Development of a Computer Model to Simulate the Injection Process

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Development of a Computer Model to Simulate the Injection Process of a Diesel Engine Jose M. Palomar,† Fernando Cruz,† Antonio Ortega,† Francisco J. Jimenez-Espadafor,‡ Gregorio Martıńez,§ 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, Department of Energetic Engineering and Fluid Mechanics, ESI, Universidad de Sevilla, Camino de los Descubrimientos s/n, Isla de la Cartuja, 41092 Sevilla, Spain, and Ingenieria y desarrollo de las instalaciones, S.L, Poligono industrial Los Olivares, C/Villatorres no. 30, 23009 Jae´ n, Spain Received January 7, 2005. Revised Manuscript Received April 7, 2005

The diesel engine injection system is greatly responsible for the emission of pollutants to the atmosphere, noise generation, and engine performance. Proper synchronization of the exact amount of fuel that can be burnt according to each crankshaft position is essential to achieve maximum power, consumption economy, and combustion without residues. In this sense, when designing an injection system, all of these factors must be taken into account. During the optimization, however, the improvement of some of them may result in detriment to others. For this reason, to achieve the system optimization, the relationship between the aforementioned variables and the parameters that characterize the injection must be established. According to this, it is important to design a strategy that allows the named “reverse modeling of the injection system”, that is to say, to determine the injection system dimensions according to the system requirements. In the present work, an iterative computer model based on the continuity and energy equations, solved by finite differences algorithms, has been proposed. This model has been designed for a Bosch rotary pump and makes it possible to plot the performance of the pressure lines in the different combustion chambers, the valve lift curves, and the fuel flow injected per cylinder and cycle. In this sense, we can conclude that the proposed computer model predicts accurately the opening and closing processes in the injector needle, as well as the pressure curves in the last division of the high-pressure tube.

Introduction The most important part in diesel engines is the fuel injection system. This system is greatly responsible for the engine performance and pollutants emission.1,2 The injection system measures out the right amount of fuel to be injected, synchronizes the injection action with the engine operation, adjusts the injection speed and the fuel flow, splits up the fuel into very small particles, and distributes the fuel appropriately in the combustion chamber. The timing and pressure of the injection process vary proportionally to the engine dimensions. In this sense, small direct injection engines present injection periods of approximately 25° of the crankshaft angle and injection pressures over 400 bar. However, large engines have injection periods up to 40° of the crankshaft angle and injection pressures higher than 1000 bar. * Author to whom correspondence should be addressed. Address: EUP de Linares, C/. Alfonso X el Sabio, 28, 23700 Linares (Jaeń). Phone: +34 953 648526. Fax: +34 953 648508. E-mail: [email protected]. † Universidad de Jae ´ n. ‡ Universidad de Sevilla. § Poligono industrial Los Olivares. (1) Beck, N. J.; Uyehara, O. A.; Johnson, W. P. The effect of fuel injection on diesel combustion; SAE Paper No. 880299, 1988. (2) Itoh, S.; Sasaki, S.; Arai, K. Advanced in-line pump for mediumduty diesel engines to meet future emission regulations; SAE Paper No. 910182, 1991.

To increase the efficiency of the combustion process, decreasing gradients and combustion peaks, it is important to determine the exact point to start the injection process, as well as the specific law that rules the process. Also, to decrease pollutants emission such as HC, NOx, or PM, it is important to deliver the exact amount of fuel that can be burnt according to each crankshaft position, thus controlling the pressure increase. On the other hand, an adequate fuel atomization speeds up the combustion starting process, causing fuel particles to give off steam. Thus, a higher surface is exposed to oxygen to combine with the fuel. In this sense, an adequate fuel distribution all over the combustion chamber is guaranteed, otherwise, part of the mixture could remain unburnt, thus decreasing engine power. Current interest for small diesel engines on cars and the increasing demand for higher accuracy of injection control systems have led to the development of computer models which are able to predict the performance of these systems under different operating conditions. However, the sole use of mathematical calculations makes it extremely difficult and complex to establish the existing relationships between parameters and variables related to the injection. Therefore, to deal with the subject, two possibilities are considered. The first one involves the development of long experimental tests that allow tabulating the performance under different

10.1021/ef0500080 CCC: $30.25 © 2005 American Chemical Society Published on Web 05/11/2005

Simulation of the Diesel Engine Injection Process

operating conditions. However, this method incurs high costs and is time-consuming. The second method requires the development of computer models to simulate the system performance to avoid the tedious task of applying the mathematical model each time, thus accelerating the investigation process.3-6 Another important phenomenon to take into consideration is cavitation. There may be points of the system where fuel pressure can be equal or lower to its saturation pressure. Should this happen, the fuel would start boiling, thus steam bubbles would be dragged by the flow and displaced to areas of higher pressure than saturation pressure, where the steam would condense abruptly. Cavities previously occupied by the steam would leave spaces, which would be abruptly filled by the surrounding fuel. These impacts may produce very high pressures at high frequency, which damage the system and reduce its useful life. This phenomenon is known as cavitation and should be avoided all through the piping system since the system may be damaged and may incur in overall poor performance, as well as a reduction in the engine life. In this sense, several authors have developed mathematical models to carry out calculations of cavitating flows in hydraulic components of diesel injection systems.7,8 Likewise, nozzle dripping and leaking can affect the performance of the system and must be avoided. In the present work, the main objective was to assess and validate a computational model that comprises a pump-tube-injector system. This simulation model was designed for a Bosch rotary pump, model VE.9 The mathematical model was based on an iterative model, mainly using the continuity equation, the energy equation, and the model proposed by Yamaoka.10 Moreover, this model took into consideration the changing characteristics of the fuel, the inlet and outlet discharge coefficients in the chambers, losses in the piping system, and possible cavitation. Materials and Methods 1. Injection System. The injection system used during this study was equipped with a Bosch rotary pump of the type VE. In four-stroke engines, the rpm of the rotary pump matches exactly the rpm of the camshaft; therefore, the pump turns at half of the rpm of the crankshaft. The Bosch rotary pump presented only one pumping device for engines with several cylinders. The fuel pumped by the piston is distributed through a slot to all the cylinders in the engine. To validate the proposed model, tests were carried out on a MAGASA F-60 test bench (La Electro Motor Diesel S.A., Seville, Spain), for maximum injector pressures in a range of (3) Arau´jo, A.; Lima, J. L. F. C.; Gracia, J.; Poch, M.; Alonso, J.; Bartrolı´, J.; del Valle, M. Anal. Chim. Acta 1995, 310, 289-296. (4) Manzie, C.; Palaniswami, M.; Ralph, D.; Watson, H.; Yi, X. Trans. ASME 2002, 124, 648-658. (5) Kegl, B. J. Mech. Des. 2004, 126, 1-8. (6) Kegl, B. Proc. Inst. Mech. Eng. 1995, 209, 135-141. (7) von Dirke, M.; Krautter, A.; Ostertag, J.; Mennicken, M.; Badock, C. Oil Gas Sci. Technol. 1999, 54, 223-226. (8) Catania, A. E.; Dongiovanni, C.; Mittica, A.; Negri, C.; Spessa, E. Trans. ASME 1999, 121, 186-196. (9) Palomar, J. M., Disenõ y simulacioń del comportamiento funcional de la bomba rotativa Bosch de inyeccioń diesel; VIII Congreso Internacional de Ingenierıá Gra´ fica; Universidad de Jaeń: Jaeń, Spain, 1996; pp 593-611. (10) Yamaoka, K.; Saito, A. Computer technique for evaluation of cavitation characteristics of certain phases of fuel injection in fuel injection system; SAE Paper No. 730663, 1973.

Energy & Fuels, Vol. 19, No. 4, 2005 1527 130-150 bar, pipe lengths of 500, 600, and 800 mm, and 2 mm internal diameter for all cases. 2. System Modeling. 2.1. Direct Problem. The computer model was based on the continuity and energy equations. The differential equations presented in this model were solved by means of a discretization technique using finite differences.11 The present method provides information about the relationship between the involved factors. The solution can be obtained without automatic control of the temporal increase by comparing the admissible error, previously defined, during the calculation cycle. As a convergence criterion, the relative error between the calculated variable and the presumed one has been minimized. The considered constants to evaluate the convergence criterion for lift have been 10-4 and 10-3 for volume variations. When a cycle is completed, the convergence criterion for the system residual conditions is the minimum relative error. The system has been divided into several chambers, and the equations have been applied to each one of them. 2.2. Equations to Represent the Fluid Motion. To model the injection process, the injection system has been divided into three parts: pump, injection pipe, and injector. The equations used to design the injection system were the continuity equation, also named mass conservation equation (eq 1), the motion equation (eq 2), and the energy equation (eq 3):

∂F/∂t + ∇Fv b)0

(1)

F(∂v b/∂t + b v ∇v b) + ∇P ) ∇ B τ + FBf m

(2)

F(-∂e/∂t + b v ∇e) ) ∇(k∇T) - P∇v b + Φv + QR + QQ (3) where e represents the internal energy, t is the time, b v is the speed field, F is the density, T is the temperature, k is thermal conductivity, P is the pressure, Φv is the viscous dissipation, QR is the radiation heat transfer, and QQ is the heat transfer related to the chemical reaction. The first addend in eq 1 represents the density variation vs time within a given volume, while the second indicates the volume variation within the volume where it is immersed. In eq 2, the first addend indicates the product of the density times the acceleration of the fluid particle. This product must be equal to the surface and volume forces, which take place over a unitary elemental fluid volume. In this sense, fm represent the external forces per mass unit acting on the fuel and τ represents the pressure and friction forces. Finally, the principle of conservation of total energy in a fluid volume is equal to the work per time unit, meaning the external forces power (mass or surface forces) which acts on the fluid volume plus the heat received from outside per unit of time. 3. Fuel Variables Analysis. The main variables that affect the fuel performance are the viscosity, the compressibility coefficient, and the propagation speed. Viscosity indicates the resistance offered by a fluid to be cut. This resistance depends on the cohesion forces and the transfer speed of the amount of motion between molecules. Viscosity depends on pressure and temperature, although the effect of pressure is almost negligible for both liquids and gases. The viscosity of a liquid decreases with temperature, whereas in gases, it is quite the opposite. Also, the isotherm compressibility coefficient, β, is defined as the quotient between the relative volume variation and the pressure increase responsible for that volume variation when there are no temperature variations (eq 4). The reverse of the compressibility coefficient is known as the volumetric elasticity module (eq 5).

β ) - [(dV)T/V]/(dP)T ) -(dV/dP)T/V ) (∂F/∂P)T/F (4) Kv ) -dP/(dV/V) ) dP/(dF/F)

(5)

On the other hand, the propagation speed of a wave originated by a small disturbance or sudden change in a pipe

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Palomar et al.

may be calculated using the continuity and amount of motion equations. The disturbance produced in a specific point would generate a spherical wave, but at a certain distance from that point, the wave front is mainly linear or one-dimensional. According to this, the speed of sound is calculated by eq 6:

a ) xdP/dF

(6)

Also, considering the volumetric elasticity module, eq 7 can be obtained:

a ) xKv/F ) x1/(βF)

(7)

Yamaoka10 described an experimental procedure to obtain the speed of sound in fuel consisting of measuring the required time for a wave to cover a 4 m long pipe. Inside diameter/ outside diameter relationships (mm/mm) of 1.5:6, 2:6, 4:10 were used. Results showed that these relationships do not have an influence on the spreading speed of sound in the fuel. In the present study, speed is considered to be constant and the density is related to the residual pressure and operation temperature. Finally, considering the compression coefficient and sound speed values, the fuel density under different pressure and temperature conditions is worked out. Correlation of these values follows this equation (eq 8).

F ) (0.8430 + 0.00005P - 0.000665t + 3.5tP × 10-7) × 103 (8) It can be seen that the relationship between density and pressure is negligible. Therefore, only the fuel density variation related to the temperature will be considered. 4. Hypotheses. To simplify the mathematical model, some hypotheses are proposed. These are as follows. (1) The force of gravity will not be considered since the height variations in the injection system are negligible; (2) the elastic train in the injection pipe will not be considered due to the relationship between external and internal diameter (6 mm/2 mm), which means high stiffness; (3) the average temperature of the system will be considered instead of the temperature variation vs pressure or time during an operation cycle; (4) the fuel viscosity will be considered to be constant and related to residual pressure and operation temperature of the system; (5) if cavitation appears, the pressure will remain constant and equal to the fuel steam pressure, while steam bubbles will be uniformly distributed in the control volume; (6) dead volumes in the injection system (piston chamber, impulsion valve chamber, nonreturn valve chamber, and injector chamber) will not be considered in relation to inertia and friction effects due to the fuel slower speed in these chambers, in comparison with the injection pipe; (7) due to the short length of the dead volumes related to the pipe length, it will be considered that the pressure wave will spread instantly, meaning pressure increases will take place all through the volume simultaneously; (8) no air is mixed or dissolved in the fuel; (9) the control volume pressure is considered to be uniform for each time interval; (10) in relation to the friction in the pipe, it is considered that, since the duration of an injection cycle is very short, the flow in the pipe does not reach a turbulent state, thus the flow is laminar although with variable speed distribution all through the pipe; and (11) although Ecomard12 developed an expression to correlate leaking to many factors, among them are the piston chamber pressure and the dynamic viscosity of the fuel, in the present work, leaks between piston and jacket will not be considered (11) Catania, A. E.; Dongiovanni, C.; Mittica, A.; Badami, M.; Lovisolo, F. Trans. ASME 1994, 116, 814-830. (12) Ecomard, A. Elements de calcul de l′injection; Institut Francais du Pe´trole (IFP); 1971; ref 19401.

Figure 1. Injection pipe to calculate the W factor. since leaking in a pumping device in good condition has little influence on the operation system. To obtain the friction force equation, the Hagen-Poiseuille equation and the motion equation are used. The HagenPoiseuille is applied to circular pipes full of an uncompressible fluid, laminar flow, and parabolic speed distribution. On the other hand, the motion equation is applied to the parallel and axilsymmetrical flow of an uncompressible fluid, considering a transversal section of the pipe where pressure and density remain constant, while speed depends on the radial coordinate. With the aid of Laplace’s transform, the equation for the friction force is obtained. This equation is composed of two members, one stationary and the other transitory, as shown in eq 9.

F)

∫ [∂V(u)/∂t]FP(t - u) du ) 8πFv∆LV(t) + 4πFv∆L∫ [∂V(u)/∂t]FP(t - u) du (9)

4FvπD∆LV(t)/R + 2FvπD∆L/R

t

0 t

0

5. Calculation Method. The most common method to analyze nonstationary motion in a pipe is the characteristics method. This method has the disadvantage of using the pressure as the main variable, which implies that if cavitation appears, other equations that do not contain that variable must be used. To solve this problem, cavitation is considered negligible by some authors,13 while some others consider cavitation as an alternative factor, though in this case, the boundary conditions applied to the injection pipe extremes do not provide satisfactory results.14-16 In the present work, to avoid this problem, we have applied the methodology proposed by Yamaoka.10 According to this, when the injection system is in operation, fuel may be either in a compression state or in a cavitation state. Therefore, liquid volume variation in the control volume may be described as follows (eq 10). This volume variation (W) varies from positive to negative values.

W ) βVP + Vc

(10)

6. Calculation of the W Factor. To perform this task, the pipe is divided into Nd divisions, with ∆L length, and a volume St∆L (see Figure 1). Through the limit of each division of the pipe and every certain time interval, a defined amount of fuel goes through. This fuel will be noted as accumulated flow, Q (J,I), that is, the accumulated flow in one I division until J instant of time. On the other hand, the fluid in each one of the divisions can be found whether in a state of compression or not and the variations in the initial stages may be the cause or not of unbalance. For each division, the balanced and unbalanced initial volume variations will be noted as WRE (I) and WRT (I), respectively. To obtain the volume variation W(J,I), the continuity equation through any pipe division is applied (see Figure 2). (13) Burman, P. G.; De Luca, F. Fuel injection and control; Simmons-Boardman: New York, 1962. (14) Whiley, E. B.; Bolt, J. A.; El-Erian, M. F. Diesel fuel injection system, simulation and experimental correlation; SAE Paper No. 710569, 1971. (15) Becchi, G. A. Analytical simulation of fuel injection in diesel engines; SAE Paper No. 710568, 1971. (16) Matsuoka, S.; Yokota, K.; Kamimoto, T.; Igoshi, M. A study of fuel injection system in diesel engines; SAE Paper No. 760551, 1976.


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Figure 4. Valve chamber. Figure 2. Pipe division.

Figure 3. Control volume between two divisions. Also, to obtain the accumulated flows required to calculate W(J,I), the motion equation is applied to the control volume placed between two divisions, as shown in Figure 3, thus obtaining eq 11:

F(J,I) ) mR(J,I) + FF

(11)

where F(J,I) is the force related to the pressure, R(J,I) is the fluid acceleration in the control volume, FF is the force obtained from friction, and m is the fuel mass in the control volume. Replacing pressure force and friction force values in eq 11, stating acceleration and speed as finite differences and stating speed as a function of the flow, eq 12 is obtained.

Si[P(J, I - 1) - P(J, I)] ) m[Q(J + 1, I) - 2Q(J, I) + Q (J - 1, I)]/(Si∆T2) + 8πFv∆L[Q(J + 1, I) - Q(J, I)]/Si∆T + J-1

4πFv∆L

∑[V(J - Z + 1, I) - V(J - Z - 1, I)]FP(Z, ∆T) z)3

(12) With the aid of the equations described above and stating the speeds as a function of the accumulated flows, we obtain eq 13:

Q(J + 1, I) ) Q(J, I + 1)/(1 + 8πv∆T/Si + 4πv∆TFP∆T/Si) + Q(J, I - 1) + Q(J, I)(8πv∆T/Si + 4πv∆TFP∆T/Si) - Q(J - 1, I)(1 - 4πv∆TFP∆T/Si) - Q (J - 2, I)4πv∆TFP∆T/Si + Vc(I) - Vc(I - 1) + Wr(I - 1) J-1

Wr(I) - 4πv∆T

∑[Q(J - Z + 2, I) - Q(J - Z + 1, I) - Q z)3

(J - Z, I) + Q(J - Z - 1, I)]FP(Z,∆T)/Si (13) where Z varies from 3 to (J - 1) and I varies from 2 to (Nd 1). This equation allows the calculation of the accumulated flows in the limits of the pipe divisions from the accumulated flows in the previous time instants and the bubble volumes that are located in the divisions in the same periods of time.

Figure 5. Nozzle chamber. If the accumulated flow in the limit of each division is known, the volume variation can be calculated. Thus, the pressure can be also worked out. However, this equation does not permit us to obtain the accumulated flows in the first and last division. As a result, and according to Yamaoka,10 it may be adopted as a boundary condition that the relationship between the volume variation in the nozzle chamber, injector chamber, and their respective volumes are the same that in the adjacent divisions (Figures 4 and 5). According to this, eq 14 is written

W(J + 1, 1)/(Si∆L) ) Wv(J + 1)/Vv(J + 1) and W(J + 1, N)/(Si∆L) ) Wl(J + 1)/Vl(J + 1) (14) where Wv represents the volume variation in the impulsion valve chamber, and WI represents the volume variation in the injector chamber. Considering that the liquid volume variations for the first and last pipe division, are given by eq 15,

W(J + 1, 1) ) Q(J + 1, 1) - Q(J + 1, 2) + Wv(1) W(J + 1, N))Q(J + 1, N) - Q(J + 1, N + 1) + Wv(N) (15) and combining eqs 14 and 15, the flows in the first and last pipe division are obtained (see eq 16):

Q(J + 1, 1) ) Wv(J + 1)St∆L/Vv(J + 1) + Q(J + 1, 2) - WR(1) Q(J + 1, N) ) WI(J + 1)St∆L/VI(J + 1) + Q(J + 1, N + 1) - WR(N) (16) where WR is the residual variation of the liquid volume. In case at the end of the pipe was a chamber with a volume considerably bigger than one of the pipe divisions, the fluid conditions could vary in both the chamber and the division. In this case, the boundary conditions would be the resulting ones from considering that the pipe had an extra division which would penetrate in the chamber and where the fluid would be in the same conditions as in the pipe.

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Palomar et al. (two types of valves have been studied and shown in Figure 7); cross section of fuel through the nonreturn valve; and cross section of fuel through the injector. Three types of injector nozzles have been analyzed: needle valve injector with nozzle tip angle equal to the seat one; needle valve injector with nozzle tip angle higher that the seat one; and pintle injector nozzle. All these areas have been evaluated considering the smallest transversal section to the flow.

Mathematical Model, Results and Discussion 1. Equations To Define the Simulation Process. 1.1. Piston Chamber. As seen from Figure 6, geometrical volume variation (due to the piston movement) per unit of time is equal to the addition of the liquid volume variation per unit of time in the piston chamber plus the flows that go through the inlet and the exhaust ports, the flow which passes to the valve chamber, and the leaks between the piston and its jacket (eq 17): Figure 6. Injection system. A, feeding chamber; D, discharge chamber; P, piston chamber; V, impulsion valve chamber; AR, nonreturn valve chamber; T, injection pipe; I, injector chamber; C, combustion chamber.

UpSp ) dWp/dT + SaCaUa + SvCvUv + Qfp (17) 1.2. Impulsion Valve Chamber. The inlet flow from the piston chamber (that is, the flow that goes through the drill orifice bypass plus the flow that goes through the valve) must be equal to the sum of the volume variation, per unit of time, of the liquid contained in the valve chamber plus the geometrical volume variation per unit of time of the valve chamber and the outlet flow to the nonreturn (AR) chamber (eq 18):

CBPSBPUBP + CvSvUv ) dWv/dT + dVv/dT + CARSARUAR (18) Figure 7. Types of impulsion valves under study (Dv, diameter of the impulsion valve, DBP, diameter of the bypass orifices, DVI, diameter of the impulsion valve orifices; Lvc, critical lift of the impulsion valve). Table 1. Main Characteristics of the Bosch Rotary Pump piston diameter (mm) number of slots injection pipes (mm) pressure stage (mm) maximum number of revolutions (rpm) stroke (mm) air intake volume (mm3) maximum injection pressure (bar) maximal needle lift (mm)

8 4 6‚2‚500 6‚2 2250 2.2 40 130 0.78

7. Simulation Process. Once the starting hypotheses and the calculation method have been established, the mathematical model that simulates the performance of the rotary injection pump is explained. As previously mentioned, in the present work, the injection system is equipped with a Bosch rotary pump of the type VE, with four cylinders and a common feeding chamber for all of them. The pump is installed in a VW Polo 1.4 LSD engine, 1400 cm3 and 36.7 kW at 4500 rpm. The main characteristics of the pump are shown in Table 1. Figure 6 represents the injection system. To facilitate the calculation process, the system is divided into the following components: piston chamber, impulsion valve chamber, nonreturn valve chamber, and injector chamber. Also, the injection pipe will be divided in Nd volumes. To calculate the flow through these chambers, it is important to consider the cross section of flow at the restrictions. A number of parameters related to the restrictions geometry and the elements motion have been defined. The modeled areas are as follows. Cross section of fuel through the inlet and the exhaust ports; cross section of flow through the impulsion valve

To solve these equations, it is necessary to find out another equation that governs the valve movement. In this equation, the force due to the pressure is equal to the sum of inertia forces plus cushioning forces and the spring force (eq 19):

Mv d2Lv/dT2 + Av dLv/dT + KvLv ) S(PF - Pv) - Fvo (19) 1.3. Nonreturn Valve Chamber. As in the previous cases, the continuity equation will be applied to the restriction composed of the nonreturn valve in the impulsion record of the pump. The inlet flow from the impulsion valve chamber is equal to the volume variation per unit of time of the nonreturn valve chamber plus the outlet flow through the first division in the pipe and the geometrical volume variation, according to the following expression (eq 20):

CARSARUAR ) dWAR/dT + dVAR/dT + UiSi (20) As a complementary equation to obtain the lift, the dynamic equilibrium equation of the nonreturn valve is considered (eq 21). This equation suggests that the inertia forces, cushioning forces, and the spring force get balanced with the liquid pressure forces applied on the valve. The equation takes the following form:

MAR d2LAR/dT2 + AAR dLAR/dT + KARLAR ) (Pv - PAR)SARO - FARO (21) 1.4. Injector Chamber. Proceeding as in the previous case for the injector, it can be seen that the inlet flow



from the high-pressure tube is equal to the sum of the volume variation vs time of the liquid in the nozzle chamber plus the geometrical volume variation vs time in the nozzle chamber and the outlet flow through the injection orifices (eq 22):

StU(N + 1) ) dWi/dT + dVi/dT + zπD2o UiCi/4 (22) Also, to obtain the liquid volume variation in the injector, the equation that rules the needle movement in the injector nozzle is applied (eq 23). This equation states that the sum of the inertia forces, cushioning forces, and limiter repulsion, plus the seating cushioning and repulsion forces, is equal to the sum of the forces due to the pressure inside the injector chamber and inside the combustion chamber minus the forces due to the pressure of the surplus fuel and to the spring precompression.

Mi d2Li/dT2 + Ai dLi/dT + KiLi + [Ais dLi/dT + Kis(Lis - Lis0)]Ji ) PiSi + PcSc - PsDi12/4 - Fi0 (23) In relation to the injector, it could be necessary to consider the needle seat elasticity and the lift limiter elasticity due to the strong spring preloads and the fast locks of the needle as a result of the sudden pressure changes, which provoke rebound of the needle on its seat. 2. Discharge Coefficients in the Restrictions. These coefficients provide information about the real flows according to the ideal flows values. Therefore, its determination is of great importance in the injection system simulation with regard to the accuracy of the results. The most accurate way to work out this coefficient is by obtaining the exact value of the real flow through the restrictions, together with the pressure gaps and the cross-area restrictions. With these data, the discharge coefficient could be indirectly obtained. Unfortunately, there are no reliable methods for measuring the instantaneous flow due to the restrictions, especially for such small flows. Next, the discharge coefficients used for the various restrictions are explained in detail. 2.1. Discharge Coefficients in the Inlet and Exhaust Ports. Measuring the discharge coefficients in the inlet and exhaust ports is of great difficulty due to the constructive characteristics of the injection system. Moreover, its determination is aggravated since the inlet and exhaust ports are partially closed. According to Saderra17 and Yamaoka,18 the coefficients were considered to be constant, and an average value of them was taken. The value of these coefficients changed depending on whether the process was related to impulsion or discharge. In fact, in the discharge process, the flow passing through the piston slots implies an additional loss of load, which must be considered. In this sense, in impulsion process, C ) 0.70, while in discharge process, C ) 0.65. (17) Saderra, L. Caracterı´sticas de la estabilidad de la inyeccioń en motores diesel, Ph.D. Thesis, ETSII, University of Tarrasa, Spain, 1983. (18) Yamaoka, K.; Saito, A.; Abe, N.; Okazaki, M. Analysis of bypass control fuel injection systems for small diesel engines by digital computer; SAE Paper No. 730664, 1973.

Figure 8. Valve discharge coefficient related to the lift (4, increment of pressure up to 10 bar; 9, increment of pressure up to 20 bar; 0, increment of pressure up to 25 bar; O, increment of pressure up to 35 bar).

2.2. Discharge Coefficient in the Impulsion Valve. Munõz 19 carried out some studies on an impulsion valve where different pressure stages were considered (see Figure 8). According to this, it can be appreciated that only for small lift values there are substantial changes in the discharge coefficient, remaining constant from 0.4 mm needle lift. Therefore, the discharge coefficient could be considered to be constant. However, to avoid discontinuities in the computer model, an expression adjusted by means of least-squares from the experimental data will be adopted (eq 24):

Cv ) Cro{1 - 1/[K(Lv - Lvc + 1)]}

(24)

where K ) 117.3 and Cro ) 0.727. 2.3. Discharge Coefficient in the Injector. In this case, the pressure in the injector chamber is the main value to be determined. According to this, in addition to working out the discharge coefficient in the nozzle orifice, the loss of load in the area between the needle and its seat must be taken into account. Therefore, a discharge coefficient which includes the restrictions from the injector chamber to the nozzle itself must be used. In this way, the injected flow will depend on the pressure increment between the injector chamber and the combustion engine chamber, the cross section in the nozzle, and this global coefficient, which includes the transversal cross sections of the fluid. With regard to the cross sections, the injector may be substituted by the one in Figure 9, which is composed of four cavities where to each one is given a fuel cross section value, a speed value, and a discharge coefficient. Starting from the loss of pressure between the inlet and the outlet and considering the continuity equation, the amount of motion equation and the energy equation, as well as the equation that governs the delivered fuel by the injector, the following expression for the global discharge coefficient is obtained (eq 25). As can be seen, (19) Munõz, A. Ana´lisis y simulacioń de los sistemas de inyeccioń de combustible en motores diesel con bomba en lıńea, Ph.D. Thesis, ETSII, Madrid, Spain, 1987.

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Palomar et al. Table 2. Simulated and Experimental Injected Fuel Flows According to the Number of Pump Revolutions pump speed (rpm) Qreal (mm3/cycle) Qsimul (mm3/cycle) E ) 100(Qsimul - Qreal)/Qsimul

Figure 9. Injector to calculate the discharge coefficient.

Figure 10. Graphical representation of the relationship between Re and the discharge number for nozzles with any geometrical form.

this equation depends on the cross sections of the restrictions and their respective discharge coefficients.

Ci ) x1/{(1/C0)2 + [S0/(CRSR) - S0/(CsSS)]2}

(25)

The determination of coefficients CR and CS is of great difficulty since the system geometry changes when the needle is lifted. However, the previous equation suggests that, for large needle lift, the CR and CS coefficients values must be very close to one, given that sections SR and SS are much bigger than SO. The discharge coefficient of the injector nozzle orifices, CO, will be obtained considering the factors that represent the discharge of a liquid through an orifice. These factors are speed, viscosity, density, and orifice diameter, in other words, the Reynolds number. According to this, it can be seen the existing relation between the discharge number and the Reynolds number (C ) f(Re)). Function f depends on the particular orifice geometry, and it can only be calculated by means of experimental studies. In this sense, Hodgson20 experimented with the previous equation for nozzles with any geometrical shape and found curves similar to the ones shown in Figure 10. According to Figure 10, if the Reynolds number, Re, falls nearby the area where C1/Re remains constant, the flow will be laminar and the discharge coefficient will (20) Hodgson, J. L. The orifice as a basis of flow measurement. In Transactions, Institution of Civil Engineers, Selected Engineering Paper No. 31; Institution of Civil Engineers: London, 1924.

2000 26.4 26.23 -0.64

1500 26.4 26.59 0.71

1000 24.6 24.8 0.80

depend on the viscosity. If Re falls between Re1 and Re2, the flow will be a mixture of laminar and turbulent and the discharge coefficient, in addition to viscosity, will also depend on the density. Above Re2, the flow is totally turbulent and the coefficient will practically be constant since it will be virtually independent from viscosity. Gelalles21 carried out experimental tests with injection nozzles from diesel engines of different geometrical shape to determine the effect of the nozzle design on the discharge coefficients in permanent regime. Results showed that the coefficient of discharge depends on the geometrical shape of the nozzle. These coefficient values have been used in the present study. The injector nozzle diagrams show that the injection orifices have inlets with sharp edges. For this reason, in this study, the appropriate discharge coefficient, using the average value, will be C ) 0.67. 3. Experimental Evaluation and Validation of the Model. To evaluate the proposed model, the following parameters have been chosen. Pressure in the injector inlet (last division in the pipe); the injector needle lift; and amount of fuel injected per cycle. Most of the geometrical characteristics of the pump have been supplied by Robert Bosch. The rest of them were determined after dismantling some components to obtain information about their geometry, which was necessary for the computer model. The position of the adjusting ring was determined by the position of the throttle, considering the distance to the exhaust port to be constant for full load conditions. As previously mentioned, tests were carried out on a MAGASA F-60 test bench (La Electro Motor Diesel S.A., Seville, Spain) for maximum injector pressures of 130150 bar, pipe lengths of 500, 600, and 800 mm, and 2 mm internal diameter for all cases. Table 2 shows the amount of fuel injected per cycle, both measured and calculated, for pump spinning speeds of 2000, 1500, and 1000 rpm, maximum injector pressure of 130 bar, pipe length of 500 mm, and the relative error between the real flow and the simulated one. In all experiments, the maximum variation was less than 1 mm3/cycle, thus, the error was less than 4%. Figures 11 and 12 show the pressure data from the impulsion valve chamber (Pv), nonreturn chamber pressure (PAR), injector chamber pressure (PI), last division pipe pressure (PPIP), impulsion valve lift (Lv), injector needle lift (Li), and instantaneous injected flow (Q), all of them in function of the angle turned by the pump shaft. Y0 represents the final state of the system (cavitation or overpressure), and SI indicates whether secondary injection exists or not. Data registration begins when the pumping piston starts moving upward from the bottom dead center and finishes when the pump shaft turns 90°. At that moment, the injection process finishes completely. In addition to the simulated curves, Li-R and PPIP-R, (21) Gelalles, A. G. Coefficients of discharge of fuel injection nozzles for compression-ignition engines; NACA Report 373, 1932.



Figure 11. Simulated and experimental results at full load and 1000 rpm pump revolutions (Q ) 24.8 mm3/cycle, Y0 ) -13.69; SI ) No). Y0 represents the final state of the system (cavitation or overpressure), and SI indicates whether secondary injection exists or not.

the obtained curves from the pump experimental tests are shown. These curves were obtained from a pump revolution of 1000 and 1500 rpm, although the computer program admits a range from 500 to 2250 rpm.

Conclusions A new mathematical model of an injection system for diesel engines has been developed and validated. As can be seen from Figures 11 and 12, the computer model

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Palomar et al.

Figure 12. Simulated and experimental results at full load and 1500 rpm pump revolutions (Q ) 26.59 mm3/cycle, Y0 ) -35.48; SI ) No).

predicts with great accuracy the opening and closing processes in the injector needle and, as a result, the starting and finishing point of the fuel injection process. Similarly, the model predicts the experimental and calculated pressure curves in the last division of the

high-pressure tube. Though there is good agreement obtained between the computed and the test-derived data, it is important to emphasize the following matters. The small difference between the modeled and experimental data is due to the fact that the speed of


sound corresponds to fuel in state of no cavitation. This is clear in Figures 11 and 12, where the difference tends to increase when cavitation does it proportionally. Considering the cushioning coefficient in the needle motion equation (only dependent on the viscous friction) contributes to the existing difference in the signals of the injector needle lift. In sum, by means of the system parameters under different engine speeds, we can conclude that the model makes it possible to plot the performance of the pressure


lines in the different combustion chambers, the valves lift curves, and the fuel flow injected per cylinder and cycle. Acknowledgment. The authors thank the firm La Electro Motor Diesel S.A. (Sevilla, Spain), which provided the test bench to carry out the tests, and Prof. Pedro Luque Escamilla and Prof. Mario Fernańdez Pantoja for their support. EF0500080

Development of a Computer Model to Simulate the Injection Process

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