# Mathematical Modeling of Wet Magnesia Flue

Authors: M. K. Mondal
Journal: Modelling and Simulation in Engineering
 (2008)

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## Abstract

Desulphurization of flue gases from various
chemical industries in a techno-econo-enviro manner is a demanding
technology. The concentrations of sulphur dioxide in and around
these plants overshoot the danger point. In recent years, the
process analysis of chemical absorption in a slurry has become
important in rational design and development of wet scrubbing
processes for the removal ofSO2 from flue gases. The elementary
steps encountered in wet scrubbing by slurries are diffusion and
reaction of gaseous species and solid dissolution in liquid film.
In the present work, the process of the absorption of sulphur
dioxide into wet magnesia slurry was theoretically analyzed
according to the two-reaction plane model incorporating the solid
dissolution promoted by the reactions with absorbed sulphur
dioxide in the liquid film. A model based on Fick's second law
has been developed to calculate enhancement factor for absorption
of Sulphur dioxide into   Mg(OH)2 slurry. The concentration of
accumulated species in the bulk of the liquid phase (sulphite ions
for this case) which substantially control the absorption rates
was included in the model for the prediction of theoretical
enhancement factor. The values of theoretical enhancement factors
obtained from model were compared with experimental enhancement
factors available in literature. The model values of enhancement
factors agreed well with the values of experimental enhancement
factor available in literature.


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## Body

## 1. Introduction

It has long been
known that besides being a hazard to human health, emission of sulphur dioxide
contributes to the acidification of soil and waterways. Sulphur
dioxide has also been reported to
support the reactions which create ozone depletion in the stratosphere. Many
countries have therefore adapted strict regulations regardingSO2 emissions from coal and oil fired boilers which are one of the primary sources
of SO2 emissions. Many different methods are available in the market
for reducing the emissions of SO2 from coal and oil fired boilers
[1]. The dominating flue gas desulphurization (FGD) technology is based on wet
scrubbing specially slurry scrubbing [2].Slurry reactions
have a widespread application in chemical and biochemical industries [3]. The gas absorption with chemical
reaction in a slurry containing fine particles has become important in the
development of processes for the removal of acidic pollutants. At present, most
wet scrubbing processes for the removal of SO2 are by lime and
limestone slurry. The major disadvantage of lime/limestone slurry processes is
the problem of disposal of calcium sulphate sludge. Wet magnesia FGD system can
eliminate scrubber sludge disposal problem and also provide saleable byproduct.
In India,
most of the sulphur is imported, and hence this process will be very useful to
meet the sulphur need in the country.Several models
available for absorption ofSO2 are based on lime/limestone slurry [4, 5]. Some models are also available in literature regarding the absorption of SO2 in sea water [6, 7]. A little work has been done on the modeling of wet
magnesia FGD slurry. This paper focuses the model for wet magnesia FGD slurry
process in terms of the two- reaction-plane model incorporating the solid
dissolution promoted by the reactions with SO2 in the liquid film.
The objective of this work has been to develop a model of the absorption of SO2 into wet magnesia slurry using an unsteady state mass transfer theory (i.e., Fick’s
second law) and to use the model to quantify the enhancement of the absorption
rate of SO2 into a slurry containing small reactive Mg(OH)2 particles.


## 2. Model Description

### 2.1. Model of Absorption through Reaction Planes

Regarding theSO2 absorption
mechanism into wet magnesia slurry, the following processes can be considered [8]: (1)SO2diffusionthroughthegasfilm(Diffusion)(2)SO2(g)⟶SO2(aq.)(Dissolution)(3)SO2(aq.)+H2O⟶H2SO3(Dissociation)(4)H2SO3⟶H++HSO3−(Dissociation)(5)HSO3−⟶H++SO32−(Dissociation)(6)MgO(s)+H2O⟶Mg(OH)2(aq.)(Dissolution)(7)Mg(OH)2(aq.)⟶Mg2++2OH−(Dissolution)(8)Diffusionofdissolvedchemical(Diffusion)speciesinliquidfilm(9)SO2+2OH−⟶SO32−+H2O(Reaction)(10)SO2+SO32−+H2O⟶2HSO3−(Reaction)(11)HSO3−+OH−⟶SO32−+H2O(Reaction). Initially,
SO2 gas diffuses from bulk of the gas phase to the gas-liquid film, dissolves into
the bulk liquid and dissociates. Simultaneously, solid MgO also dissolves and
dissociates. After that, the sulphite and hydroxyl ions react with each other
in the liquid film according to (9)–(11). The principal
chemical species of the reaction are subjected to acid dissociation; it is
necessary to consider the pH of the absorbent liquid, which largely influences
the degree of dissociation.


### 2.2. Basic Equations

In the process ofSO2 absorption in
magnesia slurry with no suspended particles, HSO3− cannot
coexist with OH−, so that reaction (9) never takes place directly. The
above condition shows that reactions (10) and (11) take place at two different
located planes in the two-reaction-plane model. However, in the slurry process,
to be considered here, both dissolve SO2 and the HSO3−,
to be produced by reaction (10), can react with OH− which is fed by
the dissolution of the solid particles in the liquid film. So in this case, dissolved
SO2 can be consumed by reactions (9) and (10) simultaneously. The
concentration of SO32− in the bulk liquid increases as
the absorption proceeds. For a saturated solution of magnesia, a plausible
sketch of the concentration profile is given in Figure 1(a). When the particles
are suspended in the liquid film, the concentration profiles shift as shown in Figure 1(b)
[9].Concentration profile forSO2/Mg(OH)2 slurry: (a) no suspended solids, (b) in the presence of suspended solids.
(a)(b)The model
considers the presence of 4 basic species:SO2, OH−, SO3−2,
and HSO3−. The basic equations were derived to form the
mass balance of perspective chemical species as shown below. The first terms of
(12)–(15) deal with the
diffusion of the species according to Fick’s second law, while the second term
shows the consumption due to their reaction with Mg(OH)2 within the
liquid film. The subscripts A,B,E,F in the equations, respectively, indicate
the chemical species SO2, OH−, SO3−2
and HSO3−.Liquid
film—Region I: it
holds that(12)DA⋅d2dz2CA−Ks⋅[1+2⋅((CA⋅DA)/CBs⋅DB)]⋅Ap⋅CBs2=0,(13)DF⋅d2dz2CF−Ks⋅(1+CF⋅DFCBs⋅DB)⋅Ap⋅CBs=0.Liquid
film—Region II: it
holds that(14)DEd2dz2CE−Ks⋅(1+CF⋅DFCBs⋅DB)⋅Ap⋅CBS=0,DF⋅d2dz2CF−Ks⋅(1+CF⋅DFCBs⋅DB)⋅Ap⋅CBs=0.Liquid
film—Region III: it holds that(15)DB⋅d2dz2CB+Ks⋅AP⋅(CBs−CB)=0,(16)DE⋅d2dz2CE=0.
The boundary conditions imposed are
as follows: (17)atz=0,CA=CAiddzCF=0,atz=z1,CA=CE=0,CF=CF*,−DF⋅ddzCF=DE⋅ddzCE−DF⋅ddzCF∣z=z1−+DF⋅ddzCF∣z=z1+=2⋅DE⋅ddzCE∣z=z1+atz=z2,CB=CF=0,CE=CE*,DB⋅ddzCB=−DF⋅ddzCF=DE⋅ddzCE∣z=z2−−DE⋅ddzCE∣z=z2+atz=zL,CB=CBs,CE=CE0=0. Here, CE* and CF*
represent the concentrations of E at z2 and F at z1, respectively.The mass balance equations in the
dimensionless form reduce to the following.Region I:(18)d2dx2YA−N⋅YA=N2⋅rA⋅qA,(19)d2dx2YF−N⋅YF=NrF,Region
II:(20)d2dx2YE−N⋅rFrE⋅YE=−NrE,(21)d2dx2YF−N⋅YF=NrF,Region III: (22)d2dx2YB−N⋅YB=−N,d2dx2YE=0  subject to (23)x=0;YA=1,dYFdx=0,x=x1;YA=YE=0,YF=YF*,−rA⋅qA⋅ddxYA=rE⋅ddxYE,−rF⋅ddxYF∣(X=X1−)+rF⋅ddxYF∣(X=X1+)=2⋅rE⋅ddxYE∣(X=X1+),x=x2;YB=YF=0,YE=YE*,ddxYB=−rF⋅ddxYF=rE⋅ddxYE∣(X=X2−)−rE⋅ddxYE∣(X=X2+),x=1;YB=1,YE=0. When the concentration of gaseous
species is extremely low and the enhancement factor for the solid dissolution
can be regarded as unity, (18)–(21) are
simplified as follows:Region I:(24)d2dx2YA=N2⋅rA⋅qA,(25)d2dx2YF=NrF.Region
II:(26)d2dx2YE=NrE,(27)d2dx2YF=NrF,Equations (24)–(27), (22) give the enhancement factor as (28)E=N⋅x14⋅rA⋅qA+1x1.E0 represents the
enhancement factor for a clear solution saturated with Mg(OH)2 and
is defined by(29)E0=1+12⋅rA⋅qA. The dimensionless positions of two
reaction planes, x1 and x2, can be determined by the
following equations [10]: (30)(x22−x14)⋅NrAqAx1+1−x21−x1⋅[Ntanh[N⋅(1−x2)]+N⋅(x2−x1)2]=0,(32⋅x1−x2)⋅N+2x2−x1−11−x2⋅[rA⋅qAx1−(x22−x14)⋅N]=0.


## 2.1. Model of Absorption through Reaction Planes

Regarding theSO2 absorption
mechanism into wet magnesia slurry, the following processes can be considered [8]: (1)SO2diffusionthroughthegasfilm(Diffusion)(2)SO2(g)⟶SO2(aq.)(Dissolution)(3)SO2(aq.)+H2O⟶H2SO3(Dissociation)(4)H2SO3⟶H++HSO3−(Dissociation)(5)HSO3−⟶H++SO32−(Dissociation)(6)MgO(s)+H2O⟶Mg(OH)2(aq.)(Dissolution)(7)Mg(OH)2(aq.)⟶Mg2++2OH−(Dissolution)(8)Diffusionofdissolvedchemical(Diffusion)speciesinliquidfilm(9)SO2+2OH−⟶SO32−+H2O(Reaction)(10)SO2+SO32−+H2O⟶2HSO3−(Reaction)(11)HSO3−+OH−⟶SO32−+H2O(Reaction). Initially,
SO2 gas diffuses from bulk of the gas phase to the gas-liquid film, dissolves into
the bulk liquid and dissociates. Simultaneously, solid MgO also dissolves and
dissociates. After that, the sulphite and hydroxyl ions react with each other
in the liquid film according to (9)–(11). The principal
chemical species of the reaction are subjected to acid dissociation; it is
necessary to consider the pH of the absorbent liquid, which largely influences
the degree of dissociation.


## 2.2. Basic Equations

In the process ofSO2 absorption in
magnesia slurry with no suspended particles, HSO3− cannot
coexist with OH−, so that reaction (9) never takes place directly. The
above condition shows that reactions (10) and (11) take place at two different
located planes in the two-reaction-plane model. However, in the slurry process,
to be considered here, both dissolve SO2 and the HSO3−,
to be produced by reaction (10), can react with OH− which is fed by
the dissolution of the solid particles in the liquid film. So in this case, dissolved
SO2 can be consumed by reactions (9) and (10) simultaneously. The
concentration of SO32− in the bulk liquid increases as
the absorption proceeds. For a saturated solution of magnesia, a plausible
sketch of the concentration profile is given in Figure 1(a). When the particles
are suspended in the liquid film, the concentration profiles shift as shown in Figure 1(b)
[9].Concentration profile forSO2/Mg(OH)2 slurry: (a) no suspended solids, (b) in the presence of suspended solids.
(a)(b)The model
considers the presence of 4 basic species:SO2, OH−, SO3−2,
and HSO3−. The basic equations were derived to form the
mass balance of perspective chemical species as shown below. The first terms of
(12)–(15) deal with the
diffusion of the species according to Fick’s second law, while the second term
shows the consumption due to their reaction with Mg(OH)2 within the
liquid film. The subscripts A,B,E,F in the equations, respectively, indicate
the chemical species SO2, OH−, SO3−2
and HSO3−.Liquid
film—Region I: it
holds that(12)DA⋅d2dz2CA−Ks⋅[1+2⋅((CA⋅DA)/CBs⋅DB)]⋅Ap⋅CBs2=0,(13)DF⋅d2dz2CF−Ks⋅(1+CF⋅DFCBs⋅DB)⋅Ap⋅CBs=0.Liquid
film—Region II: it
holds that(14)DEd2dz2CE−Ks⋅(1+CF⋅DFCBs⋅DB)⋅Ap⋅CBS=0,DF⋅d2dz2CF−Ks⋅(1+CF⋅DFCBs⋅DB)⋅Ap⋅CBs=0.Liquid
film—Region III: it holds that(15)DB⋅d2dz2CB+Ks⋅AP⋅(CBs−CB)=0,(16)DE⋅d2dz2CE=0.
The boundary conditions imposed are
as follows: (17)atz=0,CA=CAiddzCF=0,atz=z1,CA=CE=0,CF=CF*,−DF⋅ddzCF=DE⋅ddzCE−DF⋅ddzCF∣z=z1−+DF⋅ddzCF∣z=z1+=2⋅DE⋅ddzCE∣z=z1+atz=z2,CB=CF=0,CE=CE*,DB⋅ddzCB=−DF⋅ddzCF=DE⋅ddzCE∣z=z2−−DE⋅ddzCE∣z=z2+atz=zL,CB=CBs,CE=CE0=0. Here, CE* and CF*
represent the concentrations of E at z2 and F at z1, respectively.The mass balance equations in the
dimensionless form reduce to the following.Region I:(18)d2dx2YA−N⋅YA=N2⋅rA⋅qA,(19)d2dx2YF−N⋅YF=NrF,Region
II:(20)d2dx2YE−N⋅rFrE⋅YE=−NrE,(21)d2dx2YF−N⋅YF=NrF,Region III: (22)d2dx2YB−N⋅YB=−N,d2dx2YE=0  subject to (23)x=0;YA=1,dYFdx=0,x=x1;YA=YE=0,YF=YF*,−rA⋅qA⋅ddxYA=rE⋅ddxYE,−rF⋅ddxYF∣(X=X1−)+rF⋅ddxYF∣(X=X1+)=2⋅rE⋅ddxYE∣(X=X1+),x=x2;YB=YF=0,YE=YE*,ddxYB=−rF⋅ddxYF=rE⋅ddxYE∣(X=X2−)−rE⋅ddxYE∣(X=X2+),x=1;YB=1,YE=0. When the concentration of gaseous
species is extremely low and the enhancement factor for the solid dissolution
can be regarded as unity, (18)–(21) are
simplified as follows:Region I:(24)d2dx2YA=N2⋅rA⋅qA,(25)d2dx2YF=NrF.Region
II:(26)d2dx2YE=NrE,(27)d2dx2YF=NrF,Equations (24)–(27), (22) give the enhancement factor as (28)E=N⋅x14⋅rA⋅qA+1x1.E0 represents the
enhancement factor for a clear solution saturated with Mg(OH)2 and
is defined by(29)E0=1+12⋅rA⋅qA. The dimensionless positions of two
reaction planes, x1 and x2, can be determined by the
following equations [10]: (30)(x22−x14)⋅NrAqAx1+1−x21−x1⋅[Ntanh[N⋅(1−x2)]+N⋅(x2−x1)2]=0,(32⋅x1−x2)⋅N+2x2−x1−11−x2⋅[rA⋅qAx1−(x22−x14)⋅N]=0.


## 3. Result and Discussion

### 3.1. Model Parameter Estimation

In order to have
the numerical results of the above model regarding enhancement factor, it is
necessary to know the values of the dimensionless parametersrA, qA, and N. The values of rA, qA, and N have been considered
as same as the experimental data by various workers available in the
literature. To show the contribution of the presence of the solids to the absorption
rate, the ratio of enhancement factor into the slurry to that into saturated
solution (E/E0) is plotted against qA in Figures 2(a)–2(c). The ratio
E/E0 represents the degree of enhancement owing to the presence of
solid particles in the slurry. The parameter regarding the solid dissolution,
N, is fixed (1.0–5.5), and the parameter rA regarding the ratio of
diffusivity of SO2 to that of liquid is also fixed (0.5–1.5). From
the figures, it is observed that E/E0 initially decreases with the increase
in qA, and then it is almost independent of qA. It
indicates that when CAi is kept constant, the enhancement in gas
absorption rate due to the presence of the suspended reactant particles is
almost independent of the gas-phase concentration. The influences in gas-phase
concentration and N improve the absorption characteristics.Enhancement factors as
a function ofqA for the SO2–Mg(OH)2 slurry
system. Enhancement factors as a
function of qA for the SO2–Mg(OH)2 slurry
system. Enhancement factors as
a function of qA for the SO2–Mg(OH)2 slurry
system.
(a)(b)(c)To show the
effect of solid dissolution(N) on absorption rate, the ratio of enhancement factor
(E/E0) is plotted against N (see Figures 3(a)–3(c)) for
constant value of rA. It is seen from the figures that the
difference of the enhancement factors is very small with the variation of qA from 4 to 10, and E/E0 is almost independent on N.Enhancement factors as a function ofN for the SO2–Mg(OH)2 slurry system. Enhancement factors
as a function of N for the SO2–Mg(OH)2 slurry system. Enhancement factors
as a function of N for the SO2–Mg(OH)2 slurry system.
(a)(b)(c)For the contribution
ofrA on absorption rate, a plot is drawn between E/E0 versus
rA by considering both qA and N to be the same as shown in Figure 4. In this case, the
diffusivity of SO2 in the slurry was assumed to be the same as that
of the pure water, and the concentration of SO2 at the gas-liquid
interface was assumed to be equal to the solubility of the hydroxide in the
water. Under these circumstances, Figure 4 shows that E/E0 decreases as rA increases.Figure 4
Enhancement factor as a
function ofrA for SO2–Mg(OH)2 slurry system.


### 3.2. Model Verification

Figure5 shows
the comparison between the experimental values available in literature [3, 11]
and the values (obtained from model) of enhancement factor owing to the
presence of solid particles for magnesium hydroxide in the slurry. It is
evident that the experimental values available in literature and the model
values are in good agreement.Figure 5
Comparison between experimental values available in literature and model
values of the enhancement factor for theSO2–Mg(OH)2 slurry system.


## 3.1. Model Parameter Estimation

In order to have
the numerical results of the above model regarding enhancement factor, it is
necessary to know the values of the dimensionless parametersrA, qA, and N. The values of rA, qA, and N have been considered
as same as the experimental data by various workers available in the
literature. To show the contribution of the presence of the solids to the absorption
rate, the ratio of enhancement factor into the slurry to that into saturated
solution (E/E0) is plotted against qA in Figures 2(a)–2(c). The ratio
E/E0 represents the degree of enhancement owing to the presence of
solid particles in the slurry. The parameter regarding the solid dissolution,
N, is fixed (1.0–5.5), and the parameter rA regarding the ratio of
diffusivity of SO2 to that of liquid is also fixed (0.5–1.5). From
the figures, it is observed that E/E0 initially decreases with the increase
in qA, and then it is almost independent of qA. It
indicates that when CAi is kept constant, the enhancement in gas
absorption rate due to the presence of the suspended reactant particles is
almost independent of the gas-phase concentration. The influences in gas-phase
concentration and N improve the absorption characteristics.Enhancement factors as
a function ofqA for the SO2–Mg(OH)2 slurry
system. Enhancement factors as a
function of qA for the SO2–Mg(OH)2 slurry
system. Enhancement factors as
a function of qA for the SO2–Mg(OH)2 slurry
system.
(a)(b)(c)To show the
effect of solid dissolution(N) on absorption rate, the ratio of enhancement factor
(E/E0) is plotted against N (see Figures 3(a)–3(c)) for
constant value of rA. It is seen from the figures that the
difference of the enhancement factors is very small with the variation of qA from 4 to 10, and E/E0 is almost independent on N.Enhancement factors as a function ofN for the SO2–Mg(OH)2 slurry system. Enhancement factors
as a function of N for the SO2–Mg(OH)2 slurry system. Enhancement factors
as a function of N for the SO2–Mg(OH)2 slurry system.
(a)(b)(c)For the contribution
ofrA on absorption rate, a plot is drawn between E/E0 versus
rA by considering both qA and N to be the same as shown in Figure 4. In this case, the
diffusivity of SO2 in the slurry was assumed to be the same as that
of the pure water, and the concentration of SO2 at the gas-liquid
interface was assumed to be equal to the solubility of the hydroxide in the
water. Under these circumstances, Figure 4 shows that E/E0 decreases as rA increases.Figure 4
Enhancement factor as a
function ofrA for SO2–Mg(OH)2 slurry system.


## 3.2. Model Verification

Figure5 shows
the comparison between the experimental values available in literature [3, 11]
and the values (obtained from model) of enhancement factor owing to the
presence of solid particles for magnesium hydroxide in the slurry. It is
evident that the experimental values available in literature and the model
values are in good agreement.Figure 5
Comparison between experimental values available in literature and model
values of the enhancement factor for theSO2–Mg(OH)2 slurry system.


## 4. Conclusion

In spite of
noticeable progress in conventional flue gas desulphurization (FGD) process
development, claims for more efficient, more economic, and nonpollutant
technological innovations become more and more important. The new technology
essentially is the wet magnesia FGD process which is technically and
economically competitive with once-through FGD process. In this article, a
model based on Fick’s Second law has been developed for dynamic absorption rate
of sulphur dioxide intoMg(OH)2 slurry with the help of two reaction
planes.Numerical
solutions for the absorption ofSO2 in aqueous slurries of Mg(OH)2 are presented in Figures
2–5. The dissolution of fine solid Mg(OH)2 particles has significant effect on enhancement factor which contributes to the
absorption rate of SO2 from bulk of the gas phase to the liquid. It
is efficient to understand that the reaction of SO2 with accumulated
species SO32− promotes the absorption rate. The
theoretical enhancement factors obtained from present model were compared well
with experimental data available in literature. The model presented can be
applied to any highly complicated reactive absorption processes. It should be
stressed that analytical approximations are often oversimplified and cannot be
expected to predict the absorption rates for a wide range of conditions.



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*Source: 871479-2008-05-15.xml*