Design of Systems with INTERNAL CONVECTION

P M V Subbarao

Associate Professor

Mechanical Engineering Department

IIT Delhi

An Essential Part of Exchanging Heat……..

Evolution of Macro Flow Parameters

Energy Balance : Heating or Cooling of fluid

• Rate of energy inflow

Tm Tm + dTm

dx

qmpTcm

• Rate of energy outflow mmp dTTcm

Rate of heatflow through wall:

dAqTTdAhq sms''

Conservation of energy:

mpmmpms TcmdTTcmTTdAhq

mpms dTcmTTdxPh

msp

m TTcm

Ph

dx

dT

This expression is an extremely useful result, from which axialVariation of Tm may be determined.The solution to above equation depends on the surface thermal

condition.

Two special cases of interest are:

1. Constant surface heat flux.2. Constant surface temperature

Constant Surface Heat flux heating or cooling

• For constant surface heat flux:

imomps TTcmLPqq ,,''

For entire pipe:

For small control volume:

mps dTcmqdxP ''

)(''

xfcm

Pq

dx

dT

p

sm

Integrating form x = 0 xcm

PqTxT

p

simm

''

,)(

The mean temperature varies linearly with x along the tube.

mpms dTcmTTdxPh

For a small control volume:

dx

dT

Ph

cmTT mp

ms

The surface temperature variation depends on variation of h.

x

xT

x

xT

x

xTxT

x

h msmsx

)()(

0)()(

0

For a thermally developed flow with constant wall flux:

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

Determination of Heat Transfer Coefficient

Ti Ts(x)q’’

Cold Wall & Hot Fluid

T(r,x)

mpmsx dTcmTTdxPh

ms

mpx TTdxP

dTcmh

Computation of Temperature Distribution

For axi-symmetric flow & Heat Transfer :

r

Tcv

x

Tcu

x

Tk

r

Tr

rr

kpp

2

2

x

Tcu

x

T

r

Tr

rrk p

2

21

For high Prandtl number fluids, the flow can be approximated as hydrodynamically developed and thermally developing flow.

x

T

k

cu

x

T

r

Tr

rrp

2

21

Thermally Developed Flow: Constant Heat Flux

0)()(

),()(

,

tfdms

s

xTxT

xrTxT

x

x

Tu

x

T

r

Tr

rr

2

21

dx

xdT

x

xT

x

xTxT

x

h msmsx )()(0

)()(0

tfd

mtfd dx

dT

x

xrT,,

,

k

q

x

xrT

x

T

dx

dT

x

xrTtfd

otfd

stfd

mtfd

'',

,,,,,

dx

dTu

r

Tr

rr

1

Similarly for constant wall temperature:

dx

dT

TT

TTu

r

Tr

rr mw

w

1

Solution : Constant Heat Flux : Fully Developed

dx

dTu

r

Tr

rr

1

Boundary conditions:

ow rratTT

0at 0

rr

T

For hydrodynamically developed flow:

2

2

12o

m r

ruu

dx

dT

r

ru

r

Tr

rr o

m

2

2

121

Integration of above equation with substitution of boundary conditions:

41616

32 2

2

42 r

r

rr

dx

dTuTT

o

ommw

or

mom uTrdr

urT

02

2

Substitute T &u and integrate

22

96

11o

mmwm r

dx

dTuTT

2

48

11'' o

mmmw r

dx

dTuhTThq

Tm Tm + dTm

dx

q

mpmmpo TcmdTTcmxrq 2''

xdx

dTcmTcmx

dx

dTTcmxrq m

pmpm

mpo

2''

xdx

dTcmTcmx

dx

dTTcmxrq m

pmpm

mpo

2''

xdx

dTcurxrq m

pmoo 22''

dx

dTcurq mpmo

2''

2

48

11'' o

mm rdx

dTuhq

op rh

c

48

11

2 p

op ck

rh

c

48

11

2

k

Dh

k

rh oo

48

11

2

1

48

11

2

1

148

11

k

Dh o 364.4

11

48

k

hDo

mpms dTcmTTdxPh

dxcm

Ph

TT

dT

pms

m

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

,

,

For a small control volume:

Constant Surface Temperature heating or cooling

pims

oms

cm

LPh

TT

TT

,

,ln

p

surface

ims

oms

cm

Ah

TT

TT

,

,ln

ims

oms

surface

p

TT

TT

A

cmh

,

,ln

h : Average Convective heat transfer coefficient.

The above result illustrates the exponential behavior of the bulk fluid for constant wall temperature.

It may also be written as:

p

surfaceavg

ims

oms

cm

Ah

TT

TT

exp

,

,

p

avg

ims

ms

cm

xPh

TT

xTT

exp,

p

avg

ims

ms

cm

xPh

TT

xTT

,

ln

Constant Surface Temperature heating or cooling

mT

sT

T

x

mT

sT

T

x

is TT if is TT if

p

avg

ims

oms

cm

LPh

TT

TT

,

,ln

To get this we write:

iopimsomspimomp TTcmTTTTcmTTcmq

,,,,

iop TT

qcm

to get the local variation in bulk temperature.

For practical use, it important to relate the wall temperature, the inlet and exit temperatures, and the rate of heat transfer one single expression.

Constant Surface Temperature heating or cooling

mT

sT

T

x

mT

sT

T

x

is TT if is TT if

iTiT

oT

oT

p

surfaceavg

i

o

cm

Ah

T

T

ln

iop TT

qcm

iosurfaceavg

i

o TTq

Ah

T

T

ln

i

o

iosurfaceavg

TT

TTAhq

ln

Define Log Mean Temperature Difference :

i

o

ioLMTD

TT

TTT

ln

LMTDsurfaceavg TAhq

LMTDsurfaceavgconvection TAhq

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 Know

LMTDsurfaceavgconvection TAhq

p

surfaceavg

ims

oms

cm

Ah

TT

TT

exp

,

,

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,

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

Convection correlations: laminar flow in circular tubes

• The entry region : for the constant surface temperature condition

3/2

PrReL

D04.01

PrReL

D0.0668

3.66

D

D

DNu

thermal entry length

Convection correlations: laminar flow in circular tubes

for the combined entry length

14.03/1

/

PrRe86.1

s

DD DL

Nu

2/)/Pr/(Re 14.03/1 sD DL

All fluid properties evaluated at the mean T

2/,, omimm TTT

CTs

700,16Pr48.0

75.9/0044.0 s

Valid for

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 or if Re Pr D/L < 33 also for Pr > 7.

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

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.

Effects of property variation with temperature

Liquids, laminar and turbulent flow:

Subscript w: at wall temperature, without subscript: at mean fluid temperature

Gases, laminar flow Nu = Nu0

Gases, turbulent flow

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 coefficient

Introduce surface roughness to enhance turbulence.

Induce swirl.

• Increase the convection surface area

Longitudinal fins, spiral fins or ribs.

Heat transfer enhancement

• Helically coiled tube

• Without inducing turbulence or additional heat transfer surface area.

• Secondary flow