chapter 10

1

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 Cylinders Spheres

(2) Internal forced convection through channels

(3) External free convection over:

Cylinders Plates

Spheres

2

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

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.

sq

sT

TV

V

10.1 Fig.

3

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:

(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

4

(2) Identify problem classification:

Forced convection

Free convection

External flow

Internal flow

Entrance region

Fully developed region

Boiling

CondensationEtc.

(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

5

(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

(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

6

Three regions:

(1) Laminar

(2) Transition

(3) Turbulent

txRe = Transition or

critical Reynolds

number:

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

5105

tx

xVRe

t

•

(10.3)

For flow over a flat plate:

Examples of correlation equations for plates:

Laminar region, x < xt :

10.2 Fig.

transitionturbulentlaminar

x tx

V

T

7

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 sT5 105 < xRe < 107

0.6 < Pr < 60properties at fT

(10.4b)

Average h

t

t

x

x

L

tL

Ldxxhdxxh

Ldxxh

Lh

)()()(

00

11(10.5)

8

Lh = local laminar heat transfer coefficient

th = local turbulent heat transfer coefficient

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

31

0 51

54

21

21029603320 /

/

/

/

/.. Pr

x

dxV

x

dxV

L

kh

t

t

x L

x

(10.6)

Integrate

31545421 03706640 //// .. PrReReReL

kh

tt xLx (10.7a)

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 TV

txx

10.3 Fig.

t

insulation0sTox

9

Two cases:

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

local h

tx oxTurbulent flow, < : The local Nusselt number is

91109o

3154

1

02960//

//

)/(

.

xx

PrRe

k

hxNu x

x

(10.8)

(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 //. PrRek

hxNu xx (10.9)

TV

tx

sq x

10.4 Fig.

0

10

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

5485

4132

3121

0002821

41

62030

//

//

//

,/

..

DD

LRe

Pr

PrRe

k

DhNu (10.10a)

Flow normal to cylinder2.0 PrRePe D

properties at fT

Limitations:

(10.10b)

Pe = Peclet number = ReD Pr

TV

10.5 Fig.

11

For Pe < 0.2, use:

Pek

DhuN D ln.. 5082370

1

(10.11a)

flow normal to cylinderPrRePe D= < 0.2

properties at fT

Limitations

(10.11b)

10.5.3 External Flow over a Sphere

41403221060402

/.//..

sPrReRe

k

DhNu DDD

(10.12a)

flow over sphere3.5 < ReD < 7.6 104

0.71 < Pr < 380

2.31 s

properties at T , s at sT

Limitations:

(10.12b)

12

Chapter 7:Analytic solutions to h for

fully developed laminar flow

Correlation equations for h in theentrance and fully developed regions for laminar and turbulent flows

10.6 Internal Forced Convection Correlations

Transition or critical Reynolds number for smooth tubes:

2300 Du

RetD (10.13)

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:

32)(0401

06680663 /

.

)/(..

PrReL/D

PrReLD

k

DhNu

D

DD

(10.14a)

sT

10.6 Fig.insulation

T

FDV 0

u developing

xetemperatur

t

u

14

entrance region of tubesuniform surface temperature sTlaminar flow (ReD < 2300)fully developed velocitydeveloping temperatureproperties at 2/)( momim TTT

Limitations:

(10.14b)

(2) Developing Velocity and Temperature: Laminar flow

140

31861.

/)(.

s

PrReL/Dk

DhNu DD

(10.15a)

15

entrance region of tubeuniform surface temperature sTlaminar flow (ReD < 2300)developing velocity and temperature0.48 < Pr < 16700

0.0044 < s

< 9.75

properties at mT , s at sT

Limitations:

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

kDhNu

DD (10.16a)

Limitations:

16

fully developed turbulent flowsmooth tubesReD > 104

0.7 < Pr < 160L /D > 60properties at mT

(10.16b)

Accuracy: Errors can be as high as 25%

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

322/31/2

11)()12.7(1

1000))(( /)( L/DPr8f

PrRe8fNu DD

(10.17a)

Valid for: developing or fully developed turbulent flow

17

2300 < ReD < 5 106

0.5 < Pr < 20000 < D/L <1properties at mT

Limitations:

(10.17b)

The D/L factor in equation accounts for entrance effects For fully developed flow set D/L = 0

The Darcy friction factor f is defined as

2

2u

L

Dpf (10.18)

For smooth tubes f is approximated by

2641790 ).ln.( DRef (10.19)

18

10.6.3 Non-circular Channels: Turbulent Flow

Use equations for tubes. Set eDD (equivalent diameter)

P

AD f

e4

(10.20)

fA

P

= flow area

= wet perimeter

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

19

4141

21 9534884443524

3 //

/ ...xx Ra

PrPr

Pr

k

hxNu

(10.21a)

Average Nusselt number:

k

LhNuL 1/4

1/4

1/2 4.9534.8842.435LRa

PrPr

Pr

(10.21b)

vertical plateuniform surface temperature sT

laminar, 94 1010 LRa0 < Pr < properties at fT

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

Limitations:

(10.21c)

20

(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 plateuniform surface temperature sTlaminar, transition, and turbulent

121 1010 LRa

0 < Pr < properties at fT

(10.22b)

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

Local Nusselt number:

21

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

42

xk

qgGr s

xν

* (10.25)

51

421 1094//

))((

x

k

q

gPr

PrPrTxT s

s (10.26a)

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

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

22

Pr0

94 1010 laminar, PrGrx*

sq flux, surface uniform

plate vertical

(10.26b)

Properties in (10.26a) depend on surface temperature sTknown. Solution is by iteration

(x) which is not

(4) Inclined plates: Constant surface temperature

Use equations for vertical plates

Modify Rayleigh number as:

v )(

TTg

Ra sx

cos (10.27) g

(a)

Fig. 10.9

(b)T

T Ts

T Ts

23

plate inclined

sT etemperatur surface uniform910 Laminar, LRa

o600

Limitations:

(10.28)

(5) Horizontal plates: Uniform surface temperature:

(i) Heated upper surface or cooled lower surface

41540 /)(. LL RaNu 64 108102 LRa,

(10.29a)

31150 /)(. LL RaNu 96 1061108 .LRa,

(10.29b)

plate horizontaldown surface coldor up surfacehot

fTat ,except ,properties all gasesfor ,liquidsfor at sf TT

Limitations:

(10.29c)

24

(ii) Heated lower surface or cooled upper surface

41270 /)(. LL RaNu 105 1010 LRa, (10.30a)

Limitations:

Characteristic length L:

perimeter

reaa surfaceL (10.31)

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

4135

/LGrL

D for Pr 1 (10.32)

(7) Horizontal Cylinders:

(10.30b)

horizontal platehot surface down or cold surface up all properties, except, β, at Tf β at Tf for liquids, Ts for gases

25

2

278169

61

.559/01

0.3870.60

//

/

Pr

Ra

k

DhNu D

D (10.33a)

horizontal cylinderuniform surface temperature or flux

125 1010 DRa

properties at fT

Limitations:

(10.33b)

(8) Spheres 94169

41

46901

58902

//

/

.

.

Pr

Ra

k

DhNu D

L

(10.34a)

sphereuniform surface temperature or flux

1110DRa

7.0Prproperties at fT

Limitations:

(10.34b)

26

10.7.2 Free Convection in Enclosures

Examples: Double-glazed windows Solar collectors Building walls Concentric cryogenic tubes Electronic packages

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:

27

Configuration Orientation Aspect ratio Prandtl number Pr

Rayleigh number Ra

(1) Vertical Rectangular Enclosures

Rayleigh number

Pr)TT(g

Ra ch2

3

(10.36)

Several equations:

Fig. 10.10

L g

cTcT

28

290

20180

.

RaPr.

Pr.

k

hNu

(10.37a)

enclosure rectagular vertical

at properties 2)( /TTT hc

21 L

53 1010 Pr

31020

Ra

Pr.Pr

Valid for

(10.37b)

250280

20220

.. LRa

Pr.

Pr.

k

hNu

(10.38a)

102 L

510Pr103 1010 Ra



Valid for

(10.38b)

29

310460 /Ra.k

hNu

(10.39a)



401 L

201 Pr96 1010 Ra

Valid for

30

0120 250420.

. LRaPr.

k

hNu .

(10.40a)

Valid forenclosure rectagular vertical


41021 Pr74 1010 Re

4010 L

(10.40b)

(10.39b)

30

(2) Horizontal Rectangular Enclosures

Enclosure heated from below

Cellular flow pattern develops at critical Rayleigh number 1708cRa

Nusselt number:

0740310690 ./Ra.k

hNu Pr

(10.41a)

Valid for


75 107103 Ra

enclosurer rectangula horizontalbelow from heated

(10.41b)

Fig. 10.11

L

g

hT

cT

31

(3) Inclined Rectangular Enclosures

Applications: Solar collectors Nusselt number:correlations depend on: Inclination angle Aspect ratioPrandtl number Pr

Rayleigh number Ra

For:

oo 900 : heated lower surface, cooled upper surface

:18090 oo cooled lower surface,

heated upper surface

Nusselt number is minimum at

Fig. 10.12

Lg hTcT

angletilt critical10.1 Table

/L

c1212631

o25 o35 o06 o76 o07

32

a critical angle c : Table 10.1

*

118

)cos(

cos

)sin8.1(17081

cos

1708144.11

3/1

6.1*

Ra

RaRak

hNu

(10.42a)


enclosurer rectangula inclined12/L

c 0

negativewhen0set

Valid for

(10.42b)

33

c

cNu

NuNu

k

hNu

/25.0)(sin

)0(

)90()0(

o

oo

(10.43a)

Valid for


enclosurer rectangula inclined12/L

c 0(10.43b)

25.0sin)90(

oNuk

hNu (10.44a)


enclosurer rectangula inclined

/L allo09c

Valid for

(10.44b)

34

sin1)90(1 oNu

k

hNu (10.45a)

Valid for


enclosurer rectangula inclined

/L alloo 01809

(10.45b)

(4) Horizontal Concentric Cylinders

Flow circulation for oi TT

Flow direction is reversed for .oi TT

Circulation enhances thermal conductivity

)()/ln(

2oi

io

eff TTDD

kq

(10.46)

Fig. 10..13

oDoT

iT

iD

5

35

Correlation equation for the effective conductivity effk :

4/1*

861.0386.0

RarP

rP

k

keff (10.47a)

Ra

DD

DDRa

oi

io5

5/35/33

4*

)()(

)/ln(

(10.47b)

2

DD io

(10.47c)

72 1010 *Ra


cylinders concentricValid for

(10.47d)

36

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 Liquid metals Enhancements Finned geometries Irregular geometries Non-Newtonian fluids Etc.

Consult textbooks, handbooks, reports and journals

chapter 10

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