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A mass transfer model for limestone dissolution from a rotating cylinder
Chemical Engineerrng Science, Vol. 44, No. Printed in Great Britain.
1, pp. 61-67,
1989. 0
A MASS TRANSFER MODEL FOR LIMESTONE FROM A ROTATING CYLINDER MATS Department ofchemical
WALLIN
and INGEMAR
OCCE-2509/89 $3.00 +O.CNl 1989 Pergamon Press PI.2
DISSOLUTION
BJERLE
Engineering It, University of Lund, Chemical Center, PO Box 124, S-221 00 Lund,
Sweden
(First received 22 December 1987; accepted in revised form 2 June 1988) Abstract-The film theory model was used to model limestone dissolution from a plane surface. In the experimental design a limestone cylinder rotated in an aqueous solution. The effects of pH, cylinder speed, partial pressure of carbon dioxide and temperature on the dissolution rate were investigated. The results show that the mass transfer mode1 describes very well the dependence of the four parameters within the ranges investigated. Compared with dissolution in a nitrogen atmosphere, a high partial pressure of carbon dioxide increases the dissolution rate by up to a factor of 10. This is attributed to the carbonate system acid-base reactions. The dissolution rate in a nitrogen atmosphere can with good accuracy be written Ra a$ +. The results show that q is a function of pH and film thickness. The film theory model predicts and describes this behavior which previously has not been explained theoretically. The activation energy for dissolution in a nitrogen atmosphere is 16.2 & 1.5 kJ/mol. The activation energy increases for dissolutionin a
carbon dioxide atmosphere.
dissolution:
BACKGROUND
Limestone dissolution in an aqueous solution is a phenomenon of interest in several areas:
R = kraH+ + k,, aH,coJ + km aHa
m Liming of acid lakes. l Flue gas desulfurization. l Preservation of buildings.
--,v
l
l
l
(1)
where k,, k,, and k,, are rate constants, and k, is dependent on both pm, and temperature. The experiments showed that
Knowledge of limestone dissolution dependence on hydrodynamic and chemical conditions can optimize the efforts carried out in the areas listed above. .Limestone dissolution has previously been studied from a rotatihg disc or particles. Sjiiberg and Rickard (1984a, b, 1985) studied dissolution from a rotating limestone disc in an aqueous solution and a nitrogen atmosphere. Their results were treated according to the theory for rotating discs (Levich, 1962). The following were observed: l
%a~ + uHCo;
l
l
l
Dissolution of limestone is controlled by mass transfer at low pH. In this region the dissolution rate can be expressed as R mu&+, where q =0.90 + 0.04 (Sjiiberg and Rickard, 1984a, b). The activation energy is 13 kJ/mol for the transport rate constant (Sj8berg and Rickard, 1984b). The presence of Ca * + decreases the dissolution rate, which makes the dissolution process more transport-controlled (SjBberg and Rickard, 1985). In the region controlled by mass transfer, the product of the surface activities a,-,2+ acot. approaches the value of the solubility product constant (SjSberg and Rickard, 1984a).
At low pH the dissolution rate is controlled by the rate of mass transfer. In a nitrogen atmosphere the dissolution rate can be expressed as Rcca,+, that is q= 1. The activation energy of k, is 8.4 kJ/mol in an acidic environment and a nitrogen atmosphere. A high partial pressure of carbon dioxide increases the dissolution rate at pH < 5.5.
In these investigations (Plummer et al., 1978; .!ijiiberg and Rickard, 1984a, b, 1985) the varying value of the exponent (y) of the hydrogen ion dependence was not explained. No one has used any of the tools provided by mass transfer modelling in order to achieve a better understanding at the microscopic level. Chan and Rochelle (1982) investigated the dissolution of limestone particles in both nitrogen and carbon dioxide atmospheres. They successfully used a mass transfer mode1 where limestone particles are assumed to be spheres in an infinite stagnant solution. Bjerle and Rochelle (1982, 1984) investigated dissolution from a rotating disc in both nitrogen and carbon dioxide atmospheres. A mass transfer model, which is also the basis for the present work, was developed for dissolution from a plane limestone surface. The following were observed:
Plummer et al. (1978) investigated the dissolution of particles in an aqueous solution, and in both nitrogen and carbon dioxide atmospheres. Their findings resulted in the following rate equation for calcite
61
62
MATS
WALLIN
and INGEMAR
* A high partial pressure of carbon
l
l
dioxide can increase the dissolution rate (Bjerle and Rochelle, 1982, 1984; Chan and Rochelle, 1982). The dissolution rate in an acidic environment is controlled by mass transfer (Bjerle and Rochelle, 1982, 1984; Chan and Rochelle, 1982). The activation energy is 17 k.l/mol, pH -z 5, in a nitrogen atmosphere (Chan and Rochelle, 1982).
Using an interpretation of experimental results based on dissolution studies of a rotating limestone cylinder and mass transfer modelling, it is the aim of the present study to give: l
l
A microscopic understanding of dissolution from a plane surface in the region of mass transfer control. An understanding of how the dissolution depends on temperature, pH and carbon dioxide partial pressure in the region of mass transfer control.
Dissolution from a yotating cylinder provides a well-defined hydrodyna&c environment and is therefore adequate for limestone dissolution studies.
BJERLE
ment in the liquid. In the bulk solution the conditions are constant and all species are in equilibrium. Assume a steady-state diffusion through a stagnant film of thickness 6 from a plane surface of calcium carbonate. A material balance for species i leads to
(2) with the boundary conditions at
x=0:
ci = c+
at
x=6:
Ci=Ci+
In the following analysis Di is regarded as independent of the spatial location. The flux for species i at any position x0 can be written as Ni=
-_Di-
Total carbon: D co1
d2CCOJ +
d’[HCO;]
D
HCO;
dx*
dx’ + Dco: -
Reactions The net reaction for the dissolution of limestone in an acidic environment can be written as (Plummer et al., 1978; Sverdrup and Bjerle, 1982). + 2H + +Ca’+
dx X=X0.
The material balances are written as follows:
THEORY
CaCO,,,,
dC,
+HzOo,+
COz(s)
Because the carbonate species is present in the system the carbonate system acid-base reactions occur (Sverdrup and Bjerle, 1982): H,O(,, + CO,(,,,+H HCO;+H++CO$-.
* + HCO;
(Rl) (R2)
The reaction rate is slow for the forward reaction (Rl) (Bjerle and Rochelle, 1982, 1984; Chan and Rochelle, 1982). This implies that the amount of hydrogen ions formed by reaction (R 1) is dependent on the length of reaction time. Mass transfer model Chan and Rochelle (1982) developed a general mass transfer model for the dissolution of limestone particles in an aqueous environment. The model is based on diffusion from spherical particles into a stagnant aqueous solution. This model was modified for plane geometry by Bjerle and Rochelle (1982, 1984). The model takes the finite reaction rate of reaction (Rl), as well as the instantaneous reaction (R2) and the autoprotolysis of water into consideration. It is assumed that there is phase equilibrium at the liquid-solid interface (Sjijberg and Rickard, 1984a). It is also assumed that mass transfer takes place through a liquid film surrounding the surface. This film has a thickness dependent on the hydrodynamic environ-
d*[CO; dx2
-1
=o
(4)
Charge: D
Ii+
d=W+l
+2D
Ca2+
dx’
d*[Ca*+]
d*[HCO;] HCO;
dx2
d*[CO$ -1 _D dx2
-2D,,;-
_ D
OH
_d2CoH-1 dx2
dx2 =O
’
(5)
Equation (5) states that the net charge production is zero. Since the diffusivities of the species are different (DH+ is large compared to, for example, DC-z+) there would normally be an electrical potential gradient in the liquid. A large excess of calcium chloride was used to avoid the development of such gradients: Calcium: D ca*+
d*[Ca*+]
= 0
dx2
Carbon dioxide:
d*CCO,I
Qx32
dx’
where k, is the rate constant for the forward reaction (Rl), and K, is the equilibrium constant. The use of activities in the rate expression is due to the introduction of the equilibrium constant K,. The carbon dioxide flux at x = 0 must be zero, since no reaction of carbon dioxide occurs at the solid surface. The boundary conditions for eq. (7) are then: at
x-o:
at
x =6:
dCCOz1 EO
dx [CO*]
= [C02&.
63
Mass transfer model for limestone dissolution The final form of the model is obtained by integrating eqs (4)-(6). By reaction stoichiometry the flux of calcium is equal to the flux of lolal carbon. The net flux of charged species is zero. These conditions are valid throughout the film: Dcoz
CCC’,1+ D,,,; WCC’;1 + %o: - [CO:-]=/~x+cc, D Cn2+ [Ca*+]=Bx+cr,
D,+
[H+]+2DCaZ+
(9)
[Ca2+]-DHCo;
[CO;-]
-2 D,,:-
-Do,-
(8)
[HCO;]
[OH-]=a,.
(17) (10)
The integration constants (clj and fi) are determined via boundary conditions. Two equilibria are assumed to be valid throughout the film: K,=a,+
aoHuH+
K,=
the model, a 0.1 mol/l calcium chloride solution was used. Physical constants are shown in Table 1. The constants at 25°C were taken from prior works (Bjerle and Rochelle, 1982, 1984; Chan and Rochelle, 1982). Since we used a large excess of CaCl,, the diffusion constants for ions in absence of a potential gradient were used. The diffusion constants were extrapolated from 25 to 40°C using the Stokes-Einstein equation:
(11)
The rate constant (k,) at 4O”C, taken from Pinsent et al. (1956), is 0.068 s-l. The constant was assumed to be independent of ionic strength. The activation energy for the reaction is 50 kJ/mol. The activation energy was calculated according to the Arrhenius equation:
%O;-
d(ln k) -=-
(12) aHCO;
Phase equilibrium
dT
is assumed at the surface: L=aCa2+
acoz-.
(13)
The dissolution rate is given by - fi in kmol/m2 s. The composition of the bulk solution is constant and all species are in equilibrium. In the bulk, eqs (11) and (12) apply, as well as uH+
K,=
aHCO;
(14)
acO,
and Henry’s
law:
(1% If pH7 pco, and Cc, 2+ are known the bulk composition
is defined. In the model the Debye-Hiickel compute the activity coefficients:
logyi=
-OSz~-
equation is used to
Ji
(16)
l+Ji
where yi is defined by ai = yi [i], z is the ionic charge and I is the ionic strength. In order to achieve a constant ionic strength in the experiments, as well as
E R,T=’
The equilibrium constants were extrapolated to 40°C using van’t Hoffs’ equation: d(ln K)
AH,
dT
R,T2’
L 4
KW Qi+
DCd + Dco*
D
DHCog co: DonHe
CES We:1
-E
(kmol/m3)
(kmol/m3) (kmol/m3)2 (s-1) (kmol/m”)z (m*/s) (m*/s) W/s) W/s) W/s) (m*/s) (atm mj/kmol)
(19)
EXPERIMENTAL
Design
The main purpose was to investigate how carbon dioxide partial pressure, pH, temperature and speed of rotation affect the dissolution rate of calcium carbonate from a cylinder rotating in an aqueous solution. The cylinder was made of almost pure calcium carbonate (Carrara marble) as checked by EDXA. The pH-stat method (Plummer et al., 1978) was used to measure the dissolution rate of calcium carbonate at constant pH and constant bulk compo-
25°C K2
from 25
AH, was calculated from standard tables (Weast, 1982-1983) for AH:5. The model was solved for the dissolution rate under the conditions used in the experimental matrix. In Fig. 1 the dissolution rate at 25°C is shown as a function of film thickness, pH and carbon dioxide partial pressure.
Table 1. Physical constants
Kl
(18)
4.45 x 4.69 x 4.80 x 2.60 x 1.00 x 9.30 x 0.79 x 2.00 x 1.20 x 0.70 x 5.27 x 29.5
10-l lo-” iO-g 10-z 10-14 lo- 9 10-g 10-g 10-y to-9 lo- 9
40°C
5.16 x 6.25 x 3.74 x 6.80 x 2.94 x 13.3 x 1.13 x 2.86 x 1.72 x 1.00 x 7.54 x 41.9
lo-’ LOO-” lo- 9 10-l lo- I4 10-g 10-g 1O-9 lo- 9 10-v 10-g
MATS WALLIN and INGEMAR BJERLE
64
100
Fitm
thickness
(@III)
Fig. 1. Dissolution rate as a function of film thickness for various pH values at T=25”C: (- - -) pco2= 1 atm, (---) PCoI =O.OOl atm.
sition. The pH was kept constant to within +0.005 units, through autotitration with 0.2 mol/l hydrochloric acid via the stoichiometry CaCO,:HCl= 1:2. The dissolution rate R in kmol/m* s will then be R=OS---
1 dV
C ,+c,dV A
(20)
-=O%dt dt
where A is the cylinder casing surface area in II? and dV/dt is the added volume per time unit in m3/s. The pH-stat was connected to a personal computer where the volume was recorded. d V/dt was computed from the slope of the curve. A statistical analysis of the slope was performed on the personal computer after every experiment. In order to achieve a constant ionic strength during experimentation a 0.1 mol/l calcium chloride solution was used as bulk in the reactor. This, in addition to the fact that the reactor volume was kept constant by an that neither dissolved overflow outlet, ensured calcium carbonate nor added hydrochloric acid affect the ionic strength even if the experiments are run for a long period. In order to maintain a constant concentration of carbonate species, water-saturated carbon dioxide or nitrogen was sparged through a glass column (Fig. 2) at a flow rate of 1 I/min. This method is preferable to sparging into the reactor, which may affect the mass transfer rate in an uncontrollable way because the
T
Gas
1
RESULTS AND DISCUSSION
Model-experiments correlation In order to compare the model with the experiments, the hydrodynamic parameter of the model, i.e. the film thickness, must be translated into the parameters describing the hydrodynamic environment in the experiments. If the temperature and the cylinder diameter are constant the hydrodynamics is only a function of the cylinder speed (Eisenberg et al., 1955; Sherwood et al., 1975).
Gas =I-
pump Gas
bubbles disturb the flow pattern. The liquid in the glass column was circulated by a Master-flex pump. In all the experiments the temperature was controlled to within +-0.2”C by a water bath. The baffled dissolution reactor is shown in Fig. 2. The limestone cylinder was rotated by a DC-motor whose rotation speed was controlled to within _t 1 rpm by an electronic unit. The limestone cylinder was turned from a block of Carrara marble and had dimensions of h = 134 mm and d = 38.4 mm. The cylinder diameter was measured after every experiment. The diameter had, after the 14 experiments, decreased by 2%. The diameter’s effect on the mass transfer coefficient for rotating cylinders can be estimated by using an empirical correlation based on the Re, Sh and SC (Sherwood et al., 1975). A 2% error in the diameter gives a less than 2% error in the mass transfer coefficient. From the empirical correlation, a correction factor was estimated for every experiment. This factor was used in evaluating the results. The pH-electrode was an Ag-AgCl combination electrode and was calibrated by buffers before and after every experiment at the reactor temperature. The dissolution reactor (Fig. 2) contained approximately 3 1 of 0.1 mol/l calcium chloride solution. If a new solution was used, time was required to achieve equilibrium conditions with gas and limestone. The pH was adjusted by addition of acid or base. One experiment covered eight rotational speeds (100,250,400, 600, 800, 1000, 1200 and 1400 rpm). A total of 14 experiments were performed. The 14 experiments were reproduced once with good results, giving a maximum deviation of 6%.
I
Gas
I I
I ------_-----1
Fig. 2. Dissolution reactor: (--
) liquid streams, (- - -) electric signals.
Mass transfer model for limestone dissolution
The function translating the film thickness into terms of cylinder speed is found by using the nitrogen atmosphere experiments as a reference. The translation function is then used to check the model’s applicability in a carbon dioxide atmosphere. i’he translation function should have the qualities of minimizing
rl=
c
100.250,.
i
pH=3.3,3.*..
x
____c--
A---
Rex&
PHI
PI-U + K-,&S, PW
I
**-
.d
.’
, d’ d’ <./( I’
___---
H-3.8
_--
*-•
_---
_____-c---
___--__._---
, Ye
[pH=4.3
[bH=4.8
_*__----
Jp“=5.3
__“_____*------
_-
I> (21) ’
where R is the dissolution rate in a nitrogen mosphere. Minimization at 25°C yields S = 5082 n -0.‘49
__-a__W--
,*-
?’ ,_---
Rex&, PI-U - %,,(S,
bH=3.3'
__a__-*
f
c
65
at-
Fig. 3. Dissolution rate vs cylinder speed in a nitrogen atmosphere for various pH values, T=25”C: (- - -) film theory model, (0) experiment.
(22)
where S is the film thickness in pm and n is the cylinder speed in rpm. Fitting the translation function above yields a correlation coefficient of 0.999. The function was used in the evaluation of results whenever the model was compared to the experiments. For the experiments carried out at 4O”C, a different translation function had to be used. This is because the temperature, through viscosity etc. (Eisenberg et al., 1955; Sherwood et al., 1975), affects the hydrodynamic environment in the reactor. At 40°C the relation between S and n is SE5453
n-o.755
(2%
where S is the film thickness in pm and n is the cylinder speed in rpm. Dependence ofpff, carbon dioxide and cylinder speed at constant temperature The dependence of cylinder speed in the carbon dioxide free system (Fig. 3) is very obvious. The experiments show an average cylinder speed dependence of n”.s3. For the carbon dioxide system (Fig. 4) the exponent of n is less than the exponent in the carbon dioxide free system. The exponent varies from 0.14 (p?I = 5.3) to 0.87 (pH = 3.3). This variation is due to the effect of carbon dioxide on the carbonate reaction system, which increases with increasing pH and film thickness (decreasing cylinder speed). As carbon dioxide penetrates into the film layer, reactions (RI) and (R2) produce hydrogen ions, which increase the flux of hydrogen ions towards the surface. Eisenberg et al. (1955) reported a cylinder speed dependence of nO-‘l for mass transfer from a rotating cylinder. This dependence is based upon experiments in which no chemical reactions are involved. The dissolution of timestone from a plane surface in the region controlled by mass transfer is a strong function of pH. For the carbon dioxide free case the experiments (Fig. 3) can be summarized by the equation R=(3.88
x 10P7)a~+87no.83
(241
where R is the dissolution rate in kmol/m2s, a,+ is the activity in kmol/m3 and n is the cylinder speed in rpm.
Cylinder
speed
(rpm)
Fig. 4. Dissolution rate vs cylinder speed in a carbon dioxide atmosphere for various pH values, T= 25°C: (- - -) film theory model, (0) experiment.
The correlation coefficient is 0.997. The film theory model (Fig. 1) can, for a nitrogen atmosphere, be summarized by the general equation R=(1.30
x 10-9)a$,885 ~5-‘.‘~s
(25)
where R is the dissolution rate in kmol/m* s, a”+ is the activity in kmol/m3 and S is the film thickness in m. The correlation coefficient is 0.998. The dependence of the hydrogen ion activity is R~a~,8~*~-~‘. This dependence is seen in both the experiments and the model. Other investigators measured the exponent to be 0.90&0.04 (SjGberg and Rickard, 1984a), 0.95 (Plummer et al., 1979) and 1.00 (Plummer et al., 1978). In fact, the exponent above is a mean value within the region investigated. The exponent or the slope from log R vs pH is actually dependent on both pH and cylinder speed (Fig. 5). The exponent was computed by using (-s).
for the experiments
and by using
for the film theory model. These partial derivatives were computed from original data. At a low carbon dioxide partial pressure reactions (Rl) and (R2) shift backwards and consume hydrogen ions. This effect is most obvious at a high activity of hydrogen ions and at large film thickness. This is a
MATS
--- 1
-.
--*_3--
-o-
..___
0 0
I’
, ’
_-
9--e--0
.
___--
WALLIN
and INGEMAR
BJERLE
2
0
- - -_-~-_-_---_-_--g_-+_-_-_~ g
_- _-
,-
#
Q
*
(iI i
0
Fig. 5. Exponent vs cylinder speed in a nitrogen atmosphere for various pH values, T=25”C: (- - -) film theory model; (o), (0) and (*) experiments.
contributing factor to the exponent dependence on pH and cylinder speed in a nitrogen atmosphere. The film theory model shows good agreement with the experiments. The exponent dependence of pH and cylinder speed have never before been explained theoretically. A carbon dioxide partial pressure of 1 atm increases the dissolution rate up to a factor of 10 compared with dissolution in a nitrogen atmosphere. The enhancement effect from carbon dioxide on the dissolution rate increases with increasing pH and decreasing cylinder speed within the experimental limits. The enhancement factor dependence of pH and cylinder speed is predicted by the film theory model. The amount of hydrogen ions formed by reaction (RI) is dependent on the distance penetrated, which in the model corresponds to the film thickness. The dependence on pH can be explained in terms of the carbonate reactions. At low values of pH reactions (Rl) and (R2) consume hydrogen ions and thereby decrease the hydrogen ion flux. At high pH values, carbon dioxide penetrates into the film layer and reactions (Rl) and (R2) produce hydrogen ions, which increase the flux towards the surface. Temperature dependence The nitrogen atmosphere experiments (Fig. 6) give an activation energy of 16.2 & 1.5 kJ/mol. The activation energy constants were computed from the Arrhenius equation [eq. (18)]. The activation energy is independent of cylinder speed and pH. The value corresponds to the temperature influence on the diffusion constants. The activation energy for the diffusion constant, computed from the Stokes-Einstein equation [eq. (17)] and the Arrhenius equation [eq. (18)], is 18.5 kJ/mol. For a nitrogen atmosphere the film theory model gives an activation energy of 17.5 f 1 kJ/mol for 3.3 c pH < 5.3 and 10 pm < 6 c 200 pm. Other investigators reported the following activation energy constants: 13 kJ/mol (Sj6berg and Rickard, 1984b), 8.4 kJ/mol (Plummer et al., 1978) and 17 kJ/mol (Chan and Rochelle, 1982). These activation energy constants are based on experimental data. The experiments in a carbon dioxide atmosphere (Fig. 6)
Fig. 6. Dissolution rate vs cylinder speed in both nitrogen and carbon dioxide atmospheres for various pH values, T= 40°C: (---) film theory model (nitrogen), (0) exper) film theory model (carbon dioxide), iment (nitrogen); (-(0) experiment (carbon dioxide).
give an activation energy of 19.3 f0.5 kJ/mol for pH = 3.8 and an activation energy of 25.2 + 1 kJ/mol for pH = 4.8. The mass transfer model describes very well the dissolution rate in a nitrogen atmosphere at 40°C (Fig. 6). However, there are some discrepancies between the model and experiments for dissolution in a carbon dioxide atmosphere. The experiments at 40°C were run at those pH values where the largest discrepancies between the film theory model and the
experiments rate (Fig.
exist when considering
the dissolution
4) at 25°C. If the experiments
out at pH=3.3 and 5.3 a better probably be achieved.
were carried agreement would
CONCLUSIONS
Limestone dissolution from a rotating cylinder has been studied. The effect of pH, cylinder speed, partial pressure of carbon dioxide and temperature on the dissolution rate has been considered. The film theory model has been used to make a theoretical mass transfer study of the dissolution rate. In the model, a phase equilibrium at the liquid-solid interface has been assumed. The mass transfer model describes well the dependence of pH, carbon dioxide, cylinder speed and temperature at the conditions investigated. The good model-experiment agreement indicates that there is phase equilibrium at the liquid-solid interface.
(1) Limestone dissolution from a plane surface can
be described by a mass transfer model. The film theory model describes well the dependence of pH, carbon dioxide, hydrodynamic environment and temperature. The model describes the dissolution rate in a nitrogen atmosphere with very good accuracy, giving a maximum error of 10% at pH = 5.3 and n = 100 rpm. The model describes the dissolution rate in a carbon dioxide atmosphere with good accuracy, giving a maximum error of 30% at pH =4.3 and n = 100 rpm. (2) Limestone dissolution from a plane surface in a nitrogen atmosphere can, with good accuracy,
67
Mass transfermodel for limestone dissolution
be written as R oca&+ for constant cylinder speed (film thickness) and temperature. Careful experiments show that q is a function of pH and film thickness. The experiments indicate also that the exponent (q) does not equal 1 even at low pH values. The film theory model predicts and describes this characteristic of the system, which has never before been explained theoretitally. (3) Carbon dioxide enhances the dissolution rate up to a factor of 10 compared with dissolution in a nitrogen atmosphere. The enhancement factor increases with increasing pH and decreasing cylinder speed (increasing film thickness). This characteristic is described by the film theory model. (4) The activation energy for dissolution in a nitrogen atmosphere is 16.2 f 1.5 kJ/mol. The activation energy increases for dissolution in a carbon dioxide atmosphere. The film theory model describes the temperature dependence for dissolution in a nitrogen atmosphere and predicts an activation energy of 17.5 f 1 kJ/mol. The agreement between model and experiments is not as close for dissolution in a carbon dioxide atmosphere.
x I [
]
distance, L ionic charge, concentration, mol/L3
Greek letters ml, aZ, a3, P integration constants heat of reaction, E/mol AH, heat of formation at 25°C AH;5 film thickness, L S activity coefficient y dynamic viscosity, Pt p kinematic viscosity, L2/t v
E/mol
Subscript3 b bulk experiment cxp species i i model mod any point o surface s w water Superscripts +, charge REFERENCES
Acknowledgements-The authors are grateful to MS Charlotte Stromblad for valuable help during the work and to Dr Hans T. Karlsson for useful advice. This work was financially supported by STUF. NOTATION A a C D
d E He
h I K k k, L N n P 4 R R, Re r SC Sh T V v
cylinder casing surface area, L* activity, mol/L3 concentration, mol/L3 diffusivity, L2/t cylinder diameter, L activation energy, E/mol Henry’s law constant, PL3/mol cylinder height, L ionic strength, mol/L3 equilibrium constant, mol/L3 rate constant, l/t mass transfer coefficient, physical system, L/t solubility product, (mol/L’)’ flux, mol/L’t cylinder speed, rpm partial pressure, P exponent dissolution rate, mol/L’t ideal gas constant, 8.314 J/mol K Reynolds number, dv/v production, mol/L3t Schmidt number, v/D Sherwood number, kJv temperature, T volume added of acid, L3 velocity, L/t
Bjerle, I. and Rochelle, G. T., 1982, Limestone dissolution in acid lakes. V&ten 38, 156163. Bjerle, I. and Rochelle, G. T., 1984, Limestone dissolution from a plane surface. Chem. Engng Sci. 39, 183-185. Chan, P. K. and Rochelle, G. T., 1982, Limestone dissolution: effects of pH, CO, and buffers modeled by mass transfer. _ Am. them. SOc. Sjmp. Ser. 188, 75-97. Eisenberg, M., Tobias, C. W. and Wilke, C. R., 1955, Mass transfer at rotating cylinders. Chem. Engng Prog. Symp. Ser. 51, 1-16. 1962, Physiochemical Hydrodynamics. Levich, -V. G., Prentice-Hall, Englewood Cliffs, NJ. Pinsent, R. W., Pearson, L. and Roughton, F. J. W., 1956, The kinetics of the combination of carbondioxide with hvdroxide ions. Trans. Faraday Sot. 52, 1512-1520. Plummer. L. N.. Parkhurst. D. L. and Wielev. T. M. L.. 1979.
A criticalreviewof the kineticsof cal&eUhissolutionand precipitation.Am. them. Sot. Symp. Ser. 93, 538-573.
Plummer, L. N., Wigley, T. M. L. and Parkhurst, D. L., 1978, The kinetics of calcite dissolution in CO,-water systems at 560°C and O.&l.0 atm CO,. Am. J. Sci. 278, 179-216. Sherwood, T. K., Pigford R. L. and Wilke C. R., 1975, Muss Transfer. McGraw-Hill, New York. SjGberg, E. L. and Rickard, D. T., 1984a, Calcite dissolution kinetics: surface speciation and the origin of the variable pH dependence. Chem. Geol. 42, 119-136. SjBberg, E. L. and Rickard, D. T., 1984b, Temperature dependence of calcite dissolution kinetics between 1 and 62°C at pH 2.7 to 8.4 in aqueous solutions. Geochim. cosmochim. Acta 48, 485-493. Sjoberg, E. L. and Rickard, D. T., 1985, The effect of added dissolved calcium on calcite dissolution in aqueous solutions at 25°C. Chem. Geol. 49, 405413. Sverdrup, H. and Bjerle, I., 1982, Dissolution of calcite and other related minerals in acidic aqueous solution in a pHstat. Vatten 38, 59-73. Weast, R. C., Ed., 1982-1983, CRC Handbook of Chemistry and Physics, 63rd edition. CRC Press, Boca Raton, FL.
1, pp. 61-67,
1989. 0
A MASS TRANSFER MODEL FOR LIMESTONE FROM A ROTATING CYLINDER MATS Department ofchemical
WALLIN
and INGEMAR
OCCE-2509/89 $3.00 +O.CNl 1989 Pergamon Press PI.2
DISSOLUTION
BJERLE
Engineering It, University of Lund, Chemical Center, PO Box 124, S-221 00 Lund,
Sweden
(First received 22 December 1987; accepted in revised form 2 June 1988) Abstract-The film theory model was used to model limestone dissolution from a plane surface. In the experimental design a limestone cylinder rotated in an aqueous solution. The effects of pH, cylinder speed, partial pressure of carbon dioxide and temperature on the dissolution rate were investigated. The results show that the mass transfer mode1 describes very well the dependence of the four parameters within the ranges investigated. Compared with dissolution in a nitrogen atmosphere, a high partial pressure of carbon dioxide increases the dissolution rate by up to a factor of 10. This is attributed to the carbonate system acid-base reactions. The dissolution rate in a nitrogen atmosphere can with good accuracy be written Ra a$ +. The results show that q is a function of pH and film thickness. The film theory model predicts and describes this behavior which previously has not been explained theoretically. The activation energy for dissolution in a nitrogen atmosphere is 16.2 & 1.5 kJ/mol. The activation energy increases for dissolutionin a
carbon dioxide atmosphere.
dissolution:
BACKGROUND
Limestone dissolution in an aqueous solution is a phenomenon of interest in several areas:
R = kraH+ + k,, aH,coJ + km aHa
m Liming of acid lakes. l Flue gas desulfurization. l Preservation of buildings.
--,v
l
l
l
(1)
where k,, k,, and k,, are rate constants, and k, is dependent on both pm, and temperature. The experiments showed that
Knowledge of limestone dissolution dependence on hydrodynamic and chemical conditions can optimize the efforts carried out in the areas listed above. .Limestone dissolution has previously been studied from a rotatihg disc or particles. Sjiiberg and Rickard (1984a, b, 1985) studied dissolution from a rotating limestone disc in an aqueous solution and a nitrogen atmosphere. Their results were treated according to the theory for rotating discs (Levich, 1962). The following were observed: l
%a~ + uHCo;
l
l
l
Dissolution of limestone is controlled by mass transfer at low pH. In this region the dissolution rate can be expressed as R mu&+, where q =0.90 + 0.04 (Sjiiberg and Rickard, 1984a, b). The activation energy is 13 kJ/mol for the transport rate constant (Sj8berg and Rickard, 1984b). The presence of Ca * + decreases the dissolution rate, which makes the dissolution process more transport-controlled (SjBberg and Rickard, 1985). In the region controlled by mass transfer, the product of the surface activities a,-,2+ acot. approaches the value of the solubility product constant (SjSberg and Rickard, 1984a).
At low pH the dissolution rate is controlled by the rate of mass transfer. In a nitrogen atmosphere the dissolution rate can be expressed as Rcca,+, that is q= 1. The activation energy of k, is 8.4 kJ/mol in an acidic environment and a nitrogen atmosphere. A high partial pressure of carbon dioxide increases the dissolution rate at pH < 5.5.
In these investigations (Plummer et al., 1978; .!ijiiberg and Rickard, 1984a, b, 1985) the varying value of the exponent (y) of the hydrogen ion dependence was not explained. No one has used any of the tools provided by mass transfer modelling in order to achieve a better understanding at the microscopic level. Chan and Rochelle (1982) investigated the dissolution of limestone particles in both nitrogen and carbon dioxide atmospheres. They successfully used a mass transfer mode1 where limestone particles are assumed to be spheres in an infinite stagnant solution. Bjerle and Rochelle (1982, 1984) investigated dissolution from a rotating disc in both nitrogen and carbon dioxide atmospheres. A mass transfer model, which is also the basis for the present work, was developed for dissolution from a plane limestone surface. The following were observed:
Plummer et al. (1978) investigated the dissolution of particles in an aqueous solution, and in both nitrogen and carbon dioxide atmospheres. Their findings resulted in the following rate equation for calcite
61
62
MATS
WALLIN
and INGEMAR
* A high partial pressure of carbon
l
l
dioxide can increase the dissolution rate (Bjerle and Rochelle, 1982, 1984; Chan and Rochelle, 1982). The dissolution rate in an acidic environment is controlled by mass transfer (Bjerle and Rochelle, 1982, 1984; Chan and Rochelle, 1982). The activation energy is 17 k.l/mol, pH -z 5, in a nitrogen atmosphere (Chan and Rochelle, 1982).
Using an interpretation of experimental results based on dissolution studies of a rotating limestone cylinder and mass transfer modelling, it is the aim of the present study to give: l
l
A microscopic understanding of dissolution from a plane surface in the region of mass transfer control. An understanding of how the dissolution depends on temperature, pH and carbon dioxide partial pressure in the region of mass transfer control.
Dissolution from a yotating cylinder provides a well-defined hydrodyna&c environment and is therefore adequate for limestone dissolution studies.
BJERLE
ment in the liquid. In the bulk solution the conditions are constant and all species are in equilibrium. Assume a steady-state diffusion through a stagnant film of thickness 6 from a plane surface of calcium carbonate. A material balance for species i leads to
(2) with the boundary conditions at
x=0:
ci = c+
at
x=6:
Ci=Ci+
In the following analysis Di is regarded as independent of the spatial location. The flux for species i at any position x0 can be written as Ni=
-_Di-
Total carbon: D co1
d2CCOJ +
d’[HCO;]
D
HCO;
dx*
dx’ + Dco: -
Reactions The net reaction for the dissolution of limestone in an acidic environment can be written as (Plummer et al., 1978; Sverdrup and Bjerle, 1982). + 2H + +Ca’+
dx X=X0.
The material balances are written as follows:
THEORY
CaCO,,,,
dC,
+HzOo,+
COz(s)
Because the carbonate species is present in the system the carbonate system acid-base reactions occur (Sverdrup and Bjerle, 1982): H,O(,, + CO,(,,,+H HCO;+H++CO$-.
* + HCO;
(Rl) (R2)
The reaction rate is slow for the forward reaction (Rl) (Bjerle and Rochelle, 1982, 1984; Chan and Rochelle, 1982). This implies that the amount of hydrogen ions formed by reaction (R 1) is dependent on the length of reaction time. Mass transfer model Chan and Rochelle (1982) developed a general mass transfer model for the dissolution of limestone particles in an aqueous environment. The model is based on diffusion from spherical particles into a stagnant aqueous solution. This model was modified for plane geometry by Bjerle and Rochelle (1982, 1984). The model takes the finite reaction rate of reaction (Rl), as well as the instantaneous reaction (R2) and the autoprotolysis of water into consideration. It is assumed that there is phase equilibrium at the liquid-solid interface (Sjijberg and Rickard, 1984a). It is also assumed that mass transfer takes place through a liquid film surrounding the surface. This film has a thickness dependent on the hydrodynamic environ-
d*[CO; dx2
-1
=o
(4)
Charge: D
Ii+
d=W+l
+2D
Ca2+
dx’
d*[Ca*+]
d*[HCO;] HCO;
dx2
d*[CO$ -1 _D dx2
-2D,,;-
_ D
OH
_d2CoH-1 dx2
dx2 =O
’
(5)
Equation (5) states that the net charge production is zero. Since the diffusivities of the species are different (DH+ is large compared to, for example, DC-z+) there would normally be an electrical potential gradient in the liquid. A large excess of calcium chloride was used to avoid the development of such gradients: Calcium: D ca*+
d*[Ca*+]
= 0
dx2
Carbon dioxide:
d*CCO,I
Qx32
dx’
where k, is the rate constant for the forward reaction (Rl), and K, is the equilibrium constant. The use of activities in the rate expression is due to the introduction of the equilibrium constant K,. The carbon dioxide flux at x = 0 must be zero, since no reaction of carbon dioxide occurs at the solid surface. The boundary conditions for eq. (7) are then: at
x-o:
at
x =6:
dCCOz1 EO
dx [CO*]
= [C02&.
63
Mass transfer model for limestone dissolution The final form of the model is obtained by integrating eqs (4)-(6). By reaction stoichiometry the flux of calcium is equal to the flux of lolal carbon. The net flux of charged species is zero. These conditions are valid throughout the film: Dcoz
CCC’,1+ D,,,; WCC’;1 + %o: - [CO:-]=/~x+cc, D Cn2+ [Ca*+]=Bx+cr,
D,+
[H+]+2DCaZ+
(9)
[Ca2+]-DHCo;
[CO;-]
-2 D,,:-
-Do,-
(8)
[HCO;]
[OH-]=a,.
(17) (10)
The integration constants (clj and fi) are determined via boundary conditions. Two equilibria are assumed to be valid throughout the film: K,=a,+
aoHuH+
K,=
the model, a 0.1 mol/l calcium chloride solution was used. Physical constants are shown in Table 1. The constants at 25°C were taken from prior works (Bjerle and Rochelle, 1982, 1984; Chan and Rochelle, 1982). Since we used a large excess of CaCl,, the diffusion constants for ions in absence of a potential gradient were used. The diffusion constants were extrapolated from 25 to 40°C using the Stokes-Einstein equation:
(11)
The rate constant (k,) at 4O”C, taken from Pinsent et al. (1956), is 0.068 s-l. The constant was assumed to be independent of ionic strength. The activation energy for the reaction is 50 kJ/mol. The activation energy was calculated according to the Arrhenius equation:
%O;-
d(ln k) -=-
(12) aHCO;
Phase equilibrium
dT
is assumed at the surface: L=aCa2+
acoz-.
(13)
The dissolution rate is given by - fi in kmol/m2 s. The composition of the bulk solution is constant and all species are in equilibrium. In the bulk, eqs (11) and (12) apply, as well as uH+
K,=
aHCO;
(14)
acO,
and Henry’s
law:
(1% If pH7 pco, and Cc, 2+ are known the bulk composition
is defined. In the model the Debye-Hiickel compute the activity coefficients:
logyi=
-OSz~-
equation is used to
Ji
(16)
l+Ji
where yi is defined by ai = yi [i], z is the ionic charge and I is the ionic strength. In order to achieve a constant ionic strength in the experiments, as well as
E R,T=’
The equilibrium constants were extrapolated to 40°C using van’t Hoffs’ equation: d(ln K)
AH,
dT
R,T2’
L 4
KW Qi+
DCd + Dco*
D
DHCog co: DonHe
CES We:1
-E
(kmol/m3)
(kmol/m3) (kmol/m3)2 (s-1) (kmol/m”)z (m*/s) (m*/s) W/s) W/s) W/s) (m*/s) (atm mj/kmol)
(19)
EXPERIMENTAL
Design
The main purpose was to investigate how carbon dioxide partial pressure, pH, temperature and speed of rotation affect the dissolution rate of calcium carbonate from a cylinder rotating in an aqueous solution. The cylinder was made of almost pure calcium carbonate (Carrara marble) as checked by EDXA. The pH-stat method (Plummer et al., 1978) was used to measure the dissolution rate of calcium carbonate at constant pH and constant bulk compo-
25°C K2
from 25
AH, was calculated from standard tables (Weast, 1982-1983) for AH:5. The model was solved for the dissolution rate under the conditions used in the experimental matrix. In Fig. 1 the dissolution rate at 25°C is shown as a function of film thickness, pH and carbon dioxide partial pressure.
Table 1. Physical constants
Kl
(18)
4.45 x 4.69 x 4.80 x 2.60 x 1.00 x 9.30 x 0.79 x 2.00 x 1.20 x 0.70 x 5.27 x 29.5
10-l lo-” iO-g 10-z 10-14 lo- 9 10-g 10-g 10-y to-9 lo- 9
40°C
5.16 x 6.25 x 3.74 x 6.80 x 2.94 x 13.3 x 1.13 x 2.86 x 1.72 x 1.00 x 7.54 x 41.9
lo-’ LOO-” lo- 9 10-l lo- I4 10-g 10-g 1O-9 lo- 9 10-v 10-g
MATS WALLIN and INGEMAR BJERLE
64
100
Fitm
thickness
(@III)
Fig. 1. Dissolution rate as a function of film thickness for various pH values at T=25”C: (- - -) pco2= 1 atm, (---) PCoI =O.OOl atm.
sition. The pH was kept constant to within +0.005 units, through autotitration with 0.2 mol/l hydrochloric acid via the stoichiometry CaCO,:HCl= 1:2. The dissolution rate R in kmol/m* s will then be R=OS---
1 dV
C ,+c,dV A
(20)
-=O%dt dt
where A is the cylinder casing surface area in II? and dV/dt is the added volume per time unit in m3/s. The pH-stat was connected to a personal computer where the volume was recorded. d V/dt was computed from the slope of the curve. A statistical analysis of the slope was performed on the personal computer after every experiment. In order to achieve a constant ionic strength during experimentation a 0.1 mol/l calcium chloride solution was used as bulk in the reactor. This, in addition to the fact that the reactor volume was kept constant by an that neither dissolved overflow outlet, ensured calcium carbonate nor added hydrochloric acid affect the ionic strength even if the experiments are run for a long period. In order to maintain a constant concentration of carbonate species, water-saturated carbon dioxide or nitrogen was sparged through a glass column (Fig. 2) at a flow rate of 1 I/min. This method is preferable to sparging into the reactor, which may affect the mass transfer rate in an uncontrollable way because the
T
Gas
1
RESULTS AND DISCUSSION
Model-experiments correlation In order to compare the model with the experiments, the hydrodynamic parameter of the model, i.e. the film thickness, must be translated into the parameters describing the hydrodynamic environment in the experiments. If the temperature and the cylinder diameter are constant the hydrodynamics is only a function of the cylinder speed (Eisenberg et al., 1955; Sherwood et al., 1975).
Gas =I-
pump Gas
bubbles disturb the flow pattern. The liquid in the glass column was circulated by a Master-flex pump. In all the experiments the temperature was controlled to within +-0.2”C by a water bath. The baffled dissolution reactor is shown in Fig. 2. The limestone cylinder was rotated by a DC-motor whose rotation speed was controlled to within _t 1 rpm by an electronic unit. The limestone cylinder was turned from a block of Carrara marble and had dimensions of h = 134 mm and d = 38.4 mm. The cylinder diameter was measured after every experiment. The diameter had, after the 14 experiments, decreased by 2%. The diameter’s effect on the mass transfer coefficient for rotating cylinders can be estimated by using an empirical correlation based on the Re, Sh and SC (Sherwood et al., 1975). A 2% error in the diameter gives a less than 2% error in the mass transfer coefficient. From the empirical correlation, a correction factor was estimated for every experiment. This factor was used in evaluating the results. The pH-electrode was an Ag-AgCl combination electrode and was calibrated by buffers before and after every experiment at the reactor temperature. The dissolution reactor (Fig. 2) contained approximately 3 1 of 0.1 mol/l calcium chloride solution. If a new solution was used, time was required to achieve equilibrium conditions with gas and limestone. The pH was adjusted by addition of acid or base. One experiment covered eight rotational speeds (100,250,400, 600, 800, 1000, 1200 and 1400 rpm). A total of 14 experiments were performed. The 14 experiments were reproduced once with good results, giving a maximum deviation of 6%.
I
Gas
I I
I ------_-----1
Fig. 2. Dissolution reactor: (--
) liquid streams, (- - -) electric signals.
Mass transfer model for limestone dissolution
The function translating the film thickness into terms of cylinder speed is found by using the nitrogen atmosphere experiments as a reference. The translation function is then used to check the model’s applicability in a carbon dioxide atmosphere. i’he translation function should have the qualities of minimizing
rl=
c
100.250,.
i
pH=3.3,3.*..
x
____c--
A---
Rex&
PHI
PI-U + K-,&S, PW
I
**-
.d
.’
, d’ d’ <./( I’
___---
H-3.8
_--
*-•
_---
_____-c---
___--__._---
, Ye
[pH=4.3
[bH=4.8
_*__----
Jp“=5.3
__“_____*------
_-
I> (21) ’
where R is the dissolution rate in a nitrogen mosphere. Minimization at 25°C yields S = 5082 n -0.‘49
__-a__W--
,*-
?’ ,_---
Rex&, PI-U - %,,(S,
bH=3.3'
__a__-*
f
c
65
at-
Fig. 3. Dissolution rate vs cylinder speed in a nitrogen atmosphere for various pH values, T=25”C: (- - -) film theory model, (0) experiment.
(22)
where S is the film thickness in pm and n is the cylinder speed in rpm. Fitting the translation function above yields a correlation coefficient of 0.999. The function was used in the evaluation of results whenever the model was compared to the experiments. For the experiments carried out at 4O”C, a different translation function had to be used. This is because the temperature, through viscosity etc. (Eisenberg et al., 1955; Sherwood et al., 1975), affects the hydrodynamic environment in the reactor. At 40°C the relation between S and n is SE5453
n-o.755
(2%
where S is the film thickness in pm and n is the cylinder speed in rpm. Dependence ofpff, carbon dioxide and cylinder speed at constant temperature The dependence of cylinder speed in the carbon dioxide free system (Fig. 3) is very obvious. The experiments show an average cylinder speed dependence of n”.s3. For the carbon dioxide system (Fig. 4) the exponent of n is less than the exponent in the carbon dioxide free system. The exponent varies from 0.14 (p?I = 5.3) to 0.87 (pH = 3.3). This variation is due to the effect of carbon dioxide on the carbonate reaction system, which increases with increasing pH and film thickness (decreasing cylinder speed). As carbon dioxide penetrates into the film layer, reactions (RI) and (R2) produce hydrogen ions, which increase the flux of hydrogen ions towards the surface. Eisenberg et al. (1955) reported a cylinder speed dependence of nO-‘l for mass transfer from a rotating cylinder. This dependence is based upon experiments in which no chemical reactions are involved. The dissolution of timestone from a plane surface in the region controlled by mass transfer is a strong function of pH. For the carbon dioxide free case the experiments (Fig. 3) can be summarized by the equation R=(3.88
x 10P7)a~+87no.83
(241
where R is the dissolution rate in kmol/m2s, a,+ is the activity in kmol/m3 and n is the cylinder speed in rpm.
Cylinder
speed
(rpm)
Fig. 4. Dissolution rate vs cylinder speed in a carbon dioxide atmosphere for various pH values, T= 25°C: (- - -) film theory model, (0) experiment.
The correlation coefficient is 0.997. The film theory model (Fig. 1) can, for a nitrogen atmosphere, be summarized by the general equation R=(1.30
x 10-9)a$,885 ~5-‘.‘~s
(25)
where R is the dissolution rate in kmol/m* s, a”+ is the activity in kmol/m3 and S is the film thickness in m. The correlation coefficient is 0.998. The dependence of the hydrogen ion activity is R~a~,8~*~-~‘. This dependence is seen in both the experiments and the model. Other investigators measured the exponent to be 0.90&0.04 (SjGberg and Rickard, 1984a), 0.95 (Plummer et al., 1979) and 1.00 (Plummer et al., 1978). In fact, the exponent above is a mean value within the region investigated. The exponent or the slope from log R vs pH is actually dependent on both pH and cylinder speed (Fig. 5). The exponent was computed by using (-s).
for the experiments
and by using
for the film theory model. These partial derivatives were computed from original data. At a low carbon dioxide partial pressure reactions (Rl) and (R2) shift backwards and consume hydrogen ions. This effect is most obvious at a high activity of hydrogen ions and at large film thickness. This is a
MATS
--- 1
-.
--*_3--
-o-
..___
0 0
I’
, ’
_-
9--e--0
.
___--
WALLIN
and INGEMAR
BJERLE
2
0
- - -_-~-_-_---_-_--g_-+_-_-_~ g
_- _-
,-
#
Q
*
(iI i
0
Fig. 5. Exponent vs cylinder speed in a nitrogen atmosphere for various pH values, T=25”C: (- - -) film theory model; (o), (0) and (*) experiments.
contributing factor to the exponent dependence on pH and cylinder speed in a nitrogen atmosphere. The film theory model shows good agreement with the experiments. The exponent dependence of pH and cylinder speed have never before been explained theoretically. A carbon dioxide partial pressure of 1 atm increases the dissolution rate up to a factor of 10 compared with dissolution in a nitrogen atmosphere. The enhancement effect from carbon dioxide on the dissolution rate increases with increasing pH and decreasing cylinder speed within the experimental limits. The enhancement factor dependence of pH and cylinder speed is predicted by the film theory model. The amount of hydrogen ions formed by reaction (RI) is dependent on the distance penetrated, which in the model corresponds to the film thickness. The dependence on pH can be explained in terms of the carbonate reactions. At low values of pH reactions (Rl) and (R2) consume hydrogen ions and thereby decrease the hydrogen ion flux. At high pH values, carbon dioxide penetrates into the film layer and reactions (Rl) and (R2) produce hydrogen ions, which increase the flux towards the surface. Temperature dependence The nitrogen atmosphere experiments (Fig. 6) give an activation energy of 16.2 & 1.5 kJ/mol. The activation energy constants were computed from the Arrhenius equation [eq. (18)]. The activation energy is independent of cylinder speed and pH. The value corresponds to the temperature influence on the diffusion constants. The activation energy for the diffusion constant, computed from the Stokes-Einstein equation [eq. (17)] and the Arrhenius equation [eq. (18)], is 18.5 kJ/mol. For a nitrogen atmosphere the film theory model gives an activation energy of 17.5 f 1 kJ/mol for 3.3 c pH < 5.3 and 10 pm < 6 c 200 pm. Other investigators reported the following activation energy constants: 13 kJ/mol (Sj6berg and Rickard, 1984b), 8.4 kJ/mol (Plummer et al., 1978) and 17 kJ/mol (Chan and Rochelle, 1982). These activation energy constants are based on experimental data. The experiments in a carbon dioxide atmosphere (Fig. 6)
Fig. 6. Dissolution rate vs cylinder speed in both nitrogen and carbon dioxide atmospheres for various pH values, T= 40°C: (---) film theory model (nitrogen), (0) exper) film theory model (carbon dioxide), iment (nitrogen); (-(0) experiment (carbon dioxide).
give an activation energy of 19.3 f0.5 kJ/mol for pH = 3.8 and an activation energy of 25.2 + 1 kJ/mol for pH = 4.8. The mass transfer model describes very well the dissolution rate in a nitrogen atmosphere at 40°C (Fig. 6). However, there are some discrepancies between the model and experiments for dissolution in a carbon dioxide atmosphere. The experiments at 40°C were run at those pH values where the largest discrepancies between the film theory model and the
experiments rate (Fig.
exist when considering
the dissolution
4) at 25°C. If the experiments
out at pH=3.3 and 5.3 a better probably be achieved.
were carried agreement would
CONCLUSIONS
Limestone dissolution from a rotating cylinder has been studied. The effect of pH, cylinder speed, partial pressure of carbon dioxide and temperature on the dissolution rate has been considered. The film theory model has been used to make a theoretical mass transfer study of the dissolution rate. In the model, a phase equilibrium at the liquid-solid interface has been assumed. The mass transfer model describes well the dependence of pH, carbon dioxide, cylinder speed and temperature at the conditions investigated. The good model-experiment agreement indicates that there is phase equilibrium at the liquid-solid interface.
(1) Limestone dissolution from a plane surface can
be described by a mass transfer model. The film theory model describes well the dependence of pH, carbon dioxide, hydrodynamic environment and temperature. The model describes the dissolution rate in a nitrogen atmosphere with very good accuracy, giving a maximum error of 10% at pH = 5.3 and n = 100 rpm. The model describes the dissolution rate in a carbon dioxide atmosphere with good accuracy, giving a maximum error of 30% at pH =4.3 and n = 100 rpm. (2) Limestone dissolution from a plane surface in a nitrogen atmosphere can, with good accuracy,
67
Mass transfermodel for limestone dissolution
be written as R oca&+ for constant cylinder speed (film thickness) and temperature. Careful experiments show that q is a function of pH and film thickness. The experiments indicate also that the exponent (q) does not equal 1 even at low pH values. The film theory model predicts and describes this characteristic of the system, which has never before been explained theoretitally. (3) Carbon dioxide enhances the dissolution rate up to a factor of 10 compared with dissolution in a nitrogen atmosphere. The enhancement factor increases with increasing pH and decreasing cylinder speed (increasing film thickness). This characteristic is described by the film theory model. (4) The activation energy for dissolution in a nitrogen atmosphere is 16.2 f 1.5 kJ/mol. The activation energy increases for dissolution in a carbon dioxide atmosphere. The film theory model describes the temperature dependence for dissolution in a nitrogen atmosphere and predicts an activation energy of 17.5 f 1 kJ/mol. The agreement between model and experiments is not as close for dissolution in a carbon dioxide atmosphere.
x I [
]
distance, L ionic charge, concentration, mol/L3
Greek letters ml, aZ, a3, P integration constants heat of reaction, E/mol AH, heat of formation at 25°C AH;5 film thickness, L S activity coefficient y dynamic viscosity, Pt p kinematic viscosity, L2/t v
E/mol
Subscript3 b bulk experiment cxp species i i model mod any point o surface s w water Superscripts +, charge REFERENCES
Acknowledgements-The authors are grateful to MS Charlotte Stromblad for valuable help during the work and to Dr Hans T. Karlsson for useful advice. This work was financially supported by STUF. NOTATION A a C D
d E He
h I K k k, L N n P 4 R R, Re r SC Sh T V v
cylinder casing surface area, L* activity, mol/L3 concentration, mol/L3 diffusivity, L2/t cylinder diameter, L activation energy, E/mol Henry’s law constant, PL3/mol cylinder height, L ionic strength, mol/L3 equilibrium constant, mol/L3 rate constant, l/t mass transfer coefficient, physical system, L/t solubility product, (mol/L’)’ flux, mol/L’t cylinder speed, rpm partial pressure, P exponent dissolution rate, mol/L’t ideal gas constant, 8.314 J/mol K Reynolds number, dv/v production, mol/L3t Schmidt number, v/D Sherwood number, kJv temperature, T volume added of acid, L3 velocity, L/t
Bjerle, I. and Rochelle, G. T., 1982, Limestone dissolution in acid lakes. V&ten 38, 156163. Bjerle, I. and Rochelle, G. T., 1984, Limestone dissolution from a plane surface. Chem. Engng Sci. 39, 183-185. Chan, P. K. and Rochelle, G. T., 1982, Limestone dissolution: effects of pH, CO, and buffers modeled by mass transfer. _ Am. them. SOc. Sjmp. Ser. 188, 75-97. Eisenberg, M., Tobias, C. W. and Wilke, C. R., 1955, Mass transfer at rotating cylinders. Chem. Engng Prog. Symp. Ser. 51, 1-16. 1962, Physiochemical Hydrodynamics. Levich, -V. G., Prentice-Hall, Englewood Cliffs, NJ. Pinsent, R. W., Pearson, L. and Roughton, F. J. W., 1956, The kinetics of the combination of carbondioxide with hvdroxide ions. Trans. Faraday Sot. 52, 1512-1520. Plummer. L. N.. Parkhurst. D. L. and Wielev. T. M. L.. 1979.
A criticalreviewof the kineticsof cal&eUhissolutionand precipitation.Am. them. Sot. Symp. Ser. 93, 538-573.
Plummer, L. N., Wigley, T. M. L. and Parkhurst, D. L., 1978, The kinetics of calcite dissolution in CO,-water systems at 560°C and O.&l.0 atm CO,. Am. J. Sci. 278, 179-216. Sherwood, T. K., Pigford R. L. and Wilke C. R., 1975, Muss Transfer. McGraw-Hill, New York. SjGberg, E. L. and Rickard, D. T., 1984a, Calcite dissolution kinetics: surface speciation and the origin of the variable pH dependence. Chem. Geol. 42, 119-136. SjBberg, E. L. and Rickard, D. T., 1984b, Temperature dependence of calcite dissolution kinetics between 1 and 62°C at pH 2.7 to 8.4 in aqueous solutions. Geochim. cosmochim. Acta 48, 485-493. Sjoberg, E. L. and Rickard, D. T., 1985, The effect of added dissolved calcium on calcite dissolution in aqueous solutions at 25°C. Chem. Geol. 49, 405413. Sverdrup, H. and Bjerle, I., 1982, Dissolution of calcite and other related minerals in acidic aqueous solution in a pHstat. Vatten 38, 59-73. Weast, R. C., Ed., 1982-1983, CRC Handbook of Chemistry and Physics, 63rd edition. CRC Press, Boca Raton, FL.