Chapter 7: External Forced...
Transcript of Chapter 7: External Forced...
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Chapter 7:External Forced Convection
Yoav PelesDepartment of Mechanical, Aerospace and Nuclear Engineering
Rensselaer Polytechnic Institute
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
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ObjectivesWhen you finish studying this chapter, you should be able to:• Distinguish between internal and external flow,
• Develop an intuitive understanding of friction drag and pressure drag, and evaluate the average drag and convection coefficients in external flow,
• Evaluate the drag and heat transfer associated with flow over a flat plate for both laminar and turbulent flow,
• Calculate the drag force exerted on cylinders during cross flow, and the average heat transfer coefficient, and
• Determine the pressure drop and the average heat transfer coefficient associated with flow across a tube bank for both in-line and staggered configurations.
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Drag and Heat Transfer in External flow• Fluid flow over solid bodies is responsible for numerous
physical phenomena such as– drag force
• automobiles • power lines
– lift force• airplane wings
– cooling of metal or plastic sheets.
• Free-stream velocity─ the velocity of the fluid relative to an immersed solid body sufficiently far from the body.
• The fluid velocity ranges from zero at the surface (the no-slip condition) to the free-stream value away from the surface.
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Friction and Pressure Drag
• The force a flowing fluid exerts on a body in the flow direction is called drag.
• Drag is compose of:– pressure drag,– friction drag (skin friction drag).
• The drag force FD depends on the – density ρ of the fluid, – the upstream velocity V, and – the size, shape, and orientation of the body.
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• The dimensionless drag coefficient CD is defined as
• At low Reynolds numbers, most drag is due to friction drag.
• The friction drag is also proportional to the surface area.
• The pressure drag is proportional to the frontal areaand to the difference between the pressures acting on the front and back of the immersed body.
2 1 2D
DFC
V Aρ= = 阻力 (7-1)
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• The pressure drag is usually dominant for blunt bodiesand negligible for streamlined bodies.
• When a fluid separates from a body, it forms a separated region between the body and the fluid stream.
• The larger the separated region, the larger the pressure drag.
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Heat Transfer• The phenomena that affect drag force also affect heat
transfer.
• The local drag and convection coefficients vary along the surface as a result of the changes in the velocity boundary layers in the flow direction.
• The average friction and convection coefficients for the entire surface can be determined by
,0
1 LD D xC C dxL
= ∫(7-7)
0
1 Lxh h dxL
= ∫ (7-8)
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Parallel Flow Over Flat Plates
• Consider the parallel flow of a fluid over a flat plate of length L in the flow direction.
• The Reynolds number at a distance x from the leading edge of a flat plate is expressed as
RexVx Vxρ
µ ν= = (7-10)
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• In engineering analysis, a generally accepted value for the critical Reynolds number is
• The actual value of the engineering critical Reynolds number may vary somewhat from 105 to 3 x106.
5Re 5 10crcrVxρ
µ= = × (7-11)
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Local Friction Coefficient
• The boundary layer thickness and the local friction coefficient at location x over a flat plate
– Laminar:
– Turbulent:
, 1/ 25
, 1/ 2
4.91Re
Re 5 100.664Re
v xx
x
f xx
x
C
δ ⎫= ⎪⎪ < ×⎬⎪=⎪⎭
(7-12a,b)
, 1/55 7
, 1/5
0.38Re
5 10 Re 100.059Re
v xx
x
f xx
x
C
δ ⎫= ⎪⎪ × ≤ ≤⎬⎪=⎪⎭
(7-13a,b)
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Average Friction Coefficient• The average friction coefficient
– Laminar:
– Turbulent:
• When laminar and turbulent flows are significant
51/ 2
1.33 Re 5 10Ref LL
C = < × (7-14)
5 71/5
0.074 5 10 Re 10Ref LL
C = × ≤ ≤ (7-15)
, laminar , turbulent0
1 cr
cr
x L
f f x f xx
C C dx C dxL
⎛ ⎞= +⎜ ⎟⎜ ⎟
⎝ ⎠∫ ∫ (7-16)
5 71/5
0.074 1742 5 10 Re 10Re Ref LL L
C = × ≤ ≤- ( ) (7-17)
5Re 5 10cr = ×
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Heat Transfer Coefficient
• The local Nusselt number at location x over a flat plate
– Laminar:
– Turbulent:
• hx is infinite at the leading edge(x=0) and decreases by a factor of x0.5 in the flow direction.
1/ 2 1/30.332 Re Pr Pr 0.6xNu = > (7-19)
(7-20)0.8 1/30.0296 Re Prx xNu = 5 70.6 Pr 605 10 Re 10x
≤ ≤
× ≤ ≤
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Average Nusset Number• The average Nusselt number
– Laminar:
– Turbulent:
• When laminar and turbulent flows are significant
, laminar , turbulent0
1 cr
cr
x L
x xx
h h dx h dxL
⎛ ⎞= +⎜ ⎟⎜ ⎟
⎝ ⎠∫ ∫ (7-23)
( )0.8 1 30.037 Re 871 PrLNu = − (7-24)
5Re 5 10cr = ×
0.5 1/3 50.664 Re Pr Re 5 10LNu = < × (7-21)
(7-22)0.8 1/30.037 Re PrLNu = 5 70.6 Pr 605 10 Re 10x
≤ ≤
× ≤ ≤
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Uniform Heat Flux
• When a flat plate is subjected to uniform heat flux instead of uniform temperature, the local Nusseltnumber is given by
– Laminar:
– Turbulent:
• These relations give values that are 36 percent higher for laminar flow and 4 percent higher for turbulent flow relative to the isothermal plate case.
0.5 1/30.453Re Prx LNu = (7-31)
(7-32)0.8 1/30.0308Re Prx xNu = 5 7
0.6 Pr 605 10 Re 10x
≤ ≤
× ≤ ≤
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Flow Across Cylinders and Spheres• Flow across cylinders and spheres is frequently encountered in
many heat transfer systems– shell-and-tube heat exchanger,– Pin fin heat sinks for electronic cooling.
• The characteristic length for a circular cylinder or sphere is taken to be the external diameter D.
• The critical Reynolds number for flow across a circular cylinder or sphere is about Recr=2 x 105.
• Cross-flow over a cylinder exhibits complex flow patterns depending on the Reynolds number.
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• At very low upstream velocities (Re≤1), the fluid completely wraps around the cylinder.
• At higher velocities the boundary layer detaches from the surface, forming a separation region behind the cylinder.
• Flow in the wake region is characterized by periodic vortexformation and low pressures.
• The nature of the flow across a cylinder or sphere strongly affects the total drag coefficient CD.
• At low Reynolds numbers (Re5000) ─ pressure drag dominate.
• At intermediate Reynolds numbers ─ both pressure and friction drag are significant.
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Average CD for circular cylinder and sphere
• Re ≤ 1 ─ creeping flow• Re ≈ 10 ─ separation starts• Re ≈ 90 ─ vortex shedding
starts.
• 103 < Re < 105– in the boundary layer flow is
laminar– in the separated region flow is
highly turbulent
• 105 < Re < 106 ─ turbulent flow
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Effect of Surface Roughness• Surface roughness, in general, increases the drag coefficient in
turbulent flow.
• This is especially the case for streamlined bodies.
• For blunt bodies such as a circular cylinder or sphere, however,an increase in the surface roughness may actually decrease the drag coefficient.
• This is done by tripping the boundary layer into turbulence at a lower Reynolds number, causing the fluid to close in behind the body, narrowing thewake and reducing pressure drag considerably.
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Heat Transfer Coefficient• Flows across cylinders and spheres,
in general, involve flow separation, which is difficult to handle analytically.
• The local Nusselt number Nuθ around the periphery of a cylinder subjected to cross flow varies considerably.
Small θ─ Nuθ decreases with increasing θas a result of the thickening of the laminar boundary layer.
80º
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θ >90º laminar flow─Nuθ increases with increasingθ due to intense mixing in the separation zone.
90º
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Average Heat Transfer Coefficient• For flow over a cylinder (Churchill and Bernstein):
Re·Pr>0.2• The fluid properties are evaluated at the film temperature
[Tf=0.5(T∞+Ts)].
• Flow over a sphere (Whitaker):
• The two correlations are accurate within ±30%.
( )
4 55 81 2 1/3
1 42 /3
0.62 Re Pr Re0.3 1282,0001 0.4 Pr
cylhDNuk
⎡ ⎤⎛ ⎞= = + +⎢ ⎥⎜ ⎟
⎝ ⎠⎡ ⎤ ⎢ ⎥+ ⎣ ⎦⎣ ⎦
(7-35)
1 41 2 2 3 0.42 0.4 Re 0.06 Re Prsph
s
hDNuk
µµ
∞⎛ ⎞⎡ ⎤= = + + ⎜ ⎟⎣ ⎦⎝ ⎠ (7-36)
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• A more compact correlation for flow across cylinders
where n = 1/3 and the experimentally determinedconstants C and m are given in Table 7-1.
• Eq. 7–35 is more accurate,and thus should be preferredin calculations whenever possible.
Re Prm ncylhDNu Ck
= = (7-37)
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Flow Across Tube Bank
• Cross-flow over tube banks is commonly encountered in practice in heat transfer equipment such as heat exchangers.
• In such equipment, one fluid moves through the tubes while the other moves over the tubesin a perpendicular direction.
• Flow through the tubes can be analyzed by considering flow through a single tube, and multiplying the results by the number of tubes.
• For flow over the tubes the tubes affect the flow pattern and turbulence level downstream, and thus heat transfer to or from them are altered.
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• Typical arrangement– in-line– staggered
• The outer tube diameter D is the characteristic length.• The arrangement of the tubes are characterized by the
– transverse pitch ST, – longitudinal pitch SL , and the– diagonal pitch SD between tube centers.
Staggered (交錯排列)In-line (直線排列)
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• As the fluid enters the tube bank, the flow area decreases from A1=STL to AT= (ST-D)L between the tubes, and thus flow velocity increases.
• In tube banks, the flow characteristics are dominated by the maximum velocity Vmax.
• The Reynolds number is defined on the basis of maximum velocity as
• For in-line arrangement, the maximum velocity occurs at the minimum flow area between the tubes
max maxReDV D V Dρ
µ ν= = (7-39)
maxT
T
SV VS D
=− (7-40)
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• In staggered arrangement,
– for SD>(ST+D)/2 :
– for SD
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• Zukauskas has proposed correlations whose general form is
where the values of the constants C, m, and n depend on Reynolds number.
• The average Nusselt number relations in Table 7–2 are for tube banks with 16 or more rows.
• Those relations can also be used for tube banks with NL provided that they are modified as
• The correction factor F values are given in Table 7–3.
( )0.25Re Pr Pr Prm nD D shDNu Ck
= = (7-42)
, LD N DNu F Nu= ⋅ (7-43)
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Pressure drop
• the pressure drop over tube banks is expressed as:
• f is the friction factor and χ is the correction factor.
• The correction factor (χ) given in the insert is used to account for the effects of deviation from square arrangement (in-line) and from equilateral arrangement (staggered).
2max
2Lf VP N ρχ∆ = (7-48)
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流體通過等溫管壁之管排時溫度之變化
x
T
Ts
Te
Ti
Ti Te
dx
dQ
dT
ST
管外徑 = D
L’
管長 = L(垂直版面)
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.
" " ,
(i) Energy Balance (cold fluid) : ( )
(ii) Heat Transfer (tube surface to fluid) : ( )( )
(i) and (ii),
(
.( ) ( ) total heat transfer rate
e
i
p c
s
T
T
dx
dQ m c dT
dQ h pdx T T
Combined
dT
Q mc T Tp c e i
=
= −
−
= − = =
∫
考慮 間之微體積
管排之總熱傳量
'
. .0
'[ ] ; , ln( )) ( ) ( )
' exp exp( ) ( )
( ) '' exp exp exp
( ) ( )
Le s
s i sp c p c
e s s
i s p c p c
fr p T T p
hp T T hpLdxT T T Tm c m c
T T hpL hAT T mc mc
DNLh L h DNLLU A c U N S L c
ππ
ρ ρ∞ ∞
−= − =
− −
⎡ ⎤ ⎡ ⎤− − −= =⎢ ⎥ ⎢ ⎥
− ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎡ ⎤− ⎡ ⎤⎢ ⎥ − −
= = =⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦⎢ ⎥
⎣ ⎦
∫
( )T T p
DhNU N S c
πρ ∞
⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦
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))(()ln(
)()()'( LMTDAh
TTTT
TTTTpLhQ s
es
is
esis =
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
−−
−−−=
direction)- xular to(perpendic rowoneintubesofnumber lengthtubetubesofnumbertotal
)(areatransferheattotal')(
===
==×==
==
TNLN
NDLpLL'p
(垂直版面)管長
總管數單一管之表面積
x軸方向管排長度
長(平均)方向)上熱傳表面之周每單位管排長度(x軸
π
ln Difference eTemperaturMean Log )ln(
)()( T
TTTT
TTTTLMTD
es
is
esis ∆≡=
−−
−−−≡
其中定義:
.'ln( )
( )s i
s ep c
T T hpLT T mc
−=
−ratetransferheattotal
)().
(
=
−= iTeTcpcmQ
合併上兩式
由前一頁得知:
總熱傳面積
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Example 7-7