Introduction to convection heat transfer

University of Hail
Faculty of Engineering
ME 315 – Heat Transfer
Lecture notes
Chapter 6
Introduction to convection heat transfer
Prepared by : Dr. N. Ait Messaoudene
Based on:
“Introduction to Heat Transfer”
Incropera, DeWitt, Bergman, and Lavine,
5th Edition, John Willey and Sons, 2007.
2nd semester 2011-2012
The Convection Boundary Layers
• Velocity Boundary Layer
– A consequence of viscous effects
associated with relative motion
between a fluid and a surface.
– A region of the flow characterized by
shear stresses and velocity gradients.
– A region between the surface
and the free stream whose
thickness δ increases in
the flow direction.
– Manifested by a surface shear
stress τs
local friction coefficient Cf
and drag force, FD .
FD    s dAs
• Thermal Boundary Layer
– A consequence of heat transfer
between the surface and fluid.
– A region of the flow characterized
by temperature gradients and heat
– A region between the surface and
the free stream whose thickness
increases in the flow direction.
– Manifested by a surface heat
flux qs” and a convection heat
transfer coefficient h .
t 
Ts  T  y 
Ts  T
 0.99
Local and Average Convection Coefficients
• Local Heat Flux and Coefficient:
• Average Heat Flux and Coefficient for a Uniform Surface Temperature:
q  hAs Ts  T 
q  As qdAs  Ts  T  A hdAs
 hdAs
As As
• For a flat plate in parallel flow:
h  oL hdx
Per unit width
Laminar and Turbulent Flow
Laminar and Turbulent Velocity Boundary Layers
• Transition criterion for
a flat plate in parallel
Laminar and Turbulent Thermal Boundary Layers
• Effect of transition on boundary layer thickness and local convection coefficient:
105  Re x ,c  3 x 106
The Boundary Layer Equations
Consider concurrent velocity and thermal boundary layer development for steady, twodimensional, incompressible flow with constant fluid properties and negligible body forces.
boundary layer approximations.
Velocity Boundary Layer:
The velocity equations are
from the energy
(temperature) equation.
Proper BC
Thermal Boundary Layer:
Solve for u, v and T
Cf and h
Solving these equations requires methods that go beyond the scope of the present
Laminar compressible BL involve even more complicated equations which are
moreover coupled
The case of turbulent BL gets even more complicated with the introduction of new
terms (due to turbulent mixing) requiring special mathematical models.
Boundary Layer Similarity
As applied to the boundary layers, the principle of similarity is based on determining similarity
parameters that facilitate application of results obtained for a surface experiencing one set of
conditions to geometrically similar surfaces experiencing different conditions.
• Dependent boundary layer variables of interest are:  s and q or h
Functional Form of T, u
and ND parameters
N.D variables
Nusselt Number
ND Equations and BC
Physical Interpretation of the Dimensionless Parameters
Key similarity parameters may be deducted from the ND form of the momentum and energy
The Prandtl number provides a measure of
the relative effectiveness of momentum and
Reynolds Number
Prandtl Number
energy transport by diffusion in the velocity
and thermal boundary layers, respectively.
Pr ≈ 1 for gases
Pr << 1 for liquid metals
Pr >> 1 for oils
Inertia forces
Viscous forces
Momentum diffusivity
Thermal diffusivity
Pr strongly influences the relative growth of
the velocity and thermal boundary layers.
Momentum and Heat Transfer Analogy:
The Reynolds Analogy Momentum and Heat Transfer
Equivalence of dimensionless momentum and energy equations for negligible pressure gradient
(dp*/dx*~0) and Pr~1:
1  2u *
* u
Re y*2
Convection terms
1 T
* T
y* Re y*2
u*  T *
u *
y *
y* 0
T *
 *
Cf and Nu are analogous
y* 0
If the velocity parameter is known, the analogy may be used to obtain the
heat transfer parameter, and vice versa.
But this analogy is restricted to (dp*/dx*~0) and Pr~1.
A modified form may be applied over a wide range of Pr:
Applicable to laminar flow if dp*/dx* ~ 0. But Generally
applicable to turbulent flow without restriction on dp*/dx*.
Colburn j factor for heat transfer
Problem 6.19: Determination of heat transfer rate for prescribed
turbine blade operating conditions from wind tunnel data
obtained for a geometrically similar but smaller
blade. The blade surface area may be assumed to be
directly proportional to its characteristic length  As  L .
ASSUMPTIONS: (1) Steady-state conditions, (2) Constant properties, (3) Surface area A is
directly proportional to characteristic length L, (4) Negligible radiation, (5) Blade shapes are
geometrically similar.
h 2  1 h1 
q2 
L2 A1 Ts,1  T
Ts,2  T
 400  35
q1 
1500 W 
Ts,1  T
Problem 6.26: Use of a local Nusselt number correlation to estimate the
surface temperature of a chip on a circuit board.