Model for pulverized coal-fired reactors

Eighteenth Symposium (International) on Combustion

The Combustion Institute, 1981

MODEL FOR PULVERIZED COAL-FIRED REACTORS PHILIP J. SMITH, THOMAS H. F L E T C H E R , L. DOUGLAS SMOOT Chemical Engineering Department, 270 Clyde Building, Brigham Young University, Provo, Utah 84602

An axisymmetric, elliptic model has been developed for analysis of confined, turbulent, coal-laden diffusion flames. The scheme is Eulerian for gases and Lagrangian for particles. The approach emphasizes the turbulent fluid mechanics of the mixing-limited gas phase reaction processes. The two-equation (k - ~) turbulence model is used for closure. Particle drag and turbulent particle diffusion are also modeled. Gaseous combustion is modeled with a probability density function for the mixture fraction9 Fluctuations in mixing of inlet streams and coal off-gas are considered. Coal pyrolysis and oxidation reaction processes are assumed to be slow with respect to the turbulent time scale. Particle and gas radiation are incorporated by a flux method. Predictions emphasize the importance of turbulent particle dispersion9

Thermal radiation is solved with a four-flux model, which includes anisotropic and multiple scatter.'2

Introduction Attempts to calculate the detailed performance of turbulent combustion chambers have only been undertaken during the last decade. Previously, both the understanding of these flames and the computational ability to undertake calculations were very limited; the details of the mixing process, local fluctuations, etc. were computed with a macroscopic approach by means of simplified equations. In most cases, these calculations did not give local quantitative details about combustion processes. A great deal of recent effort has been spent on modeling multidimensional, gaseous combustion processes. '-'~ These approaches have been only very recently extended to coal-fired turbulent flames, as reviewed by Smoot." This paper discusses the formulation of an elliptic, axisymmetric, turbulent, pulverized coal conversion model. The model encompasses combustion and entrained-flow gasification. The gaseous fluid mechanics are solved using steady-state techniques a'~'7 and the k-~ turbulence model s for closure. Particle mechanics are solved along Lagrangian trajectories using the PSI-CELL techmque. Gaseous reachons are micrornixing hmlted, " " ~owith transport equations for the mixture fraction and its mean square fluctuation. The shape of a statistical probability density function (PDF) is assumed. Particle property changes are assumed to be slow with respect to the turbulent time scale, and thus devolatilization and heterogeneous rate processes are mcluded. 9

Physical Model Gas Phase Fluid Mechanics The equations for the gas and particle phases are solved separately, with coupling between phases. The time-averaged gas equations are solved in an Eulerian framework. The steady-state, axisymmetrie equation of continuity is: 0 --

Ox

(r~)

=g.~

(U

r Or

The time-mean, axial momentum equation is reduced to: 0

1 b

Ox

r 3r 20 {

11

1285

(r~ ~ ) 3 t / ' ~ + 1 d (~ r 0 ~ / )

7o-7

,o(02)

9

9

10 (~,7) + - - -

+ ----

r~,

r 3r I~,

3 3x

'

Or /

- --

ax

+ - - ~ + ~ k

r

Or + g.,, +

ag

(2)

1286

COAL COMBUSTION

A similar equation is solved for the radial component of momentum. All variables are defined in the nomenclature. The gas-particle coupling is included through the Sp terms. S,m is the mean net addition of mass to the gas phase due to the presence of particles. Sp,, is the mean net addition of axial momentum to the gas phase due to particle drag. These equations are Reynolds-averaged, ignoring terms involving the fluctuating density. This form is essentially the same as the Favre-averaged form with the exception of the derivation of the mean density.~3 Since questions remain about which form of averaging is experimentally measured, ~-~6 both approaches were tried. All equations are shown in the Reynolds-averaged style.

Turbulence Model Turbulence modeling still retains unsolved problems, but of the models presented to date, the k-e 8 model has the widest application for confined, turbulent jets. It has been tested in several reacting and non-reacting flow environments."3'5 It accounts for convective transport of energy while retaining computational efficiency. Defects include the assumed isotropy of the turbulent viscosity which is sometimes unsatisfactory (i.e., strong swirl). The effects of particles, buoyancy, and combustiongenerated turbulence remain unaccounted for. The turbulent viscosity (1~,) is related to the turbulent kinetic energy (k) and its rate of dissipation (~) by:

it, = C~ pk2/~

(3)

Axisymmetric, time-steady transport equations 3.8 are solved for k and ~. The model "constants" in the equations are those of Khalil et al.3 The effect of the particles on the gas phase turbulence t7 is incorporated by decreasing the eddy viscosity by a factor of [1 + p~/p~]-~12 and amounts to only a slight (1-2%) correction in lightly loaded coal dust systems.

ment of this particle history is of major significance. Interactions between the gas and particles add complexities to the particle turbulent diffusion. A lack of understandingof the physical process renders an exact representation of the governing equations impossible. Two basic approaches are being used in recent coal codes. Some TM have treated the particle phase as Eulerian like the gas. The history effect is introduced with several discrete particle classifications. Properties of major concern such as diameter, composition, temperature, or velocity are established for a group of particles. Eulerian equations for each classification with source and sink terms allow particles to change from one group to the next. The second approach treats the particle phase as Lagrangian. Particle trajectories are followed through the flow field and solid properties are changed continuously for each trajectory. Interactions between gas and particle phases are accounted for by source terms in the Eulerian gas field which are updated by the Lagrangian trajectories. 9 Disadvantages in the Eulerian approach include the large number of transport equations needed to handle particle history effects and the amount of computer storage required. The Lagrangian approach introduces problems in obtaining the turbulent diffusion velocity and in obtaining average particle properties for comparison with test results. Our approach is based on the PSI-CELL technique.9 The method does not account for particleparticle interactions and thus would not be applicable to highly loaded, particle-gas flow. Particle velocities, trajectories, temperatures and composition are obtained by integrating the equations of motion, energy and component continuity for the particles in the gas flow field while recording the momentum, energy and mass of the particles on crossing cell boundaries. The net difference in the particle properties between leaving and entering any given cell provides the particle source terms for the gas flow equations. Particle Lagrangian Equations. The Lagrangian equation of motion for a single particle is:

a, d(~.Jdt

= (AppCD(6~

Particle Phase Mechanics Approach. A distinguishing feature among coal combustion models is the treatment of the particulate phase. In most coal reactors, fluid motion is elliptic in nature. In such systems, the gas phase can be considered a continuum. The particulate phase has a typical void fraction near unity, resulting in few particle-particle interactions, and cannot be considered a continuum. Different particle sizes and types may exist at the same location with different properties due to different particle histories. Treat-

- t3<,) 9

le. - ,~,,~1/2 +a ,,~

(4)

The first term on the right-hand side of Eq. (4) depicts the aerodynamic drag force, and the last term represents the body forces. A correlation for the drag coefficient is used after correction for mass efflux. H The total particle velocity is modeled as being composed of a convection velocity and a turbulent diffusion velocity:

,~. = 6

+ ,~.~

(5)

THEORY FOR PULVERIZED COAL CONVERSION MODEL

1287

The convection velocity is calculated from Eq. (4), The diffusion velocity is approximated by a gradient diffusion law:

The Reynolds-mean value is found for any variable [3, which is only a function o f f :

," 1,, = p~b t~,~- - F . ~ ' p~b

~=ap13.+a~ + I i ~ ( f ) P ( f ) d f

(6)

where p~, b.is the bulk particle density. Melville and Bray ~ have proposed the following relationship between the particle and gas phase eddy viscosities:

v.,/v.a = [1 + (t,/tL)]-'

(7)

where t , is the particle relaxation time and tL is the Lagrangian time scale of the gas phase turbulence. An integration of the velocity equation yields the particle trajectory in space and time. The Lagrangian equation of energy for a single particle is:

d(h.a.)/dt = Q.. - Q. - r.h..

(8)

Q,,, is the net radiative heat transfer rate to the particle from the particle cloud, the gases and the walls. Q . is the net rate of convective/conductive heat transfer to the particle and is corrected for particle transpiration. The last term represents the energy lost from the particle due to mass efflux. The particle is composed of specified amounts of raw coal, char and ash: a . = a ~ + a h. + ct.v

(9)

The ash is inert and thus a op is constant. The continuity equations for raw coal and char are:

d % J d t = r~.

(10)

dah~/ dt = rh~

(11)

where rcp and rnp are the reaction rates of raw coal and char. An arbitrary number of particle types is allowed, since approximating a pulverized coal system with a non-dispersed particle size can lead to significant error, m'27 Each particle type may have its own properties such as composition or reaction rate.

Gas Phase Reactions Gaseous fuel comes~from one of two sources: (a) inlet carrier fuel gas, and (b) fuel products from coal devolatilization or heterogeneous combustion. Inlet Fuel Gas. No attempt is made to model the complex turbulent gaseous reaction processes. ~~'~'~9 Instead, mixing-rate-limited reactions2*~~176 are assumed. This requires the solution of a differential equation for the mixture fraction (f). The fluctuations of the mixture fraction are accounted for by choosing a PDF shape and by solving a transport equation for gl (the variance in f), which is modeled after the equations for k and ~.3.8ao

(12)

To obtain Favre-mean variables, a Favre PDF 2'31"a2 must be used. The Reynolds-mean density may also be recovered from the Favre PDF 2.32when required. The gaseous properties (temperature, density, and all species mole fractions) are obtained by straightforward equilibrium calculation on the grounds that the micro-mixing is the limiting rateJ ~ The only required information for a Gibbs free energy reduction scheme for chemical equilibrium is the energy level (h), pressure and the elemental composition.~ In many situations, the standard gas enthalpy is a conserved scalar and in the absence of particles can be calculated from f.32 Even when heat losses are significant, this approach can be used with minor adjustments, a2 The elemental composition is only a function of f. Thus, for gas-fired reactors, or in regions where the presence of particles makes an insignificant contribution, instantaneous gaseous properties are calculated from equilibrium and the mean properties are obtained from Eq. (12). Coal Gas. This source to the gas phase is treated in one of two ways. In the first case, the effect of the turbulent fluctuations on the evolved coal gas is ignored. In the second case the coal gas is allowed to fluctuate according to a prescribed PDF in an approach similar to that already described for gaseous systems. In both cases the coal gas mixture fraction (-q) is defined to follow the evolution of the coal gas throughout the flow field: -q = (local mass of gas originating from the coal)/ (total local gas mass)

(13)

A transport equation of continuity is derived for -q analogous to that for f. However, "q is not a conservative scalar like f, since a particle source term exists which is obtained from the Lagrangian trajectories. This term is the source of gaseous mass from reacting particles and is the term used in the overall continuity equation. The continuity equation for ~qbecomes:

Ox

rar

~x \ ~ r #x /

If the elemental composition of the off-gas from the coal is constant, b k can be calculated directly from lq and f:

1288

COAL COMBUSTION b, = b ~ l + (I - n ) [ f b ~ . + (1 - f ) b ~ l

(15)

where bk~, bk,,, and bk~ represent the mass fraction of element k in the coal, primary and secondary streams respectively. For 13(f, ~1) the time mean properties can be obtained. When "q fluctuations are ignored, the procedure reduces to Eq. (12) but with each 13 calculated from the local value of "n. Where -q fluctuations are included, convolution over the joint probability distribution is required. Since the amount of coal gas at any point is independent of the amount of primary gas, the joint probability density function can be separated:

terchange. The model describes the response of a coal particle to its thermal, chemical and physical environment. Analytical treatment of pulverized coal-char reactions is based largely on independent experiments. Since there are still unresolved questions regarding the kinetics of coal reaction, a general reaction scheme is formulated to accommodate results of future measurements and improved kinetic parameters. An overview of the reaction submodel is presented; further details are available.H'a2 The reacting particle is composed of specified amounts of raw coal, char, and ash. The raw coal, (i.e., the dry, ash-free portion of the virgin coal), devolatilizes by one or more reactions of the form: (raw coal) j---~ Y~ (volatiles) jm + (1 - Yjm)(char) (17)

"~II~l'l~'f J~)PIflPIflI~l'll)dfd'~116)ll Questions can be raised about the PDF for 0, since the coal gas progress variable is affected by the heterogeneous chemistry. In the absence of experimental evidence, a clipped Gaussian distribution with spike function intermittency is used. The variance of the coal gas mixture fraction (g~) is obtained from a transport equation, analogous to that for gs' Intermitteney can be calculated directly using the error function. Gas phase properties are a function of bk, h, p. The enthalpy fluctuations due to the presence of the particles can be accounted for by partitioning the total gas enthalpy into the contributions by "q, by f, and by the residual contributions. The residual fluctuations are ignored and the fluctuations due to "q and f are considered. Variations in coal gas elemental composition can be considered by defining a new "qm for each coal reaction; "qm would then be the local mass fraction of gas originating from the m 'h reaction. Instantaneous local gaseous properties can be obtained from f, ~q~, ~q~ ... 11~. The local mean properties must be obtained by convoluting over the joint PDF for all mixture fractions, P(f, ,q~, "qn.... ~1,,). Intermittency of each mixture fraction must be properly included as before. The added complexity in solving transport equations for each new "q,, and g. increases computational time and storage, which probably nullifies advantages in specification of coal gas composition unless "q fluctuations can be ignored.

Particle Reactions The description of coal reaction processes includes devolatilization, char oxidation, and gas-particle in-

The volatiles react further in the gas phase. A two-step devolatilization mechanism34 has been selected as a compromise between a simple one-step mechanism (e.g., Badzioch, et al.35), which will not account for the effect of particle history on volatiles yield, and a more complex, multistep mechanism~ which would increase computational time, and where rates and processes remain uncertain. The char reacts heterogeneously with reactions (i.e., O n, CO n, HnO, Hn) at the particle surface by one or more reactions of the form: ~br(char) + (oxidizer)/ --->(gaseous products),, (18) The gaseous oxidizer diffuses to the particle surface, absorbs, reacts with char, desorbs and diffuses back into the bulk gas phase. Rate-limiting steps considered for this process are gas phase diffusion and heterogeneous reaction. Recommendations in Smoot and Pratt H are used for values of the experimental rate constants. The required mass transfer coefficient is corrected for particle mass efflux. The heterogeneous reaction is assumed zero order with respect to char concentration, permitting heterogeneous ignition.

Radiation Mathematical calculations of industrial pulverized fuel furnaces have historically centered on computations of radiative heat transfer,n~ The flame is influenced by non-uniform, emitting, reflecting, absorbing surfaces whose optical coefficients are difficult to obtain.37 The governing equations for radiative heat transfer are not amenable to direct solution by finite difference techniques. The zone method is not economically feasible in terms of computational time and storage for incorporation into overall models. Flux methods ~n.~.3o have been developed to en-

THEORY FOR PULVERIZED COAL CONVERSION MODEL hance computational efficiency for use in finite difference methods. A four-flux axisymmetric model that accounts for anisotropic and multiple scattering has been reduced from a general six flux model. "~" The four resulting first order differential radiation equations are combined by the methods of Gosman and Lockwood 4~to form two second order differential equations which are solved by the same technique used for the gaseous fluid mechanics equations.

Solution Procedure

Eulerian Gas Phase The steady-state, second order, non-linear, elliptical partial differential equations to be solved were written in common form, thus requiring one solution technique. The solution of the flow equations in the primitive variables incorporated in TEACH 4~ has been used in this model. It is an iterative, steady-state, finite difference scheme. The hybrid approach 41 uses central differencing when the absolute value of the cell Reynolds Number is less than 2. Otherwise, upwind differencing or donor-cell differencing is used for the convection terms.

Lagrangian Particle Phase The particle-source-in-cell (PSI-CELL) 9 approach has been followed directly. First, the gaseous Eulerian equations are solved without the presence of the particles. The Lagrangian particle equations are then solved for a representative number of particle trajectories. Typically, fifteen trajectories have been chosen, with five starting locations and three particle sizes at each starting location. As the particles traverse any given cell in the flow field, the corresponding source terms are calculated and stored. The Eulerian gas field is then updated by solving the appropriate equations with the computed particle source terms. The Lagrangian particle equations are then solved again in the improved gaseous flow field. The procedure is repeated until convergence is achieved on the particle source terms. The only significant storage for the particle phase is for the source terms. In addition, economy is realized by integrating the particle momentum equation once analytically9 to obtain the particle velocity. Euler's method was sufficient to integrate the particle velocity for the trajectory. Integration of the particle energy and continuity equations is accomplished by standard predictor-corrector techniques. With the incorporation of the particle turbulent diffusion velocity, the trajectories do not follow gaseous streamlines. During the computations, some particles are dispersed .into the recirculation zone and become "trapped." To meet overall continuity,

1289

these particles are allowed to recirculate until they are reacted to ash particles or are of no further interest; they are then artificially allowed to "escape" by setting the turbulent diffusion velocity to zero.

Computations

The model has been applied to reacting and non-reacting systems with and without particles. Smith and Smoot ~o show model theory and evaluation applied to gaseous combustion. They compare theory and probe measurements for most major species in natural gas combustion. H 2 and CO measurements were much lower than the predicted levels in the hot regions of the reactor. Gaseous reactions may be more important in some regions for these species. Reynolds and Favre-averaged predictions for this case have also been compared? 2

0.0

(a)

(b)

r/R

r/R

0.5 '

'

1.0

0.0

0.5

1.0 ).0

-0.0

0.2

).2

0.4

).4

x/L - 0.6

0.8

:- 1 . 0

x/L ).6

9.8

1.0

FIG. l. Plot of particle trajectories showing the effect of turbulent particle diffusion. (a) Mean drag only, turbulent diffusion velocity neglected. (b) Mean drag and turbulent diffusion included. Each computation shows 15 trajectories (5 starting locations, 3 particle sizes at each starting location, 20 Ixm, 50 Ixm, 100 txm).

1290

COAL COMBUSTION

The individual coal reaction submodel has also been evaluated in a one-dimensional framework. 4~ Comparisons with experimental measurements indicate that the model can satisfactorily describe the burnout rate of a multisized coal dust. Figure 1 shows predicted particle trajectories for a pulverized coal-laden system, without combustion. The importance of a turbulent diffusion term for the particles is emphasized by comparing Fig. la with Fig. lb. The method of ejecting "trapped" particles from the recirculation zone is evident in Fig. lb. During the course of the reverse flow path of the third recirculation loop, the turbulent diffusion velocity is set to zero. The particles then follow the gas flow near the wall, until they penetrate the secondary jet, and are swept from the reactor. Predictions for a coal combustor 43are demonstrated in Figs. 2 and 3. Figure 2 shows a perspective of gas temperature throughout the computational domain. The inlet coaxial jets are located in the upper right hand corner of the figure. Mixing of the coal-containing, primary jet with the coaxial secondary air stream is observed before ignition of the coal particles. A fuel rich cold pocket is seen on the centerline downstream of the ignition point. The history of a 40 p.m coal particle passing down the centerline is plotted in Fig. 3a. After particle heatup, devolatilization is rapid, causing a peak in gas temperature. Char burnout progresses slower, with an increase in reaction rate near the reactor exit when the oxidizer concentration on the centerline has increased. The 40 p.m coal particle history of Fig. 3b follows the high temperature ridge of Fig. 2. The increased char oxidation rate is noticeable.

flailS1X';"

will

raw

1.0

coal

.8

"- " ,- e ~ ~, ~ g oE " -

.6

[

.4

9

total char

I

I

I

.25

.50 x/L

.2

aoh

n

t

9

1.0

.75

i 1.0

lal ,.0

x ~ ~ ~ o ~

E lu

" -~

~raw

.8

-

,6

-

,4

-

.2

-

0

coal

total

t 0

.25

.SO x/L ib]

FIG. 3. Particle mass histories for 40 Ixm coal particles along: (a) centerline trajectory and (b) trajectory near the peak gas temperature ridge of Fig. 2, Conclusion

Past modeling of gaseous combustion processes has provided the basis for a time-steady, two-dimensional pulverized coal flame model. The history effect of the discrete particle phase has been incorporated by following Lagrangian trajectories. This effect was shown to be particularly important for recirculating systems. Gaseous combustion is modeled by assuming mixing-limitedreactions with local chemical equilibrium. Statistical PDF's are used to account for turbulent fluctuations of both the inlet fuel gas and the coal off-gas. Coal particle devolatilization and heterogeneous reaction kinetics have been included. The inlet fuel gas combustion submodel and the coal reaction submodel have been independently evaluated. Turbulent particle dispersion was shown to have a dramatic effect on the particle flow field. Additional validation studies must be performed by comparing experimental measurements with model predictions.

Nomenclature

F[c. 2. Three-dimensional projection of gas temperature in an axisymmetric, laboratory-scale, pulverized coal combustor.

I

.75

A b

Cross sectional area Element mass fraction

THEORY FOR P U L V E R I Z E D COAL CONVERSION M O D E L C f g h / k L 19 p Q r R S t T u v x Y a 13 F ~q IL v p o 6

Constant, drag coefficient Mixture fraction Gravitational acceleration, mean square fluctuation Enthalpy Mass flux Kinetic energy of turbulence Total reactor length Pressure Probability density function Heat transfer rate Radial direction, reaction rate Total reactor radius Source term Time Temperature Axial velocity component Velocity vector, radial velocity component Axial direction Stoichiometric coefficient Mass, intermiRency Arbitrary property Diffusion coefficient Dissipation rate of turbulent energy Coal gas mixture fraction Viscosity E d d y viscosity Density T u r b u l e n t Schimdt or Prandil Number Stoichiometric coefficient

Subscripts and Superscripts a b c d D e f g h i j k g L m p r s t u ~1 I~ --* "

Ash Bulk Convective, raw coal Diffusive Drag Eddy Mixture fraction Gas Enthalpy, char Intermittency of inlet fluid Particle type or size index Element Heterogeneous reaction index Lagrangian Mass, reaction index Particle, primary Radiation Secondary Turbulent Due to axial velocity Coal gas mixture fraction Viscosity Vector Reynold's Mean Relaxation

1291

Acknowledgments This work was supported financially by the U.S. Department of Energy, and the Electric Power Research Institute. Significant contributions of Dr. David T. Pratt (University of Michigan) and Dr. John J. Wormeck (University of Utah) are gratefully acknowledged.

REFERENCES 1. SPALDING,D. B., Combustion Sci. and Technol., 13, 3 (1976). 2. BILGER, R. W., Prog. Energy Combust. Sci., 1, 87 (1976). 3. KHALIL, E. E., SPALDING, D. B., AND WHITELAW, J. H., Int. 1. Heat Mass Transfer, 18, 775 (1975). 4. BUTLEB, T. D. AND O'RoUBKE, P. J., Sixteenth Symposium (International) on Combustion, p. 1503, The Combustion Institute, 1976. 5. GOSMAN, A. D., LOCKWOOD, F. C., AND SALOOJA, A. P., Seventeenth Symposium (International) on Combustion, p. 747, The Combustion Institute, 1978. 6. SMOOT, L. D., Eighteenth Symposium (International) on Combustion, The Combustion institute, 1980. 7. GOSMAN, A. D., PUN, W. M., RUNCHAL, A. K., SPALDING, D. B., AND WOLFSHSTEIN, a., Heat and Mass Transfer in Recirculating Flows, Academic Press, 1969. 8. LAUNDEB,B. E. AND SPALDING, D. B., Mathematical Models of Turbulence, Academic Press, 1972. 9. CnowE, C. T., SHARMA, M. P. AND STOCK, D. E., Fluids Eng., 99, 325 (1977). 1O. SMITH, P. J. AND SMOOT L. D., submitted to Combustion and Flame (1979). 11. SMOOT, L. D. AND PaATT, D. T., Pulverized Coal Combustion and Gasification, Plenum, 1979. 12. VAaMA,S. A., Pulverized Coal Combustion and Gasification, (L. D. Smoot and D. T. Pratt, Eds.), p. 83, Plenum, 1979. 13. BmGEB, R. W., Combustion Sci. and Technol., 11, 215 (1975). 14. GOSMAN, A. D., LOCKWOOD, F. C. AND SYED, S. A., Sixteenth Symposium (International) on Combustion, p. 1543, The Combustion Institute, 1976. 15. BaAY, K. N. C., Seventeenth Symposium (International) on Combustion, p. 223, The Combustion Institute, 1978. 16. BILGEH,R. W., Prog. Astronautics and Aeronautics, 53, (L. A. Kennedy, Ed.), p. 49, AIAA, 1977. 17. OWEN, P. R., 1. Fluid Mech., 39, 107 (1968). 18. BLAKE, T. R., BROWNELL, D. H. JB., GARG, S. K., HERLINE, W. E., PRITCHETT, J. W., AND SCHNEYER,

G. P., "'Computer Modeling of Coal Gasification

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19.

20. 21.

22. 23. 24.

25. 26.

27.

28. 29. 30.

31.

COAL COMBUSTION

Reactors, Year 2," Annual Report for the U.S. Energy Research and Development Administration, Report No. FE-1770-32, 1977. HINZE, J. O., Prog. Heat Mass Transfer, 6, (G. Hetsroni, S. Sideman, and J. P. Hartnett, Eds.) p. 433, Pergamon Press, 1972. HINZE, J. O., Turbulence, Second Edition, McGraw-Hill, 1975. GOLDSCHMIDT, V. W., HOUSEHOLDER, M. K., AHAMADI, G., CHUAN, S. C., Prog. H e a t M a s s Transfer, 6, (G. Hetsroni, S. Sideman, and J. P. Hartuett, Eds.), p. 487, Pergamon Press, 1972. Yuu, S., YASUKOUCHI,N., HIBOSAWA,Y. ANDJOTAKI, T., A1ChE lournal, 24, 509 (1978). LILL'Z, G. P., Ind. Eng. Chem. Fundamentals, 12, 268 (1973). PUN, W. M.: ROSTEN H., SPALDING, D. B., ANn SVENSSON,U., "Calculation of Two-Dimensional Steady Two-Phase Flows," presented at International Symposium on Two Phase Systems, Florida, 1979. MELVILLE,E. K. AND BRAY,K. N. C., Int. 1. Heat Mass Trans., 22, 647 (1979). FIELD, M. A., GILL, D. W., MORGAN B. B., AND HAWKSLEV,P. G. W., Combustion of Pulverized Coal, the British Coal Utilization Research Association, 1967. GEORGE, P. E., LENZFB, R. C., THOMAS, J. F., BABNHARD,J. S., AND LAURENDEAU,N. M., "Gasification in Pulverized Coal Flames," Second Annual Progress Report, ERDA-FE-2029-6, 1977, BORGHI, R., Adv. in Geophys., 1813, 349 (1974). SPALDING,D. B., 1. Energy, 2, 16 (1978). BECKER, H. A., Fifteenth Symposium (International) on Combustion, p. 601, The Combustion Institute, 1974. KENT,J. H. AND BILGEB, R. W., Sixteenth Sympo-

slum (International) on Combustion, p. 1643, The Combustion Institute, 1976. 32. SMITH, P. J., "Theoretical Modeling of Coal and Gas Fired Turbulent Combustion and Gasification Processes," Ph.D. Dissertation, Brigham Young University, Provo, Utah, 1979. 33. PRATT, D. T., Studies in Convection, (B. E. Launder, Ed.), Vol. 2, p. 191, Academic Press, 1977. 34. UBHAYAKAR,S. K., STICKLER, D. B., VON ROSENBERG, C. W., AND GANNON, R. E., Sixteenth

Symposium (International) on Combustion, p. 427, The Combustion Institute, 1976. 35. BADZIOCH, S., HAWKSLEY,P. G. W., AND PETER,

C. W., Ind. Eng. Chem. Process Design and Development, 9, 521 (1970). 36. HOTTEL, H. C. AND SAROFIM. A. F., Radiative Transfer, McGraw-Hill, 1967. 37. SAROFIM, A. F. AND HOTTEL, H. C., "Radiative Transfer in Combustion Chambers: Influence of Alternative Fuels," Sixth International Heat Transfer Conference, Toronto, Canada, 1978. 38. CHU, C. M. ANDCHURCHILL,S. W., 1. Phys. Chem., 59, 955 (1955). 39. GOSMAN,A. D. AND LOCKWOOD,F. C., Fourteenth

Symposium (International) on Combustion, p. 40.

41. 42. 43.

661, The Combustion Institute, 1972. GOSMAN,A. D., ANt)PUN, W. M., "Lecture Notes for Course Entitled 'Calculation of Recirculating Flow,' " Imperial College, London, England, 1973. SPALnING,D. B., Int. J. Num. Methods in Eng., 4, 551 (1972). SMITH, P. J. AND SMOO'r, L. D., to appear in Combustion Sci. and Technol. (1980). THUBGOOO, J. R., SMOOT, L. D. AND HEt)MAN, P. 0., Combustion Sci. and Technol., 21, 213 (1980).

COMMENTS C. 1. Lawn, C.E.G.B., Matchwood Engineering Labs, England. Your model for particle dispersion implies that once a particle gets into the recirculation zone, it is trapped and must burn out there. As a consequence the residence time distribution at furnace exit must be far from realistic. Do you see this as a significant defect of the model?

Author's Reply. The method for ejecting "trapped" particles from the recirculation zone discussed under Computations in the paper is admittedly somewhat arbitrary. In all computations performed to date the mass proportion of recirculated particles has been small enough that errors in overall

residence time distribution at the furnace exit have been insignificant. This will obviously not be the case for all flow configurations. The need for an improved turbulent particle dispersion model is thus emphasized.

F. C. Lockwood, Imperial College, England. Since the volatiles are mainly released in a relatively cool, fuel rich region, they will not immediately combust, Some premixing will occur. Does your method handle this premixing and the ignition of and subsequent burning of the mixture.

THEORY FOR PULVERIZED COAL CONVERSION MODEL

Author's Reply. The present gas phase c o m b u s tion model does not account for any premixing of fuel and oxidizer as would an "eddy break-up" model. Rather, we have e m p h a s i z e d turbulent fluctuations in the a m o u n t of coal off-gas. A model w h i c h included both would obviously be more comprehensive.

P. Wood, Michigan State University, USA. Is your only modification to the k - e model due to the presence of the particles incorporated in the eddy viscosity, or do you alter the k - eqn (for example,

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by including an additional dissipation term linear in the particle loading).

Author's Reply. Our only modification to the k ~ gas phase turbulence model due to the presence of the particles is in the eddy viscosity as noted in the discussion on the turbulence model in the paper. An additional dissipation term in the k-equation is certainly warranted. However, we do not have the necessary fundamental m e a s u r e m e n t s to establish the constant ("universal" or otherwise) for s u c h a term, nor are we aware of any s u c h data available in the literature. Such information will hopefully be forthcoming. -