1. Introduction

There is a problem to study physical and chemical characteristics of colloidal silica in hydrothermal heat carrier in the context of silica extraction technology and increasing of effectiveness of heat carrier using (Potapov V.V., Karpov G.A., Podverbny V.M., 2002; Potapov V.V., 2003).

Colloidal silica forms in the hydrothermal solution during several stages (Potapov V.V., 2003). Silica comes into the solution as molecules of silicic acid as a result of water chemical interaction with silica-alumina rocks of hydrothermal fields at the depth 1.0-3.5 km in thermal anomaly zones under a high temperature (up to 250-350℃) and pressure (4.0-20 МPa). Hydrothermal solution is multicomponent. Compounds with Na, K, Si, Ca, Mg, Al, Fe, Cl, S, C, B, Li, As, Cu, Zn, Ag, Au and others are present at the solution as ions and molecules.

Under the temperature 250-350℃ when solution contacts with rocks minerals a total silica content C_t (mole/kg) in water can be estimated by α-quartz solubility (Crerar D.A., 1971):

(1)

Equation (1) gives the following values of quartz solubility SiO₂ (mg/kg): 25℃ – 3.46, 50℃ – 10.29, 100℃ – 47.6, 200℃ – 256.0, 250℃ – 415.6, 300℃ – 592.5. Over rising filtration in fractured-and-porous rocks or during solution movement through the producing wells of geothermal electric power plants (GeoPP) pressure and temperature of the solution reduce and solution is divided in steam and liquid phases. Total silica content C_t in a liquid phase can reach 700-1500 mg/kg. Consequently water solution becomes supersaturated relative to a solubility of amorphous silica C_e (Marshall W.L., 1980). In accordance with experimental data (Marshall W.L., 1980) C_e value (mole/kg) for pure water depends on an absolute temperature T in the following way:

(2)

Under the temperature 200℃ solubility C_e is 940.8 mg/kg, under the temperature 150℃ – 651.8 mg/kg, 100℃ – 405.3 mg/kg, 25℃– 130.8 mg/kg.

Such condition of monomeric silicic acid in water solution is unstable. Solution supersaturation S_m which is equal to the difference (C_s-C_e) of orthosilicic acid concentration C_s and solubility C_e causes nucleation and polymerization of silicic acid molecules with condensation of silanol groups and formation of siloxane links and partial dehydration in accordance with the following reactions (Iler R., 1982):

(3)

(4)

2. Chemical Composition of Hydrothermal Solution

It is necessary to study the process of orthosilicic acid nucleation to work out the technology of extraction of silica-containing materials with given characteristics from the hydrothermal heat carrier. Final concentration and the size of colloidal particles before their extraction from aqueous solution depend on the parameters of the process nucleation of orthosilicic acid. Numerical simulation in different conditions corresponding to the heat carrier characteristics of Verkhne-Mutnovsky GeoPP was made to study the process of nucleation.

Table 1 presents physical and chemical characteristics of liquid phase of hydrothermal heat carrier (separate) of GeoPP. This phase is separated from two-phase water-steam flow in separators of GeoPP. Aqueous solution of separate refers to natrium-potassium-chloride-sulphate type of thermal waters. Total silica content in solution varies in the range of 500-1000 mg/kg, pH=8.0-9.4, ionic strength I_s = 0.01-0.02 mole/kg.

Table 1. Ions and cations concentrations in hydrothermal separate from the pipe-line of reverse pumping of Verkhne-Mutnovskaya GeoPP, ionic strength Is = 14.218 mmole/kg, conductivity σ = 1250 microS/cm, n/d means that concentration was not determined, (-) means that values were not estimated Component mg/l mg•equ/l mg•equ/l %mg•equ/l Na+ 239.4 10.413 88.044 K+ 42.0 1.074 9.080 Ca2+ 1.6 0.0798 0.6747 Mg2+ 0.72 0.0592 0.5005 Fe2, 3+ <0.1 <0.0053 0.0448 Al3+ 0.27 0.033 0.2790 NH4+ 1.1 0.0609 0.5149 Li+ 0.71 0.102 0.8624 Cations sum 285.9 11.827 100.0 Cl- 198.5 5.591 47.664 HCO3- 81.0 1.327 11.312 CO32- 19.9 0.663 5.652 SO42- 192.1 3.9995 34.096 HS- 4.95 0.15 H2S0 5.92 - - F- n/d n/d - Anions sum 496.5 11.73 100.0 H3BO3 106.9 - (H4SiO4)t 1190 - (H4SiO4)s 222 - Mineralization Mh, mg/kg 1638.9 pH 9.35

3. Mathematical Model

Mathematical model (Weres 0., Yee A., Tsao L., 1981) worked out by Weres, Yee and Tsao in Lawrence Berkeley Laboratory (USA) was used for numerical simulation. Supersaturation S_N(T) which is equal to C_s/C_e and pH are main factors defining nucleation rate I_N of silicic acid in aqueous solution:

(5)

where ΔF_cr is a free energy change connected with nuclei formation of critical radius R_cr, A_cr is the surface area of critical nuclei, A_cr=4•π•R_cr², σ_sw is the surface tension on silica-water boundary, R_md is the rate of molecular precipitation of SiO₂ onto a solid surface, g•(cm²•min)^-1, k_B is Boltzmann constant, M_Si is the molar mass SiO₂, N_A is Avogadro constant, Q_LP is Lothe-Paund factor, Q_LP =3,34•10²⁵ kg^-1, Z is Zeldovich factor, n_cr is the quantity of molecules of SiO₂ in nuclei of critical size.

Critical radius, quantity of molecules in nuclei of critical radius, free energy and Zeldovich factor were estimated in accordance with the following formulas (Weres 0., Yee A., Tsao L., 1981):

(6)

(7)

(8)

(9)

ρ is the density of amorphous silica.

The dependence of the function R_md of the rate of SiO₂ molecules deposition on the temperature and pH solution in a given model is shown in the following equations (Weres 0., Yee A., Tsao L., 1981):

(10)

(11)

(12.1)

(12.2)

(12.3)

(13.1)

(13.2)

(13.3)

The dependence of surface tension on the temperature and pH coefficient σ_SW is given by the function I(pH, pH_nom):

(14.1)

(14.2)

where S_a = (1-α_i)•S_N, α_i is a fraction of silicic acid in ionic form, pH_nom = pH + lg([Na⁺]/0.069), [Na⁺] is an ion activity [Na⁺], mole/kg, pK_i=6.4, f(7.0) = 0.119, h_f=0.45, Hσ, Sσ are specific enthalpy and entropy of silica surface in water, Hσ = 63.68•10^-3 Joule/m², Sσ = 0.049•10^-3 Joule/m²•К, n_o = 6.84 nm^-2.

Equation (5) shows maximum rate IN of nucleation for the particles with the radius that is higher critical. Its molecules quantity of SiO₂ is equal to n = ncr+0.5/Z. IN(t) value reaches maximum in induction time τin that is necessary to grow and form stable population of the particles which sizes are close to critical. Time dependence IN(t) has got the form (Weres 0., Yee A., Tsao L., 1981):

(15)

To calculate time the following formula is used:

(16)

4. Program for Numerical Simulation

The system of equations (1), (2), (5)-(16) was used to develop MSANUC.FOR program. The solution algorithm in accordance with MSANUC.FOR was the following:

1) Initial values entry: temperature T, pH, concentration of ions and ionic strength of solution, the assignment of total silica C_t content of initial time “TIME” and time step DT, the assignment of initial radii R(I) and particles quantity MP(I) of every class “I” which are present in the solution before nucleation.

2) The calculation of colloidal silica content in solution by summation of particles of all classes:

(17)

3) The calculation of concentration of monomeric orthosilicic acid as difference between total content and colloidal silica content:

(18)

4) The calculation of current value of supersaturation solution S_N:

(19)

5) The calculation of values of σ_sw, R_md, R_cr, Z, I_N(t) corresponding to current values S_N, pH and T by equations (5)-(16).

6) The calculation of the quantity of new particles appearing during the time DT in accordance with current value of nucleation rate I_N on a given program step and educing a new class of particles N+1:

(20)

7) The calculation of summary concentration of particles:

(21)

8) The calculation of particle mass difference DPM of every class during the time DT in accordance with the equation:

DPM(I)=4•π•R²(I) •R_md•DT

9) The calculation of particle radius R(I) corresponding to a new value of particle mass in every class:

(22)

And the calculation of mean values of R_ср, R²_ср, R³_ср in all classes of particles:

(23)

(24)

(25)

10) The calculation of new temperature values, pH, ionic strength of solution if these values change with time. After the tenth step we must return to the second step of the program to make the next cycle. Registration of calculations results in a separate file is provided for: mean radius, summary concentration of particles and concentration of monosilicic acid for given time.

5. Results

Table 2 a,b presents the results of numerical simulation of nucleation with MSANUC.FOR program under different temperatures and pH solution. During the initial time colloidal silica content supposed to be zero and all silica was in a monomeric orthosilicic acid form:

C_s(t=0) = C_t

Table 2. Final mean sizes of silica particles under a constant temperature and different рН; CS=700 mg/kg; (-) means that it was not estimated рН t, ℃ 20 40 50 60 80 100 4 5.289 19.247 48.067 156.551 6342.132 - 5 5.290 19.054 47.305 153.158 5858.727 - 6 2.983 16.823 40.691 128.018 4684.754 - 7 2.978 7.667 15.282 38.175 701.463 - 8 0.988 1.219 1.508 1.775 3.488 18.77 8,5 0.645 0.731 0.784 0.841 1.013 1.268

Simultaneous results show that critical radius R_cr increases due to supersaturation reduce when temperature becomes higher. Consequently the tendency for decrease of nucleation rate I_N and increase of induction time τ_in takes place; it leads to reduce of particles concentration N_p and increase of final mean radius of the particles R_f. Under the temperature 120-150℃ and higher nucleation hasn’t time to develop during a real residence time of hydrothermal solution in wells and thermal equipment of GeoPP.

After nucleation and polymerization completion a part of silica continues to be as molecules of orthosilicic acid H₄SiO₄ which concentration is close to solubility of C_e. It is in equilibrium with colloidal silica. Small amount of silicic acids ions (H₃SiO₄^-, H₂SiO₄^2-, HSiO₃^-, etc.) and macromolecules of polysilicic acids are in the solution parallel with colloidal particles and molecules of silicic acids. Data processing obtained by Rothbaum and Rohde (Rothbaum H.P., Rohde A.G., 1979) has showed the concentrations of dimmers C_dimer and trimers C_trimer of silicic acid are approximated by the equations (mole/kg):

(26.1)

(26.2)

The dependence of constants of orthosilicic acid ionization of the first K₁ = [H⁺]•[H₃SiO₄^-]/[H₄SiO₄] and second K₂ = [H⁺]•[H₂SiO₄^2-]/[H₃SiO₄^-] stages on the temperature has the following form:

(27.1)

(27.2)

In accordance with equations (26-1, 26-2) under the temperature 20-180℃ and pH=7.0-9.2 a fraction (portion) of dimmers with respect to orthosilicic acid which concentration is close to a solubility of C_e(T) doesn’t exceed 1.0%, a fraction(portion) of trimers is 0.1%, a fraction(portion) of tetramers and low-molecular circular polymers (up to 6 units SiO₂) (Rothbaum H.P., Rohde A.G., 1979) is < 0.1%. A fraction (portion) of ions H₃SiO₄^- and H₂SiO₄^2- under such conditions is not more than 14.0%.

The calculations by the method of photon correlation spectroscopy show that particles mean radius of the polymerized silica in hydrothermal solution has values from 7.0 up to 30.0 nm, values of particles radii vary from 1.0 up to 50.0 nm (Potapov V.V., 2003).

With the help of MSANUC.FOR final mean size of silica particles under different constant temperatures 20℃, 40℃, 50℃, 60℃, 80℃, 100℃ and different рН=4-8.5 were estimated. Table 2 presents estimated final mean sizes of the particles (nm).

When pH=5-6 and the temperature is 50℃ the calculations were made with step by рН 0.1. It made the possibility to study the influence of pH solution on the size of silica particles more detailed. The estimated results are shown in Table 3 and on Figure 1.

Table 3. Final mean sizes of silica particles under the temperature 50℃ and different рН; CS=700 mg/kg рН RA, nm рН RA, nm 5 47.305 5.5 45.423 5.1 46.995 5.6 44.854 5.2 46.771 5.7 44.007 5.3 46.42 5.8 43.077 5.4 45.954 5.9 42.028

Figure 1. The dependence of final mean size of silica particles on рН. t=50℃, CS = 700 mg/kg

On Figure 2 the dependence of concentration of dissolved silicic acid C_S (mg/kg) on time T_p (min) is shown for t=50℃, initial concentration C_S=700 mg/kg and different values рН; Figure 3 presents the dependence of supersaturation S_N on time T_p (min); Figure 4 (а-f) presents the dependence of distribution of final sizes of silica particles.

Figure 2. The dependence of concentration of dissolved silicic acid on time. t=50℃, initial concentration CS=700 mg/kg

Figure 3. The dependence of supersaturation SN on time. t=50℃, initial concentration CS=700 mg/kg

Figure 4. The distribution of particles sizes under the temperature t=50℃, initial concentration CS=700 mg/kg. a – рН=4; b – рН=5; c – рН=6; d – рН=7; e – рН=8; f – рН=8.5

The influence of initial solution concentration C_S (mg/kg) on a final mean size of silica particles (nm) is shown in Table 4 and Figure 5. Calculations were made under different solution temperatures: 20℃, 40℃, 60℃, 80℃, 90℃; different initial solution concentrations C_S (mg/kg): 500, 600, 700, 800, 1000.

Table 4. Values of final mean size of silica particles (nm) under different temperatures t and initial concentrations of solution CS, рН=8; (-) - means that it was not estimated CS,mg/kg t, ℃ 20 40 50 60 300 93.57 - - - 350 12.46 1019.9 - - 400 4.35 49.11 586.3 - 500 1.79 4.088 8.968 32.71 600 1.246 1.846 - 3.808 700 0.988 1.219 - 1.775 800 0.876 1.054 - 1.311 1000 0.739 0.836 - 0.952 CS,mg/kg t, ℃ 70 80 85 90 300 - - - - 350 - - - - 400 - - - - 500 - - - - 600 7.56 24.14 56.14 169.8 700 - 3.488 - 6.496 800 - 1.815 - 2.308 1000 - 1.108 - 1.212

Figure 5. The dependence of a final mean size of silica particles on different temperatures t and different initial concentrations of the solution CS; рН=8

Figure 2 shows that curves reflecting the dependence of concentration of dissolved silicic acid on time have two areas: convex curve reflects homogeneous nucleation, that is the stage of formation of new particles centres, concave part reflects heterogeneous nucleation, i.e. the growth of centres formed earlier.

Figure 6 (a–c) presents the dependence of quantity of the formed particles PNS (nucl) on time T_P (min) and dependence of nucleation rate R_nuc (nucl/kg*s) on time Т_Р (min) under the temperature t=50℃, initial concentration C_S=700 mg/kg and different рН.

Figurer 6. Dependences of formed particles quantity PNS on time ТР and dependences nucleation rate Rnuc on time ТР; t=50℃, initial concentration CS=700 mg/kg. а – рН=5; b – рН=7; c – рН=8

On Figure 4 left curve part of particles sizes distribution is concave and right curve part is convex.

Figure 7 presents the dependence of concentration of dissolved silicic acid C_S (mg/kg) on time T_P (min) for t=60℃, рН=8 and different initial; Figure 8 shows the dependence of supersaturation S_N on time T_P (min); Figure 9 (a-e) presents the dependence of distribution of final sizes of silica particles.

Figure 7. The dependence of dissolved silicic acid concentration on time. t=60℃, рН=8

Figure 8. The dependence of supersaturation SN on time. t=60℃, рН=8

Figure 9. The distribution of particles sizes at the temperature t=60℃, рН=8. а – CS=500 mg/kg; b – CS=600 mg/kg; c – CS=700 mg/kg; d – CS=800 mg/kg; e – CS=1000 mg/kg

Final mean sizes of silica particles (nm) for initial concentration C_S=300, 350 mg/kg, рН=8 and ionic strength corresponding to the solution characteristics of Pauzhetsky geothermal field are in Table 5.

Table 5. Values of final mean size of silica particles (nm) under different temperatures t and different initial solution concentration CS. (-) - means that it was not estimated CS, mg/kg t, ℃ 20 30 40 50 300 13.99 129.9 9095.69 - 350 3.59 10.26 63.95 2103.6

The results of simulation show that critical radius (nm) R_cr and induction time τ_in (min) increase when рН is stated and temperature becomes higher due to supersaturation reduce S_N. Consequently nucleation rate I_N reduces; it causes to increasing of final mean particles size R_A (nm).

At the same temperature when рН reduces the surface tension σ_sw increases and rate of molecular deposition of silicic acid reduces. Owing to it critical radius of particles and induction time increase, nucleation rate reduces and final mean size of particles increases.

6. Conclusions

1. The results of numerical simulation show that in hydrothermal solutions the transformation of orthosilicic acid into colloidal silica happens as a result of two sequential processes: 1) homogeneous nucleation during which the transformation of orthosilicic acid into colloidal silica happens mainly as a result of increasing of new centers quantity; 2) heterogeneous nucleation which happens due to increasing of sizes of centers formed earlier if their quantity is practically constant. Durations of these stages depend on process parameters: temperature, pH, ionic strength, total silica content and supersaturation ratio.

2. The results of numerical simulation show that the nucleation of orthosilicic acid molecules and formation of colloidal particles in hydrothermal solution actively develop under the temperature below 120-150℃. Under the temperature 120-150℃ and higher the induction time becomes so long that colloidal partiles are not formed time of geothermal water treatment in heat equipment of GeoPP.

3. pH solution is an another factor considerably influencing the nucleation kinetics. When pH is low critical radius of particles increases considerably, the nucleation rate reduces and induction time increases. When pH is 7.0 and lower the induction time considerably exceeds the time of of geothermal water treatment in heat equipment of GeoPP and colloidal particles are not formed.

4. The method based on of numerical simulation carried out in this paper can be applied to project equipment for flow sheet of silica extraction from the hydrothermal solution during the stage of ageing. Such method gives a possibility to take into account how temperature profile, pH, ionic strength, concentration of solution components, duration of ageing during solution movement through the wells, pipe lines and equipment influence on final concentration and size of colloidal particles.