Modeling of dynamics, heat transfer, and combustion in two-phase turbulent flows: 2. Flows with heat transfer and combustion

ELSEVIER

Modeling of Dynamics, Heat Transfer, and Combustion in Two-Phase Turbulent Flows: 2. Flows with Heat Transfer and Combustion L. I. Zaichik V. A. Pfrshukov M. V. ~ o z e l e v A. A. Vinberg

Institute for High Temperaturesof the Russian Academy of Sciences, Moscow, Russia

• The objective of this part of the paper is to summarize the information concerning the authors' works in the field of simulation of two-phase gas-particle turbulent flows with heat transfer and combustion. A kinetic equation had been derived for the probability density function (PDF) of the particle velocity, temperature, and mass distributions in turbulent flows. This PDF equation is used for the construction of the governing conservation equations of mass, momentum, and heat transfer in the dispersed particle phase. The numerical scheme incorporates two-phase fluid dynamics, convective and radiative heat transfer, and combustion. The proposed models have been applied to the calculation of various particle-laden turbulent flows in jets, combustion and gasification chambers, and furnaces. © Elsevier Science Inc., 1997 Keywords: mathematical model, kinetic equation, probability density function, turbulence, particle, gas, combustion, gasification, radiation, jet, chamber, furnace

INTRODUCTION This paper is a continuation of an earlier paper by Zaichik et al. [1] and is based on its results. The purpose of the paper is to present the mathematical models for the calculation of two-phase gas-particle turbulent flows with chemical reactions and combustion. Theoretical models for heat transfer simulation of twophase turbulent flows have been developed to a lesser degree in comparison with momentum transfer modeling. As a rule, the simulation of nonisothermal particle-laden flows is carried out by using the Lagrangian approach or the algebraic local equilibrium Eulerian models--see, for example, Derevich et al. [2] and Shraiber et al. [3]--for a description of turbulent heat transfer in the dispersed phase. The second-moment order modeling of particle heat transfer was applied by Vinberg et al. [4] to the calculation of turbulent gas-particle jet flows. The models presented in this paper are based on a kinetic equation for the probability density function (PDF) of the particle velocity, temperature, and mass distributions in turbulent flows. Integration of the PDF equation with respect to all particle velocities, temperatures, and masses gives the conservation equations of mass, momentum, and heat transfer for the dispersed phase. The heat flux in the particle phase is expressed by means of the

Boussinesq approximation as well as the second-moment closure. The turbulent characteristics of the gaseous phase are calculated on the basis of the k-e turbulence model accounting for the modulation effects due to the addition of particles and combustion. Because of high temperatures induced by combustion, the governing system of equations has to include the radiation transfer. So, the numerical scheme incorporates two-phase fluid dynamics, convective and radiative heat transfer, and combustion. The proposed models have been used for calculations of various gas-particle turbulent nonisothermal flows in jets, combustion and gasification chambers, and furnaces.

EQUATIONS OF H E A T TRANSFER Variations in the temperature and mass of a single particle are described as dOp dr

t -- Op - + Q, Zt drop

d~-

= n,

(1)

(2)

Address correspondence to Dr. Leonid Zaichik, Institute for High Temperatures, Russian Academy of Sciences, Krasnokazarmennaya 17a, Moscow 111250, Russia.

Experimental Thermal and Fluid Science 1997; 15:311-322 © Elsevier Science Inc., 1997 655 Avenue of the Americas, New York, NY 10010

0894-1777/97/$17.00 PII S0894-1777(96)00201-4

312

L.I. Zaichik et al.

where the particle thermal relaxation time is given by

Cppp#~ Tt

6ANUp

2 + 0.6Relp/ZPr 1/3

Nup

(3)

Here Op and m_ are the temperature and mass of the particle, t IS the actual temperature of the gaseous phase, pp and Cp_ are the density and the heat capacity of the particle material, dp is the particle diameter, A is the heat conduction of the gaseous phase, N u p a n d Re_ are the particle Nusselt and Reynolds numbers, Pr is th~ Prandtl number, and ~ is the rate of particle mass variation due to combustion• The first term on the right-hand side of Eq. (1) defines a particle temperature variance due to interphase heat transfer, and the second term, Q, is the thermal source caused by combustion and radiation. The thermal particle-particle interaction as well as the combustion rate and radiation flux fluctuations are neglected here• Thus, it is supposed that the existence of the particle temperature fluctuations is induced only by the thermal interaction between particles and random gas temperature field. Equation (2) describes a variance of the particle mass due to combustion• The PDF of the particle velocity, temperature and mass distributions is introduced as •

.

kI

tJ

.

P(t,.?,J,O,m) : (6(?-/~p)6(F

f . = 1 - exp( - Tu/%),

ft = 1 - e x p ( - Tt/7"t) , (6) gu = Tu/% - 1 + e x p ( - T u / % ) ,

gut = Tut//Tu ] d- e x p ( - Tut/~'u)• i' The conservation equations of mass, momentum, and heat transfer of the dispersed phase are derived from the appropriate moments of Eq. (5) taken over all particle velocities, temperatures, and masses. The equations describing the mean and pulsating motion of particles-namely, momentum and the second and third moments of fluctuating velocity--were given in the paper by Zaichik et al. [1]. The balance equations of mass and enthalpy are written as --

az

a z vk

--

O.r

-- mp)>.

(4)

Assuming the turbulent gas velocity and temperature fields by Gaussian random processes with a finite time of correlation and taking into account Eqs. (1), (2), and (5) from the paper by Zaichik et al. [1], we can write a kinetic equation for the PDF in the form

]Cut = 1 - exp( - Tut/7"u) ,

ftu = 1 - e x p ( - T u t / T t ) ,

Up)

-

X t S ( O -- O p ) 8 ( m

Integrating Eq. (5) over all particles temperatures and masses, one can obtain Eq. (6) from the paper by Zaichik et al. [1] for the particle velocity distribution• The coefficients fu, )rut, ftu, ft, gu, and gut in Eq. (5) determine the response of particles to the turbulent velocity and temperature fluctuations of the continuous phase• By means of the stepwise approximation of the temporal correlation functions of gas velocity and temperature fluctuations along the particle trajectories, the coefficients are presented as [5]

+

ce,

Ox k

aZH~ aZV~H. -{- _ _

-

0~"

Oxk = _ aZCpp(V'kO')

Oxk +

CppTUgut(U,t, ) OZ

Tt

ZCpp(T - 6))

3Xk Oqpk

te A H - - - ,

"1"t

--+v,--

0~"

+--

Ox k

- - + F

Ov k

+

k P

"ru

+ Q P

+

-=-(riP)

= f° a2P Tu

Ov i Ovk

+

+

-aU k O0

T u

ft

,2

oZP

+ "~tt( t ) "~'~ + g u( U'iU'k) 02P x

-

-

OV~

02P

+

O0

02P )

+ - -

O X i 0 Uk

OX k

0 Ui 0 Un

OXk Ov i O0

+ --gut

where ce is the rate of interracial mass transfer due to combustion, H p is the averaged enthalpy of the dispersed phase, (v'kO') and (u'~t') are the turbulent heat fluxes in the dispersed and gaseous phases, A H is the change of enthalpy by transition of substance from the solid phase into the gaseous phase due to combustion, and qpk is the radiation heat flux in the dispersed phase• It is seen from Eq. (8) that variations in the particle enthalpy are attributed to a turbulent heat flux in the dispersed phase because of the entrainment of particles into pulsating motion, the diffusion transfer of temperature fluctuations, the interphase heat transfer, and the combustion and radiation transfer• The equation for the correlation between the pulsations of velocity and temperature (i.e, for the particle turbulent heat flux) is given by

%

o2P X

-

+

Ox k O0

O'ik

+

-

oV~

O

2p

ru Ovi Ovk

d2p

Ox k Ov n O0

= -z

O0 02P ) + - - - -

(8)

OX k

( O) Z O-c + Vk Oxk

%

(7)

o6) -

Ox k

z

3x k aVe

Oxl,

Ox k 002

+z(ft--'~u+~t)-z(TU+~'tt" Tu \ "/'u"/'t ] + J[P].

(5)

(9)

Nonisothermal Two-Phase Turbulent Flows 313 The equation for the third moments of particle and temperature pulsations included in Eq. (9) is given by using the representation of the fourth moment of fluctuations as the sum of the products of the second moments:

o(@;o')

a(@;o')

+ v~

O~"

a(1 - dP)pCp(U'it') Ox k

ovj + (v~v'ko')--

Ox k

+(1

Ox k

-

~)

ZCpp(T-

av,. ao + (~'~o')ox----k~ + (~i~'~)ax~

+ U k - -

07"

OX k

6))

Cpp'ru +

Tt

OX k

Tt

r

3qgi

+(1 - (I))pe + a~AH +((viva)

gu(~',u', ) ) a( ~jo'______~)

+

o(v;O') axk

ox,

+ -

+

+

1)

, , ,

TuTt

(v~v~O') = O.

C~(U;Or)

[

aXk

--

+

OZ

OqXi.

(1

Uk

OXk

(1 - - ~ ) ) p

( u'iu'D

(16)

-

-

OX k au,.

(I))p /'t O"k

-

Ox k

OXk

2 T~k ap aPg

o 1

.....

Prt

The balance equations for the turbulence energy and its dissipation rate of the gaseous phase with combusting particles are written in the form

a(vfO') ,.

(15)

(10)

Tu + 2~-----~ t ((v~v~) + gu(U'iU~))

+((vjv'~) + g,(u)u'k))

.

vt OT (U'it')

The algebraic approximation for the third moments obtained from Eq. (10) has the form (viv)O )

3x k

Here P_ is the pressure in the gas, H is the averaged enthalp~ of the gaseous phase, and qgi r is the radiation heat flux in the gaseous phase. The turbulent stresses in Eq. (14) are determined by means of Boussinesq's approximation--see Eqs. (49) and (50) in the paper of Zaichik et al. [1]. The turbulent heat flux in Eq. (15) is given by

3x k +((v;v'k) + gu(U)U'~))

aZ gut(U'kt')--

+ (V'kO') + -~tgu~,Ukt 2 ) ~ ] -

3 p2 ax k ax k

2(1-f~)a~ ( 1 - (I))p '

Ak

(11)

Algebraic expression for the particle turbulent heat flux can be written as a result of a simplification of Eq. (9) in the form of the Boussinesq approximation: = _[(~'ufut_+ rtftu)Vt 2"ru'rtkp ] d O (v~O') t (~'u + "rt)Prt + 3(% + %) ~x~" (12)

(17) O~ Or

3~ 3x k

1 (1-¢P)p

(c \

a [

_.

vt a e ]

3x k ( 1 - - q ) ) p - ~ - ~ X k ]

, , age el(UiUk) Ox k

+

2(1 - L ) e t e The conservation equations of mass, momentum, and enthalpy for the gaseous phase are a(1 - (I))o

+

o(1 - ~)ou~

Or

ae,

Ox k

o(1 - ~I,)pUi

o(1 - ~)pU~Uk

+

Or

3x k

'I))o(u'~u'k)

0(1 -

- (1 - O ) OPg

Ox k z(u,

-

Zu

vi)

Ox k OZ + gu(U'iU'k) - Ox k

+ ( 1 - a,)pF~ + , e E ,

0(1 - q ) ) p H O~"

(13)

+

a(1 - Cb)pUkH Ox k

(14)

- A,

(1 - (I))p

(18)

For isothermal flow Eqs. (17) and (18) reduce to the k-e turbulence model given by Zaichik et al. [1]. The last source terms in Eqs. (17) and (18) account for the turbulent modulation effect because of particle combustion. The boundary condition for the dispersed phase energy [Eq. (8)] has to take into consideration a possibility of particle temperature change because of particle-wall interaction. So a probability density relating the particle temperature transition from a situation before a collision with the surface (index 1) into a situation after a collision with the surface (index 2) is defined as 7ro(2/1) = 6 [ 0 2 - Tw - 4 ' o ( O 1 - T w ) ] , (19) where the thermal particle-wall interaction is accounted for by means of the temperature restitution coefficient 4,0. Using Eq. (19) and assuming a quasi-normal distribution of the normal velocity component in the near-wall space, we can obtain the following boundary condition for

314

L.I. Zaichik et al.

the dispersed phase temperature:

1 + X~bo

-

I+X]\

-

-

~

]

particle surface is determined by means of the relations (O -

T w) =

-(v'yO').

(20) As a boundary condition for the gaseous phase energy [Eq. (15)], the wall function method is applied [6].

fly

DjShpj dp

'

Shpj

= 2

+

1/2 1/3 0.6 R e p Scj .

(26)

The coefficient ~0j in Eq. (25) describes a contribution of the internal reacting surface of porous coal particle to the combustion process [8]. The combustion rate in the kinetic regime is calculated by using the Arrhenius law:

COMBUSTION AND GASIFICATION Here we consider the combustion and gasification processes of coal particles. The gas-particle medium is supposed to be sufficiently dilute to calculate the combustion of isolated particles in the surrounding gas. The effect of turbulent fluctuations on the combustion is neglected. The reacting particles are assumed to be composed of three components: ash, volatiles, and char. Ash is supposed to be inert in the devolatilization and combustion processes. So the rate of interracial mass transfer between phases is defined as the sum of the devolatilization rate and the char combustion (gasification) rate: te = a~v + aeC.

(21)

The devolatilization rate is determined on the basis of the two-parallel destruction scheme suggested by Kobayashi et al. [7]. Volatiles are assumed to be composed of three gaseous components: CO2, H 2 0 , and CnH m. We take into consideration the following heterogeneous chemical reactions:

C + H 2 0 --~ CO + H2,

C + 2 H 2 - * CH 4

(22)

The coal char combustion rate is gained as the sum taken over all heterogeneous reactions: j = O2,CO2,H20,H

~C = E~Cj' J

).

(27)

For a monodispersed system of particles, the rate of interfacial mass transfer is connected with the rate of particle mass variation by a relation ae

= NI~,

(28)

where N is the number density of particles. The combustion of a polydispersed system of particles is simulated on the basis of the fractional method. Within the framework of this method, the whole spectrum of particles is divided into separate fractions, and the particle transitions from large-size fraction to small-size fraction as a result of combustion are taken into account. The mass, momentum, and heat transfer of the dispersed phase is calculated by solving the corresponding conservation equations for each particle fraction. In the gaseous phase, we consider only the following reactions: m

C + CO 2 ~ 2 CO

C -q--0 2 ~ C O 2 ,

kj = kjoexp( - ~

2.

(23)

The rate of each heterogeneous reaction is calculated

CnH m +

n +

0 2 ~ n C O 2 -{- ~ - H 2 0

CO+H2OoCO

,

(29)

2 + H 2.

The rates of homogeneous reactions are calculated by taking into account both turbulent mixing and chemical kinetics [9]. The mass fractions of gas components are determined as a result of solution of the turbulent diffusion equations.

as

6ap

°ecj = bjk;fpcjfj dp '

(24)

where bi is the stoichiometric coefficient by means of which ttie influence of other reactions on the examined reaction is taken into account, Cj is the mass fraction of the gas component participating in the heterogeneous reaction, fj is the function describing a deviation from a linear dependence of the reaction rate on the reacting gas fraction due to an adsorption/desorption mechanism according to Langmuir law [8]. The effective rate of heterogeneous reaction per unit of the particle external surface is defined by accounting for both the diffusional resistance and the chemical kinetics on the particle external surface and in the particle volume: 1

k~f

1

flj

1

+

kj(i + Wj)"

(25)

The mass transfer coefficient of the reacting gas component transport in the surrounding gas mixture to the

R A D I A T I O N TRANSFER The radiation transfer is predicted in the approximation of gray medium. The equation of radiation transfer is ~ V / ( ~ , s ~) + ( a + f l ) I ( ~ , s ~)

= aI b + --~f4J(,~,~')y(g',~")dw'.

(30)

In two-phase gas-particle medium, the absorption and scattering coefficients and the radiation intensity of black body are determined by the following relations: O/ = Otg + a p ,

t = tip,

Otlb = a g / b g + C~plbp, (31)

where the radiation intensities of black body for the gaseous and particle phases are Ibg = ° ' T 4 / 7 r ,

Ibp = orO4//'/T'

(32)

Solution of Eq. (30) is derived by using the six-flux method or the moment method. The six-flux method is

Nonisothermal Two-Phase Turbulent Flows 315 written in the form

OXk[ fOxk

The scattering phase function is approximated with the help of the delta-Eddington formula [10]:

+mfot(Ib--li)

mfflC "~

I j +2l k '

Ii

(33) where

i,j,k =x,y,z Ff = 1 / m f [ a + fl - fl(A - B)/47r], (34) mf =

y(~o) = 2f6(1 - cos ~) + (1 - f ) ( 1 + 3gcos ~). (41) For this approximation, the coefficients in Eqs. (33), (34), (36), and (37) are becoming equal: A = 47rf+ --~-(1 - f )

, (42)

(27r/3)[2V~-arctg(v~-/2)] -I = 1.20.

The coefficients A, B, and C in Eqs. (33) and (34) characterize the scattering angles in forward, rearward, and lateral directions. These angles satisfy the condition A + B + 4C = 4~'. In isotropic scattering, the relation A = B = C = 27r/3 holds. The boundary condition for Eq. (33) is described as 4Ff 0Iy 3mf Oy

1+

(~mf[

(3 Iy + 4mf

1) 2

I .,1/1

X(Ix + Iz) - aw b J / a w

2

(35)

aI b

+

= [or + (1 - mml)fl](3Ii -- Ij -- I k) -- ram2 fl(I i + Ij + Ik) ,

(36)

Fm = [a + (1 - ram,)/3] 1

(37)

where The coefficients mini and mn~2 in Eqs. (36) and (37) are determined by a form of the scattering phase function 3'The boundary condition for Eq. (36) has the form

4Fm OIY 3 Oy

(

"wlb)(1 w aw

ly

1+

,

m m l = f + g(1 - f ) ,

C=--~-(1-f),

mm2 =

(1 - f ) / 3 .

(43)

PREDICTION EXAMPLES The present models have been applied to the calculation of gas-particle turbulent flows with heat transfer and combustion.

As an approximation of the moment method, Eq. (30) transforms into the following equation:

OXk°9 _1 OIik ) [Fm~-~x

B=--~-(1-f)

1) -1 2

(38)

The radiative heat fluxes in Eqs. (8) and (15) for the energy conservation of the dispersed and gaseous phases are defined as

Heat Transfer in Straight Round Jet

The simulation of heat transfer in free jet flows was performed by using both the differential [Eq. (9)] and the algebraic approximation [Eq. (12)] for determination of the particle turbulent flux in energy [Eq. (8)]. The hydrodynamic characteristics of the jet flow considered were calculated in the paper by Zaichik et al. [1] and corresponded with the experiments by Modarress et al. [11] and Tsuji et al. [12]. The inlet gas and particle temperatures were different and given by uniform profiles. The results of calculating the thermal characteristics of jet flows are presented in Fig. 1, which shows the dependence of the dimensionless temperatures of the gas and particles on the axial coordinate. For the large particles, the dispersed phase temperature is almost constant along the jet length, whereas the gas temperature decreases in accordance with the law of a single phase jet, which is confirmed by comparison with the experimental data by Golubev and Klimkin [13]. As the particle size decreases, T-To e-to To-T.'To-T .

V. ~p = 4=ap

Ibp

3

'

..

(

V" ~g -- 4'n'Otg Ibg

3

4

"

To predict the radiation transfer and radiative heat fluxes, it is necessary to determine the radiative properties of the gas-particle medium. The particle absorption and scattering coefficients are assumed to be proportional to the volume concentration of the dispersed phase: 3~ 3~ O~p = " ~ p a a ,

o-

(39)

tip =

~"pp as'

(40)

where the absorption and scattering factors Qa and Qs are determined on the basis of the theory by Mie for a single particle [10].

0

to

I

~

X/D

1. Variation in (1) gas and (2, 3) particle temperatures along the jet a x i s : a - d P = 5. 0 0 / , m . , m 0 = 0 8" 6 ; b - d p =50 . /,m, m 0 = 0.32; (2) calculation using Eq. (12); (3) calculation using Eq. (9); (4) experimental data by Golubev and Klimkin [13]. Figure

316

L.I. Zaichik et al.

the interphase heat transfer is observed to intensify, which leads to the fairly rapid equalization of the gas and particle temperatures, after which a self-similar spatial distribution of temperature in accordance with the law x-1 holds. It is interesting to note that, in the self-similar region, the dimensionless temperature of the dispersed phase always exceeds the gas temperature, which is a consequence of the particle thermal inertia. Calculations made by using the balance equation for the particle turbulent heat flux [Eq. (9)] shows that the effect of the convective and diffusion transfer of (u~O') is not pronounced. Taking these transfer mechanisms into consideration leads to a somewhat more rapid equalization of the temperatures of the carrier and dispersed phases. As can be seen, the particle temperature distributions obtained by using both Eq. (9) and approximation (12) are sufficiently close. Therefore, formula (12) may be recommended for the simulation of the particle turbulent heat flux in two-phase free flows. Pulverized Coal Combustion in Cylindrical C h a m b e r Next, we direct our attention to the modeling of the combustion process in a cylindrical chamber with a sudden expansion. The primary gas-particle jet is injected through the central hole and mixed with the secondary coaxial air jet. The flow configuration of the combustion chamber is chosen in the same way as in the experimental investigation described by Jang and Acharya [14]. The predicting dynamics and heat transfer of gas-particle turbulent flow in the combustion chamber is based on Eqs. (7), (8), (12)-(18), as well as (11), (15), (16), (49), and (50) from the paper of Zaichik et al. [1]. The heterogeneous chemical reaction between carbon and oxygen is assumed to produce carbon dioxide--that is, only the first reaction (22) is taken into consideration. The combustion rate of coal particles is defined by means of the Arrhenius law [Eq. (27)], taking into account the diffusion resistance to oxygen transport according to Eqs. (25) and (26). The radiation transfer in the two-phase medium is calculated with the help of the moment method in the gray approximation. The gas and particle temperature maps are presented in Fig. 2. The difference between the gas and particle temperatures reaches 100-150 in Kelvins. This temperature difference is induced by the energy source due to particle combustion, and its maximum is situated in the ignition

T,K 2300

1400

500

3.0

0

Figure 3. Variation in gas temperature along the chamber axis: line, calculation; symbols, experiment by Jang and Acharya [14].

zone. On the whole, the gas and particle temperature distributions are similar. In Figs. 3 and 4, the axial distributions of the gas temperature and oxygen and carbon dioxide concentrations along the centerline of the combustion chamber are plotted. The initial sharp increase in temperature and the decrease in oxygen concentration corresponding to the growth of carbon dioxide concentration are explained by the ignition of coal particles. The subsequent increase in oxygen concentration after the initial decrease is caused by the turbulent diffusion and the turbulent mixing. The calculated distributions of the gas temperature and oxygen concentration are found in rather good agreement with the experiment by Jang and Acharya [14]. However, the considerable deviation of the predicted carbon dioxide concentration from the experimental data is observed. This fact is explained as follows. First, the applied scheme for the description of the devolatilization process is rather crude and does not consider CO as a product of the pyrolysis. Second, the sharp ignition of volatiles (CH 4) leads to a local value of the ratio of air to fuel of less than 1.0, and so carbon monoxide can be formed. This conclusion is corroborated by the presence of CO in the experimental measurements presented by Jang and Acharya [14].

X

e-O 2

--, [

T,K 2700

.

-. .

.

DI

, k -co 2

.

~ -.. A

2260 2O4O

X/D

A

~-

0.12_5

1820 1600 ¢:~:: 1380 1160

94O 720 5O0 0

2000

4000

X, rfltll

Figure 2. Maps of (a) gas and (b) particle temperatures in the combustion chamber.

0

3.0

X/D

Figure 4. Variation in oxygen and carbon dioxide concentrations along the chamber axis: lines, calculation; symbols, experiment by Jang and Acharya [14].

Nonisothermal Two-Phase Turbulent Flows 317 Oxygen-Steam Gasification in Cylindrical C h a m b e r

Pulverized Coal Combustion in Boiler Furnaces

In this section, we consider the process of oxygen-steam gasification of pulverized coal fuel. The calculation model is the same as that used in the preceding section, but now all four heterogeneous chemical reactions (22) are taken into consideration. The purpose of these calculations is to show a possibility for the developed method to be applied to the simulation of the gasification process, in spite of the fact that, in this section, no experimental data are presented for comparison with the computations. The geometry of the gasification chamber is shown schematically in Fig. 5. The central primary flow consists of the coal, air, oxygen, and methane. The coaxial secondary flow consists of the steam and coal. The pressure is equal to 30 bars, and the particle size is 50 ~m. The temperatures of the primary and secondary flows are 297 K and 773 K, respectively. Figure 6 shows the predicted distribution of the temperature. The sharp increase in the temperature is a result of the ignition and combustion of methane. Because oxygen is virtually completely consumed by the methane combustion, the coal carbon reacts with the H 2 0 and CO 2 that forms in the combustion zone. The coal conversion in the secondary flow is limited by the mixing of the cold secondary and hot primary flows. The gas temperature in the vicinity of the wall does not exceed 1300-1400 K; it characterizes the lack of the slag wall regime. The analogous regime had been also observed in experimental investigations. Figure 7 shows the distribution of the gasified products. Maximums of concentrations of CO and H 2 are observed in the zone of mixing between the high temperature core of the flow and the surrounding cooling stream. The degree of the carbon conversion does not exceed 70-75%. This result is determined by the small amount of time of the particle being in the gasification chamber.

Because industrial boiler furnaces are characterized by very large sizes, the particle inertia parameter ~'u/Te has a small value, and the dynamics of the coal particles can be described by the diffusion-inertia model [1]. The temperatures of the gaseous and dispersed phases are supposed to be equal; that is, the heat transfer of the gas-particle medium is described in the framework of an one-temperature approach. The radiation transfer is calculated on the basis of the six-flux method. The inlet conditions are specified in the outlet plane of the burners. Thus, the profiles of all parameter in the inlet sections are assumed to be uniform. We first consider a two-dimensional simulation of the aerodynamic and thermophysic processes. The particle concentration in the furnace with slagging wall is shown in Fig. 8. The highest concentration of the particles is observed both near the nozzles and in the core region of the flow. The particle deposition flux, G w, is plotted on the furnace perimeter (see Fig. 8). The maximum deposition rate is realized on the surfaces both above and below the nozzles. The particle sedimentation induced by the gravity force determines approximately 40% of the total deposition process. Another part of the particle deposition flux is caused by the turbulent transfer (diffusion and migration). Simulations of the coal combustion were carried out in the furnace chamber of a P-57 boiler, which has power of 500 MW and is designed for burning the coal of the Ekibazstuzsky coal field. The results of the calculated temperature distributions are presented in Fig. 9 for two types of coal--namely, with the normal ash content (36%) and with the high ash content (50%). It is seen that the temperature fields for these two cases are similar; however, the maximum temperature for burning coal with the normal ash content is 1720°C, and that for burning coal with the high ash content reaches 1760°C. A comparison of calculated and experimental data for the temperature in the near-wall region of the furnace and for the wall heat flux is presented in Fig. 10 for burning coal with the normal ash content. Agreement between the results of modeling and measurements of Ivanov et al. [15] is reasonable. Figure 11 shows the carbon concentration fields in the furnace space related to a corresponding value in the inlet sections. The calculation results of the carbon concentration coincide approximately with the experimental data by Ivanov et al. [15].

CU4+N2+O 2 + COAL

Figure 5. Scheme of the gasification chamber.

F,

T,K 4000 3630 3260 2890 2520 2150 1780 1410 1040 670 3O0 0

200

400

600

gl]O

X, rrttrl

Figure 6. Maps of (a) gas and (b) particle temperatures in the gasification chamber.

318

L.I. Zaichik et al.

r~ 'l'Cl",/12•

Z j Tla

270.

90-

40 0 -~0-

-270

30 200

400

600

800

X, ITtl~

Figure 7. Profiles of mass concentration of CO and H 2 in the gasification chamber. Three-dimensional calculations were performed for the furnace chamber of the P-67 boiler, which has power of 800 MW and is intended for burning the coal of the Beresovsky coal field. The following major geometrical and design features of the actual furnace were taken into account in the numerical simulations: the presence of four burner tiers with eight burners on each (two burners on each wall) creating swirling flow, the presence of eight recirculation nozzles twisting the flow in the opposite direction, an inclination of the forward and back walls of a cold funnel, and narrowing and subsequent expansion in the top region of the furnace (aerodynamic pinches). The computational domain was limited downstream by a section corresponding to the beginning of the gas-turn channel. The number of grid nodes was equal to 40 x 40 × 60. This number of grid nodes affords a satisfactory accuracy, and the grid dependence of the calculation results was virtually absent when the number of grid points was increased. The velocity vectors in vertical and horizontal sections are shown in Fig. 12. Such "visualization" presents the aerodynamics of the flow as a whole. An intensive vortical

10

5

0

10

15

X, rrt

Figure 9. Distributions of temperature in furnace P-57: (a) normal ash content; (b) high ash content. core, well observed in the horizontal sections on the levels of the first and fourth burner tiers, is formed in the central region of the chamber at the level of nozzles. The flow has a jet character in the vicinity of burner tiers, and, as seen from Figs. 12c and 12d, jets retain their individuality until joining the vortex core. The computational results of the velocity profiles in various horizontal sections in comparison with the experimental data by Alekseenko et al. [16] are shown in Figs. 13 and 14. The experimental setup was a 1/122-scale model of the P-67 boiler furnace in the form of a vertical chamber of a 188 x 188 mm= cross section (i.e., the semiwidth, i , of the furnace was equal to 94 mm). Evidently, the sharply expressed maximum is typical of the tangential Z, m

:H

I',

|

tI

og,.]I o,8""',

,:1)/t N

40 30 20 10

Y i

%

I

" ,. ,'-Gw Figure 8. Concentration of particles, ~/~o, and deposition rate, Gw, on the furnace wall.

1000

1300

T, °C

100 200

I

qw,kW/mz

Figure 10. Temperature on the centerline of the furnace and wall heat flux: lines, calculation; symbols, experiment by Ivanov et al. [15].

Nonisothermal Two-Phase Turbulent Flows 319 computational results and experimental data shows that they are in satisfactory agreement. The computational results of the temperature distributions in various sections of the furnace are shown in Fig. 15. In general, the simulated field of the temperature agrees with the real temperature distribution in the furnace.

4O

PRACTICAL SIGNLFICANCE 30

20

i0

!

0

5

I0

15

X,m

Figure 11. Distributions of carbon concentration in furnace P-57: (a) normal ash content; (b) high ash content.

velocity component both in computations and in experiments for the sections passing through the burners. The tangential velocity component is abruptly decreased near the wall. Above the burner tiers, the profiles of the tangential velocity become more sloping and the maximum less expressed. Near the pinches, flow swirl decreased owing to its compression. The vertical velocity profile (see Fig. 14) in the region of burners has a complex structure, which is typical of the jet mixing region. The reverse flow velocity maxima, whose locations coincide with the tangential velocity maxima, reveal trajectories of the jets exhausting from the nozzles. A comparison of

'7

The design of power-generating equipment is impossible without studies of organic fuel combustion in chambers and furnaces. Under modern conditions, the physical modeling, considered until quite recently the most reasonable way of solving this problem, increasingly gives way to a numerical simulation. This change is related to the fact that, in the course of experimental studies, it is very difficult, and sometimes impossible, to ensure the similarity of a model and actual object even for small groups of similarity criteria. An adequate mathematical model permits one to reduce the number of physical tests, to correctly formulate their concepts, as well as to complement experimental results at the expense of a large number of "points of measurement" and "accessibility" of any spatial point in the furnace chamber. Besides, the mathematical model enables one to move easily from one investigated design to another, permitting an evaluation of design ideas before the construction of an expensive physical model of the object of studies. The models presented in this paper may be used for calculations of solid fuel combustion and gasification apparatuses of various types. We believe that these models will be useful for designing new power-generating equipment. CONCLUSIONS The kinetic equation for the PDF of the velocity, temperature, and mass distributions for a particle ensemble in a turbulent flow has been obtained. On the basis of this kinetic equation, the Eulerian models for the simulation of heat transfer in gas-particle turbulent flows are derived. The calculation scheme incorporates two-phase fluid dynamics, heat transfer, radiation transfer, and combustion.

m

60- ttt

tt

t . .

tttttttttt tl t ttttltttttttttl

'

40-

20-

: 1'1

~, rr~

i-iiii i I

...........

Figure 12. Velocity vectors in various sections of furnace P-67: (a) vertical section across the center of the furnace; (b) vertical section near the furnace wall; (c) horizontal section at the level of the first burner tier; (d) horizontal section at the level of the fourth burner tier. Numbers represent the velocity scales.

320

L . I . Zaichik et al.

UY I

I

-1.01

-t.0

°_/h

'

I

°% t

"%/

0

4P°@

I

×/e

-10

0.

"'/ °

×/e

-t.0

0

•

x/Q

Figure 13. Comparison of tangential velocity distributions in the furnace with the experimental data of Alekseenko et al. [16]: sections at the level of (a) the first, (b) the second, and, (c) the fourth tier of burners.

I

I

Uym~

b

8 e °e

•

o

0. •

°

-

.'°° 41

°

410°

Figure 14. Comparison of vertical velocity distributions in the furnace with experimental data of Alekseenko et al. [16]: sections at the level of (a) the first and (b) the second tier of burners. z,m

OI

-1.0

I

-1.o

o.

×/e

-I.0

O.

x/~

"b

60-

40-

20-

¢/////f///J~/J/J///////////////////J//f////J/f/J//f//////////AI

lnl

x, m

Figure 15. Temperature distributions (K) in various sections of furnace P-67: (a, b) vertical sections across the center of the furnace; (c) horizontal section at a level of the fourth burner tier.

i

Nonisothermal Two-Phase Turbulent Flows 321 The proposed models have been applied to calculation of hydrodynamics, heat transfer, and combustion in various particle-laden turbulent flows. The computational results are in rather satisfactory agreement with experimental data in combustion chambers and furnaces. The authors would like to acknowledge the INTAS Foundation (grant no. 94-4348) as well as the Russian Foundation of Basic Investigation (grant no. 96-02-18768).

NOMENCLATURE A Ak

A~ aw B bj C Cj

Cp, Cpp

scattering indicatrix parameter, dimensionless modulation term in Eq. (17), mZ/s 3 modulation term in Eq. (18), m 2 / s 4 wall absorptivity, dimensionless scattering indicatrix parameter, dimensionless stoichiometric coefficient, dimensionless scattering indicatrix parameter, dimensionless mass fraction of gas component, dimensionless heat capacity of gas and particles, J/(kg

K)

C,1,C,2 D Do dp

Ej Fi f fy

L,Lu, L , L , , L Cw g

gu, gut

n , np I Ib

J k

turbulence constants, dimensionless chamber diameter, m nozzle diameter, m diffusivity coefficient, m2/s particle diameter, m activation energy of combustion, J / m o l external force acceleration, m / s 2 scattering indicatrix parameter, dimensionless coefficient accounting for a deviation from a linear law in Eq. (24), dimensionless entrainment coefficients, dimensionless particle deposition flux, kg/(m 2 s) scattering indicatrix parameter, dimensionless entrainment coefficients, dimensionless enthalpy of gaseous and dispersed phases, J / k g radiation intensity, W / m 2 radiation intensity of black body, W / m 2 collision operator, m -3 s 2 turbulent energy of the gaseous phase, m2/s 2

combustion rate in kinetic regime, m / s combustion constant, m / s effective rate of heterogeneous reaction per unit of particle surface, m / s g semiwidth of the furnace, m m particle mass, kg mf, mml , ram2 scattering indicatrix parameters, dimensionless m p mass of a particle, kg kjo k; f

m 0 mass loading ratio [= (pp~/P)o], dimensionless N number density of particles, [= 6~/~d~], m -3 Nup Nusselt number, dimensionless P probability density function, m -3 s 3 Pg pressure in the gaseous phase, kg/(m s 2) Pr Prandtl number, dimensionless Pr t turbulent Prandtl number, dimensionless Q density of heat source due to combustion and radiation, K / s Qa absorption factor, dimensionless Qs scattering factor, dimensionless qr radiation heat flux, W / m 2 Rpi particle position vector, m R 0 universal gas constant, J / ( m o l K) Rep Reynolds number, dimensionless r radial coordinate, m Scj Schmidt number, dimensionless Shpj Sherwood number, dimensionless s', 9' directions of incident and scattered radiation fluxes, dimensionless T, t, t' averaged, actual, and pulsation temperature of the gaseous phase, K T L integral Lagrangian scale of turbulence, s times of interaction of particles with Tu, Zut, Tt high-energy gas pulsations of velocity and temperature, s ro gas temperature at the nozzle exit, K T= surrounding gas temperature, K (t '2 ) intensity of gas temperature pulsations, K2

UD u i, uPi averaged, actual, and pulsation velocity of the gaseous phase, m / s ( u'it') gas turbulent heat flux, m K / s ( u'iu~) gas turbulent stress, m2/s 2 Vi, ~i, vi averaged, actual, and pulsation velocity of the dispersed phase, m / s Upi velocity of a particle, m / s second moment of particle velocity pulsations, m E / s 2

(v;o') particle turbulent heat flux, m K / s third moment of particle velocity pulsations, m3/s 3 third moment of particle velocity and temperature pulsations, m2/(K s 2) X volumetric gas concentration, dimensionless x distance in the longitudinal direction, m x i coordinate, m y distance in the normal direction to the wall, m Z mass concentration of the dispersed phase [= ppqb], k g / m 3 z distance in the transversal or vertical direction, m

322 L.I. Zaichik et al. Greek symbols c~ absorption coefficient, m-1 /3 scattering coefficient, m 1 /3j mass transfer coefficient of reacting gas component transport to the particle surface, m / s r~,rm radiation coefficients, dimensionless y scattering indicatrix, dimensionless A H change of enthalpy by transition of substance from the solid phase into the gaseous phase due to combustion, J / k g 6 Dirac function rate of turbulence dissipation, m2/s 3 ew wall emissivity, dimensionless 0, O, 0' averaged, actual, and pulsation temperature of the dispersed phase, K temperature of a particle, K #g combustion rate, kg/(m 3 s) #gv devolatilization rate, kg/(m 3 s) char combustion rate, kg/(m 3 s) ~c k coefficient of heat conductivity, W / ( m

K) vt

coefficient of gas turbulent viscosity,

~e

probability density relating particle temperature by particle-wall interaction density of gas and particles, k g / m 3 Stefan-Boltzmann constant, W / ( m z K 4) tensor of fluid-dynamic particle interaction, m2/s 2 turbulence constants, dimensionless time, s particle thermal and dynamic relaxation times, s volume particle concentration, dimensionless angle between the directions of incident and scattered radiation fluxes, rad coefficient of restitution of temperature by particle-particle collisions, dimensionless coefficient of reflection of particles from the wall, dimensionless coefficient accounting for a contribution of internal reacting surface of porous particle to combustion, dimensionless rate of particle mass variation due to combustion, kg/(m 3 s) solid angle in direction s~', rad

P, Pp or

o'ij O'k, 0"~ "i"

rt, ~-u qb

~b0 X ~. fl to' r '

m2/s

Superscripts radiation pulsation

Subscripts g gas i , j , k coordinates P particle

t w x, y 0 1 2

turbulent at wall parallel and normal to the wall value in the inlet section before particle-wall collision after particle-wall collision REFERENCES

1. Zaichik, L. L., Pershukov, V. A., Kozelev, M. V., and Vinberg, A. A., Modeling of Dynamics, Heat Transfer, and Combustion in Two-Phase Turbulent Flows: 1. Isothermal Flow. Exp. Thermal Fluid Sci. 15(1), XX-XX, 1997. 2. Derevich, I. V., Yeroshenko, V. M., and Zaichik, L. I., Hydrodynamics and Heat Transfer of Turbulent Gas Suspension Flows in Tubes: 2. Heat Transfer. Int. J. Heat Mass Transfer 32(12), 2341-2350, 1989. 3. Shraiber, A. A., Gavin, L. B., Naumov, V. A., and Yatsenko, V. P., Turbulent Flows in Gas Suspensions. Hemisphere, New York, 1990. 4. Vinberg, A. A , Zaichik, L. I., and Pershukov, V. A., Calculation of the Momentum and Heat Transfer in Turbulent Gas-Particle Jet Flows, Fluid Dynamics, 27(3), 353-362, 1992. 5. Derevich, I. V., and Zaichik, L. I., An Equation of Probability Density of Velocity and Temperature of Particles in a Turbulent Flow Modeled by a Random Gaussian Field, Appl. Math and Mech., 54, 722-729, 1990. 6. Patankar, S., Numerical Heat Transfer and Fluid Flow, Hemisphere Publ. Corp., Washington D.C., 1980. 7. Kobayashi, H., Haward, J. B., and Sarofim, A. F., Coal Devolatilization at High Temperature, Sixteenth Int. Symp. on Combustion, Pittsburgh, pp. 411-425, 1976. 8. Laurendau, N. M., Heterogeneous Kinetics of Coal Char Gasification and Combustion, Progr. Energy Combust., 4(4), 221-270, 1978. 9. Magnussen, B. F., and Hjertager, H., On Mathematical Modeling of Turbulent Combustion with Special Emphasis on Soot Formation and Combustion, Sixteenth Int. Symp. on Combustion, Pittsburgh, pp. 747-759, 1976. 10. Viskanta, R., and Mengilc, M. P., Radiation Heat Transfer in Combustion Systems, Progr. Energy Combust. Sci., 13, 97-160, 1987. 11. Modarress, D., Tan, H. and Elghobashi, S., Two-Component LDA Measurement in a Two-Phase Turbulent Jet, AL4A J., 22(5), 624-630, 1984. 12. Tsuji, Y., Morikava, Y., Tanaka, T., Kariminc, K., and Nishida, S. Measurement of an Axisymmetric Jet Laden with Coarse Particles, Int. J. Multiphase Flow, 14(5), 565-574, 1988. 13. Golubev, V. A., and Klimkin, V. F., Investigation of Turbulent Submerged Gas Jets of Different Density, Inzh.-Fiz. Zh., 34(3), 493-499, 1978. 14. Jang, D. S., and Acharya, S., Improved Modeling of Pulverized Coal Combustion in a Furnace, J. Energy Res. TechnoL, 110(6), 124-132, 1988. 15. Ivanov, A. G., Kiselman, L A., Luzhnov, M. I., et al., Experimental Burning of Ekibazstuzsky Coal with More than 50% Ashes in Furnace P-57 with 500 MW of Power, Teploenergetika, 1, 4-11, 1980. 16. Alekseenko, S. V., Borisov, V. I., Goryachev, V. D., and Kozelev, M. V., Three-Dimensional Numerical and Experimental Simulation of Aerodynamics in Furnace Chambers of Advanced Steam Generators at Isothermal Conditions, Thermophysics and Aeromechanics, 1(4), 325-331, 1994.