progress variables for cfd - strömningsteknik · generic curves from synchronization of events can...

Edward S. Blurock, Energy Sciences, Lund University Progress Variable Fundamentals

Progress Variablesfor CFD

Trying to get something for nothing

Edward S. BlurockDivision of Fluid DynamicsDepartment of Energy SciencesLund University

Work supported by Swedish Science Ministry (VR project)

Wednesday, March 17, 2010


Outline

•Progress Variables as representation of chemical source terms•Standard Enthalpy as progress•Synchronizing chemical events to improve progress chemistry•Generic chemical behavior•Clustering configurations to improve range of validity•Summary


Edward S. Blurock, Dept. of Energy Sciences Lund University Chemical Source Terms in CFD

3

Kinetic Information and Complexity

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 ExpertiseComplexity



Interaction with CFD cells

Exchange ofPhysical properties (T,P,...)

andchemical composition ( Y )

Mixing Differential Equations:Physical Properties

+Chemical source term ( ὠ )

(Ti,Pi,.., Yi)

(Tl,Pl,.., Yl)

(Tk,Pk,.., Yk)

(Tj,Pj,.., Yj)

(Tm,Pm,.., Ym)



Chemical Source Termsω = f(T,P,Y).

Zero dimensional adiabatic constant V (or P)

SystemDifferential Equations

Differential equation in n + 2 variables where n is the number of species

Discretized

Δω = f(T,P,Y,Δt)Complexity highly dependent on number of species

(operator splitting)

(T,P,Y)

Δt(T,P,Y)+ Δω



Interaction with CFD

CF

D Evaluate f(T,P,Y,Δt)

(T´,P´,Y´)

Sol

veD

iffe

rent

ial E

quat

ions

Calculation of source terms(conditions supplied by CFD cell)



Δω = f(T,P,Y,Δt)Substitute differential equations

with a simple (tabulated) vector functionx(t+Δt) = f(x(t),Δt) where x is (T, P, Y1, Y2, ..., Yn)

A mapping from xi to xj

Principle of Tabulation



xi = f(xj)

Need to cover(tabulate)

all f(xi)from ignition to equilibrium

Principle of Tabulation

(vector function)



TabulationInteraction with CFD

CF

DEvaluate f(T,P,Y,Δt)

(T´,P´,Y´) Ret

riev

e F

rom

Tab

ulat

ion

Differential Equations replaced bySearch into a table

Assumption:Time to search table << Time to solve differential equations



Do not need to span the entire n dimensional space

Chemical events are happening in unisonThe n dimensions are not totally independent

neffective < nBasis must at least span a set of

zero dimensional adiabatic constant volume ignition sets

What Needs to be Tabulated?



Single Ignition rununder a single condition T

ime

(T,P,Y)

(Full) Tabulationcombines

set of conditions

Vary:Starting Temperature

PressureEquivalence Ratio

Equilibrium Configurations: EGR

Simple Tabulation



12

Exchange ofPhysical properties (T,P,...)

andchemical composition ( Y )

Physical Properties+

Chemical source term ( ὠ )

(Ti,Pi,.., Yi)

(Tl,Pl,.., Yl)

(Tk,Pk,.., Yk)

(Tj,Pj,.., Yj)

(Tm,Pm,.., Ym)

However dimension is increased by MixingPrinciple of Tabulation



xi´

xjxiTwo points in configuration space

This moves them off the ‘normal’ configurations

xj´Should accountfor these extraconfigurations

also

Additional Tabulation Points



A simple compensationAddition of End Gas Residue (EGR)

(percentages of Equilibrium configurations)

+ p* =

Mix in a percentage ofEquilibrium state

Varynot only equivalence ratio

but alsoamounts of EGR

Additional Tabulation Points



What Should be Tabulated?What information

from the range of conditionsshould be tabulated?

Form of Δω = f(T,P,Y,Δt)

Progress Variables where

Y is a single parameter



Progress Variables

Regardless of what the starting conditionsbehavior of species is similartheir progress is also similar

A single progress variablecan tell you where you are in the ignition process

Advantage for CFDOnly have to transport, mix, etc. a single parameter

Reactive Progress



Use of progress(T,P) used to access

specific ignition condition

c used to access ignition conditionConditions:

time-step, Δt, correspondence to Δc

(Tj,Pj,cj)

(T,P)

CFD cell

Reactive Flow Conditions

Non-Reactive Flow Conditions



Prerequisites of Progress Variable

Represents the ‘progress’ of the combustion process

Monotone along this progress

A given progress value, under varying conditions,represents the same state of the ignition process

Representative of the ‘chemistry’ and ‘thermodynamics of the process



Standard Enthalpy?

ΔH = ΔH298K,1atm + ∫ Cp dt = 0 under adiabatic conditions

ΔH298K,1atmAdiabatic



Monotonicity Requirement

Non-Monotonic under equilibrium and rich conditions

Under Lean conditions(sort of)

Monotonic Behavior



(Sort of) Monotonicity Under Lean Conditions

Initially non-montonic

Assume Low Significance:0.15% of Progress

0.7% of Standard Enthalpy Value



Standard Enthalpy

!H298(t)!mint(!H298(t))maxt(!H298(t))!mint(!H298(t))

Normalization

!H298(inf) = mint(!H298(t))!H298(0) = maxt(!H298(t))

If monotonic, then

otherwise, correction for min and max



Standard Energy as Progress

Monotonic over process The progress value representsthe same progress in ignition



Thermodynamic ViewRepresents (related to) the inherent ‘energy’

bound up in the molecules

This is released to the environmentthrough the combustion process

Due to the transformationfrom reactants to products

Reactants

Products



Chemical View

Combustion processdoes not span the entire n dimensional space

of configurationsOnly a small subset

Standard Enthalpy is single parameter representationof the weighted sum of all the components

(almost like a hash code)one number represents one configuration

h = ∑ Yi hi



Chemical EventsPrerequisite:

A given progress value represents a given chemical event in ignition process

Ethanol Oxygen CO2 H2O

CH4OHOCH2O



Choice of ProgressSearch among the available parameters

Monotonic Parameterthat represents the chemical/thermodynamic progress of combustion

This work:Given a progress variable

activelyimprove its definition to better meet requirements

For a single parameter: Standard Enthalpy a good choice



Synchronizing Chemical Events

Synchronized Ignition

Progress Morphing: Define the ignition event to be at 0.5

OH (and other chemical events)events do not align

Actively satisfying the prerequisite that progress synchronizes chemical events



SynchronizingChemical EventsBasic Principle:

However, the timing of the states may change:Morphing synchronizes the timing of these states

Qualification:Similarity of mechanistic properties

(follows same pathways: Only the timing of important pathways changes)

An ignition process goes through a similar set of reactive states

Under a given condition (a given starting condition)



Multiple SynchronizationEvents

Synchronization EventsMaximaMinimaInflection Points

(any well defined mathematical characteristic)

Prerequisite:Set of events occur in same order

(if not, a way to characterize different mechanistic behavior)

Progress value taken as average of events

Try to distribute eventsthroughout combustion process



Relationship Δc and ΔtΔc = f(Δt)

f normally a linear equation(from normalization of ΔH298

Morphing:Δc = f(Δt)

can be any functional form

in this case,piecewise linear

(each range between synchronized events is linear)



Generic CurvesFrom synchronization of events

Can determine average or generic behavior

Without progress synchronization, this is not possible

Generic curvesand deviation from generic curves

offers a more compact representation of curves over a range of conditions

Average

Deviations

More S

ync Points

Generic Characterization of Ignition Behavior

Towards Ignition Curve Parameterization: Non-Linearity of Ignition Progress and Generic Ignition CurvesEdward S. BlurockCombustion Science and Technology, 2010



Piecewise Polynomial FitDeviations from Generic Curve

Synchronization Points

Error with Polynomial Fit

1-2% error in values

Compact Representation

Perturbation from ‘average’ valuesleads to more accurate results



Mixing of States

This moves them off the ‘normal’ configurations

Should account for these extra configurations also

Thermodynamically:Standard Enthalpy Progress

Coolerconfiguration

Hotterconfiguration

Chemically(?)

In this extreme case,burnt and unburnt gases mixing

This is not in the basis of configurationsdescribing chemistry

Makes physical sense,CFD interactions more physical in nature



Mixing(T,P) used to access

specific ignition condition(Tj,Pj,cj)

(T,P)

CFD cell



Even if mixing of EGR was in basis

Not accessed by this procedure

Chemically:The set of states represented

in the tabulationdoes not include mixing states



Mixing Configurations

To a certain extent,covered by

EGR (defined as equilibrium config)mixed states



Inclusion of mixing states

This Work:Expansion and modification of tabulation basis to be able to take the mixing states into account

Basis:Tabulation based on phases/regions of chemical behavior

Chemically significant regions systematically generated

Mixed states are not included in the set of tabulated states(cannot leave manifold of a single ignition run)



Determination of Regions1st Level Clusters

2nd Level ClustersFiner Divisions

Rough Devisions

Clustering Procedure

Regions formed with similar properties

Each time step is a configuration

Cluster the set of configurations

Parameter Value

First Derivative

Approximately a MaximumApproxim

ently Zero

This study used Fuzzy Logicto determine maxima, etc.



Phase (Region) Determination Studies

Ignition Process

Stochastic Reactor Model (SRM) in an EngineAutomatic Characterization of Ignition Processes with MachineLearning Clustering Techniques, Blurock, Edward S.; International Journal of Chemical Kinetics, 2006.

Characterizing Complex Reaction Mechanisms using Machine Learning Clustering TechniquesBlurock, Edward S. International Journal of Chemical Kinetics, 2004.

Phase Optimized Skeletal Mechanisms for Engine Simulations,Blurock, Edward S., Tuner, Martin, Mauss, Fabian, Combustion Theory and Modeling, 2010

Machine Learning ClusteringMachine Learning Decision Trees



Transformation

Original Set of Ignition RunsTabulated

Reactive Phase Cluster

Tabulation

C1

Ci

Cn

Ci

Clusterw.r.t

time-step configuration

Each Cluster Phasecharacterized by a

center configurationCi = {Xi,j}ith Cluster,

jth species mass/mole fraction



Cell Configuration

(Tj,Pj,cj)

(T,P)

CFD cell



Replacement of Non-reactive Flow condition

Cell Configuration

Ccell,m = ∑ai,m Ci ai,m = contribution of cluster i to configuration

Ccell determines which cluster tableand chemistry of progress variable

Ccell,n assigned to nearest cluster configuration: i such that distance(Ccell,n , Ci ) = minimum



Cell Configuration

Ccell,m Ccell,n C´cell,m

C´cell,m = Ccell,m + α Ccell,n α = extent of mixing

Mixing

Cell Configuration

Ccell,m = ∑ai,m Ci ai,m = contribution of cluster i to configuration

Progress Transition

cm in cluster Ci cm + Δc in cluster Cj

aj,m ai,m

Ci determined from Ccell,m at each time stepCi determines progress variable behaviorand chemical/thermodyanamic meaning



Summary

Improvement of Chemistry of Progress variables

Progress morphing to synchronize chemical events

A given progress represents a consistent chemistry

Cell configurations as Cluster configurations

Expands the basis of configurations especially in regard to mixing



Other workAccumulative ISAT (also under VR work)

•Polynomial approximation only done when enough points accumulated•No wasted computations

•Only selected number approximations kept in memory (rest on disc)•Solves memory constraint problems

Automatic Generation of Large Hydrocarbon Mechanisms

•Reaction classes and pathways defined•REACTION software system

•Working toward combining Europe's three automatic generation systems•Sweden, France and Italy (under COST network)

•Net-based tool for thermodynamic constants based on structure•Cooperation with CNRS Nancy, France


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