383 NO02 Review
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Lecture Objectives:
• Review - Heat transfer– Convection – Conduction – Radiation
Analysis of a practical problem
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What can you see from this picture?
Infrared image
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What do you think about this statement?
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Example Problem –radiant barrier in attic
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Convection
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Convection coefficient – h [W/m2K]
khLNu
[W] )()( TThATThAQ wairwall
Conduction
Convection
Natural convection Forced convection
wT
T
L – characteristic length
wT
T
[W/m2] )( TThq w
h – natural convectionk – air conductionL- characteristic length
or
Nusselt number:
area Specific heat fluxHeat flux
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How to calculate h ?
What are the parametrs that affect h ?
What is the boundary layer ?
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Laminar and Turbulent Flowforced convection
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Which surface in this classroom has the largest forced convection
A. Window B. CeilingC. WallsD. Floor
Which surface has the largest natural convection
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Forced convection governing equations
v 2
2
yuv
yu
xuu
0 v
yxu
1) Continuity
2) Momentum which includes gravitational forceu, v – velocities – air viscosity
oooo UUuuLyyLxx v v;; ; ****
2*
*2
*
**
*
** 1 v
yu
LUyu
xuu
oo
Non-dimensionless momentum equation
Using
L = characteristic length and U0 = arbitrary reference velocity
ReL Reynolds number
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Forced convection governing equations
TTTTT
w
*
2
2 v
yT
yT
xTu
Energy equation for boundary layer
Non-dimensionless energy equations
2*
*2
*
**
*
**
.Pr.Re1 v
yT
yT
xTu
L
T –temperature, – thermal diffusivity =k/cp,
k-conductivity, - density, cp –specific cap.
Wall temperature
Air temperature outside of boundary layer
LU
LRe Inertial force
Viscous force a
Pr Momentum diffusivity
Thermal diffusivity
Reynolds number Prandtl number
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Simplified Equation for Forced convection
Pr) (Re, fNu
LU
LRe 3/1PrRe LCNu
5/4PrRe TCNu
For laminar flow:
For turbulent flow:
For air: Pr ≈ 0.7, = viscosity is constant, k = conductivity is constant
khLNu
General equation
mnmforced UCLUfh ),(
Simplified equation:
mforced ACHCh
Or:
RoomVolumeACH rate flow Volume
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Natural convection
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GOVERNING EQUATIONSNatural convection
Continuity
• Momentum which includes gravitational force
• Energy
v 2
2
yuvTTg
yu
xuu
0 v
yxu
2
2 v
y
TyT
xT
u
u, v – velocities , – air viscosity , g – gravitation, ≈1/T - volumetric thermal expansion T –temperature, – air temperature out of boundary layer, –temperature conductivity T
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Characteristic Number for Natural Convection
TTTTTUU
uuLyyLxxw
***** ;v v;; ;
2*
*2*
2*
**
*
**
Re1 v
yuT
ULTTwg
yu
xuu
L
Non-dimensionless governing equations
Using
L = characteristic length and U0 = arbitrary reference velocity Tw- wall temperature
The momentum equation become
2
3
LTTg w
Multiplying by Re2 number Re=UL/
Gr
2*
*2*2
*
**
*
** )Re/1()Re/( v
yuTGr
yu
xuu LL
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Grashof number Characteristic Number for Natural Convection
2
3
LTTwgGr
The Grashof number has a similar significance for natural convection as the Reynolds number has for forced convection, i.e. it represents a ratio of buoyancy to viscous forces.
Buoyancy forces
Viscous forces
Pr) ,( GrfNu
General equation
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Even more simple
Natural convection simplified equations
4/1Pr GrCNu L
3/1Pr GrCNu T
For laminar flow:
For turbulent flow:
For air: Pr ≈ 0.7, = constant, k= constant, = constant, g=constant
),(),)(( nmnmforced LTfLTTwfh
Simplified equation:
mforced TCh
Or:
T∞ - air temperature outside of boundary layer, Ts - surface temperature
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Forced and/or natural convection
Gr) Pr, (Re, 1Re2 fNuGr LL
Pr) (Re, 1Re2 fNuGr LL
Pr) ,( 1Re2 GrfNuGr LL
In general, Nu = f(Re, Pr, Gr)
natural and forced convection
forced convection
natural convection
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Combined forced and natural convention
nnforced
nnatural
nnncombined hhhhh /1/1
21 )()(
0 1 2 3 40
1
2
3
4
5h
T or ACH
n=2
n=3
n=6
h2
h1
hcombined
Churchill and Usagi approach :
This equation favors a dominant term (h1 or h2), and exponent coefficient ‘n’ determines the value for hcombined when both terms have the same order of value
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Example of general forced and natural convection
8.019.1 ACHh forced
3/138.0333.0 )19.1()12.2( ACHThcombinbed
333.0 )12.2( Thnatural
Equation for convection at cooled ceiling surfaces
n
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What kind of flow is the most common for indoor surfaces
A. Laminar B. TurbulentC. TransitionalD. Laminar, transitional, and turbulent
What about outdoor surfaces?
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Conduction
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Conductive heat transfer• Steady-state
• Unsteady-state
• Boundary conditions
– Dirichlet Tsurface = Tknown
– Neumann
)(/ 21 SS TTLkq
sourcep
qxT
ckT
2
2
)( surfaceair TThxT
L
Tair
k - conductivity of material
TS1 TS2
h
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Boundary conditions
Biot number
solidkhLBi
convention
conduction
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0 1 2 3 4 5 6 7 8 9 100.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Analytical solution Numerical -3 nodes, =60 min Numerical -7 nodes, =60 min Numerical -7 nodes, =12 min
(T-T
s)/(T
o-Ts
)
hour
Ts
0
T
-L / 2 L/2
h
h
h
T o
T
h omogenous wa ll
L = 0.2 mk = 0 . 5 W/ m Kc = 9 20 J/kgK
= 120 0 k g/mp
2
Importance of analytical solution
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What will be the daily temperature distribution profile on internal surface
for styrofoam wall? A.
B.
External temperature profile
T
time
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What will be the daily temperature distribution profile on internal surface
for tin glass? A.
B.
External temperature profile
T
time
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Conduction equation describes accumulation
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Important numbers
LUo
L Re Inertial force
Viscous forceReynolds number
a
Pr Momentum diffusivity
Thermal diffusivityPrandtl number
2
3
LTTsgGr
Buoyancy forcesViscous forces
khLNu
Conduction
Convection Nusselt number
solidkhLBi thermal internal resistance
surface film resistance
Grashof number
Biot number
Reference book: Fundamentals of Heat and Mass Transfer, Incropera & DeWitt