External Flow 1

Chapter 7

Convection: External Flow

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 =

External Flow 3

Laminar and Turbulent Flow

Transition criterion:

5,Re 5 10cx c

u xρµ∞= = ×

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)

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

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)

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)

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

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 ==

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)

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+

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