Mean multiple-phase boundary

By process, we mean what occurs inside the reactor. If the material in the reactor is single phase and homogeneous, then the process is a reaction. Such a reaction can occur in a batch, a semi-batch, or a continuous reactor, depending upon our design. However, if the material in the reactor is multiphase, e.g., gas—liquid or two immiscible liquids, then it is a process. In other words, conversion of reactant to product involves more than chemical reaction it involves multiple steps, some of which are physical, such as diffusion across a phase boundary. If diffusion across a phase boundary or diffusion through one of the phases in the reactor is slower than the chemical reaction, then we define the process as diffusion rate limited. If physical diffusion occurs at a much higher rate than chemical reaction, then we define the process as reaction rate limited. ... 

In order to find the combined effect of a pile of thin layers it should be noted that, approximately, the vector E has a constant length and direction throughout the system. This means that fi decreases by p for the successive layers, since we make use of a rotating co-, ordinate system. Consequently fi increases by p (see equation (3)), and relative to a fixed direction e rotates 2p at subsequent boundaries. When the waves arrive at the surface they have relative orientations as given in Fig. 3. (Multiple refiexions are neglected.) The phase difference of two successive waves is 2n6, b being the thickness of one layer. It will be clear from Fig. 3 that there will be a maximum of intensity when E[ and E fi E 2 and JFg, etc., are in phase. This requires that... 

Exact solutions. It is possible to obtain some exact results for mean residence times even for channels with large numbers of particles although the results are typically cumbersome [90, 91]. Here, we briefly sketch the main points of the derivation for the case of single-file transport in a uniform channel in equilibrium with a solution of particles [90]. Most generally, the system of multiple particles in a channel is described by the multi-particle probability function P(x,t y) that the vector of particles positions is x at time t, starting from the initial vector y [53, 90, 92]. The crucial insight is that because the particles cannot bypass each other, the initial order of the particles is conserved if y < y for any two particles at the initial time, it implies that x < for all future times. That is, the parts of the phase space accessible to these particles are bounded by the planes defined by the condition = x in the vector space x. This implies a reflective boundary condition at the x = plane for any two different particles m and n,... 

---