Coal pyrolysis II. Species transport theory

C O M B U S T I O N A N D F L A M E 5 5 : 1 8 1 - 1 9 4 (1984)

181

Coal Pyrolysis H. Species Transport Theory G I R A R D A. S I M O N S Physical Sciences Inc., Andover, MA 01810

Volatile transport during pyrolysis is described. Volatiles are generated at the walls of the pores and are transported through the pore tree via both diffusion and convection. The model is parametric in the intrinsic pyrolysis rate k and suggests that for k in excess of I s - 1 , 1 0 0 - ~ m - d i a m e t e r particles will experienceinternal pressures in excess of 103aim. Since a large fraction of the volatiles experience this limiting pressure, vast potential exists for secondary chemistry within the pores. Several phenomena are identified which may limit the gas pressure within the porous solid. The limiting pressure is represented parametrically by Plimit' It is shown that the fluid mechanicaltransport of the volatiles decreases monotonicallywith decreasingvaluesofPtimitand the devolatilizationrate is significantlyreducedby the fluid transport.

1. INTRODUCTION Whenever coal of any rank is injected into a gasifier or combustor, whether fixed bed, fluidized bed, or entrained flow, the volatile matter within the coal particle is released due to the heat addition. The volatile products can amount to 50% or more of the original mass, including nearly all of the hydrogen and oxygen content. The process of devolatilization is generally very fast compared with the particle dwell time within the reactor, but the events that occur during this relatively short time can change the overall efficiency of the reactor. The chemical species composition of the off gas and the reactivity of the remaining char are dependent upon the particle temperature, heating rate, pressure, coal rank, particle size, and the chemical nature of the background gas. Optimal efficiency and productivity of any coal utilization process require accurate knowledge of the devolatilization stage. The devolatilization stage consists of three distinct physical processes: pyrolysis (the decomposition chemistry), volatile transport through the pores, and secondary reactions which can change Copyright © 1984 by The CombustionInstitute Published by Elsevier Science PublishingCo., Inc. 52 VanderbiltAvenue, New York, NY 10017

the chemical products of the gas and/or cause deposition of volatile products on the walls of the pores. Each process must be described by a fundamental mechanistic model before the voluminous devolatilization data base can be clearly understood. This work is intended solely to describe the fluid mechanical environment within the pore structure during pyrolysis. It is shown that internal pressures may reasonably be expected to exceed 1000 arm, fluid mechanics may retard devolatilization, and devolatilization is a coupled fluid mechanical-chemical problem. Two recent descriptions of volatile transport during pyrolysis have been reported by Russel et al. [1] and Gavalas and Wilks [2]. While both models treat the three coupled process described above, volatile transport is treated as a bulk macroscopic process occurring on the length scale of the particle with reduced diffusivity to account for the pore structure. These models predict relatively low internal gas pressures @1 atm above ambient). When the pore tree theory [3] of the pore structure is invoked, the volatiles must pass through a sequence of fine branches before leaving the particle. While the pressure

O010-2180/84/$03.00

182

GIRARD A. SIMONS

buildup in the tree trunk is very small, that in the fine branches is quite significant. It is this effect and its consequences that distinguishes the present model from other descriptions of volatile transport. 2. VOLATILE TRANSPORT EQUATION An equation describing the transport of volatiles through the pore tree during pyrolysis must be compatible with the source of the volatiles as prescribed by the theory of pore evolution [3]. The following derivation utilizes the pore structure and pore evolution theories described in Sections 2 and3 of Part I [3]. A schematic of the pore tree is illustrated in Fig. 1. The mean velocity of the volatiles ~ and the skewed coordinate y are positive out of the tree. At each location y, there are n pores of radius rp within a given pore tree of trunk radius r t.

n -- rt2/rp 2.

(1)

(2)

or

rp =rp(O)(OlO(O))~/a.

(3)

where 0 is the porosity and t is time. The skewed coordinate y is related to rp by

drp/dy

=

rpllt,

(4)

where It is the length of the tree trunk. The length of a pore (or trunk) is proportional to rp (or rt) and is given by

lp=Korp/OX/a,

as

d

np~ Unrp z) = nthw '

(5)

where K o is a constant, approximately equal to 5. Equations (3) and (5) show that the pore length is constant during pyrolysis. Pores are connected end to end in the pore tree and only the radius increases during pyrolysis.

(6)

where P6 is the mean density of the volatiles. The mass removal rhw at the walls of the pores is due to the growth of the pores and is given by

rhw = ps27rmrp(drp/dt),

The pore growth during pyrolysis is given by

drp__= rp dO , dt 30 dt

The flux of volatiles through the pore tree is described in a quasi-steady limit. This is possible because volatile matter is released from the walls of every pore in the pore tree and not just from the leaves of the pore tree (smallest pores). The density of the volatile material in solid phase is so much greater than that in the gas phase that the gas flux will be established "instantaneously" when compared with the overall pyrolysis time. Using the continuous branching description of the pore tree, the continuity equation describes the flux of volatile matter through n branches of radius rp due to the mass removal rh w from the walls of each of the n pores. This is expressed

(7)

where Ps is the density of the solid material and m is a shape factor of order unity which accounts for the fact that pores are not perfectly round. The value of m was determined from an analysis of pore combination [4] and was shown to be m = 3/2.

(8)

The solid density Ps in Eq. (7) is the helium density of the material. While it is known that the helium density increases with pyrolysis, it is not appropriate to include this detail in th w because the corresponding expression for drp/dt is a consequence of the preservation of the O/rp a distribution, which, itself, is not precise. Total mass balance is preserved with Ps constant provided that drp/dt is related to the rate of change of porosity, dO/dt, via Eq. (2), and the rate of change of porosity is related to the intrinsic pyrolysis rate k by

dO/dt = k(Of -- 0),

(9)

where Of is the final porosity of the sample after

SPECIES TRANSPORT IN COAL PYROLYSIS pyrolysis and 1/k represents the time required for 1/e of the volatile mass to pyrolyze. The pyrolysis rate k is a function of both the particle temperature T and the organic composition of the coal. For simplicity in assessing the role of fluid transport during pyrolysis, k is taken to be a function of temperature only. The value of k (in 1/ seconds) is assumed to be

183 where P c is the pressure of the perfect gas,

PG = PGRT, R is the gas constant, T is the isothermal temperature of both the particle and the gas, and Pref is defined as psg2~'(0 f -- 0) (12)

k = 2 X 107 exp(--14000/T(°K))

Pref

and corresponds to the maximum observed values of k as reported by Anthony and Howard [5]. All lower values of k reported [5] are assumed to be limited by either the external heating rate, thermal conductivity of the particle, fluid transport through the pores, or some yet unidentified mechanism. The true kinetic rate of decomposition will always represent an upper limit to the observed values of k. While a single bulk rate (k) of mass release is insufficient to describe individual pyrolysis species, it is appropriate to the description of mass transport. In reality, each species is generated at a different rate and the mean molecular weight and viscosity of the pyrolysis gas is space and time dependent. However, within the approximation of bulk transport through smooth walled cylinders, precise descriptions of the bulk properties of the volatile gas are superfluous. Equations (1)-(9) are used to express the transport equation [Eq. (6)] in terms of the rate constant k, the instantaneous value of porosity 0, the tree size lt, and the pore size rp :

where V is the mean thermal speed of a gas molecule,

d drp --

=

V=

80

RT

Typical values of these and other variables are listed in Table 1. If the volatfles exit the trunk of the pore tree at subsonic velocity, the pressure PG(rt) must match the background pressure p**. Otherwise, the velocity will be choked at the speed of sound and PG is obtained from Eq. (11) with u - a . Combining these two limits in an approximate manner, the pressure at rp = r t becomes PG(rt) = P** +

k/tPref

Orp

(13)

TABLE 1

(I0)

Typical Values of Variables

Equation (10) is the working form of the volatile transport equation. Since the volatiles are introduced into the pore tree along the walls of the pores, the mass flux must be zero at rp = mmin (the leaves of the tree). Integration of Eq. (10) subject to this boundary condition yields kltPref =- In (rp/rmin), Pc

ln(rt/rmin).

The first integral of the species transport equation was possible because the mass flux, #GU, must balance the production term, rhw,

Psltk(Of- O) (pG ~) =

'

(11)

Variable

Symbol

Value

Porosity Final porosity Material density Aspect constant Thermal speed Gas viscosity Minimum pore radius Thermal diffusivity

0 of Ps K_0 V #G rmin c~

Reference pressure

Pref

0.1 0.4 2 g/cm3 5 I0 5 cm/s 5 X 10---4 g/s cm 1A 0.02 cm2/s 3 X 104 atm

184

GIRARD A. SIMONS

independently of the fluid mechanical process controlling ft. Identification of the fluid transport mechanism will yield an additional relation between the gas velocity ~ and the pressure gradient dpG/dy. Eliminating ~ via Eq. (11) and integrating the resulting equation subject to the boundary condition on PG (rt) [as given by Eq. (13)] will yield the solution for the gas pressure and velocity everywhere in the pore tree. 3. FLUID TRANSPORT MECHANISMS

(14)

PG ay where D is the diffusion coefficient. The gas flux through pores whose diameter is smaller than a gaseous mean free path is controlled by gaswall collisions (Knudsen diffusion): otherwise, diffusion is controlled by gas-phase collisions (continuum diffusion). The corresponding diffusion coefficients are DKN = 2 rpV,

(15)

and De - 4 / . t G / 3 P G ,

(16)

respectively, where #G is the gas viscosity and "-V is the mean thermal speed of a molecule. For larger pores, the mean velocity ~ is controlled by subsonic convection. The velocity may be limited by the pressure gradient acting against viscous drag,

dpG, 8#G dy

--Fp2

CONTINUUM DIFFUSION

1Xy

VISCOUS DRAG VOLATI LE DRAG

CHOKED FLOW

Fig. 1. Volatile flux through the pore tree.

The flux of volatile matter through the pores may be controlled by either diffusion or convection. Diffusion will control the smaller pores, while convection will limit the flux through the larger pores. The mean velocity due to diffusion is given by D dPG

KNUDSEN DI FFUSION

(17)

or, for still larger pores, the drag may be due to the volatile matter introduced into the gas stream.

This is referred to as volatile drag and ~ is given by = --~rp 2 ~PG

rnw dy

(18)

Equation (14), (17), or (18) may be combined with Eq. (11) to obtain a first order differential equation for dpG/drp which must be integrated subject to the boundary condition on PG [Eq. (13)] at rp = rt. Each equation for dpG/drp will differ as prescribed by the fluid mechanics limiting the gas velocity ~. In general, each of these four mechanisms will limit the gas flux through pores of various sizes in the pore tree. The most general situation is illustrated in Fig. 1. As the volatiles flow out the pore tree, they encounter larger and larger pores. The smallest pores are limited by Knudsen diffusion, the next largest by continuum diffusion, then viscous drag, volatile drag, and, possibly, choked flow (ff = a-) in the trunk of the tree. The pressure must be continuous at every location in the pore tree where the fluid mechanical description of ff changes from one mechanism to another. In this way, a general description of the gas pressure and velocity in the pore tree may be constructed. Fortunately, many simplifications are possible. 4. TREES CONTROLLED BY KNUDSEN DIFFUSION By restricting the analysis to trees sufficiently small that Knudsen diffusion limits the fluid transport throughout the entire tree, ff in an isothermal

SPECIES TRANSPORT IN COAL PYROLYSIS particle is obtained from Eqs. (14), (15), and (4): U'=

2Wrp2

dp~

3pGlt

drp

(19)

Eliminating K between Eqs. (11) and (19) yields

dpG drp

3klt2Pref ln(rp/rmin)

.2 Vrp 2

which, upon integration, becomes

PG(rp, rt) =

3klt2pref

Plirnit ~ 3000 atm.

3klt2pre~ 2Vrt

[ 1 + ln(rt/rmila)] .

The interesting feature of pG(rp, rt) is that the internal pressure is independent of the background environment for a pore whose radius is much smaller than the trunk radius. Hence, for r p ' ~ r t,

3klt2pref PG(rP' rt) --> 2Vrp [1 + ln(rp/rmin) ] .

(20)

This solution illustrates that the pressure is a maximum at rp = main. While the concept of fluid mechanics may be questionable at main = 1 A, it is worth noting that rmi n itself represents the radius of transition from Knudsen diffusion to activated diffusion through a solid lattice. The value of the activated diffusion coefficient varies exponentially with rp and, for all practical purposes, the diffusion process is blocked below 1 A. Therefore, it is reasonable to use the concept of fluid mechanical mass transport for all pores down to 1 A in size. The maximum pressure in the leaves of the pore tree becomes

3kltZPre~ Pmax(¥t)= 2WFmin '

and may be used to estimate maximum internal pressures in trees whose fluid transport is limited by Knudsen diffusion. Equation (21) predicts that the maximum internal pressure increases with the tree size. There is clearly an upper limit (Plimit) for Praax which corresponds to the situation where the solid pyrolyzes but the gas remains in the pore. This limit corresponds to a gas of density 1/2 g/cm a which, at 1500K, generates a pressure of approximately 3000 atm:

[1 + ln(rp/rmin) ] +A 1,

where A1 is an arbitrary constant of integration. The boundary condition at rp = r t prescribes p6(rt) as given by Eq. (13) and A t becomes

A 1 = PG(rt)

185

(21)

If Plimit occurs homogeneously througlaout the porosity 0, fracture will occur when the internal force/area [Plirait0] exceeds the material yield strength [Omax(1 - 0)]. Using Oraax ~ 1000 atm, fracture is possible for O > 0.25. The very fact that particles have been known to fracture during pyrolysis suggests that these pressures are attainable. Hence, one basic test of Eq. (21) is its ability to predict pressures of this magnitude. Using the maximum value of lt(2/3 of the particle radius [6]), the values of Pmax corresponding to various particle sizes and pyrolysis rates are illustrated in Fig. 2. Internal pressures of the order of 3000 atm are predicted to occur over the temperature range and particle sizes of interest. Suuberg et al. [7] have noted that equilibrium gas phase chemistry at 1000 atm internal pressure would be consistent with their observed product compositions for 75-/am-diameter Montana lignite heated to 800-1000°C at 10aK/s. Using the maximum observed values of k indicated above, Suuberg's conditions correspond to internal pressures as high as 3000 atm. Hence, the present theory is not inconsistent with experimental observations. The range of validity of the Knudsen transport model is illustrated in Fig. 2 and will be derived in the following section. It suffices to Say that the Knudsen transport model is valid when the pressure in the leaves of the large trees equals Plimit. The region within the pore tree for which PG = Plirait corresponds to ~ = 0 and dpG/dx = O. Asserting that there is some pore radius rjK that represents the edge of this stagnant region,

186

GIRARD A. SIMONS lmml---

I

I

I I IIII I

~- ~

I

I I I III i

[

I

I I II1-1-

Pmax = Plimit ffi 3000 atm

~ ~ . ~ " M. ttl I-ttl

I

: . ~ ~ / 4

LIMIT OF KNUDSEN

_-

,¢ tlJ

ra< IL.

I0~ 10- 1

(S00K) ~ l li~Itll

t

[

(900K) I IIIII11

1

(1000K) (1100K) I I IIII 10

102

k - INTRINSIC PYROLYSIS RATE (s- 1 )

Fig. 2. Pyrolysisinduced pressures.

Eq. (20) may be rederived subject to the boundary condition that ~ -- 0 and dp6/dx = 0 at rp = tiE. It follows that

3klt2pref PG(rP' rt) =

2Vrp

[1 + ln(rp/riK)]

(22)

cal phenomena, assume a spectrum of values between 1 and 3000 aim. In the following sections, the role of fluid transport during pyrolysis is assessed through parametric variation of the value of Plimit. 5. FLUID MECHANICAL REGIMES

for FjK ~ rp '~ r t. The value of FjK is chosen such that PG is continuous (i.e., PG = P l i m i t ) at rp = tiE. Thus, tie becomes

3klt2pref riK(rt) = 2~Pnmi----~ ,

(23)

and the solution for pG(rp, rt) is rewritten as

The large pressures calculated from the Knudsen diffusion model suggest that continuum and convective processes are likely to occur during pyrolysis. The boundary between Knudsen diffusion and continuum diffusion may be determined by equating Eqs. (15) and (16). For a perfect gas, Transition, Knudsen/Continuum:

r v = (3~r/64)rp*, PG(rp, rt) = P l i m i t

\rp /

[1 + ln(rp/rjK)].

(25)

(24)

Equations (20) and (24) represent the solutions for pG(rp, rt) when rjK ~ rmin(PG ~

rmin, respectively. The region of validity of Eq. (24) is quite narrow (see Fig. 2). However, if Plim it were 10-100 atm, the region of validity of Eq. (24) would widen considerably. The gas pressure in the pores may be limited by particle deformation (for softening coals), condensation (tar), or to the value at which the volatfles exist in gas phase at near solid densities (3000 atm). Hence, the limiting pressure may, depending upon the physi-

where rp* = 16# G V/(3PG).

(26)

Similarly, the transition from continuum diffusion to viscous convection is determined by equating Eqs. (14) and (17) for D = De. Therefore, Transition, Continuum/Viscous: rp = (37r/64)l/Zrp *.

(27)

Comparison of Eqs. (25) and (27) indicates that continuum diffusion limits the gas velocity

SPECIES TRANSPORT IN COAL PYROLYSIS

187

only in a very narrow range of pore radii from 0.15rp* to 0.28rp* and, for all practical purposes, may be disregarded from further consideration. The fluid mechanical process then undergoes transition directly from Knudsen diffusion to viscous convection. This transition radius is obtained by equating Eqs. (14) and (17) for D = DEN. Therefore, Transition, Knudsen/Viscous: rp = rp*.

(28)

The transition radius r* is given by Eq. (26) and may be evaluated by utilizing the Knudsen solution for PG(rp). Hence, Eqs. (20) and (26) yield

where all pore trees whose trunk radius is less than rt* are limited entirely by Knudsen diffusion and all r t > rt* are limited entirely by convection. This conclusion is based solely on an assessment of viscous convection, and the limiting regimes are illustrated in Fig. 3. The role of volatile drag in the convection solution for rt >>rt* is now addressed. The boundary between viscous drag and volatile drag is denoted r** and is determined by equating Eqs. (17) and (18). Therefore, Transition, Viscous/Volatile: rp = r**,

where rp* _ exp [ 32#GV2

9klt2pref

rmtn

11 .

(n)l/2Ko , r **=

The argument of the exponent will vary over several orders of magnitude and rp* will vary from rmi n to rma x while the argument of the exponent varies from 1 to 10. Thus, for all practical purposes, the entke pore tree will switch from Knudsen control to viscous control when the argument of the exponent is of order 3. The transition radius rt* is defined by rt* =

Ko

10o A

=L v Z a.. I-Ukl W

az i-

< eel

I

~,== rt = r* I ~ I ~ 10~ . I

~

r t -20rt*.

Every pore tree whose trunk radius is less than r** contains only pores whose radius rp is less than r**. Thus, in the size range rt* < r t < r**, viscous convection limits the fluid transport in the entire pore tree. When r t > r**, the pore tree contains both rp > r** and rp < r**. The volatile drag solution (rp > r**) must match the viscous drag solution (rp < r**) at rp = r**. This solution is called the General convection solution and is applicable for all r t > r**. The

I

I

GENERAL CONVECTION • VOLATILE DRAG: rp ~ r** " VISCOUSDRAG : r p < r * " ~CONDUCTION LIMITED _

rt" rt

~~

~j(

ALL VISCOUS CONVECr'ON

,-

~

10

102

(30)

~~i=m~i=t ~

=30OOo~.

103

104

k -- INTRINSICPYROLYSISPATE (=-1) Fig. 3. F l u i d m e c h a n i c a l r e g i m e s .

105

188

GIRARD A. SIMONS

three fluid mechanical regimes are summarized by rmi n ~ < r t ~ < r t * :

All Knudsen Diffusion,

rt* ~
All viscous Convection,

r** ~
General Convection.

The range of applicability of these three regimes is illustrated in Fig. 3. Note that all three fluid processes may dominate the mass transport in some range of trunk radius and pyrolysis rate (k). However, the heat conduction through the coal will be shown to prevent the pyrolysis rate from ever achieving the level necessary to allow volatile drag to dominate the convection solution. At time t after a heat source is supplied, thermal conduction will transport the thermal energy a distance 6 from the surface of the coal particle. The thermal depth 6 is given by 8 = (2at) z/2, where t~ is the thermal diffusivity (0.02 cm2/s for coal). The pyrolysis chemistry is occurring in an isothermal environment if and only if the thermal depth is greater than a characteristic tree height (5 > It) in a time shorter than the chemical time (t < l/k). This criterion is expressed as

1/2 0113 r t < r*** =

is controlled by Knudsen diffusion and that where the entire tree is controlled by viscous convection. The Knudsen solution was obtained in Section 4. The value of r t corresponding to Pmax = Plimit is determined from Eqs. (21) and (5). By normalizing this value o f r t by rt*,

rt(Plimit) = ( 2Plimitrmin~112, rt*

\

6. TREESCONTROLLEDBY VISCOUS CONVECTION For those pore trees whose trunk radii are of sufficient size that the mass transport in the entire tree is limited by viscous convection, the mean flow velocity is given by Eqs. (17) and (4): =

rP 3

dp G

8lda 6

drp

drp

Ko V \ lr/ae /

where riv

8#6klt2pref ln(rp/rjv) rp 3

,

(32)

represents the value of rp where

pG(rp, rt) ~ Pnmit- Integration of Eq. (32) yields F41aGklt2pr'~ (2 ln(rp/ryv) + 1} PG(rp, rt) = L rp2

r * * * _ 01__/_ a (2otPre___._~ f y/2 r*--~

(31)

Eliminating ~ between Eq. (31) and the transport equation, Eq. (11), yields PG ---

and establishes an upper limit on the values of k and r t that are of practical importance. The ratio oft*** to r** becomes

]

it is shown in Figs. 2 and 3 that the limiting pressure of 3000 atm (or less) is always attained in the Knudsen regime. Hence, the viscous regime will always correspond to conditions under which the gas pressure attains Plimit in the leaves of the tree. The viscous solution, subject to this constraint, is developed below.

dpG

Ko

3PGV

~ 1

and demonstrates that the general convection solution is not applicable due to the finite thermal conductivity of coal. Analysis of the volatile mass flux during pyrolysis has thus been reduced to two relatively simple idealizations: the case where the entire tree

+A21 x/2,

(33)

where A 2 is an arbitrary constant of integration to be determined by the boundary condition on the gas pressure at rp -- rt.

189

SPECIES TRANSPORT IN COAL PYROLYSIS Just as in the Knudsen controlled regime, the internal pressure is independent of the background environment if the pore radius is much smaller than the trunk radius. Thus, for rp ,~ rt,

which is Eq. (11) revised to allow ~ = 0 at rp = rj. The mass flux of a perfect gas becomes pG~rrr p 2

-

8kltPrefrp2

~2

ln(rv/ri)"

PG(rp, rt) ~ (21t/rp)(kllGPref) 112 × [2 ln(rp/rjv ) + 1] 1/2.

(34)

Comparison of Eqs. (20) and (34) demonstrates that the solutions for the gas pressure in the Knudsen and viscous regimes are nearly compatible at the transition radius rt*. Knudsen PG(rt*)

3 [1 + ln(rp/rjK)]

Viscous pG(rt* )

411 + 2 ln(rp/rjv)] x/2

which, when evaluated at rp = r t, represents the total volatile flux per pore tree,)l;/t:

)flit =

Mv =

Thus, the e½act solutions are consistent with the approximate analysis of the fluid mechanical regimes. The pore radius at which pG(rp, rt) is equal to Puraa is denoted riv and is obtained from Eq. (34): 21t =

~2

ln(rt/5)"

The total volatile flux per particle, 3~/v, is obtained from the integral o f M t over all r t. Hence,

0.75 -- 1.5.

r/v(rt)

8kPrefltrt 2

rfrmnax /~tg(rt)41ra 2 dr t in

(37)

where ~(rt)dr t represents the number of pores per unit area in size range rt to rt + drt [3, 6],

g(rt) = O/(27r~rt3), a is the particle radius, and = ln(rra.x/rrain).

(klAGPref) 112 .

(35)

Plim it

The largest value of rp is related to the particle radius by [3, 6],

The solution for pG(rp, rt) is now rewritten as

rra ax = 201/3a/(3Ko),

PG(rp, rt) = Plirait [ "_LL.*] [2 ln(rp/rjv )

+

1] x/2

\rp /

and rmin is the smallest value Ofrp [6] (36)

and demonstrates the same behavior with rp as the Knudsen solution. However, the dependence of pG(rp, rt) on the trunk radius and the pyrolysis rate is different in the Knudsen and viscous regimes. 7. FLUID MECHANICAL RETARDATION OF THE DEVOLATILIZATION RATE The mean velocity of the volatiles in the pore tree is given by

=

kltPref PG

main = 20/(OPsSp), where Sp is the specific internal surface area (typically several hundred mg-/g). Normalizing ~/v to the total mass of the volatiles, M v = ~ ~raaps(O f -- 0),

the total volatile mass flux becomes

My_ Mv

3 k K 0 rfrmn ax 2a/30x/a ln(rt/rj) dr t.

(38)

in

ln(rp/ri) ,

It is readily shown that when rj =/'rain, i.e., when

190

GIRARD A. SIMONS

PG ~ Plirait for all (rp, rt), Eq. (38) yields

M v / M v = k,

(39)

which is the basic definition of the pyrolysis rate k introduced in Eq. (9). The role of Plirait is now obvious. Whenever PG ~ Plimit, ff is zero in those pores and the volatile mass flux is reduced. This is readily illustrated by Eq. (38): ~,lv/M v is less than k for all ri i> main. Integration of Eq. (38) requires specification of r~ for all rt. If PG < Plimit for all (rp, rt), then rs = rmin and the entire pore tree is outgasing without fluid mechanical retardation. Once Plimit is exceeded, r/assumes the values of rSk and rjv in the Knudsen and viscous regimes, respectively. However, if PG ~ Plimit in the trunk of the tree, ~ = 0 everywhere and rj is set equal to r t so that 3~/t is zero. Similarly, if r t > rt***, thermal conduction within the particle limits ~/t and r~ is set equal to r t. In this manner, values of ~lv/Mv are determined as functions of three parameters: intrinsic pyrolysis rate, particle diameter, and Plimit" Figure 4 illustrates the fluid mechanical retardation of the devolatilization rate (3;/v/Mv). Increasing values of Pltmit result in the increasing ability of the fluid mechanics to keep up with the intrinsic pyrolysis rate. Bituminous coals are known to exhibit reduced tar yields at background pressures greater than 1-10 atm. This may be ex1000

I l ilillt

I

plained on the basis of tar condensation pressures of the order of 1-10 atm [8]. In addition, bituminous coals do deform and may not be structurally capable of supporting internal pressures greater than 10 atm. If PUmit of 1-10 arm is associated with bituminous coals, then Fig. 4 predicts that the maximum observed devolatilization rate would be I 0 - I 0 0 ( s - l ) . This is consistent with observation [5, 9]. If Pllmit = 3000 atm is associated with lignite (no deformation or tar condensation), then Fig. 4 predicts that the devolatilization rate is nearly equal to the intrinsic pyrolysis rate. Increased ambient pressure would not influence the devolatilization of a lignite, whereas it would strongly influence that of a bituminous coal. This is consistent with observation [5]. The effect of particle size on the devolatilization rate is illustrated in Fig. 5. For small values of Plimit(<30 arm), the devolatilization rate is strongly dependent on particle size, whereas this dependence does not occur for Pumit = 3000 atm. This implies that the devolatilization of a lignite is not particle size dependent, whereas that of a bituminous coal is. An interesting feature illustrated in Fig. 5 is that the devolatilization rate is independent of the thermal conductivity of the particle, even for 1 mm particles. This is due to the low value of Plimit- The fluid transport terminates the volatile mass flux before thermal conduction

I I Ililii I

I I III1~1~)00111111~

n-

N 3 s~

--

.>,

1 - ' ~ / ~ l 1iIIII1 2 0 I 0 I K Illlld 1 10 100

I I I IIIIII 114~. K) , I III~IIKI~I 103 104

k - INTRINSIC PYROLYSIS RATE (=--1)

Fig. 4. Fluid mechanical retardation of the devolatilization rate.

SPECIES TRANSPORT IN COAL PYROLYSIS 103

I

< Z 102 _o t< N .J

__--

< ..>.

I IIIIII

I

I

Plimit = 3 atm O~=0.02cm2/s

I

to

191 I IIIIII

I

I

co

I IIIIll

/

/ /

I

I

I

IIIIH

10/z DIAMETER

~

--

S ~ D I. A M ~ E T E R

_--

10

I

/

~ , ,,n,.,

..~?~,ooK)..~.,.%ooK), , ,,,,,,., , .,,,,,,i

10

102

.(1.~) , .,,..,,

103

104

k - INTRINSIC PYROLYSISRATE (s-1) Fig. 5. Effect of particle size on the devolatilization rate.

1 0 a S- 1 . If the particle heating rate were in excess of 10~ S- 1 , any devolatilization rate observed to proceed slower than 10 a s- 1 would suggest transport limitations. Predictions corresponding to these conditions are illustrated in Fig. 7. The devolatilization rate is directly proportional to Plirait and inversely proportional to particle diameter. These calculations suggest that rate measurements at these conditions could be used to test empirically the present theory. The fluid mechanical retardation occurs because a large fraction of the volatiles stagnate in the pores at pressure Plimit- Since (.,flv/Mv)k is the

limits k. As Plirait is increased, fluid transport increases and thermal conduction can become rate limiting. An example of this limit is illustrated in Fig. 6 for Plimi t = 300 atm and 1 mm particles. The examples illustrated in Figs. 5 and 6 suggest that the particle size dependence of the transport limited devolatilization rate may be used to scrutinize empirically the pore structure, pore evolution, and volatile transport models. Volatile transport will limit the devolatilization rate if the heating rate and the intrinsic pyrolysis rate are sufficiently large. At temperatures in the range of 1700-2000K, k may be in excess of

A

103

I

I IIIIII

I

I

I IIIIHI

I

I IIIIII

Plimit = 300 atm lmm DIAMETERPARTICLE

I

I

~-/ / ~ !oo

I IIIli:l:

"~ ---------

/// Z

102

N

.3

_

,,.o.o....,. I

I Illllll

I

10.

I |llllll

102

I

I II||lll

I

103

i I Illlll

104

k - INTRINSICPYROLYSISRATE ($--1) Fig. 6. Influence of thermal conductivity on the devolafilization rate.

192

GIRARD A. SIMONS 103

Section 4 and suggests vast potential for secondary chemistry. 8. CONCLUSIONS

i u.i <

cc

:p O I<

10 2

N_

,,J I< .J 0 w e~

10

I

ff ,IE

1 10

10 2

103

PARTICLE DIAMETER (#)

Fig. 7. Prediction of transport limited devolatilization rate.

A model has been developed to describe volatile transport during pyrolysis. The rate of volatile release is described parametrically in terms of a single intrinsic pyrolysis rate k. The pore evolution model of Part I [3] prescribes the origin of the volatfles on the walls of every pore in every tree in the particle. The species are transported through the pore tree by diffusion and/or convection. Within reasonable limits of approximation, small trees are controlled primarily by Knudsen diffusion and large trees by viscous convection. The largest pressures occur in the smallest branches of the largest trees. Internal pressures are limited by several phenomena, and the limiting internal pressure is represented parametrically by Plimit- Decreasing values of Plirait result in increased fluid mechanical retardation of the volatile flux. Values of Plimit

fraction of volatiles that escape the pores, 1 must represent the fraction of the volatiles that exist at pressure Plimlt. Figure 8 illustrates that a very large fraction of the volatiles experience pressures of 300-3000 atm. Indirect observation of this prediction was discussed in

Mv/kMv

= -

I

I tlllll

I

I I Illlll

approximately 1 - 1 0

I

I

I IIIlill

I

I III1~

_

-

100~ D I A M E T E R PARTICLES Ol= 0.02 cm2/s

1

/ -

of

1

3 ate....... ~ "

o.,yS -

-10-2,,

I lllllilJ

atm

correspond to bituminous coals. The predicted devolatilization rates for these values of Plimit are sensitive to particle size and background pressure but will not exceed 10-100 s- 1 . Such predictions are consistent with observations for bituminous coals [5, 9].

/ (1000K) 11200K1 11400K1 11800K I Iaaatlii I IIl[Zlal I IIIIII 10 102 103 104

k - INTRINSIC PYROLYSIS RATE (s-1)

Fig. 8. Fraction o f gas experiencing pressure p during pyrolysis.

SPECIES TRANSPORT IN COAL PYROLYSIS Values of Plimit of the order of 1000 atm correspond to 1ignites. At these high pressures, no particle size or background pressure dependence is predicted. This is consistent with observation [5]. These high pressures are shown to be in reasonable agreement with the pressures suggested by Suuberg et al. [7] and clearly indicate vast potential for secondary chemistry, It is noted that the present study has concentrated only on the role of fluid transport, treats the intrinsic pyrolysis rate and the limiting pressure parametrically, and in no way represents a complete devolatilization model. Mechanistic chemical models (or definitive empirical models) for k and Plirait must be developed and coupled to the fluid transport model. The model for k must include the finite thermal conductivity and the finite external heating rate. While this work does not describe such a complete model, it does demonstrate that fluid transport plays a dominant role in devolatilization and that devolatflization models must include the coupled effects of chemistry and fluid transport.

193 Pref p~. rp mmin rmax rt r*

r** r*** rj rjK rjv R sp t T V Y

reference pressure-Eq. (12) background pressure radius of pore radius of smallest pore radius of largest pore radius of the pore that represents the trunk of a tree pore radius corresponding to transition from Knudsen diffusion to viscous convection pore radius corresponding to transition from viscous to volatile drag pore radius corresponding to the onset of conduction limitations. radius of pore for which U(rp, rt) = 0 value ofrj in Kundsen regime value ofri in viscous regime gas constant specific internal surface area (area/mass) time temperature bulk velocity of volatiles mean thermal speed of a molecule skewed coordinate in pore tree

NOMENCLATURE a

radius of porous particle speed of sound in gas diffusion coefficient D Knudsen value of D DKN continuum value of D Dc pore distribution function per sample area intrinsic pyrolysis rate k constant ~ ratio of pore length to diamKo eter ~- 5 length of pore of radius rp length of the pore that represents the it trunk of a tree pore roughness factor (3/2) m rate of mass removal from walls of pores rnw volatile flux per pore tree volatile flux per particle Mv total mass of volatiles Mv nfy) number of pores of radius rp at location y in pore tree gas pressure at (rp, rt) PG Plim it upper limit on PG in the pore P m ilx value of PG at (rmin, rt)

Greek Symbols

t~

thermal diffusivity In (rmax/rmin) 0 porosity Of final porosity after pyrolysis /zG gas viscosity PG gas density Ps density of nonporous material amax material yield stress

Work supported by U.S. Department o f Energy, Pittsburgh Energy Technology Center, under con tract DE-A C22-80PC-30293. REFERENCES 1. Russel, W. B., Saville, D. A., and Greene, M. I., AIChE

J., 25:65 (1979). 2. Gavalas, G. R., and Wilks, K. A., AIChE J. 26:201

(1980). 3. Simons, G. A., Combust. Flame, 53:83 (1983). 4. Simons, G. A., Comb. $ci. Tech. 19:227 (1979).

194

G I R A R D A. SIMONS

5. Anthony, D. B,, and Howard, J. B., AIChE J. 22:625 (1976). 6. Simons, G. A., and Finson, M. L., Comb. Scl Tech. 19:217 (1979). 7. Suuberg, E. M., Peters, W. A., and Howard, J. B., i & EC Proc. Des. Dev. 17:37 (1978). 8. Suuberg, E. M., Peters, W. A., and Howard, J. B., Seventeenth

Symposium

(International)

on Corn-

bustion, The Combustion Institute, Pittsburgh, 1979,

p. 117. 9. Solomon, P. R., and Colket, M. B,, Seventeenth Symposium (International} on Combustion, The Combustion Institute, Pittsburgh, 1979, p. 131.

Received 7 October 1982; revised 26 August 1983