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.

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.

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.

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.

• 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)

• 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.

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)

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)

• 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)

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)

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 = ×

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

≤ ≤

× ≤ ≤

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

≤ ≤

× ≤ ≤

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

≤ ≤

× ≤ ≤

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.

• 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.

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

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.

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

θ >90º laminar flow─Nuθ increases with increasingθ due to intense mixing in the separation zone.

90º

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)

• 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)

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.

• 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)

• 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)

• In staggered arrangement,

– for SD>(ST+D)/2 :

– for SD

• 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)

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)

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)

.

" " ,

(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 ∞

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

))(()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

Example 7-7