Ind. Eng. Chem. Res. 1997, 36, 3889-3897
3889
A Model for Prediction of Limestone Dissolution in Wet Flue Gas Desulfurization Applications Charlotte Brogren*,† and Hans T. Karlsson‡ Department of Chemical Engineering II, Lund University, Center for Chemistry and Chemical Engineering, P.O. Box 124, S-221 00 Lund, Sweden, and ABB Corporate Research, S-721 78 Va¨ stera˚ s, Sweden
A model has been developed to predict the dissolution rate of a limestone slurry as a function of particle size distribution and limestone conversion. The model is based on basic mass-transfer theory and includes a factor allowing the flux of calcium ions from the limestone surface to vary with the fraction dissolved. Changes in the flux with the fraction dissolved have been reported to be caused by the presence of sulfite but can also be caused by accumulation of inerts at the liquid-solid interface and/or by changes in the effective mass-transfer area. Calculations show that the decrease in flux reported for sulfites can have a significant impact on the slurry conditions within the reaction tank, i.e., impact on the limestone conversion and the relationship between liquid and solid alkalinity. In the absence of sulfites, the flux from limestone particles has been assumed to be constant with respect to the degree of dissolution. The modeling results have been found to be in good agreement with the measured values of a continuous stirred tank reactor. The model was able to accurately predict the impact of both the particle size distribution and reaction tank residence time on limestone conversion and dissolution rate. Introduction Limestone-based wet flue gas desulfurization (WFGD), has since the early 1970s been considered the most cost effective and reliable method for removal of sulfur dioxide from coal- and oil-fired power plants. Limestone scrubbing is based on the absorption of SO2 into a limestone slurry where dissolved limestone neutralizes the absorbed SO2 to form calcium sulfite and/or gypsum. The wet FGD process is generally represented by the following overall reactions:
SO2 + CaCO3(s) f CaSO3 + CO2
(1)
CaSO3 + 0.5O2 f CaSO4(s)
(2)
Several different types of scrubber designs are available for limestone scrubbing, including spray, venturi, static, and mobile packed beds. However, the most commonly used is the countercurrent spray scrubber, in which the gas enters at the bottom of the tower and the liquid is distributed at different levels in the tower by nozzles positioned on horizontal spray header, Figure 1. The slurry leaving the absorber is collected in a reaction tank which normally is integrated as the bottom part of the absorber where the limestone feed is also added. To oxidize sulfite to sulfate, air is sparged into the bottom of the reaction tank. The reaction tank must be large enough to provide sufficient liquid residence time to enable precipitation of the waste solids as well as dissolution of the limestone. From the reaction tank, a bleed-off stream is sent to a dewatering equipment for gypsum recovery. Hydrocyclones are often used for primary dewatering and for separation of limestone and gypsum particles in the stream leaving the reaction tank. * Author to whom all correspondence should be addressed. Telephone: +46-21-323028. Fax: +46-21-323090. Email:
[email protected]. † ABB Corporate Research. ‡ Lund University. Telephone: +46-46-2228244. Fax: +4646-149156. Email:
[email protected]. S0888-5885(97)00030-4 CCC: $14.00
Figure 1. Principal drawing of a wet FGD plant.
Prediction of the limestone dissolution rate is important for the design and operation of WFGD plants. The limestone dissolution rate affects, for example, the relationship between the liquid and solid alkalinity, the required residence time in the reaction tank, and the mass-transfer rate of SO2 within the spray zone. All these factors affect the capital and operating costs of a WFGD unit. A number of investigators have studied the dissolution rate of limestone in aqueous solutions. In general, it has been found that limestone dissolution is mass transfer controlled at low pH, whereas surface kinetics are of importance at high pH values (Plummer and Wigley, 1976; Sjo¨berg and Rickard; 1984). Chan and Rochelle (1982) and Wallin and Bjerle (1989) studied the dissolution rate of limestone under various conditions using a pH-stat. They successfully modeled the dissolution rate by mass transfer with equilibrium of acid-base reactions and finite-rate reaction between CO2 and H2O. Jarvis et al. (1988) showed that the impact of the rate of the CO2 reaction was insignificant for limestone dissolution in wet FGD systems where other buffering species such as sulfite are present. Gage and Rochelle (1992) showed that limestone dissolution in the presence of sulfite under wet FGD conditions is controlled by a © 1997 American Chemical Society
3890 Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997
combined surface kinetic/mass-transfer mechanism. A surface rate correlation was developed to account for the observed inhibition of sulfite including a limestone typespecific constant. Toprac and Rochelle (1982) developed a mass-transfer model to estimate the impact of particle size distribution on the limestone dissolution rate. The mass-transfer model was able to predict dissolution rates with an error of less than 30%, and it was claimed that the particle size distribution was the most important reactivity characteristic of ground limestone of reasonable purity. Chang et al. (1982) showed that finer limestone grinding could have a considerable effect on SO2 scrubber efficiency using a turbulent contact absorber. Brogren and Karlsson (1997b) modeled the SO2 absorption into a limestone slurry droplet and showed that limestone dissolution within the spray zone has a significant impact on the SO2 absorption rate. Recently, Ukawa et al. (1993) developed a mass-transfer model to study the effects of particle size distribution on limestone dissolution in both batch and continuous systems. The changes in particle size distribution (PSD) as a function of limestone utilization were studied, and the dissolution rate constant was found to be proportional to the activity of the hydrogen ions. The objective of this work was to develop a model for prediction of limestone conversion and limestone particle size distribution in a WFGD system. The modeling results have been compared to experimental results from both laboratory experiments and from actual plant data. Model Reactions. The following equilibria are assumed to be valid throughout the limestone slurry:
H2O f H+ + OH-
1 ∇(Jkb2) ) rk b2
(4)
HSO3- f H+ + SO32-
(5)
HCO3- f H+ + CO32-
(6)
-
The model takes into account nine ion pairs: CaHSO3+, CaSO3°, CaHCO3+, CaSO4°, CaCO3, MgHSO3+, MgSO3°, MgHCO3+ and MgSO4°. Instantaneous equilibrium is assumed for the formation of these ion pairs. Equilibrium constants, activity coefficients, and diffusivities are calculated by correlations from Bechtel Modified Radian Equilibrium Program, BMREQ (Epstein, 1977) with the exception of the ion pairs: CaHSO3+ and MgHSO3+ and the calcite solubility (Gage and Rochelle, 1992):
log Ksp CaCO3 ) -171.9065 - 0.077993T +
Jk ) -Dk∇Ck
(9)
where the diffusion constant Dk is constant. To avoid using rate constants fixed at arbitrarily high values of the equilibrium reactions, mass balances for individual concentrations are combined to balances for total sulfite, total carbonate, total calcium, total magnesium, and total sulfate as follows: n
0)
2
∑ -Dk∇2Ck - bDk∇Ck k)1
(10)
Species involved in the combined balances are for the following. For total sulfite: SO2, HSO3, SO3, CaSO3°, and MgSO3°. For total carbonate: HCO3-, CO32-, CaHCO3+, MgHCO3+, and CaCO3°. For total calcium: Ca2+, CaSO3°, CaHCO3+, CaSO4°, and CaCO3°. For total magnesium: Mg2+, MgSO3°, MgHCO3+, and MgSO4°. For total sulfate: SO42-, CaSO4°, and MgSO4°. Similarly, a balance for the net charge is requested:
2839.319 + 71.595 log T (7) T
n
0) The diffusion constants in aqueous solution at 25 °C were taken from the work of Gage (1989). Extrapolation of the values from 25 to 55 °C was done using the Stokes-Einstein equation. Limestone Dissolution Rate. The dissolution rate of limestone is equal to the product of the flux from the
(8)
where k indicates component k, b is the distance from the center of the particle, and r is the reaction rate of component k. It is assumed that neither precipitation nor dissolution of calcium sulfite and sulfate take place in the liquid film around the limestone particle. Furthermore, oxidation of sulfite to sulfate in the film is disregarded. Dissolved CO2 can hydrolyze and enhance the limestone dissolution rate by assisting in the transport of hydrogen ions to the limestone surface. Jarvis et al. (1988) have, however, shown that the first hydrolysis reaction of CO2 is insignificant to the dissolution rate when buffering species such as sulfite are present. It has previously been shown (Rochelle, 1992; Brogren and Karlsson, 1997a) that the impact of any electric potential gradient on the flux of ions may be disregarded under FGD conditions as long as the mass flux equations are combined with a flux of charge equation. The flux J can, therefore, be expressed as follows:
(3)
SO2 + H2O f H + HSO3 +
limestone surface and the effective mass-transfer area of the limestone. The flux is calculated by setting up boundary conditions at the limestone surface and solving the mass balance equations over the film domain surrounding the particle. The effective mass-transfer area is calculated from the limestone particle size distribution. The flux of calcium ions from the limestone surface is calculated by a steady-state mass-transfer theory, the film theory, assuming the limestone particles to be spheres surrounded by a stagnant film. The differential mass balance in spherical coordinates around a limestone particle is:
2
-zkDk∇2Ck - zkDk∇Ck ∑ b k)1
(11)
where the species involved are: H+, OH-, Ca2+, Mg2+, SO42-, HSO3-, SO32-, HCO3-, CO32-, CaHCO3+, MgHCO3+, and Cl-. By multiplying the material balances by b2, the balances turn into simple differential equa-
Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 3891
tions which can easily be integrated to the following expressions:
JCa2+ + JCaSO3° + JCaHSO3+ + JCaCO3° + JCaHCO3+ + JCaSO4° ) ξ1/b2 (12) JHCO3- + JCO32- + JCaCO3° + JCaHCO3+ + JMgHCO3+ ) ξ2/b2 (13) JSO2 + JHSO3- + JSO32- + JCaSO3° + JCaHSO3+ + JMgSO3° + JMgHSO3+ ) ξ3/b2 (14) JMg2+ + JMgSO3° + JMgHSO3+ + JMgHCO3+ + JMgSO4° ) ξ4/b2 (15) JSO42- + JCaSO4° + JCaSO° + JMgSO4° ) ξ5/b2 (16)
the film thickness at particle diameters above 20 µm is relatively constant and that the film thickness approaches d/2 (Sh ) 2) for small particles. The constant k was found to be a strong function of the agitation speed in the absence of sulfite, but independent of the agitation speed if sulfite was present. As a particle dissolves, it should dissolve more and more quickly per unit area due to variations in film thickness with shrinking particle diameter. However, Jarvis et al. (1988) measured a decrease in the flux as the limestone dissolves, which is the opposite result compared to mass-transfer theory. This behavior was especially significant when sulfite was present and was referred to as an aging effect. The decrease in the flux was experimentally examined, and it was claimed that the aging effect could be modeled as a function of the fraction dissolved. Other factors that could also affect the dissolution over time might be inhibition, accumulation of inerts at the interface, and changes in effective area. In this work the change in flux has been modeled using a correction factor, R:
n
zkJk ) ξ6/b2 ∑ k)1
where k indicates the species of the charge balance. There is no net flux of S(IV), Mg, and S(VI) species, and the flux of charge is equal to zero. Hence, the constants ξ3, ξ4, ξ5, and ξ6 are equal to zero. The flux of calcium species must be equal to the flux of total carbonate. Hence, ξ1 ) ξ2 ) ξ. It has been shown that the overall rate of dissolution of limestone in the presence of sulfite is covered by mass-transfer and surface kinetics. Gage and Rochelle (1992) have derived a rate expression of the surface kinetics:
JCatot ) kc
xaCaCo °
3 (eq)
- aCaCo3°(s)
aCaSo3°(s)aCaCo3°(s)
(18)
where kc is a surface rate constant dependent on the limestone type. Values given by Gage for kc fall in the range 6-60 × 10-8 mol5/2/(m13/2 s). The subscript eq corresponds to the activities at the limestone surface when the solubility product of CaCO3 is met and the subscript s corresponds to the actual activities at the limestone surface. The boundary conditions for the limestone flux equations are:
b ) d/2 + δ b ) d/2
J(d) ) J(d0)
R
(20)
1 E(t) ) e-t/θ θ
2
JCatot ) ξ/b
d 2 + kRe0.5Sc0.33
d d0
where J(d) is the flux of a particle with a diameter d and an initial diameter equal to d0, J(d0) is the initial flux from the particle, and R is the correction factor. J(d0) is calculated by solving the flux equations using the film thickness from eq 19. The limestone dissolution rate is calculated by integrating the product of the flux from the limestone surface and the available mass-transfer area of limestone. The parameters usually known in a slurry system are the sieve data of the limestone feed and the limestone utilization. Within the wet FGD system, the particles dissolve and change size. The size of each limestone particle within the slurry therefore depends on the initial size and the age of the particle within the system. The scrubber together with the reaction tank can be regarded as a continuous stirred tank reactor (CSTR) due to the large recirculation ratio of slurry over the scrubber. Thus, the particle size distribution, and thereby also the limestone mass-transfer area of the slurry, will be determined by the conditions of the scrubber system, such as residence time distribution and the overall limestone utilization. The residence time distribution function of a CSTR is:
C ) Cbulk
The procedure is to iterate the hydrogen ion concentration at the surface until the boundary conditions at the surface are met. The iteration is done according to Gauss-Newton in multiple dimensions (Burden and Faires, 1989). The most widely used correlation between the film thickness and the particle size has the following form:
δ)
()
(17)
(19)
Jarvis et al. (1988) determined the constant k to be 0.95 at an agitation speed of 1800 rpm, which means that
(21)
where θ is the mean residence time. The particle size distribution of the limestone feed has been modeled with the log-gamma density function, which has been successfully used by Gage (1989) to predict the particle size distribution of limestone for wet FGD applications. The cumulative distribution curve, P(Y), is found by integrating the density function:
P(Y) )
[
λ - 1 (λ - 1)(λ - 2) e-YYλ-1 + 1+ + ... + Y (λ - 1)! Y2 (λ - 1)(λ - 2)...2 × 1 Yλ-1
]
(22)
3892 Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997
to
where
Y)
( )
d100 3 ln β d
The cumulative distribution curve is characterized by three parameters, λ, β, and d100. Gage (1989) showed by statistical analysis that the λ value could be adequately represented by assuming integer values of 4 and 6. Using a fixed λ, the β and d100 values can be determined by two sieve measurements. The mass of limestone added to the system, M0,tot, can be considered as a sum of fractions with different diameters: ∞
M0,tot )
rd,i )
(23)
M0,i ∑ i)1
ni )
(25)
∞
M0,i′ ∑ i)1
(26)
This can also be expressed in terms of the mass fraction of the spent sorbent:
6M0,tot FQ
n
∑ i)1
Mj ∑ j)1
(27)
()
∂d 2M ˜ d )J(d0) ∂t F d0
(28)
2M ˜ t (1 - R)J(d0) R F d
R ) 1: d ) d0e-[2M˜ J(d0)/Fd0]t
0
)
1/(1-R)
(29) (30)
Consider one fraction of the limestone feed, M0,i. The dissolution rate of this fraction in the system is equal
]
di2 1 e-t/θ dt (33) 3 θ d 0,i
where eq 29 or eq 30, is used to estimate di from t. Limestone Conversion. The limestone conversion, f, of the system can be expressed in terms of the ratio between the limestone of the slurry, Mtot, and of the feed, M0,tot:
f ) 1 - Mtot/M0,tot
(34)
The mass of one fraction of the limestone feed, M0,i, consists of ni particles with a mass of m0,i and a diameter of d0,i. The remaining mass of the ith fraction in the slurry, M0,i′, can be calculated according to the following equation:
∫0∞nimiE(t) dt
(35)
Expressing mi by the following equation:
( )
mi ) m0,i
di d0,i
3
(36)
and ni by eq 32 and by adding up all fractions, the total conversion of limestone can be calculated as a function of the average residence time, θ, and the particle size distribution: n
R
where M ˜ is the molar weight of limestone and F is the density of limestone. The integral can be solved analytically:
(
()
∫0∞J(d0,i) ×
a
di
M0,i′ )
where M0,i' is the remaining mass of the ith fraction of the feed with diameter between d0,i-1 and d0,i and Mj is the mass of the jth fraction of the slurry with diameter between dj-1 and dj. The limestone particles are assumed to be spherical. By expressing the mass and the area of the particle in terms of the diameter, the following relationship can be derived for a particle:
R * 1: d ) d01-R -
[
(P(Y)d0,i - P(Y)d0,i-1)
d0,i
∞
Mtot )
(32)
The total dissolution rate of all fractions can now be expressed as:
Limestone particles within the liquid leaving the tank, Mtot, is defined as:
Mtot )
M0,i M0,i )6 m0,i pπd 3 0,i
rd )
M0,i ) P(Y)d0,i - P(Y)d0,i-1 M0,tot
(31)
where Ai is the area of a particle with diameter di and ni is the number of particles with initial diameters between d0,i-1 and d0,i. The latter parameter may be expressed in terms of the initial diameter:
(24)
where M0,i is the mass of the ith fraction of the feed with diameter between d0,i-1 and d0,i. Each fraction of the feed can be considered to consist of ni particles with a diameter d0,i. The relationship between the fraction of limestone M0,i and the cumulative distribution curve is:
niAi E(t) dt Q
∫0∞J(di)
f)1-
∑ i)1
[
(P(Y)d0,i - P(Y)d0,i-1)
() di
∫0∞ d
0,i
3
1 -t/θ e dt θ
]
(37) Particle Size Distribution of the Slurry. M0,i′ is the remaining mass of the mass of the ith fraction of the feed, M0,i. Due to the residence time distribution of a CSTR, the diameters of the particles of M0,i′ will range from 0 up to d0,i. This means that the particles corresponding to fraction M0,i′ will be divided on all fractions of the slurry, Mj where j e i (Figure 2). Mj is thus equal to the sum of the parts of M0,i′ with diameter dj-1 to dj. Wi,j is defined as the remaining mass of the ith fraction of the feed with a diameter less than dj (see
Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 3893
Figure 2. Schematic drawing of the change of the particle size distribution due to dissolution.
Figure 4. Cumulative distribution curves of the limestones used in the experiments measured by laser diffraction.
Figure 3. Test apparatus for a continuous stirred tank reactor.
Figure 2). The mass of the ith fraction of the feed with a residence time longer than t is equal to
M)
∫t∞nimiE(t) dt
(38)
The mass, mi, can be expressed with eq 36 and ni with eq 32. If t is substituted with d, using eq 28, the remaining mass of the ith fraction of the feed with a diameter less than dj, Wi,j, can be calculated by the following equation:
Wi,j ) -M0,i
1 F ∫d0 2M ˜ J(d
( )
d d 0,i 0,i)
j
3-R1
e-t(d)/θ dd
θ
(39)
Adding the contribution from all the feed fractions, the total mass of the jth fraction of the slurry is obtained: ∞
Mj )
(Wi,j - Wi,j-1) ∑ i)1
(40)
The particle size distribution of the slurry is calculated as follows:
P(Y)dj ) P(Y)dj-1 + Mj/Mtot
(41)
Experimental Section A continuous stirred tank reactor, CSTR, was used to simulate a wet FGD as shown in Figure 3. A limestone slurry with a solids concentration of 10 wt % was fed to the reaction tank from a slurry feed tank. The CSTR had a fixed volume of 6.5 L and by varying the limestone slurry feed rate from 10 to 40 mL/min, solids residence times ranging from 2 to 8 h were simulated. pH-stat equipment, TITRINO 718, was used to maintain the pH value of the reaction tank at a constant value by adding 18 M H2SO4. The added H2SO4 reacted with the dissolved limestone, forming gypsum, CaSO4‚2H2O, which precipitated in the CSTR. The content of the CSTR was mixed by powerful agitation. The contact area between the air and the liquid in the CSTR formed by the vortex from the agitation was assumed to be sufficient to maintain the
liquid carbonate concentration at a constant low value by stripping. Each experiment was run at least 3 times the mean residence time, θ, in order to obtain steady state before any analysis was made. The temperature was kept at 300 ( 2 K. The pH electrode was calibrated before each experiment, and the pH was controlled within (0.04 pH units from the set point throughout the experiments. A computer, connected to the pH stat, recorded the time, the pH value, the volume of added acid, and the flow rate of acid. By the end of each experiment, a slurry sample was taken from the reaction tank for solids analysis. The solids concentration was measured by gravimetric analysis. The content of limestone was analyzed by dissolving a dry slurry sample in excess hydrochloric acid followed by backtitration with sodium hydroxide. The dissolution rate of the slurry was measured by the flow rate of added acid. A limestone from France, with three different particle sizes, was used in the tests: limestone A, d50 ) 22.3 µm and d90 ) 46.6 µm; limestone B, d50 ) 7.1 µm and d90 ) 37.2; limestone C, d50 ) 6.0 µm and d90 ) 18.1. The limestone purity was g98% CaCO3 with e0.5% MgCO3 and e0.2% acid insoluble. Figure 4 shows the cumulative distribution curves of the limestones measured by using laser diffraction. All the dissolution experiments were operated with zero sulfite concentration and with a chloride concentration of 0.2 mol/L. The chloride was added as CaCl2 with the limestone feed. The presence of sulfite in wet FGD systems has a very strong impact on the limestone dissolution rate with respect to surface kinetics, aging, and mass transfer. Due to difficulties in maintaining a constant level throughout the experiments and since the main of objective of this work was to study the impact of limestone particle size distribution, it was decided not to add sulfite to the reaction tank. Results Modeling the Impact of the Correction Factor. The R value indicates how the flux from the limestone changes with the fraction dissolved according to eq 20. An R value equal to zero means that the flux is independent of the change in particle diameter, whereas an R value > 0 corresponds to decreasing flux with decreasing particle diameter and an R value < 0 corresponds to increasing flux with decreasing particle diameter. The corresponding values between limestone dissolution rate, limestone conversion and reaction tank residence time have been calculated for R values equal to -0.5, 0, and 0.5. The calculations have been made for one coarse limestone, d50 ) 15 µm and d90 ) 90 µm,
3894 Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997
Figure 5. Calculated limestone dissolution rate and required residence time as a function of limestone conversion: T ) 320 K, pH ) 5.6, sulfite concentration ) 0.5 mM, d50 ) 8 µm, and d90 ) 25 µm.
Figure 7. Measured and modeled values of the dissolution rate as a function of limestone conversion: T ) 300 K and pH ) 5.8.
tank. In this work, however, no experimental studies of different values of the correction factor have been performed. When modeling the CSTR experiments, it was decided to assume the R value to be zero, primarily since the experiments were performed without any addition of sulfite. This assumption is in accordance with the results by Jarvis et al. (1988), who found the flux to be constant in the absence of sulfite. The integrals in eqs 33, 37, and 39 can thereby be solved analytically:
rd )
6M0,tot FQ
Figure 6. Calculated limestone dissolution rate and required residence time as a function of limestone conversion, T ) 320 K, pH ) 5.6, sulfite concentration ) 0.5 mM, d50 ) 15 µm, and d90 ) 44 µm.
and for one fine-ground limestone, d50 ) 8 µm and d90 ) 25 µm. The corresponding values of the limestone dissolution rate, solids residence time, and limestone conversion are plotted in Figures 5 and 6. The calculations were made for T ) 320 K, pH ) 5.6, sulfite concentration ) 0.5 mM, and a carbonate level corresponding to a CO2 concentration in the gas phase of 7%. Figures 5 and 6 show how the dissolution rate varies with the conversion of the limestone and how the conversion of limestone varies with the solids residence time, i.e., the size of the reaction tank. The dissolution rate decreases slightly with increasing conversion. This can be explained by the change in limestone particle size distribution with conversion. As the conversion increases, the particle size distribution is shifted toward larger diameters since the small particles dissolve first. Increasing the R value from -0.5 to +0.5 leads to decreasing limestone conversion. With the fine-ground limestone (Figure 5), the limestone conversion has been calculated to 95% at a solids residence time of 6 h (R value equal to -0.5). If the R value is increased to 0.5, the solids residence time needs to be increased up to 8 h to obtain the same limestone conversion, i.e., a 30% increase in the reaction tank volume. Changing to the coarser limestone, the dissolution rate decreases by about 50%, which is why the limestone conversion decreases from 95% down to 91% at a solids residence time of 6 h (R ) -0.5) or 8 h (R ) 0.5). The calculation results indicate that the change in flux with fraction dissolved could have a significant impact on limestone conversion within the reaction
n
f)1-
[
n
[ (
(P(Y)d ∑ i)1
- P(Y)d0,i-1) 0,i
() ( ()
- P(Y)d0,i-1) 1 - 3 0,i 6
( )
d0,i
×
)]
(42)
)]
(43)
κiθ 2 κiθ 1-2 +2 (1 - e-d0,i/κiθ) d0,i d0,i
(P(Y)d ∑ i)1
Wi,j ) M0,i
J(d0,i)
[
κiθ
d0,i
κiθ + d0,i
(e-d0,i/κiθ - 1)
(( ) ( ) ( ) )]
κiθ 3 -d0,i/κiθ dj e 6 + e-dj/κiθ d0,i κiθ 6
3
-3
dj κiθ
dj -6 κiθ
2
+ (44)
where
κi )
2M ˜ J(d0,i) F
(45)
The flux, J(d0), has been calculated under the assumption of steady-state conditions between limestone dissolution and desorption of CO2 from the liquid to the gas. Equations 42-44 have been solved by using the calculated flux expression and the measured particle size distribution of the unreacted limestones: A, B, C. Limestone Dissolution Rate. The dissolution rate as a function of limestone conversion has been measured within the continuous reactor for limestones A, B, and C, where A represents a coarser limestone and C a finely ground limestone with a very narrow particle size distribution (Figure 4). The pH has been equal to 5.8, and the solids residence time was varied from 2.2 to 8 h. The measured dissolution rates are given in Figure 7 and Table 1. The modeled limestone dissolution rates correspond very well to the measured values both with respect to the trend of decreasing dissolution rate with
Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 3895
Figure 8. Measured and modeled values of limestone conversion as a function of the residence time: T ) 300 K and pH ) 5.8.
Figure 10. Calculated particle size distribution of limestone A at different limestone conversions in a continuous system.
Figure 9. . Modeled values of conversion of limestone C versus the measured values: T ) 300 K and θ ) 2.2-8 h. Table 1. Experimental Data limestone
pH
residence time (h)
A
5.8 5.8 5.8 5.8 5.8 5.8 5.8 5.8 5.4 5.4 5.6 5.6 5.8 5.8 5.8 5.8 6.0 6.0 6.0
2.3 3.4 4.8 6.9 2.2 3.5 5.1 7.2 2.3 3.8 2.2 4.0 2.2 3.7 5.2 7.4 2.3 3.9 5.8
B
C
conversion (%)
dissolution rate (µmol/g‚s)
49 58 64 64 61 70 76 84 87 94 70 87 53 82 90 93 47 67 77
1.2 × 10-3 1.2 × 10-3 7.0 × 10-4 6.1 × 10-4 2.0 × 10-3 1.9 × 10-3 1.7 × 10-3 2.0 × 10-3
1.5 × 10-3 3.6 × 10-3 5.1 × 10-3 4.4 × 10-3
increasing conversion and to the absolute values, except for the low conversion data point for limestone C. However, it is obvious that this data point is an outlier since the dissolution rate is lower than that for the coarser B limestone. The dissolution rates of limestone A were found to be about 50% and 25% of the dissolution rates of limestones B and C, respectively. Limestone Conversion. Figure 8 shows the corresponding values between limestone conversion and residence time for limestones A, B, and C at a pH value of 5.8. The agreement between the model and the measured values is very good, except for the outlier of limestone C. The limestone conversion increases with increasing residence time, and, as anticipated, as the conversion approaches unity, the rate of conversion decreases. Figure 9 shows the correlation between measured and calculated limestone conversions for limestone C at pH
Figure 11. Calculated particle size distribution of limestone C at different limestone conversions in a continuous system.
values ranging from 5.4 to 6.0 and residence times between 2.2 and 8 h. The agreement is good, especially for limestone conversions exceeding 70%. Limestone Particle Size Distribution. Figures 10 and 11 show the calculated particle size distribution of limestones A and C in a continuous system as a function of the limestone conversion. As the conversion increases, the particle size distribution is shifted toward larger diameters since the small particles dissolve first. The particle size distribution within a continuous system is consequently determined by the large end of the cumulative distribution curve. No measurements of the particle size of the unreacted limestone were done, but the modeled result corresponds very well to the distributions reported by Ukawa et al. (1993). To verify eq 44 measurements of the particle size distribution of a slurry (gypsum, limestone, and inerts), a full-size wet FGD plant was made. The plant uses hydrocyclones as a primary dewatering step and for separation of limestone from gypsum particles. Most of the limestone particles are recirculated back to the reaction tank. Since the mass flow rate of the recirculated limestone is very small compared to the limestone feed rate, the tank could still be considered as a CSTR. The limestone dissolution model has been used to estimate the particle size distribution of the unreacted limestone particles of the slurry. The particle size distribution of gypsum and inerts has been calculated as the difference between the measured distribution and the calculated distribution. By changing the setup of the hydrocyclones, different cuts between the particles distributed in the overflow and underflow were achieved. By using the calculated distribution curves of gypsum and limestone together with the solids separation data from the hydrocyclone, the distribution of limestone and gypsum between the overflow and the underflow could
3896 Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997
and the possibilities for optimization of the limestonebased flue gas desulfurization process. Acknowledgment The authors are grateful to Mr. Karl Christiansson for valuable help during the work and to Dr. Mats Wallin for useful advice. This work was financially supported by ABB and A° FORSK. Notation Figure 12. Predicted and measured carbonate content of the overflow and underflow from a hydrocyclone installed at a fullsize wet FGD plant.
be calculated. The agreement between modeled and analyzed carbonate content was found to be very good (Figure 12). Hence, eq 44 can be used to accurately predict the particle size distribution of the unreacted limestone in the slurry. Conclusions Prediction of the limestone dissolution rate is important for the design and operation of wet FGD plants. The limestone dissolution rate affects the relationship between liquid and solid alkalinity of the reaction tank as well as the mass-transfer rate of SO2 within the spray zone, both having a significant impact on capital and operating costs of a wet FGD unit. A model has been developed to predict the dissolution rate of a limestone slurry as a function of particle size distribution and limestone conversion. The model includes a factor allowing the flux to vary with the fraction dissolved. This factor accounts for effects such as aging, blinding, accumulation of inerts at the solid/liquid interface, changes of active area, etc. A sensitivity analysis in which the dissolution rate was calculated for a correction factor from -0.5 to +0.5 shows that the decrease in flux that, for instance, is reported for sulfites may have a significant impact on the slurry conditions within the reaction tank. This requires a lower pH value of the slurry in order to maintain a high level of gypsum purity. In the absence of sulfite, the dissolution rate has been found to be accurately modeled assuming the correction factor to be equal to zero. The modeling results have been compared to the limestone conversion and dissolution rate measured in a CSTR. The model was able to accurately predict the impact of both particle size distribution and reaction tank residence time on limestone conversion and dissolution rate. According to mass-transfer theory, the flux from a particle should increase with the fraction dissolved due to changes in the film thickness with shrinking particle size. The good agreement between the model when assuming constant flux, on the one hand, and the experimental results, on the other hand, therefore indicates that the limestone particles are affected to some extent also in a sulfite-free environment. The decrease in flux could, for example, be due to changes in the effective area or accumulation of inerts at the liquid/solid interface. Another reason for the good fit with the correction factor assumed to 0 could be that only dissolution of the large particles, for which the flux is independent of particle size, is of importance. The factors impacting the flux should be further studied to increase the understanding
A ) area of one particle, m2 b ) distance from center of the limestone particle, m C ) molar concentration, mol/m3 d100 ) constant in the cumulative distribution curve (eq 23), m d ) limestone particle size, m D ) diffusivity, m2/s E(t) ) residence time distribution f ) limestone conversion J ) flux, mol/m2‚s k ) constant (eq 12) kc ) surface rate constant, kmol5/2/(m13/2‚s) m ) mass of one limestone particle, kg Mj ) mass of the jth fraction of the slurry with diameter between dj and dj-1, kg/s M0,i ) mass of the ith fraction of the feed with diameter between di and di-1, kg/s M0,i′ ) remaining mass of the ith fraction of the feed with diameter between di and di-1, kg/s M0,tot ) amount of limestone added to the reaction tank, kg/s Mtot ) amount of limestone leaving the reaction tank in the waste, kg/s M ˜ ) molar weight, kg/mol n ) number of particles P(Y) ) cumulative distribution curve Q ) volumetric flow (see Figure 1), m3/s r ) reaction rate, mol/m3‚s rd ) limestone dissolution rate, mol/m3‚s Re ) Reynolds number Sc ) Schmidt number Sh ) Sherwood number t ) time, s Wi,j ) remaining mass of the ith fraction of the feed with a diameter less than dj, kg Y ) help variable (eq 23) z ) charge Greek Letters R ) correction factor (eq 20) β ) constant in the cumulative distribution curve δ ) film thickness, m λ ) constant in the cumulative distribution curve θ ) mean residence time, s F ) density, kg/m3 ξ ) integration constant, (eqs 12-17), mol/s κ ) help variable (eq 46) ∇ ) first spatial derivative Subscripts 0 ) corresponding to the limestone feed b ) liquid bulk eq ) equilibrium i ) ith fraction (feed), diameter between di-1 and di j ) jth fraction (slurry), diameter between dj-1 and dj k ) species k L ) liquid (slurry) s ) surface
Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 3897 Superscripts ° ) ion pairs +/- ) charge
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Jarvis, J. B.; Meserole, F. B.; Selm, T. B.; Rochelle, G. T.; Gage, C. L.; Moser, R.E. Development of a predictive model for limestone dissolution in wet FGD systems. EPA/EPRI combined FGD and dry SO2 control symposium, St. Louis, Oct 1988. Plummer, L. N.; Wigley, T. M. L. The dissolution of calcite in CO2saturated solutions at 25 °C and 1 atmosphere total pressure. Geochim. Cosmochim. Acta 1976, 40, 191. Rochelle, G. T. Comments on SO2 absorption into aqueous systems. Chem. Eng. Sci. 1992, 47, 3169. Sjo¨berg, E. L.; Rickard, D. T. Calcite dissolution kinetics: surface speciation and the origin of the variable pH dependence. Chem. Geol. 1984, 42, 119. Toprac, A. J.; Rochelle, G. T. Limestone dissolution in stack gas desulfurization. Environ. Prog. 1982, 1, 52 Ukawa, N.; Takashina, T.; Shinoda, N. Effects of particle size distribution on limestone dissolution in wet FGD process applications. Environ. Prog. 1993, 12, 238. Wallin, M.; Bjerle, I. A mass transfer model for limestone dissolution from a rotating cylinder. Chem. Eng. Sci. 1989, 44, 61.
Received for review January 2, 1997 Revised manuscript received May 19, 1997 Accepted May 21, 1997X IE970030J
X Abstract published in Advance ACS Abstracts, July 15, 1997.
