G • COOLING BY PROTECTIVE FLUID FILMS 

 For low values of velocity, Darcy's law 



f = const ^ (2-1) 



gives the relationship between the pressure difference Ap acting on the 

 two surfaces of a plain porous wall of thickness L, the viscosity /x, the 

 velocity v^ of the coolant flowing through the porous wall, and the length d 

 characterizing the pore openings. This law is valid only if the pressure 

 drop is the result of viscous shear in laminar flow. It gives a linear rela- 

 tionship between pressure drop and velocity analogous to Poiseuille flow 

 in a pipe. For high Reynolds numbers the pressure drop is proportional 

 to the density of the coolant and the square of the velocity which can be 

 expressed as follows: 



^ = const ?^- (2-2) 



L a 



In the flow through a porous medium, unlike the flow in pipes, there is 

 no definite small range of Reynolds number to distinguish the laws given 

 in Eq. 2-1 and 2-2. The gradual transition from the Darcy regime is due 

 to the inertia of the fluid contracting and expanding through the pores. 

 The inertia factor becomes progressively more important with increasing 

 velocity. Hence, in the pressure drop equation, the loss due to both viscous 

 shear and inertia effects must be included. The two foregoing equations 

 can be combined in the following manner if the weight rate of flow G is 

 introduced to take account of the compressibility effect. It becomes 



^ = , f ?P*) G + » (2po) (J. (2-3) 



in which co is the specific weight of the fluid at a reference pressure po- 

 The two coefficients, a and i8, defined by Eq. 2-3, are independent of the 

 nature of the fluid and have only the dimension of some unknown length 

 characterizing the structure of the porous medium itself. 



Fig. G,2a gives typical curves of pressure-squared difference vs. the 

 weight rate of flow from experimental results made with fine iron and 

 fine ammonium bicarbonate powders. Fig. G,2b gives the relation between 

 the strength, the flow rate, the relative density, and the pressure for 

 Poroloy stainless steel with a 35" crossing angle, where pi and p2 are in 

 absolute pressure (lb/in. 2) and L is the thickness in inches. The viscous 

 resistance coefficient a and the inertia resistance coefficient jS of Eq. 2-3 

 can be determined from these experimental curves. The viscous resistance 

 coefficient a is found to be inversely proportional to approximately the 

 seventh power of the porosity. The variation of the inertial resistance 

 coefficient /3 with porosity is rather complex and the only conclusion to be 

 drawn is that it decreases with increasing porosity. The relation between 



< 432 > 



