Convection in Flat Plate Turbulent Boundary Layers

P M V Subbarao

Associate Professor

Mechanical Engineering Department

IIT Delhi

An Extra Effect For Same Dose ……

Transition to Turbulence

• When the boundary layer changes from a laminar flow to a turbulent flow it is referred to as transition.

• At a certain distance away from the leading edge, the flow begins to swirl and various layers of flow mix violently with each other.

• This violent mixing of the various layers, it signals that a transition from the smooth laminar flow near the edge to the turbulent flow away from the edge has occurred.

Turbulent Flow Regime

• For a flat place boundary layer becomes turbulent at Rex ~ 5 X 105.

• The local friction coefficient is well correlated by an expression of the form

7x

51

, 10Re Re059.0

xxfC

Local Nusselt number: 60 0.6 Re029.0 3/154

prprNu xx

Mixed Boundary Layer

• In a flow past a long flat plate initially, the boundary layer will be laminar and then it will become turbulent.

• The distance at which this transitions starts is called critical distance (Xc) measured from edge and corresponding Reynolds number is called as Critical Reynolds number.

• If the length of the plate (L) is such that 0.95 Xc/L 1, the entire flow is approximated as laminar.

• When the transition occurs sufficiently upstream of the trailing edge, Xc/L 0.95, the surface average coefficients will be influenced by both laminar and turbulent boundary layers.

Flat Plate Boundary Layer Trasition

Important point:

–Typically a turbulent boundary layer is preceded by a laminar boundary layer first upstream

need to consider case with mixed boundary layer conditions!

L

xcturb

xc

lamx dxhdxhL

h 1

0

Xc

L

LeadingEdge Trailing

Edge

L

x

xturb

x

xlamLavg

c

c

dxhdxhL

h ,

0

,,

1

31

51

54

0 21

21

, 0296.0332.0 prdxx

dxu

x

dxu

L

kh

L

x

x

Lavg

c

c

On integration:

31

54

54

21

, ReRe037.0Re664.0 prNucxLcxLavg

31

54

, Re037.0 prANuLLavg

For a smooth flat plate: Rexc = 5 X 105

31

54

, 871Re037.0 prNuLLavg

For very large flat plates: L >> Xc, in general for ReL > 108

31

54

, Re037.0 prNuLLavg

Convection for Cylinder in Cross Flow

P M V Subbarao

Associate Professor

Mechanical Engineering Department

IIT Delhi

A Concept That Changed the Lifestyle of the World….

Industrial Applications

• Many thermal industries used the equipment, where convection from cylinders in cross flow is present.

• Super heaters, economizers in Power plant steam generators.

• Condensers and feed water heaters in a power plant.• Radiators in automobiles.• Condenser and evaporator in a refrigerator or air

conditioner.• Two most common equipment are:• Shell and tube heat exchanger.• Fin and tube heat exchanger.

Power Plant Steam Generator : Establish Heat Transfer from Hot Flue Gas to High Pressure Water

Basic Geometry of A Furnace

1500 – 17000C

1000 – 12000C

700 – 8000C

300 – 4000C

Burner

Flame

Hot Exhaust gases

Furnace Exit

Heat Radiation & Convection

1500 – 17000C

Structure of Furnace Wall

Furnace Wall

Convective Superheater (Pendant)

• Convective super heaters are vertical type (Pendant ) or horizontal types.

• The Pendant SH is always arranged in the horizontal crossover duct.

• Pendant SH tubes are widely spaced due to high temperature and ash is soft.

• Transverse pitch : S1/d > 4.5

• Longitudinal pitch : S2/d > 3.5.

• The outside tube diameter : 32 – 51mm

• Tube thickness : 3 – 7mm

S1

S2

Convective Superheater (Horizontal)• The horizontal SH are located in the back pass.• The tubes are arranged in the in-line configuration.• The outer diameter of the tube is 32 – 51 mm.• The tube thickness of the tube is 3 – 7 mm.• The transverse pitch : S1/d = 2 – 3.• The longitudinal pitch :S2/d = 1.6 – 2.5.• The tubes are arranged in multiple parallel sets.• The desired velocity depends on the type of SH and operating steam

pressures.• The outside tube diameter : 32 – 51mm• Tube thickness : 3 – 7mm

S1

S2

Fin - tube Heat Exchanger

Anatomy of Fin & Tube Heat Exchanger

Gas Flow

Tube

Liquid Flow

Plate

HVAC Fin and Tube Heat Exchangers

• Fin and tube heat exchangers are used widely in residential, commercial and industrial HVAC applications.

• HXs of different fin spacing are found in HVAC applications.

Cylinder in Cross Flow

Generally the overall average Nusselt number for heat transfer with the entire object is important.

As with a flat plate, correlations developed from experimental data to compute Nu as a f(Rem,Prn)

Overall Average Nusselt number

6

4131

10Re1500Pr7.0

:for Valid

Pr

Pr Pr Re

D

s

mDD C

k

DhNu

•All properties are evaluated at the freestream temperature, except Prs

which is evaluated at the surface temperature.

Values for C and m

Expect an accuracy within 20% with these correlations

ReD C m

1 -40 0.75 0.4

40 - 1000 0.51 0.5

1000 – 2X 105 0.26 0.6

2X 105 - 106 0.076 0.7

Cylinder in Cross Flow

3/1PrRemDD C

k

DhNu

The empirical correlation due to Hilpert

ReD C m

0.4 -4 0.989 0.330

4 - 40 0.911 0.385

40 -- 4000 0.683 0.466

4000 -- 40000 0.193 0.618

40000 -- 400000 0.027 0.805

Square Cylinder in Cross Flow

D

3/1588.0 PrRe246.0 DDk

DhNu

Valid for 5 X 103 < ReD < 105

D3/1675.0 PrRe102.0 DD

k

DhNu

Valid for 5 X 103 < ReD < 105

Hexagonal Cylinder in Cross Flow

D

3/1638.0 PrRe160.0 DDk

DhNu

Valid for 5 X 103 < ReD <1.95X104

3/1782.0 PrRe0385.0 DDk

DhNu

Valid for 1.95X104 < ReD < 105

D3/1638.0 PrRe153.0 DD

k

DhNu

Valid for 5 X 103 < ReD < 105

Vertical Plate in Cross Flow

D

3/1731.0 PrRe228.0 DDk

DhNu

Valid for 4 X 103 < ReD < 1.5 X104

Convection heat transfer with a sphere

External flow and heat transfer relations are similar to those around a cylinder.Numerous correlations proposed from lab experiments, one being:

2.30.1

106.7Re5.3380Pr71.0

:for Valid

Pr)Re 0.06 Re (0.4 2

4

414.03221

s

D

sDDD k

DhNu

All properties except s are evaluated at T∞.

Special case: Free falling liquid drops

3121 Pr Re 0.6 2 DDNu

Convection heat transfer with banks of tubes

TS

• Typically, one fluid moves over the tubes, while a second fluid at a different temperature passes through the tubes. (cross flow)

• The tube rows of a bank are staggered or aligned. The configuration is characterized by the tube diameter D, the transverse pitch ST and longitudinal pitch SL.

•For tube bundles composed of 10 or more rows

3/11 PrRe13.1 max,

mD DCNu

10

0.7r

104Re2000

:for valid

4max,

L

D

N

P

All properties are evaluated at the film temperature.

•For Reynolds number

DV

Dmax

max,Re

VDS

SV

T

T

max V

DS

SV

D

T

)(2max

If staggered and 2

DSS T

D

or

If number of tubes are less than 10, a correction factor is applied as:

)10(2

)10(

LL ND

ND NuCNu

And values for C2 are from table

•More recent results have been obtained by Zhukauskas.

4/1

36.0

Pr

PrPrRe max,

s

mD DCNu

20

500r7.0

102Re1000

:for valid

6max,

L

D

N

P

All properties except Prs are evaluated at the arithmetic mean of the fluid inlet and outlet temperatures.Values for C and m.

)20(2

)20(

LL ND

ND NuCNu

Array of Cylinders in Cross Flow : A Shell

• The equivalent diameter is calculated as four times the net flow area as layout on the tube bank (for any pitch layout) divided by the wetted perimeter.

For square pitch:

For triangular pitch:

Number of tube centre lines in a Shell:

Ds is the inner diameter of the shell.

Flow area associated with each tube bundle between baffles is:

where A s is the bundle cross flow area, Ds is the inner diameter of the shell, C is the clearance between adjacent tubes, and B is the baffle spacing.

the tube clearance C is expressed as:

Then the shell-side mass velocity is found with

s

shellshell A

mG

Shell side Reynolds Number:

Shell-Side Heat Transfer Coefficient