Solute Transport in a Temperature Gradient

Growth from High-Meltinp Solutions 17.2.4.2.3. Solute Transport in a Temperature Gradient. 

Eq. 2.18) and Ohm s law for electrical conduction (where the electrical conductivity is also always positive, according to Exercise 2.1 s solution). However, this is not necessarily the case for the cross-effect tensors /3 and 7. For example, in mass diffusion in a thermal gradient, the heat of transport can be either positive or negative the direction of the atom flux in a temperature gradient can then be in either direction. The anisotropic equivalent to the heat of transport relates the direction of the mass diffusion to the direction of the temperature gradient. There is no physical requirement that these quantities could not be in reversed directions, and indeed, sometimes they are. 

The water transport in OMD is a simultaneous heat and mass transfer process. Evaporation cools the feed and condensation warms the brine solution. This results in a temperature gradient across the membrane, which adversely affects the driving force and in turn the mass flux. 

To transport solute in a temperature gradient, undissolved nutrient is maintained at some temperature (E in Fig. 2 of 17.2.4.2), and a flow of solution occurs to a lower temperature (F in the same figure), where a seed is located and growth occurs because the solution is supersaturated. The growth rate depends primarily on the temperature difference AT = ( — F)S and can be controlled by adjustment to this parameter. 

Thermal diffusion is a process in which solute is driven through solvent by the action of a temperature gradient rather than by a concentration (or chemical potential) gradient [46]. It is a natural outgrowth of the laws of irreversible thermodynamics (Section 3.2) in which all driving forces are expected to be associated with some transport of matter. 

In general, the functions Hi and Fjc are nonlinear. These norUinearities are usually due to the exponential activation of the electrochemical rate constants by the potential (see Section 5.5). In addition, even for time-invariant electrochemical systems, equations (14.2) can comprise either differential equations, when only kinetic equations are considered to be involved at the interface, or partial differential equations, when distributed processes occur in the bulk of the solution (such as may result from transport of the reacting species or a temperature gradient in the solution). 

The gas motion near a disk spinning in an unconfined space in the absence of buoyancy, can be described in terms of a similar solution. Of course, the disk in a real reactor is confined, and since the disk is heated buoyancy can play a large role. However, it is possible to operate the reactor in ways that minimize the effects of buoyancy and confinement. In these regimes the species and temperature gradients normal to the surface are the same everywhere on the disk. From a physical point of view, this property leads to uniform deposition - an important objective in CVD reactors. From a mathematical point of view, this property leads to the similarity transformation that reduces a complex three-dimensional swirling flow to a relatively simple two-point boundary value problem. Once in boundary-value problem form, the computational models can readily incorporate complex chemical kinetics and molecular transport models. 

Let us consider the transport of one component i in a liquid solution. Any disequilibration in the solution is assumed to be due to macroscopic motion of the liquid (i.e. flow) and to gradients in the concentration c,. Temperature gradients are assumed to be negligible. The transport of the solute i is then governed by two different modes of transport, namely, molecular diffusion through the solvent medium, and drag by the moving liquid. The combination of these two types of transport processes is usually denoted as the convective diffusion of the solute in the liquid [25] or diffusion-advection mass transport [48,49], The relative contribution of advection to total transport is characterised by the nondimensional Peclet number [32,48,49], while the relative increase in transport over pure diffusion due to advection is Sh - 1, where Sh is the nondimensional Sherwood number [28,32,33,49,50]. 

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