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The instructional version of the cycle simulation used constant specific heats as compared to using variable properties and composition for the complete simulation.. For the proper selec

Comparisons of instructional and complete versions of

thermodynamic engine cycle simulations for spark-

ignition engines

JERALD A CATON, Texas A&M University, Department of

Mechanical Engineering, College Station, TX 77840, USA.(jcaton@ mengr.tamu.edu)

Received 6th July 1999

Revised 21st March 2000

Instructional and complete versions of thermodynamics engine cycle simulations for spark- ignition engines were compared The instructional version of the cycle simulation used constant specific heats as compared to using variable properties and composition for the complete simulation For the proper selection of constant properties, the global engine performance parameters obtained from the instructional version of the cycle simulation were

in close agreement to the values obtained from using the complete version of the simulation The specific values of items such as maximum pressure and temperature, however, were not exactly duplicated Examples are given based on a commercial, spark-ignition engine For the cases studied here, the brake power and thermal efficiency as obtained from the constant property version of the simulation (for a ratio of specific heats of 1.30 and a gas constant of 0.287 kJ/kg K) were in excellent agreement with the same parameters from the complete simulation for a range of operating conditions

Key words: engine, simulation, modeling, thermodynamics

INTRODUCTION

In teaching the subject of internal combustion engines, the use of engine cycle simulations has proved to be an effective way to help students understand engine thermodynamics and operation By integrating the development and use of engine simulations with the other class material, the students acquire a much deeper and fuller appreciation of the subject

Thermodynamic engine cycle simulations have been developed and used since at least the early 1960s [1-5] Over the years, these simulations have evolved into complex and computationally intensive computer programs A major component of the complexity is the algorithms for determining the composition and thermodynamic properties of the working fluid As an alternative to these complex simulations, for instructional purposes, simpler

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simulations may be developed Most often these instructional simulations are based on constant specific heats to avoid the complexity of variable properties and composition Such instructional engine cycle simulations have been used for many years as part of courses on internal combustion engines (see e.g Sorenson [3], Caton [6~-8])

The motivation for the instructional version of the simulation is not shorter computational times Today’s computers are able to execute the more complex programs in reasonably

short times Rather, the use of the instructional versions is motivated by the desire to provide

students the opportunity to program their own simulation and to use this simulation The

benefits of this are described below

The current author has assigned, as a special project, the development and use of an instructional version of an engine cycle simulation to students in an internal-combustion engine undergraduate class since 1982 Each student selects an individual topic to investigate, and develops (and programs) their own individual simulation By carefully structuring and monitoring the students’ development of the cycle simulation, the students have been successful in completing the project during one semester (approximately 15 weeks) The majority of the students have expressed appreciation for the insight and understanding that they have derived from this project

The use of engine models or simulations has many educational benefits The process of developing their own simulation allows the students to apply a number of the capabilities they have obtained from previous classes These previous capabilities include the use of knowledge from their engineering courses (particularly thermodynamics), numerical methods, and computer programming In addition, the use of the simulation permits the students to conduct meaningful parametric of optimization studies within a set of stated constraints Finally, the students must summarize the relevant literature, prepare a full technical paper, and give a technical presentation

The goal of these instructional simulations is not to be predictive, but rather to assist in

understanding engine operation and in categorizing performance trends Although relatively basic, such simulations still retain most of the important features of engines In particular, these features include time (or crank-angle), varying quantities such as the cylinder gas pressure and temperature, heat release, heat loss, intake and exhaust flow rates, and cylinder

mass

The disadvantages of using such a basic cycle simulation is that the lack of time-varying thermodynamic properties results in a loss of accuracy As an example, the maximum cylinder pressure and temperature would not be expected to be accurately predicted even if the overall average properties were used Additionally, the effect of varying the inlet fuel—air ratio can not be described in an exact fashion Finally, the lack of time-varying thermodynamic properties (and the related Jack of variable composition) precludes the ability

to accurately predict exhaust emissions

The objective of the current paper is to compare the results from a complete version of an engine cycle simulation with an instructional version In addition, this paper will specify the conditions which accentuate the agreement between these two versions This paper will first describe the two versions of the thermodynamic cycle simulation: with and without variable properties and composition Next, results will be compared from the two simulations for a number of global and detail parameters

MODEL DESCRIPTION

The two versions of the cycle simulation share many of the same features The major

difference, as described above, is the treatment of the thermodynamic properties Many of the features of the overall cycle simulation have been well documented elsewhere (see e.g [1, 2, 4, 5, 9]) The major assumptions and approximations which were common for the two simulations include the following:

(1) _ the thermodynamic system is the cylinder contents which are assumed to be spatially homogeneous;

(2) all gases obey the ideal gas equation of state;

(3) _ the flow rates are determined from quasi-steady, one-dimensional flow rate equations; (4) _ the intake and exhaust manifolds are assumed to be infinite plenums containing gases

at constant temperature and pressure;

(5) _ the fuel is assumed to be completely vaporized and mixed with the incoming air; (6) the combustion efficiency was assumed to be 100% (i.e no unburnt fuel), and (7) _ the blow-by was assumed to be zero

The final sets of assumptions and approximations are related to the properties For the complete version of the simulation:

(8a) the composition of the cylinder gases is determined using well-accepted algorithms, and for temperatures above a specified temperature (such as, 1000 K), the concentrations of the combustion products are determined from chemical equilibrium

considerations

For the instructional version of the simulation,

(8b) the working fluid is assumed to be a single component with constant specific heats and gas constant

Other assumptions and approximations will be described below as they are introduced in the development

Thermodynamic formulation—complete version

Fig 1 is a schematic of the engine cylinder which shows cylinder heat transfer, work, and intake and exhaust flows For simplicity, the combustion chamber was assumed to be a cylindrical shape For this work, all cylinders of a multiple-cylinder engine are assumed to

be identical, to possess the same thermodynamics, and to operate with identical conditions This means, therefore, that overall results for a multiple-cylinder engine are obtained by multiplying the results from the single-cylinder analysis by the number of cylinders

The primary feature used in the development of this cycle simulation is the first law of thermodynamics which is utilized to derive an expression for the time (crank-angle) derivative of the overall gas temperature in terms of engine design variables, operating conditions, and sub-model parameters Due to space limitations, only a brief overview of this development may be provided Complete details are available from Caton [9, 10]

The first law of thermodynamics for the one-zone formulation for this system is

GE _ 8Qro0_ AW

where E is the total energy, Q,,, is the total net heat input, W is the net work done by the system, m is the mass flow rate into or out of the system, and h is the specific enthalpy

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Fig 1 Schematic of the thermodynamic system

either into or out of the system, and ¢ is time The total net heat input (Q,,,) is different for the two formulations For the complete version, total net heat input is only the heat transfer from the cylinder walls For the instructional version, the total net heat input includes both

the heat transfer from the cylinder walls and the heat release due to combustion This is

The only significant energy of the system is internal energy (u), the only significant work term is due to the piston motion (system boundary motion) So equation (1) becomes

d(mu)

dt

= Orot _ pv + Min Nin — MousMout (2) _ where m is the mass in the cylinder, p is the cylinder pressure, and V is the cylinder volume

time rate of change Expanding,

mu+um+ Qrot — pŸ + Minhin ~ MoutMout (3)

Since, in general, the thermodynamic properties are functions of temperature, pressure and equivalence ratio (composition), the derivative of u may be written as

To obtain a derivative of the gas temperature which is independent of the pressure, an

expression for the derivative of the pressure, p, is needed This may be obtained from the

derivative of the ideal gas equation of state

The derivative of R may be given by

=—T+—p+— OR, OR OR

6

Combining equations (3) through (6), the following expression is obtained for T

mC,(1+ P,)

where,

G= (Qrot — pŸ+ Minhin — Moutlout —um) (8)

where the term du/0T has been identified as the specific heat at constant volume, Cụ

An expression similar to equation (7) is needed for the cases where two different substances exist such as during combustion (bumed and unburned masses) and during intake (fresh charge and residual gases) These expressions are somewhat more complicated, but in general, have the same form For both of these cases, a mass rate of mixing is required For the combustion process, this is given by the derivative of the mass fraction burned (described below), and for the inlet process is given by the inlet flow rate Complete details for these latter two cases are provided by Caton [9]

Thermodynamic formulation—instructional version

For the instructional version of the simulation, the composition and thermodynamic properties are constant For this case, the temperature differential (equation (7)), reduces to

An alternative differential equation for the constant property version of the simulation may

be obtained by using equation (5) and property relations between + Cp, Cụ, and R This results in an expression for the pressure differential equation

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(y¥-DI- xo 2

p= CD ~ (pv MinC Tin — Tuy Cp 1u | (14)

This latter differential equation (equation (14) is often preferred for instructional purposes

since the integration of the equation results directly in cylinder pressure For both equations

(13) and (14), the appropriate mean values of the properties (C,, C,, R and y) are required Items for the gas derivative equations

To solve the above differential equations (equations (7), (13), or (14)), several items are needed These items include the thermodynamic properties, the total net heat input, the mass fraction burned, the convective heat transfer, and the mass flow rates These items are described next

(a) Thermodynamic properties

The thermodynamic properties needed for solving the first law for the complete version of the simulation include the instantaneous specific internal energy, specific enthalpy, specific gas constant, molecular mass, and six (6) property derivatives [9, 11] All the properties are needed as a function of time (crank angle) for the cylinder contents and for any matter entering the cylinder,

Depending on the specific processes during a cycle, the thermodynamic properties may

be for mixtures of air, fuel vapour, and combustion products The concentrations of the combustion products may be ‘frozen’ for the lower temperatures, or these concentrations may be based on an instantaneous determination of chemical equilibrium (see e.g., Olikara and Borman [12]) for higher temperatures Complete descriptions of the algorithms used for determining the compositions are presented in a number of references (e.g [4, 9])

Once the composition is known, the individual thermodynamic properties may be determined The properties of each species in the mixture are first determined for the given temperature and pressure by the use of polynomial curve fits [4] to the thermodynamic data [13] The overall mixture properties are then determined by suitable expressions [9]

(b) Thermodynamic properties

For the constant property formulation, two of the four properties must be specified The four

properties that are relevant for the constant property formulation are the specific heats (C,

and C,), the gas constant (R) and the ratio of specific heats (7) Note that only two of these are independent

Total heat input

The forms for the rate of total heat input are different for the two versions of the simulation,

and are given by

Complete model : đại = Oheat

(15) Instructional model: Q,,) = Qapp + Oneat

where app is the apparent heat release from combustion, and ;„„ is the heat transfer to

the cylinder gases For the complete version of the simulation, the apparent heat release from combustion 1s automatic dụe to the use of absolute values of enthalpy for the reactants and products

For the instructional version of the simulation, the apparent heat release may be determined from an expression for the burned fuel mass The sub-model chosen for the burned fuel mass fraction is given below Using this rate of burning yields the following

apparent heat release rate due to combustion

_ LHV(m)

where LHV is the lower heating value of the fuel, m is the total charge mass, AF is the mass

air to fuel ratio, and x, is the rate of fuel mass burning Note that the total fuel mass is

included in equation (6) by the following group of parameters

m

Mass fracHon burned

The mass fraction burned may be represented by a number of expressions [4] For this work,

the Wiebe function has been used for the mass fraction burned

where Ø, Øp, and @, are the instantaneous crank angle, the crank angle for the start of

combustion, and the combustion duration, respectively Also, a and m are parameters that are selected to provide a match with experimental information For this work

as recommended by Heywood ef al [1] Also, the start of combustion and the combustion duration are specified

Cylinder heat transfer

The heat transfer to the cylinder gases, is

A heat transfer correlation developed for diesel engines, but commonly used for both spark-

ignition and compression-ignition engines, was proposed by Woschni {14] and was selected

for use in the current work

To complete the heat transfer expression (equation (20)), the surface area for the heat transfer is needed The total surface area, which varies with crank angle, consists of the cylinder walls, the piston top, and the bottom of the cylinder head

2

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where B is the bore, V_, is the clearance volume, / is the connecting rod length, a is the crank

radius, and s(Ø) is the instantaneous distance between the crank and piston

1/2

Cylinder volume

Finally, the cylinder volume and the derivative of the cylinder volume as functions of crank

angle are needed The cylinder volume is obtained from the classic slider crank formula

* The flow is quasi-steady

* The flow is one-dimensional

* The flow is reversible

* The flow is adiabatic

* The flow is incompressible

* The flow discharge coefficients are assumed constant.”

Since a real flow would not conform to the above assumptions and approximations, these considerations are corrected by use of an empirical discharge coefficient (C 'p)

_ actual mass flow rate

a typical average value [4]

The flow rates may be determined from the following relationships The flow rates may

be either subsonic of sonic The subsonic flow rates are given by

2(p 2/7 y p (Y-D/y

Note that within the scope of this work, the flow đischarge coefficients easily could be a function of instantaneous valve lift

where A, is the actual open flow area, p,, is the upstream pressure T,, is the upstream gas temperature, and p, is the throat (or downstream) pressure

On the other hand, the flows may be sonic (also known as choked or critical) if the upstream pressure is sufficiently high relative to the downstream pressure such that

The instantaneous cylinder temperature as a function of crank angle (or time) is obtained

by the numerical integration of the temperature differential equation At each time step, all parameters are recalculated The numerical method chosen was a basic Euler technique with

a predictor—corrector adjustment

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where 7p{",9 is the new estimated temperature using the (dT/d®)g based on values at the

current 9, and A@ is the computational time step (crank angles) This estimated temperature

is then used to determine all other values (such as pressures, flow rates, volumes, areas, and

so forth), and then the (d7/d®)g, 49 is recomputed A final (dT/d6)§'*, is determined by averaging the two values

This average (d7/ d6) ^ø is used to determine the final cylinder temperature

Tạ+Ao = Tg (4) A@ (34)

đØ/2+à

Once the cylinder temperature is determined for each calculation step, the ideal gas equation

of state is used to determine the corresponding average cylinder gas pressure

Boundary and initial conditions

To complete the required input information, the boundary conditions for the inlet (temperature and pressure) and for the exhaust (pressure) are needed For the current work, these parameters are specified (either from values obtained from reference [1] or from engineering estimates)

To begin a particular engine cycle calculation, several parameters are now known The initial amount of exhaust gases retained in the cylinder (residual), and the initial cylinder gas temperature and pressure are not known and must be assumed After 720 crank angles, the final values of these three parameters are compared to the initial guesses If the final values are not within an acceptable tolerance to the initial values, the calculation is repeated using the final values for the initial values All else the same, this procedure usually finds convergence within about three (3) complete cycles

ENGINE AND OPERATING SPECIFICATIONS

To conduct the comparison, both versions of the simulation were executed for a commercial spark-ignition engine The selected engine was a V-8 configuration with a compression ratio

of 8.i:1, and with a bore and stroke of 101.6 and 88.4 mm, respectively A part load

operating condition at 1400 rpm with an equivalence ratio of 1.0 was selected for the base case condition In total, nine combinations of engine load and speed are examined This engine was also used by Heywood et al [1] in a set of related studies

Table | lists the engine specifications as used in the present work The valve timings used differ slightly from the original values, and a multiplier was used for the heat transfer These

modifications were used to provide good agreement with the published data, and are

explained in detail elsewhere [9, 10]

Table 2 lists other parameters (and their source) which were needed in this work Although most of the significant engine parameters were duplicated in an exact fashion by the current study, a few parameters were estimated, inferred or interpreted from figures, or modified The parameters affected included items such as the valve timings, the valve lift profiles, the exhaust and inlet pressures, the unused fuel, and the cylinder wall temperature

In general, modest changes in these parameters will not significantly change the results of the current study

Table 1 Engine specifications

diameter (mm) 50.8 max lift (mm) 10.0 opens (°CA aTDC) 357 closes (°CA aTDC) ~136 Exhaust valves:

diameter (mm) 39.6 max lift (mm) 10.0 opens (°CA aTDC) 116 closes (°CA aTDC) 371 Valve overlap (degrees) 14°

Heat transfer multiplier 1.33

Table 2 Engine input and other parameters

Displaced volume (dm?) 5.733 Computed

Frictional MEP (kPa) 72.4 From reference [1]

Inlet pressure (kPa) 52 Inferred from Fig 10 [1] Exhaust pressure (kPa) 105 Estimated

Engine speed (rpm) 1400 From reference [1]

CA of combustion start -22 From reference [1]

Combustion duration CCA) 60 From reference [i]

Inlet temperature (K) 319 From reference [1]

Cylinder wall temp (K) 450 Estimated

Valve discharge coefficient 0.7 Estimated

For the instructional version of the simulation, the ratio of specific heats and the gas constant are specified The values selected for these two properties spanned the complete range of possible instantaneous values for typical engine operating conditions [4] For example, the values for the ratio of specific heats were selected to vary from 1.20 to 1.40 For unburned mixtures, this represents a temperature range of from ambient to over 1000 K, equivalence ratios from 0.0 to over 1.5, and all values of residual fraction For burned mixtures, this represents a temperature range of from ambient to over 2200 K, and equivalence ratios from 0.0 to over 1.5

Similarly, the values for the gas constant were selected so as to vary from 0.26 to 0.32 kJ/kg K This range includes values of all reasonable possible gas mixtures for typical

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engine operation [4] This range of values for the gas constant is equivalent to a range of

values for the molecular mass from about 31 to 26 kg/kmol

RESULTS AND DISCUSSION

The two versions of the simulation were used to compute engine performance for the above

conditions Fig, 2 shows the cylinder pressure as a function of cylinder volume for the variable property case and for three constant property cases with three different values of the

ratio of specific heats with a common gas constant’ of 0.287 kJ/kg K The general trend of

the results for the constant property cases agree with the general trend of the results of the variable property case The agreement of the absolute values depends on the value of the ratio of specific heats For the two cases with a ratio of specific heats of 1.20 and 1.40 the

This value for the gas constant, 0.287 kJ/kg K, is for pure air, and is a common assumption for the

gas constant when constant properties are used

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