CHEMICAL ENGINEERING

MOMENTUM, HEAT AND MASS TRANSFER 311

Solution

The Universal Velocity Profile is discussed in detail in Section 12.4, and in the region
where eddy transport dominates (yC>30) and making all the stated assumptions:

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

If, in the laminar sub-layer (from equation 12.40),uCDyCthen:

yCD 2 .5lnyCC 5. 5

and, solving by trial and error:yCD 11. 6 DusŁυb/ (from equation 12.44)

SinceuŁD

p

R/, then:υbD 11. 6 /uŁD 11. 6 /

p

R (i)

But, from the Blasuis equation:R/u^2 D 0. 0228 usυ/^0.^25

and substituting for R in equation (i), υbD 11. 6 /
p
 1 /us
p
 1 / 0. 0228 ^0.^5
usυ/^0.^125

and: υb/υD 76. 8 usυ/^0.^875

Using Prandtl’s one seventh power law,ub/usDυb/υ^1 /^7 Dυb/υ^0.^143

But: RDub/υbD 0. 0228 us^2 usυ/^0.^25

∴ /υbusυb/υ^0.^143 D 0. 0228 us^2 usυ/^0.^25

and: υb/υD 82. 38 usυ/^0.^875

The ratio of the values obtained using the two approaches to the problem is:

 82. 38 / 76. 8 D 1. 073

PROBLEM 12.19

Obtain the Taylor – Prandtl modification of the Reynolds analogy between momentum
transfer and mass transfer (equimolecular counterdiffusion) for the turbulent flow of a
fluid over a surface. Write down the corresponding analogy for heat transfer. State clearly
the assumptions which are made. For turbulent flow over a surface, the film heat transfer
coefficient for the fluid is found to be 4 kW/m^2 K. What would the corresponding value
of the mass transfer coefficient be, given the following physical properties? Diffusivity
DD 5 ð 10 ^9 m^2 /s. Thermal conductivity,kD 0 .6 W/m K. Specific heat capacityCpD
4 kJ/kg K. Density,D1000 kg/m^3. Viscosity,D1mNs/m^2.
Assume that the ratio of the velocity at the edge of the laminar sub-layer to the stream
velocity is (a) 0.2, (b) 0.6.
Comment on the difference in the two results.

Solution

The discussion of the Reynolds analogy is presented in Section 12.8 and consideration of
mass transfer in Sections 12.8.2 and 12.8.3 leads to the modified Lewis relation which