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Ind. Eng. Chem. Res. 2007, 46, 3170-3179
Reactive Precipitation in Gas-Slurry Systems: The CO2-Ca(OH)2-CaCO3 System Sajan Kakaraniya, Ankur Gupta, and Anurag Mehra* Department of Chemical Engineering, Indian Institute of Technology, Bombay, Powai, Mumbai-400076, India
The present work examines the chemical absorption of carbon dioxide (CO2) gas in aqueous calcium hydroxide (Ca(OH)2) slurries, which leads to the precipitation of calcium carbonate. Carbonation of the Ca(OH)2 slurry is an important method for the industrial manufacture of precipitated calcium carbonate (PCC), wherein the hydroxide particles are gradually converted to carbonate. Experimental data have been generated by contacting CO2 with an aqueous Ca(OH)2 suspension in a model gas-liquid, stirred contactor, under different operating conditions. It is not possible to distinguish between the different types of particles present in the system (hydroxide, carbonate, mixed particles) “visually”. Therefore, the particle size distribution of all the particles present has been obtained using microscopy and optical image analysis techniques. The conversion of hydroxide-to-carbonate is determined by titration. A modeling approach that assumes that the hydroxide particles get converted by the diffusion of dissolved CO2 into its internal parts (as is observed in a “shrinking core” scenario), coupled with a population balance to describe the size distribution of the unreacted particle cores has been formulated. The experimental and theoretical conversions agree reasonably well. Introduction Slurry reactors have many applications in chemical and biochemical industries. Precipitated calcium carbonate (PCC) is manufactured in industry, using slurry reactors. Limestone lumps are broken into small pieces and decomposed to calcium oxide (CaO, quick lime) and carbon dioxide (CO2) at 10001100 °C. This CaO is further reduced in size and slaked in a hydrater to obtain a calcium hydroxide (Ca(OH)2) suspension (slaked lime), and this slaked lime is carbonated in slurry reactors to get PCC.1 PCC is one of the most widely used compounds in the chemical and metallurgical industries. It is used as a filler in talcs, paints, varnishes, plastics, lubricants, etc. and also used in the manufacture of textiles, adhesives, dentrifices, and pharmaceuticals. The present work focuses on the behavior of the slurrycarbonator batch system. The CO2 gas that is bubbled through the Ca(OH)2 suspension enters the liquid phase through the gas/ liquid interface and diffuses into the interfacial liquid, where it may react with the Ca(OH)2 that is present. Residual CO2 is transferred to the bulk liquid and may react with the Ca(OH)2 present here. Thus, the reaction between the CO2 and the Ca(OH)2 may occur in the “film” and “bulk” regions. The reaction of CO2 with Ca(OH)2 involves the dissolution of the reactant particles and the simultaneous appearance of the product CaCO3 particles. It is likely that the slurry-carbonator batch system will have, at any time, an assortment of particles: unreacted hydroxide, product carbonate, hydroxide particles coated with the precipitated carbonate, mixed agglomerates, etc. The absorption rate of CO2 and the consequent conversion of Ca(OH)2 to CaCO3 may be expected to be dependent on the particle mix prevailing in the slurry, insofar as this will influence the access of the CO2 to the Ca(OH)2. A general, schematic scenario is shown in Figure 1. Here, a fast reaction between dissolved OHand CO2 is shown, along with all types of particles (hydroxide, carbonate, hydroxide coated with carbonate). In this study, an attempt is made to develop a detailed model for this complex slurry reactor. The model results are compared * To whom correspondence should be addressed. Tel: 0091-2225767750. Fax: 0091-22-25721210. E-mail address:
[email protected].
Figure 1. Schematic representation of the general case of mass transfer with fast (almost instantaneous) reaction near the gas/liquid interface (penetration element) in a slurry of fine particles. All possible types of particles (reactant, product, reactant coated with product) are shown.
with actual experiments performed by contacting CO2 with aqueous Ca(OH)2 suspensions in a stirred contactor, under various operating conditions. Literature Review Various studies on the gas absorption in the slurries that contain fine particles of the solid reactant have appeared in the literature. Fine particles refer to particles whose size is smaller than that typical length scale of diffusion. Although the coarse particles can affect the rate behavior through the bulk liquidphase concentrations, fine particles cause significant diffusional gradients of reactive species within the film by direct reaction with the diffusing (absorbed/dissolved) species. Because coarse particles are not present within the film, their effect is restricted to changing only the bulk concentrations. Studies in this field were reported initially by Ramchandran and Sharma,2 who used the film theory of mass transfer, combined with instantaneous reaction, to model such processes.
10.1021/ie060732l CCC: $37.00 © 2007 American Chemical Society Published on Web 01/18/2007
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They considered both cases, i.e., solid dissolution in the film is negligible due to coarse particle size and the case when solid dissolution in the film occurs because of the fine size of the reactant particles. In the latter case, reaction and diffusion in the film becomes parallel steps and the absorption rate of the gas is enhanced. Uchida et al.3 considered the case of enhancement of solid dissolution that is due to the presence of fine particles, as an extension to the work of Ramchandran and Sharma,2 and they introduced another enhancement factor in the solid-liquid transport rate, because of reaction. Sada et al.4-6 have reported experimental investigations on the absorption of sulfur dioxide (SO2) into slurries of fine magnesium hydroxide (Mg(OH)2) and theoretical extensions for the cases when the bulk aqueous phase is not saturated with the dissolved solid as well as when the reaction is fast but not instantaneous. Later, Yagi and Hikita7 adopted a different approach. Instead of using a mass-transfer coefficient for the solid dissolution, they used two-dimensional diffusion of the gaseous solute in a regular lattice of particles fixed in the liquid phase. A penetration theory analog of Ramchandran and Sharma2 was proposed by Uchida et al.,8 which concluded that, for equal diffusivities of reactants, an analytical solution could be found for the problem of a moving reaction front in the presence of fine, sparingly soluble solid reactant particles. The predictions of rate values from this model have been demonstrated to be within a few percent of the film model computations. In this context, Mehra9 theoretically examined the problem of dissolution of solid particles in the penetration element, using population balances and an explicit particle size distribution. The evolving particle size distributions in the bulk phase were estimated by accounting for convective mixing of the particles arriving from and departing to the gas/liquid interface. The particle size was observed to decrease rapidly as we move toward the gas/liquid interface and, in some instances, there is even a zone that has no particles, close to the interface, where all the particles have dissolved. The author concluded that all previous models that use the concept of a fixed particle size overestimate the rates of absorption. It was also shown that previously computed rates can result in drastic underestimations for the actual batch time required. Even the theory proposed by Mehra9 is applicable only to systems where the product is not a solid, such as, for example, the case of SO2 absorption in Mg(OH)2 slurries. Here, the product, magnesium sulfite (MgSO3), is soluble in water, so the model can be used. However, for the case of CO2 absorption in Ca(OH)2 slurries, the product, CaCO3, is sparingly soluble in water and precipitates as a solid product, which is likely to interfere with the access of the dissolved CO2 for the Ca(OH)2. The objective of the present study is to propose a model for this complex scenario when a solid reactant and solid product are present together in a slurry absorber. Experimental Section Considering the limited amount of experimental data available in the literature, it was thought desirable to obtain the particle size distribution (PSD) and hydroxide-to-carbonate conversion data over a range of operating conditions that involve masstransfer coefficients and initial Ca(OH)2 holdups. All the experiments were performed in a stirred contactor at room temperature (25 °C) and pressure. The experimental setup included an acrylic stirred contactor with an internal diameter of 90 mm and a height of 260 mm, which was mounted between two steel disks with the help of
Figure 2. Schematic sketch of the stirred contactor apparatus. Table 1. Reactor Dimensions parameter
value
reactor diameter reactor height gas-liquid contact area, aGL liquid impeller type
90 mm 260 mm 6.46 x 10-3 m2 flat-blade turbine (4-bladed, 0.014 m in diameter) flat-blade turbine (6-bladed, 0.014 m in diameter)
gas impeller type Table 2. Experimental Conditions parameter
value
temperature liquid volume, VL stirring speed initial Ca(OH)2 holdup, lo (volumetric) CO2 partial pressure
25 °C 5 × 10-4 m3, 10 × 10-4 m3 2.16 rev/s, 1.16 rev/s 1%, 0.5% 1 atm (Ptotal)1atm)
four rods (see Figure 2). The specifications of the reactor are listed in Table 1. The upper plate and the lower plate carried the shafts of impellers of the gas and liquid phase, respectively, which, in turn, were connected to a two-stepped pulley that was driven by geared motors. Mechanical seals were provided for both the stirrers to prevent any leakage. The four-bladed steel impellers each were 7 mm in length. The two-stepped pulley of the liquid-phase stirrer can provide speeds of 1.16 and 2.16 rev/s only. Baffles were placed in the liquid phase to prevent vortex formation. The top and bottom plates of the reactor were provided with two ports each, for the entry and exit of gas and liquid phases, respectively. Two sample collection ports were also provided on the acrylic reactor, 30 and 40 mm above the reactor bottom, respectively. These ports were packed with a latex septum. A polythene balloon was filled with pure CO2 and connected to the gas inlet port of the contactor. The balloon was used to absorb CO2 at atmospheric pressure. The purging rate was measured by a rotameter that was connected to the CO2 cylinder. The various possible values of the operating parameters in the experiments conducted here are listed in Table 2. First, the mass-transfer coefficients (kl) at different stirring speeds were determined by conducting the physical absorption of CO2 into pure distilled water and measuring the absorption rates using a soap film meter (connected between the CO2
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Table 3. Mass-Transfer Coefficient (kl) Values at Different Stirring Speeds in the Stirred Contactora
a
stirring speed (rev/s)
kl (m/s)
1.16 2.16
3.72 x 10-5 5.66 x 10-5
VL ) 5 × 10-4 m3, aGL ) 6.36 × 10-3 m2.
balloon and the gas inlet port of the reactor). The values are listed in Table 3, and the procedure followed is described elsewhere.10 For a normal run, ∼10 × 10-4 m3/s to 11 × 10-4 m3/s of distilled water was stirred with an excess amount of analyticalgrade Ca(OH)2 powder for ∼1/2 h, so that the water was saturated with Ca(OH)2. The solution then was filtered using No. 42 grade Whatman filter paper under vacuum. A known amount of Ca(OH)2 powder was added to this saturated solution to obtain the desired holdup. Before commencement of the experimental run, the reactor, as well as all the slides, cover slips, and sampling bottles, were thoroughly washed with a dilute acid and then with fresh water and then dried. The reactor was then loaded with 10 × 10-4 m3 of a Ca(OH)2 suspension and then purged with pure CO2 at a flow rate of 1.33 × 10-4 m3/s for 30 s, to remove air that may be present in the contactor. After purging, all the ports were closed and the port that was connected to pure CO2 stored in the balloon, was opened. The stirrers were started and samples were drawn every 1800 s, up to a lapsed time of 9000 s, with the help of a syringe. Two and, in some cases, three runs for each operating condition were performed. Preliminary imaging slides show that it is desirable to dilute the suspension before microscopic observation. Therefore, the samples were diluted with a saturated Ca(OH)2 solution, to avoid any dissolution of Ca(OH)2. The extent of dilution is determined by trial and error, and a dilution of twice the sample volume was determined to give sufficient and correct information. A few drops of the slurry were placed on a slide from the sampling bottles and immediately covered by a cover slip. The slide was placed on the microscope table and magnified; multiple monochrome images were grabbed and stored on a personal computer (PC). Approximately 50 images of each slide at different positions were taken and analyzed. The image analysis system consisted of a transmission microscope (Olympus) that was connected to a digital camera and a grabber card. ImagePro software was used to grab and analyze the images stored on the PC. The image analysis works by detecting the difference in the optical properties of the continuous phase, relative to those of the particles. For one batch time, multiple slides were prepared. Also, an empty slide image was taken, to provide a background reference. This background image was subtracted from the image of the sample to remove the background objects and particles. A watershed filter (available in the ImagePro software) was used to break apart the loosely agglomerated particles into individual particles for each slide. Approximately 8000 particles were grabbed for each batch time, for a given set of operating conditions. A calibrated scale was loaded and the diameter of all the objects in the image were measured and stored. This data was used to generate the PSD and number-average size data. Some sample images are shown in Figure 3 at different batch times. The hydroxide-to-carbonate conversion was determined during the runs by titrating 0.1 × 10-4 m3 (10 cm3) of the slurry with standard hydrochloric acid (HCl) solution, using phenolphthalein and methyl orange indicators, for every sample drawn.
Figure 3. Image of sample taken from stirred contactor at a stirring speed of 2.16 rev/s, at (a) 0 s, (b) 5400 s, and (c) 9000 s. Other conditions: lo ) 0.01, kl ) 5.66 × 10-5 m/s.
Figure 4. Experimental size distribution function for all the particles present, at different batch times, for a typical experiment; Conditions: lo ) 0.01, kl ) 5.66 × 10-5 m/s.
Model Development Figure 4 shows the experimental PSD as obtained from microscopy and optical image analysis techniques. Also, the number-average size is shown in the legend. It is evident from the figure that the PSD does not change much at different batch times, even though the particle mixes (lime, limestone, mixed) present are different. The average size of the particles also almost remains the same. This is possible only in two cases: (1) As the precipitation proceeds, exactly the same number of product particles enter the population within a specific size range (bin) distributions as the reactant particles exiting it. (2) The product forms a layer on the reactant particle itself as the unreacted reactant core shrinks. The first hypothesis is highly improbable, because the dissolution and disappearance of hydroxide particles and the nucleation and formation of carbonate particles obey different rates. Thus, it is not possible to maintain a stationary distribution via this mechanism. The second explanation seems to be reasonable, where the CO2 diffuses into the hydroxide particle and reacts to form carbonate, which, because of its extremely low solubility, gets deposited on the hydroxide particle as soon as it is precipitated. This suggests that one can apply a “shrinking core” mechanism for this solid-liquid reaction. Figure 5 represents the schematic of this model and shows the different stages of the particle, from the beginning of the reaction to the time when it is totally consumed. We assume that the outer
Ind. Eng. Chem. Res., Vol. 46, No. 10, 2007 3173
∫0R
RA ) 4πRp2
p
kls(CA - CSA)η(Rc,x,t) dRc
(4)
Substituting this into eq 1, we get the mass-balance equation:
DA
Figure 5. Schematic of the shrinking-core model.
particle size (Rp) remains the same as the hydroxide layers convert to carbonate. We are considering a situation in which Ca(OH)2 (product B) does not dissolve in the liquid and a layer of product (CaCO3) continues to be deposited on the surface of the shrinking reactive core as more and more CO2 (product A) gas diffuses through the product layer to the reaction surface. The following assumptions are made in formulating the model: (1) Higbie’s penetration theory of mass transfer is used as a basis for formulating the new model. (2) Dissolution of the solid particles is neglected, in the sense that no Ca(OH)2 dissolves into the liquid to react with the CO2 present there. Thus, the profile of OH- shown in Figure 1 will vanish. A dominant surface reaction/nucleation step, as compared to homogeneous nucleation in the liquid phase, is likely to prevail, because of the much lower energy barriers associated with nucleation on an existing surface. (3) Particles are assumed to be spherical and the slurry is assumed to be sufficiently dilute to invoke a pseudo-homogeneous model.11,12 (4) The outer radius of the particles is assumed to be fixed at R p. (5) The reaction occurs instantaneously at the surface of the unreacted shrinking core. The species balance for the diffusing, dissolved gaseous solute, A, in a penetration element is given by
∂2CA
∂CA DA 2 ) + RA ∂t ∂x
)
∂x
∂CA + 4πRp2 ∂t
∫0R
kls(CA - CSA)η(Rc,x,t) dRc
p
(5)
Here, CSA must be computed from the shrinking-core model that is applied to a single particle. The diffusion of CO2 into the Ca(OH)2 particles is described by assuming that no accumulation occurs for the solid-liquid mass transfer; i.e., invoking the quasi-steady state assumption,
(
P
)
d 2 dCA r D°A )0 dr dr
(6)
The boundary conditions for the aforementioned equation are
R ) Rc, CPA ) 0
(7)
R ) Rp, CPA ) CSA
(8)
The first condition occurs by assuming an instantaneous reaction between the diffusing CO2 and the reactive core in the solid. The second condition connects the model to the external world through the surface concentration of A. Solving eq 6 and using the aforementioned boundary conditions, we get
CPA CSA
(1/Rc) - (1/R)
)
(9)
(1/Rc) - (1/Rp)
The rate of consumption of A, by a particle, thus becomes
(
)
dCAP ) 4πr D°A dr 2
)
r)Rp
4πD°ACAS (1/Rc) - (1/Rp)
(10)
(1) Also, 2 S Rone A ) 4πRp ksl(CA - CA) )
(2)
(1/Rc) - (1/Rp)
∂2CA ∂x2
)
∂CA + ∂t
)
∂CA + ∂t
∫0R
[
4πD°ACSA
p
(1/Rc) - (1/Rp)
]
n(Rc,x,t) dRc
(11)
(12)
or
DA
∂2CA 2
∂x
4πD°A
[
∫0
CSA
Rp
(1/Rc) - (1/Rp)
]
n(Rc,x,t) dRc (13)
Substituting for CSA, by equating the two right-hand side terms in eq 11, yields
(3)
where η(Rc,x,t) is the core size distribution function (and CSA will be dependent on η). Finally, for all the particles that have a core size from Rc ) 0 to Rc ) Rp, we integrate and get
4πD°ACSA
and from eqs 2 and 11, we get
DA
where ksl is the solid-liquid mass-transfer coefficient and CSA is the surface concentration of A for a particle that has an unreacted core radius Rc. The overall mass transfer, from all the particles present, which have different core sizes, can be obtained by multiplying Rone A with the number concentration (number per unit volume of liquid phase) of solid reactant particles that have a core size between Rc and Rc + dRc, located at a distance x from the gas/liquid interface, within a surface penetration element, at time t, i.e.,
dR′A ) ksl(CA - CSA)η(Rc,x,t) dRc
2
Rone A
where RA represents the rate at which A is transferred from the liquid to the particles. The concentration of A at a distance x from the interface (CA(x)), as well the concentration of A at the surface of particles (CSA), varies along the penetration thickness. The rate of transfer of A from the dissolved phase to a particle is given by S 2 Rone A ) ksl(CA - CA)4πRp
∂2CA
DA
∂2CA 2
∂x
)
∂CA + 4πD°ACA ∂t
{
∫0R
p
}
1 n(Rc,x,t) dRc [D°A/(kslRp )] + [(1/Rc) - (1/Rp)] 2
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which can be tranformed as 2
DA
∂ CA 2
∂x
)
∂ η (R ,θ) ) ∂θ b c
∂CA Rp + 4πRpD°ACA 0 ∂t 1 n(Rc,x,t) dRc (14) (D°A/kslRp) + [(Rp/Rc) - 1]
∫
{
}
where D°A/(kslRp) ) ShR, which is assumed to be equal to 1.9 The initial and boundary conditions for the aforementioned equation are given as follows:
t ) 0, all x, CA ) 0
(15)
x ) 0, all t, CA ) C*A
(16)
t > 0, x f ∞, CA ) 0
(17)
The population balance in the penetration element, over the core sizes in particles present at any x, at any time, t, is given by considering the fact that there are no birth events or death events that occur in the particle size range 0 to Rp:
∂η(Rc,x,t) ∂η(Rc,x,t)G(Rc,x,t) + )0 ∂t ∂Rc
(18)
∫0∞[ηb(Rc,x,tc) - ηb(Rc,θ)] dxtac
(24)
with the initial conditions
θ ) 0, ηb(Rc,θ) ) η0b(Rc)
(25)
where η0b(Rc) describes the initial state of the core size in the particles fed as slurry to the batch reactor. This integral term in the bulk population balance represents the exchange of material between the interfacial region and the bulk: whereas the first term in the integral is the number concentration of particles that have cores in the size range between Rc and Rc + dRc, that are being carried away from interface by the departing penetration elements, and the second term gives the equivalent number of particles that are brought to the interface from the bulk. Solution Strategy: Define a variable
n(Rc,x,t)
ω(Rc,x,t) )
pRc2 + Rc
where
p)
where G is the growth rate of the core (G ) (dRc/dt)). The boundary conditions for the population balance equation are as follows:
DA 2
1 Rp
-
kslRp
(26)
so that
t ) 0, η(Rc) ) ηb(Rc)
(19)
t > 0, Rc ) Rp, η(Rc,x,t) ) 0
(20)
The growth rate G may be deduced from
-
(
)
d FB 4 πR 3 ) Rone A dt Mw 3 c
[
DACSA Mw dRc )dt FBRc2 (1/Rc) - (1/Rp)
(21)
]
∂ω ∂ω )0 +G ∂t ∂Rc
(28)
The left-hand side of this equation is equal to dω/dt, according to the definition given in eq 27. Hence, ω(Rc,x,t) is a constant and, therefore,
(22)
which defines the rate of shrinkage (negative growth) of the core in any particle. Because the bulk concentration of A is taken to be fixed (CbA ) 0), the bulk material balance for this species is not required. The bulk PSD function ηb(R,θ) must be computed versus the batch time (θ), which is much larger than the contact time (tc). The bulk population balance on this slurry reactor may be written as
∂ η (R ,θ) ) ∂θ b c
(27)
If we substitute this definition in the population balance (eq 18), then the population balance becomes
which states that the rate of consumption of A is equal to the rate of consumption of B. Using eq 10,
G)
∂ω dω ∂ω dRc ∂ω ∂ω ) + +G ) dt ∂t dt ∂Rc ∂t ∂Rc
∫0∞ [ηb(Rc,x,tc) a ∂ ηb(Rc,θ)] dx [η (R ,θ)Gb(Rc,θ)] (23) tc ∂Rc b c
where the last term is related to particle reaction in the bulk phase. However, because CbA ) 0, no CO2 is available for diffusion into the hydroxide particles in the bulk and the last term in the aforementioned equation may be dropped. Equation 23 thus reduces to
ω(Rc,x,t) )
η(Rc,x,t) pRc + Rc 2
)
η(R0,x,0) pR0 + R0 2
)
ηb(R0) pR02 + R0
(29)
To eliminate the dummy variable, R0, we integrate eq 22 with the initial condition, R ) R0 at t ) 0. This yields the cubic equation
(31)pR + (21)R ) (31)pR + (21)R 3
0
2
0
3
c
2 c
+K
∫0t CA dt′
(30)
(where K ) DAMw/FB) which may be substituted in eq 29. Thus,
η(Rc,x,t) )
ηb(R0,θ)(pRc2 + Rc) pR02 + R0
(31)
with R0 given by eq 30. We now use orthogonal collocation to convert the nondimensionalized form of the partial differential equations (PDEs) to a set of ordinary differnetial equations (ODEs) in dimensionless time. Equation 14 can be transformed as
Ind. Eng. Chem. Res., Vol. 46, No. 10, 2007 3175
∂Cj ∂φ
)
1
N
∑
L2 i)1
BjiCi + A1Cj
∫01
(31)p[R (j) ] + 21[R (j) ] ) (31)pR + (21)R 3
0
ηj dr m + (1/r)
(32)
2
0
3
c
η(Rc,j,t) )
2 c
+K
∫0t CA(j) dt′
ηb(R0,θ)(pRc2 + Rc) pR0(j)2 + R0(j)
(33)
∫0t
1 tc
c
(
-DA
)
∂CA ∂x
x)0
dt
property
value
source/remarks
C*A DA Fp Fpp Mw aGL/VL
2 x 10-2 kmol/m3 2 x 10-9 m2/s 2200 kg/m3 2750 kg/m3 74 kg/kmol 12.72 m2/m3 6.36
Perry and Chilton13 Jones et al.14 Carr and Frederick1 Carr and Frederick1 measured
(34)
where C ) CA/C*A, φ ) t/tc, r ) Rc/Rp, A1 ) 4πD°ARptc and m ) [D°A/(kslRp)] - 1. This set of equations is solved for each collocation point, denoted by the subscript j, successively and the solution is reached. The values of various parameters used in the model is given in Table 4. A solution of the aforementioned equations, along with the relevant conditions, provides us the concentration profile of A (CA) and the core size distribution function (η(Rc,x,t)). Also,
RA )
Table 4. Property and Parameter Values Used in the Model, at 25 °C
(35)
initial loadings (l0) and mass-transfer coefficients (kl) are shown in Figures 6-9. Figures 6 and 7 show the conversion versus batch time trajectories at initial loadings of l0 ) 1% and 0.5%, respectively, at different mass-transfer coefficients. We can clearly see that higher conversions are obtained at higher stirring speeds (masstransfer coefficients). Figures 8 and 9 show plots of the conversion versus the batch time at different holdups, at two different mass-transfer coefficient values, respectively. As expected, higher conversions are obtained at lower holdups at any given time. Core Size Distributions. The computed core size distributions, at different batch times, are shown in Figure 10a. In all
where tc ) 4DA/(πkl2). In addition, the following quantities are important: core number concentration,
N0(x,t) )
∫0R η(R,x,t) dRc p
(36)
the average size of the reactive core in all particles present,
∫0R Rcη(Rc,x,t) dRc p
Rmean (x,t) c
)
N0(x,t)
(37)
the local, volumetric ratio of solid (reactive) core to liquid,
l(x,t) )
∫0R Rc3η(Rc,x,t) dR
4π 3
p
(38)
and the conversion of hydroxide-to-carbonate can be calculated as
XB )
lo - l(t) lo
Figure 6. Variation of hydroxide-to-carbonate conversion with batch time, at different mass-transfer coefficient (kl) values. Data used: Rp ) 2.5 × 10-6 m, lo ) 1%; other parameter values have been taken from Table 4.
(39)
The collocation equations were solved by a combination of an algebraic equation root finder (for eq 30) and an ODE solver for the main differential equation (eq 32) from the MATLAB package. Results and Discussion Conversion-Time Trajectories. The theory proposed has first been validated by comparing the hydroxide-to-carbonate conversions (XB) as obtained from theory and experiments, respectively. Figures 6 and 7 show that the experimental conversion data match the computed values reasonably well. No explicit effort was made to fit the experimental data to the theory. A default value of the solid-liquid mass-transfer coefficient (ksl), first observed in eq 2, was taken to be given by the corresponding Sherwood number that is equal to 1 (based on the particle radius Rp). The conversions obtained at different
Figure 7. Variation of the hydroxide-to-carbonate conversion with batch time, at different mass-transfer coefficient (kl) values. Data used: Rp ) 2.5 × 10-6 m, lo ) 0.5%; other parameter values have been taken from Table 4.
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Figure 8. Variation of the hydroxide-to-carbonate conversion with batch time, at different hold-up values. Data used: Rp ) 2.5 × 10-6 m, kl ) 5.66 × 10-5 m/s; other parameter values have been taken from Table 4.
Figure 10. Theoretical evolution of core size distribution at different batch times. Data used: Rp ) 2.5 × 10-6 m, lo ) 1%, kl ) 5.66 × 10-5 m/s; other parameter values have been taken from Table 4.
Figure 9. Variation of the hydroxide-to-carbonate conversion with batch time, at different hold-up values. Data used: Rp ) 2.5 × 10-6 m, kl ) 3.72 × 10-5 m/s; other parameter values have been taken from Table 4.
our computations, a monodisperse initial distribution, where all the particles are of size Rp, corresponding to the median size value of the experimental data (2.5 µm), has been used (see eq 40). The number concentration can be observed to decrease rapidly with increasing batch times. Figure 10b is an enlarged version of Figure 10a. Penetration Element Profiles. The initial core size distribution, as previously referenced, can be described by an impulse function, namely,
ηb(Rc) ) Noδ(Rc - Rp)
(40)
This implies that, when the penetration element arrives at the interface from the bulk region, at the start of the batch process, all slurry particles have a core size, Rp. This represents unconverted (fresh) particles. As the reaction proceeds, the outer radius remains the same while the core shrinks. Typical values of Rp and an initial holdup lo are chosen to get the value of the initial number of particles (No), given by
No )
3lo 4πRp3
(41)
Figure 11. Concentration of dissolved gaseous solute versus distance from the gas/liquid interface in a penetration element, at different contact times. Data used: Rp ) 1 µm, lo ) 1%, kl ) 5.66 × 10-5 m/s; other parameter values have been taken from Table 4.
The standard data used are shown in Table 4. The concentration profiles of dissolved CO2 in the penetration element are plotted in Figure 11. The average core size, as a function of distance from the gas/ liquid interface, is shown in Figure 12, for different Rp values, at t ) tc. We observe that a zone that consists of particles that have zero core size (i.e., fully converted) forms, for smaller particles. This is because the smaller particles get converted faster than bigger particles. Figure 13 shows a similar variation in volume of unreacted material available (core volumes) versus distance from the gas/liquid interface.
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Figure 12. Average size of core in particles versus distance from the gas/ liquid interface in a penetration element, at different particle sizes (Rp). Data used: lo ) 1%, kl ) 5.66 × 10-5 m/s, t/tc ) 1; other parameter values have been taken from Table 4.
Figure 13. Holdup of core in particles versus distance from the gas/liquid interface in a penetration element, at different particle sizes (Rp). Data used: lo ) 1%, kl ) 5.66 × 10-5 m/s, t/tc ) 1; other parameter values have been taken from Table 4.
Figure 14. Variation of the core radius (Rc) at the gas/liquid interface with contact time, at different particle sizes (Rp). Data used: lo ) 1%, kl ) 5.66 × 10-5 m/s; other parameter values have been taken from Table 4.
Figure 14 shows the variation of core radius (Rc) at the gas/ liquid interface with time t, for different Rp values. As may be observed, the core shrinks rapidly for smaller particles, vanishing
Figure 15. Variation of the specific rate of gas absorption with contact time, for different models. Data used: Rp ) 1 µm, lo ) 0.5%, kl ) 5.66 × 10-5 m/s; other parameter values have been taken from Table 4.
Figure 16. Variation of the specific rate of gas absorption with contact time, for different models. Data used: Rp ) 10 µm, lo ) 0.5%, kl ) 5.66 × 10-5 m/s; other parameter values have been taken from Table 4.
completely within a fraction of the contact time tc for Rp ) 1 µm. Figure 15 shows a comparison between the flux of A (CO2), at the gas/liquid interface, for different models (without particles, with particles but no change in core size, and the current model). We find that, with increasing tc, there is a substantial difference in rates. The rate computed from the constant core size model are observed to be overpredicted. Similar graphs are plotted for particles that have a larger initial core size of Rp ) 10 µm. Because the cores in these particles undergo a relatively insignificant change in size, the difference between the estimates of constant and changing core sizes is not much (Figure 16); in fact, the lines coincide. A comparison of the specific rates of absorption predicted by the proposed shrinking-core model versus the fixed-core model is shown in Figure 17 for different Rp values. The extent of overprediction for fine particles by the fixed-core models is substantial. Bulk Liquid Profiles. In this section, a systematic simulation of the outputs is repeated, with respect to the major operating variables, namely, the mass-transfer coefficient (kl) and the initial holdup of the reactant particles (lo). Figure 18 shows the theoretical hydroxide-to-carbonate conversion (XB) versus batch time (θ), at different kl values. As discussed previously, higher kl values lead to higher conversions.
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Figure 17. Variation of the specific rate of gas absorption with radius, for Rc ) Rp and for changing Rc. Data used: lo ) 0.5%, kl ) 5.66 × 10-5 m/s, t/tc ) 1; other parameter values have been taken from Table 4.
Figure 18. Variation of the theoretical hydroxide-to-carbonate conversion with batch time, at different mass-transfer coefficient (kl) values. Data used: lo ) 1%, Rp ) 2.5 µm; other parameter values have been taken from Table 4.
Figure 20. Variation of the theoretical hydroxide-to-carbonate conversion with batch time, at different values of initial calcium hydroxide (Ca(OH)2) holdup. Data used: kl ) 5.66 × 10-5 m/s, Rp ) 2.5 µm; other parameter values have been taken from Table 4.
Figure 21. Variation of the theoretical hydroxide-to-carbonate conversion with batch time, at different Rp values. Data used: lo ) 1%, kl ) 5.66 × 10-5 m/s; other parameter values have been taken from Table 4.
Figure 21 gives similar comparisons but at different Rp values. The conversion rates are higher for lower particle sizes. This shows that, in smaller particles, the cores get exhausted faster, in comparison to larger particles. Conclusion
Figure 19. Variation of the theoretical average CO2 absorption rate with batch time, at different mass-transfer coefficient (kl) values. Data used: lo ) 1%, Rp ) 2.5 µm; other parameter values have been taken from Table 4.
Figure 19 shows the computed, average CO2 absorption rates (RA). The observed behavior is expected. Similarly, Figure 20 shows XB versus θ, at different values of the initial Ca(OH)2 holdup (lo).
This study presents a theory for the carbonation of a suspension of fine calcium hydroxide (Ca(OH)2) particles where the product calcium carbonate, which is also a very sparingly soluble substance, precipitates on the reacting hydroxide particle itself, forming a shell around the unreacted hydroxide particle core. Therefore, a moving reaction front model (or shrinkingcore model) is proposed. This model is solved, and the computed results are compared with the experimentally observed conversions. The theory and experiment match very well, using the “default” values of the parameters listed in Table 4. The diffusion coefficient for carbon dioxide (CO2) in the product layer (ash) surrounding a particle is taken to be identical to the diffusivity value in water, thereby assuming that the solid product is fairly porous. In principle, this parameter could be fitted to the experimental data, if necessary.
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Further simulations show the effect of various operating parameters on the overall particle conversion and core size distributions. This is the first model that attempts to compute particle conversions in precipitated calcium carbonate (PCC) formation, wherein the product coats the reactant. (All the prior models2-9 involve the precipitation of a solid from a solution or dissolution of a solid reactant.) The model can be further improved by taking into account multiparticle distribution (on the Rp space), instead of the median size (Rp).
Greek Letters
Nomenclature
Literature Cited
A ) gas being absorbed or the concentration of species A, CO2 (kmol/m3) Ap ) specific solid-liquid interfacial area (m2/m3) aGL ) gas-liquid interfacial area (m2) 3 C* A ) concentration of A at the gas-liquid interface (kmol/m ) CsA ) surface concentration of A on the particle in penetration element (kmol/m3) P CA ) concentration of A inside the particle (kmol/m3) DA ) diffusivity of A in liquid phase (m2/s) D°A ) diffusivity of A inside the particle (m2/s) dp ) particle diameter (m) G ) growth rate for single particle (in units of m/s) between A and B (m3/kmol) kl ) liquid-side mass-transfer coefficient for gas-liquid contact (m/s) ksl ) mass-transfer coefficient for gas transfer into the particle (m/s) l ) volumetric solid loading Mw ) molecular weight of solid reactant (kg/kmol) m ) distribution coefficient of solute partitioning between the solid and liquid phases N ) number concentration of the solid reactant particles (1/ m3) Ptotal ) total gas pressure in the contactor (atm) R ) radius of single particle (m) RA ) specific rate of absorption (kmol/m3s) RAi ) instantaneous specific rate of absorption (kmol/m3s) Rc ) radius of core (m) Rp ) radius of shell (m) ) mean core radius (m) Rmean c ShR ) Sherwood number for particle dissolution, based on radius t ) time on the penetration element contact time scale (s) tc ) contact time of a penetration element (s) x ) distance from the gas-liquid interface within the penetration element (m) XB ) conversion of B
(1) Carr, F. P.; Frederick, D. K. Calcium carbonate. In Encyclopedia of Chemical Technology, Vol. 4; Wiley-Interscience: New York, 1996; p 796-801. (2) Ramchandran, P. A.; Sharma, M. M. Absorption with fast reaction in a slurry containing sparingly soluble fine particles. Chem. Eng. Sci. 1969, 24, 1681-1686. (3) Uchida, S.; Koide, K.; Shindo, M. Gas absorption with fast reaction into a slurry containing fine particles. Chem. Eng. Sci. 1975, 30 (5-6), 644-646. (4) Sada, E.; Kumazawa, H.; Butt, M. A.; Sumi, T. Removal of sulphur dioxide by aqueous slurries of magnesium hydroxide particles. Chem. Eng. Sci. 1977, 32 (8), 972-974. (5) Sada, E.; Kumazawa, H.; Butt, M. A. Simultaneous gas absorption with reaction in a slurry containing fine particles. Chem. Eng. Sci. 1977, 32 (12), 1499-1503. (6) Sada, E.; Kumazawa, H.; Butt, M. A. Absorption of sulphur dioxide into aqueous slurries of sparingly soluble fine particles. Chem. Eng. Sci. 1980, 35 (4), 771-777. (7) Yagi, H.; Hikita H. Gas absorption into a slurry accompanied by chemical reaction with solute from sparingly soluble particles. Chem. Eng. J. 1987, 36, 169-174. (8) Uchida, S.; Miyachi, M.; Ariga, O. Penetration model of gas absorption into slurry accompanied by an instantaneous irreversible chemical reaction. Can. J. Chem. Eng. 1981, 59, 560-561. (9) Mehra, A. Gas absorption in reactive slurries: particle dissolution near gas-liquid interface. Chem. Eng. Sci. 1996, 51, 461-477. (10) Venugopal, B. V.; Mehra, A. Gas absorption accompanied by chemical reaction in water-in-oil emulsions. Chem. Eng. Sci. 1994, 49 (19), 3331-3336. (11) Mehra, A. Advances in Reactor Design and Combustion Science. In Handbook of Heat and Mass Transfer, Vol. 4; Gulf Publishing: Houston, TX, 1990. (12) Mehra, A. Heterogeneous modeling of gas absorption in emulsions. Ind. Eng. Chem. Res. 1999, 38, 2460-2468. (13) Perry, H. P.; Chilton, C. H. Chemical Engineers’ Handbook; McGraw-Hill; Tokyo, 1973. (14) Jones, A. G.; Hostomsky, J.; Zhou, L. On the effect of liquid mixing rate on primary crystal size during the gas-liquid precipitation of calcium carbonate. Chem. Eng. Sci. 1992, 47, 3817-3824.
Fp ) density of solid reactant (kg/m3) θ ) batch time (s) η ) particle size distribution function (1/m4) Subscript A ) pertaining to A Superscripts s ) surface b ) pertaining to bulk
ReceiVed for reView June 9, 2006 ReVised manuscript receiVed November 14, 2006 Accepted November 20, 2006 IE060732L
