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Forced Convection Heat Transfer
Convection is the mechanism of heat transfer through a fluid in the presence of bulk fluid
motion. Convection is classified as natural (or free ) and forced convection depending on
how the fluid motion is initiated. In natural convection, any fluid motion is caused by
natural means such as the buoyancy effect, i.e. the rise of warmer fluid and fall the cooler
fluid. Whereas in forced convection, the fluid is forced to flow over a surface or in a tube
by external means such as a pump or fan.
Mechanism of Forced Convection
Convection heat transfer is complicated since it involves fluid motion as well as heat
conduction. The fluid motion enhances heat transfer (the higher the velocity the higher
the heat transfer rate).
The rate of convection heat transfer is expressed by Newton’s law of cooling:
(^)
Q hA T T W
q hT T W m
conv s
conv s
2 /
The convective heat transfer coefficient h strongly depends on the fluid properties and
roughness of the solid surface, and the type of the fluid flow ( laminar or turbulent ).
Fig. 1: Forced convection.
It is assumed that the velocity of the fluid is zero at the wall, this assumption is called no‐
slip condition. As a result, the heat transfer from the solid surface to the fluid layer
adjacent to the surface is by pure conduction, since the fluid is motionless. Thus,
Solid hot surface, Ts
Qconv
Qcond
V∞
T∞ Zero^ velocity
at the surface.
V∞
W m K T T
y
T
k
h
q hT T
y
T
q q k
s
y
fluid
conv s
y
conv cond fluid /.
(^02) 0
The convection heat transfer coefficient, in general, varies along the flow direction. The
mean or average convection heat transfer coefficient for a surface is determined by
(properly) averaging the local heat transfer coefficient over the entire surface.
Velocity Boundary Layer
Consider the flow of a fluid over a flat plate, the velocity and the temperature of the fluid
approaching the plate is uniform at U∞ and T∞. The fluid can be considered as adjacent
layers on top of each others.
Fig. 2: Velocity boundary layer.
Assuming no‐slip condition at the wall, the velocity of the fluid layer at the wall is zero.
The motionless layer slows down the particles of the neighboring fluid layers as a result of
friction between the two adjacent layers. The presence of the plate is felt up to some
distance from the plate beyond which the fluid velocity U∞ remains unchanged. This
region is called velocity boundary layer.
Boundary layer region is the region where the viscous effects and the velocity changes are
significant and the inviscid region is the region in which the frictional effects are negligible
and the velocity remains essentially constant.
The friction between two adjacent layers between two layers acts similar to a drag force
(friction force). The drag force per unit area is called the shear stress:
2
0
N / m y
V
y
s
where μ is the dynamic viscosity of the fluid kg/m.s or N.s/m
2 .
Viscosity is a measure of fluid resistance to flow, and is a strong function of temperature.
The surface shear stress can also be determined from:
The Reynolds number at which the flow becomes turbulent is called the critical Reynolds
number. For flat plate the critical Re is experimentally determined to be approximately Re
critical = 5 x
5 .
Prandtl number: is a measure of relative thickness of the velocity and thermal boundary
layer
k
C p
moleculardiffusivityofheat
moleculardiffusivityofmomentum Pr
where fluid properties are:
mass density : ρ, ( kg/m
3 ) specific heat capacity : Cp ( J/kg ∙ K )
dynamic viscosity : μ, (N ∙ s/m
2 ) kinematic viscosity : ν, μ / ρ (m
2 /s)
thermal conductivity : k, (W/m∙ K) thermal diffusivity : α, k/(ρ ∙ Cp) (m
2 /s)
Thermal Boundary Layer
Similar to velocity boundary layer, a thermal boundary layer develops when a fluid at
specific temperature flows over a surface which is at different temperature.
Fig. 3: Thermal boundary layer.
The thickness of the thermal boundary layer δt is defined as the distance at which:
s
s
T T
T T
The relative thickness of the velocity and the thermal boundary layers is described by the
Prandtl number.
For low Prandtl number fluids, i.e. liquid metals, heat diffuses much faster than
momentum flow (remember Pr = ν/α <<1) and the velocity boundary layer is fully
contained within the thermal boundary layer. On the other hand, for high Prandtl number
fluids, i.e. oils, heat diffuses much slower than the momentum and the thermal boundary
layer is contained within the velocity boundary layer.
Flow Over Flat Plate
The friction and heat transfer coefficient for a flat plate can be determined by solving the
conservation of mass, momentum, and energy equations (either approximately or
numerically). They can also be measured experimentally. It is found that the Nusselt
number can be expressed as:
m n C L k
hL Nu Re Pr
where C, m , and n are constants and L is the length of the flat plate. The properties of the
fluid are usually evaluated at the film temperature defined as:
T T
T
s f
Laminar Flow
The local friction coefficient and the Nusselt number at the location x for laminar flow
over a flat plate are
, 1 / 2
1 / 2 1 / 3
Re
- 332 Re Pr Pr 0. 6
x
fx
x x
C
k
hx Nu
where x is the distant from the leading edge of the plate and Rex = ρV∞x / μ.
The averaged friction coefficient and the Nusselt number over the entire isothermal plate
for laminar regime are:
1 / 2
1 / 2 1 / 3
Re
- 664 Re Pr Pr 0. 6
L
f
L
C
k
hL Nu
Taking the critical Reynolds number to be 5 x
5 , the length of the plate xcr over which the
flow is laminar can be determined from
cr cr
V x
5 Re 5 10
Turbulent Flow
The local friction coefficient and the Nusselt number at location x for turbulent flow over a
flat isothermal plate are:
Example 1
Engine oil at 60°C flows over a 5 m long flat plate whose temperature is 20°C with a
velocity of 2 m/s. Determine the total drag force and the rate of heat transfer per unit
width of the entire plate.
We assume the critical Reynolds number is 5x
5
. The properties of the oil at the film
temperature are:
m s
k W mK
kg m
C
T T
T
s f
Pr 2870
6 2
3
The Re number for the plate is:
ReL = V∞L / ν = 4.13x
4
which is less than the critical Re. Thus we have laminar flow. The friction coefficient and
the drag force can be found from:
N
kg m m s m
V
F C A
C
D f
f L
- 328 Re 0. 00653
3 2 2
2
- 5
The Nusselt number is determined from:
Q hA T T W
mK
W
h
Then
k
hL Nu
s
L
0664 Re Pr 1918
2
- 5 1 / 3
oil
T∞ = 60°C
V∞=2 m/s
L = 5 m
Ts=20°C
Q°
Flow across Cylinders and Spheres
The characteristic length for a circular tube or sphere is the external diameter, D , and the
Reynolds number is defined:
V D
Re
The critical Re for the flow across spheres or tubes is 2x
5
. The approaching fluid to the
cylinder (a sphere) will branch out and encircle the body, forming a boundary layer.
Fig. 4: Typical flow patterns over sphere and streamlined body and drag forces.
At low Re ( Re < 4 ) numbers the fluid completely wraps around the body. At higher Re
numbers, the fluid is too fast to remain attached to the surface as it approaches the top of
the cylinder. Thus, the boundary layer detaches from the surface, forming a wake behind
the body. This point is called the separation point.
To reduce the drag coefficient, streamlined bodies are more suitable, e.g. airplanes are
built to resemble birds and submarine to resemble fish, Fig. 4.
In flow past cylinder or spheres, flow separation occurs around 80° for laminar flow and
140° for turbulent flow.
:frontalarea 2
2
D D N N A N
V
F C A
where frontal area of a cylinder is AN = L×D , and for a sphere is AN = πD
2 / 4.
which is valid for 3.5 < Re < 80,000 and 0.7 < Pr < 380. The fluid properties are evaluated
at the free‐stream temperature T∞ , except for μs which is evaluated at surface
temperature.
The average Nusselt number for flow across circular and non‐circular cylinders can be
found from Table 10 ‐ 3 Cengel book.
Example 2
The decorative plastic film on a copper sphere of 10 ‐mm diameter is cured in an oven at
75°C. Upon removal from the oven, the sphere is subjected to an air stream at 1 atm and
23°C having a velocity of 10 m/s, estimate how long it will take to cool the sphere to 35°C.
Assumptions:
- Negligible thermal resistance and capacitance for the plastic layer.
- Spatially isothermal sphere.
- Negligible Radiation.
Copper at 328 K Air at 296 K
ρ = 8933 kg / m
3
k = 399 W / m.K
Cp = 387 J / kg.K
μ∞ = 181.6 x 10 ‐ 7 N.s / m
2
v = 15.36 x 10 ‐ 6 m
2 / s
k = 0.0258 W / m.K
Pr = 0.
μs = 197.8 x 10 ‐ 7 N.s / m
2
The time required to complete the cooling process may be obtained from the results for a
lumped capacitance.
T T
T T
h
C D
T T
T T
hA
VC
t f
p i
f
P i ln 6
ln
^
Whitaker relationship can be used to find h for the flow over sphere:
1 / 2 2 / 3 0. 4 1 /^4
NuSph hD / k 2 0. 4 Re 0. 06 Re Pr / s
where Re = ρVD / μ = 6510.
Hence,
P∞= 1 atm.
V = 10 m/s
T∞= 23°C
Copper sphere
D = 10 mm
Ti = 75°C
Tf = 35°C
W m K D
k h Nu
Nu (^) Sph hD k
2
1 / 4
7
7 1 / 2 2 / 3 0. 4
The required time for cooling is then
- 2 sec 35 23
ln 6 122 /.
2
3
W m K
kg m J kgK m t
