Viscous transport

Viscous Transport. Low velocity viscous laminar dow ia gas pipes is commonplace. Practical gas dow can be based on pressure drops of <50% for low velocity laminar dow ia pipes whose length-to-diameter ratio may be as high as several thousand. Under laminar dow, bends and fittings add to the frictional loss, as do abmpt transitions. 

From this equation it is clear that there is an instantaneous relationship between the velocity field and the pressure field that is described by a Poisson equation. It does not depend directly on the viscosity nor involve any viscous transport terms. Note, again, that this result is only for incompressible flows. 

Viscous transport is laminar or streamline, if molecules in the (imaginary) planes of fluid in any geometry, flowing along a velocity gradient, do so with the same translational and rotational velocities. Streamlines are occa-... 

It turns out that Eq. (5-56) can also be applied to turbulent flow over a flat plate and in a modified way to turbulent flow in a tube. It does not apply to laminar tube flow. In general, a more rigorous treatment of the governing equations is necessary when embarking on new applications of the heat-trans-fer-fluid-friction analogy, and the results do not always take the simple form of Eq. (5-56). The interested reader may consult the references at the end of the chapter for more information on this important subject. At this point, the simple analogy developed above has served to amplify ouf understanding of the physical processes in convection and to reinforce the notion that heat-transfer and viscous-transport processes are related at both the microscopic and macroscopic levels. 

The transport of a sub-critical Lennard-Jones fluid in a cylindrical mesopore is investigated here, using a combination of equilibrium and non-equilibrium as well as dual control volume grand canonical molecular dynamics methods. It is shown that all three techniques yield the same value of the transport coefficient for diffusely reflecting pore walls, even in the presence of viscous transport. It is also demonstrated that the classical Knudsen mechanism is not manifested, and that a combination of viscous flow and momentum exchange at the pore wall governs the transport over a wide range of densities. 

It is clear from the above results that all three simulation techniques yield identical results for the transport coefficient in pores with diffusely reflectmg walls. Further, a combination of momentum transfer at the wall and viscous transport in the fluid suffices to explain the transport behavior of pure component fluids in mesopores. 

The Stefan-Maxwell (Maxwell, 1860 Stefan, 1872) equation gives implicit relations for the fluxes when the system is isothermal and the wall effects are negligible, this means negligible viscous transport (i.e. constant pressure) and Knudsen diffusion. For multicomponent mixture the equation has the form ... 

The catalyst structure is assumed to be macro-porous, so that transport mechanisms like viscous transport or Knudsen diffusion can be neglected. It is assumed that the component mass transport inside the particle is described by Pick s law of diffusion (due to the relatively low concentrations of the relevant components). 

Besides the diffusive transport, the viscous transport due to pressure gradient also contributes to the total flux, and can be conveniently represented by Darcy s formula. Therefore, choosing the right model for transport and reaction in porous medium is highly important in... 

One of the main draw backs of GMS formulation from SOFC modeling perspective is its inability to account for Knudsen diffusion. And furthermore, GMS formulation guarantees that the sum of diffusive fluxes vanishes, which is not the case for fuel cell operating conditions due to the viscous transport. The viscous transport can be introduced as in the case of Pick... 

The model accounts for three different transport mechanisms, molecular diffusion, Knudsen diffusion, and viscous transport. The total diffusive flux in DGM results from molecular diffusion acting in series with Knudsen diffusion. The viscous porous media flow (Darcy flow) acts in parallel with diffusive flux. The DGM can be written as an implicit relationship among molar concentrations, fluxes, concentration gradients and pressure gradient as... 

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. 

From Figure 5.3(a) we can see that steady state at j) = 1 is reached when f 3> 1, i.e. i = DtjL 1 or tL fD. It is also very convenient to define dimensionless time as = tlx using the time constants x = L jD, x = pCpL fX, and x = pL /ti for diffusion, heat conduction, and the viscous transport of momentum, respectively. In Figure 5.3(a), y is close to 1 when f = 1, and in this case we can, with reasonable confidence, assume steady state when t > x. 

In this case study we assiune a constant even flow and concentration at the inlet and only consider steady state. However, the concentration will have a radial variation downstream due to the reaction and the radial difference in residence time. Assuming an even inflow and a minor effect of gravity in the radial and tangential directions, the convective radial transport terms can be removed. Axial viscous transport, diffusion, and conduction can also be neglected because the convective axial transport is much larger. After a sufficiently long period of time, the transient terms become very small, and we obtain a model with steady axial convection, radial diffusion, and conduction with a reaction soiuce term. 

To account for the viscous drag along bounding walls, the Brinkman equation is often used. The Brinkman equation describes the flow in porous media in cases where the transport of momentum by shear stresses in the fluid is not ignored. The model extends to include a term that accounts for the viscous transport in the momentum balance and introduces the velocities in the spatial directions as dependent variables. 

Brinkman equations extend Darcy s law to iiKlude a term that accounts for the viscous transport in the momentum balance and introduce the velocities iu the spatial directious as depeudeut variables. This approach is more robust thau using Darcy s law alone, since it can be applied in a wider range of flow rates and porous package permeabihties. 

Collisional contributions Explicit expressions for the collisional contributions to the viscous transport coefficients can be obtained by considering various choices for k and a and in (25), (27), and (29). Taking k in the y-direction and a = 15 = I yields... 

Other choices lead to relations between the colhsional conffibutions to the viscous transport coefficients, namely... 

---