Transport equations viscous flow

All transport processes (viscous flow, diffusion, conduction of electricity) involve ionic movements and ionic drift in a preferred direction they must therefore be interrelated. A relationship between the phenomena of diffusion and viscosity is contained in the Stokes-Einstein equation (4.179). 

If these assumptions are satisfied then the ideas developed earlier about the mean free path can be used to provide qualitative but useful estimates of the transport properties of a dilute gas. While many varied and complicated processes can take place in fluid systems, such as turbulent flow, pattern fonnation, and so on, the principles on which these flows are analysed are remarkably simple. The description of both simple and complicated flows m fluids is based on five hydrodynamic equations, die Navier-Stokes equations. These equations, in trim, are based upon the mechanical laws of conservation of particles, momentum and energy in a fluid, together with a set of phenomenological equations, such as Fourier s law of themial conduction and Newton s law of fluid friction. When these phenomenological laws are used in combination with the conservation equations, one obtains the Navier-Stokes equations. Our goal here is to derive the phenomenological laws from elementary mean free path considerations, and to obtain estimates of the associated transport coefficients. Flere we will consider themial conduction and viscous flow as examples. 

Control volume method Finite element method Boundary element method and analytic element method Designed for conditions with fluxes across interfaces of small, well-mixed elements - primarily used in fluid transport Extrapolates parameters between nodes. Predominant in the analysis of solids, and sometimes used in groundwater flow. Functions with Laplace s equation, which describes highly viscous flow, such as in groundwater, and inviscid flow, which occurs far from boundaries. 

Further the pressure and temperature dependences of all the transport coefficients involved have to be specified. The solution of the equations of change consistent with this additional information then gives the pressure, velocity, and temperature distributions in the system. A number of solutions of idealized problems of interest to chemical engineers may be found in the work of Schlichting (SI) there viscous-flow problems, nonisothermal-flow problems, and boundary-layer problems are discussed. 

These equations are called the Navier-Stokes equations, and when supplemented by the state equation for fluid pressure and species transport equations, they form the basis for any computational model describing the flows in fires. For simplicity, several approximations are inherent (see Equation 20.3) (no Soret/Dufour effects, no viscous dissipation, Fickian diffusion, equal diffusion coefficients of all species, unit Lewis number). 

Before looking at how Gaussian peaks arise from these transport equations we will devote our attention, by way of the next chapter, to the details of another form of transport, flow transport, and some related viscous phenomena. 

Our theoretical calculations indicated viscous flow to be dominant at high densities, as depicted the agreement of the theory in the inset in Figure 1. A viscous flow model may be used over length scales larger than the mean free path, which is largely satisfied for mesopores. To obtain the theoretical transport coefficient we solved die Navier Stokes equation... 

Instead of using a lumped diffusivity to account for transport through the membrane as in Equation (10>5), Sloot et al. [1990] assumed two mechanisms of transport viscous flow and molecular diffusion according to Pick s law. These two mechanisms are assumed to be independent of each other and thus their fluxes are additive ... 

When A < dpoie, the transport mechanism is Poiseuille (viscous) flow and the molar flux NJ from Equations 19.37 and 19.39 can be written as [61]... 

However, if convective transport of heat and species mass in porous catalyst pellets have to be taken into account simulating catal3dic reactor processes, either the Maxwell-Stefan mass flux equations (2.394) or dusty gas model for the mass fluxes (2.427) have to be used with a variable pressure driving force expressed in terms of mass fractions (2.426). The reason for this demand is that any viscous flow in the catalyst pores is driven by a pressure gradient induced by the potential non-uniform spatial species composition and temperature evolution created by the chemical reactions. The pressure gradient in porous media is usually related to the consistent viscous gas velocity through a correlation inspired by the Darcy s law [21] (see e.g., [5] [49] [89], p 197) ... 

Microfluidics is a concept that describes the science and technology of design, fabrication and operation of systems of microchannels that conduct liquids and gases. T q)ically, the channels have widths of tens to hundreds of micrometers and the speed of flow of the fluids is such that the viscous forces dominate over inertial ones. The resulting - linear - equations of flow and its laminar character provide for extensive control the speed of flow obeys the simple Hagen-Poiseuille equation that relates the speed linearly to the pressure drop through the particular channel and to its inverse hydraulic resistance, which in term is a function of the dimensions of the channel and the viscosity of the fluid. This property, when combined with t5q)ically large values of the Peclet number [1] that reflect the fact that diffiisional transport is t5q)ically slow in comparison to the flow, it is possible to control the profiles of concentration [2] of chemicals and... 

Because amorphous materials, such as glasses, have no grain boundaries, their neck growth and densification are caused by viscous flows and the deformation of the particles. In practice, the paths of matter flows are not clearly defined. The geometrical changes caused by the viscous flow could be complex, in which the equations for matter transport can only be established with significantly simplified assumptions. The sintering mechanisms of polycrystalline and amorphous solids are summarized in Table 5.2. 

Given the importance of low-Re, viscous flow on microscale aerodynamics, it is possible to take advantage of the dominant heat transfer effects to enhance microrotorcraft flight. These heat effects can be characterized using the standard transport equations and a Navier-Stokes solver. In order to accurately apply the physical properties, it is important to include the effect of temperature on the viscosity (using, e.g., Sutherland s, Wilke s, or Keyes laws), thermal conductivity, and specific heat of the surrounding fluid (air). 

We have shown the essential features of the time lag in Section 12.2 using the simple Knudsen diffusion as an example, and a direct method of obtaining the time lag in Section 12.3. The diffusion coefficient dealt with in the Frisch s method in Section 12.3 is concentration dependent. In this section we will deal with a case where the transport through the porous medium is a combination of the Knudsen diffusion and the viscous flow mechanism. We shall see below that this case will result in an apparent diffusion coefficient which is concentration dependent, and hence it is susceptible to the Frisch s analysis as outlined in the Section 12.3. This means that the results of equations (12.3-21) are directly applicable to this case. 

The viscous flow mechanism is important when the pressure of the system is reasonably high. When this is the case, the constitutive flux equation describes a combined transport of Knudsen diffusion and viscous flow as ... 

A model, frequently referred to as dusty-gas model [1-3], can be used to describe multi-component diffusion in porous media. This model is based on the Stefan-Maxwell approach for diluted gases which is an approximation of Boltzmann s equation. The pore walls are considered as consisting of giant molecules ( dust ) distributed in space. These dust molecules are treated as the n+l-th pseudo-species in a n-component gaseous mixture. The dust particles are kept fixed in space, and are treated like a gas component in the Stefan-Maxwell equations. This model analyzes the transport problem by distinguishing three separate components 1) diffusion, 2) viscous flow and 3) structure of the porous medium. 

Hereby, B, A and Tq are material-dependent parameters. The parameter is proportional to the activation energy of ionic transport. In a system with a strict coupling between dynamic viscosity and conductivity, as described by the Stokes-Einstein equation, the parameter B in (8.8) is equal to the parameter B in (8.10). In a system with a higher probability for the motion of ionic charge carriers than for viscous flow events, as it can be found in case of cooperative proton transport mechanisms, the strict coupling between dynamic viscosity and conductivity does not hold [56-58]. In this case the parameter Bg in (8.10) will be smaller than B in (8.8). Combining (8.8) and (8.10) and considering the concentration dependence of cr, by introduction of the molar conductivity one will yield a fractional Walden rule (-product) as shown in (8.11). 

The flow in the gas channels and in the porous gas diffusion electrodes is described by the equations for the conservation of momentum and conservation of mass in the gas phase. The solution of these equations results in the velocity and pressure fields in the cell. The Navier-Stokes equations are mostly used for the gas channels while Darcy s law may be used for the gas flow in the GDL, the microporous layer (MPL), and the catalyst layer [147]. Darcy s law describes the flow where the pressure gradient is the major driving force and where it is mostly influenced by the frictional resistance within the pores [145]. Alternatively, the Brinkman equations can be used to compute the fluid velocity and pressure field in porous media. It extends the Darcy law to describe the momentum transport by viscous shear, similar to the Navier-Stokes equations. The velocity and pressure fields are continuous across the interface of the channels and the porous domains. In the presence of a liquid phase in the pore electrolyte, two-phase flow models may be used to account for the interaction between the gas phase and the liquid phase in the pores. When calculating the fluid flow through the inlet and outlet feeders of a large fuel cell stack, the Reynolds-averaged Navier-Stokes (RANS), k-o), or k-e turbulence model equations should be used due to the presence of turbulence. 

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