A Numerical Method for Combustion Chemistry

8/7/2019 A Numerical Method for Combustion Chemistry

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LDC-2011-0013/9/2011

A LINEARLY IMPLICIT ALGORITHM FOR INTEGRATING

THE EQUATIONS OF CHEMICAL KINETICS

Lawrence D. Cloutman

[email protected]

Abstract

The rate equations of chemical kinetics are a system of nonlinear ordinary differ-ential equations. Explicit numerical integration methods tend to have restrictive timestep limitations, so there is an incentive to use implicit methods with better stabilityproperties. However, the non-linear character of these equations makes numerical so-lution challenging. We present a method of linearizing the difference equations in theadvanced-time terms that will provide improved numerical performance over explicitmethods while remaining numerically tractable. The applications include regular chem-ical reaction networks, nuclear chemistry networks, and the Lotka-Volterra equationsof theoretical ecology.

c2011 by Lawrence D. Cloutman. All rights reserved.

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1 Introduction

The equations of chemical kinetics are a system of nonlinear ordinary differential equations.

In many combustion applications, these rate equations are incorporated into the multicompo-

nent partial differential equations of fluid dynamics. In that case, the rate equations typically

are operator-split from the fluid equations and solved as a system of ordinary differentialequations.

Many methods have been used to solve the rate equations. Explicit integration meth-

ods tend to have restrictive time step limitations, so there is an incentive to use implicit

methods with better stability properties. However, the non-linear character of these equa-

tions makes numerical solution challenging. We present a method of linearizing the difference

equations in the advanced-time terms that will provide improved numerical performance over

explicit methods while remaining numerically tractable. The method described here is a spe-

cialization of a commonly used approach described in Appendix A [1]. This report documents

its implementation into an updated version of an existing program [2].

The proposed method is not limited to combustion research. It is applicable to any

system of coupled nonlinear ordinary differential equations. Other applications include non-

combustion chemical reaction networks, nuclear chemistry networks (including astrophysical

applications), and the Lotka-Volterra equations of theoretical ecology [3].

In Section 2, we describe the governing equations and the constitutive relations used

in combustion research. Section 3 details the rate expressions for chemical reaction networks.

Section 4 defines and describes some of the simple and commonly used methods for solving

the rate equations, plus a discussion of some of their numerical properties. Section 5 describesthe linearized implicit algorithm for integrating the rate equations. Sections 6 and 7 present

some numerical examples, plus the analytic solution for a single-reaction example. Section

8 contains the summary and conclusions. There are two appendixes providing additional

details on selected topics. All units are CGS and the temperature T is in K unless otherwise

noted.

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2 Governing Equations

To solve reactive flow problems, we assume the fluid is a mixture of species described by

the single-velocity (mass weighted) representation. The equations outlined in this section

have been incorporated into an updated version of the COYOTE computer program [2] to

produce the numerical results presented here.Mass conservation is expressed by the continuity equation for each species :

t

+ (u) = J + R, (1)

where is the density of species , t is time, u is the velocity, and R is the rate at which

species is created by chemical reactions. The exact diffusional mass flux is given by an

extremely complicated expression [4, 5, 6]. A clear summary of the mass transport equations

and an efficient numerical algorithm for solving them are given by Ramshaw [7, 8] and by

Ramshaw and Chang [9]. Equation (1) may be summed over species to obtain the total

continuity equation

t+ (u) = 0. (2)

The momentum equation is

(u)

t+ (uu) = g P + T, (3)

where g is the gravitational acceleration, P is the pressure, and T is the stress tensor

T =

u + (u)T 2

3 u U

+ b u U, (4)

where U is the unit tensor, is the coefficient of viscosity, and b is the bulk viscosity.

We can express energy conservation in terms of the specific thermal internal energy

I:(I)

t+ (Iu) = P u + T : u q +

HR, (5)

where q is the diffusional heat flux, and H is the heat of formation of species (per unit

mass). The heat flux is a complicated function, and for many applications it is adequate to

use the sum of Fouriers law and enthalpy diffusion:

q = KT +

h

J

, (6)

where K is the thermal conductivity.

In most cases, we prefer to use the total energy density E = I+ 0.5u u + , where

is the gravitational potential.

E

t+ [ (E + P)u u T] =

t q +

HR. (7)

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For a constant g, = g x, where x is the position vector.

The thermal equation of state is the sum of partial pressures for each species:

P =

RT

M, (8)

where R is the universal gas constant, and M is the molecular weight of species .The caloric equation of state is

I =

I(T), (9)

where I is the species specific thermal internal energy. In the present application, we assume

I = CvT =RT

( 1)M, (10)

where is the ratio of specific heats for species .

These same equations for chemical combustion apply to nucleosynthesis in stellar

interiors, although the stress tensor and constitutive relations may need to be modified to

include additional physical processes such as radiative effects and electron degeneracy [10,

11].

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3 Chemical Reaction Networks

Chemical reactions are symbolized by

ar

br, (11)

where represents one mole of species , and ar and br are the dimensionless stoichio-

metric coefficients for the rth reaction. It is assumed that ar and br are integers. The

chemical source term in the species continuity equation is given by

R = Mr

(br ar)r, (12)

where r is the rate of progress of the rth reaction:

r =

kfr

M

ar kbr

M

brCrM moles(of reaction)/cm

3s. (13)

Here kfr and kbr are the forward and reverse rate coefficients for reaction r. For an elementary

reaction, ar = ar and b

r = br. The product operations are over all species for which

the stoichiometric coefficients ar and br are nonzero. For combustion, the coefficients kfr

and kbr are assumed to be of a generalized Arrhenius form:

kfr = AfrTfr exp(Efr/RT), (14)

kbr = AbrTbr exp(Ebr/RT), (15)

where Efr and Ebr are the activation energies. The expressions for nuclear reaction rates are

similar but tend to be a bit more complex [12].

In most cases, the constants Abr, br, and Ebr are computed from Afr, fr, Efr, and

the reactions equilibrium constant KCr . In equilibrium, r = 0, so

kbrkfr

=

M

ar| /

M

br= KCr . (16)

The factor CrM requires some explanation. Some chemical combustion reactions in-

clude the modified total molar density

CM =N=1

n

M, (17)

where is the third body efficiency (or chaperon coefficient) for species (normally unity

unless otherwise specified). An example of such a reaction is 2 H + M H2 + M, where M

represents all of the species in the mixture. The parameter r is zero or unity depending on

whether M occurs in the reaction.

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4 Some Commonly-Used Numerical Methods

The chemical reaction terms are usually time-split from the rest of the fluid dynamics equa-

tions so the numerical chemistry problem reduces to a system of ordinary differential equa-

tions in each computational zone on each time step. We shall consider several simple methods

for numerically integrating the system of ordinary differential equations

dx

dt= f(x, t) (18)

for the solution vector x.

The simplest algorithm is the first-order Eulers method, the fully explicit method for

advancing from time ti1 to ti,

xi = xi1 + t f(xi1, ti1), (19)

where t = ti ti1 and the subscript i denotes the value of the function at time ti.Next consider the fully implicit first-order method

xi = xi1 + t f(xi, ti). (20)

In general, we must solve nonlinear algebraic or transcendental equations for xi. A solution

algorithm is outlined in Appendix A.

Now we shall consider some second-order methods. First, the frequently-used Crank-

Nicholson scheme is

xi = xi1 +t

2 [f(xi1, ti1) + f(xi, ti)] . (21)

Next we consider a two-step second-order Runge-Kutta scheme based on the midpoint

rule.

xi = xi1 +t

2f(xi1, ti1)

xi = xi1 + t f(x, ti1/2), (22)

where ti1/2 = 0.5(ti + ti1).

Finally we consider a second-order Runge-Kutta scheme based on the trapezoidal

rule.

xi = xi1 + t f(xi1, ti1)

xi = xi1 +t

2[f(xi1, ti1) + f(x

, ti)] . (23)

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For the limit of vanishing t to make sense, we must choose the minus sign regardless of

the sign of K. For K < 0, the solution decays and stays positive for all values of t, as

it should. However, for K > 0 and sufficiently large t, the argument of the square root

becomes negative.

Now we shall consider some second-order methods. First, the Crank-Nicholson scheme,

equation (21), is

xi = xi1 +K t

2

xmi1 + x

mi

. (31)

For m = 1,

xi =xi1 (1 + 0.5 K t)

1 0.5 K t. (32)

Regardless of the sign ofK, the solution will alternate its sign from one time step to the next

if t exceeds twice the explicit limit. The same thing happens for the diffusion equation in

spite of a formal stability analysis showing that the method is unconditionally stable. That

is, the solution remains bounded, but it is bounded garbage. For m = 2, the Crank-Nicholsonmethod has qualitatively the same issues as the fully implicit method.

Next we consider the two-step Runge-Kutta scheme based on the midpoint rule,

equation (22).

xi = xi1 +K t

2xmi1

xi = xi1 + K t (x

i )m . (33)

These may be combined algebraically to obtain

xi = xi

1

1 + 2 + 2

2(m = 1) (34)

and

xi = xi11 + 2xi1 + 4(xi1)

2 + 2(xi1)3

(m = 2), (35)

where = 0.5 K t. We note the following behaviors:

For K > 0, both solutions solution grow along with t, just as they should.

For K < 0 and m = 1, the method is only conditionally stable (in the sense of

maintaining xi 0). The polynomial in has a minimum at = 0.5 (that is,

t = 1/K). Physically, we expect xi to decrease more on each time step as t is

increased. This fails to happen for < 0.5 (remember that becomes more negative

with increasing t if K < 0).

Things are even worse for m = 2 and K < 0: The polynomial has two extrema (at

= 1/3 and 1) and begins to behave badly for xi1 < 1/3.

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Finally we consider the Runge-Kutta scheme based on the trapezoidal rule, equation (23).

xi = xi1 + K t xmi1

xi = xi1 +K t

2

xmi1 + (x

i )m

. (36)

The combined equations are

xi = xi1

1 + 2 + 22

(m = 1) (37)

and

xi = xi11 + 2xi1 + 4(xi1)

2 + 4(xi1)3

(m = 2). (38)

The behavior is qualitatively the same as for the previous method. For m = 1, the two

Runge-Kutta methods are identical. For m = 2, the polynomial factor goes negative for

xi1 less than about 0.75.

The method of most interest for this report is linear in the advanced time variables.

For the case m > 1, it is obtained by by approximating the fully implicit equation (27) with

xmi mxixm1i1 (m 1)x

mi1. The solution may be written as

xi = xi11 2(m 1) xm1i1

1 2 m xm1i1. (39)

The method fails if mxm1i1 > 0.5. All seems well for negative values of . However, as

xm1i1 approaches , the ratio xi/xi1 saturates at a value of (m 1)/m rather than

zero. This means that while the method may be stable for very large time steps, transientsolutions may be quite inaccurate.

A second method of linearization is xmi xixm1i1 . The solution may be written as

xi =xi1

1 2xm1i1. (40)

While this method has the correct limit for large t, it is not obvious how to apply it to a

term such as xiyi, where x and y are the molar concentrations of two different species.

There are two main lessons from this discussion. First, implicitness is not always the

solution to numerical stability issues. Second, numerical stability does not imply accuracy.Indeed, explicit stability limits often play the additional role of accuracy conditions. We note

that there is another, more subtle failure mode for nonlinear difference equations, which is

described in Appendix B.

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5 Linearized Implicit Chemistry Algorithm

The species continuity equation (1) is solved in COYOTE by operator splitting. That is,

the advection, diffusion, and chemical terms are computed independently. The chemistry

problem involves the reactions

ar

br. (41)

The operator-split chemical rate equations are

(x, t)

t= R[T(x, t), 1(x, t), . . . , N(x, t)]

= Mr

(br ar)

kfr(T)

y(r)=react

Cayry kbr(T)

y(r)=prod

Cbyry

CrM, (42)

where there are N species and 1 N. The algebraic product operations are over

chemical products and reactants in an obvious notation. For elementary reactions, ar = ar

and br = br. The parameter r is zero unless the reaction involves species that do not

change during the reaction (typically denoted by M in reactions such as H2 + M 2 H +

M), in which case it is unity. The energy released per mole of reaction is

Qr =

(ar br)HM, (43)

where H is the heat of formation per unit mass.

The original version of the program used the explicit method

t

=n+1

n

t= R (T

n, n1 , . . . , nN) , (44)

where t is the time step and the superscript n denotes the solution at time tn. Both stability

and (possibly) accuracy require a time step that is smaller than the time scale for the fastest

reaction. If we replace all occurrences of n on the right-hand side of equation (44) with

n + 1, we have a fully implicit method. While this method should be much more stable, it

is more expensive and difficult to solve. We have also tried two second-order Runge-Kutta

methods. While these perform somewhat better than the explicit method, they still have

similar accuracy and stability requirements.We propose a method that is linear in advanced-time quantities. It should be more

stable than the fully explicit method, but simpler to solve than the fully implicit method.

First, we use the temperature at time level n. It is more convenient to do the numerical

computations in terms of the species molar concentrations C = /M, which we take at

time level n + 1. Finally, we linearize the products of concentrations to make them linear

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in time level n + 1 quantities. If we assume Cn+1 = Cn + C, then for a product of Np

species,Npy=1

Cn+1y

my

Npy=1

Cny

my1 + Np

y=1

myCyCny

, (45)

where my are constants, usually positive integers.2 Note that linearization is equivalent to

that used in equation (39).

The rate equations are discretized fully implicitly in the species densities, but explic-

itly in temperature:

1

M

t

=r

(br ar)

kfr(Tn)

y(r)=react

n+1yMy

ayr kbr(T

n)

y(r)=prod

n+1yMy

byr CrM.(46)

Next we use equation (45) to linearize the equations and to produce a linear system in the

. Since the total molar concentration of the mixture will change little during the time

step, we compute CM at time n:

CM =N=1

n

M, (47)

where the are the third body efficiencies (also called the chaperon coefficients). Rewriting

equation (46) in terms of concentrations,

Ct

=r

(br ar)

kfr(Tn)

y(r)=react

Cn+1y

ayr kbr(T

n)

y(r)=prod

Cn+1y

byr CrM. (48)Linearizing by using equation (45), we obtain

C t CrM

r

(br ar)

kfr(Tn)

y(r)=react

Cny

ayry

ayrCyCny

kbr(Tn)

y(r)=prod

Cny

byry

byrCyCny

= t CrM

r(br ar)

kfr(T

n)

y(r)=react Cny

ayr

kbr(Tn)

y(r)=prod Cny

byr

. (49)

This linear system is not singular. The mass conservation condition

=

M C = 0 (50)

2Occasionally one encounters rate expressions that involve non-integer powers. Those cases are generallyin simplified global rate expressions (for example, [14]), and they can be handled using the above linearexpansion in the . However, those will not be considered explicitly here.

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is redundant and need not be included in the linear system. However, should we ever

encounter conditions where the matrix is ill-conditioned, we might want to try replacing one

of the species equations with this one.

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6 Example 1: A Single Elementary Chemical Reaction

Consider the single two-body elementary reaction H2 + O2 2 OH. The rate equations

(r = 1) aredH2

dt= RH2 = MH2 kf1CH2CO2 kb1C

2OH , (51)

dO2dt

= RO2 =MO2MH2

RH2 , (52)

anddOH

dt= ROH = RH2 RO2. (53)

To simplify the notation, let us assign k = 1, 2, and 3 for H2, O2, and OH, respectively.

Then we may rewrite our three rate equations as

dC1dt

= kf1C1C2 kb1C

23

, (54)

dC2dt

= kf1C1C2 kb1C

23

, (55)

anddC3dt

= 2kf1C1C2 kb1C

23

. (56)

6.1 Analytic Solution

We begin solving equations (54) through (56) by finding invariants of the problem. First,

adding the equations shows that the total molar density

C = C1(t) + C2(t) + C3(t) = C1(0) + C2(0) + C3(0) (57)

is constant. 3 Comparing pairs of equations shows that

dC1dt

=dC2dt

= 1

2

dC3dt

. (58)

Integrating these equalities gives us the constraints

C1(t) C1(0) = C2(t) C2(0) = 1

2[C3(t) C3(0)] . (59)

If we solve equation (59) for C1(t) and C2(t) in terms of C3(t), we can eliminate

these functions from equation (56). If we let bi = Ci(0) + 0.5 C3(0), i = 1 or 2, thenCi(t) = bi 0.5 C3(t). Then equation (56) becomes

dC3dt

= 2kf1b1b2 kf1 (b1 + b2) C3 +

kf1

2 2kb1

C23 B0 + B1C3 + B2C

23 . (60)

3Total molar concentration is not always conserved; for example 2 H + M H2 + M changes the numberof moles.

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The constants B0, B1, and B2 may be evaluated from the initial conditions.

Equation (60) may be solved by using the following integrals. If B21 > 4B0B2, then

dx

B2x2 + B1x + B0=

1

(B21 4B0B2)1/2

ln

2B2x + B1 (B21 4B0B2)1/2

2B2x + B1 + (B21 4B0B2)1/2

. (61)

If B21 < 4B0B2, then

dx

B2x2 + B1x + B0=

2

(4B0B2 B21)1/2

arctg

2B2x + B1

(4B0B2 B21)1/2

. (62)

If B21 = 4B0B2, then dx

B2x2 + B1x + B0=

2

2B2x + B1. (63)

The corresponding solutions for equation (60) are easily found. First, let

A =B21 4B0B21/2 . (64)

If B21 > 4B0B2, then let

D =1

Aln

2 B2 C3(0) + B1 A

2 B2 C3(0) + B1 + A

. (65)

Then the solution is

C3(t) =(B1 + A) exp [A (t + D)] B1 + A

2 B2 {1 exp[A (t + D)]}. (66)

If B21 < 4B0B2, let

E =2

Aarc tg

2 B2 C3(0) + B1

A

. (67)

Then the solution is

C3(t) =A tan[A (t + E) /2] B1

2 B2. (68)

If B21 = 4B0B2, then

C3(t) = 12B2

12B2C3(0) + B1

t2

1 B1

. (69)

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8.28624D-01 analytic

Species Mass Fractions, ncyc = 2000 t = 1.000000D-061.56861D-01 9.88203D-03 8.33257D-01 Euler1.56863D-01 9.88213D-03 8.33255D-01 midpoint RK1.56864D-01 9.88223D-03 8.33254D-01 linearized


This problem was solved for t = 0.5 ns using three methods: the first-order explicit Eu-

lers method, the second-order midpoint Runge-Kutta method, and the first-order linearized

implicit method. The analytic solution is also shown in Table 1. All three methods agree

with one another and with the analytic solution. To further demonstrate grid-independence

of the solution, the calculation was repeated with the time step cut in half. The results are

shown in Table 2.

+++++ Table 2. Cut delt in half (0.25 ns): jchem2.0.5 +++++








Species Mass Fractions, ncyc = 4000 t = 1.000000D-061.56862D-01 9.88208D-03 8.33256D-01 Euler

1.56863D-01 9.88213D-03 8.33255D-01 midpoint RK1.56863D-01 9.88218D-03 8.33254D-01 linearized8.33255D-01 analytic

We see that the agreement is even better. The solution at late times should approach the

equilibrium condition,

KC =kbkf

=C1C2

C23. (74)

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Indeed, the solution is converging toward the correct equilibrium value C3(t) = 0.833333.

Next we try solutions at three larger time steps. The first solution uses t = 5.0 ns

and is shown in Table 3.

+++++ Table 3. Increase delt by 10 (5 ns) +++++





Species Mass Fractions, ncyc = 200 t = 1.000000D-06

1.56848D-01 9.88122D-03 8.33271D-01 Euler1.56863D-01 9.88215D-03 8.33255D-01 midpoint RK1.56880D-01 9.88320D-03 8.33237D-01 linearized



Although accuracy is slightly degraded, it is still acceptable for most purposes for all three

methods.


Species Mass Fractions, ncyc = 1 t = 5.000000D-08-2.05086D-02-1.29201D-03 1.02180D+00 Euler

7.20625D-01 4.53984D-02 2.33976D-01 midpoint RK6.24729D-01 3.93570D-02 3.35914D-01 linearized


Species Mass Fractions, ncyc = 10 t = 5.000000D-071.52707D-01 9.62035D-03 8.37672D-01 Euler1.67403D-01 1.05461D-02 8.22051D-01 midpoint RK

1.70697D-01 1.07537D-02 8.18549D-01 linearized8.28644D-01 analytic



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Increasing the time step another factor of 10 produces a poor transient, as shown in the

results on cycle 1. The worst method is Euler, which produces negative mass fractions of O2and H2 on the first cycle. However, all three methods converge to the correct equilibrium.


Species Mass Fractions, ncyc = 1 t = 5.000000D-07-8.66921D+00-5.46148D-01 1.02154D+01 Euler

1.09750D+01 6.91408D-01-1.06664D+01 midpoint RK4.92171D-01 3.10061D-02 4.76823D-01 linearized



1.80428D+05 1.13667D+04-1.91794D+05 midpoint RK2.81624D-01 1.77419D-02 7.00634D-01 linearized



--- --- --- midpoint RK1.56789D-01 9.87750D-03 8.33333D-01 linearized

Increasing the time step to 500 ns finally shows failure of both the Euler method and the

Runge-Kutta method. Both produce large negative concentrations on cycle 1, and things get

worse as the calculation proceeds. The Runge-Kutta method fails catastrophically, producing

a NaN on cycle 4. Even though the linearized implicit method is stable, the transient has

substantial truncation errors.

These results are a reminder that a stable solution is not necessarily an accurate

solution. The linearized implicit model has excellent stability properties, but it should

include a time step control that prevents any concentration or the temperature from changing

too much on any one time step.

6.2.2 Case 2

Case 1 was about the simplest case possible, and next we wanted to try something a bit morechallenging. Case 2 differs from Case 1 in having different initial concentrations plus a fourth

inert species (argon, MAr = 39.948). The initial concentrations are C1(0) = 1.60529 105,

C2(0) = 2.40161 105, C3(0) = 9.43882 109, and C4(0) = 1.20554 108. In addition,

this time we included the heats of formation so we could track the evolution of temperature.

The initial temperature was 1500 K. The results for t = 0.5 ns are summarized in Table 6.

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Table 6. Numerical Solution for Case 2

Species Mass Fractions, ncyc = 0 t = 0.000000D+001 2 3 4 T

9.12825D-01 8.60336D-02 2.85269D-04 8.55806D-04 1.50000D+03 t=0.0

Species Mass Fractions, ncyc = 1 t = 5.000000D-10

9.01864D-01 8.53430D-02 1.19371D-02 8.55806D-04 1.48590D+03 Euler9.01974D-01 8.53499D-02 1.18206D-02 8.55806D-04 1.48604D+03 midpoint RK9.02079D-01 8.53566D-02 1.17082D-02 8.55806D-04 1.48617D+03 linearized


Species Mass Fractions, ncyc = 100 t = 5.000000D-083.70361D-01 5.18591D-02 5.76924D-01 8.55806D-04 7.39364D+02 Euler3.72251D-01 5.19782D-02 5.74915D-01 8.55806D-04 7.42286D+02 midpoint RK3.74113D-01 5.20955D-02 5.72936D-01 8.55806D-04 7.45164D+02 linearized


Species Mass Fractions, ncyc = 200 t = 1.000000D-072.05712D-01 4.14864D-02 7.51946D-01 8.55806D-04 4.78566D+02 Euler

2.06935D-01 4.15635D-02 7.50646D-01 8.55806D-04 4.80536D+02 midpoint RK2.08147D-01 4.16398D-02 7.49357D-01 8.55806D-04 4.82488D+02 linearized






All three numerical methods agree quite well and with the analytic solution. All three

seem to be converging to the correct equilibrium OH mass fraction, 0.909598. Note that

this reaction is strongly endothermic and would be quenched if we had used the physical

temperature dependence of the rates instead of the same constant rates used in Case 1.

The behavior of the three methods is the same as for Case 1 as we vary the time step.

Again, the linearized implicit method lets us run with the largest time step. Table 7 is a

short summary of results for t = 5.0 10

7 s.

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Table 7. Numerical Solution for Case 2

Species Mass Fractions, ncyc = 1 t = 5.000000D-073.91721D-01 5.32048D-02 5.54218D-01 8.55806D-04 7.72349D+02 linearized

Species Mass Fractions, ncyc = 1000 t = 5.000000D-045.74031D-02 3.21432D-02 9.09598D-01 8.55806D-04 2.40613D+02

The results for cycle 1 of Table 7 are at the same time as cycle 1000 in Table 6. There is a

big difference between the two solutions. Although accuracy of the transient is poor for the

larger time step, the solution is converging to the correct equilibrium by cycle 1000.

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7 A 21-Species Example

In this section, we consider a more complex kinetics mechanism. It uses 21 species and

30 reactions. Ar, H2O2, C2H2, and CH2O are treated as inert. No hydrocarbon fuels are

included in the initial conditions so the chemistry is mainly that of oxygen and hydrogen,

minus any H2O2 chemistry. The initial conditions are P = 1.013 106

dynes/cm2

, =1.25029104 g/cm3, and T = 1700 K. Table 8 lists the molecular weights, heats of formation,

and equilibrium constants for the 21 species. Table 9 is a list of the forward rate constants

for the 30 reactions.

Table 10 is a summary of computational results. It shows temperatures and species

mass fractions at 10 s for two numerical methods and three values of t (2.5, 5.0, and

20.0 ns). The second-order Runge-Kutta method is denoted by midpoint RK, and the

first-order linearized implicit method is denoted by linearized.

Consider first the inert species Ar, H2O2, C2H2, and CH2O. Only Ar is inert in the

sense of not normally being chemically reactive. However, the other 3 species are not included

in any of the chemical reactions in this mechanism (except all species are included in the

chemical symbol M). Therefore the mass fractions of this set of species should be constant

in time. This is true for all four inert species for both numerical methods and for all three

time steps.

Next consider the reactive hydrocarbon species CH4, C2H4, and C3H8. Both the

initial mass fractions and the reverse rates of their global oxidation reactions were set to

zero. The mass fractions remained zero for the Runge-Kutta integration for all three time

steps. For the linearized method, the mass fractions are all tiny negative numbers. Theseare due to rounding errors in the linear-system solver. 4 The worst case is a mass fraction of

1.0 1024 for methane. This represents roughly one methane molecule in a mole of fluid,

which is physically insignificant. These small negative numbers are merely a minor cosmetic

issue, as long as they stay small.

For the 0.25 ns time step, the two numerical methods agree to about one part in 104

for all species except N, where the difference is about 1 percent. This is a trace species, so

the agreement between the two methods suggests that this solution is time-step-independent

for all practical purposes.

For the Runge-Kutta method, doubling the time step to 5.0 ns makes a little differ-

ence, and the agreement is still mostly at the 1 part in 104 level. The exceptions are N and

CHO. The N mass fraction is still fairly accurate, but that of CHO now is in error by about

30 percent. For the first-order linearized implicit method, the worst discrepancy between

4All calculations were done in double precision (real*8).

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the two time steps is about one percent in the mass fractions. This includes N, and there is

almost no change in CHO. Note also that the predicted temperatures for the four solutions

discussed so far agree to within 5 K.

Next the time step was increased by a factor of 4 to 20.0 ns. The Runge-Kutta

solution is now totally unacceptable. The mass fraction of OH is -0.1. OH is a key radical

for the combustion of all hydrocarbons, and such a large negative mass fraction renders

the solution physically meaningless. On the other hand, the linearized implicit method

is still well-behaved. The only negative mass fractions are the tiny rounding errors for

methane, ethylene, and propane. With few exceptions, the truncation errors are on the

order of 3 percent. The mass fraction of N has decreased almost 10 percent for N with the

Runge-Kutta method, but has increased by almost the amount with the first-order linearized

implicit method. N2O and CHO are significantly more accurate for the first order method.

The second-order mass fraction of CHO is off by 3 orders of magnitude (possibly influenced

by the massive error in OH). The final temperatures increased by 45 K for the Runge-Kuttasolution, but only by 18 K for the first-order method.

In closing, we make the following observation: The formal order of accuracy for

a numerical method says little about its accuracy. Even for the largest time step that

produced a large negative mass fraction of OH, the Runge-Kutta method kept on running.

It was not unstable in the sense of the solution becoming unbounded, but the solution is

physically unrealistic. The first-order method seems to be less prone to producing negative

mass fractions, and its accuracy at large time steps for species such as CHO may be related

to its ability to keep fast reactions in near-equilibrium under conditions that would cause

stability problems with explicit methods.

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Table 8. Species Thermodynamic Parameters

Species M H A b E

1 CH4 1.604296D+01 -1.59921128D+01 1.51456D-06 0.00000D+00 -2.19712D+042 O2 3.199880D+01 0.00000000D+00 1.00000D+00 0.00000D+00 0.00000D+003 N2 2.801348D+01 0.00000000D+00 1.00000D+00 0.00000D+00 0.00000D+004 CO2 4.401000D+01 -9.39653441D+01 1.12780D+01 -2.86036D-01 -9.37359D+045 H2O 1.801528D+01 -5.71034894D+01 2.03999D-02 -3.56445D-01 -5.86397D+046 H 1.007940D+00 5.16336042D+01 1.61801D+01 5.14866D-01 5.21438D+04

7 H2 2.015880D+00 0.00000000D+00 1.00000D+00 0.00000D+00 0.00000D+008 O 1.599940D+01 5.89842256D+01 2.64100D+02 2.86036D-01 5.98602D+049 N 1.400674D+01 1.12528680D+02 1.27475D+02 3.69647D-01 1.13213D+05

10 OH 1.700734D+01 9.17543021D+00 2.46399D+01 -1.80423D-01 9.48250D+0311 CO 2.801060D+01 -2.72000478D+01 4.27854D+07 -8.75711D-01 -2.49755D+0412 NO 3.000614D+01 2.14567399D+01 4.60300D+00 0.00000D+00 2.16200D+0413 Ar 3.994800D+01 0.00000000D+00 1.00000D+00 0.00000D+00 0.00000D+0014 C2H4 2.805416D+01 1.45760038D+01 3.81096D-04 -2.50832D-01 9.41822D+0315 C3H8 4.409712D+01 -1.94400000D+01 8.86700D-15 -5.90400D-01 -2.76300D+0416 HO2 3.300674D+01 1.19646271D+00 3.74583D-03 -3.08039D-02 -3.35439D+02

17 H2O2 3.401468D+01 -3.10248565D+01 2.07494D-05 -3.34443D-01 -3.28504D+0418 N2O 4.401288D+01 2.04304493D+01 1.44700D+00 -3.75100D-01 3.42000D+0419 C2H2 2.603828D+01 5.63467973D+01 5.35000D+02 0.00000D+00 5.30100D+0420 CH2O 3.002648D+01 -2.67822657D+01 9.05000D-03 0.00000D+00 -3.03700D+0421 HCO 2.901854D+01 1.03133365D+01 1.41000D+02 0.00000D+00 8.67800D+03

The heats of formation H are in units of kcal/mol, and E is in cal/mol (R = 1.9872).

The species equilibrium constants are KP = ATb exp(E/RT) atm (use R = 82.06 atm-

cm3/(K mol) to convert to KC). 4.184 J/cal.

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Table 9. Chemical Mechanism

No. Reaction Aj bj Ej/R (K) Ref.1. O2 + H O + OH 2.00 1014 0.0 8.4551 103 [15]2. H2 + O H + OH 1.80 1010 1.0 4.4416 103 [15]3. H2 + OH H2O + H 1.17 109 1.3 1.8245 103 [15]4. 2 OH H2O + O 6.04 108 1.3 2.766 101 [15]5. H + HO2 2 OH 1.5 10

14 0.0 5.0514 102 [15]6. H + OH + M H2O + M 2.20 1022 -2.0 0.0 [15]7. 2 H + M H2 + M 1.80 1018 -1.0 0.0 [15]8. 2 O + M O2 + M 6.17 1015 -0.5 0.0 [15]9. H + O + M OH + M 6.20 1016 -0.6 0.0 [15]10. H + O2 + M HO2 + M 2.30 1018 -0.8 0.0 [15]11. CO + OH CO2 + H 1.5 107 1.3 3.855 102 [16]12. CO + O2 CO2 + O 2.53 10

12 0.0 2.40036 104 [16]13.1 CO + O + M CO2 + M 1.35 1024 -2.79 2.1085 103 [16]

14.2

C2H4 + O2 2 CO + 2 H2 1.6 1013

0.0 1.5 104

15.3 N2 + M 2 N + M 3.70 1021 -1.6 1.132 105 [17]16. O + N2 NO + N 1.80 1014 0.0 3.8368 104 [18]17. O2 + N NO + O 6.40 10

9 1.0 3.1512 103 [18]18. OH + N NO + H 3.0 1013 0.0 0.0 [18]19.4 N2O + M N2 + O + M 9.13 1014 0.0 2.9036 104 [18]20. N2O + O 2 NO 1.0 10

14 0.0 1.409 104 [16]21. N2O + O N2 + O2 1.0 1014 0.0 1.409 104 [16]22. N2O + H N2 + OH 2.53 1010 0.0 2.2897 103 [16]23. N2O + H N2 + OH (pt. 2) 2.23 1014 0.0 8.4541 103 [16]24.5 CH4 + O2 CO + 2 H2 + O 3.00 1013 0.0 1.500 104

25.6 C3H8 + O2 CH4 + 2 CO + 2 H2 2.30 1013 0.0 1.5000 10426. CO + H2 CHO + H 1.30 1015 0.0 4.52811 104

27. CO + H2O CHO + OH 2.80 1015 0.0 5.28186 104

28. CO + HO2 CHO + O2 6.70 1012 0.0 1.62451 104

29. CO + H + M CHO + M 1.14 1015 0.0 1.20031 103

30. N2O + OH N2 + HO2 2.00 1012 0.0 1.05978 104

kf = AjTbj exp(Ej/RT), units are cm, mol, kJ, and K (R = 8.31451 103).

1Third body efficiencies: CO2 = 3.8, H2O = 12.0, H2 = 2.5, CO = 1.9.2 [C2H4]

0.75[O2]

3Third body efficiency: N = 4.5.4Third body efficiencies: Ar = 0.63, H2O = 7.5.5 [CH4]

0.75[O2]6 [C3H8]

0.75[O2]

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8 Summary and Conclusions

We investigated a first-order method for integrating the rate equations for chemical kinetics.

It is based on an implicit method that is linear in advanced-time quantities. It is compu-

tationally less work than the fully implicit method. The amount of computational effort is

approximately the same as for one iteration of the fully implicit method when solved by thestandard Newton-Raphson method. When compared with the Euler method and a second-

order Runge-Kutta method base on the mid-point rule, the linearized-implicit method has

improved accuracy and stability properties at larger time steps.

Work is in progress to see if this translates into being able to simulate combustion in

simple burners more quickly and with more accuracy than is possible with the explicit meth-

ods. At the time of this writing, a detailed two-dimensional simulation of a laminar Bunsen

burner flame is in progress that couples the full fluid dynamics equations with the skeletal

mechanism for methane combustion and NOx production given in Table 1 of Glarborg, et

al. [19]. This mechanism has 27 species and 77 reactions. I supplemented the reaction set

with 9 additional reactions to aid in thermal ignition. So far the linearized implicit method

is performing well. The midpoint Runge-Kutta method was numerically unstable for t of

20 ns, but the linearized implicit method is stable at 200 ns.

In addition, a one-dimensional laminar H2-air flame has been simulated with the GRI

3.0 mechanism for methane combustion [20], which includes 53 species and 325 reactions.

There is excellent agreement between the linearized implicit method and Runge-Kutta when

run with a time step for which the Runge-Kutta method performed well. Further numerical

experimentation will be needed to find the limits of accuracy and stability in this method.A task for future code development will be to extend this linearly implicit method

to second order by using the linearly implicit algorithm in both the predictor and corrector

steps of the midpoint-rule Runge-Kutta method.

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References

[1] F. X. Timmes, Integration of nuclear reaction networks for stellar hydrodynamics,

Ap. J. Suppl. 124, 241 (1999).

[2] L. D. Cloutman, 1990, COYOTE: A Computer Program for 2D Reactive Flow Simu-

lations, Lawrence Livermore National Laboratory report UCRL-ID-103611, 1990.

[3] D. G. Cloutman and L. D. Cloutman, A Unified Mathematical Framework for Popu-

lation Dynamics Modelling, Ecol. Modelling 71, 131 (1993).

[4] L. H. Aller and S. Chapman, Diffusion in the sun, Ap. J. 132, 461 (1960).

[5] R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport Phenomena (Wiley, New

York, 1960).

[6] S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases

(Cambridge University Press, London, 1952).

[7] J. D. Ramshaw, Self-consistent effective binary diffusion in multicomponent gas mix-

tures, J. Non-Equilib. Thermodyn. 15, 295 (1990).

[8] J. D. Ramshaw, Hydrodynamic theory of multicomponent diffusion and thermal dif-

fusion in multitemperature gas mixtures, J. Non-Equilib. Thermodyn. 18, 121 (1993).

[9] J. D. Ramshaw and C. H. Chang, Ambipolar Diffusion in Multicomponent Plasmas,

Plasma Chem. Plasma Proc. 11, 395 (1991).

[10] J.-L. Tassoul, Theory of Rotating Stars, (Princeton U. Press, Princeton, 1978).

[11] D. D. Clayton, Principles of Stellar Evolution and Nucleosynthesis (McGraw-Hill, New

York, 1968).

[12] G. R. Caughlin and W. A. Fowler, Thermonuclear reaction rates V, At. Data Nucl.

Data Tables 40, 284 (1988).

[13] J. D. Ramshaw, Elements of Computational Fluid Dynamics (Imperial CollegePress/World Scientific, London/Singapore, 2011).

[14] C. K. Westbrook and F. L. Dryer, Simplified reaction mechanisms for the oxidation of

hydrocarbon fuels in flames, Combustion Sci. Tech. 27, 31 (1981).

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[15] C. J. Montgomery, G. Kosaly, and J. J. Riley, Direct numerical simulation of turbulent

reacting flow using a reduced hydrogen-oxygen mechanism, Comb. Flame 95, 247

(1993).

[16] M. T. Allen, R. A. Yetter, and F. L. Dryer, High pressure studies of moist carbon

monoxide/nitrous oxide kinetics, Combust. Flame 109, 449 (1997).

[17] C. Park, Assessment of a two-temperature kinetic model for dissociating and weakly

ionizing nitrogen, J. Thermophysics 2, 8 (1988).

[18] J. Warnatz, Concentration-, pressure-, and temperature-dependence of the flame ve-

locity in hydrogen-oxygen-nitrogen mixtures, Combust. Sci. Technol. 26, 203 (1981).

[19] P. Glarborg, N. I. Lilleheie, S. Byggstoyl, and B. F. Magnussen, A Reduced Mech-

anism for Nitrogen Chemistry in Methane Combustion, Twenty-Fourth Symposium

(International) on Combustion, The Combustion Institute, 1992, pp. 889-898.

[20] G. P. Smith, D. M. Golden, M. Frenklach, N. W. Moriarty, B. Eiteneer, M. Goldenberg,

C. T. Bowman, R. K. Hanson, S. Song, W. C. Gardiner, Jr., V. V. Lissianski, and Z.

Qin. http:www.me.berkeley.edugri mech

[21] L. D. Cloutman, A Note on the Stability and Accuracy of Finite Difference Approx-

imations to Differential Equations, Lawrence Livermore National Laboratory report

UCRL-ID-125549, 1996.

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A Fully Implicit Chemistry Algorithm

Timmes [1] outlines three implicit methods for integrating nuclear reaction networks. The

simplest one is equivalent to the method outlined in Section 5, although he describes it in

slightly different language. Following Timmes, we consider the system of equations

y = f(y). (75)

Define the Jacobian

J =f

y. (76)

Let a superscript n denote the solution at time step n. Then we assume

yn+1 = yn + t f(yn+1) yn + tJ

yn+1 yn

, (77)

where t is the time step. If we let = yn+1 yn, then equation (77) may be rearranged to

(U t J) = t f(yn), (78)

where U is the unit matrix. This linear system may be solved for using any standard

linear system solver, and then yn+1 may be computed.

A fully implicit method may be similarly derived. Define a function g(y()) such that

g(yn+1) = yn+1 yn t f(yn+1) = 0. (79)

If we have an estimate ofyn+1

, sayy()

, then

g(y(+1)) = g(y()) + G(+1)

y(+1) y()

= 0. (80)

The last equality provides the basis for an iterative solution method for yn+1. Unfortunately,

evaluation ofG(+1) = g/y(+1) is as expensive as evaluating J. However, it is likely that

the iterations will converge ifG(+1) is held fixed after being initialized on the first iteration.

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B The Logistic Equation: A Warning

The logistic equation is the basis of a crude but popular and useful model of population

dynamics [3]. If N is the population, then the hypothesis is that it obeys the logistic

equationdN

dt = ALN BLN2 = ALNKL N

KL

, (81)

where KL = AL/BL is the carrying capacity of the environment. The first term on the right

hand side represents the familiar exponential growth of the population via constant difference

between birth and mortality rates. The nonlinear term is a simple ad hoc representation of

an increase in mortality or decrease in fertility due to crowding.

The logistic equation has a very simple general solution. At first glance this equation

appears to have two free parameters, but it can be reduced to a dimensionless equation that

has a self-similar solution with no parameters. Let us make the linear changes of variables

N = N/KL and = ALt. Then equation (81) becomes

dN

d= N N2. (82)

We assume that the value of N(0) is given. Equation (82) is easily solved by separation of

variables:

N() =N(0)

N(0) + [1 N(0)] exp(). (83)

There are two equilibrium solutions, N = 0 and 1.

The equilibrium solution N() = 1 is stable. If the solution is perturbed such that

N() > 1, the solution decays monotonically to unity. If the equilibrium is perturbed by

decreasing the population, N grows monotonically back to unity.

The equilibrium solution N = 0 is unstable. For a positive perturbation, the solution

follows equation (83) to unity as increases. For a negative perturbation, the formal solution

diverges to in a finite time. A negative perturbation is not realizable physically.

The ecologically interesting case is 0 < N(0) < 1. In this case, the solution rises

monotonically to unity at late times. The solution does not overshoot the carrying capacity.

The nature of the solution is independent of the parameters A and B. These parameters

determine the carrying capacity and the time scale for population growth, but not the shapeof the curve of N versus t.

We note that populations obeying equation (81) do not exhibit chaos. Ordinary

differential equations that are local in the independent variable and have no external forcing

terms must be of at least third order to have chaotic solutions.

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7. For A greater than 4, the solution diverges to .

When using the approximation equation (84) to solve numerically equation (82), the

first possibility, A < 1, is not physically meaningful. The second is the range of A that

produces meaningful approximations to equation (83), although accuracy decreases as A

increases. Possibilities 3 through 6 represent bounded solutions to equation (84) that arequalitatively different than the solution to equation (82). The final possibility represents a

traditional numerical instability.

Other ways of differencing the logistic equation show similar types of behavior, al-

though the details can differ considerably. Each particular difference approximation has its

own behavior, but typically there are a region of stability for small time steps, then oscilla-

tory and periodic solutions, chaotic solutions, and eventually solutions that become infinite.

In particular, it is noteworthy that the two second-order Runge-Kutta methods discussed in

previous sections can converge to the wrong steady state value for a limited range of time

steps [21].

So what does all this have to do with combustion simulations? Simple: the finite

difference equations for combustion simulations, with or without coupling with fluid dynam-

ics, are simply very complicated nonlinear iterated maps with t as a free parameter. The

present result suggests that we should not be surprised to see the same progression of solution

types in combustion simulations as in the logistic map: smooth solutions, various oscillatory

solutions, chaotic solutions, and finally instability as t is increased. This behavior has in

fact been observed [21].

Documents

A Numerical Method for Combustion Chemistry