Capillary flow hydraulic radius

The flow in a packed bed, where the packing may be spherical, c indrical, etc., is quite complex (Figure 6.1.1(b)). However, it is often modeled as a collection of cylindrical capillaries of hydraulic radius R/, and length L, which is the packed bed length. Let e be the fractional void volume of the bed, Op be the total particle surface area per particle volume, v be the actual interstitial velocity in the void volume between particles and dp (= 2rp) be the mean particle diameter. Then the superficial velocity Vo based on the empty cross section of the packed bed is defined as... 

The characteristic length scale, a, is the hydraulic radius of the channel at some reference point, the velocity, U, is the gas velocity at that point, and a is the surface tension. The starred quantities are dimensional. The capillary number, Ca, is based on the reference gas velocity and the liquid phase viscosity. The flow coefficient is defined as... 

Another measure of length is the equivalent hydraulic radius, rh, determined inversely from saturated or unsaturated flow experiments. By comparing Darcy s equation with Poiseuille s law, and invoking a capillary bundle model, one obtains the following definition of rh (Kutflek Nielsen, 1994),... 

The heat pipe is a device that utilizes evaporation heat transfer in the evaporator and condensation heat transfer in the condenser in which the vapor flow from the evaporator to the condenser is caused by the vapor pressure difference, and the liquid flow from the cmidenser to the evaporator is produced by capillary force, gravitational force, electrostatic force, or other forces directly acting on it. A micro heat pipe is so small that the mean curvature of the liquid-vapor interface is comparable in magnitude to the reciprocal of the hydraulic radius of the total flow channel. Mathematically, the definition of micro heat pipe can be expressed as... 

The simplest approaches estimate the capillary pressure in porous media by adapting the Young-Laplace equation [Eq. (8)], assuming more or less uniform flow front with no voids behind the front [92], Thus there is no dependence of on the saturation. Either the relation proposed by Carman, or the notion of the hydraulic radius, or the direct use of Eq. (8) leads to [17,63,93] ... 

In an early paper, Sadowski and Bird (1965) recognised that using a bulk viscosity function for the fluid together with a capillary-hydraulic radius model for the porous media in the manner described above did not take into account any time-dependent elastic phenomena. They suggested that, in a tortuous channel of a porous medium, elastic effects would not be seen provided that the fluid s relaxation time was small compared with the transit time through the contraction/expansion. The fluid would have enough time to readjust to the changing flow conditions. However, if the transit time is small compared with the fluid s relaxation time, then the elasticity of the fluid would have an effect. 

Note that the liquid flow rate during electroosmotic motion is = na U, while the hydraulic flow rate of liquid due to a given pressure gradient be equal to Qkfdr = na dp/dx)/%n. Hence, Qa/Qhydr 1/a. It means that as the capillary radius gets smaller, the motion of the liquid becomes predominantly electroosmotic, rather than hydraulic. This conclusion is true only for a Ad. 

Note Equation (6.118) shows that the ratio of EO flow to hydraulic flow is independent of the capillary radius. Hence, in terms of flow rates achievable, there is no particular advantage in using an electric field rather than a pressure gradient for large Debye length case. 

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