Mathematical modelling of spark-ignition engines

Mathematical modelling of spark-ignition engines

M. H. Carpenter and J. I. Ramos

Department of Mechanical Engineering, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213, USA

fReceived May 1984)

The flow field, scavenging efficiency, power output, heat transfer losses, and unburned hydrocarbon emissions have been numerically studied by means of a two-equation model of turbulence in a four-stroke, homogeneous-charge, spark-ignition engine. The engine is equipped with an intake valve, an exhaust valve, and a constant rate heat source which simulates the spark plug. Combustion has been modelled by means of a one-step irreversible chemical reaction whose rate is controlled by an Arrhenius-type expression. The numerical results indicate that the intake stroke is characterized by the formation of two eddies which persist in the compression stroke. Turbulence is generated at the shear layers of the air jet drawn into the cylinder, but its level decreases in the compression stroke. Due to the heat released by the spark plug and the chemical reaction, a spherical flame kernel is formed. This kernel evolves into a cylindrical flame when the flame front reaches the piston. Fuel remains unburnt at the corner between the cylinder head and the cylinder wall due to heat transfer losses. The numerical results also indicate that despite uncertainties about the turbulence and heat transfer models, an engine model such as the one studied here can be used to understand the flow field, heat transfer losses, scavenging efficiency, and power output in conventional spark-ignition engines. Such capabilities are very helpful in the development and optimization stages of engines. For example, here the engine model thermal and scavenging efficiencies are 15.69% and 94%, respectively. The peak pressure is 33 atm and occurs at 6’ ATDC. The unburnt hydrocarbon emissions are 7.41% of the total fuel admitted into the cylinder.

Key words: mathematical model, finite difference, combustion, internal combustion engines

The purpose of this paper is to study numerically the turbulent flow field in a piston-cylinder configuration equipped with a spark plug and two valves. The intake valve is an annular orifice located at the cylinder head which opens and closes instantaneously when appropriate; the exhaust valve consists of an infinitesimally thin disc which can penetrate into the cylinder and which has a valve overlap of 36” crank angle with the intake valve. A constant rate heat source located at the cylinder centreline and cylinder head simulates the spark plug. The configuration models a homogeneous charge, spark-ignition engine whose thermal and scavenging efficiencies, power output, flow field, heat transfer losses, and unburnt hydrocarbon emissions are studied in this paper.

The flow field in the same configuration but under motored conditions has been studied by Carpenter and Ramos’ who showed the importance of vortex interactions in establishing the flow field and turbulence levels within the engine cylinder at the moment of ignition. The numerical calculations reported here have been performed in an axisymmetric configuration without swirl by means of a mean Arrhenius kinetics model for the reaction rate. The calculations have been performed under stoichiometric conditions in a homogeneous-charge, spark-ignition model. Similar calculations have been previously reported in references 2-4. Ahmadi-Befrui et al.2 studied combustion in an idealized homogeneous-charge spark-ignition engine having a disc-shaped combustion chamber equipped with a

40 Appl. Math. Modelling, 1985, Vol. 9, February 0307-904X/85/010040-13/$03.00 0 1985 Butterworth & Co. (Publishers) Ltd

Mathematical modelling of spark-ignition engines: M. H. Carpenter and J. I. Ramos

centrally located spark plug and an inlet/exhaust valve. They employed a two-equation model of turbulence and the Magnussen and Hjertager eddy-break-up model’ where the chemical reaction rate is proportional to a turbulence frequency and the minimum concentration of fuel, oxidizer and products. The model assumes that the time scale of the turbulence energy dissipation controls the reaction rate. Ramos and Sirignano3 used a chemical kinetics reaction model in which the contributions of the temperature and concentration fluctuations to the reaction rate were neglected. Both Ahmadi-Befrui et al.’ and Ramos and Sirignano3 employed a two-equation turbulence model and studied the four strokes of the engine cycle. In both calculations an intake/exhaust valve was studied. Ahmadi- Befrui et al.’ used a transformation of coordinates to accommodate the piston motion,6 while Ramos and Sirignano3 used a two-domain technique to transform the moving boundary value problem associated with the piston motion into a fixed boundary problem.7 In addition, Ramos and Sirignano3 introduced a transformation of coordinates to study the valve motion. This coordinate transformation permits the air motion between the intake/exhaust port and the intake/exhaust valve to be studied. The air motion in this region was not studied by Ahmadi-Befrui et al.2

Griffin et a1.4 have also studied a four-stroke engine cycle. They considered a three-dimensional inviscid flow field in which combustion was simply modelled by constant volume heat addition. They also performed two-dimensional calculations using finite rate chemical reactions for the oxidation of gasoline, but did not consider the turbulence effects on the flow field. Multidimensional flow field calculations in internal combustion engines have also been reported by Gupta et aZ.,* Grass0 et aZ.’ and Diwakar.“. l1 Gupta et aL8 studied the two-dimensional flow field in a divided-chamber, stratified-charge engine using a constant eddy diffusivity model and a one-step irreversible chemical reaction. The numerical calculations were compared with experimental data and adequate agreement between the computed and measured results was obtained. Syed and Bracco12 studied the same configuration by means of a one- step overall chemical reaction rate and a two-equation turbulence model. Heat transfer losses were assumed to be proportional to the heat release. The numerical results were shown to compare better with the experimental data than those obtained with the constant eddy diffusivity model of Gupta et a1.8 Grass0 and Braccog used the eddy-break-up model of Magnussen and Hjertage? and compared their numerical results with the experimental data of reference 8; good agreement was obtained despite uncertainties about both the measured quantities and the model.

Diwakar” studied a two-dimensional, direct-injection, stratified-charge engine by means of a constant eddy diffusivity and a one-step irreversible chemical reaction. He also studied the nitric oxide formation and found good agreement between the model and experimental results for the trends in the cylinder pressure and in the unburnt hydrocarbon and nitric oxide emissions. Diwaker” assumed adiabatic walls, but in a later publication (cf: reference 11) he employed a model to compute wall heat transfer rates and wall shear stresses. His computations indicated that improved models for the wall heat transfer rates and shear stresses improved the overall agreement between the model results and the experimental data. He also found that the two-equation turbulent model employed in his calculations was sensitive to the prescribed swirl velocity profiles.

Markatos and Mukerjeer3 also used a two-equation model of turbulence to analyse the three-dimensional flow field in internal combustion engines using a one-step irreversible chemical reaction whose rate was controlled by an Arrhenius- type expression. Benjamin et aZ.14 have also simulated the two-dimensional flow field within internal combustion and assumed that, at the leading edge of the flame front, the reaction rate is governed by the engulfment rate of the unburnt mass into the eddies and their further turbulent breakdown. In the tail flame zone, however, the reaction rate was supposed to be governed by the chemical kinetic action within dissipated eddies. Thus, this model combines a mean Arrhenius kinetics and an eddy-break-up model and was employed to study combustion in a stratified-charge engine.

In this paper studies are made on a homogeneous-charge, spark-ignition engine. The study employs a two-equation model of turbulence, a spark plug model, and a one-step irreversible chemical reaction whose rate is controlled by an Arrhenius-type expression in a piston-cylinder configuration equipped with two valves. Calculations have been performed for two numerical cycles in order to assess the periodicity of the flow and the influence of the residual gas mass and temperature on the computed results. It is shown that two-dimensional models of internal combustion engines can yield very important information on the flow field, flame propagation and emissions. This information can then be used to interpret and extrapolate data from real engines.

Problem formulation

We consider a piston-cylinder configuration such as the one in Figure 1, consisting of a piston of diameter (bore) 7.67 cm; the piston is connected to the crankshaft through a connecting rod of length 20.32 cm. In this configuration the clearance is 1.27 cm, the compression ratio is 7, and the engine was operated at 1000 rev/min. The engine is equipped with an annular orifice located at the cylinder head; the inner and outer radii of this annulus are 1.375 cm and 1.875 cm, respectively. This annular orifice serves as an intake valve which instantaneously opens at 342’ ATPC (after top-dead-centre of the compression stroke) and closes at 180’ BTDC (before top-dead-centre). The exhaust valve consists of a centrally located infinitesimally thin disc of 1.57 cm diameter which can penetrate into the cylinder. The exhaust valve opens at 180’ ATDC and closes at 342”

i

Figure 1 Schematic of the piston-cylinder configuration

Appl. Math. Modelling, 1985, Vol. 9, February 41


BTDC. The exhaust valve stem is infinitesimally thin and is located at the cylinder centreline. We consider a cylindrical polar coordinate system whose origin is located at the intersection point between the cylinder centreline and the cylinder head. We assume that the flow is axisymmetric so that the tangential or azimuthal derivatives of the flow variables are identically equal to zero; in addition, the flow is assumed to come into the cylinder without swirling so that the azimuthal velocity component is identically equal to zero.

With reference to the axes r and z, where r is the radial distance from the cylinder centreline and z is the axial distance from the cylinder head, we can write the instantaneous conservation equations of mass, momentum and energy. The instantaneous flow variables are then decom- posed into the sum of ensemble-averaged and fluctuating quantities which are then substituted into the Navier-Stokes equations; an ensemble average of the governing equations is then taken. In this averaged equations velocity correlations, velocity-temperature correlations, and velocity-species mass fraction correlations are modelled as gradients of the averaged values of velocity, temperature, and species mass fraction, respectively, i.e. a Boussinesq approximation is

Table 1 Values of a, Sa and ‘ym

used in the ensemble-averaged equations of mass, momentum and energy. These equations can be written in general form as:

L (Pa) = S@

where the operator L is:

(2)

and where t is time, r and z are the radial and axial coordinates, p is the density, u is the ensemble-averaged axial velocity, v is the ensemble-averaged radial velocity, Qi is an ensemble-averaged flow variable, and yQ is a diffusion coefficient. The values of a, y@, and SD, i.e. the source term in the equation for @, are given in Table 1.

A two-equation model of turbulence, k/e, where k is the turbulent kinetic energy and E the dissipation rate of turbulence kinetic energy, has been used in the calculations; the model uses an isotropic eddy diffusivity /.Q, employs the Boussinesq approximation to model the triple correlations

Variable @ Equation Diffusion

coefficient y,#,

1 Continuity cl

” Ensemble-averaged axial Pe=Lr+!.G momentum

” Ensemble-averaged radial cc, momentum

k Turbulence kinetic energy &

E Dissipation rate of !&?

turbulence kinetic energy , 2

Yf Fuel mass fraction &

160 ysz = Yf - 44 y* Shvab-Zeldovich variable &. 0

e + $(u’ + 2) Total energy

Source term So

0

F’-_pe

f [1.44P- 1.92pel

09

o”s,, = Z/+;pk+J’~v

+$(;+g] -;V.r[/.@v+pkl

au ia s~~==(-H(OL-(YEND)+H((Y--,GN)I v’.v=-+--(Ty)

az r ar

K = 3.125 x lO”cms/g/s; E = 30000 Cal/mole; Pr = 0.73; Pr, = 1.00; LY,G,V = 12” BTDC; (YEND = 6.5’ BTDC; o = 3000 J/ems/s; W = 28.91 g/mole;

p = 1.983 x 10-4g/cm/s; R = 8.3143 J/mole/K

42 Appl. Math. Modelling, 1985, Vol. 9, February


of the velocities, and, in the form presented here, includes the effects of compressibility in the production term of turbulence kinetic energy, i.e. includes the divergence of the velocity, V-v. The k/e model was developed for incompressible thin shear flows, e.g. boundary layers and shear layers; its validity in compressible recirculating flows is a subject of current interest. Even for incompressible flow situations the validity of the e-equation can be questioned since the modelling of some of the terms appearing in this equation is not very convincing.” For round jets, the k/e model is known to overpredict the spreading rate so the constants of the model are not universal. Despite these difficulties the k/e model is used here, and the effects of compressibility on the Reynolds stresses are included in both the k- and e- equations. It should be pointed out that some authors include a dilatational term (V v) in the e-equation2”6 which has been questioned by others;6 for example, Cosman et al.* included such a term and then performed numerical experi- ments on the values of the constant which multiplies the dilatation term.2 Morel and Mansou? compared the models of Ramos and Sirignano’ and Gosman et aZ.* with their model and concluded that the Ramos and Sirignano model produces results which are in between those obtained by the Gosman et al.* model and the Morel and Mansou? model. The effects of compressibility on the k/e model as well as the applicability of incompressible turbulence models to compressible flow situations are subjects of current interest.

Due to the piston motion, the flow field within an internal combustion engine is characterized by the presence of a moving boundary. A transformation of coordinates has been made to transform the moving boundary value problem into a fixed boundary problem. This is accomplished by defining:

77 = z/W) (3)

where 6(t) is the distance from the cylinder head to the piston. In previous studies3 a two-domain technique was introduced to account for both the motion of the piston and the motion of the valve. It was found that that transformation gave rise to discontinuous temperature profiles because the transformation was itself discontinuous. In order to have continuous temperature profiles, it was necessary to introduce a global temperature correction which accounted for the compression of the air due to the piston motion. With the transformation given by equation (3) it was found that the temperature profiles are continuous and that a global temperature correction is not necessary. However, it is still necessary to account for the pressure rise due to the piston compression by including a global pressure perturbation. Introducing equation (3) into equation (1) and defining:

t* =t

we have

L*(pQ) = so

where the operator L* is given by:

(4)

(5)

1 (6)

where :

dS c=u-_17-

dt

Equation (6) together with the values given in Table I con- stitute a system of parabolic equations which yield the values of p, U, U, e, k, Yf, Y,, and e (these variables are defined in Table 1). Once the internal energy of the system is known the temperature can be calculated. In the calculations it is assumed that the specific heat at constant volume is constant and equal for all the species; C, = 5 Cal/mole/K. The pressure can be calculated from the equation of state which was assumed to be that of an ideal gas with fixed molecular weight, W:

,=+ (8)

where p is the pressure, T the temperature, and R” the universal gas constant.

The following one-step irreversible chemical reaction was considered:

CsHs + 5(02 f 3.76Ns) + 3CO2 + 4H20 + l8.8N2

The activation energy and pre-exponential factor for this (9)

chemical reaction are given in Table I. This reaction has also been used to study the minimum ignition time and energy of confined gaseous mixtures” From the transport equations for the fuel mass fraction (Yf> and Shvab-Zeldovich variable :

160 ysz= Y-- - Y,

44

we can calculate the mass fractions of oxygen (Y,), carbon dioxide and water; the nitrogen is assumed to be uniformly distributed throughout the engine cylinder so that this mass fraction is spatially uniform.

Equations (2) and (6) were provided with appropriate initial and boundary conditions. At the cylinder centreline, the radial gradients of the flow variables were set to zero, except for the radial velocity which is identically equal to zero at the centreline. At any solid wall, including the infinitesimally thin exhaust valve, the normal component of the velocity to the boundary was set equal to the boundary velocity; the tangential velocity at the boundary was calculated by using the wall model or a linear velocity profile depending on the location of the grid nodes with respect to the wall; the Reynolds analogy was employed to calculate the temperature profiles; the temperature of the solid walls was kept equal to 350 K; the component of the gradient of turbulence kinetic energy normal to a solid boundary was set to zero; the dissipation rate of turbulence kinetic energy near a solid boundary was calculated by assuming that the turbulence was in equilibrium and that the production of turbulence kinetic energy is equal to its dissipation rate; the component of the gradient of species mass fractions normal to a solid boundary was set to zero.

Initially the calculations were started 12” BTDC using adiabatic and uniform temperature and pressure profiles, and turbulence kinetic energy and dissipation rate profiles. At 12” BTDC a constant heat rate spark plug model was employed at the centreline to simulate the ignition process and to establish a flame kernel; the heat rate of the spark plug model was 3000 J/cm3/s and was activated from 12” BTDC to 6.5” BTDC. This crankshaft angle interval


Mathematical modeling of spark-ignition engines: M. H. Carpenter and J. 1. Ramos

and heat rate corresponds to 2.75 J/cm3. The heat released by the spark plug was supplied to a computational cell whose outer radius is 0.250 cm and whose height varied with time due to the piston motion and the transformation of coordinates (CL equation (3)); an average value of the spark computational cell depth is 0.058 cm. The heat given by the spark plug is a small fraction of the heat of combustion.

The first cycle was then started with adiabatic conditions and with ignition. Once the combustion was believed to be completed, the mass fractions of fuel and oxidizer were assumed to be frozen. At 180’ ATDC, the exhaust valve opened and a Bernoulli equation was employed to account for the pressure difference between the exhaust manifold and the engine cylinder. The discharge coefficiency was taken equal to 0.29. Once this pressure difference was less than 0.13 atm, the velocity at the exhaust port was calculated from the continuity equation by considering the mass of gases displaced by the piston and the areas of the piston and exhaust port. In the intake stroke, the continuity equation was used to calculate the axial velocity at the intake port; the radial velocity was set equal to zero; the air drawn into the cylinder was assumed to be at 300 K; a turbulence intensity of 0.3% and a turbulence length scale equal to half the engine bore, i.e. 3.835 cm, were used to specify the values of the turbulence kinetic energy and its dissipation rate at the intake port. At the exhaust port the axial components of the gradients of temperature, turbulence kinetic energy and dissipation rate of turbulence kinetic energy were set to zero, and the flow was assumed to leave the cylinder axially.

The valve overlap was also accounted for by means of discharge coefficients in such a way that the intake and exhaust valves always give a flow into and out of the engine cylinder, respectively. The calculations were repeated over three cycles in order to assess the effects of scavenging efficiency, residual gas mass and temperature, and combustion duration on the peak pressure, engine efficiency, and heat losses. In the second and third cycles the spark plug model was also activated at 12’ BTDC and turned off at 6.5” BTDC.

The numerical calculations were performed using an implicit finite difference algorithm of the control volume variety in an unequally spaced 22 x 22 grid; central or upwind differences were used depending on the local value of the mesh Reynolds number. The finite difference equations of axial and radial momentum components were solved first; then solutions were found for the turbulence kinetic energy, dissipation rate of turbulence kinetic energy, energy, and species mass fraction equations. The continuity equation was then brought into balance by locally altering the pressure field; the global pressure variations due to the piston motion were accounted for by integrating the continuity equation throughout the entire computational domain and by globally perturbing the pressure field. The sum of the local and global pressure corrections and the assumed pressure is equal to the new pressure which, through the equation of state, gives the new density. The set of finite difference equations was solved iteratively using a line-by-line procedure because the governing equations, except the continuity equation, are elliptic in space. Convergence within the time step occurred when a specified mass imbalance of less than lo-* g was achieved. The crankshaft angle steps used in the calculations were 0.075” during combustion and 6” in the rest of the cycle.


It was found that due to the chemical reaction, the equations for the fuel, oxidizer, nitrogen, water and carbon dioxide mass fractions did not sum to give the continuity equation. This was corrected for by solving the equations of the partial densities of fuel and oxidizer, i.e. pYf and pY,, and then dividing these densities by the density which is obtained from the equation of state.

Presentation of results

The results presented in this section correspond to the second numerical cycle; the first cycle was computed by starting at 12’ BTDC using adiabatic pressure and temperature conditions, and assuming zero velocity and turbulence fields. Thus, the second cycle contains information about the residual gas mass and temperature. Only some of the most representative figures of the second engine cycle are presented here; further details on the flow field and flame propagation can be found in reference 18. Figure 2 shows the ensemble-averaged axial velocity and turbulent kinetic energy profiles at 300’ BTDC. Four axial velocity and turbulent kinetic profiles are shown in this and some of the following figures. The axial locations of the velocity profiles are also shown in these figures with respect to the cylinder head. For convenience, the figures have been drawn in a fixed ‘cylinder’ although the axial velocity locations are indicated. For reasons of symmetry, only a half cylinder is plotted. At 300” BTDC an air jet is drawn into the cylinder through the intake valve, strikes the piston, and forms a recirculation zone near the cylinder centreline; this recirculation is referred to as the valve vortex. A small recirculation zone is also created at the corner between the cylinder head and the cylinder wall; this recirculation zone is referred to as the cylinder vortex. Only the valve vortex can be seen in Figure 2. This vortex is about 3.445 cm long and has an outer radius equal to

4Q 12.00m/s

K 3 00 (m/s+

0 0.382 1.330 2.087 3.035 3.445

Figure 2 Mean axial velocity and turbulence kinetic energy profiles

at 300” BTDC

Mathematical modeling of spark-ignition engines: M. H. Carpenter and J. I. Ramos

the intake valve inner radius. Turbulence generation occurs at the shear layers of the indrawn annular jet; turbulence is also diffused radially and axially throughout the engine cylinder.

Similar characteristics are observed at 240’ BTDC (Figure 3). This figure shows a turbulence kinetic energy peak near the cylinder wall; this peak seems to be due to the wall-generated turbulence. Figure 3 also indicates that due to the piston velocity, the velocity profiles near the piston are positive, i.e. they do not show the presence of a recirculation zone. Turbulence generation occurs near the intake valve port and is clearly visible in Figure 3. This turbulence diffuses and dissipates; its level is small near the piston.

Figures 3 and 4 indicate that the turbulence kinetic energy levels decrease as the piston decelerates and approaches 180” BTDC. The magnitude of the axial velocity at 240” BTDC and at 180” BTDC is smaller than at previous crankshaft angles since the indrawn jet velocity is smaller. The valve vortex length is smaller than in previous figures. The levels of turbulence are diffused and dissipated in the compression stroke (Figures 4-6). There is a small turbulence peak near the piston due to the larger flow velocity there. The turbulence kinetic energy profiles become more uniform when the piston approaches TDC.

At 12” BTDC the spark plug is activated and the flow field is characterized by the presence of a clockwise rotating vortex and a counterclockwise vortex which are located near the cylinder centreline and cylinder wall, respectively. The temperature profiles at 12’ BTDC are almost uniform although the temperature of the fluid drops sharply to 350 K, the temperature of the solid walls. The temperature increases near the spark plug due to the heat addition and the heat released by the chemical reaction. Elsewhere in the cylinder the temperature is almost uniform except near the solid walls. Fuel is consumed near the spark plug. Due to the temperature rise near the spark, the flame kernel can be visualized as a permeable piston which moves into the fresh (unburnt) mixture; behind the piston, the burnt mixture expands. The compression of the fresh mixture produces small positive axial velocities near the spark plug


at 240’ BTDC

Figure 4

0 0.967 3.432 7.632 6.690

Mean axial velocity and turbulence kinetic energy

at 180” BTDC

K

3.00 (m/sf

I -

profiles

0 0.605 2.600 4.396 6.391 7.255


et 120’ 8TDC

and the centreline; elsewhere in the cylinder the clockwise and counterclockwise rotating vortices are still present, i.e. the effects of the flame kernel on the flow field are local. Initially, the turbulent kinetic energy profiles are almost unaffected by the flame kernel and heat release, while the temperature profiles show a peak of approximately 1500 K near the spark plug due to the heat release and the heat given by the spark plug model.


Mathematical modelling of spark-ignition engines: M. H. Carpenter and J. Ramos

L-- _u_ 5CQm/s

K r-i 3.00(m/s)’

:1 .44!


at 60” BTDC

, ” 4 00 m/s

K LA

6.00 (m/s)*

i 1.157 1.


at 8” BTDC

The propagation of the flame produces flow accelerations and destroys the clockwise rotating vortex. The acceleration of the unburnt gases can be seen in Figure 7. This figure indicates that turbulence is generated by the flame at the axial locations of 0.507 cm and 0.796 cm. This flame- generated turbulence is due to the flow acceleration and velocity gradients.

In order to illustrate more clearly the flame propagation phenomenon, only a sample of the fuel mass fraction profiles are given here. Figure 8 indicates that, near the spark plug, the fuel concentration has been reduced to almost zero. The flame thickness is about 0.30 cm at this crankshaft angle. (A rough idea of the flame thickness can be obtained from Figure 8 by multiplying the axial distances shown in the fuel mass fraction profiles by 1.27 cm, i.e. the clearance). It should be pointed out that in Figure 8 the axes labelled ‘axial distance’ and ‘radial distance’ correspond to the cylinder centreline and piston, respectively. These axes have been normalized by the piston location and half bore, respectively. The spark plug is located at the intersection of the vertical axis and the ‘axial distance’ axis.

0.06

Figure 8 Mean fuel mass fraction profiles at 8” BTDC

u

4.00 m/s

K

6 00 (mls12

0 0.143 0.497 0.780 1.134 1.207


at 5” BTDC


Mathematical modelling of spark-ignition engines: M. H. Carpenter and J. 1. Ramos

At 6.5” BTDC the spark plug is turned off. The flame advances through the combustion chamber accelerating the flow and generating turbulent kinetic energy. Figures 9 and 10 show the velocity, turbulence kinetic energy and fuel mass fraction profile at 5” BTDC. Turbulence continues being generated by the combustion and flame propagation. The flow is also accelerated. However, this acceleration is smaller than at previous crankshaft angles due to the presence of the piston (Figure 9). The flame front is now located near the piston and cylinder wall, and the flame peak temperature is about 2300 K (Figure IO). This figure indicates that the flame is near the piston but not so near the cylinder wall (there is a scale factor of about 3 between the axial and radial axes).

Figure I I indicates that at 6” ATDC the counterclockwise rotating vortex is still present and that the turbulence levels are higher near the cylinder wall where the flame is located. This can be clearly seen in Figure 12 which shows the fuel mass fraction profiles. This figure indicates that the fuel near the piston is burnt before the fuel located at the corner between the cylinder head and cylinder wall. This may be due to the cold walls, but it may also be due to the deceleration that the flame experiences when it approaches a cold wall. It should be pointed out that the heat transfer rates through the solid walls may be in error because of the use of the Reynolds analogy in unsteady compressible flows. It is well known that heat transfer rates to solid walls are controlled by the rates of molecular diffusion; in the present calculations, the thermal boundary layers have not been adequately resolved. More accurate heat transfer rates can be obtained by resolving the boundary layers. Unfortunately, this requires the use of very refined grids which are extremely expensive in terms of both computer storage and execution time.

The chemical reaction proceeds until 30” ATDC, i.e. the combustion angle is about 42”. The fuel consumption is governed by the amount of unburned fuel and oxidizer, and the mixture temperature. The latter drops quickly in the power stroke. Some representative velocity and fuel mass fraction profiles are shown in Figures I3 and I4 before the combustion is almost complete. Figure 13 indicates that the turbulence levels within the engine decrease in the power stroke because of the absence of any

0.06

Figure 10 Mean fuel mass fraction profiles at 5” BTDC

0 0.144 0.500 0.785 1.141 1.295


at 6” ATDC

Figure 12 Mean fuel mass fraction profiles at 6’ ATDC

turbulence-generating mechanism. Turbulence is still generated near the cylinder wall but is also dissipated there. The velocity profiles are being controlled by the piston motion which drives the flow. Some flow acceleration can be observed near the cylinder wall due to the flame front. Figure 14 shows the fuel mass fraction profiles and indicates that fuel remains unburnt at the corner between the cylinder head and the cylinder wall.

In the power stroke and after combustion has been completed, the piston drives the flow which is almost unidirectional (Figure 1.5). Some turbulence is generated near the cylinder wall; turbulence, however, decreases in the power stroke. Heat is being lost at a considerable rate through the solid walls, but especially at the corners between the piston


Mathematical modelling of spark-ignition engines: M. H. Carpenter and J. 1. Ramos

c-_u_ 3OOmls

0 0.168 0.653 1.026 1.491 1693


at 25” ATDC

0.06

figure 74 Mean fuel mass fraction profiles at 25” ATDC

and cylinder wall, and the cylinder head and the cylinder wall. The heat losses through the latter corner explain the amount of unburnt fuel which remains there.

Figures 16 and 17 show the presence of a counterclockwise rotating vortex. This occurs at 180’ ATDC when the piston stops and the exhaust valve opens. The velocities throughout the cylinder are small and the turbulence profiles are almost uniform as are the temperature profiles at this crank angle. After 42’ crank angle, the amount of fuel left unburnt was assumed frozen and the species equations were not solved afterwards. In the absence of combustion, the species equations become uncoupled from the velocity and energy fields; the fuel diffuses from the cylinder wall where its concentration is greatest.

In the exhaust stroke, the piston drives the flow, which is radially deflected towards the cylinder wall by the exhaust valve. Between the exhaust valve and the exhaust port, the flow is radially directed towards the cylinder centreline and the axial velocity presents a peak off the centreline (Figures 18-20). Turbulence is generated between the exhaust valve and the exhaust port because of the flow acceleration. Figure 18 shows that at 240” ATDC the

K _.__ --I

0.80 h/s?

0 0.605 2.600 4.396 6.391 7.255


at 60’ ATDC

u

3COmls

0 0.362 1.330 2.087 3.035 3.445


at 120’ ATDC


Mathematical modeling of spark-ignition engines: M. H. Carpenter and J. 1. Ramos

through the intake port. This can be clearly seen in the axial velocity profile located at 0.141 cm. The turbulence levels at 360’ ATDC are smaller by a factor of 15 than at 300” ATDC and are very large near the exhaust port. (Note the change in the scales of the velocity and turbulence kinetic energy profile when comparing Figures 19 and 20).

The fuel mass (g) profile versus the crankshaft angle (degrees) is shown in Figure 21. This figure indicates that the fuel mass fraction profile is a monotonically decreasing function of the crankshaft angle; it also indicates that it takes about 6” crank angle for the fuel consumption to be

0 0 987 3 432 5.387 7.832 8.890


at 180” ATDC


at 240” ATDC

velocity profiles are almost uniform except near the exhaust port. The locations of the exhaust valve in Figures 18-20 are 1.005 cm, 0.595 cm, and 0.455 cm, respectively. Turbulence generation occurs near the valve and between the valve and the piston because of the axial flow deceleration there. Between the exhaust port and the valve, the flow is accelerated because of the difference of areas between the exhaust manifold and the cylinder. At 360” ATDC (Figure 20) the valve is moving towards the cylinder head. The flow is still accelerated between the exhaust port and the exhaust valve and some air is drawn into the cylinder

H

90 00 (mkJ2

0 0 382 13M 2 087 3 035 3 445


at 300” ATDC

K ;_-- _ _

600 (m/s)’

0 0 141 0.490 0.770 1.119 1.270


at 360’



01 1 I I I

346 354 360 366 372 378 364 390

Cronkstmft angle (‘1

Figure 21 Mass of fuel versus crankshaft angle

650 t Legend

550 x second cycle total

0 3X cylinder head 105s

. First cycle tota,

0 3x piston ,oce loss 450

-507 I I I I I 160 240 300 360 420 460 5

Crankshaft o”9le (“)

Figure 22 Heat transfer losses versus crankshaft angle

appreciable. By the end of combustion the mass of fuel remaining in the cylinder is 0.002511 g compared with the initial fuel mass of 0.027 120 g; thus 92 59% of the fuel is

burnt. The total mass of the charge, i.e. the mass of fresh air, fresh fuel, and residual gases, is 0.4937 g and the scavenging efficiency of the engine is 94%.

Figure 22 shows the heat transfer (Cal/s) through the cylinder wall, cylinder head and piston versus the crankshaft angle; this figure also includes the total heat losses in the first and second engine cycles. The heat transfer losses in the second engine cycle are higher than in the first cycle because of the higher gas temperature due to the residual gases. The heat transfer losses through the piston and cylinder head are smaller than those through the cylinder wall; the heat losses through the cylinder wall decrease when the piston approaches TDC because of the smaller cylinder area for heat transfer, and increase rapidly after TDC because of the temperature rise due to combustion. Beyond 30’ ATDC, i.e. from when the combustion is almost complete, the heat transfer losses decrease quickly in the expansion stroke.

As mentioned before the heat transfer losses calculated here may not be very accurate because of the use of the Reynolds analogy in highly transient and compressible flows. They may also be in error because a wall model was used to match the solid wall temperature of 350 K with that of the gases near the walls, i.e. the thermal boundary layer was not resolved. Our calculations show that the heat transfer losses between 180’ BTDC and 180’ ATDC are 345.074 J; the total available chemical energy is 1270 J but due to the fuel that remains in the cylinder only 1175.9 J are released. Thus the total heat losses through the piston, cylinder wall, and cylinder head are about 29% of the total heat released which is in good agreement with the heat losses found in real internal combustion engines.

The pressure profile versus the piston position, i.e. a p-V diagram in Figure 23. The pressure curve shows some oscillations in the intake stroke because of the use of the continuity equation to define the mass flow rate through the intake port. From 180” BTDC to 12” BTDC the pressure increases almost adiabatically; at about 9” BTDC the pressure curve slope is discontinuous because of the pressure increase due to the spark plug and heat released by the chemical reaction. Figure 23 also indicates that the pressure peak is about 33 atm and occurs at about 6’ ATDC. From when combustion is almost complete, the pressure decreases almost adiabatically but shows a sharp drop when the

exhaust valve opens at 180” ATDC. In the interval from 180” BTDC to 180” ATDC the integral of the pressure curve gives an engine work of 180.761 J, which corresponds to a thermal efficiency of 15.69% (the thermal efficiency is defined here as the work performed by the piston in the compression and power strokes divided by the heat released) which is in good agreement with the efficiency found in real internal combustion engines operating under the same load conditions.

The calculations reported here show that if one knew the pressure and temperature values at the moment of ignition, combustion calculations would be very useful in establishing the engine power, efficiency and unburnt hydrocarbon emissions. Thus, if the residual gas temperature and mass were known, one could approximately calculate the average gas temperature when the intake valve closes. Then one could use an adiabatic compression law to establish

35

Pmton to cylinder head distance

Figure 23 Pressure versus piston location


Mathematical modeling of spark-ignition engines: M. H. Carpenter and J. I. Ramos

the initial conditions at the moment of ignition; it is believed that these thermodynamic conditions are very close to those existing in a production internal combustion engine. The residual gas temperature, however, depends on the mixture stoichiometry, the fresh mixture temperature, and the details of the intake stroke. In addition, the residual gas temperature is a function of the pressure and temperature at the instant of ignition. Thus, the residual gas mass and temperature, intake air mass and temperature, and stoichiometry contribute to the numerical ‘cycle-to-cycle’ variations, which are also a function of the discharge coefficient through the exhaust port. In the calculations reported here, a Bernoulli equation was used up to 270” ATDC at the exhaust port; afterwards the continuity equation was used to establish the boundary conditions. Calculations were also performed for a third numerical cycle but did not show any substantial differences when compared with the second cycle calculations reported here. In particular, at the moment of ignition, i.e. at 12” BTDC, the velocity profiles of the third cycle differ by less than 0.1% from those computed in the second cycle.

Conclusions

The turbulent flow field in a spark-ignition internal combustion engine equipped with two valves has been studied by means of a two-equation turbulent model, a one-step irreversible chemical reaction governed by a mean Arrhenius kinetics expression, and a constant heat rate spark plug. The calculations have been performed under stoichiometric conditions in a piston-cylinder configuration provided with an annular intake port located at the cylinder head which opens and closes when appropriate. The exhaust valve consists of an infinitesimally thin disc provided with an infinitesimally thin stem which can penetrate into the cylinder. The parabolic partial differential equations which govern the flow within the cylinder were transformed into a system of equations with fixed boundaries and written in finite difference form using a control volume approach. The resulting set of difference equations was solved line-by-line in an iterative manner until a specified convergence criterion was reached. The numerical calculations indicate that the intake stroke is characterized by the presence of a clockwise rotating vortex which occupies most of the distance between the cylinder head and the piston. A counterclockwise rotating vortex is also present in the intake stroke; this vortex is located near the cylinder wall. Contrary to previous studies, the cylinder wall vortex strength decreases in the compression stroke due to the piston motion but persists until the moment of ignition. A constant rate heat source was used to simulate the spark plug and flame kernel formation; this heat source is located at the cylinder centreline and at the cylinder head. Once a flame kernel is estab- lished a flame propagates through the cylinder increasing the gas temperature and pressure, and consuming the fuel; fuel consumption is small at the beginning of the flame propagation phenomenon due to the small surface area of the flame. The flame is about 3 mm thick and is characterized by very steep fuel mass fraction gradients; the temperature profiles are, however, much smoother. Initially the flame has an almost spherical shape but, when it reaches the piston, becomes almost cylindrical. The maximum temperature within the cylinder is about 3 100 K but decreases fairly rapidly in the expansion stroke; the temperature profiles also show very sharp gradients near the

solid walls which were assumed isothermal and at a temperature of 350 K. After 30” ATDC the reaction was almost complete and the fuel and oxidizer were assumed frozen. The calculations at the end of the combustion process show that fuel remains unburnt at the corner between the cylinder head and cylinder wall due to the large heat transfer losses through these solid walls. The heat transfer losses are significant in the power stroke where the flow field is driven by the piston and where the velocity and turbulence kinetic profiles are almost uniform; i.e. the flow is almost unidirectional. Turbulence generation occurs at the cylinder wall during the power stroke. This generation may be due to numerical errors since in the calculations reported here the turbulent boundary layers were not adequately resolved, and wall laws were used to match the conditions at the solid walls with those in the fluid.

The exhaust stroke is characterized by an almost unidirectional flow except near the exhaust valve and exhaust port; near the exhaust valve the flow is radially deflected towards the cylinder wall. Between the exhaust valve and the exhaust port the flow is radially deflected towards the cylinder centreline and axially deflected towards the exhaust port; this flow presents a mean axial velocity peak off the centreline. Turbulence generation occurs near the exhaust valve and exhaust port due to the flow acceleration. This turbulence generation does not contribute much to the turbulence levels within the cylinder in the intake stroke; this is because turbulence generation in the exhaust stroke approaches zero as the piston approaches 36O’ATDC.

Although the configuration studied here is not very realistic in that an annular orifice is used as an intake valve, it does show some of the qualitative trends that could be expected in real internal combustion engines. For example, the calculations show that the turbulence levels within the cylinder at the instant of ignition are controlled by the geometry of the intake port; turbulence is generated at the shear layers of the indrawn air jet and is very little influ- enced by the turbulence generated in the exhaust stroke or the turbulence remaining in the cylinder. It also shows that the residual gas mass and temperature, heat transfer losses and vortex structures created in the intake stroke play an important role in establishing a flame kernel and in defining the flame propagation. The numerical calculations also yield information about unburnt hydrocarbons, power output, engine efficiency,.and heat transfer losses. How- ever, some of these results are not accurate, especially those concerning the heat transfer losses, since the boundary layers near the solid walls were not adequately resolved. Furthermore, the Reynolds analogy was used in highly unsteady, compressible situations: the validity of this analogy in internal combustion engine calculations should be examined in the future.

It should also be pointed out that the amount of unburnt hydrocarbons predicted by the numerical model is based on a one-step irreversible chemical reaction whose rate is controlled by the mean temperature and concentra- tions. It is well known, however, that in premixed reacting flows the reaction rate can increase because of the reactant concentration correlations and temperature fluctuations; these effects have been neglected in the present calculations.

More work is also needed in the turbulence models used in unsteady, compressible flows and in spark plug models. Most of the turbulence models used in compressible situations are extensions of those obtained under incompressible flow conditions and do not account for the effects of



compressibility and pressure-velocity correlations; these effects may be very important in compressible flows. Finally, better spark plug models are needed to simulate the ignition phenomena in a reasonable way. Despite the limitations mentioned above, two-dimensional models can be used to understand the flow field, heat transfer losses, power output, emissions, and scavenging efficiency in conventional spark-ignited engines. Such capabilities are very helpful in the development and optimization stages of engines.

Acknowledgements

The calculations reported here were performed under Grant NAG 3-2 1 from the NASA Lewis Research Center with Dr Harold J. Schock as technical monitor.

References

Carpenter, M. H. and Ramos, J. I. ‘Turbulent flow field calculations in an internal combustion engine equipped with two valves’, Numer. Methods in Laminar and Turbulent Flow (eds C. Taylor, J. A. Johnson and W. R. Smith), Pineridge Press, Swansea, UK, 1983, p. 1137 Ahmadi-Befrui, B., Gosman, A. D., Lockwood, F. C. and Watkins, A. P. ‘Multidimensional calculation of combustion in an idealized homogeneous charge engine: a progress report’, SAE Paper No 810151,198l Ramos, J. I. and Sirignano, W. A. ‘Turbulent flow field in homogeneous-charge spark-ignition engines’, Eighteenth Symp. (Int.) on Combustion, The Combustion Institute, Pittsburgh, PA, 1980, p. 1825 Griffin, M. D., Diwaker, R., Anderson, J. D. and Jones, E. ‘Computational fluid dynamics applied to flows in an internal combustion engine’, AIAA Paper-No 78-57, AIAA 16th Aero- snace Sciences Meeting. Huntsville. Alabama. 1978 Magnussen, B. F. and Hjertager, B.‘H. ‘On mathematical modelling of turbulent combustion with special emphasis on soot formation and combustion’, Sixteenth Symp. (Int.) on Combustion, The Combustion Institute, Pittsburgh, PA, 1976, p. 719

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Ramos, J. I., Humphrey, J. A. C. and Sirignano, W. A. ‘Numeri- cal prediction of axisymmetric laminar and turbulent flows in motored reciprocating internal combustion engines’, SAE Trans. 1980,88,1217 Ramos, J. I. and Sirignano, W. A. ‘Axisymmetric flow model in a piston-cylinder arrangement with detailed analysis of the valve region’, SAE Paper No 800286, 1980 Gupta, H. C., Steinberger, R. L. and Bracco, F. V. ‘Combustion in a divided chamber, stratified charge, reciprocating engine: initial comparisons of calculated and measured flame propagation’. Combust. Sri. and TechnoL 1980.22.27 Grasso, F. and Bracco, F. V. ‘Evaluation of a mixing-controlled model for engine combustion’, Combust. Sci. and Technol. 1982,28,185 Diwakar, R. ‘Multidimensional modeling applied to the direct- injection stratified-charge engine: calculation versus experi- ment’, SAE Paper No. 810225,198l Diwakar, R. ‘Direct-injection stratified-charge engine: computations with improved-submodels for turbulence and wall heat transfer’. SAE Pauer No. 820039. 1982 Syed, S. ‘A. and Byacco, F. V. ‘Further comparisons of computed and measured divided-chamber engine combustion’, SAE Paper No. 790247,1979 Markatos, N. C. and Mukerjee, T. ‘Three-dimensional computer analysis of flow and combustion in automotive internal combustion engines’, Math. and Comput. in Simulation 1981, XXIII, 354 Benjamin, S. F., Weaving, J. H., Glynn, D. R., Markatos, N. C. and Spalding, D. B. ‘Development of a mathematical model of flow, heat transfer and combustion in a stratified charge engine’, Stratified Charge Engines 1.Mech.E. Conference Publi- cations 1980-9,1980, p. 91

Ramos, J. I. ‘The k/e model of turbulence in the limitina case Re --f --‘, Report CO/80/3, Department of Mechanical Engin- eering, Carnegie-Mellon University, Pittsburgh, PA, 1980 Morel, T. and Mansour, N. N. ‘Modeling of turbulence in internal combustion engines’, SAE Paper No. 820040,1982 Ramos, J. I. ‘Ignition of confined gaseous mixtures by hot surfaces and hot wires’, AIAA Paper No. 83-0240, AIAA 21st Aerospace Sciences Meeting, Reno, Nevada, 1983 Carpenter, M. H. and Ramos, J. I. ‘Turbulent combustion studies in a spark-ignition engine equipped with two valves’, Report CO/83/4, Department of Mechanical Engineering, Carnegie-Mellon University, Pittsburgh, PA, 1983


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Mathematical modelling of spark-ignition engines