Porous media boundary layer

If the Darcy assumptions are used then with forced convective flow over a surface in a porous medium, because the velocity is not assumed to be 0 at the surface, there is no velocity change induced by viscosity near the surface and there is therefore no velocity boundary layer in the flow over the surface. There will, however, be a region adjacent to the surface in which heat transfer is important and in which there are significant temperature changes in the direction normal to the surface. Under many circumstances, the normal distance over which such significant temperature changes occur is relatively small, i.e., a thermal boundary layer can be assumed to exist around the surface as shown in Fig. 10.9, the ratio of the boundary layer thickness, 67, to the size of the body as measured by some dimension, L, being small [15],[16]. 

The left-hand side of this equation is of order 1. Considering the right-hand side, 8t/L is by assumption small so, since d2B dX2 and S2B/dY2 are both of order l, the first term on the right-hand side of the above equation will be much less than the second and the first term is. therefore, negligible compared to the second. Hence, the following form of the energy equation applies in two-dimensional boundary layer flow in a porous medium ... 

The approximate integral equation method that was discussed in Chapters 2 and 3 can also be applied to the boundary layer flows on surfaces in a porous medium. As discussed in Chapters 2 and 3, this integral equation method has largely been superceded by purely numerical methods of the type discussed above. However, integral equation methods are still sometimes used and it therefore appears to be appropriate to briefly discuss the use of the method here. Attention will continue to be restricted to two-dimensional constant fluid property forced flow. 

As discussed in Chapters 2 and 3, in the integral method it is assumed that the boundary layer has a definite thickness and the overall or integrated momentum and thermal energy balances across the boundary layer are considered. In the case of flow over a body in a porous medium, if the Darcy assumptions are used, there is, as discussed before, no velocity boundary layer, the velocity parallel to the surface near the surface being essentially equal to the surface velocity given by the potential flow solution. For flow over a body in a porous medium, therefore, only the energy integral equation need be considered. This equation was shown in Chapter 2 to be ... 

FIGURE 10.19 Boundary layer on a surface in a porous medium. 

An approximate model of the flow in a vertical porous medium-filled enclosure assumes that the flow consists of boundary layers on the hot and cold w alls with a stagnant layer between the two boundary layers, this layer being at a temperature that is the average of the hot and cold wall temperatures. Use this model to find an expression for the heat transfer rate across the enclosure and discuss the conditions under which this model is likely to be applicable. 

Bejan, A., On the Boundary Layer Regime in a Vertical Enclosure Filled with a Porous Medium", Lett. Heat Mass Transfer, Vol. 6, pp. 93-102, 1979. 

Bejan, A. and Poulikakos. D.. The NonDarcy Regime for Vertical Boundary Layer Natural Convection in a Porous Medium , Int. J. Heat Mass Transfer, Vol. 27, pp. 717-722, 1984. 

In structured disperse systems, where particles of the dispersed phase form united spacial networks, as well as in porous media with open porosity, the existence of double layers at interfacial boundaries results in some peculiarities in the processes of substance transfer and electric current transport. We will devote most of our attention to the discussion of transfer phenomena in an individual capillary, which is the simplest element of any structured disperse system, and then only qualitatively address the peculiarities related to complex structure of porous medium. 

Darcy-like flows obtain except in a thin boundary layer near solid walls, where Darcy s equation is unable to satisfy the no-slip boundary condition [cf. Brinkman s treatment (B35) of flow through a porous medium bounded externally by a circular cylinder with solid walls]. 

Figure 2.1. Structure and composition of catalyst layers at three different scales At the nanoparticle level, anode and cathode processes are depicted, including possible anode poisoning by CO. At the agglomerate level, ionomer functions as binder and proton-conducting medium are indicated, and points with distinct electrochemical environments are shown (double- and triple-phase boundary). At the macroscopic scale, the interpenetrating percolating phases of ionomer, gas pores, and solid Pt/Carbon are shown, and the bimodal porous structure is indicated.

Most porous electrodes are made up of a mixture of ionomer (for proton conduction), high surface area carbon (for electron conduction), and nanoparticles of catalyst all mixed together to form an ink-like random medium. This ink is then either sprayed or doctored onto either a microporous layer on the gas diffusion media, or applied directly to the polymer electrolyte. In either case, the density of triple phase boundaries is to a large extent left determined by the random nature of the ink and application process. 

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