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Transcript of heat transfer
# 1
Heat Transfer Su Yongkang
School of Mechanical Engineering
HEAT TRANSFER
CHAPTER 6
Introduction to convection
# 2
Heat Transfer Su Yongkang
School of Mechanical Engineering
CH6 – INTRODUCTION
Where we’ve been ……Basic Conduction Heat Transfer Finished
Fourier’s law:
Where we’re going:Begin study of convective heat transfer.
Newton’s law of cooling:
dx
dtkq
tkq
)( TThq s
# 3
Heat Transfer Su Yongkang
School of Mechanical Engineering
Convective transfer problem
# 4
Heat Transfer Su Yongkang
School of Mechanical Engineering
CH6 INTRODUCTION
KEY POINTS THIS CHAPTER• What are the key variables when analyzing
convection heat transfer?
• Review boundary layer concept and significance
• General idea of relationship between velocity and thermal profiles in a boundary layer.
• Effect of laminar versus turbulent flow on heat transfer potential
• Boundary layer similarity
• This chapter will be taught in two lectures: the first includes text book sections §6.1 to 6.4the other includes text book sections §6.5 to 6.10
# 5
Heat Transfer Su Yongkang
School of Mechanical Engineering
Convection overview
• Consider a flat plate of length L, in air flow with velocity u and temperature T
• Local heat flux is: where h is the local heat transfer coefficient
• Total heat transfer rate:
haverage heat transfer coefficient
Determination of ‘h’ will rely on analytical as well as empirical data
)( TThq s
ss A ssA s hdATTdAqq )(
)( TTAhq ss
# 6
Heat Transfer Su Yongkang
School of Mechanical Engineering
Convection overview (Cont’d)
• Same principal applies to any arbitrary shape, not just a flat plate
• Average convection heat transfer coefficient:
So, we need to know how h varies with x, the distance from the leading edge……..
What do you think key parameters that might influence h?
q sdA
ss TA ,
or, for unit width: sA s
s
hdAA
h1
Lhdx
Lh
0
1
# 7
Heat Transfer Su Yongkang
School of Mechanical Engineering
Key parameters
• Transfer potential: forced flow or free flow
• Phase change: boiling and condensation
• Flow conditions: laminar or turbulent flow
• Geometries: shape, size, position and roughness.
• Properties: density, viscosity, thermal conductivity, specific heat, and so on.
# 8
Heat Transfer Su Yongkang
School of Mechanical Engineering
Example
Given• Experimental results for measured local heat
transfer coefficient h for flow over a flat plate with a rough surface
• where: a = coefficient x = distance from leading edge
– Find expression for average heat transfer coefficient, and the relation of average heat transfer coefficient to the local coefficient
)( 3.0 axxhx
7.0
0
3.0
0
3.0
0.7
1
1
xx
ah
dxxx
adxxa
xh
x
xx
x
# 9
Heat Transfer Su Yongkang
School of Mechanical Engineering
The Convection Boundary Layers
Velocity Boundary Layer
• For fluid flow over a flat plate, which disturbs the fluid flow:– As y: where u is velocity in
x-direction– As y0: (no-slip condition)– The boundary layer thickness is defined as
the value at which:– The boundary layer thickness varies with
x• Shear Stress
• Local friction coefficient
0
y
s y
u
2
2u
C sf
uu
0u
uyu 99.0)(
Dynamic viscosity
# 10
Heat Transfer Su Yongkang
School of Mechanical Engineering
The Convection Boundary Layers
Thermal Boundary Layer
• A hot or cold plate alters the temperature distribution in the air– As y:– As y0:– The thermal boundary layer thickness is
defined as the value at which:
– The thermal boundary layer thickness, t
also varies (increases) with x
99.0)(
TT
yTT
s
airs
TTs
TyT )(
sTyT )(
# 11
Heat Transfer Su Yongkang
School of Mechanical Engineering
The Convection Boundary Layers
Thermal Boundary Layer (Cont’d)
Heat Flux• Heat flux analogous to shear stress in velocity
boundary layer• Heat flux proportional to the temperature
gradient at the surface,ANDsince u(y=0) =0, energy transfer to/from fluid occurs by conduction only!
• Since thermal boundary layer gets larger along x direction, the temperature gradient changes with x, and therefore
TTs
fluid thermal conductivity
wall temperature gradient
____________________
0
y
fs y
Tkq
# 12
Heat Transfer Su Yongkang
School of Mechanical Engineering
The Convection Boundary Layers
Thermal Boundary Layer (Cont’d)Heat Flux (Cont’d)• Using Newton’s law of cooling:
We obtain
• While increases with increasing x, temperature gradients in the boundary layer must decrease with increasing x.
• Accordingly, and h decrease with increasing x.
)( TThq s
TT
yTkh
s
yf 0/
sq
# 13
Heat Transfer Su Yongkang
School of Mechanical Engineering
The Convection Boundary Layers
Laminar Versus Turbulent Flow
Characterization of laminar flow• Low amount of “mixing” of fluid within the
boundary layer (smooth flow)
Characterization of turbulent flow• High amount of mixing of fluid within the
boundary layer (irregular flow)• High amount of mixing means increased surface
friction as well as convection transfer rates (heat and mass)
# 14
Heat Transfer Su Yongkang
School of Mechanical Engineering
Convection Heat Transfer Variations Along Flat Plate
• Consider flat plate with:and all laminar flow
• Thermal boundary layer defined by
Compare temperature gradients at points 1 and 2 to evaluate the heat flux rate (and hence the heat transfer coefficient)
C0 and C100 TTx
21
C0T
C100sT1 2
99.0
TT
TT
s
s
# 15
Heat Transfer Su Yongkang
School of Mechanical Engineering
Convection Heat Transfer Variations Along Flat Plate (Cont’d)
• Consider flat plate with: C0 and C100 TTx
2
1
C100C1C100C1
at x1 at x2
To determine the location that transition begins, we define the Reynolds number,
And the critical Reynolds number,
xu
xRe
5, 105Re
c
cx
xu
# 16
Heat Transfer Su Yongkang
School of Mechanical Engineering
Convective Transfer Equations
Topic of the Day
Project Teams
# 17
Heat Transfer Su Yongkang
School of Mechanical Engineering
Convective Transport Equations
Where we’ve been ……Last section:Overview of the topic of convective transport of heat.
Where we’re going:
• Convection transfer detailed equations
• Heat and mass transfer analogy
• Eventually get to applications in external and internal flow.
# 18
Heat Transfer Su Yongkang
School of Mechanical Engineering
Convective Transport Equations
KEY POINTS THIS SECTION
• Detailed development of boundary layer equations for velocity, temperature and species concentrations
• What approximations can be made in the boundary layers?
# 19
Heat Transfer Su Yongkang
School of Mechanical Engineering
Recall the convection overview
• Local heat flux is:
where h is the local heat transfer coefficient
)( TThq s
# 20
Heat Transfer Su Yongkang
School of Mechanical Engineering
Develop convection transfer equations
• Consider steady, 2-D flow of a viscous, incompressible fluid with constant properties.
• Key point to remember:At each point in the fluid, conservation of mass, energy and momentum must be satisfied.
• Appendix E contains detailed development of the full boundary layer equations, for example:
Conservation of mass (continuity)
0
y
v
x
u
# 21
Heat Transfer Su Yongkang
School of Mechanical Engineering
The magnitude of variables in the thermal boundary layer
variables x(main flow direction)
y u v t
magnitude 1 1 1
x
T
y
T
Thermal boundary layer
y
v
y
u ,
x
v
y
u ,
x
u
y
u and ;
vu
Velocity boundary layer
Boundary Layer Approximations
# 22
Heat Transfer Su Yongkang
School of Mechanical Engineering
Continue convection transfer equations
• Quick overview of fluid equationsConservation of mass (continuity):
OR
x-momentum equation:
y-momentum equation:
0 0
y
v
x
u
y
v
x
u
0
y
v
x
u
forces"Body " 2
2
2
2
y
u
x
u
x
P
y
uv
x
uu
x
u
y
u 0
0
2
2
1
y
u
x
P
y
uv
x
uu
:where
forces"Body " 2
2
2
2
y
v
x
v
y
P
y
vv
x
vu
0 0 0 0 0
0
y
PSo:
# 23
Heat Transfer Su Yongkang
School of Mechanical Engineering
Continue convection transfer equations
Energy equation:
qy
v
x
u
x
v
y
u
y
T
x
Tk
y
Tv
x
Tuc p
222
2
2
2
2
2
0 0 0 0
0
2
y
u
c p
NOTE: is usually small unless u is high
(as in sonic flows) or is high (such as flow of oils).
Result is 4 equations and 4 unknowns:
Unknowns are: u, v, P, and T
Since:
P(x) can be obtained from free stream flow.
dx
dP
x
P and only, )x(fP then 0
y
P
x
T
y
T Why?
x
v
y
u Why?
2
2
2
y
u
cy
T
y
Tv
x
Tu
p
# 24
Heat Transfer Su Yongkang
School of Mechanical Engineering
• Review boundary layer concept.• General relationship between velocity and
thermal boundary layers.• Convective heat transfer is dependent on
the temperature gradient at the fluid/solid interface
• Boundary layer grows with distance from the leading edge, and this decreases the local heat transfer rates.
• Turbulent boundary layers have much greater potential for heat and mass transfer due to velocity fluctuations.
• Convection transfer equations (4).
Go back and review fluids course notes!
SUMMARY