This document describes a multiphysics model of deposit formation in diesel fuel injectors. The model simulates the oxidation of diesel fuel in a thin evaporating film, leading to the phase separation and deposition of insoluble oxygenated products. Reaction kinetics are modeled using a detailed chemical mechanism generated by RMG. Mass transfer during fuel injection washing is also modeled. Parameter studies using the model provide insight into deposit formation and guide further experimental work.

1. Multiphysics Model of Diesel
Injector Deposit Formation.
Richard H. West
Amrit Jalan
William H. Green
Riccardo Rausa
Massachusetts Institute of Technology

2. Why do diesel engines lose power over time?
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Caprotti et al. (Infineum UK) SAE 2006-01-3359

3. Deposits form on diesel fuel injectors,
blocking the nozzles
Leedham et al. (Infineum UK) SAE 2004-01-2935
More efficient engines and less pollution → smaller nozzles
Smaller nozzles → deposit more problematic

4. Fuel additives is big busine$$!
bp.com
bp.com
Dirty and clean injector nozzles, from BP Ultimate Diesel advertisement
Nobody understands the deposit-forming process.
Current development is based on expensive engine tests.

5. We want a predictive model
of deposit formation.
Aim to gain insight into the effect of
•different conditions
•different fuels and fuel blends
•different detergents

6. Model system of interest:
a thin film of fuel, evaporating.
vapor diffusion
evaporating film

7. Oxygen diffuses into the fuel from the air.
vapor diffusion
evaporating film
O2 diffusion

8. Free-radical autoxidation reactions
gradually oxidize the fuel.
vapor diffusion
evaporating film
O2 diffusion
chemical
reaction

9. Over time the reaction continues;
the diesel becomes more oxidized.
vapor diffusion
evaporating film
O2 diffusion
chemical
reaction

10. Heavily oxidized reaction products are
insoluble in non-polar diesel & phase separate.
vapor diffusion
evaporating film
O2 diffusion
chemical
reaction insoluble
oxygenates

11. This polar oxygenate phase forms the deposit.
vapor diffusion
evaporating film
O2 diffusion
chemical
reaction insoluble
oxygenates
deposit

12. Fuel injections wash the walls
fresh diesel
species
diffusion
insoluble oxygenates

13. Model overview: React & evaporate,
equilibrate, wash, replenish...
partially oxidized
diesel phase equilibration
and separation
vapor diffusion
evaporating film
O2 diffusion
chemical diesel
reaction
insoluble oxygenates
washing with clean diesel
add fresh diesel to
replace evaporated partially washed
diesel

14. chemical
reaction
Chemistry depends on starting species
Diesel contains thousands of species.
Choose simple mixture to represent surrogate diesel.

15. Oxidation is driven by free-radical
chain reactions
O
O O O HO O
H

16. Oxidation is driven by free-radical
chain reactions
O
O O O HO O
H
O
O O O O
O

17. Oxidation is driven by free-radical
chain reactions
O
O O O HO O
H
O
O O O O
O
O O HO
O H O O

18. Oxidation is driven by free-radical
chain reactions
O
O O O HO O
H
O
O O O O
O
O O HO
O H O O
HO O OH OH
O O O O
O O O O

19. Oxidation is driven by free-radical
chain reactions
O
O O O HO O
H
O
O O O O
O
O O HO
O H O O
HO O OH OH
O O O O
O O O O
OH OH OH
O
O O O O
O H O
O OH

20. Oxidation is driven by free-radical
chain reactions
O
OH O
OH H2O
HO O
H
O
O O O O
O
O O HO
O H O O
HO O OH OH
O O O O
O O O O
OH OH OH
O
O O O O
O H O
O OH

21. Detailed kinetic modeling is complex
Estimating all the reactions is tedious and error prone.

22. Detailed kinetic modeling is complex
Estimating all the reactions is tedious and error prone.
Teach the chemistry to a computer!
Reaction Mechanism Generator
•free and open source software
•version 3.3 released in February
rmg.sourceforge.net
facebook.com/rmg.mit

23. Reaction families propose all possible
reactions with given chemical species
bond breaking and
hydrogen abstraction
intramolecular
H-abstraction

24. Reaction families propose all possible
reactions with given chemical species

25. Octane autoxidation has many pathways

26. Detailed kinetic modeling is complex
For each chemical reaction
A+B C+D
we need:
•forward rate coefficient
r = k f [A][B]
− Ea
k f = A exp
RT
•equilibrium constant
kf −∆G
= Keq = exp
kr RT
∆G = ∆H − T∆S

27. We can estimate thermochemistry of solvation
from the molecular structure of the solute
Solvent
parameters
Abraham’s Partition
coefficient,
method
K K=exp(-∆G/RT) ∆G
Platts’ group Solute solvation
contributions parameters
Molecular ∆S
Mintz model ∆G=∆H-T∆S
structure ∆H solvation
for alkanes
RMG group solvation
gas phase
contributions
H(T), S(T)
See (551d) “Progress towards Capturing Solvent Effects In Automatic Mechanism Generation”
Amrit Jalan, Richard H. West and William H. Green.
Reaction Path Analysis II, Wednesday, 1:30pm

28. RMG with solution-phase corrections was
used to study autoxidation of surrogate diesel
rmg.sourceforge.net
facebook.com/rmg.mit

29. Current model contains 252
species and 5185 reactions. 0.951
•n-decylbenzene is most reactive component
and dominates kinetics 1 0.00518 0.048
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30. Model overview: React evaporate,
equilibrate, wash, replenish...
partially oxidized
diesel phase equilibration
and separation
vapor diffusion
evaporating film
O2 diffusion
chemical diesel
reaction
insoluble oxygenates
washing with clean diesel
add fresh diesel to
replace evaporated partially washed
diesel

31. vapor diffusion
evaporating film
Evaporation O2 diffusion
•Abraham model gives gas/solvent concentration ratio
•Diffusivity estimated from molecular structure
Fuller, Schettler, Giddings. IEC, 58, 1966.
HO
O
HO HO
O
O

32. Model overview: React evaporate,
equilibrate, wash, replenish...
partially oxidized
diesel phase equilibration
and separation
vapor diffusion
evaporating film
O2 diffusion
chemical diesel
reaction
insoluble oxygenates
washing with clean diesel
add fresh diesel to
replace evaporated partially washed
diesel

33. Abraham’s Method predicts partition coefficients
for different solutes and solvents
Solvent
parameters
Partition
Abraham’s
coefficient,
method
K
Solute
parameters
log K= c + eE + sS + aA + bB + lL
gas /solvent partitioning

34. Abraham’s Method predicts partition coefficients
for different solutes and solvents
Solvent
parameters
Partition
Abraham’s
coefficient,
method
K
Solute
parameters
log K= c + eE + sS + aA + bB + lL
gas /solvent partitioning
log K= c + eE + sS + aA + bB + vV
solvent /solvent partitioning

35. Model overview: React evaporate,
equilibrate, wash, replenish...
partially oxidized
diesel phase equilibration
and separation
vapor diffusion
evaporating film
O2 diffusion
chemical diesel
reaction
insoluble oxygenates
washing with clean diesel
add fresh diesel to
replace evaporated partially washed
diesel

36. Washing is mass transfer problem
Look up a correlation in Perry’s Handbook:
0.0097NRe NSc 1.10 + 0.44NSc1/3 − 0.70NSc1/6
9/10 1/2 − −
NSh,avg =
1 + 0.064NSc 1.10 + 0.44NSc1/3 − 0.70NSc1/6
1/2 − −
kavg dt µ ρubulk dt
NSh,avg = NSc = NRe =
D ρD µ
TABLE 5-19
Mass-Transfer Correlations for Flow in Pipes and Ducts—Transfer Is from Wall to Fluid.
W: Tubes, turbulent, smooth tubes, constant surface concentration

37. Current model overview:
React, equilibrate, wash, replenish...
partially oxidized
diesel phase equilibration
and separation
vapor diffusion
evaporating film
O2 diffusion
chemical diesel
reaction
insoluble oxygenates
washing with clean diesel
add fresh diesel to
replace evaporated partially washed
diesel

38. Implementation
•Python: chosen for speed of development.
•PyDAS: Python interface to Fortran DAE solver DASSL.
•Cython: gives orders of magnitude reduction in CPU time.
•Parallel computing: simple use of array jobs.

39. Python chosen for speed of development.
•General-purpose, high-level programming language.
•Free, open-source, extensible.
•Easy to learn.
•Fast (and fun) to develop in.
•Multi-paradigm programming language,
but this project mostly object-oriented.

40. PyDAS created to access specialized
ODE solvers from Python code.
•Chemical reactions
→ very stiff systems of differential equations
→ require specialized ODE solvers.
•VODE (provided by SciPy) is not always robust enough.
•Fortran-based DASSL (Petzold, 1982) is a stiff DAE solver
widely used in kinetic modeling.
•Python interface to DASSL developed and released:
https://github.com/jwallen/PyDAS

41. Cython used to speed up the slow parts
by orders of magnitude.
•Develop your code in Python
•Identify the slow parts by profiling (right hand side of ODE)
•Add some static type declarations (“cdef double x”)
•Compile to C using Cython
•50x speed up in overall computation time!

42. Running simulations with different parameters
gives a variety of results.
Parameters:
•Reaction T
•Phase separation T
•Nozzle diameter
•Nozzle length chemical
reaction
•Film thickness
•Injection velocity
•Injection duration
•Reaction time

43. Parallel computing is easy and useful
for parameter studies and sensitivity analysis.
Simulate
with parameter Simulate Simulate
run job 1
set 1 with parameter with parameter
run job 2 set 2 set 3
submit jobs
1-1000 run job 3
run job 4...
Head Server
Compute nodes
•Use cluster’s queuing system for “array” jobs.
•Your code translates job number into set of parameters:
•Random (Uniform Distribution)
•Global Sensitivity Analysis (Modified Morris Method)

44. Global Sensitivity Analysis
identifies most significant model parameters.
•Identifies sub-models needing refinement.
•Guides laboratory experiments.
•Gives insight to underlying processes.

45. Acknowledgements
•Project collaborators at MIT:
Amrit Jalan and Prof. William Green (ChE)
Yinchun Wang and Prof. Wai Cheng (MechE)
•RMG developers:
rmg.sourceforge.net
facebook.com/rmg.mit
•Industrial sponsors:
Eni S.p.A.

46. Contributions to industrial problem of
deposit formation in diesel fuel injectors:
Multiscale, multiphysics model gives insight into
•chemical reactions
OH
O
•evaporation O H
O
•phase separation chemical
reaction
•washing
Computational experiments reveal important parameters
•guide model development
•guide experiments

47. Suggestions for multi-scale modeling of
other chemical engineering systems.
•Build detailed kinetic model to capture
complicated T,P-dependence of real chemistry (RMG).
•Write software in Python: easy, fast, pleasant to write.
•Use specialized DAE solvers when necessary (PyDAS).
•Use Cython to speed up the slow parts.
•Use parallel computing environment for Global Sensitivity
Analysis (eg. modified Morris method)
•Identify sensitive parameters,
then refine models and experiments accordingly.

48. Any questions?
r.west@northeastern.edu
Massachusetts Institute of Technology