Instructional use of a single-zone, premixed charge, spark-ignition engine heat release simulation
Abstract Modeling and computer simulation of an internal combustion engine’s operating processes offers a valuable tool for enhancing our understanding of real physical phenomena and contributes significantly to optimizing and controlling the engine’s operation to meet different objectives. This paper illustrates the use of engine modeling in the educational setting through the development and use of a single-zone, premixed charge, spark-ignition engine heat release simulation. The paper begins by describing the operation of an engine. A heat release simulation is then discussed in depth, and a description is given of how it can be used to offer an understanding of thermodynamic fundamentals in an internal combustion engine. In particular, a comprehensive examination of the thermodynamic properties of the engine working fluid and in-cylinder gas-to-wall heat transfer demonstrates the need for accurate physical-chemical sub-models when performing a high-fidelity heat release analysis. Overall, this study demonstrates the power of such an engine simulation tool in an educational setting.
Keywords IC engine; thermodynamic modeling; heat release analysis
(ProQuest: … denotes formulae omitted.)
A^sub s^ heat transfer surface area (m^sup 2^)
B cylinder bore (m)
c Sitkei and Ramanaiah’s heat transfer calibration constant
C Kornhauser and Smith’s heat transfer calibration constant
C^sub 1^ Woschni heat transfer calibration constant
C^sub 2^ Woschni heat transfer calibration constant
c^sub p^ constant-pressure specific heat (J/kg/K)
c^sub v^ constant-volume specific heat (J/kg/K)
EGR exhaust gas recirculation
EVO exhaust valve opening (degrees)
h^sub c^ convective heat transfer coefficient (W/m^sup 2^/K)
h^sub out^ specific enthalpy of species exiting the control volume (J/kg)
h^sub r^ radiative heat transfer coefficient (W/m^sup 2^/K^sup 4^)
IVC intake valve closing (degrees)
k thermal conductivity (W/m/K)
m mass of contents in control volume (kg)
m^sub a^ air mass flow rate (kg/cylinder/s)
m^sub f^ fuel mass flow rate (kg/cylinder/s)
MFB mass fraction burned
m^sub out^ mass exiting the control volume (kg)
N engine speed (rev/s)
P pressure of contents in control volume (kPa)
P^sub 0^ pressure in the intake manifold (kPa)
P^sub IVC^ pressure of cylinder contents at intake valve closing (kPa)
P^sub m^ motoring pressure (kPa)
Pr Prandtl number
q Heat transfer rate flox (w/m^sup 2^)
Q^sub ch^ apparent fuel heat release (J)
Q^sub ht^ heat transfer from control volume (J)
R gas constant of contents in control volume (J/kg/K)
r radial distance from cylinder centerline (m)
Re Reynolds number
s entropy (J/kg/K)
S cylinder stroke (m)
SP instantaneous piston speed (m/s)
SP mean piston speed (m/s)
T temperature of contents in control volume (K)
t time (s)
T^sub 0^ temperature in the intake manifold (K)
T^sub IVC^ temperature of cylinder contents at intake valve closing (K)
T^sub w^ wall temperature (K)
u instantaneous velocity of working fluid nearest to flux measuring point
U^sub cv^ internal energy of control volume (J)
V volume of control volume (m3)
V^sub d^ displaced volume (m3)
V^sub IVC^ volume of contents at intake valve closing (m3)
W^sub cv^ work from control volume (J)
α thermal diffusivity (m^sup 2^/s)
εg emissivity of in-cylinder gas
εw emissivity of cylinder walls
φ equivalence ratio
γ ratio of specific heats
θ engine crankshaft angle (degrees)
μ dynamic viscosity (Ns/m^sup 2^)
υ kinematic viscosity (m^sup 2^/s)
σ Stephan-Boltzman constant (5.67 × 10-8W/m^sup 2^/K^sup 4^)
ω swirl and engine angular velocity (rad/s)
ωp rotational speed of the swirl paddle wheel (rad/s)
ωs rotational velocity of cylinder contents (rad/s)
Analysis of the rate and cumulative amount of heat released from the fuel within an internal combustion (IC) engine generates considerable information on how to control several engine characteristics, including fuel economy, power, harmful emissions, and noise. A trained engine researcher or designer preferentially attempts to optimize the fuel heat release, as opposed to any other engine diagnostic, since more information about the combustion process lies within heat release curves. Hence, students must learn to diagnose and comprehend heat release curves to improve engine design.
Unfortunately, fuel heat release is not directly measured and must be calculated from other measured engine parameters. Fortunately, in-cylinder diagnostics, such as cylinder pressure, provide enough thermodynamic information to calculate the amount of energy released by fuel combustion. Numerous authors have laid a strong foundation for the accurate and effective computation of fuel heat release [1-14]. Inevitably, disagreement exists between so many different models and approaches to calculating fuel heat release.
Two examples of different modeling approaches are thermodynamic property treatment and heat transfer correlation. For thermodynamic property treatment, options range from a constant value over the engine cycle  to an equilibrium package where cylinder temperature, pressure, and species concentration affect the mixture’s thermal properties . For heat transfer correlation, one can assume negligible heat transfer, or instead model heat transfer based on engine geometry, firing and motoring cylinder pressures, and air motion [16-24]. The choice of model affects the successful analysis of fuel heat release, and hence the potential improvements in engine design. As a result, students must understand the effects thermodynamic property, heat transfer and other correlations can have on heat release analysis.
Similar in nature to the engine cycle simulation created by Caton , the heat release analysis of an engine’s combustion process takes students’ understanding to a more advanced level of in-cylinder physics. The present paper satisfies this educational need by describing a computer program, which features a graphical user interface (GUI), that includes the most popular heat release models and demonstrates the significant differences between them. In order to maintain flexibility, the program has been written in sub-levels, each of which involves a specific model of a physical parameter (for example, heat transfer). In this way, the complexity of the code is minimized, and the program could contain as many subroutines as necessary to encompass the various models and correlations. Linking all the subroutines together is the main program, which references only those necessary to satisfy the user’s preferences. In order to make the program user-friendly, it is linked to the GUI with separate form-based interfaces, allowing the user to pick and choose from various models and calculation methods.
Consequently, the goal of assembling physical models and correlations related to fuel heat release during an engine’s combustion process in one user-friendly environment has been satisfied. This way, students quickly and accurately observe which models best predict fuel heat release for their given engine application. Above and beyond this necessary goal, the program has additional educational benefits. Since it is written as a series of subroutines, modifications or additions are made with relative ease. If students or researchers wish to develop their own correlation for a physical phenomenon, only the subroutine that encompasses that model needs to be modified. Additionally, verifying the new model’s validity against previous models is contained within one program.
Often, heat release analysis is avoided simply because of the effort needed to choose the physical models and to program these models into a code. This may involve more than the student’s basic objectives. Therefore, opportunities to explore the fundamental nature of the engine’s combustion processes are missed. Now, with the ease of a user-friendly environment that alleviates issues associated with model selection, students, researchers, and faculty alike will be able to carry out their fundamental study of engine combustion processes.
The following sections outline the physical-chemical processes associated with heat release analysis, as well as the method used to develop a user-friendly program to compute it. The paper then demonstrates its power as an educational tool with which to explore the performance differences resulting from the use of various model treatments and alternative physical correlations. Following publication of this paper, the heat release program generated will be made open source, to allow other institutions to build upon its framework.
Overview of engine cycles
Before explaining the details of heat release, one must fully understand the basic processes of engine operation. Most engines utilize a piston/cylinder arrangement, where the piston reciprocates from a minimum cylinder volume to a maximum cylinder volume. An IC engine is a reciprocating device that uses combustion to maintain motion. Its primary purpose is to produce torque, which can be used in a wide assortment of applications. For instance, an automobile engine produces torque which is translated into linear motion by the vehicle’s transmission, drive-train, and wheels.
An engine cycle begins with the combustion chamber at its minimum volume, and the piston moving such that the volume is increasing (see Fig. 1). Afresh mixture of fuel and air is inducted as the volume of the combustion chamber increases. After complete induction has occurred, the mixture is compressed by reducing the combustion chamber volume so that both pressure and temperature increase. The mixture ignites near a point where the combustion chamber volume is smallest. In a spark-ignition (SI) engine, a spark plug ignites the fuel/air mixture. Ignition leads to a rapid combustion process that significantly increases the mixture’s temperature and pressure. It releases enough energy within the working fluid to perform piston work, while also producing torque through the shaft. The combustion process expands the volume of the chamber, until it has reached its maximum point. At this time, the burned products are discharged from the chamber and the volume decreases to its minimum point, purging the cylinder. Once the chamber reaches its minimum volume, the cycle repeats itself.
For piston/cylinder engines, these processes usually take four strokes of the piston, or two crankshaft revolutions. Therefore, this engine cycle would be classified as a four-stroke cycle. Two-stroke cycles exist but are losing popularity due to fuel conversion efficiency and pollution issues.
Heat release overview
Heat release analysis of in-cylinder pressure of an IC engine has been performed by many researchers [1-7, 10-14]. The goal in calculating heat release is to determine, on a crank-angle resolved (or time) basis, the energy released by fuel during the combustion stroke of the engine cycle. Since the pressure rise due to combustion inside the cylinder is linked to the fuel energy release, one can calculate this energy release by analyzing the cylinder pressure trace. Cylinder pressure measurement is now a standard diagnostic in any research engine. Note that the heat release calculation is computed only during the closed part of an engine cycle, that is, from intake valve closing (IVC) to exhaust valve opening (EVO).
To calculate heat release from cylinder pressure, one must begin with the First Law of Thermodynamics. The First Law fundamentally states that energy can be neither created nor destroyed. This means that the energy entering the cylinder, mostly in the form of fuel chemical potential, must account for (1) the change in internal energy stored in the cylinder, (2) the piston work output, (3) the heat transfer losses to the walls, and (4) the energy exiting the cylinder with lost mass due to blow-by or crevice flow losses. In mathematical terms, calculation of heat release decomposes into the following equation:
To satisfy the First Law of Thermodynamics, the solution to equation 1 should exactly equal the energy provided by the fuel. To solve this equation, one must accurately determine each of the four terms listed on the right-hand side. Since engine cycles comprise complex physical processes, it is not possible to assign a closed-form equation to all of these terms. Therefore, each term is approximated through models of physical processes and appropriate numerical methods. This immediately indicates that the heat release calculation can only be approximated and never solved exactly.
Notice that equation 1 is written on a per degree basis, or a crank-angle (q) basis. This is because the pressure data are experimentally measured on a crank-angle basis. It can be converted to a time scale by considering the engine speed and the number of crankshaft revolutions per cycle (two for a four-stroke engine).
The first term shown on the right-hand side of equation 1 relates to the change in internal energy of the cylinder mixture. As a mixture’s temperature and pressure change, so does its ‘stored energy’, or internal energy. Intermolecular potential energy, Molecular Kinetic energy, and intramolecular energy are modes of energy storage within a substance. The relationship between internal energy, temperature, and pressure of a substance depends on the complexity of these modes.
Assuming the cylinder mixture is an ideal gas (PV = mRT) and has constant mass, m, during the heat release portion, the crank-angle rate of change of internal energy can be calculated with equation 2:
The mean temperature determined from the ideal gas law is close to the mass-averaged cylinder temperature during combustion, because the molecular weights of the burned and unburned gases have been found to be nearly the same [5, 7].
The mass of the mixture is calculated by simply adding the measured mass of fuel to the measured mass of air. For multi-cylinder engines, the total mass of fuel and air supplied to the engine is usually measured. However, the mass of fuel to each cylinder must be known, since heat release analysis is done for one cylinder only (one thermodynamice process). To maneuver around this, the total mass is divided by the number of cylinders, providing a close but still approximated mass to each cylinder. In general, the constant-mass assumption is applicable to SI engines since a premixed charge of fuel and air is inducted into the cylinder. However, significant blow-by and crevice flow (discussed later) can corrupt the constant-mass assumption.
The temperature in the cylinder is directly calculated from the ideal gas law and the experimentally measured pressure. Cylinder pressure is measured in situ by dynamic piezo-electric pressure transducers. Accurate methods to measure cylinder pressure are described by Lancaster et al. . Cylinder volume is geometrically computed by knowing the cylinder bore, stroke, compression ratio and connecting rod wrist pin offset (if one exists).
The temperature derivative term in equation 2 cannot be computed exactly, since a closed-form equation of temperature versus crank-angle dose not exist. Several numerical methods exist to approximate the derivative of a sample of data. These numerical methods are employed in heat release analysis, which again introduces an approximation.
The treatment of the specific heat value is perhaps more contentious. Specific heat for pure air is a function of temperature and, to a much lesser degree, pressure. However, the chamber mixture is composed of both fuel and air, making the mixture’s specific heat value different from that of air. Numerous approaches have been suggested to accurately depict the true specific heat value [1-3, 5, 12, 17]. A simple assumption which provides first-order accuracy is that the specific heat has a constant value throughout the cycle (which is not true, since changing temperatures or fuel/air ratios change the specific heat value). More complex models use correlations for temperature dependence, or fuel/air ratio dependence, or both. The most sophisticated models use speciation of the mixture, with detailed temperature-dependent correlations for each species, so that a very accurate assessment of specific heat is captured [27-30]. Since the change in internal energy is such a large contributor to heat release, obtaining an accurate value for specific heat is very important (see data comparison later in paper).
The mixture gas constant used in the ideal gas law, R, is closely connected to the treatment of the specific heat values (R = cp – cv). The two specific heat values will have the same functional dependence on temperature and pressure, such that the difference between the two is always the same (i.e. R is not a function of temperature or pressure). However, R will change as the fuel/air ratio changes and/or if dissociation occurs. If the mixture is homogeneous during induction (premixed) one simply needs to accurately determine R based on the measured fuel/air ratio. Accuracy will be lost if the same approach is used for a heterogeneous mixture, where global fuel/air ratios change during the fuel injection process. Similarly, even when the global fuel/air ratio remains constant (i.e. after fuel injection stops), the mixture remains heterogeneous, resulting in spatial variations in R.
Accurately capturing the spatial variations in R throughout the mixture would involve extensive computational work. Assuming the fuel/air charge is mixed perfectly and instantaneously as the fuel/air ratio is changed simplifies the computation. This assumption is known as a single-zone model and is an overriding feature throughout this heat release analysis (i.e. it pertains to specific heat computation as well). A number of papers describe multi-zone modeling approaches [9, 11, 31, 32]. The use of a single-zone model, while not as accurate as multi-zone codes, obtains the same shape, form, and first-order approximation of the assumed true heat release. Therefore, it is deemed acceptable for instructional purposes.
Employing the ratio of specific heats, g, the constant-volume specific heat is determined as:
This simplification is applied because it is easier to fix a value for the gas constant and determine a relationship for the ratio of specific heats based on temperature and/or composition. Analysis of the above equation shows that a small change in the gas constant results in a small change in the constant-volume specific heat. However, a small change in the ratio of specific heats can have a large impact on the constant-volume specific heat because the denominator will be less than one.
Experimental data provide curve fits for g based on mixture temperature and composition [2, 3, 5]. The mixture g is that of the fresh charge before combustion, and as combustion progresses the average temperature in the cylinder increases, yielding lower values for g . The ratio of specific heats varies less on the rich side of combustion . Ideally, g would be varied with fuel specification, air/fuel ratio, exhaust gas recirculation (EGR), and charge pressure and temperature. However, for general-purpose applications, g is usually assumed to be a function of temperature only .
To summarize the discussion related to the mixture gas constant, a correlation usually is applied only if the mixture’s fuel/air ratio changes throughout the cycle. Since R does not depend on temperature or pressure, it only changes when the mixture’s composition changes. Note that this relates only to the computation of the mixture’s temperature from the ideal gas law. The computation of internal energy depends on specific heat values, which depend on the temperature, pressure and composition of the mixture. In the Results section, a comprehensive literature search to find existing correlations for R, cv, and g is documented, along with examples of their effect on a heat release calculation.
The second term on the right-hand side of equation 1 is the work term. This is the term engine researchers are most interested in, since it directly relates to the purpose of the engine. The work produced by the engine is given mathematically by equation 4:
The work computation is perhaps the least contested term of heat release analysis, since pressure is accurately measured and volume is accurately calculated. Equation 4 in essence provides a numerical approximation to the true work term, which is the integral of Pd V. Because the volume of the cylinder can be calculated exactly on a per degree basis, the first derivative of the volume can be determined analytically .
While work is perhaps the least contested term, heat transfer is perhaps the most debated term in heat release analysis. In mathematical form, the heat transfer term is calculated by:
where the surface area of the cylinder can be computed analytically .
Several authors have suggested methods to compute the heat transfer coefficient [16-24, 33-40]. Engine processes are so complex that one universal heat transfer coefficient, even though it has been proposed , does not truly exist. As an example, SI engines have flames of very low luminosity, whereas conventional compression ignition engines have very luminous, radiating flames. Therefore, the SI heat transfer correlation involves only convective heat transfer, while the compression-ignition heat transfer correlation involves both convective and radia-tive effects. Similarly, air motions within the cylinder such as swirl, tumble, and squish all affect heat transfer. As a result of the complexity of the various heat transfer mechanisms, heat transfer coefficients are largely empirical and very specific to individual engine applications. In the Results section, a comprehensive study of heat transfer correlations used for IC engines is given. While correlations that have been proposed for radiation heat transfer have been included for completeness, they are not applicable to the SI simulation created here.
The wall temperature is as contentious as the heat transfer coefficient, since it is rarely measured. Often, the wall temperature value is assumed constant, with ‘best guess’ estimates, or correlated to previous cycle operating temperatures . Even if the wall temperature is computed from a correlation, it is assumed to be spatially constant. In an actual engine, differences in cooling passages and the heterogeneity of combustion processes cause variations in wall temperature. Wall temperature measurements provide an opportunity for more accurate assessments; however, this requires complex instrumentation of the cylinders.
Blow-by and crevice flow effects
The fourth term in equation 1 relates to the blow-by and/or crevice flow effect in the engine. Due to imperfect sealing of the piston rings, gases flow into the ring region (crevice flow) and possibly past the rings (blow-by). For homogeneous charge engines, this results in a loss of fuel mass. Without accurate modeling of blow-by flows, net heat release sums to a value less than the gross heat delivered by the fuel. For heterogeneous charge engines, air mass is usually the only lost mass. Although fuel mass may remain nearly constant, the loss of air mass reduces the total trapped mass, which has implications for the heat release calculations, as well.
Since the program development presented in the following sections does not include a model for blow-by or crevice flows, no more detail will be given. However, it should be noted that this can be a significant effect, especially when attempting to calculate heat release very accurately. There are models in the literature which describe these effects and which can be easily implemented in the program, if desired [5, 7, 11-13]. Note that including blow-by and crevice flow effects changes the constant mass treatment used to simplify equation 2.
As this section has identified, heat release analysis is just as much an art as it is a science. The exact computation of heat release is not possible, since many approximations are embedded in the calculation, ranging from the use of numerical integration to mathematically modeling physical processes. Validating a computed heat release trace is not straightforward, since direct measurement of heat release is not possible. Nevertheless, pure modeling efforts predict heat release results similar to those generated from experimental data; therefore, there is strong confidence in the approach taken.
The heat release program contains two distinct components: the GUI, written in Visual C++, and the heat release code itself, written in FORTRAN. The GUI serves as a straightforward means by which the user inputs data, selects a number of parameters, and views the program output. Once the user has defined the desired inputs to the program, the GUI will call the FORTRAN heat release code. All the computation work is done by the FORTRAN portion of the program. When the final results have been calculated, they are sent back to the GUI to be reported to the user. The FORTRAN portion of the code is stand-alone, meaning that it can be easily included in other programs and called as a subroutine. This characteristic of the code gives it the flexibility to adapt to the needs of a variety of users. The GUI was selected as the interface between the user and the heat release code because of its simplicity and compatibility with Windows-based menus, commands, and graphics.
The FORTRAN heat release code was written with a modular structure, where the bulk of the main program consists of subroutine call commands. In Fig. 2, each box represents a subroutine that is called by the main program. Most of the computation in the heat release code takes place in these subroutines. This architecture allows the code to be understood and modified easily by future contributors. To initialize a heat release calculation, the operational parameters must be defined through the ‘Input/Output Parameters’ window. Here, the location of the source data (pressure versus crank angle) is identified, as well as the destination of the heat release output.
The subroutines that are first called by the main program are used either to prepare the input data for calculation, or to determine important parameters that are used in the heat release calculation. Once these parameters have been calculated, the cylinder heat transfer, work, and internal energy are determined and then summed to arrive at the heat release. Additional subroutines compute the mass fraction burned and rate of heat release. The heat release program uses implicit iteration techniques to converge on final heat release values. To initialize the procedure, constant gas properties (g, R, cp, cv) are assumed to generate values for the mass fraction burned at each step of crank angle resolution. Once the initial values are obtained, the program utilizes user-specified fluid properties, chemical equilibrium or empirical equations to account for the actual variation of these properties. The program recalculates the mass fraction burned at each crank angle and compares its value at EVO with its value from the previous iteration. If the difference between the two is greater than some user-defined small value, the program will continue the iteration process. Otherwise, the program ceases to iterate and the final output is reported.
To demonstrate the usefulness of the program, a number of computational studies were done with two of the most contested terms: internal energy and heat transfer. A comprehensive literature search found several correlations related to each of these terms. In this section, the correlations are given along with the resultant effects on a heat release calculation. A six-cylinder, four-stroke SI engine was used to generate the experimental pressure data used in the heat release analysis. Some of the correlations provided were not explicitly developed for SI engines; however, they have been included in our analysis to demonstrate the variance that the terms give and the overall power of the simulation. Specifications of the engine and operating condition are shown in Table 1, while the pressure trace associated that operating condition is shown in Fig. 3.
As mentioned earlier, correlations can be developed for the gas constant, the ratio of specific heats and the constant-volume specific heat. Commonly used models for each of the above gas properties are described in the next section, to allow for a comprehensive understanding of their overall effect on heat release. In addition, a comparison of each model is performed against an equilibrium properties calculation, also implemented within the heat release program. The discussion will begin with a description of this equilibrium program, and proceed to a comparison with previously published correlations.
A good approximation in engine simulations is to regard the burned gases produced by the combustion of fuel and air as being in chemical equilibrium. This assumption may lose its validity late in the expansion stroke and during the exhaust processes because of the lack of time needed for chemical kinetics. The idea of chemical equilibrium is that any given chemical reaction ceases when specific concentrations of reactants and products are reached. Chemical equilibrium will persist until the system is disturbed, for example by a change in temperature.
Thermodynamically, chemical equilibrium is represented by the minimization of the Gibbs free energy, derived from the First and Second Laws of Thermodynamics. There are a few programs that calculate chemical equilibrium based on this principle, with the most common being the NASA program CEA [42, 43]. This program has the potential to determine the chemical equilibrium of up to 600 species of a fuel/air combustion process. Because of the large number of species available and because this program is well documented, as well as widely used, it is generally considered to be the benchmark program (i.e. all other chemical equilibrium programs are typically compared with this program to assess their accuracy).
In an engine application, there is no need to include this full chemical equilibrium program to determine the burned gas species. This is because there are two smaller chemical equilibrium programs geared towards the IC engine community. In an applied thermodynamics textbook , Ferguson describes a practical chemical equilibrium formulation containing 10 species, while a paper by Olikara and Borman  illustrates a similar program with 12 species in the products. Depcik  found, when comparing these two programs to the NASA CEA program, that NO2 cannot be neglected in the lean region of chemical equilibrium for methanol and nitromethane. Even though the percentage of NO2 is small, it is still important to consider because of its toxicity. In diesel engines, NO2 can be about 10-30% of the total exhaust oxides of nitrogen emissions . Since previous practical mechanisms did not allow for production of NO2, the Olikara and Borman program was modified here to incorporate NO2 using a mechanism introduced by Merryman and Levy . This modified equilibrium program was then incorporated into the heat release program. If the mixture temperature is below 600 K at EVO, the equilibrium program by Ferguson is used; elsewhere, the modified equilibrium program is used. A separate subroutine calculates the thermodynamic properties of the mixture based on the ideal gas law.
The properties of the fuel/air mixture are calculated using their mole fractions (not including all mass, just the fuel and the air). The burned gas properties are then calculated using the fuel input and thermodynamic conditions. The mass fraction burned (MFB) diagram is then used to determine the overall mixture composition (when MFB(q) > 1, it is set equal to one for calculation purposes only).
It has been assumed that any exhaust gas recirculation (EGR) and/or residual fraction do not change composition in the cylinder. The validity of this assumption may be subject to scrutiny, especially under high-EGR operating regimes. Current work within our laboratory aims to improve this formulation. Fortunately, any modifications can easily be implemented, due to the program’s modular design.
As previously stated, heat release analysis using the above equilibrium calculation will serve as a ‘best’ computation within the framework of the single-zone code. Therefore it is included in the following comparisons with other correlations, which are generally easier to apply to heat release analysis.
Krieger and Borman  calculated curve fits for the gas constant during lean and rich operations of IC engines. The lean curve fit was meant to be used for a diesel engine heat release calculation and the rich curve fit for a gasoline engine. They used the equilibrium thermodynamic properties of the products of combustion of Cn H 2 n and air calculated by Newhall and Starkman , using data from the JANAF tables .
No other correlations were found for the gas constant, so the Krieger and Borman correlation was compared with a non-varying gas constant assumption (R = 300 J/kg/K) and the equilibrium formulation. The ratio of specific heats was set constant to 1.25 for both the Krieger and Borman correlation and the constant R solution.
From Fig. 4 it is apparent that the Krieger and Borman gas constant does not vary during the engine cycle. Comparing the non-varying gas constant assumption with the Krieger and Borman correlation, it was found that the gas constant does not have much of an effect on the overall mass fraction burned (see Fig. 4). The Krieger and Borman curve falls right on top of the non-varying gas constant assumption. The equilibrium program does calculate a change in the gas constant during the engine cycle due to dissociation effects, but this is small. In the next section, these discrepancies are shown not to result from the differing gas constant values.
Thus, from the experimental pressure data used here, the Krieger and Borman correlation provides mass fraction burned results nearly identical to those obtained by using a non-varying gas constant until the crank angle is about 0° (i.e. the crank is top dead center). Thereafter, the equilibrium formulation provides slightly higher final mass fraction burned values (by around 2%).
Ratio of specific heats
It has been mentioned that the most important thermodynamic property used in calculating the heat release rates for engines is the ratio of specific heats . Quite often, an approximate approach is to model g as a linear function of temperature for a representative engine fuel/air composition. This is because the behavior does not depart greatly from a linear decrease with temperature. A listing of relationships for the ratio of specific heats available in the literature is given in Table 2.
Gatowski et al.  determined the relationship for g for an indolene-fueled Ricardo engine which operated with a stoichiometric mixture. They also determined values of g for a square piston engine, using the NASA equilibrium program [42, 43] for propane/air products of reaction at the particular equivalence ratio used.
Brunt et al.  found that the relationship between g and temperature is almost linear and that the variation with equivalence ratio is significant but much smaller than the effect of temperature. A second-order correlation between g and temperature was obtained which gave a reasonable fit to all of the data for a gasoline engine. Brunt and Platts  extended these results for the analysis of simulated diesel data. A small offset was applied to the earlier fit to accommodate the leaner mixtures in a diesel engine.
In Fig. 5, the correlations provided in Table 2 are compared with a constant g assumption (g = 1.25) and the equilibrium formulation. The effects on the mass fraction burned are also presented. The gas constant was set at 300 J/kg/K for the correlations and the constant g assumption to distinguish the effects of the different correlations.
The Gatowski curves predict a significantly lower minimum g, likely the result of using atypical fuels. The equilibrium formulation provides a dependence on both temperature and equivalence ratio, making it most suitable for applications where the fuel/air ratio can change from data-set to data-set (such as heterogeneous, direct-injection gas, and homogeneous charge compression-ignition engines). To use any of the correlations, one must choose the appropriate one for any particular engine.
Notice from Fig. 5 that the ratio of specific heats has a large effect on the mass fraction burned. As previously discussed regarding equation 3, g has one subtracted from it in the denominator; hence, any change in its value will have a more dramatic change in the constant-volume specific heat. However, any change in the gas constant, R, will result in an equivalent change in the constant-volume specific heat. By evaluating equations 1 and 2, one will notice that the constant-volume specific heat is an important variable in the computation of heat release.
While the overall g values may be subject to further scrutiny, the intention is to demonstrate the effect of the ratio of specific heats on the overall mass fraction burned results. The ratio of specific heat correlations applied to the experimental pressure data resulted in a 10% variation from the highest mass fraction burned value to the lowest. The correlations in Table 2 all had a final mass fraction burned value approximately 5% lower than the run involving a constant ratio of specific heats.
This analysis makes evident the usefulness of such a program in an educational setting, as students are able to quickly visualize how the choice of certain ‘constants’ can have dramatic effects on thermodynamic outcomes. Beyond blindly choosing constants from published tables, students learn to appreciate the need to scrutinize both the equations they use and the researchers’ methods to develop these equations.
Constant-volume specific heat
Instead of calculating the gas constant and g separately, some authors [12, 27] have calculated the constant-volume specific heat directly. Hohenberg and Killman  determined that the constant-volume specific heat is a linear function of temperature, as shown in Table 3. In their correlation, the first and second terms related to air only; the third term is added to account for fuel and residual contents in the cylinder. The factor A in the third term was determined empirically by them based on their own data, and differs for gasoline and diesel fuel.
On the other hand, Krieger and Borman  decided to directly calculate the internal energy in the cylinder using the same reference data as they did for the gas constant. By taking into account a reference temperature (298 K), their constant-volume specific heat value was inferred as a function of the crank angle for comparison within this paper.
The correlations provided by Hohenberg and Killman and Krieger and Borman are compared in Fig. 6 against the constant R and g assumption (R = 300 J/kg/K and g = 1.25), as well as with the equilibrium formulation. The effects on the mass fraction burned are also illustrated. Clearly, the specific heat correlations have the same shape as the equilibrium formulation. Furthermore, the results show that the correlations by Hohenberg and Killmann are practically equivalent to each other and over-predict the mass fraction burned with values greater than one. The First Law analysis described in a previous section dictates that the mass fraction burned should be less than or equal to one. When using the Hohenberg and Killmann correlation, the hump in the mass fraction burned curve for crank angles between 0 and 50 degrees indicates that the correlation over-predicts specific heat when temperatures are high, and under-predicts it as temperatures fall (since the mass fraction burned curve falls to around 1). Krieger and Borman’s correlation similarly exhibits a hump, but not as dramatic. The differences among the final mass fraction burned values computed from the measured pressures using the various alternative correlations are within 10%.
It should be noted that most published research indicates a preference for utilizing curvefits for the ratio of specific heats instead of directly calculating the constant-volume specific heat and/or internal energy. This presumably is because it is easier to linearly correlate g to temperature and equivalence ratio, as evident by the equations in Table 2.
Over the past 80 years, a number of correlations have been developed for predicting the heat transfer from the cylinder gases to the cylinder walls in an IC engine. This mode of heat transfer reduces the potential work of the combustion process and, as a result, needs accurate modeling. Unfortunately, the changing volume of the cylinder, the non-instantaneous combustion process, and the changing wall temperature of the cylinder create a very complex physical situation to model. The following discussion provides a snapshot of some of the more popular engine heat transfer correlations developed in the past.
One of the first heat transfer correlations was developed by Nusselt in 1923 . This correlation was developed from empirical observations in a spherical bomb simulating conditions close to those observed in an engine combustion process. The correlation has two heat transfer contributions (convection and radiation) and applies them to the one-dimensional heat transfer equation. Both Briling [taken from 48] and van Tyen  later modified the Nusselt convection correlation by adjusting the constants in front of the mean piston speed.
Eichelberg developed a correlation based on experimental measurements of the instantaneous heat flux on naturally aspirated two- and four-stroke diesel engines . Henein evaluated the Eichelberg model against experimental data collected from a Lanova-type diesel engine . He concluded that a theoretically estimated instantaneous gas velocity, as opposed to the mean piston speed, must be used to accurately capture heat transfer in the compression stroke.
Elser’s model represented the first attempt to apply theoretical and dimensionless terms to the heat transfer correlations . However, his model bundles two major physical effects into one mathematical term, where radiation and combustion are lumped as ‘change in entropy divided by specific heat’. Providing little physical meaning, this term makes the Elser model both controversial and rarely used. Furthermore, in Elser’s model, the thermal and fluid properties of the working fluid were simplified to be those of air, as given from Incropera and DeWitt . Oguri later modified the Elser model such that it was a function of engine crank angle q . Similarly, Taylor and Toong relied on dimensionless parameters to theoretically develop an engine heat transfer correlation . Their correlation was time averaged for in-cylinder convection and defined for one engine cycle.
Overbye et al. made considerable contributions to the study of engine heat transfer . By studying motored engine operation, they pointed out that a physical phase lag occurs between the heat flux and the bulk gas/wall temperature difference. This phase lag develops from the unsteady nature of the thermal boundary layer, and imposes a problem concerning the physical meaning of the heat transfer coefficient. At times during the cycle, the heat transfer coefficient could reflect an unreasonable value (such as infinite or negative) due to this physical phase lag. As such, they developed a correlation to accommodate this phase lag by considering wall temperature. They similarly investigated the effects of engine deposits on in-cylinder heat transfer. Their experimental results show that engine heat transfer stabilizes after 12 hours of operation in the presence of wall deposits, whose growth depends on the chemical structure and boiling point of the fuel.
Annand also made significant contributions to the heat transfer modeling effort, by proposing a simplified relation for the non-dimensional Nusselt correlation . The correlation is similar to those describing turbulent flow in pipes or over plates, and depends on the value of the Reynolds number. This correlation further includes a radiation heat transfer coefficient for application in equation 5. However, this coefficient uses empirically developed constants. While still widely used, Annand’s approach is perhaps the most significant contribution. Since its publication, most heat transfer models were developed in similar fashion from the Nusselt/Reynolds numbers relationship. Knight later adopted the approach proposed by Annand and further defined an energy-mean velocity . Many researchers continue to use this idea of an energy-mean velocity.
Subsequently, Annand and Ma introduced an unsteady convection term in addition to redefining a characteristic velocity based on energy-mean velocity, as per Knight’s methodology . The unsteady convection term attempts to empirically correct for the phase lag discovered by Overbye et al. In this analysis, the engine’s angular velocity is calculated from the engine speed. Continuing the improvements to Annand’s original correlation, Annand and Pinfold further revised the correlation beyond that of Annand and Ma . They added an unsteady correction based on empirical data from a motored engine. Kornhauser and Smith revisited the original issues discovered by Overbye et al. regarding the phasing of the heat flux and gas/wall temperature difference . After theoretical analysis, they developed a complex variable formulation that utilizes an unsteady correction term that nearly equals Annand and Pinfold’s empirically based term.
Perhaps the most widely used engine heat transfer correlation is that proposed by Woschni . Continuing the development of heat transfer from the Reynolds and Nusselt numbers, Woschni’s correlation depends on the cylinder bore, cylinder pressure, bulk gas temperature, and a characteristic velocity. The characteristic velocity includes effects from piston motion as well as combustion. For the analysis in this paper, the engine speed, N, was converted from (rev/s) to (rad/s) and used for the swirl parameter wp. Woschni’s definition of characteristic velocity, which includes piston effects as well as combustion effects, distinguishes it from other heat transfer coefficients. Similarly, it uses embedded constants to calculate properties of the gas mixture, as opposed to computing these prior to execution of the model.
In a parallel effort, Sitkei and Ramanaiah used an equivalent diameter for the characteristic length and the mean piston speed for the characteristic velocity in order to develop a new exponent of the Reynolds number . They added a constant to account for potential turbulence, and varied the constant depending on the shape of the combustion chamber (c varies from 0 to 0.35). Subsequently, Hohenberg revised Woschni’s model based on time-averaged heat flux measurements in a direct-injection diesel engine with swirl . Modifications to Woschni’s model include revisions to the characteristic length, changes to the effective gas velocity, and modifications to the exponents in the temperature terms. More recently, Han et al. developed a heat transfer correlation resembling that of Woschni and Hohenberg; however, they estimated the effects combustion on in-cylinder gas velocity and turbulent intensity by using a zero-dimensional heat release model . They claim previous models underestimate this effect, and by considering these effects improve the issues with phase lag.
Lefeuvre et al. took a different approach by basing their model on a rotating system, to simulate radial variations within the cylinder . However, their analysis revealed that the assumption of a uniform gas temperature may impose physical errors. Similarly, they raise suspicion over the use of a mass-averaged temperature. In this analysis, the radial distance was set to half the bore value and engine speed, N, was converted from (rev/s) to (rad/s) for the rotational velocity parameter. Similar to the work by Lefeuvre et al., Dent and Suliaman developed a radial steady turbulent convective heat transfer correlation for flow along a flat surface . They utilized a combustion zone temperature as opposed to a bulk mean gas temperature for more accurate results. Similarly, they accounted for swirl enhancement of combustion.
It should be acknowledged that several other heat transfer correlations have been reported, which primarily contain turbulence sub-models to more accurately determine characteristic velocity and length scales . However, these models are not currently incorporated into the heat release program since they require considerable additional programming effort. Nevertheless, the above discussion covers most of the heat transfer correlations that are commonly used in engine research. Consequently, all those models have been incorporated as user options in our heat release program, and their relative performance is illustrated below. Table 5 summarizes the parameters that had to be specified in the models used for this comparative illustration.
From Fig. 7, it appears that the various alternative convective heat transfer correlations produce coefficients which vary widely in both shape and magnitude. As a result, selection of the heat transfer correlation can alter the final values of the mass fraction burned, shown in Fig. 8, by up to 24% for the same set of measured pressure data. This demonstrates that the choice of heat transfer correlation plays a significant role in estimating heat release. Ideally, the ultimate goal is to have enough accuracy in modeling and computing fuel heat release for the mass fraction burned to equal one at the end of combustion. However, as described throughout this paper, the heat release analysis method has several inherent errors that prevent an accurate assessment. These inherent errors can range from experimental error in measuring pressure traces and flow rates, to incorrect mathematical modeling of various physical processes, such as heat transfer coefficients and thermodynamic properties. Therefore, it may be inappropriate to tweak the heat transfer coefficient until the cumulative mass fraction burned equals one. Nevertheless, with the inclusion of nearly all heat transfer correlations, students are well equipped to visualize the variations imposed by differing assumptions and to make educated decisions about which correlations sufficiently model their application.
This paper has described a comprehensive single-zone, premixed charge, SI heat release simulation. While the theory of such a model already exists in the literature, a number of important improvements have been made. By building a flexible simulation framework, the assessment of various property treatments and heat transfer correlations is quickly demonstrated. For instance, it was shown that the ratio of specific heat correlations applied to the experimental pressure data resulted in a 10% variation, from the highest to the lowest, in the value for the mass fraction burned. Similarly, selection of the heat transfer correlation altered the final mass fraction burned values by up to 24%. Clearly, the choice of these parameters significantly affects the calculated result.
It is possible to extend the model presented herein utilizing the following considerations:
* multiple zones [9, 11, 31, 32];
* variable mass (i.e. fuel injection) [10, 52, 53].
While this model was not intended for a variable-mass system (diesel, gasoline direct injection), it has been used in the past for such a case . There will be some error introduced in doing so; however, this may be within the tolerance of the assumptions used in the model (heat transfer correlations, for example).
The educational aspects of this model allow students to gain a better fundamental understanding of the effects of thermodynamic properties, heat release and IC engines. By utilizing actual experimental data, the students are able to expand their theoretical knowledge into real-world phenomena.
The authors wish to acknowledge Dr Zoran Filipi, research scientist at the University of Michigan, for his contributions and comments regarding model selection and implementation. Furthermore, Dr Pin Zeng, who worked along with the authors at the University of Michigan, is thanked for his contributions to the heat transfer correlations. Dr Stani Bohac is thanked for providing experimental data. Finally, the authors would like to thank William Lim for his contributions to improved imagery in the paper.
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Christopher Depcik,1 Tim Jacobs (corresponding author),2 Jonathan Hagena1 and Dennis Assanis1
1 University of Michigan, 1116 W. E. Lay Automotive Laboratory, 1231 Beal Avenue, Ann Arbor, MI 48109-2121, USA
2 Texas A&M University, 3123 TAMU, College Station, Texas, 77843-3123, USA
Copyright Manchester University Press Jan 2007
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