CHAPTER 6: INTRODUCTION TO CONVECTION
Objectives:
1. To obtain an understanding of the physical mechanisms that underlie convection heat transfer.
2. To develop the means to perform convection transfer calculations.
6.0 Introduction
 Convection is energy transfer that takes place between a surface and a fluid moving over it
when they are at different temperatures.
 Convection is comprised of two mechanisms: diffusion and advection.
 Analogous to convection heat transfer is convection mass transfer. For example, when air
moves past the surface of a pool of water, the liquid water will evaporate and water vapor will
be transferred into the air stream.
6.1 The Convection Boundary Layers
 The concept of boundary layers is central to the understanding of convection heat and mass
transfer between a surface and a fluid flowing past it.
6.1.1 The Velocity Boundary Layer
 Consider flow over a flat plate as shown below:

Due to the no-slip condition on the surface, the velocity profile in the boundary layer is as
shown.

The velocity (or hydrodynamic) boundary layer is defined as the layer of fluid within
which velocity gradients and shear stresses are significant.

The boundary layer thickness, δ, is typically defined as the value of y for which u =
0.99U. δ increases with distance from the leading edge.

The velocity gradient in the boundary layer is associated with shear stresses τ acting in
planes that are parallel to the fluid velocity. For a Newtonian fluid, the surface shear
stress is
s  
u
 y y 0
(1)
where μ = dynamic viscosity.

The frictional drag on the plate may be determined from the local friction coefficient,
given as
Cf 

s
 U 2 2
(2)
Since δ = δ(x), u y y  0 depends on the distance x from the leading edge of the plate.
Therefore, τs and Cf also depend on x.
6.1.2 The Thermal Boundary Layer
 Consider flow over an isothermal flat plate as shown below:

Due to the difference in temperature between the fluid in the free stream and the surface
of the plate, temperature gradient develops in the fluid as shown.

The region of fluid in which the temperature gradient exists is called the thermal
boundary layer.

The thermal boundary layer thickness, δt, is typically defined as the value of y for which
the ratio [(Ts-T)/(Ts-T)] = 0.99. δt increases with distance from the leading edge.
 The condition in the thermal boundary layer determines the convection heat transfer
coefficient. This is illustrated below.

At any location x on the plate, there is no fluid motion and energy transfer occurs only by
conduction. Therefore, the local heat flux is obtained from Fourier’s law as
qs   k f
where kf = thermal conductivity of fluid.
T
 y y 0
(3)

Recall the Newton’s law of cooling,


(4)
 k f T  y
y 0
T s  T
(5)
qs  h T s T

Combining (3) & (4), we obtain
h 

Since (Ts-T) is constant while δt increases with x, T y y  0 decreases with increasing x.
Therefore, q''s and h decrease with increasing x.
6.2 Local and Average Convection Coefficients
 Consider flow over a surface of arbitrary shape as shown below. The surface is presumed to
be at uniform temperature Ts, where Ts ≠ T.

Both heat flux and convection heat transfer coefficient vary along the surface, i.e. q'' =
q''(x) and h = h(x).

The total heat transfer rate over the surface is
q 
 A s q dA s

 T s  T
 A h dA s
s
(6)
where h is the local heat transfer coefficient.

Defining an average convection coefficient h for the entire surface, the total heat transfer
rate may also be expressed as

q  h A s T s  T


(7)
Equating (6) and (7), it follows that
h 

1
h dA s
As As
(8)

For the special case of flow over a flat plate (see figure 6.4b of textbook),
h 
1 L
h dx
L 0

(9)
6.3 Laminar and Turbulent Flow
 The velocity boundary layer development on a flat plate is shown below.

The flow starts from the leading edge as laminar flow.

Due to propagation of disturbances in the flow, transition to turbulence later occurs.

The parameter used to characterize the flow is the Reynolds number, defined as
Re x 
 U x

(10)
where x is the distance from the leading edge.

The location xc at which transition begins is determined by the critical Reynolds number,
Rex,c, which is
Re x, c 
 U  xc
 5  105

(11)
6.4 The Boundary Layer Equations
 The velocity, temperature, and concentration fields in a fluid are obtained by solving the
conservation laws for mass, momentum, energy, and concentration. For laminar flow, these
are equations 6.27-6.30 of the textbook.
 Once the velocity, temperature, and concentration fields are obtained, the wall shear stress
and the heat transfer coefficient may be determined from equations 1 and 5 of this note.
6.5 Boundary Layer Similarity
 Heat and mass transfer analysis results are usually presented in terms of dimensionless
similarity parameters.
 This allows us to apply results obtained for a surface experiencing one set of convective
conditions to geometrically similar surfaces experiencing entirely different conditions.
 Some dimensionless similarity parameters are:

Reynolds number: It may be interpreted as the ratio of inertia to viscous forces in a
region of characteristic dimension L with characteristic velocity V. It is expressed as
Re L 

V L

Nusselt number: It may be interpreted as the ratio of convection to pure conduction heat
transfer. It provides a measure of the convection heat transfer occurring at the surface. It
is expressed as
Nu 
hL
kf
where L is the characteristic dimension and kf is the thermal conductivity of the fluid.
Note the difference between the Nusselt number and Biot number.

Prandtl number: It is the ratio of the momentum diffusivity to the thermal diffusivity,
i.e.
Pr 
cp 



k
 Other useful dimensionless numbers can be found in Table 6.2 of the textbook.