Thermal Development of Internal Flows.ppt

We present research that has been conducted over the last three years.

1. Development of Flow

2. q’’
T
i
T
s(x)
T
i T
s
(x)
q’’
Hot Wall & Cold Fluid
Cold Wall & Hot Fluid
Temperature Profile in Internal Flow
T(x)
T(x)
•The local heat transfer rate is:  xTTAhq
mwallxx 

3. We also often define a Nusselt number as:
 
fluid
mwall
x
fluid
x
D
k
D
xTTA
q
k
Dh
xNu

)(

4. Mean Velocity and Bulk Temperature
Two important parameters in internal forced convection are the
mean flow velocity u and the bulk or mixed mean fluid
temperature T
m
(z).
The mass flow rate is defined as:
while the bulk or mixed mean temperature is defined as:
p
A
cp
m
Cm
TdAuC
xT
c




)(


cA
c
c
m uTdA
Au
xT
1
)(
For Incompressible Flows:

5. Mean Temperature (T
m
)
•We characterise the fluid temperature by using the mean
temperature of the fluid at a given cross-section.
•Heat addition to the fluid leads to increase in mean
temperature and vice versa.
•For the existence of convection heat transfer, the mean
temperature of the fluid should monotonically vary.

6. First Law for A CV : SSSF
T
m,in T
m,exit
dx
q
z
 
inmexitmmeanpz
TTCmq
,,,

No work transfer, change in kinetic and potential energies are negligible
    CV
exit
exit
in
in
CV WgzVhmgzVhmq


22
exit
exit
in
in
CV hmhmq
~~


 
inexitz
hhmq
~~


7. THERMALLY FULLY DEVELOPED FLOW
•There should be heat transfer from wall to fluid or vice
versa.
•Then What does fully developed flow signify in Thermal
view?
 0
,,,

inmexitmmeanpz
TTCmq
 0 xTTAhq
mwallxz

8. FULLY DEVELOPED CONDITIONS (THERMALLY)
(what does this signify?)
Use a dimensionless temperature difference to characterise the
profile, i.e. use
)()(
),()(
xTxT
xrTxT
ms
s


This ratio is independent of x in the fully developed region, i.e.
0
)()(
),()(
,











tfdms
s
xTxT
xrTxT
x

9.  
 
 
 
0
)()(
),()(
),()(
)()( 






x
xTxT
xrTxT
x
xrTxT
xTxT
ms
s
s
ms
 
 
 
 
0
)()(
),()(
),()(
)()( 






x
xTxT
xrTxT
x
xrTxT
xTxT
ms
s
s
ms
    0
)()(
),()(
),()(
)()( 
























x
xT
x
xT
xrTxT
x
xrT
x
xT
xTxT
ms
s
s
ms
     0),()(
)(
)()(
),(
)(,
)(









xrTxT
x
xT
xTxT
x
xrT
xTxrT
x
xT
s
m
msm
s

10. Uniform Wall Heat flux : Fully Developed Region
 

tfd
m
tfd
dx
dT
x
xrT
,,
,



  Temp. profile shape is unchanging.
 )()(constant'' xTxThq
msx 
x
xT
x
xT
ms




 )()(
     0),()(
)(
)()(
),(
)(,
)(









xrTxT
x
xT
xTxT
x
xrT
xTxrT
x
xT
s
m
msm
s
   0),()(
)(
)()(
),(






xrTxT
x
xT
xTxT
x
xrT
ms
m
ms
 0)()(
)(),(












xTxT
x
xT
x
xrT
ms
m

11.  
 
dx
cm
Ph
TT
TTd
pms
ms





Integrating from x=0 (T
m = T
m,i) to x = L (T
m = T
m,o):
 
 
dx
cm
Ph
TT
TTd
L
pms
ms
T
T
om
im





0
,
,

Constant Surface Heat Flux : Heating of Fluid

12. Temperature Profile in Fully Developed Region
 
Uniform Wall Temperature (UWT)
 
)(0 x
dx
dT
s

tfd
m
ms
s
tfd
dx
dT
TT
TT
x
T
,,
)(
)(






 axial temp. gradient is not independent of r and shape of temperature
profile is changing.

13. The shape of the temperature profile is changing, but the
relative shape is unchanged (for UWT conditions).
Both the shape and the relative shape are independent of x for
UWF conditions.
At the tube surface:




























)(
][

but
)(
"
00
"
0
0
xf
TTk
q
r
T
k
y
T
kq
xf
TT
r
T
TT
TT
r
ms
s
rrys
ms
rr
rr
ms
s
)(xf
k
h


14. i.e. the Nusselt number is independent of x in the thermally fully
developed region.
Assuming const. fluid properties:-
tfdxx
xfh
,
)(


This is the real significance of thermally fully developed

15. Evolution of Macro Flow Parameters

16. Thermal Considerations – Internal Flow
T
fluid
 T
surface
a thermal boundary layer develops
The growth of 
th
depends on whether the flow is
laminar or turbulent
Extent of Thermal Entrance Region:
Laminar Flow:
PrRe05.0
,






D
x
tfd
Turbulent Flow:
10
,






D
x
tfd

17. Energy Balance : Heating or Cooling of fluid
•Rate of energy inflow
T
m
T
m
+ dT
m
dx
Q
mp
Tcm
•Rate of energy outflow 
mmp
dTTcm 
Rate of heatflow through wall:
 
msTTdAhQ 


Conservation of energy:
   
mpmmpms TcmdTTcmTTdAhQ 


18.  
mpms dTcmTTdxPh 
 
ms
p
m
TT
cm
Ph
dx
dT




This expression is an extremely useful result, from which axial
Variation of T
m
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

19. Constant Surface Heat flux heating or cooling
•For constant surface heat flux:
 
imomps TTcmLPqQ
,,
''


For entire pipe:
For small control volume:
mps dTcmqdxPh 
''
)(
''
xf
cm
Pq
dx
dT
p
sm





20. Integrating form x = 0
x
cm
Pq
TxT
p
s
imm 

''
,)(
The mean temperature varies linearly with x along the tube.
 
mpms dTcmTTdxPh 
For a small control volume:











dx
dT
Ph
cm
TT
mp
ms

The mean temperature variation depends on variation of h.

21.  
 
dx
cm
Ph
TT
TTd
pms
ms





Integrating from x=0 (T
m = T
m,i) to x = L (T
m = T
m,o):
 
 
dx
cm
Ph
TT
TTd
L
pms
ms
T
T
om
im





0
,
,

Constant Surface Heat Flux : Heating of Fluid

22.  
mpms
dTcmTTdxPh 
 
dx
cm
Ph
TT
dT
pms
m




 
 
dx
cm
Ph
TT
TTd
pms
ms





Integrating from x=0 (T
m = T
m,i) to x = L (T
m = T
m,o):
 
 
dx
cm
Ph
TT
TTd
L
pms
ms
T
T
om
im





0
,
,

For a small control volume:
Constant Surface Heat flux heating or cooling

23. pims
oms
cm
LPh
TT
TT















,
,
ln
p
surface
ims
oms
cm
Ah
TT
TT















,
,
ln













ims
oms
surface
p
TT
TT
A
cm
h
,
,
ln

h : Average Convective heat transfer coefficient.

24. The above result illustrates the exponential behavior of
the bulk fluid for constant wall temperature.
It may also be written as:
to get the local variation in bulk temperature.
It important to relate the wall temperature, the inlet
and exit temperatures, and the heat transfer in one
single expression.









 



p
surfaceavg
ims
oms
cm
Ah
TT
TT

exp
,
,










 



p
avg
ims
ms
cm
xPh
TT
xTT

exp
,

25. Constant Surface Heat flux heating or cooling
m
T
sT
T
x
mT
sT
T
x
is
TT if
is
TT if

26. To get this we write:
      
iopimsomspimomp
TTcmTTTTcmTTcmQ 

,,,,

27. which is the Log Mean Temperature Difference.
The above expression requires knowledge of the exit
temperature, which is only known if the heat transfer rate
is known.
An alternate equation can be derived which eliminates the
outlet temperature.We Know

28. Thermal Resistance:

29. Dimensionless Parameters for Convection
Forced Convection Flow Inside a Circular Tube
All properties at fluid bulk mean
temperature (arithmetic mean of
inlet and outlet temperature).

30. Internal Flow Heat Transfer
•Convection correlations
–Laminar flow
–Turbulent flow
•Other topics
–Non-circular flow channels
–Concentric tube annulus

31. Convection correlations: laminar flow in circular tubes
•1. The fully developed region
from the energy equation,we can obtain the exact
solution.
for constant surface heat fluid
36.4
k
hD
Nu
D
Cq
s

66.3
k
hD
Nu
D
for constant surface temperature
Note: the thermal conductivity k should be evaluated at average T
m

32. Convection correlations: laminar flow in circular tubes
•The entry region : for the constant surface temperature
condition
3/2
PrRe
L
D
04.01
PrRe
L
D
0.0668
3.66




















D
D
D
Nu
thermal entry length

33. Convection correlations: laminar flow in circular tubes
for the combined entry length

14.03/1
/
PrRe
86.1















s
D
D
DL
Nu


  2/)/Pr/(Re
14.03/1

sD DL 
All fluid properties evaluated
at the mean T
 2/
,, omimm
TTT 
CT
s

700,16Pr48.0 
75.9/0044.0 
s

Valid for

35. Thermally developing, hydrodynamically
developed laminar flow (Re < 2300)
Constant wall temperature:
Constant wall heat flux:

36. 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.

37. Convection correlations: turbulent flow in circular tubes
•A lot of empirical correlations are available.
•For smooth tubes and fully developed flow.
heatingFor PrRe023.0
4.05/4
DD
Nu
coolingfor PrRe023.0
3.05/4
DD
Nu
)1(Pr)8/(7.121
Pr)1000)(Re8/(
3/22/1



f
f
Nu
D
d
•For rough tubes, coefficient increases with wall roughness. For fully developed flows

38. 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.

39. 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 = Nu
0

Gases, turbulent flow

40. Noncircular Tubes: Correlations
For noncircular cross-sections, define an effective
diameter, known as the hydraulic diameter:
Use the correlations for circular cross-
sections.

42. Selecting the right correlation
•Calculate Re and check the flow regime (laminar or turbulent)
•Calculate hydrodynamic entrance length (x
fd,h
or L
he
) to see
whether the flow is hydrodynamically fully developed. (fully
developed flow vs. developing)
•Calculate thermal entrance length (x
fd,t or L
te) 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 h
i
or h
o
.
•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.

43. 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.

44. Heat transfer enhancement
•Helically coiled tube
•Without inducing turbulence or additional heat transfer
surface area.
•Secondary flow