Ind. Eng. Chem. Res. 2002, 41, 5187-5195
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Gas Absorption with Instantaneous Irreversible Chemical Reaction in a Slurry Containing Sparingly Soluble Fine Reactant Particles Fabrizio Scala* Istituto di Ricerche sulla Combustione, CNR, Piazzale Tecchio, 80-80125 Napoli, Italy
The problem of gas absorption followed by instantaneous irreversible chemical reaction in a slurry containing sparingly soluble fine reactant particles is considered. Models are developed for an infinitely deep quiescent liquid and, using the film theory, for an agitated liquid when the reactant particle dissolution in the liquid film next to the gas-liquid interface is nonnegligible. Simple analytical expressions for the rate of gas absorption are derived under the assumption of slow particle dissolution, and conditions are given for applicability of the models. A numerical example for the system SO2-Ca(OH)2 in water is discussed. Introduction Gas absorption into a slurry containing finely suspended reactant particles in which chemical reaction takes place between the dissolved gas and the solute dissolving from the sparingly soluble particles is frequently encountered in the chemical industry. In particular, the case of instantaneous irreversible chemical reaction between the dissolved reactants has a remarkable practical relevance, for example, in gas purification processes. Slurry operation is advantageous because the suspended solid particles provide a continuous supply of reactant to the liquid, thus enhancing the gas absorption rate. Ramachandran and Sharma1 first theoretically discussed this problem and proposed two film models, one for the case of insignificant solid dissolution in the liquid film and one for the case where solid dissolution in the liquid film is important. Analytical solutions were derived, and the results showed that the gas absorption rate could be increased considerably compared to that obtained in the absence of solids. The dissolution of the fine particles near the gas-liquid interface modifies the reactant diffusional gradients and increases the absorption rate. Later, Uchida and co-workers extended the analysis to account for the enhancement of solid dissolution in the liquid film due to the chemical reaction2 and considered the case when the concentration of the reactant dissolving from the particles in the bulk liquid phase is lower than the saturation solubility (slow dissolution rate compared to gas absorption rate).3 Uchida and Wen4 classified the problem of gas absorption into a slurry into six cases according to the reactant concentration profiles established in the liquid phase. For each case, a model based on film theory was given, and an analytical solution was provided. The solutions, however, are not straightforward to use because of their implicit nature, and a trial-and-error iterative procedure is needed. In accordance with their own experimental results, Sada et al.5 further modified the theory to account for the presence of an inert region near the gasliquid interface in which there are no particles. Yagi and Hikita6 criticized all of the previous models, pointing out that a parameter representing the average inter* Tel.: +39 081 7682969. Fax: +39 081 5936936. E-mail:
[email protected].
particle spacing is necessary to characterize the system and that the use of a mass-transfer coefficient for particle dissolution is inappropriate. The qualitative heterogeneous two-dimensional model that they developed, however, yielded unreliable results.7 Recently, Mehra7 examined the process of particle dissolution near the gas-liquid interface and pointed out that a change in the particle size should be taken into account when the particles are small enough to dissolve significantly. He also criticized the approach of Sada et al.,5 indicating that there is no physical reason that a particle-free zone should exist near the gas-liquid interface. The author proposed a penetration model based on a transient population balance near the gasliquid interface that was solved numerically for some representative cases. In this work, the problem of gas absorption followed by instantaneous irreversible chemical reaction in a slurry containing sparingly soluble fine reactant particles is considered under the assumption of slow reactant particle dissolution. This assumption implies that the shrinkage of particles is negligible, so that a constant particle size can be assumed. Models are developed for an infinitely deep quiescent liquid and, using the film theory, for an agitated liquid when the reactant particle dissolution in the liquid film next to the gas-liquid interface is nonnegligible. Solid dissolution is treated in a rigorous way, with the average spacing between the particles being taken into account. Simple analytical expressions are provided for the gas absorption rate. Theory The problem can be represented by the following scheme
A(g) f A(aq) B(s) f B(aq) A(aq) + B(aq) f products
(1)
The theoretical treatment is based on the following simplifying assumptions: (1) Both gaseous and solid reactants have a low solubility in the liquid. (2) The reaction between the two species is irreversible and instantaneous. (3) The influence of the product species
10.1021/ie020110+ CCC: $22.00 © 2002 American Chemical Society Published on Web 09/13/2002
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Figure 1. Schematic representation of gas absorption with instantaneous reaction in a slurry of fine particles. Case A, infinitely deep quiescent liquid.
on the process is negligible. (4) The solid particle dissolution is slow, so that particles shrinkage can be neglected. (5) The solid particles are spherical, uniform in size, and uniformly dispersed in the liquid phase and do not agglomerate. (6) Solid particles are “fine”, i.e., their size is smaller than the typical reactant diffusional length scale. (7) There is no surface kinetic resistance to particle dissolution, which is entirely controlled by mass transfer. (8) The solid particles are stationary (nondiffusing), and the liquid around the particles can be considered stagnant, as discussed by Mehra.7 (9) The heats of reaction and dissolution are small and can be neglected, and the system is isothermal. (10) Interfacial effects are negligible. Two cases have been taken into account, namely, an infinitely deep quiescent liquid and an agitated liquid. The gas-liquid interface is planar, and the concentrations of diffusants are uniform over any plane parallel to the interface. Reactants diffuse only in the direction perpendicular to the interface. A. Infinitely Deep Quiescent Liquid. The system is schematized in Figure 1. A sharp “macroscopic” reaction plane parallel to the gas-liquid interface is established at steady state at a distance λ from the interface, where the reactant concentrations fall to zero. In the zone between the interface and the reaction front (0 < x < λ) in the liquid phase, only reactant A exists. Because of the presence of the dissolving solid particles, a small “microscopic” spherical reaction front is established around each particle in this zone, as shown in Figure 2a. Beyond the macroscopic reaction front (x > λ) in the liquid phase, only reactant B exists. Around each dissolving particle in this zone, a gradient of reactant B is established, as shown in Figure 2b. If rP is the solid particle radius and P is the solids volume fraction in the slurry, the number of particles per unit volume of slurry is
N)
3P 4πrP3
Figure 2. Schematic representation of reactant concentration profiles around one fine particle: (a) particles between the gasliquid interface and the macroscopic reaction front, (b) particles beyond the macroscopic reaction front.
of slurry equal to 1/N in its cell. The radius of each cell is
rC )
rP
(3)
3
xP
Let us now consider the particles present in the zone between the gas-liquid interface and the macroscopic reaction front (0 < x < λ) (Figure 2a). At the surface of the particles, the concentration of B (given by B) is at equilibrium with the solid (saturation solubility), namely, BS. On the other hand, at the cell external surface, the concentration of reactant A (given by A) is that in the liquid phase, namely, AL. Somewhere between the particle surface and the cell surface (rP < r < rC), a microscopic reaction front is established, at a distance rλ from the particle center, where the reactant concentrations falls to zero. This reaction front divides the spherical cell into two zones: a zone near the particle surface where only reactant B is present (rP < r < rλ) and a zone near the cell external surface where only reactant A is present (rλ < r < rC). The material balances in the two zones give
( ) ( )
{ {
d 2 dB r )0 dr dr d 2 dA r )0 dr dr
r ) rP w B ) BS r ) rλ w B ) 0
(4)
r ) rλ w A ) 0 r ) rC w A ) AL
(5)
The solutions of the two equations are
( (
) )
B ) BS
rλ/r - 1 rλ/rP - 1
(6)
A ) AL
1 - rλ/r 1 - rλ/rC
(7)
(2)
It is supposed that the unit slurry volume can be divided into N nonoverlapping cells (assumed spherical) with a solid particle at their center. Each particle has a volume
respectively. At the reaction front, the fluxes of the two reactants must equal each other
DB
|
|
dB dA ) -DA dr rλ dr rλ
(8)
Ind. Eng. Chem. Res., Vol. 41, No. 21, 2002 5189
Equation 8 allows for the calculation of rλ
can be carried out with reference to Figure 1 (using eqs 11 and 16)
[ ( )]
DBBS DA A L r λ ) rP rP DBBS 1+ rC DAAL 1+
(9)
The flux of reactant A per unit external surface of the cell is
-DA
|
DAAL + DBBS dA ) dr rC rC(rC/rP - 1)
DAAL + DBBS rC2(rC/rP - 1)
(1 - P)DB
d2AL dx2
1 rC
) (1 - P)R2(DAAL + DBBS)
) RB ) -(1 - P)R2DB(BS - BL)
{
x ) λ w BL ) 0 x f ∞ w BL ) BS (18)
(11)
ϑA ) 1 +
DAAL DB B S
(19)
ϑB ) (BS - BL)/BS
x
x
3 1 ) (1 - P)(rC/rP - 1) rP
3P 3
d2ϑA
(12)
Let us now shift attention to the particles suspended in the zone beyond the macroscopic reaction front (x > λ) (Figure 2b). In this zone, no reactant A is present. At the surface of the particles, the concentration of reactant B is BS, whereas at the cell external surface, it is equal to that in the liquid phase, namely, BL. The material balance for reactant B in the cell gives
( )
d 2 dB r )0 dr dr
{
r ) rP w B ) BS r ) rC w B ) BL
) (
rC/r - 1 1 - rP/r + BL rC/rP - 1 1 - rP/rC
)
|
d2ϑB 2
dx
{ {
) R2ϑA ) R2ϑB
x ) 0 w ϑA ) ϑ* x ) λ w ϑA ) 1
(21)
x ) λ w ϑB ) 1 x f ∞ w ϑB ) 0
(22)
where
(13)
DAA* DB B S
(23)
The last boundary condition can be equivalently written as x f ∞ w dϑB/dx ) 0. Noting that the boundary condition at the reaction front
(14)
The flux of reactant B per unit external surface of the cell is
DB(BS - BL) dB )-DB dr rC rC(rC/rP - 1)
2
ϑ* ) 1 +
The solution is
(
(20)
Equations 17 and 18 then become
(1 - P)(1 - xP)
dx
B ) BS
x ) 0 w AL ) A* x ) λ w AL ) 0 (17)
where the term (1 - P) on the LHS accounts for the reduced volume for diffusion due to the presence of the solid particles. The last boundary condition can be equivalently written as x f ∞ w dBL/dx ) 0. Let us make the following change of variables
where (using eq 3)
R)
{
d2BL dx2
) RA ) (1 - P)R2(DAAL + DBBS)
(10)
The consumption of reactant A per unit volume of slurry is given by
RA ) 3
(1 - P)DA
(15)
The generation of reactant B per unit volume of slurry is given by
DB(BS - BL) ) -(1 - P)R2DB(BS - BL) RB ) -3 2 rC (rC/rP - 1) (16) At this point, the material balances in the liquid phase before and after the macroscopic reaction front
DB
|
|
dBL dAL ) -DA dx λ dx
λ
(24)
implies the identity
|
|
dϑA dϑB ) dx λ dx
λ
(25)
the problem can be simplified by solving the single equation
{
d 2ϑ ) R2ϑ 2 dx
x ) 0 w ϑ ) ϑ* xf∞wϑ)0
(26)
The solution of eq 26 is
ϑ ) ϑ*e-Rx
(27)
where ϑ ) ϑA for 1 < ϑ < ϑ* and ϑ ) ϑB for 0 < ϑ < 1.
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Figure 3. Schematic representation of gas absorption with instantaneous reaction in a slurry of fine particles. Case B, agitated liquid.
Location of the reaction front can be found by imposing the condition ϑ ) 1
λ)
ln ϑ* R
in the liquid film (of thickness δ) at a distance of λ from the interface. In the zone between the interface and the reaction front (0 < x < λ) in the liquid phase, only reactant A exists. In this zone, a small microscopic spherical reaction front is established around each dissolving particle, as shown in Figure 2a. Beyond the macroscopic reaction front (λ < x < δ) in the liquid phase, only reactant B exists. Around each dissolving particle in this zone, a gradient of reactant B is established, as shown in Figure 2b. The theoretical treatment for the suspended particle dissolution rate is exactly the same as that described in the previous section (eqs 2-16) and will not be repeated here. Equations 17 and 18 still hold with only a change in the last boundary condition that now reads x ) δ w BL ) B∞, where B∞ is the bulk liquid concentration of reactant B. The bulk liquid is supposed to be perfectly mixed and to have negligibly small inlet and outlet streams. After the change of variables reported in eqs 19 and 20, the subsequent treatment (eqs 2126) still holds with the new boundary condition x ) δ w ϑ ) ϑB ) ϑ∞, where ϑ∞ ) (BS - B∞)/BS. Solution of eq 26 gives
(28) ϑ)
ϑ∞ sinh(Rx) + ϑ* sinh[R(δ - x)] sinh(Rδ)
The gas absorption rate per unit interface area is
J/A ) -(1 - P)DA
|
dAL ) (1 - P)R(DAA* + DBBS) ) dx 0 DBBS (1 - P)RDAA* 1 + (29) DAA*
(
)
Equation 29 is very easy to handle; for example, if Henry’s law is assumed to hold at the interface (p* ) HA*), then eq 29 can be combined with the gas-phase diffusion resistance to give
J/A
)
DApG/H + DBBS 1/(1 - P)R + DA/(kGH)
As before, location of the reaction front can be found by imposing the condition ϑ ) 1 when x ) λ in eq 32. The gas absorption rate per unit interface area is
J/A ) -(1 - P)DA
|
dAL dx
0
) (1 - P)R × [(DAA* + DBBS) cosh(Rδ) - DB(BS - B∞)] sinh(Rδ)
(33)
(30)
where pG is the bulk gas-phase partial pressure of reactant A and kG is the gas-side mass-transfer coefficient. The bulk gas is assumed to be perfectly mixed. It is interesting to study the asymptotic behaviors: if pG e(1 - P)RDBBS/kG or if H f ∞ the surface concentration of reactant A goes to zero, the reaction front shifts to the gas-liquid interface (λ ) 0), and the absorption rate is entirely controlled by the gas-phase resistance (reactant B is assumed to be nonvolatile), i.e.
J/A ) kGpG
(32)
(31)
This would be the case also when rP f 0, but in this case, the initial assumption that solid particle dissolution is slow would not be valid and particles shrinkage can not be neglected. On the other hand, for a very dilute suspension (P f 0), no steady-state solution can be found (λ f ∞). The physical reason is that, in this case, solid particles are not able to provide a continuous supply of reactant B to the liquid phase. B. Agitated Liquid. The system is schematized in Figure 3 according to film theory. As in the previous case, a sharp macroscopic reaction plane parallel to the gas-liquid interface is established at the steady state
To use eq 33, the bulk liquid concentration B∞ must be expressed as a function of the other variables. This can be done by equating the flux of reactant B through the film (at x ) δ) to the generation of reactant B from particle dissolution in the liquid bulk, that is (using eq 16)
JδB ) -(1 - P)DB
|
dBL dx
δ
) (1 - P)R × DB(BS - B∞) - (DAA* + DBBS) cosh(Rδ) sinh(Rδ) )
RB∞ a
2
R DB(BS - B∞) a
) -(1 - P)
(34)
where a is the gas-liquid interfacial area per unit volume of liquid. Solution of eq 34 gives
(
B∞ ) BS - BS +
)
DA a A* (35) DB R sinh(Rδ) + a cosh(Rδ)
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Substituting eq 35 into eq 33 and solving gives
J/A ) (1 - P)β(DAA* + DBBS) )
(
Equation 34 now reads (using eq 11)
)
DBBS (36) (1 - P)βDAA* 1 + DAA* where
[
β)R
]
JδA ) -(1 - P)DA
) (1 - P)R × [(DAA* + DBBS) - (DAA∞ + DBBS) cosh(Rδ)]
(37) )
R A∞ a
From eq 36, the gas absorption enhancement factor can be derived as
(
)
DApG/H + DBBS
(38)
(39)
1/(1 - P)β + DA/(kGH)
It is interesting to note that eqs 36 and 39 are equal to eqs 29 and 30 if R is replaced by β. The asymptotic behaviors are similar to those studied in the previous section: If H f ∞ or if the following condition is satisfied
pG e (1 - P)βDBBS/kG
(40)
the reaction front shifts to the gas-liquid interface (λ ) 0), and the absorption rate is given by eq 31. The same holds if rP f 0, but in this case, the initial assumption that solid particle dissolution is slow is not valid, and particle shrinkage can not be neglected. On the other hand, if the following condition is satisfied
pG g
[
]
DBBS γHkG (1 - P)β(γ - 1) + kG DA
(41)
where
γ ) cosh(Rδ) - 1 +
R sinh(Rδ) a
(42)
then reactant A is able to reach the liquid bulk phase, and no reactant B is present in the liquid film. In eq 41, the equality represents the case when the reaction front distance from the interface coincides with the film thickness (λ ) δ). If pG is larger than the quantity on the RHS of eq 41, then the problem must be slightly reformulated. In this case, in fact, in the second boundary condition, ϑ∞ ) 1 + (DAA∞/DBBS), where A∞ is the bulk liquid concentration of reactant A. The gas absorption rate per unit interface area is
J/A ) -(1 - P)DA
|
dAL dx
0
) (1 - P)R × [(DAA* + DBBS) cosh(Rδ) - (DAA∞ + DBBS)] sinh(Rδ)
R2(DAA∞ + DBBS) a
) (1 - P)
(44)
Solution of eq 44 gives
where kL is the liquid-side mass-transfer coefficient. As for the previous case, if Henry’s law is assumed to hold at the interface, eq 36 can be combined with the gasphase diffusion resistance to give
J/A )
δ
sinh(Rδ)
R + a tanh(Rδ) R tanh(Rδ) + a
J/A DBBS E/A ) ) (1 - P)βδ 1 + kLA* DAA*
|
dAL dx
(43)
A∞ )
(
)
DB DB a BS + A* B DA R sinh(Rδ) + a cosh(Rδ) DA S (45)
Substituting eq 45 into eq 43 and solving gives
J/A ) (1 - P)β(DAA* + DBBS) )
(
(1 - P)βDAA* 1 +
)
DBBS (46) DAA*
Equation 46 is identical to eq 36, with β given by eq 37. This indicates that eqs 36 and 39 are able to handle all possible cases provided that the condition in eq 40 is not satisfied. Finally, it is interesting to note that, in the case of a very dilute suspension (P f 0) and for B∞ ) BS (that is, when a f 0), eq 36 becomes the familiar expression for gas absorption accompanied by instantaneous reaction when solid dissolution is unimportant, namely
J/A )
(
)
(
DA DBBS DBBS ) kLA* 1 + A* 1 + δ DAA* DAA*
)
(47)
Comparing eq 47 with eq 36, it can be noted that the enhancement to the gas absorption rate due to the presence of the dissolving solid particles is given by (1 - P)βδ. C. Conditions for Applicability of the Models. Let us now discuss the limits of applicability of the agitated liquid model as regards the size of the suspended reactant particles. The maximum size is imposed by the initial assumption that the particle size should be smaller than the typical reactant diffusional length scale, that is
rP < δ )
DA kL
(48)
On the other hand, the minimum size is determined by the initial assumption that particle dissolution is slow, so that particle shrinkage can be neglected. One way to estimate this minimum size is the following. The maximum rate of dissolution of one particle is calculated
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from eq 10 and is given by
RP,max ) -4πrC2DA
|
DAA* + DBBS dA ) 4π dr rC,max (1/rP - 1/rC)
(49)
The particle volume change per unit time is given by
∆VP M M DAA* + DBBS ) RP,max ) 4π ∆t FP FP (1/rP - 1/rC)
(50)
where M is the molecular weight and FP is the particle density. Let us assume that the fractional volume change is lower than a certain value, i.e., ∆VP/VP < ξ. This implies (using eq 3)
(
) (
3
FP rP3(1/rP - 1/rC) FP rP2(1 - xP) ∆t < ξ )ξ 3M DAA* + DBBS 3M DAA* + DBBS
)
(51) The average residence time of any particle in the film (or, equivalently, near the gas-liquid interface) can be estimated by means of the surface renewal theory as ∆t ≈ DA/kL2. Combining this result with eq 51 and rearranging, the condition that the volume change of the particle due to dissolution be low during its residence time near the gas-liquid interface can be given as
rP >
x (
)
3MDA DAA* + DBBS
ξFPkL2
3
1 - xP
(52)
Equation 52 sets the minimum particle size for applicability of the model. As concerns the quiescent liquid model, applicability is determined by eq 51, that is, the model is valid only for short times, when solid particles near the gas-liquid interface do not dissolve significantly. This is because, in this case, the liquid phase near the interface is still and is not renewed as in the case for agitated liquid. One last consideration should be added. Along the lines discussed by Yagi and Hikita,7 one could raise the objection that the liquid film around the solid particles and the liquid film for gas absorption overlap, causing a distortion of the concentration profiles. Whereas the net effect of the solid particles on the process is very low for very low solids volume fractions, for large solids volume fractions, eq 9 indicates that the microscopic reaction front practically coincides with the particle surface. This would indicate that the mathematical development reported in this work is likely to be suitable for treating the entire range of solids volume fractions of interest. Numerical Example As a representative case, the absorption of SO2 (reactant A) into an aqueous slurry of Ca(OH)2 (reactant B) at ambient conditions was chosen. The ionic reaction in the liquid phase between the two dissolved reactants can be considered irreversible and instantaneous.8 A stirred-tank reactor was considered. Physical param-
Figure 4. Condition in eq 40 for the particle size range of interest at three different slurry solids volume fractions.
eters values for this system have been estimated as follows
DA ) 1.8 × 10-9 m2/s DB ) 1.6 × 10-9 m2/s BS ) 2.0 × 10-2 kmol/m3 H ) 7.3 × 10-1atm‚m3/kmol M ) 74 kg/kmol FP ) 2.2 × 103 kg/m3 kL ) 2.0 × 10-4 m/s kG ) 1.0 × 10-3 kmol/atm‚s‚m2 a ) 2.0 × 102 m-1 The first step is to establish the conditions for the applicability of the model. According to eqs 48 and 52 (a value of ξ ) 0.1 was used), the model is applicable for approximately 1 µm < rP < 10 µm. This is the typical range of particle sizes used in practice, so the model has physical relevance. Figure 4 shows the condition in eq 40 for the particle size range of interest at three different solids volume fractions in the slurry. The areas below the curves represent the ranges of sulfur dioxide partial pressures in the gas phase for which the gas absorption is completely controlled by the gas-side diffusion resistance, i.e., for which eq 31 holds. It is clearly seen that this would be the case for most flue gas purification processes of practical interest. Analysis of the order of magnitude of the two terms in the denominator of the RHS of eq 39 shows that, even for the case of sulfur dioxide partial pressures above the curves in Figure 4, the gas-side resistance can never be neglected, apart, of course, from the case of pure sulfur dioxide. On the other hand, the condition in eq 41 is satisfied only for sulfur dioxide partial pressures in the gas phase well above 1 atm for all particle sizes and solids volume fractions of interest. Thus, under practical conditions, the macroscopic reaction front never reaches the bulk liquid phase, but always stands in the liquid film near the gas-liquid interface.
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Figure 5. Reactant concentration profiles in the liquid film for three different particle sizes.
Figure 6. Sulfur dioxide absorption rate per unit interfacial area as a function of SO2 bulk gas partial pressure for three different particle sizes.
Figure 5 reports the reactant concentration profiles in the liquid film for three different particle sizes and for a sulfur dioxide partial pressure value above the curves in Figure 4. As expected, the reaction front approaches the gas-liquid interface as the particle size decreases. It is interesting to note that the surface SO2 concentration is quite lower than the bulk gas concentration, confirming the importance of the gas-phase resistance. Moreover, it is noted that the calcium hydroxide concentration at the film edge (and in the liquid bulk) is practically equal to the equilibrium solubility under all conditions. Similar profiles can be calculated for different solids volume fractions and gasphase SO2 partial pressures, provided that the condition in eq 40 is not satisfied. The effect of increasing the solids volume fraction or decreasing the sulfur dioxide partial pressure is to bring the reaction front nearer the gas-liquid interface. Figure 6 reports the gas absorption rate per unit interface area as a function of the sulfur dioxide partial pressure for three different particle sizes. For comparison, the maximum gas absorption rate (i.e., complete gas-side control, eq 31), the physical gas absorption rate (J/A ) kLA* combined with gas-phase resistance), and the chemical gas absorption rate when no solids are present (eq 47 combined with gas-phase resistance) are also reported. Analysis of the figure shows that, whereas at low SO2 partial pressures, the absorption rate is the maximum possible, at high partial pressures, the curves tend toward the physical absorption one. In particular,
Figure 7. Enhancement factors for sulfur dioxide absorption as a function of SO2 bulk gas partial pressure for three different particle sizes. Nonnegligible gas-side resistance.
the larger the particle size, the closer the curve to the physical absorption one. When no solids are present, the absorption rate curve coincides with the physical absorption curve at large partial pressures (the term in brackets in eq 47 tends to 1). Similar figures can be derived for different solids volume fractions: the effect of increasing the solids volume fraction is to enhance the gas absorption rate. Figure 7 reports the “real” enhancement factors for sulfur dioxide absorption (derived from Figure 6) as a function of the sulfur dioxide partial pressure for three different particle sizes. Two enhancement factors are reported, namely, the ratio between the actual absorption rate and the physical absorption rate (E/A) and the ratio between the actual absorption rate and the chemi/ ). cal absorption rate when no solids are present (EA,P Whatever the particle size, at low SO2 partial pressures, E/A ) 1 + kGH/kL, and E/A,P ) 1. This means that, under these conditions, there is no benefit in suspending solid reactant particles, provided that the saturation solubility of calcium hydroxide can be maintained in the liquid bulk phase. The only way to enhance sulfur dioxide absorption is to increase the gas-side mass-transfer coefficient. At high SO2 partial pressures, the two enhancement factors tend to the same asymptotic values, because the physical absorption rate and the chemical absorption rate when no solids are present coincide. In this case, the lower the particle size, the larger the enhancement to sulfur dioxide absorption. It is interesting to consider also the case when gasside resistance can be excluded (for example for absorption of pure sulfur dioxide). Figure 8 reports the two enhancement factors defined before for sulfur dioxide absorption for this special case as a function of the sulfur dioxide interfacial concentration for three different particle sizes. In this case, very large enhancement factors (E/A) can be obtained for low SO2 concentrations (in particular, for A* f 0 w E/A f ∞), whereas at large concentrations E/A tends to asymptotic values. The enhancement with respect to the chemical absorption rate when no solids are present is a constant at each particle size (E/A,P ) (1 - P)βδ), equal to the asymptotic value of E/A at large SO2 concentrations. For this case, regardless of the sulfur dioxide concentration, the lower the particle size, the larger the enhancement to sulfur dioxide absorption.
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Figure 8. Enhancement factors for sulfur dioxide absorption as a function of SO2 bulk gas partial pressure for three different particle sizes. Negligible gas-side resistance.
Conclusions The problem of gas absorption followed by instantaneous irreversible chemical reaction in a slurry containing sparingly soluble fine reactant particles has been considered. This problem has considerable industrial importance, for example in gas purification processes. Two models have been developed for (a) an infinitely deep quiescent liquid and (b) for an agitated liquid (using the film theory) when the reactant particle dissolution next to the gas-liquid interface is nonnegligible. Solid dissolution is treated in a rigorous way that takes into account the average spacing between the particles in the slurry. Simple analytical expressions for the rate of gas absorption and the enhancement factor are derived under the assumption of slow particle dissolution. The range of particle sizes for which the models are applicable is determined. A numerical example for SO2 absorption into an aqueous slurry of Ca(OH)2 at ambient conditions is presented and discussed. The particle size range for model applicability turns out to be that typically used in practice, so that the model has practical relevance. Results show that for most flue gas purification processes the gas absorption would be completely controlled by gas-side diffusion resistance. Even for very high sulfur dioxide partial pressures, the gas-side resistance can never be neglected, except for operation with pure sulfur dioxide. The analysis points out that, at high SO2 partial pressures, the chemical absorption rate tends toward the physical absorption rate. In particular, the smaller the particle size and/or the larger the solids volume fraction, the higher the absorption rate. On the other hand, at low sulfur dioxide partial pressures, there is no benefit in suspending solid reactant particles, provided that the saturation solubility of calcium hydroxide can be maintained in the liquid bulk phase. The only way to enhance sulfur dioxide absorption in these conditions is to increase the gas-side mass-transfer coefficient. Acknowledgment Useful discussions with Mr. M. D’Ascenzo are gratefully acknowledged. Notation A ) liquid-phase concentration of reactant A, kmol/m3
A* ) interfacial liquid-phase concentration of reactant A, kmol/m3 a ) gas-liquid interface area per unit volume of liquid, 1/m B ) liquid concentration of reactant B, kmol/m3 D ) diffusion coefficient, m2/s E* ) gas absorption enhancement factor E/A,P ) gas absorption enhancement factor with respect to no solids H ) Henry’s constant, atm‚m3/kmol J* ) gas absorption rate per unit interface area, kmol/m2‚ s Jδ ) flux of reactant through the film, kmol/m2‚s kG ) gas-side mass-transfer coefficient, kmol/m2‚atm‚s kL ) liquid-side mass-transfer coefficient, m/s M ) molecular weight, kg/kmol N ) number of particles per unit volume of slurry, 1/m3 p ) partial pressure, atm R ) consumption (generation) rate per unit volume of slurry, kmol/m3‚s RP,max ) maximum dissolution rate of one particle, kmol/s r ) radial coordinate in the particle cell, m t ) time, s x ) axial coordinate from the interface, m V ) volume, m3 Greek Letters R ) coefficient defined in eq 12, 1/m β ) coefficient defined in eq 37, 1/m γ ) coefficient defined in eq 42 δ ) film thickness, m ) volume fraction λ ) macroscopic reaction front distance from the interface, m F ) density, kg/m3 ϑ ) variable defined in eqs 19 and 20 ϑ* ) interface value of ϑ defined in eq 23 ξ ) coefficient in eq 51 Subscripts A ) reactant A B ) reactant B C ) cell G ) gas phase L ) liquid phase P ) solid particles S ) saturation λ ) reaction front ∞ ) bulk liquid
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Received for review February 6, 2002 Revised manuscript received July 9, 2002 Accepted July 9, 2002 IE020110+
