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Energy & Fuels 2007, 21, 110-120
Computer Model to Simulate the Injection Process in a Rotary Injection Pump: The Inverse Problem Jose´ M. Palomar,† Fernando Cruz-Perago´n,† Francisco J. Jimenez-Espadafor,‡ and M. Pilar Dorado*,§ Department of Mechanics and Mining Engineering, EPS de Jae´ n, UniVersidad de Jae´ n, Campus Las Lagunillas s/n, 23071 Jae´ n, Spain, and Department of Energetic Engineering and Fluid Mechanics, ESI, UniVersidad de SeVilla, Camino de los Descubrimientos s/n, Isla de la Cartuja, 41092 SeVilla, Spain ReceiVed May 5, 2006. ReVised Manuscript ReceiVed October 24, 2006
The design of conventional injection systems requires the use of trial and error processes (with a high experimental load). Mathematical models, such as the one described in a previous work [Palomar et al. Energy Fuels 2005, 19, 1526-1535], helps in the design process. However, these models do not allow us to approach the inverse problem; that is to say, given a target law, the model cannot provide the system’s dimensional characteristics which allow the injection system to reproduce the objective law. In this way, this study confronts a first approach to the resolution of the inverse problem, by means of an optimization algorithm over the direct problem. For this purpose, a second-order Newton’s method is proposed to solve this problem. In this problem, a small number of parameters are maintained as variables to be used in the optimization algorithm. Thus, the main goal of the procedure is to generate a prearranged needle lifting law. The defined laws have been geometrically designed. They are indicative, without relation to a reference parameter, such as velocity, load, or injected fuel. As an initial approach, an unconstrained optimization has been carried out. Herein, we have demonstrated the efficiency of the proposed methodology to optimize the design of the engine fuel injection system according to the appropriate working conditions of the engine related to parameters such as combustion, performance, exhaust emissions, and noise.
Introduction The increasing compromise with the reduction of engine exhaust and noise emissions are leading engine manufacturers to develop appropriate technical solutions. According to this, the fuel injection system plays an important role because it is greatly responsible for the engine performance and pollutants emission.2-4 To increase the efficiency of the combustion process, thus decreasing exhaust emissions, efforts have to be made in the selection of appropriate fuel injection design. The application of common rail fuel injection equipment is an example of a technical solution that makes controlling the fuel injection timing, pressure, and quantity possible.5,6 * Corresponding author. Dep. Chemistry Physics and Applied Thermodynamics, EPS, C/. Marı´a Virgen y Madre s/n, 14071 Co´rdoba. Phone: +34 957 218332. Fax: +34 957 218417. E-mail:
[email protected]. † Universidad de Jae ´ n. ‡ Universidad de Sevilla. § Present address: University of Cordoba, Department of Chemistry Physics and Applied Thermodynamics, EPS, C/. Maria Virgen y Madre s/n, 14071 Cordoba, Spain. (1) Palomar, J. M.; Cruz, F.; Ortega, A.; Jimenez-Espadafor, F. J.; Martinez, G.; Dorado, M. P. Energy Fuels 2005, 19, 1526-1535. (2) Beck, N. J.; Uyehara, O. A.; Johnson, W. P. The effect of fuel injection on diesel combustion. SAE Paper No. 880299, 1988. (3) Itoh, S.; Sasaki, S.; Arai, K. Advanced in-line pump for mediumduty diesel engines to meet future emission regulations. SAE Paper No. 910182, 1991. (4) Garret, T. K. AutomotiVe fuel and fuel systems; Pentech Press & SAE, Inc.: Warendale, PA, 1994. (5) Edwards, S. P.; Pilley, A. D. The optimisation of common rail FIE equipped engines through the use of statistical experimental design, mathematical modeling and genetic algorithms. SAE Paper No. 970346, 1997; 143-161. (6) Needam, J. R.; Bouthenet, A. Competitive fuel economy and low emissions achieved through flexible injection control. SAE Paper No. 931020, 1993.
To develop an appropriate technology, long experimental tests that make tabulating the injection system performance under different engine operating conditions possible are needed. However, this method is highly costly and time-consuming. For this reason, a more efficient way to perform this task can be accomplished by means of optimal design.7-9 In this sense, several simulation models of diesel fuel injection equipment have been developed over the last years.10,11 In the beginning, those models used linearized wave equations to calculate primary parameters, and the injection pressure was low enough to ignore the nonlinearity of the fuel flow without incurring significant error. When very high injection pressure became necessary, nonlinearity and secondary parameters began to affect the process.12 Nowadays, McKenzie13 found that computational fluid dynamic (CFD) model software programs could be used to acquire reliable results of the fluid parameters within a unit injector. The data results from this model were used to design better-performing unit fuel injectors. Also, other authors have applied mathematical models to design conventional fuel injection equipment for a diesel engine. Ficarrella et al.14 developed a mathematical model based on mass momentum (7) Kegl, B. Proc. Inst. Mech. Eng. 1995, 209, 135-141. (8) Kegl, B. J. Mech. Des. 2004, 126, 703-710. (9) Manzie, C.; Palaniswami, M.; Ralph, D.; Watson, H.; Yi, X. Trans. ASME 2002, 124, 648-658. (10) Becchi, G. A. Analytical simulation of fuel injection in diesel engines. SAE Paper No. 710568, 1971; Vol. 80, p 30. (11) Wylie, E. B.; Bolt, J. A.; El-Erian, M. F. Diesel fuel injection system, simulation and experimental correlation. SAE Paper No. 710569, 1971. (12) Lee, H. K.; Russell, M. F.; Bae, C. S. Proc. Inst. Mech. Eng. 2002, 216, 191-204. (13) McKenzie, E. A., Jr. A transient computational fluid dynamic model of a unit injector using methanol fuel. Ph.D. Thesis, 228 pp; Mechanical and Aerospace Engineering Department, West Virginia University, 1998.
10.1021/ef0602022 CCC: $37.00 © 2007 American Chemical Society Published on Web 12/10/2006
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conservation laws to simulate the injection system behavior with rotative pump, while Lee et al.12 studied diesel fuel injection equipment incorporating nonlinear fuel injection. Kegl7,15 proposed an optimal design of conventional in-line fuel injection based on a dynamic multibody system and continuous/discrete design variables, respectively. A previous work describing a mathematical model to analyze the system response in terms of injection rate, total flow per cycle, residual pressure, etc. helped the authors in the design process of a rotary pump-based injection system.1 However, this model was not enough to undertake the inverse problem. On the basis of this idea, in the present work, the main objective was to approach the inverse problem, thus to reproduce the objective law by means of the system dimensional characteristics provided by the target law. The injection system used during this study was equipped with a Bosch rotary pump of the type VE.1 Materials and Methods 1. Optimization of the Injection System. 1.1. Problem Approach. The design problem within an optimization strategy may be generally defined as a nonlinear problem in the following form:16 min φ(x)
(1)
subject to φi[y(t, x), x, t] e 0;
0 e t e T; i ) 1, ..., j
(2)
and the state equation h(t, y, y˘ , x) ) 0
(3)
where φ(x) is a scalar function called the objective function and the set of functions (eq 2) are noted as constraint functions. The subscript j denotes the number of constraints, whereas T is the length of the relevant time intervals. The vector x ∈ Rn is the assemblage of all the structural parameters, which may be modified independently whereas the vector y ∈ Rm is the set of all the possible mechanical system’s responses. Finally, eq 3 corresponds to the mathematical model of the injection system,1 resulting in the target law which is used to generate the objective function. In order to solve the problem, it must be found for x ∈ Rn that while satisfying the constraints (eq 2) the quantity φ(x) is minimized. The generic process for solving this inverse problem by a parameter optimization technique is shown in Figure 1. The comparison of both the modeled and target curves results in the objective function. To determine whether the algorithm has reached the objective or else the structural parameters must be redefined, a termination criterion must be fixed. First, this convergence criterion considers the difference between the objective functions of two iterations. Then, it considers the difference between the model parameters of those two iterations. Finally, the value of the gradient of the current step “k” is calculated. In each of these steps, the error “” is insignificant, being 10-6. The algorithm used in this study is known as the Newton method. The main reasons for its use lie in the fact that this is a quadratic model of great simplicity and efficiency, besides its having good convergence properties. The numerical difficulties and errors found with this method usually occur due to the fact that the quadratic model is a poor approximation outside the proximity of the current point. In general, the errors in finite-differences approximations depend on factors such as the large number of function derivatives, which cannot be evaluated. (14) Ficarella, A.; Laforgia, D.; Cipolla, G. J. Eng. Gas Turbines Power 1990, 112, 317-323. (15) Kegl, B. Trans. ASME 1996, 118, 490-493. (16) Gill, P. E.; Murray, W.; Wright, M. H. Practical optimization, 11th ed.; Academic Press: London, 1997.
Figure 1. Generic optimization algorithm to solve the inverse problem.
1.1.1. ObjectiVe Law. One of the main characteristics of the injection system is the performance of the injection rate1 q˘ [y(t, x), x, t]. Thus, it can be stated that this law influences the combustion process and therefore the specific consumption, the emission of pollutant gases, and the combustion noise.17 Therefore, it is appropriate to involve the shape of the curve from the injection rate in the objective function. In general, it is accepted that the beginning of the injection process uses a reduced volume of fuel in order to maintain low levels of noise and NOx. On the other hand, the supplied fuel in the last stage of the injection process must be very small since the fuel in this stage is introduced at a low atomization level, which contributes to the increase in the emissions of HC and soot particles.18 Nevertheless, it should be noted that, using the injection rate as the objective law, the discontinuity of the first and second derivatives of the real injection rate may produce inexact results,19 more complex procedures being necessary to solve this problem successfully.20 Another important characteristic from the injection system is the needle lifting law,1L[y(t, x), x, t]. The form of this law is related to the injection rate, which in addition affects the system components dimensioning in that it defines the magnitude of the impacts between the needle and the seating; therefore, it may be included as an objective function. One of the main advantages that provides the shape of the chosen ideal law (the one associated to the needle lifting) comes from the continuity of the first and second derivatives,1 which implies that the optimization algorithm to be used becomes stronger than using an injection rate law. For these purposes, an ideal or objective law Lp(t) related with the injection system objective should be defined. As the system presents rotary cyclic behavior, the temporal variable t has been replaced by θ (cam turn angle). Both are related to the mean pump angular speed. 1.1.2. Variables to be Optimized. With regard to the number of variables to be included in the optimization strategy, in principle, there are no constraints. However, the computational load derived from the optimization process, and the mathematical model of the injection system itself, makes a large number of variables unfeasible. (17) Challen, B.; Baranescu, R. Diesel engine reference book, 2nd ed.; Elsevier: New York, 1999. (18) Heywood, J. B. Internal combustion engine fundamentals; McGraw Hill: New York, 1988. (19) Kegl, M.; Butinar, B. J.; Kegl, B. Commun. Numer. Methods Eng. 2002, 18, 363-371. (20) Kegl, B. J. Mech. Des. 1999, 121, 159-165.
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Table 1. Values of the Cam Law Polynomial (H ) b0 + b1θ + b2θ2... + b6θ6; H in mm, θ in rad) term order sixth fifth fourth third second first zero
approx to the measured values
maximum values
minimum values
optimized values (ramp slope increase, Figure 3)
optimized values (ramp slope decrease, Figure 7)
3.843 × 10-2 -5.26 × 10-2 -9.8315 × 10-3 -3.314 × 10-2 -1.1258 -3.7486 × 10-2 2.2048
4.6116 × 10-2 -4.208 × 10-2 -7.8652 × 10-3 -2.6512 × 10-2 -0.9 -2.9989 × 10-2 2.6457
3.071 × 10-2 -6.31 × 10-2 -11.7978 × 10-3 -3.9768 × 10-2 -1.351 -4.4983 × 10-2 1.764
3.9398 × 10-2 -5.4794 × 10-2 -8.309 × 10-3 -2.8581 × 10-2 -1.2584 -4.0554 × 10-2 1.8705
3.843 × 10-2 -5.2619 × 10-2 -9.8627 × 10-3 -3.2394 × 10-2 -1.1531 -3.7462 × 10-2 2.1948
The following two groups of variables, treated separately, have been used: (a) The first is the cam’s law. In this case, a sixth-order polynomial has been used to model the cam’s law (see Table 1). Therefore, the seven coefficients which define the geometry of the cam are the independent variables which set the x ∈ R7 vector present in the initial approach (eqs 1-3; xi are the polynomial coefficients to be optimized). The equations for the cam rise and fall were selected according to MacKenzie,13 avoiding more complex approximations.20 This method is defined for a reduced number of parameters, up to seven. Increasing the number of parameters to be optimized over ten, the proposed method does not model the system accurately. (b) The second group is the system’s parameters. Only those parameters which do not determine the loss of load or discharge coefficients have been considered. The variables are the following: the injector needle mass (Mi), the spring constant of the impulsion valve (Kv), injector spring constant (Ki), tube diameter (Dt), and impulsion valve mass (Mv). These parameters constitute the x vector that appears in eqs 1-3. In this case, xi are the system parameters to be optimized. 1.1.3. ObjectiVe Function. The objective function measures the lack of agreement between the objective law and the one given by the injection system model. It is clear that there is not just one way to quantify the lack of agreement between bothacurves; thus, the use of a certain function obeys the gained experience in its performance in similar structures. The adopted objective function is defined as follows: φ(x) )
1 θf - θ i
∫
θf
θi
[LP(θ) - L(y(x, θ), x, θ)]2 dθ
(4)
where (θi, θf) is the interval where the optimization process is carried out and where LP(θ) and L[y(x, θ), x, θ] are defined. It should be noted that the chosen rule (quadratic function) is the one that presents the best performance in a great diversity of optimization mathematical problems in different engineering fields.21-25 Nevertheless, there are no limitations to use any other objective rule as long as the associated errors are considered in both the positive and negative parts of the objective function. 1.2. Methodology Used in the Optimization Process. There are two main reasons to choose a quadratic model instead of a linear one: simplicity and efficiency. If the first and second derivatives of the φ function are known, a quadratic model of the objective function can be obtained by taking the first three terms from the Taylor-series expansion of the current point (vector x), thus: 1 T φ(x bk + b p ) ≈ φk + b g kTb p+ b B kb p p G 2
(5)
(21) Rizzoni, G.; Zhang, Y. Mech. Syst. Signal Process. 1994, 8, 275297. (22) Fantozzi, F.; Desideri, U. Proc. Inst. Mech. Eng. 1998, 212, 299313. (23) Edwards, S. P.; Grove, D. M.; Wynn, H. P. Statistics for engine optimization; John Wiley and Sons: Canada, 2000. (24) Chuawittayawuth, K.; Kumar, S. Renewable Energy 2002, 26, 431448. (25) Cruz, F. Analisis de metodologias de optimizacion inteligentes para la determinacion de la presion en camara de combustion para motores alternativos de combustion interna por metodos no intrusivos. Ph.D. Thesis, University of Seville, Seville, Spain, 2005.
This may help to formulate the quadratic function in terms of p (the step to the minimum). The minimum of the right-hand side in eq 7 may be obtained if pk is a minimum of the quadratic function 1 T p G p+ b B kb p φ(p) ) b g kTb 2
(6)
where a stationary point pk (vector p in the iteration k) from eq 8 satisfies the following linear equations system: G B kb p k ) -g bk
(7)
where the vector b pk is the unknown quantity in the system and it represents the search direction. This method works with discrete data because the present functions are not analytical. 1.3. Algorithm Structure. The objective is to find the value of the necessary parameters that will make it possible to find the needle lifting law. The law to be defined will provide some desired results such as minimizing the combustion noise (ascending ramp reduction) or to produce needle closings with the maximum possible slope. 1.3.1. Variables of the Algorithm. The variables, also considering the previously described needle lifting laws L(θ) and LP(θ), are the following: θ instantaneous angle turned by the cam; PYk parameter column vector at iteration k in the process of calculation (normalized so they take values close to unity at the beginning of the algorithm). This is the variable to be optimized, composed of the typified variables of xi to be optimized, ai (i ) 1, 2, ... n), defined as follows:
[
∀ xi ∈ xj f ai ∈ [0, 1] x ) [x1 ... xn] PYk ) [a1 a2 a3....an]k
i ) 1...n xi - xi,min ai ) xi,max - xi,min
xst ) [a1... an]
]
(8)
Where, pk is the search direction vector and R is the updating coefficient of the solution. Few calculation steps are required in this algorithm to take values close to unity. H and D are diagonal matrices of n-order whose elements are Hj ) (xj,max - xj,min)-1 and Dj ) Hj-1; K and C are vectors with elements equal to Kj ) -xj,min/ xj,max - xj,min and Cj ) xj,min. These last four terms are used for normalization purposes,16 according to xst ) Hx + K and x ) Dxst + C. Here, n is the number of parameters to be optimized; φ is the objective function defined in eq 5; xi (i ) 1, 2, ..., n) are variables to be optimized; P0Yki is the parameter’s column vector which is equal to zero in all its components, except in row i, which takes the same value as the component i of the vector PYk; P0Ykj is the parameter’s column vector which is equal to zero in all its components, except in row j, which takes the same value as the component j of the vector PYk; p0yki is the ith component of column vector P0Yki; p0ykj is the jth component of column vector P0Ykj; β is the parameter whose value is lower than unity and is used to evaluate the derivative by incremental quotient. Its value is related to the quality of the results. Also, b gk is a column vector with ncomponents that contains the gradients at point k. The gradient is
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Figure 2. Increase in ascending slope of the injector needle lifting law (1500 rpm).
calculated by incremental quotient for each one of the variables, such as b gk )
[
]
∂φk ∂φk ∂φk ∂φk ... ∂a1 ∂a2 ∂a3 ∂a4
(9)
G B k is the Hessian is the matrix formed by the second-order derivatives evaluated at point k. Its value must be estimated from the finite differences of the objective function. It is defined as follows: ∂φk2 ∂φk2 ∂φk2 ∂φk2 ‚‚‚ ∂a1∂a1 ∂a1∂a2 ∂a1∂a3 ∂a1∂a4
∂φk2 ∂φk2 ∂φk2 ∂φk2 ‚‚‚ ∂a3∂a1 ∂a3∂a2 ∂a3∂a3 ∂a3∂a4
F2 ) φ[D(PYk + βP0Yki - βP0Ykj) + C] F3 ) φ[D(PYk - βP0Yki + βP0Ykj) + C] F4 ) φ[D(PYk - βP0Yki - βP0Ykj) + C] F1 - F2 - F3 + F4 4β2p0yki p0ykj
(13)
And finally, the Hessian is evaluated as follows: (10) G ˇ )
∂φk2 ∂φk2 ∂φk2 ∂φk2 ‚‚‚ ∂a4∂a1 ∂a4∂a2 ∂a4∂a3 ∂a4∂a4 l l l l 1.3.2. Algorithm. The algorithm is designed and calculated as follows: (1) Start. Initial values of the scaled values (xst) of the parameters (x) are defined so all of them have a unique standardized value equal to unity: PY0 ) [ 1 1 1 1...1 1 1 1 ],
that is ∀ i ) 1...n w ai,k ) 1 being k ) 0 (11)
(2) Calculation of the Gradient Vector. The gradient is calculated by incremental quotient for each one of the parameters i to be optimized (polynomial’s coefficients or system’s parameters). If required, the given expression may be replaced by others with higher numbers of terms. The component of the resulting derivative vector is the following: ∂φk φ[D(PYk + βP0Yki) + C] - φ[D(PYk - βP0Yki) + C] ≈ ∂ak 2βp0yki
F1 ) φ[D(PYk + βP0Yki + βP0Ykj) + C]
G ˆ kij )
∂φk2 ∂φk2 ∂φk2 ∂φk2 ‚‚‚ ∂a2∂a1 ∂a2∂a2 ∂a2∂a3 ∂a2∂a4 G Bk )
(3) Hessian Calculation. The Hessian is estimated from the following incremental expression:
(12)
G ˆk + G ˆ kT 2
(14)
This latter step is necessary because even for any analytical function it verifies that the Hessian elements placed in symmetrical sites are equal, that is G ˆ kij ) G ˆ kji. When carrying out the numerical approximation of the second derivative by increments, there are always differences that are reduced with the previous expression. (4) Search Direction Calculation pk. The search direction pk is evaluated by solving the linear equation system defined in eq 8, where vector pk is the system’s unknown quantity. (5) Updating of Vector PYk. The updating of vector PYk is carried out by adding to the previous value the new one multiplied by a coefficient R: PYk+1 ) PYk + Rpk
(15)
Next, the objective function and the convergence criterion are evaluated. Then, if the error goes beyond the fixed bounds, we must go back to step 2 until convergence is achieved, as shown in Figure 1. 1.4. Practical-Type Considerations. We can see that the different options for the finite-differences intervals occasionally produce acceptable approximations although they may give bad results due to the presence of bad scaling or due to an atypical starting point.
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Figure 3. Effect of increasing the ascending slope of the injector needle lifting law on the cam’s law (1500 rpm).
Figure 4. Effect of increasing the ascending slope of the injector needle lifting law on the injection rate (1500 rpm).
It is worth mentioning the different types of error and particularities, which may arise. (a) Errors in approximations by finite differences. In general, errors in approximations by finite differences depend on factors such as the large number of derivatives of the function which cannot be evaluated. The most important errors are the truncation error (caused by the neglected terms of the Taylor series) and the condition error caused by inaccuracies in the calculated values of the function. When f′(x) and f′′(x) are calculated, the relative errors produced have values between β2 and β4, β being the incremental step. (b) Effect of modifying the step length (R) of the search direction. In order to ensure the convergence, the step length must yield a sufficient decrease in the function. The usual agreement for descent step methods is to take the value of unity in order to achieve the best convergence. Anyway, in the present study, different values were tested: 0.2, 0.5, 0.8, and 1.0. The model showed the highest effectiveness with the last value. (c) Effect of the scale difference. The purpose of variables scaling is to ensure that all the variables have similar weight or importance during optimization. If the Hessian matrix is ill-conditioned, the function
will vary much faster in some directions than in others leading to bad scaling, since changes for a given point will not produce similar changes in the function. When the variables are scaled by the linear transformations described before, that is, xst ) Hx + K and x ) Dxst + C, the derivatives are also scaled. This assures that the magnitude of a variable does not vary substantially during minimization.16 Once the cam’s law is optimized, the upper limit of vector x in the optimization algorithm (with maximum standardized value of 1) is fixed to a value 20% higher than initial data, while the lower limit (with minimum standardized equal to 0) is fixed to a value 20% lower than the initial value. In another case, these limits will vary up to 50% above and below the upper and lower limit, respectively, compared to the initial values; thereby, a solution for the structural parameters close to the initial values is guaranteed (Tables 1 and 2). (d) Unconstrained optimization. Local convergence properties make Newton’s method a very attractive algorithm in the case of unconstrained minimization. To accomplish the optimization, the sufficient conditions can be verified as follows: in the solution, the gradient vector must be zero or less
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Figure 5. Accumulated initial and optimized flows (1500 rpm).
Figure 6. Reduction of the ascending slope of the injector needle lifting law (1500 rpm).
than a certain error (as explained in the termination criteria) and the Hessian matrix must be positive definite. Additionally, for a generic nonlinear function, the method converges to the target law if the step lengths converge to unity and the initial values of structural parameters are close to the solution (as previously mentioned in particularities b and c). So, in this study, the optimization process will be carried out without constraints. According to this, eq 2 will not be taken into consideration.
Results and Discussion First, the viability of obtaining a law of needle lifting similar to the lifting in the double spring injector (lifting curve with “step”) was studied by using the cam’s law as an independent
variable. The optimization process was verified to be unfeasible, but due to fact that the injection system itself is unable to give that answer; that is, the problem lies within the physics associated with the pumping system. Next, once the optimization methodology was applied to the simulation process, the obtained results are analyzed. All these cases were carried out by testing with different values of the updated coefficient of the solution R. The best trained values were obtained considering R ) 1, since the differences in the results obtained considering lower values were not significant. On the other hand, convergence was achieved with a fewer number of steps. With regards to β (parameter for the evaluation of the derivative by incremental quotient), several values were
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Figure 7. Effect of decreasing the ascending slope of the injector needle lifting law on the cam’s law (1500 rpm).
Figure 8. Effect of decreasing the ascending slope of the injector needle lifting law on the injection rate (1500 rpm). Table 2. System Parameter Values To Be Optimized parameter
real initial value
max value
min value
optimized value
injector needle mass (g) injector spring const (N/mm) tube diameter (mm)
23.77 155.62 2
35.65 233.24 1
11.9 80 3
29.72 214.8 2.06
checked, and it was observed that the results were not sensitive to important variations of β. Thus, the adopted value of β was 0.01. The cases under study were the following: 1. Objective Law: Ramp Slope Increase of the Needle Lifting Law. Second, by using the cam’s law as an independent variable, the viability of obtaining an injector needle lifting law with the ramp slope increasing with regard to the initial value was studied. Figure 2 shows that the optimization algorithm works perfectly well, since it is able to significantly increase
the ramp slope of the needle lifting law in relation to the original one. The singularity which appears at the end of the ascending ramp obeys the calculation method since the objective function is discontinuous. The same effect can be observed in Figures 6 and 10. Figure 3 shows the initial and optimized needle laws graphs, where the higher ascending ramp of the cam can be appreciated. The initial, maximum, minimum, and optimized values of the cam’s law polynomial are shown in Table 1. There, the target values are very close to the initial ones. This assures the convergence and demonstrates the effectiveness of the unconstrained optimization applied to the cases under study. Figure 4 shows the optimized injection rate as a result of applying the algorithm when considering the needle lifting as the objective law, with an increase in the ascending ramp with respect to the initial value. It is important to underline that the
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Figure 9. Accumulated initial and optimized flows (1500 rpm).
Figure 10. Effect of modifying the parameters of the injection system (Mi injector needle mass, K spring injector constant, and Dt internal tube diameter; 1500 rpm).
use of the injection rate as the objective law would have been more convenient, since this law is the one which most determines the combustion process in diesel engines; however, the bad performance (discontinuity in the first and second derivative) in the injection rate makes the optimization results, according to Newton’s algorithm, less valid. Nevertheless, despite using the needle lifting law as the objective function, it affects the injection rate, thus achieving great gradients in the rate when the needle lifting ascending slope grows, as shown in Figure 5. It can be seen that, for a given revolution angle of the pump shaft, the optimized injection rate is always larger than the initial one. Finally, it should be noted that the global injected flows are notably modified because constraints are not included in the algorithm. For this reason, the net injected flow is a parameter which is not controlled in the current algorithm structure.
2. Objective Law: Decrease in the Ascending Ramp Slope of the Needle Lifting Law. Results are related to section 1.1.2 group a in the Materials and Methods section (cam’s law). By using the cam’s law as an independent variable, the viability of obtaining an injector needle lifting law with less ascending slope than the initial one was studied. Figure 6 shows how the optimization algorithm is able to reduce the ascending ramp slope with respect to the initial curve. Figure 7 shows the similarities between initial and optimized cam law curves. The optimized values of the cam law polynomial are shown in Table 1. Results are very close to the initial values, thus avoiding the need for a constrained optimization procedure. Figure 8 shows the resulting optimized injection rate after applying the algorithm when the needle lifting law is applied as the objective law, that is, with the ascending ramp with less slope than the initial one. As in the previous case, the use of the injection rate as the
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Figure 11. Effect of modifying the parameters of the injection system on the injection rate (1000 rpm).
Figure 12. Effect of modifying the parameters of the injection system on the injection rate (1500 rpm).
objective law would have been more convenient; nevertheless, despite using the needle lifting law as the objective function, it can be appreciated that the injection rate is reduced when the lifting ascending slope decreases. This effect can be appreciated in Figure 9 where, for a given pump shaft rotary angle, the optimized injection rate is always smaller than the initial one. 3. Objective Law: Progress of the Needle Lifting Law with Regard to the Initial One. The system parameters were taken as the optimization variables to evaluate the viability of the optimization model (results related to section 1.1.2 group b in the Materials and Methods section). The objective was to obtain an injector needle lifting law at 1500 rpm which started the lifting movement in the same pump shaft rotary angle that is produced when the injection system model (without any sort of optimization process) is carried out at 1000 rpm.
According to this, the target was to optimize all the parameters (x ∈ R5). However, mainly due to the unconstrained optimization, results appeared to be physically impossible and incompatible with the system characteristics. Then, the level of exigence of the procedure was reduced, decreasing the number of parameters to be optimized. Finally, acceptable results were reached for the following parameters: injector needle mass, injector spring constant, and tube diameter, being x ∈ R3. Table 2 shows the initial, maximum, minimum, and optimized values of these three parameters. It can be seen that the target values are very close to the initial ones. Once the parameters (massoptimal, Koptimal, diameteroptimal) were obtained, the injection system was evaluated at 1000 rpm, and it was verified that the needle lifting started slightly earlier (in terms of pump shaft rotary degrees) than the model solution with the same parameters at 1500 rpm. Thus, the difference
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Figure 13. Injection rate at different revolutions with the initial parameters.
Figure 14. Injection rate at different revolutions with the parameters provided by the optimization process.
between the initial angle at 1500 rpm and the optimized one at 1000 rpm is smaller than the difference between the initial angle at 1500 rpm and that at 1000 rpm without optimization: optimal unoptimized Roptimal - Runoptimized ini,1500 - Rini,1000 < Rini,1500 ini,1000
(15)
The aim of decreasing the gap is to eliminate the advance to the injection, thus the injection pump manufacturing cost would decrease significantly. Figure 10 shows the response of the model by modifying the injection system parameters by advancing around 3.8° at the beginning of the injection process. Figure 11 shows the effect on the injection rate at 1000 rpm, where we can see that the injection starting angle is not modified (approximately 0.2°). Figure 12 shows the same effect for 1500 rpm where an important advance in the injection (3.8°) can be observed. Figure 13 shows the injection rate related to the initial parameters, whereas Figure 14 shows the injection rate results including the optimized parameters. The number of iterations varied between 300 and 750 (maximum fixed k value), considering the convergence criterion. The approximate CPU time of the model calculation using a
Pentium IV 2.3 GHz microprocessor was 5 s. According to this, the number of parameters to be optimized affects the time requested to perform each of the iterations. Thus, the total computation time to solve the algorithm varied between 3.5 and 50 h. Conclusions We can conclude that the proposed methodology can be used to accomplish the optimization of the parameters associated with the performance of a fuel injection system. The inverse problem, in addition to the identification a parameter technique, has shown its effectiveness. The optimization of the system parameters by means of an objective function based on the needle lifting law bring several advantages compared to the use of the injection rate; i.e., constraints are not needed. On the other hand, other parameters different than the presented ones do not give any satisfactory result. This is due to their small incidence over the injection model, the existence of a strong correlation between the parameters, and the existence of a high nonlinear relation between those parameters and the output variable. This finding highlights the importance of finding out the more distinctive
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parameters of the model with the help of a sensitive analysis and justifies the need for using an optimization technique, i.e., genetic algorithms, independent of the derivatives of the objective function. However, according to our results, it seems that the parameters evaluated during this study are some of the most important ones related to the injection model. This study shows the suitability of the proposed optimization technique toward the improvement of the design of the injection system parameters. First of all, an appropriate modeling as well as the definition of more characteristic parameters is needed. This is independent of the operation rule of the injection system, either common rail or injection pump. In any case, the first step
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must consist of defining the desired outputs related to engine combustion, emissions, etc. Note Added after ASAP Publication. There was an error in reference 8 in the version published ASAP on December 10, 2006; the corrected version was published ASAP December 15, 2006. Acknowledgment. The authors thank the firm “La Electro Motor Diesel S.A.” (Sevilla, Spain), which provided the test bench to carry out the tests. EF0602022
