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
u2 / 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