Flux expressions

Material Balances Whenever mass-transfer applications involve equipment of specific dimensions, flux equations alone are inadequate to assess results. A material balance or continuity equation must also be used. When the geometry is simple, macroscopic balances suffice. The following equation is an overall mass balance for such a unit having Nm bulk-flow ports and Nn ports or interfaces through which diffusive flux can occur  

For tlie /tli component, mIf = m,w is the component mass flow rate in stream i, , is the mass fraction of componentj in stream i and q is the net reaction rate (mass generation minus consumption) per unit volume V that contains mass M. If it is inconvenient to measure mass flow rates, the product of density and volumetric flow rate is used instead. 

In addition, most situations that involve mass transfer require material balances, but the pertinent area is ambiguous. Examples are packed columns for absorption, distillation, or extraction. In such 

Flux Expressions Simple Integrated Forms of Fick s First 

Law Simplified flux equations that arise from Eqs. (5-189) and (5-190) can be used for unidimensional, steady-state problems with binary mixtures. The boundary conditions represent the compositions xAl and xA(l at the left-hand and right-hand sides of a hypothetical layer having thickness Az. The principal restriction of the following equations is that the concentration and diffusivity are assumed to be constant. As written, the flux is positive from left to right, as depicted in Fig. 5-25. 

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A major achievement [71, 82, 83, 84, 85, 86, 87 and 88] was the development of a simple quantum ( flux-flux ) expression for the cumulative reaction probability, N E), with the final result... 

The flux-flux expression and its extensions have been used to calculate reaction probabilities for several important reactions, including H2+02 H + H2O, by explicit calculation of the action of G in a grid representation with absorbmg potentials. The main power of the flux-flux fomuila over the long mn will be the natural way in which approximations and semi-classical expressions can be inserted into it to treat larger systems. 

Flux Expressions Simple Integrated Forms of Fick s First Law. 5-47... 

The first equahty (on the left-hand side) corresponds to the molar flux with respect to the volume average velocity while the equahty in the center represents the molar flux with respect to the molar average velocity and the one on the right is the mass flux with respect to the mass average velocity These must be used with consistent flux expressions for fixed coordinates and for Nc components, such as ... 

When the flux expressions are consistent, as in Eq. (5-182), the dif-fusivities in Eq. (5-181) are identical. As a result, experimental diffu-sivities are often measured under constant volume conditions but may be used for applications involving open systems. It turns out that the two versions are veiy nearly equivalent for gas-phase systems because there is negligible volume change on mixing. That is not usually true for hquids, however. 

Crystal growth rate may be expressed either as a rate of linear inerease of eharaeteristie dimension (i.e. veloeity) or as a mass deposition rate (i.e. mass flux). Expressed as a veloeity, the overall linear erystal growth rate, G (=dL/dt where L is the eharaeteristie dimension that is inereasing). The rate of ehange of... 

Here, J (a, t) is the instantaneous interfacial flux expressed as a state vector whose components are the diffusional and heat fluxes. Therefore, the expected value < J(a)> takes into account the variations in residence time among the entire bubble population. 

Williams (1987) has pointed out that there are two main approaches to the treatment of such couplings. TTre first is to apply flux expressions like the one described for CO2 in Section 4.4, in Equation (33), and let both the rate coefficient k) and the upper limit of the biota reservoir size... 

Two equivalent flux expressions define such a steady-state transport model [41]... 

There are obvious similarities in the derived relative flux expressions for the MMHS model and the precursor HWPH model. The second term on the right-hand side of Eq. (11) accounts for the increase in dissolution observed near the pKa of the acid. Addition of the ionization reaction [Eq. (9)] provides the added flexibility and accountability so that dissolution at low pH s can be accurately predicted. The agreement between theoretical predictions and experimental results for three carboxylic acids with intrinsic solubilities ranging from 10 2 to 10 M over a pH range of 2-12 was good (see Fig. 2). Computationally, the MMHS model was also quite reasonable to use the roots for a quadratic expres-... 

Fig. 7 Specific TDS flux expressed in t km-2 year-1, together with mean value, and standard deviation for Ebro monitoring stations. Data are reported versus the distance from the source of the Ebro River...

Figure 9.9 Summary of rate or flux expressions for gas-liquid reactions (two-film model)...

Maintenance requirements can be included in the uptake flux expressions by the addition of a constant term [30]. This term expresses the minimum flux necessary for the organism to survive. 

Since interaction phenomena due to simultaneous diffusion of several components play an important role, the Maxwell-Stefan theory has been selected to describe the mass transfer processes. The general form of the flux expressions can be represented by (Taylor and Krishna, 1993)... 

In the very short time limit, q (t) will be in the reactants region if its velocity at time t = 0 is negative. Therefore the zero time limit of the reactive flux expression is just the one dimensional transition state theory estimate for the rate. This means that if one wants to study corrections to TST, all one needs to do munerically is compute the transmission coefficient k defined as the ratio of the numerator of Eq. 14 and its zero time limit. The reactive flux transmission coefficient is then just the plateau value of the average of a unidirectional thermal flux. Numerically it may be actually easier to compute the transmission coefficient than the magnitude of the one dimensional TST rate. Further refinements of the reactive flux method have been devised recently in Refs. 31,32 these allow for even more efficient determination of the reaction rate. 

In environmental systems fluxes are usually expressed as mass per unit area and per time (dimension ML 2 "1) and the occupation numbers as concentrations (dimension ML 3). Then, the constant appearing in the flux expression must have the dimension of a velocity (LT"1). Later we will use the term transfer or exchange velocity and designate it as vA/B or vex to discuss the speed with which mass is moved from subsystem A to subsystem B. To summarize, the mass transfer model takes the form ... 

Here we have assumed that the solution pH and temperature are the same throughout the boundary region. Effectively, we have obtained the same flux expression as Eq. 

Looking back to our molar flux expression (Equation 3.43),... 

Conductive heat transfer does not require any motion of atoms or molecules, as only the interactions between atoms or molecules transfers heat. The heat flux expressed in W rn 2 can be described using Fourier s law ... 

DeLancey GB. Multicomponent film-penetration theory with linearized kinetics— I. Linearization theory and flux expression. Chem Eng Sci 1974 29 2315-2323. 

For such micromolecularly selective processes, an additional partitioning phenomenon must also be considered in the flux expression to enable describing the... 

For an ideal gas flowing through a column or channel we have modified Boyle s law, pV= constant, to the form p(v) = constant. Prove that this modification is valid at constant temperature under steady flow conditions using the general flux expression (Eq. 3.14) in the form... 

Data obtained at NETL for hydrogen flux through the ANL-1 dense ceramic membranes were correlated to yield a flux expression. The flux of hydrogen through these... 

Combining the continuity equation and flux expressions together with particle and iiuid velocities, then writing the result in terms of dimensionless variables, yields the transport equation ... 

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