Abstract

Despite three decades of experimentation, no general, quantitative description of the polymerization of silicic acid in aqueous solutions has emerged. An accurate mathematical model could have a seminal effect on the solution of many problems in the utilization of geothermal resources, e.g., in the prevention of silication in boreholes, energy conversion equipment and reinjection wells. In addition, such a model might advance understanding of hydrothermal alteration in geothermal reservoirs.

We report here an analysis that divides the overall reaction into successive regimes that are limited in rate by nucleation; chemical reaction; and diffusion of monomer to growing polymer particles.

In the case of homogeneous nucleation, an analogy with the theory of condensation suggests that RN, the rate of formation of critical nuclei, is

  • Equation 2

where Î3 and Vo are the surface tension and specific volume of the solid, respectively; S C/Ce where C is the concentration of monomer and Ce is its solubility; and A is a constant. Exploration of the properties of equation (2) leads to the prediction of maxima and minima in the polymerization rate as a function of temperature and of the temperature-dependent induction times that have been observed in the laboratory and in nature.

The dimensionless ratio rfko/D serves as a useful index of whether diffusion or chemical reaction exerts the dominant influence over silica particle growth. rf is a characteristic length; ko is the surface specific, forward, rate constant; and D is the monomer diffusivity. For rfko/D 1, the reaction rate is very rapid, diffusion is limiting, and particle growth is described by

  • Equation 36

where Rp is the particle radius divided by rf; Co=Ciâ?'Ce, where Ci is the initial monomer concentration; and Ï? tD/rf2.

In the chemical reaction-limited regime, rfko/D 1 and the appropriate solution becomes

  • Equation 42

For both diffusion-limited and reaction-limited regimes, the relationship

  • Equation 37

holds and allows the calculation of C vs. time from equation (36) or (42).

The agreement between theory and extant laboratory data on natural and synthetic silica solutions is excellent. The analysis indicates, moreover, that a priori evaluation of the ratio rfko/D would permit the wily experimentalist to avoid diffusion limitations whilst attempting to measure chemical kinetic rates.

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