Mass-energy flux

Each energy flux consists of an externally supplied part and an internally generated part. For example, the heat flux dQ is decomposed into the heat flow d Q supplied from outside the system and the internally generated heat flow d Q the mechanical power dW and the mass-energy flux dC are decomposed in the same manner ... 

Affinity at reaction stage r Acceleration Activity of species a Left Cauchy-Green tensor Body force per unit mass Diffusive body force Right Cauchy-Green tensor Mass-molar concentration Mass concentration of species a Mass-energy flux of the system... 

The tliree conservation laws of mass, momentum and energy play a central role in the hydrodynamic description. For a one-component system, these are the only hydrodynamic variables. The mass density has an interesting feature in the associated continuity equation the mass current (flux) is the momentum density and thus itself is conserved, in the absence of external forces. The mass density p(r,0 satisfies a continuity equation which can be expressed in the fonn (see, for example, the book on fluid mechanics by Landau and Lifshitz, cited in the Furtlier Reading)... 

V is the material velocity. a is the stress tensor. g is the acceleration of gravity. e is the internal energy per unit mass. h is the energy flux. 

In addition, Turner and Trimble defined a slip equation of state combination as the specification of mass flux, momentum flux, energy density, and energy flux as single-valued functions of the geometric parameters (area, equivalent diameter, roughness, etc.) at any z location, and of mass flux, pressure, and enthalpy,... 

Chang, S. H., and K. W. Lee, 1988, A Derivation of Critical Heat Flux Model in Flow Boiling at Low Qualities Based on Mass, Energy, and Momentum Balance, Korea Advanced Institute of Science and Technology, Taejeon, Korea. 

This expression applies to the transport of any conserved quantity Q, e.g., mass, energy, momentum, or charge. The rate of transport of Q per unit area normal to the direction of transport is called the flux of Q. This transport equation can be applied on a microscopic or molecular scale to a stationary medium or a fluid in laminar flow, in which the mechanism for the transport of Q is the intermolecular forces of attraction between molecules or groups of molecules. It also applies to fluids in turbulent flow, on a turbulent convective scale, in which the mechanism for transport is the result of the motion of turbulent eddies in the fluid that move in three directions and carry Q with them. 

The mass and heat transport model should be able to predict mass and energy fluxes through a gas/vapour-liquid interface in case a chemical reaction occurs in the liquid phase. In this study the film model will be adopted which postulates the existence of a well-mixed bulk and a stagnant transfer zone near the interface (see Fig. 1). The equations describing the mass and heat fluxes play an important role in our model and will be presented subsequently. 

In addition to the expression for the mass and energy fluxes, conservation equations for mass and energy are required to enable the calculation of concentration and temperature profiles. From these profiles the mass and heat transfer rates through the va-pour/gas-liquid interface can subsequently be obtained. The species conservation equations for the liquid and the vapour/gas phase are respectively given by... 

In the gas/vapour phase the dimensionless distance tj ranges from 0 to 1, where tj — 1 corresponds to the position of the interface. In the liquid phase this parameter ranges from 0 to 1 for the mass transfer film and from 0 to Le for the heat transfer film. Hence, rj = 0 corresponds to the position of the interface and rj = I and t] = Le correspond, respectively, to the boundaries of the mass and heat transfer film. The mass and energy fluxes can now be calculated by solving the differential equations (4) and (8)-(12) subject to the boundary conditions (15). Due to the non-linearities a numerical solution procedure has been used which will be discussed subsequently. 

Theoretical investigations of the problem were carried out on the base of the mathematical model, combining both deterministic and stochastic approaches to turbulent combustion of organic dust-air mixtures modeling. To simulate the gas-phase flow, the k-e model is used with account of mass, momentum, and energy fluxes from the particles phase. The equations of motion for particles take into account random turbulent pulsations in the gas flow. The mean characteristics of those pulsations and the probability distribution functions are determined with the help of solutions obtained within the frame of the k-e model. 

To close the model, one needs the expressions for mass, momentum, and energy fluxes from the other phases. 

It should be emphasized that the flux vectors for which expressions have been given in Eqs. (28) through (36) are all defined here as fluxes with respect to the mass average velocity. Not all authors use this convention, and considerable confusion has resulted in the definition of the energy flux and the mass flux. Mass fluxes with respect to molar average velocity, stationary coordinates, and the velocity of one component (such as the solvent, for example) are all to be found in the literature on diffusional processes. Research workers in the field of diffusion should be meticulous in specifying the frame of reference for fluxes used in writing up their research work. In the next section this important matter is considered in detail for two-component systems. 

The approach pursued in this and the next chapter is focused on the common mathematical characteristics of boundary processes. Most of the necessary mathematics has been developed in Chapter 18. Yet, from a physical point of view, many different driving forces are responsible for the transfer of mass. For instance, air-water exchange (Chapter 20), described as either bottleneck or diffusive boundary, is controlled by the turbulent energy flux produced by wind and water currents. The nature of these and other phenomena will be discussed once the mathematical structure of the models has been developed. 

Fig. 11.7 Schematic of the mass and energy flux balances at the gas-surface interface.

As the star expands, the photosphere becomes deeper as the recombination front proceeds through the hydrogen-rich envelope deeper in mass. At the same time a heat wave is propagating out from the interior. At a certain stage, energy flux due to radioactive decays exceeds that from shock heating. The dates when the radioactivity starts to dominate and when the luminosity reaches its peak depend on the above three factors as follows. 

Due to the chemical conversion in the liquid film, the molar fluxes at the interface and at the boundary between the film and the bulk of the phase differ. The system of equations is completed by the conservation equations for the mass and energy fluxes at the phase interface and the necessary linking conditions between the bulk and film phases (see Refs. 57, 59, and 84). 

The tidal flats of marine environments are areas of extreme complexity and biological activity. They serve as both sources and sinks of a wide variety of compounds and materials. They are in a constant state of mass, energy and momentum flux with the surrounding environment In these areas sulfur plays a major role in biological processes, principally because of the relatively nigh concentration of sulfate ion in marine waters. 

Available energy Mass flow rate Mass flux Pressure Gas Constant Entropy per unit mass Entropy flux Entropy production Time... 

Internal energy per mass Internal energy flux Volume... 

The correlations for the mass and heat transfer coefficients and interfacial also take into account packing or tray geometries for the actual column. The total mass and energy rates are calculated from intergrating the mass and energy fluxes across the total surface area, a.j. 

Jx, Jy, Jz are the vector components in the x, y and z axis directions of the coordinate system, Jx, Jy, Jz are their contributions and i,jand k are the corresponding unit vectors. Given a mass quantity m that is transported during time t through an area A, then let/ represent the contribution of the mass flux. For energy transport, then J is the contribution of the energy flux with the dimensions J/m2s (where J = Joule). 

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