AME$60634$$ Int.$HeatTrans.$ External Convection: …sst/teaching/AME60634/lectures/AME... ·...
Transcript of AME$60634$$ Int.$HeatTrans.$ External Convection: …sst/teaching/AME60634/lectures/AME... ·...
AME 60634 Int. Heat Trans.
D. B. Go
External Convection: Laminar Flat Plate For a constant property, laminar flow a similarity solution exists for the flow field u(y)
€
δx =5xRex
12
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Cf ,x = 0.664Rex− 12
local boundary layer thickness
local skin friction coefficient
Cf ,x =τ s,x12ρu∞
2⇒ τ s,x =
1x
τ s,x dx0
x
∫average skin friction coefficient
Major flow parameters:
⇒Cf ,x =1.328Rex−12
AME 60634 Int. Heat Trans.
D. B. Go
External Convection: Laminar Flat Plate For a constant property, laminar flow a similarity solution exists for the flow field u(y)
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Nux = 0.332Rex12 Pr
13local Nusselt number
(Pr > 0.6)
local thermal boundary layer thickness δx δt,x = Pr13
average Nusselt number Nux =hxxk⇒ hx =
1x
hx dx0
x
∫ ⇒ Nux = 0.664Rex12 Pr
13
uniform surface temperature, Ts
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Nux = 0.453Rex12 Pr
13
Ts −T∞( ) =##qsLNuL
=##qL
xkNux
dx0
L
∫ ⇒ NuL = 0.680ReL12 Pr
13
uniform surface heat flux, q”s
Major heat transfer parameters:
local Nusselt number (Pr > 0.6)
average Nusselt number
AME 60634 Int. Heat Trans.
D. B. Go
External Convection: Turbulent Flat Plate
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Cf ,x = 0.0592Rex− 15local skin friction coefficient
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Nux = 0.0296Rex45 Pr
13
local Nusselt number (Pr > 0.6)
average skin friction coefficient
average Nusselt number
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NuL = 0.037ReL45 Pr
13
For xc= 0 or L >> xc (Rex,L >> Rex,c)
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C f ,L = 0.074ReL− 15
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Nux = 0.0308Rex45 Pr
13
uniform surface temperature, Ts
uniform surface heat flux, q”s
average skin friction coefficient
average Nusselt number
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NuL = 0.037ReL45− 871#
$ % &
' ( Pr
13
assuming xc for Rex,c = 5×105
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C f ,L = 0.074ReL− 15−1742ReL
−1
uniform surface temperature, Ts
uniform surface temperature, Ts
For turbulent flow, only empirical relations exist
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τ s,L =1L
τ s,x,lamdx0
xc
∫ + τ s,x,turbdxxc
L
∫$
% & &
'
( ) )
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h L =1L
hx, lamdx0
xc
∫ + hx,turbdxxc
L
∫#
$ % %
&
' ( (
Average parameters
AME 60634 Int. Heat Trans.
D. B. Go
External Convection: Starting Length • The effect of an unheated starting length (USL) can be represented
on the local Nusselt number as:
• Parameters a, b, C, & m depend on – thermal boundary condition: uniform surface temperature (UST) or
uniform heat flux (UHF) – flow conditions: laminar or turbulent
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Nux =Nux ξ = 0
1− ξx
$ % & '
( ) a*
+ ,
-
. /
b where
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Nux ξ = 0 = CRexm Pr
13 for Pr > 0.6
LAMINAR TURBULENT
a 3/4 3/4 9/10 9/10
b 1/3 1/3 1/9 1/9
C 0.332 0.453 0.0296 0.0308
m 1/2 1/2 4/5 4/5
€
" " q s
€
Ts
€
" " q s
€
Ts
AME 60634 Int. Heat Trans.
D. B. Go
External Convection: Starting Length • Uniform Surface Temperature (UST)
• Uniform Heat Flux (UHF)
• The Nusselt number (and heat transfer coefficient) are functions of the fluid properties (ν,ρ,α,cp,k) – the effect of variable properties may be considered by evaluating all
properties at the film temperature
– most accurate solutions often require iteration on the film properties
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Tf =Ts + T∞2
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NuL = NuLξ = 0
LL −ξ
1− ξ L( ) 2p+1( ) 2p+2( )[ ]2p( ) 2p+1( ) p = 1 (laminar throughout)
p = 4 (turbulent throughout)
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h L =1L
hx, lamdxξ
xc
∫ + hx,turbdxxc
L
∫$
% & &
'
( ) )
numerical integration for laminar/turbulent flow
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" " q x = hx Ts −T∞( )⇒ q = h L As Ts −T∞( )
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Ts x( ) = T∞ +# # q s
hx
⇒ q = # # q sAs
AME 60634 Int. Heat Trans.
D. B. Go
External Convection: Cylinder in Cross Flow • As with flat plate flow, flow conditions determine heat transfer • Flow conditions depend on special features of boundary layer
development, including onset at stagnation point, separation, and onset of turbulence
• Stagnation point: location of zero velocity and maximum pressure – boundary layer development under a favorable pressure gradient è
acceleration of the free stream flow
• There is a minimum in the pressure distribution p(x) and toward the rear of the cylinder, the pressure increases. – boundary layer development under an adverse pressure gradient €
dpdx < 0→ du∞
dx > 0
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dpdx > 0
AME 60634 Int. Heat Trans.
D. B. Go
External Convection: Cylinder in Cross Flow • Separation occurs when the momentum of the free stream flow is
insufficient to overcome the adverse pressure gradient – the velocity gradient reduces to zero – flow reversal occurs accompanied by a downstream wake
• Location of separation depends on boundary layer transition
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ReD =VDν
note:
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V ≠ u∞
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dudy y= 0
= 0
AME 60634 Int. Heat Trans.
D. B. Go
External Convection: Cylinder in Cross Flow • Force (FD) imposed by the flow on the cylinder is composed of
two phenomena – friction è boundary layer shear stress – form drag (pressure drag) è pressure differential due to wake
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CD ≡FD
Af ρV2 /2( )
drag coefficient
Af is the area projected perpendicular to free stream
AME 60634 Int. Heat Trans.
D. B. Go
External Convection: Cylinder in Cross Flow • Thermal considerations: uniform surface temperature, Ts
– heat transfer a function of the angel of separation θ – empirical correlations are used to determine average Nusselt
numbers
• Hilpert correlation: Pr ≥ 0.6 – also suitable for non-circular cylinders
• Churchill and Bernstein: ReDPr > 0.2
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NuD =h Dk
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NuD = CReDm Pr
13
ReD C m 0.4-‐4 0.989 0.330
4-‐40 0.911 0.385
40-‐4000 0.683 0.466
4000-‐40,000 0.193 0.618
40,000-‐400,000 0.027 0.805
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NuD = 0.3+0.62ReD
1 2 Pr1 3
1+ 0.4 Pr( )2 3[ ]1 4 1+
ReD282,000"
# $
%
& ' 5 8(
) * *
+
, - -
4 5
AME 60634 Int. Heat Trans.
D. B. Go
External Convection: Sphere in Cross Flow • Similar flow issues as cylinder in cross flow arise
• Thermal considerations: uniform surface temperature, Ts – heat transfer again defined by empirical correlations
• Whitaker correlation: – 0.71 < Pr < 380 – 3.5 < ReD < 7.6×104
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NuD = 2 + 0.4ReD1 2+ 0.06ReD
2 3( )Pr0.4 µµs
"
# $
%
& '
1 4
evaluate fluid properties at T∞ except for µs which is evaluated at Ts
AME 60634 Int. Heat Trans.
D. B. Go
External Convection: Impinging Jet • Impinging jet consists of a high speed flow impacting a flat surface
– generates large convection coefficients • The flow and heat transfer are affected by a number of factors
– shape/size of jet, velocity of jet, distance from plate, … • Significant hydrodynamic features:
– mixing and velocity profile development in the free jet – stagnation point and zone – velocity profile development in the wall jet
AME 60634 Int. Heat Trans.
D. B. Go
External Convection: Impinging Jet • Local Nusselt number distribution:
• Average Nusselt number based on empirical correlations for single nozzles and arrays of nozzles – function of Reynolds number, Pr, distance along wall (r or x), height of
jet (H)
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Nu = f ReDh,Pr, r or x( ) Dh ,H Dh( )
AME 60634 Int. Heat Trans.
D. B. Go
External Convection: Impinging Jet • Martin correlation: uniform surface temperature, Ts
– single round nozzle
• Martin correlation: uniform surface temperature, Ts – single slot nozzle
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NuPr0.42
=G r D,H D( )F1(ReD )
F1(ReD ) = 2ReD1 2 1+ 0.005ReD
0.55( )1 2⇒ ReD =
VexitDν
G =D r( ) 1−1.1 D r( )( )
1+ 0.1 H D− 6( ) D r( )
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2000 ≤ ReD ≤ 400,0002 ≤ H D ≤122.5 ≤ r D ≤ 7.5
valid for
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NuPr0.42
=3.06
x W + H W +2.78(ReDh
m )
⇒ ReDh=VexitDh
ν=Vexit (2W )
ν
m = 0.695 − x2W%
& '
(
) * +
H2W%
& '
(
) * 1.33
+ 3.06+
, -
.
/ 0
−1
€
3000 ≤ ReDh≤ 90,000
2 ≤ H W ≤104 ≤ x W ≤ 20
valid for