Initial solid-forming condition

Figure I. Flow diagram showing computation procedure used for determining initial solid forming condition...

Equilibrium The physical process (reaction) of adsorption or ion exchange is considered to be so fast relative to diffusion steps that in and near the solid particles, a local equilibrium exists. Then, the so-called adsorption isotherm of the form q = f(Ce) relates the stationary and mobile-phase concentrations at equilibrium. The surface equilibrium relationship between the solute in solution and on the solid surface can be described by simple analytical equations (see Section 4.1.4). The material balance, rate, and equilibrium equations should be solved simultaneously using the appropriate initial and boundary conditions. This system consists of four equations and four unknown parameters (C, q, q, and Ce). 

If the initial solid substance is a chemical compound (an intermetallic, a silicide, etc), then its oxidation can proceed via two different mechanisms, depending on the experimental conditions. Two oxides are formed in the severe oxidation (combustion) usually resulting in the disintegration of the compact solid phase. In the partial (soft) oxidation the chemical compound undergoes a partial decomposition giving another chemical compound of the same class and an oxide. 

Figure 17. Polymorph screen for carbamazepine using one solvent, cumene temperature-solubility curves (forms I and III) and solid forms initially crystalbzed at various temperature-concentration conditions. The blue (solid) line and the green (dotted) line represent the temperature-solubility curves of form III and form I respectively. From Getsoian et al., 2006.

Carslaw, H. S and Jaeger, J. C. (1959) Cottduciion of Heat in Solids, Oxford University Press. Oxford, 2nd ed. This work includes a compilation of solutions to the equation of unsteady heat conduction in the absence of flow for many different geometries, initial and boundaiy conditions. The basic equation is of the same form as the diffusion equation with the thermal diffusivity. K/pC, in place of the diffusion coefficient. (Here k, p, and C are the thermal conductivity, density, and specific heatof the continuous fluid.) Like D, the thcnnal diffusivity has cgs- dimensions of cm"/sec,... 

In crystallization processes involving a material that displays polymorphism, it is quite common for an unstable polymorph to appear first and then transform into a stable form. This observation is summarized by Ostwald s step rule, sometimes referred to as the Law of Successive Reactions, which says that in any process, the state which is initially obtained is not the stablest state but the least stable state that is closest in terms of free energy change, to the original state. What this means, therefore, is that a crystallization process, the initial solid phase, can be the least stable polymorph that will then transform into successively more stable forms until the stable form, at the conditions of the system, is reached. With some systems this can mean the formation of an... 

The second of these boundary conditions requires some explanation, since this situation obviously is not often encountered in practice. The set of conditions (9-65) corresponds to those of (9-18), which is the convenient form for solution of problems using Laplace transforms. All temperatures, thus, are scaled with respect to an initial solids temperature of zero. The heat-transfer problem is now mathematically... 

The isothermal flow of incompressible liquid is described by equations (5.13) and (5.21), and the viscosity coefficient n = const. Hence, there are four equations for four unknowns - the pressure p and three velocity components u, v, and w. Thus, the system of equations is a closed one. For its solution it is necessary to formulate the initial and boundary conditions. Let us discuss now possible boundary conditions. Consider conditions at an interface between two mediums denoted as 1 and 2. The form and number of boundary conditions depends on whether the boundary surface is given or it should be found in the course of solution, and also from the accepted model of the continuum. Consider first the boundary between a non-viscous liquid and a solid body. Since the equations of motion of non-viscous liquid contain only first derivatives of the velocity, it is necessary to give one condition of the impermeability u i = u 2 at the boundary S, where u is the normal component of the velocity. The equations of motion of viscous liquid include the second-order derivatives, therefore at the boundary with a solid body it is necessary to assign two conditions following from the condition of sticking u i = u 2, Wii = u i where u is the tangential to S component of the velocity. If the boundary S is an interface between two different liquids or a liquid and a gas, then it is necessary to add the kinematic condition Ui = U2 =... 

To the left is the heat flux vector Jq (heat flow rate per unit area A) and to the right is the spatial change of the temperature field. The thermal conductivity X combines both of these vectors, which makes it a tensor. In general, X depends on the direction inside a solid (depending on its crystal structure) moreover, it is a function of temperature. Equation (4.1) can be solved in a closed form in only a few special cases because the initial and boundary conditions, that is, the temperature field at the initial time and the temperature of the boundary, as well as the geometry of the arrangement are included in the solution. Somewhat less complex are the relationships in the one-dimensional treatment under such circumstances, Eqs. (4.1) and (4.2) become... 

In the HCS, combustible dust hazards must be addressed on labels and SDSs. Label elements are provided for combustible dust in the HCS and include the signal word warning and the hazard statement May form combustible dust concentrations in the air. For chemicals in a solid form that do not present a combustible dust hazard, but may form combustible dusts while being processed in normal downstream uses, the HCS allows the chemical manufacturer some flexibility in labeling requirements. The manufacturer or importer may transmit the label to the customer at the time of the initial shipment, but the label does not need to be included with subsequent shipments vmless it changes. This provides the needed information to the downstream users on the potential hazards in the workplace, while acknowledging that the solid metal or other materials do not present the same hazards that are produced when these materials are processed under normal conditions of use. 

It is not essential that, for combustion synthesis, the reactants are elements and initially solid (at least one of them). Refractory carbides and nitrides with complex chemical compositions (metal halogenides, organometallics, etc.) are also formed in combustion of gaseous systems. The special features of combustion synthesis in the gas phase were analyzed in the recent review by Brezinsky (5). The synthesis in a gaseous system proceeds at a stationary combustion front with moving flows of reactants and product. Consequently, these processes differ in essence from combustion of condensed systems by the conditions of nucleation and growth of the new phase, which manifest themselves in the product morphologies. 

For a case in which eqn. (4.2) holds, if a boat full of liquid of initial composition C, is solidified to a bar under the conditions considered above, the final solute distribution is as shown in Fig. 4.10, making allowance for the initial and terminal transients. At points A and B on these two curves the compositions of the solid formed instantaneously are Co o and C respectively. The average composition (ignoring the small overall average increase resulting from the localised interface pile-up) in both cases is C,. Thus the iilcciive distribution coefficients are k, and 1 for the two cases. The distribution coefficientis therefore seen to vary from the equilibrium value /cq to the value 1 as the steady-state is approached. 

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