Computational Optimization of Split Injections and EGR in ... - CiteSeerX

2006-01-0059

Computational Optimization of Split Injections and EGR in a Diesel Engine Using an Adaptive Gradient-Based Algorithm Seshasai Srinivasan and Franz X. Tanner Michigan Technological University Jan Macek and Milos Polacek Czech Technical University

c Copyright 2006 Society of Automotive Engineers, Inc.

ABSTRACT The objective of this study is the development of a computationally efficient CFD-based tool for finding optimal engine operating conditions with respect to fuel consumption and emissions. The optimization algorithm employed is based on the steepest descent method where an adaptive cost function is minimized along each line search using an effective backtracking strategy. The adaptive cost function is based on the penalty method, where the penalty coefficient is increased after every line search. The parameter space is normalized and, thus, the optimization occurs over the unit cube in higherdimensional space. The application of this optimization tool is demonstrated for the Sulzer S20, a central-injection, non-road DI diesel engine. The optimization parameters are the start of injection of the two pulses, the duration of each pulse, the duration of the dwell, the exhaust gas recirculation rate and the boost pressure. A zero-dimensional engine code is used to simulate the exhaust and intake strokes to predict the conditions at the closure of the inlet valves. These data are then used as initial values for the three-dimensional CFD simulation which, in turn, computes the the emissions and specific fuel consumption. Simulations were performed for two different cost functions with different emphasis on the fuel consumption. The best case showed that the nitric oxide and the particulates could be reduced by over 83% and almost 24%, respectively, below the EPA mandates while maintaining a reasonable value of specific fuel consumption. Moreover, the path taken by the algorithm from the starting point to the optimum has been investigated to understand the influence of each parameter on the process of optimization.

INTRODUCTION Over the years, engine research has focused on improving the engine performance by optimizing various design and operating parameters. For diesel engines, the common rail injection

system in conjunction with other injection strategies, such as the split injection, (e.g. [1–7]), have contributed to the reduction of fuel consumption as well as emissions. These injection techniques in combination with exhaust gas recirculation (EGR) (e.g. [8–10]) and water injection strategies (e.g. [11–13]) have been investigated in many experimental and computational studies. The above cited studies have identified a variety of engine parameters which influence the formation of pollutants such as soot, (i.e. particulates PM), and nitric oxide (NOx). Unfortunately, different parameters influence the behavior of these pollutants differently. Typically, a reduction in nitric oxide is associated with an increase in the soot formation, referred to as the soot-NOx trade-off, which usually occurs at the expense of the fuel consumption. In view of this complex dependence of the engine input and output data, and due to the fact that changes in the experimental setup can be very costly, computational techniques seem to be a natural choice for finding optimal engine operating conditions. The search for optimal engine operating conditions leads to an optimization problem, where an appropriate cost function is minimized over a high-dimensional parameter space which reflects the engine’s input data. In order to obtain accurate emission values, the computations are generally performed by means of a CFD code which simulates the engine’s compression and combustion phase. This simulation is computationally very expensive and is the main contribution to the overall computational costs of the optimization process. A well-suited optimization method for input-output systems with unknown dynamics, is based on the genetic algorithm (GA). GAs are modeled on the principle of natural selection, where an optimal state is determined over many generations of successful outcomes, subject to possible mutations (cf. [14]). If iterated over enough generations, GAs are likely to produce global optima. However, pure GA methods require thousands of cost function evaluations and are therefore too expensive for CFD-based engine optimization. A variation of the GA, called the micro-genetic-algorithm

(µGA), has successfully been developed and applied in engine optimizations by Reitz and co-workers (cf. [15–21]). The main feature of the µGA is the efficient selection process used in the determination of the next generation, which allows a drastic reduction in the population size. Typically, the population can be reduced from at least thirty, as required by the standard GA, down to as low as five. Consequently, the number of costly engine simulations is greatly reduced, which makes the µGA applicable to engine optimization using present-day computers. In the aforementioned computations, NOx and soot emissions in conjunction with fuel consumption have been optimized with respect to a variety of engine parameters. These parameters include various patterns of pulsed injections, start of injection, combustion chamber geometry and others. The cost function used in the µGA is derived from the widely used penalty method in constrained optimization problems. In this approach, the quantities which need to be optimized, i.e. emissions and fuel consumption, are combined in one expression which then is minimized. A different approach to the evaluation of the performance criteria is taken by the multi-objective GAs. In these algorithms, the optimal performance is determined by extreme points of an associated set, called the Pareto set, and the selection of the new generations involves a special type of fitness or cost function. Multi-objective µGAs have been used by de Risi et al. [22] in the optimization of engine combustion chamber geometries using a three-dimensional CFD code. Also, a multi-objective µGA utilizing a phenomenological engine model has been developed and implemented in the study of Hiroyasu et al. [23]. The big disadvantage of GA-based optimization methods is their enormous computational costs, especially in the context of CFD engine simulations. More efficient approaches are gradient-based minimization methods. In gradient-based methods, an optimum is obtained by minimizing the cost function along a sequence of search directions. These methods approach their target monotonically and, therefore, are computationally much more efficient than GA-based strategies. The main drawback of gradient methods is the fact that they are less likely to reach a global minimum in the presence of local minima. In such situations, the obtained minimum is determined by the starting point. The efficiency of gradient-based optimization often outweighs their drawback of only finding a local minimum. In an experimental study of Lee and Reitz [24], a response surface method has been successfully utilized to find optimal engine operating conditions. In response surface methods, the gradient is determined from a plane which is fitted through neighboring points of a pivot using a least squares approach. The determination of the gradient by this method is necessary because the engine optimization parameters are subject to experimental fluctuations, which can adversely influence the cost function evaluation. In previous studies by these authors (cf. [25, 26]), a conjugate gradient method in conjunction with a backtracking algorithm had been introduced and tested for an experimental, non-road Sulzer S20 DI diesel engine utilizing a KIVA-3-based

CFD code [27]. The first study [25] explored the effect of using different cost function weights, when the conventional injection parameters were optimized with respect to fuel consumption and emissions. The second paper [26] was a preliminary investigation of a two pulse split injection strategy and its effect on emissions reduction. Both studies showed that the conjugate gradient method is computationally extremely efficient and that considerable improvements in terms of emissions can be achieved. However, this approach suffered from the disadvantage that the monotonic cost function descent could lead to very low emissions but an unacceptably high fuel consumption. In the present study, an adaptive steepest descent method in conjunction with a modified backtracking strategy in the associated line search is used. The standard backtracking algorithm is modified to use a dynamic first step, depending upon the steepness of the search direction. The idea is to reduce the number of steps taken along the line search. The entire optimization has been performed on a normalized unit cube in n-dimensional space, i.e., the range of each optimization parameter is mapped onto the interval [0, 1]. In this study, a Sulzer S20 stationary diesel engine has been optimized for two different cost function formulations. The three-dimensional engine simulations used to predict the emissions and the SFC have been performed from the closure of the inlet valves to the opening of the exhaust valves with a KIVA-3-based code. A zero-dimensional engine code has been used to simulate the exhaust and inlet strokes to predict the equilibrium conditions at inlet valve closure for the CFD code. The parameters that are optimized are the start of injection of the first pulse, injection duration of the first pulse, duration of dwell, injection duration of the second pulse, EGR rate and the boost pressure. The emission mandates used are the ones prescribed by the United States Environmental Protection Agency (EPA) for stationary engines [28].

OPTIMIZATION METHOD The optimization approach taken in this study is based on the steepest descent method which utilizes an adaptive cost function in conjunction with a backtracking strategy for the line search. The backtracking algorithm utilizes quadratic and cubic polynomials to accelerate the convergence, and the initial backtracking step employs an adaptive step size mechanism which depends on the steepness of the search direction. Following a precise problem statement, these algorithms and their implementations are then described in more detail. Problem Formulation In this study we consider the constrained optimization problem where the specific fuel consumption is minimized subject to prescribed emission levels. More formally, this is expressed as minimizing the objective function, i.e., the specific fuel consumption, s : X → R subject to the constraints g(x) = 0. Here, X ⊂ Rm represents the admissible set of engine input parameters and g : X → R2 denotes the emissions which in this case are the nitric oxide and the particulate mass.

Table 1 Adaptive steepest descent algorithm. xo = initial guess compute ρ0 for k=0, 1, 2 ... pk = −∇ f (xk , ρk ) xk+1 = minx∈X f (x, ρk ) in direction pk compute ρk+1 ρk+1 = max{ρk , ρk+1 } end

This problem can be reformulated as an unconstrained optimization problem by introducing the penalty term g(x)T Dg(x), where D is an appropriate positive definite matrix, usually taken to be diagonal. This leads to the unconstrained optimization problem of minimizing the cost function (also called merit or penalty function)over the admissible parameter set X, i.e., min f (x; ρ) = min{s(x) + ρg(x)T Dg(x)}. x∈X

x∈X

(1)

Here, ρ > 0 is a penalty parameter such that if x∗ρ = minx∈X f (x; ρ) then limρ→∞ x∗ρ = x∗ , where x∗ is the solution of the original constrained optimization problem (cf. [29]). In practice this means that the choice of ρ yields a solution x∗ρ which is as close to the optimum, x∗ , as desired. The Adaptive Steepest Descent Method In each iteration step, k, the steepest descent method determines a search direction, pk = −∇ f (xk ; ρk ), at the pivot, xk . (The gradient, ∇, is taken with respect to x, keeping ρ fixed.) The cost function f is then minimized along this search direction, using the backtracking algorithm described below. This minimization process, called line search, yields a new pivot xk+1 and a new penalty parameter ρk+1 (see below), which allows the determination of the new search direction pk+1 . This iteration process is continued until either the target is reached or a minimum is encountered. The algorithm for this adaptive steepest descent method is given in Table 1. The Backtracking Algorithm The backtracking algorithm used in this study is a modified version of the one presented in [30]. It is equipped with an additional adaptive initial step size for the first backtracking step and minimizes either a quadratic or a cubic polynomial fit to find the subsequent step sizes. Note that during the backtracking phase the penalty parameter ρ remains constant and, therefore, in order to simplify the notation, the dependence of f on ρ is ignored in this subsection, i.e., f (x) = f (x; ρ). This algorithm is summarized in Table 2. In this algorithm, a new pivot, xk+1 , is determined such that the cost function, f , is minimized in the descent direction, p k ,

(starting point) (initialize penalty parameter at xo ) (repeat until optimum is reached) (search direction) (line search gives new pivot xk+1 ) (new penalty parameter) (insure that {ρk } is non-decreasing)

starting from the present pivot, xk . The endpoint of the first backtracking interval, xs , is given by xs = xk + λ/|δk |pk /||pk ||, where λ is a user input constant and δk =< ∇ f (xk ), pk > /||pk || is the derivative of f at xk in the direction pk , i.e. δk = ∂ f (xk )/∂pk. If xs lies outside the parameter space (the unit hypercube in this study) then it is projected onto its boundary. Now, using the slope δk at xk , a parabola is fitted through the three points {(xk , f (xk )), (xk , δk ), (xs, f (xs ))} and its minimum, xm , is determined. Again, if xm lies outside the parameter space then it is projected onto its boundary. This gives the new backtracking step ∆x = ||xm − xk ||. If f (xm ) is larger than the difference between f (xk ) and the tolerance, β|δk |∆x, where β is a constant, then the backtracking step is changed (usually reduced) as follows: a cubic polynomial is fitted through the four points {(xk , f (xk )), (xk, δk ), (xm, f (xm ), (xs, f (xs ))}, and then its minimum point xc is determined. Once more, if xc lies outside the unit hypercube then it is projected onto its boundary. Next, set the new interval endpoint xs = xm and the new pivot candidate xm = xc . This new pivot candidate xm determines the new step size ∆x = ||xm −xk || and the associated change in the tolerance. This iterative process is continued until f (x m ) is smaller than the difference between f (xk ) and the tolerance; this then qualifies xm as the new pivot for the next search direction. The first step in the backtracking algorithm is normally the largest distance used for finding a new pivot. In each subsequent iteration this distance is usually reduced. The backtracking algorithm in this study is equipped with an adaptive initial step size control to make this first step as large as possible. This is achieved by linking the first step size to the steepness of the cost function derivative in the search direction, ∂ f (xk )/∂pk = δk . This has the effect that if δk is very steep then a small step size is chosen, and if the gradient is flat, then the first step size is taken to be large. This behavior is achieved by taking the initial backtracking interval as ∆x = λ/|δk |. Note that the parabola fit during the first backtracking step serves the purpose of accelerating the cost function minimization. A heuristic argument shows that in most cases, except possibly in some pathological situations, the cost function at the parabola minimum xm satisfies f (xm ) < f (xk ) and f (xm ) < f (xs ). The cubic polynomial fit is used in the sub-

Table 2 Backtracking algorithm. At each pivot xk perform a line search in direction pk δk =< ∇ f (xk ), pk > /||pk|| (derivative in search direction) xs = xk + (λ/|δk |)pk /||pk|| (initial backtracking step) if xs lies outside the unit hypercube then project x s onto the boundary fit a parabola through {(xk , f (xk )), (xk, δk ), (xs, f (xs ))} find the minimum xm of the parabola (first new pivot candidate) if xm lies outside the unit hypercube then project x m onto the boundary ∆x = ||xm − xk || (set step size) while f (xm ) ≥ f (xk ) − β|δk |∆x do (backtracking condition) fit a cubic polynomial through {(xk , f (xk )), (xk, δk ), (xm , f (xm )), (xs, f (xs ))} find the minimum xc of the cubic if xc lies outside the unit hypercube then project x c onto the boundary set xs = xm (update new interpolation point) set xm = xc (update new pivot candidate) set ∆x = ||xm − xk || (update new step size) end xk+1 = xm (found new pivot xk+1 )

sequent backtracking steps to increase the speed of the cost function minimization even further. The positive constant β helps to control the tolerance used in the search of the new pivot, and λ, also positive, provides a user-control for the initial backtracking step. Both of these constants influence the convergence rate and the number of function evaluations. The values taken in this study are β = 0.01 and λ = 0.5.

Table 3 EPA mandates and target values used in the optimization.

PMo [g/KW-hr] (NOx)o [g/KW-hr] SFCo [g/KW-hr]

EPA Mandate

Target

0.12 4 –

0.03 3.5 194.13

The Adaptive Cost Function An optimal engine performance means that the fuel consumption is minimized subject to prescribed emission targets. As discussed in the Problem Formulation, this constrained optimization problem can be expressed as an unconstrained optimization problem by minimizing the cost function given in Eq. (1). In this study, the fuel consumption is expressed in terms of the normalized specific fuel consumption SFC/SFC 0 , as is discussed below in more detail. The emissions, which constitute the penalty terms, are the nitric oxides, NO x , and the particulates, PM. Therefore, based on Eq. (1), the cost function used in this study is given by c   S SFC s ρ PM − PM0 + f (x; ρ) = C 2 SFC0 2 PM0 n ! NOx − (NOx)0 + N (2) , (NOx)0 where x denotes the engine input, ρ the penalty parameter, and the subscripts 0 denote the target values. The positive weights C, N and S, and the positive exponents c, n and s, determine the importance of each of the expressions involved. In this study, all the weights and exponent were set to one except for s = 2. (As discussed below, the factors of 12 are used

to set f (xk ; ρk ) = 1 at the start of each line search.) Notice that the dependence between the engine output quantities, SFC, PM and NOx , and the engine operating parameters, x, are not known explicitly. As discussed in the Problem Formulation, the penalty parameter ρ determines the closeness of the minimum x∗ρ = minx∈X f (x; ρ) to the actual solution of the original constrained optimization problem. This suggests that ρ can be updated after every line search. This ρ-update uses the appropriate values at the new pivot xk+1 according to the following expression !s c SFC(xk+1 ) PM(xk+1 ) − PM0 / C ρk+1 = S SFC0 PM0 n ! NOx(xk+1) − (NOx)0 (3) + N . (NOx)0

With this expression for ρk+1 , the cost function in Eq. (2) satisfies f (xk+1 ; ρk+1 ) = 1. Note that with the choice of ρk+1 = max{ρk , ρk+1 } used in the adaptive steepest descent algorithm (cf. Table 1), the sequence {ρk } is non-decreasing which gives the penalty terms more and more weight, and

thus the optimization parameters approach the constrained minimum. The EPA mandated values for NOx and PM were chosen to meet the Tier 3 2006 (Blue Sky Series) standards of stationary engines as given in Ref. [28]. These mandates are PM0 = 0.12 g/KW-hr and (NOx)0 = 4 g/KW-hr1 . The specific fuel consumption is normalized with the tuning case value of SFC0 = 194.13 g/KW-hr. These values are summarized in Table 3. In the simulations the SFC has been determined as SFC =

m˙ f P

,

where m˙ f is the injected fuel mass rate and the power output P has been computed from P=

RPM 120

Z

EVO

pdV − L. IVC

In this expression, RPM is the engine speed in revolutions per minute, p is the cylinder pressure, dV is the volume change increment, IVC denotes the inlet valve closure and EVO is the exhaust valve opening. L denotes the power losses due to friction, scavenging, and operation of peripherals such as the turbocharger and the injection system. The power losses, L, have been estimated from the difference of the experimental power output of 157.62 KW and R EVO the tuning case integral RPM 120 IVC pdV = 169.62 KW as L = 12 KW. This value of L has been used in all the simulations in this study. It should be noted that the computations have been performed at full engine load where changes to L are small in comparison to the power output. Therefore, using a constant L in all the simulations is expected to have a negligible effect on the final optimization results. Note that the above expression for the power output has been used throughout this study in the normalization of the emissions and the SFC with respect to KW-hr. Normalization of the Parameter Space The determination of the cost function gradients involves the computation of the partial derivatives. The partial derivatives are approximated by difference quotients which requires an appropriate discretization of the parameters. Therefore, to make this optimization method less dependent on the nature of the parameters, the optimization is performed on the unit cube in n-dimensional space X = {x = (x1 , x2 , . . . , xn ) : 0 ≤ xi ≤ 1, i = 1, 2, . . . , n}. This means that the actual parameter set has been mapped onto X by the transformations xi =

pi − pi,min pi,max − pi,min

,

1 The regular Tier 3 2006 EPA mandates are PM = 0.20 g/KW-hr 0 and (NOx)0 = 4 g/KW-hr.

where pi is the i-th parameter bounded by pi,min and pi,max . Therefore, a point x ∈ X has to be mapped into the true parameter space by the inverse transformation before an actual computer simulation can be performed. This normalization has the additional advantage that the constants β and λ, which influence the error criterion and the backtracking step sizes, are less sensitive to changes in the optimization parameters or cost functions. Thus, the normalization of the parameter space makes the optimization method more universal. The parameter range together with the increments used in the computations of the difference quotients are listed in Table 7. Temperature Constraint A constraint on the average cylinder temperature at the exhaust valve opening has been used in analyzing the final optimum point predicted by the simulations. This value has been set to 1250 K. If, in a computation, this constraint is violated then that particular parameter set is marked and later eliminated as a possible optimum. More precisely, at the end of each run, possible solution points which violate the temperature constraint are eliminated, and the optimum is chosen to be the best solution point which does not violate the constraint. Note that in this study the constraints have never been violated. Computational Costs The computational costs are measured in terms of the number of cost function evaluations needed until the optimum or target point is reached. In the steepest decent method, the total number of function evaluations consists of contributions from the gradient approximations and the line searches. Once a new pivot is determined, in order to compute the gradients, one additional function evaluation per optimization parameter is needed. The number of function evaluations along each line search varies from case to case. The last line search yields a pivot which corresponds to the optimum point, and therefore, the gradients do not need to be computed there. Consequently, the total number of cost function evaluations, #F, can be expressed as #F = 1 + n(m − 1) +

m−1 X

lk ,

(4)

k=1

where n is the dimension of the parameter space, m is the number of pivots (including the final point) and l k is the number of cost function evaluations in the k-th line search.

ZERO-DIMENSIONAL ENGINE CODE Changes in the engine operating conditions lead to changes in the output, which, by means of the turbocharger and the EGR, affect the engine input conditions. Such changes require many engine cycles until the new equilibrium is reached. The CFD-code utilized in this study simulates only the incylinder process from the inlet valve closure to the exhaust

valve opening, assuming that the engine system is in equilibrium. Therefore, in order to account for the EGR and the turbocharger feed-back, an additional simulation is required which can predict the correct engine input conditions for a given engine operating point. This task is usually performed with a so-called zero-dimensional simulation code. The zero-dimensional engine simulation code utilized in this study is a version of the OBEH code, developed at the Czech Technical University in Prague in the research group of J. Macek. This code has been designed primarily for turbocharged medium and large-bore DI diesel engines, equipped with an EGR system. OBEH is interfaced with the CFD code KIVA-3 in such a way that a given engine operating condition, i.e, the injection timing, the EGR rate and the desired boost pressure, provides the appropriate equilibrium engine input data for KIVA-3 at inlet valve closure. These input data include the temperature, pressure, and the cylinder composition due to EGR. The OBEH models are outlined in the following. A more detailed description can be found in [31]. The highlights of the OBEH code include • a cumulative ignition delay model with a user defined function for induction time in a quasi-dimensional approach, • a simple empirical model for combustion with three cumulated Vibe functions based on Woschini’s ideas, • a heat transfer model with user defined heat transfer coefficients and an adaptive heat-resistance model, fitting surface temperatures of a piston, a cylinder and a head with valves, • a zero-dimensional exhaust manifold model calibrated by experiments, • a detailed turbine model using normalized maps for waste gate, variable geometry turbine (VGT) or by-pass boost control, two stage turbocharging, parallel and serial power gas turbine, mechanical or electrical super combined with a turbocharger, etc. • an EGR model for low and high pressure loops including venturis for pressure difference enhancement, • a simplified experimentally validated mechanical loss model which includes losses due to crank gears, piston, rings and bearings. In addition, the code is equipped with many iterative control procedures used for the acceleration of the code calibration or optimization, e.g., control of a constant peak pressure, VGT area control for constant boost pressure, air excess control etc.

MULTI-DIMENSIONAL ENGINE MODELS The multi-dimensional engine computations for determining the pollutants and the power output have been performed with a modified version of the KIVA-3 code [27]. This code is equipped with the RNG k −  turbulence model as

implemented by Han and Reitz [32], the CAB atomization and drop breakup model [33, 34] and the LIT auto ignition model [35] of Tanner. The heat release is modeled using the laminar-turbulent-laminar (LTL) characteristic time combustion (CTC) combustion model that is described in the following paragraphs, along with the emission models for soot and nitric oxide. All other models used in the simulations are the standard KIVA-3 models. The LTL-Characteristic Time Combustion Model The LTL-CTC model is based on the CTC model of Abraham et al. [36] as adapted to diesel combustion by Kong et al. [37], but it employs only one global reaction to model the heat release. As in the original CTC model, the LTL-CTC model uses a laminar reaction rate for the precombustion and a turbulence reaction rate for the spray combustion, but once the fuel injection is terminated, the LTL-CTC model gradually shifts back to the laminar reaction rate. This last step is motivated by the fact that the turbulence in a diesel engine is dominated by the spray-induced flow during fuel injection, but it is considerably reduced in the later combustion phase, as is discussed in more detail in a study by Tanner and Reitz [38]. The gradual shift from the mixing-controlled combustion back to the laminar combustion improves the reaction rate in the late combustion phase. As a consequence, the notorious under-prediction of the heat release rate is improved, which results in a reduction of the unburned fuel at the end of the combustion. Further details of the LTL characteristic time combustion model can be found in [39]. Emission Models In diesel engines, nitric oxide formation is dominated by the high-temperatures. This high-temperature NOx formation is modeled using the extended Zeldovich mechanism is described in detail in [25]. The forward and the backward reaction rates of the Zeldovich mechanism as a function of the temperature, T, are computed from the Arrhenius relation k = KTb exp(To /T),

(5)

where the constants K and b along with the activation temperature To are listed in Table 4. The H, O and OH radicals involved in the Zeldovich mechanism are obtained from the set of equilibrium reactions listed in Table 5. In this table the constants {A, B, C, D, E} are the ones in the expression for the equilibrium concentration Ke , which is given by ln Ke = A ln Tr + B/Tr + C + DTr + ETr2 , where Tr = T/1000. For additional details see [40]. The net soot density is modeled according to Hiroyasu and Kadota [41] by dρs f dρso dρs = − , dt dt dt where ρs is the net soot density, ρs f is the density of soot formed and ρso is the density of soot oxidized.

Table 4 Constants used in the Zeldovich mechanism. Reaction O + N2 ←→ N + NO N + O2 ←→ O + NO N + OH ←→ H + NO

Forward Reaction K b T0 1.71E+14 0 37996 1.80E+10 1 4680 4.10E+13 0 0

Backward Reaction K b T0 3.80E+13 0 425 2.36E+09 1 19446 1.34E+14 0 24750

Table 5 Equilibrium reactions and rate constants. Reaction H2 ←→ 2H O2 ←→ 2O O2 + H2 ←→ 2OH O2 + 2H2 O ←→ 4OH O2 + 2CO ←→ 2CO2

A 0.990 0.431 –0.653 1.16 0.981

The soot formation is modeled according to (cf. [41]) dρs f dt

= C f ρv p0.5 exp −

Ef RT

!

B –51.8 –59.7 –9.82 –76.8 68.4

C 0.993 3.5 3.93 8.53 –10.6

D –0.343 –0.340 0.163 –0.868 0.574

E 0.01117 0.01587 –0.01429 0.04635 –0.04146

Table 6 Engine specifications and tuning case operating conditions for the Sulzer S20.

,

where ρv is the fuel vapor density, p is the pressure in bar, T is the temperature in Kelvin, E f =52300 J/mol is the activation energy, and R=8.3143 J/(mol K) is the universal gas constant. C f is a tuning constant and the value used in this study is C f = 4.7. The oxidation model used is the one by Nagle and Strickland-Constable (NSC) [42], as presented by Chan et al. [10]. The overall oxidation rate of the chemical reactions represented by this model is given by dρso m c Rt = Co ρs , dt σd where Co =1.215 is a tuning constant, mc =12 g/mol is the molecular weight of carbon, σ=2 g/cm3 is the density of a soot particle, and d = 3 × 10−6 cm is the average diameter of a soot particle. Rt represents the overall reaction rate of this system. Further details can be found in [39].

COMPUTATIONAL DETAILS The optimization algorithm has been programmed in C. This code also served as the fully automated control interface for the optimization runs. The zero-dimensional engine simulations were used to predict the gas composition at the closure of the inlet valves. This data is then used in the threedimensional engine simulations, that have been performed with a KIVA-3-based code using the models described in the previous section, to determine the emissions and SFC. The engine used in the simulations is the four-stroke Sulzer S20 DI diesel engine with a central injector equipped with 12 nozzle orifices. The experimental data used in the tuning case

Type Bore [mm] x stroke [mm] Compression ratio Engine speed [rev/min] Injector Orifices x diameter [mm] Swirl ratio Inlet valve closure [CA ATDC] Outlet valve opening [CA ATDC] Injection system Injection angle (vertical) [deg] Injection start [CA ATDC] Injection duration, ∆tin j [CA] Injected fuel mass [g] Injection pressure (avg.) [MPa] Power output [kW/cylinder]

four-stroke 200 x 300 13.6 1000 one (central) 12 x 0.285 0 –144 129 conventional 80 –10.5 32 1.02 95 157.62

has been obtained from a stationary nine-cylinder production engine [43]. The engine specifications are listed in Table 6. The cylinder flow is assumed to be periodic with respect to the number of nozzle orifices and hence only one sector of the engine cylinder corresponding to one injection orifice is simulated. The mesh used for the engine simulation had 23×7×14 cells in radial, azimuthal and vertical direction, respectively at TDC. This corresponds to a computational domain of 30 deg. As discussed at the end of the Model Tuning subsection, this mesh gave adequate mesh independent results. All the computations have been performed from the closure of the inlet valves at 144 CA before TDC to the opening of the exhaust valves at 129 CA after TDC. A common rail injection system was used which was simulated with constant injection

Table 7 Parameter ranges used in the optimization and the corresponding increments used in the computation of the gradients. Parameter Start of injection of pulse 1 [CA ATDC] Injection duration of pulse 1 [CA] Duration of dwell [CA ATDC] Injection duration of pulse 2 [CA] EGR [%] Boost Pressure [bars]

velocities. The simulations were done at full engine load at 1000 RPM, with a total injected fuel-mass of 1.02 g. Optimization Values and Parameters In the present study the following parameters have been optimized with respect to NOx, PM and SFC: the start of injection of the first pulse, duration of the first pulse, duration of dwell, duration of second pulse, % EGR and the boost pressure. The individual parameters together with the corresponding increments used in the computation of the gradients are listed in Table 7. The gradients have been approximated with a forward difference quotient. This approach has been chosen because in past investigations (c.f. [26]), optimal split injections have been characterized by a short pilot injection followed by a longer second pulse. If, in this situation, the short injection duration is of the same size as the differential step size, then the computation of a backward difference or a centered difference can involve pulse durations which are shorter than the step size (or even negative) and, therefore, the gradient approximations become meaningless. A similar argument can be made for the gradient approximation involving the dwell. Also, the gradient step sizes have been determined by conducting systematic step size variations in the previous studies. An initial analysis has shown that the step sizes chosen are reasonable in the sense that small variations don’t have a significant influence on the gradients. The start of injection of first pulse has a range from -15 CA to 5 CA. The other three parameters of the split injection, viz., duration of the first pulse, duration of dwell and the duration of second pulse have a range from 0 CA to 30 CA. In order to avoid a very high injection pressure, a constraint of 20 CA has been placed on the sum of the injection duration of the two pulses. More precisely, a condition ∆t1 + ∆t2 ≥ 20 was imposed, where ∆t1 and ∆t2 represent the injection durations (in crank angles) of the two pulses. An injection duration of 20 CA corresponds to an injection pressure of approximately 2100 bars. Exhaust gas recirculation rates were limited from 0 % to 50 %, while the boost pressure was given a range from 2 to 5 bars. The main motivation to choose the boost pressure as an optimization parameter is the desire to compensate the

Min.

Max.

Increments

–15 0 0 0 0 2

5 30 30 30 50 5

2 2 2 2 10/3 0.2

relatively low cylinder pressure caused by a delayed start of injection. This typically results in higher temperatures and pressures at the end of the combustion cycle. This excess energy, which cannot be directly converted into piston work, can be recovered via the turbocharger in the form of an increased boost pressure, which then leads to an increased cylinder pressure. Investigations of other authors (c.f [8–10,15,23]) have shown that an increased boost pressure in combination with EGR and split injection, can be a very effective means of reducing emissions while keeping the specific fuel consumption in an acceptable range. Note that an increased boost pressure, hence higher cylinder pressures and temperatures, also leads to a reduction in the soot formation via the following mechanism. The use of EGR aids in the reduction of NOx but at the same time it usually increases the amount of PM. This increase in PM can be curbed by increasing the cylinder pressure using a higher initial boost pressure, if possible. A high cylinder pressure generally leads to higher cylinder temperatures, which then increase the soot oxidation and result in a decrease of the net soot formation. Model Tuning The soot and nitric oxide models described in one of the previous sections have been tuned to match the experimental data of the S20 engine from Stebler [43], for the tuning case. The engine operating conditions for the tuning case are summarized in Table 6. Unlike the optimization cases, the tuning of the S20 engine’s soot and NOx models were done with a conventional injection system. Figures 1, 2 and 3 represent the pressure, NOx and soot curves of the tuning case. As seen from Figure 1, the experimental and tuned pressure curves are in close agreement. The most sensitive parameter of the Zeldovich mechanism for predicting the NOx proved to be the forward reaction constant, K, of the first reaction listed in Table 4. Hence, only this constant was tuned to obtain the experimental NOx value of 10.99 g/KW-hr. The result of this tuning case is shown in Figure 2, where the experimental value lies in the center of the big circle. In the present study the value of K = 1.71E + 14. In the tuning of the soot model, the soot formation constant, C f , and the soot oxidation constant, C0 , were determined to obtain the experimental soot value. The value of

Experiment Computation

Experiment Computation

0.2

15

Soot [g/KW−hr]

Cylinder Pressure [MPa]

20

10

0.1

5

0 −150

−100 −50 0 50 Crank Angle [deg TDC]

0.0 −20

100

Figure 1 Cylinder pressures for the S20 tuning case.

0

Figure 3 Net soot for the S20 tuning case.

15

10

5

0 −20

0

20 40 60 80 100 120 Crank Angle [deg TDC]

Figure 2 Nitric oxide for the S20 tuning case.

the other parameters of the soot model are as reported in Chan et al. [10]. In this study, the tuning constants, C f and C0 had values of 4.7 and 1.215, respectively. The parameters obtained from the tuning case have been kept constant in all the optimization computations. Further, the constants used in all the other KIVA-3 models are the ones reported in the respective publications cited in the Engine Models section. A mesh dependency analysis of the cylinder pressure is shown in Figure 4. Three meshes have been analyzed in this study, viz: the fine mesh, the standard mesh and the coarse mesh. The fine mesh had a resolution of 23 × 14 × 14 cells at the TDC, the standard mesh had 23 × 7 × 14 cells at the TDC and the resolution of the coarse mesh was 23 × 4 × 14 cells at TDC. As can be seen from Figure 4, the pressure curve of the standard mesh is closer to that of the fine mesh. In view of the enormous computational costs of the optimization runs, the standard mesh used in this study is considered to give adequate mesh-independent results.

Fine Mesh Tuning Case Standard Mesh Tuning Case Coarse Mesh Tuning Case

20 Cylinder Pressure [MPa]

NOx [g/KW−hr]

Experiment Computation

20 40 60 80 100 120 Crank Angle [deg TDC]

15

10

5

0 −10

0

10

20 30 40 50 60 70 Crank Angle [deg TDC]

80

90

Figure 4 The pressure curves of the tuning case using the the fine mesh, the standard mesh and the coarse mesh.

RESULTS AND DISCUSSION Optimizations have been performed for two different cost functions for the starting point EGR rates of 5% and 10%, respectively. The motivation for choosing two different cost functions comes from a previous study by Tanner and Srinivasan [26] in which it was shown that the search direction, and thereby the optimum, are influenced by the form of the cost function. In the present study, the first set of computations use a quadratic SFC term in Equation (2), i.e., s = 2, while s = 1 for the second set. The weights {C, N, S} and the exponents {c, n} were set equal to one in both formulations. The starting point values of all the parameters, except the EGR rate, were the same in all the optimization runs. These values are summarized in Table 8 for the 5% EGR case and in Table 9 for the 10% EGR case. Note that because of the differences in the cost functions, the optimization paths are also different.

270

0.2

0.1

EPA Mandate Quadratic SFC (5% EGR) Linear SFC (5% EGR)

250

SFC [g/KW−hr]

Soot [g/KW−hr]

EPA Mandate Quaratic SFC (5% EGR) Linear SFC (5% EGR)

(Start)

230 210 190

(Start)

170 150

0

130

0

2

4 6 NOx [g/KW−hr]

8

10

Figure 5 NOx and soot values at the pivots of the search path for the 5% EGR case. 270

EPA Mandate Quadratic SFC (5% EGR) Linear SFC (5% EGR)

SFC [g/KW−hr]

250 230 210 190

(Start)

170 150 130

0

2

4 6 NOx [g/KW−hr]

8

10

Figure 6 NOx and SFC values at the pivots of the search path for the 5% EGR case. Optimization Using a Quadratic SFC Term This optimization run has been performed with a quadratic SFC term, i.e., f (x; ρ) = +

  1 SFC 2 2 SFC0 ρ  NOx − NOx0 PM − PM0  + . 2 NOx0 PM0

This larger exponent results in a higher priority for the reduction of the fuel consumption. The results of the 5% EGR starting point are shown in Figures 5−7, and those of the 10% EGR starting points in Figures 8−10. The data at the pivots obtained via the quadratic SFC term are marked with circles in the respective figures. For the simulation with a starting point EGR rate of 5%, the optimum was reached after one line search. As seen from Figure 5, only the NOx target is met. At the optimal point, the emission values are PM = 0.145 g/KW-hr and NOx =

0

0.1 Soot [g/KW−hr]

0.2

Figure 7 Soot and SFC values at the pivots of the search path for the 5% EGR case. 1.822 g/KW-hr, respectively. As can be seen from Table 8, at the optimum point, duration of the first pulse is zero, thus the injection degenerates into a single pulse starting at 4.69 CA ATDC with a short injection duration of 20 CA. This delay in the start of injection is the main cause for the increase in soot and the reduction in NOx. The specific fuel consumption at the optimal point is predicted as SFC = 186.4 g/KW-hr. This is below the idealized target of 194.13 g/KW-hr that was used in the cost function. This can be explained by the high initial boost pressure which raises the peak cylinder pressure to almost 172 bar, thus resulting in a high power output. In the simulation with a starting point EGR rate of 10%, the optimum is reached after two line searches. Once again, as in the 5% EGR case, the emission targets are not met (PM = 0.135 g/KW-hr and NOx = 4.30 g/KW-hr). The idealized SFC target is reached with SFC = 178.9 g/KW-hr. The results of this simulation are listed in Table 9. Unlike in the 5% EGR case, the optimum of the 10% EGR case exhibits an actual split injection. In comparison with the starting point, the injection of the first pulse is delayed by 1.17 CA, the dwell duration is increased from 6.0 CA to 9.13 CA and the pulse durations are reduced from 13 CA to 11.4 CA and 9.64 CA, respectively. The overall effect on the soot formation of this optimum injection timing, despite the shorter pulse durations, is an increase from 0.129 g/KW-hr to 0.135 g/KW-hr. The increase in the NOx formation, despite the (short) injection delay and the increased dwell, is the result of the high cylinder pressure of 17.6 MPa, hence higher cylinder temperature. This in turn is a consequence of the increased boost pressure from 3 bar to 3.75 bar, the reduction of the EGR rate from 10% to 8.48%, and the reduced injection durations of the two pulses. Compared to their starting points, the SFC is reduced in both simulations. In the 5% EGR case, the SFC is reduced by 1.5% and in the 10% EGR case by 7%. Thus, the high priority given to the SFC term in the cost function, s = 2, steers the search in a direction which is favorable for the reduction of SFC but fails to meet the emission targets. Further, the

Table 8 Starting and optimum data obtained from the simulations with the starting point EGR rate of 5%. Starting Point Parameters SOI of first pulse [CA ATDC] Duration of 1st pulse [CA] Duration of dwell [CA] Duration of 2nd pulse [CA] EGR rate [%] Boost pressure [bars]

Optimal Points Quad. SFC Lin. SFC

–10.5 13 6 13 5 3.0

-3.70 0 8.40 20 2.095 5.00

-6.57 3.12 15.86 16.88 1.64 4.098

Particulates [g/KW-hr] NOx [g/KW-hr] SFC [g/KW-hr]

0.118 5.753 189.2

0.145 1.822 186.4

0.124 1.439 193.1

Max. cyl. pressure [MPa] Computation # function evaluations

15.04

17.18

16.48

10

28

Engine output

reduced SFC values are the result of the high peak cylinder pressures which, besides the increased NOx, also might negatively affect the life span of the engine. Therefore, in the subsequent simulations, the cost function will be modified to accommodate a reasonable increase in the SFC, thus yielding acceptable peak cylinder pressures and reduced emission levels. Finally, it should be noted that the two different starting EGR rates have resulted in two different optima. This indicates that the algorithm has most likely converged to different local minima. As discussed previously, this is a characteristic of gradient-based algorithms in the presence of several local minima. Optimization Using a Linear SFC Term In this optimization run the priority of the SFC term is lowered by setting its exponent s = 1 in Equation (2). Thus the resulting cost function is   1 SFC f (x; ρ) = 2 SFC0 ρ  NOx − NOx0 PM − PM0  . + + PM0 2 NOx0

Again, two optimizations for the starting point EGR rates of 5% and 10% have been performed. The data of these runs are marked with the squares in Figures 5−7 for the 5% EGR case, and Figures 8−10 for the 10% EGR case. In the first simulation with an EGR of 5% at the starting point, the optimum was reached after three line searches. The emissions were obtained as PM = 0.124 g/KW-hr and NOx = 1.439 g/KW-hr. As seen from Table 8, at the optimum a split injection has been obtained with a short first pulse of 3.12 CA, followed by a long dwell of 15.86 CA. About 84%

of the fuel is injected after this long delay. Thus, most of the combustion takes place in the expansion stroke, which explains the increase in the soot formation and the reduction in the NOx. On comparing the optimum of this 5% EGR case with the corresponding quadratic SFC case, it is apparent that the emissions, soot in particular, are considerably improved. However, as expected, the SFC is only marginally reduced over the target value of 194.13 g/KW-hr, namely by 0.53%. The peak cylinder pressure in this case is 16.48 MPa which is more acceptable than the 17.18 MPa obtained in the corresponding quadratic SFC case. It should be noted that none of the three cases considered so far meet both of the emission targets. The best case so far has been the linear SFC case with a starting EGR rate of 5%. In the last simulation, characterized by a starting EGR rate of 10%, the optimum was again reached after three line searches. Unlike in the previous three cases, the EPA emission mandates for this simulation were met with PM = 0.091 g/KW-hr and NOx = 0.676 g/KW-hr. Again, a split injection was obtained at the optimum point, in which a short pilot injection of 5.24 CA is followed by a long dwell of 22.94 CA, after which the remaining fuel is injected over a duration of 14.76 CA. This injection characteristic is similar to the one of the linear 5% EGR case. In fact, as in the linear 5% EGR case, the short pilot injection followed by a long dwell moves the main combustion phase into the expansion stroke which is responsible for the considerable reduction in NOx. Interestingly, despite the long delay of the second pulse, the soot EPA mandate is also met. This, on the one hand, is a consequence of the high boost pressure of 4.5 bar which leads to higher cylinder temperatures in the late combustion phase and hence higher soot oxidation. On the other hand, the total fuel is injected over

270

0.2

EPA Mandate Quadratic SFC (10% EGR) Linear SFC (10% EGR)

250

SFC [g/KW−hr]

Soot [g/KW−hr]

EPA Mandate Quaratic SFC (10% EGR) Linear SFC (10% EGR) (Start)

0.1

230 210 190

(Start)

170 150

0

130

0

2

4 6 NOx [g/KW−hr]

8

10

Figure 8 NOx and soot values at the pivots of the search path for the 10% EGR case.

EPA Mandate Quadratic SFC (10% EGR) Linear SFC (10% EGR)

250

SFC [g/KW−hr]

0.1 Soot [g/KW−hr]

0.2

Figure 10 Soot and SFC values at the pivots of the search path for the 10% EGR case. Computational Costs

270

230 210

(Start)

190 170 150 130

0

0

2

4 6 NOx [g/KW−hr]

8

10

Figure 9 NOx and SFC values at the pivots of the search path for the 10% EGR case.

only 20 CA which corresponds to the maximum allowed injection pressure of approximately 2100 bar. This high injection pressure leads to an excellent fuel atomization, hence a low soot formation. In comparison with the other cases, the optimum of the linear 10% EGR case meets the EPA emission standards. In fact, the PM lies 24.2% and the NOx 83.1% below the EPA mandates. However, unlike in the other cases, the specific fuel consumption has increased by 17.9% over the target value of 194.13 g/KW-hr. Thus the emissions are met at the expense of the specific fuel consumption. Also, the peak cylinder pressure in this case is 16.98 MPa, which is 12 bar higher than in the tuning case, but still acceptable. Note that, as in the quadratic SFC case, the two different starting EGR rates have resulted in two different optima. Again, this is an indication that the algorithm has probably converged to different local minima.

The computational cost is measured in terms of the number of cost function evaluations given by Eq. (4). As can be seen from Tables 8 and 9, these numbers lie between ten for the 5% quadratic SFC case and thirty-two for the best, the 10% linear SFC case. Note that these numbers are directly related to the number of line searches, namely one for the 5% quadratic SFC case and three for the 10% linear SFC case. Typical numbers of cost function evaluations used in the µGA optimization lie between 250 (cf. Senecal and Reitz [16]), and 400, as reported in Wickman et al. [18]. The adaptive steepest descent method implemented in this study needs considerably less function evaluations. It must be pointed out, however, that in the cited µGA studies some of the parameters optimized were different from the ones optimized here. Also, the cost function used in the µGA optimization was different, and, as mentioned earlier, the target values found are likely to correspond to a global optimum. Nevertheless, this study shows that the adaptive gradient methods are computationally very effective in finding improved operating conditions for engines. The Optimal Path In order to understand the influence of the individual optimization parameters on the optimum point (linear SFC, 10% starting EGR rate), additional engine simulations were performed. In these simulations, each run had only one additional parameter changed, when moving from the starting point to the optimum point. Two different sequences of changes were studied. Table 10 lists the changes introduced along with the emissions and SFC values at each step. The soot and NOx values of these two paths are shown in Figure 12. In this figure, the steps A1 through D1 constitute the first optimal path, while A2 through D2 correspond to the second optimal path. With the first step, A1, in path one, where the EGR rate is reduced from the starting value of 10% to the

Table 9 Starting and optimum data obtained from the simulations with the starting point EGR rate of 10%. Starting Point Parameters SOI of first pulse [CA ATDC] Duration of 1st pulse [CA] Duration of dwell [CA] Duration of 2nd pulse [CA] EGR rate [%] Boost pressure [bars]

Optimal Points Quad. SFC Lin. SFC

–10.5 13 6 13 10 3.0

-9.33 11.40 9.13 9.64 8.48 3.75

-0.056 5.24 22.94 14.76 8.81 4.498

Particulates [g/KW-hr] NOx [g/KW-hr] SFC [g/KW-hr]

0.129 3.907 192.4

0.135 4.302 178.9

0.091 0.676 228.9

Max. cyl. pressure [MPa] Computation # function evaluations

15.06

17.60

16.98

23

32

Engine output

optimal value of 8.81%, there is an increase in NOx and a decrease in soot. Next, the start of injection was delayed from −10.5 CA ATDC to −0.056 CA ATDC in step B1. This step decreased NOx while the soot remained almost unchanged. As expected, a delay in the start of injection decreases the power output. This is reflected in the SFC which increases to 211.3 g/KW-hr. With the introduction of the optimum fuel injection profile in step C1, there was a further decrease in NOx. This is because the dwell is much longer than the dwell in the step B1 which corresponds to the dwell of the starting point, and most part of the fuel is injected in the second pulse. Thus, most of the combustion takes place in the expansion stroke, leading to a lower peak cylinder pressure and temperature, which is favorable for the reduction of NOx. This NOx reduction occurs at the expense of the fuel consumption which is increased to SFC = 251.6 g/KW-hr. In the final step D1, the boost pressure was raised to the value of 4.5 bars. This resulted in a very low soot value and unexpectedly, a decrease in the NOx. The value of the SFC was also reduced to 228.9 g/KW-hr. In order to investigate this peculiar behavior of the reduction in NOx, a second optimal path was investigated where the sequence in which the optimal values were introduced was changed. This is represented as optimal path two in the Figure 12. As in the first optimal path, a series of simulations were conducted by changing one parameter at the time from the starting value to the same optimum value. In the first step the boost pressure was raised to the optimum value of 4.5 bars. This is indicated by the step A2 in Figure 12. Once again, an unexpected result in which the NOx is reduced while the soot is increased, was obtained. It should be noted that both the simulations, viz., the starting point and the step A2, had an injection profile of 13−06−13. In the second step, the EGR rate was decreased. This re-

duced soot from 0.182 g/KW-hr to 0.176 g/KW-hr, while the NOx increased to 3.255 g/KW-hr. Next, the start of injection was delayed to the optimum value. As expected, the NOx decreased even further to 0.786 g/KW-hr and the SFC increased to 197.7 g/KW-hr. Finally, when the optimum injection shape was introduced, the emissions, PM in particular, were considerably lowered while the SFC increased to 228.9 g/KW-hr. To understand the counterintuitive behavior of the NOx decrease under the influence of the boost pressure, the fuelmass, heat release and the NOx curves from the starting point and step A2 were studied in detail. From the graph of the total fuel-mass present in the cylinder in the two simulations, shown in Figure 14, we find that the total fuel in the cylinder at a given time is lower in step A2. Further, the liquid fuel present in the cylinder in both cases is almost the same. The area between the curves representing the total fuel in the cylinder in the two cases is the amount of extra fuel that has been turned into combustion products in the step A2. This indicates a higher amount of combustion in step A2. This is also supported by the heat release curves shown in Figure 13. Further, a high boost pressure in the step A2 raises the cylinder pressure and temperature, resulting in a better atomization which causes an earlier and increased combustion in the first pulse (c.f. Figure 13). This results in an internal EGR effect where these newly formed combustion products suppress NOx formation. Besides, they also raise the amount of soot in the cylinder. As can be seen from the NOx curves in Figure 11, the initial value of NOx in step A2 rises when the first pulse is being injected from −10.5 CA ATDC to 2.5 CA ATDC. The second pulse is injected from 8.5 CA ATDC, but the combustion products from the first pulse begin to suppress the NOx formation which is reflected as a bend in the NOx curve at about 10 CA ATDC. This internal EGR effect is not as prominent at the starting

Table 10 Changes made to the engine operating conditions from the starting point along the optimal paths and the corresponding emissions and SFC values. Parameter Varied Starting point Path One A1 B1 C1 D1

EGR EGR EGR EGR

Path Two A2 B2 C2 D2

Boost Boost Boost Boost

(10% to 8.81%) + SOI + SOI + Injection rate shape + SOI + Injection rate shape + boost pressure pressure pressure + EGR (10% to 8.81%) pressure + EGR + SOI pressure + EGR + SOI + Injection rate shape

PM

NOx

SFC

0.129

3.907

192.4

0.126 0.124 0.122 0.091

4.53 1.17 0.864 0.676

191.2 211.3 251.6 228.9

0.182 0.176 0.175 0.091

3.092 3.255 0.786 0.676

181.5 180.9 197.7 228.9

6

Starting point Step A2

Optimal path one Optimal path two

0.2

Soot [g/KW−hr]

NOx [g/KW−hr]

5 4 3 2

B2

C2

A2

D2 C1

(Start) A1

B1 D1

0.1

(Optimum)

1 0 −20

0.0 0

20

40

60

80

100

120

Crank Angle [deg ATDC] Figure 11 NOx values at the starting point and in step A2. the 10% EGR case with linear SFC term.

0

1

2

3 4 5 NOx [g/KW−hr]

6

7

Figure 12 Influence of each parameter on the emissions, along the two optimal paths for the 10% EGR case with linear SFC term.

point, which has a lower boost pressure.

SUMMARY AND CONCLUSIONS

From the analysis of the optimal paths it is easy to identify some of the key parameters that contribute extensively towards the reduction of emissions. This is done by looking at the size of the step taken by the individual parameter towards the optimum, in the two paths. Thus, from the steps B1 and C2 the effect of SOI is noted to be substantial. Likewise, the steps A2 and D1 clearly identify boost pressure as a major contributor in reducing the SFC. The injection rate shape has a considerable influence on the emissions in the second path, where it is denoted by the step D2. However, in the step C1 of the first path, it does not seem to contribute much in reducing the emissions. This indicates an interdependence of these parameters in influencing the emissions. In other words, the effect of a particular emission reduction measure depends on the engine operating conditions, i.e., for some measures the effect is large, for others it is small.

A gradient-based optimization tool has been developed and utilized in the search of new optimal engine operating conditions. The approach taken uses a steepest descent method with an adaptive cost function, where the line search is performed with a backtracking algorithm. The backtracking algorithm utilizes quadratic and cubic polynomials to accelerate the convergence, and the initial backtracking step employs an adaptive step size mechanism which depends on the steepness of the search direction. The adaptive cost function is based on the penalty method, where the penalty parameter is increased after every line search. The parameter space is normalized and thus, the optimization occurs over the unit cube in higher-dimensional space. This algorithm has been tested on the Sulzer S20 engine, where the optimizations were carried out with respect to emissions and engine performance, using a KIVA-3-based code. A zero-dimensional code has been integrated into the optimiza-

0.6

1000

Total fuel [g] / Liquid fuel [g]

ROHR [J/ca]

800

Starting point Step A2

600

400

200

0 −20

0

20

40

0.4 0.3 0.2 0.1 0 −20

60

Crank Angle [deg ATDC]

Figure 13 The rate of heat release at the starting point and in step A2. tion process in order to close the engine cycle and to compute the equilibrium input conditions for KIVA-3. The parameters that were optimized are the start of injection, the injection duration of the first pulse, the duration of the dwell, the injection duration of the second pulse, the EGR and the boost pressure. Optimizations have been performed for two different starting points using two different adaptive cost functions. The starting points varied only in their initial EGR rate which was either 5% or 10%; the cost functions used a linear or a quadratic SFC term. Both quadratic SFC cases and the linear 5% EGR case produced optima with excellent SFC values, but did not meet the EPA emission standards. On the other hand, the linear 10% EGR case satisfied both, soot and NOx EPA standards. In fact, the soot had been reduced below 24% and the NOX below 83% of the EPA mandates. However, unlike in the other three cases, the SFC had been increased by 18% over the 194.13 g/KW − hr target. These computations have once more demonstrated the difficulties in reducing emissions while maintaining high fuel efficiency. The results of the optimization cases have shown that the different EGR starting points have led to different optimum points. This is an indication that the obtained optima are most likely local minima of the cost function. However, the main objective of obtaining improved engine operating points which meet the EPA mandates and simultaneously yield high fuel efficiency, is still realized. Thus, this study has illustrated that the adaptive gradient method can be applied to engine optimization problems while keeping the computational costs low. In fact, less than thirtyfive engine simulations were required until an optimum was reached in each of the four optimization runs. An analysis was performed to study the influence of each parameter on the process of optimization by performing a series of simulations in which one parameter was varied at a time, beginning from the starting point to the optimal point. The analysis showed that the boost pressure and the start

Starting point liquid fuel Step A2 liquid fuel Starting point total fuel Step A2 total fuel

0.5

0 20 40 60 Crank Angle [deg ATDC]

80

Figure 14 The total fuel and liquid fuel present in the cylinder at the starting point and in step A2. of injection were the most influential parameters in reducing the emissions. Further, the increased boost pressures were mainly responsible for the reduction of the SFC. An unexpected result was the decrease in NOx when the boost pressure was increased. This counterintuitive behavior is a result of the internal EGR effect experienced by the second pulse of the split injection. This internal EGR results in a decrease of the nitric oxide production and can even increase the soot formation. The study of the two different optimization paths has shown that the amount of emissions reduction achieved by a particular measure depends on the engine operating conditions. This indicates that there is a strong interdependence between the parameters to be optimized. This phenomenon warrants a detailed study of the correlation between the different parameters which will be the focus of a future investigation.

NOMENCLATURE ATDC CA EGR EVO GA IVC NOx PM S20 SFC SOI TDC

After top dead center Crank angle Exhaust gas recirculation Exhaust valve opening Genetic algorithm Inlet valve closure Nitric-oxide Particulate matter Sulzer S20 Specific fuel consumption Start of injection Top dead center

A, B, C, D, E C, N, S Co Cf

Equilibrium reaction rate constants Preconstants of the cost function Soot oxidation constant Soot formation constant

Ef K, b Ke L P R T c, n, s d f (xm ) k mc p pk xm

Activation energy Arrhenius reaction rate constants Equilibrium concentration Losses in KW Power output in KW Universal gas constant Temperature Exponents of the cost function Average diameter of the soot particle Cost function value at xm Arrhenius reaction rate Molecular weight Pressure in bar Search direction Set of input parameters

Greek Symbols ∆t1 , ∆t2 β λ ρfv ρ, ρk ρs ρs f ρso σ

Injection duration of the 2 pulses Backtracking algorithm constant Backtracking algorithm constant Fuel vapor density Penalty parameter Net soot density Density of soot formed Density of soot oxidized Density of soot particle

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