Particle modelling of combustion

Donald Greenspan

Depurtment of Mathematics, University of Texas ut Arlington, Arlington, Texas, USA

Keywords: combustion, flame, molecular model

Introduction

Interest in combustion is as old as humankind’s ac- quaintance with fire. However, only relatively recently have chemists, physicists, mathematicians, and engi- neers made extensive analytical and experimental studies of the subject. Of late, environmentalists have become additional interested participants.

Current theories of combustion are primarily con- tinuum theories, even though it is understood clearly that combustion is a noncontinuum, molecular phe- nomenon (see, for example, Refs. l-3 and the numer- ous references contained therein). The equations stud- ied are relatively small systems of differential equations that relate the reaction of a physical system to the release of chemical energy. Major areas or inquiry re- late to fuels, ignition, heat transfer, mass transfer, flames, explosions, equations of state, enthalpy, equilibrium, turbulence, and conservation laws.

With the advent of modern computer technology there has, however, been growing interest in molecular

modelling4.5 and particle (molecular type) modelling.6,7 The equations of these models are large systems of nonlinear, ordinary differential equations that incor- porate or approximate the dynamics of classical mo- lecular mechanics. These systems have to be solved numerically.’ Our purposes in this paper are to dem- onstrate qrralitatively the feasibility of simulating com- bustion phenomena by this new type of modelling and to derive intuition for later quantitative modelling.

In general, the parameter choices discussed throughout are results of computer experiments that yielded flame motions similar to those described in physical experiments.”

Model formulation

Consider a premixed system of 1080 particles P,, i = 1, 2, . . . . 1080, each of unit mass, arranged in nine rows of I20 particles each, with respective coordinates x(i), y(i) given by

x(l) = 0.0, y(l) = 0.0, x( 121) = 0.5, y(l21) = 0.866

x(i + I) = I.0 + x(i), y(i + I) = y(l), i= 1,2,...,119

x(i + I) = I.0 + x(i), y(i + I) = y(l21), i= 121,122,....239

x(i) = x(i - 240), y(i) = 1.732 + y(i - 240), i=241,242,...,1080

Such a system can be studied economically and within reasonable time frames on the computational and graphics equipment available to us. These positions are the vertices of a regular triangular grid whose basic triangular unit has edge length unity. The particles, shown in Figure I, are bounded by a rectangular region that is 120 units long and 6.928 units high.

Address reprint requests to Dr. Greenspan at the Department of Mathematics, University of Texas at Arlington, Box 19408, Arling- ton, TX 76019-0408, USA.

Received 5 March 1990; accepted 11 September 1990

To ensure that the triangular particle arrangement is of no significance in the simulations to be described, the initial particle velocities ux(i), uy(i) are assigned in a relatively random fashion in terms of parameters u and V as follows:

ux(i) = uy(i) = u, i= 1,2,...,119

ux(i) = uy(i) = u, i = 121,122,. . . ,239

ox(i) = -uy(i) = -u, i = 241,242,. . . , 1080

which are then modified by

ux(5i) = -uy(4i) = ux(4i) = -uy(5i) = u, I = 1,2,...,200

104 Appl. Math. Modelling, 1991, Vol. 15, February 0 1991 Butterworth-Heinemann

Particle modelling of combustion: D. Greenspan

Examples

In each of the following examples the simulation was carried out on a CRAY X-MP/24 through tsooo. Each example required only 274 seconds of CPU time.

and which are further modified by

ox(i) = v, oy(i) = 0 if -u(i) 5 2. I

The choice of u will be much smaller than that of V. The particle velocities in the range x(i) 5 2.1, which is the shaded region in Fig:lrre 2, will be chosen to simulate ignition.

We assume that any particle Pi is acted upon only by those particles Pi that are within 2 units of P,. The force of interaction will be of a local molecular type, with magnitude F given by

(1)

in which r,; is the distance between P; and P,. When u and V have been prescribed, the motion of the system will be determined by Newtonian dynamics, with the resulting 1080 second-order, nonlinear vector equa- tions being solved numerically by the leapfrog formulas’ with At = 0.0001.

A major computational advantage of formulas of the form (I), in which the exponents of attraction and re- pulsion are both odd, is that no square root routine is ever required.

During the simulation the particles will fall into three distinct categories: combustion. fuel, and burned par- ticles. Before ignition, all 1080 particles are assumed to be fuel particles. If and when a fuel particle’s speed exceeds I2 units, it will be considered a combustion particle. At the time of ignition the parameters L’ and V will be prescribed so that particles in the range 0 5 x 5 2.1 will be combustion particles while all others are fuel particles. A combustion particle will have its velocity increased by the factor C every T time steps until its speed exceeds 300, at which time its velocity will be reset to zero and it will be considered to be a burned particle.

Finally, we will allow symmetric particle reflection from the walls. However, the velocity of each reflected particle will be multiplied by the positive parameter d. A choice d < I will imply transmission of heat to the walls.

We consider next examples for various choices of parameters v, V, C, T, and d.

Figure 1. The initial particle configuration

Y

X

Figure 2. The initial ignition area at the left-hand end

Case I

Consider the parameter choices u = 3, V = 60, C = 1.75, T = 50, d = 0.90. Figures 3(a)-3(h) show the combustion particles at every 1000 time steps through ~~~~~~~~ at which time there remains only one particle. Figure 3(i) shows the fuel particles at txooo. One can say that the combustion was relatively efficient, since only IO fuel particles have escaped combustion. Figure 3(j) shows the burned particles at time txooo. This con- figuration is typical of all the cases yet to be discussed.

Crlse 2 Consider this time the parameter choices u = I,

V = 80, C = 1.5, T = 70, d = 1.0. All the parameters are different from Case I. The motion is shown in Figures 4(a)-4(h). The number of combustion particles is greater at each time step than in Case I, one factor

(a) t1000

i .

(b) t2000

I [ (c) t:OOO

i I

(d) t4cOo

I I (e) t5000

1

* rD I- l .

(d t?OOO

i .

Ch) tsooo

1 _ . -. . 1

(1) t8000

(3) tgooo Figure 3. v = 3.0, V = 60.0, C = 1.75, T = 50.0, d = 0.9

Appl. Math. Modelling, 1991, Vol. 15, February 105

Particle modeling of combustion: D. Greenspan

(a) t1000

Cb) t2000

t 1

(c) t3000

I

Cd) tbm

I t u - . l 0

(f) t6m

I r

l . r- . l &oh ’ 3i

(g) t7000 . I

Ch) %Oco

. . .’ .

(I) 53000

Figure4. v=l.0,V=80.0,C=1.5,T=70.0,d=1.0

being that the parameter C is greater in the present case. Figure 4(i) shows the remaining fuel particles at ~sooo.

Cuse 3 The parameters this time are u = 1, V = 80, C =

1.75, T = 70, d = 0.9. The combustion particles are shown in Figures 5(a)-5(f) through f6,j0,1, after which there is burnout. Figure 5(g) shows the remaining fuel particles after the rapid and efficient combustion of this case.

Case 4 The parameters this time are u = 3, V = 80, C =

1.75, T = 70, d = 1 .O. The rate of combustion increases over that of Case 3 and is shown in Figures 6(a)-6(d) through tdooo, after which time there is burnout. Figure 6(e) shows the remaining fuel particles at ts(j(j,).

Case 5

Figures 7(a)-7(b) show the most rapid combustion exhibited thus far and the fewest fuel particles re- maining after burnout at time tSooo. The combustion in this case is relatively explosive, yet the parameters were the same as for Case 1 with the single exception of d = 1.0, indicating the significance of the material structure of the walls.

. . . l l ___ .+= Cd) ‘m

(g) t@JocJ

Figure 5. Y = 1.0, V = 80.0, C = 1.75, 7 = 70.0, d = 0.9

I .*C -. ~ . . ..* _

- 9 .* ‘0 ._

(c) t3000

. I

Cd) t,,m

I - . I

Figure6. v=3,V=880,C=1.75,7=70,d=l.O

(a) t1000

a . . . .

l .

(b) t2OOo

i 4

(cl tg3oo

Figure 7. v = 3.0, V = 60.0, C = 1.75, 7 = 50.0, d = 1.0

106 Appl. Math. Modelling, 1991, Vol. 15, February

g= ‘^ i Cc) t3000

i l 8 . . I

Ce) t5000

I-- -- . a-- T

L..___ l ‘_ -3 j (8) t7000

i ---‘.‘t

Ch) %CO

_._.._. . . -.:. - __. *.. .

: *. . . 1. ;.. . _ . :* :‘. . . . -sI 1: ‘4’

(I) t8000

Figure 8. v = 1.0, V = 60.0, C = 1.75, T = 30, d = 0.9

Casr 6

In this final case the parameters are c = I. V = 60. C = 1.75, T = 30, d = 0.9. As is seen in Figrlrcs 8(u)-&(h), the result is a relatively small combustion areaat each time step through fXO,H,, after which burnout occurs. Figure cl(i) shows the fuel particles that remain at tnocx, and indicates the relative inefficiency of the parameter combination L’ = 1 with V = 60.

The totality of computer examples run consisted of

Particle modelling of combustion: D. Greenspan

all 48 possible cases with L’ = I, 3; V = 60. 80; C = I .5. 1.75; T = 30, 50, 70; and d = 0.9. 1 .O. In general, Cases l-6 were typical of the results.

Remarks

Though the general ideas developed in this paper can be applied to combustion in solids, liquids, and gases. the examples were motivated by the wide interest in cylindrical combustion engines. Mathematically, the related vapor combustion is exceptionally difficult to model because it is of a turbulent nature and not usually accessible through continuum models. However, the turbulent character of the particle combustion in Fig- I4rc.v 3-8 is in agreement with the turbulent profiles described by Kanury.’

It is unfortunate that Figurrs 3-8 represent only frames of system motions. The actual motions should be shown dynamically as very rapid successions of many such frames, that is, as moving pictures. Journal publication is not the best vehicle to display the rele- vant dynamics. Indeed, color motion pictures would be best.

Acknowledgments

Computations were performed at the University of Texas Center for High Performance Computation.

References

Hirschfelder. J. 0.. Curti\s. C. F. and Bird. K. B. Mo/v(,r,/tr/ T/rcory (!/‘G‘tr.tc.y od Liyltitls. John Wiley. New York. 1965 Kanury. A. M. In/txduc.tiorr IO C‘o~~~hr~vtim Phc~twr~c~utr. Gor- don and Breach. New York. 1975 Zeldovich. Y. B.. Barenhlatt. G. I.. Lehrovich. V. B. and Makh- viladze. G. M 7%ca Matlwttrcrticwl 7’/rcory o/’ Comhrr.sticm cod Erplo.~io~.s. Consultants Bureau. New York. 19X5 Greenapan. D. Supercomputer simulation of colliding micro- drops of water. C‘c~tnprrt. Mrrrlt. ApIt/. IWO. 19, 91 Hoover. W. G. Computer simulation of many body dynamics. Pfrv.tic~\ T~xlo~. January 1’384. 44 tircenspan. D. /)i,s(.wrc Muxlc~lv. Addison-Wesley. Keading, Mass.. 1972 Greenzpan, D. Ari/hmc,tic~ Applird Mtr/hc,mrr/ic~.s. Pergamon, Ox- ford. En&ind. 1%)

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