Transcript Chapter 6
Heat Transfer Coefficient
Recall Newton’s law of cooling for heat transfer between a surface of
arbitrary shape, 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 a local convection
coefficient.
The total heat transfer rate is
q
AS
where
q" dAS (TS T )
1
h
AS
Introduction to Convection
AS
h dAS
AS
h dAS h AS (TS T )
is the average heat transfer coefficient
14
Heat Transfer Coefficient
• For flow over a flat plate:
1 L
h
h dx
L 0
How can we estimate heat transfer coefficient?
Introduction to Convection
15
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 d 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
For Newtonian
fluids:
Introduction to Convection
u
S
y
and
y 0
Cf
S
where Cf is the local
u2 / 2 friction coefficient
16
The Thermal Boundary Layer
Consider flow of a
fluid over an
isothermal flat
plate:
• The thermal boundary layer is the region of the fluid in which
temperature gradients exist
T T
• Its thickness is defined as the value of y for which the ratio: S
0.99
TS T
At the plate surface (y=0) there is no
fluid motion – Conduction heat
transfer:
qS"
T
k f
y
Introduction to Convection
and
y 0
h
k f T / y y 0
TS T
17
Boundary Layers - Summary
• Velocity boundary layer (thickness d(x)) characterized by the presence
of velocity gradients and shear stresses - Surface friction, Cf
• Thermal boundary layer (thickness dt(x)) characterized by
temperature gradients – Convection heat transfer coefficient, h
• Concentration boundary layer (thickness dc(x)) is characterized by
concentration gradients and species transfer – Convection mass
transfer coefficient, hm
Introduction to Convection
18
Laminar and Turbulent Flow
Transition criterion:
u xc
Re x
5 105
Introduction to Convection
19
Boundary Layer Approximations
Need to determine the heat transfer coefficient, h
• In general, h=f (k, cp, , , V, L)
• We can apply the Buckingham pi theorem, or obtain exact solutions by
applying the continuity, momentum and energy equations for the
boundary layer.
• In terms of dimensionless groups:
Nu x f ( x*, Re x , P r)
where:
(x*=x/L)
Nu f (Re L , Pr)
hL
hL
Nu
, Nu
kf
kf
Local and average Nusselt numbers (based
Pr
Prandtl number
u x
Re x
Introduction to Convection
on local and average heat transfer coefficients)
Reynolds number
(defined at distance x)
20
Example (Problem 6.27 textbook)
An object of irregular shape has a characteristic length of L=1 m and is
maintained at a uniform surface temperature of Ts=400 K. When
placed in atmospheric air, at a temperature of 300 K and moving with
a velocity of V=100 m/s, the average heat flux from the surface of the
air is 20,000 W/m2. If a second object of the same shape, but with a
characteristic length of L=5 m, is maintained at a surface temperature
of Ts=400K and is placed in atmospheric air at 300 K, what will the
value of the average convection coefficient be, if the air velocity is
V=20 m/s?
Introduction to Convection
21
Summary
• In addition to heat transfer due to conduction, we considered for the
first time heat transfer due to bulk motion of the fluid
• By applying the overall energy balance, we derived the thermal energy
equation, which can be used for solving problems involving
convection.
Useful in problems were we need to know the temperature
distribution within a fluid
• We discussed the concept of the boundary layer
• We defined the local and average heat transfer coefficients and
obtained a general expression, in the form of dimensionless groups to
describe them.
• In the following chapters we will obtain expressions to determine the
heat transfer coefficient for specific geometries
Introduction to Convection
22