Viscous Flow Transport

The viscous transport in membrane pores is generally calculated from the Hagen-Poiseuille equation for stationary Newtonian flow in a cylindrical capillary. This leads to the following equations for calculating permeance. 

Viscous flow is nonselective for molecules. However, the flow field around particles, smaller than near the pore entrance may lead to a certain level of size exclusion. Mesoporous membranes that have a charged pore surface in salt solutions may exhibit significant ion retention by a space charge effect if is smaller than the Debye length of the solution. 

Holt and et al. [3] showed that their DWNT membranes are virtually gas and liquid dense nntil the caps of the pores are etched open. This means that there are no major processing defects introduced in the early processing stages. They also demonstrated the size exclusion effect by showing the permeation of 1.3 run [Ru(bpy)3] + complexes while Au particles with 2 run diameter were completely blocked by the pore. Therefore, it can be assumed that 

The unique gas permeation properties of the DWNT membranes stem from the very weak interaction between the gas molecules and the tube wall when the molecule flows to the axial (z) direction. Skouhdas et al. [4] demonstrated that the potential energy barrier, m, for axial translation of Ql. inside a (10, 10) 

In pure, non-hydrocarbon gas permeation experiments (H, He, Ne, N2, O2, Ar, CO2, Xe), Holt et al. foimd an dependence in permeabihty. That is, light gases diffuse faster, in proportion to the molecule s thermal velocity va k T/Mf . The above relationship is usually associated with Knudsen diffusion for the following reasons The self diffusion eoefficient, ZX, of a gas molecules is simply vA, where A is its mean free path. Thus the permeability could be explained if one assumes  

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Jakobsen HA, Lindborg H, Handeland V (2002) A numerical study of the interactions between viscous flow, transport and kinetics in fixed bed reactors. Computers and Chemical Engineering 26 333-357... 

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. 

FIGt 22-48 Transport mechanisms for separation membranes a) Viscous flow, used in UF and MF. No separation achieved in RO, NF, ED, GAS, or PY (h) Knudsen flow used in some gas membranes. Pore diameter < mean free path, (c) Ultramicroporoiis membrane—precise pore diameter used in gas separation, (d) Solution-diffusion used in gas, RO, PY Molecule dissolves in the membrane and diffuses through. Not shown Electro-dialysis membranes and metallic membranes for hydrogen. 

Rheology deals with the deformation and flow of any material under the influence of an applied stress. In practical apphcations, it is related with flow, transport, and handling any simple and complex fluids [1], It deals with a variety of materials from elastic Hookean solids to viscous Newtonian liquid. In general, rheology is concerned with the deformation of solid materials including metals, plastics, and mbbers, and hquids such as polymer melts, slurries, and polymer solutions. 

Although the transport properties, conductivity, and viscosity can be obtained quantitatively from fluctuations in a system at equilibrium in the absence of any driving forces, it is most common to determine the values from experiments in which a flux is induced by an external stress. In the case of viscous flow, the shear viscosity r is the proportionality constant connecting the magnitude of shear stress S to the flux of matter relative to a stationary surface. If the flux is measured as a velocity gradient, then... 

In Figure 2 we presented the permeability coefficient K of oxygen as a function of the mean gas pressure experimentally obtained for a sample of porous material from acetylene black modified with 35% PTFE. The experimental linear dependence is obtained. The intercept with the abscissa corresponds to the Knudsen term DiK. The value obtained is 2,89.1 O 2 cm2/s. The slope of the straight line is small, so that the ratio K,/ Dik at mean gas pressure 1 atm. is small ( 0.1) which means that the gas flow is predominantly achieved by Knudsen diffusion and the viscous flow is quite negligible. At normal conditions (1 atm, 25°C) the mean free path of the air molecules (X a 100 nm) is greater than the mean pore radii in the hydrophobic material (r 20 nm), so that the condition (X r) for the Knudsen-diffusion mechanism of gas transport is fulfilled. 

Hence the heat transport, in this case, depends on the dimension and shape of the liquid container. As we can see in Fig. 2.13, the thermal conductivity (and the specific heat) of liquid 4He decreases when pressure increases and scales with the tube diameter. At temperatures below 0.4 K, the data of thermal conductivity (eq. 2.7) follow the temperature dependence of the Debye specific heat. At higher temperatures, the thermal conductivity increases more steeply because of the viscous flow of the phonons and because of the contribution of the rotons. 

The following compilation is restricted to the transport coefficients of protonic charge carriers, water, and methanol. These may be represented by a 3 X 3 matrix with six independent elements if it is assumed that there is just one mechanism for the transport of each species and their couplings. However, as discussed in Sections 3.1.2.1 and 3.2.1, different types of transport occur, i.e., diffusive transport as usually observed in the solid state and additional hydrodynamic transport (viscous flow), especially at high degrees of solvation. Assuming that the total fluxes are simply the sum of diffusive and hydrodynamic components, the transport matrix may... 

The coefficients Lu, L2A, and L34 describe the viscous flow contributions of the transport of all three species in a total pressure gradient totai- Because a pressure gradient also imposes a chemical potential gradient on each species (eq 24), experimentally, there is always a superposition of diffusive and viscous flow e.g., for the description of the water flux in a total pressure gradient, all coefficients must be included, i.e.. 

The transport properties that are most significantly affected by changes of the water volume fraction are the water/methanol electro-osmotic drag and permeation, both of which have significant contributions from viscous flow (see Section 3.2.1.1). For DMFC applications (where the membrane is in contact with a liquid water/methanol mixture), this type of transport determines the crossover, which is only acceptably low for solvent volume fractions smaller than 20 vol % (see Figures 14 and 15). Consequently, recent attempts have been focused on strengthening... 

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. 

Such expressions can be extended to permit the evaluation of the distribution of concentration throughout laminar flows. Variations in concentration at constant temperature often result in significant variation in viscosity as a function of position in the stream. Thus it is necessary to solve the basic expressions for viscous flow (LI) and to determine the velocity as a function of the spatial coordinates of the system. In the case of small variation in concentration throughout the system it is often convenient and satisfactory to neglect the effect of material transport upon the molecular properties of the phase. Under these circumstances the analysis of boundary layer as reviewed by Schlichting (S4) can be used to evaluate the velocity as a function of position in nonuniform boundary flows. Such analyses permit the determination of material transport from spheres, cylinders, and other objects where the local flow is nonuniform. In such situations it is not practical at the present state of knowledge to take into account the influence of variation in the level of turbulence in the main stream. 

Atoms are transported by viscous flow by differences in the capillary pressure at nonuniformly curved surfaces. 

If sintering occurs by grain-boundary diffusion, the ratio of rates will be the same as for the surface-diffusion case, A-4. A A-1 scaling law can be derived for viscous flow and a A-2 law applies for vapor transport [32],... 

As discussed above, a thermodynamically unstable surface will reduce its total surface energy by forming facets. From the point of view of kinetics, gradients in the chemical potential on a nonequilibrium surface will drive the movement of surface materials toward equilibrium. The transport mechanisms are the same as those that can operate during sintering (47) (a) surface diffusion, (b) bulk diffusion, (c) evaporation-condensation, and (d) plastic or viscous flow. 

Forced Convection. An additional complication arises from convection in the melt forced by the motion of the slider and only marginally assisted by the gas flow above the melt. Forced convection will transport solute across the substrate from the back edge. Moving a solid horizontal boundary across the bottom of an initially stagnant and semiinfinite liquid is a classical problem of unsteady viscous flow (91). The ratio of the velocity of the fluid in the direction of motion, v(y), to the solid-boundary velocity, V, is given by... 

Selective barrier structure. Transport through porous membranes is possible by viscous flow or diffusion, and the selectivity is based on size exclusion (sieving mechanism). This means that permeability and selectivity are mainly influenced by membrane pore size and the (effective) size of the components ofthe feed Molecules... 

As noted above, as the size difference between the solvent and solute become progressively smaller, viscous flow rapidly becomes less important, and molecular interactions become dominant factors. In this limit, molecular solution (or sorption) and diffusion phenomena control the relative transport rates of the solute and solvent. This transition region is an area of ongoing discussion regarding what is a pore and what is not a pore ... 

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