increasing RAFC during the end of combustion. This can be attributed to high relative airfuel ratio in the combustion chamber due to small mass of injected fuel during this low load operating point. Therefore, only a part of the incylinder charge is mixed with the fuel and thus being subjected to combustion at low loads. ROHR (Fig. 4b) trace is also reflected in the traces of mean incylinder temperature (Fig. 4e), whereas it also dictates the shape of burned zone temperature and NO_{x} traces (Fig. 4f).
Within the proposed modeling framework, equilibrium NO_{x} mass and NO_{x} concentration in the burned zone feature increased values during the combustion of the fuel injected during pilot injection. Afterwards, there is a slight drop in both traces due to no heat addition by combustion and presence of heat extraction through the heat transfer to the combustion chamber walls. During the combustion of the fuel injected during main injection there is first a slight increase in equilibrium concentration, which is followed by its drop as the temperature of the burned zone is decreasing as the piston moves towards the BDC. Unlike equilibrium concertation in the burned zone, equilibrium NO_{x} mass is increasing until approx. 17 deg. CA, as mass and volume of the burned zone are still increasing. As NO kinetics according to Zeldovich mechanism is slow, actual NO_{x} values significantly deviates from the equilibrium ones. It can thus be seen that within the proposed modeling framework actual NO_{x} concentration in the burned zone peaks in the decreasing phase of the equilibrium concentration as at that instant there are favorable conditions for NO_{x} formation. During the combustion of the fuel injected in the main injection actual NO_{x} concentration in the burned zone is thus decreasing, however actual NO_{x} mass still increases, which is again related to increasing mass of the burned zone. Although, NO_{x} mass can in the present study be experimentally validated only as engineout emissions, i.e. at the EVC, the predicted trend qualitatively coincides well with the published results of incylinder NO_{x} traces, e.g. [38, 41], which also indicate nonnegligible NO_{x} formation during the premixed burning period. This confirms meaningfulness of keeping RAFC above value 1.0 during the initial combustion phase, which ensures surplus of oxygen in the burned zone being required for NO_{x} formation.
In Figure 5, the same thermodynamic properties are shown for operating point with higher engine load. Consequently, this operating point is characterized by higher incylinder pressure (Fig. 5a) and higher ROHR values (Fig. 5b) corresponding to higher injected mass of fuel. Therefore, higher portion of mass of unburned zone is transferred to burned zone. In comparison with Figure 4c where less than 50% mass of unburned zone was used during combustion, at this higher load around 80% of mass of unburned zone took part in the combustion process within the proposed modeling framework. This is due to low relative airfuel ratio in the combustion chamber and also due to more intense combustion in the later combustion phases, which are characterized by larger values of the RAFC (Fig. 5d). Relatively long period of high burned zone temperature (Fig. 5e) is the main reason for approximately 8 times higher NO_{x} mass (Fig. 5f) in this high load operating point compared to the low load point analyzed in Figure 4.
Fig. 4
a) Experimental p, b) calculated dQ, c) mass of the zones, d) RAFC, e) temperature of the zones and f) NO_{x} mass and concentration in burned zone for operating point at 3000 min^{−1} and 20 Nm. 
Fig. 5
a) Experimental p, b) calculated dQ, c) mass of the zones, d) RAFC, e) temperature of the zones and f) NO mass and concentration in burned zone for operating point at 3000 min^{−1} and 160 Nm. 
4.4 Model validation
As presented in the Introduction, RAFC represents the main calibration parameter of the virtual NO_{x} sensor. To analyze the influence of this calibration parameter two approaches were analyzed: 1. RAFC function (Eq. (13) featuring a single set of parameters was used for the entire engine operating range denoted as “const_PAR”, and 2. parameters of the RAFC function (Eq. (13) are operating point dependent denoted as “var_PAR”.
4.4.1 LD engine
In the model calibration phase, sensitivity analysis on the parameters of the RAFC function Equation (13) was performed. Based on this knowledge, parameters of the RAFC function were determined manually as this study is aimed to demonstrate capabilities of the modeling framework and thus automated model parametrization was not the scope of the research.
For the case with constant RAFC parameters, initial value of RAFC was thus set to 1.01 and constant of linear function k_{lam} was set to 0.02 (Tab. 5). Good agreement of the NO_{x} model results with the measurement even without calibration (Tab. 5) was demonstrated in the low and middle load operating points (e.g. 20 Nm and 100 Nm) at all engine speeds with the most pronounced deviation in the operating point 1200_20. In this operating point, constant RAFC parameters lead to higher burned zone temperature compared to the case with variable RAFC (Tab. 6) thus yielding too high NO_{x} mass. At high engine load (e.g. 160 Nm) deviation between the simulated and measured values is slightly higher. Compared to the case with variable RAFC (Tab. 6) this can be attributed to the more intense transport of mass from unburned zone to burned zone in the late combustion phase as a result of a relatively high k_{lam} parameter.
This case represents a very limiting case, where entire engine operating range is covered by a single set of model parameters of the emission formation model within a mechanistically based modeling framework, which is to the best of author's knowledge unique in this modeling depth. By additionally considering the fact that extremely complex physicochemical phenomena in diesel engines are treated with a very simplistic approach, it can be concluded that the modeling framework provides satisfactory results of the NO_{x} emissions.
As most of the studies related to virtual NO_{x} sensors use operating point specific calibration parameters [17, 43], an operating point specific input of the RAFC function was also applied in this study. As presented in the Introduction it is expected that RAFC, as a physically motivated calibration parameter, allows for high NO_{x} emission prediction capability while using only RAFC as a calibration parameter. In Table 6, the results for the LD engine are shown for this case. It can be seen from Table 6 that only relatively small spread of parameters of the RAFC function was required for the parameter λ_{com_init} and larger spread is characteristic for the parameter k_{lam}. At certain engine speed the highest value of k_{lam} was set at lower load and was decreased with increasing load. That is in agreement with the change of the relative airfuel ratio in the unburned zone and confirms plausibility of the applied approach. Additional graphical interpretation of the NO_{x} results presented in Tables 5 and 6 is given in Appendix A.
LD engine results with constant RAFC parameters (const_PAR).
LD engine results with variable RAFC parameters (var_PAR).
4.4.2 HD engine
The main difference in the hardware configuration between the LD and the HD engine is in fuel injection systems influencing the preparation of the air and fuel mixture. Fuel injection pressure is lower in the HD engine and HD engine does not feature a pilot injection. In addition, HD engine generally operates with leaner mixture than the LD engine. These differences have impact on the concentration and the temperature at the SOC and also later during combustion phase. This leads to different engine specific set of calibration parameters used for the case with constant RAFC parameters. As can be seen from Table 7, for the case with constant RAFC parameters, parameters of the RAFC function feature larger values for the HD engine compared to the LD one (Tab. 5). Initial value of RAFC was set to 1.1 and constant of linear function k_{lam} was set to 0.04.
On the HD engine, differences between predicted and measured values of mass of the NO_{x} in the case with constant parameters are even smaller (Tab. 7) than for the LD engine. One of possible explanations for this trend is related to the absence of the pilot injection in the HD engine. Current approach, due to the sake of generality and simplicity, assumes equal RAFC values for all injections, which is in general not the case in real engines. As dwell time between injections varies this simplification impacts accuracy of the NO_{x} prediction for the engines with multiple injections as in the analyzed case presented for the LD engine.
At lower engine speed, the matching of the predicted values with measurements is very good at low to middle load where even without additional tuning results were within the desired accuracy (Tab. 7). At high engine load were predicted values of the NO_{x} mass quite higher than the measured in the case with constant parameters which is similar pattern that was already observed at LD engine at middle and high engine speed where the RAFC at late combustion period were decreased to match the measured values. In this case, where the RAFC is already higher due to initialization value it should be further increased in order to increase mass transport from unburned zone to burned zone and consequently decrease temperature in burned zone. This difference between the engine specific tuning procedures is the result of much higher relative airfuel ratio in combustion chamber in HD engine. At higher engine speed a constant shift in the predicted values compared to measured data can be observed which can be attributed to selected initial value of RAFC and was effectively solved by increasing initial RAFC value as can be seen from Table 8. Additional graphical interpretation of the NO_{x} results presented in Tables 7 and 8 is given in Appendix B.
HD engine results with constant RAFC parameters.
HD engine results with variable RAFC parameters.
4.5 Computational times and memory consumption
The aim of this work was to develop a model of such a format and with sufficiently low demands of computational power that it could be calculated on an embedded system and is potentially compliant with next generation large series commercial hardware. The use of lookup tables was crucial to obtain a fast and computationally efficient algorithm since the FPGA architecture does not allow calculation of nonlinear dependencies. Despite that, commercial engine ECUs are also using lookup tables to model nonlinear and operating point dependent behaviors [20].
Precision for every mathematical operating on the FPGA was manually determined in order to improve computational times and reduce memory consumption. Total flash memory used for implementation of the virtual NO_{x} sensor on FPGA was 2.34 MBs (14.6% of total flash memory available on FPGA), which proves that the modeling framework could be implemented on a commercial ECU of which the most powerful version has 8 MB of flash memory [44].
Calculation time for one cycle was determined by measuring time for calculation of 100 consecutive cycles. The calculation time of one engine operating cycle is 21.8 ms, which means that the FPGA could perform the complete thermodynamic and NO_{x} calculation of one engine cycle up to engine speed at 5500 min^{−1}, which is higher than the maximum speed of most of the modern automotive diesel engines. Such computational time thus ensures ample margin to adapt the control parameters in the subsequent cycle. Since the commercial ECUs consist of a microcontroller with integrated flash memory and Application Specific Integrated Circuit (ASIC) chip architecture [45] and it is generally excepted that the FPGA is not as fast as ASIC [23], it is therefore concluded that the calculation could also be executed on an commercial ECU with similar or even shorter computational times.
5 Conclusion
A realtime capable virtual NO_{x} sensor for diesel engines was presented in the paper. The main characteristics of the modeling framework can be summarized as: (1) the thermodynamic modeling framework was developed in a way that the thermodynamic equations could be solved in a closed form while applying a detailed twozone modeling framework and variable gas properties, (2) the NO_{x} submodel relies only on one innovative physically motivated calibration parameter being relative airfuel ratio during combustion (RAFC). Both features are crucial to effectively combine computational efficiency and sufficient modeling depth to achieve high level of predictiveness. Besides computational efficiency and high level of predictiveness, an important advantage of the proposed modeling approach is also relatively low calibration effort of the RAFC function. It was shown in the paper that the model features good prediction capability if only unique RAFC input is used for the entire engine operating regime. It was also shown that prediction capability can be further improved with additional operating point specific calibration of the RAFC input. This parameter can be later stored in the lookup table and programed on the ECU. It was shown that demands on computational resources are within the specifications of the current ECUs. High prediction capability of the virtual NO_{x} sensor was demonstrated on two significantly different engines featuring also two different fuel injection systems, which proves general applicability of the method.
Abbreviations
CFD: computational fluid dynamics
CLD: chemiluminescence detector
CNO, multi: NO scaling parameter
EGR: exhaust gas recirculation
FPGA: field programmable gate array
H_{LHV}: lower heating value [J/kg]
PSG: glow plug pressure transducer
RAFC: relative airfuel ratio during combustion
c: species concentration [mol/cm^{3}]
c_{e}: equilibrium species concentration [mol/cm^{3}]
k_{lam}: constant of the linear function of RAFC
n_{lam}: constant of the linear function of RAFC
v_{m}: mean piston speed [m/s]
v_{u}: circumferential velocity [m/s]
α_{HT}: heat transfer coefficient
α_{NO}: ratio of actual NO concentration and its equilibrium concentration
Acknowledgments
The research is cofunded by the Republic of Slovenia and the European Union from the European Regional Development Fund within the framework of the EVA4GREEN program. The authors also acknowledge the financial support from the Slovenian Research Agency (research core funding No. P20401 and young researcher program with evidence No. 35395).
Appendix A
Fig. 6
Calculated vs. experimental NO_{x} mass at 1200 min^{−1} for LD engine. 
Fig. 7
Calculated vs. experimental NO_{x} mass at 2000 min^{−1} for LD engine. 
Fig. 8
Calculated vs. experimental NO_{x} mass at 2000 min^{−1} for LD engine. 
Appendix B
Fig. 9
Calculated vs. experimental NO_{x} mass at 1500 min^{−1} for HD engine. 
Fig. 10
Calculated vs. experimental NO_{x} mass at 2400 min^{−1} for HD engine. 
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All Tables
All Figures
Fig. 1
Variation of representative RAFC function. 

In the text 
Fig. 2
Comparison of NO_{x} mass calculated on a PC and on a FPGA in operating point at 3000 min^{−1} and 160 Nm. 

In the text 
Fig. 3
Temperature trace in the burned zone in operating point at 3000 min^{−1} and 160 Nm for different modeling depths. 

In the text 
Fig. 4
a) Experimental p, b) calculated dQ, c) mass of the zones, d) RAFC, e) temperature of the zones and f) NO_{x} mass and concentration in burned zone for operating point at 3000 min^{−1} and 20 Nm. 

In the text 
Fig. 5
a) Experimental p, b) calculated dQ, c) mass of the zones, d) RAFC, e) temperature of the zones and f) NO mass and concentration in burned zone for operating point at 3000 min^{−1} and 160 Nm. 

In the text 
Fig. 6
Calculated vs. experimental NO_{x} mass at 1200 min^{−1} for LD engine. 

In the text 
Fig. 7
Calculated vs. experimental NO_{x} mass at 2000 min^{−1} for LD engine. 

In the text 
Fig. 8
Calculated vs. experimental NO_{x} mass at 2000 min^{−1} for LD engine. 

In the text 
Fig. 9
Calculated vs. experimental NO_{x} mass at 1500 min^{−1} for HD engine. 

In the text 
Fig. 10
Calculated vs. experimental NO_{x} mass at 2400 min^{−1} for HD engine. 

In the text 