CHEMICAL ENGINEERING

MOMENTUM, HEAT AND MASS TRANSFER 309

Given that for a smooth surface,uCD14 atyCD30, then:

B^0 D 14  2 .5ln30D 5. 5

and: uCD 2 .5lnyCC 5. 5 (equation 12.37)

For molecular transfer in the laminar sub-layer near the wall, from Section 12.4.2:

RyDux/y

or: Ry/DuŁ^2 Dux/y

∴ ux/uŁDyuŁ/anduCDyC (equation 12.40)

If the buffer zone stretches fromyCD5toyCD30 at whichuCis 5 and 14 respectively,
then in equation 12.41:

uCDalnyCCa^0

or: 5 Daln 5Ca^0 and 14Daln 30Ca^0

and: uCD 5 .0lnyC 3. 05 (equation 12.42)

From equation 12.46, the velocity gradient, duC/dyCD 5 /yC

From equation 12.61:RyDCEdux/dy

and substitutinguŁD

√

Ry/:

dux/dyDuŁ^2 /EC/ (equation 12.62)

∴ duC/dyCD/ 1 /[EC/]D 5 /yC

and: E//DyC/ 5  1 (equation 12.63)

Hence asyCgoes from 5 to 30, the ratio of the eddy kinematic viscosity to the kinematic
viscosity goes from 0 to 5.

PROBLEM 12.17

Derive the Taylor – Prandtl modification of the Reynolds analogy between heat and
momentum transfer and express it in a form in which it is applicable to pipe flow.
If the relationship between the Nusselt numberNu, Reynolds numberReand Prandtl
numberPris:

NuD 0. 023 Re^0.^8 Pr^0.^33

calculate the ratio of the velocity at the edge of the laminar sub-layer to the velocity at
the pipe axis for waterPrD 10 flowing at a Reynolds number of 10,000 in a smooth
pipe. Use the pipe friction chart.