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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