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SAE TECHNICAL
PAPER SERIES
Heat Transfer and Performance Characteristics
of a Dual-Ignition Wankel Engine
M.S.Raju
Sverdrup Technology, Inc.
NASA Lewis Research Center Group
2001 Aerospace Parkway
Brook Park, OH-44142
=For
The Engineering Society
Advancing Mobility
and sea Air and Spacem
International Congress & Exposition
Detroit, Michigan
February 24-28,1992
-
400
C O M M O N W E A L T H
DRIVE,
W A R R E N D A L E ,
PA
15096-0001 U . S . A .
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ISSN 0148-7191
Copyright 1992 Society of Automotive Engineers, Inc.
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Heat Transfer and Performance Characteristics
of a Dual-Ignit ion Wankel Engine
M. S. Raju
Sverdrup Technology, Inc.
NASA Lewis Research Center
2001 Aerospace Parkway
Brook Park, Ohio-44142
ABSTRACT
A computer code, AGNI-3D, was developed for
the modeling of turbulent, reacting flows with sprays
occurring inside of a Wankel engine based on unsteady, three-dimensional computations. The primary objective of the present study is to assess the
limitations and capabilities of AGNI-3D in predicting the combustion characteristics of the stratifiedcharge rotary engine (SCRE) that is being developed
at the John Deere Rotary Engine Division. This engine has been modified recently with the inclusion
of a second ignition source to supplement the standard pilot ignitor. Experimental tests of the modified
dual-ignition Wankel engine demonstrated a 7.5% reduction in brake specific fuel consumption (BSFC)
at low loads. Additional reductions in BSFC at high
loads were limited by the onset of combustion instability. Since our interest in this engine lies a t higher
loads, we have limited making pressure comparisons
to those few cases where experimental pressure traces
have shown normal combustion behavior and cycleto-cycle repeatability. The numerical results show
excellent agreement with those pressure traces obtained from the Bottom Top Center (BTC) pressure
transducer. Combustion in a dual-ignition engine appears to be dominated by the contribution from the
trochoid ignition. The computations also provide instantaneous spatially-averaged and local heat fluxes
on the rotor, side walls, and the rotor housing. The
heat flux from the gas to the rotor housing near the
vicinity of top dead center (TDC) is observed to be
higher than the corresponding flux to the rotor since
the sliding motion of the rotor near T D C generates
higher velocity gradients near the rotor housing similar to Couette flow. Comparisons indicate a need
for significant improvement in the Woschni model, a
widely-used heat transfer correlation in the performance analysis of a Wankel engine. The results of
a limited-systematic study conducted with the variation of the overall fuel/oxidizer equivalence ratios,
fuel-composition, intake temperature, fuel-injection
and spark timings are also summarized.
STARTING FROM the beginning of the mid1980s, continuing research and development sponsored by the NASA Rotary Engine Technology Enablement Program has been aimed a t reducing the
cruise BSFC from a value of above 0.50 lb/bhp-hr to
0.35 or less by the end of 1992.'~' The expected improvement in rotary combustion engine (RCE) performance was envisioned to be brought about based
on a combination of computational fluid dynamics
(CFD)-driven fuel injection, nozzle, and spray optimization studies, improved ignition strategies, and
rotor pocket optimization and relocation studies. A
BSFC value of 0.375 lb/bhp-hr is achieved at present
through a combination of both CFD work and other
related modifications - (1) an enlarged exhaust port
area, (2) turbocharger matching and optimization
work, and (3) an increased compression ratio.
So far three different design modifications based
~~~
comon Abraham and B r a c c o ' ~three-dimensional
putations have been studied: (1) An improvement
was made in the main fuel injection spray pattern
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by developing what is known as a non-shadowing or
"fan" spray pattern, which has resulted in performance gains of a t least 2% at low loads and 6% at
high loads; (2) The dual spray or "rabbit ears" pilot
nozzle when implemented together with the improved
main fuel spray pattern has resulted in performance
gains of about 10%; and (3) The third improvement
is based on the idea of providing a second ignition
source to give faster combustion and therefore better
BSFC. This concept has so far failed to yield better
BSFC at high loads even after extensive refinements
to the test setup with the measured pressure traces
showing large cycle-to-cycle variability with the appearance of unstable combustion due t o knock.1°
The schematic of the Wankel engine studied is
shown in Fig. 1. The initial test engine configuration
consisted of a cavity located at about 1.5 cm after the
minor axis within which a pilot injector and a
dard pilot igniter were placed and a main injector
was located a t about 1.5 cm before the minor axis.
Usually, 5-10% of the total fuel is injected through
the pilot injector and the rest from the main injector. After ignition, the combustion of fuel from the
pilot injection causes a high-temperature region to
develop in the immediate vicinity of the pilot which
eventually ignites and burns the fuel from the main
injection. Especially at higher loads and speeds, heat
release is reported to be very slow with the mixture still burning at the exhaust port opening. This
characteristic feature of a SCRE typically leads t o
a delayed occurrence of peak pressure well beyond
TDC. Based on their CFD computations Abraham
and Bracco predicted the formation of a fuel-rich
region containing a sizable fraction of the unburnt
charge near the main injector location. The combustion of fuel from the main injection is found to be
particularly slow until after TDC when the higher
turbulence intensity generated around TDC leads to
greater mixing of fuel and air. And then combustion
extends t o the whole mixture after the local effective
flammability limit of the mixture increases at prevailing combustion-chamber gas temperature, pressure
and turbulence level. The concept of dual-ignition derived from the CFD studies of Abraham and Bracco
predicted an improvement in the BSFC as the second
ignition source located upstream of the main injector
which when fired later than the standard igniter, ignites and burns the unburnt charge upstream of the
main injector, while the other flame front originating
from the pilot burns the unburnt charge downstream
of the main injection. To test this concept, the en-
gine configuration was modified with the inclusion of
a trailing, trochoid surface mounted ignitor (trochoid
ignitor) located upstream of the main injector nozzle. Initial testing of the dual-ignition engine indicated no combustion contribution from the trochoid
spark. Since then the engine has been modified by
providing spray clearance relief in the housing surface
for two "lightoff" sprays of the main injection spray
pattern to direct fuel towards the trochoid igniter.''
Even with this modification both the spark timings
have to be advanced prior to 70 deg TDC to get ignition off of the trochoid igniter as the areas around the
trochoid spark were reported to be wetted with liquid
fuel." Since these advanced timings are at near the
optimum for low loads, this modification apparently
produced successful gains of about 7.5% in BSFC at
low 'peeds and loads. However, at high loads these
timings were too far advanced for good performance
and resulted in erratic engine behavior.'
As part of the CFD development program a
second code, AGNI-3D, was developed based on an
Eulerian-Lagrangian approach where the unsteady,
three-dimensional Navier-Stokes for a perfect gasmixture with variable properties are solved in generalized, Eulerian coordinates on a moving grid by making use of an implicit finite-volume, Steger-Warming
flux vector splitting scheme, and the liquid-phase
equations are solved in Lagrangian coordinate^."-'^
The motivating factor behind the development of
AGNI-3D is the need for exploring different numerical
techniques based on computational efficiency and accuracy considerations. Complete mathematical and
numerical details of the solution procedure are described in the sections on Gas-Phase Equations in
Generalized Coordinates, Liquid-Phase Equations,
Details of Fuel Injection, Details of Turbulence and
Combustion models, and Details of the Numerical
Met hod.
Although this code has not undergone a very
extensive validation process, recent computations indicate proper predictions for the single igniter configuration as indicated by the comparisons made
with the experimentally-observed flow patterns during intake," and measured ~ r e s s u r e s . 'The
~ characteristic development of a non-uniform pressure distribution near TDC and its attendant effect on the
torque generated due to pressure non-uniformity under both motoring and firing conditions was discussed in Raju and illi is.^^^^^ The code also was
assessed through verification of some of the conclusions reached by Abraham and Bracco through their
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CFD computations on the flow, turbulence, and combustion characteristics of a Wankel enginel3tl4. The
present study deals with the application of AGNI-3D
to the modeling of a dual-ignition Wankel engine by
presenting detailed results on the effect of a second
ignition source on the performance and heat transfer
characteristics of SCRE. Results and major conclusions of the present study are summarized in the sections on Results and Discussion, and Summary and
Conclusions.
(
(v+st)
PD
P D ~
PDV
pDw
st) + p ~ s ,
st) + P D Q ,
st) + P D %
p ~e (V
+ 7,) + PDV
+ vt)
( v+ st)
P D Y ~(V
GAS-PHASE EQUATIONS IN GENERALIZED
COORDINATES
P DYo
p Dk
The governing unsteady equations based on the
conservation of mass, momentum, energy, and species
for turbulent, reacting, and compressible flows are
presented in strong conservation law form. The exchanges of mass, momentum, and energy through
liquid-phase interaction are considered by the inclusion of appropriate source terms. The Reynoldsaveraged equations are formulated in generalized coordinates to accommodate the time variation of the
complex combustor geometry.
P
(v+st)
(V + st)
\
P DY ( v + ' 1 t )
(
P~(w+ct)
P Du ( ~ + ( t +
) PDG
( w + c )+ p D e y
P D W (w+ct)+ P D G
pDv
p De (W
+ ~ t +) PDW
(w + ( t )
p DYO ( w + c t )
P Dk ( w + c t )
P DYj
P Df
where
\
(u + t t )
(u + t t ) + P D t,
(0. + &) + P D t,
( a + & )+ P D L
PD
D.
D.
/ fvl
f v2
fv3
fv4
D~(U+G)+PDU
( u + G)
P DYO(u + G)
P D (u+tt)
~
P D f (u + t t )
P D (~u + c )
P Dyj
fv5
fv6
fv7
fv8
fv9
\
fvl0
(w+ct)
pDg(~+it)
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turbulent Prandtl number; Sct (= 0.90) is the turbulent Schmidt number; Dim represents the turbulent
diffusivity of the ith species in a multi-component
mixture; A and E, are the pre-exponential coefficient
and activation energy of a given Arrhenius reactionrate term; Cg is a constant used in the eddy breakup
model17; the subscripts f , o, I, !, c, m, and k represent fuel, oxidizer, liquid-phase, laminar, chemical
reaction, gaseous mixture, and droplet or Lagrangian
characteristic representing a group of liquid droplets,
respectively.
The pressure and temperature are calculated iteratively from the following procedure:
where p, e, yi, k,
E,
and g are the fluid density, internal
energy, mass fraction of the ith species, turbulence kinetic intensity, dissipation rate of turbulence kinetic
energy, and variance of the fuel mass fraction fluctuations, respectively; x, y, and z are the Cartesian
coordinates in the physical space; u, v, and w are the
velocity components in Cartesian coordinates; (, 7,
and (' are the coordinates in the computational space;
D is the determinant of the matrix, J in Eq. 4, and
is also a measure of the volume of a computational
cell; yj is the mass fraction of the ith species; gcis
a vector representing the finite reaction rate terms of
species equations and also the source terms of turbulence model; W iis the molecular weight of the species;
vi is the stoichiometric ratio of the ith species participating in a given reaction step; R f u is the combustion
production rate of the turbulence kirate; \E is the -,
netic energy; Sl is a vector representing the source
terms arising from the liquid-phase interaction; nk is
the number of droplets in a kth characteristic representing a group of droplets; mk is the vaporization
rate of a droplet belonging to the kth characteristic;
rk is the droplet radius; h f , and lk,,f are the enthalpy of the fuel vapor at the droplet surface, and
the effective latent heat of vaporization; the terms involving T represent the nine components of a stress
tensor; pt is the turbulent viscosity; K t , and /itm are
the thermal conductivity and laminar viscosity of the
gas mixture and are determined using Wilke's mixing rule with fourth-order polynomial fits based upon
temperature dependence15; Cp, is the specific heat
of the gas mixture at constant pressure and is also
determined from fourth-order polynomial fits involving temperature dependence16; P r t (= 0.90) is the
where
where h;i is the heat of formation of ith species, and
R,, is the universal gas constant. Eq. 3 is the equation of state for a gas mixture of N, species.
The Jacobians of the coordinate transformation
are given by
and
where the elements of J and J-l are known as the
metric coefficients.
The intake port conditions are given by
P = P i n t , T = z n t , Yi = Yi,int
p = pint, k = 0.03vkt,
E =A , c : . ~ ~ ~ ~
u .=
~w
, = g = 0,
v = -Cd
[2 (A;
- P)]
O"
(6)
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where C,, (=0.09) and A, are turbulence model con- in the liquid-phase is not considered, although this
stants.
factor might become important when the droplets vaThe exhaust conditions are given by
porize near the critical conditions.23~24
Cdc(= 0.9) is the discharge coefficient; and subscript i and n represent species and the normal component of the boundaries, respectively.
LIQUID-PHASE EQUATIONS
Some of the advantages of formulating the
liquid-phase equations in a Lagrangian-coordinate
system over an Eulerian-coordinate system are: ( 1 )
No numerical diffusion is introduced when the governing equations are finite-differenced; ( 2 ) Its ability to
handle multivaluedness of solutions in a natural way;
and ( 3 ) The computations are restricted to the region
where the droplets are present so that the EulerianLagrangian approach can be used for fine resolution
where required.
spray
model
The
used in Raju and Sirignano18119 has been extended
in the present study from two-dimensional to threedimensional computations with the consideration of
additional effects arising from the droplet dispersion
due to turbulence. The spray model is based on a dilute spray assumption which is valid in the regions of
the spray where the droplet loading is low. The liquid fuel is assumed to enter the combustor as a fully
atomized spray comprised of spherical droplets. The
spray characteristics are determined based on isolated
droplet behavior. The present model does not take
into account the details of the phenomena arising
from the local droplet breakup and coalescence processes which may become important in a dense spray
situation. Although O'Rourke and Bracco2', Reitz
and Diwakar21, and Asheim et a1.22 made some attempts to model the liquid sprays by including these
effects, these studies should be considered preliminary since the assumptions invoked in the modeling
of these processes are not well established. In the
present computations, the effect of variable properties
where
where s represents the conditions at the droplet
surface.
The droplet regression rate is determined from
three different correlations depending upon the
droplet-Reynolds-number range. When Rek > 20,
the regression rate is determined based on a gas-phase
boundary-layer analysisz5 which is valid over an intermediate Reynolds-number regime. The other two
correlations which are valid when Rek 5 20 are taken
from Clift et
where B k is the Spalding transfer number defined in
Eq. 22. The function f ( B k ) is obtained from the
solution of Ernmon's problem. The range of validity
of this function was extended in Raju and
S i r i g n a n ~ l ~to
? ' consider
~
the effects of droplet
condensation.
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Based on a vortex modelz5 for the internal where Pn is the normal pressure, la is a constant,
and subscript b refers to the boiling conditions of
droplet temperature
the liquid fuel.
In Eq. 14 the molecular viscosity is evaluated at a
reference temperature using Sutherland's equation
where
where
where a represents the streamline of a Hill's Vortex
in the circulating fluid and C(t) represents a
nondimensional form of the droplet regression rate.
The boundary conditions for Eq. 17 are given by
The droplet dispersion due to turbulence is
determined following the method of Gosman and
1oannidesz7 and Shuen et a1.28 where the particle motion is tracked as it interacts with a succession of eddies, each having a life-time (t,), length (L,), and
isotropic velocity fluctuations with a standard deviation of (2k/3)1I2 The droplet is assumed to interact
with an eddy for a time which is taken t o be the minimum of either the eddy life-time (t,) or the time (tt)
it takes for the particle to traverse the eddy.
where a = 0 refers to the vortex center and cr = 1
refers t o the droplet surface.
tt = - ~ 1 n ( l- L,/(T /Ug- Uk1))
The Spalding transfer number is given by
(30)
where
where yj, is the fuel mass fraction a t the droplet
surface, lk is the latent heat of fuel vaporization,
lk,,f is the effective latent heat of vaporization as
modified by the heat loss to the droplet interior, mk
is the droplet vaporization rate, Wa is the molecular
weight of the gas excluding fuel vapor, and x is the
mole fraction of the species.
Based on the assumption that phase equilibrium
exists a t the droplet surface, the Clausius-Clapeyron
relationship yields
The velocity fluctuations u', v', and w' associated
with a particular eddy are generated from a Gaussian probability density distribution having a standard deviation of (2k/3).lI2
The droplets may evaporate, move along the
wall surfaces, and/or reflect with reduced momentum
upon droplet impingement with the combustor walls.
In our present computations it is assumed subsequent
to the impingement with the walls that the droplets
flow along the wall surfaces with a velocity equal to
that of the surrounding gas.29
DETAILS O F FUEL INJECTION
The success of a spray model depends a great
deal on the correct specification of the injector exit
conditions. The location of the main and pilot injectors are shown in Fig. 1. The main injector has
five holes and fuel from it emerges in a fan-shaped
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pattern consisting of five sprays. The initial particle location is determined from the known location
of the fuel injector. The initial particle velocity and
temperature are specified from the experimental conditions. The fuel injection time-step is determined
based on the resolution permitted by the size of the
computational mesh and the initial droplet velocity.
The liquid fuel injection is simulated by injecting a
discretized parcel of liquid mass from each one of
the injector holes a t the end of the the fuel-injection
time step. The droplet-size distribution within the
injected liquid-fuel mass is generated from the following c ~ r r e l a t i o n . ~ ~
where n is the total number of droplets and dn is
the number of droplets in the size range between D
and D dD. The Sauter mean diameter (032) is
estimated from the following correlation3':
+
unsteady analysis. The k - 6 model, therefore, might
only be applied for the moderate frequency turbulent
components. The implementation of the boundary
conditions at the solid wall would become straightforward if the grid mesh could be made fine enough
to resolve the turbulent boundary-layer structure in
the vicinity of the wall. Instead, the source terms
in the governing equations of momentum, kinetic energy, dissipation rate, and energy are modified with
the introduction of standard wall functions. The procedure used for wall functions is similar to that given
in Ref. 33. Since the shear stress near the wall remains nearly constant, a zero gradient boundary condition is used for the k equation.
The dissipation at the wall is given by
c=-
~!k;
(35)
nnw
where n, is the normal distance from the wall to the
2ra,
D32 = BdPX*
(33) nearest grid point. Eq. 35 is based on the assumption
that the production of turbulence within the log-law
~gl/T2
layer is approximately equal t o its dissipation. The
where Bd is a constant, a, is the surface tension, VT
momentum and energy fluxes are evaluated as
is the average relative velocity between the liquid
interface and the ambient gas, and A*, is a function
of the Taylor number, ( ~ l o ~ ) / ( ~ ~ ~ ~ V , " ) .
A typical droplet size distribution obtained
from the above correlation in terms of the cumulative
percentage of droplet number and mass as a function
of the droplet diameter is shown in Fig. 2.
The pilot injector is a two-hole configuration. A
fine spray emerging from the pilot injector provides
enough vaporized fuel a t gaseous temperatures of 500
K to 650 K near the end of the compression event even
before spark ignition occurs. The droplet distribution
for the pilot injector is assumed to be monodisperse.
DETAILS O F TURBULENCE AND
COMBUSTION MODELS
The turbulent shear stresses are evaluated using
a two equation k - c turbulence model of Launder
and Spalding.32 Use of this model implies that the
influence of droplets on turbulence structure is negligible, and that the possible oscillatory motions have
a low frequency which does not appreciably alter the
turbulence properties. However, the possible low frequency oscillations can be simulated directly with the
where U is the gas velocity component parallel to
wall, u* is the wall friction velocity, K is the Von
Karman constant, E is the wall constant, and n$ =
pou*n,/pw is the non-dimensional distance from the
wall.
The combustion model is based on an analogous treatment of turbulent diffusion flames with the
assumption that the liquid fuel acts as a distributed
source of fuel vapor within the spray. The combustion rate, R f , of Eq. 1, is determined by taking into
consideration the minimum of either the Arrhenius
kinetic rate as determined from a single-global rate
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expression of Westbrook and Dryer35 for hydrocarbon
fuel/oxidizer combustion or the mixing rate as determined from the eddy breakup model of Spalding.17
Note that in the regions of interest for flame stability studies the chemical kinetics should be slower
than molecular mixing within a turbulent eddy. The
conversion rate will be given there by a kinetic formula. Use of the eddy breakup model is by no means
well established in a spray environment but its application in the modeling of gas-turbine combustor
flows with sprays is quite w i d e ~ ~ r e aand
d ~prc~~
vides some useful results.
So far, we have modeled the combustion of three
different fuels, n-decane, n-octane, or n-hexane in a
Wankel engine. For example, the overall reaction representing the oxidation of the n-decane fuel is given
by
By assuming that the effective diffusivities for
all species in a multi-component mixture are equal,
the mass fractions of N2, C 0 2 , and H 2 0 can be determined from simple algebraic relationships based
on the atomic balance of the constituent species, after the mass fractions of fuel and oxidizer are determined from the solution of the two gas-phase e q u a
tions based on the conservation of fuel and oxidizer.
ignition computations, the area of the hot spot for
the standard ignition is maintained at 1 cm2 and for
the trochoid ignition at 1.4 cm2 and the temperature
of both hot-spots is maintained at 1200 K. Perhaps,
with this model a glow plug could be simulated more
accurately rather than spark ignition but it does introduce some relevant physical structure. As the process evolves the gases near the hot spot continue being heated and some heat is also lost to the walls due
to conduction. And eventually combustion process
~starts
~
as the local equivalence ratio of the mixture
reaches the flammable limits.
This model differs from from the spark ignition
model used in Ref. 36, where spark ignition is simulated by increasing the total internal energy of a specified region of ignition cells next to the ignition source
at a predetermined rate during a specified crank angle
interval. Even within this specified time period, the
energy deposition is brought to an end if the temperature within the ignition cells reaches 1600 K. This
model precludes the simulation of advanced spark ignition until after the beginning of fuel injection.
Both these models proved to be useful in a
number of engine applications but their applicability should be viewed with caution since none of these
models takes into account of the various complex processes associated the spark ignition starting with the
formation of a high-temperature plasma kernel created by the electric discharge produced between the
electrodes to the point where chemical reactions initiate.
DETAILS OF T H E NUMERICAL METHOD
where IC1 = 4.29, I(2= 0.08723, and K3 = 2.222.
The spark ignition in the present study is modeled by a crude physical model that maintains a
known, high-temperature in a specified number of ignition cells next to the trochoid surface at the selected ignition location. The size of hot-spot and its
temperature are determined by a parametric study
by matching the measured and predicted pressures.
Once after the optimum values are determined for
one single case, the hot-spot size and its temperature
are kept the same in all other subsequent cases. For
the single ignition computations, the area of the ignition cells on the trochoid surface is maintained at
1.8 cm2 and its temperature at 1375 K. For the dual
Solution of the gas-phase equations is obtained
using a finite-volume, two-factor (Lower-Upper, LU)
decomposition scheme. There exist many computer codes which are constructed based on a finitevolume, LU ~ c h e m e . ~ ~ Yu
- ~ Oet a1.37 developed a
code, RPLUS-3D for steady-state computations of reacting flows for H 2 / 0 2 combustion with a Baldwin
and Lomax41 turbulence model where inviscid Jacobian matrices are split similar to the Jameson and
Turke14' splitting. Y o k ~ t adeveloped
~~
a diagonally
inverted LU implicit multigrid scheme for steadystate computations with a k - E turbulence model.
Both these formulation^^^^^^ require the addition of
explicit artificial dissipation terms for stability purposes.
The present finite-difference formulation is
based on an upwind scheme because of its superior
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numerical stability, and efficiency properties compared to those of a centered difference scheme.43
The most widely used flux-vector splitting methods
, ~
are those of Steger and
van ~ e e rand
Roe.45 We have chosen the Steger and Warming fluxvector splitting because of its demonstrated success
in the modeling of complex turbomachinery problem
based on unsteady three-dimensional inviscid computations on dynamic grids.39 Because of a recent interest in the modeling of reacting flows, various fluxvector splitting methods originally developed for an
ideal gas have been extended for a real gas with variable properties.46147The derivation of the Steger and
Warming flux-vector splitting for a perfect gas mixture with variable properties which is applicable for
the present formulation can be found in Raju and
Willis."
Since the flux vector F(Q) of Eq. 1 retains its
homogeneous property for the equation of state considered, the flux vector can be split into two parts,
~
also^^ = (e+<i+(:)+,
a = J(L)
cvm5 is the
N
speed of sound, CPm= Gill
yiCpj, Cum= C'pm - R,
Ns
8 = Ru Ci,l ,, , - Cpm/Cym, & = &, fu = %,
& = &, and 0 = Qu + fyv + fz w. The correspondand
ing split fluxes associated with the vectors G(Q) and
H(Q) can be written in a similar way.
The governing equations are linearized in a
delta form such that the nonlinear convective terms
and the source terms associated with finite-rate chemistry, turbulence, and the variance of the fuel concenwhere F+ is the subvector associated with the non- tration fluctuations can be discretized into an implicit
negative eigenvalues of A, F- is the subvector asso- approximation. The computational effort required is
ciated with the non-positive eigenvalues of A, and A kept to a minimum by casting the diffusion terms
is the Jacobian matrix,
and the source terms arising through liquid-phase interaction
into an explicit approximation. The timeThe resulting components of the split fluxes Ff are
linearized
governing equations in delta form are
given by
%.
6:~-
f
L
CPmp D [A:&u
+
$(u
-&a)]
LPD
Cpm
k:&v
+
+ %(u
*
+ +(v
A
+ 6:~-
- L)] AQ = - Q " A D + A ~ ~(44)
+(;a)
3,
w h e r e ~ f= a
0 9E t f l g af9= a 8 f ,ac
9 *=e,~=
At is the time step size, 6+ and 6- are forward and
backward differences, respectively, and
+&,a)
$(v-&a)]
AQ = @AD
f
L
CP
~[ AD
f&w+~(w+~;a)
+ D"+~AQ,
m
+
g(w-ea)]
L p D [Af&(h
Cpm
+
*
- C p m ~+) g ( h + Do)
%(h - ~ a ) ]
Y j Fc
YO
9
EFI
SF:
= -~-j?+
t
- 6-G+ - 6 - 2 +
rl
- 6:g-
-6;G-
+ Sc+ S,
+bcffV
C
- 6+$C
+ 6,Pu + 6,Gu
(45)
The system of algebraic equations resulting
from the discretized form of the unfactored implicit
approximation of Eq. 44 has a very large bandwidth.
(42) It is not possible to solve this system of equations on
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any existing computers because of the excessive CPU
and memory requirements. Steges and Warming43
have reported several possible ways of factoring this
implicit delta operator. A two-factor LU factorization, which is based on one-sided, implicit, spatial
differences, is used in the present study due to its
better stability properties compared to that of a sixfactor method.39 Upon factoring Eq. 44 we obtain
the following sequence:
I
+ a t (a;
A+ + a; B+
+ a;
backward Euler time differencing which is only firstorder in time. The formulation can be made secondorder time-accurate with little programming effort
by switching to a three point backward scheme.39
However, at present we have not pursued this implementation into our code because of the marginal
improvement reported from its use in some unsteady
computations.39
It is noteworthy that the following equations
represent the metric invariant terms arising from the
coordinate transformation:
]
ct) AQ =
Solution of Eq. 46 is obtained by a simple
When the governing equations are formulated
forward substitution and solution of Eq. 47 is ob- in strong conservation form, it is essential that the
tained by a simple backward substitution. During left-hand side of Eqs. 48 to 51 vanish identically when
both forward and backward sweeps, LU factorization the derivatives are approximated by finite-differences;
requires the solution of a block triangular operator, otherwise spurious source terms may result from geowhich can be reduced to the problem of solving a metrically induced errors.48 Thomas and om bard^^
10x10 matrix a t every computational cell through showed that the discretized form of Eqs. 49 to 51
back substitution. By adopting an algorithm taken will be satisfied identically when central differences
from the RPLUS-3D code,37 the present code is vec- are used to evaluate the spatial derivatives and also
torized rather efficiently by operating on all points in when the metric coefficients are formulated in the fola diagonal plane of the computational space, simul- lowing conservative form:
taneously. The diagonal plane is one on which i+j+k
= constant. The discretized counterpart of the governing equations differs in some ways from that used
in RPLUS-3D37: (1) RPLUS-3D formulation requires
only scalar diagonal inversions for the flow equations
and block diagonal inversions for the species equai
tions; and (2) Also the manner in which both the
split-flux differences and the metric coefficients of the
coordinate transformation are implemented.
It is noteworthy that to be consistent with the
objective of deriving a finite-volume code, the split- V X , vy, 77t, Cx, Cy, it,rlt, and Ct can be written in a
flux differences in Eqs. 46 t o 47 are implemented ac- similar way. However, the determinant of the coorcording to Monotone Upstream-Centered Schemes for dinate transformation is computed numerically from
? ~ ' solution of Eq. 48 in order to avoid grid-motion
Conservation Laws (MUSCL)-type d i f f e s e n ~ i n g . ~ ~the
The fluxes at the cell faces are first obtained by a induced errors.48 For the dynamic grid calculations,
fully upwind first-order accurate interpolation, and the metric quantities are evaluated at time level n + l ,
then centered differences are used for both the for- and DnS1 is evaluated from the solution of Eq. 48
ward and backward spatial operators evaluated at the by using an explicit method.
The numerical grid is generated by an algebraic
cell centers. Centered differences are also used for
evaluating the spatial operators associated with the technique with the help of the grid-generation code
viscous terms. The present formulation is based on a taken from the LEWIS-3D code.49
Downloaded from SAE International by University of British Columbia, Tuesday, September 25, 2018
The boundary conditions are implemented explicitly by defining a layer of phantom cells outside
the boundaries of the computational domain. In all
the cases that we have considered the temperature
of the walls is specified. The momentum and energy
fluxes are determined from the standard wall functions. For equations involving species, mass fraction
fluctuations and turbulence kinetic energy, the normal flux to the wall is set to zero. The pressure at
the boundary is determined by assuming there the
normal gradient of pressure to be zero. For example,
on a boundary aligned with q - C plane the pressure
is determined from the following relationship.
After the particle location in a computational
cell is established, the gas-phase properties at the
particle location are evaluated by using an interpolation method involving volume-weighted averaging.
Fig. 3 shows a grid cell in the transformed domain
surrounding a characteristic for the dependent variable $. The gas-phase properties are extracted from
the grid generated in the previous time step. The
gas-phase properties at the characteristic location are
interpolated from the computational cell as follows:
($:(I,
Finally, the density is determined from the equation
of state, Eq. 3.
The interaction between the gas- and liquidphases is determined by making several modifications to the solution procedure developed in Raju and
Sirignano.1811g These modifications are introduced
since the gas-phase computations in the present study
are performed in a three-dimensional generalized coordinate system on a moving grid, whereas in the
previous computations they were performed in a twodimensional fixed-rectangular coordinate system.
In order to obtain the solution of the liquidphase equations, it is first necessary to search for
the computational cell of the gas-phase equations in
which the particle is located so the gas-phase properties a t the particle location, which are needed in
the solution of the liquid-phase equations, could be
evaluated. It becomes a trivial task t o search for the
appropriate computational cell in rectangular coordinates. However, a search for the particle location
becomes a complicated problem when the computa
tional cells are no longer rectangular in the physical
domain. An efficient search method is developed in
the present solution procedure by limiting the search
to the local region where the particle is found during
the previous step. Although this procedure requires
an additional effort of saving the coordinates of the
computational cell, the savings in the overall computational effort could become significant especially
when the spray model requires consideration of several hundreds of particles.
* 1/01 I1+ $:(I + 1, J, Ii') * 1/01 V I
+$;(I + 1,J, Ii' + 1) * Vol V I I
+$:(I, J,I<+ 1)* Vol 111
+$:(I, J + 1,Ii')* Vol I
+$!(I+
1, J + 1, I<) * Vol V
+$:(I+ 1, J + 1,Ii' + 1)* Vo1 V I I I
+$:(I, J + 1, Ii' + 1) * Vol I V )
J,Ii')
/(Total cell volume)
(54)
After the gas-phase properties are evaluated at
the particle location, the ordinary differential equations describing particle size, position, and velocity
are solved by a second-order accurate Runge-Kutta
method. The partial differential equation describing the transient temperature variation within the
droplet interior is solved by an implicit method. Because of the prohibitively small time-step restriction
imposed by the stability criterion of the numerical
scheme used in the solution of the liquid-phase equa) ' used
~ where
tions a fractional time-step r n e t h ~ d ' ~ is
the liquid-phase equations are advanced in time a t a
fraction of the time-step used in the integration of the
gas-phase equations. A fractional time-step method
of this nature is useful only for those cases where v a
porization rate is not the rate-controlling process. For
the other cases both the liquid- and gas- equations
should be advanced in time with an equal time-step.
After the liquid-phase equations are solved, the
source terms evaluated at the particle location are
redistributed within the eight nodes of a computai
tional cell surrounding the particle by using volumeweighted averaging. These source terms are redistributed on a grid generated at the new time-step.
The source terms at the cell centers due to liquidphase interaction are given by
Downloaded from SAE International by University of British Columbia, Tuesday, September 25, 2018
-
*
S1,char V OII
~
Sl,(I, J, K ) = Total cell volume
(55)
and the source terms at the remaining seven cell
centers are determined in a similar way. However,
4, are the source terms evaluated at Atl,, which
is the time-step used in the integration of the
liquid-phase equations.
All the three steps involving the interpolation
of gas-phase properties at the particle location, the
integration of liquid-phase equations, and the determination of source terms are repeated until the liquidphase equations are advanced over a time period
equal to that of the gas-phase time-step, At,. The
+
time-avzraged contribution of these source terms, Sl,
yields Sl of Eq. 1.
where
C atl,
= at,
RESULTS AND DISCUSSION
The details of engine specifications-and some of
the operatings conditions common to all the cases examined are given in Table 1. All the computations
were performed for an engine speed of 6000 rpm while
maintaining the exhaust pressure at 0.85 atm and the
wall temperatures a t 330 K. The overall equivalence
ratio (a),fuel composition, pilot and trochoid spark
timings, pilot and main injection timings, the amount
of fuel injected from the pilot injector as a percentage of total fuel flow, intake prePsure and intake temperature for the cases studied are listed in Table 2.
The operating conditions of Cases 1 and 2 correspond
t o the engine conditions for which measured pressure data are a~ailable.~'The operating conditions
of Cases 3 to 8 reflect the variations from Case 2
which is chosen as the baseline. The effect of changing the fuel rates is considered in Cases 3 to 4; the
effect of changing the octane rating of fuel is considered by varying the fuel composition from n-decane
in Case 2 to a more volatile fuel, n-hexane, in Case
5; the effect of increasing the intake temperature is
addressed by making comparisons between Cases 2
and 6; and the effect of retarding both fuel-injection
and spark-ignition timings is considered in Case 7.
The effect of second ignition source is considered by
making detailed comparisons between the results obtained for the single ignition case of Case 8 and the
dual ignition case of Case 2.
The computations were performed with a fixed
time step size requiring about 20,000 steps to cover
one cyclic period on a grid of Ni=31, Nj=16, and
Nk=20. Here, N is the number of grid points and
i, j, and k represent the coordinate surfaces in the
direction extending from the trailing-edge surface to
the leading-edge surface of the combustor, from the
rotor t o housing surface, and from the side wall to the
symmetry plane of the domain between the end-toend side walls, respectively. The perspective view of a
typical grid used in the computations a t a crank angle
(CA) of 6.7 rad is shown in Fig. 4. The computations
were initiated just before the the exhaust port opens
and were terminated just after the intake port closes
during the next cycle.
PRESSURE COMPARISONS - Figs. 5 and
6 show the comparisons of measured and computed
pressures for Cases 1 and 2 of Table 2, respectively.
The pressure around the top center position is measured by two different pressure transducers, one, the
after top center (ATC) transducer, is located at 6.8
cm ATC and the other, the before top center (BTC)
transducer, at 9.84 cm BTC. The BTC transducer
measures pressures between 320 deg BTC and 45
deg ATC and the ATC transducer measures pressures between 10 and 375 deg ATC. Each of these two
pressure traces represent an ensemble average of 250
cycles. A certain amount of spark interference was
reported with the signal from the BTC transducer.
Comparisons during the CA interval of 10 and 45
deg ATC, when both the transducers are measuring
the pressure simultaneously from the same combustion chamber, indicate substantial deviation between
the two measured pressure traces with the ATC transducer consistently showing higher pressures. The reason for this deviation is not known, especially since
the computations indicate that the pressure distribution near the top center position becomes slightly
non-uniform across the combustion chamber with the
pressure near the trailing apex region slightly higher
than the pressure near the leading apex region. The
results on the pressure non-uniformity are not re-
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ported here since the present results are similar t o
those reported in the earlier papers of Abraham et
a14 and Raju and W i l l i ~ The
. ~ predicted
~ ~ ~ ~ volumeaveraged pressures match very well with the pressures
measured by the BTC transducer between -50 and 45
deg ATC showing a correct value and location for the
peak pressure. The predicted pressures fall slightly
below the pressures measured by the ATC transducer
during the expansion period but the comparisons indicate very similar trends.
Fig. 7 shows pressure comparisons between the
Cases 2 t o 4. Case 2 has an overall equivalence ratio
of 0.51, and its value in Case 3 is 0.60 and 0.47 in Case
4. The pressures of Case 2 as measured by both the
ATC and BTC transducers are also shown in Fig. 7.
Changing the fue1:air ratio seems to have a negligible
effect on the location of peak pressure but the peak
pressure magnitude seems to increase almost linearly
with an increase in the fuel rate. Surprisingly, the
computed pressures for Case 3 matches well with the
pressures measured by the ATC transducer of Case
2.
Fig. 8 shows the pressure comparisons between
Cases 2, 5, and 6. Changing the fuel composition
from n-decane in Case 2 to n-hexane in Case 5 is
shown to yield considerable gains in power output
as shown by the pressure comparisons with the peak
pressure location advancing from 27.6 to 16 deg ATC
and an accompanied rise in peak pressure from 66 to
75 atm. The octane rating of n-hexane is very low
compared t o n-decane, which leads to faster burning
in Case 5, and n-hexane is also more volatile than ndecane, which leads to faster conversion of liquid to
vapor. The effect of changing the intake temperature
from 330 K in Case 2 to 400 K in Case 6 also seems to
have a significant effect on the resulting pressure distribution. The peak location is advanced from 26.5
to 16 deg ATC while the magnitude of peak pressures
remains approximately the same. Following the peak,
the pressures in Case 6 fall below Case 2 during the
remaining period. The initial rise in pressures is due
to a substantial increase in burning resulting from an
increase in the intake temperature. But, the power
output in Case 6 still falls slightly below Case 2, since
(the only difference between Cases 2 and 6 is in the
intake temperature) an increase in intake temperature invariably leads t o a decrease in volumetric efficiency while keeping all other parameters constant,
and therefore resulting in a considerably lower intake
charge. Experimentally, both the use of a low-octane
rating fuel and increasing the engine intake temper-
ature was found t o lead to faster combustion and
improved efficiency at low loads, and an increased
propensity towards erratic combustion at high loads.
Also, the use of high-octane fuel was found to provide
greater resistance to auto-ignition than Jet-A fuel.1°
The pressure comparisons between Cases 2, 7
and 8 are shown in Fig. 9. Both Cases 7 and 8 involving delayed timings and single ignition predict a
loss of fuel efficiency, the loss being more pronounced
in the single ignition case. With delayed timings, the
peak moves from 27.6 t o 36 deg ATC with an accompanied decrease in its magnitude from 66 to 56 atm.
The pressure distribution shows two peaks when the
computations are performed by turning off the trochoid ignition source for Case 8. Sometimes, Case 8
is referred to as the single igniter case. The reasons
for the notably different behavior observed between
the single- and dual-ignition cases will be discussed
in the following sections by making a detailed examination of the flow and combustion characteristics.
FUELING AND REACTION-RATE HISTORIES - Figs. 10a and lob show the crank angle variation of liquid fuel injection, vaporized fuel, and the
burnt fuel for Cases 2 and 8, respectively. The total
amount of fuel injected per engine per cycle is equal
to 0.052 g. Liquid fuel vaporization is found to be
a fairly rapid process with most of the vaporization
occurring before 15 deg in Case 2 and 30 deg in Case
8. Combustion in both cases is slow prior to TDC
despite advanced ignition timings. Combustion becomes very rapid after 5 deg ATC in Case 2 with
more than 80% of the burning taking place between
5 to 30 deg ATC. The burning curve in Case 8 exhibits two distinct regimes in which 35% of burning is
shown to occur between 8 t o 30 deg ATC followed by
a slight decrease in combustion rate before increasing
again around 36 deg ATC. Overall combustion with
single ignition is much slower than with dual ignition.
Figs. l l a and l l b show the crank angle variation of cumulative reaction rates for Cases 2 and 8,
respectively. The contribution of the reaction rate
arising from Arrhenius kinetics is shown separately
from that arising from the eddy breakup term. This
distinction aids in determining the rate-controlling
step which shows that chemical kinetics are slower
than molecular mixing within a turbulent eddy, as
expected during the initial ignition phase of combustion process. The results of Figs. l l a and l l b further
elucidate the observations made on the burning rate
histories of Figs. 10a and lob. The reaction rate
variation of the single-ignition case is clearly charac-
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terized by two distinct peaks and a single peak characterizes the dual-ignition case. The absence of the
second peak in Case 2 suggests the significance of
a combustion contribution from the trochoid spark
ignition. Surprisingly, the behavior of the single ignition case is in agreement with the heat release analTheir experiments on a
ysis of Lawton and Milla~-.~l
carburatted rotary engine with early ignition showed
that ignition timing has a significant effect on the
shape of the heat release curve with two peaks appearing with early ignition and a single peak with late
ignition. From these findings, it appears that combustion in the single ignition case resembles closely
that of a premixed RCE with early ignition. The
reasons given by Lawton and Millar for the engine behavior with early ignition is summarized herelaThis
may be due to the separate effects of pressure and
temperature, both of which increase combustion rate
and have separate maxima after top dead centre. Alternatively, the two peaks in the heat release pattern
may be caused by purely geometrical conditions relating to the transfer of charge between the two major
sections of a Wankel engine combustion chamber lying either side of the minor axis."
DETAILS OF FLOWFIELD DURING COMBUSTION - The side and frontal views of gaseous
fuel/oxidizer equivalence ratio (a) and temperature
contours of Case 2 a t the crank angle intervals of 38.7, -24.3, 5, 18.6, 40, and 90.25 deg ATC are shown
in Figs. 12a to 12f, respectively. The corresponding
contours of Case 8 are shown in Figs. 13a t o 13f. In
all these figures, as shown in Figs. 12a and 13a, the
rotor motion is in counter clockwise direction. During the initial stages of fuel injection, it is obvious
from the contours of Fig. 12a and 12b that the unburnt mixture is mostly concentrated near the fuel
injectors. And the highest level of fuel concentration
is observed t o coincide with one of the fuel injector
locations. The observed values are well within the
flammability limits of n-decane, however, the flame
speed upon ignition will be substantially lower in the
regions where the equivalence ratio of mixture exceeds far greater than one. The results differ from
those reported by Abraham and Bracco8 where the
fuel concentration near the main fuel-injector location
was observed t o fall outside of the rich-flammability
limits during the earlier stages of fuel evaporation.
As can be seen from Figs. 13a and 13b, a similar fuel concentration distribution seems t o develop
for the single ignition case. The temperature contours show that the center of the highest temperature
contours coincides with the location of the spark igniters, two in Case 2 and one in Case 8. And the lowest temperature contour corresponds with the walls
which are maintained at 330 K. Fig. 12c shows the
contour levels prior to the beginning of the period in
which maximum combustion activity is observed t o
occur. The contour levels show less stratification of
fuel near the main injection. A significant portion of
the unburnt charge is located in a 1000 K tempera
ture region with equivalence ratios ranging from 0.5
to 1.5, indicating that the state of reactive medium
is such that it would soon lead to rapid burning.
Fig. 13c represents the conditions prior to the
beginning of combustion activity as related to the
first of the two peaks that seem to characterize the
combustion in Case 8. In this case, only a small
fraction of the total unburnt mixture is found to be
located in a high-temperature region near the pilot.
And the remaining unburnt mixture near the main
injection is found to be located in a relatively lowtemperature region unlike Case 2 in which a significant fraction of unburnt flammable mixture near the
main injection is found to be in a high temperature
region produced by the second ignition source. Fig.
12d shows that during the middle of intense combustion activity the highest temperature region of 2000
K occurs roughly near the location of main injection
with equivalence ratios of the mixture ranging between 0.12 and 0.38. However, Fig. 13d of the single
ignition case shows that a significant fraction of the
flammable mixture remains unburnt while the high
temperature region is just beginning to advance upstream past the minor axis. And the equivalence r a
tios of unburnt mixture range between .2 to .7. Comparisons between Figs. 12d and 13d illustrate the
fact that the combustion in the single ignition case is
an extremely slow process even though the unburnt
mixture mostly falls within the flammability limits as
evidenced from Figs. 13a to 13d.
The slowness in flame propagation upstream of
the rotor direction may not be a unique property of
the stratified-charge rotary engines. Similar findings
were reported earlier for the premixed rotary combustion engines by Lawton and Millar.51 The reasons
for the slow flame propagation are as follows: (1)
The flow velocities remain essentially unidirectional
in the direction of rotor during most of the combustion period. Especially, at high engine speeds, the
relative speed of flame becomes very low as it has to
travel upstream of the rotor-induced bulk fluid motion which is reinforced by strong squish; and (2) The
Downloaded from SAE International by University of British Columbia, Tuesday, September 25, 2018
flame propagation properties are also adversely affected by the heat transfer characteristics of a Wankel
engine due to convective heat transfer. The advancing flame front gets quenched near the walls as the
convective heat transfer losses to the walls increase as
a result of higher "squish" flow velocity near TDC.
Some of these results will be discussed in a section on
the heat transfer characteristics.
The slow flame propagation restricts the high
temperature region near the pilot injection from expanding further upstream until after TDC, when
greater mixing of fuel and air occurs as a result of
higher turbulent intensity generated near TDC and
the rotor-induced fluid motion brings in more unburnt mixture downstream in contact with the high
temperature region. Once this occurs the flame advances upstream consuming most of the unburnt mixture as evidenced by Fig. 13e. Near the end of
combustion, the equivalence ratios observed within
the combustion chamber fall near or below the lean
flammability limits as shown in Figs. 12e, 12f and
13f. A positively uniform gradient in fuel concentration appears to develop in the direction opposite to
the rotor motion. And during the expansion after the
combustion is nearly completed, the temperature becomes fairly uniform within the combustion chamber.
A seduction in both gas temperature and fuel concentration levels takes place during expansion period as
the CA position changes from 40 to 90.25 deg.
~h~ perspective views of the droplet trajectories at CA positions of -38.7, -24.3, 5, and 18.6 deg
ATC for Cases 2 and 8 are shown in Figs. 14 and 15,
respectively. The polydisperse character of the spray
is represented by four different initial droplet sizes
corresponding to a Sauter mean diameter of 14 pm.
T . , ~liquid fuel is injected through a staggered fivehole main and a two-hole pilot. Two cclightopsprays
from the main are directed towards the trochoid ignition at an angle very close to the trochoid surface.
The purpose of lightoff sprays is to create a sufficient
build-up of flammable mixture near the trochoid ignition in order t o get combustion off of that ignition.
The particle trajectories initially seem to retain their
direction as defined by their initial conditions for a
short duration before they become small
to
be deflected by the rotor-induced fluid motion in a
counter clockwise direction. The drag-induced particle motion, in particular, of the two lightoff sprays
away from the their intended location may be one
of the reasons why it was so difficult experimentally
to get ignition off of the trochoid igniter. At 5 deg
ATC, just after the completion of main injection, particles are found t o be finely dispersed within the rotor pocket. At 18.6 deg ATC, more liquid droplets
still remain within the combustion chamber in Case
8 than in Case 2.
HEAT-TRANSFER CHARACTERISTICS Accurate prediction of engine heat transfer from the
combustion chamber gas to the walls is of paramount
importance Since it affects engine efficiency, performance, and emissions.52 Higher heat transfer to the
walls reduces the work available to the piston by lowering the gas Pressure and temperature, heat transfer
from hot spots surrounding the spark plugs might
lead to the onset of knock, and also the exhaust
temperature determines the work recovery from turbocharging devices. Changes in the gas temperature
due to heat transfer also affects the emission formation processes. From the modeling point of view, it
is very important to accurately predict heat transfer
to the walls since the combustion characteristics will
be strongly influenced by the heat transfer through
its effect on the evaluation of the local properties
of flowfield such as gas temperature, composition,
flow velocity, and turbulence intensity. The engine
heat transfer in the present computations is modeled
through the use of k-c turbulence model with standard wall functions. The details of which were presented earlier in the paper. And the instantaneous
local heat transfer fluxes, which may be required for
thermal stress calculations, are also presented. Unfortunately,
data On heat
reliable
fluxes are
for comparisons.
The instantaneous spatially-averaged heat
fluxes, which are required for engine perforrnance analysis, are compared with the Woschni
~ o r r e l a t i o n ,which
~ ~ is presently the most widelyused in Wankel engine performance analysis.54 The
presently-used heat transfer correlations were originally developed for reciprocating engine applications
and have 'erY little
to be useful for the Wankel engine
The torof Mcadams,52 W0schni,"3 AnnandyS5 and
based On the
that the
Eiche'berg56 are
functional relationship between Nusselt, Reynolds,
and ""ndtl numbers
the same
kind of relationship that was found for turbulent
pipe and flat-plate boundary-layer flows. Woschni
a
of the form:
Nu = 0 . 0 3 7 ~ e ~ . '
(58)
where Nu is the Nusselt number and Re is the
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Reynolds number. The average gas velocity w , used
in the Reynolds number'is given by:
nomena in the absence of combustion is similar to unsteady Couette flow, which is subjected to a mostly
favorable non-uniform pressure gradient.
Fig. 17 further amplifies these differences for
VdTref ( P - pmot)]
W = [Clsrotor +c2.
(59) the spatially-averaged heat transfer coefficients of the
ref r e f
rotor and housing walls. The steep rise in heat flux
near TDC is due to higher gas temperature generand
ated by combustion. The peak has a value of about
~Ncr-ankR
3 MW/m2 and its location occurs slightly after the
Srotor =
(60) peak pressure location. The predicted peak fluxes
90
fall below those reported for a ~ Y P where Srotor
is the average rotor tip speed, Vd is the are
displaced volume, Tref, pPe
f , and Vref are the av- ical diesel engine operating under normal conditions
but above those reported for a typical spark-ignition
erage combustion gas temperature, pressure and
ume at some reference state, P is the average cham- engine.52 Comparison with the Woschni correlation
her pressure, pmot
is the corresponding motored pres- for heat fluxes and heat transfer coefficients are shown
sure at the same crank angle, N~~~~~is the crank in Figs. 18 and 19, respectively. The correlation overrevolutions per second, and R is the generating r& predicts the peak heat flux by a substantial measure
dius of the Wankel engine. The constant Cl, has a but gives the correct magnitude for the heat transfer
value of 0.75 and the value chosen for c2is 0 for coefficient. However, the correlation predicts the lothe compression period and 0.011 for the com~ustion cation of heat transfer coefficient peak to occur much
and expansion period. pressureused in the corre- closer to TDC. The comparison with the correlation
lation is evaluated on a ~ ~ l ~ m basis
e from
- ~ could
~ ~probably
~ ~ be~ improved
~ d by changing the values
the three-dimensional computations and temperature used for the constants, C1 and C2. But no such effort
on a mass-averaged basis. The specific heat CP, and Was made in the present study since the comparisons
cases showed sometimes OverPrethe transport property p , are calculated based on the made in some
properties of a five-species fluid as in the 3- diction and sometimes underprediction. The crank
D computations. But these properties are calculated angle variation of instantaneous spatially-averaged
based on a spatially-averaged sense unlike the 3-D heat transfer coefficients and its comparisons with
compu~ationswhich take into account the local vari- the Woschni correlation are shown in Figs. 20 and
ations within the combustion chamber.
21 for Case 8 and those for Case 6 in Figs. 22 and
Figs. 16 represents the crank angle variation of 23.
Fig. 24 shows the instantaneous heat flux on
instantaneous spatially-averaged heat fluxes for Case
2. Also shown in this figure are the heat losses, the rotor surface, and the instantaneous gas temperas averaged separately, to the rotor surface, hous- ature, turbulence kinetic energy, and velocity distriing, and side walls. Fig. 17 shows the correspond- bution near the rotor surface at a CA of -24.3 deg
ing spatially-averaged heat transfer coefficients. The ATC for Case 2. The heat flux distribution on the
computations show that most of the heat loss occurs rotor surface is found to be highly non-uniform and
through the rotor and housing surfaces and the loss the local maximum of 1.4 M W / m 2 is observed to octhrough the side walls is relatively low since the rotor cur after the minor axis near the side walls away from
and housing surfaces are more directly exposed to the the rotor pocket. The maximum heat transfer may
high-temperature combustion-chamber gas. During not necessarily occur in the high temperature region
the initial stages of compression up to CA of -70 deg as expected. The higher turbulence intensities and
ATC heat flux through the rotor and housing sur- flow velocities are observed t o occur near the leading
faces remains approximately the same. During the apex-seal region which is where the gas is subjected
remaining compression period prior to TDC the heat to higher shear stresses. These shear stress effects
loss through the housing surface increases more than extend all the way from the leading apex-seal region
the rotor surface because of higher velocity gradients to the position where the heat flux is maximum.
Both the local turbulent intensities and flow vegenerated there by the rotor motion as it slides over
the housing surface. Higher temperature generated locities within the rotor pocket are found to be lower
by spark ignition and to a minor degree by combus- which is not very conducive for the effective mixing
tion would also contribute to this increase. This phe- of fuel with the oxidizer. The computations suggest
Downloaded from SAE International by University of British Columbia, Tuesday, September 25, 2018
that the heat transfer distribution is determined by
a combination of several other parameters including
the gas temperature. The corresponding phenomena
a t CA of 5 deg ATC are shown in Fig. 25. The location of the maximum heat flux of 2.5 MW/m2 is now
shifted to after the minor axis as the region of the
high turbulence intensities and flow velocities shift to
the trailing apex-seal region. The instantaneous heat
fluxes demonstrate the uncertainty associated with
the use of simplified correlations since the engine heat
transfer appears to be far more complicated for it to
be analyzed by simple flat-plate boundary-layer flow
type of analysis.
SUMMARY AND CONCLUSIONS
The concept of dual-ignition offers a potential
means of improving the fuel efficiency in a Wankel engine by promoting faster combustion. Analysis of the
energy release rate of a single-ignition Wankel engine
with advanced ignition timings shows the characteristic rapid energy release associated with the burning
of pilot fuel injection followed by a period of considerably slow energy release because of the inability of
the flame to propagate upstream and burn effectively
the fuel near the main fuel injector where a sizable
portion of the unburnt mixture forms even though the
equivalence ratio of the unburnt mixture falls mostly
within the flammability limits. The slowness in flame
propagation especially at high engine speeds could be
attributed to the following reasons: (1) The flame
speed becomes very slow as it has to travel upstream
of the rotor-induced bulk fluid motion reinforced by
strong squish.; and (2) The flame propagation properties are also adversely affected by the heat transfer characteristics of a Wankel engine due to convective heat transfer. Following this slow combustion
event, another rapid energy release occurs when the
higher turbulent intensity generated near TDC leads
to greater mixing of fuel and air and also the bulk
fluid motion bring in more unburnt mixture in contact with the high temperature region near the pilot
for the combustion to progress rapidly. Thus, the energy release rate in the single-ignition case is found
to be characterized by two peaks.
By providing a second ignition source upstream
of the main fuel injection, the energy release following dual ignition becomes very rapid and combustion
rapidly extends to the whole rnixture.The second ignition source seems to have a significant effect on the
emerging combustion process showing a single peak
in its energy release variation with time. Combustion
in a dual ignition engine appears to be dominated by
the contribution from the trochoid ignition. Increasing either fuel loads or intake temperature and, also,
the use of low octane-number fuels have shown to promote rapid combustion and therefore contribute to
better engine efficiency whereas delayed ignition timings seem to result in slower combustion. However,
the failure to experimentally attain combustion off
of the trochoid ignition without advancing the spark
and fuel timings too far prior to TDC was hampered
by the reported incidence of knock at higher loads.
Noting that the combustion was modeled using a single-step global reaction rate for determining
the laminar-characteristic time and an eddy breakup
model for determining the turbulent-characteristic
time, it is recognized that modeling of abnormal combustion behavior such as knock is beyond the scope
of our present model since it requires consideration of
detailed kinetics.57 However, the rapid pressure rise
and faster energy release rate observed in some of the
cases considered with either low octane number fuel
or higher intake temperature provide an indication
for potential incidence of abnormal c o m b u s t i ~ n . ~ ~
Our computations showed that it is difficult to
build a flammable mixture of adequate quantity near
the trochoid spark for achieving ignition off of that
igniter when the second spark is located upstream
of the main fuel injection. Analysis of the particle
motion shows that the liquid fuel from the lightoff
sprays is deflected from their intended location near
that spark away towards the direction of combustion
chamber gas because of the drag forces acting on the
particle motion. This leads us to the speculation that
for achieving consistent ignition under a wide range of
operating conditions, the trochoid spark should either
be provided with its own fuel supply similar to the
one used for the pilot ignition by providing a housing
within which the igniter be placed with another pilot
injector of its own or the location of the second spark
should be moved downstream of the main fuel injection. Since all the injectors and igniters are located
on the axis of symmetry, the space requirements may
preclude the second igniter from being placed downstream of the main injector. This leads to a suggested
configuration where both the pilot and second sparks
are located on either sides of the axis of symmetry
close to and downstream of the main injector.
The magnitude of peak heat transfer losses to
the walls in a Wankel engine fall somewhere between
Downloaded from SAE International by University of British Columbia, Tuesday, September 25, 2018
those measured for spark-ignition and diesel engines.
The convective heat transfer correlations of Woschni,
Annand, and Eichelberg which are the most widelyused in Wankel engine performance analysis should
be used with caution since the instantaneous local
heat fluxes show significant variation across the combustion chamber for the spatially-averaged heat correlations to be very useful.
ACKNOWLEDGEMENTS
The author was supported by the NASA Lewis
Research Center under contract NAS3-25266. He
would like to extend his sincere appreciation to Dr.
E.A. Willis for his constant encouragement and support of this work. Thanks also go to Mr. J.J. McFadden of the NASA LeRC Propulsion Systems Division.
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Table 1. Engine Specifications and Common Operating Conditions
Engine Parameters
Generating Radius(R) = 0.1064 m
Eccentricity(E) = 0.01542 m
(See Fig. 1)
Clearance(C) = 0.000635 m
Chamber Width(W)= 0.077114 m
Port Width(Wp)= 0.05 m
Displaced Volume = 662.5 cm3
Minimum Volume = 84 cm3
Geometric Compression Ratio = 7.54
6000 rpm
Engine Speed
BStart = -1.26 rad, Bend = 5.96 rad,
Intake Port
Turbulence Parameters Are
Specified, Yf,s,t = 0
BStart = -5.96 rad, Bend = 1.07 rad,
Exhaust Port
Pezh= 0.85 atm
Temperature of Rotor
And Housing Surfaces Th = T,. = 330 K
Case
Overall
Equivalence
Ratio
Fuel
1
2
3
4
5
6
7
8
.54
.51
.60
.47
.51
.51
.51
.51
n-decane
n-decane
n-decane
n-decane
n-hexane
n-decane
n-decane
n-decane
Pilot
Igniter
(deg ATC)
Start End
-72
-11
-72
-11
-72
-11
-72
-11
-72
-11
-72
-11
-55
-10
-72
-11
Table 2. Operating Conditions
Trochoid
Pilot
Main
Igniter
Injector
Injector
(deg ATC)
(deg ATC) (deg ATC)
Start End Start End Start End
-72
4
-49
-18 -47
10
-72
0
-49
-15 -48
3
-72
0
-49
-15 -48
3
-72
0
-49
-15 -48
3
-72
0
-49
-15
-48
3
-72
0
-49
-15 -48
3
-55
10
-33
-8
-32
17
-49
-15 -48
3
Pilot/Total
Flow %
Intake
Pressure
(at4
Intake
Temperature
(K)
8.05
9.1
9.1
9.1
9.1
9.1
9.1
9.1
1.93
1.78
1.78
1.78
1.78
1.78
1.78
1.78
330
330
330
330
330
400
330
330
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Fig. 2
Initial droplet-size distribution curve.
Fig. 3
Grid cell surrounding a characteristic.
Fig. 4
A perspective view of the computational grid a t a crank angle of 6.7 rad.
Crdnk d n g l 0 , d e g A T C
Fig. 5
Measured and computed pressures for
Case 1.
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C r a n k a n g l e , d e g ATC
Fig. 6
E
l
-
C r a n k a n g l e , d e g ATC
Measured and computed pressures for
Case 2.
,
,
,
1
'
1
! , #
1
,
,
,
1
,
,
,
,
,
Fig. 8
Pressure comparisons between Cases
2, 5 , a n d 6 .
8
BTC t r a n s d u c e r
o ATC t r a n s d u c e r
7
-
b
-
5
-
4
-
3
-
2
-
-Case
2
-6
.
C r a n k a n g l e , d e g ATC
Fig. 7
Pressure comparisons between Cases
2, 3, and 4.
C r a n k a n g l e , d e g ATC
Fig. g
Pressure comparisons between Cases
2 , 7, and 8.
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C r d n k a n g l p , U e g ATC
Fig. 1 0
Fueling histories for Cases 2 and 8.
Fig. 11 Cumulative reaction rate histories
for Cases 2 and 8.
26
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Fe 12d Computed flowfield for Case 2 at a crank angle of 18.6
@) Temperature contours.
(a) Fueltoxidizer equivalence ratio contours.
--
deg ATC.
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@) Temperature contours.
Fuelloxidizer equivalence ratio contours.
Fig.12f Computed flowfield For Case 2 at a crank angle of 90.25 deg ATC.
(a)
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Crank a n g l e .
deg A T C
Fig. 16 Variation of surface heat flux with
crank angle for Case 2.
Crank a n g l e ,
Fig. 1 7
C r a n k a n g l e , deg ATC
Fig. 18
deg ATC
Variation of surface heat transfer coefficient with crank angle for Case 2.
Comparison of surface heat flux wit h Woschni correlation for Case 2.
C r a n k a n g l e , deg ATC
Fig. 19
Comparison of surface heat transfer
coefficient with Woschni correlation
for Case 2.
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Crank a n g l e , deg ATC
Fig. 20
Crank angle, deg ATC
Comparison of surface heat flux wit h Woschni correlation for Case 8.
Fig. 22
Comparison of surface heat flux wit h Woschni correlation for Case 6.
3800T,,I,,,,,,,,,1,,,,1,,,,I,,,,I,,,,I,,,,I,,,,,,,,,
3600 -
I
Y
I 3200 N
ornputatIons
2 3000 -- --3-- D-cWoschnl
modal
3400
-
:
2000 1
3
2600
4-
-
-
2400 -
0
""'""""'""'""'"""""'"""""""
-200
-150
-100
-50
0
50
100
150
200
250
300
Crank angle, deg ATC
Fig. 21
Comparison of surface heat transfer
coefficient with Woschni correlation
for Case 8.
Crank angle, deg ATC
Fig. 23
Comparison of surface heat transfer
coefficient with Woschni correlation
for Case 6.
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u
;r
ru-
E
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