Page 136 - Separation process principles 2

3.5 Mass Transfer in Turbulent Flow 101

where the "overbarred" component is the time-averaged Equation (3-180) is a highly accurate quantitative represen-
(mean) local velocity and the primed component is the local tation of turbulent flow because it is based on experimental
fluctuating component that denotes instantaneous deviation data and numerical simulations described by Churchill and
from the local mean value. The mean velocity in the perpen- Zajic [70] and in considerable detail by Churchill [7 11. From
dicular z-direction is zero. The mean local velocity in the x- (3-142) and (3-143), the shear stress at the wall, T,, is
direction over a long period O of time 0 is given by related to the Fanning friction factor by

The time-averaged fluctuating components uk and u: equal
where ix is the flow-average velocity in the axial direction.
zero.
Combining (3- 179) with (3-1 81) and performing the required
The local instantaneous rate of momentum transfer by
integrations, both numerically and analytically, lead to the
turbulence in the z-direction of x-direction turbulent mo-
following implicit equation for the Fanning friction factor as
mentum per unit area at constant density is
a function of the Reynolds number, NR, = 2ai,p /p:
The time-average of this turbulent momentum transfer is
equal to the turbulent component of the shear stress, T,,, ,

Because the time-average of the first term is zero, (3-177)
reduces to
- This equation is in excellent agreement with experimental
Tzx, = ~(uiu:) (3-178)
data over a Reynolds number range of 4,000-3,000,000 and
which is referred to as a Reynolds stress. Combining (3-178) can probably be used to a Reynolds number of 100,000,000.
with the molecular component of momentum transfer gives Table 3.14 presents a comparison of the Churchill-Zajic
the turbulent-flow form of Newton's law of viscosity, equation, (3-182), with (3-174) of Drew et al. and (3-166)
of Chilton and Colburn. Equation (3-174) gives satis-
factory agreement for Reynolds numbers from 10,000 to
10,000,000, while (3-166) is useful only from 100,000 to
If (3-179) is compared to (3-151), it is seen that an alterna-
1,000,000.
tive approach to turbulence is to develop a correlating equa-
- Churchill and Zajic [70] show that if the equation for the
tion for the Reynolds stress, (uiu:), first introduced by
conservation of energy is time averaged, a turbulent-flow
Churchill and Chan [73], rather than an expression for a tur- form of Fourier's law of conduction can be obtained with the
bulent viscosity pt . This stress, which is a complex function -
to
a
fluctuation term (uLT'). Similar time averaging leads -
of position and rate of flow, has been correlated quite accu-
turbulent-flow form of Fick's law of diffusion with (uica).
rately for fully developed turbulent flow in a straight, circu-
To extend (3-180) and (3-182) to obtain an expression for
lar tube by Heng, Chan, and Churchill [69]. In generalized
the Nusselt number for turbulent-flow convective heat trans-
form, with a the radius of the tube and y = (a - z) the dis- fer in a straight, circular tube, Churchill and Zajic employ an
tance from the inside wall to the center of the tube, their
analogy that is free of empircism, but not exact. The result
equation is
Table 3.14 Comparison of Fanning Friction Factors for Fully
Developed Turbulent Flow in a Smooth, Straight Circular Tube
f, Drew et al. f, Chilton-Colburn f, Churchill-Zajic
NR~ (3-174) (3- 166) (3-182)

where