Correlations for INTERNAL CONVECTION

P M V Subbarao

Associate Professor

Mechanical Engineering Department

IIT Delhi

An Essential Part of Exchanging Heat……..

dx

cm

Ph

TT

TTd

pms

ms

Integrating from x=0 (Tm = T m,i) to x = L (Tm = Tm,o):

dx

cm

Ph

TT

TTd L

pms

ms

T

T

om

im

0

,

,

Constant Surface Heat Flux : Heating of Fluid

Constant Surface Temperature heating or cooling

mT

sT

T

x

mT

sT

T

x

is TT if is TT if

Constant Surface Temperature heating or cooling

mT

sT

T

x

mT

sT

T

x

is TT if is TT if

iTiT

oT

oT

i

o

iosurfaceavg

TT

TTAhq

ln

Define Log Mean Temperature Difference :

i

o

ioLMTD

TT

TTT

ln

LMTDsurfaceavg TAhq

LMTDsurfaceavgconvection TAhq

Constant Surface Temperature heating or cooling

mT

sT

T

x

mT

sT

T

x

is TT if is TT if

iTiT

oT

oT

The above expression requires knowledge of the exit temperature, which is only known if the heat transfer rate is known and vice versa.

An alternate equation can be derived which eliminates the outlet temperature.

We already Know

LMTDsurfaceavgconvection TAhq

p

surfaceavg

ims

oms

cm

Ah

TT

TT

exp

,

,

imompconvection TTCmq ,,

p

surfaceavgimsimimoms cm

AhTTTTTT

exp,,,,

p

surfaceavgimsimomims cm

AhTTTTTT

exp,,,,

p

surfaceavgims

p

convims cm

AhTT

cm

qTT

exp,,

p

surfaceavgims

p

conv

cm

AhTT

cm

q

exp1,

p

surfaceavgimspconv cm

AhTTcmq

exp1,

System Thermal Resistance:

conv

imsth q

TTR ,

p

surfaceavgp

th

cm

Ahcm

R

exp1

1

dx

dT

TT

TTu

r

Tr

rrm

mw

w

1

Constant wall temperature : Thermally Developed Flow

Boundary conditions:

ow rratTT constant

0at 0

rr

T

For hydrodynamically developed flow:

2

2

12o

m r

ruu

dx

dT

TT

TT

r

ru

r

Tr

rrm

mw

w

o

m

2

2

121

This problem has been solved by Bhatti (1985):

0

2

2i

n

on

mw

w

r

rC

TT

TT

Where, 8284.14

1 ,1 2

020 CC

704364.2 4

022422

20

2 nnn CC

nC

Convection correlations: laminar flow in circular tubes

• 1. The fully developed region for constant surface heat flux

36.4k

hDNuD Cqs

66.3k

hDNuD

for constant surface temperature

Note: the thermal conductivity k should be evaluated at average Tm

Thermally developing, hydrodynamically developed laminar flow (Re < 2300)

Constant wall temperature:

Constant wall heat flux:

Simultaneously developing laminar flow (Re < 2300)

Constant wall temperature:

Constant wall heat flux:

which is valid over the range 0.7 < Pr < 7

Fully developed turbulent and transition flow (Re > 2300)

Constant wall Temperature:

Where

Constant wall temperature: For fluids with Pr > 0.7 correlation for constant wall heat flux can be used with negligible error.

Convection correlations: turbulent flow in circular tubes

• A lot of empirical correlations are available.

• For smooth tubes and fully developed flow.

heatingFor PrRe023.0 4.05/4DDNu

coolingfor PrRe023.0 3.05/4DDNu

)1(Pr)8/(7.121

Pr)1000)(Re8/(3/22/1

f

fNu D

d

•For rough tubes, coefficient increases with wall roughness. For fully developed flows

f is friction factor.

Heat Transfer in Entry Length

A general expression for the ratio of the local heat transferCoefficient to the fully developed value is

693.0ln113.0

1

Dxh

hx

Variable properties

• Wall temperature Ts or Tw

• Fluid temperature Tb (mean bulk temp)

• For small changes Ti or To may also be used

• For example there may be a large radial temperature gradient in circular duct. At what temperature properties are evaluated matters.

• There may be a need for temperature correction in correlations.

• Indices cp and vp correspond to constant and variable properties.

• Some properties are strong functions of temperature. Convention:– For liquids lump all property variations to (dynamic

viscosity). Sometimes variations are lumped to Pr.

– For gases use temperature dependence directly (everything depends on T)

• Fluids:

• Gases:

where n and m depends on the case.

,

n m

b b

cp w cp w

Nu f

Nu f

, n m

w w

cp b cp b

T TNu f

Nu T f T

Turbulent Liquid Flow in Ducts

Petukhov reviewed the status of heat transfer in fully developed turbulent pipe flow with both constant and variable physical properties.

liquids. of heatingfor 11.0

b

w

CP

VP

Nu

Nu

liquids. of coolingfor 25.0

b

w

CP

VP

Nu

Nu

Validity range: 104 < Reb < 5 x 106 , 2 < Prb < 140 and 0.08 < w/b) < 40

Turbulent Gas Flow in Ducts

Petukhov & popov reviewed the status of heat transfer in fully developed turbulent pipe flow with both constant and variable physical properties.

gases. of heatingfor 36.0log3.0

b

w

T

T

b

w

CP

VP

T

T

Nu

Nu

gases. of coolingfor 36.0

b

w

CP

VP

T

T

Nu

Nu

Validity range: 104 < Reb < 4.3 x 106 and 0.37 < w/b) < 3.1 for air

Validity range: 104 < Reb < 5.8 x 106 and 0.37 < w/b) < 3.7 for hydrogen.

Noncircular Tubes: Correlations

For noncircular cross-sections, define an effective diameter, known as the hydraulic diameter:

Use the correlations for circular cross-sections.

Selecting the right correlation

• Calculate Re and check the flow regime (laminar or turbulent)• Calculate hydrodynamic entrance length (xfd,h or Lhe) to see

whether the flow is hydrodynamically fully developed. (fully developed flow vs. developing)

• Calculate thermal entrance length (xfd,t or Lte) to determine whether the flow is thermally fully developed.

• We need to find average heat transfer coefficient to use in U calculation in place of hi or ho.

• Average Nusselt number can be obtained from an appropriate correlation.

• Nu = f(Re, Pr)• We need to determine some properties and plug them into the

correlation. • These properties are generally either evaluated at mean (bulk)

fluid temperature or at wall temperature. • Each correlation should also specify this.

Heat transfer enhancement

• Enhancement• Increase the convection coefficientIntroduce surface roughness to enhance turbulence. Induce swirl.• Increase the convection surface areaLongitudinal fins, spiral fins or ribs.

Heat Transfer Enhancement using Inserts

Heat Transfer Enhancement using Inserts

Heat transfer enhancement :Coiling

• Helically coiled tube

• Without inducing turbulence or additional heat transfer surface area.

• Secondary flow