Ind. Eng. Chem. Res. 2007, 46, 1079-1090
1079
Attrition-Enhanced Sulfur Capture by Limestone Particles in Fluidized Beds J. J. Saastamoinen* VTT Technical Research Centre of Finland, P.O. Box 1603, FIN-40101 JyVa¨skyla¨, Finland
T. Shimizu Department of Chemistry and Chemical Engineering, Niigata UniVersity, 2-8050 Ikarashi, Japan
Sulfur capture by limestone particles in fluidized beds is a well-established technology. The underlying chemical and physical phenomena of the process have been extensively studied and modeled. However, most of the studies have been focused on the relatively brief initial stage of the process, which extends from a few minutes to hours, yet the residence time of the particles in the boiler is much longer. Following the initial stage, a dense product layer will be formed on the particle surface, which decreases the rate of sulfur capture and the degree of utilization of the sorbent. Attrition can enhance sulfur capture by removing this layer. A particle model for sulfur capture has been incorporated with an attrition model. After the initial stage, the rate of sulfur capture stabilizes, so that attrition removes the surface at the same rate as diffusion and chemical reaction produces new product in a thin surface layer of a particle. An analytical solution for the conversion of particles for this regime is presented. The solution includes the effects of the attrition rate, diffusion, chemical kinetics, pressure, and SO2 concentration, relative to conversion-dependent diffusivity and the rate of chemical reaction. The particle model results in models that describe the conversion of limestone in both fly ash and bottom ash. These are incorporated with the residence time (or reactor) models to calculate the average conversion of the limestone in fly ash and bottom ash, as well as the efficiency of sulfur capture. Data from a large-scale pressurized fluidized bed are compared with the model results. 1. Introduction Different types of fluidized-bed combustors (FBCs) are in operation. These include atmospheric combustors, pressurized combustors (PFBCs), and circulating fluidized-bed combustors (CFBCs). Systems that use oxygen as the oxidizing gas instead of air and flue gas recycling are also under development, to reduce the plant size or to achieve a high flue gas concentration of CO2 for the purpose of recovery. In furnace injection processes with small particles, the residence time of the particles is short. In this case, the sulfur capture efficiency is dependent on the rapid initial sulfation stage. However, in fluidized-bed systems, a large particle size and longer residence times are generally applicable. Therefore, long-term sulfation behavior becomes influential. In sulfur capture in FBCs, an unreactive limestone particle can be good, with respect to conversion (but not with respect to rate of conversion), because the internal parts have time to undergo reaction before pore plugging occurs on the surface of the particle. Three sulfation patterns were observed in different limestones: the unreacted core, the uniformly sulfated, and the network.1 The CaO or, under pressurized conditions, CaCO3 particles can be considered to consist of small grains that may swell. The micrograins will grow in volume with an increasing degree of sulfation and eventually fill the micropores at ∼50% sulfation.2 Any further reaction in the grains is accompanied by an increase in grain volume. Extensive literature exists that involves sulfur capture and its modeling.2,3 The gas-solid reaction models can be divided into three categories:3 the unreacted shrinking-core models, the grain models, and the pore models. Various methods of * To whom correspondence should be addressed. Tel.: +358 20 722 2547. Fax: +358 20 722 2597. E-mail address:
[email protected].
describing the local reduction of the chemical reaction rate inside a particle have been presented. One way is to use exponential decay due to deactivation, sintering, and the decrease of reaction surface area.4 This treatment would predict different reaction rates for different particle sizes, because diffusion into larger particles requires more time. If the rate is decreased due to the formation of a product layer on a grain, then a grain-modeltype relation,2,5-10 where the reaction rate is dependent on conversion, would be better-suited to describe the decrease in reactivity. In these types of models, the particle is considered to consist of small grains, in which the conversion occurs via the shrinking-core concept. An alternative approach is to use semiempirical models for reaction rates of single particles,11-15 where the available reaction surface area or reaction rate is described as exponentially decreasing with time. The exponential decrease of the reaction rate, with respect to conversion of single particles, has also been used.16 Many papers involve the modeling of transport, reaction, and formation of the product layer on the particle surface.5,8,17 Some models also include attrition.18-22 The ability of attrition to enhance calcium utilization via the removal of impervious sulfate layers that build up on the particle surface was determined to be rather limited.19 However, only time periods up to 300 min were considered in this study, which is far shorter than the average residence time of particles in a boiler. Longterm sulfation tests, which last for up to 24 or 60 h in a thermobalance, have been conducted recently.23,24 The effect of attrition can be considerable in the long term and enhances sulfation. Attrition rates of w ) 0.2-0.7 nm/s in PFBC22 and w ≈ 0.29 nm/s in a laboratory-scale FBC (as calculated by their formula25) have been reported. The diffusivities and reaction rates have been reported to vary greatly. Under some conditions, the effective diffusivity and intrinsic reaction rate had only a minor role, in contrast to attrition, in regard to determining the rate of sulfur capture.22 Attrition rates increase as the fluidization
10.1021/ie060570t CCC: $37.00 © 2007 American Chemical Society Published on Web 01/23/2007
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Ind. Eng. Chem. Res., Vol. 46, No. 4, 2007
velocity increases and can then be considerably higher in CFBCs. A particle that has an attrition rate of 0.5 nm/s and a diameter of 500 µm would lose 74% of its mass via attrition within 50 h, which contributes greatly to sulfation. In principle, it could be possible to enhance the attrition of a size class by artificially taking a solid flow from the boiler and recycling it back into the boiler after mechanical abrasion. The diffusivities of the reacting solid and product layer on the particle are important factors that affect the reaction rate of the particle. They also determine the importance of attrition. In the limiting case, with low product-layer diffusivity, sulfur capture may be totally induced by attrition. The effective diffusivity of SO2 in the direct sulfation of limestone at high CO2 concentration or at high pressure, is higher than that in the sulfation of precalcined limestone,26-30 which is probably the reason for the reported increase in sulfur capture with increasing pressure.14 The diffusivity under atmospheric conditions is on the order of 10-12 m2/s and, under pressurized conditions, on the order of 10-10 m2/s;31,32 yet, the range of reported diffusivities is rather broad. It has been reported that the diffusivity values of the product layer decrease as the conversion increases33,34 and the concentration of SO2 increases.32 The effective diffusivity is dependent on the porosity, and several models of it have been presented.7,18,35,36 Pore structure and particle size have effects on the capacity of limestone for SO2 removal.37 The kinetics and reactivity of calcined limestone have been studied.38-40 In steady-state operation, the efficiency of sulfur removal can be expressed using the mass-flow rates of the sulfur (S), in the flue gas (g), fuel, fly ash (f), and bottom ash (b), as
η ) 1 - m˘ S,g/m˘ S,fuel ) (m˘ S,f + m˘ S,b)/m˘ S,fuel
(1)
The first form of this equation is applied in direct measurement of the efficiency and also in some models, where the outlet SO2 concentration is calculated based on the SO2 reacted in the bed.41 The right-hand side of eq 1 could also be used to measure the efficiency indirectly; however, it is cumbersome, because of the need to analyze the sulfur content in the fly ash and bottom ash. However, this relation can be useful in regard to the study and modeling of sulfur capture. To calculate the effectiveness of sulfur capture, the sulfur content in the fly ash and bottom ash particles should be estimated. Fly ash particles that are formed by attrition are fragments from the surface of the sorbent particles. Therefore, knowing the conversion profile on the surface of the mother particle provides an estimate for the degree of sulfation of these particles. The conversion of the fines in the feed, in the few seconds that they are in the boiler before leaving as fly ash, can be estimated based on the rapid initial reaction stage. The interactions between attrition, diffusion, and reaction are complicated. In continuous attrition models, the size of fines generated is assumed to be infinitesimal but, in reality, small particles with a finite size are abraded from the surface. An increase in molar fraction of Ca as CaSO4 in the elutriated fines with an increase in conversion has been measured.19 If the surface is only partially sulfated after a short residence time, the unsulfated part is more easily selectively abraded.25 If the particle consists of small grains, the size of the fines generated may be approximately the size of the grain.42 The size distribution of the fines was not dependent on the size of the mother particles, according to measurements for one limestone in cold tests.42 In laboratory trials, the general trend of a decrease in the degree of conversion with an increasing in particle size has been
widely reported. However, in a large-scale boiler, this is not always the case.32 Small particles may soon escape the bed as fly ash, whereas larger ones have longer residence and reaction time in the furnace. Primary fragmentation and attrition can be more intense for large particles, which have a high momentum. Sulfur capture under oxidizing and reducing conditions has been discussed.43,44 The attrition rate after sulfation is reduced by an order of magnitude or more.44,45 In a reducing atmosphere, the surface may release SO2, making it again more prone to attrition. If zones with reducing conditions exist in a commercial plant, it can promote attrition. In this case, the extent of the sulfur capture is improved, because a new accessible area is revealed for sulfation in the mother particles and the ”pore plugging” effect will be diminished. However, for this to occur, the residence time of the fines generated should be long enough allow for resulfation. Attrition can be an important factor in sulfur capture. In the first relatively short period after a limestone particle is fed to the furnace, the rate of sulfur capture is relatively high, but soon it decreases significantly, because of the decrease in the diffusion and chemical reaction rates. A product layer is formed on the particle surface, making the capture process very slow or even stopping it, if attrition would not remove the layer. Although the rate of sulfur capture in the latter period is slow, its contribution to the total sulfur capture can be large. The particles that are experiencing the latter slow regime constitute the major part of mass of the sorbents in the bed, because the residence time of the particles is very long, compared to the short rapid reaction period. Thus, this slow later stage can be important in sulfur capture, because of this long residence time in the boiler. If the reaction rate is high, a significant increase in the density and diffusion resistance on a surface layer on the particle may occur before much SO2 capture has happened. This may cause pore plugging, which prevents diffusion of SO2 into the particle and further capture of SO2 before the interior of the particle has undergone much reaction. The rate of sulfur capture becomes dominated by attrition, when the formation of a dense product layer on the particle surface greatly limits diffusion through it. The simultaneous attrition, diffusion, and reaction, as well as the relevance of attrition under different operational conditions, is studied in the present paper. By solving the conservation equation of SO2 inside a limestone particle with distributed reaction model in a transient state, Shimizu et al.20 have shown that the equivalent product layer thickness and surface conversion became steady after ∼2500-5000 s, which is sufficiently shorter than the residence time of the limestone particle in the 71 MWe PFBC.20 Therefore, most fine particles formed by attrition consist of the surface layer after reaching the steady state.20 An analytical solution for this asymptotic state for the conversion of the limestone particles, where the rates of attrition, diffusion, and reaction are balanced, is derived in the present paper for the general conversion-dependent diffusivity and reaction rate. The limestone fed into the reactor is eventually removed as fly ash and bottom ash. Solutions for residence time distributions of fly ash and bottom ash particles under attrition conditions are derived. Equation for sulfur capture efficiency is found by coupling these with the particle conversion model. In earlier work,20-22 PFBCs have been considered. Extension of the treatment under atmospheric conditions is discussed. 2. Modeling In laboratory-scale fluidized-bed experiments, the bed may mainly consist of sand and a continuous small feed of limestone
Ind. Eng. Chem. Res., Vol. 46, No. 4, 2007 1081
boundary conditions at the surface of the particle and the symmetry conditions in the center of the particle are
hm(FG,g - FG,r)R) ) D
( ) ∂FG ∂r
r)R
(5a)
and
( ) ∂FG ∂r
Figure 1. Reacting particle with decreasing particle radius R due to attrition.
is applied (see, e.g., ref 41). The duration of the experiments is not very long, and the reactivity of the sorbent has an effect on the sulfur capture. In real operation, after startup, when the operation continues for a long period, the sand is mostly replaced by the mixture of sorbent and ash. The mass and surface area of limestone can become large. The long-term reaction rate is low but, because of the large amount of sorbent material and attrition, its effect can become important in sulfur capture. Average residence times of 28-280 h in 71 MW PFBC,20,22 14 h in 12 MW and 165 MW CFBC, and 40 h in 40 MW CFBC16 have been reported. In this case, the degree of final conversion is more important than the chemical reactivity of limestone. The right-hand side of the efficiency equation (eq 1) can be expressed as
η ) βX h t ) β[zX h f + (1 - z)X h b]
(2)
where β is the Ca/S mole ratio and z is the fraction of the fly ash of the total mass removed. Thus, in principle, if the average degree of conversion (X h ) of sorbent particles in fly ash and bottom ash can be estimated, it is possible to calculate efficiency of the sulfur capture. The average degree of conversion in an output from a reactor is46
X h)
∫0∞ X(t)E(t) dt
(3)
The exponential residence time distribution E(t) ) exp(-t/τ)/τ has been used as an approximation for fluidized-bed reactors; however, attrition affects the residence time. 2.1. Reacting Particle with Surface Attrition. Sulfur capture experiments with different particle sizes show the particle-size dependency of the rate of conversion and of the extent of conversion (see, e.g., refs 13 and 47). A model, which accounts for radial diffusion and reaction inside a lime particle, can be used to analyze the effects of concentration of reacting gas, its pressure, particle size, and attrition. The diffusion and reaction of reactant gas (G ) SO2) inside the particle (Figure 1) can be described by
(
)
∂FG ∂FG 1 ∂ DrΓ + F˘G ) Γ ∂t ∂r r ∂r
(4)
The term on the left-hand side of the equation describes the storage. The first term on the right-hand side of the equation describes diffusion inside the particle. The diffusivity, D ) D0f(X), may be dependent on the local conversion X. This model predicts pore plugging with conversion Xc at the surface if f(Xc) ) 0. The second term describes the local disappearance of the reacting gas due to reaction with the solid. The chemical reaction rate F˘G is dependent on the local concentration of reactant gas and it also may be dependent on the local conversion. The
r)0
)0
(5b)
respectively. In the later slow regime, the rate of external mass transfer is high, compared to diffusion inside, and the surface condition FG,r)R ) FG,g can be applied. During attrition, the radius of the particle decreases with velocity (w ) -dR/dt) or piecewise with a layer ∆R being removed at certain time instants.20 In the case of continuous attrition (where particles removed at the surface are approximated as being infinitesimal), the conversion of the fines is the same as the surface conversion. If a fine particle with a diameter d, corresponding to a surface layer thickness ∆R ≈ (π/6)1/3d, is released from the surface, the average conversion of this fragment would be
Xa )
3 R - (R - ∆R)3 3
∫RR- ∆R Xr2 dr
(6)
The local density of the reacted solid Fs will increase with local conversion, because of the reaction with the gas:
-F˘G )
MG ∂Fs MGFs,0 ∂X ) Mp - Mm ∂t Mm ∂t
(7)
The following reactions are considered:
1 CaO + SO2 + O2 f CaSO4 2
(a)
1 CaCO3 + SO2 + O2 f CaSO4 + CO2 2
(b)
SO2 is captured by calcined limestone in the sulfation reaction (reaction a) under atmospheric conditions. Under pressurized conditions, the partial pressure of carbon dioxide (CO2) is so high that CaCO3 is not calcined. In this case, SO2 capture occurs and the compound reacts directly with calcium carbonate (see reaction b). There also exists other analogous gas-solid reactions, in which a product layer is formed on the particle surface.48 The reaction rate is described by nth order reaction,
F˘G ) -kFnG
(8)
where the reaction rate constant is dependent on the degree of conversion, k ) k0g(X). The local conversion rate is obtained from eq 7:
dX ) k(Mm/MG)FnG/Fs,0 dt
(9)
The storage term, ∂FG/∂t ≈ 0, can be neglected as small. The pseudo-steady-state profile then can be numerically calculated more rapidly with a moving surface.20 If the reaction zone on the particle surface is very thin, computations can be accelerated by making calculations only in the reaction zone, where the concentration of the reacting gas is over a certain limit (for example, FG/FG,g > 10-5). 2.2. Transition Period. Shimizu et al.20 studied the transition period using a distributed reaction model with n ) 0.16 and g(X) ) 1 - X. Analytical solutions are derived here for constant
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values for diffusivity D and reaction rate coefficient k (f(X) ) g(X) ) 1) and assuming an oxygen order of n ) 1. The simplified equations are obtained from eqs 4 and 8:
D
d2θ - kθ ) 0 d2x
(10a)
dX ) γkθ dt
(10b)
where θ ) FG/FG,g and γ ) (Mm/MG)(FG,g/Fs,0). The initial conversion is X(t ) 0) ) 0. The surface can be moving continuously with a velocity w or stepwise as discussed by Shimizu20 because of erosion by attrition. In the continuous case, θ(x ) wt) ) 1 and eqs 10 can be solved:
θ ) exp[-(x - wt)xk/D]
γk{exp[-(x-wt)xk/D] - exp(-xxk/D)} wxk/D
(11b)
Equations 11 show that the penetration thickness of the concentration profile is ∼xD/k. The conversion at the moving surface, x ) wt, is
Xx)wt )
γk[1 - exp(-wtxk/D)] wxk/D
(12)
The transition time constant then is τ ) xD/k/w. At large times, when t f∞, an asymptotic regime is attained and
Xx)wt )
γxkD w
(13)
This relation gives w < γxkD/Xmax for the existence of a fully converted product layer. This relation is the same as that previously derived49 using a different method. The small difference is due to the inclusion of the effect of the storage term ∂FG/∂t earlier, which has been neglected here. The difference becomes negligible, when 2xkD/w . 1. Intermittent attrition has been studied using a numerical method and using a shrinking-core model.20,21 Here, an analytical solution of eqs 10 with constants k and D is presented. If a layer (thickness ∆x) at the surface is erased at time intervals ∆t, then the boundary condition is θ(x ) n∆x) ) 1. Equations 10 can be solved. When n∆t < t < (n + 1)∆t,
θ ) exp[-(x - n∆x)xk/D]
(14a)
n-1
X ) γk∆t
∑ exp[-(x-i∆x)xk/D] +
k)0
γk(t - n∆t) exp[-(x - n∆x)xk/D] (14b) The average conversion at the time of breakoff can be determined by integrating eq 14:
X h (t ) n∆t) ) γxkD
( ) ∆t
∆x
exp(-∆xxk/D)]
[1 -
n-1
exp[-(n - 1 - i)∆xxk/D] ∑ i)0
When n f ∞, we get, as the limit, the simple formula
X h∞ )
γxkD∆t ∆x
(16)
(11a)
and
X)
Figure 2. Moving asymptotic steady zones in simultaneous attrition, diffusion, and reaction: (left) a particle, and (right) a schematic cut of the particle surface region approximated as a semi-infinite solid.
(15)
Equation 16 becomes identical to eq 13 when ∆x f 0 and ∆x/∆t f w. According to eq 16, a fully converted layer would be formed, if ∆x/∆t < γxkD. In this case, the surface consists of two layers: the product layer and the reaction zone. 2.3. Asymptotic Regime. In the capture of SO2, a thin dense layer is formed on the particle surface, slowing the diffusion and reaction. Attrition will erase the surface, thus increasing the sulfation rate. The process then will be a balanced interplay between attrition, product layer diffusion, and reaction. In SO2 capture, the residence time of the sorbent particle in the FBC is long and the diffusion through a product layer is slow, so that the slow attrition which removes the product layer can be of importance. Here, this long-term behavior is considered. A product layer and reaction zone, which are thin, compared to the size of the particle (Figure 2), are considered. In such a case, a semi-infinite planar geometry can be assumed20 (Γ ) 0 in eq 4). Shimizu et al.20 found that asymptotic moving profiles for the concentration and conversion on the particle surface are attained after some time. These profiles maintained constant shapes. This can also be seen through inspection of eqs 11, when t becomes large. At this stage, the interplay between attrition, reaction, and diffusion has reached equilibrium and the profiles move with the same velocity as attrition consumes the material. This finding was obtained by solving the quasi-steady-state diffusion equation numerically, as a function of time and one space coordinate with a chemical reaction term and moving boundary due to attrition.20 The knowledge of the existence of such an asymptotic regime, where the shapes of the moving wave profiles do not change with time, poses a challenge for finding a mathematical solution to this regime. In analogy with a moving solitary wave problem by applying a moving coordinate, it is possible to simplify the problem by eliminating the time coordinate. In this way, it is possible to find an analytical solution to the model equations formulated by Shimizu et al.20 in the equilibrium state. When the particle is fed into the boiler, a product layer starts to grow on the particle surface. The rate of the growth decreases as the thickness increases, because of an increase in the diffusion resistance of the layer. Finally, under some operational conditions, an asymptotic state is approached, in which the rate of the growth of the product layer and reaction zone due to diffusion and reaction is equal to the abrasion velocity w of the surface by attrition. Here, it is assumed that the process has attained this asymptotic steady state with a pseudo-steady profile moving in the particle with the same velocity as the surface is erased. The parameter x the is coordinate from the surface of the particle inward. By applying a new moving coordinate x*
Ind. Eng. Chem. Res., Vol. 46, No. 4, 2007 1083 Table 1. Integral G(X) ) ∫X0 [X/(fg)] dX for Typical Diffusivity (f) and Reaction Rate (g) Relationsa g(X) ) 1 f(X) ) 1
X2 2
f(X) ) 1 - bX
-
X ln(1 - bX) b b2
f(X) ) e-bX
ebX
a
(
)
(for a * b)
1 X/a + ln(1 - aX) 1 - aX a2
(for a)b)
eb/a b b E1 - bX - E1 a a a2
]
En(z) ) ∫∞1 (e-zt/tn) dt is the exponential integral.
( ) [
dFG dFG d + D + F˘G ) dx* dx* dx*
( ) ]
for the reaction zone (X < Xmax). No chemical reactions occur in the fully converted product layer (F˘G ) 0, X ) Xmax). The form on the right-hand side of the first equality sign was obtained using eq 7. The concentration of reacting gas FG(x* ) ∞) ) 0, its gradient, and local conversion are all zero at distances far from the particle’s surface (or, in fact, at infinity). Equations 9 and 17 give two coupled ODEs:
Mmk dX F n )dx* MGFs,0w G
(18a)
for local conversion X and gas concentration FG in the moving
() ( ) ( )
MG dFG w w + F + F X)0 dx* D G Mm s,0 D
(18b)
coordinate frame. These equations are nonlinear, because D ) D0f(X) and k ) k0g(X) and n may differ from unity. The term (w/D)FG is small, compared to the others in eq 18, because it stems from the storage term ∂FG/∂t in eq 4. By ignoring it, eqs 18 can be solved to give an explicit relationship between the concentration FG and conversion X profiles in the reaction layer:
[ ] G(X) Π
1/(n+1)
2
where
G(X) )
X dX ∫0X f(X)g(X)
(19)
where the dimensionless parameter is given as
Π)
MmxD0k0 FG,g(n+1)/2 MGxn + 1Fs,0w
(20)
In special cases, with constant diffusivity and reaction rate coefficient, f(X) ) g(X) ) 1 and eq 19 becomes θ ) X/(Πx2). The decay of chemical reactivity and diffusion can be described by linear or exponential models, and some cases are presented
)
X 1 1 + 2 a a2 a
ea/b eaX 1 {E [(a/b) aX] E (a/b)} + 1 1 ab ab b2
bX
1 1 e(a+b)X X+ a+b a+b (a + b)2
( )} + ab1 - eab
(
)
in Table 1. The conversion and concentration profiles in the solid can be found by applying the relationship between a dimensionless coordinate ξ ) (Mm/MG)k0(FG,gn/Fs,0)x*/w, and conversion:
ξ)
dFG MG d wFG + D + F wX ) 0 (17) dx* dx* Mm s,0
θ ) FG/FG,g )
eaX
ln(1 - aX) ln(1 - bX) + a(b - a) b(a - b)
{ [( )
1 X 1 + 2 b b2 b
(
X ln(1 - aX) a a2
) x - wt (eq 4) is transformed into the ordinary differential equation (ODE)
w
g(X) ) e-aX
g(X) ) 1 - aX
∫XX
x*)δ
dX ) θ g(X) n
∫XX
x*)δ
dX [G(X)/Π2]1/(n+1) g(X)
(21)
Attrition under atmospheric conditions may have a stronger effect than in pressurized conditions during this slow regime, because the diffusivity of the product layer or reaction zone is a magnitude lower under atmospheric conditions. However, the SO2 partial pressure is lower under atmospheric conditions, because of the lower system pressure, which makes the rate of formation of the product layer slower. Under atmospheric conditions, the conversion in the asymptotic regime is calculated in the same way, but the conversion far from the surface is not zero. During the initial stage, when the diffusion resistance of the surface is still low, a conversion profile X0(r) has been developed in the particle, because CaO is porous (see X at 30 min in Figure 3a). The conversion in the initial stage makes the progress, of reaction zone on the surface, faster during the slow regime. Under atmospheric conditions, instead of the condition X(x* ) ∞) ) 0, the conditions X(x* ) ∞) ≈ X0(R0 wt) are valid. The asymptotic analysis can be applied in this case by replacing X with X - X0(R0 - wt). X0(R0 - wt) can be approximated as a constant at a specific time t in pseudo-steady state asymptotic analysis, because the thickness of the reaction zone is small. 2.4. Effect of Attrition on the Mass Balance of the Reactor. In FBC, the bed mass is usually maintained at a chosen level, depending on the rate of fuel feed. Fragmentation and attrition of the fed material then will determine the particle size distribution and the proportion of the material flow, which escapes as fly ash. The rest is removed as bottom ash to maintain the chosen bed mass. In this section, a model to calculate fly ash and bottom ash rates under attrition conditions is presented. This reactor model (or residence time distribution) is needed to calculate the average conversion of limestone particle and sulfur capture enhanced by attrition. Under attrition, the particles are divided into two parts: fines, which are generated by attrition, and original mother particles, which are moved in the bottom ash. The mother particles may also leave in the fly ash, if they survive a long enough time to reduce to the critical size Rc of the system. Residence-time distributions, combined with the conversion to evaluate average conversion, are derived for these fractions. The residence-time distributions can be found as the normalized responses for an impulse.46 A batch of
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Ind. Eng. Chem. Res., Vol. 46, No. 4, 2007 Table 2. Transient Response of Reactor to a Mass Impulse m0 Fed at Time t ) 0 under Attrition, When t e tca quantity
transient response
particle number/initial number mass in bed/initial mass in bed attrition mass flow rate/initial mass bottom ash mass flow rate/initial mass instantaneous mass release (at t ) tc)/ initial mass
N/N0 ) exp(-t/τb) m/m0 ) (R/R0)3 exp(-t/τb) m˘ f/m0 ) 3(R2/R03)w exp(-t/τb) m˘ b/m0 ) (R/R0)3 exp(-t/τb)/τb mA/m0 ) (Rc/R0)3 exp(-t/τb)
a
When t > tc, all responses are zero.
An interesting question is what would be the mass in the reactor in the limiting case, when no bottom ash is removed (corresponding to τbf∞) and steady state is reached. The solution for this is obtained by integrating the impulse mass (Table 2) with respect to time, when τb ) ∞:
mτb)∞ m0
)
∫0∞
()
R 3 dt R0
(23)
The upper limit ∞ can be replaced with tc, because R/R0 ) 0, when t > tc. The integration gives the results presented in Table 3. The proportions of masses released by attrition and as bottom ash are
mf ) m0
∫0t
c
m˘ f dt m0
and
m f mA mb )1m0 m0 m0 Figure 3. Calculated concentration of SO2 (θ, represented by dashed lines) and conversion (X, represented by solid lines) inside a particle at various time instants under (a) atmospheric conditions (FSO2 ) 0.003 kg/m3, n ) 1, k0 ) 250 s-1, p ) 101.3 kPa, Fs,0 ) 1120.6 kg/m3, and d ) 300 µm) and (b) pressurized conditions (FSO2 ) 0.00343 kg/m3, n ) 0.16, k ) 5.509 kg0.84 m-2.52 s-1, p ) 1 MPa, Fs,0 ) 2000 kg/m3. Note the different scales in the radial coordinate.
particles with constant size R0 and total mass m0 is introduced to the reactor. Bottom ash particles are removed from the reactor at a rate that is assumed to be proportional to the volume or mass of calcium (m˘ b ) m/τb). Relationships for the number of particles and mass of the batch, attrition rate, rate of bottom ash and ash drain can be derived by simple unsteady number and mass balances for the reactor. The particle size decreases due to attrition, and, finally those mother particles (mass mA) not yet lost in bottom ash will carried in the fly ash instantaneously at time tc, when their size is reduced to the critical one Rc. The results are shown in Table 2. The change of particle radius due to attrition is given as
dR ) w ) kaRna dt
(22)
Different values for the exponent na (0, . . ., 2.6) have been reported in the literature for limestone.50 but there seems to be no systematic research on attrition of sulfated limestone using a wide range of particle sizes. Some results for different exponents are summarized in Table 3.
(24)
respectively. In continuous flow, the proportions of mass released by attrition, as small mother particles and as bottom ash, are the same as those released from a batch: mf/m0 ) m˘ f/ m˘ 0, mA/m0 ) m˘ A/m˘ 0, mb/m0 ) m˘ b/m˘ 0, respectively. The two first are presented in Table 4. The bed mass and the proportion of bottom ash of the total mass mb/m0 ) m˘ b/m˘ 0 are shown in Table 5. The residence time distributions for fines generated by attrition and for bottom ash are Ef(t) ) m˘ /mf and Eb(t) ) m˘ b/ mb, where the mass-flow rates and masses are defined in Tables 2 and 4, respectively. The particle size distributions in the bed, which are due to the change of size by attrition, are also presented in Table 5. They have been derived from a more general equation:51
p(d) ) -
d3m˘ 0
∞ ∫ 0 mR(d)
K(d,x) )
∫xd
p0(x) x3
exp(-K(d,x)) dx
(m˘ b/m)ψ(u) R(u)
du
(25a) (25b)
where R(d) is rate of change of particle diameter, which, here, is due to attrition. Here, for the nonselective case, ψ(u) ) 1 and, for monosize particles, p0(d) ) δD(d - d0). The model development was done for a constant initial feed particle size d0. The theory can be applied to a practical feed with a wide size distribution, which is presented as size fractions. Then, using the average diameter of the fractions and by summing the contribution of each fraction, multiplied by their mass fraction, the total attrition mass flow rate, bottom ash rate,
Ind. Eng. Chem. Res., Vol. 46, No. 4, 2007 1085 Table 3. Time Constant τa, Critical Time tc, and Particle Size R, as a Function of Time and Steady-State Bed Mass m(τb ) ∞) with No Bottom Ash Removal na
τa
tc/τa
R/R0 ) F(t)
mτb ) ∞/(m˘ 0τa)
0 1 2 n*1
R0/ka 1/ka 1/(R0ka) 1/[(n - 1)Rn-1 0 ka]
1 - Rc/R0 ln(R0/Rc) R0/Rc - 1 (R0/Rc)n-1 - 1
1 - t/τa exp(-t/τa) 1/(1 + t/τa) (1 + t/τa)1/(1-n)
[1 - (Rc/R0)4]/4 [1 - (Rc/R0)3]/3 [1 - (Rc/R0)2]/2 [|n - 1|/(4 - n)][1 - (R0/Rc)n-4 ) 3 ln(R0/Rc) if n f 4
Table 4. Mass Proportion Released by Attrition (mf/m0) and in Mother Particles (mA/m0) Reaching Critical Size Rc in a Continuous Flow n
mf/m0 ) m˘ f/m˘ 0
0
R)2
mA/m0 ) m˘ A/m˘ 0
3{1 + (1 - exp[R(Rc/R0 - 1)] × [(1 - RRc/R0)2 + 1]}/R3 3+R 3[1 - (Rc/R0) ]/(3 + R) 3eR[E4(R) - (Rc/R0)3E4(RR0/Rc)]
1 2
(Rc/R0)3exp[R(Rc/R0 - 1)] (Rc/R0)3+R (Rc/R0)3 exp[R(1 - R0/Rc)]
Table 5. Mass of Calcium in the Reactor (m), Mass Flow Rate of Bottom Ash (m3 b), and Particle Size Distribution in Steady State under Attrition, When the Inlet Size is d0 ) 2R0 m/(τbm˘ 0) ) m˘ b/m˘ 0 ) mb/m0
n
mp(d)/m˘ 0a
R)2]/R3
0 1 - 3[1 + (1 + exp[R(Rc/R0 - 1)] × (τa/d0)(d/d0)3exp[R(d/d0)- 1] {3[(1 - RRc/R0)2 + 1]/R3 - (Rc/R0)3} 1 R[1 - (Rc/R0)3+R]/(3 + R) (τa/d0)(d/d0)2+R 2 ReR[E3(R) - (Rc/R0)2E3(RR0/Rc)] (τa/d0)(d/d0)exp[R(1 - d0/d)] a
When d < dc, p(d) ) 0.
Table 6. Diffusivity Values Used in Calculationsa Diffusivity, D (m2/s) conversion, X
atmospheric
pressurized
0 0.7 1
4 × 10-7 (CaO) 1 × 10-9 1 × 10-12 (CaSO4)
1 × 10-9 (CaCO3)
a
1 × 10-10 (CaSO4)
A linear decrease with conversion is assumed between the given values.
residence times, bed mass, and particle size distribution in the bed can be calculated. 3. Results and Discussion 3.1. Initial Stage of SO2 Capture by CaO or CaCO3 Particles. Under atmospheric conditions, CaO is porous and the reaction can occur inside the entire particle until the conversion on the surface is high in the initial stage. A dense surface layer then is formed and the process will be slow. Under pressurized conditions, CaCO3 is not as porous as CaO and diffusivity is lower in the initial stage. The concentration and conversion profiles on the surface then will be steeper also in the initial stage. A similar situation occurs in systems using O2 instead of air. A high CO2 concentration prevents the calcination and, therefore, SO2 is directly captured by CaCO3.
Figure 4. Dependence of surface conversion in the continuous attrition on parameter Π, when g(X) ) 1 - aX and f(X) ) 1. This figure also applies for the symmetric case g(X) ) 1 and f(X) ) 1 - aX.
The reaction rate of small particles is dominated by chemical kinetics and the spatial conversion profile is smooth. Eventually, however, even for very small particles, pore plugging may occur at a critical conversion Xmax. For a larger particle, the average critical conversion Xc is less than Xmax, because the reactions inside the particle cease when the conversion at the surface reaches Xmax. This prevents any further diffusion of SO2 inside the particle. In the following illustration, a pseudo-steady assumption for diffusion of gas, ∂FG/∂t ) 0 in eq 4, is applied. Values of diffusivities at given conversions used in the calculations are presented in Table 6. For other values of X, it is assumed that diffusivity decreases linearly with conversion between the given values. In the calculations, w ) 0, hm is large, and Xmax ) 1. Simulation results for SO2 concentrations and conversions inside a particle in atmospheric and pressurized conditions are shown in Figure 3. A great difference can be observed between the calculation results under atmospheric conditions versus pressurized conditions, as shown in Figure 3. In pressurized conditions, the conversion proceeds in the shrinking-core manner in a very narrow zone, whereas under atmospheric conditions, the conversion occurs inside the entire particle until it practically ceases. After 30 min under atmospheric conditions, the SO2 concentration inside the particle decreases and the reactions will practically cease, whereas in the pressurized case, a thin reaction zone proceeds inward in a shrinking core manner. The profile X0(r) formed during the initial stage in the atmospheric case and shown in Figure 3 also will influence in the long term. 3.2. Asymptotic Slow Regime. Equation 19 gives the asymptotic conversion at the surface, if no fully converted product layer exists:
Xx*)0 ) G-1(Π2)
(26)
where G-1 is the inverse function of G. The conversion at the surface can be expressed using a single dimensionless parameter Π, which is defined by eq 20. It combines the effects of diffusivity, the reaction rate coefficient, reaction order, attrition rate, particle density, pressure, and SO2 concentration in a single dimensionless parameter. Equation 26 is conceptually valuable. In mathematical physics, if the starting equations are correct and exact mathematical methods are applied in the derivation, then the solutions can also considered to be correct. If it is not, then any criticism should be directed toward the validity of the starting equations and assumptions. In the present case, because the equation for diffusion is widely accepted, the criticism should be aimed at the “concept of attrition”. Can it be described reasonably as a continuous process? If so, then eq 26 should be valid. If not, its accuracy may be poor, but as shown earlier in the comparison of eqs 13 and 16 there is a definite similarity between the conversion of the surface in continuous and discrete models of attrition. The diffusivity or reaction rate can decrease and approach zero at high conversions. The cases, where either f(X) ) 1 aX and g(X) ) 1 or f(X) ) 1 and g(X) ) 1 - aX, are shown in Figure 4. In Shimizu’s special case, g(X) ) 1 - X, f(X) ) 1,
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δ)
Figure 5. Dependence of surface conversion in continuous attrition on parameter Π, when g(X) ) 1 - X and f(X) ) 1 (or g(X) ) 1 and f(X) ) 1X) (denoted by curve A) and f(X) )1 - X and g(X) ) 1 - X (denoted by curve B).
( )( )( )[ ( ) ] Mm FG,g DP G(1) 1MG Fs,0 w Π2
and G(X) ) -X - ln(1 - X). Equation 26 then can be approximated as
X ≈ 0.119Π3 - 0.701Π2 + 1.424Π (for Π e 2.06) (27a) and
X ≈ 1 (for Π > 2.06)
Π > Πc )xG(1)
∫0∞ X dx*
(28)
The critical value is shown in Figure 6 for the case f(X) ) 1 aX and g(X) ) 1. In this case G(1) ) -1/a - ln(1 - a)/a2. If a ) 1, as in the model calculations by Shimizu et al.,20 the surface conversion remains always below full conversion (X < 1), but it may approach a value close to 1. The reactions can cease or pore plugging can occur at lower conversions. This is described, for example, by g(X) ) 1 - aX or f(X) ) 1 - bX, respectively, where a > 1 and b > 1. If conversion can proceed to full conversion, then a < 1 and b < 1. The conversion at the surface will increase with decreasing attrition velocity until a fully converted product layer is obtained. If a product layer exists (i.e., δ > 0) and Xmax ) 1, its thickness can be calculated from the relation
(30)
The effect of parameter Π on the effective dimensionless thickness of the reacted layer Λ ) (Mm/Mg)k0(FnG,g/Fs,0)δR/w is shown in Figure 8. The effective thickness of the reacted layer is much greater for the case in which the reaction rate coefficient decreases with conversion, instead of the diffusivity, because the reactions can occur in a broader area. 3.3. Optimization of Attrition Velocity or Particle Size. Attrition can explain some of the discrepancies between the results in laboratory scale and those found from large units, but other explanations, such as alternating oxidizing and reducing conditions, have also been presented.58 Alternating oxidizing and reducing conditions promote the diffusion of sulfur into the solid.58 The level of limestone utilization, especially under atmospheric conditions in a boiler, is determined by many factors other than attrition, because conversion during the short initial stage can be high. Under pressurized conditions, the effect of attrition becomes important. In an asymptotic state, the rate of the capture of sulfur dioxide per unit of surface area of particles is given as
(27b)
It corresponds to line a ) 1 in Figure 4 and case A in Figure 5. Equation 27 was tested by comparing its results to the numerical calculations of Shimizu et al.20 If both the diffusivity and reaction rate decrease with conversion, then the degree of conversion at surface is lower (Figure 5, case B). To get a fully converted product layer, eq 19, with θ ) 1 and Xmax ) 1, provides the relation
(29)
which was derived from the boundary condition for a flux of sulfur dioxide (SO2) at the boundary x* ) δ. If a fully converted product layer is formed and an asymptotic state is attained, then chemical reaction kinetics and diffusion will have a minor role in the capture of SO2, provided that the size of the fines is smaller than the thickness of the product layer. The functions f and g have a similar effect on the conversion at the surface; however, there is some difference in the profiles inside the solid when comparing the case f(X) ) 1- X and g(X) ) 1 to the case f(X) ) 1 and g(X) ) 1 - X (see Figure 7). The differences in the profiles become greater for larger values of Π. The effective thickness of the reacted layer on the particle is defined by
δR )
Figure 6. Critical value for the attrition parameter for the existence of fully converted product layer for the case f(X) ) 1 - aX and g(X) ) 1 or g(X) ) 1 - aX and f(X) ) 1.
1/(n+1)
m˘ ′′G ) wFs,0Xx*)0
( ) MG Mm
(31)
Too-slow attrition is not favorable for SO2 capture. Under slow attrition, rate conversion at the surface (Xx*)0) reaches a high value, but the SO2 capture removal rate (eq 31) is also directly proportional to attrition velocity w. Because of the limited amount of limestone in the reactor, the available surface area is limited. Thus, SO2 capture is enhanced by attrition. If the attrition rate is so low that the total SO2 capture is not sufficient, the limestone feed should be increased so that fresh surface can capture SO2. For such cases, the bed-ash drain rate is increased (to maintain a constant bed mass) and the degree of limestone utilization will be low. If the attrition rate is too high, the limestone utilization efficiency will also be reduced, because the surface conversion will then decrease. Thus, there is an optimum attrition rate that gives the maximum degree of sorbent utilization, minimizing the limestone use to achieve a specific efficiency η (see eq 2). Optimization will give useful information for the selection of limestone, when comparing their properties. Usually, the attrition rate is determined by the process conditions and the properties of limestone, but it could be possible to artificially enhance attrition in a recycled flow of the bed material. The
Ind. Eng. Chem. Res., Vol. 46, No. 4, 2007 1087
Figure 7. Conversion (X, solid lines) and concentration (θ, solid lines) profiles in solid, when Π ) 0.303, n ) 0.16 (with n ) 1, the results are quite the same): f(X) ) 1 - X, g(X) ) 1 (denoted by curve A), and Λ ) 0.092, and f(X) ) 1 - X, g(X) ) 1, Λ ) 0.123 (denoted by curve B).
Figure 8. Dependence of dimensionless effective thickness of reacted layer Λ on parameter Π: (s) case where f(X) ) 1 - X, g(X) ) 1 and (- - -) case where f(X) ) 1, g(X) ) 1 - X.
attrition velocity then could be considered to be an adjustable variable that is optimized. The optimum attrition rate is illustrated by simplified calculations. The sulfur mass flow rate (m˘ S) in coal into the boiler and the efficiency η for the sulfur capture are known or specified. We further simplify the problem by assuming coal ash to be fragile and produce fines that escape the reactor immediately. The problem is further simplified so that a constant particle size R0 for the inlet feed of limestone and constant attrition velocity w (i.e., na ) 0) are assumed. The optimum value for the attrition rate w needed to reach the minimum use of calcium in sulfur capture for achieving the specified η and bed mass is sought. The degree of conversion of fines generated by attrition is higher than that of mother particles removed as bottom ash. Under optimum attrition conditions, all material then would be removed as fines with no bottom ash removal. Then, z ) 1 in eq 2 and η ) (m˘ Ca/ m˘ S)(MS/MCa)Xf. The mass of calcium in the bed is obtained from Table 3: mCa ≈ m˘ CaR0/(4w). By eliminating m˘ Ca from these equations, the relation wX h f ) ηR0(MCa/MS)m˘ s/(4mCa) ) C is determined, where C is a known constant. The dimensionless parameter defined by eq 20 is denoted as Π ) B/w. The conversion of the fines (X h f) is estimated by eq 27. The optimum attrition velocity then can be determined as w ) B/(2.945 x(8.403C/B)-3.291), provided that 0.392 < C/B < 1.42 and w > C. For example, when η ) 0.95, R0 ) 1.5 mm, m˘ S ) 0.018 kg/s, and mCa ) 22 000 kg (mCaCO3 ) 55 000 kg), the constant C becomes C ) 1.82 × 10-10 m/s. When SO2 concentration is FG,g ) 0.001 kg/m3 and the limestone properties are Fs,0 ) 2700 kg/m3, D0 ) 1 × 10-10 m2/s, k0 ) 5.509 kg0.84 m-2.52 s-1, n ) 0.16, and the parameter B ) 2.29 × 10-10 m/s, the optimum attrition velocity becomes 2.1 × 10-10 m/s.
It is seen that the optimum attrition velocity w is dependent on the concentration of SO2, the system pressure in the boiler, and parameters Fs,0, D0, k0, and n of limestone. Their effect is combined in the single parameter B ) (Mm/MG) xD0k0FG,g(n+1)/2/(xn+1Fs,0). In addition, the initial particle size, the required efficiency level, the feed rate of sulfur in the fuel, and the bed mass also affect. Their effect is combined in the constant C. An increase in the particle size increases the optimum attrition velocity. Instead of optimizing attrition velocity, which is difficult in practice, the particle size could be optimized, when the attrition velocity and its dependence on the particle size are known. The bed mass consists of ash and limestone. Selectively separating the limestone particles from a recycle flow of mixture of limestone and ash through the use of differences in particle size or density and by returning limestone to the furnace would increase its residence time in the boiler. An increase in the share of limestone in the total mass then would increase the rate of sulfur capture. This could be studied by the use of population balance modeling with particle-size and density-dependent bottom ash, fly ash, and recycle mass removal. Recirculating the fly ash will also increase the residence time and degree of sulfation of the fines formed by attrition. Coals can be classified by their requirement of limestone for SO2 capture. The best coals have low sulfur content or their ash contains compounds that can capture sulfur. Coal with a low ash content or a small ash particle size is also good, with respect to sulfur capture, because ash is removed rapidly as fly ash and, consequently, the proportion and the residence time of limestone in the bed can be higher. Limestones can also be classified by the requirement of limestone (ratio β in eq 2). In addition to their chemical reactivity, diffusivity, and maximum degree of conversion, the attrition rate may have a significant role in SO2 capture. 3.4. Degree of Conversion of Bottom Ash Particles. In the pressurized case, if the product and reaction zones are narrow, compared to the particle size (δR as defined by eq 30 is small), the conversion of the mother particles drained in bottom ash flow can be expressed approximately by
Xb ≈
3δR R
(32)
In the atmospheric case, the initial short period is important when also considering the long-term attrition and conversion. The conversion profile X0(r) formed during the initial period will remain inside the particle, because, at later stages, changes occur only close to the surface in a narrow zone. Then, when the attrition rate is known, the average conversion of the remaining mother particle can reasonably be estimated by
Xb ≈
3 R3
∫0R X0(r)r2 dr
(33)
when the effect of the narrow reaction zone on the particle surface is negligible, because of its small thickness. Interestingly, eq 33 predicts a decreasing average conversion of a remaining mother particle as function of time. This is because those parts that have been reacted more in the initial transient period closer to the surface are being erased away by attrition, resulting in a decrease in the average conversion of the remaining particle. In reality, there may be continued slowly progressing conversion inside the particle23,24 (for example, because of cracks), but this equation gives the “worst-case scenario”. The particle size R will decrease as function of time due to attrition (Table 3). The
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conversion is calculated numerically as a function of the radius R. The maximum conversion level attained at the initial period decreases as the particle size increases (see, e.g., ref 53). Then, naturally, the profile X0(t) is dependent on the initial size and, to find the total conversion of the material, the analysis should be made for size fractions and then sum up the contributions of each size. 3.5. Combination of Residence Time Distributions and Conversion of Calcium in Fly Ash and Bottom Ash to Calculate the Average Conversion and Efficiency of Sulfur Capture. Equation 2 can be applied to fly ash and bottom ash separately. By combining the residence time distributions Ea(t) and Eb(t) with the time-dependent conversion, one obtains the average conversions of the fines generated by attrition and of particles in the bottom ash drain,
X hf )
∫0t XfEf(t) dt
(34a)
X hb )
∫0t XbEb(t) dt
(34b)
c
c
respectively. The conversion Xf is calculated by eq 26 or, if a fully converted product layer is formed, Xf ) Xmax. The conversion of bottom ash is estimated by eq 32 in the pressurized case and by eq 33 in the atmospheric case. 3.6. Comparison to Measurements. Operational conditions of a 71 MWe PFBC power plant have been reported earlier.21,22 The Phase 1 experiments are used in the comparison here. In these tests, there was no fly ash recycle and the minimum particle size (2Rc) that corresponded to the elutriation size was ∼200 µm. Smaller particles were carried away as fly ash. The ash from the coal used was fragile and is easily broken into small particles (
