Transport of mass

At this point it is appropriate to introduce the particular flux vectors associated with the transport of mass, momentum, and kinetic energy. 

---

Diffusion is the molecular transport of mass without flow. The diffu-sivity (D) or diffusion coefficient is the proportionality constant between the diffusion and the concentration gradient causing diffusion. It is usually defined by Fick s first law for one-dimensional, binary component diffusion for molecular transport without turbulence shown by Eq. (2-149)... 

High-pressure systems are characteristicaly the opposite of lows. Since the winds flow outward from the high-pressure center, subsiding air from higher in the atmosphere compensates for the horizontal transport of mass. 

Mass transfer Irreversible and spontaneous transport of mass of a chemical component in a space with a non-homogeneous field of the chemical potential of the component. The driving force causing the transport can be the difference in concentration (in liquids) or partial pressures ( in gases) of the component. In biological systems. 

In Table 13-2, the numerator of a well-known group is identified by the arrowhead, while the denominator is assigned by the tail of die aiTow. For example, the Damkdhler I represents the division of the dimensions representing die generation per unit volume by die dimensions representing die transport of mass by eonveetion. This is defined as ... 

The Damkdher II is determined by dividing die dimensions representing die generation per unit volume by die dimensions representing die transport of mass by moleeular proeess as ... 

The chemical engineer is concerned with the industrial application of processes. This involves the chemical and microbiological conversion of material with the transport of mass, heat and momentum. These processes are scale-dependent (i.e., they may behave differently in small and large-scale systems) and include heterogeneous chemical reactions and most unit operations. Tlie heterogeneous chemical reactions (liquid-liquid, liquid-gas, liquid-solid, gas-solid, solid-solid) generate or consume a considerable amount of heat. However, the course of... 

If we take a thin slice of the cylinder of thickness dx we can write an expression for the transport of mass through this slice at steady state. What goes in either comes out or reacts, i.e. 

Biofilms adhere to surfaces, hence in nearly all systems of interest, whether a medical device or geological media, transport of mass from bulk fluid to the biofilm-fluid interface is impacted by the velocity field [24, 25]. Coupling of the velocity field to mass transport is a fundamental aspect of mass conservation [2]. The concentration of a species c(r,t) satisfies the advection diffusion equation... 

Convection—the transport of mass or energy as a result of streaming in the system produced by the action of external forces. These include mechanical forces (forced convection) or gravitation, if there are density gradients in the system (natural or free convection). 

More complicated models account for the transport of mass or heat into or out of the system, so that its composition or temperature, or both, vary over the course of the calculation. The system s initial equilibrium state provides the starting point... 

Although mass transfer across the water-air interface is difficult in terms of its application in a sewer system, it is important to understand the concept theoretically. The resistance to the transport of mass is mainly expected to reside in the thin water and gas layers located at the interface, i.e., the two films where the gradients are indicated (Figure 4.3). The resistance to the mass transfer in the interface itself is assumed to be negligible. From a theoretical point of view, equilibrium conditions exist at the interface. Because of this conceptual understanding of the transport across the air-water boundary, the theory for the mass transport is often referred to as the two-film theory (Lewis and Whitman, 1924). 

In this paper a transfer model will be presented, which can predict mass and energy transport through a gas/vapour-liquid interface where a chemical reaction occurs simultaneously in the liquid phase. In this model the Maxwell-Stefan theory has been used to describe the transport of mass and heat. On the basis of this model a numerical study will be made to investigate the consequences of using the Maxwell-Stefan equation for describing mass transfer in case of physical absorption and in case of absorption with chemical reaction. Despite the fact that the Maxwell-Stefan theory has received significant attention, the incorporation of chemical reactions with associated... 

This exciton diffuses to the donor/acceptor interface via an energy-transfer mechanism (i.e., no net transport of mass or charge occurs). (3) Charge-transfer quenching of the exciton at the D/A interface produces a charge- transfer (CT) state, in the form of a coulombically interacting donor/acceptor complex (D A ). The nomenclature used to describe this species has been relatively imprecise, and has... 

The physical transport of mass is essential to many kinetic and d3mamic processes. For example, bubble growth in magma or beer requires mass transfer to bring the gas components to the bubbles radiogenic Ar in a mineral can be lost due to diffusion pollutants in rivers are transported by river flow and diluted by eddy diffusion. Although fluid flow is also important or more important in mass transfer, in this book, we will not deal with fluid flow much because it is the realm of fluid dynamics, not of kinetics. We will focus on diffusive mass transfer, and discuss fluid flow only in relation to diffusion. 

Chapter 4 Mass, Heat, and Momentum Transport Analogies. The transport of mass, heat, and momentum is modeled with analogous transport equations, except for the source and sink terms. Another difference between these equations is the magnitude of the diffusive transport coefficients. The similarities and differences between the transport of mass, heat, and momentum and the solution of the transport equations will be investigated in this chapter. 

Prandtl s mixing length hypothesis (Prandtl, 1925) was developed for momentum transport, instead of mass transport. The end result was a turbulent viscosity, instead of a turbulent diffusivity. However, because both turbulent viscosity and turbulent diffusion coefficient are properties of the flow field, they are related. Turbulent viscosity describes the transport of momentum by turbulence, and turbulent diffusivity describes the transport of mass by the same turbulence. Thus, turbulent viscosity is often related to turbulent diffusivity as... 

Recall from the beginning of this chapter that the transport of mass at steady state is governed by Pick s Law, given in one-dimension by Equation (4.4) ... 

Transport of mass, energy, and momentum in porous media is a key aspect of a large number of fiber-reinforced plastic composite fabrication processes. In design and optimization of such processes, computer simulation plays an important role. Recent studies [1-14] have... 

It is extremely difficult to model macroscopic transport of mass, energy, and momentum in porous media commonly encountered in various fields of science and engineering based on microscopic transport models that account for variation of velocity and temperature as well as other quantities of interest past individual solid particles. The basic idea of porous media theory, therefore, is to volume average the quantities of interest and develop field equations based on these average quantities. 

The first step in applying volume averaging is to consider a representative volume for every point A in the porous medium. This volume must be large enough to contain sufficient amount of each phase such that continuum theory for transport of mass, energy, and... 

In this section a specialized set of equations governing transport of mass, momentum, and energy in resin transfer molding, injected pultrusion, and autoclave processing are obtained from the general balance equations presented in Section 5.3. This involves eliminating unimportant terms in the general balance equations based on the specific nature of the process. 

Momentum transfer can be described by Equation 5.29 provided Rep < 1 (which is a reasonable assumption in majority of RTM processes [16]). Finally, combining all of the preceding assumptions plus the assumption of a local equilibrium allows us to simplifyjiquation 541 significantly and obtain an energy equation for this process (i.e., (Uf) = V (Ur) = V (Uf) = 0). In summary, the appropriate governing equations for transport of mass, momentum, and energy in the RTM process are ... 

In summary, the appropriate governing equations for transport of mass, momentum, and energy in the IP process are ... 

Throughout this summary we have neglected the effect of dispersion on the overall transport of mass and heat. This is due to the fact that if dispersion is included, dispersion tensors must be determined before the equation can be solved. This can be done by solving the appropriate transport equation within a unit cell. Because a unit cell cannot be defined in most reinforcements used in polymer matrix composites, however, dispersion tensors cannot be accurately determined, so we have left dispersion effects out of our equations. In general, we anticipate dispersion to play a minor role in the IP, AP, and RTM processes. This assumption can be checked, however, by evaluating the dispersion terms using an approach similar to [16] where experiments and correlations are used to determine the importance of dispersion. 

It should be noted that the effect of fluid viscoelasticity on transport of mass, momentum, and heat in porous media has not been discussed in this summary. Although some preliminary studies have been performed in this area [21], no definitive governing equations exist. 

Axial transport of mass and energy in the gas phase by convection only. 

It should be mentioned here that, in living systems the transport of mass sometimes takes place apparently against the concentration gradient. Such uphill mass transport, which usually occurs in biological membranes with the consumption of biochemical energy, is called active transport, and should be distinguished from passive transport, which is the ordinary downhill mass transport as discussed in this chapter. Active transport in biological systems is beyond the scope of this book. 

Occasionally, some residual homopolar bonds remain in metals, for example a small per cent of the molecules Li—Li, Na—Na, etc. are found in the vapours of these metals, analogous to the hydrogen molecule, but there is no trace of them in the solid state. The most characteristic property of metals, in which the smallest potential difference produces an electric current, is their electrical conductivity. Since no transport of mass takes place in a metallic conductor, a metal must contain free electrons, from which it follows that positive ions must also be present. The picture of a metal is thei efore one in which the lattice is composed of positive ions held together by electrons which move freely in the space between. It is as though the ions were cemented together by an electronic gas. 

In this text we are concerned exclusively with laminar flows that is, we do not discuss turbulent flow. However, we are concerned with the complexities of multicomponent molecular transport of mass, momentum, and energy by diffusive processes, especially in gas mixtures. Accordingly we introduce the kinetic-theory formalism required to determine mixture viscosity and thermal conductivity, as well as multicomponent ordinary and thermal diffusion coefficients. Perhaps it should be noted in passing that certain laminar, strained, flames are developed and studied specifically because of the insight they offer for understanding turbulent flame environments. 

It is important to realize that there cannot be a net transport of mass by diffusive action within a homogeneous multicomponent fluid. The transport of some species in one direction must be balanced by transport of other species in the other direction. The reasons for this behavior will be discussed later. For now we simply note that... 

Collisions between molecules occur in the gas phase. These collisions can transfer momentum and energy between the collision partners, or lead to net transport of mass from one part of the system to another. 

The so-called diffusion theories of flame propagation, as exemplified by the work of Tanford and Pease 38), emphasize the transport of mass, in that concentration of an active radical is assumed to be the rate-controlling property. Its use seems to be fairly limited in that only a few specific reactions have been successfully studied with this theory. What is more interesting, however, is that this theory forms the counterpart to the thermal theory. These two extreme views bracket the actual case, and their study allows a consideration of each of two of the basic flame mechanisms, unencumbered by the other. Actual deflagration depends on both the transport of heat and the transport of mass, and a successful theory should contain both phenomena. 

Finally, mechanisms besides diffusional transport of mass between internal interfaces can contribute to diffusional creep. For instance, single crystals containing dislocations exhibit limited creep if the dislocations act as sources and sinks, depending on their orientation with respect to an applied stress (see Exercise 16.3). 

Process-scale models represent the behavior of reaction, separation and mass, heat, and momentum transfer at the process flowsheet level, or for a network of process flowsheets. Whether based on first-principles or empirical relations, the model equations for these systems typically consist of conservation laws (based on mass, heat, and momentum), physical and chemical equilibrium among species and phases, and additional constitutive equations that describe the rates of chemical transformation or transport of mass and energy. These process models are often represented by a collection of individual unit models (the so-called unit operations) that usually correspond to major pieces of process equipment, which, in turn, are captured by device-level models. These unit models are assembled within a process flowsheet that describes the interaction of equipment either for steady state or dynamic behavior. As a result, models can be described by algebraic or differential equations. As illustrated in Figure 3 for a PEFC-base power plant, steady-state process flowsheets are usually described by lumped parameter models described by algebraic equations. Similarly, dynamic process flowsheets are described by lumped parameter models comprising differential-algebraic equations. Models that deal with spatially distributed models are frequently considered at the device... 

Fig. 2 Layout of isotope separator in beam line C. The numbers indicate (1) He-jet recoil chamber, (2) beam-stdpper, (3) ion-source, (4) analysing magnet, (3) Einzel lens for transport of mass-separated beam, (6) tape-transport and counting station, (7) power/control rack of tape-transport, (8) power/control of mass-separa tor.

The permeability P of a gas through a polymer can be measured directly by determining the transport of mass through a membrane per unit time. The sorption constant S can be measured by placing a saturated sample into an environment, which allows the sample to desorb and measure the loss of weight. As shown in Fig. 2.63, it is common to plot the ratio of concentration of absorbed substance c t) to saturation coefficient Coo with respect to the root of time. 

---