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MEMORANDUM REPQRtT ARBRL-MR-0310
A COMPUTER CODE FOR THE SOLUTION OF.THE
EQUATIONS GOVERNING A LAMINAR, PREMIXED,ONE-DIMENSI-ONAL FLAME
r Terence P. Coffee
April 1982
QIR
82 04 28 ~
-- 7.
Do not rotuqa it to the orIginator.
Secondary distribution of this report by origsintifgor sponsoring activity is prohibited.
Additional copies of this "eport may be obtaim 6from the National Technical Information Service,U.S. Department of Commerce, Springfield, Virginia22161.
*The findings In this report are not to be onstrsed asan official Department of the Army position, unlessso* designated by other authorized documets.
thev O 't~m Mwime i~MWef Of~~ 4V 4MMW eP 111616.
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It EPORT NiUIseR t* GOVT ACCEgsowN 10S.E: PIN' CATALOG nummER
Memorandum Report ARBRL-MR-03165 L4-Ait4 TITLE (and Subtitle) S. TYPE OF REPORT & PERIOD COVERED
A COMPUTER CODE FOR THE SOLUTION OF THE EQUATIONSGOVERNING A LAMINAR, PREMIXED, ONE-DIMENSIONAL g Z~~MN R.RPR UOCF LAM.PRFRIGOGRPOTNMR
7. AUTNORfe) S. CONTRACT OR GRANT NUMBER~e)
Terence P. Coffee
S. PERFORMING ORGANIZATION NAME AND ADDRESS to. PROGRAM ELEMENT POROCT. TASK
US Army Ballistic Research Laboratory AE OKUI UDR
ATTN: DRDAR- BLI 1L161102A143Aberdeen Proving Ground, MD 21005ISREOTDE
11. CONTROLLING OFFICE NAME AND ADDRESSI.REOTAE
U.S. Army Armament Research & Development Command April 1982U.S. Army Ballistic Research Laboratory (DRDAR-BL) IS. NUMBEfR OF PAGES
Aberdeen Proving Ground, MD 21005 141 M f111081-14. MONITORING AGENC;Y NAME & ADDRESS(II Effereii eROM Canke~llind OffIO0) '0S. SECURIY CLAS.(fNt ~et
UnclassifiedIS. DEfrASIICATIoNIDOWNGRADING
IS. DISTRIBUTION STATEMENT (at this Rebport)
Approved for public release; distribution unlimited
17. DISTRIUUTION STATEMENT (of th. abstract enterd to Wek"S. It afithret hemn Report)
IS. SUPPLEMENTARY NOTES
1S. KEY WORDS (Contine en reverse side If aa~aam and ids.,aft1 bl eek asailg)
Laminar Flame Species Profiles* -One Dimensional Flame Temperature Profiles
Premixed FlameFinite Elements MethodMon-uniform Grid
2L AM. Acre( somr ato mewww~ mdai Uby blek iumbe (raj)~ A computer program for numerically solving the equations governing a
laminar, premixed, one-dimensional flame is described. Both unbounded flamest and burner stabilized flames are discussed. Some numerical conside~rations in
successfully using the code are presented.
DD Or' 193 ESIIo@orI Nov is II@ LETE
SEOIIrITV CLUPCATROF TWiS PAM(Mt W Ifts 4ae
TABLE OF CONTENTS
. ....... ... Page
I. INTRODUCTION. .. .. ........ ........ ......
II. THE UNBOUNDED FLAME EQUATIONS .. .. ..... ......... 6
III. THE NUMERICAL NETHOD-PDECOL. .. .. ........ ...... 13
IV. MODIFICATION OF PDECOL .. .. ..... ........ ..... 1
V. ADDITIONS TO PDECOL .. .. ..... ........ .......1
VI. BURNER STABILIZED FLAMES .. .. ..... ........ ... 25
VII. NUMERICAL CONSIDERATIONS .. .. ..... ........ ... 31
REFERENCES .. .. ..... ......... ..........34
APPENDIXA .. .. ..... ........ ...........37
GLOSSARY.. .. .. .............. ..... 13
Aoefsianf For
OTTO TAB
7-A lablity Codes
3Dist special
Mum""
I. INTRODUCTION
We are interested in determining validated seti of elementarychemical reactions for use in predictive combustion models. For thispurpose, a simulation of the laminar, premixed, one dimensional, steadystate flame has been implemented. This has the advantage of being arelatively simple combustion problem that also yields predicted tempera-ture and species profiles that can be compared with suitable burnerexperiments.
The basic idealism is of an infinite column of premixed gas. Atsome point the gas is ignited. A flame front forms and propagates alongthe column of gas. All effects are assumed to be one-dimensional.Eventually, the flame will reach steady state. A basic characteristicof the solution is the flame speed, the velocity with which the flamefront moves with reference to the undisturbed gas mixture. The flamefront is marked by steep temperature and chemical species profiles. The
*above formulation is referred to as an unbounded, or adiabatic flame.In practice, a flame is usually stabilized on a burner. That is,
the flame will propagate to the burner surface. Loss of heat to the
burner will stabilize the flame. This creates different boundaryconditions.
This report describes a method of numerically solving the equationsgoverning a laminar, premixed, one-dimensional, steady state flame.Both unbounded flames and burner stabilized flames are discussed. To
4solve such systems, we have modified a standard package for integratingone-dimensional partial differential equations so that it efficientlyhandles flame equations. The basic procedure is to integrate theequations in time until the steady state solution is reached. Changeshave been made in the basic structure of the code, as well as adding anumber of subroutines. The program has been implemented for the ozoneflame and the H2-02-N2 flame.
Section II describes the partial differential equations governingthe flames. In Section III, the basic code used to solve the equationsis introduced. Section IV describes the modifications made in the code,and Section V the additional subroutines added to the program. SectionVI discusses the changes in the equations and the code necessary tomodel burner stabilized flames. Section VII reports some of the numericalconsiderations in successfully using the code. Finally, the Appendixgives a complete listing of the code, plus the output for a run modelinga typical H2-02-N2 flame.
'= X
>i! .--
II. THE UNBOUNDED FLAME EQUATIONS
The equations for a multicomponent reacting ideal gas mixture can*i be found in the literature. 1 4 We are interested in the equations that
describe a one-dimensional, laminar, premixed flame that propagates inan unbounded medium. The effects of radiation, viscosity and body forcesare ignored. The momentum equation
at 3x ax
can be eliminated. Here v is the fluid velocity, p is the density, andp is the pressure. The independent variables are time t (sec) and dis-tance x (cm). For flamesA the fluid velocity is much less than thespeed of sound, that is, v2 <<p/p. Forsteadystate-flow, this implies
that the pressure gradient is negligibly small. So the pressure isassumed to be constant.
The pertinent equations are then, overall continuity.
ap a(pv)()
at ax
That is, any change in density with respect to time is due to the overall
r convection of the mixture, which equals the gradient of the total mass
flux Pv.
Continuity of species:ay ak a- AA k a 3
p -. p= - - -. (PYx( k Vk) + RkMk' k 1 1,2,...N (2)at ax ax
1F. A. WilZians, Combustion Theory Addison-WesZey, Reading, MA, Chapter1. 1965.
2R. B. Bird, W. E. Stewart and E. N. Lightfoot, Transport PhenomenaJohn Wiley and Sons, NY, Chapter 18, 1960.
3 R. M. Fristram and A. A. Westenberg, Flame Structure, McGrca-HiZ,NY,Chapter V-i, 1965
4 J.0. Hirshfelder, C. F. Curtiss and R. B. Bird, 1964.]hooGae an ~9 Liquid, 2nd Printing corrected, with noeJh ie nSons, NY, hapter 11.1, 1964.
6
".."
where Y is the mass fraction of the kth species, Vk is its diffusionvelocityM k is its molecular weight, and is the rate at which the
species is produced or consume by the chefical reactions. So changesin the concentration of the ktf species can be due to the convection ofthe species, the diffusion of the species in the mixture, 9 ;he product-ion or consumption of the species by chemistry. The term PYkVk is themass flux of ttg kth species (relative tO v). Note that the total massflux for the k species is P(v + V) Yk"
Conservation of Energy.
A A T BA T a 3 AT Mh- +--A - Rk~k
St ax ax ax kal
C YV aT_! P k= I- pk ax (3)
A A thwhere T is the temperature, Ck is the specific heat of the k species,c is the specific heat of the mixture, A is the thermal conductivity of
the mixture, and hk is the specific enthalpy of the kth species. Sochanges in the temperature can be due to the convection of heat, theconduction of heat, the production or consumption of energy by thechemical reactions, and a small amount due to the diffusion of specieswith different specific heats. We have not written a negligibly smallterm due to the species gradients (Dufour effect).
The thermal equation of state is given by the ideal gas law
p -pR E (4)k=l
where R is the gas constant. The caloric equation of state is
kuh + cpkdT" (5)T 0
where h 0 is the specific enthalpy of the kth species at some reference
temperature T The specific heat of the mixture is given by0.
7
. - - - - - . .. . ' . .. . . . .:.. . . . . ..
N *
p k-i Cpk k. (6)
From conservation of mass we have the relation
Nkl k =I (7)
and
Nkil ikVk O. (8)
The boundary conditions are the following. For x - - -.
T a TU and = akU' (kal,2,..N),
(9)
where Tu is the temperature and the YkU are the mass fractions of the
original, undisturbed mixture. For x
T TB and Yk - YkB' k-l,2,...N . (10)
where TB is the temperature and YkB are the mass fractions of the burned
mixture. T is called the adiabatic temperature. Since we are assumingno heat lost to the surrgundings, TB and YkB depend only on the chemistry,and can be calculated in advance.
In practice, the integration will be over a finite interval
(XL, xR), where the flame front will be located roughly in the center of
the interval. The boundary conditions (9) will be applied to xHowever, the conditions (10) are not convenient. After the flake front,there is a long recombination period before the mixture reaches adiabaticconditions. Choosing the interval of integration long enough to coverthe entire recombination zone would be computationally expensive. Sonormally the flame will be cut off before it reaches adiabatic conditions,
.8
- --.------- ________________ . ,. ----------D---
anc the weaker boundary conditions
3T a' kk l2- , k-,2,...N
ax ax (II)
will be applied at xR.
At this stage the partial differential Eq. (l for the density pcan be eliminated by introducing a new coordinate such that
f. p(x t)dx. 12(x,t)= *x~ A(X ,A:dx (12)
L
Then and - = - pv * m (t), where a (t) = PVlo. With thisax at 0 0 x=o
notation Eqs. (2) and (3) become
aYk a aYk a v , j (13)" -0 - (PYkVk) + Rk Mk/p
and
at _o 4~ a(~ ;
N N. Mkhk/'P - cpk k Ek c (14)
k-l k-l
where the tilde variables are functions of t = t (sec) and (gm/cm 2
For numerical convenience, dimensionless forms of Eqs. (13) and (14)
are integrated. That is, we will define T - /T0 , t = T/t. and J =where t. and *. are chosen so as to obtain reasonable time and spacescales for a given flame, and T.. is chosen so that T is the same orderof magnitude as the larger Yk'S . Then the equations are
9
Y k ay Yk t a" (PYk Vk) t- Rk Mk/P ,(15)
and
aT t o MT t 1 a (P
at i~ . p (16)
N NRk Mk hk/(p T.) - CP Y " }V
k=l k=l
where the variables are functions of t and , The boundary conditionsare i
T = TU , k = k u k = 1,2...N (17)
at = 1 h and
aT ak .
TT = a = 0 , k =1,2...N (18)
at R'
The integration can start from any initial profiles, as long asthere is enough energy available to begin the combustion. As shown inSection V, the mass flux mo through the origin is modified as theintegration proceeds so as to keep the flame front in the center of theinterval of integration (*L, *R
). The integration proceeds until mo
approaches a constant, and the terms aYK BT approach zero. Theat and t
equation for species N is not integrated; N is found from the relation
(7).
The burning velocity, S , can be calculated at steady state. Fromthe continuity equation a(pvy /a8=O, or pv is constant with respect to4. Then we can take any of the equations (15) and integrate over anyinterval (a,b) to obtain
10
2 I
N.
pvk [Yk(b) . Yk(a)] ,f b pRkd PYk Vk ab (19)a
Then
vk(.C) = R Yab'- R kdIS -p YkVkIa b
P (C=) [Yk(b) Yk (a)] (20)
At steady state, all of the vk are equal. Since in the present coordinatesystem the flame does not move, v (--) is the speed at which theunburned gas is approaching the fAame. Conversely, S. = vk (-=) is thespeed at which the flame is propagating into the unburned mixture.
,! The parameters of the equations (15) and (16) are the specificheats cpk, the specific enthalpies hk, the chemistry production terms
Rk, and the diffusion velocities Vk for each species, plus the thermal
conductivity A of the mixture. The Cpk and hk are functions only of the
temperature, and can be evaluated very accurately using sixth degreepolynomial fits5 . The chemistry is generally the least well known ofthe input data. We need to know which species are involved, whichreactions can occur, and the rate constant for each reaction. The rate
constants k. will be of the form a b exp (c/ ), where either b or ccan be zero . We can then find the rate r. for each reaction by multiply-ing the rate constant times the concentrations of the reactants (con-centrations of the kth species = P Yk/ Mk). Each Rk is then found by
adding the rates of the reactions in which the kth species is a reactant.The transport parameters X and Vk are in general very complicated func-tions of temperature and species concentrations. Because of thenumerical complexity, various approximations to these quantities havebeen used. We have the capability to run the code with a number ofdifferent levgls of approximation. This is discussed in detail inanother paper
5S. Gordon and B. J. McBride, "Computer Program for Calculation ofComplex Chemical Equilibrium Compositions, Rocket Performance,Incident and Reflected Shocks, and Chapman-Jouguet Detonations",NASA-SP-273, 1971, (1976 program version).
6T. P. Coffee and J. M. Heimerl, "Transport Algoritke for Premixed,Laminar, Steady-State Flames", to be published in Combustion andFlame.
11
For future reference, we will discuss here the simplest transportalgorithm used in the code. First assume that Ficks law,
Yk Vk Dkm a" (21)
is valid. This is only strictly true for a binary mixture where thermaldiffusion is negligible. Dk m is a multicomponent diffusion coefficient.Equation (21) can be written as
P Y2P k m a k .(22)o k Vk = - -*a a* 22
As a first approximation, we can take p2 Dk m and p X to be constant.
Also, we can assume that c cp k' k=l,2,...N. Here c must be chosen
so as to obtain the proper adiabatic temperature T The procedure forchoosing these constants and the justification of his approximation isgiven in reference 6. This level of approximation is reasonably accurate,if the constants are properly chosen.
Equations (15) and (16) now become
k. aY k 2 Y-k- m - + p Dkm tP (23)
at - - , 0 alp km 2 kk
and
_ t6 mo aT + cp {2T N. RkMk hk 1.(24)a p *92 3a*2 kl P
Equation (5) now simplifies to
h = hk0 + Cp(T - To), (25)
so the required input parameters for this level of approximation are theconstants P2Dk m' p X, Cp, hk0 and the functions Rk(T , . YN) "
12
III. THE NUMERICAL METHOD - PDECOL
The package PDECOL, developed by Madsen and Sincovec7 , was used tosolve the equations. This package is designed to solve a general systemof N nonlinear partial differential equations of at most second order ona finite interval. In our coordinate system the appropriate form is
4. 4
a4. -). 4.
a-t f (t ,V u,u , u), (26)
%here
u= (Y1 , "' Y YN-1' T)(27)
8Y1 aYN-I aT
a Y1 N TU 2 -".2 234, 34'34
Fairly general boundary conditions of the form
t ) = , (t) (28)
are allowed, where and I are arbitrary vector valued functions withN components. Each solution component is assumed to be a known functionof * at the initial time t = to. That is, Yk(to, *), k=l,2,...N-l, and
T (t, *) are known functions. The initial conditions must be consistentwithothe boundary conditions.
The spatial discretization is accomplished by finite elementcollocation methods based on B-splines 8. The user must supply a set of
7N. K. Madsen and R. F. Sincovec, "PDFXOL: GeneraZ Collocation Softwarefor Partial Differential Equationa", Preprint UCRL-78263 (Rev 1),Lawrence Livermore Laboratory, (197?).
8C. de Boor, "Package for Calculating with B-Splinee'" Siam, J. Nwuer.
Anal. 14 441-472, (1977).
13
.- . - - --. - ........
NB breakpoints, that is, a set of strictly increasing locations wherethe polynomials are joined. He must also supply the order KORD of thesplines and the number of continuity conditions, NCC, to be applied atthe breakpoints. That is, if NCC=I, the approximating function iscontinuous; if NCC=2, it is continuous and smooth; and so on. PDECOLthen generates a set of NC = KORD(NB-l) - NCC(NB-2) basis functions andcollocation points. The basis functions B.(4) are piecewise polynomials
of order KORD - 1. The basic assumption is that the solution can bewritten in the form
uk ck i)(t) Bi(4) , k=l,...N , (29)i=l1
where the basis functions B iJ) span the solution space for any fixed t
to within a small error tolerance. The time dependent coefficientsck i) are determined uniquely by requiring that the expansions abovesatisfy the given boundary conditions and that they satisfy the partialdifferential equations exactly at the (N-2) interior (collocation)points. Since by definition a B-spline is zero except over a smallinterval, at any collocation point no more than KORD of the B-s linesare non-zero. So the system of ODE's for the coefficients ck(1M willnot be fully coupled.
The boundary conditions (28) must also be changed into ordinarydifferential equations. This can be done by taking the derivative withrespect to time, which results in
N bk dZkk + k tu, =d- , k-l,2...N. (30)
j=l a
When equation (29) is substituted into equation (30), a set of ODE's inthe ck( i) results.
Unlike a finite difference code, the program can be run with noboundary condition at either the left or the right boundary. The programsimply collocates at the boundary, using the same procedure as for theinterior points. It must be, of course, the user's responsibility todefine a mathematically meaningful PDE problem.
14
'
This system of ODE's is integrated in time, using a variant of theGear stiff integrator9 . This is a fully implicit, predictor-corrector
4 method. The required banded Jacobian is generated internally by theprogram. Once the integrator has reached a desired output time, thevalues of Yk and T can be obtained for any * by substituting into the
expansions (29). The accuracy of the time integration is determined bya user supplied error tolerance e.
For the actual integration, the code requires the values of thebasis functions and the first two space derivatives only at thecollocation points. Since computing these values is complicated, thisis done once at the beginning of the program and the values are saved.
In general, a user must supply a main program and three subroutines.The program MAIN sets the values of the required parameters, calls theintegrator, and writes any desired output. The subroutine UINIT gives
~the initial conditions.4 Given a collocation point *, the subroutineust return the vector u (to0 , ). The subroutine F evaluates the functionf given by Eq. (26). Given t, P, u, u., and u , the routine must return
the vector au/8t. The program automatically converts this to thecorresponding set of ODE's involving the c(i) The subroutine BNDRY
specifies the boundary conditions. That is, given t,ui, u, and uthe subroutine returns the quantities abk/auJ, 3bk/a j aZk/it
j,k = 1,2... N, for both *L and *R"
IV. MODIFICATIONS OF PDECOL
The time integration is controlled by a user supplied error
tolerance e. Single step error estimates divided by CHAXk (i) will be
kept less than c in the root-mean-square norm. In PDECOL, CMAXk
is initially set to the maximum of Ic (i) and 1.0. Thereafter,
kAX(k is the largest value of ck (i seen so far, or the initial
CMAXk if that is larger. However, this error criterion does notproduce the desired accuracy for flame simulations, because radicalspecies with small concentrations will control the flame, and must becomputed accurately. But mass fractions much smaller than 1.0 will notbe computed accurately using the original PDECOL criterion. An alternatecriterion is the purely relative error criterion, that is, CMAXk (i)
Ick I.. This criterion is also unacceptable because some species
v j at some locations will approach zero, and excess computation will result
9A. C. Hindmareh, "Preliminary Docwentation of GEARIB: Solution ofImplicit Syetema of Ordinary Differential Equatione with BandedJacobian", Rep. UCID-3 030, Lawrence Livermore Laboratory, (1976).
15
in accurately computing these negligible concei) rations. Consequentlya semi-relative error control is used. CM.AXk( is chosen as the
maximum of Ck'iM and a user supplied parameter SREC. Thus, mass fractions
less than SREC will be computed less accurately. We have normally usedSREC=lO -0
The original program PDECOL is fully implicit. That is, it generatesa set of NO = N X NC ordinary differential equations of the form
A c g (31)dt g t
where c= (c(1)...CN(1), c1 (2) ...cN ... ). The resulting Jacobian
ag/a is a banded NO by NO matrix. The band width is 3 X ML+l, whereML = N (KORD - 1) - 1. To advance a time step, the program firstcomputes an explicit predictor for the values of the ck i) at the nexttime step. Then it solves a set of linear algebraic equations involvingthe Jacobian to correct the values. For a stiff system, such as oneinvolving chemistry, this allows time steps orders of magnitude largerthan those of an explicit method9 . The drawback is the storage andexecution time required to work with the Jacobian.
For example, consider a system where KORD = 4, N = 10, and NC a 20.The bandwidth of the Jacobian is 88, and we have a system of 300 ODE's.
We then have 26,400 possible non-zero elements in the banded matrix.The amount of storage required also increases rapidly. For instance,if N = 20 instead of 10 the bandwidth will be 178, and we will require106,800 words to store the Jacobian. Since we want to be able to solvesystems at least this large, the storage requirements become almostprohibitive. In addition, solving such large systems of linear algebraicequations is very time consuming.
To avoid this problem, we essentially uncouple the partial differ-ential equations and solve them successively. The basic procedure isillustrated below. Suppose we are at time tn and we want to advancea time step to tn+l* We first integrate the equation for YI, under the
assumption that Y2,"" YN-1' T are constant at their tn values. This
uncouples the first PDE from the system. Subsequently, we integrate thesecond PDE for Y2, using the new value of Y at tn+l, and the old values
for the other variables. Continuing this process, we finally solve for*. T(tn+l), using updated values for all the mass fractions.
16
In general, this method is restricted to smaller step sizes than afully implicit method. However, as steady state is approached, all thevariables approach constants with respect to time, and the temporalcoupling vanishes.
Now consider how this assumption affects the associated system (31)of ODE's. In the Jacobian, we will have
Mgi)/ac (j) = 0 if k 0 A.
That is, changes in the th PDE will not affect the kth PDE if k 0 t.So' most of the terms in the Jacobian will become zero.
The remaining nonzero elements are not in-banded frm. However, we
can accomplish this by rearranging the vector c = (C1 J3, c1(2),
c(NC) , c ) c2C)...). That is, the coefficients for each PDE
are grouped together instead of the coefficients for each collocationpoint.
The Jacobian matrix can now be decomposed into N smaller NC X NCmatrices on the main diagonal. Moreover, the band width of thesematrices is only 3XML+l, where ML - KORD - 2. The Jacobian is essentiallyN separate Jacobians, each for one PDE with NC collocation points.
The savings in storage space is dramatic. For our previous example
of KORD - 4, N = 10, and KC = 30, we have 2100 nonzero elements insteadof 26,400. If N = 20, we have only 4200 nonzero elements instead of106,800.
The following procedure is used to actually integrate a time step.The.predictor of the pre4itor-corrector method is ied to obtain firstestimates for all the ck kJ. Then the corrected cl are computed,
using the first small Jacobian and assuming that the other c k() do notchange. Then the corrected values for c1(i) and the predictedchanepredctedvalues
for ckCi) k > 2, are used, and the corrected c2 (') are computed. This
process continues through the N PDE's. The estimated error is calculated.If necessary, the above procedure is iterated. It is more efficient touse the predicted values rather than the values at the previous timestep. In using the procedure, it is more efficient to integrate theminor species first, since they change most rapidly, then the majorspecies, and finally the temperature.
The above procedure is similar to one developed by Spalding andStephenson for use in a finite difference code
10 D. B. Spalding and P.L. Stephenon, "Laminar F e Pro pagat ion in idogen+ Bromine Mfitaeet, Proo. R. Soo. Lond. A. 3,24, 315-337 (19?1).
17
PDECOL was rewritten to use the successive calculation method. Thisalso means that we must change the user supplied routines F and BNDRY.
Because of the rearrangement of the vector c, the core integrater willcall F first at each collocation point, asking for the values of 3Y1 /3t.It will then repeat for each PDE. The routine F is written so as toreturn only the desired time derivative.
Since the problem can no longer directly handle coupling terms,the boundary conditions (28) must be uncoupled, that is, the boundaryconditions must be of the form
bk(uk' uko,) = Zk(t) (32)
The rewritten subroutine BNDRY evaluates the quantities abk/auk, abk/
SUk*, a Zk/at, k = 1,2.. .N.
Comparisons of the execution time of the fully implicit method versusthe successive calculation method have not been made. However, our mainpurpose was not to reduce the execution time but to reduce the storagerequirements. This iterative procedure accomplishes this goal andsimultaneously gives accurate results in reasonable run times.
V. ADDITIONS TO PDECOL
The computer program PDECOL requires a set of user supplied subroutines.We have written a set of subroutines that casts the equations into acomputationally efficient form and which generates the required output.These routines are the first seven listed in the appendix; namely MAIN,F, UINIT, FLSP, BNDRY, BKPT, and RT. In addition, several auxiliaryprograms are mentioned, but without providing a listing. These routinesgenerate sets of input data or actual subroutines that are used frequently,or analyze the output of the flame code.
The code can be run with several different options. We first describethe case of an unbounded flame (NBURN = 0) using the simplest constanttransport algorithm (NTRAN = 1). There is no information available aboutthe flame speed or the solution profiles (NSTART = 1). Later in thissection we describe restarting the integration (NSTART = 2), and usinga more complicated transport algorithm (NTRAN - 2). Burner stabilizedflames (NBURN = 1) are discussed in the next section.
Initially, a chemistry scheme must be chosen. In the appendix an
H2 - 02 - N2 system is used, with nine chemical species (H, OH, 0, HO2,
H202 , H2, 020 H20, N2) and a set of thirty reactions involving these
species, each with a rate constant of the form a Tb exp (c/T) (either bor c can be zero). This information is used as input to an auxiliary
18
code. This code writes the subroutine RT that computes the chemistryterms Mk/ ?./. The procedure is analogous to that tised to write thetranspo tubroutines6 . This subroutine is attached to the flame codein the job stream and can be used for any problem involving this set ofkinetics.
The actual subroutine RT has three main parts. The rate constantsare evaluated for the current temperature and stored in the vector RK.Since the code uses a successive calculation method, the subroutine willbe called N times at each time step and each collocation point. Therate constants are only recomputed if the temperature has been changed.
* Otherwise this section is skipped. Each rate constant is multipliedby the concentrations of the appropriate reactants divided by p toobtain r./p. The terms Rk Mk/p (stored in the vector R) are calculated
by adding the rates for the reactions in which Yk is a product,* subtracting the rates for which Yk is a reactant, and multiplying by Mk.
The choice of a transport algorithm determines the form of thesubroutine F. The version given in the appendix is for constanttransport. Common statements are used to make the appropriate constantsavailable for all subroutines. The chemistry terms required are foundby calling RT.
The initial temperature Tu and mass fractions YkU of the unburnedgas are input quantities. Anotner auxiliary code determines theadiabatic temperatures TB and mass fractions YkB' and the constants
p2D kmfhk, pX, and cp needed by the constant transport algorithm (see
reference 6). This information is saved on a data file and attached toTAPE 11 when the code is run.
The remaining data necessary to run the program is read in on cards(TAPE 5). The pressure p and the normalizing constants t* , *0, and Tare specified. The optioi parameters NSTART, NTRAN, and NBURN are c~osen.The numerical parameters required by the code are specified; that is,
*L and *R1 the final integration time tFINAL' the time integration error
control parameters e and SREC, and a set of breakpoints.
The major difficulty in efficiently solving the flame equations is
choosing an appropriate set of breakpoints. These must be close enoughthat spatial errors are minimized and yet not so dense that one'scomputer resources are exceeded. The breakpoints should be densest inthe flame front, where the gradients are very steep.
The technique in the code is to use a static mesh, with the break-points most closely spaced near the center of the interval of integration.The flame front is then forced to remain near the center of the interval.This is done by adjusting o , the mass flux through the origin. Atsteady state, the mass flux through the flame is a constant. So mo is
19
. ... . ...- - .. . .. ,-- _ , -_ - .. . _' .. .- ...
iteratively modified to match the steady state mass flow through theflame. This leads to a coordinate system in which the flame front isat rest. The transient behavior can cause the flame 'front to drift awayfrom the center. This can also be corrected by modifying mo .
This method requires a procedure for tracking the flame front. Todo this the position of a specific temperature Tcn is monitored. Thecode attempts to keep this temperature located at the center of theinterval of integration. As a heuristic rule this temperature is definedby
Tcn= TU + C.4 (TB - Tu) (33)
The average of TU and TB is not used. This is because the mixture willnot normally reach the adiabatic temperature at the end of the flame front.Rather, there is a radical overshoot, and the recombination of theseradicals will very slowly raise the temperature. Tcn as defined willusually be close to the center of the flame front. The details of thisiterative procedure are discussed in reference 11.
So the breakpoints may be chosen to be densest in the center of theinterval of integration. However, it is tedious to have to choose anentire breakpoint sequence for each problem. We have developed a procedureto generate an appropriate type of breakpoint sequence from a smallnumber of parameters. By varying three parameters, a wide variety ofbreakpoint sequences can be generated.
The user must supply NINT, the number of intervals (NB - NINT+l),NCN, the number of intervals of equal length that will be at the center
of the interval, and FC, the ratio between the longest intervals (on theboundaries) and the shortest intervals. Also let L be the total lengthof the interval of integration, that is, L = VR - *L- The programgenerates a set of intervals whose lengths increase by a constant factorz, where
a = log -1 [2(log FC)/(NINT-NCN)] . (34)
The common length LC of the NCN center shortest intervals is
LC = L/[NCN + 2 m(a (NINT-NCN)/21 )/(1-)]. (35)
The procedure can best be seen by example. Suppose we have *L = 0,
T1.p. Coffee and J.M. BeimerZ, "A Method for Computing the FZwe Speed
of a Laminar, Premixed, One Dimeneaonal Fiwne", BRL TechnicaZ ReportARBRL-TR-02212, January 1980.
20
q i i
, = 10, NINT = 12, NCN = 4, and FC = 6. Then L = 10, a = 1.5651, and= .3155. The resulting breakpoint sequence is giyen in Table 1. Note
that the 4 center intervals are of the same length, the length of theintervals then increases by a factor of a, and the two intervals by theboundaries are 6 times as long as the central intervals. So the procedureautomatically generates a set of breakpoints that are closest togethernear the center, with the spacing increasing smoothly toward theboundaries.
Some experimentation is necessary to choose the proper value of theabove parameters. However, choosing NINT = 12, NCN = 4, and FC between4 and 8 has worked in m3st of the cases we have tried. Normally weexperiment using the constant transport algorithm, and then use a morerealistic transport subroutine once we have a good breakpoint sequence.
TABLE 1. THE SET OF BREAKPOINTS GENERATED BY L = 10, NINT = 12,NCN = 4 and FC = 6.
Breakpoints Interval Lengths
0.01.8929
1.89291.2094
3.1024.7728
3.8752.4937
4.3689.3155
4.6844.3155
5.0000.3155
5.3155.3155
5.6310.4937
6.1247.7728
6.89751.2094
8.10691.8929
10.0000
21
The breakpoint sequence is written by the subroutine BKPT. It alsogenerates a larger set of evaluation points by interpplating between thebreakpoints. Using Eq. (29), the subroutine VALUES can evaluate the Yand T at this larger set of points. This information is useful in
generating detailed output, such as graphs, or in performing numericalintegrations (see below).
The subroutine UINIT writes the initial profiles, where
*1 = *L + 0.24 (R - L
(36)
2 = *L + 0.64 R -
and
Y kU" *L *1 @.
Yk(to '0) = YkU + CYkB - YkU) sin 3] (37)
VkB '2 * - *R.
The temperature T is defined similarly.
This particular definition will give us T =Tcn at *cn = 0.5(*L+*R).The choice of the particular function that defines the Yk and T between*1 and *2 is not important. We have used a straight line with success,but defining a smooth function is sl',.ghtly more efficient.
A requirement of PDECOL is that the initial profiles satisfy theboundary conditions, in our case given by Eqs. (17) and (18). The aboveinitial profiles have the proper unburned values at 1L and are constant(space derivative zero) near
*R"
For this case the subroutine BNDRY has a simple form. At *L, thesubroutine returns the values abk/auk = 1, abk/auko = 0, and aZk /at - 0.
At * R' the conditions are Dbk/au. 0, abk/au k* - 1, and aZk/at - 0.
22
* . 1- ! 7
To begin the integration, a starting value for m0 , the mass fluxthrough the origin is required. To do this, the code ignores the timedependent terms, assumes that the mass flux pv is constant, and uses
4Eq. (20) to obtain a value of pv. This gives a reasonable starting valuefor m .
The evaluation of Eq. (20) is carried out in the subroutine FLSP.The integral is approximated using the trapezoidal rule, where theintegrand Rk Mk/p is calculated at the evaluation points. The flame
speed vk(--) is evaluated for each species k and for several intervals
(a,b), where a =L The initial value for m0 is based on species N-1
integrated over the entire interval (* L' *R) "
The integration is performed over several intervals (a,b) becauseof the behavior of the minor species. These are normally close to zeroat L1 reach a peak in the flame front, and are close to zero again at
*R" For these species, integrating from the left boundary to the flame
front is much more accurate.
The chemistry terms in Eq. (20) are found by calling RT. Thediffusion terms for this case are found using Picks law, Eq. (22). Sincewe are assuming that p2Dkm is constant, this value is stored in the
vector R2D. The subroutine FLSP also computes the x values from therelation
x (t, *) = 0 J p dip, (38)
using the trapezoidal rule, and computes an estimate of the flamethickness.
As the time integration proceeds, control returns to MAIN at aseries of output times. The present spatial location of the temperatureTcn is found, and this is used to find the average value of the mass flux
since the last output time. The function MO(t) is redefined at these
times so as to keep the flame front in roughly the same position. Theoutput times are chosen by the program so that the flame front will notdrift too far between evaluations. Also FLSP is called so the user cansee if the flame speeds computed from the different species profiles areapproaching a common value.
At the final time tFINAL' FLSP also writes an output file. It
consists of all the evaluation points *i, the corresponding xi, the
23
values of Yk and T, plus their first and second derivatives with respectto *. This file can be attached to an output routine. By also attachingF and RT, we can compute and print out any quantity in the steady statesolution in which we are interested. This file can also be attached toa graphics routine.
Similarly, the program MAIN writes a restart file. This consists
of the present location of Tcn, the present value of mo, and the values
of Yk and T at the collocation points. This file can be used to restart
the time integration. It is attached to TAPEl, and the parameter NSTARTis set equal to 2. The input parameters read in on cards can now bechanged if desired. UINIT will translate these input values to centerthe flame front, and will use interpolation to find the appropriatevalues of the starting profiles at the new collocation points. The oldvalue of m0 is used to start the integration.
So far only the constant transport case (NTRAN = 1) has beenconsidered. For more realistic algorithms F is written by an auxiliarycode and attached to the flame code NTRAN) = 2). Several differentlevels of approximation can be used.
6
In these more complicated algorithms, the diffusion velocities Vkare coupled, and they must be computed simultaneously. But since thecode uses successive calculation, only the value of one Vk is required
on each call to F. Because the computation is time-consuming, it ispreferable not to compute the VV N times at each time step for eachcollocation point. To economize computer time, all the thermodynamicand transport quantities required at all the collocation points fork = 1 are computed, and stored in vectors. For k > 1, we use the samevalues, even though some of the Yk terms have changed slightly. Onlythe chemistry terms are reevaluated. Since the chemistry normallychanges much more rapidly than the transport, this will still be a goodapproximation.
In generating an approximation to the Jacobian (using finitedifferences) it is necessary to recompute the transport each time F iscalled, since what is of interest is the effect of changes in Yk and T on
the time derivatives. Ignoring the changes in transport leads to aninaccurate Jacobian. However, the rate constants k. and the thermodynamicJquantities cp and h k need be recomputed only if T is changed, since they
only depend on temperature.
As will be seen in the next section it will still be useful toformulate the mass flux in a Ficks law form Eq. (22). For the morecomplicated transport subroutines, we define
p2 = Ykk (39)
24
This new quantity P Dkm is no longer constant with respect to space or
time.
VI. BURNER STABILIZED FLAIES
So far we have only discussed flames that propagate in an unboundedmedium. In actual experiments, the flame will usually be stabilized bya burner. It is useful to be able to model this type of experiment sothat we can compare experimental and calculated profiles.
Our basic idealization is of a cylindrical, porous plug burner.The premixed gas exits the plug with a constant velocity v, but the plugprevents back diffusion of the products into the burner. When the gasis ignited, a flame front develops and propagates toward the burner.The gas velocity v at the burner must be less than the flame velocity,or the flame will be blown off the burner. In the model this will looklike an unbounded flame. In an experiment the flame will be extinguishedby the surrounding atmosphere. As the flame approaches the burner, theburner surface acts as a heat sink for the flame. The loss of heat slowsdown the flame velocity, until the flame stabilizes near the surface ofthe burner at the gas velocity. As the gas velocity v is decreased, theflame loses more heat to the burner, and stabilizes closer to the burnersurface. If v is made too small in an experiment, the flame can flashback into the burner.
For this kind of burner, air is entrained along the outside edgesof the flame. However, the center of the flame will correspond closelyto a premixed, laminar, one-dimensional flame.
The only change in the equations is in the left boundary conditionat the surface of the burner. Because of back diffusion from the flameto the burner surface, the mass fractions of the unburned mixture arenot conserved. However, due to conservation of mass, the mass fluxfractions, defined as
Ykv + 'Yk Vk Y + + Y k Vk (40)
k v k P v
are conserved. Within the burner, the diffusion velocities are zero,and ekU = YkU" So the appropriate boundary conditions at the burner
surface are
C k : kU (41)
The boundary conditions for the temperature equation depends on how heat
25
t .......... _
is extracted. We assume that the burner is maintained at a constanttemperature. Then the boundary condition is the same as for an unboundedflame,
T = TU. (42)
12This idealization is discussed by Hirshfelder, Curtiss and Bird
To implement this in the flame code, the input parameter NBURNis set equal to 1. The fluid velocity at the burner must be specified.Then m is a predetermined constant instead of an adjustable parameter.All thg other input data remains the same. However, the code will handlethis data differently.
The breakpoints are generated differently, since the flame frontwill be near the left boundary instead of in the center of the intervalof integration. NCN is the number of intervals of equal length at theleft boundary. FC is the ratio between the longest interval (at theright boundary) and the shortest interval (at the left boundary). Theconstant a is now defined by
a = log - [(log FC)/(NINT-NCN)]. (43)
The common length LC of the NCN shortest intervals is now
LC = L/[NCN + a(a(NINT-NCN)-I)/(,-I)]. (44)
As an example, consider the same input parameters that we used foran unbounded flame; *L = 0, -,R = 10, NINT = 12, NCN =4, and FC = 6.
Then L = 10, a = 1.2510 and LC = .4181. The resulting breakpoint sequenceis given in Table 2. The shortest intervals are near the burner surface,where fairly steep gradients are expected.
In general, it is possible to choose a shorter interval of integrationL for a burner stabilized flame. There is no longer a need for arelatively long interval to the left of the flame front in order toapproach the unburned conditions. The number of breakpoints can thenalso be reduced.
The initial profiles are also chosen differently. The code sets
12J. 0. Hirshfelder, C.F. Curtiss, and R.B. Bird, op. cit., pp. 761-763.
26
..- =o
TABLE 2. THE SET OF BREAKPOINTS GENERATED BY L-1O, NINT - 12,
NCN = 4 and FC = 6 (NBURN = 11
Breakpoints Interval Lengths
0.3458
.3458.3458
.6916.3458
1.0374.3458
1.3832.4326
1.8159.5412
2.3571.6771
3.0342.8471
3.88121.0597
4. 94091.3257
6.26661.6585
7.92512.0749
10.0000
*1 -*L + 0.1 N - *L )
(4S)
2= 'L + 0.3 { R -L )
and then uses Eq. (37) as before. This flame front is more likely to beclose to the position of the final steady state stabilized flame. Also,the flame front is chosen to be narrower. If Eq. (36) is used, theinitial transient flame velocity may be less than the velocity of thefluid. The flame then drifts toward the right, and can be carriedcompletely outside the interval of integration before it stabilizesChoosing a steeper flame front leads to a larger initial flame velocity.
The boundary condition, Eq. (41), can cause some difficulty. It isfairly straightforward for the constant transport case. Using Fick'slaw, Eq. (22), it can be rewritten as
27
i"d
2 -Yk p m2 km--k Y . (46)
Recall that PDECOL converts this to a time dependent equation by takingthe time derivative. In the form used by PDECOL, Eq. (46) becomes
ar a k p D k m B ayk (47)at *. mo 0 t T I *L 0.(
So the subroutine BNDRY must now return the values 3bk/aUk = I2abk/auk - p Dkm/*. io, and aZk/@t = 0 at pL for k = 1,2... N - 1.
The mass flux fractions will then have the proper values as theintegration proceeds.
For more realistic transport algorithms the mass flux p Yk Vk is a
complicated 2function of all the mass fractions. But by using Eq. (39)to define P Dk , the boundary condition can be put in the same form as
Wq. (46), except that p 2Dk is not a constant. Taking the time derivativewe obtain
ak P Dkm a Yk ID) 2 0. (8
0*m ° t Dkm L
The time derivative of p2 Dkm can not be evaluated. It is not
possible to write it as an explicit function of the Yk and 3Yk/a*.
Moreover, since the code uses successive calculation, cross couplingterms between the different species are not allowed. By necessity, thelast term in Eq. (48) must be ignored.
It is possible to just use the expression
2rk P Dkm a Yk0 (49).- o -- ) I = 0
L
as the boundary condition. At steady state, the mass flux fractions2will approach constants. But because of the transient behavior of P Dkm,
28
the mass flux fractions will converge to incorrect values.
What is required is a correction term that will' cause the code toconverge to the proper steady state limit. Moreover, it must be of aform that can be used in the code. This can be done by choosing theboundary condition
2
aYk p Dkm a ak aZ (50)
at - t- (a*) L
where
az Y kU - (51)at 0.1 tFINA L
Therationale is that p Dkm changes rather slowly. As it changes themass flux fraction at * L will change slowly from the proper value. We
use a heuristically defined function az/at to modify the value of kuntil it again equals Y So as the time integration proceeds, thevalue of £k at * L will Vary slightly, but will approach the proper
steady state limit.
It is necessary to check that the boundary conditions are stillconsistent with the initial conditions. For the initial profiles definedby Eq. (37) this will be the case. The mass fractions Yk are constant
near *L' so aYk/a4 at *L will be zero. Also aT/a3 at * L is zero. For
any transport algorithm all the Vk will be zero at L' and Ek U Yk" So
the initial profile is consistent with the boundary conditions.
When the integration is restarted (NSTART = 2), the space derivativesat *L are normally non-zero. It is necessary to modify the way PDECOL
determines the initial values of c(i) . Normally this is done by thesubroutine 1NITAL. It calls the uter supplied subroutine UINIT toobtain the values uk(to, *j), where the *j , j - 1,2... NC are the
collocation points. Then by substituting into the expansion (29), itobtains a set of linear algebraic equatios of the form
NC
k ( c(i)(t o ) Bi(*.) k 1,2...N, j -1,2...NC. (52)i=
29
.. . . . .
These systems of equations are solved to obtain the initial ck (to).
For a burner stabilized flame, the appropriate values of the massfractions at *1 = h are not known but the values of the mass fluxfractions ek are. Thus, we use the boundary condition (46), substitutethe expansion (29), and obtain
NC C (t)ld.. 2 D. 11i*Yku = c (i) to B ) _ k, k-l,2...N-1. (53)
2The values of p Dkm from the last run of the code are used. Thesevalues are also saved in the restart file. So for a burner stabilizedflame, the subroutine INITAL has been changed so that it generates thisnew set of equations for j = i, k = 1,2.. .N - 1.
Note that the above procedure is not strictly necessary. Since wehave a correction term 8Z/at, the boundary conditions will approach theproper values, even if they are incorrect at the start of the integration.But it is more efficient to begin with the boundary conditions as accurateas possible.
To actually make comparisons with experiments, some of the boundaryconditions may have to be further modified. For instance, evidence existsthat hydrogen atoms H combine very rapidly on the burner surface to formH .13 For practical purposes, we consider this to be instantaneous. Then5 our first species is H and our second species is H2, their boundaryconditions are
YI = 0 at *L (54)
and,
I + C2 = Y2u at *L (55)
These changes can easily be made in UINIT and INITAL.
As the time integration proceeds, control returns to the routineMAIN at five equally spaced output times. The code calls subroutineFLSP so the integration process can be monitored. There is no need toadjust m0, since this is now a predetermined constant.
3J. Wrnatz, "Calculation of the Structure of Lawinaz Flat FtZia III:
Structure of Burner-Stabilized Hydrogen-Oxygen and Hydrogen-FluorineFZ=neel, Verlog Chemie GmbH, BdB 8/78 E 4018.
30
A, 1
VII. NUMERICAL CONSIDERATIONS
Thil code has been applied to the H -O -N system, with nine chemicalspecies. An earlier versIgn of the code his seen applied to the ozonesystem, with three species" . Several calculations have been performedfor methane-air flames. In each case the code integrates in time untilthe steady state solution has been reached. What is desired is accuratevalues for the flame speed and accurate temperature and species profiles.Results are not discussed here, since they are given in the papersreferenced above. Instead, we will discuss some of the considerationsnecessary to obtain accurate results, based on our experiences with theabove systems.
The flame speed can be calculated from any of the species profilesusing Eq. (20). Normally the time integration proceeds until the flamespeeds based on the major species (reactants and products) agree towithin a small fraction of a percent. At this point the solution isvery close to steady state. The values of the flame speed calculatedfrom the minor species (radicals) are very sensitive, and require moreaccuracy (both spatial and temporal) in order to achieve the sameagreement.
In general, it is easier to obtain an accurate solution for a fastflame than for a slow flame. The slow flame requires many moreintegration steps before all the oscillations die out and a steady statesolution is achieved.
To obtain an accurate solution, both the spatial and temporal accuracymust be sufficient. If the temporal accuracy is low, the calculated flamespeeds can oscillate around the correct value. If the spatial accuracyis low, the problem may come very close to convergence, but to the wrongvalue, since the communication between the collocation points isinadequate. As a consequence of the spatial or temporal accuracy becomingtoQ small, the integration may break down completely.
Spatial accuracy is more important in the flame front, where wehave steep gradients. Hence, our algorithms for choosing breakpointsconcentrates them in the region to be occupied by the flame front.
The need for spatial and temporal accuracy is connected. Supposethe number of breakpoints is increased, but the temporal error tolerancee is not decreased. The calculated solutions will oscillate, and eventuallythe solution will break up. On the other hand, suppose e is decreasedbut the breakpoint sequence is not changed. We can still obtain an
14J.M. Heimerl and T.P. Coffee, "The Detailed Modeling of Praemixed,Laninar Steady-State Flawes. 1. Ozone". Combuetion and Flame,Volume 39, pp. 301-315, 1980.
31
answer, but the integration will take longer, and the accuracy of thesolution is not increased. At this stage, the appropriate value of Cfor a given breakpoint sequence is a matter of trial and error.
An additional problem can occur with very slow flames. If theinterval of integration is chosen too small, there may be a noticeablegradient in the temperature profile at the cold boundary. Since the gasis entering through the cold boundary quite slowly, this can result ina substantial heat loss through the boundary. This heat loss will slowdown the flame. The integration will converge, but the computed flamespeed will be too low. This effect can occur for any flame. However,for most flames, a noticeable heat loss is due to a fairly steep tempera-ture gradient at the boundary. It is then obvious that the intervalof integration must be increased. But for a slow flame (less than 20cm/sec) a noticeable error in the flame speed can result (5 to 10 percent),even when the gradient at the cold boundary appears negligible.
Most of the cases we have run have been with NINT - 12, KORD = 4 andNCC = 2 (13 breakpoints, 26 collocation points). We have let e = I0- ,
and SREC = 10-6. The number of central intervals NCN was 4, and theratio FC between the longest and shortest intervals was between 4 and 8.
The length of the interval of integration L, as well as the optimumvalue for FC, is a matter of trial and error. However, a reasonable setof breakpoints can be found fairly easily by experimenting with theabove scheme, using the constant transport subroutine. The code can thenbe run with a more realistic transport algorithm, using the restartoption.
To make sure that convergence was obtained, the code was also runwith NINT = 16 and e = 3 x 10-4 for a number of cases. Only negligibledifferences occurred.
The program was run on the BRL CYBER 76. For hydrogen-oxygenflames, the run time varied from about 30 seconds with the simpliesttransport subroutine to 5 minutes or more for the most complicated case.
An earlier version of this code was applied to the solution of anunbounded ozone flame1 4 . The successive calculation procedure was notimplemented at that time. However, since the number of species is so
small, the savings in reducing the size of the Jacobian is not veryimportant.
32
At that time we reported using NINT = 57 to NINT = 70, KORD a 6and NCC - S. The number of collocation points was tben between 62 and75. We have now been able to solve this case with equal accuracy usingNINT a 12, KORD a 4, and NCC - 2. Partly this is due to choosing NCC a 2.The program then generates a special choice of collocation points(Gauss - Legendre quadrature points in each subinterval) which givesincreased spatial accuracy. More important is the improved algorithmfor choosing breakpoints. By using a little more care, a much smallerset of breakpoints can still reproduce the flame with sufficient accuracy.So the careful choice of breakpoints is probably the most importantconsideration in running the code efficiently.
33
REFERENCES
1. F. A. Williams, Combustion Theory, Addison-Wesley, Reading, MA,Chapter 1, 1965.
2. R. B. Bird, W. E. Stewart and E. N. Lightfoot, Transport Phenomena,John Wiley and Sons, NY, 1960, Chapter 18.
3. R. H. Fristram and A. A. Westenberg, Flame Structure, McGraw-Hill,NY, Chapter V-1, 1965.
4. J. 0. Hirshfelder, C. F. Curtiss and R. B. Bird, Molecular Theoryof Gases and Liquids, 2nd Printing corrected, with notes, JohnWiley and Sons, NY, 1964, Chapter 11.1.
5. S. Gordon and B. J. McBride, "Computer Program for Calculation ofComplex Chemical Equilibrium Compositions, Rocket Performance,Incident and Reflected Shocks, and Chapman-Jouguet Detonations",NASA-SP-273, 1971, (1976 program version).
6. T. P. Coffee and J. M. Heimerl, "Transport Algorithms for Premixed,Laminar, Steady-State Flames", to be published in Combust. and Flame.
7. N. K. Madsen and R. F. Sincovec, "PEDCOL: General Collocation Soft-ware for Partial Differential Equations", Preprint UCRL-78263(Rev I), Lawrence Livermore Laboratory, (1977).
8. C. de Boor, "Package for Calculating with B-Splines", Siam. J. Numer.Anal. 14, pp. 441-472, 1977.
9. A. C. Hindmarsh, "Preliminary Documentation of GEARIB: Solution ofImplicit Systems of Ordinary Differential Equations with BandedJacobian", Rep. UCID-30130, Lawrence Livermore Laboratory, 1976.
10. D. B. Spalding and P. L. Stephenson, "Laminar Flame Propagation inHydrogen + Bromine Mixtures", Proc. R. Soc. Lond. A., Volume 324,315-337, 1971.
11. T. P. Coffee and J. M. Heimerl, "A Method for Computing the FlameSpeed of a Laminar, Premixed, One Dimensional Flame", BRL TechnicalReport ARBRL-TR-02212, January 1980.
12. J. 0. Hirshfelder, C. F. Curtiss, and R. B. Bird, op. cit., pp. 761-763.
13. J. Warnatz, "Calculation of the Structure of Laminar Flat FlamesIII: Structure of Burner-Stabilized Hydrogen-Oxygen and Hydrogen-Flourine Flames", Verlog Chemie GmbH, BdB 8/78 E 4018.
34
L i__ _ _ _ _ _ _ _ _ _ _ _-
REFERENCES (continued)
14. J. M. Heimeri and T. P. Coffee, '"The Detailed Modeling of Premixed,Laninar Steady-State Flames. 1. Ozone", Combustion and Flame,Volume 39, pp. 301-315, 1980.
35
APPENDIX A
37
APPENDIX A
A listing of the computer code follows. The subroutines MAIN, F,UINIT, FLSP, BNDRY, BKPT, and RT are written for laminar flame problems.The subroutine F given is for the constant transport assumption. Thesubroutine RT given is for a H 2-0 2-N2 flame. There are nine chemical
species and thirty reactions.
The rest of the subroutines are from PDECOL. They have beenmodified as discussed in the text.
After the listing, the job stream and output is given for a typicalhydrogen-oxygen flame. The initial unburned gas is 50% H2 and 50% air,where air is 21% 0 and 79% N . The nitrogen is considered to be adiluent, and does not react. The required starting information isattached to TAPEll. The main program and the subroutine RT are attachedin a compiled form. The integration begins from the initial profiles(37). At the final output time, the flame speed and the temperature andspecies profiles are given, as well as the corresponding x values. Therestart file is written on TAPE 2, and the output file on TAPE 9. Bothfiles are catalogued for possible future use.
39 FHLC~ a,,a p~ jj " " gw SLU
x X
4 A' 3D A' A. a
2.~~r LA f -I#4
If 0X. .9
T 1 -j-.x 0 LA LA z -1O)
0X Off. 4 Xt . A.)wU41 it 003
N - x4 Jr .) - dl *in )4 . j1 3. 4)a )a X j A 1 Z X 1 0 NI v v 7 4]
X(0. 0 MLAb Z 0 I. CIOLA LAUU -- U=M *
3e - I eII of om- 4- o w u O -4-
If - .*LA -j m LA -. *LujJ4 L U4jJ) A (.3.1 wOAJin ,X N I2xx 00011 0 UuU w 0444 -
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GLOSSARY
Cpk - specific heat of species k, cal-g - K
Cp z specific heat of the mixture, cal-g - K
2 -1D km = diffusion coefficient of species k in the mixture, cm -s
hk = specific enthalpy of species k, cal-g1
k. = rate constant for reaction j in centimeter - mole - second units.J
m 0 = mass flux of the mixture through the origin, g-cm 2-s"1 .
Hk = molecular weight of species k, g-mole 1
N = number of chemical species (also number of PDE's).
NO = number of ODE's.
p = pressure, atmos.
r. = rate of reaction j, mole-cm- -s"I .
R = gas constant = 82.05 cm3-atoms-mole' -K-1.
Rk = rate of production of species k, mole-cm -- s-1
Sv = velocity of the flame relative to the unburned mixture, cm-s
t = temporal coordinate, s. (t, x coordinate system).
t = temporal coordinate, s. (t, coordinate system).
t = non dimensional temporal coordinate, with respect to t (t,coordinate system).
T = temperature of the mixture, K.
T = temperature of the mixture, K.
T = non dimensional temperature of the mixture, with respect to
T - temperature of the unburned mixture.
133
* - -- -V - - '
GLOSSARY (continued)
TB = temperature of the burned mixture (adiabatic temperature).
v = fluid velocity of the mixture, cm-s-l
Vk = diffusion velocity of species k, cm-s .
x = spatial coordinate, cm.
Y Yk = mass fraction of species k.
YkU = mass fraction of species k in the unburned mixture.
Y kB = mass fraction of species k in the burned mixture.
" •= mass flux fraction of species k.
X = heat conductivity of the mixture, cal-cm 1-s 1-KI .
P = density of the mixture, g-cm "3 .
-2= transformed distance coordinate, g-cm -
= non dimensional transformed distance coordinate, with respect to
134
.... - : ---: -- : o," ~ ~ ., -- 2 :: : .-- -.. ... .. .
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