Convective crystal dissolution

Although diffusive crystal dissolution is seldom encountered in nature, its theoretical development is instructive for understanding convective crystal dissolution, and it is often encountered in experimental studies. Such experiments are easy to conduct, and can be applied to infer diffusion coefficients, to establish equilibrium conditions, and to investigate the rate of diffusive crystal dissolution. Furthermore, the interface-melt composition and diffusivity obtained from diffusive crystal dissolution experiments are of use to estimate convective crystal dissolution rates (Section 4.2.3). 

Convective crystal dissolution means that crystal dissolution is controlled by convection, which requires (i) a high interface reaction rate so that crystal dissolution is controlled by mass transport (see previous section), and (ii) that mass transport be controlled by convection. In nature, convective crystal dissolution is common. In aqueous solutions, the dissolution of a falling crystal with high solubility (Figure 1-12) is convective. In a basaltic melt, the dissolution of most minerals is likely convection-controlled. 

Some convective crystal dissolution problems can be treated by combined consideration of dissolution kinetics and hydrodynamics. Hydrod5mamic consideration is necessary because it is necessary to know how rapidly the interface fluid is removed. In considering the problems, steady state is assumed. Irregular and unsteady convection is not treated. The following problems have been tackled ... 

For convective crystal dissolution, the dissolution rate is u = (p/p )bD/8. For diffusive crystal dissolution, the dissolution rate is u = diffusive boundary layer thickness as 5 = (Df), the diffusive crystal dissolution rate can be written as u = aD/5, where a is positively related to b through Equation 4-100. Therefore, mass-transfer-controlled crystal dissolution rates (and crystal growth rates, discussed below) are controlled by three parameters the diffusion coefficient D, the boundary layer thickness 5, and the compositional parameter b. The variation and magnitude of these parameters are summarized below. 

The boundary layer thickness 5. For convective crystal dissolution, the steady-state boundary layer thickness increases slowly with increasing viscosity and decreasing density difference between the crystal and the fluid. It does not depend strongly on the crystal size. Typical boundary layer thickness is 10 to 100/rm. For diffusive crystal dissolution, the boundary layer thickness is proportional to square root of time. 

By the above definition, b is positive for crystal dissolution, and negative for crystal growth. During convective crystal dissolution, the dissolution rate u is directly proportional to b. During diffusive crystal dissolution, the dissolution rate is proportional to parameter a, which is positively related to b. Hence, for the dissolution of a given mineral in a melt, the size of parameter b is important. The numerator of b is proportional to the degree of undersaturation. If the initial melt is saturated, b = 0 and there is no crystal dissolution or growth. The denominator characterizes the concentration difference between the crystal and the saturated... 

Kinetics of convective crystal dissolution and melting, with applications to... 

The scope of kinetics includes (i) the rates and mechanisms of homogeneous chemical reactions (reactions that occur in one single phase, such as ionic and molecular reactions in aqueous solutions, radioactive decay, many reactions in silicate melts, and cation distribution reactions in minerals), (ii) diffusion (owing to random motion of particles) and convection (both are parts of mass transport diffusion is often referred to as kinetics and convection and other motions are often referred to as dynamics), and (iii) the kinetics of phase transformations and heterogeneous reactions (including nucleation, crystal growth, crystal dissolution, and bubble growth). 

Diffusive crystal dissolution means that crystal dissolution is controlled by diffusion, which requires high interface reaction rate and absence of convection. In nature, diffusive crystal dissolution is rarely encountered, because there is almost always fluid flow, or crystal falling or rising in the fluid. That is, crystal dissolution in nature is often convective dissolution, which is discussed in the next section. One possible case of diffusive crystal dissolution is for crystals on the roof or floor of a magma chamber if melt produced by dissolution does not sink or rise. For these... 

In convective dissolution, the crystal dissolution rate is again denoted by u (or -da/df) for consistency with earlier sections, and the ascent or descent velocity of the crystal is denoted by U. 

Diffusive crystal growth at a fixed temperature would not result in a constant crystal growth rate (see below). However, under some specific conditions, such as continuous slow cooling, or in the presence of convection with diffusion across the boundary layer, time-independent growth rate may be achieved. Similarly, time-independent dissolution rate may also be achieved. 

Convective dissolution of a falling or rising crystal in an infinite liquid reservoir... 

Convective dissolution of a falling or rising single crystal in an infinite fluid reservoir. The theory has been developed by Kerr (1995) and Zhang and Xu (2003). 

Based on the above results, the following is a summary of steps to calculate the convective dissolution rate of a single falling or rising crystal in an infinite melt reservoir ... 

For the calculation of convective dissolution rate of a falling crystal in a silicate melt, the diffusion is multicomponent but is treated as effective binary diffusion of the major component. The diffusivity of the major component obtained from diffusive dissolution experiments of the same mineral in the same silicate melt is preferred. Diffusivities obtained from diffusion-couple experiments or other types of experiments may not be applicable because of both compositional effect... 

Dissolution distance in 18,000 s would be 174/im, greater than the diffusive dissolution distance of 48 ixm obtained earlier. There are no experimental data to compare. The convective dissolution rate can be applied only when the diffusion distance (Dt) is greater than the boundary layer thickness. If diffusion distance (Dt) is smaller than the boundary layer thickness (86.4 fim), i.e., if t< 1408 s, the dissolution would be controlled by diffusion even for a falling crystal, and the method in Section 4.2.2.3 should be used. 

Convective dissolution of many rising or sinking crystals... 

In nature, it is likely to encounter convective dissolution of many crystals. In this case, if their boundary layers do not overlap and the flow velocity fields do not overlap, each crystal may be viewed as dissolving individually without interacting with other crystals. However, if their boundary layers overlap or their flow velocity fields overlap, the above treatment would not be accurate. Furthermore, when there are many crystals, the whole parcel of crystal-containing fluid may sink or rise (large-scale convection), leading to completely different fluid dynamics. Such problems remain to be solved. 

Statement of the problem. Now let us consider mass transfer from a solid wall to a liquid film at high Peclet numbers. Such a problem is of serious interest in dissolution, crystallization, corrosion, anodic dissolution of metals in some electrochemical processes, etc. In many practical cases, dissolution processes are rather rapid compared with diffusion. Therefore, we assume that the concentration on the plate surface is equal to the constant Cs and the incoming liquid is pure. As previously, we introduce dimensionless variables according to formulas (3.4.5). In this case, the convective mass transfer in the liquid film is described by Eq. (3.4.1), the boundary condition (3.4.2) imposed on the longitudinal variable x, and the following boundary conditions with respect to the transverse coordinate ... 

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