A mass transfer model for dissolution and precipitation of calcite from solutions in turbulent motion

Chemical Geology, 90 ( 1991 ) 107-122 Elsevier Science Publishers B.V., A m s t e r d a m

107

[6]

A mass transfer model for dissolution and precipitation of calcite from solutions in turbulent motion Wolfgang Dreybrodt and Dieter Buhmann Fachbereich 1, Institut.~r Experimentelle Physik, Universitiit Bremen, D-2800 Bremen 33, Federal Republic of Germany (Received January 3, 1989; revised and accepted October 24, 1990 )

ABSTRACT Dreybrodt, W. and Buhmann, D., 1991. A mass transfer model for dissolution and precipitauon of calcite from solutions in turbulent motion. Chem. Geol., 90: 107-122. A film theory model for dissolution and precipitation of calcite from a plane surface in a H 2 0 - C O 2 - C a C O 3 solution under turbulent flow conditions is presented. It takes into account: ( 1 ) molecular diffusion across a diffusion boundary layer of thickness e adjacent to the liquid-solid interface; (2) slow conversion of CO2 into HCO3 and H+: and (3) heterogeneous chemical reactions at the surface of solid C a C O 3. The rates depend heavily on the thickness of the boundary layer. The parameter ~ is determined by the hydrodynamic conditions of the problem. Its value can be found either experimentally by dissolution experiments entirely determined by diffusional mass transport, such as dissolution of CaSO4, or by hydrodynamic correlations. We have calculated dissolution and precipitation rates of calcite for various values of Ca 2+ and CO2 concentrations in the solution, and various values ofe. The results are compared to data of calcite dissolution by rotational disc experiments and show satisfactory agreement. Finally, we discuss the consequences of our results with respect to dissolution of limestone in karst terranes.

1. Introduction

The kinetics of calcite dissolution and precipitation is of importance in many geological fields, e.g. denudation of limestone areas, development of karst aquifers or growth of travertine (Dreybrodt, 1981a, b, c, 1987). In a series of recent papers (Baumann et al., 1985; Buhmann and Dreybrodt, 1985a, b, 1987, Dreybrodt, 1987, 1988) a comprehensive model of calcite dissolution kinetics for geologically relevant situations was established. This model is used to predict dissolution and precipitation rates of calcite for thin layers of solution in contact to a CO2-containing atmosphere, which flow laminarly or turbulently on calcite surfaces (open system), or for water with laminar or turbulent flow in fractures,

modelled by parallel planes, without contact to an atmosphere (closed system). The model considers the concerted action of three processes which determine the rate of dissolution simultaneously: (a) The kinetics of the heterogeneous surface reaction of CaCO3 release from the crystalline surface into the solution, using the rate equation established by P l u m m e r et al. ( 1978, 1979). (b) The kinetics of conversion of aqueous CO2 into the aggressive agents H 2 C O 3 , H + and HCOy (Kern, 1960; Usdowski, 1982). (c) Mass transport of the dissolved species Ca 2+, CO~- and HCO3 from the solid surface into the bulk of the solution and the flux of H +, H 2 C O 3 and C02 towards the calcite surface. Laminar flow was approximated by plug flow with constant velocity. Diffusional mass trans-

108

port under these flow conditions is governed by molecular diffusivity. To model turbulent flow in a first crude approximation the molecular diffusivity, Dm, was replaced by eddy diffusivity, De, higher by about a factor 104, De = 104Din, everywhere in the solution. Thus boundary layers and the spatial distribution of eddy diffusivity, dying out at the surface of the solid, were neglected entirely. The predictions of the model have been proved experimentally over a wide range of temperature, CO2 pressure and Ca 2+ concentration for conditions similar to laminar flow at various thickness ~ of the water layer in contact to the calcite (0.04 cm ~<6 ~<0.15 cm ). Conditions of eddy diffusion as they exist in turbulent flow were simulated by stirring vigorously a water layer of ~ 1-cm thickness on top of a calcite surface. In comparison to dissolution rates observed at laminar flow conditions, where mass transport is due to molecular diffusion, at turbulence a steep increase of these rates by a factor of ~ 10 was observed experimentally in accordance to the predictions of the model. This behaviour was postulated by White and Longyear (1962) and termed dissolutional "hydraulic j u m p " . It plays a significant role in processes of karstification. In view of the encouraging results of our previous work an improvement for modelling dissolution and also precipitation under conditions of turbulent flow by considering hydrodynamical and diffusion sublayers (boundary layers) adjacent to the surface of the solid seems to be promising. Different experimental setups have been used to study calcite dissolution under turbulent flow conditions in the laboratory. One is to rapidly stir a suspension of calcite particles in distilled water or in seawater of given CO2 pressure (Berner and Morse, 1974; P l u m m e r et al., 1978 ). The advantage of these batch experiments is to provide a large n u m b e r of dissolution sites at the calcite surface (kinks and steps) and thus the dissolution proceeds rapidly. However, the main disadvantage is the

W. DREYBRODT AND D. BUHMANN

undefined m o v e m e n t of the calcite particles relative to the surrounding water. Therefore, the problem is hydrodynamically not well defined and cannot be treated easily (Levich, 1962). Experiments closer to natural conditions have been performed by Rauch and White (1977) and by Buhmann and Dreybrodt (1985a, b) as mentioned above. The former authors recirculated water through tubes bored into limestone at Reynolds numbers of 1800, where turbulent flow is established. They measured dissolution rates for a variety of natural limestones. Hydrodynamically these experiments are much better defined. Another type of experiments makes use of a rotating disc apparatus. This system is hydrodynamically well known and suitable for calculations (Levich, 1962; Pleskov and Filinovskii, 1976). It consists of a calcite disc rotating at a known angular velocity in an aqueous solution of known composition (Sj~berg, 1978; Herman, 1982; Rickard and Sj6berg, 1983; C o m p t o n and Daly, 1984). The disadvantage of these experiments is that as a consequence of the small surface of calcite ( ~ 1 cm 2) exposed to the dissolving solution with a volume of ~ 500 cm 3, long times are needed to observe the dissolution process experimentally. Although most of these experiments are conducted at laminar flow conditions for the bulk, mixing in the bulk by fast convective transport is so effective that similarity to turbulent flow can be assumed, as we will show later. Recently Wallin and Bjerle ( 1989a, b) measured dissolution rates of limestone at constant pH by using rotating cylinders of Carrara marble with rotational speeds between 100 and 1500 r.p.m. These experiments were conducted at low pH < 5.3, where dissolution rates are governed almost entirely by diffusional transport o f H + ions to the surface of the solid. In all these investigations a diffusion boundary layer with concentration gradients between the solid surface and the bulk is postulated.

DISSOLUTIONAND PRECIPITATIONOF CALCITEFROM SOLUTIONS1N TURBULENTMOTION

This layer determines the concentration of H +, Ca 2+, CO 2-, H C O ; at the solid surface and therefore influences the dissolution rates resulting from the heterogeneous reaction. To compare all these different types of experiments a model giving information on diffusional boundary layers to mass transport is required. In the following sections we will develop such a model of dissolution under turbulent flow conditions in the bulk. Then we will show some results of calculations for open-system conditions in geologically relevant situations. Finally, we will discuss several experimental results as described in the literature and compare them to the theoretical predictions. 2. Film model of mass transfer

All these experimental situations can be described by a film model, the theoretical basis of which is discussed in detail by Levich (1962) and Kay and N e d d e r m a n n (1985). This model is especially suitable if mass transfer and chemical reactions occur simultaneously as in the case in the dissolution of calcite. The liquid is divided into two regions. In the thin diffusion boundary layer of thickness e, mass transfer is entirely by molecular diffusion. In the region outside of this layer fully developed turbulence exists and eddy diffusivity is so high that a constant concentration, c, of all species exists everywhere. Fig. 1 shows this schematically for a plane surface covered by a fluid layer of thickness ( 8 + e). For the region 0>z>-8 the concentration of all species is constant due to perfect mixing by eddies. For the diffusion boundary layer (film) 0 < z < e the concentration varies approximately linearly in z such that the mass flux from the dissolving wall into the bulk is given simply by: F=

( D m / ~ ) (Cw --CB)

( 1)

where Cw is the concentration at the solid; and cs that in the bulk. However, this equation is no progress unless one knows the value of e. This can be done by dissolution experiments

109

diffusion boundary toyer

/-)/furbutent //\ -

core

Q i ( - i

-5

0

c

Z

Fig. 1. CaCO3-H20-CO2 system with turbulent core and diffusion boundary layer (3>>~). 6+e represents the thickness of water layer in contact to the solid.

where mass transport is entirely by diffusion as in dissolution of CaSO4 or CaSO4-2H20. In this case Cw= Ceq is equal to the saturation concentration and CB and F can be measured experimentally. Therefore for a given experimental situation, e can be determined (Opdyke et al., 1987). This value of ~ is then used for the dissolution of calcite, where the chemical reactions at the surface and of CO2 conversion are included into the film model. Another way of estimating the value of e in a given hydrodynamic environment is by use of hydrodynamic semi-empirical correlations. This is discussed in detail by Levich (1962). ~ is the thickness of the diffusion boundary layer, which is related to the thickness of the hydrodynamic boundary (viscous sublayer) layer eh by:

6_=f_h'Sc-l/3"~

Sc=v/D

(2)

where Sc (Schmidt number) is the ratio of viscosity, v, and the constant of diffusion, D. This boundary layer concept goes beyond the assumption of the simple Nernst model, since it includes implicitly convective diffusion resulting from the flow field in the viscous sublayer into the value ~, which accordingly depends on the flow velocity (Levich, 1962 ). In turbulent flow in a tube, where the inner

I 10

W. DREYBRODT AN[) D. BUHMANN

walls are the dissolving material, and in turbulent flow past a plane dissolving plate, it is possible to estimate ~ by semi-empirical correlations of the Sherwood number, Sh, which is related to e by:

Sh=Lc/~

(3)

where Lc is a characteristic geometrical dimension as the diameter of the tube or the length of the plate. SkeUand (1974) reports Sherwood numbers in correlation to the Reynolds n u m b e r and the Schmidt number:

Sh=O.O23"Re°S3.Scl/3;

Re=vd/v

(4)

for smooth circular tubes of diameter d, where v is the average flow velocity; and v the kinematic viscosity. For flat plates of length L one has:

Sh=O.O37.Re°8.Sc~/3;

Re=vL/v

(5)

Using these correlations and eq. 3 it is possible to estimate e for the above-mentioned cases. In the case of a rotating disc one obtains (Levich, 1962): Sh=0.01

.Re°9.ScJ/4;

Re=aeog/v

(6)

in case of turbulent flow, where a is the radius of the disc and ~o the angular velocity. In the case of laminar flow below Re~ 104 the diffusion boundary layer is given by: f:'~ 1.8(P/0))1/2"Sc-1/3

(7)

Finally, we want to mention the work of Bjerle and Rochelle (1984) and Wallin and Bjerle (1989a, b), who investigated calcite dissolution at low constant pH by a similar film model as discussed here and found for a rotating cylinder of given dimension a correlation to its rotational speed n (r.p.m.). fm5.103n

3/4

(am)

(8)

Using this value of e these authors have developed a film model of mass transport at low pH which is kept constant. They have included

slow conversion of CO2 and have assumed that at the calcite surface the solution is saturated. Dissolution rates are calculated for this model for various pH, CO2 concentrations, and Ca -~+ concentrations of 10- l mol 1- ~. The results are in excellent agreement to the experiment and show that the film model in its simple form is a useful approximation. The investigations of Wallin and Bjerle aim at the geological situation of dissolution of limestone in acid lakes, since pH is constant during the process of dissolution. Our investigations try to give answers to dissolution of limestone by karst waters. In this case pH values are ~ 7 and change with uptake of calcite into the solution. Furthermore, the reaction kinetics of CO2 are more complicated at high pH (Kern, 1960; Usdowski, 1982), and finally the boundary condition of a saturated solution at the calcite surface breaks down. 3. Dissolution of

CaCO3in the film model

In modelling CaCO3 dissolution one has to include the slow chemical reactions of C O 2 conversion into H + and HCO3 into the film model described in section 2. The geometry of the problem is shown in Fig. 1. A calcite surface is covered by a water layer of total thickness ( ~ + e ) . Adjacent to this surface we have a diffusion boundary layer of thickness e. In all cases e << ~ (Levich, 1962 ). In the region 0>~z>~ - 3 motion of the fluid is turbulent and diffusion is characterized by high eddy diffusivity De >> D .... such that concentration gradients vanish and the zone is well mixed with constant concentration everywhere. This region is also called turbulent core. Dissolution of calcite takes place at z = e, resulting in fluxes of Ca 2+, CO~- and HCO~ ions from the surface into the bulk. We assume that the well-mixed zone is at rest on average with respect to the dissolving surface as is the case in experiments such as a stirred solution or rotating samples. The problem of fluid flow past a surface, e.g. in a tube,

DISSOLUFIONAND PRECIPITATIONOF CALCITEFROM SOLUTIONSIN TURBULENTMOTION

as it occurs under natural conditions will be discussed at the end of this section. To calculate the dissolution rates, we have to solve the transport equations for each dissolved species (Buhmann and Dreybrodt, 1985a, b):

-O[C02]/Ot+ Dco2( O2[CO2]/Oz 2) = R c 0 2 (9a)

since each Ca 2+ ion released is accompanied by one carbon atom. Thus: -

D(O [HCOT ]/Oz)l~ - D ( O [CO~- ] / O z ) = - D ( O[Ca2+]/Oz)=F (11)

Using these boundary conditions eq. 9d can be solved: [Ca 2+ ] (z) =

3[H + ] [ C O 2 - ] - k + 3 [ H C O ; ] (9b)

-O[CO~- ]/OI+ D( O2[CO2- ]/Oz2)= -k

3[H+ ] [CO2- ] + k + 3 [ H C O ; ]

(9c)

])[CO2]

- (k_, [H+] + k _ 2 ) [ H C O ; ]

(9e)

F=a:l ( H + ) , +K2(H2CO;)~ 2+ )~

0~
FeO-l(a+e)-'(le+(~)

+ [Ca 2+]~; =F(fi+e)-~t

(12)

z~0

The second step is the solution of the coupled differential equations ( 9 a ) - (9d). Adding these equations leads to: 0[COz]/at+O[HCO;]/at+O[CO~-]/at

The left-hand sides of these equations represent mass transport, whereas the right-hand sides express chemical reaction rates. The expression R with rate constants k+ l, k+2, k and k__2 in eqs. 9a, 9b and 9e describes the slow conversion of CO2 into H + and H C O £ . This implies that during dissolution of CaCO3, CO2 and H C O ; are not in equilibrium with each other. The reaction of H + + C O 2- a H C O ; is rapid and is only given for completeness. Square brackets denote concentrations. Details of the constants used and their temperature dependence are given in the work of Buhmann and Dreybrodt ( 1985a, b). The boundary conditions of the problem are as follows. The flux of Ca 2+ ions released from the surface at z = e is given by the PWP equation ( P l u m m e r et al., 1978, 1979) as:

nt-K3 -t- K4 ( H C O ; ) t ( C a

+ [Ca2+]~; [Ca 2+ ] (z) = [Ca]¢

-O[Ca2+ ]/Ot+D(O2[Ca2+ ]/Oz2)=O (9d) Re02 = ( k + t -t-k+2 [ O H -

-FD-1 ( fi+ ~) --1 x [ ~1 ( z"~- - ~ 2 ) + a ( z - ~ ) ]

-O [HCO; ]/Ot+ D( O2[HCO; ]/Oz 2) = -Rco,+k

I 1I

(10)

where ( ) , denote the corresponding activities at z = e, i.e. at the solid surface. The total flux of carbon atoms away from the surface has to be equal to F, the flux of Ca 2 +,

(O2[C02]/Oz :) + D ( O 2 [ H C O ; ] / O z ~)

=DcQ

+D(O 2[CO~- ]/Oz 2)

( 13 )

If the system is closed to exchange of CO2: 0[CO2]/0t+0[HCO3 =F/(fi-e)

]/Ot+O[CO3- ]/Ot

~F/6

(14)

because the increase in concentration of carbon atoms is related by stoichiometry to the flux of Ca z+ from the solid surface. Thus eq. 13 can be integrated: D(:o~ (O [COz ]/Oz) +D(O [ H C 0 3

]/Oz)

+ D ( a [ c o ~ - l/az) = F z / ( f i + e ) + C,

(15)

The integration constant C~ can be calculated from the flux of carbon atoms across the boundary at z = 0: C 1= F O / ( a + e )

(16)

Second integration ofeq. 15 yields: [CO2] "[- [ H C O ; ] + [ CO2- ]

=½Fz2/(~+e)+F&/(~+e)+C2

(17)

1 12

W. DREYBRODT AND D. BUHMANN

where C2 is calculated from the concentrations at z = e. Thus: C2 = [CO21~+ [HCO~-]¢+ [CO~-]¢

+ lF~2/(~+e)+F&/(6+c)

(18)

Using mass action equations: Kw=yHyoH[H+][OH -] K2-

(19a)

7H Yco3[H + ] [CO 2- ] ~HCO3 [ H C 0 3 ]

(19b)

2[Ca 2+1+ [H +1 -]

(20)

Inserting mass action equations ( 19 ) into eq. 20 yields an expression for [H + ]: [H + ] -- ½ (2 [Ca 2+ ] - [HCO~- ] ) + {¼(2 [Ca 2+ ] - [HCO~- ] )2 + K w / y n 7OH

+(2K27HCO3/THTCO3)[HCO;]} '/2

(21)

The activity coefficients y are calculated by extended Debye-Hiickel theory using a z-independent average ionic strength: I=½{4[Ca 2+] + 2 [ H C O ; ] + 4 [ C O 3 1

+ [H+]+ [OH-l} ~ 3 [ C a 2+ ]

(22)

The concentration of CO 2- can be calculated as a function of HCO~- and Ca 2+ by inserting eq. 21 into the mass action equation (19). Inserting this into eq. 17 yields a cubic relation for [HCO~-] as a function of [Ca 2 + ] (z), [CO2 ] (z) and z only: L [ H C O y ]3+M[HCO~- ]2 +N[HCO~- ] + P = 0

L = K2 (THCO3/YH7CO3) M = (Kw/YH ~ o u ) - L 2 + 3 B L + 2 L [ C a 2+ ] U = 2B{ (Kw/ P = YHYOH)+L[Ca 2+] +BL} B 2(Kw/TH Yon )

(24)

and the function B which is dependent of [ CO2 ] and reads:

B = D - ' { - F / ( d + e ) (½z2-dz+ ~1 ¢ " + & )

and the condition of electroneutrality it is possible to express all the quantities on the righthand side of eq. 9a, i.e. [H+], [ O H - ] , [HCO;-] and [CO2] as a function of [CO2] and z. Thus eq. 9a can be decoupled and solved numerically. The equation of electroneutrality reads:

=[HCOT]+2[CO2-]+[OH

with the coefficients:

(23)

- Dco2 ( [ CO21~ - [ CO2 ] ) - D [ C O ~ - 1~- D [ H C O ; 1¢}

(25)

The solution of this cubic equation expresses [HCO~- ] (z) as a function: [HCO~- ] (z) = f ( [Ca 2+ ] (z), [COz ] (z),z) (26) where [Ca 2+ ] (z) is given by eq. 12. From [ H C O 3 ] ( z ) one can calculate [ H + ] (z) by use of eq. 21. Thus the reaction rate in the right-hand side in the transport eq. 9a for CO2 which depends on [H ÷] and [ H C O ; ] can be expressed as function depending on z and on [CO2] (z). The differential equation can now be solved numerically by a Runge-Kutta procedure (Kamke, 1967 ). The calculation of dissolution rates is performed following the flow diagram in Fig. 2. We start with fixed reasonably chosen values of temperature T, thicknesses e and & Ca 2+ concentration [Ca2+]~, CO2 concentration [CO2]~. We also select a plausible value of Ca 2+ flux, F. From this we calculate the ionic strength and the activity coefficients of the individual species. Then we compute the concentration profile of Ca 2+ (eqs. 12). We then change the bicarbonate concentration at z = e until the PWP equation (10) yields the chosen flux F of Ca 2+ ions. With the value of [HCO3 ]~ we can obtain [H+]~ by eq. 21 and [CO2- ]~ at z = e by application of mass action equations ( 19 ) and the equation of electroneutrality (20) (cf. Buhmann and Dreybrodt, 1985a, b).

DISSOLUTION AND PRECIPITATION OF CALCITE FROM SOLUTIONS IN TURBULENT MOTION

]nput T, 6, C [Ca 2.1~ , [CO21~ , F Compute I,'~i ,[Co2+](Z) Change [HCO]]E unfit the PWP equation yietds the stated flux F

ICompute tH'l , Eco -l l I

Runge - Kuffa procedure [HCO] ]o , [H*] o , [CO~-]ol[CO2 ]o, Fco2 = / Rco2d z Change [ CO2 ]c

IYes

l

End

1

Fig. 2. Flow diagram of the numerical calculations. T h e index 0 indicates c o n c e n t r a t i o n values at z = 0.

Starting with these concentrations at z = E we calculate the concentration profiles of the carbonate species and H + using the R u n g e - K u t t a procedure, in the region 0 ~
Fco2 =

--5

Rco2"dz

(27)

where Rco2 is given by eq. 9e. This value must be equal (but oppositely directed) to the flux F o f C a 2+ from the calcite surface into the bulk, since each Ca 2+ ion released into the solution uses one molecule o f CO2 for converting the accompanying CO ] . ion into H C O ; - . Therefore, we change [CO2]~ in an iterative procedure until Fco2 agrees with the chosen value of F w i t h i n < + 1% (cf. Fig. 2).

1 13

As a result we obtain the C O 2 concentrations at z = 0 and z = e for a given Ca 2+ concentration of the solution and a given Ca 2+ flux, F. Changing F over a wide range leads to a function of [CO2]o as shown in Fig. 3, where the CO2 concentration is given as Pea2, the pressure in equilibrium to this concentration. In the following figures instead of the term flux we use now dissolution rate which is synonymous. The different lines in Fig. 3 are characterized by different thicknesses e of the boundary layer; the thickness 5 of the turbulent core is assumed to be fixed ( 5 = 1 c m ) . From such graphs as in Fig. 3 we derive Ca 2+ fluxes for any CO2 pressure (cf. dashed lines for three different CO2 pressures) and [Ca 2 + 1. These results are valid for the closed system with defined actual CO2 concentration in the turbulent zone. They are, however, also valid for an open system with identical CO2 concentration. This is the case because the chemical composition is then the same for both systems due to equal CO2 concentrations. Therefore, the dissolution rates must be identical. As dissolution proceeds, the dissolution rates become different in both systems, since under closed conditions CO2 concentration drops. Therefore, it should be stressed that the dissolution rates in this context depend only on the CO2 concentration in the bulk and are the same, whether the system is closed or open to CO2 exchange. In the following we will therefore give dissolution rates for fixed CO2 concentrations, related to the CO2 pressure Pc~ in equilibrium with the solution. This corresponds to an open system. If one wants to calculate dissolution rates for closed systems with given initial concentration [CO2]i one has to consider that after dissolution starting at [Ca 2+ ]i = 0 to the concentration [Ca 2+ 1 the CO2 concentration is given by (Dreybrodt, 1988 ): [ C O 2 ] = [ C O 2 ] ~ - [Ca 2+1

(28)

So far we have discussed calcite dissolution

1 14

W. DREYBRODT AND D. BtJHMANN

1.0

0.005

0.0002

(00) 1.10 ~'

0.0001

10 1

E

4-C3 OJ t/I t,q

10 2

CL

5.10 j otn~

[Co2"]~ : 3.10 ~ mmo[. cm 8 : l/m T = 20%

c~ u 10 J

I, 10~ atm

3.10 ~ at m

10 ~.J~ 0

5

10 dissotufion

rote

15 ( 10 -7 m m o t .

20

25

crn - 2 . s -1 )

Fig. 3. Calculated CO2 pressure in open systems as function of the dissolution rate of calcite at different thicknesses of the b o u n d a r y layer ( n u m b e r s on the curves in c m ) . The three indicated CO: pressures ( d a s h e d lines) refer to conditions of Fig. 4.

as it occurs in laboratory experiments. In this situation the concentrations of the species increase in time as given by eqs. 14 and 12. Because of perfect mixing in the core the gradient of concentrations is zero everywhere and advective terms v.grad c do therefore not occur in the transport equation for z.%<0. Of course advection does occur in the hydrodynamic boundary layer. However, this is taken into account by the thickness of the diffusion boundary layer, which is the region where to a reasonable approximation the concentrations vary linearly in z (Levich, 1962). In the case of flow passing a plane surface, or along a circular tube in the direction x a stationary state is established such that at each fixed position x all the time derivatives of the concentrations vanish. Instead of this, gradients of concentration must develop. For [Ca 2÷ ] (x) mass balance requires:

vx{O[Ca 2+ ] (x)/Ox}

(29)

=Fl(d+e) ~F/5 This equation states that the advective mass flux at position x results from the mass removed by dissolution between ( x - d x ) and x with rate F from the surface, v,. is the flow velocity in the turbulent core. Therefore instead Oc/Ot in eqs. 9a-9d we have to consider advective terms vxOc/Ox, which replace the derivatives with respect to time in the region of turbulence. In the diffusion boundary layer equation (9d) now reads: D(02[Ca 2+]/0z 2)=v~(O[ca 2 + ] / 0 x )

(9d')

Further, the solutions are those of eq. 12 with [Ca 2+ ]~ replaced by: [ C a 2+ ]~ =

(x/u,-) [FI(5+{)]

(12')

DISSOLUTION AND PRECIPITATION OF CALCITE FROM SOLUTIONS IN TURBULENT MOTION

Eq. 13 is then replaced by:

C O 2 "]- C 0 2 3 - + H 2 0

v,.(0 [C02 ]/Ox) + vx(O [ H C O ; ]/Ox)

CH + +HCO3 +CO~-

+v,(0[co l/Ox)= Dco2(O2[CO2l/Oz2)

~2HCO3

+ D ( 0 2 [ H C 0 3 ]/Oz 2)

+ D(O2[CO~- ]/Oz 2)

(13')

Correspondingly, mass balance changes eq. 14 into: /fl,c(0 [CO2]/Ox) -Jf-l).<(O[HCO3 l/Ox)

Because the first step in this reaction is slow CO2 conversion, while the second reaction is instantaneous, the conversion rate of CO~ into HCO~- is determined by slow CO2 conversion. Thus the distance travelled by a CO3 ion before it vanishes by conversion into HCO3 is given by the diffusion length:

2 = (Din/kco2 ) '/2

l/Ox)= F~ ( fi+e) "~F/fi

1[5

(14')

Therefore, eqs. 15-27 remain unchanged and their results are also valid for dissolution in flow along dissolving surfaces as is the case under natural conditions. 4. Results

Fig. 4 shows dissolution rates calculated from the film model for open-system conditions, at various Pco2 pressures in equilibrium with the solution, as a function of the concentration of dissolved Ca 2+. The different curves in Fig. 4a-c are calculated for different thicknesses of the diffusion boundary layer. The uppermost curves have been calculated by assuming = 0, i.e. neglecting the existence of a diffusion boundary layer. This is the extreme situation of highest possible dissolution rate (Buhmann and Dreybrodt, 1985a, b). With increasing thickness e, the rates decrease until they crowd into a lower limit for a value ~ of ~ 0.02 cm. The existence of this lower limit can be explained by examining the concentration profiles across the boundary layer as shown by Fig. 5. For e = 0.02 cm, the concentration profile of CO 2- shows a steep decrease away from the surface. At a distance of ~ 0.01 cm the bulk concentration is attained. The concentration of H ÷ mirrors CO~- and decreases toward the solid. This behaviour is due to the reaction:

where kco2 is an effective rate constant for CO2 conversion (Buhmann and Dreybrodt, 1985a, b). If e >/2, mass transport is determined by the chemical reactions, which take place in the layer of thickness 2 and the dissolution rate becomes independent of e. This is the case for the concentration profiles for e=0.02 cm and e=0.01 cm in Fig. 6. For the case ~ << ;t mass transport of C a 2 + ions via molecular diffusion across the boundary layer becomes rate limiting. The concentration profiles are also shown in Fig. 5. Because now most of the CO 2- ions can diffuse across the boundary layer, without reacting to HCO3, conversion to HCOy takes place in the bulk and the dissolution rates are controlled by diffusion of CO ] - across the boundary layer. If the diffusion boundary layer vanishes, i.e. --0, the rates are entirely determined by the heterogeneous surface reaction as given by eq. 10 and constitute the maximal possible dissolution rates. Because the PWP equation is a rate equation, containing forward and backward reactions, and is valid also for precipitation of CaCO3 (Inskeep and Bloom, 1985 ), the transport equations can also be solved in principle for supersaturated solutions. The sign of F in eq. 10 has to be changed to obtain deposition (precipitation) of CaCO3. Fig. 7 gives an example of the deposition rates from supersatur-

1 16

W DREYBRODT

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5

6

7

8 0

5

10

15

Ca:" concenfrofion (!0 ~ mmoL cm3) Fig. 4. Dissolution rates o f calcite in an open system as function o f the Ca 2+ concentration in the solution at three different CO2 pressures. Crossed squares on the curves refer to calculations o f Fig. 3. The arrow in (c) indicates the situation for which the concentration profiles o f Fig. 5 are calculated. The numbers on the curves are related to the thickness ~ o f the boundary layer as listed in ( c ) . Thickness o f the water layer: 6 = 1 cm.

10'

BI 6

E ~,E z

ld

o

F-

lC0~ ] * 10s

N Z 8

T =

20°C

[Co2*]~ = 5.10-~mmol.cm-3 Pcoz = 5 "10-3otto 10"1 20

18

16

1'1.

1~2

10

8

6

--4

[ 1 ~

DISTANCE FROM SOLID SURFACE

Fig. 5. Concentration profiles o f H +, CO 2- , HCO~- and CO2 across the diffusion boundary layer for ~ = 0.02 cm, 0.01 cm and 0.002 cm. N o t e that the corresponding values in the bulk are these at z = 0 , i.e. where the distance from the solid surface is ~.

ated solutions with initial Ca 2 ÷ concentrations up to 3 . 1 0 - 3 mol l - 1 under open-system conditions at different CO2 pressures. The upmost curves in Fig. 6a-c again show the limiting situation o f turbulence without boundary layer, i.e. e = 0, resulting in the highest rates possible under the given boundary conditions. With increasing thickness c of the boundary layer the rates decrease and in the case o f Pco2 = 5- 10- 3 atm. again converge towards a lower limit at e ~ 0 . 0 1 cm. Thus, the behaviour of the rates with respect to the layer thickness c is exactly analogous for deposition and dissolution. In the following we will therefore restrict ourselves to dissolution rates. To obtain more insight into the dissolution processes, we investigated the effect of arbitrarily changing either the rate constants of the PWP equation ( 1 0 ) , the CO2 conversion rate (eq. 9d) or the coefficient of molecular diffusion. This procedure can help to find the individual contributions of CO2 conversion, diffusion or heterogeneous calcite dissolution

DISSOLUTION A N D PRECIPITATION OF CALCITE FROM SOLUTIONS IN T U R B U L E N T MOTION

1 17

I : 20%

-10 ~7 E u

-6

@

©

pco2 : 5.103o.fm

Pco2 : 1.103arm

Pea : 3 10~c:ffm

~- Icrn]

0

-8- 0 0

i 0 000! 2 0 0002

E E

0 0003

3 [

OOO: C,,C,O~ 0 005 00,I 602

GJ

L.

c, f-O 4--. O

2 6

-2

i

s

i

67

0--.-1

2

3

1

2

Co ~" c o n c e n t r a t i o n

3

0

1

"

~

3

( t0 ~ mmo[. crn 3 )

Fig. 6. D e p o s i t i o n rates o f calcite in an open system as function o f the Ca 2+ concentration in the solution at different atmospheric CO2 pressures. The numbers on the curves are related to the thickness ~ o f the boundary layer as listed in (a). Thickness o f the water layer: 5 = 1 cm.

15

'"tCo,2*lc = 5.10~'mmol • cm 3

't/)

Pc02 { =1 20%'10 3Qfm

E

r:= E

10

t---

o o..i -4---

oL

5

CO -.I--

6:1Cm : 0.001 cm

0

/ 0

0,1

10

1

fQcfor

100

n

Fig. 7. Dissolution rate of calcite as function of the factor n by which the PWP equation is multiplied. (= 0.001 cm. ( P W P e q u a t i o n ) to the overall d i s s o l u t i o n rates. Fig. 7 s h o w s the c h a n g e o f d i s s o l u t i o n rates resulting from m u l t i p l y i n g in the P W P equat i o n ( 1 0 ) all rate c o n s t a n t s Ki by a c o m m o n

factor n. This m e a n s that the rates o f the three e l e m e n t a r y reactions c o m p r i s i n g the heterogen e o u s reaction at the CaCO3 surface are increased in p r o p o r t i o n to n if n > 1 and are decreased c o r r e s p o n d i n g l y if n < 1. T h u s a factor

1 18

W. D R E Y B R O D T

of n = 0 would stop the reaction entirely. All the other parameters of the dissolution process are kept fixed. One can see from Fig. 7 that small variations in the rate constants xi, 0.5 < n < 2, result in a linear change o f the overall dissolution rates. Larger variations (n > 5 ) in the rate constants, however, induce changes in the dissolution rate, which converge to an upper limit for very large n. For n - , ~ , the surface reaction becomes so fast that the surface concentration of [Ca 2+ ] is at saturation, resulting in mass transport determined entirely by diffusion across the b o u n d a r y layer into the bulk with concentration [Ca 2+ ]. Therefore, we have: F=De

'([Ca2+]~q-[Ca2+])

(30)

as limiting rate. For the data in Fig. 7 this value is ~ 30 times larger than the value at n = 1. This shows that for sufficiently small e diffusion becomes so fast, that the surface reactions become rate limiting and determine the dissolution rates entirely. Fig. 8 demonstrates the influence of molecular diffusion and chemical conversion of CO2. ]'he upper curve shows the change in dissolution rates, which results from an increase

E : Ocm 15

T c~

-~Rcoz=

E t_J

-6 E E

@.Rco2

10

o

D c_

5-

ro .4.-

[C02÷]E = 5 ' 1 0 ¢ m m o l • cm 3

~ = I cm

pco2 = 2.45 • 10 .2 o,fm T : 20%

~: = 0 . 0 0 0 1 cm

u,i ~

0-

~b

16o

~o'oo

facfor n,x Fig. 8. Dissolution rate of calcite as function of factors n and X by which the CO2 conversion rate and the diffusion coefficient, respectively, are multiplied, e =0.0001 cm.

A N I ) D. B U H M A N N

of the molecular diffusivity Dm by a factor X. In the case of very large X the value of the diffusivity in the diffusion boundary layer approaches that of eddy diffusivity in the bulk. This is equivalent to a vanishing boundary, layer and the rates are maximal and given by the P W P equation (10). In contrast to this behaviour, an acceleration of the CO2 conversion (eq. 9d) by multiplying all rate constants ki in eq. 9e by a factor n up to n = 100 yields only a slight increase in the rates. At higher values of n the rates increase slowly which indicates that delivering H + by CO2 conversion inside the boundary layer becomes important. At normal conversion ( n = 1 ), diffusion o f H ~ from the bulk into the boundary layer is the dominant source of H +. With increasing conversion rate H + is produced in the b o u n d a r y layer close to the surface and therefore diffusion from the bulk is no longer important. Finally, at very high n the dissolution is governed exclusively by the surface chemistry ( P W P equation; dashed line in Fig. 8 ). In summary, we find that the inhibiting effect of the boundary layer is mainly due to the slow diffusion of the ionic species and at small layer thickness e due to the heterogeneous chemical reactions at the calcite surface ( P W P equation), cf. Fig. 6. The conversion of CO2 into H C O 3 and H + is of minor influence but becomes important for thick boundary layers. At that point we would like to stress the significance of the value of 6 in our model. Conversion of CO: into H + and H C O 3 and the reaction H + + C O 2- ~ H C O ; occur in both, the boundary layer and the turbulent bulk, cf. eq. 27. Since the a m o u n t of CO2 converted depends on the volume of the solution CO2 conversion becomes rate limiting at small fi ( ~<0.1 cm ) and low CO2 pressures (Pco2 ~ 3 . 1 0 - 4 atm. ). For our case of 5 = 1 cm, this is not observed because the bulk volume is sufficiently large ( B u h m a n n and Dreybrodt, 1985a, b). 5. Discussion Calcite dissolution and precipitation rates

[ 19

I ) I S S O L U T I O N .AND P R E C I P I T A T I O N O F C A L C I T E F R O M S O L U T I O N S IN T U R B U L E N T M O T I O N

under various conditions of flow can be calculated by the m e t h o d described in the preceding section. However, the thickness e of the diffusion b o u n d a r y layer is a parameter which must be known a priori in our model. Therefore, it is necessary to restrict our attention to hydrodynamic situations where the value of e can be predicted by known correlations between Sherwood, Reynolds and Schmidt numbers (cf. eqs. 3-8 ). In more complicated situations it is possible to determine the value of e by measuring mass transfer coefficients for an hydrodynamically equivalent problem where mass transport is determined entirely by diffusion ( O p d y k e et al., 1987). One tool for studying dissolution in the presence of, diffusion b o u n d a r y layers is the rotating disc. Levich (1962), and Pleskov and Filinovskii (1976) have solved the mass transport problem of a rotating disc with smooth surface and sufficiently large radius such that edge effects b e c o m e negligible in a viscous incompressible fluid for laminar flow. Our model can also be applied to the rotating disc in laminar flow (Levich, 1962). Since depends on the angular velocity co of the rotation, the rotating disc experiment can be used to change the thickness of the diffusion boundary layer and to observe its influence on the dissolution of calcite. This experiment was performed by H e r m a n (1982) and H e r m a n and White (1985) who examined the relation between dissolution rates of calcite and dolomite and the rotational speed co. By use of eq. 7 we have calculated the diffusion b o u n d a r y layer thickness. This value and the data of the experimental conditions (Herman, 1982 ) were used to predict the rates by our model. A comparison between the rates thus calculated to those observed experimentally is shown in Fig. 9 as a function of ~. The upmost curve in Fig. 9 shows the measured initial rates of calcite dissolution at [Ca 2+ ] ~ 0. They increase sharply at e ~<0.003 cm, i.e. at a Reynolds n u m b e r of R e > 15,000. At higher Ca 2+ concentrations this increase is less pro-

pc02 = 1 arm 6 = l cm

T = 25% E u E

E

2

11

o

O L C 0

I-

×

izl

@

FT~ :L~,'

~5 00

0.01

tayer

fhickness

0('2

c

(cm)

Fig. 9. Dissolution rates of calcite as function of the layer thickness at different Ca 2+ c o n c e n t r a t i o n s of the solution. Dashed curves=experimental values from H e r m a n ( 1 9 8 2 ) , o b t a i n e d by rotating disc experiments: solid squares= initial rates ( [ C a 2+ ] ~< 1-10 -4 m m o l c m - ~ ) ; crosses, [Ca 2+ ] = 2 . 2 - 1 0 3 mmol cm 3; open triangh,s, [Ca 2+ ] = 5 - 1 0 -3 m m o l cm-~. Solid curves=calculaled rates; the numbers on the curves refer to the Ca 2+ concentration in the solution (in mmol cm ~): 1 = 1 - 1 0 s. 2 = 1 . 1 0 4 ; 3 = 2 . 1 0 3 : 4 = 5 - 1 0 ~.

nounced as can be seen from both the experimental and theoretical data. The solid curves 1 and 2 in Fig. 9 show the calculated rates for Ca :+ concentrations of [Ca 2+] = 1-10 -s mmol cm -3 and [Ca 2 + ] = 1 ° 1 0 - 4 mmol cm -3. These rates must be compared with the u p m o s t experimental dashed curve with [Ca 2+ ] ~< 1 ° 10 - 4 mmol cm Curves 3 and 4 show calculated data for higher Ca 2+ concentrations and have to be compared to the corresponding lower dashed curves. The overall correspondence between the experimental data and the calculated curves is satisfactory. The difference of a factor of two in the absolute magnitude of the numbers cannot be taken too seriously in view of uncertainties of similar magnitude in the rate constants of the P W P equation ( P l u m m e r et al., 1979; C o m p t o n and Daly, 1984) and the approximation used in the film model. The steep increase in the upmost curve at

120

e ~<0.003 cm has been interpreted by Herman to result from transition to turbulent flow which she assumes to set in at 900 r.p.m., corresponding to Re>_-15,000. If this is the case, one would expect that also the dissolution rates with [ Ca 2+ ] = 2.1- 10- 3 mmol c m - 3, which are sufficiently far from equilibrium, should also show a similar behaviour, since effective transport in the turbulent region does not depend on the concentration of Ca 2+ in dilute solutions. Therefore, we believe that the reason for the steep increase results from the fact that at this rotational speed the diffusion boundary layer has become sufficiently small in comparison to the diffusion length, thus increasing the rates as discussed in the last section. Furthermore, the transition of turbulent flow for wellcentered discs with a smooth surface starts at higher Reynolds number, Re=2.7-105 (Daguent and Robert, 1967). Experiments with rotating discs to investigate calcite dissolution at low pH ( < 4 ) by Sj6berg and Rickard (1984) show that laminar flow is retained up to the highest rotational speed of 3000 r.p.m. This can be seen from the fact that in these experiments, where dissolution is controlled by diffusional mass transport of H + ions across the diffusion layer towards the surface, rates of dissolution are proportional to o~1/2. In turbulent flow, however, rates should be proportional to co°9 as in eq. 6. In dolomite dissolution experiments performed with an identical experimental setup as those on calcite Herman and White (1985) have also observed a rate dependence on ~o1/2 for the initial rates, indicating laminar flow conditions. Finally, one would expect mass transport by convection outside the diffusion boundary layer to be equally effective in laminar flow as in turbulent flow, since in a closed vessel both produce perfect mixing. From the results of our model, one can predict that the dissolution rates will be rate limited by the chemistry of the surface reactions only in the limiting case e << 2. On the other hand, for e >/2, CO2 conversion is dominant in limiting the rates. Thus experimental

W. DREYBRODT AND D. BUHMANN

investigations of surface reactions in a H20CO2-CaCO3 system must assure that E is as small as possible. From Figs. 4 and 6 e-values of < 10-3 cm are appropriate. Rotating disc experiments at rotational speeds of > 3000 r.p.m, can meet this requirement. At a rotational speed of 6000 r.p.m., e = 6 - 10 -4 cm. Therefore, they are most suitable for such investigations. Due to the small ratio of the CaCO3 surface to the volume of the solution these experiments, however, are extremely time consuming. Batch experiments using small crystals of CaCO3 in the size of a few micrometres up to diameters of several 100 pm suspended in a turbulently stirred solution are therefore preferred by many experimenters. For particles with sizes from 60 up to 630 /lm in diameter Plummer and Wigley ( 1976 ) have estimated Sherwood numbers from which diffusion boundary layers with e,.~ 10 #m are calculated. Recently Armenante and Kirwan (1989) have investigated mass transfer from such microparticles and have established a correlation for the Sherwood number. From this one can infer that boundary layers of ~ 10 #m can be established in turbulently agitated systems. In view of the sensitivity of rates upon changes of e as can be visualized by Figs. 4 and 6 the deviations in the result of differing experiments can be understood. To summarize: in any kind of experiment on calcite dissolution or precipitation one has to keep the thickness of the diffusion boundary layer well below 10-3 cm in order to obtain reliable results. In geological situations, such as flow in a phreatic cave conduit, it may be questionable to estimate the thickness of the diffusion boundary layer by hydrodynamic correlations, even if the necessary parameters are known. Natural conduits cannot easily be considered as smooth and the effects of surface roughness have to be considered. In any case roughness of the dissolving surface will tend to increase dissolution rates by diminishing the influence

DISSOLUTION AND PRECIPITATION OF CALCITE FROM SOLUTIONS IN TURBULENT MOTION

of the diffusion boundary layer (Levich, 1962). Where the sites of geological interest are accessible it may be possible to determine e by dissolution experiments o n CaSO4 or

12 1

ische Vorg~inge im Wasserkreislauf in der ungesiittigten und gesiittigten Zone" ). References

CaSO4-H20.

But even without the knowledge of ~ an order of magnitude estimation for dissolution rates can be given. The curves in Fig. 5 show that the range of dissolution rates is limited by the upper curves, which give the maximal possible dissolution rate and the lower curves, where e >> 2. This is a variation by a factor of ~ 4. Dissolution rates occurring in laminar flow for 3 < 0 . 1 cm ( B u h m a n n and Dreybrodt, 1985a, b) will in any case be lower or equal than those in the lowest curves in Fig. 5. To model karstification in its initial state one has to consider dissolution under the condition of laminar flow. After onset of turbulence, however, one has to use the rates as they are given in turbulent flow conditions. The importance of boundary layers on the dissolution in karst environments has recently been demonstrated by S.E. Lauritzen (pers. c o m m u n . , 1987). He investigated dissolution rates in a single isolated phreatic cave conduit by analyzing the chemical composition of the water at the inlet and the exit of the conduit. Increases in hardness were assumed to originate from dissolution of marble from the cave walls. At low discharges (1 m 3 s - l ) the observed dissolution rates are small, then are increasing steeply to reach constant values at discharges above 10 m 3 s - 1 . The maximal dissolution rates are in the order of 1-10 - 7 m m o l cm 2 s - I (S.E. Lauritzen, 1987, pers. c o m m u n . ) , which agrees well with the maximal rates calculated from our model. Acknowledgement

The authors thank the Deutsche Forschungsgemeinschaft for financial support ( Schwerpunktprogramm: "Hydrogeochem-

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122 Herman, J.S.and White, W.B., 1985. Dissolution kinetics of dolomite: Effects of lithology and fluid flow velocity. Geochim. Cosmochim. Acta, 49:2017-2026. lnskeep, W.S. and Bloom, P.R., 1985. An evaluation of rate equations for calcite precipitation at p less than 0.01 atm and pH greater than 8. Geochim. Cosmochim. Acta, 49: 2165-2180. Kamke, E., 1967. Differentialgleichungen, L6sungen und L6sungsmethoden I. Akademische Verlagsgesellschaft, Leipzig, 000 pp. Kay, J.M. and Neddermann, R.M., 1985. Fluid Mechanics and Transfer Processes. Cambridge University Press, Cambridge, 602 pp. Kern, D.M., 1960. The hydration of carbon dioxide. J. Chem. Educ., 37: 14-23. Lauritzen, S.E., 1986. Hydraulics and dissolution kinetics ofa phreatic conduit. Proc. 9th Int. Congr. on Speleology, Barcelona, 1: 20-22. Levich, V.G., 1962. Physiochemical Hydrodynamics. Prentice Hall, Englewood Cliffs, N J, 700 pp. Nielsen, A.E., 1964. Kinetics of Precipitation. Pergamon Press, Oxford, 151 pp. Opdyke, B.N., Gust, G. and Ledwell, J.R., 1987. Mass transfer from smooth alabaster surfaces in turbulent flows. Geophys. Res. Lett., 14:1131-1134. Pleskov, Y.V. and Filinovskii, V.Y., 1976. The rotating disc electrode. Stud. Sov. Sci., Consult. Bur., New York, N.Y., 402 pp. Plummer, E.N. and Wigley, T.L.M. and Parkhurst, D.L., 1978. The kinetics of calcite dissolution in CO2-water systems at 5°C to 60"C and 0.0 to 1.0 atm CO. Am. J. Sci.. 278: 179-216. Plummer, L.N., Parkhurst, D.E. and Wigley, T.E.M., 1979. Critical review of calcite dissolution of precipitation. In: E.A. Jenne (Editor), Chemically Modelling in Aqueous Systems. Am. Chem. Soc., Syrup. Ser., 93: 537 -573.

W. DREYBRODT AND D. B U H M A N N

Plummer, L.N. and Wigley, T.L.M., 1976. The dissolution of calcite in CO2-saturated solutions at 25 °C and 1 atmosphere total pressure. Geochim. Cosmochim. Acta, 40:191-202. Rauch, H.W. and White, W.B., 1977. Dissolution kinetics of carbonate rocks, 1. Effects of lithology on dissolution rate. Water Resour. Res., 13:381-394. Rickard, D.T. and Sj6berg, E.L., 1983. Mixed kinetic control of calcite dissolution rates. Am. J. Sci., 283: 815830. Sj6berg, E.L., 1978. Kinetics and mechanism of calcite dissolution in aqueous solutions at low temperatures. Stockholm Congr. on Geology, 32( 1 ): 1-92. Sj6berg, E.L. and Rickard, D.T., 1984a. Temperature dependence of calcite dissolution kinetics between 1 and 62cC at pH 2.7 to 8.4 in aqueous solutions. Geochim. Cosmochim. Acta, 48: 485-493. Sj6berg, E.L. and Richard, D.T., 1984b. Calcite dissolution kinetics: surface speciation and the origin of the variable pH dependence. Chem. Geol., 42:119-136. Skelland, A.H.P., 1974. Diffusional Mass Transport. Wiley, New York, N.Y., 510 pp. Usdowski, E., 1982. Reactions and equilibria in the systems CO2-H20 and C a C O 3 - C O 2 - H 2 0 - A review. Neues Jahrb. Mineral. Abh., 144:148- t 71. Wallin, M. and Bjerle, I., 1989a. A mass transfer model for limestone dissolution from a rotating cylinder. Chem. Eng. Sci., 44:61-67. Wallin, M. and Bjerle, 1., 1989b. Rate models of limestone dissolution: A comparison. Geochim. Cosmochim. Acta, 53:1171-1176. White, W.B. and Longyear, J., 1962. Some limitations on speleogenetic speculations imposed by the hydraulics of groundwater flow in limestone. Nittany Grotlo Newl., 10: 155-167.