Energy & Fuels 1988,2,430-437
430
crease, decrease, or remain constant, depending on the region of the spectrum being considered. Pyrometry measurements in this wavenumber interval are subject to errors due to nongray effects. The errors in temperature measurement vary from a few hundred degrees to many thousands, depending on the wavenumbers chosen for the channels of the pyrometer. Operating one channel of the pyrometer at a high wavenumber reduces the chance for error.
The effect of nongray emissions on heat-transfer rates will depend on the particular combustor and flow field being used. Total particle emittance ranged from about 0.6 to 0.95 for 115-pm-sized particles, depending on particle temperature and size, coal rank, and radiative environment. Smaller particles from low-rank coals could have lower emittances. Partially devolatilized samples of bituminous coal emitted as gray bodies over a large portion of the infrared, with an emittance of about 0.8.
Intraparticle Nonisothermalities in Coal Pyrolysis+ Mohammad R. Hajaligol, William A. Peters, and Jack B. Howard* Energy Laboratory and Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Received December 21, 1987. Revised Manuscript Received April 13, 1988
Time and spatial temperature gradients within pyrolyzing coal particles can exert strong effects on devolatilization behavior including apparent pyrolysis kinetics. This paper mathematically models transient and spatial nonisothermalities within a single, spherical coal particle, with temperatureinvariant thermal and physical properties, pyrolyzing by a single first-order reaction. Analyses were performed for two surface heating conditions of practical interest, a constant surface heating rate and a constant surface heat flux density. The treatment provides three distinct indices of heat-transfer effects by quantitatively predicting the extent of agreement between (a) center-line and surface temperature, (b) volume-averaged pyrolysis rate [or (c) volume-averaged pyrolysis weight loss], and the corresponding quantity calculated by using the particle surfac,etemperature for the entire particle volume. Regimes of particle size, surface heating rate or surface heat flux density, and reaction time, where particle "isothermality" according to each criterion (a-c) is met to within prescribed extents, are computed for conditions of interest in entrained gasification and pulverized-coal combustion, including pyrolysis under nonthermally neutral conditions.
Introduction Many coal combustion and gasification processes involve particle sizes and surface heating conditions producing temporal and spatial temperature gradients within the coal particles during pyrolysis. These gradients may strongly influence volatiles yields, compositions, and release rates and can confound attempts to model coal pyrolysis kinetics with purely chemical rate expressions. Mathematical modeling of particle nonisothermalities is needed to predict reaction conditions (viz. particle dimension, surface heating condition, final surface temperature, and reaction time) for which intraparticle heat-transfer limitations do not significantly influence pyrolysis kinetics and to predict pyrolysis behavior when they do. When pyrolysis is not thermally neutral, the analysis is further compounded by the fact that local temperature fields are coupled nonlinearly to corresponding local heat release (or absorption) rates and hence to local pyrolysis kinetics. Much of the pertinent literature has addressed pseudo-steady-state models for spatial temperature gradients within catalyst particles playing host to endothermic or exothermic reactions, including, for some cases, mathe*Towhom correspondence should be addressed at the Department of Chemical Engineering, MIT. Presented at the Symposium on Coal Pyrolysis: Mechanisms and Modeling, 194th National Meeting of the American Chemical Society, New Orleans, LA, August 31-September 4, 1987.
matical treatment of the attendant limitations on intraparticle mass-transfer of reactants or products (see ref 1 and references cited therein). There appear to have been few analyses of nonisothermalities within a condensed phase material simultaneously undergoing nonthermally neutral chemical reaction(s). Previous work includes rather empirical approaches to fitting coal weight loss kinetics (see reviews by Howard2 and Gavalas3 and more refined analyses of spatial nonisothermalities within exploding solid^.^,^ Gavalas3calculated regimes of coal particle size where pyrolysis kinetics should be free of heat-transfer effects, and Simmonse provided similar information for cellulose pyrolysis. Valuable contributions are also emanating from the laboratories of Essenbigh' and Freihaut and Seery.8 (1)Froment, G. F.; Bischoff, K. B. Chemical Reactor Analysis & Design; Wiley: New York, 1979. (2) Howard, J. B. In Chemistry of Coal Utilization, Second Supplementary Volume; Elliott, M. A., Ed., Wiley-Interscience: New York, 1981; Chapter 12. (3) Gavalas, G. R. Coal Pyrolysis; Elsevier Scientific: New York, 1982. (4) Boddingtan, T.; Gray, P.; Scott, S. K. J . Chem. SOC.,Faraday Trans. 1982,2, 1721. ( 5 ) Scott, S. K.; Boddington, T.; Gray, P. Chem. Eng. Sci. 1984,39, 1079. (6) Simmons, G. M. Prepr. Pap.-Am. Chem. SOC., Diu. Fuel Chem. 1984. 29(2). 58. (7) Errsenhigh, R. H., Department of Mechanical Engineering, The Ohio State University, Work in Progress, 1987.
0887-0624188l 2 ~ 0 2 - 0 4 ~ 0 $ 0 l .,I5O0 0 1988 American Chemical Society I
,
Nonisothermalities in Coal Pyrolysis
Energy & Fuels, Vol. 2, No. 4, 1988 431
There is need for a generic, quantitative formalism to reliably predict transient intraparticle nonisothermalities and their effects on pyrolysis, as a function of operating conditions of interest in modern fuels utilization technologies. To this end the present paper presents, for an isolated spherical coal particle pyrolyzing by a single first-order reaction, quantitative predictions of three distinct indices of particle nonisothermality-namely the extent of agreement between (1) temperatures at the particle surface and centerline, (2) the pyrolysis rate [or (3) the pyrolysis weight loss] averaged over the particle volume, and the corresponding quantity calculated by using the particle surface temperature for the entire particle volume. Each index is explicitly dependent on time and thus accounts for the temporal as well as the spatial nonuniformities in particle “isothermality”.
the heat release or absorbtion by the pyrolysis reaction. The quantity (V* - V) reflects the effects of reactant concentration changes within the particle as a function of time. This term is approximately constant at low extents of substrate conversion and is unity when the pyrolysis kinetics are zero order. However, in general (V* - V), which is given by
Method of Analysis Only a brief summary of the thoeoretical approach is given here. Hajaligol et aL9 provide a more detailed description with broader applications. The analysis is applied to a reasonable first approximation of coal devolatilization-an isolated spherical coal particle with temperature invariant thermal physical properties, pyrolyzing by a single first-order endothermic or exothermic reaction with an Arrhenius temperature dependency, heated at its surface, and transmitting heat internally only by conduction, (or by processes well described by an apparent isotropic thermal conductivity). Effects of devolatilization weight loss on changes in particle density and thermal diffusivity are neglected. These approximations reduce the mathematical complexity of the problem while retaining sufficient rigor to provide results of practical and scientific interest. When the additional mathematical and numerical effort can be justified, the analysis could be extended by investigating other mechanisms of intraparticle heat transmission and other models for devolatilization kinetics,”-15 including those predicting an explicit effect of heating rate on volatiles yield. The availability of straightforward procedures to correct the single-reaction model for heating rate effectsle reinforces its appeal for the present calculations. Subject to the stated assumptions, a standard heat balance gives the following partial differential equation for the time and spatial dependence of the intraparticle temperature field: cr-’(dT/dt) = r-2(d/dr)(r-2(dT/dr)) (-M,,,)p~-l( V* - V)koe-E/RT (1) The second term on the right-hand side of eq 1 expresses
where 6* is a dimensionless parameter given by
+
(8) Seery, D. J.; Freihaut, J. D.; Proscia, W. M. Technical Progress Reports on Contract No. DE-RP22-84PC-70768to the Pittsburgh Energy Technoloty Center, US.DOE, 1985-1988; United Technologies Research Center: East Hartford, CT. (9) Hajaligol, M. R.; Peters, W. A.; Howard, J. B., to be submitted for publication. (10) Hajaligol, M. R.; Peters, W. A.; Howard, J. B. Prepr. Pap.-Am. Chem. SOC.,Diu. Fuel Chem. 1987, 32(4), 8. (11) Anthony, D. B.; Howard, J. B.; Hottel, H. C.; Meissner, H. P. Fifteenth Symposium (International)on Combustion; The Combustion Institute: PitGburgh, 1975; p 1303. (12) Kobayashi, H.; Howard, J. B.; Sarofim, A. F. Sixteenth Symposium (International)on Combustion: The Combustion Institute: Pitbburgh, 1977; p. 411. (13) Solomon, P. R.; Hamblen, D. G.; Carangelo, R. M.; Serio, M. A., Deshuande. G. V. PreDr. Chem. Soc., Diu. Fuel Chem. 1987, - PaD.-Am. 32(3); 83. . (14) Niksa, S.;Kerstein, A. R. Combust. Flame 1986, 66,95. (15) Niksa, S., Combust. Flame 1986, 66, 111. (16) KO,G. H.; Sanchez, D. M.; Peters, W. A,; Howard, J. B. Prepr. Pap-Am. Chem. SOC.,Diu. Fuel Chem. 1988,33(2), 112.
(V* - V) = exp(-ltkoe-E/RTdt)
(1A)
decreases with pyrolysis time and thus attenuates the heat of pyrolysis release or absorption rates, in effect damping their impact compared to the case of constant (V* - V), i.e., a zeroth-order reaction. Equation 1may be rewritten in dimensionless form as de/& = r2(d/df)(i-2(de/df))
+ 6* exp[e/(P + 4 1
6* = 6 e x p ( - l r 4 exp[-p/(pc
+ c2e)]
d7)
(2)
(2A)
where the quantity in the bracket is the dimensionless form of the damping factor given in eq 1A and 6 is also dimensionless. The physical significance of 6 and 6* are discussed below. Other symbols are defined in the nomenclature section at the end of the paper. Solution of eq 1or 2 requires specification of one initial condition and two spatial boundary conditions. The initial condition prescribes the temperature field throughout the coal particle at the instant heating begins. T(r,O) = To for t < 0 (3) One boundary condition is the mathematical expression for center-line symmetry of the particle temperature field at all pyrolysis times. dT(r,t)/dr = 0 at r = 0
(4)
Four cases are of interest for the secondary boundary condition in the present analysis. Case 1is a finite rate of heat transmission to the particle surface describable in terms of an apparent heat-transfer coefficient: -A(dT/dr) = heff(T- To) at r = R, (5) This case would be applicable to heating of coal particles in a fluidized bed or by molecular conduction from a high-temperature gas. Case 2 is a prescribed constant rate of increase in the particle surface temperature: T(r,t) = To mt at r = R, (6)
+
This case is applicable to screen heater reactors or other apparatus where surface heating rates are maintained essentially constant. Case 3 is a known, constant surface heat flux density: -X(dT(r,t)/dr) = q at r = R, (7) This case is especially applicable to fires and furnaces under conditions where the particle surface temperature remains well below the temperature of the surroundings and sample heating is dominated by radiation. Case 4, a special limiting case of 1-3 above, is an infinitely rapid surface heating rate: T(r,t) = T8,f= constant at r = R, (8) This case would approximate the heat-transfer characteristics of systems providing very rapid surface heating of the coal particles, for example shock tubes, laser and
432 10
a
-
I
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Haating R a t e
.lo4
1
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8
.
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'
-0-m
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n
dp = 500pm
f W 3
0.5
-
-
2 w n. 2
w
LL
10-2 PYROLYSIS
16'
1
Figure 1. Effects of particle diameter on the temperature index at lo4 K/s heating rate.
flash-lamp reactors, and coal dust explosions, where surface heating rates are estimated to exceed 105-107 K/s. Equations 1 and 2 were solved numerically for each of the above four cases of boundary condition, by using a procedure based on the method of lines and on Gear's method. Dimensional results for the constant surface heating rate and surface heating rate and surface heat flux density cases are presented here. Results in dimensionless form and for other surface boundary conditions are presented by Hajaligol et al.9 The solutions to eq 1and 2 are predictions of the spatial variation of the intraparticle temperature field with pyrolysis time. This information was used to compute three distinct indices of particle nonisothermality. (1) Temperature Index. This is a time-dependent factor that shows the extent of agreement between the surface and center-line temperature of the particle: VT(t) =
[Ts(t) - Tcl(t)l/T,(t)
(9)
(2) Rate Index. This is a time-dependent effectiveness factor for pyrolysis rate, defined as the ratio of a local pyrolysis rate averaged over the particle volume to the pyrolysis rate calculated by using the particle surface temperature for the entire particle volume: u-l
Luke exp[-E/RT(u,t)]
du
For a zeroth-order reaction this definition represents a ratio of actual rates. For a first-order reaction [7&)] represents a ratio of normalized rates defined as d[ln (V* - V)]/dt, the rate of change of unaccomplished fractional conversion. (3) Conversion Index. This is a time-dependent effectiveness factor for overall pyrolytic conversion (i.e. weight loss or total volatiles yield) defined as the ratio of the volume-averaged weight loss to the weight loss calculated by using the particle surface temperature for the entire particle volume: dt)= 1 - u - l i U exp(-Jt[k0
exp(-E/RT(u,t))] dt) du (11)
1 - exp(-Jt[h
1 0 ' 4
163
TIME.SEC
exp(-E/RT,(t))l dt)
Results and Discussion The above analysis was used with boundary conditions 2 (constant surface heating rate) and 3 (constant surface heat flux density) to predict the effects of various pyrolysis parameters on vT(t),v,(t), and v,(t) as a function of py-
162 10-1 PYROLYSIS T I M E , SEC
1
Figure 2. Effects of particle diameter on the rate index at lo4 K/s heating rate.
Heating Roto io4 *K /Set Dashed Ljne ldaal Conversion
LL
0
10'4
162
10-3 PYROLYSIS
16'
1
T I M E , SEC
Figure 3. Effects of particle diameter on the conversion index at lo4 K/s heating rate.
rolysis time. Unless otherwise stated, the following values of thermal physical and chemical properties of the coal were employed: p = 1.3 g/cm3, X = 0.0006 cal (cm s "C), C, = 0.4 cal/(g "C), AH = 0 cal/g, k, = lo1 s-l, and E = 50 kcal/(g mol). T i e effects of activation energy, nonzero heat of pyrolysis, and thermal diffusivity are discussed later in this section. Constant Surface Heating Rate (K/s). Figures 1-3 respectively show the effects of particle diameter on a&), T&), and &), for a surface heating rate and a final temperature of lo4 "C/s and 1000 "C. For particle diameters greater than 50 pm, the time required to relax internal temperature gradients [Le., for qT(t) to decline from its maximum value to about zero] increases with roughly the square of the particle diameter, as expected. Initially the spatial non-uniformities in temperature (Figure 1) increased with increasing pyrolysis time, because the time for the surface to reach the final temperature [TSs/m] is much less than the particle thermal response time. As a corollary the intraparticle temperature field is unable to keep pace with the rapidly rising surface temperature. The magnitude and duration of this initial temperature transient increases with particle diameter because the particle thermal response time increases with particle size. The rate index of nonisothermality, a,(t) (Figure 2) tracks the nonuniformities in particle temperature. For a given particle diameter and thermally neutral reactions, ar(t)indicates spatially nonisothermal kinetic behavior [i.e., a,(t) < 11 over a broader range of pyrolysis times than does q T ( t ) [i.e., for which v T ( t ) > 0, Figure 13,because the rate index expresses the influence of the temperature nonuniformities on the predicted volume-averaged pyrolysis rates via an exponential function. Thus, the magnitude and duration of the temperature nonuniformities are amplified. Furthermore, when release rates for volatiles are of interest, ar(t) is clearly a more reliable index of particle
3/
Energy & Fuels, Vol. 2, No. 4, 1988 433
Nonisothermalities i n Coal Pyrolysis +
c
1.0
.
.
' sq'-.', . . ' s a , P o r t i c l e D i o m a t a r =lo0um 1
9
.
I
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I
I
I
I
I
0
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.
I 1 1 1 1 1 1 1
.
I
,,,,,#I
,
, 1,1,1,1
I
I , , , , , ,
I
I
,
I
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TIME, S E C
Figure 4. Effects of heating rate on the temperature index for 100 pm particle diameter.
nonisothermality than is qT(t). The conversion index of nonisothermality, qc(t) (Figure 3), reflects the nonuniformities in rate and (for a first-order reaction) the exponential dependence of conversion on cumulative pyrolysis time and the reaction constant. For small pyrolysis times, most of the reaction is yet to occur, so rate nonuniformities are of little consequence. At long pyrolysis times, the larger cumulative reaction time counterbalances the lower rate constants due to the nonuniformities expressed by q r ( t )(Figure 2). Thus, for each particle diameter there is an intermediate range of reaction times to which the strong nonuniformities in pyrolysis rate (Figure 2) contribute major nonuniformities to the conversion index (Figure 3). The exponential dependencies of conversion on rate and reaction time cause the magnitude of q,(t) to change rapidly with pyrolysis time, resulting in the sharp variations in this index depicted in Figure 3. Clearly, in light of the differences in Figures 3.-3, when total yield of volatiles is of interest, q,(t) is the preferred index of nonisothermality over either qr(t)or qr(t)*
The q,(t) and q,(t) indices should be used in conjunction with information on the absolute magnitude of predicted conversion. For example, for the kinetic parameters used in the present analysis, the conversion index (for zero AHpy, Figure 3) registers a maximum nonuniformity at a pyrolysis time (0.05 s) for which little actual conversion has occurred. Similarly for some particle diameter, q,(t) reapproaches uniformity (Le., value of unity) well before conversion reaches completion. Figure 4 shows that, at a fixed particle diameter, increasing the surface heating rate increases the magnitude but decreases the duration of the initial nonuniformities in the particle temperature field. The first arises because, with increasing surface heating rate, the surface temperature increases so rapidly during a time equal to the thermal resonse time of the particle that the intraparticle temperature field lags further and further behind the surface temperature. The shorter relaxation time arises because the higher initial temperature gradients generated at higher surface heating rates cause a more rapid attenuation of the initial disturbance. With increasing heating rate, these sharp initial intraparticle temperature gradients translate into strong nonuniformities in the local pyrolysis rates and, hence, into significant departures of q,(t) from unity (Figure 5). With declining heating rates (Figure 5 ) , the magnitude of these nonuniformities is attenuated, but they remain significant over increasing ranges of pyrolysis time. These effects, respectively, arise because decreasing the surface heating rate reduces the differences between the surface and internal temperatures (Figure 4) and because the resulting intraparticle temperature gradients are
w
105.K/Sac Particla Diameter r 1 0 0 p m
PYROLYSIS TIME I S E C
Figure 6. Effects of heating rate on the conversion index for 100 g m particle diameter.
lower and thus provide less driving force for temperature relaxation, thereby extending the time over which the intrasample pyrolysis rates exhibit significant nonuniformities. Increasing particle diameter at a fixed heating rate exacerbates the above nonidealities (Figures 1and 2). The impacts of the intrasample rate variations on the conversion index of nonisothermality q,(t) are attenuated strongly in both magnitude and duration (Figure 6). This is due to the interplay of rate and cumulative pyrolysis time discussed above. The magnitude of the nonuniformities in qc(t) are worsened with increasing heating rate, because surface temperature and surface pyrolysis rate more and more rapidly outpace the corresponding quantities within the particle, thus expanding the differences between the extents of conversion predicted for these two regions at smaller and smaller reaction times. Conversely, the nonuniformities in q c ( t ) decrease when heating rate decreases, because the particle temperature field (Figure 4) and average pyrolysis rate (Figure 5 ) track the surface temperature more and more closely and because the greater time required for the surface temperature to attain its final value allows intraparticle conversion to proceed further to completion. Nonisothermality can also be defined by comparing the time needed for complete conversion of a nonisothermal particle to the corresponding time if the particle were totally isothermal. The inverse ratio of these times has been estimated for a variety of conditions, by using in part the data of Figures 1-6. At a constant heating rate, increasing the particle diameter decreases this ratio (Figure 7), indicating that the particle is even more nonisothermal. A similar trend is seen if, for a fixed particle diameter, the surface heating rate is increased. Constant Surface Heat Flux Density. Figures 8 and 9 respectively show the effects of particle diameter on the temperature index [&)I, and on the rate and conversion indices [qr(t),q,(t)] for a constant surface heat flux density of 100 W/cm2 and a final surface temperature of 1000 "C.
434 Energy & Fuels, Vol. 2, No. 4, 1988
Hajaligol et al. 10,
u
el
u
10-2
E
10-4
I 1 1 1 1 1 1 1
.
I
, 1 1 1 , 1 1 1
10-2
I
, , , , , / a , ,
I
10-1
,,""',
,
1.
, ,, ,1.0
5
PYROLYSIS T I M E , SEC
Figure 10. Effects of surface heat fluxdensity on the temperature index and the surface temperature for 100 fim particle diameter.
HEATING RATE. W S e c
Figure 7. Effects of heating rate and particle diameter on the ratio of times for complete conversion.
# #
10.3
162
PYROLYSIS TIME
10.1 SEC
Figure 8. Effects of particle diameter on the temperature index and the surface temperature at 100 W/cm2 surface heat flux density.
0.4 -
02 -
1 6 ~
10-3 PYROLYSIS
10-2
16'
1.
TIME, SEC
Figure 9. Effects of particle diameter on the rate and conversion indices at 100 W/cm2surface heat flux density. A nondimensional surface temperature [ T,(t)/T,,f]is also plotted in Figure 8. The trend of the results in Figure 8 is very similar to those for the constant surface heating rates given in Figure 1, for which the stated surface heating rate would be roughly equivalent to a surface heat flux density of 100 W/cm2. The time to raise the surface temperature to its final value increases linearly with particle diameter, but the time required to relax the internal temperature gradients increases roughly with the square of the particle diameter. The large difference in the particle response time and the time for the surface heating is reflected by the significant spatial nonuniformity during the initial phase of heating. However, for smaller particle diameters the initial temperature gradient is not significant since the two characteristic times are comparable. As before (Figure 2) the nonuniformities in the rate index [vr(t)] reflect the intraparticle temperature gradient
Nonisothermalities in Coal Pyrolysis 10
I
O
1
,,,,,
I
Particle Diameter l O O u m Heatlrq Rate: 105.Klsec 0 8 - D a s h e d LinesZero.0 Concentra t ion Solid L m e x R r s t - O r d e r in F Concentmiton
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.
I
. I . . .
I
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.
I
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.
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1 1 1 . 1 . 1
I
I
lilllll
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Figure 12. Effects of activation energy on the rate and conversion
indices.
1d4
10-2
16'
1.
10
PYROLYSIS TIME SEC
Figure 13. Effects of heat of pyrolysis on the indices at constant reactant concentration.
are smaller but remain significant over a longer period of time. As a consequence of the combined effects of pyrolysis time and these variations in the local rates, the nonuniformity in the conversion index strongly increases in magnitude but decreases in duration with an increasing surface heat flux. However, at lower heat flux densities, the smaller temperature gradients (Figure 10)and consequent lower nonuniformities in the local pyrolysis rates (Figure ll),cause the corresponding nonuniformities in the conversion index to virtually disappear (Figure 11). Effects of Activation Energy. For a thermally neutral reaction, a change in the activation energy for pyrolysis has no effect on intraparticle temperature gradients [and hence on qT(t)], but Figure 12 shows that increasing E increases the magnitude of the nonuniformities in T&), as would be expected since larger E's imply stronger dependencies of rate on temperature. Since ?&) is unaffected by changing E, any effect of E on qe(t)will reflect only E-induced changes in q,(t). Such effects should be small since a t this heating rate (lo3OC/s) quite large changes in T,I,(~)at an E of 50 kcal/(g mol) (Figure 5) are strongly attenuated in vc(t)(Figure 6) and the E-induced variations in q r ( t ) (Figure 12) are by comparison, rather small. Effects of activation energy for reactions that are not thermally neutral and effects of frequency factor (ko) are presented by Hajaligol et al.9 Effects of Heat of Pyrolysis. Figure 13 shows the effects of pyrolysis time on vT(t),qr(t),and q,(t) for cases where pyrolysis is not thermally neutral. Heat of pyrolysis in this analysis enfolds all effects of enthalphy release or absorption associated with pyrolysis, including heat of reaction and latent heat for tar evaporation. It proves convenient to discuss the results in terms of the dimensionless parameters 6* and 6 introduced in eq 2 and 2A.
The quantity 6 is interpretable as a ratio of an average rate of heat generation (or absorption) at the particle surface at Ts,fdue to pyrolysis to an average rate of heat conduction from the particle surface into the particle. The quantity 6* is a corrected 6, which accounts for the damping of the surface heat release (or absorption) rate when the pyrolysis kinetics are other than zeroth order in yet to be formed volatiles. The correction becomes necessary because under these conditions the local instantaneous release (or absorption) rates for the heat of pyrolysis now depend not only on the rate constant (as treated in our previous analysislO) but also on the local concentration of yet to be volatilized substrate. Thus the rates of heat release or absorption compared to zero-order kinetics are damped especially in regions of the particle where devolatilization has been more extensive. To simplify the discussion, the case of zeroth-order pyrolysis kinetics is first examined, allowing use of 6 rather than 6*. The results (Figure 13) clearly demonstrate additional nonuniformity in all three indices due to the heat of pyrolysis. For reasonable exothermicities or endothermicities, q T ( t ) and q r ( t )do not approach perfect uniformity, even at long pyrolysis times, while v,(t) goes virtually to unity (isothermal behavior) in reasonable times. The magnitude and duration of the nonuniformities expressed by &) depend on the ratio of the pyrolysis time to the particle heatup time (Hajaligol et al.9) Heating rate and particle size both affect the magnitude and duration of the transients in vT(t)and Tr(t). However at long py( t ar(t) ) attain 6-specific plateaus that rolysis times ~ ~ and are independent of heating rate and particle size, because pyrolysis attains completion before the temperature and rate nonisothermalities can be relaxed. When the more general case of non-zeroth-order kinetics is considered (i.e., use of 6* rather than 6) as would typically be required in describing coal pyrolysis under gasification and combustion conditions, heat of pyrolysis effects are attenuated in magnitude and duration. The point is illustrated in Figure 14 for vr(t)and different 6 values. The quantity 6* decreases rapidly with pyrolysis time, because of the depletion of the reacting material. Consequently, the heat of pyrolysis effects disappear and the nonisothermality behavior becomes similar to that for a thermoneutral reaction. As noted, for zeroth-order reactions, 6* is equal to 6, which remains nonzero and therefore leads to 6-specific-plateau behavior of results. At low heating rates (100K/s) and smaller particle diameters, where the particle is essentially spatially isothermal, heat of pyrolysis effects would also disappear. Effects of Physical Properties. Thermal conductivity and thermal diffusivity are important physical properties in nonisothermality problems. For constant p and C,,
436 Energy & Fuels, Vol. 2, No. 4, 1988
?
0.5
i
t W
Particla Diainutar~lOOum Heating Ratu=lO3 'Klsec
\
\
L
Dashad Line ? C
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1 1 , t ' a 1
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Figure 15. Effects of thermal diffusivity on rate and conversion
indices.
.
10 2
lo3
lo4
lo5
Figure 16. Isothermal regions defined by different indices.
decreasing either the thermal conductivity or thermal diffusivity of the coal strongly increases internal temperature gradients. As expected, this increases the magnitude and duration of non-uniformities in q,(t) and vc(t) (Figure 15). A qualitatively similar effect would be seen for a nonthermoneutral reaction. If it is noted (Figure 13)that 6 is inversely proportional to A, an increase in thermal conductivity decreases the nonisothermality. For example, for an endothermic reaction the values of q T ( t ) and q,(t) at longer pyrolysis times would be improved with increasing values of either of these thermal parameters, i.e. for AHpy= +500 cal/g of coal, these two indices would be improved 10 to 15% if a or X increased by a factor of 10. The present analysis assumed constant, i.e., temperature-independent thermal physical properties. The effects of avoiding this simplification could be treated by the same general formalism, but the mathematics and numerical analysis would be more complex. The probable magnitude of the effects can be scoped out from the present analysis by using Figure 15 to estimate the influence of parametric changes in a or X on vr(t)and qc(t). Domains of Isothermality. Combinations of particle diameter and surface heating rate (or surface heat flux density), where the particle is isothermal according to one of the criteria, define isothermality domains. Figure 16 shows, for thermally neutral pyrolysis, domains of particle size and surface heating rate where the particle is at least 95% "isothermal" according to each of the above indices [qT(t),q,(t), and q,(t)]. The temperature index admits a broader range of compliant particle sizes and heating rates because it is uninfluenced by devolatilization kinetics. For a non thermally neutral reaction it is more difficult to define domains of particle size and heating rate where qdt) meets the 95% uniformity criterion; note the 6 values required for q T ( t ) to approach 0 in Figure 13 and 14. The rate index [q,(t)] obviously enfolds kinetic effects and consequently presents a more narrow domain of iso-
1
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101
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Figure 17. Isothermal domains defined by different indices. 105
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HEATING RATE, k/sec Figure 18. Isothermal domains defied by the conversion index.
thermality (Figure 16). Clearly this index, rather than vdt) alone, should be considered in evaluating the role of heat-transfer effects in devolatization kinetic data and in designing experiments to probe intrinsic chemical rates. The conversion index, vc(t),provides a somewhat broader isothermality domain than I&), due to the damping effect of pyrolysis time discussed above. Figure 17 shows for thermally neutral pyrolysis, particle diameter/surface heat flux density isothermality domains defined by the three indices. Effects of activation energy and thermal physical properties on the isothermality domains can be inferred from Figures 12 and 15, respectively. Increasing or decreasing E , as in Figure 12, has no effect on the regions prescribed by qdt) and q c ( t )but respectively decreases or increases the domains defined by vr(t). The above criteria prescribe heat-transfer-free regions that are somewhat conservative, since they require 95% compliance at all values of conversion. However, for many conditions of practical interest, much of the nonisothermality precedes significant conversion. Less conservative and for many cases more practical isothermality domains can be defined by determining combinations of particle size and heating rate where, according to one of the indices, the particle is at least 95% isothermal for all values of conversion greater than some fixed value. An example based on v,(t) is given by the dotted line in Figure 18 for conversions of >1% . When pyrolysis is not thermally neutral, the domains of particle isothermality may, depending on the magnitude of AHpy,shrink from the boundaries defined in Figures 16 and 17. Regimes of D , and m (Figure 16) and D, and q (Figure 171, meeting each of the above criteria, can still be prescribed by calculating compliant families of curves (Figures 16 and 17) using AHpyas a parameter. Alternatively, the parameter 6 can be used to advantage in a more efficient computation procedure that accounts explicitly for all parameters contributing to heat of pyrolysis effects.
Nonisothermalities in Coal Pyrolysis
1.2
Energy & Fuels, Vol. 2, No. 4, 1988 437
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3. Domains of isothermality were constructed in terms of particle diameter and surface heating condition for three different indices. Compliant particle diameter scales approximately inversely with heat flux density and inversely with square root of heating rate. 4. The analysis showed that diagnosing a pyrolyzing coal particle as “isothermal” based upon close agreement between its surface and center-line temperature can lead to serious errors in estimates of corresponding volatiles release rates and total volatiles yields. 5. Each isothermality index exhibits significant temporal variations, showing that pyrolysis time and hence extent of conversion must be considered in assessing particle nonisothermality and its impact on pyrolysis behavior. 6. When pyrolysis is not thermally neutral, the effects of heat of pyrolysis (AHpy)on the nonisothermality must be judged relative to the effects of other parameters. The nonisothermality depends linearly on the heat of pyrolysis but exponentially on the activation energy for pyrolysis. Glossary particle diameter, cm activation energy, cal/ (g mol) effective heat transfer coefficient, cal/ (cm2s “C) reaction frequency factor, s-l surface heating rate, K/s surface heat flux density, cal/(s cm2) or W/cm2 gas constant: 1.987 cal/(g mol K) radial coordinate, cm particle radius, cm temperature, K initial temperature, K surface temperature, K final surface temperature, K ambient temperature, K time, s particle volume, cm3 particle density, g/cm3 particle thermal conductivity, cal/(cm s “C) particle thermal diffusivity, cmz/s heat of pyrolysis, cal/g dimensionless temperature: defined as [ T(r,t) T,(t)l/(To - Ts,3 dimensionless time: defined as a t / R dimensionless length: defined as r/gP temperature index rate index conversion index dimensionless parameter: defined as (-AH,y)pR,2X-1(To- Ts,f)-lkoe-E/RTnf dimensionless parameter: defined as R T , f / E dimensionlessparameter: defined as tTd/(?o - TBJ dimensionless parameter: defined as R,2ko/cu Acknowledgment. Prime U.S. DOE support for this work was provided via Contract No. DE-AC22-84PC70768, from the AFt & TD Combustion Prograrp of the Pittsburgh Energy Technology Center, via a subcontract to MIT from United Technologies Research Center, East Hartford, CT. Additional U. S. DOE support was provided under Contract No. DE-AC21-85MC 22049, from the Advanced Gasification Program of the Morgantown Energy Technology Center.
