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External Flow 1
Chapter 7
Convection: External Flow
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External Flow 2
Introduction In Chapter 6 we obtained a non-dimensional form for the heat transfer coefficient, applicable for problems involving the formation of a boundary layer:
Pr),(Rexx fNu =
• In this chapter we will obtain convection coefficients for different flow geometries, involving external flows: – Flat plates – Spheres, cylinders, airfoils, blades
Ø In such flows, boundary layers develop freely • Two approaches:
– Experimental or empirical: Experimental heat transfer measurements are correlated in terms of dimensionless parameters
– Theoretical approach: Solution of boundary layer equations.
Pr),Re*,( xx xfNu =
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External Flow 3
Laminar and Turbulent Flow
Transition criterion:
5,Re 5 10cx c
u xρµ∞= = ×
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External Flow 4
Flat Plate in Parallel Flow Ø For Laminar Flow:
0.6Pr , PrRe664.0 3/12/1 ≥== xx
xkxhNu
Ø For Turbulent Flow: 60Pr0.6 , PrRe0296.0 3/15/4 <<= xxNu
Ø Fluid properties are usually evaluated at the film temperature:
2∞+
=TTT s
f
(7.1a)
(7.2a)
(7.3)
0.6Pr , PrRe332.0 3/12/1 ≥== xx
x kxhNu
PrRe037.0 3/15/4LxNu =
(7.2b)
(7.1b)
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External Flow 5
Flow around Cylinders and Spheres • Flow around cylinders and spheres is characterized by boundary layer
development and separation. • Heat transfer coefficients are strongly influenced by the nature of
boundary layer development at the surface.
Ø Laminar boundary layer for 5102Re ×<µ
ρ= VDD
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External Flow 6
Crossflow around Cylinders 1. Zhukauskas correlation:
NuD =
hDk
=CReDmPrnPrPrs
⎛
⎝⎜⎞
⎠⎟
1/4
0.7<Pr <500,1<ReD <106
where C and m are listed in Table 7.4, (n=0.37 for Pr>10) and (n=0.36 for 10<Pr). Properties evaluated at , except Prs which is evaluated at Ts. ∞T
(7.4)
2. Churchill and Bernstein correlation, for all ReD and Pr>0.2
NuD =0.3+ 0.62ReD1/2Pr1/3
1+ 0.4/Pr( )2/3⎡⎣⎢
⎤⎦⎥
1/4 1+ ReD282,000
⎛
⎝⎜⎞
⎠⎟
5/8⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
4/5(7.5)
3. Hilpert correlation, can be used for cross flow around other non-circular shapes – see Table 7.3 for values of C and m
NuD =CReDmPr1/3 (7.6)
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External Flow 7
Crossflow around Spheres
• Whitaker correlation: 4/1
4.03/22/1 Pr)Re06.0Re4.0(2 ⎟⎟⎠
⎞⎜⎜⎝
⎛µµ++=s
DDDNu 4106.7Re5.3
380Pr71.0
×<<
<<
D
where properties are evaluated at , except ms which is evaluated at Ts ∞T
(7.7)
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External Flow 8
Procedure for Calculations
• Begin by recognizing the flow geometry (i.e. flat plate, sphere, cylinder etc.)
• Specify appropriate reference temperature for evaluation of fluid properties (usually film temperature, equation 7.3)
• Calculate Reynolds number – determine whether flow is laminar or turbulent Ø Reminder: Transition criteria:
5105Re ×=µ
ρ= ∞ cx
xu
• Decide whether a local or average heat transfer coefficient is required • Use appropriate correlation to determine heat transfer coefficient • Proceed with other calculations, such as determination of heating or
cooling rate
5102Re ×<µ
ρ= VDDFlat plates
Cylinders and spheres
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External Flow 9
Example 7.6
The decorative plastic film on a copper sphere of 10-mm diameter is cured in an oven at 75oC. Upon removal from the oven, the sphere is subjected to an airstream at 1 atm and 23oC, having a velocity of 10 m/s. Estimate how long it will take to cool the sphere to 35oC.
CT
smVo23
/10
=
=
∞
CtTCT ooi 35)(,75 ==
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External Flow 10
Tube Banks
Flow Across Tube Banks
• A common geometry for two-fluid heat exchangers.
maxSTV V
S DT=
−
• Aligned and Staggered Arrays:
Aligned:
Staggered: ( ) ( )if 2maxSTV V S D S DD TS DT
= − ≥ −−
( ) ( ) ( )if 2max 2
STV V S D S DD TS DD= − ≤ −
−or,
(@ A1)
(@ A1 or @A2)
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External Flow 11
Tube Banks (cont.)
• Flow Conditions:
• Flow around tubes in the first row of a tube bank corresponds to that for a single cylinder in cross flow. For subsequent rows flow depends strongly on the tube bank arrangement. • Typically the convection coefficient of a row increases with increasing row number until approximately the fifth row. After that there is a little change in the turbulence and hence in the convection coefficient. • Average Nusselt Number for an Isothermal Array:
( )1/ 40.362 ,maxRe Pr Pr/ Prm
D D sNu C C⎡ ⎤= ⎣ ⎦
2
, Table 7.7Table 7.8
C mC
→→
All properties are evaluated at except for Prs. ( ) / 2i oT T+
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External Flow 12
Summary
• Convection heat transfer coefficients in external flows depend on the nature of boundary layer development.
• There are numerous correlations available for describing convection heat transfer for external flows
• Technologically important cases include flows around flat plates, cylinders, spheres, tube banks, and etc.
Ø Comprehensive summary of correlations provided in Table 7.9, textbook