chapter 10 correlation equations: forced and free...

CHAPTER 10

CORRELATION EQUATIONS:

FORCED AND FREE CONVECTION

10.1 Introduction

• Correlation equations: Based on experimental data

Chapter outline: Correlation equations for:•

(1) External forced convection over:

Plates

1

Plates

Cylinders

Spheres

(2) Internal forced convection through channels

(3) External free convection over:

Cylinders

Plates

Spheres

10.2 Experimental Determination of Heat Transfer Coefficient h

Newton's law of cooling defines h:

∞∞∞∞−−−−

′′′′′′′′====

TT

qh

s

s (10.1)

sq ′′′′′′′′ = surface flux

sT = surface temperature

∞∞∞∞T = ambient temperature

••••

••••

sq ′′′′′′′′

T

V∆∆∆∆

−−−−

++++

2

∞∞∞∞T = ambient temperature

Example: Electric heating

sT ∞∞∞∞TMeasure: Electric power, ,

Use (10.1) to calculate h

• Form of correlation equations:

• Dimensionless: Nusselt number Is a dimensionless heat transfer

coefficient.

••••

••••sT

∞∞∞∞T

∞∞∞∞V

++++

10.1Fig.

1.Example: Forced convection with no dissipation

k

hxNux ==== )PrRe,xf ;( *= (2.52)

Use (2.52) to plan experiments and correlate data

10.3 Limitations and Accuracy of Correlation Equations

! slimitation have equations ncorrelatio All

• Limitations on:

3

• Limitations on:

(1) Geometry

(2) Range of parameters: Reynolds, Prandtl, Grashof, etc.

(3) Surface condition: Uniform flux, uniform temperature, etc.

• Accuracy: Errors as high as 25% are not uncommon!

10.4 Procedure for Selecting and Applying Correlation Equations

(1) Identify the geometry

(2) Identify problem classification:

Forced convection

Free convection

External flow

Internal flow

Entrance region

Fully developed region

Boiling

Condensation

4

Condensation

Etc.

(3) Define objective: Finding local or average heat transfer coefficient

(4) Check the Reynolds number:

(a) Laminar

(b) Turbulent

(c) Mixed

(5) Identify surface boundary condition:

(a) Uniform temperature

(b) Uniform flux

(6) Note limitations on correlation equation

(7) Determine properties at the specified temperature:

(a) External flow: at the film temperature fT

2/)( ∞∞∞∞++++==== TTT sf (10.2)

(b) Internal flow: at the mean temperature mT

(c) However, there are exceptions

5

(8) Use a consistent set of units

(9) Compare calculated values of h with Table 1.1

10.5 External Forced Convection Correlations

10.5.1 Uniform Flow over a Flat Plate:

Transition to Turbulent Flow

• Boundary layer flow over a semi-infinite flat plate

Three regions:

(1) Laminar

(2) Transition

(3) Turbulent

txRe = Transition or

critical Reynolds

number:

xRe depends on: Geometry, surface finish, pressure gradient, etc.

•10.2Fig.

transitionturbulentlaminar

x•••• tx

∞∞∞∞V

∞∞∞∞T

6

txRe depends on: Geometry, surface finish, pressure gradient, etc.

5105 ××××≈≈≈≈==== ∞∞∞∞

νt

xxV

Ret

For flow over a flat plate:

• Examples of correlation equations for plates:

Laminar region, x < xt :

Use (4.72a) or (4.72b) for local Nusselt number to obtain local h

Turbulent region, x > xt :

Local h:

(((( )))) (((( )))) 315402960 //. PrRe

k

hxNu xx ======== (10.4a)

Limitations:flat plate, constant sT

5 ×××× 105 < xRe < 10

7

0.6 < Pr < 60(10.4b)

7

0.6 < Pr < 60

properties at fT

Average h

++++======== ∫∫∫∫ ∫∫∫∫∫∫∫∫

t

t

x

x

L

tL

L

dxxhdxxhL

dxxhL

h

)()()(

00

11(10.5)

Lh = local laminar heat transfer coefficient

th = local turbulent heat transfer coefficient

(4.72b) and (10.4a) into (10.5):

(((( )))) 31

051

54

21

21

029603320 /

/

/

/

/

.. Prx

dxV

x

dxV

L

kh

t

t

x L

x

++++

==== ∫∫∫∫ ∫∫∫∫∞∞∞∞∞∞∞∞

νν(10.6)

Integrate

(((( )))) (((( )))) (((( ))))[[[[ ]]]]{{{{ }}}} (((( )))) 31545421 03706640 ////.. PrReReRe

L

kh

tt xLx −−−−++++====

8

(((( )))) (((( ))))[[[[ ]]]]{{{{ }}}}L tt xLx

Dimensionless form:

(((( )))) (((( )))) (((( ))))[[[[ ]]]]{{{{ }}}}(((( )))) 3154542103706640 ////

.. PrReReRek

LhNu

tt xLxL −−−−++++======== (10.7b)

(2) Plate at uniform surface temperature

with an insulated leading section

x0=Length of insulated section

∞∞∞∞T

∞∞∞∞V

txx

10.3Fig.

tδδδδ

insulation0

sTox•••••••• ••••

Two cases:

tx ox• Laminar flow, > : Use (5.21) for the local Nusselt number to obtain

local h

tx ox•Turbulent flow, < : The local Nusselt number is

[[[[ ]]]] 91109o

3154

1

02960

//

//

)/(

.

xx

PrRe

k

hxNu x

x

−−−−

======== (10.8)

(3) Plate with uniform surface flux

9

(3) Plate with uniform surface flux

Two regions:

• Laminar flow, 0 < x < xtUse (5.36) or (5.37) for the local

Nusselt number to obtain local h

•Turbulent flow, txx >>>> :

31540300 //. PrRe

k

hxNu xx ======== (10.9)

••••∞∞∞∞T

∞∞∞∞V

tx

sq ′′′′′′′′x

10.4Fig.

0

2/)( ∞∞∞∞++++==== TTT sf sTProperties at and is the average surface temperature

10.5 External Flow Normal to a Cylinder

• For uniform surface temperature or uniform

surface flux

54853121620/

///. RePrReDh

∞∞∞∞T

∞∞∞∞V θθθθ

10.5Fig.

10

(((( ))))[[[[ ]]]]

5485

4132

3121

0002821

41

62030

//

//

//

,/

..

++++

++++

++++======== DDL

Re

Pr

PrRe

k

DhNu (10.10a)

Flow normal to cylinder2.0>>>>==== PrRePe D

properties at fT

Limitations:

(10.10b)

Pe = Peclet number = ReD Pr

For Pe < 0.2, use:

Pek

DhuN

Dln.. 5082370

1

−−−−======== (10.11a)

flow normal to cylinder

PrRePe D= < 0.2

properties at fT

Limitations

10.5.3 External Flow over a Sphere

11

[[[[ ]]]] (((( )))) 41403221

060402/

.//..

s

PrReRek

DhNu DDD µ

µ++++++++======== (10.12a)

flow over sphere

3.5 < ReD < 7.6 ×××× 104

0.71 < Pr < 380

2.31 <<<<<<<<sµµµµ

µµµµ

properties at ∞∞∞∞T , sµµµµ at sT

Limitations:

(10.12b)

Chapter 7:

Analytic solutions to h for

fully developed laminar flow

Correlation equations for h in the

entrance and fully developed regions

10.6 Internal Forced Convection Correlations

12

entrance and fully developed regions

for laminar and turbulent flows

• Transition or critical Reynolds number for smooth tubes:

2300≈≈≈≈====νDu

RetD (10.13)

10.6.1 Entrance Region: Laminar Flow Through

Tubes at Uniform Surface Temperature

• Two cases:

(1) Fully Developed Velocity, Developing Temperature: Laminar Flow

• Solution: Analytic

Correlation of analytic

results:

•sTT

FDV 0

u developing

••••

xu

13

[[[[ ]]]]{{{{ }}}}32

)(0401

06680663 /

.

)/(..

PrReL/D

PrReLD

k

DhNu

D

DD

++++

++++======== (10.14a)

10.6Fig.

insulation

FDV 0 x

etemperaturt

δδδδ

entrance region of tubes

uniform surface temperature sT

laminar flow (ReD < 2300)

fully developed velocity

developing temperature

properties at 2/)( momim TTT ++++====

Limitations:

(10.14b)

14

(2) Developing Velocity and Temperature: Laminar flow

[[[[ ]]]]140

31861

./

)(.

========

s

PrReL/Dk

DhNu

DD µµ

(10.15a)

entrance region of tube


laminar flow (ReD < 2300)

developing velocity and temperature

0.48 < Pr < 16700

0.0044 < sµµµµ

µµµµ < 9.75

properties at mT , sµµµµ at sT

Limitations:

10.6.2 Fully Developed Velocity and Temperature in Tubes: Turbulent Flow

15

10.6.2 Fully Developed Velocity and Temperature in Tubes: Turbulent Flow

• Entrance region is short: 10-20 diameters

• Surface B.C. have minor effect on h for Pr > 1

• Several correlation equations for h:

(1) The Colburn Equation: Simple but not very accurate

1/34/50.023 PrRe

k

DhNu

DD======== (10.16a)

Limitations:

fully developed turbulent flow

smooth tubes

ReD > 104

0.7 < Pr < 160

L /D > 60

properties at mT

(10.16b)

• Accuracy: Errors can be as high as 25%

(2) The Gnielinski Equation: Provides best correlation of experimental

16

(2) The Gnielinski Equation: Provides best correlation of experimental

data

[[[[ ]]]][[[[ ]]]]322/31/2

11)()12.7(1

1000))(( /)( L/D

Pr8f

PrRe8fNu DD ++++

−−−−++++

−−−−==== (10.17a)

• Valid for: developing or fully developed turbulent flow

2300 < ReD < 5 ×××× 106

0.5 < Pr < 2000

0 < D/L <1

properties at mT

Limitations:

(10.17b)

• The D/L factor in equation accounts for entrance effects

• For fully developed flow set D/L = 0

17

• For fully developed flow set D/L = 0

The Darcy friction factor f is defined as

2

2u

L

Dpf ρ

∆∆∆∆==== (10.18)

For smooth tubes f is approximated by

2641790 −−−−−−−−==== ).ln.( DRef (10.19)

10.6.3 Non-circular Channels: Turbulent Flow

Use equations for tubes. Set eDD ==== (equivalent diameter)

P

AD

fe

4====

fA

P

= flow area

= wet perimeter

10.7 Free Convection Correlations x

18

10.7 Free Convection Correlations

10.7.1 External Free Convection Correlations

(1) Vertical plate: Laminar Flow, Uniform Surface

Temperature

• Local Nusselt number:

••••sT

g

∞∞∞∞T

x

y

u

Fig. 10.7

(((( )))) 4141

21 9534884443524

3 //

/...

xx RaPrPr

Pr

k

hxNu

++++++++======== (10.21a)

• Average Nusselt number:

========k

LhNuL (((( ))))1/4

1/4

1/24.9534.8842.435

LRaPrPr

Pr

++++++++

(10.21b)

(10.21a) and (10.21b) are valid for:

19

vertical plate


laminar, 94 1010 <<<<<<<< LRa

0 < Pr < ∞∞∞∞properties at fT

(10.21a) and (10.21b) are valid for:

Limitations:

(10.21c)

(2) Vertical plates: Laminar and Turbulent, Uniform Surface

Temperature

(((( ))))[[[[ ]]]]

2

8/279/16

1/6

0.4921

0.3870.825

++++

++++========

/Pr

Ra

k

LhNu L

L (10.22a)

Limitations:vertical plate


laminar, transition, and turbulent121 1010 <<<<<<<<−−−− Ra

(10.22b)

20

121 1010 <<<<<<<<−−−−LRa

0 < Pr < ∞ ∞ ∞ ∞ properties at fT

(10.22b)

(3) Vertical Plates: Laminar Flow, Uniform Heat Flux

• Local Nusselt number:

51

1/2

2

1094

/

*

++++++++======== xx Gr

PrPr

Pr

k

hxNu

Determine surface temperature: Apply Newton’s law:

(10.23)

∞∞∞∞−−−−

′′′′′′′′====

TxT

qxh

s

s

)()( (10.24)

where *xGr is defined as

4

2x

qgGr s

x

′′′′′′′′====

ββββ* (10.25)

21

2x

kGrx

νννν

==== (10.25)

(((( ))))51

4211094

//

))((

′′′′′′′′++++++++====−−−− ∞∞∞∞ x

k

q

gPr

PrPrTxT s

s β

να (10.26a)

(10.23) and (10.26a) are valid for:

(10.24) and (10.25) into (10.23) and solve for ( )sT x T∞−

∞∞∞∞<<<<<<<< Pr0

941010 laminar, <<<<<<<< PrGrx

*

sq ′′′′′′′′ flux, surface uniform

plate vertical

• Properties in (10.26a) depend on surface temperature sT

known. Solution is by iteration

(x) which is not

22

(4) Inclined plates: Constant surface temperature

• Use equations for vertical plates

• Modify Rayleigh number as:

vvvvαααα

θθθθββββ )( ∞∞∞∞−−−−====

TTgRa

sx

cos(10.27) g

(a)

Fig. 10.9

(b)T∞∞∞∞

T Ts<<<< ∞∞∞∞

T Ts >>>> ∞∞∞∞θθθθ

θθθθ

plate inclined

sT etemperatur surface uniform9

10 Laminar, <<<<LRao

600 ≤≤≤≤≤≤≤≤ θθθθ

Limitations:

(10.28)

(5) Horizontal plates: Uniform surface temperature:

(i) Heated upper surface or cooled lower surface

23

41540 /)(. LL RaNu ==== 64 108102 ××××<<<<<<<<×××× LRa,

31150 /)(. LL RaNu ==== 96 1061108 ××××<<<<<<<<×××× .LRa, (10.29b)

plate horizontal

down surface coldor up surfacehot

fTat ,except ,properties all ,,,,ββββgasesfor ,liquidsfor at sf TT ββββ

Limitations:

(10.29c)

(ii) Heated lower surface or cooled upper surface

41270 /)(. LL RaNu ==== 105 1010 <<<<<<<< LRa,

Limitations:

Characteristic length L:

(10.30b)

horizontal plate

hot surface down or cold surface up

all properties, except, β, at Tf

β at Tf for liquids, Ts for gases

24

Characteristic length L:

perimeter

reaa surface====L

(6) Vertical Cylinders. Use vertical plate correlations for:

(((( )))) 41

35

/LGrL

D>>>> for Pr ≥≥≥≥ 1 (10.32)

(7) Horizontal Cylinders:

(((( ))))

(((( ))))[[[[ ]]]]

2

278169

61

.559/01

0.3870.60

++++

++++========//

/

Pr

Ra

k

DhNu D

D (10.33a)

horizontal cylinder

uniform surface temperature or flux125 1010 <<<<<<<<−−−−

DRa

properties at fT

Limitations:

(8) Spheres

25

(8) Spheres (((( ))))

(((( ))))[[[[ ]]]] 94169

41

46901

58902

//

/

.

.

Pr

Ra

k

DhNu D

L

++++

++++======== (10.34a)

sphere

uniform surface temperature or flux1110<<<<DRa

7.0>>>>Prproperties at fT

Limitations:

10.7.2 Free Convection in Enclosures

Examples:

• Double-glazed windows

• Solar collectors

• Building walls

• Concentric cryogenic tubes

• Electronic packages

Fluid Circulation:

26

Fluid Circulation:

• Driving force: Gravity and unequal surface temperatures

Heat flux:

Newton’s law: )( ch TThq −−−−====′′′′′′′′ (10.35)

Heat transfer coefficient h:

Nusselt number correlations depend on:

• Configuration

• Orientation

• Aspect ratio

• Prandtl numberPr

• Rayleigh number δδδδRa

(1) Vertical Rectangular Enclosures

Rayleigh number

δ

cTcT

27

Pr)TT(g

Ra ch2

3

νννν

δδδδββββδδδδ

−−−−==== (10.36)

Several equations:

Fig. 10.10

L g

cTcT

290

20180

.

RaPr.

Pr.

k

hNu

++++======== δδδδδδδδ

δδδδ(10.37a)

enclosure rectagular vertical

at properties 2)( /TTT hc ++++====

21 <<<<<<<<δδδδ

L

53 1010 <<<<<<<<−−−− Pr

31020

>>>>++++

δδδδRaPr.

Pr

Valid for

(10.37b)

28

250280

20220

..L

RaPr.

Pr.

k

hNu

−−−−

++++========

δδδδ

δδδδδδδδδδδδ (10.38a)

102 <<<<<<<<δδδδ

L

510<<<<Pr

103 1010 <<<<<<<< δδδδRa



Valid for

(10.38b)

[[[[ ]]]] 310460

/Ra.

k

hNu δδδδδδδδ

δδδδ======== (10.39a)



401 <<<<<<<<δδδδ

L

201 <<<<<<<< Pr

96 1010 <<<<<<<< δδδδRa

Valid for

30.−−−−δδ

(10.39b)

29

[[[[ ]]]] [[[[ ]]]]30

0120 250

420

.. L

RaPr.k

hNu

.

−−−−

========

δδδδ

δδδδδδδδδδδδ (10.40a)

Valid forenclosure rectagular vertical


41021 ××××<<<<<<<< Pr

74 1010 <<<<<<<< δδδδRe

4010 <<<<<<<<δδδδ

L

(10.40b)

(2) Horizontal Rectangular Enclosures

• Enclosure heated from below

Cellular flow pattern develops at critical Rayleigh number 1708====cRa δδδδ

Nusselt number:

•

•

[[[[ ]]]] [[[[ ]]]] 0740310690

./Ra.

k

hNu Prδδδδδδδδ

δδδδ========

δδδδ

L

g

T

cT

30

k(10.41a)

Valid for


75 107103 ××××<<<<<<<<×××× δδδδRa

enclosurer rectangula horizontal

below from heated

(10.41b)

Fig. 10.11hT

(3) Inclined Rectangular Enclosures

• Applications: Solar collectors

• Nusselt number:correlations depend on:

• Inclination angle

• Aspect ratio

•Prandtl number Pr

• Rayleigh number δδδδRaθθθθ

δδδδ

Lg hT

cT

31

For:

oo900 <<<<<<<< θ : heated lower surface, cooled upper surface

:18090oo <<<<<<<< θ cooled lower surface,

heated upper surface

• Nusselt number is minimum at

Fig. 10.12

angletilt critical

10.1Table

δδδδ/L

cθθθθ

12>>>>12631

o25

o35

o06

o76

o07

a critical angle cθθθθ : Table 10.1

*

−−−−

++++

−−−−

−−−−++++========

118

)cos(

cos

)sin8.1(17081

cos

1708144.11

3/1

6.1*

θθθθ

θθθθ

θθθθ

θθθθ

δδδδ

δδδδ

δδδδδδδδδδδδ

Ra

RaRak

hNu

(10.42a)

Valid for

32


enclosurer rectangula inclined

12≤≤≤≤δδδδ/L

cθθθθθθθθ ≤≤≤≤<<<<0

[[[[ ]]]] negativewhen0set ====∗∗∗∗

Valid for

(10.42b)

c

cNu

NuNu

k

hNu

θθθθθθθθ

δδδδ

δδδδδδδδδδδδ θθθθ

δδδδ/

25.0)(sin)0(

)90()0(

========

o

oo

(10.43a)

Valid for



12≤≤≤≤δδδδ/L

cθθθθθθθθ ≤≤≤≤<<<<0(10.43b)

33

[[[[ ]]]] 25.0sin)90( θθθθδδδδ

δδδδδδδδoNu

k

hNu ======== (10.44a)



δδδδ/L allo09<<<<<<<< θθθθθθθθc

Valid for

(10.44b)

[[[[ ]]]] θθθθδδδδ

δδδδδδδδ sin1)90(1 −−−−++++======== oNuk

hNu (10.45a)

Valid for



δδδδ/L all

oo 01809 <<<<<<<< θθθθ

(10.45b)

(4) Horizontal Concentric Cylinders oD

34

(4) Horizontal Concentric Cylinders

• Flow circulation for oi TT >>>>

• Flow direction is reversed for .oi TT <<<<

• Circulation enhances thermal conductivity

)()/ln(

2oi

io

effTT

DD

kq −−−−====′′′′

ππππ(10.46)

Fig. 10..13

oD

oT

iT

iD

5

Correlation equation for the effective conductivity effk :

4/1*

861.0386.0

++++==== Ra

rP

rP

k

keff(10.47a)

[[[[ ]]]]

[[[[ ]]]]δδδδ

δδδδ

Ra

DD

DDRa

oi

io5

5/35/33

4*

)()(

)/ln(

−−−−−−−− ++++

==== (10.47b)

DD −−−−

35

2

DD io −−−−====δδδδ (10.47c)

721010 <<<<<<<< *Ra


cylinders concentricValid for

(10.47d)

10.8 Other Correlations

The above presentation is highly abridged.

There are many other correlation equations for:

• Boiling

• Condensation

• Jet impingement

• High speed flow

• Dissipation

36

• Dissipation

• Liquid metals

• Enhancements

• Finned geometries

• Irregular geometries

• Non-Newtonian fluids

• Etc.

Consult textbooks, handbooks, reports and journals

chapter 10 correlation equations: forced and free...

Documents