Fluxes definitions

The combined diffusivity is, of course, defined as the ratio of the molar flux to the concentration gradient, irrespective of the mechanism of transport. The above equation was derived by separate groups working independently (8-10). It is important to recognize that the molar fluxes (Ni) are defined with respect to a fixed catalyst pellet rather than to a plane of no net transport. Only when there is equimolar counterdiffusion, do the two types of flux definitions become equivalent. For a more detailed discussion of this point, the interested readers should consult Bird, Stewart, and Lightfoot (11). When there is equimolal counterdiffusion NB = —NA and... 

An analytical expression for the heat flux vector can be derived in a similar manner using the Enskog approach. That is, we introduce the first order approximation of the distribution function from (2.246) into the heat flux definition (2.72) and thereafter substitute the partial solution for flux vector integrand as follows [39] ... 

In the literature the net momentum flux transferred from molecules of type s to molecules of type r has either been expressed in terms of the average diffusion velocity for the different species in the mixture [77] or the average species velocity is used [96]. Both approaches lead to the same relation for the diffusion force and thus the Maxwell-Stefan multicomponent diffusion equations. In this book we derive an approximate formula for the diffusion force in terms of the average velocities of the species in the mixture. The diffusive fluxes are introduced at a later stage by use of the combined flux definitions. 

The design of a complete set of governing equations for the description of reactive flows requires that the combined fluxes are treated in a convenient way. In principle, several combined flux definitions are available. However, since the mass fluxes with respect to the mass average velocity are preferred when the equation of motion is included in the problem formulation, we apply the species mass balance equations to a (/-component gas system with q — independent mass fractions Wg and an equal number of independent diffusion fluxes js. However, any of the formulations derived for the multicomponent mass diffusion flux can be substituted into the species mass balance (1.39), hence a closure selection optimization is required considering the specified restrictions for each constitutive model and the computational efforts needed to solve the resulting set of model equations for the particular problem in question. 

Again, it is mentioned that the binary mass flux definitions given above are commonly used also for pseudo-binary systems. In these particular cases the above relationships are only approximate. 

Wherein the definition of the thennodynamic fluxes and forces of (A3.2,131 and (A3.2.141 have been used. Onsager defined [5] the analogue of the Rayleigh dissipation fiinction by... 

But from the definition of the diffusion fluxes given above it follows immediately that... 

Equation (9.23) is to be compared with the Feng and Stewart relations (9.4) which describe the fluxes in the same system under non-reactive conditions, The factor (BA coth BA - 1) has the form sketched in Figure 9.2. From the definition of B given by equation (9.19) it is seen chat 9- 0 as and each tend to zero, their ratio remaining equal to Che... 

Hugo s approach can be extended without difficulty to apply throughout the whole range of pore sizes, but to accomplish this a specific and complete flux model must be used. To be definite we will assume that the dusty gas model is adequate, but the same reasoning could be applied to certain other models if necessary. The relevant flux relations are now equations (5.4). Applied to the radial flux components In one of our three simple geometries they take the form... 

Several additional terms related to the absorption of x-radiation require definition energy of a x-ray photon is properly represented in joules but more conveniently reported in eV fluence is the sum of the energy in a unit area intensity or flux is the fluence per unit time and the exposure is a measure of the number of ions produced in a mass of gas. The unit of exposure in medicine is the Rn ntgen, R, defined as the quantity of radiation required to produce 2.58 x C/kg of air. The absorbed dose for a tissue is a measure of energy dissipated per unit mass. The measure of absorbed dose most... 

Problem Solving Methods Most, if not aU, problems or applications that involve mass transfer can be approached by a systematic-course of action. In the simplest cases, the unknown quantities are obvious. In more complex (e.g., iTmlticomponent, multiphase, multidimensional, nonisothermal, and/or transient) systems, it is more subtle to resolve the known and unknown quantities. For example, in multicomponent systems, one must know the fluxes of the components before predicting their effective diffusivities and vice versa. More will be said about that dilemma later. Once the known and unknown quantities are resolved, however, a combination of conservation equations, definitions, empirical relations, and properties are apphed to arrive at an answer. Figure 5-24 is a flowchart that illustrates the primary types of information and their relationships, and it apphes to many mass-transfer problems. 

Tbe mass-transfer coefficients k c and /cf by definition are equal to tbe ratios of tbe molal mass flux Na to tbe concentration driving forces p — Pi) and (Ci — c) respectively. An alternative expression for tbe rate of transfer in dilute systems is given by... 

The area of the surface across which they jump is I2, so the net flux of atoms, using the definition given earlier, is ... 

In vertical downward flow as well as in upward and downward inclined flows, the flow patterns that can be observed are essentially similar to those described above, and the definitions used can be applied. Experimental data on flow patterns and the transition boundaries are usually mapped on a two dimensional plot. Two basic types of coordinates are generally used for this mapping - one that uses dimensional coordinates such as superficial velocities, mass superficial velocities, or momentum flux and another that uses dimensionless coordinates in which some kind of dimensionless groups are used as coordinates. The dimensional coordinates maps are inherently limited to the range of data and flow conditions under which the experiments were conducted. In spite of this limitation, it is widely used because of its simplicity and ease of use. Figure 24 provides an example of such a map. 

Considered are mass conservation of air and species (contaminants and humidity). Momentum equations are not considered on a global scale but have been used in some cases for the definition of the airflow-pressure relation of the individual links. Heat fluxes and thus energy conservation equations are not considered. 

X-ray production by electron excitation, 5-9, 27, 28, 98-102, 176-179 X-ray radiance, definition, 6 X-ray radiant energy, definition, 6 X-ray radiant flux or power, definition, 6... 

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