Prepared by
Kak ina
FKK, UITM
FUNDAMENTAL CONCEPTS OF
CONVECTION
• Convection denotes energy transfer
between a surface and a fluid moving
over the surface.
• Recall Newton’s law of cooling for heat
transfer between a surface of area As and
temperature Ts and a fluid:
q = h(TS − T )
• Generally flow conditions will vary along
the surface, so q” is a local heat flux and
h is a local convection coefficient.
THE VELOCITY BOUNDARY LAYER
• Consider flow of a fluid over a flat plate:
• The flow is characterized by two regions:
– A thin fluid layer (boundary layer) in which
velocity gradients and shear stresses are large.
– Its thickness δ is defined as the value of y for
which u = 0.99 u∞
– An outer region in which velocity gradients and
shear stresses are negligible
From Newtonian fluids:
u
S =
y y =0
and
S
Cf = 2
u / 2
where Cf is the local friction coefficient
THE THERMAL BOUNDARY LAYER
• Velocity boundary layer develops when there is fluid
flow over a surface
• Thermal boundary layer develops if the fluid free
stream and surface temperature differ.
• Consider flow of a fluid over an isothermal flat plate:
• The thermal boundary layer is the region of the fluid
in which temperature gradient exist
• Its thickness is defined as the value of y for which the
ratio: TS − T
= 0.99
TS − T
• At the plate surface (y=0), there is no fluid
motion
– Conduction heat transfer:
T
qS" = −k f
y y =0
- From Newton’s Law (convection):
qs’’ = h (Ts - T∞)
- Combining these equations:
− k f T / y y =0
h=
TS − T
LOCAL AND AVERAGE
CONVECTION COEFFICIENTS
• Consider the convective heat transfer as a
fluid passes over a surface of arbitrary shape:
• Newton’s law of cooling states:
where h is the local heat transfer coefficient.
• Flow conditions will vary along the surface, both
q’’ and h also vary along the surface
• The total heat transfer rate is
• where h is the average convection coefficient for
the entire surface
• For the special case of flow over a flat plate, h
varies with the distance x
1 L
h=
L 0
h dx
Laminar and Turbulent Flow
fluid motion is orderly and highly irregular and fluctuates
regular
Flow conditions are typically The critical Reynolds number for flow
characterized by a Reynolds over a flat plate:
number, Re: ρu ∞ x c
Re x ,c = = 5 ×105
ρu ∞ x μ
Re x = For flow in a pipe:
ρυD
μ Re D =
μ
Laminar : ReD < 2300
Transition: 2300 < ReD < 4000
Turbulent: ReD > 4000
SUMMARY – BOUNDARY LAYERS
• Velocity Boundary Layer (thickness δ(x))
characterized by the presence of velocity
gradients and shear stresses
– Surface Friction, Cf
• Thermal Boundary Layer (thickness δt(x))
characterized by temperature gradients.
– Convection heat transfer coefficient, h
Problem 6.1
In a flow over a surface, velocity and temperature
profiles are of the forms
u(y) = Ay + By2 – Cy3
T(y) = D + Ey + Fy2 – Gy3
where the coefficients A through G are constants.
Obtain a expressions for the friction coefficient Cf
and the convection coefficient h in terms of u∞ , T∞
and appropriate profile coefficients and fluid
properties.
KNOWN: Temperature distribution in boundary layer for air flow
over a flat plate.
FIND: Variation of local convection coefficient along the plate and
value of average coefficient.
PROBLEM 6.3
In a particular application involving airflow over a
heated surface, the boundary layer temp.
distribution may be approximated as
T – Ts = 1 – exp ( - Pr u∞y)
T∞ - Ts υ
where y is the distance normal to the surface and
the Prandtl number, Pr = cp μ/k = 0.7 is a
dimensionless fluid property. If T∞ = 400k ,
Ts =300k and u∞ /υ = 5000m-1 , what is the surface
heat flux?
PROBLEM 6.4
For laminar flow over a flat plate, the local heat
transfer coefficient hx is known to vary as x-1/2 ,
where x is the distance from the leading edge
(x=0) of the plate. What is the ratio of the average
coefficient between the leading edge and some
location x on the plate to the local coefficient at x?
