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Combustion of coal particles in wall-burning swirl combustors
03th5442/79/04OI-O3 15/502.00/O
Energy Vol. 4. pp. 315-327 @ Pergamon Press Ltd., 1979. Printed in Great Britain
COMBUSTION OF COAL PARTICLES IN WALL-BURNING SWIRL COMBUSTORSt PAUL M. CHUNG University of Illinois at Chicago Circle, Chicago, IL @6J30,U.S.A.
and R. SCOJT SMITH Argonne National Laboratory, Argonne, IL 60439, U.S.A. (Receiued 7 November 1978)
Abstract-An approximate analysis is made of the performance of the wall-burning coal combustors. The results show that the swirl number plays a more important role than the char gasification chemical kinetics in determining the overall gasification rate. At high pressures, the wall-burning combustors may not be as efficient as the gas-stream combustors. The situation, on the other hand, may be reversed at the lower pressures. NOTATION particle Damkahler number defined by Eq. (42) coal combustion rate per unit combustor cross-sectional area functions of Dm and r respectively linear and angular momentum fluxes respectively char combustion rate per unit combustor wall area dimensionless combustion rate per unit combustor area defined by Eq. (31) surface reaction rate 2-4 for moderately rough, and >4 for rough surface specific wall reaction rate for the ith species combustor length molecular weight species mass fraction species mass fraction along the combustor center line coal particle radius combustor radius Reynolds number, uOR,/v radial coordinate (K,/2)“‘s0
wda
swirl number defined by Eq. (34) x-component of gas velocity species diffusion velocity r-component of gas velocity tangential component of gas velocity x/R, beginning of developed region defined by Eq. (22) axial coordinate x-x, normal distance from combustor wall Damkohler numbers defined by Eqns. (21) and (27) respectively boundary layer thickness density of coal species mass flux molecular kinematic viscosity gas density; gas-coal density in Eqns. (40), (4l), and (44) parameter defined by Eq. (30) shear stress Subscripts c carbon i ith species
s w 0 I, 2,3 m
solid particle combustor wall boundary layer edge for X X, 4, I&O, and CO* respectively inlet to the combustor
tThis Paper was presented at the 1978Midwest Energy Conference, Chicago, Illinois, 19-21 November, 1978. 315
316
p.hi.CHUNG andR.Scorr SMITH 1.INTRODUCTION
With the projected construction of open-cycle MHD-power-generation systems using coal as the fuel, rather intensive interest has arisen in recent years in the design of efficient coal combustors which would satisfy the particular demands of the MHD systems. The primary demand is to generate combustion products in the order of 5000°F which are reasonably free of the ash vapor, at a difficient rate to meet the power requirement. High pressure and temperature, swirling combustors of different configurations have been proposed to meet the demand in the past (see Syred and Beer,’ Heywood and Womack,’ Stickler et al.? Lewellen et ai.: Way? and Baer6). The swirl is introduced so that combustion and gasification of the coal particles will largely take place along the combustor walls. The wall combustion is considered to be advantageous for the following two reasons. The first is that this ensures trapping of the slag along the wall for its eventual removal. The second is that by transporting the coal particles to the wall it is thought (see Syred and Beer’) that the coal residence time necessary for the combustion and gasification is lengthened and, consequently, the combustor e5ciency is enhanced. In chemical reactions where two or more of the initially unmixed reactants participate, arbitrarily increasing the residence time of one of the reactants does not necessarily enhance the reactions. Efficiency of a coal combustor depends on many complicated factors, and it can not be estimated simply from the individual residence times of the reactants. Previously, one-dimensional flow models have been employed to analyze the performance of a high temperature coal combustor (see, for instance, Sarofim et al.,’ and Chung and Smith’). Inherent in such models is the assumption that all reactants including the coal particles are uniformly mixed at any position along the direction of the flow. Swirl quickly separates the coal particles from the gaseous species and destroys the uniformity. In the present study, it is considered that the swirl is sufficiently high and the coal injection system is designed such that all char combustion and gasification take place along the combustor wall. The problem analyzed here is the limiting case, opposite to the onedimensional model, wherein segregation of the solid reactant from the gaseous reactants is complete. It is a major task to describe the flow in a given swirl combustor even with the assumption of no mixing and chemical reaction (see Lewellen4). In view of the present incomplete state of knowledge of coal chemical-kinetic properties (see Chung and Smith’), an embarkation to a detailed numerical computational program of the swirl combustor doesn’t seem to be justified. In this study, a very approximate but general approach is employed. It is the objective of the analysis to elucidate the parameters controlling the combustion of coal along the wall in swirl combustors. A comparison will be made of the combustion rates between the wall burning and the completely mixed one-dimensional combustors. As an initial phase of the complex problem, rigor will be sacrificed in favor of generality, and closed-form solutions will be sought. 2.FORMULATION
We consider a rather idealized swirl combustor of cylindrical geometry for simplicity as shown in Fig. 1. The flow pattern in the combustor will depend on the manner in which the swirl is generated and on physical configuration of the combustor. As was mentioned in Section 1, it is not the present purpose to describe the details of the flow in a given combustor, but to estimate the degree of influence of the various governing parameters on the wall combustion of coal in general. With this in mind, we consider that the axial and tangential velocity components u. and wo, are given and uniform along the swirl boundary layer edge. All symbols are defined in the nomenclature. Ratio, then, of the two velocity components, So = wo/uo, is related to the combustor swirl number, SW,as will be shown subsequently. With So considered to be uniform and given, the vortex separation, recirculation, etc. are bypassed, and the flow description is greatly simplified. Since the solid particles are considered to bum only at the combustor wall, the recirculation and other flow details have only a moderate influence on the combustion. It is assumed that the coal injection system and swirl are such that the coal particles are thrown onto the wall at very small values of x after losing a portion of the volatile matter during their journey to the wall. The particles then move along the wall at small velocities being dragged by the gas streams above them. In a steady state, the combustor wall is covered by a layer of coal particles which burn and gasify as they interact with the gaseous reactants
Combustionof coal particles in wall-burningswirl combustors
317
Fig. I. idealized swirl combustor.
transported to the layer surface by turbulence. The distance x required to complete the combustion and gasification is sought in the present analysis. GOVERNING
EQUATIONS
The most important gaseous reactants for the coal combustion and gasification are the oxygen, carbon dioxide, and the water vapor. Also, with the coal particle temperature along the combustor wall estimated, energy equation is not required. The starting point of the analysis is then the following set of the conservation equations: (continuity) (momentum)
spur ax
I apvr _ o ar ’
pu~+pu$=-;$r).
(chemical species)
(1)
(2) (3)
In the above equation, i = 1.2, and 3 signify 02, HzO, and CO2 respectively. 7 and 01denote the shear stress and the mass ditfusion flux of the ith species respectively. BOUNDARYCONDITIONS The standard boundary conditions are applicable to Eqs. (1) and (2). The boundary conditions
for Eq. (3) are as follows. mi(O,r) = mjm
(4)
&(X9 l?c) = Jim
(5)
The reaction rates, Ji, between the ith gaseous reactant and the wall layer of the coal particles are discussed below.
318
P. M. CHUNG and R. SCOITSMITH CHEMICAL
REACTIONS
It is considered that all hydrogen constituent volatilizes before the coal particles reach the wall and is oxidized to HzO. The particles at the wall then mostly consist of carbon and ash. The following combustion and gasification reactions are considered to take place: 02tc++coz,
co* t c 2
(6) 2c0,
(7)
H,OtCLZ"-COtH,.
63)
The specific reaction constants, kj,, are Arrhenius type functions of temperature; however, their functional relationships are of no importance here since these values are considered to be given based on the estimated value of the wall particle temperature. A considerable amount of published data is available for the char surface oxidation rate, k,, (see for instance, Field, Smith,” and Howard and Essenhigh”). Estimating the particle temperature on the combustor wall to be about 4OOO“R(see Chung and Smith*), these data%*’ give k,, of between 100 and lOOOftper sec. On the other hand, the available experimental data on the k,, and k3, are very scarce. It can be deduced from Johnson’s work” that k,, and k3, at the temperature range of 4000”R are about two orders of magnitude smaller than k,, It should be noted that the present definition of ki, is based on the surface area of the particle layer as it is with that of the reactant mass flux fli, The reaction rate between the particle layer and the adjacent gas layer may be substantially greater than those measured in the experiments.e’2 As it will be seen later, the values of lOO-lOOOft/sec for k,, is already sufficiently large so that the reaction of Eq. (6) is diffusion controlled in a typical combustor. We shall, therefore, employ, k,, = 500 ftlsec
(9)
in the subsequent computation. For the values of kzw and k3,,., the following range of the values will be employed. klw = ksw = l-100 ftlsec.
(10)
With the ki,‘s chosen, the reaction rate, Jr, is expressed by (see Chung13) Ji = p&iwmiw for i = 1 and 2, (11)
J3 = p,hmw - $&hmlw. 1 The second term in the above J3 equation results from the reaction of Eq. (6) which is a CO2 source for the reaction of Eq. (7). In constructing Eqs. (ll), first order reactions are assumed for Ji and J3 for lack of better information. SHEAR
STRESS
AND MASS FLUX
Effect of the swirl on the surface reactions enters into the problem through its effect on the shear stress, 7, and the reactant mass flux, 0, 7 and & denote the turbulent transport rates, and the swirl enhances the turbulence and, consequently, the transport rates. With the subsequent closed-form solution in mind, we derive the expression for rw as follows. For a high Reynolds number, turbulent flow over a straight surface, Sch1ichting14gives rw =
KpolP4
0 ;
114 ,
where K is 0.0225 for a smooth surface, and increases with the surface roughness. This
Combustion of coal particles in wall-burning swirl combustors
319
expression is modified, after the manner of Rott and Lewellen,” and Bloor and Ingham,16 for the present range of interest where So B 1 as, rw = Kpow3’4u
0 11
I/4 .
Y
(13)
Expression for the species mass flux at the wall, fli,, is then constructed from Eq. (13) by exploiting the general analogy between the momentum and mass transport. Postulating a similitude between the transports of the quantities (mi - miw)/(mio- miw)and u/uo, we write, 6i, = K&W3’4
0 5
l/4 (??ti - ??Z+)*
(14)
Formulation of the problem is now complete. Solution of the governing equations, Eqs. (l)-(3), constitutes the next section. 3. ANALYSIS
The Pohlhausen integral scheme14 is employed in the solution. The solutions are obtained separately for 0 x,, and are matched at x = x, (see Fig. 1). Region 0 < x 5 xc
Equations (1) and (2) are first combined and integrated with respect to r to give, pu(uo- u)r dr = R‘r,.
(15)
As usual,14 a seventh-power profile is assumed for u/uO. Equation (13) is substituted for r,,, in Eq. (15) with y = 6. The resulting first order differential equation is then solved to give,
(16) This is a transcendental algebraic equation for 8 which is difficult to use. It can be put into a more convenient form in the following manner. For S/R, 4 1, Eq. (16) gives, $
= (~Kp~‘s(-!_)“5X4~5,
(17)
On the other hand, Eq. (16) gives for S/R, = 1,
(18) As was mentioned following Eq. (12), K may increase by several factors from the smooth wail values of 0.0225 when the wall is rough. Expecting at least a moderately rough gas-coal particle layer interface, and noticing the factor of 3/2 differentiating Eq. (18) from (17), we shall employ the relationship, $
= [ (~)2(o.0225)luIs”5(+-)“sx4~5,
(19)
where S = (K /2)4’3S Equation 44) wa:‘constructed by envoking the general analogy between the momentum and mass transport. We now assume that the species boundary layer thickness is the same as that of the momentum boundary layer, 8. Also, we assume for the region 0
320
P.M. CHUNG and R.SCOTT SMITH
constant, mi is locally similar (see Chung”). Then, Eq. (14) represents a solution at the wall of the species conservation Eq. (3), and no further solution of Eq. (3) is needed for the present purpose. Equations (5) and (11) are now combined with Eq. (14) where y is set equal to the S given by Eq. (19). The resulting equation is solved for mi, to satisfy the boundary condition, Eq. (4), as for i = 1 and 2,
where the Damkohler numbers, Ii’s, are defined for all i = I, 2, and 3 as,
(21) In deriving the closed form solutions, the gas density was assumed to be constant at po. This assumption is commensurate with other approximations made in the problem. Finally, X, is determined from Eq. (19) by setting S = R, as, X, = l.728(Re/S3)“4.
(22)
With the miw’sfound, Eqns. (11) give the reaction rates as functions of X. The reaction rates will be considered more in detail after the solution for x > xc. Region x > xc
Thickness of the diffusion layer in this region is simply equal to R,. Hence, ei, is given by Eq. (14) with y replaced by R,. The mass fraction mio is a function of x, and therefore, the local similarity concept may not be acceptable. The species-conservation equation, Eq. (3), will be solved by an integral method. Integrating Eq. (3) with respect to r, there results,
(234 Commensurate with the assumption of constant gas density, it is assumed that u = u. in the integrand of this equation. The following seventh-power profile is substituted for the species mass-fraction.
mi-mi, Wo -
=(Y)“‘=(R;~~~
(23b)
4,
Equation (23a) is then put into the following form with the use of Eq. (14) in which y is replaced
by Rc. l/4
$(49mio + 1lmiw) = - ~(0.045)W~‘4(+)
c
(mio
- miw).
(24)
The wall boundary condition, Eq. (5), is written with the use of Eqns. (11) and (14) as, for i = 1 and 2,
(25)
(26) where for all i, Pi= 1.115 $ “20fi. ( >
(27)
Combustionof coal particlesin wall-burningswirl combustors
321
buations (24) and (25), or (26), constitute two equations for the two unknown functions, mio and mip These equations are integrated and there result, upon satisfaction of the boundary condition, m&x,) = mica, ?ttio = ntjmexp
(> M3
m30=
E
for i = 1 and 2,
(-Jd)
Mom
(28)
rnti exp(-13R),
[exp(-~~)-exp(-5,~)lt
(29)
(14+3/F,)
where for all i, &=
(30)
5.4(z)l'*/(49 t 7). i
The wall values, mi, are obtained from Eqs. (25) and (26) with the mio given by Eqs. (28) and (29). Combustion and gasification With the appropriate mass fractions of the gaseous reactants having been obtained, the char combustion and gasification rate is determined from Eqs. (11) as, Local char combustion and gasification JCW)= rate per unit combustor wall area Pwk3wm3m p&.m3=
JXX)=
(31)
The total combustion and gasification rate is obtained, for I, & 1, by integrating K(X) as Total char combustion and gasification rate for the combustor length of X
w,(X)
wxx) = 27rR~pfl&,)
=
=
2~&*(pwmA3J
X J: dX
(324 and M kl, mco 1 =A_[1- ew (-Ml M ksw m3.sI (I+ f,)S, , MCkzwmm M2
+ --K
k3w
&
1
M3(1tfi3)
1
-[Il (I+
exp
xc)
I
12i)1+ w2, Xd}
r2M2
-M3 {[
(-
+ w,,
ml,+l
1
M,(l-f3/fI)mh 1
(1 - f,/f,)
MC 1 t j$ ~)qr3, +z ( I 300
[I
-
exp
(-
I-IAl 1[I-exp I
l3@1
53
Cl
I
Xc) for X > XC
(32b)
In the above equations, 415
x3/5
x215
p
trix”‘)].
“(ri,X)=~[~-~t~-~t~ln(l I
1
L
I
I
(33)
P. M.
322
and R.
CHUNG
SCOTT
SMITH
Solution of the equations is now completed. Typical results will be discussed in the next section. 4. RESULTS
AND
DISCUSSION
is evident from the solutions obtained that there are two dimensionless parameters that dictate the behavior of the problem. They are the Damkohler number, I’i, and the ratio S3/Re. fi also enters into the problem prominently; however, it is closely related to, and the same order of magnitude as ri as can be seen from Eqs. (21) and (27). Before the actual consideration of the results, a discussion of the relationship between the parameter S and the more commonly accepted swirl number, SW,may be useful. Swirl number is generally defined as the ratio of the axial flux of the angular momentum to that of the linear momentum multiplied by a suitable flow radius. For a cyclone, the exit radius is a meaningful quantity (see Syred and Beer’), and the swirl number can be written for the present problem as It
SW = G&G&
(34)
For the purpose of estimating SW,we approximate the tangential velocity by, (35) This expression is an approximate solution of the momentum equation for large values of S. Then, for the axial velocity of the order of uo, Eq. (34) shows that, So = 2Sw, and S = 2(K,/2)4’3Sw for r. 5 i R,.
(36)
With the physical meaning of the S clarified, let us now consider the typical numerical results shown in Figs. 2-7. These sample computations are based on the general property of a
0
I
2
3
4
5
6
7
6
9
IO
X
Fig. 2. Variations of reactant mass fractions. 0.12 010
1 s=20 -
’
K,w=500ft/s KeW=K3W=I0 CASE A
-1
.
1
. .
1
I
.
fl/s
I
--_,“I,0 .
-----_
I
. .
NW
-.
---
-
-_
c,“o
--4 --__
____r%
--_
--
-----___ \m3W
002
-
/m2w VW
0s 0
, I
III
I
I
I
I
I
I
Z&3
4
5
6
7
6
9
X
Fig. 3. Variations of reactant mass fractions.
IO
323
Combustion of coal particles in wall-burning swirl combustors
K2W=K3W=I,
SOLID
K2W*K3W’100, NOTE:
0
VALUES
t
2
CURVES
DASHED
CURVES
FOR DASHED
3
4
CURVE
5
6
8
7
9
IO
X Fig.
4. Combustion and gasification rate per unit combustor wall area.
2
I
I
I
I
I
I
I
I
I
4
5
6
7
8
9
KeW’K3W=I0
-s
\
z
CASE
“0
A- SOLID
I
CURVES
2
3
IO
Fii. 5. Combustion and gasification rate per unit combustor wall area.
_
CASE
A
K~w=K~w=I,
SOLIDCURVES
K~w=K~w=IOO, NOTE: _
0
DASHED
VALUES
HAVE
BEEN
I
2
CURVES
FORDASHEDCURVE
MULTIPLIED
3
4
BY TEN
5
6
7
8
9
IO
X
Fig. 6. Integrated combustion and gasification rate.
Montana Rosebud coal with the fuel-to-air ratio of 0.2, which corresponds to about 60% of the stoichiometric oxygen supply. The first stage of a two-stage combustion system usually operates with the oxygen supply between 40 and 60% of the stoichiometric value. Two conditions of the coal-air mixture before the wall combustion of char are considered. In the first case, “Case A”, 10% of the coal combustible is assumed to volatilize before the coal particles reach the combustor wall. In the second case, “Case B”, 30% of the combustible is assumed to volatilize before the wall. In both cases, all H and a portion of C are volatilized and are oxidized to Hz0 and CO2 respectively. At X = 0, therefore, the initial mass fractions of 02,
P. M. CHUNO and R. Scorr SMITH
324
,-
‘kw=kw=IO CASE
A-SOLID
CURVES
CASEB-DASHEDCURVES
-0
I
2
3
4
5
6
7
8
9
IO
I
Fig. 7. Integratedcombustionand gasificationrate.
HzO, and COZare computed as, Table 1.
mlm2., m3=
Case A
Case B
0.120 0.077 0.041
0.013 0.077 0.188
The three reactions given by Eqs. (6), (7), and (8) then commence along the combustor wall being dictated by the parameters Ii and S’/Re as stated earlier. The practical range of kiw’swas given in Section 2. In order to render the ensuing discussion physically meaningful, we shall fix R,, uo, and Re, and shall show the computed results as functions of kiy’s and S. Since the present solutions are in simple closed form, implication of these results to other specific combustor conditions can be readily drawn via the particular values of Ii and S3/Re to which these results correspond. For the computations of gigs. 2-7, the following values were employed. R, = l/2 ft, I(~= 80 ftlsec.,
(37)
Re = 31,746. Figures 2 and 3 show typical variations with respect to X of the gaseous reactant mass fractions along the combustor wall and core. Because of the high reaction rate of oxygen, k,, the oxygen at the wall is reduced to near zero from the initial value of 0.12 almost instantly. The wall oxidation of the coal is, therefore, turbulent diffusion controlled throughout the combustor. The initial value of m3,,,is 0.041 for Case A. Figure 2 shows that it increases immediately to 0.115 as the oxidation reaction of Eq. (6) produces CO2 at the wall according to the large k,, but the small k3w is unable to consume it. The accumulated CO2 is then slowly depleted along the wall through the gasification reaction of Eq. (7). Figure 3 shows that the initial accumulation of m3wis not as severe when ksw is 10ftlsec. When the swirl is such that S = 1, the core region of the combustor is still oblivious at X = 10 to the wall chemical reactions as seen by the undisturbed mass fractions at r = 0 (see Fig. 2). At S = 20, on the other hand, it is seen in Fig. 3 that the increased turbulent mixing begins to affect the core region of X > X,. The dimensionless local reaction rate, &, is shown in Figs. 4 and 5. For all cases, the effect of increased swirl on the tendency toward freezing of the reaction is seen. This is expected from the composition of the Damkohler number given in Eqs. (21) or (27).
Combustion of coal particles in wall-burning swirl combustors
325
The physical meaning of the Damkiihler number can be made clear at this point. The mass flux of a gaseous reactant at the combustor wall is given by Eq. (14). The order of magnitude of the reactant diffusion velocity across the combustor can be determined from Eq. (14) as, .
(38)
From Eq. (27), it is readily seen that the present Damkohler number is simply the ratio of the reaction velocity to the gaseous reactant diffusion velocity,
For a given value of ki,, the swirl increases Vd and decreases fi. According to the present solution, this tends to freeze the reaction resulting in a higher mi, This, in turn, increases the combustion and gasification rates as given by Eqs. (11). Note that the actual reaction rates are equal to the J: shown in Figs. 4 and 5 multiplied by (p&3~ti). The integrated dimensionless char combustion and gasification rates, W:, are shown in Figs. 6 and 7. Note that, in these computations, it has been assumed that 10 and 30% of the combustible are volatilized before the particles reach the wall for the Cases A and B respectively. These figures show the wall gasitications only. Figure 7 shows that for the given ki, and S, the wall combustion and gasification rate is substantially greater for the Case A than Case B. These two cases represent the same fuel to air ratio and fuel properties. The only difference is that a greater amount of O2 is converted to CO2 before the commencement of the wall combustion in Case B. The larger amount of COZ production at the wall by the diffusion of 02 and fast oxidation caused the more efficient char gasification for Case A than for Case B. However, the total combustion and gasification rate including the initial volatilization is still higher for the Case B since the volatilization is the most efficient mechanism of combustion. It may be useful, at this point, to discuss certain aspects of the efficiency of coal combustors in general. In particular, we shall briefly compare the wall-burning swirl combustor with the combustors designed for combustion in the main gas streams. For combustors with uniform diameters, one may write approximately for the combustion rate per unit volume, i,,
”
dm,=+,, dx
For one-dimensional flow model with the particle velocity assumed to be equal to the gas velocity, Eq. (40) can be transformed into the following equation (see Chung and Smith’).
(41) where, for convenience of discussion, only a single gaseous reactant i is considered. In the above equation, the particle Damkiihler number, Dmi, is defined as
D,ib!5, Y
(42)
and, the function f(Dmi) is of the form
(43) As it was the case with the wall combustors explained earlier, the combustor efficiency is
326
P. hf. CHUNGand R. SCOTTSMITH
enhanced when Dmi & 1 for a given ki,. Dmi is reduced as the particle radius is reduced and as the molecular diffusivity, v,is increased. Returning to Eq. (41), Fb represents the mass rate of coal combustion per unit combustor cross sectional area. For the given ki, and Dmi, the combustion efficiency increases (the required combustor length, L, is reduced) with p2. Consider now the case wherein the particles are made to accumulate by reducing u,. Reduction of u, implies a longer particle residence time in the combustor for a given combustor length. For the given required burning rate, & this implies the corresponding increase in the mixture density p and, therefore, the enhanced combustor efficiency. This enhancement is simply due to the increased particle surface area (reaction surface area) created by the increase in the particle number density in the combustor. The combustion efficiency, however, does not continuously increase with the particle residence time. Increased particle density quickly causes mutual interactions to set in among the particles. This increases average Dmi and reduces f. At the same time, the quantity p*/(RS,) of Eq. (41) which was based on combustion of the non-interacting single particles begins to be replaced by another quantity representing a group combustion. These new quantities are substantially smaller in magnitude than that for the Eq. (41). In the case of a vertical combustor, it will behave as a fluidized bed in which p’/(RS,) is replaced by a quantity which is much smaller. Therefore, increasing the particle residence time does not necessarily result in the corresponding enhancement of the combustor efficiency. When swirl is employed, the residence time of the particles can be made almost arbitrarily long. Let us consider the longest particle residence time which is attained when all particles burn and gasify at the wall. This is the limiting case analyzed in the present paper. Equation (40) gives for this case, approximately, as
In the first place, the present Damkijhler number, Tipis substantially larger than the Dmi, and, hence, fw(I’i) is smaller than f(Dnt)i for given ki,. Also, depending on the pressure and other factors, we can easily have p*/(R&) > pJR,. For a given &, therefore, the combustion efficiency of the wall-burning combustor could be substantially lower than that of the onedimensional combustor in spite of the fact that the particle residence time of the former is much longer than that of the latter. These comparisons are made quantitatively in the following. The relationship between the wall burning rate and the combustor length is given by Eqs. (32). For R, = l/2 ft, rn%given in Table 1, and for the p = 0.08 lb,/ft3 corresponding to the combustor pressure of about 8atm, Eqs. (32) give the following values for WC in pounds of carbon per second at X = 5. Case B is used for the illustration. (For k3, = 1 ftlsec, S = 1) W, = 2.282 x lo-* lb/set
(For k3, = 1 ftlsec, S = 20) WC= 5.825 x lo-* Ib/sec
(45)
(For k3, = 100ftlsec, S = 1) WC= 3.348 x lo-* lblsec
(For k3, = 100ftlsec, S = 20) WC= 1.918x 10-l lb/set
For comparison we estimate the corresponding WC for the one-dimensional combustor with uniform velocity from Eq (41). This gives WC= 0.1 lb/set when k3, = 1 ftlsec, and WC= 0.4 Iblsec when k3, = 100ftlsec. A comparison of these values with those of Eqs. (45) shows that the wall combustion may not be as efficient as the gas-core combustion as was alluded earlier. As was seen in Eqs. (41) and (44), the char gasification efficiency of the one-dimensional flow combustor varies approximately with p* whereas that of the wall combustor varies with p.
Combustion of coal particles in wall-burning swirl combustors
321
Therefore, at the lower pressures, the one-dimensional flow combustor may not be as efficient as the wall combustor with high S. In the illustration given here, this reverse in the relative efficiencies would take place when the pressure is reduced to below about 4 atm when S is equal to 20. It should be noted in Figs. 6 and 7 and Eqs. (45) that the swirl, S, has a much greater influence on the wall combustion and gasification than the k2,,, and k3, in the present computed regime. Also, it should be remembered that in the present analysis, a pn’on’ assumption has been made that all combustion, following the initial fractional volatilization, take place at the wall. This will not be the case in practice, and, therefore, the present results should be viewed with certain care. 5. CONCLUDING
REMARKS
Performance of the wall-burning coal combustors was studied. It was assumed that the coal injection and swirl are such that ail coal particles burn and gasify along the combustor wall following a fractional volatilization in the gas stream. With the chemical reaction confined to the wall, details of the flow field have only a moderate effect on the reaction, and, therefore, a very simplified flow description was employed. Approximate but closed form solutions have been obtained. Detailed discussion is given in the preceding section. Only a few points are made here. In a wall-burning combustor, the appropriate Damkiihler number is substantially larger than that in the one-dimensional flow combustor based on the single-particle combustion, which means that the char gasification tends more toward the diffusion controlled regime than it does in the one-dimensional flow combustor. The swirl affects both the actual mixing and the Damkohler number through turbulence generation. These facts established the swirl to be the more important parameter than the gasification chemical kinetics. It is shown that the wall-burning combustors may not be as efficient as the gas-stream combustors at high pressures. At lower pressures, on the other hand, the situation may be reversed. As was mentioned at the end of the preceding section, the present analysis was carried out with the II priori assumption that all char combustion and gasification take place at the combustor wall. The results, therefore, should be viewed with certain care. Acknowledgemenrs-This work was jointly supported by the Research Board of University of Illinois at Chicago Circle and by U.S. D.O.E. through the Argonne National Laboratory. REFERENCES I. N. Syred and J. M Beer, Combusf. Phvne 23, 143 (1974). 2. J. B. Heywood and G. J. Womack, Open Cycle MHD Power Generation. Pergamon Press, Oxford (1%9). 3. D. B. Stickler and S. K. Ubhayaker, 17th Symp. Engineering Aspects of Magnetohydrodynamics, A. I.1 (1978). 4. W. S. Lewellen, H. Segur, and A. K. Varma, Modeling Two-Phase flow in a Swirl Combustor, ARAP Report No. 310 (1977). 5. S. Way, Combustion Technology (Edited by H. B. Palmer and J. M. Beer), p. 291. Academic Press, New York (1974). 6. M. R. Baer, 17th Symp., Engineering Aspects of Magnetohydrodynamics, A. 4.1 (1978). 7. A. F. Sarofim, 1. B. Howard, A. Padia, and H. Kobayashi, Heat Transfer and Kinetic Limitafions in the Design of Sing/e-Stage Coal-Fired Combustors for Open-Cycle MHD Generafors, 6th International Conference on MHD Power Generation, Washington, DC. (1975). 8. P. M. Chung and R. S. Smith, An Analysis of a High-Temperature Coal Combustor According to a One-Dimensional Flow Mode/, Argonne National Lab. Report ANL/MHD-77-2 (1977). 9. M A. Field, Combust. Flame 13, 237 (1%9). IO. I. W. Smith, Cornbust. Flame 17, 303 (1971). I I. J. B. Howard and R. H. Essenhigh, I & EC Process Design Devel. 6, 74 (1%7). 12. J. L. Johnson, Coal Gasification (Edited by L. G. Massey), Advances in Chemistry, Series 131,ACS, Washington, DC. (1974). 13. P. M. Chung, Chemically reacting nonequilibrium boundary layers, Advances in Heat Transfer, Vol. 2, Chap. 2. Academic Press, New York (1%5). 14. H. Schlichting, Boundary Layer Theory. McGraw-Hill, New York (1955). IS. N. Rott and W. S. Lewellen, Prog. Aeronaunt. Sci. 7, 111(1966). 16. M. Bloor and D. Ingham, Trans. Instir. Chem. Engrs. 53,7 (1975).
Energy Vol. 4. pp. 315-327 @ Pergamon Press Ltd., 1979. Printed in Great Britain
COMBUSTION OF COAL PARTICLES IN WALL-BURNING SWIRL COMBUSTORSt PAUL M. CHUNG University of Illinois at Chicago Circle, Chicago, IL @6J30,U.S.A.
and R. SCOJT SMITH Argonne National Laboratory, Argonne, IL 60439, U.S.A. (Receiued 7 November 1978)
Abstract-An approximate analysis is made of the performance of the wall-burning coal combustors. The results show that the swirl number plays a more important role than the char gasification chemical kinetics in determining the overall gasification rate. At high pressures, the wall-burning combustors may not be as efficient as the gas-stream combustors. The situation, on the other hand, may be reversed at the lower pressures. NOTATION particle Damkahler number defined by Eq. (42) coal combustion rate per unit combustor cross-sectional area functions of Dm and r respectively linear and angular momentum fluxes respectively char combustion rate per unit combustor wall area dimensionless combustion rate per unit combustor area defined by Eq. (31) surface reaction rate 2-4 for moderately rough, and >4 for rough surface specific wall reaction rate for the ith species combustor length molecular weight species mass fraction species mass fraction along the combustor center line coal particle radius combustor radius Reynolds number, uOR,/v radial coordinate (K,/2)“‘s0
wda
swirl number defined by Eq. (34) x-component of gas velocity species diffusion velocity r-component of gas velocity tangential component of gas velocity x/R, beginning of developed region defined by Eq. (22) axial coordinate x-x, normal distance from combustor wall Damkohler numbers defined by Eqns. (21) and (27) respectively boundary layer thickness density of coal species mass flux molecular kinematic viscosity gas density; gas-coal density in Eqns. (40), (4l), and (44) parameter defined by Eq. (30) shear stress Subscripts c carbon i ith species
s w 0 I, 2,3 m
solid particle combustor wall boundary layer edge for X
tThis Paper was presented at the 1978Midwest Energy Conference, Chicago, Illinois, 19-21 November, 1978. 315
316
p.hi.CHUNG andR.Scorr SMITH 1.INTRODUCTION
With the projected construction of open-cycle MHD-power-generation systems using coal as the fuel, rather intensive interest has arisen in recent years in the design of efficient coal combustors which would satisfy the particular demands of the MHD systems. The primary demand is to generate combustion products in the order of 5000°F which are reasonably free of the ash vapor, at a difficient rate to meet the power requirement. High pressure and temperature, swirling combustors of different configurations have been proposed to meet the demand in the past (see Syred and Beer,’ Heywood and Womack,’ Stickler et al.? Lewellen et ai.: Way? and Baer6). The swirl is introduced so that combustion and gasification of the coal particles will largely take place along the combustor walls. The wall combustion is considered to be advantageous for the following two reasons. The first is that this ensures trapping of the slag along the wall for its eventual removal. The second is that by transporting the coal particles to the wall it is thought (see Syred and Beer’) that the coal residence time necessary for the combustion and gasification is lengthened and, consequently, the combustor e5ciency is enhanced. In chemical reactions where two or more of the initially unmixed reactants participate, arbitrarily increasing the residence time of one of the reactants does not necessarily enhance the reactions. Efficiency of a coal combustor depends on many complicated factors, and it can not be estimated simply from the individual residence times of the reactants. Previously, one-dimensional flow models have been employed to analyze the performance of a high temperature coal combustor (see, for instance, Sarofim et al.,’ and Chung and Smith’). Inherent in such models is the assumption that all reactants including the coal particles are uniformly mixed at any position along the direction of the flow. Swirl quickly separates the coal particles from the gaseous species and destroys the uniformity. In the present study, it is considered that the swirl is sufficiently high and the coal injection system is designed such that all char combustion and gasification take place along the combustor wall. The problem analyzed here is the limiting case, opposite to the onedimensional model, wherein segregation of the solid reactant from the gaseous reactants is complete. It is a major task to describe the flow in a given swirl combustor even with the assumption of no mixing and chemical reaction (see Lewellen4). In view of the present incomplete state of knowledge of coal chemical-kinetic properties (see Chung and Smith’), an embarkation to a detailed numerical computational program of the swirl combustor doesn’t seem to be justified. In this study, a very approximate but general approach is employed. It is the objective of the analysis to elucidate the parameters controlling the combustion of coal along the wall in swirl combustors. A comparison will be made of the combustion rates between the wall burning and the completely mixed one-dimensional combustors. As an initial phase of the complex problem, rigor will be sacrificed in favor of generality, and closed-form solutions will be sought. 2.FORMULATION
We consider a rather idealized swirl combustor of cylindrical geometry for simplicity as shown in Fig. 1. The flow pattern in the combustor will depend on the manner in which the swirl is generated and on physical configuration of the combustor. As was mentioned in Section 1, it is not the present purpose to describe the details of the flow in a given combustor, but to estimate the degree of influence of the various governing parameters on the wall combustion of coal in general. With this in mind, we consider that the axial and tangential velocity components u. and wo, are given and uniform along the swirl boundary layer edge. All symbols are defined in the nomenclature. Ratio, then, of the two velocity components, So = wo/uo, is related to the combustor swirl number, SW,as will be shown subsequently. With So considered to be uniform and given, the vortex separation, recirculation, etc. are bypassed, and the flow description is greatly simplified. Since the solid particles are considered to bum only at the combustor wall, the recirculation and other flow details have only a moderate influence on the combustion. It is assumed that the coal injection system and swirl are such that the coal particles are thrown onto the wall at very small values of x after losing a portion of the volatile matter during their journey to the wall. The particles then move along the wall at small velocities being dragged by the gas streams above them. In a steady state, the combustor wall is covered by a layer of coal particles which burn and gasify as they interact with the gaseous reactants
Combustionof coal particles in wall-burningswirl combustors
317
Fig. I. idealized swirl combustor.
transported to the layer surface by turbulence. The distance x required to complete the combustion and gasification is sought in the present analysis. GOVERNING
EQUATIONS
The most important gaseous reactants for the coal combustion and gasification are the oxygen, carbon dioxide, and the water vapor. Also, with the coal particle temperature along the combustor wall estimated, energy equation is not required. The starting point of the analysis is then the following set of the conservation equations: (continuity) (momentum)
spur ax
I apvr _ o ar ’
pu~+pu$=-;$r).
(chemical species)
(1)
(2) (3)
In the above equation, i = 1.2, and 3 signify 02, HzO, and CO2 respectively. 7 and 01denote the shear stress and the mass ditfusion flux of the ith species respectively. BOUNDARYCONDITIONS The standard boundary conditions are applicable to Eqs. (1) and (2). The boundary conditions
for Eq. (3) are as follows. mi(O,r) = mjm
(4)
&(X9 l?c) = Jim
(5)
The reaction rates, Ji, between the ith gaseous reactant and the wall layer of the coal particles are discussed below.
318
P. M. CHUNG and R. SCOITSMITH CHEMICAL
REACTIONS
It is considered that all hydrogen constituent volatilizes before the coal particles reach the wall and is oxidized to HzO. The particles at the wall then mostly consist of carbon and ash. The following combustion and gasification reactions are considered to take place: 02tc++coz,
co* t c 2
(6) 2c0,
(7)
H,OtCLZ"-COtH,.
63)
The specific reaction constants, kj,, are Arrhenius type functions of temperature; however, their functional relationships are of no importance here since these values are considered to be given based on the estimated value of the wall particle temperature. A considerable amount of published data is available for the char surface oxidation rate, k,, (see for instance, Field, Smith,” and Howard and Essenhigh”). Estimating the particle temperature on the combustor wall to be about 4OOO“R(see Chung and Smith*), these data%*’ give k,, of between 100 and lOOOftper sec. On the other hand, the available experimental data on the k,, and k3, are very scarce. It can be deduced from Johnson’s work” that k,, and k3, at the temperature range of 4000”R are about two orders of magnitude smaller than k,, It should be noted that the present definition of ki, is based on the surface area of the particle layer as it is with that of the reactant mass flux fli, The reaction rate between the particle layer and the adjacent gas layer may be substantially greater than those measured in the experiments.e’2 As it will be seen later, the values of lOO-lOOOft/sec for k,, is already sufficiently large so that the reaction of Eq. (6) is diffusion controlled in a typical combustor. We shall, therefore, employ, k,, = 500 ftlsec
(9)
in the subsequent computation. For the values of kzw and k3,,., the following range of the values will be employed. klw = ksw = l-100 ftlsec.
(10)
With the ki,‘s chosen, the reaction rate, Jr, is expressed by (see Chung13) Ji = p&iwmiw for i = 1 and 2, (11)
J3 = p,hmw - $&hmlw. 1 The second term in the above J3 equation results from the reaction of Eq. (6) which is a CO2 source for the reaction of Eq. (7). In constructing Eqs. (ll), first order reactions are assumed for Ji and J3 for lack of better information. SHEAR
STRESS
AND MASS FLUX
Effect of the swirl on the surface reactions enters into the problem through its effect on the shear stress, 7, and the reactant mass flux, 0, 7 and & denote the turbulent transport rates, and the swirl enhances the turbulence and, consequently, the transport rates. With the subsequent closed-form solution in mind, we derive the expression for rw as follows. For a high Reynolds number, turbulent flow over a straight surface, Sch1ichting14gives rw =
KpolP4
0 ;
114 ,
where K is 0.0225 for a smooth surface, and increases with the surface roughness. This
Combustion of coal particles in wall-burning swirl combustors
319
expression is modified, after the manner of Rott and Lewellen,” and Bloor and Ingham,16 for the present range of interest where So B 1 as, rw = Kpow3’4u
0 11
I/4 .
Y
(13)
Expression for the species mass flux at the wall, fli,, is then constructed from Eq. (13) by exploiting the general analogy between the momentum and mass transport. Postulating a similitude between the transports of the quantities (mi - miw)/(mio- miw)and u/uo, we write, 6i, = K&W3’4
0 5
l/4 (??ti - ??Z+)*
(14)
Formulation of the problem is now complete. Solution of the governing equations, Eqs. (l)-(3), constitutes the next section. 3. ANALYSIS
The Pohlhausen integral scheme14 is employed in the solution. The solutions are obtained separately for 0
Equations (1) and (2) are first combined and integrated with respect to r to give, pu(uo- u)r dr = R‘r,.
(15)
As usual,14 a seventh-power profile is assumed for u/uO. Equation (13) is substituted for r,,, in Eq. (15) with y = 6. The resulting first order differential equation is then solved to give,
(16) This is a transcendental algebraic equation for 8 which is difficult to use. It can be put into a more convenient form in the following manner. For S/R, 4 1, Eq. (16) gives, $
= (~Kp~‘s(-!_)“5X4~5,
(17)
On the other hand, Eq. (16) gives for S/R, = 1,
(18) As was mentioned following Eq. (12), K may increase by several factors from the smooth wail values of 0.0225 when the wall is rough. Expecting at least a moderately rough gas-coal particle layer interface, and noticing the factor of 3/2 differentiating Eq. (18) from (17), we shall employ the relationship, $
= [ (~)2(o.0225)luIs”5(+-)“sx4~5,
(19)
where S = (K /2)4’3S Equation 44) wa:‘constructed by envoking the general analogy between the momentum and mass transport. We now assume that the species boundary layer thickness is the same as that of the momentum boundary layer, 8. Also, we assume for the region 0
320
P.M. CHUNG and R.SCOTT SMITH
constant, mi is locally similar (see Chung”). Then, Eq. (14) represents a solution at the wall of the species conservation Eq. (3), and no further solution of Eq. (3) is needed for the present purpose. Equations (5) and (11) are now combined with Eq. (14) where y is set equal to the S given by Eq. (19). The resulting equation is solved for mi, to satisfy the boundary condition, Eq. (4), as for i = 1 and 2,
where the Damkohler numbers, Ii’s, are defined for all i = I, 2, and 3 as,
(21) In deriving the closed form solutions, the gas density was assumed to be constant at po. This assumption is commensurate with other approximations made in the problem. Finally, X, is determined from Eq. (19) by setting S = R, as, X, = l.728(Re/S3)“4.
(22)
With the miw’sfound, Eqns. (11) give the reaction rates as functions of X. The reaction rates will be considered more in detail after the solution for x > xc. Region x > xc
Thickness of the diffusion layer in this region is simply equal to R,. Hence, ei, is given by Eq. (14) with y replaced by R,. The mass fraction mio is a function of x, and therefore, the local similarity concept may not be acceptable. The species-conservation equation, Eq. (3), will be solved by an integral method. Integrating Eq. (3) with respect to r, there results,
(234 Commensurate with the assumption of constant gas density, it is assumed that u = u. in the integrand of this equation. The following seventh-power profile is substituted for the species mass-fraction.
mi-mi, Wo -
=(Y)“‘=(R;~~~
(23b)
4,
Equation (23a) is then put into the following form with the use of Eq. (14) in which y is replaced
by Rc. l/4
$(49mio + 1lmiw) = - ~(0.045)W~‘4(+)
c
(mio
- miw).
(24)
The wall boundary condition, Eq. (5), is written with the use of Eqns. (11) and (14) as, for i = 1 and 2,
(25)
(26) where for all i, Pi= 1.115 $ “20fi. ( >
(27)
Combustionof coal particlesin wall-burningswirl combustors
321
buations (24) and (25), or (26), constitute two equations for the two unknown functions, mio and mip These equations are integrated and there result, upon satisfaction of the boundary condition, m&x,) = mica, ?ttio = ntjmexp
(> M3
m30=
E
for i = 1 and 2,
(-Jd)
Mom
(28)
rnti exp(-13R),
[exp(-~~)-exp(-5,~)lt
(29)
(14+3/F,)
where for all i, &=
(30)
5.4(z)l'*/(49 t 7). i
The wall values, mi, are obtained from Eqs. (25) and (26) with the mio given by Eqs. (28) and (29). Combustion and gasification With the appropriate mass fractions of the gaseous reactants having been obtained, the char combustion and gasification rate is determined from Eqs. (11) as, Local char combustion and gasification JCW)= rate per unit combustor wall area Pwk3wm3m p&.m3=
JXX)=
(31)
The total combustion and gasification rate is obtained, for I, & 1, by integrating K(X) as Total char combustion and gasification rate for the combustor length of X
w,(X)
wxx) = 27rR~pfl&,)
=
=
2~&*(pwmA3J
X J: dX
(324 and M kl, mco 1 =A_[1- ew (-Ml M ksw m3.sI (I+ f,)S, , MCkzwmm M2
+ --K
k3w
&
1
M3(1tfi3)
1
-[Il (I+
exp
xc)
I
12i)1+ w2, Xd}
r2M2
-M3 {[
(-
+ w,,
ml,+l
1
M,(l-f3/fI)mh 1
(1 - f,/f,)
MC 1 t j$ ~)qr3, +z ( I 300
[I
-
exp
(-
I-IAl 1[I-exp I
l3@1
53
Cl
I
Xc) for X > XC
(32b)
In the above equations, 415
x3/5
x215
p
trix”‘)].
“(ri,X)=~[~-~t~-~t~ln(l I
1
L
I
I
(33)
P. M.
322
and R.
CHUNG
SCOTT
SMITH
Solution of the equations is now completed. Typical results will be discussed in the next section. 4. RESULTS
AND
DISCUSSION
is evident from the solutions obtained that there are two dimensionless parameters that dictate the behavior of the problem. They are the Damkohler number, I’i, and the ratio S3/Re. fi also enters into the problem prominently; however, it is closely related to, and the same order of magnitude as ri as can be seen from Eqs. (21) and (27). Before the actual consideration of the results, a discussion of the relationship between the parameter S and the more commonly accepted swirl number, SW,may be useful. Swirl number is generally defined as the ratio of the axial flux of the angular momentum to that of the linear momentum multiplied by a suitable flow radius. For a cyclone, the exit radius is a meaningful quantity (see Syred and Beer’), and the swirl number can be written for the present problem as It
SW = G&G&
(34)
For the purpose of estimating SW,we approximate the tangential velocity by, (35) This expression is an approximate solution of the momentum equation for large values of S. Then, for the axial velocity of the order of uo, Eq. (34) shows that, So = 2Sw, and S = 2(K,/2)4’3Sw for r. 5 i R,.
(36)
With the physical meaning of the S clarified, let us now consider the typical numerical results shown in Figs. 2-7. These sample computations are based on the general property of a
0
I
2
3
4
5
6
7
6
9
IO
X
Fig. 2. Variations of reactant mass fractions. 0.12 010
1 s=20 -
’
K,w=500ft/s KeW=K3W=I0 CASE A
-1
.
1
. .
1
I
.
fl/s
I
--_,“I,0 .
-----_
I
. .
NW
-.
---
-
-_
c,“o
--4 --__
____r%
--_
--
-----___ \m3W
002
-
/m2w VW
0s 0
, I
III
I
I
I
I
I
I
Z&3
4
5
6
7
6
9
X
Fig. 3. Variations of reactant mass fractions.
IO
323
Combustion of coal particles in wall-burning swirl combustors
K2W=K3W=I,
SOLID
K2W*K3W’100, NOTE:
0
VALUES
t
2
CURVES
DASHED
CURVES
FOR DASHED
3
4
CURVE
5
6
8
7
9
IO
X Fig.
4. Combustion and gasification rate per unit combustor wall area.
2
I
I
I
I
I
I
I
I
I
4
5
6
7
8
9
KeW’K3W=I0
-s
\
z
CASE
“0
A- SOLID
I
CURVES
2
3
IO
Fii. 5. Combustion and gasification rate per unit combustor wall area.
_
CASE
A
K~w=K~w=I,
SOLIDCURVES
K~w=K~w=IOO, NOTE: _
0
DASHED
VALUES
HAVE
BEEN
I
2
CURVES
FORDASHEDCURVE
MULTIPLIED
3
4
BY TEN
5
6
7
8
9
IO
X
Fig. 6. Integrated combustion and gasification rate.
Montana Rosebud coal with the fuel-to-air ratio of 0.2, which corresponds to about 60% of the stoichiometric oxygen supply. The first stage of a two-stage combustion system usually operates with the oxygen supply between 40 and 60% of the stoichiometric value. Two conditions of the coal-air mixture before the wall combustion of char are considered. In the first case, “Case A”, 10% of the coal combustible is assumed to volatilize before the coal particles reach the combustor wall. In the second case, “Case B”, 30% of the combustible is assumed to volatilize before the wall. In both cases, all H and a portion of C are volatilized and are oxidized to Hz0 and CO2 respectively. At X = 0, therefore, the initial mass fractions of 02,
P. M. CHUNO and R. Scorr SMITH
324
,-
‘kw=kw=IO CASE
A-SOLID
CURVES
CASEB-DASHEDCURVES
-0
I
2
3
4
5
6
7
8
9
IO
I
Fig. 7. Integratedcombustionand gasificationrate.
HzO, and COZare computed as, Table 1.
mlm2., m3=
Case A
Case B
0.120 0.077 0.041
0.013 0.077 0.188
The three reactions given by Eqs. (6), (7), and (8) then commence along the combustor wall being dictated by the parameters Ii and S’/Re as stated earlier. The practical range of kiw’swas given in Section 2. In order to render the ensuing discussion physically meaningful, we shall fix R,, uo, and Re, and shall show the computed results as functions of kiy’s and S. Since the present solutions are in simple closed form, implication of these results to other specific combustor conditions can be readily drawn via the particular values of Ii and S3/Re to which these results correspond. For the computations of gigs. 2-7, the following values were employed. R, = l/2 ft, I(~= 80 ftlsec.,
(37)
Re = 31,746. Figures 2 and 3 show typical variations with respect to X of the gaseous reactant mass fractions along the combustor wall and core. Because of the high reaction rate of oxygen, k,, the oxygen at the wall is reduced to near zero from the initial value of 0.12 almost instantly. The wall oxidation of the coal is, therefore, turbulent diffusion controlled throughout the combustor. The initial value of m3,,,is 0.041 for Case A. Figure 2 shows that it increases immediately to 0.115 as the oxidation reaction of Eq. (6) produces CO2 at the wall according to the large k,, but the small k3w is unable to consume it. The accumulated CO2 is then slowly depleted along the wall through the gasification reaction of Eq. (7). Figure 3 shows that the initial accumulation of m3wis not as severe when ksw is 10ftlsec. When the swirl is such that S = 1, the core region of the combustor is still oblivious at X = 10 to the wall chemical reactions as seen by the undisturbed mass fractions at r = 0 (see Fig. 2). At S = 20, on the other hand, it is seen in Fig. 3 that the increased turbulent mixing begins to affect the core region of X > X,. The dimensionless local reaction rate, &, is shown in Figs. 4 and 5. For all cases, the effect of increased swirl on the tendency toward freezing of the reaction is seen. This is expected from the composition of the Damkohler number given in Eqs. (21) or (27).
Combustion of coal particles in wall-burning swirl combustors
325
The physical meaning of the Damkiihler number can be made clear at this point. The mass flux of a gaseous reactant at the combustor wall is given by Eq. (14). The order of magnitude of the reactant diffusion velocity across the combustor can be determined from Eq. (14) as, .
(38)
From Eq. (27), it is readily seen that the present Damkohler number is simply the ratio of the reaction velocity to the gaseous reactant diffusion velocity,
For a given value of ki,, the swirl increases Vd and decreases fi. According to the present solution, this tends to freeze the reaction resulting in a higher mi, This, in turn, increases the combustion and gasification rates as given by Eqs. (11). Note that the actual reaction rates are equal to the J: shown in Figs. 4 and 5 multiplied by (p&3~ti). The integrated dimensionless char combustion and gasification rates, W:, are shown in Figs. 6 and 7. Note that, in these computations, it has been assumed that 10 and 30% of the combustible are volatilized before the particles reach the wall for the Cases A and B respectively. These figures show the wall gasitications only. Figure 7 shows that for the given ki, and S, the wall combustion and gasification rate is substantially greater for the Case A than Case B. These two cases represent the same fuel to air ratio and fuel properties. The only difference is that a greater amount of O2 is converted to CO2 before the commencement of the wall combustion in Case B. The larger amount of COZ production at the wall by the diffusion of 02 and fast oxidation caused the more efficient char gasification for Case A than for Case B. However, the total combustion and gasification rate including the initial volatilization is still higher for the Case B since the volatilization is the most efficient mechanism of combustion. It may be useful, at this point, to discuss certain aspects of the efficiency of coal combustors in general. In particular, we shall briefly compare the wall-burning swirl combustor with the combustors designed for combustion in the main gas streams. For combustors with uniform diameters, one may write approximately for the combustion rate per unit volume, i,,
”
dm,=+,, dx
For one-dimensional flow model with the particle velocity assumed to be equal to the gas velocity, Eq. (40) can be transformed into the following equation (see Chung and Smith’).
(41) where, for convenience of discussion, only a single gaseous reactant i is considered. In the above equation, the particle Damkiihler number, Dmi, is defined as
D,ib!5, Y
(42)
and, the function f(Dmi) is of the form
(43) As it was the case with the wall combustors explained earlier, the combustor efficiency is
326
P. hf. CHUNGand R. SCOTTSMITH
enhanced when Dmi & 1 for a given ki,. Dmi is reduced as the particle radius is reduced and as the molecular diffusivity, v,is increased. Returning to Eq. (41), Fb represents the mass rate of coal combustion per unit combustor cross sectional area. For the given ki, and Dmi, the combustion efficiency increases (the required combustor length, L, is reduced) with p2. Consider now the case wherein the particles are made to accumulate by reducing u,. Reduction of u, implies a longer particle residence time in the combustor for a given combustor length. For the given required burning rate, & this implies the corresponding increase in the mixture density p and, therefore, the enhanced combustor efficiency. This enhancement is simply due to the increased particle surface area (reaction surface area) created by the increase in the particle number density in the combustor. The combustion efficiency, however, does not continuously increase with the particle residence time. Increased particle density quickly causes mutual interactions to set in among the particles. This increases average Dmi and reduces f. At the same time, the quantity p*/(RS,) of Eq. (41) which was based on combustion of the non-interacting single particles begins to be replaced by another quantity representing a group combustion. These new quantities are substantially smaller in magnitude than that for the Eq. (41). In the case of a vertical combustor, it will behave as a fluidized bed in which p’/(RS,) is replaced by a quantity which is much smaller. Therefore, increasing the particle residence time does not necessarily result in the corresponding enhancement of the combustor efficiency. When swirl is employed, the residence time of the particles can be made almost arbitrarily long. Let us consider the longest particle residence time which is attained when all particles burn and gasify at the wall. This is the limiting case analyzed in the present paper. Equation (40) gives for this case, approximately, as
In the first place, the present Damkijhler number, Tipis substantially larger than the Dmi, and, hence, fw(I’i) is smaller than f(Dnt)i for given ki,. Also, depending on the pressure and other factors, we can easily have p*/(R&) > pJR,. For a given &, therefore, the combustion efficiency of the wall-burning combustor could be substantially lower than that of the onedimensional combustor in spite of the fact that the particle residence time of the former is much longer than that of the latter. These comparisons are made quantitatively in the following. The relationship between the wall burning rate and the combustor length is given by Eqs. (32). For R, = l/2 ft, rn%given in Table 1, and for the p = 0.08 lb,/ft3 corresponding to the combustor pressure of about 8atm, Eqs. (32) give the following values for WC in pounds of carbon per second at X = 5. Case B is used for the illustration. (For k3, = 1 ftlsec, S = 1) W, = 2.282 x lo-* lb/set
(For k3, = 1 ftlsec, S = 20) WC= 5.825 x lo-* Ib/sec
(45)
(For k3, = 100ftlsec, S = 1) WC= 3.348 x lo-* lblsec
(For k3, = 100ftlsec, S = 20) WC= 1.918x 10-l lb/set
For comparison we estimate the corresponding WC for the one-dimensional combustor with uniform velocity from Eq (41). This gives WC= 0.1 lb/set when k3, = 1 ftlsec, and WC= 0.4 Iblsec when k3, = 100ftlsec. A comparison of these values with those of Eqs. (45) shows that the wall combustion may not be as efficient as the gas-core combustion as was alluded earlier. As was seen in Eqs. (41) and (44), the char gasification efficiency of the one-dimensional flow combustor varies approximately with p* whereas that of the wall combustor varies with p.
Combustion of coal particles in wall-burning swirl combustors
321
Therefore, at the lower pressures, the one-dimensional flow combustor may not be as efficient as the wall combustor with high S. In the illustration given here, this reverse in the relative efficiencies would take place when the pressure is reduced to below about 4 atm when S is equal to 20. It should be noted in Figs. 6 and 7 and Eqs. (45) that the swirl, S, has a much greater influence on the wall combustion and gasification than the k2,,, and k3, in the present computed regime. Also, it should be remembered that in the present analysis, a pn’on’ assumption has been made that all combustion, following the initial fractional volatilization, take place at the wall. This will not be the case in practice, and, therefore, the present results should be viewed with certain care. 5. CONCLUDING
REMARKS
Performance of the wall-burning coal combustors was studied. It was assumed that the coal injection and swirl are such that ail coal particles burn and gasify along the combustor wall following a fractional volatilization in the gas stream. With the chemical reaction confined to the wall, details of the flow field have only a moderate effect on the reaction, and, therefore, a very simplified flow description was employed. Approximate but closed form solutions have been obtained. Detailed discussion is given in the preceding section. Only a few points are made here. In a wall-burning combustor, the appropriate Damkiihler number is substantially larger than that in the one-dimensional flow combustor based on the single-particle combustion, which means that the char gasification tends more toward the diffusion controlled regime than it does in the one-dimensional flow combustor. The swirl affects both the actual mixing and the Damkohler number through turbulence generation. These facts established the swirl to be the more important parameter than the gasification chemical kinetics. It is shown that the wall-burning combustors may not be as efficient as the gas-stream combustors at high pressures. At lower pressures, on the other hand, the situation may be reversed. As was mentioned at the end of the preceding section, the present analysis was carried out with the II priori assumption that all char combustion and gasification take place at the combustor wall. The results, therefore, should be viewed with certain care. Acknowledgemenrs-This work was jointly supported by the Research Board of University of Illinois at Chicago Circle and by U.S. D.O.E. through the Argonne National Laboratory. REFERENCES I. N. Syred and J. M Beer, Combusf. Phvne 23, 143 (1974). 2. J. B. Heywood and G. J. Womack, Open Cycle MHD Power Generation. Pergamon Press, Oxford (1%9). 3. D. B. Stickler and S. K. Ubhayaker, 17th Symp. Engineering Aspects of Magnetohydrodynamics, A. I.1 (1978). 4. W. S. Lewellen, H. Segur, and A. K. Varma, Modeling Two-Phase flow in a Swirl Combustor, ARAP Report No. 310 (1977). 5. S. Way, Combustion Technology (Edited by H. B. Palmer and J. M. Beer), p. 291. Academic Press, New York (1974). 6. M. R. Baer, 17th Symp., Engineering Aspects of Magnetohydrodynamics, A. 4.1 (1978). 7. A. F. Sarofim, 1. B. Howard, A. Padia, and H. Kobayashi, Heat Transfer and Kinetic Limitafions in the Design of Sing/e-Stage Coal-Fired Combustors for Open-Cycle MHD Generafors, 6th International Conference on MHD Power Generation, Washington, DC. (1975). 8. P. M. Chung and R. S. Smith, An Analysis of a High-Temperature Coal Combustor According to a One-Dimensional Flow Mode/, Argonne National Lab. Report ANL/MHD-77-2 (1977). 9. M A. Field, Combust. Flame 13, 237 (1%9). IO. I. W. Smith, Cornbust. Flame 17, 303 (1971). I I. J. B. Howard and R. H. Essenhigh, I & EC Process Design Devel. 6, 74 (1%7). 12. J. L. Johnson, Coal Gasification (Edited by L. G. Massey), Advances in Chemistry, Series 131,ACS, Washington, DC. (1974). 13. P. M. Chung, Chemically reacting nonequilibrium boundary layers, Advances in Heat Transfer, Vol. 2, Chap. 2. Academic Press, New York (1%5). 14. H. Schlichting, Boundary Layer Theory. McGraw-Hill, New York (1955). IS. N. Rott and W. S. Lewellen, Prog. Aeronaunt. Sci. 7, 111(1966). 16. M. Bloor and D. Ingham, Trans. Instir. Chem. Engrs. 53,7 (1975).