Modeling Principles

Levels Of Modeling

Electronic Structure

3D structure

C4H6 + C2H3 = C6H8 + HHow it reacts

(with labels)

General Properties

1,3.butadieneC4H6

Label

Connectivity between atoms

The models of a molecule, the basis of all chemical modeling, have many levels of complexity. The key to modeling, in general, is to decide which level of modeling is appropriate based on what is needed and what is computationally a!ordable.

At the quantum mechanical level, a molecular computation can take from hours to days. For this e!ort the model yields a high degree of detail of the electronic structure, giving fundamental information about the bonding within the molecule, including the three dimensional structure of the molecule. However, if this information is to be computed for a large number of molecules, as in some combustion models, this level of detail is computationally una!ordable.

At a simpler level, the molecule can be viewed as atoms connected together. Implicit within this representation, is some distilled information about how the molecule will look and will react. This type of model has distilled the valence structure of a molecule into a computationally simple form. This type of model is used to generate combustion mechanisms.

Even simpler, a molecule is treated as a label. No detail about its structure is given, but knowledge about how it reacts (and the associated thermochemistry) implicitly gives information about structure. This type of model is used in detailed kinetic computations in combustion.

Combustion Modeling

Where does Combustion Kinetic Modelingfit within

Chemical Modeling?

At what level of modeling is it?

Where does combustion kinetic modeling fit within chemical modeling? Modeling complex molecules and complex chemical systems with the computer can occur at many levels, including many levels of detail. Combustion modeling is no exception. The next couple of slides gives an indication from within all the chemical modeling possibilities where combustion modeling lies and why.

Combustion Modeling

Detailed Chemistry

Network of Reactions

Hundreds or Thousandsof Reactions

andMolecules

Detailed Chemistry is a term often seen in combustion modeling.

The number of molecules can be on the order of tens (for simple molecules and simplified mechanisms), hundreds (typical for hydrocarbon fuel mechanisms) or thousands (for larger fuels such as JET-A or diesel fuels).

What is important is the interaction between the molecules, i.e. the reactions.

The reactions form a network of interactions between the molecules. The molecular details about structure, which in fact could come from more complicated computations such as quantum mechanics, is implicit in which reactions are involved and the associated thermodynamics of each molecule.

This modeling decision about the level of detail is made based on what is absolutely necessary and computational power. In detailed mechanisms, the large network of reactions is of primary importance.

Kinetic Mechanisms

Rate Constants

Reactions

Molecules as

Labels

A kinetic mechanism has the complete information to do a numerical calculation of how a species combusts.

It consists of rate constants, giving how the molecules react with each other.

The molecules are represented as labels. Associated with each label is a set of thermodynamic information.

A combustion mechanism is typically a list of reactions, numbering in the hundreds(or thousands for large hydrocarbons), and the rate constants for each reaction.The thermodynamic information and the set of reactions implicitly detail the structure of the molecule. The origin of this information is either directly from experiment, from analogy with similar molecules (when no direct information is available) or even more complicated quantum mechanical, semi-empirical calculations or statistical mechanical calculations.

Combustion Kinetics

ChemistryReactivity of Species

Kineticist concentrates on

Complexity can be inChemistry

Homogeneous Models

Kinetics concentrates onThermodynamics

andGlobal Chemistry

(if at all)

CFD Models

Complexity in the Flow Models

Isolating PrimaryKinetic Features

Span of Expertise

There are many types of combustion kinetists.

At the chemical end, at the left hand side of the diagram, the combustion kinetisist is concerned with the detailed chemistry and the structure and reactivity of the individual species. This field of combustion is important, for example, determining the reactivity of new fuels such as additives or biofuels, more in-depth knowledge of combustion behavior and the production of emissions such as NOx or soot.

At the physical end, the right hand side of the diagram, chemistry is condensed to a very simple form and the primary concern is the fluid dynamics within a reactive flow. Here, the major role of chemistry is the thermal properties and some simple information about emissions.

General Concepts of Modelingas Applied to Combustion

Edward S. Blurockpresented by

State of the System

Need to translate the Real World

into the Mathematical World

of the model

Essential Concept A State is a description of a 'reality'

An essential concept in modeling is the state of the system.

This stems from the need to translate real world objects to the mathematical objects of a model.

The essential concept of the state of the system is a description of the 'reality' of the model.

This is the set of quantities that describe our reality at any particular place in time, space or whatever dimension we are modeling. The state represents the

isolated set of quantities that we choose to watch within a model.

The key to modeling is representing the real world in simple conceptual objects that a human can understand or even data objects that the computer can

understand. This involves translating the real world objects into mathematical quantities that, eventually, through a program, represents the dynamics of the

system.

Levels of Modeling

How the state is described depends on the levels of modeling.

The model acts on the stateTo produce other states

The process describes the real world

Parameters describing properties of the state

Since it is humanly and computationally impossible to include the infinite detail of reality, the art of modeling entails isolating which aspects of the real world are

relevant to what we want to describe.

Which properties are included and how much detail within each property are included established the level or complexity of modeling.

The model acts on this space of states producing the dynamics of the model.

Within engineering or systems modeling of a combustion process, it is often enough to describe the global quantities such as temperature, pressure, etc. The

particular model, meaning the physical system we are trying to describe, shows the transformations of these global properties and describe the dynamics of

the system.

In detailed kinetic modeling, the relevant information is the set of reactions and thermodynamic information of the molecules. Details about the molecules, such

as electronic structure are ignored and represented indirectly with the thermodynamic and reaction information. All the structural and electronic changes that

occur when two molecules collide are condensed into a simple reaction with a set of rate constants.

Description of State

Within a state there areIndividual objects and their descriptions

What things and what descriptionsdepends on the modeling

A global state condition:Temperature, Pressure, Density...

Several molecules 'reacting' through bond breaking and making

An isolated molecule and its description

The state of the system is the heart of a model and encompasses the objects within the model and the description of the objects

How the state is defined, depends on what the level of modeling and what the model needs.

A more global model could involve a state with just the temperature, pressure, density and other global properties of the system. A particular state would have specific values of these quantities. For example, a global description of what is going on in a engine cylinder could be the

temperature, pressure and density of the gases and the volume of the cylinder. This is enough information to describe what is going on globally within the cylinder.

A more detailed state could be that of an individual molecule, for example a quantum mechanical description (closed shell Hartree-Fock) of a molecule is the state of the electron clouds around fixed nuclei.

Or on a kinetic level, the state of the system can be the vector concentrations (mass/mole fractions) of the species, along with the global temperature and pressure. The kinetic model, described by the set of reactions between these species, gives the transformations of one vector state

to another vector state after a given time interval.

Micro- and Macroscopic View

Microscopic State:

Individual 'things' and their descriptionsAn isolated molecule

Several molecules 'reacting' through bond breaking and making

Macroscopic State:

A conglomerate of 'things' acting togetherstatistical

TemperaturePressure

Within combustion, there are two common views of how to describe the combustion process:

The microscopic and the macroscopic view.

Looking at a microscopic view,

we see individual molecules, each traveling a certain speed.

The reaction fo molecules can be seen as these molecules colliding

resulting in the breaking and making bonds forming a different set of molecules.

But since we are looking at an enormous number of these individual molecules, that we can look at conglomerate properties, the macroscopic view. Instead of

the translational speed of individual molecules, we can look at a distribution of speeds for a whole set of molecules.

This, in turn, can be interpreted into a measureable macroscopic quantity like temperature.

Individual molecules pounding against the sides of a container (at their given speeds) can, at a macroscopic level, be related to pressure.

Statistical mechanics is a bridge between the two views.

What is important, especially at an intuitive level, is that the two views should coincide and not contradict each other.

Combustion Systems

A mixture of theMacroscopic and Microscopic

Which viewdepends on the information one needs

from modeling

But intuitively the two views must coincide

If you understand the process both at the microscopic and microscopic levels

then you understand the systemCombustion modeling makes use of both the microscopic view and the macroscopic view.

Distinguishing between these views reflects the type of information one wants to extract from the modeling process.

But the two views are not exclusive, they must coincide.

Each individual view also helps in understanding the process involved and helps isolate the information that one is trying to extract from the modeling process.

Macroscopic level

Engineering Level:

Engines, Turbines, Combustors, ...

The temperature and pressure changes

(more interested in how to use these properties)

The Macroscopic view consists of more global quantities

At an engineering level,

for example in engines, turbines and combustors

modeling is done at the macroscopic level because of the simplified modeling involved,

One is interesting in more !statistical" or global quantities such as temperature and pressure changes, corresponding heat release of the process as a whole.

An engineer is not so much interested in the exact reason for these changes.

Microscopic Level

Kinetic Studies:

The changes from individual molecules

The consequences of bond making and breaking

Heat releaseTemperature

PressureIntermediates and Pollutants,

However, to understand more thoroughly the origins of the macroscopic quantities, one must look more at the microscopic level.

Because the origin of these changes is a consequence of the bond making and breaking at the molecular level. The making and breaking of bonds, can

release or absorb bond energy.

This, in turn, creates a global heat release effecting the global temperature of the system.

In addition, with bond making and breaking, the number of molecules changes and thus directly effecting the pressure.

More importantly, now with the current emphasis on reducing emissions, understanding the kinetic system at the microscopic is becoming more important. The

origin of pollutants such as soot and NOx can only be investigated at a microscopic level, identifying the individual source molecules involved. If one

understands the source of the pollutants, then one can investigate under which conditions less pollutants are formed. Explaining more thoroughly the

observations that certain pollutants are reduced under certain conditions can aid in optimizing the process to reduce emissions more.

The bottom line is that understanding a process is not just an academic exercise. It has important consequences in the real world of engineering.

Macroscopic: Ideal Gas

The modeling of the gas phase

Viewed as a conglomerate of'non-interacting' (hard sphere billiard) balls.

Good for high temperatures and low densities.

(common assumption in combustion)

The ideal gas law is an equation representing the relationship between essential macroscopic states of a gaseous system, temperature, pressure, volume and

amount of substance.

The ideal gas law models molecules as non-interacting objects with only translational energy and colliding with each other (with no molecular interaction, like

billiard balls.

The assumption is good for high pressure and low densities.

In most combustion kinetic models, the ideal gas law is assumed to hold.

It quantifies observed relationships between temperature, pressure, volume

and amount of a substance in one encompassing equation.

Ideal Gas

Macroscopic Equation of State: PV = nRT

In a volume of size VA bunch of balls (n moles)

Are zooming around (represented by temperature, T)Hitting the walls (creating a pressure, P)

R is the factor that relates them all.

The ideal gas law reflects the proportional dependencies between the macroscopic quantities of temperature, pressure, volume and number (moles) of

particles.

The ideal gas law describes a system in a volume, V,

of n moles of non-interacting molecular balls (like billiard balls)

zooming around with a given average translational energy, represented as the temperature, T,

of the system hitting the wall producing a pressure, P.

The Rydbyrg constant, R, give the proportional relationship.

Non-Ideal Gas

A gas becomes non-ideal when molecules are not billiard balls any more.

They start interactingElectron attractions and repulsion between molecules

Van der Waals forces, Electrostatic

Solids, Liquids, Plasma...High Pressure, High Density

If the gas molecules do not act like billiard balls, then it does not have ideal gas behavior.

Non-ideal gas behavior occurs when the molecules start interacting with each other.

For example with Van der Waals forces or electrostatic forces

The molecule is not a hard sphere anymore, there is interaction caused by electron attractions and repulsions.

This can happen, for example in solids, liquids, plasmas

or under high pressure and high density conditions (the opposite of the ideal gas conditions)

How Much (How Many)?

How much of a particular species is an important question

•Moles of each species

•Partial pressure of each species

•Mole Fraction of each species

•Mass Fraction of each species

In combustion, several common ways to answer this question:

How much of a particular species is an important question.

In combustion, there are several common ways to answer this question: the number of moles, the partial pressure, the mole fraction and the mass fraction

Partial Pressure

P = nRTV

n =N!

i

ni

P =N!

i

Pi

=(!N

i ni)RT

PV=

N!

i

niRT

PV=

N!

i

Pi

The definition of the partial pressure can be directly derived from the ideal gas law.Within the ideal gas law relationship substitute n with the summation.From this relationship, bring out the summation leaving the partial pressure expression for the ith species.The concept of partial pressure is useful to establish the composition of a gas mixture. It is directly proportional to the number of moles of the substance.

Molefraction

1.0 =!nspecies

i=1 Xi

Xi = ninMixtures: Mole Fraction

where n_i is the number of moles of species i

The total sum is 1.0 Gives the fraction of each component

For computations, it is often better to normalize so that the sum of the quantity is equal to one. This gives a relationship between the quantities, like a percentage.The normalized quantity in terms of moles is called the molefraction.By convention, the symbol used for mole fractions is X.The mole fraction is the fraction of number of moles of the ith species over the total number of moles.The sum of the mole fractions is equal to one.During the solving of the algebraic and di!erential equations normalized quantities provide a distinct computational advantage. In kinetic computations, it is used quite often.

Relationship

Being a percentage, the mole fraction can be directly related to the partial pressure within a gas.

The fraction of the total pressure

XiP =ni

nP =

ni

n(nRT

V) =

niRT

V= Pi

Both the mole fraction and the partial pressure measure proportions of the given species. Multiplying the mole fraction by the total pressure represents a fraction of the total pressure. Substituting the definition of the mole fraction and the ideal gas expression for pressure and canceling n, the total number of moles, we get the expression for partial pressure.The mole fraction represents a fraction of the total pressure.

Mass Fraction

Since mass is a more measurable than moles the mass percentage is often used.

1.0 =!nspecies

i=1 YiYi = mass fraction of component

MWmix =!nspecies

i=1 XiMWi

1.0 =!nspecies

i=1XiMWiMWmix

=!nspecies

i=1 Yi

Since mass is a more directly measurable than moles the mass percentage is often used.It is similar in concept to the mole fraction, in that the sum is equal to one. The di!erence is that it is the fraction with respect to weight. In the combustion literature, this is actually the most common form when talking about quantity of species within a combustion process.

The key to the conversion between mole fractions (or concentrations, pressures or moles) to mass fractions is the molecular weight of the species involved. The molecular weight of the mixture is the (mole) fractions of the weights of the individual species.

The mass fraction is the normalized value of the molecular weight multiplied by the mole fraction, measuring the fraction of the total molecular weight due to that species.

By convention, the symbol used for mass fractions is Y.

Typical Problem

Given a 50% mixture of oxygen and hydrogen at room temperature and one atmosphere pressure:

•What is the mole fraction of hydrogen?

• What is the mass fraction of oxygen?

In evaluating scientific results, it is important to be able to convert between the many representations of units that are found within the literature. Fortunately, for the most part, it is only a matter of unit conversions using proportional relationships, i.e. not complicated, just bookkeeping.

An important exercise is to be able to convert between moles, pressure or weight to mass and mole fractions.

For example, given a mixture of 50% oxygen and 50% hydrogen, what is the mole fractions and mass fractions of the species.

Typical Problem

First Assumption: Both hydrogen and oxygen are gases at room temperature, and we can safely assume that 50% is referring to pressure.

Second Assumption: We will ignore all non-ideal deviations and assume that the

Ideal Gas Law applies.

Of course, we have to make some assumptions. First, 50% of what? This is often not said explicitly. However, since we are talking of two gaseous compounds at room temperature, we can safely assume that they are referring to partial pressure. The second assumption, used for computational convenience, is that the ideal gas law holds.

Mass Fraction

•Mass (moles)(molecular weight)

•oxygen: (0.020 moles)(32.0 g/mole) = 0.640 g

•hydrogen: (0.020 moles)(2.02 g/mole) = 0.0404 g

•total: 0.0404 g + 0.640 g = 0.6804 g

•Mass Fractions (species mass)/(total mass)

•oxygen: 0.640/0.6804 = 0.941

•hydrogen: 0.0404/0.6804 = 0.059

•total: 0.941+0.59 = 1.0

To calculation the mass fractions, first the masses of the individual species needs to be calculated using the moles times the molecular weight. In this case the species are oxygen and hydrogen. The total mass is the sum of the two species.The mass fractions are calculated by taking the mass of each species divided by the total weight. This is done for both oxygen and hydrogen. By definition, the total mass fraction is one.

Mole Fraction

The partial pressure of each is 0.5 atmospheres.

By ideal gas law, PV=nRT, each has the same number of moles,

mole fraction =

or directly 0.5 since we know for an ideal gas if the partial pressures are the same, the number of moles are the same

n =PV

RT=

(0.5 atm)(1 liter)(0.082056 l atm K!1 mole!1)(298 Kelvin)

= 0.020..

0.020 moles

0.020 moles + 0.020 moles= 0.5

Fifty percent means that 50% of the pressure is due to each species, meaning that the partial pressure is 0.5 atmospheres. By the ideal gas law, each has the same number of moles. Using the ideal gas law, we calculate the number of moles of the partial pressure, 0.5, which is the same for both species. The mole fraction can then be calculated, yielding 0.5. In one sense, we could have known that the mass fraction was 0.5 because if the partial pressures are the same, the mole fractions are the same.

More Typical Example

•The conditions of the system:

•A reactor of 35cm3

•Temperature 800K

•Pressure 30 atm

•0.1% of a 90:10 mixture of n-heptane:isooctane

•5.5% air (assume 79% Nitrogen 21% oxygen)

•Diluted in Argon

•Calculations

•Partial pressures of each species

•Number of moles of each

•Mass fractions of each

•Mole fractions of each

A more typical example consists of a more complex mixture of species, each given in particular units. The calculations are not particularly di!cult, but require bookkeeping and conversion of units.