Page 95 - Modelling in Transport Phenomena A Conceptual Approach

4.3. FLOW PAST A SINGLE SPHERE 75

4.3.1 Friction Factor Correlations

For flow of a sphere through a stagnant fluid, Lapple and Shepherd (1940) pre-
sented their experimental data in the form of f versus Rep. Their data can be
approximated as

(4.3-7)

f=- '8' 2 5 Rep < 500 (4.3-8)
Re'$6
f = 0.44 500 5 Rep < 2 x lo5 (4.3-9)

Equations (4.3-7) and (4.3-9) are generally referred to as Stokes' law and Newton's
law, respectively.
In recent years, efforts have been directed to obtain a single comprehensive
equation for the friction factor that covers the entire range of Rep. Turton and
Levenspiel (1986) proposed the following fivcconstant equation which correlates
the experimental data for Rep 5 2 x lo5:

f=- 24 (1 + 0.173Re$657) + 0.413 (4.3-10)
Rep 1 + 16,300 Re;'.''

4.3.1.1 Solutions to the engineering problems

Solutions to the engineering problems described above can now be summarized as
follows:
W Calculate ut; given p and Dp
Substitution of Eq. (4.3-10) into Q. (4.3-4) gives

Ar = 18 (Rep +0.173Reg657) + 0.31 Re$ (4.3-1 1)
1 + 16,300 Re,'.''
Since Eq. (4.3-11) expresses the Archimedes number as a function of the Reynolds
number, calculation of the terminal velocity for a given particle diameter and fluid
viscosity requires an iterative solution. To circumvent this problem, it is necessary
to express the Reynolds number as a function of the Archimedes number. The fol-
lowing explicit expression relating the Archimedes number to the Reynolds number
is proposed by Turton and Clark (1987):

(4.3-12)

The procedure to calculate the terminal velocity is as follows: