Viscous Poiseuille

DGM visualises the porous medium as a collection of giant spherical molecules (dust particles) kept in space by external force. The movement of gas molecules in the space between dust particles is described by the kinetic theory of gases. Formally, the MTPM transport parameters and qr can be used also in DGM. The third DGM transport parameter characterises the viscous (Poiseuille) gas flow in pores. 

Viscous (Poiseuille) flow and molecular diffusion are non-selective. Nevertheless they play an important role in the macroporous substrate(s) supporting the separation layer and can seriously affect the total flow resistance of the membrane system. Mesoporous separation layers or supports are frequently in the transient-regime between Knudsen diffusion (flow) and molecular diffusion, with large effects on the separation factor (selectivity). 

Most attempts at describing the Knudsen-viscous Poiseuille transition involve a combination of Eqs. (9.2) and (9.6). For single gases this is a good approximation and after some reorganisation and integration of Eqs. (9.2) and (9.6) and assuming a linear pressure drop across the membrane this yields the expression for the total flux /j ... 

Knudsen diffusion [95,97-99] depending on pressure and mean free path which applies to pores between 10 A and 500 A in size [100]. In this region, the mean free path of molecules is much larger than the pore diameter. It is common to use Knudsen munber = X/d to characterize the regime of permeation through pores. When K 1, viscous (Poiseuille) flow is realized. The condition for Knudsen diffusion is T An intermediate regime is realized when 1. The Knudsen diffusion coefficient can be expressed in the following form ... 

Finally we require a case in which mechanism (lii) above dominates momentum transfer. In flow along a cylindrical tube, mechanism (i) is certainly insignificant compared with mechanism (iii) when the tube diameter is large compared with mean free path lengths, and mechanism (ii) can be eliminated completely by limiting attention to the flow of a pure substance. We then have the classical Poiseuille [13] problem, and for a tube of circular cross-section solution of the viscous flow equations gives 2... 

Equation (46) takes into consideration only the viscous drag due to Poiseuille flow inside the tube. 

The consequences of the wetting ridge in the capillary penetration of a liquid into a small-diameter tube have been evaluated. Viscoelastic braking reduces the liquid flow rate when viscoelastic dissipation outweighs the viscous drag resulting from Poiseuille flow. 

Figure 18 illustrates the difference between normal hydrodynamic flow and slip flow when a gas sample is confined between two surfaces in motion relative to each other. In each case, the top surface moves with speed ua relative to the bottom surface. The circles represent gas molecules, and the length of an arrow is proportional to the drift velocity for that molecule. The drift velocity variation with distance is illustrated by the plots on the right. When the ratio of the mean free path to the separation distance between surfaces is much less than unity (Fig. 18a), collisions between gas molecules are much more frequent than collisions of the gas molecules with the surfaces. Here, we have classical fluid flow or viscous flow. If the flow were flow in tubes, Poiseuille s law would be obeyed. The velocity of gas molecules at the surface is the same as the velocity of the surface, and in the case of the stationary surface the mean tangential velocity of the gas at the surface is zero. 

The analysis of the conductance of an orifice in the viscous flow range is quite complicated and is reported in ref. [7,8], The conductance of a long pipe for the laminar flow of an incompressible gas is given by the Poiseuille equation ... 

Laminar flow In circular tubes with parabolic velocity distribution Is known as Poiseuille flow. This special case is found frequently in vacuum technology. Viscous flow will generally be found where the molecules mean free path is considerably shorter than the diameter of the pipe X d. 

From Equation (1.2), which is derived from the Poiseuille formula for viscous flow, it can be seen that Q is directly proportional to p. For air at 25 °C, Equation (1.2) becomes Equation (1.3) ... 

Several modifications of Poiseuille s equation have been attempted by various authors to describe permeability in the transitional region between viscous and diffusional flow. The assumptions underlying these modifications are often questionable and the results obtained offer little or no theoretical or experimental advantage over the BET theory for surface area measurements. Allen" " discusses these modifications as well as diffusional flow at low pressures. 

The application of the hydrostatic pressure to the solvent of gel causes the permeation flow of the solvent. In this process, the gel behaves as a molecular sieve and imposes a frictional resistance on the flowing water. The permeation flow of water through a gel is a process analogous to the capillary flow of a viscous fluid. The frictional resistance of a single capillary is well described by the Hagen-Poiseuille s law by which the relationship between the dimension of the capillary, the applied pressure, and the flow rate is given as follows... 

If the viscosity varies during flow for some reason (decreases with rising temperature or increases as a result of a chemical reaction such as polymerization), the linear Poiseuille P-vs-Q relation is violated and the pressure drop - flow rate curve may become nonmonotonic. This effect in polymerizing reactors can be explained by the fact that the most viscous products of a reaction are swept out of the reactor with increasing flow rate and are replaced. Instead, a reactor is refilled with a fresh reactive mixture of low viscosity. This leads to a decrease of the volume-averaged integral viscosity and therefore the pressure drop decreases. This can be illustrated by the following relationship ... 

The laminar stationary flow of an incompressible viscous liquid through cylindrical tubes can be described by Poiseuille s law this description was later extended to turbulent flow. Flowing patterns of two immiscible phases are more complex in microcapillaries various patterns of liquid-liquid flow are described in more detail in Chapter 4.3, while liquid-gas flow and related applications are discussed in Chapter 4.4. 

We consider viscous dissipation in Couette flow We consider the pressure gradient in Poiseuille flow Viscous dissipation acts as heat source in Couette flow For moderate flow, no heat source within Poiseuille flow. 

Figure 4 schematically represents these processes. Poiseuille flow (or viscous flow) occurs when collisions between the gas molecules are more frequent than collisions between gas molecules and the pore walls. This mechanism, which is a pressure-driven one, is non-separative and takes place in large pores (and defects) of the membrane. 

Hagen-Poiseuille equation — predicts the laminar flow of an incompressible and uniform viscous liquid (Newtonian fluid) through a cylindrical pipe of constant cross-section. The rate of movement of a liquid volume V, during a time t, may be predicted using ... 

The flow rate of a viscous liquid through a tube is given by the Poiseuille equation ... 

Aside from this, the data on burning velocities seem to be in almost quantitative accord with the conduction equation (XIV. 10.23) when adapted to flames in finite systems such as cylinders and spheres. The velocity of flame propagation in tubes is complicated by the viscous drag exerted by the walls on the flowing gas, together with the heat losses at the walls. The resulting Poiseuille type of flow tends to make the flame fronts parabolic in these systems. 

The temperature contours for convectionless flow are shown in figure 2, which shows a hot region at the entrance of the capillary due to the combination of high viscous energy dissipation there and its distance from cool boundaries to which heat may be conducted. These isotherms are normalized on the maximum centerline temperature expected for Poiseuille flow in the capillary. 

If liquid is allowed to accumulate on one side of a diaphragm, an excess hydrostatic pressure is set up which eventually counterbalances the electro-osmotic flow. The simplest case to consider is that of electroosmosis through a single capillary tube, for it is then possible to apply the Poiseuille equation for viscous flow thus, if v is the volume of liquid of viscosity r) flowing per second through a capillary tube of length I and radius r under a difference of pressure P, then... 

Old observations on spheres falling through liquids are collected by Muncke.12 The falling sphere method seems to have been first used for viscous liquids (tar, glycerol) by Schottner, ho used 2-4 cm. diam. spheres in a vessel of 20 cm. diam. With. 13 cm. diam. tubes the rate of fall was slower. The values of 97 were larger than those found by Poiseuille s method. 

Here is a typical Leslie viscosity, AT is a Frank constant, V is a flow velocity, /i is a length scale of the flow geometry, such as the tube diameter in Poiseuille flow, and ftn is the average shear rate. The Ericksen number is the ratio of the flow-induced viscous stress 6Ye.fi = /h to the Frank stress K/h. The appropriate Leslie viscosity or Frank constant... 

Considering at first transport mechanisms in porous membranes, viscous flow (Fig. 9), also called Poiseuille flow, takes place when the mean pore diameter is larger than the mean free path of gas molecules (pore diameter higher than a few microns), so that collisions between different molecules are much more frequent than those between molecules and pore walls. In such conditions, no separation between different molecules can be attained [45]. 

Basic mechanisms involved in gas and vapor separation using ceramic membranes are schematized in Figure 6.14. In general, single gas permeation mechanisms in a porous ceramic membrane of thickness depend on the ratio of the number of molecule-molecule collisions to that of the molecule-wall collisions. In membranes with large mesopores and macropores the separation selectivity is weak. The number of intermolecular collisions is strongly dominant and gas transport in the porosity is described as a viscous flow that can be quantified by a Hagen-Poiseuille type law ... 

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]... 

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