Mar 24, 1997 - slow reaction CO2 + HZ0 f H' + HCO, and the chemical reactions at the surface ..... The rate constants of these reactions are well-known.
Geochimica et Cosmochimica Acta, Vol. 61, No. 14, pp. 2879-2889, 1997 Copyright 0 1997 Elsevier Science Ltd Printed in the USA. All rights reserved 0016.7037/97 $17.00 + .OO
Pergamon
PI1 SOO16-7037( 97)00143-9
Dissolution kinetics of calcium carbonate minerals in H20-CO2 solutions in turbulent flow: The role of the diffusion boundary layer and the slow reaction Hz0 + CO2 G H + + HCO; ZAIHUA LIU ‘J
‘Institute
and
WOLFGANG
DREYBRODT’.*
of Experimental Physics, University of Bremen, ZInstitute of Karst Geology, Guilin,
D-28334 China
Bremen,
Germany
(Received July 30, 1996; accepted in revised form March 24, 1997) and precipitation of calcium carbonate minerals in aqueous solutions with turbulent flow are controlled by a diffusion boundary layer (DBL) adjacent to the surface of the mineral, across which mass transfer is effected by molecular diffusion. A rotating disk technique was used to investigate the effect of the DBL on the dissolution rates of CaC03. This technique allows an exact adjustment of the thickness of the DBL by controlling the rotation speed of a circular sample of CaC03. Measurements of the dissolution rates in H20-CO*-CaZt -solutions in equilibrium with various partial pressures of CO2 from 1. 10e3 up to 1 atm showed a dependence of the rates R on the rotation frequency w, given by R w wn. The exponent n varies from 0.25 at low PcO, to about 0.01 at a P,,, of 1 atm. This reveals that the rates are not controlled by mass transport only, which would require n = 0.5. The experimental data can be explained employing a theoretical model, which also takes into account the slow reaction CO2 + HZ0 f H’ + HCO, and the chemical reactions at the surface (Dreybrodt and Buhmann, 1991). Interpretation of the experimental data in view of this model reveals that conversion of CO1 plays an important role in the control of the rates. At high P,-,z and large DBL thickness (e > 0.001 cm), conversion of CO* occurs mainly in the DBL and, therefore, becomes rate limiting. This is corroborated by the observation that upon addition of the enzyme carbonic anhydrase, which catalyzes CO,-conversion, the dissolution rates are enhanced by 1 order of magnitude. From our experimental observations we conclude that the theoretical model above enables one to predict dissolution rates with satisfactory precision. Since the precipitation rates from supersaturated solutions are determined by the same mechanisms as dissolution, we infer that this model is also valid to predict precipitation rates. The predicted rates for both dissolution and precipitation can be approximated by a linear rate law R = (Y * ( ceq - c) , where C~ is the equilibrium concentration with respect to calcite and (Y a rate constant, dependent on temperature, Pcoz, DBL thickness (E), and the thickness of the water sheet flowing on the mineral. Values of cr are listed that can be used for a variety of geologically relevant conditions. Copyright 0 1997 Elsevier Science Ltd Abstract-Dissolution
NOTATION R A V D r F c c=T P
[ ‘;’ ( ) t w s” a
dissolution rate surface area of rotating disk volume of the solution coefficient of diffusion radius of rotating disk diffusional flux concentration equilibrium concentration partial pressure of CO1 concentration activity thickness of DBL angular velocity kinematic viscosity of water thickness of turbulent core rate constant
mmole cm* cm3 cm’s_’ cm mmole mmole mmole atm mmole mmole cm
such as diagenesis of calcareous deep-sea sediments (Bemer, 1980; Boudreau and Canfield, 1993), the formation of karst in limestone terranes (Dreybrodt, 1988; Ford and Williams, 1989; White, 1988), the evolution of water chemistry in calcite depositing stream systems (Hermann and Lorah, 1987; Dreybrodt et al., 1992; Liu et al., 1995), and the global cycle of CO2 (Archer and Maier-Reimer, 1994; Kempe, 1977). Both the precipitation and the dissolution rates in the system HZO-C02-CaC03 are controlled by three rate determining processes: ( 1) The kinetics at the mineral surface, which is given by the mechanistic rate equation (called PWP equation) proposed for dissolution by Plummer et al. ( 1978) and which has later been verified also for precipitation (Busenberg and Plummer, 1986; Reddy et al., 1981; Inskeep and Bloom, 1985). This PWP-equation reads
crnm2 S-I
cm-’ s-l cmm3 cmm3 cm-3 cme3
SC’
cm2 ss’ cm cm ss’
R =
K,(H+)
+
K~(H$O::)
1. INTRODUCTION + Dissolution and precipitation of calcite in a CO? - HZ0 system play an important role in many geological processes,
K3
-
K4(CaL+)
(HCO,)
(1)
where the rate R is given in mmol cm-* s-’ ; and K, , K*, ~3, and K~ are temperature dependent rate constants and the parentheses denote activities in mmole cmm3 of the corresponding species at the surface of the mineral; (2) The slow
*Author to whom correspondence should be addressed. 2879
2880
Z. Liu and W. Dreybrodt
reaction Hz0 + CO* f H+ + HCOF exerts a significant influence on the rates because stochiometry requires that for each calcium ion released from the solid one CO* molecule must react to form H’ + HCO,. At large mineral surface areas and small volumes of the solution this slow reaction can be rate determining. Its kinetics has been reviewed by Kern ( 1960) and Usdowski ( 1982); (3) Mass transport of the reacting species away from and towards the mineral surface by molecular diffusion. Taking into account these three processes, Buhmann and Dreybrodt (1985a,b, 1987) put forward a model from which dissolution and precipitation rates can be calculated for a sheet of water of given thickness, covering the surface of the mineral. They have shown that due to CO, conversion the rates depend on the thickness of this water sheet. The influence of mass transport is significant when the water is stagnant or flows laminarly on the calcite. If, however, the flow becomes turbulent, molecular diffusion is enhanced by turbulent eddies, such that under otherwise unchanged conditions the rates of dissolution or precipitation, respectively, can increase by almost 1 order of magnitude. Thus the rates are also controlled to a significant extent by the hydrodynamic conditions under which dissolution or precipitation occurs. This was first observed in the field. Deposition rates of calcite from water flowing turbulently across rimstone dams were found to be higher by a factor of up to 5 compared to those observed in the corresponding pools, with identical chemical composition, but almost stagnant water (Liu et al., 1995). Recently, it has been shown experimentally (Dreybrodt et al., 1996a,b) that the model of Buhmann and Dreybrodt ( 1985a,b) is also valid for solutions contained in porous media of calcite both for dissolution and precipitation, respectively, thus proving its basic assumptions. This model, however, is too crude to treat correctly the case of turbulence. Therefore, Dreybrodt and Buhmann ( 199 1) extended it by introducing a diffusion boundary layer separating the bulk of the solution from the surface of the mineral. Mass transport through this layer proceeds by molecular diffusion, whereas in the turbulent bulk complete mixing occurs within extremely short time spans. This extended model predicts rates, which depend on the thickness of the diffusion boundary layer. An experimental means to create such diffusion boundary layers of well-defined thickness is the rotating disk technique (Levich, 1962; Pleskov and Filinovskii, 1976). In this paper we report on measurements of dissolution rates of marble and limestone for various boundary layer thicknesses under COz-pressures from 10-j atm up to 1 atm. These experiments agree satisfactorily with the rates predicted by the diffusion boundary-layer (DBL) model and offer a deeper understanding of the dissolution and precipitation rates under turbulent flow in natural systems. 2. EXPERIMENTAL
METHODS
2.1. Materials Marble and limestone (from Guilin, China) disks, 3 cm in diameter, were cored from marble slabs of 5 mm thickness. These samples were cemented into the holder of the rotating disk apparatus. This was then mounted to the shaft of the rotating disk equipment and polished during rotation by using progressively 400 (64 pm), 800
Fig. 1. Schematic of the experimental set-up: ( I ) controller for rotating speed, (2) motor and gear, (3) rotating shaft, (4) rotating disk, (5) computer for data acquisition, (6) conductometer, (7) conductivity cell, (8) thermometer, (9) temperature sensor for ( 10) solution, ( 12) CO* disperser, ( 13) thermostat, ( I I ) H,O-CO*-Ca’+ CO* + NZ-supply, (14) supply tube, ( 15) water circulation from thermostat to waterbath cooling, (16) reactor vessel.
(32 wm), 1200 (21 pm), 2400 (10 pm), and finally 4000 (4 pm) grit size waterproof silicon carbide paper. This procedure assures a continuous surface of the disk including its rim. The roughness of the surface was measured by mechanically scanning with a diamond tip. Its value was 1.2 2 0.3 pm. This value is 1 order of magnitude lower than the thickness of the smallest boundary layer used in the experiments. The surface area of the marble sample in contact with the solution was 7 cm’. The marble was white and of coarse crystalline structure. It contained less than 1% Mg. To investigate the influence of the slow reaction CO1 + H20 + H+ + HCO; carbonic anhydrase, an enzyme catalysing this reaction was introduced into the solution to obtain concentrations of 0.1 pmole. Bovine carbonic anyhdrase was purchased as lyophylized powder from Sigma and used as obtained. Some experiments were carried out with limestone from Guilin, China, and with a sample of monocrystalline Iceland spar. To obtain solutions in equilibrium with fixed CO2 partial pressures (Pro*) N,-CO2 gas mixtures with PccIi = lo-‘, 5. 10 ‘. IO -‘, 5. lo-‘, IO-‘, and 1 atm were bubbled through the solution. 2.2. Experimental
Set-Up
A schematic of the experimental apparatus is shown in Fig. 1. Runs were performed in a 1.2 L glass beaker containing I 180 mL of a solution, which was prepared by dissolution of calcite powder in bidistilled water to the wanted concentration under a specified CO, partial pressure. The beaker was immersed in a constant temperature water bath capable of maintaining the temperature to ?O.YC. The reaction vessel was covered with Plexiglas lids with holes allowing insertion of the motorized shaft, electrodes, and a gas dispersion tube. A commercial N,-CO, gas mixture with fixed I’,,,> was bubbled through a metal dispersion tube into the reaction vessel. Thus the experiment was performed as a free drift experiment under the condition of an open-system with respect to CO,. In this case the chemical composition of the solution is determined entirely by the partial pressure of CO> in equilibrium with the solution and the Ca concentration (Dreybrodt, 1988). The disk-shaped sample was held centred about 4 cm above the bottom of the reaction vessel on the end of a shaft. A motor drove the shaft, and gear reducers allowed the disk to rotate in the solution at speeds varying from 0 to 3500 rpm (rotations per min) The rotation speed was held constant by a controller within 6%. The rotating disk was formed by use of epoxy from a mold into which the limestone disk and a screw was fitted. This screw served as the rotation axis of the disk and was fixed to the shaft. Then using the rotating shaft as a lathe to which the cast was fixed, it was shaped into its final form, warranting that the whole set was free of asymetric centrifugal forces leading to disturbance of well-defined hydrodynamic conditions. The rock face was polished while the rotating disk was fixed to the rotating shaft of the apparatus.
Dissolution
T: 20°C n-
Pco,:
1x10”
kinetics
of calcium
carbonate
2881
atm
T:20”C
3.12 6
E S j
-
From Levich (1962)
/
0
3.09 /
0 c ‘5 3.06 $ c :=
?? ??
0
1
2
3
4
Experimental
5
6
7
8
2
Measurements of the dissolution rates were performed at a fixed rotation speed by measuring the increase in the electrical conductivity of the solution. The conductivity was measured by a conductivity cell, and the data were stored in a computer at intervals of one second. Then they were converted to calcium concentrations by the relation [Ca’+] (mmol/l) = 6.18. IO-‘*cond. (&cm) - 1.38. 10-Z; yz = 0.999 which is valid in the dilute solutions of our experiments. This relation was found in the following way. First we calculated the chemical composition of the solution for various calcium concentrations and specified Pco, as used in the experiments, by employing the program Equilibrium (Dreybrodt, 1988). Thus we obtained pH, [ HCO;] and [CO;-] as a function of [ Ca’+] These values were then used to calculate the conductivity by WATEQ.4. These results were also verified experimentally by titrating for Ca concentration at the beginning and the end of each run. The record of a typical run with P,oz = 1 . 10 m3atm and initial [Ca”] = 3. 10m4 mol/L is shown by Fig. 2. The straight lines give the increase in concentration at differing rotating speeds from 100 rpm up to 3500 rpm. The rates were calculated from the slopes by =
1 d[Ca*+l A
6
8
IO
12
14
time (hours)
Fig. 2. Ca’+ -concentration vs. time for a typical experiment. The straight lines show a linear increase of the concentration at constant rotating speeds. The numbers on the line denote rotating speeds listed in the inset. The corresponding dissolution rates are also given.
R
4
Exp. of gyp. disso.
dt
Fig. 3. DBL-thickness E plotted vs. w-I’*. The straight line gives the theoretical values of E by use of Eqn. 3. The points mark the experimental values using a sample of gypsum and employing Eqn. 4.
[Cazl] is the equilibrium concentration of gypsum ( 15.4 mmole/L at 20°C). Thus by measuring the dissolution rates at a given rotating speed, t can be determined experimentally. Figure 3 shows the result for four rotating speeds. The full line represents Eqn. 3. The good agreement between experiment and theory shows that the set-up works satisfactorily. 3. EXPERIMENTAL The dissolution sample
at 20°C
rates for
(cf. Eqn.
various
RESULTS 2) measured
partial
on the marble
pressures
of CO*
as a
-4.6
c;v, - -5.1 -
c( ‘E ; -5.6 -
T: 2O”C,
p6.1~~
n -
Pco,(atm)-n:
0
0.1-0.003
A v 0 0
0.05-0.08 0.01-0.13 0.005-O. 12
0.001-0.25 ?? 0.005-0.18 0
where V is the volume of the solution and A the area of dissolving surface. The experimental error obtained by repeating typical experiments for several times was +lO%. The thickness of the boundary diffusion layer (DBL) is given (Levich, 1962) by t = 1.61(Dlv)“‘~(v/w)“*
(3)
where D is the coefficient of molecular diffusion (7.1. 10 + cm’ s -’ at 20°C) and v the kinematic viscosity of water. w is the angular velocity in s -‘. In our set-up E = 5 * lO-3 cm at a rotating speed of 100 rpm, and at 3500 rpm e = 8.5 * 10m4 cm. In all experiments the Reynolds number Re = &J/V (r radius of sample) is below 2. 105, such that laminar flow can be assumed (Pleskov and Filinowskii, 1976). To assure the accuracy of our set-up, we performed dissolution experiments on gypsum. The sample was cut from a piece of a gypsum single crystal (Marienglas, Harz, Germany). In this case dissolution rates are entirely determined by molecular diffusion across the boundary layer (Opdyke et al., 1987) R = 4 ([Gag]
- [Ca”])
(4)
2.0
2.4
2.8
3.2
3.6
Log rotating speed (rpm) Fig. 4. Log of the experimental dissolution rates for various Pcol (listed by the symbols on the curve and by the first row in the inset) as function of the log of the rotating speed in rpm. All curves exhibit straight lines. Therefore, a relation R o( un is valid. The corresponding values of n are listed in the second row in the inset. Open symbols refer to a Ca concentration of 3 x 10m4 mole/L, closed symbols to a concentration of 1 X 10m3 mole/L, respectively.
2882
2. Liu and W. Dreybrodt
Table la. Experimental dissolution rates of marble at 10°C for various rotational speeds and P co2 pressures in atm. (1) The first entry gives the experimental rates, (2) the second one the rates as obtained from the DBL-model, and (3) the ratio of both. Rates are given in lo-’ mmole cm-* s-‘. The rates were measured at calcium concentrations of 3. lo-“ mmole cmm3. Rates
mm
t lcml
0.001 atm
3500
8.5. IO-”
14.40’ 7.042 2.05’ 13.20 6.74 I .96 7.83 5.44 1.45 7.32 4.62
2500
0.001
600
0.002
300
0.0029
150
0.004 1
100
0.005
0.005 atm
6.57 3.90 1.68 5.88 3.13
1 58
function plot.
of w are illustrated performed
0.05 atm
0.1 atm
5.50 3.00
15.50 8.01 1.94 14.30 7.32 I .95 11.20 6.15 I .82 11.20 6.06 1.85 9.50 5.43 2.14 9.50 5.30 I .x9
21.90 8.82 2.48 20.50 8.29 2.47 13.20 8.04 I .64 14.40 8.00 I .80 14.10 7.98 1.I7 15.70 7.98 1.97
24.10 9.95 2.40 21.80 9.58 2.28 19.80 9.22 2.15 20.10 9.21 2.18 20.70 9.20 2.25 21.80 9.20 2.25
1.83 5.32 2.73 1.95 4.46 2.05 2.18 3.30 1.52 2.17 3.55 1.52 2.33 3.04 1.45 2.1
at 1 106.0 66.8
1.59 101.0 65.1 I .55 98.6 54.7 1.80 92.7 52.8 1.76 94.7 50.2 1.89 92.70 48.80 1.90
of I . lo-’ mmole cm-’
to concentrations
We also
0.01 atm
15.20 7.52 2.02 14.30 7.15 2.00 IO.20 5.87 1.74 10.20 4.42 2.31 10.00 4.04 2.48 9.50 4.03 2.36
1.58
* Relates
0.005 atm*
by Fig. 4 in a double experiments
logarithmic
at a Ca concentration
at Pco,of 5 -10m3atm. These data are also plotted in Fig. 4. For low PC,?5 0.01 atm we found that the rates follow a power law
of 10m3 mole/L
R 3:W"
(5)
with exponents n between 0.13 and 0.25. For Pco,2 0.05 atm the dissolution rates are independent on ti. This behaviour shows clearly that dissolution of calcite is complex. If only mass transport by diffusion were rate limiting one would expect n = 0.5 (cf. Eqns. 3 and 4). Table la lists the experimental results plotted in Fig. 4 and also compares them to theoretical values obtained in section 4. Dreybrodt and Buhmann ( 199 1) in their diffusion bound-
ary layer model (DBL-model) suggested that the slow conversion of CO2 into H+ + HCOi might be rate limiting, especially at thicker boundary layers. To investigate this experimentally we performed dissolution experiments adding pure bovine carbonic anhydrase (BCA) to obtain a concentration of 0.1 pmole in the solution. This enzyme without changing the solution chemistry catalyses CO,-conversion thus increasing the conversion rate by about 2 orders of magnitude (Stryer, 1988). It has been successfully used by Dreybrodt et al. ( 1996a,b) to show the rate limiting character of CO,-conversion for dissolution and precipitation of calcite in porous media. Figure 5 shows the dissolution rates for various PC,,at Ca concentrations of about 10 m4mole/L. The open squares represent the dissolution rates at a rotating speed of 100
Table lb. Comparison of experimental dissolution rates of marble obtained in solutions with and without BCA at 3000 rpm and 100 rpm for various Pco2 at 10°C and 30°C. Rates are given in lo-* mmole cm-’ s-‘.
P i-0,
T
Rates 3000 rpm
Rates with BCA 3000 rpm
Rates 100 rpm
Rates with BCA 100 rpm
0.0003 0.00 1 0.005 0.01 0.1 1
30 30 30 30 30 30
90.0 100.0 107.0 110.0 110.0 241.0
89.9 99.9 120.0 195.0 715.0 2160.0
18.8 24.6 24.7 28.6 54.9 163.0
18.6 24.7 47.1 78.6 330.0 729.0
0.0003 0.001 0.005 0.01 0.1
10 10 10 10 10 10
28.8 36.4 37.5 38.3 55.1 142.0
31.6 39.8 111.0 129.0 671.0 1440.0
8.0 8.3 10.9 11.8 36.2 78.6
8.1 14.9 41.5 77.0 320.0 597.0
Dissolution
kinetics
of calcium
2883
carbonate 4. INTERPRETATION
16
EXPERIMENTAL
T: lo”C, [Ca2+] 0.01 atm a significant enhancement of the rates occurs when BCA is added. Measurements
were
lists the rates cal
conditions
with for
performed
and without these
two
at 10°C and 30°C. BCA
under
temperatures
Table
otherwise and
d[ C02]ldt is the change of the CO? concentration resulting from two elementary reactions, occurring neously. CO2 + HZ0 f
H2C03 f
CO?; + OHThe
rate constants
of these
in time, simulta-
H + + HCO;
= HCO, reactions
are well-known
lb
identi-
for various
Pco,. Below Pco, = 5 * 10 m3 atm BCA has no influence on the rates. But at 1 atm the rates are enhanced by a factor of 10. Enhancement at the higher temperature is lower than at 10°C. These results show that some change in the dissolution of calcite occurs at high Pco,. Therefore, one must be cautious when extrapolating from such experiments to dissolution processes in nature, which often occur at low Pco, (
I
0
E
I
I
0
1
I 2
3
4
5
DBL thickness (1 Om3cm)
Z
Fig. 6. The geometry of the DBL-model: The DBL extends from Z = 0 to Z = E. The well mixed core extends from Z = 0 to Z = -6. The arrows denote fluxes of CO:-, HCO;, and Ca*+ from the mineral surface. Note that F is given by the PWP-eqn. (Eqn. 1) employing the activities of the species at Z = E.
Fig. 7. Dissolution rates vs. the DBL-thickness E for various PCo2. The symbols represent the experimental points from Fig. 4. The full lines are obtained from the DBL-theory and are scaled by a factor of 2. The symbols denote the Pcol values used for their calculation. Concentration of Ca was 3 * 10m4 mmol/cm for the open symbols and 1. lo-’ mmole/cm’ for the closed symbol.
2884 Pa2
Z.Liu and W. Dreybrodt [H+l
(Co327
(10%?l)(l0-%lol/l)
MC03‘1
l0%01/1)
~lo-%lol/l) (1
5.8
T: 20% Pcoq:
lxl0-3atm
[Ca2+]:3x1044mmol/cm3
9.6
9.5
9.4
5.7
Fig. 8. Profiles of P,,,, [H+], [CO:-], and [HCO;] across the diffusion boundary layer for t = 0.005 cm, when Pco, = 1. lo-’ atm. F,,z* = 3.7’ IO-’ mmole cm-’ SK’.
(Kern, 1960; Usdowski, 1982). Using these boundary conditions the equations of mass transport can be solved numerically (Dreybrodt and Buhmann, 1991). For general information on chemical reactions in DBL the reader is referred to the textbook of Beek and Mutzall ( 1975). In the following we give results, which have been obtained employing this programme. As a first step we calculated the dissolution rates for the experiments as depicted in Fig. 4. They are listed in Table la. The DBL thickness t was obtained from Eqn. 3. For the experimental conditions with a disk area of 7 cm’ and a total volume V of solution of 1 180 cm3 the ratio V/A = 167 cm. This ratio determines the value of 6 = V/A of the turbulent core (cf. Fig. 6). Table la also lists the experimental values for comparison. All the experimental rates are higher by a factor of about 2. This can be seen from the ratio between experimental and theoretical values. Thus if one multiplies the theoretical rates by a scaling factor of two, they are in satisfactory agreement to the experimental data. This is shown by Fig. 7, which depicts as full lines the scaled theoretical dissolution rates as a function of c for the various CO* partial pressures as used in the experiment. The data points represent the experimental data from Fig. 4. The reason for this behaviour is not very clear. One reason may be that owing to the roughness of the surface in the order of 1 pm the effective surface area for dissolution is higher than the geometrical one which would yield correspondingly an overestimation of the experimental rates. Another reason could be the variability of the PWP-rate constants with varying natural material. An increase of these constants by a factor of two raises the dissolution rates by the same factor (Dreybrodt and Buhmann, 199 1 ). Rate constants of this magnitude have been observed by Compton and Daly ( 1984). Compton et al. ( 1985) have shown that surface roughness exerts significant influence to the rate constant K~. They have measured this constant for samples of Iceland spar polished with various grit sizes. They find K? = 2. lo-” mol cm-’ s-’ for grit sizes of 4 pm
as we have used, twice as large as the PWP-value. Schott et al. (1989) have reported that the dissolution rates on strained calcite are higher by about a factor of two in comparison to unstrained samples. Considering this, a scaling of the theoretical values by a factor of two is feasible to render them comparable to the experimental data. Figure 7 gives evidence that within such a scaling factor the experimental data and the model predictions are in satisfactory agreement. The model shows clearly that the rates depend on t only if the Pco, pressure is low, whereas almost no dependence on 6 is predicted for Pco, 2 0.05 atm and at t > 5 * 10 -a cm. To elucidate this behaviour we use the DBL-model to calculate the concentration profiles in the boundary layer. Figure 8 illustrates these profiles of H+ , HCO, , CO:-, and PCoL across the boundary layer for Pco, = 1 * lOem atm and E = 5 * 10m3 at [Ca”] = 3 * lo-“ mol/L. The boundary to the bulk is at the left-hand side at z = 0. The units for the different species are denoted at the corresponding ordinates. From the profiles it is clear that H+ and CO2 migrate from the bulk towards the solid phase on the right-hand side whereas CO:- and HCO, diffuse towards the bulk. The fluxes of these species are given by the first law of Fick as
FE_& i3Z where c is the concentration of the corresponding species. Therefore, the fluxes are proportional to the slopes of the profiles. Spatial changes of the slopes must be interpreted as changes of fluxes, which result from chemical reactions by which the concentrations of the corresponding species are changed. At the calcite surface (z = 5 *10m3cm) due to the dissolution process there is a flux of HC0.y and CO:- from the
10s
T:20”C, Pco2:lx10-3atm [Ca2+]: 3x10~4mmol/cm3
1o-7
‘v)
v g
10-a
0
E IO-9
‘;;
s
cz 1O-10
1O-1’ 0
5
10
15
20
25
30
DBL thickness ( 103cm) Fig. 9. Calcite dissolution rate (dashed curve) and CO2 conversion rates in the bulk (0) and in the layer (M) as a function of c. Pco. = 1. lo-? atm.
Dissolution
[c03~-]
kinetics
of calcium
carbonate
2885
(a)
WC03‘1
12 T: 2o"C, Pcog: O.latm
T: 2O”C, Pco,: l~lO_~atrn
0,
E(cm): 0: 0 1: 0.0001 1.069-
2-
2: 0.0002
1
b
c
6
3: 0.0005 4: 0.001
1.067-
l-
5: 0.002 6: 0.005 7: 0.01 I 0
I
I
I
1
2
3
, 4
8: 0.02
I 5
Z(10-3cm)
0
Fig. 10. Profiles of Pco2, [H+], [CO:-], and [HCO;] across the diffusion boundary layer for E = 0.005 cm, when Pco2 = 0.1 atm.
2
4
6
Ca*+ concentration
6
10
(1 O~mmol/cm3)
Fca*+ = 9.2. lo-” mmol cm-’ SK’. (W 10
In order to keep the saturation index with respect to calcite sufficiently low, such that this flux can be maintained, CO:- must react with H’ to form HCO, . Due to mass transport by diffusion towards the bulk the flux of HCO, (given by the slope of the profile) increases slightly for two reasons: First, CO:- ions react with H+ ions diffusing from the bulk into the DBL, and second, close to the solid phase boundary CO2 migrating from the bulk reacts to H + , which is then consumed by the reaction with CO:-. Nevertheless, there is still a considerable flux of CO:- into the bulk, such that most of CO:calcite
surface
towards
the bulk.
;m ‘6
a
-
5 j
6 0 =. 9 !! _
,
1 1
T: 2O”C, Pco,: 1xl Oe3atm 6: lcm
E(cm):
0: 1: 0 0.0001
2: 0.0002 3: 0.0005
4
4: 0.001
0
1
2
Ca*+ concentration
3
4
(1 0”mmol/cm3)
Fig. 12. Theoretical dissolution rates (a) and precipitation rates (b) for an open-system at Pcol = 1. lo-’ atm for various values t of the DBL, denoted on the curves by numbers. The corresponding E is listed in the inset.
0
released
-
,o_9
-
r -1 0
1
calcite
dissolution
rate
( 2
DBL thickness
3
4
5
(1 0m3cm)
Fig. Il. Calcite dissolution rate (dashed curve) and CO* conversion rates in the bulk (0) and in the layer (W) as a function of t. Pco2 = 0.1 atm.
from
the solid
reacts
to HC0.y
in the bulk
there
protons
released
by CO2 conversion.
consum-
Therefore, most of the conversion of CO, must be effective there. This situation, however, becomes different when the thickness of the boundary layer exceeds some critical value. If the DBL is sufficiently thick, the time for the species to migrate across the layer becomes so long that the chemical reactions entirely take place in the layer. To show this, we have calculated separately the total amount of CO? converted in the bulk and in the DBL for various t. This is shown by Fig. 9. At a rate in the critical thickness E = lo-* cm the conversion layer equals that in the bulk. For smaller t the rate in the bulk by far exceeds that in the DBL. This explains the variation of ing
2886
Z. Liu and W. Dreybrodt Table 2. Numerical values of 01 = 01(T, P,,,,, t, 6), cf. eqn. 9, for calcite dissolution in the open system. cy
Pcoz = 3 X lo-’ atm c (cm) 0.001 0.005 0.01 0.02 0.001 0.005 0.01 0.02 0.001 0.005 0.01 0.02
(lo-5 cm/s)
Pco, = 1
X
IO-’ atm
Pro, = 5
X
10m3atm
6 (cm)
5°C
10°C
20°C
5 (cm)
5°C
10°C
20°C
6 (cm)
5°C
10°C
20°C
0.1
2.74 2.3 1 1.93 1.40 8.95 3.92 2.40 1.48 9.62 4.02 2.43 1.47
5.02 3.84 3.25 2.37 12.71 6.19 3.90 2.47 13.49 6.23 3.92 2.47
15.67 11.32 8.79 6.3 1 25.08 14.02 9.55 6.44 26.17 14.15 9.55 6.44
0.1
3.81 2.19 1.45 1.07 5.03 2.24
12.38 7.10 5.28 4.19 13.03 7.20 5.33 4.19 13.32 7.23 5.34 4.19
0.1
1.07 5.43 2.28 1.46 1.07
6.17 3.44 2.36 1.76 7.36 3.52 2.36 1.77 7.65 3.55 2.36 1.77
2.77 1.32 1.09 1.08 2.78 1.32 1.09 1.08 2.79 1.32 1.09 1.08
3.83 2.03 1.74 1.74 3.83 2.03 1.74 1.74 3.84 2.03 1.74 1.74
6.37 4.10 3.81 3.81 6.37 4.11 3.81 3.81 6.37 4.11 3.81 3.81
6.75
6.30
5.60
10.10
9.30
8.30
17.10
16.20
14.30
1
10
[Ca’+ ] 10” mmol/cmi
the dissolution rates with decreasing conversion in the bulk is given by
0.2
1.46 1
e: The rate of CO,-
If 5 is sufficiently large only small deviations of [CO,] from equilibrium are necessary to meet the condition Fco,= F,,. Therefore, CO,-conversion is not rate limiting. Molecular diffusion across the layer, however, still plays an important role, and, therefore, the rates depend on e. This also explains why adding carbonic anyhdrase to the solution at low PC,, and for e < 10e3 cm does not enhance the rates, as shown in Fig. 5. We call dissolution under such hydro-chemical conditions bulk controlled. In contrast to bulk controlled dissolution at low Pco, the situation is quite different for high Pcoz. Figure 10 provides the concentration profiles for P,,,= 0.1atm for E = 5 * 10m3 cm and otherwise unchanged conditions. Note there is a steep decrease of the CO:- concentration within 5 - 1O-4 cm, which is mirrored by a steep decrease of H+ due to the reaction to form HCO;. HCOT shows a dramatic increase of its flux towards the bulk in this region. The carbonate ions migrating towards the bulk react with the Ht ions diffusing towards the calcite surface, and, therefore, the amounts of both fluxes must decrease. This is clearly seen by the slopes in the narrow region of 5 . 10 m4cm adjacent to the solid. Close to this region there is a steep increase in the flux of H + towards the solid, which results from conversion of CO2 diffusing from the bulk towards the surface of the calcite. The flux of H’ ions entering from the bulk is much lower than the flux in the region of CO? conversion, as can be judged from the slopes. This supply of H’ is caused by CO, conversion in this region. Outside this reaction region, the concentration of carbonate is practically zero such that no carbonate is transferred into the bulk. HCO; shows a constant flux towards the bulk. Figure 11 shows the CO+onversion rates in the bulk and in the layer for Pcoz= 0.1 atm as a function of E. In contrast to Fig. 9 the critical value of t = 10 m3 cm is lower by 1 order of magnitude. For smaller E the reaction is bulk
0.2
1
controlled. For t > 1 * 10-j cm the conversion of CO2 occurs solely in the layer. Therefore, conversion of CO, is rate limiting and the dissolution rates become independent on E. In this case addition of carbonic anyhdrase is effective in catalysis of CO,-conversion, and the calcite dissolution rates increase upon addition of this enzyme, as depicted in Fig. 5 for rotating speeds of 100 rpm (E = 5 *10e3cm) and also for 3000 rpm (E = 1.10 -3 cm). The decrease of the BCA enhancement factors from 10°C to 30°C (cf. Table lb) can also be understood. The CO,-conversion rate constants increase with temperature (Usdowski, 1982). Therefore, at higher temperature the effect of CA is smaller than at low temperature, and correspondingly the enhancement of calcite dissolution rates is also lower. Summarizing, we have found that in the presence of a diffusion boundary layer two limits exist: At low Pco, (
