Epstein drag

Pierce F, Sorensen CM, Chakrabarti A (2006) Computer simulation of diffusion-limited cluster-cluster aggregation with an epstein drag force. Phys Rev E 74 021411. 

As particle size is decreased to the point where dp tp, the drag for a given velocity becomes less than predicted from Stokes law and continues to decrease with particle size. In the range dp ip, the free molecule range (Chapter 1), an expression for the friction coefficient can be derived from kinetic theory (Epstein, 1924) ... 

Epstein [1988] develops the concept of a particle in free flight with an initial velocity of subject to Stokes drag forces as it travels through a fluid. 

Watkinson and Epstein [1971] considered that the deposition term in the Kem and Seaton [1959] relationship would be dependent upon the mass transfer to the surface coupled to the sticking probability. Watkinson and Epstein [1971] assumed that the sticking probability is a function of the adhesive forces present and the drag forces acting on the particles resident on the surface. The incorporation of a sticking probability into the assessment means that the particulate concentration at the wall could not be regarded as zero and consequently that the driving force could not be equated to the bulk concentration. 

The former problem involves a difficult choice. Either the tube length L can be simply replaced by TL in the above expressions for AP, a procedure which maintains the same fluid velocity in the equivalent tube as in the particle bed but leads to different fluid-residence times or alternatively the fluid residence times can be matched by allowing fluid velocities to differ by a factor of T. This latter procedure was proposed by Carman (1937) for packed beds, and supported more recently by Epstein (1989). The arguments presented below, however, relating particle drag to bed pressure loss, suggest the matching of fluid velocity as the more consistent alternative, and this we now adopt. 

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