CHEMICAL ENGINEERING

310 CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS

Solution

The derivation of the Taylor – Prandtl modification of the Reynolds analogy between heat
and momentum transfer is presented in Section 12.8.3 and the result is summarised as:

StDh/CpusDR/u^2 s/[1C ̨Pr 1 ] (equation 12.117)

or: R/u^2 sD[h/Cpus][1 ̨ 1 Pr]

For turbulent pipe flow,usis approximately equal toumean/ 0. 82 and:

0. 82 R/u^2 D[h/Cpu][1 ̨ 1 Pr]

WhenReD 10 ,000, then from Fig. 3.1,R/u^2 D 0 .0038, and forPrD10:

 0. 82 ð 0. 0038 DSt[1 ̨ 1  10 ]

or: St 1 C 9 ̨D 0. 0031 (i)

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

and: StDNu/ReÐPrD 0. 023 Re^0.^2 Pr^0.^67

D 0. 023  10 , 000 ^0.^2  10 ^0.^67 D 0. 000777

Hence, substituting in equation (i):

0. 000777  1 C 9 ̨D 0 .0031 and ̨Dub/usD 0. 33

PROBLEM 12.18

Obtain a dimensionless relation for the velocity profile in the neighbourhood of a surface
for the turbulent flow of a liquid, using Prandtl’s concept of a “Mixing Length” (Universal
Velocity Profile). Neglect the existence of the buffer layer and assume that, outside the
laminar sub-layer, eddy transport mechanisms dominate. Assume that in the turbulent
fluid the mixing lengthEis equal to 0.4 times the distanceyfrom the surface and that
the dimensionless velocityuCis equal to 5.5 when the dimensionless distanceyCis unity.
Show that, if the Blasius relation is used for the shear stressRat the surface, the
thickness of the laminar sub-layerυbis approximately 1.07 times that calculated on the
assumption that the velocity profile in the turbulent fluid is given by Prandtl’s one seventh
power law.
Blasius Equation:
R
u^2 s

D 0. 0228

(

usυ



) 0. 25

where,are the density and viscosity of the fluid,usis the stream velocity, andυis
the total boundary layer thickness.