Interface fixed

For example, for an experimental study of crystal growth, we measure the concentration as a function of distance away from the interface after the experiment. This would be a concentration profile in the interface-fixed reference frame. Using the new reference frame leads to... 

To simplify notation, let s use x and t for y and fnew and remember that now x means the interface-fixed reference frame (and t still has the regular meaning because fnew = f)- The above equations become... 

Figure 3-26 Quartz crystal growth and diffusion profile in (a) a laboratory-fixed reference frame and (b) an interface-fixed reference frame. At a given time, a given kind of curve is used to outline the crystal shape and plot the concentration profile.

For clarity, use w to denote mass fraction (dimensionless, i.e., the concentration unit is not kg/m or mol/m ) in the melt Wqtz denotes mass fraction in quartz. The mass flux toward the interface (in the interface-fixed reference frame) is... 

Crystal growth rate may be constant, which could happen if temperature is decreasing or if there is convection. Smith et al. (1956) treated the problem of diffusion for constant crystal growth rate. In the interface-fixed reference frame, the diffusion equation in the melt is... 

That is, the concentration in the crystal is the same as that in the initial melt at steady state. Therefore, the growth of the crystal does not affect the mass excess or deficiency in the melt anymore, meaning that the concentration profile (in interface-fixed reference frame) in the melt is at steady state. Steady state maybe reached only for elements whose concentration in a mineral can vary non-stoichiometrically. 

The new reference frame is known as the interface-fixed reference frame, and the old reference frame is called the laboratory-fixed reference frame. The melt consumption rate u depends on whether the growth is controlled by interface reaction, or by diffusion, or by externally imposed conditions such as cooling. 

No matter which reference frame we start with, when transformed into the same interface-fixed reference frame, the results should be the same. Hence, starting from Equation 4-86b, we should also arrive at Equation 3-114a. Let y be the coordinate fixed at the crystal-melt interface then... 

Transforming Equation 4-86b into the interface-fixed reference frame, then... 

One-dimensional diffusive dissolution With the above general discussion, we now turn to the special case of one-dimensional crystal dissolution. Use the interface-fixed reference frame. Let melt be on the right-hand side (x > 0) in the interface-fixed reference frame. Crystal is on the left-hand side (x < 0) in the interface-fixed reference frame. Properties in the crystal will be indicated by superscript "c". For simplicity, the superscript "m" for melt properties will be ignored. Diffusivity in the melt is D. Diffusivity in the crystal is D. The concentration in the melt is C (kg/m ) or w (mass fraction). The initial concentration in the crystal is or simplified as or if there would be no confusion from the context. It is assumed that the interface composition rapidly reaches equilibrium. In the following, diffusion in the melt is first considered, and then diffusion in the crystal. 

Note that Equation 4-99 means that the solution is an error function with respect to the lab-fixed reference frame (x = x—2aVSf). In the interface-fixed reference frame, the solution appears like an error function, and its shape is often error function shape, but the diffusion distance is not simply especially... 

Using the interface-fixed reference frame (i.e., x = 0 at the interface) and defining melt to be at the right-hand side (x > 0) and crystal to be at the left-hand side (x < 0), the diffusion profile for the major component in the melt is... 

If the diffusion of a minor or trace element can be treated as effective binary (not uphill diffusion profiles) with a constant effective binary diffusivity, the concentration profile may be solved as follows. The growth rate u is determined by the major component to be n D ff, and is given, not to be solved. Use i to denote the trace element. Hence, w, and Dt are the concentration and diffusivity of the trace element. Note that Di for trace element i is not necessarily the same as D for the major component. The interface-melt concentration is not fixed by an equilibrium phase diagram, but is to be determined by partitioning and diffusion. Hence, the boundary condition is the mass balance condition. If the boundary condition is written as w x=o = Wifl, the value of Wi must be found using the mass balance condition. In the interface-fixed reference frame, the diffusion problem can be written as... 

In the context of IMPROVE, commercial and third-party tools had to be integrated into the engineering design environment. They were mostly equipped with a COM interface (e.g. Aspen Plus, Comos PT, or Documentum). Hence, our approach concentrated on COM (Component Object Model [846])" . COM follows the object-oriented paradigm, i.e. COM components represent subsystems, consisting of a set of classes including attribute and method definitions, relationships between classes, and a set of interfaces to access the subsystem. For every COM interface, fixed by standardization, a textual description is available in form of a so-called type library or a dynamic link library, respectively. 

For a process limited by diffusion in thick layer, calling and C the matter carrier concentrations at arrival and starting interfaces (fixed by equilibriums of... 

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