Page 97 - Modelling in Transport Phenomena A Conceptual Approach

4.3. FLOW PAST A SINGLE SPHERE 77

Air bubble rising in water
In this case, the Archimedes number is

49P (PP - PI
Ar =
P2
- (5 x 10-3)3(9.8)(999)(1.2047 - 999) = - 1.219
- 106
(1001 x 10-6)2
The minus sign indicates that the motion of a bubble is in the direction opposite to
gravity, ie., it is rising. The Reynolds number and the terminal velocity are
Ar
Rep = - (1 + 0.0579Ar0.412)-1'214
18
- le219 lo6 [1+ 0.0579 (1.219 x 106)0.412]-1'214 = 1825
-
18
P Rep
ut =
PDP
(1001 x 10-6)(1825)
- = 0.37m/ s
-
(999)(5 x 10-3)
Calculate Dpj given p and vt
In this case Eq. (4.3-4) must be rearranged such that the particle diameter is
eliminated. If both sides of Eq. (4.3-4) are divided by Re;, the result is
f
-=y (4.3-14)
Rep
where Y, which is independent of Dp, is a dimensionless number defined by

(4.3- 15)

Substitution of Eq. (4.3-10) into Eq. (4.3-14) yields
Y=- 24 (1 + 0.173Re$657) + 0.413 (4.3-16)
Re$ Rep + 16,300 ReS0.O9
Since Eq. (4.3-16) expresses Y as a function of the Reynolds number, calculation
of the particle diameter for a given terminal velocity and fluid viscosity requires
an iterative solution. To circumvent this problem, the following explicit expression
relating Y to the Reynolds number is proposed by Tosun and Aksahin (1992) as

(4.3- 17)