Heat Transfer Handbook part 115. The Heat Transfer Handbook provides succinct hard data, formulas, and specifications for the critical aspects of heat transfer, offering a reliable, hands-on resource for solving day-to-day issues across a variety of applications. | BASIC PRINCIPLES 1137 d 2ộ3 150 1 - ộ 2 1 - ộ ộ3d Brinkman s 1947 modification of Üie Darcy flow model accounts for the transition from Darcy flow to highly viscous flow without porous matrix in the limit of extremely high permeability v -V p pg K V 2v Ĩ The more appropriate way to write Brinkman s equation is Nield and Bejan 1999 VP - ĩ v p V2v which is similar to eq. without the body force term and multiplied by K p. In eqs. and two viscous terms are evident. The first is the usual Darcy term and the second is analogous to the Laplacian term that appears in the Navier-Stokes equation. The coefficient p is an effective viscosity. Brinkman set p and p equal to each other but in general that is not true. The reader is referred to Nield and Bejan 1999 for a criticifl dsscussocn of toe appliaability of j. . There are situations in which it is convenient to use the Brinkman equation. One such situation is when flows in porous media are compared with those in clear fluids. The Brinkman equation has a parameter K the permeability such that the equation reduces to a form of the Navier-Stokes equation as K L2 œ and to the Darcy equation as K L2 0. Another situation is when it is desired to match solutions in a porous medium and in an adjacent viscous fluid. The two modifications of toe Darcy flow model dsscussed above toe Forchheimer model of q. 115T3 and flee Brinkman model oe eq. were ueed 811 1 16-ously by Vafai and Tien 1981 in a study of forced-convection boundary layer heat transfer. In the presence of gravitational acceleration Vafai and Tien s momentum equations would read v T v v K -VP pg K VVv None of the foregoing models account adequately for the transition from porous medium flow to pure fluid flow as the permeability K increases. Note that in the high-K limit the terms that survive in eq. or account for momentum conservation only in highly viscous flows in which the effect of fluid
