mixture fraction are specified correctly and that they are the same for both of them.
8. The reactions themselves are defined by specifying the amounts (in
kilomoles) of the participating leading reactants, reactants and products. For example, the input required for the following reaction (combustion of
methane)
(8-1) is
9. pro-STAR includes facilities for checking that mass is conserved for each reaction.
Presumed Probability Density Function (PPDF) Models
Models of this type are described inChapter 10, “Presumed-PDF (PPDF) Model for Unpremixed Turbulent Reaction”in the Methodology volume. These fall into two main groups:
• Single-fuel PPDF, where only one type of fuel and one type of oxidiser are present, though each of these may enter the combustion system through more than one inlet.
• Multiple-fuel PPDF, where two types of fuel and one type of oxidiser are present.
Single-fuel PPDF
The basic equations solved are for the mean mixture fraction and its variance (seeChapter 10, “Single-fuel PPDF”in the Methodology volume). There is a choice between equilibrium chemistry models (these assume a local instantaneous
chemical equilibrium) and a laminar flamelet model that allows for non-equilibrium effects (such as flame stretch).
Equilibrium models
In these models, the PDF integration may be performed in two ways:
1. By employing a numerical integration technique
2. By expressing all instantaneous values of the variables as polynomials of the mixture fraction and then doing the integration analytically. Polynomial coefficients may be
(a) supplied by the user
(b) read in from a built-in database stored in file ppdf.dbs
(c) calculated by the CEA (Chemical Equilibrium with Applications) program [5,6]. This is an auxiliary program that computes the chemical
Reaction (1) kmol
Leading reactant (fuel) (1) — 1
Reactant (1) — 2
Presumed Probability Density Function (PPDF) Models
equilibrium composition of a mixture. This program is included in the STAR-CD suite and is used in conjunction with the built-in PPDF model.
There is also a choice between adiabatic and non-adiabatic PPDF:
1. For adiabatic PPDF:
(a) The mixture density and temperature are calculated numerically or from polynomials in f. Note that these polynomials are based on molar
fractions.
(b) Since temperature is calculated independently, the ‘Constant’ specific heat property option with default values may be used
2. For non-adiabatic PPDF, the density is calculated from the ideal gas law and the temperature from the enthalpy transport equation.
The mass fractions of all other chemical species related to the reaction are defined as additional scalar variables and calculated numerically or from the user-supplied polynomials in f, as above. Up to forty eight such species can be specified by the user.
Laminar flamelet model
In this model, the PDF integration is always performed numerically and the results stored in a look-up table which is characterised by its mean mixture fraction, mixture fraction variance and strain rate. There is also a choice between an adiabatic and a non-adiabatic model, as above.
The setup procedure for the model is described in the on-line Help for the
“Reaction System”STAR GUIde panel. One part of this procedure is to specify the reaction mechanism, stored in a reaction definition file in CHEMKIN format. This is organized in three sections:
• Element data
• Species data
• Reaction data
The basic data are often supplemented by auxiliary data for special reactions such as third-body reactions.
Element Data
All chemical species in the reaction mechanism must be composed of chemical elements or isotopes of chemical elements. Each element and isotope must be declared using a one- or two-character symbol. The purpose of the element data is to associate the element atomic weights with their character symbol representations.
If an ionic species is used in the reaction mechanism (e.g, OH+), an electron must be declared as the element E.
Element data must start with the word ELEMENTS (or ELEM) but, following that, there are minimal restrictions on the formatting of the rest of the section. Any number of element symbols can be written on any number of lines. The symbols may appear anywhere on a line, but those on the same line must be separated by blanks. Any line or portion of a line starting with an exclamation mark (!) is considered a comment and will be ignored. Blank lines are also ignored.
If an element is in the list below, then only the symbol identifying it need appear
Presumed Probability Density Function (PPDF) Models
in the element data. The recognized elements are as follows:
H, HE, LI, BE, B, C, N, O, F, NE, NA, MG, AL, SI, P, S, CL, AR, K, CA, SC, TI,
For an isotope, the atomic weight must follow the identifying symbol and be delimited by slashes (/). The atomic weight may be given in integer, floating-point, or “E” format, but internally it will be converted to a floating-point number. For example, the isotope deuterium may be defined as D/2.014/. If desired, the atomic weight of an element in the above list may be altered by including the atomic weight as input just as though the element were an isotope.
An acceptable format for element data specification is shown below:
ELEMENTS H D /2.014/ O N END! END is optional
Species Data
Each chemical species in a reaction must be identified on one or more species line(s). Any set of up to 16 upper or lower case characters can be used, as for species names, which are case sensitive. In addition, each species must be composed of elements that have been identified in the element data section.
Species data must start with the word SPECIES (or SPEC) but, as already discussed, subsequent formatting of this section is not particularly important. An acceptable format for species data specification is shown below:
SPEC H2 O2 H O OH HO2 H2O
Reaction Data
The reaction mechanism may consist of any number of chemical reactions involving the species named in the species section. A reaction may
• be reversible or irreversible;
• be a three-body reaction with an arbitrary third body and/or enhanced third-body efficiencies;
• have one of several pressure-dependent formulations.
The rate of each reaction is defined by specifying , and from the general Arrhenius rate equation for the forward reaction, seeequation (10-65) in the Methodology volume.
Reaction data must start with the word REACTIONS (or REAC). On the same line, you may specify units of the activation energies to follow by including the word CAL/MOLE, KCAL/MOLE, JOULES/MOLE, KJOULES/MOLE, KELVINS, or EVOLTS. The default units for are cal/mole and the default units for are cm, mole, sec and K. Including the word MOLECULES on the REACTIONS line changes the units of to cm, molecules, sec and K.
The lines following the REACTIONS line contain reaction definitions together with their Arrhenius rate coefficients, as described inTable 8-1. The description is
AR βR ER
ER AR
AR
Presumed Probability Density Function (PPDF) Models
composed of reaction data and optional auxiliary information data.
Table 8-1:
Species Symbols
Each species in a reaction is described by a unique sequence of characters, as they appear in the species data (e.g. CH4).
Coefficients
A species symbol may be preceded by an integer or real coefficient. The coeffi-cient’s meaning is that there are that many moles of the particular species present as either reactants or products; e.g. 2OH, is equivalent to OH + OH.
Non-integer coefficients are allowed, but the element balance in the reaction must still be maintained.
Delimiters
+ A plus sign is the delimiter between each reactant species and each product species.
= An equality sign is the delimiter between the last reactant and the first product in a reversible reaction.
=>
An equality sign with an angle bracket on the right is the delimiter between the last reactant and the first product in an irreversible reaction.
Special Symbols
+M
An M as a reactant and/or product stands for an arbitrary third body. It should appear as both a reactant and a product. In a reac-tion containing an M, certain species can be specified as having enhanced third-body efficiencies; in which case auxiliary data (described below) must follow the reaction line. If no enhanced third-body efficiencies are specified, all species act equally as third bodies and the effective concentration of the third body is the total concentration of the mixture.
(+M)
An M as a reactant and product surrounded by parentheses indi-cates that the reaction is pressure-dependent, in which case auxil-iary information line(s) (described below) must follow the reaction to identify the fall-off formulation and parameters. A species may also be enclosed in parentheses. For example, (+H2O) indicates that water is acting as the third body in the fall-off region, not the total concentration M.
!
An exclamation mark means that all following characters on the reaction line are comments. For example, the comment may be used to give a reference to the source of the reaction and rate data.
Presumed Probability Density Function (PPDF) Models
The second field of each reaction line is used to define the Arrhenius rate
coefficients , and , in that order. At least one blank space must separate the first number and the last symbol in the reaction. The three numbers must be separated by at least one blank space and be given in integer, floating point, or “E”
format (e.g., 123, 123.0 or 123E1). Their units are as specified in the REACTIONS line above. An example of reaction data for a simple mechanism is shown below:
REACTIONS CAL/MOLE
! example of real coefficients END ! END statement is optional;
The basic rules for specifying reaction data are summarised below:
1. The first line must start with the word REACTIONS (or REAC), and may include units definition(s).
2. The reaction description can begin anywhere on the line. All blank spaces, except those between Arrhenius coefficients, are ignored.
3. Each reaction description must have = or => between the last reactant and the first product.
4. Each reaction description must be contained within one line.
5. Three Arrhenius coefficients ( , and ) must appear in order on each line, separated from each other and from the reaction description by at least one blank space; no blanks are allowed within the numbers themselves.
6. No more than six reactants or six products are allowed in a reaction.
7. Comments are any characters following an exclamation mark.
Auxiliary Reaction Data
The format of an auxiliary information line is a character-string keyword followed by a slash-delimited (/) field containing an appropriate number of parameters (in either integer, floating point, or “E” format). Different types of auxiliary reaction data are described below, followed by an example:
1. Third-Body and Pressure-Dependent Reaction Parameters
If a reaction contains M as a reactant and/or product, auxiliary information lines may follow the reaction line to specify enhanced third-body efficiencies of certain species. The keyword defining an enhanced third-body efficiency is the species name of the third body, and its single parameter is its enhanced efficiency factor. A species that acts as an enhanced third body must be declared as a species.
If a pressure-dependent reaction is indicated by a (+M) or by a species contained within parentheses, say (+H2O), one or more auxiliary information lines must follow to define the pressure-dependence parameters. For all pressure-dependent reactions, an auxiliary information line must follow to specify either the low-pressure limit Arrhenius parameters (for fall-off reactions) or the high-pressure limit Arrhenius parameters (for chemically
AR βR ER
AR βR ER
Presumed Probability Density Function (PPDF) Models
activated reactions). For fall-off reactions, the keyword LOW must appear on the line, with three rate parameters , and . For chemically activated bimolecular reactions, the keyword HIGH must appear on the line, with the three rate parameters , and .There are then three possible
interpretations of the pressure-dependent reaction:
(a) Lindemann formulation - no additional parameters are defined (b) Troe formulation - in addition to the LOW or HIGH parameters, the
keyword TROE followed by three or four parameters must be included in the following order: a, b, c and d, as defined inequation (10-77) of the Methodology volume. The fourth parameter is optional and if omitted, the last term inequation (10-77) is not used.
(c) To define an SRI pressure-dependent reaction, in addition to the LOW or HIGH parameters, the keyword SRI followed by three or five parameters must be included in the following order: a, b, c, d and e, as defined in equation (10-79) of the Methodology volume. The fourth and fifth parameters are optional. If only the first three are stated, then by default
and .
2. Landau-Teller Reactions
To specify Landau-Teller parameters, the keyword LT must be followed by two parameters — the coefficients and fromequation (10-81) in the Methodology volume. The Arrhenius parameters , , and are taken from the numbers specified on the reaction line itself. If reverse parameters are specified in a Landau-Teller reaction via REV (see item4 below), the reverse Landau-Teller parameters must also be defined, with the keyword RLT and two coefficients and for the reverse rate.
3. Logarithmic Interpolation of Pressure-Dependent Rates
This generalized way of describing the pressure dependence of a reaction rate is indicated by the PLOG keyword in auxiliary lines. In this case, the reaction description should not include (+M) in it, although this is used to indicate that the reaction is pressure dependent in other cases. This particular option for describing pressure-dependent reactions cannot be combined in any given reaction with other options for describing pressure dependence. One
supplementary line starting with the PLOG keyword needs to be supplied for each pressure in the set. The keyword is followed by slash-delimited values for the pressure (in atmospheres) and the rate parameters for that pressure.
The supplementary lines need to be in order of increasing pressure. If the rate expression at a given pressure cannot be described by a single set of
Arrhenius parameters, more than one set may be provided. Each of these should be followed by the keyword DUPLICATE, meaning the sum of the sets of rates provided for a given pressure will be used. The units of the rate parameters provided with the PLOG keyword should match the units used for the overall reaction description. Note that, in this case, although rate
parameters need to be supplied on the main reaction line to prevent an error, those values are superseded by the ones provided on the supplementary lines.
4. Reverse Rate Parameters
For a reversible reaction, auxiliary data may follow the reaction to specify Arrhenius parameters for the reverse-rate expression. Here, the three
A0 β0 E0
Presumed Probability Density Function (PPDF) Models
Arrhenius parameters ( , , and ) for the reverse rate must follow the keyword REV. This option overrides the reverse rates that would be normally computed by satisfying microscopic reversibility through the equilibrium constant, as described byequation (10-66) in the Methodology volume.
5. Reaction Order Parameters
Auxiliary data may be included to override the reaction order for a species, using the auxiliary keywords FORD or RORD, for forward and reverse reaction descriptions, respectively. Each occurrence of these keywords must be followed by the species name and the new reaction order. This option overrides the stoichiometric coefficients for the species included in the auxiliary data.
6. Reaction Units
It is sometimes convenient to specify units for a particular reaction rate fit that differ from the default units specified for other reaction expressions in the chemistry mechanism. In such a case, you should employ the auxiliary keyword UNITS. This keyword must be followed by one or more of the following unit descriptors: MOLECULE, CAL, KCAL, JOULE, KJOULE, KELVIN, or EVOLTS. The inclusion of MOLECULE would indicate that the reaction rate expression is in units of molecules/cm3 rather than mole/cm3. The remaining unit descriptors specify the energy units in the rate expression.
Note that the temperature units in the rate expression are always in Kelvin.
An example of the use of auxiliary reaction data for a three-parameter Troe fall-off reaction with enhanced third-body efficiencies is shown below:
CH3+CH3(+M)=C2H6(+M) 9.03E16 -1.18 654 LOW / 3.18E41 -7.03 2762 /
TROE / 0.6041 6927 132. / H2/2/ CO/2/ CO2/3/ H2O/5/
Multiple-fuel PPDF
1. Four equations are solved, for the progress variables (primary fuel mixture fraction), (secondary fuel mixture fraction), (primary fuel variance) and (variance of variableξ, seeChapter 10, “Multiple-fuel PPDF” in the Methodology volume).
2. Only an equilibrium chemistry model is available in this case 3. The PDF integration is always performed numerically
Other noteworthy points about PPDF models are:
1. In order to increase the efficiency of combustion systems by increasing the temperature of incoming oxidisers, the use of vitiated air containing combustion products is a viable option. The basic PPDF model, which assumes that only fuel and air enter the system, cannot be used for this kind of problem. However, STAR-CD’s implementation has been extended to allow up to four dilutants to enter the combustion system. The basic setup is the same as that used for the standard PPDF model. However, additional
transported scalars are defined to represent the dilutants; therefore, additional boundary conditions need to be defined for them.
AR βR ER
fp
fs gf
gξ