NEW APPROACH ON MATHEMATICAL MODELING OF JAKE BRAKE PROCESS FOR DIESEL ENGINES

Auxiliary braking systems, which include also the Jake brake system, have a decisive role in issues such as road safety and optimization of fuel consumption. These issues are raised especially for heavy vehicles such as trucks or buses. One of the most delicate problems encountered when designing such a braking mechanism is to accurately evaluate the thermal conditions present in the combustion chamber. Failure to know and control these conditions can lead to deteriorations of injection system components, such as the injectors, which are no longer cooled by fuel. The present paper aims to advance a simple and quite accurate mathematical model to describe the processes occurring in a Jake brake system. Such a model can be very interesting when high power diesel engines are designed.


INTRODUCTION
Engine brake systems based on compression release became known as the "Jake Brake", which is a registered trademark of "Jacobs Vehicle Systems".This term describes an auxiliary braking system which is usually installed on diesel engines to allow opening of the exhaust valve in close proximity to the moment of injection.
These types of systems have been developed by several companies, being used in a wide variety of structural forms.Compression release braking concept is the most widely used auxiliary braking system.It was developed by Cummins in the `60s and presented in [1] as a result of increased braking efficiency achieved by transforming the engine into a huge compressor.The Jake system is one of the most effective auxiliary braking systems, being able to provide braking powers of up to approximately 85 % of the rated power as well as good response time [2].

MATHEMATICAL MODEL
The mathematical model advanced in the present paper is based on a simplified model, developed by the authors, for the thermal calculus of a compression ignition engine.For the present study, the numerical model is implemented using the Mathcad environment.Pressure and temperature values after admission are determined using the abovementioned model and adopted as input data for the advanced Jake brake model.
The numerical model implemented under Mathcad, considers in a first stage that the compression process occurs according to a polytrophic transformation.The variables pertinent to this process are determined by solving the differential equations for temperature, work and heat exchange with combustion chamber walls, to which, the gas state equation is added.The differential equation of temperature ( 1) is obtained by writing the energy balance for the gasses present inside the cylinder and has the following well-known expression: Values obtained after each iteration step for the variables corresponding to the compression process are generated in a matrix form.Values placed on the last column of the resulting matrix serve as input data for the Jake brake process.
The adopted computation method uses fixed pitch iterations.A schematic representation of the algorithm implemented under Mathcad to thermally evaluate the compression process is illustrated in Figure 1.
Fig. 1.The algorithm to compute the compression process.
The Jake brake is basically a gas exchange process, which is considered to take place adiabatically.Due to the fact that during operation of a diesel engine in braking regime, the combustion process is completely suppressed, it was considered that, at the end of the compression stroke, the only working fluid present in the cylinder is air.The determination of Jake brake parameters begins at a rotation angle of the crankshaft equal to the injection advance, and finishes when the rotation angle of the crankshaft, corresponding to the end of combustion, is reached.
In order to implement the model corresponding to a braking event, the following assumptions were considered: -The Jake brake event takes place by opening of the exhaust valve; -Flow through the exhaust valve takes place initially in an critical regime; -Gas flow through the exhaust valve gate initially takes place along the cylinder-manifold direction; -The exhaust system introduces by its shape, dimensions and surface properties (roughness) a gas-dynamic resistance evaluated via the resistance coefficient of the exhaust route ( ev ); -At the beginning of the engine brake process, substance transfer taking place under the exhaust valve is considered null; -Flow speed under the exhaust valve and through the exhaust manifold are therefore considered null at the beginning of engine brake process; -No substance losses are considered to be occurring during compression; The root code was built by aid of iterative "for loops", present in the "Programming" toolbar from Mathcad.Besides these loops, it includes input values for various characteristic parameters, as well as storing and printing sequences, used for variable values yielded after each integration step.The presence of "for loops" in the iteration process requires the use of a predetermined range for the crankshaft rotation angles.To that regard, an opening advance, aDSF , and a closing delay, îISF , for the exhaust valve were considered.These values limit the braking event according to the relation ( 2) and (3).
Mathematical formula used to evaluate cylinder volume is found in literature [3].The relations for pressure, and the derivatives of the heat exchanged with the walls as well as the work done by the fluid were found in literature [4], and introduced into the Mathcad code through relations ( 4), ( 5), ( 6) and (7) .The proposed mathematical model is used to evaluate the values of the above mentioned parameters.Pressure evolution was plotted against the ratio of instant volume to maximum volume for the braking process, as illustrated in Figure 2. The number of air kilo moles present in the cylinder is determined based on cylinder capacity and molar volume as , where the molar volume also takes into consideration air density corresponding to an average environmental temperature, chosen at 298 K.
Evaluation of internal energy and working fluids enthalpy are possible by first defining two matrices that contain specific heat coefficients corresponding to temperatures up to 1273 K and above.Values needed to generate these matrices were taken from literature [5].
Due to the nonlinear dependency of specific heat, internal energy and enthalpy with temperature, these parameters must be determined via one of two branches of the computation process.Thus, by aid of an "ifotherwise" condition that verifies whether 1273 T or 1273 T , it was ensured that the appropriate formula is employed.The relation to calculate specific heat is given by equation (8).In order to compute flow permitting cross section area, the valve-lift law must be defined first.This law is implemented in MathCAD as a matrix obtained by aid of the "cspline" tool.The cspline function interpolates lifting height values during the Jake brake process, also taking into account the maximum value required for exhaust valve lift.
The instant lift height of exhaust valve is taken from the definition matrix and brought to each step of iteration, using the equation (9).Depending on the valve lift height, two branches for calculating the gas flow cross section area are defined in equation (10), by aid of the conditioning function "if-otherwise": If null values are yielded, the model proceeds to determining the derivatives of substance quantity and cylinder temperature, respectively.Otherwise, it proceeds to the calculation of parameters describing the thermo-gas dynamic conditions in the exhaust manifold.
Both direct and reversed (manifold-cylinder) flows may occur during the braking process, taking place under critical or subcritical conditions.In order to determine the flow conditions, pressure downstream from the outlet is compared to the critical pressure.If the exhaust manifold pressure is higher than the pressure in the cylinder and the flow condition is subcritical, the direct flow stops and the newly established flow direction is manifoldcylinder.
The advanced model determines the collector pressures, as well as working fluid flow rates in the exhaust valve gate and manifold, depending on flow conditions.Through equations (11), ( 12) and (13) these parameters are calculated for direct flow in the exhaust manifold.Qualitative assessment of exhaust during the braking process is done through a parameter called the exhaust degree of perfection.This represents the ratio of the number of gas kilo moles present inside the cylinder at the end of an integration step to the product of molar variation coefficient ( c ) and the number of gas kilo moles initially present in the cylinder.The equation ( 14) is used to determine the degree of evacuation.
where 0cil represent the amount of gases inside the cylinder evaluated at the end of compression.
Pressure and velocity evolutions inside the exhaust manifold were plotted according to equations ( 12) and ( 13), as shown in Figure 3. Another important quantity of the Jake brake process is the adiabatic exponent, which is computed for each iteration step by aid of a "while loop".The control condition that determines loop exit and therefore the adoption of the last computed values for the adiabatic exponent consists of imposing an error threshold given by the equation (15).The collector parameters are influenced by the amount of gases exhausted during one iteration step of the Jake brake process, as well as by their temperature.For this reason, the program developed in MathCAD also deals with managing gas tranches present in the exhaust manifold, by varying their number.Two tranches of gas are considered to be initially present in the collector, characterized by identical thermodynamic parameters.The number of gas tranches increases with each integration step, in the case of direct flow and decreases if the size of the last formed substance tranche (closest to the exhaust valve gate) is less than or equal to zero.
To fully assess the parameters describing the Jake brake, a number of five differential equations must be solved, in order to determine cylinder temperature gradient and the amount of substance corresponding to each integration step present within the cylinder or flowing under the exhaust valve.Equations used in the calculation were taken from [3], and are listed in equations ( 16), ( 17), ( 18), ( 19) and (20).Based on the results yielded by the mathematical model implemented in MathCAD to compute the Jake Brake process parameters, Figure 5 illustrates the variation of cylinder temperature during braking.Notations used: p cil -participation of cylinder gas, p ge -participation of gases from the exhaust manifold, p ga -participation of gases from manifold, cga -the amount of gas flowing from the cylinder into the intake manifold, cge -the amount of gas flowing from the cylinder to the exhaust, gec -the amount of gas flowing into the cylinder exhaust pipe, gac -the amount of gas flowing from the inlet to the cylinder, i cil -cylinder gas enthalpy, u cilinternal energy of the cylinder gas, i gec -gas enthalpy passing from exhaust manifold to cylinder, i gac -gas enthalpy passing from intake manifold to cylinder, Tps ev.ant -gate valve exhaust temperature at the previous iteration step, Tps ev -temperature in exhaust valve gate, c -molar variation coefficient, ev -degree of evacuation, ev -resistance coefficient of the exhaust route, Mcol ev -molar mass of gas in exhaust manifold, s ev -flow coefficient, M cil -molar mass of gas in cylinder, k ev -adiabatic exponent, P 0 -initial pressure, Pcol evpressure in exhaust manifold, W ge -gas velocity in the exhaust manifold, W cge -gas velocity flowing from cylinder to exhaust manifold, dsup ev -exhaust valve diameter, h cr -critical height, dcol ev -exhaust manifold diameter, ev -exhaust valve seat angle, As ev -cross-sectional area of exhaust manifold, d -injection advance, joc fm -heat clearance of exhaust valve, ampl fm -amplification ratio of the rocker, hsup .fm-opening height of exhaust valve, a -specific heat coefficient matrix, cv -heat capacity at constant volume, ach -cylinder head area, ase -exhaust valve area, asa -intake valve area, ap -piston area, D -cylinder bore, T ch -cylinder head temperature, T SF -exhaust valve temperature, T SA -intake valve temperature , T p -piston temperature, T occylinder temperature, n p -engine speed, C conv -convection coefficient, Vs -cylinder swept volume, Vcilcylinder volume, îISF -closing delay of exhaust valve, aDSF -exhaust valve opening advance, -crank angle, r -the ratio of crank radius and connecting rod length, -compression ratio, V c -minimum volume of combustion chamber, R -gas constant, P cil -cylinder pressure, T c il -cylinder temperature, L m -work done by gas, Q r -heat exchanged with the walls, cil -amount of gas in cylinder, C cil -specific heat of gas.

CONCLUSIONS
From the work presented herein, several conclusions can be drawn with regard to the braking process, as follows: -Due to high pressures generated in the outlet region of the exhaust valve opening, no subcritical flow is present; -Gas flow during engine braking process always takes place in the direction cylinder -exhaust manifold (normal flow); -Input data that characterizes the flow regime are the absence of critical flow and, as stated above, the gas flow always is always oriented from the cylinder to the manifold.

Fig. 3 .
Fig. 3.The pressure and the flow velocity in exhaust manifold.
parameter during braking process is shown in Figure4.

Fig. 5 .
Fig. 5. Cylinder temperature variation during braking.Temperatures inside the cylinder are determined by adding in each iteration step, the size of the derivative cil dT d to the previous value.