Page 265 - Academic Press Encyclopedia of Physical Science and Technology 3rd Chemical Engineering

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Encyclopedia of Physical Science and Technology En006G-249 June 27, 2001 14:7

Fluid Dynamics (Chemical Engineering) 57

which are respectively known as the Poiseuille velocity 1 + 3n 1 (1+n)/n
profile and the Hagen–Poiseuille relation. u = 1 + n F(ξ 0 , n) (1 − ξ 0 ) , ξ ≤ ξ 0
When the same operations are performed for the con-
(86)
centric annulus geometry, the results are
1/n

D n τ w
2 2 2 v = (1 − ξ 0 ) (1+n)/n F(ξ 0 , n)
u = [1 − ξ + (1 − σ )ln ξ/ln(1/σ)] (76) 2 1 + 3n k
F(σ)
(87)
v = Dτ R F(σ)/8µ (77)
2
2
2
F(σ) = 1 + σ − (1 − σ )/ln(1/σ) (78) F(ξ 0 , n) = (1 − ξ 0 ) + 2(1 + 3n)ξ 0 (1 − ξ 0 ) + 1 + 3n ξ 0 2
1 + 2n 1 + n
where σ = R i /R is the “aspect” ratio of the annulus.
(88)
2. Non-Newtonian where ξ 0 has the same significance as in the Bingham case.
Because of the extreme complexity of the expressions for
d. Casson. The pertinent results are
the velocity distributions and average velocities in con-
centric annuli for even simple non-Newtonian fluids, we 2 2 8 1/2 3/2
include here only the results for pipe flow. u = 1 − ξ + 2ξ 0 (1 − ξ) − ξ 0 (1 − ξ ) ,
G(ξ 0 ) 3
a. Bingham Plastic. The pertinent results are ξ> ξ 0 (89)

2 2 2 8 1/2 1 2
u = 1 − ξ − 2ξ 0 (1 − ξ) , ξ > ξ 0 (79) u = 1 − ξ 0 + 2ξ 0 − ξ 0 , ξ ≤ ξ 0
F(ξ 0 ) G(ξ 0 ) 3 3
u = 2(1 − ξ 0 ) 2 F(ξ 0 ), ξ ≤ ξ 0 (80) (90)
(91)
(81) v = Dτ w G(ξ 0 )/8µ ∞
v = Dτ w F(ξ 0 )/8µ ∞
4 1 4
16 1/2
1 4
4
F(ξ 0 ) = 1 − ξ 0 + ξ (82) G(ξ 0 ) = 1 − ξ + ξ 0 − ξ (92)
3 3 0 0 0
7 3 21
where ξ 0 = τ 0 /τ w . Equation (81) is a version of the well-
where ξ 0 has the same significance as in the Bingham case.
known Buckingham relation and is the Bingham plastic
It should be observed that in all cases, even the
equivalent of the Hagen–Poiseuille result. The parameter
linear Bingham plastic case, the resultant average
ξ 0 , because of the linearity of Eq. (70), also represents the
velocity expressions are nonlinear relations between
dimensionless radius of a “plug” or “core” of unsheared
v and −dp/dz. This is true of all non-Newtonian
material in the center of the pipe, which moves at the
constitutive relations. A direct consequence of this result
maximum velocity given by Eq. (80). This is a feature of
is that the friction factor relation is also nonlinear.
all fluids that possess yield stresses.
b. Power law. The pertinent results are C. Friction Factors
1 + 3n (1+n)/n In Eq. (65) the friction factor was introduced as an em-
u = 1 − ξ (83)
1 + n pirical factor of proportionality in the calculation of the
friction loss head. If Eq. (63) is applied to a length of
1/n

D n τ w

v = . (84) straight horizontal pipe with no pumps, one finds that
2 1 + 3n k
−h f = p/ρg. (93)
Note that these results reduce to the Newtonian results in
the limit n = 1, k = µ. Elimination of h f between Eqs. (65) and (93) results in
8 −D p 8τ w
c. Herschel–Bulkley. The pertinent results are f = = , (94)
ρ v 2 4L ρ v 2
1 + 3n 1 (1+n)/n
u = (1 − ξ 0 ) which may be looked on as an alternate definition of the
1 + n F(ξ 0 , n)
friction factor. From Eq. (66) it is evident that Eq. (94) with
the numeric factor 8 replaced by 2 defines the Fanning

(1+n)/n
−(ξ − ξ 0 ) , ξ > ξ 0 (85)
friction factor.