The Trajectories and Distribution of Particles in a Turbulent Axisymmetric Gas Jet Injected into a Flash Furnace Shaft Y.B. HAHN and H. Y. SOHN Numerical computations have been performed for the behavior of a vertical turbulent particle-laden gas jet exemplified by the shaft region of a flash-smelting furnace. The two-equation (k-e) model was used to describe turbulence. Model predictions for the gas and solid flow fields give a satisfactory representation of experimental data taken from the liter~Lture. The predictions of flow properties of the two phases under flash-smelting conditions have been obtained for various inlet conditions, particle sizes, particle loading, and oxygen enrichment. Model predictions show that the axial velocity of the particle phase is substantially higher than that of the gas phase. The presence of solid particles causes the axial velocity of the gas phase to be greater near the centerline and lower in the outer region than in a single-phase gas jet. A more uniform distribution of particles was obtained by introducing a strong radial velocity of the distribution air at the inlet. The implications of the behavior of a particle-laden gas jet on flash-smelting processes are discussed.
I.
INTRODUCTION
IN the flash-smelting process, fine particles of sulfide concentrates and flux are injected with oxygen-enriched air into the furnace. The major component of the process is the furnace shaft in which sulfur is removed by oxidation of the mineral particles in a turbulent gas jet. Therefore, it is important to understand better the subprocesses taking place in such a gas jet. The mathematical modeling of the particle-laden turbulent flows in confined systems has been a difficult problem not only due to the difficulties in solving the nonlinear elliptic partial differential governing equations but also due to the difficulties associated with mathematical description of complex turbulent characteristics of the interaction between the dispersed particle phase and the continuous fluid phase. Crow et al. I~l analyzed gas-droplet flows by accounting for the mass, momentum, and energy coupling between phases. They developed the particle-source-in-cell (PSICELL) model to predict the behavior of the particle-laden turbulent system. Spalding I2'3J developed numerical procedures for multiphase flows. Melville and Bray I41studied the turbulent two-phase jet and proposed a correlation to describe the effect of the presence of particles on turbulence by using the ratio of the mean particle bulk density to the mean gas density. Elghobashi and Abou-Arab I51 developed a two-equation turbulence model for two-phase flows. They derived the equations of the turbulent kinetic energy and the rate of its dissipation using third-order correlations involving fluctuating components of variables. They compared model predictions of the particle-laden jet in a vertical cylindrical system with experimental data and obtained good agreement) 5'61 However, they mentioned that more tests are required to establish the universality of the coefficients used in their correlations. I5'61 Smoot and coworkers 17-~61have extensively investigated two-phase turbulent jets in confined systems such as pulverized-coal
Y. B. HAHN, Graduate Student, and H. Y. SOHN, Professor, are with the Department of Metallurgy and Metallurgical Engineering, University of Utah, Salt Lake City, UT 84112-1183. Manuscript submitted July 10, 1987. METALLURGICALTRANSACTIONS B
combus~:ors in terms of both experimental and modeling efforts. Weber e t a / . [171did simulation work on the dispersion of heavy particles in a confined flow. They neglected all exteraal forces except for the drag force. Since its development by Outokumpu Oy and INCO in the late 1940's and early 1950's, the flash-smelting process has been a dominant process for smelting various sulfide minerals, including those of copper and nickel. The attractiveness of the flash-smelting process has greatly increased with the increasing availability of inexpensive tonnage oxygen and because of major advantages of substantial reduction in fuel requirement, efficient sulfur dioxide recovery, and rapid smelting rates. In spite of the increasing industrial stature cf the flash-smelting process, it is only in recent times that much attention has been focused on the mathematical modeling of that process. Such mathematical modeling for the flash-smelting process incorporating the turbulent flow of tluid, chemical kinetics, and transport of heat and mass is ~ very complex problem. Only one such study has been published. Hs] The author used a one-equation model to describe turbulence under the assumption of constant turbulent mixing length, contrary to the fact that it strongly depends on local conditions. He did not account for the effect of the presence of particles on the physical properties of the fluid such as eddy viscosity of fluid and turbulent Schmidt number, and the contribution of turbulent diffusion of particles to the change of momentum of the particle phase. In the present work, as part of an overall investigation of flash-sme lting processes, numerical computations have been performed to predict the behavior of a turbulent particleladen gas jet at room temperature. The trajectories and distribution of solid particles are important factors in flash smelting from the viewpoints of the effective utilization of furnac,~ volume and the avoidance of severe refractory wear. The particles should be dispersed as widely as possible throughout the furnace volume, but not too closely to the furnace wall. The two-equation (k-e) model which takes into consideration the local variation of the turbulent mixing length was used for the gas phase. The effect of the presence of particles on the turbulence was described in detail. Lagrangian treatment of the particles was used, representing the particle field as a series of trajectories. VOLUME 19B, DECEMBER 1988--871
The previous investigators [5'6'n'~4] who worked on mathematical modeling of a particle-laden turbulent jet in a confined system have predicted the system behavior for the case of straight axial feeding of the primary gas stream with zero radial-velocity component at the inlet. Elghobashi and coworkers [5'6] treated the particle phase in the Eulerian framework, which needs larger computer storage than the Lagrangian treatment. Weber et a/. [17] neglected the effect of the gravitational force on the calculation of particle momentum. In this work, in order to predict the behavior of the particle-laden turbulent gas jet under flash-smelting conditions, the effect of the radial as well as the axial velocity component of the primary gas (or distribution gas) stream at the inlet was studied. The particle phase was treated in the Lagrangian framework, and the drag and the gravitational forces acting on the particle were included. The correlation equation proposed by the present authors tl9] for the dissipation rate of turbulent kinetic energy at the inlet and correlation equations proposed by Melville and Brayf4] to account for the effect of the presence of particles on the fluid properties were used throughout. The predictions of flow properties of the turbulent particle-laden flow have been obtained for various inlet conditions, particle sizes, particle loading, and oxygen enrichment in a flash furnace. The computer program used in this work is named FSMELT-2.
II.
(3) Contribution to the gas-phase momentum due to the dilation of gas, which describes the volume change of a fluid element due to expansion or compression, can be neglected. The properties of a turbulent flow are expressed in timeaveraged and fluctuating components as follows: 4' = ~
+ 4''
[31
where 4' and 4'' represent any variable and its fluctuating component, respectively. The time-averaged value of 4' is defined as follows: 1
f,+,o 4' dt
[4]
4' = to -t
Primary air wifh particles
I Secondaryair
MODEL EQUATIONS
The confined system considered in this work is shown schematically in Figure 1. The primary flow of particleladen gas with or without a secondary flow enters the system through the nozzle and expands radially. The modeling equations to describe a turbulent nonreacting compressible flow in such a confined system can be expressed by the equations of continuity and momentum for each phase as discussed below. A. Equations for Velocity Field in the Gas Phase
The gas phase is viewed from the Eulerian framework. The continuity and momentum equations combined with the effect of the presence of particles can, respectively, be expressed as: op + ~ . (p,~) = o
[1]
at
a ,-,~p~'~+ ~ -
0t
-,-, 9 (pvv) = -~p - ~.
F B(x,r) I
-' -'v ~ + pg + sp
[2] where the arrow and the double bar represent a vector and a second-order tensor, respectively. The term ~; represents the coupling between the gas phase and the particle phase, i.e., momentum source or sink to the gas due to the presence of solid particles. In order to reduce the above equations to simpler forms, the following assumptions are made: (1) The system is at steady state and axisymmetric in cylindrical coordinates. (2) Body forces, such as the gravitational force, can be neglected. 872--VOLUME 19B, DECEMBER 1988
I
L
R
Fig. 1 --Particle-laden turbulent gas jet in a confined system. METALLURGICAL TRANSACTIONS B
where the time interval to is sufficiently large compared to the turbulence time scale. Substituting Eq. [3] in Eqs. [1] and [2] and taking the time average together with the above assumptions, the continuity and momentum equations are expressed with the time-averaged quantities and terms involving fluctuating components such as -fi u ' v ' , u o'v ' , p ' u ' v ' , etc. Neglecting the fluctuating components of density and turbulent viscosity, the governing equations become much simpler. I14'16'2~ The only remaining terms are of the form p u 'v' which are related to the turbulent viscosity and the gradients of timeaveraged velocity components, as discussed in Section II-C. The time-averaged equations for two-dimensional elliptic flows without swirl in cylindrical coordinates are now written as follows: Continuity: O ~)+ Ox
1 O(r-p~) r
0
[51
that this mass fraction also represents the fraction of the primary fluid in the mixture with the secondary air stream. By defining this mixture fraction, any conserved scalar such as the local enthalpy, temperature, or mass fraction of gas species can be calculated from the local value off: s = f S ~ + (1 - f ) S 2
where subscripts 1 and 2 represent the primary and secondary streams, respectively. The differential equation for the transport of the averaged scalar variablef is given by: [~<2~ Or
r
Ox \o"I Ox/
10(rlZ~cgJ ) r O r \ o"s ~ r = 0
[121
o-I = v,/D~.
L L s (r-~;~)_ Ox ~-ff-ff) + r Or O
er -
~X ~k/,.te._~X/] _
Or
e + ---
~r
r Or
+ Sp
[6]
The same value of o-r can be used for a reacting system. In Eq. [11], the first two terms represent the transport by bulk flow, and the last two terms that by turbulent eddy motion. The predicted centerline profiles o f f can also be used for determining the unknown value of turbulence intensity at the inlet by comparing with measured data. IJ6~ C. Turbulence Model
Radial momentum: 0S ('P'U'V) + - - - - (F'-fi'V'V) -/.Ze r Or ~x \ Ox] r Or
[11]
where ~rI is the Schmidt number for f, which may be defined as:
Axial momentum:
1 r
[101
er
--
+ -r- - Or -
--
~r
0--~
e
- 2tz~-;5 + Spv
[71
The presence of solid particles affects the turbulent field significantly. The kinetic energy spectrum of the turbulent field is ,damped by the presence of solid particles. These effects ihave been studied by Melville and Bray L4~and Elghobashi and coworkers.t5'61 The approach to the turbulence closure problem can be credited to the Boussinesq analogy (1877) given by: 04'2~
where/z~ represents the effective viscosity, given by: /a/~e =
/,Z t -[- /a/, l .
u'v'= [8]
Sp and Sp represent the momentum addition to the gas phase due to the presence of particles. The method for computing these terms is shown in Section IV on Numerical Methods. B. Equations for Mbcing in the Gas Phase
The mixing of gas species due to turbulent fluctuation plays an important role in describing a system in which gasphase reactions are as important as gas-particle reactions. Describing the mixing extent in a conservative quantity makes the problem much simpler compared to solving several conservation equations for gas species. For nonreacting systems where there are two separate inlet streams each of which has uniform properties, it is convenient to define the mass fractionf as follows: rh, { mole fraction of the fluid "~ f--- rnl-~ rh2 = !koriginating in the primary stream//
[9]
where rh~ and ~/2 are the mass flow rates of the fluid in the primary and secondary streams, respectively. It is noted METALLURGICALTRANSACTIONSB
[13]
-Vg~-~r +
where the prime represents a fluctuating velocity component, and V'g is the kinematic eddy viscosity of the fluid in the presence of particles. The turbulence closure model relates the turbulent kinematic eddy viscosity to the turbulent kinetic energy (k) and its dissipation rate (e) by: Im'''2~ v'~ = c~k2/e [14] where the turbulent kinetic energy is defined by: 1 .-~
k=~(u
+v
,----~
+w'2).
[15]
The turb~alent kinetic energy and its dissipation rate can be expressed in terms of time-averaged variables. This turbulence closure is referred to as the "two-equation (k-e)" model, and k and e can, respectively, be obtained from the following equations: 1~4'2~ r Or O[-fiuk + Ox
o-k IXeOk] o'~
G
-fie
[16]
VOLUME 19B, DECEMBER1988--873
The first and second terms on the right-hand side of Eq. [20] represent, respectively, the steady aerodynamic drag force and the gravitational force acting on the particle. The drag coefficient can be expressed as: I8a~
and
r Or
r
~13 - - -
o-e
+ 7x
=
- Go-
24(1 + 0.15Re ~
[171
where
[22]
Co = Re where
(0~']2 (_~_)2] G = la.~l L~ox J + \Or] + f2r{O-~2
+
+ Ox/j"
[181
The constants C,, C1, C2, o t , and 0-~ are known as the "universal constants." Their values are given in Table I.[14'221 Although the k-e model is the most widely-used one for turbulent closure for both gaseous and particle-laden systems, better models need to be developed which account for combustion-generated turbulence and gas/particle interaction.j14'231 Melville and Bray I41 proposed a correlation equation to describe the effect of particles on the gas-phase turbulence by using the ratio of the densities of the gas-particle mixture (Pbe) and pure gas (p), as follows: t
t
(Ug)particles = (Ug)noparticles[l Av ~ b e / F ) ] -0"5 .
[19]
Equation [19] suggests that the kinematic eddy viscosity of the particle-laden fluid decreases as the particle bulk density increases. The present work used the above correlation to take account of the effect of particles on the gas-phase turbulence in the k-e turbulence model.
D. Particle-Phase Equations To treat the particle phase, the Lagrangian approach, which has the advantages of reduced computer storage and computation time compared with the Eulerian approach, was used together with the following assumptions: (1) The pressure gradient, virtual mass effect, and Basset force are negligible compared with the steady-state aerodynamic drag.[9'l~ (2) The interaction between particles is negligible because the particle concentration in the flash-smelting furnace is very low. The particle momentum equation for a single particle in Lagrangian framework can be written as: [1.10 . . I/14 . 21]
m ~}-'e P
= F d ( V _ Ve ) + meg
dt
[201
Fd Table I.
=
->
Vp = Vec + Veal.
Turbulence Model Constants
874--VOLUME 19B, DECEMBER1988
[24]
The convective velocity can be defined as the velocity that would arise in the abs%nce of turbulence, or that based on the mean gas velocity. Vpc along a trajectory can be obtained from Eq. [20] by numerical integration. The turbulent diffusion velocity accounting for turbulent fluctuations can be modeled using the mean particle bulk density gradient: [u'14'21]
Je +
=
(Ve +
= ~ed-Pbe = O ' f l -m .
v+e t ) m-e
-
9
[25]
The transport coefficient Dte is defined as the turbulent particle diffusivity that can be expressed by: [4d1'14'211 t
De
~
t
l
Up/Orp
[26]
where ue' and o'te are, respectively, the turbulent particle kinematic viscosity and the turbulent Schmidt number for particles. t The turbulent kinematic viscosity of particles, up, must account for the degree of turbulence and particle size. Much work is currently being performed by several investigators [4'6'21'241 on how to obtain ue. Although no reliable model based on sound theory has been developed yet, a model by Melville and Bray I4~ gave satisfactory results for the pulverized-coal combustion process, tu.14] For the present work, the following relation by Melville and Bray t4J was selected for the expression of u'e, because it is more complete than other models and is applicable to the particle size range of interest in the flash-smelting processes: vet
~---
dg/[l + (rp/t,)]
[27]
where r e and t, are the particle relaxation time and the time scale of turbulence, respectively. The particle relaxation time is related to the Stokes particle drag by: [4'u'21'24'251
mp/(3~'lxgde).
[281
The time scale of turbulence can be related to the local turbulence by: [4a4'211
-~
- f c o o & l v - vel
Constant C, C1 C2 o-, o'~
[23[
In order to account for turbulence in the equation of particle momen~m, the particle velocity is broken down i~to a convective (V-c) and a turbulent diffusive component (Veal), as follows: [u'f~'211
r e =
where I
e14/ .
Re = pl -
Value 0.09 1.44 1.92 0.9 1.22
[211
t, = u ~ g / ~ .
[29]
By assuming isotropic turbulence, Eq. [29] can be expressed as
t, = 1.5Cr
[30]
The bulk particle density in Eq. [25] can be related to the bulk particle number density as: -fibe = me-ff p "
[31]
METALLURGICALTRANSACTIONSB
Under the assumption of no breakage or coagulation of particles by collision, it is convenient to replace Pbp by ~p in Eq. [25] as follows: ")' - -
t
gpdn p : D p~np .
[32]
However, n--~cannot be calculated from the Lagrangian particle-phase information and can only be approximated using the Eulerian gas-phase information, ml The continuity equation for the particle number density of the fh size particle in a turbulent flow at steady state can be expressed as: 0 O--x
r Or
-~x V 1
0
(
'
Ox ,]
rD
r Or \
= 0
[33]
Or]
The diffusion coefficient Dj in Eq. [33] is the same as in Eq. [25]. For very small particles completely following the turbulent fluctuations, the turbulent particle Schmidt number o-'p can be approximated by unity; for large particles failing to follow the turbulent fluctuations, o-'p is less than unity. I4'2u For the present work, the value of 0.35 was used for O-'p, as recommended by Smoot and Smith. ml
III.
BOUNDARY CONDITIONS
A complete specification of boundary conditions is necessary for solving the governing e~uations. The previous work done by the present authors I~gjexamined extensively the effects of boundary conditions, especially those at the inlet, on the numerical results. They also tested various correlations for the dissipation rate of turbulent kinetic energy at the inlet and obtained the relation yielding the best results as: = C~k~5/(0.015 de)
[34]
where d e = 4 x hydraulic radius. They found that the uniform profile of the axial velocity at the inlet yielded the best predictions in the downstream region in which fluid phenomena are of greater importance in the flash-smelting process. The detailed descriptions of boundary conditions of fluid properties except the mixture fraction (f) can be found in Reference 19. In the inlet region, the mixture fraction is specified by the definition, Eq. [9], asfj = 1 and f2 = 0 for the primary and secondary streams, respectively. At the side and the inlet walls, Of/Or = 0 and Of/Ox = 0, respectively. At the centerline, Of/Or = 0 and at the outlet Of/Ox = O. IV.
NUMERICAL METHODS
Modeling equations except for the momentum equation of particles can be expressed in the following form: Ox (-p -ff ~b) + r
( rp ~ 49)
Ox F~
1 o (rre o~] = S~
r Or \
Or]
135]
where ~b represents a dependent variable and S 6 is the "source term" which includes all other terms in the govMETALLURGICAL TRANSACTIONS B
erning equations not embodied in the first four terms in Eq. [3511. Fe denotes an effective transport exchange coefficient in a turbulent flow. Equation [35] can be cast into finite difference equations to be solved by the line-by-line or tri-diagonal matrix algorithm. The TEACH code developed by Gosman and Pun t26j was used to solve the gas-phase equations of a recirculating flow in a confined system. The SIMPLER algorithm devised by Patankarl271was used to calculate the pressure field. Detailed descriptions of the method are given in Reference 27. The l_agrangian particle equations of motion, which are easier to solve than the Eulerian gas-phase equation, were solved by the particle-source-in-cell (PSI-CELL) technique developed by Crowe and coworkers.lU The PSI-CELL technique has been followed directly to take account of the particle field in the Eulerian gas field. The procedure can be outlined as: (1) Solve the Eulerian gas field without particles. (2) Calculate the Lagrangian particle field; i.e., particle trajectories are calculated. (3) Evaluate the particle source terms. (4) Solve the gas field with the updated particle source terms. (5) Repeat Step (2) until convergence is achieved. The PSI-CELL technique is very efficient in terms of computer sterage and computational time required. Steps (2) through (4) are termed "one-particle iteration." Convergence is achieved when the gas field does not change between particle iterations. The momentum source term in Eqs. [6] and [7], ~p, added to the gas phase due to the presence of particles in a cell (or a control volume) can be described by: I~'~41 v
(~p)ce l = Vcel I
hii[(Vpijmpij)o. , - ( i,mpo)i.
~r [36]
for particles of the i th s~ze injected at thej th starting location at the inlet. Vce,, hij, Vpi~, and mpi~ represent the volume of the cell, the number flow rate of particles, the velocity vector of pa~'ticle, and the mass of particle of the i th size injected through the j,h starting location.
V.
RESULTS AND DISCUSSION
The present work used a staggered grid s y s t e m tl4'22'271 where fine grid sizes are concentrated in the upstream region near the inlet stream. The present authors conducted grid-dependence tests in their previous work II91 and noted that the virtual grid independence was achieved with grid points more than 27 x 27 in a confined system they tested. For the present work, grid points of 32 • 32 were used throughout. In this section, the model predictions of the behavior of the particle-laden gas jet in a confined system were first compared with experimental data taken from the literature for the purpose of validating the model. The boundary conditions which were determined in referenced experiments were used for calculation whenever possible. Table II shows the boundary conditions for various flow systems. VOLUME 19B, DECEMBER 1988
875
Table II.
Tice and Inlet Geometry dj (m) d2 (m) ds (m)
S m o o t [151
Experimental Flow Conditions
Memmott and
double entry 0.0255 0.130 0.206
S m o o t [71
**Elghobashi et al. t6]
Sharp I281
double entry 0.0255 0.130 0.130
double entry 0.0255 0.130 0.260
30.5
30.5
double entry 0.02 0.60 0.60
Primary Stream Velocity (m/s)
3.5
Mass flow rate (kg/s) Air Argon Particles
0.0052 0.0169 0.0331
Pct (mass) solids Pct (mole) argon Temperature (K) Turbulence intensity
0.0054 0.0174 0.0152
2r],j66
12.6 1 - d-~l]
0.0053 0.017 0.0149
60 70 283 0.15"
40 70 283 0.15"
40 70 283 0.15"
0.00376 -0.0012, 0.0032 24.2, 46.0 -293 0.04 + O.l(r/dO
38.1 0.54 283 0.18"
38.1 0.54 283 0.18"
38.1 0.52 283 0.18"
0.05 0.01663 293 0.1
Secondary Stream Velocity (m/s) Mass flow rate (kg/s) air Temperature (K) Turbulence intensity Particle Properties Particle type Mean diameter (/xm) Density (kg/m 3)
silicon 54 2330
silicon 19, 54 2330
silicon 46 2330
glass beads 50 2990
* E s t i m a t e d v a l u e s due to lack o f m e a s u r e m e n t s . ** F l o w conditions at 0.1 d~ d o w n s t r e a m o f n o z z l e exit.
The predictions of flow properties of the turbulent particleladen air jet under flash-smelting conditions were then performed for various inlet geometries, inlet velocities, particle sizes, and particle loading.
Comparison between Predictions and Experimental Data Memmott and Smoot, 171Sharp, 1281and Tice and Smoot fjSl measured flow properties of the nonreacting particle-laden gas jets in systems described in Table 11. They all used isokinetic collection probes for gas and particle samples and obtained gas velocities from the measured static and stagnation pressures. They also measured the profiles of mixture fraction, particle mass flux, and gas velocity. As described in Reference 19, the numerical results are substantially affected by the value of the turbulence intensity (I) at the inlet. Turbulent intensity is defined as 114't9l I =
[371
where 171 denotes the magnitude of velocity. Unfortunately, however, the inlet value of I is usually not reported in the literature due to the experimental difficulty in measuring it. Instead, the estimated value of I which yields the best predictions of the centerline axial velocities or mixture fractions compared with experimental data is usually used as the inlet boundary condition, l~6'22JFigure 2 shows the effect of turbulence intensities at the inlet on the centerline mixture fraction. Either I~ = 0.15 and 12 = 0.18 or I1 = 12 = 0.18 shows the best agreement with the experimental data obtained by Sharp. I281The former values were 8 7 6 - - V O L U M E 19B, DECEMBER 1988
used as inlet values of I for predictions of the systems studied by Memmott and Smoot 17] and Tice and Smoot t~Sj as well as Sharp, I28j since inlet velocities and geometries in their systems were the same. The effect of the Schmidt number for the mixture fraction is tested in Figure 3. The first figure (a) is for Sharp's data and shows that the centerline mixture fraction is not greatly affected by the values of o7, but o7 = 0.8 shows the best agreement. In the second figure (b), the radial profiles of mixture fraction obtained by Tice and Smoot I15j for the case of 60 pct (mass) particles loading in the chamber of 1.0
C
..-I
,,-
0.5
X
I
0.0
i
i
I
I
0.2
I
I
i
I
I
I
0.4
i
t
J
0.6
AXIAL DISTANCE (m) Fig. 2 - - E f f e c t o f t u r b u l e n c e i n t e n s i t y at the inlet: A . 1~ = 12 = 0 . 1 ; B. li = 12 = 0 . 1 5 ; C. 11 = 0 . 1 5 , 1 x = 0 . 1 8 ; D. 11 = I2 = 0 . 1 8 ; E. I~ = 12 = 0 . 1 9 ; F. li = 12 = 0 . 2 ; G . I I = 12 = 0.17. 9 S h a r p ' s datat281; - - prediction by F S M E L T - 2 .
METALLURGICAL TRANSACTIONS B
I.O
(o)
Z
~~
I.l.J
Z .~I I.a.. l-td.J z 1.4.3
0.5
I--X
Z
0.0
0.2
0.6
0.4
AXIAL
OISTAltCE (m)
(a) 0.5
x/R t ./e
-o-
o'f O.9
0.4 ' ~
-:<-
(b)
24
0.3 0.2
_
,~/o-f
0.9
: "._" ,~%N
O.i
x~"~y~
9
22
I 0.1
i 0.2
I 0.3
~'~
~;"~r'e~"=?" - - i - - - i -'@~r,-- -j 0.4 0.5 O.l] 0.7 0.8 0.9 1.0 r/Rf
(b) Fig. 3 - - Effect of the Schmidt number on mixture fraction. (a) Profile of centerline mixture fraction, experiment by Sharp. pSI (b) Radial profile of mixture fraction, experiment by Tice and Smoot; t~SI1~ = 0.15, 12 = 0.18 for both cases.
0.206 m diameter were compared with predicted results by FSMELT-2. It is shown that the prediction using o-i = 0.8 also gives good results. Memmott and Smoot 17] tested the effect of particle size on the mixture fraction and particle mass flux. Computed results are compared with their experimental data in Figure 4. The dotted lines are for centerline gas mixture fraction; the solid lines are for the centerline normalized particle mass flux. It is shown that the dispersion rate of small particles is faster than that of large particles; the gas mixing rate is increased only slightly by increasing the particle size. Computations by FSMELT-2 show some under-prediction, but the agreement with experimental data is quite satisfactory. Weber eta/. tl7] did simulation work on the dispersion of heavy particles in confined flows. They treated the particle phase in a Lagrangian way by assuming that all external forces except for the drag force were negligible, which is an unreasonable assumption for heavy particles, especially in a vertical confined chamber. They further did not account for the effect of the presence of particles on the fluid eddy viscosity, which was incorporated in the present work. They compared their prediction with experimental data obtained by Tice and Smoot tlSj for the case of 60 pet (mass) solids loading in the chamber of 0.206 m diameter. Comparisons of their prediction and the present work toMETALLURGICAL TRANSACTIONS B
gether with experimental data are shown in Figure 5. It is shown that their computation substantially under-predicts
In(x/R 1 ) 0
1.2 l
~
2.4
3.6
.8
\ ;
\
I=
,4
,,..o -2.0
.
-
,_~"B
-:
o,,,, I
-3.0
i
I
I
I
i
i
i
i
Fig. 4 - - E f f e c t of particle size on the centerline mixture fraction (fc) and normalized particle mass flux (mc/mo): A. small particles (19 ~ m ) ; B. large par:icles (54 ixm): - - - mixture fraction, - - particle mass flux; 9 A, i , 9 experiments by Memmott and Smoot. 171 VOLUME 19B, DECEMBER 1988--877
In (x/R 1 ) 1.2
0.0
2.4
3.6
4.8
-I.0 t=
=
-2.0
-3.0 Fig. 5 - Profileof normalizedcenterlineparticlemass flux:-- prediction by FSMELT-2(this work), - - - predictionby Weberet al., [171and 9 experiment by Tice and Smoot.t~Sj the results, whereas the present FSMELT-2 satisfactorily predicts the measured data. Sharp IzSl performed similar experiments and obtained similar results to those by Memmott and Smoot I71 and by Tice and Smoot. E151His measured radial profiles of axial velocity of the gas phase are shown in Figure 6 compared with predictions by FSMELT-2. It is shown that, although Sharp's data were rather scattered, FSMELT-2 predicts the experimental results quite well. Elghobashi and coworkers IS'6Jhave studied the two-phase flows in terms of modeling and experimental efforts. Elghobashi and Abou-Arab TM have recently developed a two-equation turbulence model for two-phase flows. They O
40 -
~'
o
o
x=O./TSm
20
E
~ i.u
0
I
I
I
20 O
I
I
I
0.25
0.50
0.15
1.0
r/Rf
Fig. 6--Radial profilesof axial velocityof the gas phase: 9 experiment by Sharp,1281-- predictionby FSMELT-2(this work). 878--VOLUME
19B, D E C E M B E R
1988
derived the exact equations of the turbulent kinetic energy and its dissipation rate, and developed third-order turbulent correlations resulting from time-averaging. Their twoequation model accounts for the interaction between the two phases and its influence on the turbulence structure. In order to validate their proposed model, Elghobashi et al.[6] studied a turbulent axisymmetric air jet laden with spherical, uniformly sized solid particles in a vertical cylindrical container described in Table II. They treated both the gas and the particle phases in the Eulerian framework in their modeling work. They compared their model predictions with experimental data measured using a laser-Doppler anemometer (LDA) and obtained good agreement between them. However, they mentioned that considerable testing for a wide range of turbulent two-phase flows is necessary to establish the universality of the coefficients used in their model. They used the flow conditions measured at 0.1 d 1 downstream of the nozzle exit as inlet boundary conditions (Table II). The present work also performed predictions using the same inlet conditions for their system described in Table II. The predicted results by FSMELT-2 are compared with their predictions and experimental data in Figures 7(a) and (b), in which the radial profiles of mean axial velocities are shown. In these figures, V, u , and Uc.s.phrepresent, respectively, the mean axial velocity of the particles, that of the gas phase, and the centerline velocity of the singlephase jet. The particle mass loading 4)0 was obtained by dividing the mass flow rate of particles by that of the primary gas. Good agreement between predictions by both groups and measured results was obtained for both cases of 4)o = 0.85 (Figure 7(a)) and 0.32 (Figure 7(b)). However, for the profile of the gas-phase velocity, their prediction is somewhat better than the present work. This better agreement may be credited to the fact that they modeled turbulence using third-order correlations involving fluctuating components of variables in contrast to the second-order correlations as shown in Eq. [13]. For the profile of the particle velocity, it is hard to distinguish which prediction is better except that the present work shows under-prediction and over-prediction of the centerline velocity, respectively, for the cases of heavy loading (Figure 7(a)) and light loading (Figure 7(b)), while Elghobashi e t a / . 161under-predicted it for both cases. According to them, the substantial increase of velocities of the two-phase flow compared with the single-phase flow can be explained by the fact that large particles do not respond well to the fluid turbulent fluctuations; thus the main force that accelerates a particle in the radial direction is the viscous drag exerted on the particle by the fluid radial velocity. The resulting drag force will be directed inward, thus limiting the radial spread of particles. Conservation of momentum of each phase then results in the solid-phase axial velocity being much higher than that of the fluid, and in turn the particles continue to be a source of momentum for the fluidJ 61 They also mentioned that an increase in the number of particles (or particle loading) resuits in increasing the momentum sources of the fluid, thus reducing the rate of decay of its centerline velocity. VI.
PREDICTIONS OF T W O - P H A S E F L O W FIELD IN A FLASH FURNACE SHAFT
In order to predict flow properties of the two phases under flash-smelting conditions, the system described in METALLURGICAL
TRANSACTIONS
B
2.4
2.4
A o Single phase 8 A Gas r2Two-phase C 9 ParticleJ
Uc.s.ph 2.0
u c.s.p-~h 2.0
x,O.4m
at
Uc.s.ph
A o Single phase B ZX Gas.,, ~Two-phase
Uc.s.ph
1.5
C 9
t'orTIcle J
at x = 0.4 m
. , ~ 1.5 ~-
1.0 -
0.5
-o,-.~.k.X - ~ ,
"',
0.5 %
,%
0
0.04
0.08
0.12
0.16
I
I
I
0.04
0.08
0.12
r/x
(b)
(a)
0.16
r/x
Fig. 7--Radial ~rofiles of mean axial velocity (V = particle velocity; u = gas velocity): - - prediction by FSMELT-2 (this work), - - - prediction by Elghobashi et al., i ] and 9 9 experiments by Elghobashi et al. [6] (a) Particle mass loading cb0 = 0.85, d~ = 50/xm. (b) Particle mass loading q50 = 0.32, dl = 50 txm.
Table III. Geometry:
Boundary Conditions for Flash Furnace dl -- 0.36 m dz = 1 d:= 4
m m
Lf=-7 m 4300 k g / m 3 45, 93, 1 5 0 / z m temperature = 298 K volumetric flow rate = 0.085 m3/s linear velocity: ul = 0.84 and vl = 0 m / s for axial feeding ul = 0 and vl = 260 m / s for radial feeding (equivalent slit width = 0.29 mm) 0 2 content = 21 pct turbulence intensity, 11 = 0.02 Secondary stream: temperature = 298 K Particle density: Particle size: Primary stream:
Simulation N u m b e r
1
2
3
4
20.5 30 21 0.15
20.5 30 50 0.15
20.5 30 100 0.15
10.2 15 50 0.08
26.3
62.4
124.7
31.3
Secondary Stream Volumetric flow rate (m3/s) Linear velocity ( m / s ) 0 2 content (pct) Turbulence intensity, 12 Primary Stream Particles feeding (kg/s)
METALLURGICAL TRANSACTIONS B
VOLUME 19B, DECEMBER 1988--879
~ 1.0
Table III was used. The dimensions of the system were chosen to be close to those of a commercial flash-smelting system. The primary flow of normal air (or the distribution air stream) laden with chalcopyrite concentrates and the secondary flow of an oxygen-enriched air (or the process air stream) enter the system through the nozzle at room temperature and expand radially. The effects of distribution air, inlet velocities, particle sizes, and particle loading or oxygen enrichment on the flow field were tested as discussed below.
(I-o}
-J~/O 2
0.5 -
(l-b)
A. Effects of the Distribution Air and Inlet Velocities The distribution air (or the primary air) plays an important role in the dispersion of solid particles in a flash furnace. In this work, two possible modes of feeding the distribution air were tested: the axial flow of this air with zero radial-velocity component (Figure 8(a)) and the radial flow with zero axial-velocity component at the inlet (Figure 8(b)). The latter represents a simplified approximation of an industrial flash-smelting burner system. [29]Two values for the inlet velocity of the process air, 15 and 30 m / s , [18'3~ were used to test its effect. A turbulence intensity of 0.15 was used for the inlet value of the process air at 30 m / s , and a value of 0.08 was assumed for the case of 15 m/s. To determine the inlet value of turbulence intensity for the distribution air (I 0, several values of I l were tested for both cases of the axial flow only and the radial flow only at the inlet. The results were carefully compared in terms of axial velocities. It was found that the overall profile of the axial velocity was not significantly affected by the initial value of 11 of the distribution air, and virtual independence was observed with values in the range of 0.01 to 0.1. This is due to the fact that the amount of the distribution air used in the flash-smelting system is much less than that of the process air (see Table III). Hence, the value of 0.02 was used throughout as the inlet value for the distribution air. Figure 9 shows the particle trajectories and the contours of axial velocities of the gas phase for different feeding modes of the distribution air (primary stream). The first two figures illustrate the particle trajectories and contours of axial velocity of the gas phase for the case of the axial flow of the distribution air with zero radial-velocity component at the inlet (i.e., u~ = 0.84 m / s and v I = 0 ) , while the last two figures are for the case of the radial flow with zero axial-velocity component (i.e., u~ = 0 and Vx = 260 m/s; equivalent slit width = 0.29 mm). For the former case, a
u2
u2
It,,lil
u2
u2
tjjJt
tt 7
"7
(o)
(b)
Fig. 8--Feeding modes of the distribution air at the inlet: (a) u~ with zero radial-velocitycomponent, (b) vI with zero axial-velocitycomponent. 880--VOLUME 19B,DECEMBER1988
- -
r
eo'~ ~, 1.0
,"
~,~1~ IL_
0.5
-~
(2-a)
(.o Z
--
I
i
0 '
~
I
~
I
/
I 0.5
0.5
t.z
(2-b)
_
t
2
3
4
5
AXIAL DISTANCE (m) Fig. 9--Particle trajectories and contours of axial velocities of gas phase ( d l = 45 /.Lm, 50 pct 02 enriched air): u I = 0.84, vl = O, and u2 =
15 m/s; (l-a) particle trajectory, (l-b) contour of axial v e l o c i t y , u I = 0 , vl = 260, and us = 15 m/s: (2-a) particle trajectory, (2-b) contour of axial velocity.
small, strong recirculation zone was formed between the two streams near the inlet region (l-b). It is of interest to observe that a field of increased axial velocity was formed near the tip of the nozzle. In this region, the maximum centerline velocity was 65 m / s at x = 0.08 m. These results may be explained in terms of the effects of gravitational force and particle distribution, as described in the next several paragraphs. Since particles gain momentum due to the gravitational force, and the inertia of particles is much greater than that of the gas due to the large difference of densities (pp/p = 3500) after the particle-laden distribution air is injected into the furnace, the velocity of the particle phase is greater than that of the gas phase. These factors result in the continuous momentum transfer from the particle to the gas. The effect of the gravitational force is illustrated in Figure 10 in terms of the centerline axial velocity of the gas in the two-phase flow: Curve A ( - - line) with the gravity effect and Curve A" ( - . - line) without it for the case of u I = 0.84, vl = 0, and us = 15 m/s. It is seen that the centerline axial velocity of the gas obtained by neglecting the gravitational force in the particle momentum is a little greater than that of the singlephase flow (Curve A ' , - 9 9line), while it is much lower than that obtained by including the gravitational force, especially near upstream. These results indicate that the particle is accelerated mainly by the gravitational force. They also METALLURGICALTRANSACTIONSB
indicate that the jet is less expanded radially due to the much lower inertia of that phase compared to the particle phase and the reduction of the turbulent kinetic energy due to the presence of particles. This can be seen in Figure 11, which shows the radial distribution near the inlet of the turbulent kinetic energy of the two-phase flow compared to the single-phase flow. The momentum addition to the gas phase due to the presence of particles can be obtained by Eq. [36]. This equation also suggests that the momentum source to the gas phase increases as the number of particles increases. Since the velocity ratio of the secondary to the distribution air stream at the inlet is 18, most particles are concentrated near the end of the distribution air stream. This high concentration of particles results in augmenting the momentum source to the gas phase, thus substantially increasing the velocity of that phase compared to the single-phase flow in this region. From the plot of particle trajectory for the former case ((l-a) in Figure 9), it is seen that most particles first hit the
1.0
0.5 Z
"-"
=
50
,~o
'4[i A i0 ,,
"
J 1')'7
x'/m
._
0.0
I
1.0
I.I.J
,,,
0.5 x'2m
" ,-,
.~
~~
0.0 1.0
~S_~.
l
r,,..d
__
_._1
_
"~
0.5
z o.o
~.
......._
x "3m I
0.0 -20 Fig. 1 0 - - P r o f i l e s o f centerline axial velocity (d~ = 45 p~m, 50 pct 0 2 enriched air): A , A ' , a n d A": ui = 0 . 8 4 , v~ = 0, and u2 = 15 m / s . B and B': u~ = 0, v~ = 260, and u2 = 15 m / s ; - - - single-phase flow. G a s phase in the t w o - p h a s e flow: - - with gravity effect, - . - without gravity effect.
10
0.5
x-- O.Olm
-~ _~
0.0 1.0
"~
0.5 x = O.32 m
0.0
0.25
0.50
0.T5
1.0
r/R 1 F i g . 11 1 R a d i a l p r o f i l e o f the t u r b u l e n t k i n e t i c e n e r g y : 50 p c t 0 2 , uj = 0 . 8 4 , vl = 0, and u2 = 15 m / s , d l = 4 5 / x m ; - - - single phase, - two-phase. METALLURGICAL TRANSACTIONS B
0.5
1.0
r/Rf Fig. 1 2 - - E f f e c t o f inlet velocities on the n o r m a l i z e d particle n u m b e r density: dj = 4 5 / z m and 50 pct 02. - - Ul = 0 . 8 4 , v~ = 0, u2 = 15 m / s ; - . - u~ = 0 . 8 4 , vl = 0, u2 = 30 m / s ; - - - ul = 0, v~ = 260, u2 = 15 m / s ; - - - Ul = 0, v I = 260, u 2 = 30 m / s .
centerline near the inlet and then move back toward the inlet of the secondary stream before traveling in the positive axial direction. For the latter case (i.e., ul = 0, vl = 260, and u 2 = 15 m/s), a wide recirculation zone was observed at the upstream region ((2-b) in Figure 9). No recirculation was observed in the comer region, unlike in the former case (l-b). The results show that for both cases the velocity field in the upstream region is very much affected by the vigorous turbulent motion of fluid elements and the large momentum difference. From Figure 9 (2-a), it is seen that most particles are dispersed radially due to the strong radial velocity near the inlet, and some particles are recirculated in the upstream zone. The profile of the centerline axial velocity for the latter case is shown in Figure 10. The dotted line (B') and the solid line (B) represent, respectively, the velocities of the single phase and the gas phase in the two-phase flow. It is seen that both the single- and the two-phase flows show the strong central recirculating zone up to x = 0.35 m, and the centerline axial velocity of the gas phase in the twophase flow is greater than that of the single phase, especially downstream. These results may be attributed to the same reasons as discussed previously. The effects of inlet velocities of the process air on the particle dispersion are shown in Figure 12. The particle number density was normalized by the centerline value to VOLUME 19B, DECEMBER 1988--881
illustrate the extent of dispersion of particles. For the case of ul = 0.84 m/s with zero radial-velocity component at the inlet, the dispersion of particles with u 2 = 30 m/s is wider than that with u2 = 15 m/s. For the case of strong radial velocity of vl = 260 m/s with zero axial-velocity component at the inlet, more uniform dispersion of particles is observed for both u2 = 15 and 30 m/s. The particle dispersion for the case of u2 = 30 m/s is somewhat more uniform than that for u2 = 15 m/s. The above model predictions imply that, in flash-smelting processes, the distribution air plays an important role in the dispersion of solid particles and more uniform dispersion of particles can be obtained by the radial feeding of the distribution air at the inlet. This uniform dispersion of chalcopyrite particles may provide better contact with gaseous reactants and, consequently, enhance the smelting rate.
B. Effect of Particle Size In order to analyze the effect of particle size on dispersion, three different particle sizes were tested at a fixed particleloading level. The computed results are shown in Figure 13(a) for particle sizes of 45, 93, and 150/zm for the case of 50 pct 02, u~ = 0.84, vl = 0, and u2 = 15 m/s. It is seen that smaller particles show better dispersion compared with larger particles, as expected. Since smaller particles follow the turbulent fluctuations more closely than larger particles, they tend to be more uniformly dispersed. The better dispersion of smaller particles is also in part due to the smaller drag force on the smaller particles. Fig-
ure 13(b) shows the computed results for the case of 50 pct 02, ul = 0, vl = 260, and u2 = 30 m/s. It is seen that the value of particle number density normalized by the centerline value is greater than unity for particles of 150/xm, while it is not for smaller particles. This again confirms the fact that larger particles are not dispersed as readily as smaller particles. Figure 14 shows the effect of particle size on the predicted centerline particle mass flux normalized by the inlet value for the case of axial feeding of the distribution air. It is seen that, since the velocity ratio of the process air to the distribution air stream at the inlet is 18, most particles are concentrated near the centerline after they are injected into the furnace; thus the centerline particle mass flux has a maximum value near the inlet. The figure also shows a smaller normalized mass flux for smaller particles. The larger negative values of particle mass flux for smaller particles indicate that they more closely follow the centralrecirculating flow field than the larger particles. Although not illustrated, a lower particle mass flux was observed for the larger particles along the centerline for the radial feeding mode of the distribution air. This is due to the fact that the larger particles radially injected into the furnace by the radial flow of distribution air are not dispersed as readily as smaller particles. It was also observed that the computed result for the case of mixed particles of 50 wt pct of 37/zm 2.0
/ /
Z
1.0
=
Z
~
i 0.5
I
1.0
x.lm
X* /m z
,.,
"~,,..
""
0.0 2.0
I
l.,u
0.0
I
1.0
I 7
~"
1.0
I
w _..I
=
0.5
w
\
x=2m I
O0
0.0
~-
1.0
2.0 11
I.w
!
Id.I
/ _._1
=
0.5
I
1.0
xz
x=Jm
%:(
0.1
0.2
.
.
.
.
0.3
, 0.4
0
I
0..0
0.5
F/RI (a)
r/Rf (b)
Fig. 1 3 - - R a d i a l profiles o f particle n u m b e r density for various particle sizes: (a) 50 pct 0 2 , u~ = 0 . 8 4 , v~ = 0, and u 2 9 3 / x m , - - 1 5 0 / z m . (b) 50pct O+, ut = 0, v~ = 260, and u2 = 30 m / s . - - - 1 5 0 / x m , - - 9 3 / z m , - . - 4 5 / z m .
882--VOLUME 19B, DECEMBER 1988
1.0
=
15 m / s . - - - 45 /xm, - . -
METALLURGICAL TRANSACTIONS B
8",,iI 6 IE
2
,? P
0.6 4)
/f
0.4 0.2
r N
0.0 1
2
-0.2
Axial
i. 0
3
4
5
Distance(m)
-O.Z
Z
1. A more uniform dispersion of particles can be obtained by introducing a strong radial velocity of the distribution air at the inlet. 2. An increase in the velocity of the process air results in a somewhat more uniform dispersion of particles. 3. Smaller particles show better dispersion since they follow the fluid turbulent fluctuations more closely than larger particles. 4. The particle-loading level in the range of the flashsmelting conditions does not significantly affect the normalized particle mass flux. 5. Model predictions imply that, in the flash-smelting process, more uniform dispersion of chalcopyrite particles obtained by the radial feeding mode of the distribution air at the inlet may provide better contact with gaseous reactants and, consequently, enhance the smelting rate of the sulfide mineral.
Fig. 1 4 - - E f f e c t o f particle size on the centerline particle m a s s flux norm a l i z e d b y the inlet v a l u e : 50 p c t 0 2 , u] = 0 . 8 4 , v] = 0, a n d u2 = 15 m / s . - - 4 5 / x m , - . - 9 3 / x m , - - - 1 5 0 / z m .
NOMENCLATURE projected area of a particle Cl, C2, C. constants in the turbulence model, Eqs. [14] and [17] drag coefficient defined in Eq. [22] Co dj diameter of the furnace dl diameter of the primary inlet diameter of the secondary inlet d2 turbulent diffusion coefficient of species i Dpt turbulent particle diffusivity mixture fraction f F~ defined in Eq. [21 ] g gravitational acceleration I turbulence intensity k turbulent kinetic energy Lj length of furnace m particle mass flux mp mass of a particle n particle number density pressure P r radial distance from the axis of symmetry Re Reynolds number, defined in Eq. [23] RI radius of furnace Ri radius of the primary stream S scalar quantity S~ source or sink term for any variable, Eq. [35] t time tt time scale of turbulence U axial velocity U~ fluctuating component of axial velocity centerline velocity of single-phase jet //c.s.ph average axial velocity of the primary and the t/l,U 2 secondary streams, respectively V radial velocity Vt fluctuating component of radial velocity V1 average radial velocity of the primary stream mean axial velocity of the particles g_, v,v velocity vectors of the gas phase and the particle phase, respectively, Eq. [20] Wr fluctuating component of tangential velocity X axial distance from the nozzle exit Ap
and 50 wt pct of 150/zm shows almost the same result for the particle size of 93 pm, which is the average value of 37/zm and 150/zm.
C. Effect of Particle Loading Particle loading can be adjusted by varying the oxygen enrichment. The oxygen contents of 21, 50, and 100 pct are equivalent to particle loadings of 0.0257, 0.061, and 0.12 vol pct, respectively, under the feeding condition of 0.25 kg O2/kg copper concentrate. 13~ Computed results indicated that particle-loading level does not significantly affect the normalized particle mass flux under flash-smelting conditions, although the results may be different at very high solid loadings. Tice and Smoot t~Sjalso obtained similar resuits in their system.
VII.
CONCLUSIONS
The numerical predictions of the particle-laden flow field with the two-equation (k-e) turbulence model show in general good agreement with experimentally measured data. Model predictions show that the presence of solid particles increases the axial velocity of the gas in their vicinity by providing additional momentum. Agreement of the predicted results with experimental data is satisfactory, but some improvement needs to be made in order to overcome the shortcomings of the turbulence model. A turbulence model based on a firmer theoretical foundation for the interaction between two phases may provide this needed improvement. In spite of this limitation, the accuracy of prediction obtained by the present work represents a considerable improvement over that by Weber et alfl 7] or can be achieved with greater computational economy than that by Elghobashi et al. 161 Model predictions of the behavior of a particle-laden turbulent jet under flash-smelting conditions point to the following conclusions:
METALLURGICAL TRANSACTIONS B
VOLUME 19B, DECEMBER 1 9 8 8 - - 8 8 3
Greek Symbols Fe e 0v
effective transport exchange coefficient dissipation rate of turbulent kinetic energy volume fraction of particles (= volume of particles/volume of gas) /x viscosity l) kinematic viscosity density of fluid O Ork , Ore , O'~p Prandtl-Schmidt numbers for k, 8, and particle diffusivity, respectively shear stress particle relaxation time % general dependent variable, Eq. [35] 4, particle mass loading (= particle mass flow 4,0 rate/gas mass flow rate) Subscripts
bp c e f g l o p
pc, pd t 1 2
particles in bulk phase centerline effective furnace gas laminar inlet particle phase convective and diffusive velocity of the particle phase, respectively turbulent primary stream secondary stream
Superscripts t
turbulent
Overline time-averaged
ACKNOWLEDGMENTS The authors wish to express their appreciation to Dr. P. J. Smith of the Department of Chemical Engineering, Brigham Young University, Provo, Utah, for providing their computer program based on which the computer code FSMELT-2 was developed. This work was supported in part by the National Science Foundation under Grant No. CPE8204280, United States Bureau of Mines under Grant No. USDI-G1125129-UTAH, the State of Utah Mineral Leasing Fund, and a Student Research Fund from the University of Utah Computer Center. REFERENCES 1. C.T. Crowe, M. P. Sharma, and D. E. Stock: Journal of Fluid Engineering, Trans. ASME, 1977, pp. 325-32.
884--VOLUME 19B, DECEMBER 1988
2. D.B. Spalding: Numerical Computation of Multiphase Flows, Lecture Notes, Purdue University, Lafayette, IN, Thermal Sciences and Propulsion Center, 1979, pp. 161-90. 3. D.B. Spalding: paper presented at American Nuclear Society Meeting on Nuclear-Reactor Thermal Hydraulics, Saratoga, NY, 1980. 4. E.K. Melville and N. C. Bray: Int. J. Heat Mass Trans., 1979, vol. 22, pp. 647-56. 5. S.E. Elghobashi, T.W. Abou-Arab: Phys. Fluids, 1983, vol. 26, pp. 931-938. 6. S.E. Elghobashi, T.W. Abou-Arab, M. Rizk, and A. Mostafa: Int. J. Multiphase Flow, 1984, vol. 10, pp. 697-710. 7. V.J. Memmott and L.D. Smoot: AIChE Journal, 1978, vol. 24, pp. 466-73. 8. P.J. Smith and L.D. Smoot: Combustion Sci. Technol., 1980, vol. 23, pp. 17-31. 9. J.R. Thurgood, L.D. Smoot, and P.O. Hedman: Combustion Sci. Technol., 1980, vol. 21, pp. 213-23. 10. L.D. Smoot and D. T. Pratt: Pulverized Coal Combustion and Gasification, Plenum Press, New York, NY, 1979, pp. 107-19, 263-95. 11. P.J. Smith, T.H. Fletcher, and L.D. Smoot: 18th Syrup. (Int.) on Combustion, The Combustion Institute, 1980, pp. 1285-93. 12. L.D. Smoot: 18th Syrup. (Int.) on Combustion, The Combustion Institute, 1980, pp. 1185-1202. 13. P.J. Smith, S.C. Hill, and L. D. Smoot: paper presented at the 19th Symp. (Int.) on Combustion, The Combustion Institute, 1982. 14. L.D. Smoot and P.J. Smith: User's Manual for a Computer Program for 2-Dimensional Coal Gasification or Combustion (PCGC-2 ), Combustion Laboratory, Chem. Eng. Department, Brigham Young University, Provo, UT, 1983. 15. C.L. Tice and L. D. Smoot: A1ChE Journal, 1978, vol. 24, pp. 1029-35. 16. E J. Smith: Ph.D. Dissertation, Brigham Young University, Provo, UT, 1979. 17. R. Weber, E Boysan, W.H. Ayers, and J. Swithenbank: AIChE Journal, 1984, vol. 30, pp. 490-92. 18. S. Ruottu: Combustion and Flame, 1979, vol. 34, pp. 1-11. 19. u B. Hahn and H. Y. Sohn: Chem. Eng. Commun., 1987, vol. 61, pp. 39-57. 20. B.E. Launder and D.B. Spalding: Mathematical Models for Turbulence, Academic Press, London, 1972. 21. A.S. Abbas, S.S. Koussa, and F.C. Lockwood: 18th Symp. (Int.) on Combustion, The Combustion Institute, 1980, pp. 1427-37. 22. E.E. Khalil, D.B. Spalding, and J. H. Whitelaw: Int. J. Heat Mass Trans., 1975, vol. 18, pp. 775-91. 23. D.B. Spalding: in Turbulent Mixing in Nonreactive and Reactive Flows, S. N. B. Murthy, ed., Plenum Press, New York, NY, 1975, pp. 85-130. 24. F.C. Lockwood, A.P. Salooja, and S.A. Syed: Combustion and Flame, 1980, vol. 38, pp. 1-15. 25. G.P. Lilly: Ind. Eng. Chem. Fundam., 1973, vol. 12, pp. 268-75. 26. A.D. Gosman and W.M. Pun: Lecture Notes for Course Entitled "Calculation of Recirculating Flows", Imperial College, London, 1973. 27. S.V. Patankar: Numerical Heat Transfer and Fluid Flow, McGrawHill Book Co., New York, NY, 1980, pp. 79-135. 28. J.L. Sharp: M.S. Thesis, Brigham Young University, Provo, UT, 1981. 29. Outokumpu Oy, Engineering Division, Flash Smelting, FINNAD/ A & M, 3831-948-4HE-9, printed in Finland, 1981. 30. D.B. George: personal communication, Kennecott, Salt Lake City, UT, 1986. 31. N.J. Themelis and H.H. Kellogg: in Advances in Sulfide Smelting, H.Y. Sohn, D. B. George, and A. D. Zunkel, eds., 1983, vol. 1, pp. 1-29. 32. A.A. Mostafa and S.E. Elghobashi: Int. J. Multiphase Flow, 1985, vol. 11, pp. 515-33.
METALLURGICALTRANSACTIONS B
