Transport momentum transfer

Bubble-column slurry operations are usually characterized by zero net liquid flow, and the particles are held suspended by momentum transferred from the gas phase to the solid phase via the liquid medium. The relationships between solids holdup and gas flow rate is of importance for design of bubble-column slurries, and some studies of this aspect will be reviewed prior to the discussion of transport phenomena. 

The topics of heat, mass and momentum transfer, known collectively as transport processes, are fully examined in the books by Welty et al. [21] and Bird et al. [22]. There is a useful introduction to fluid mechanics and heat transfer by Kay and Nedderman [23], while mass transfer is fully discussed by Treybal [24] and Sherwood et al. [25]. Coulson and Richardson [26] also give clear introductions to these subjects. 

Of the three general categories of transport processes, heat transport gets the most attention for several reasons. First, unlike momentum transfer, it occurs in both the liquid and solid states of a material. Second, it is important not only in the processing and production of materials, but in their application and use. Ultimately, the thermal properties of a material may be the most influential design parameters in selecting a material for a specific application. In the description of heat transport properties, let us limit ourselves to conduction as the primary means of transfer, while recognizing that for some processes, convection or radiation may play a more important role. Finally, we will limit the discussion here to theoretical and empirical correlations and trends in heat transport properties. Tabulated values of thermal conductivities for a variety of materials can be found in Appendix 5. 

What makes the fabrication of composite materials so complex is that it involves simultaneous heat, mass, and momentum transfer, along with chemical reactions in a multiphase system with time-dependent material properties and boundary conditions. Composite manufacturing requires knowledge of chemistry, polymer and material science, rheology, kinetics, transport phenomena, mechanics, and control systems. Therefore, at first, composite manufacturing was somewhat of a mystery because very diverse knowledge was required of its practitioners. We now better understand the different fundamental aspects of composite processing so that this book could be written with contributions from many composite practitioners. 

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 ... 

The driving forces, or driving potentials, for transport phenomena are (i) the temperature difference for heat transfer (ii) the concentration or partial pressure difference for mass transfer and (iii) the difference in momentum for momentum transfer. When the driving force becomes negligible, then the transport phenomenon will cease to occur, and the system will reach equilibrium. 

Just as diffusive momentum transfer depends on a transport property of the fluid called viscosity, diffusive heat transfer depends on a transport property called thermal conductivity. This section provides a brief discussion on the functional forms of thermal conductivity, with the intent of facilitating the understanding of the heat-transfer discussions in the subsequent sections on the conservation of energy. 

As a layer of gas at one velocity is pulled across an adjacent layer of gas at a slightly different velocity, gas in the faster layer tends to be slowed down by the interaction, and gas in the faster layer tends to speed up. There is velocity or, more precisely, momentum transfer between the layers. Thus it takes a force to maintain the velocity gradient across the fluid. (Recall the definition that force is the time rate of change of momentum.) Fundamentally, the viscosity is a transport property associated with momentum transfer. 

The Prandtl number, Cu/k = (p/p)/(k/pC), compares the rate of momentum transfer through friction to the thermal diffusivity or the transport of heat by conduction. 

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... 

While some reconciliation models only have material balance relationships, more meaningful reconciliation results are obtained with models that include material balances, heat balances, equilibrium constraints (both in the separation and reaction domains), rate relationships (heat transfer, mass transfer, momentum transfer, and kinetics), as well as equipment-specific relationships. In other words, one should include more than just material balance constraints when reconciling a model. Heat balance, kinetics, transport relationships—if needed for the... 

Fluid flow, also known as fluid mechanics, momentum transfer and momentum transport, is a wide-ranging subject, fundamental to many aspects of chemical engineering. It is impossible to cover the whole field here, so we will concentrate upon a few aspects of the subject. 

There are many standard texts of fluid flow, e.g. Coulson Richardson,1 Kay and Neddermann2 and Massey.3 Perry4,5 is also a useful reference source of methods and data. Schaschke6 presents a large number of useful worked examples in fluid mechanics. In many recent texts, fluid mechanics or momentum transfer has been treated in parallel with the two other transport or transfer processes, heat and mass transfer. The classic text here is Bird, Stewart and Lightfoot.7... 

Studies with many types of porous media have shown that for the transport of a pure gas the Knudsen diffusion and viscous flow are additive (Present and DeBethune [52] and references therein). When more than one type of molecules is present at intermediate pressures there will also be momentum transfer from the light (fast) molecules to the heavy (slow) ones, which gives rise to non-selective mass transport. For the description of these combined mechanisms, sophisticated models have to be used for a proper description of mass transport, such as the model presented by Present and DeBethune or the Dusty Gas Model (DGM) [53], In the DGM the membrane is visualised as a collection of huge dust particles, held motionless in space. 

The rate at which the momentum transfer takes place is dependent on the rate at which the molecules move across the fluid layers. In a gas, the molecules would move about with some average speed proportional to the square root of the absolute temperature since, in the kinetic theory of gases, we identify temperature with the mean kinetic energy of a molecule. The faster the molecules move, the more momentum they will transport. Hence we should expect the viscosity of a gas to be approximately proportional to the square root of temperature, and this expectation is corroborated fairly well by experiment. The viscosities of some typical fluids are given in Appendix A. 

A necessary condition for the two-term expansion of the distribution function of equation (2) to be valid is that the electron collision frequency for momentum transfer must be larger than the total electron collision frequency for excitation for all values of electron energy. Under these conditions electron-heavy particle momentum-transfer collisions are of major importance in reducing the asymmetry in the distribution function. In many cases as pointed out by Phelps in ref. 34, this condition is not met in the analysis of N2, CO, and C02 transport data primarily because of large vibrational excitation cross sections. The effect on the accuracy of the determination of distribution functions as a result is a factor still remaining to be assessed. 

Equation (2.26) for heat conduction and Eq. (2.3) for momentum transfer are similar, and the flow is proportional to the negative of the gradient of a macroscopic variable the coefficient of proportionality is a physical property characteristic of the medium and dependent on the temperature and pressure. In a three-dimensional transport, Eqs. (2.27) and (2.15) differ because the heat flow is a vector with three components, and the momentum flow t is a second-order tensor with nine components. 

This model assumes the diffusive flows combine by the additivity of momentum transfer, whereas the diffusive and viscous flows combine by the additivity of the fluxes. To the knowledge of the authors there has never been given a sound argument for the latter assumption. It has been shown that the assumption may result in errors for certain situations [22]. Nonetheless, the model is widely used with reasonably satisfactory results for most situations. Temperature gradients (thermal diffusion) and external forces (forced diffusion) are also considered in the general version of the model. The incorporation of surface diffusion into a model of transport in a porous medium is quite straightforward, since the surface diffusion fluxes can be added to the diffusion fluxes in the gaseous phase. 

Thus, when a particle jumps, it leaves behind a hole. So then, instead of saying that a transport process occurs by particles hopping along, one could equally well say that the transport processes occur by holes moving. The concept is commonplace in semiconductor theory, where the movement of electrons in the conduction band is taken as being equivalent to a movement of so-called holes in the valence band. It has in fact already been assumed at the start of the viscosity treatment (Section 5.7.1) that the viscous flow of fused salts can be discussed in terms of the momentum transferred between liquid layers by moving holes. 

In the case of momentum transfer, we have a particular situation where the property transport occurs towards the walls and its transformation is controlled by the geometry of the wall. 

Interphase transfer kinetics. At this point, we need to characterize the process that leads to the transfer of the property through the interphase. The transport of the momentum from one phase to another is spectacular when the contacting phases are deformable. Sometimes in these situations we can neglect the friction and the momentum transfer generates the formation of bubbles, drops, jets, etc. The characterization of these flow cases requires some additions to the momentum equations and energy transfer equations. 

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