Fundamentals of Gas–Particle Flow by G. Rudinger

By G. Rudinger

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Reprinted from ref. [ 5 0 ] with permission of the American Society of Mechanical Engineers. and in polar coordinates relative t o t h e cylinder R = (x + y) = R (l θ =φ-ίζη- (γ/χ) = 2 2 112 0 + AB) βχρ(Αφ) (3-37) ί 2Α φ 2 In t h e limit for large particles — A » trajectory reduce t o 1 — t h e e q u a t i o n s for t h e particle χ =*Ro + J B J R 0 o Ρ , (3-38) Relative t o t h e cylinder, t h e y b e c o m e R^Ro[φ 2 + (l+Bφ) ] 2 1/2 (3-39) θ ^0-tan" 1 [0/(1 +Βφ)] In a s e c o n d e x a m p l e o f rotating f l o w , let gas b e injected tangentially at t h e periphery o f a h o l l o w cylinder and provide for discharge at t h e cylinder 34 axis.

Such e x t r e m e situations are rare and go b e y o n d t h e s c o p e o f this v o l u m e . These c o n c l u s i o n s also can be illustrated b y considering t h e average spac­ ing b e t w e e n particles. For t h e purpose o f estimating, assume t h a t t h e parti­ cles are arranged in a regular cubic array o f side s. O n e such c u b e c o n t a i n s I of a particle in each of its 8 corners. 1 and p / p = 1 0 , or ζ = 1 0 , t h e average spacing b e t w e e n particles b e c o m e s a b o u t 2 0 particle diameters, and direct 3 P 4 42 interaction b e t w e e n particles t h e n is rare.

Since f(Re) = 1 for S t o k e s drag, it is clear t h a t e q n . ( 3 - 2 4 ) gives t h e penetra­ t i o n for S t o k e s drag Y b y setting a = 0 ; t h u s , st Yst = T i;p,o = Ρρ2? ι;ρ,ο/18μ (3-26) 2 v T h e ratio Y/Y t is p l o t t e d in Fig. 1 0 as a f u n c t i o n o f t h e parameter a. This figure d e m o n s t r a t e s that c o m p u t a t i o n o f t h e m a x i m u m p e n e t r a t i o n o n t h e basis o f S t o k e s drag can b e m u c h t o o large. N o t e that t h e p e n e t r a t i o n o f a particle injected i n t o a gas is finite, b u t t h e t i m e required for c o m p l e t e pene­ tration is infinite as is evident from e q n .

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