PE GATE : Reservoir : Klinkenberg Effect

Klinkenberg (1941) discovered that permeability to gas is relatively higher than that to liquid, and he interpreted this phenomena as “slip flow” between gas molecules and solid walls. 

Gas molecules collide each other and to pore-walls during traveling through the pore medium. When the pore radius approaches to the mean free path of gas molecules, the frequency of collision between gas molecules and solid walls increases. Therefore this additional flux due to the gas flow at the wall surface, which is called “slip flow”, becomes effective to enhance the flow rate. This phenomenon is called Klinkenberg effect, and its effect is expressed as follows, 

where kg is permeability to gas (m2), kl is permeability to liquid (m2), l is mean free Path of the gas molecules (m), r is pore radius (m), κ is Boltzmann’s constant (JK−1), T is temperature (K), c is constant, p is pore pressure (Pa), b is Klinkenberg slip factor (Pa). The third term in Eq. (above) is given by the following relationship,

First Equation indicates that if pore radius and gas pore pressure are small and temperature of gas is high, kg becomes much larger than kl, and kg approaches to kl when pore pressure goes to infinite. 

When Klinkenberg defined the interactions to be considered, he supposed the existence of a layer (sometimes called Knudsen layer), thinner than molecular mean free path, adjacent to the pore's wall where only molecules-wall collisions would occur and collisions among molecules could be ignored. 

Thus the slippage velocity, as obtained from the Klinkenberg's approach, captures the contribution of molecule-wall interactions and when this velocity is zero, the Poiseuille velocity profile (which results from molecule-molecule interactions) is recovered. 

However, Klinkenberg's formulation ignores the transition flow region, where neither molecule-molecule nor molecule-wall interactions can be neglected because both are playing a relevant role. The feasibility of Klinkenberg linear function of the reciprocal pressure depends on the Knudsen number. For Knudsen numbers from 0.01 to 0.1 the Klinkenberg approach is acceptable. 

A more accurate correction factor can be obtained using Knudsen correction. 

To see more on klinkenberg graphs : Click here

  

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