Complex Diesel Engine Simulation with Focus on Transient Operation

Nov 20, 2007 - Wayne State UniVersity, Detroit, Michigan 48202, and U.S. Army, TARDEC .... the state parameters of the gas in the exhaust manifold. It...
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Energy & Fuels 2008, 22, 1411–1417

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Complex Diesel Engine Simulation with Focus on Transient Operation† Dinu Taraza,*,‡ Naeim A. Henein,‡ Radu Ceausu,‡ and Walter Bryzik§ Wayne State UniVersity, Detroit, Michigan 48202, and U.S. Army, TARDEC ReceiVed August 4, 2007. ReVised Manuscript ReceiVed NoVember 20, 2007

The paper presents a complex simulation model for multicylinder, common rail diesel engines developed on a SIMULINK platform. The model consists of the main modules simulating the processes in the engine cylinders, the exhaust and intake manifolds, and the turbocharger. It is developed on the of filling and emptying concept and uses zero-dimensional combustion models considering multiple injection events, characteristics of modern diesel engines. The transient nature of the simulation model is enhanced by the use of a lumped mass elastic model of the crankshaft and a detailed friction model of the engine. In this way, more realistic simulation of the crankshaft dynamics and engine transients becomes possible. The model is validated by comparison of the simulation results with experimental data measured on a heavily instrumented 2.5 L, fourcylinder, common rail diesel engine.

1. Introduction The advent of high speed computers had stimulated the development of simulation codes for many engineering applications and their use has helped in improving design and reducing the design cycle. Internal combustion engines (ICEs) are very complex machines, and simulation has played a major role in advancing their design and predicting their performance in specific applications. Depending of the major goal of the design, different types of simulation codes have been developed. By far the most complex processes in ICEs is combustion, and very sophisticated 3D commercial codes have been developed (KIVA, FIRE, STAR-CD, etc.). These codes require a large amount of time to construct the input model and to run the code. They are used in the development of the combustion chamber architecture and the injection system optimization. More simplified and capable to predict exhaust emission are such codes as BOOST (AVL), Wave (Ricardo), and GT-POWER (Gamma Technologies). Transient engine simulation has been pioneered by Benson,1 Watson et al.,2,3 and Winterbone.4 In their models, the behavior of the turbocharger was simulated using look-in tables based on the turbine and compressor maps assuming quasi-steadystate conditions. Also, the crankshaft was considered rigid and friction losses were estimated by simple friction models. † Presented at the 10th International Conference on Energy and Environment. * Corresponding author. ‡ Wayne State University. § U.S. Army, TARDEC. (1) Benson, R. S.; Ledger, J. D.; Whitehouse, N. D.; Walmsley, S. Comparison of Experimental and Simulated Transient Responses of a Turbocharged Diesel Engine; SAE Paper No. 730666, Society of Automotive Engineers: Warrendale, PA, 1973. (2) Watson, N.; Marzouk, M. A Non-Linear Digital Simulation of Turbocharged Diesel Engines under Transient Conditions; SAE Paper No. 770123, Society of Automotive Engineers: Warrendale, PA, 1977. (3) Watson, N. Transient Performance Simulation and Analysis of Turbocharged Diesel Engines; SAE Paper No. 810338, Society of Automotive Engineers: Warrendale, PA, 1981. (4) Winterbone, D. E.; Tennant, D. W. H. The Variation of Friction and Combustion Rates During Diesel Engine Transients; SAE Paper No. 810339, Society of Automotive Engineers: Warrendale, PA, 1981.

If engine dynamics is the main interest, more detailed models must be developed for the crankshaft and engine friction. Also, a dynamic model of the turbocharger is necessary to determine the correct evolution of the pressure in the intake manifold. The simulation model DETRANS (Diesel Engine TRANsient Simulation), presented in this paper has been developed using a modular approach and the SIMULINK environment. Detailed presentation of the combustion, the turbocharger, and the crankshaft models can be found in ref 5. The updating of the friction model and validation of the transient capabilities of DETRANS are the main features of this paper. 2. Simulation Model DETRANS The main modules of DETRANS are the cylinder block, the intake and the exhaust manifolds, the turbocharger, the control unit, the friction, and the dynamics modules (Figure 1). The modules are interconnected by gas flow (exhaust manifoldturbocharger-intake manifold-cylinder block) and by mechanical links (cylinder block-friction-engine dynamics). The control unit meters the fuel amount injected in the cylinders, and depending of the operation conditions that are simulated, this amount could be limited by the air fuel ratio in the cylinders or the engine speed. The filling and emptying concept is used to model the cylinders and the intake and exhaust manifolds. According to this concept, the pressure and temperature, while varying during the engine operation, are uniform in the whole volume of the corresponding control volumes (cylinders and intake and exhaust manifolds). Cylinder Block Module. The cylinder block module groups the four cylinders of the engine considered in this simulation (Figure 2). In each cylinder, the energy conservation law (first principle of thermodynamics) is integrated to yield the instantaneous gas temperature and the corresponding cylinder pressure. During the gas exchange phase, the enthalpies of the out-flowing and inflowing gas, together with the heat exchanged with the walls determines the temperature of the gas and the cylinder pressure. The four modules representing the cylinders are connected to the exhaust and, respectively, intake manifold modules. The

10.1021/ef700472x CCC: $40.75  2008 American Chemical Society Published on Web 02/12/2008

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Figure 1. General structure of the engine simulation model DETRANS.

Figure 2. Cylinder block module.

gas mass rate of flow through the exhaust valves, with the corresponding gas composition and enthalpy, is conveyed between the four cylinder modules and the exhaust manifold module. In the same way, the gas flow through the intake valves with the gas composition and enthalpy is conveyed between the intake manifold module and the four cylinder modules.

The exhaust manifold module simulates the gas exchange between the manifold and the turbocharger turbine and calculates the state parameters of the gas in the exhaust manifold. It links these data to the four cylinder modules such as to allow each cylinder module to calculate the gas exchange through the exhaust valves. The inputs to the exhaust manifold module are

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Figure 3. Structure of the “friction” module.

the mass rate of flow, the composition and enthalpy of the gas exchanged with the cylinders. The outputs are the exhaust manifold gas pressure, temperature, composition, enthalpy, and adiabatic exponent. The intake manifold module is connected to the turbocharger compressor to determine the mass flow rate of air into the manifold and its temperature, and it calculates the state parameters of the gas in the intake manifold linking these data to each cylinder module to determine the gas exchange through the intake valves. The inputs to the intake manifold module are the mass rate of flow, the composition and enthalpy of the gas exchanged with the cylinders. The outputs are the intake manifold gas pressure, temperature, composition, enthalpy, and adiabatic exponent. Both the intake and the exhaust manifold models are considered as constant volume enclosures that exchange mass and enthalpy with the cylinders and the turbocharger, assuming uniform temperature, pressure, and gas composition inside the manifold. The friction module calculates the friction torque produced by the piston-ring assembly (PRA) of the four cylinders and the friction torque produced by the connecting rod and the main bearings. The inputs to this module are the crank angle of the crank slider mechanism of each cylinder, the speed of the lumped masses representing each cylinder, and the pressure in each cylinder of the engine together with the pressures behind the first and second rings. The output is the total friction torque corresponding to the PRA and engine bearings. In the “cylinder block” module of DETRANS the outputs corresponding to the camshaft torque of each cylinder are added to yield the total camshaft torque. A simple module “auxiliaries” calculates the torques required to drive the high pressure fuel pump and the oil and water pumps. The inputs to this module are the crankshaft’s speed and the amount of fuel injected per cycle. Finally, the friction torque from the output of the module “friction” is added with the camshaft torque and the auxiliaries torque to yield the total

torque corresponding to the internal mechanical losses of the engine. This torque is input to the “Crankshaft’s dynamics” module together with the gas pressure torque calculated in each cylinder module and the load torque to determine the instantaneous crankshaft speed and the torsional deflection at the location of each crank slider mechanism. Engine Frction Module. In order to be able to accurately predict engine transients, the friction torque must be calculated for each crank angle considering the major friction components of the engine: the piston-ring assembly and the engine bearings. Several models have been proposed in the literature,6–8 the most used in engine simulation codes being the Rezeka-Henein model.9 These models require calibrations for every distinct application relying on calibration coefficients.10 The model proposed in the paper is a generic one depending only on geometric dimensions and operating parameters. The piston ring assembly (PRA) friction is a complex process characterized by periodic transitions from mixed to hydrodynamic lubrication and much work has been done in modeling (5) Ceausu, R.; Taraza, D.; Henein, N. A.; Bryzik, W. A Generic Transient Model of a Turbocharged, Multi-Cylinder, Common-Rail Diesel Engine. Proceedings of ICES 2005, ASME Internal Combustion Engines DiVision 2005 Spring Technical Conference, Chicago, IL, Apr 5–7, 2005. (6) Zweiri, Y. H.; Whidborne, J. F.; Senvirante, L. D. Instantaneous Friction Components Model for Transient Engine Operation. Proc. Inst. Mech. Eng., Part D 2000, 214, 809–824. (7) Zweiri, Y. H.; Whidborne, J. F.; Senvirante, L. D. Detailed analytical model of a Single Cylinder Diesel Engine in the Crank Angle Domain. Proc. Inst. Mech. Eng., Part D 2001, 215, 1197–1216. (8) Kouremanos, D. A.; Rakopoulos, C. D.; Hountalas, D. T.; Zannis, T. K. DeVelopment of a Detailed Friction Model to Predict Mechanical Losses at EleVated Maximum Combustion Pressures; SAE Paper 2001-010333 Society of Automotive Engineers: Warrendale, PA, 2001. (9) Rezeka, S. F.; Henein, N. A. A New Approach to EValuate Instantaneous Friction and its Components in Internal Combustion Engines; SAE Paper 840179, Society of Automotive Engineers: Warrendale, PA, 1984. (10) Tucillo, R.; Arnone, L.; Bozza, F.; Nocera, R.; Senatore, A. Experimental Correlation for Heat Release and Mechanical Losses in Turbocharged Diesel Engines; SAE Paper 932459, Society of Automotive Engineers: Warrendale, PA, 1993.

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this process.11–14 The model used in this work is based on the results presented in ref 15. As the piston moves, its velocity varies between zero and a maximum value and the lubrication regime changes between boundary and hydrodynamic. The friction coefficient in the hydrodynamic domain can be expressed as12 follows: (1) f ) C × Dm In this equation, D is a nondimensional parameter called the duty parameter D)

η|VP| FN/L

(2)

and the coefficient C and exponent m, for the usual geometry of the ring profile, take the following values:12 C ) 1.9-2.25 and m ) 0.45-0.525. In the boundary and mixed lubrication domains, the variation of the friction coefficient was assumed as a linear function of the duty parameter: f ) f0(1 - D/Dcr) + fcr(D/Dcr)

(3)

In this equation, f0 is the dry friction coefficient (f0 = 0.2 for cast iron over cast iron); correlating the oil film thickness (OFT) with the combined asperities of the rubbing surfaces, the critical value of the duty parameter is estimated to be Dcr ) 1 × 10-4 and the corresponding critical value of the friction coefficient fcr ) 0.0225.15 The calculation of the instantaneous friction forces between the three rings and the liner requires the knowledge of the piston speed VP and the values of the gas pressures behind the first and the second rings (pC1 and pC2) at each crank angle. The following expressions are used to calculate the friction forces:

( (

) )

η|VP| m πdC[p1 + pC1(di/dC)]h1 (FN/L)1 η|VP| m Ff2 ) C πdC[p2 + pC2(di ⁄ dC)]h2 (FN ⁄ L)2 η|VP| m Ff3 ) C πdC p3h3 (FN ⁄ L)3 Ff1 ) C

(

)

(4)

The lubrication of the piston skirt and liner is considered always hydrodynamic, and the friction coefficient is calculated in the same way as for the rings. Equation 2 takes the following form for m ) 0.5 and C ) 5



fs ) 5.0

η|VP| FN/Ls

The engine bearings model takes into consideration the significant load variation during each engine cycle and, in general, the lubrication is hydrodynamic. The first thing that must be determined is the load applied on a bearing at each crank angle. Considering a rigid structure of the crank slider mechanism of each cylinder, the force loading the connecting rod bearing can be calculated from the gas pressure force and the reciprocating inertia force. The loading of the main bearings is more complex. The crankshaft of multicylinder engines is a statically undetermined structure, and the bearing load results as a contribution of all engine cylinders. In fact, the contribution of the cylinders, situated far from the considered bearing, are negligible and the load may be calculated as the contribution of only the two cylinders between which the bearing is situated. Once the load is known, the orbit of the shaft in the bearing must be calculated to determine the variation of the minimum oil film thickness (OFT) and the corresponding friction force. The usual way to calculate the shaft orbit is the mobility method.16 The method assumes the bearing to be absolutely rigid and gives only an estimate of the minimum OFT. To further simplify the calculation a quasi-steady-state (QSS) method is developed. The proposed method assumes that for every value of the variable load the position of the shaft inside the bearing corresponds to the one obtained for a constant load of the same value. The correlation between the shaft position and the load line (direction of the force acting in the bearing) is shown in Figure 4. According to the theory of the short bearing,16 the friction force in the bearing and the attitude angle φ are the following:

(5) Ff )

and the friction force is the following: Ffs ) 5.0√FNη|VP|Ls

Figure 4. Instantaneous shaft position inside the bearing.

(6)

In these formulas, FN is the thrust force applied to the piston skirt by the gas pressure and the reciprocating inertia forces, and LS is the length of the piston skirt. (11) Furuhama, S. A Dynamic Theory of Piston-Ring Lubrication - 1st Report, Calculation. Bull. JSME 1959, 2, 423–428. (12) Yang, Q.; Keith, T. G., Jr. Two-Dimensional Piston Ring Lubrication - Part I Rigid Piston Ring and Liner Solution. Tribology Trans. 1996, 39, 757–880. (13) Tian, T.; Wong, W. V.; Heywood, J. B. A Piston Ring- Pack Film Thickness and Friction Model for Multigrade Oils and Rough Surfaces; SAE Paper 962032, Society of Automotive Engineers: Warrendale, PA, 1996. (14) Stanley, R.; Taraza, D.; Henein, N. A.; Bryzik, W. A Simplified Friction Model of the Piston Ring Assembly; SAE Paper 1999-01-0974, Society of Automotive Engineers: Warrendale, PA, 1999.

2πηR2ωL c√1 - ε

2

+

eF sin φ 2R

(7)

The first term in this equation represents the shearing force of the oil film. The second term is the equivalent of the moment produced by the load force F due to the shaft’s eccentricity e and is negligible with respect to the first term. The only unknown variable in eqs 7 is the specific eccentricity ε. This is an implicit function of the load, the oil viscosity, and the bearing geometry:16 πε √0.62ε2 + 1 ) F/L c Rωη R (1 - ε2)2

( ) ( 2RL ) 2

2

(8)

(15) Taraza, D.; Henein, N. A.; Bryzik, W. Friction Losses in MultiCylinder Diesel Engines; SAE Paper 2000-01-0921, Society of Automotive Engineers: Warrendale, PA, 2000. (16) Taylor, C. M., Ed. Engine Tribology; Elsvier Science Publishers: New York, 1993; pp 89–112.

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Figure 5. Forces acting on the cam.

In order to speed up the calculation, approximation functions were developed to yield directly the value of ε as a function of the ratio (2R/L)2 and the Sommerfeld number S: F/L c 2 (9) Rωη R To keep the errors less than 1%, two approximation functions were developed to cover all possible values of the Sommerfeld number.17 The structure of the “friction” module is shown in Figure 3. There are four Matlab functions “fricpra”, one for each cylinder of the engine. The inputs to each function are the corresponding crank angle and speed, the cylinder pressure, and the pressure behind the first and second rings. The function calculates the forces acting in the crank slider mechanism and the forces and torques corresponding to the PRA and connecting rod bearing friction. The output represents the total friction torque of the PRA and connecting rod bearing. The second and third outputs are the horizontal and vertical reactions in the two main bearings between which the cylinder is situated. These forces are used to calculate the friction torques in each main bearing of the engine. Finally, all friction torques are added to yield the total friction torque due to all piston-ring assemblies and all engine bearings, which is the output of the friction module (Figure 3). Valve Train Torque Module. In the case of the valve train, it is necessary to determine the torque required to drive the camshaft TVT. This torque is produced by two forces (Figure 5): the force acting along the axis of motion of the tappet FV and the friction force FfV. S )

()

TVT ) eFV + hVFfV

(10)

In this relation, FV is the force required to move the tappet. It is determined by the valve spring stiffness, valve lift spring preload, the tappet acceleration, and the mass of the valve system, all reduced at the tappet motion. The valve train friction model is based on the theory developed in ref 18. According to this theory, the friction force has two components: a boundary component Fb and a viscous component Fη. FfV ) Fb + Fη

(11)

The boundary friction force will result from the shearing of an extremely thin oil film when non-Newtonian behavior of the oil prevails:19 Fb ) τ0Aa + mPa

(12)

In this equation, τ0 is the Eyring stress of the oil (about 2 MPa), and m, the pressure coefficient of the boundary shear strength (17) Taraza, D.; Henein, N. A.; Ceausu, R.; Bryzik, W. Engine Friction Model for Transient operation of Turbocharged, Common Rail Diesel Engines; SAE Paper 2007-01-1460, Society of Automotive Engineers: Warrendale, PA, 2007.

Figure 6. Injection pressure as a function of speed and load.

(about 0.17). The first term in eq 12 may be neglected because the asperity contact area is less than 1.4% of the Hertzian contact area, and the boundary component of the friction force becomes Fb ) mPa. The load carried by asperities, Pa, may be estimated by the following relation:16 123√RLFV < Pa < 185√RLFV

(13)

For Newtonian oil behavior, the viscous friction component is the following: Fη )

ηV A h0

(14)

If non-Newtonian behavior prevails, the viscous friction component may be calculated as follows: Fη = 0.99[τ0A + γ(FV - Pa)]

(15)

Formula 15 was written with the assumption that the asperity contact was 1% of the Hertzian contact area.14 Injection System Module. Modern diesel engines are using common rail injection systems with electronically controlled injectors capable of very high injection pressure and multiple injection events per cycle. A large torque is necessary for driving the high pressure fuel pump to achieve the required injection pressure. As the engine speed and load vary, the optimum injection pressure also varies such as to ensure both low particulate emissions and a minimum possible value of the power consumed by the pump. The variation of the injection pressure with speed and load for a typical 2.5 L automotive engine is presented together with an analytical approximation of the measured surface in Figure 6. An analytical function approximating the injection pressure was developed to replace the look-in table and speed up the calculation. To estimate the power required by the high pressure fuel pump at different speeds and loads, the following experiments have been performed. (18) Teodorescu, M.; Taraza, D.; Henein, N. A.; Bryzik, W. Simplified Elasto-Hydrodynamic Friction Model of the Cam-Tappet Contact; SAE Paper 2003-01-0985, Society of Automotive Engineers: Warrendale, PA, 2003. (19) Wilkiamson, B. P.; Bell, J. C. The Effects of Engine Oil Rheology on the Oil Film Thickness and Wear Between a Cam and Rocker Follower; SAE Paper 962031, Society of Automotive Engineers: Warrendale, PA, 1996.

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Table 1. Estimation of the Ratio Q/ωηP n [rpm] 1000 1250 1500 1750 2000 2250 2500

Q/(ηPn)

ω

Q/(ηPω)

9.08 × 10–3 8.768 × 10–3 8.713 × 10–3 8.770 × 10–3 8.845 × 10–3 9.270 × 10–3 9.410 × 10–3

104.7 130.9 157.1 183.2 209.4 235.6 261.8

86.7 × 10–3 83.7 × 10–3 83.2 × 10–3 83.8 × 10–3 84.5 × 10–3 88.5 × 10–3 89.9 × 10–3

The engine was motored by the AC dynamometer with the ECU switched on, but with injectors decoupled from the power source. The rail pressure was maintained by the ECU at the low value required for idling. The motoring torque and the common rail pressure were measured for different motoring speeds. Then, the engine was motored with the ECU turned off, the rail pressure being controlled only by the relief valve. The motoring power PM represents the power consumed by all internal mechanical losses of the engine. This power could be expressed as the power corresponding to all internal losses except the power required to drive the high pressure fuel pump Pf plus the power required to pump the fuel. This power depends on the fuel flow rate Q, the rail pressure prail, and the efficiency of the pump ηpump: PM ) Pf + Qprail/ηP

Figure 7. Engine speed variation during a transient under constant load torque of 200 Nm.

(16)

Assuming that, at the same motoring speed, Pf, Q, and ηP are the same for the two cases, the following relation is obtained: (PM)W⁄O.ECU - (PM)W.ECU Q ) ηP (prail)W⁄O.ECU - (prail)W.ECU

(17)

The rate of flow increases linearly with the speed of the pump, which is proportional to the engine speed ω. If the previous assumptions are correct, the following condition must be met: Q = const. ωηP

(18)

Processing the experimental data, the results shown in Table 1 are obtained. Analysis of the data in Table 1 shows that the ratio Q/ωηP varies very little and could be considered constant. In the limits of error of (5%, it could be estimated as the mean value of measurement results. The power required to drive the high pressure fuel pump Pinj may be calculated by the formula: Pinj ) 85.76 × 10-3ωprail

(19)

Model Validation. The validation of the model developed in the paper was done by comparing simulation results with experimental data measured on a 2.5 L, four-cylinder, common rail diesel engine. The engine was heavily instrumented and coupled to an AC dynamometer capable to run controlled transients. Pressure was measured in each cylinder, in the intake and exhaust manifolds, and in the common rail. The injector of cylinder no. 1 was equipped with a needle lift transducer, and the turbocharger speed was measured with a eddy current transducer. Temperatures in different locations in the engine were also measured. A first set of data were obtained under steady-state operating conditions. On the basis of the pressure traces in the cylinder, the mean indicated pressure and the indicated power Pi were calculated. The internal losses of the engine were calculated as: Pf ) Pi - PB

(20)

where PB is the break power measured by the dynamometer and Pf is the power lost by friction and drive of the engine auxiliaries.

Figure 8. Turbocharger speed variation during the engine transient under constant load torque of 200 Nm.

The errors of the simulated values of Pf with respect to the measured ones were in the range of (4%, the larger errors being registered at higher engine speeds. After validating the model for steady-state operating conditions, several transients were run on the dynamometer. The transients were performed under constant engine torque for given values of the pedal setting. Figures 7 and 8 show the results of measurements and the corresponding simulations for the engine and turbocharger speed variation. Figures 9 and 10 show the results of the measurements and simulation for the intake manifold pressure variation. The engine speed variation and the turbocharger speed variations are fairly well-simulated, especially in the first part of the transient. The intake manifold pressure variation is also quite well-simulated, but it seems that the volume of the manifold has been slightly smaller than the one considered in the simulation. Conclusion The capability of DETRANS to correctly simulate the behavior of the engine was proved by the fairly good agreement between the simulated parameters and the measured ones.

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lag effect. The model could be also used to estimate the torsional vibrations of the crankshaft to decide if a vibration damper is necessary and evaluate the efficiency of a particular damper. The modular structure of DETRANS makes possible the addition of new modules and modification of the existing ones to improve the capacity to accurately simulate different processes characterizing diesel engine operation. Nomenclature

Figure 9. Measured intake manifold pressure during the engine transient under constant load torque of 200 Nm.

Figure 10. Simulated intake manifold pressure during the engine transient under constant load torque of 200 Nm.

The engine simulation model DETRANS is especially developed for the simulation of transients, but it is capable of simulating both transient and steady-state operation conditions. One reason to develop the model was to allow the determination of the means to improve engine acceleration and reduce turbo-

Aa ) area carried by asperities C ) radial clearance D ) duty parameter Dcr ) critical value of duty parameter di ) inner diameter of the ring dC ) cylider diameter f ) friction coefficient f0 ) dry friction coefficient Ff1, Ff2, Ff3 ) friction forces of the first, second and third ring, respectively FN ) normal force Fb ) boundary friction force Fη ) viscous friction force FV ) normal force on the tappet h1, h2, h3 ) axial thickness of the first, second and third ring, respectively hV ) valve lift L ) characteristic length, bearing length Ls ) length of the piston skirt p1, p2, p3 ) elastic pressure of the first, second and third ring, respectively pC1, pC2 ) gas pressure behind the first and the second ring, respectively prail ) fuel pressure in the common rail Pa ) load carried by asperities PB ) engine break power Pi ) indicated engine power Pf ) power dissipated by friction PM ) power required to drive the high pressure fuel pump Q ) fuel rate of flow R ) radius of the bearing or base circle of the cam TVT, TV, TfV ) total camshaft torque, torque required to push the tappet, and friction torque between cam and tappet, respectively Vp ) piston velocity γ ) slope of the oil limiting shear stress-pressure relation ε ) specific eccentricity η ) oil viscosity ηP ) efficiency of the high pressure fuel pump φ ) atitude angle ω ) angular velocity τ0 ) Eyring stress EF700472X