Progress Variables
for CFD
Trying to get something for nothing
Edward S. Blurock
Division of Fluid Dynamics
Department of Energy Sciences
Lund University
Work supported by Swedish Science Ministry (VR project)
Edward S. Blurock, Energy Sciences, Lund University
Wednesday, March 17, 2010
Progress Variable Fundamentals
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, Energy Sciences, Lund University
Wednesday, March 17, 2010
Progress Variable Fundamentals
Kinetic Information and Complexity
Homogeneous Models
CFD Models
Complexity can be in
Chemistry
Kineticist concentrates on
Chemistry
Reactivity of Species
Complexity in the Flow Models
Isolating Primary
Kinetic Features
Span
of Expertise
Complexity
Kinetics concentrates on
Thermodynamics
and
Global Chemistry
(if at all)
3
Edward S. Blurock, Dept. of Energy Sciences Lund University
Wednesday, March 17, 2010
Chemical Source Terms in CFD
Interaction with CFD cells
Mixing
(Tj,Pj,.., Yj)
Exchange of
Physical properties (T,P,...)
and
chemical composition ( Y )
(Tm,Pm,.., Ym)
(Ti,Pi,.., Yi)
Differential Equations:
Physical Properties
+
Chemical source term ( ὠ )
(Tk,Pk,.., Yk)
(Tl,Pl,.., Yl)
Edward S. Blurock, Energy Sciences, Lund University
Wednesday, March 17, 2010
Progress Variable Fundamentals
Chemical
Source
Terms
.
ω = f(T,P,Y)
Differential Equations
Zero dimensional adiabatic
constant V (or P)
System
(operator splitting)
Discretized
Δω = f(T,P,Y,Δt)
Complexity highly dependent on number of species
Differential equation in n + 2 variables
where n is the number of species
(T,P,Y)
(T,P,Y)+ Δω
Δt
Edward S. Blurock, Energy Sciences, Lund University
Wednesday, March 17, 2010
Progress Variable Fundamentals
Evaluate f(T,P,Y,Δt)
(T´,P´,Y´)
Solve
Differential Equations
CFD
Interaction with CFD
Calculation of source terms
(conditions supplied by CFD cell)
Edward S. Blurock, Energy Sciences, Lund University
Wednesday, March 17, 2010
Progress Variable Fundamentals
Principle of Tabulation
Δω = f(T,P,Y,Δt)
Substitute differential equations
with a simple (tabulated) vector function
x(t+Δt) = f(x(t),Δt)
Edward S. Blurock, Energy Sciences, Lund University
Wednesday, March 17, 2010
where x is (T, P, Y1, Y2, ..., Yn)
A mapping from xi to xj
Progress Variable Fundamentals
Principle of Tabulation
xi = f(xj)
(vector function)
Need to cover
(tabulate)
all f(xi)
from ignition to equilibrium
Edward S. Blurock, Energy Sciences, Lund University
Wednesday, March 17, 2010
Progress Variable Fundamentals
Evaluate f(T,P,Y,Δt)
(T´,P´,Y´)
Retrieve From
Tabulation
CFD
Tabulation
Interaction with CFD
Differential Equations replaced by
Search into a table
Assumption:
Time to search table << Time to solve differential equations
Edward S. Blurock, Energy Sciences, Lund University
Wednesday, March 17, 2010
Progress Variable Fundamentals
What Needs to be Tabulated?
Do not need to span the entire n dimensional space
Chemical events are happening in unison
The n dimensions are not totally independent
neffective < n
Basis must at least span a set of
zero dimensional adiabatic constant volume ignition sets
Edward S. Blurock, Energy Sciences, Lund University
Wednesday, March 17, 2010
Progress Variable Fundamentals
Simple Tabulation
Single Ignition run
under a single condition
Time
(T,P,Y)
Starting Temperature
Pressure
Vary:
Equivalence Ratio
Equilibrium Configurations: EGR
(Full) Tabulation
combines
set of conditions
Edward S. Blurock, Energy Sciences, Lund University
Wednesday, March 17, 2010
Progress Variable Fundamentals
Principle of Tabulation
However dimension is increased by Mixing
(Tj,Pj,.., Yj)
Exchange of
Physical properties (T,P,...)
and
chemical composition ( Y )
(Tm,Pm,.., Ym)
(Ti,Pi,.., Yi)
Physical Properties
+
Chemical source term ( ὠ )
(Tk,Pk,.., Yk)
(Tl,Pl,.., Yl)
12
Edward S. Blurock, Energy Sciences, Lund University
Wednesday, March 17, 2010
Progress Variable Fundamentals
Additional Tabulation Points
xi
Two points in configuration space
xj
This moves them off the ‘normal’ configurations
xi´
xj´
Should account
for these extra
configurations
also
Edward S. Blurock, Energy Sciences, Lund University
Wednesday, March 17, 2010
Progress Variable Fundamentals
Additional Tabulation Points
A simple compensation
Addition of End Gas Residue (EGR)
(percentages of Equilibrium configurations)
+ p*
=
Mix in a percentage of
Equilibrium state
Vary
not only equivalence ratio
but also
amounts of EGR
Edward S. Blurock, Energy Sciences, Lund University
Wednesday, March 17, 2010
Progress Variable Fundamentals
What Should be Tabulated?
What information
from the range of conditions
should be tabulated?
Form of Δω = f(T,P,Y,Δt)
Progress Variables
where
Y is a single parameter
Edward S. Blurock, Energy Sciences, Lund University
Wednesday, March 17, 2010
Progress Variable Fundamentals
Progress Variables
Reactive Progress
Regardless of what the starting conditions
behavior of species is similar
A single progress variable
their progress is also similar
can tell you where you are in the ignition process
Advantage for CFD
Only have to transport, mix, etc. a single parameter
Edward S. Blurock, Energy Sciences, Lund University
Wednesday, March 17, 2010
Progress Variable Fundamentals
Use of progress
(T,P) used to access
specific ignition condition
CFD cell
Reactive Flow Conditions
(Tj,Pj,cj)
Non-Reactive
(T,P)
Flow Conditions
Conditions:
c used to access ignition condition
time-step, Δt, correspondence to Δc
Edward S. Blurock, Energy Sciences, Lund University
Wednesday, March 17, 2010
Progress Variable Fundamentals
Prerequisites of Progress
Variable
Representative of the ‘chemistry’ and ‘thermodynamics of the process
Represents the ‘progress’ of the combustion process
A given progress value, under varying conditions,
represents the same state of the ignition process
Monotone along this progress
Edward S. Blurock, Energy Sciences, Lund University
Wednesday, March 17, 2010
Progress Variable Fundamentals
Standard Enthalpy?
Adiabatic
ΔH = ΔH298K,1atm +
Edward S. Blurock, Energy Sciences, Lund University
Wednesday, March 17, 2010
ΔH298K,1atm
∫
Cp dt = 0 under adiabatic conditions
Progress Variable Fundamentals
Monotonicity Requirement
Non-Monotonic
under equilibrium and rich conditions
Under Lean conditions
(sort of)
Monotonic Behavior
Edward S. Blurock, Energy Sciences, Lund University
Wednesday, March 17, 2010
Progress Variable Fundamentals
(Sort of) Monotonicity
Under Lean Conditions
Assume Low Significance:
0.15% of Progress
0.7% of Standard Enthalpy Value
Initially non-montonic
Edward S. Blurock, Energy Sciences, Lund University
Wednesday, March 17, 2010
Progress Variable Fundamentals
Standard Enthalpy
Normalization
∆H298 (t) − mint (∆H298 (t))
maxt (∆H298 (t)) − mint (∆H298 (t))
If monotonic, then
∆H298 (inf) = mint (∆H298 (t))
∆H298 (0) = maxt (∆H298 (t))
otherwise, correction for min and max
Edward S. Blurock, Energy Sciences, Lund University
Wednesday, March 17, 2010
Progress Variable Fundamentals
Standard Energy as
Progress
Monotonic over process
Edward S. Blurock, Energy Sciences, Lund University
Wednesday, March 17, 2010
The progress value represents
the same progress in ignition
Progress Variable Fundamentals
Thermodynamic View
Represents (related to) the inherent ‘energy’
bound up in the molecules
This is released to the environment
through the combustion process
Reactants
Products
Edward S. Blurock, Energy Sciences, Lund University
Wednesday, March 17, 2010
Due to the transformation
from reactants to products
Progress Variable Fundamentals
Chemical View
Combustion process
does not span the entire n dimensional space
of configurations
Only a small subset
Standard Enthalpy is single parameter representation
of the weighted sum of all the components
h = ∑ Yi hi
(almost like a hash code)
one number represents one configuration
Edward S. Blurock, Energy Sciences, Lund University
Wednesday, March 17, 2010
Progress Variable Fundamentals
Chemical Events
Prerequisite:
A given progress value represents a given chemical event in ignition process
Ethanol
CH2O
Edward S. Blurock, Energy Sciences, Lund University
Wednesday, March 17, 2010
Oxygen
O
CO2
OH
H2O
CH4
Progress Variable Fundamentals
Choice of Progress
Search among the available parameters
Monotonic Parameter
that represents the chemical/thermodynamic progress of combustion
For a single parameter: Standard Enthalpy a good choice
This work:
Given a progress variable
actively
improve its definition to better meet requirements
Edward S. Blurock, Energy Sciences, Lund University
Wednesday, March 17, 2010
Progress Variable Fundamentals
Synchronizing Chemical Events
OH (and other chemical events)
events do not align
Synchronized Ignition
Progress Morphing: Define the ignition event to be at 0.5
Actively satisfying the prerequisite that progress synchronizes chemical events
Edward S. Blurock, Energy Sciences, Lund University
Wednesday, March 17, 2010
Progress Variable Fundamentals
Synchronizing
Chemical Events
Basic Principle:
Under a given condition (a given starting condition)
An ignition process goes through
a similar set of reactive states
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)
Edward S. Blurock, Energy Sciences, Lund University
Wednesday, March 17, 2010
Progress Variable Fundamentals
Multiple Synchronization
Events
Progress value taken as average of events
Synchronization Events
Maxima
(any well defined mathematical characteristic)
Minima
Prerequisite:
Inflection Points
Set of events occur in same order
Try to distribute events
throughout combustion process (if not, a way to characterize different mechanistic behavior)
Edward S. Blurock, Energy Sciences, Lund University
Wednesday, March 17, 2010
Progress Variable Fundamentals
Relationship Δc and Δt
Δc = 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)
Edward S. Blurock, Energy Sciences, Lund University
Wednesday, March 17, 2010
Progress Variable Fundamentals
Generic Curves
From synchronization of events
More Sync Points
Average
Can determine
average or generic behavior
Without progress synchronization,
this is not possible
Generic Characterization
of Ignition Behavior
Deviations
Towards Ignition Curve Parameterization:
Non-Linearity of Ignition Progress and Generic Ignition Curves
Edward S. Blurock
Combustion Science and Technology, 2010
Edward S. Blurock, Energy Sciences, Lund University
Wednesday, March 17, 2010
Generic curves
and deviation from generic curves
offers a more compact representation
of curves over a range of conditions
Progress Variable Fundamentals
Piecewise Polynomial Fit
Compact Representation
Deviations from Generic Curve
Error with Polynomial Fit
1-2% error in values
Synchronization Points
Edward S. Blurock, Energy Sciences, Lund University
Wednesday, March 17, 2010
Perturbation from ‘average’ values
leads to more accurate results
Progress Variable Fundamentals
Mixing of States
This moves them off the ‘normal’ configurations
Should account for these
extra configurations also
Thermodynamically:
Standard Enthalpy Progress
Hotter
configuration
Cooler
configuration
Makes physical sense,
CFD interactions more physical in nature
Chemically(?)
In this extreme case,
burnt and unburnt gases mixing
This is not in the basis of configurations
describing chemistry
Edward S. Blurock, Energy Sciences, Lund University
Wednesday, March 17, 2010
Progress Variable Fundamentals
Mixing
CFD cell
Reactive Flow Conditions
(Tj,Pj,cj)
Non-Reactive
Flow Conditions (T,P)
(T,P) used to access
specific ignition condition
Even if mixing of
EGR was in basis
Not accessed by
this procedure
Chemically:
The set of states represented
in the tabulation
does not include mixing states
Edward S. Blurock, Energy Sciences, Lund University
Wednesday, March 17, 2010
Progress Variable Fundamentals
Mixing Configurations
To a certain extent,
covered by
EGR (defined as equilibrium config)
mixed states
Edward S. Blurock, Energy Sciences, Lund University
Wednesday, March 17, 2010
Progress Variable Fundamentals
Inclusion of mixing states
Mixed states are not included in the set of tabulated states
(cannot leave manifold of a single ignition run)
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
Edward S. Blurock, Energy Sciences, Lund University
Wednesday, March 17, 2010
Progress Variable Fundamentals
Determination of Regions
Clustering Procedure
1st Level Clusters
Rough Devisions
2nd Level Clusters
Finer Divisions
Approximately a Maximum
Parameter Value
Regions formed with similar properties
First Derivative
Cluster the set of configurations
Edward S. Blurock, Energy Sciences, Lund University
Wednesday, March 17, 2010
Approximently Zero
Each time step is a configuration
This study used Fuzzy Logic
to determine maxima, etc.
Progress Variable Fundamentals
Phase (Region)
Determination Studies
Ignition Process
Automatic Characterization of Ignition Processes with Machine
Learning Clustering Techniques,
Blurock, Edward S.;
International Journal of Chemical Kinetics, 2006.
Characterizing Complex Reaction Mechanisms using
Machine Learning Clustering Techniques
Blurock, Edward S.
International Journal of Chemical Kinetics, 2004.
Edward S. Blurock, Energy Sciences, Lund University
Wednesday, March 17, 2010
Stochastic Reactor Model (SRM) in an Engine
Phase Optimized Skeletal Mechanisms for Engine Simulations,
Blurock, Edward S., Tuner, Martin, Mauss, Fabian,
Combustion Theory and Modeling, 2010
Machine Learning Clustering
Machine Learning Decision Trees
Progress Variable Fundamentals
Transformation
Cluster
w.r.t
time-step configuration
C1
Ci
Cn
Original Set of Ignition Runs
Tabulated
Each Cluster Phase
characterized by a
center configuration
Ci = {Xi,j}
ith Cluster,
jth species mass/mole fraction
Edward S. Blurock, Energy Sciences, Lund University
Wednesday, March 17, 2010
Reactive Phase
Cluster
Tabulation
Ci
Progress Variable Fundamentals
Cell Configuration
CFD cell
Reactive Flow Conditions
(Tj,Pj,cj)
Non-Reactive
Flow Conditions (T,P)
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 table
and chemistry of progress variable
Ccell,n assigned to nearest cluster configuration:
i such that distance(Ccell,n , Ci ) = minimum
Edward S. Blurock, Energy Sciences, Lund University
Wednesday, March 17, 2010
Progress Variable Fundamentals
Cell Configuration
Cell Configuration
Ccell,m = ∑ai,m Ci
ai,m = contribution of cluster i to configuration
Mixing
Ccell,n
Ccell,m C´cell,m
C´cell,m = Ccell,m + α Ccell,n
α = extent of mixing
Progress Transition
cm
in cluster Ci
cm + Δc in cluster Cj
ai,m
aj,m
Ci determined from Ccell,m at each time step
Ci determines progress variable behavior
and chemical/thermodyanamic meaning
Edward S. Blurock, Energy Sciences, Lund University
Wednesday, March 17, 2010
Progress Variable Fundamentals
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
Edward S. Blurock, Energy Sciences, Lund University
Wednesday, March 17, 2010
Progress Variable Fundamentals
Other work
Accumulative 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
Edward S. Blurock, Energy Sciences, Lund University
Wednesday, March 17, 2010
Progress Variable Fundamentals