CHEMICAL ENGINEERING

MOMENTUM, HEAT AND MASS TRANSFER 303

the surface. Obtain the relation betweenuCandyCin the laminar sub-layer. Outside the
laminar sub-layer, the relation is:

uCD 2 .5lnyCC 5. 5

At what value ofyCdoes the transition from the laminar sub-layer to the turbulent zone
occur?
In the “Universal Velocity Profile”, the laminar sub-layer extends to values ofyCD 5
and the turbulent zone starts atyCD30 and the range 5<yC<30, the buffer layer,
is covered by a second linear relation betweenuCand lnyC. What is themaximum
difference between the values ofuC, in the range 5<yC<30, using the two methods
of representation of the velocity profile?
Definitions:uCDux/uŁ,yCDyuŁ/anduŁ^2 DR/whereuxis the velocity at
a distanceyfrom the surface,Ris the wall shear stress andandare the density and
viscosity of the fluid respectively.

Solution

As discussed in Section 12.4.2, if the velocity gradient dux/dyapproaches a constant
value near the surface,d^2 ux/dy^2 approaches zero andRDux/y.

∴ uŁ^2 Dux/y

and, as given in equation 12.39:

ux/uŁDyuŁ/DyC

Hence: uCDyC (equation 12.40)

SinceuCD 2 .5lnyCC 5 .5andyCD 2 .5lnyCC 5 .5, then solving by trial and error,
the transition from the laminar sub-layer to the turbulent zone occurs when:

yCD 11. 6 anduCD 2 .5ln11. 6 C 5. 5 D 11. 62

For the buffer layer,uCDalnyCCa^0 (equation 12.41)

WhenyCD5,uCD5andwhenyCD30,uCD 2 .5ln30C 5. 5 D 14

∴ 5 Daln 5Ca^0

∴ a^0 D 5 aln 5D 5  1. 609 a

and: 14 Daln 30Ca^0 Daln 30C 5  1. 609 a

∴ 9 D 3. 401 a 1. 609 aandaD 5. 02

and: a^0 D 5  1. 609 ð 5. 02 D 3. 08

The difference between the two values ofuCis a maximum whenyCD 11 .6.

From the two-layer theory:uCD 11. 6

From the buffer-layer theory:

uCD 5 .02 ln 11. 6  3. 08 D 9. 2

and hence, the maximum difference is 11. 6  9. 2 D 2. 4