CHEMICAL ENGINEERING

THE BOUNDARY LAYER 289

PROBLEM 11.7

Explain the concepts of “momentum thickness” and “displacement thickness” for the
boundary layer formed during flow over a plane surface. Develop a similar concept to
displacement thickness in relation to heat flux across the surface for laminar flow and heat
transfer by thermal conduction, for the case where the surface has a constant temperature
and the thermal boundary layer is always thinner than the velocity boundary layer. Obtain
an expression for this ‘thermal thickness’ in terms of the thicknesses of the velocity and
temperature boundary layers.
Similar forms of cubic equations may be used to express velocity and temperature
variations with distance from the surface.
For a Prandtl number,Pr, less than unity, the ratio of the temperature to the velocity
boundary layer thickness is equal toPr^1 /^3. Work out the ‘thermal thickness’ in terms of
the thickness of the velocity boundary layer for a value ofPrD 0 .7.

Solution

Consideration is given to the streamline portion of the boundary layer in Section 11.3
where, assuming:
uxDuoCayCby^2 Ccy^3 (equation 11.10)

it is shown that the equation for the velocity profile is:

ux/us D 1. 5 

y/υ 0. 5 

y/υ^3 (equation 11.12)

The equivalent equation for the thermal boundary layer will be:

/s D 1. 5 

y/υt  0. 5 

y/υt 3

whereυtis the thickness of the thermal boundary layer.
The heat flow is given by:

QD

∫l

0

CpuxTdy

DusTsCp

∫l

0

[1. 5 

y/υt  0. 5 

y/υt 3 ][1. 5 

y/υ 0. 5 

y/υ^3 ]dy

This is made up of two components:
the heat flow through the thermal boundary layer:

DusTsCp

∫

υt/υ

0

f 

2. 25 y^2 /υÐυt  0. 75 y^4 /

υ^3 υt  0. 75 y^4 /

υÐυ^3 t

C 

0. 25 y^6 /υ^2 υ^5 t gd

y/υ

and the heat flow through the velocity boundary layer betweenyDυtandyDυ:

DusTsCpυ

∫ 1

υt/υ

[1. 5 y/υ 0. 5 

y/υ^3 ]d

y/υ