Convection Part1

External Flow

IntroductionRecall: Convention is the heat transfer mode between a fluid and a solid or a 2 fluids of different phases

In order to simplify the process we used Newton’s correlation

Where h is the convective heat transfer coefficient also called the film coefficient.

h is a function of: Fluid flowFluid propertiesGeometry of the solid

ThA

q

There are four means to evaluate the heat transfer coefficient

1) Dimensional analysis2) Exact analysis of boundary layer3) Approximate integral analysis of the boundary

layer4) Analogy between energy and momentum transfer

Significant Parameters:

Nusselt Number Nu

v TTs

TTs x

y

The heat transfer rate between the surface and the fluid is

At the surface itself

Where k is the thermal conductivity of the fluid. Therefore:

TThAq sy

0

y

sy TTy

kAq

TThATTy

kA s

y

s

0

TT

TTy

k

h

s

y

s

0

LTT

TTy

k

hLNu

s

y

s

0

Momentum Diffusivity

Thermal Diffusivity

The ratio of the momentum diffusivity over the thermal diffusivity is a combination of fluid properties and is also thougth of as a property (Named Prandtl Number Pr).

Dependent on fluid and temperature

pC

k

k

C pPr

Prandtl Number Pr

Dimensional Analysis of Convective Heat Transfer

Forced Convection: movement dictated by v

Variable Symbol Dimensions

Tube Diameter D L

Fluid density ρ M L-3

Fluid viscosity μ M L-1 t-1

Fluid heat capacity Cp Q M –1 T –1

Fluid thermal conductivity k Q t –1 L –1 T –1

Velocity v L t –1

Heat transfer coefficient h Q t –1 L –2 T –1

Using the Buckingham method we group the variables in dimensionless number:

This dimensional analysis for a forced convection in a circular conduit indicates the possibility of correlating the variables as

Similarly we could have developed the Stanton number instead of the Nusselt

Dv

vkD dcba Re1

k

CCpvkD phgfe

Pr2

k

hLNuhvkD lkji 3

PrRe,1fNu

vCp

hSt

Free Convection: movement dictated by buoyancy

Given the coefficient of thermal expansion β: T 10

TggFbuoyant 00

Variable Symbol Dimensions

Significant length D L

Fluid density ρ M L-3

Fluid viscosity μ M L-1 t-1

Fluid heat capacity Cp Q M –1 T –1

Fluid thermal conductivity k Q t –1 L –1 T –1

Fluid Coef. Therm. Exp. β T –1

Gravitational acceleration G L t –2

Temperature difference ΔT T

Heat transfer coefficient h Q t –1 L –2 T –1

Using the Buckingham method we group the variables in dimensionless number:

Define the Grashof number as

This dimensional analysis for a forced convection in a circular conduit indicates the possibility of correlating the variables as

k

CpCpgkL edcba Pr1

Pr,2 GrfNu

TTgkL onmlk 3

k

hLNuhgkL tsrqp 4

2

23

2 gL

gkL jihgf

Gr32

Nu vs Re

0

25

50

75

100

100 1000 10000

Re

Nu

Pr = 2

Pr = 1

Pr = 0.5

Nu vs f ( Re,Pr)

0

25

50

75

100

10 30 50 70

Re0.5 Pr 0.33

Nu

Pr = 2

Pr = 1

Pr = 0.5

Group Symbol Definition Interpretation

Grashof Number Gr Ratio buoyancy to viscous forces

Colburn Factor jH Dimensionless heat transfer coefficient

Nusselt Number Nu Dimensionless surface temperature gradient

Prandtl Number Pr Ratio momentum to thermal diffusivity

Reynolds Re Ratio inertia to viscous forces

Stanton Number St Modified Nusselt number

Peclet Number Pe RePr Independent heat transfer parameter

Selected Dimensionless Groups

k

C p

k

hL

Dv

32

PrSt

2

23

TgL

vCp

h

Flat Plate in Parallel Flow

Laminar FlowTurbulent Flow

Tra

nsit

ion

Reg

ion

δ(x)

xL

Lv

L Re

xv

x Re

Properties of fluid evaluated at the film temperature Tf

2s

f

TTT

Forced ConvectionFlat Plate in Parallel FlowLaminar flow: Re<2 x 105

Prandtl number >0.6The local Nusselt number is

The average Nusselt number

All Prandtl number and Pe >100The local Nusselt number is

The average Nusselt number

xL

31

21

PrRe664.0 xL

k

LhNu

31

21

PrRe332.0 xx

k

xhNu

41

32

31

21

Pr/0468.01

PrRe3387.0

xNu

41

32

31

21

Pr/0468.01

PrRe6774.0

xNu

Forced ConvectionFlat Plate in Parallel FlowTransition flow: Rec=5 x 105

60>Prandtl number >0.63 x 106 >Re > 2 x 105

The average Nusselt number

31

54

Pr871Re037.0 LL

k

LhNu

L

Forced ConvectionFlat Plate in Parallel FlowTurbulent flow: Re>3x106

60>Prandtl number >0.6107 >Re >3 x 106

The average Nusselt number

The local Nusselt number

31

54

PrRe037.0 LNu

L

31

54

PrRe0296.0 xNu

Cylinder in a Cross Flow

5102Re D Separation

v D

TransitionLaminar

Turbulent

D

Separation5102Re D

v

Properties of fluid evaluated at the film temperature Tf

2s

f

TTT

Dv

D Re

Forced ConvectionCylinder in a Cross Flow

The average Nusselt number

If ReDPr>0.2

54

85

32

31

21

282000

Re1

Pr4.01

PrRe62.03.0

41

DDDNu

ReD C m

0.4-4 0.989 0.330

4-40 0.911 0.385

40-4000 0.683 0.466

4000-40,000 0.193 0.618

40,000-400,000 0.027 0.805

31

PrRemDD CNu

Forced ConvectionVarious Object in a Cross Flow

The average Nusselt number 31

PrRemDD CNu

Geometry ReD C mSquare 5x103-105 0.246 0.588Square 5x103-105 0.102 0.675

Hexagon 5x103-1.95x104

1.95x104 -105

0.160

0.0385

0.638

0.782Hexagon 5x103-105 0.153 0.638

Vertical Plate 4x103-1.5x104 0.228 0.731

D

D

D

D

D

Sphere in a Cross Flow

All properties of fluid evaluated at temperature , except μs at TsT

Dv

D Re

41

4.032

21

PrRe06.0Re4.02

sDDDNu

Restrictions 0.71 < Pr < 3803.5 < ReD < 7.6x104

Bank of Tubes in a Cross Flow

V

Fluid in cross flow over tube bank

Aligned Bank of Tubes in a Cross Flow

ST

DA1

SL

Tv,

Properties of fluid evaluated at the film temperature Tf

max

max,ReDv

D vDS

Sv

T

T

max

Staggered Bank of Tubes in a Cross Flow

ST

A1

Tv,

Properties of fluid evaluated at the film temperature Tf

If

else

max

max,ReDv

D

vDS

Sv

T

T

max

SL

D

DSDS TD 2

vDS

Sv

D

T

2max

Number of row (NL) greater or equal to 102000 < ReD,max < 40000Pr > 0.7

C1 in table 7.5

If number of row is smaller than 10

C2 in table 7.6

31

max,1 PrRe13.1 mDD CNu

10210

LL NDND NuCNu

Number of row (NL) greater or equal to 201000 < ReD,max < 2x106

500 > Pr > 0.7

C in table 7.7

If number of row is smaller than 10

C2 in table 7.8

41

Pr

PrPrRe 36.0

max,

s

mDD CNu

20220

LL NDND NuCNu

All properties of fluid evaluated at the average temperatureexcept Prs at Ts

2outin TT

In this case the temperature difference in the convective heattransfer equation is defined as the log-mean temperature difference ΔTlm

Where Ti is the temperature of the fluid entering the bankTo is the temperature of the fluid leaving the bank

And the outlet temperature can be estimated using

Where N is the total number of tube and NT the transverse number of tube. Finally the heat transfer rate per unit length is

os

is

osislm

TTTT

TTTTT

ln

CpSvN

hDN

TT

TT

TTis

os

exp

lmTDhNq '

Packed Bed

v

Properties of fluid evaluated at the the average temperature

ε is the porosity or void fraction of the bed (0.3 to 0.5)

Valid for

gas flow

Dv

D Re

2outin TT

575.0Re06.2 DHj

4000Re90

7.0Pr

D

Ap,T is the total area of the particles and Ab,c is the bed cross sectional area

CpvA

hA

TT

TT

bc

Tp

is

os

,

,exp

lmTp TAhq ,