Page 266 - Academic Press Encyclopedia of Physical Science and Technology 3rd Chemical Engineering

P1: GLM/GLT P2: GLM Final
Encyclopedia of Physical Science and Technology En006G-249 June 27, 2001 14:7

58 Fluid Dynamics (Chemical Engineering)

1. Newtonian b. Power law. The pertinent results are

Equation (94) provides the means for rearranging all of f = 16/Re PL (100)
the theoretical expressions for v given above into ex- n 2−n n
D v ρ n
pressions involving the friction factor. For example, when Re PL = 2 3−n (101)
Eq. (75) for Newtonian pipe flow is so rearranged and k 1 + 3n
one eliminates v in terms of the Reynolds number, Historically, Re PL was invented to force the form of
Re = D v ρ/µ, one obtains Eq. (100).

f = 64/Re. (95)
c. Herschel–Bulkely. The pertinent results are
Equation (95) is the source of the laminar flow line on the f = 16 Re HB (1 − ξ 0 ) 1+n F(ξ 0 , n) n (102)

Moody chart (Fig. 3).
In the case of the concentric annulus the problem is  n 2n  1/(2−n)
)
somewhat ambiguous, because there are two surfaces of  He n HB 1 + 3n (2 3−n 2 
different diameter and hence the specification of a length ξ 0 = 2   (103)


Re 2
in Re is not obvious as in the case of the pipe. For example, f  HB 
one could use D i , D,or D − D i or a host of other possi-
bilities. Obviously, for each choice a different definition 2
D ρ
of Re arises. Also, the specification of τ w in Eq. (94) is He HB = (τ 0 /k) 2/n (104)
ambiguous for the same reason. Here, we list only one of τ 0
many possible relations, and Re HB is identical in definition to Eq. (101). Indeed,
Eqs. (102)–(104) reduce to Eqs. (100) and (101) for the
2

f = 2τ R ρ v = 16/F(σ)Re D , (96) limit τ 0 = 0. In Eq. (104) He HB is the Herschel–Bulkley
R
equivalent of the Bingham plastic Hedstrom number He.

where both f and Re D are based on τ w and D for the
R
outer pipe. The function F(σ) in Eq. (96) is the same as
d. Casson. The pertinent results are
given by Eq. (78).
f = 16/Re CA G ( f , Ca, Re CA ) (105)

√
2. Non-Newtonian 16 2 (Ca/f )
1/2

G ( f , Ca, Re CA ) = 1 −
The ambiguity of definition of Re encountered in the con- 7 Re CA
centric annulus case is compounded here because of the 4
8 (Ca/f ) 16(Ca/f )

fact that no “viscosity” is definable for non-Newtonian + 2 − 8 (106)
3 Re 21 Re
fluids. Thus, in the literature one encounters a bewilder- CA CA
ing array of definitions of Re-like parameters. We now 2 2
Ca = D ρτ 0 µ (107)
present friction factor results for the non-Newtonian con- ∞
stitutive relations used above that are common and con- Re CA = D v ρ/µ ∞ (108)
sistent. Many others are possible.
The parameter Ca is called the Casson number and is anal-
ogous to the Hedstrom number He for the Bingham plastic
a. Bingham plastic. The pertinent results are
and Herschel–Bulkley models.
16 8 He 16 He 4

f = + 2 − 3 8 (97)
Re BP 3 Re 3 f Re V. TURBULENT FLOW
BP BP
2

He = D ρτ 0 µ 2 (98)
∞
A. Transition to Turbulence
(99)
Re BP = D v ρ/µ ∞
As velocity of flow increases, a condition is eventually
Note that a new dimensionless parameter He, called the reached at which rectilinear laminar flow is no longer sta-
Hedstrom number, arises because in the constitutive re- ble, and a transition occurs to an alternate mode of motion
lation there are two independent rheological parameters. that always involves complex particle paths. This motion
Parameter He is essentially a dimensionless τ 0 . This mul- may be of a multidimensional secondary laminar form, or
tiplicity of dimensionless parameters in addition to the Re it may be a chaotic eddy motion called turbulence. The
parameter is common to all non-Newtonian constitutive nature of the motion is governed by both the rheological
relations. nature of the fluid and the geometry of the flow boundaries.