Diffusional Flux Equations

For theoretical analysis of sintering, it is necessary to establish the equations for diffusional mass transport. These equations can be solved when subject to 

In an elemental solid, if the influence of the flux of the neutral atoms can be neglected, the flux of the atoms in one dimension can be expressed as  

According to the relation between chemical potential and concentration given in Eq. (5.68), there is  

For an ideal system, a is independent of concentration, i.e.. In a does not vary with In C, so that the second term in the brackets in Eq. (5.90) is zero. By putting Eq. (5.89) into Eq. (5.86), the atomic flux equation can be derived as follows  

In a pure elemental solid, if the point defects are only vacancies, the total number of lattice sites will not be changed, when the atoms or vacancies diffuse from one region to another. Within a given region, the changes in number of atoms and the 

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Kingery adopted a diffusional flux equation similar to that assumed by Coble for the intermediate stage of solid-state sintering (see Chapter 8). In this case, the flux from the boundary per unit thickness is given by... 

For small values of y. Equation 1.32 reduces to a simple diffusional flux equation ... 

The estimation of the diffusional flux to a clean surface of a single spherical bubble moving with a constant velocity relative to a liquid medium requires the solution of the equation for convective diffusion for the component that dissolves in the continuous phase. For steady-state incompressible axisym-metric flow, the equation for convective diffusion in spherical coordinates is approximated by... 

The term hID is often called the diffusional resistance, denoted by R. The flux equation, therefore, can be written as... 

In the particular case dealt with now (fully labile complexation), due to the linearity of a combined diffusion equation for DmCm + DmlL ml, the flux in equation (65) can still be seen as the sum of the independent diffusional fluxes of metal and complex, each contribution depending on the difference between the surface and bulk concentration value of each species. But equation (66) warns against using just a rescaling factor for the total metal or for the free metal alone. In general, if the diffusion is coupled with some nonlinear process, the resulting flux is not proportional to bulk-to-surface differences, and this complicates the use of mass transfer coefficients (see ref. [II] or Chapter 3 in this volume). 

Equation 1.70 shows that the molar diffusional flux of component A in the y-direction is proportional to the concentration gradient of that component. The constant of proportionality is the molecular diffusivity 2. Similarly, equation 1.69 shows that the heat flux is proportional to the gradient of the quantity pCpT, which represents the. concentration of thermal energy. The constant of proportionality klpCp, which is often denoted by a, is the thermal diffusivity and this, like 2, has the units m2/s. 

Equation 3.23 gives the velocity of the local C-frame with respect to the V-frame (i.e., the velocity of local mass flow measured by the velocity of an embedded inert marker relative to the ends of a diffusion couple such as in Figs. 3.3 and 3.4). The measurement of and D at the same concentration in a diffusion experiment thus produces two relationships involving Di and D2 and allows their determination. In the V-frame, the diffusional flux of each component is given by a simple Fick s-law expression where the factor that multiplies the concentration gradient is the interdiffusivity D. In this frame, the interdiffusion is specified completely by one diffusivity. 

Gas Phase Diffusion. The gas phase diffusion limitation arises when the diffusional flux of molecules to the surface of the droplet is less than the maximum possible flux of gas across the surface as given by Equation 2. Under these circumstances the gas density near the surface of the droplet (n ) is smaller than average volume density (n). The situation can be simply analyzed by writing the rate equation for the total number of molecules N in the neighborhood of the droplet ... 

For the total mass conservation of a single-phase fluid, / represents the fluid density p. jr represents the diffusional flux of total mass, which is zero. For flow systems without chemical reactions, d> = 0. Therefore, from Eq. (5.12), we have the continuity equation as... 

In the general case of varying thickness of the ApBq layer, the summation of dl and dt e m makes it possible to match the diffusional flux of the atoms across its bulk with the flux of the same atoms combined at the corresponding interface into the chemical compound. Actually, this is used instead of the continuity equation of any kind, which can hardly be employed in the case under consideration. 

Hence, the current (at any time) is proportional to the concentration gradient of the electroactive species. As indicated by the equations above, the diffusional flux is time-dependent. Such dependence is described by Fick s second law (for linear diffusion) ... 

LP is the hydraulic conductivity coefficient and can have units of m s-1 Pa-1. It describes the mechanical filtration capacity of a membrane or other barrier namely, when An is zero, LP relates the total volume flux density, Jv, to the hydrostatic pressure difference, AP. When AP is zero, Equation 3.37 indicates that a difference in osmotic pressure leads to a diffusional flow characterized by the coefficient Lo Membranes also generally exhibit a property called ultrafiltration, whereby they offer different resistances to the passage of the solute and water.14 For instance, in the absence of an osmotic pressure difference (An = 0), Equation 3.37 indicates a diffusional flux density equal to LopkP. Based on Equation 3.35, vs is then... 

The mass transfer flux law is analogous to the laws for heat and momentum transport. The constitutive equation for Ja, the diffusional flux of A resulting from a concentration difference, is related to the concentration gradient by Pick s first law ... 

With this equation and Fick s law the diffusional flux can be discovered. For a binary mixture of components A and B, (1.161) this is... 

An exclusively analytical treatment of heat and mass transfer in turbulent flow in pipes fails because to date the turbulent shear stress Tl j = —Qw w p heat flux q = —Qcpw, T and also the turbulent diffusional flux j Ai = —gwcannot be investigated in a purely theoretical manner. Rather, we have to rely on experiments. In contrast to laminar flow, turbulent flow in pipes is both hydrodynamically and thermally fully developed after only a short distance x/d > 10 to 60, due to the intensive momentum exchange. This simplifies the representation of the heat and mass transfer coefficients by equations. Simple correlations, which are sufficiently accurate for the description of fully developed turbulent flow, can be found by... 

Diffusion is the macroscopic result of the sum of all molecular motions involved in the sample studied. Molecular motions are described by the general equation of dynamics. However, because of the enormous difference in the orders of magnitude of the masses, sizes, and forces that characterize molecules and macroscopic solids, it can be shown [1] that, when a force field (e.g., an electric field to an ionic solution) is applied to a chemical system, the acceleration of the molecules or ions is nearly instantaneous, molecules drift at a constant velocity, and, in the absence of an external field and of internal forces acting on the feed components, which is the case in chromatography, the diffusional flux, /, of a chemical species i in a gradient of chemical potential is given by... 

If the pressure at the downstream (permeate) side is in the transition or continuum regime and is not negligible, there is a back-diffusional flux into the membrane decreasing the value of a. Equation 9.38 gives the effect of back diffusion on the actual separation factor [23,24]. 

The water flux, J, which is normally expressed as kg (or L) m h is proportional to the water vapor pressure gradient, Apm, between the feed-membrane and strip-membrane interfaces, and the membrane mass transfer co-efficient K, [Eq. (3)]. The vapor pressure gradient between the two interfaces depends on the water activity, a, in the bulk feed and strip streams, and the extent to which concentration polarization reduces that activity at each interface. Whilst can be estimated using established diffusional transport equations, it is more difficult to estimate values for the water vapor pressure at the membrane wall for use in Eq. (3). However, an overall approach using the vapor pressures of the bulk solutions and semi-empirical correlations that take account of the different conditions near the membrane wall can be used to estimate J. 

A more realistic approximate theory for the SECM with a tip shaped as a cone or spherical segment was presented in Ref. 9. The surface of the nonplanar tip electrode was considered to be a series of thin circular strips, each of which is parallel to the planar substrate. The diffusional flux to each strip was calculated using approximate equations for a disk-shaped tip over a conductive or an insulating substrate. The normalized current to the nonplanar tip was obtained by integrating the current over the entire tip surface. Two families of working curves for conical tips over conductive (Fig. 8A) and insulating substrates (Fig. 8B) illustrate the effect of the tip geometry. 

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