Thermal Development of Internal Flows.ppt

q’’
Ti
Ts(x)
Ti Ts(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:  
 
x
T
T
A
h
q m
wall
x
x 


We also often define a Nusselt number as:
 
 
fluid
m
wall
x
fluid
x
D
k
D
x
T
T
A
q
k
D
h
x
Nu



)
(

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 Tm(z).
The mass flow rate is defined as:
while the bulk or mixed mean temperature is defined as:
p
A
c
p
m
C
m
TdA
uC
x
T c




)
(


c
A
c
c
m uTdA
A
u
x
T
1
)
(
For Incompressible Flows:

Mean Temperature (Tm
)
• 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.

First Law for A CV : SSSF
Tm,in Tm,exit
dx
qz
 
in
m
exit
m
mean
p
z T
T
C
m
q ,
,
, 
 
No work transfer, change in kinetic and potential energies are negligible
    CV
exit
exit
in
in
CV W
gz
V
h
m
gz
V
h
m
q










 2
2
exit
exit
in
in
CV h
m
h
m
q
~
~






 
in
exit
z h
h
m
q
~
~

 

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
,
,
, 

 in
m
exit
m
mean
p
z T
T
C
m
q 
 
  0


 x
T
T
A
h
q m
wall
x
z

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

        0
)
(
)
(
)
,
(
)
(
)
,
(
)
(
)
(
)
( 









x
x
T
x
T
x
r
T
x
T
x
x
r
T
x
T
x
T
x
T m
s
s
s
m
s
        0
)
(
)
(
)
,
(
)
(
)
,
(
)
(
)
(
)
( 









x
x
T
x
T
x
r
T
x
T
x
x
r
T
x
T
x
T
x
T m
s
s
s
m
s
    0
)
(
)
(
)
,
(
)
(
)
,
(
)
(
)
(
)
( 

























x
x
T
x
x
T
x
r
T
x
T
x
x
r
T
x
x
T
x
T
x
T m
s
s
s
m
s
 
      0
)
,
(
)
(
)
(
)
(
)
(
)
,
(
)
(
,
)
(












x
r
T
x
T
x
x
T
x
T
x
T
x
x
r
T
x
T
x
r
T
x
x
T
s
m
m
s
m
s

Uniform Wall Heat flux : Fully Developed Region
 
t
fd
m
t
fd
dx
dT
x
x
r
T
,
,
,



Temp. profile shape is unchanging.
 
)
(
)
(
constant
'
' x
T
x
T
h
q m
s
x 


x
x
T
x
x
T m
s




 )
(
)
(
 
      0
)
,
(
)
(
)
(
)
(
)
(
)
,
(
)
(
,
)
(












x
r
T
x
T
x
x
T
x
T
x
T
x
x
r
T
x
T
x
r
T
x
x
T
s
m
m
s
m
s
    0
)
,
(
)
(
)
(
)
(
)
(
)
,
(








x
r
T
x
T
x
x
T
x
T
x
T
x
x
r
T
m
s
m
m
s
  0
)
(
)
(
)
(
)
,
(













x
T
x
T
x
x
T
x
x
r
T
m
s
m

 
 
dx
c
m
P
h
T
T
T
T
d
p
m
s
m
s






Integrating from x=0 (Tm = T m,i) to x = L (Tm = Tm,o):
 
 
dx
c
m
P
h
T
T
T
T
d
L
p
m
s
m
s
T
T
o
m
i
m







0
,
,

Constant Surface Heat Flux : Heating of Fluid

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

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
)
(
"
0
0
"
0
0
x
f
T
T
k
q
r
T
k
y
T
k
q
x
f
T
T
r
T
T
T
T
T
r
m
s
s
r
r
y
s
m
s
r
r
r
r
m
s
s
)
(x
f
k
h 


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

Evolution of Macro Flow Parameters

Thermal Considerations – Internal Flow
T fluid
 Tsurface
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: Pr
Re
05
.
0
,







D
x t
fd
Turbulent Flow:
10
,







D
x t
fd

Energy Balance : Heating or Cooling of fluid
• Rate of energy inflow
Tm Tm + dTm
dx
Q
m
pT
c
m

• Rate of energy outflow  
m
m
p dT
T
c
m 

Rate of heatflow through wall:
 
m
s T
T
dA
h
Q 





Conservation of energy:
    m
p
m
m
p
m
s T
c
m
dT
T
c
m
T
T
dA
h
Q 

 








  m
p
m
s dT
c
m
T
T
dx
P
h 





 
m
s
p
m
T
T
c
m
P
h
dx
dT




This expression is an extremely useful result, from which axial
Variation 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:
 
i
m
o
m
p
s T
T
c
m
L
P
q
Q ,
,
'
'






For entire pipe:
For small control volume:
m
p
s dT
c
m
q
dx
P
h 



 '
'
)
(
'
'
x
f
c
m
P
q
dx
dT
p
s
m





Integrating form x = 0
x
c
m
P
q
T
x
T
p
s
i
m
m 


'
'
,
)
(
The mean temperature varies linearly with x along the tube.
  m
p
m
s dT
c
m
T
T
dx
P
h 





For a small control volume:












dx
dT
P
h
c
m
T
T m
p
m
s

The mean temperature variation depends on variation of h.

  m
p
m
s dT
c
m
T
T
dx
P
h 





 
dx
c
m
P
h
T
T
dT
p
m
s
m




 
 
dx
c
m
P
h
T
T
T
T
d
p
m
s
m
s






Integrating from x=0 (Tm = T m,i) to x = L (Tm = Tm,o):
 
 
dx
c
m
P
h
T
T
T
T
d
L
p
m
s
m
s
T
T
o
m
i
m







0
,
,

For a small control volume:

p
i
m
s
o
m
s
c
m
L
P
h
T
T
T
T

















,
,
ln
p
surface
i
m
s
o
m
s
c
m
A
h
T
T
T
T
















,
,
ln














i
m
s
o
m
s
surface
p
T
T
T
T
A
c
m
h
,
,
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:
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
surface
avg
i
m
s
o
m
s
c
m
A
h
T
T
T
T

exp
,
,
 









 





p
avg
i
m
s
m
s
c
m
x
P
h
T
T
x
T
T

exp
,

m
T
s
T
T
x
m
T
s
T
T
x
i
s T
T
if  i
s T
T
if 

To get this we write:
     
   
i
o
p
i
m
s
o
m
s
p
i
m
o
m
p T
T
c
m
T
T
T
T
c
m
T
T
c
m
Q 











,
,
,
,

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

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).

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

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
NuD
C
qs 


66
.
3


k
hD
NuD
for constant surface temperature
Note: the thermal conductivity k should be evaluated at average Tm

• The entry region : for the constant surface temperature
condition
3
/
2
Pr
Re
L
D
04
.
0
1
Pr
Re
L
D
0.0668
3.66





















D
D
D
Nu
thermal entry length

for the combined entry length
14
.
0
3
/
1
/
Pr
Re
86
.
1 














s
D
D
D
L
Nu


   
  2
/
)
/
Pr/(
Re
14
.
0
3
/
1

s
D D
L 

All fluid properties evaluated
at the mean T
  2
/
,
, o
m
i
m
m T
T
T 

C
Ts 
700
,
16
Pr
48
.
0 

  75
.
9
/
0044
.
0 
 s


Valid for

Thermal Development of Internal Flows.ppt

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

Simultaneously developing laminar flow (Re < 2300)
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.
heating
For
Pr
Re
023
.
0 4
.
0
5
/
4
D
D
Nu 
cooling
for
Pr
Re
023
.
0 3
.
0
5
/
4
D
D
Nu 
)
1
(Pr
)
8
/
(
7
.
12
1
Pr
)
1000
)(Re
8
/
(
3
/
2
2
/
1




f
f
Nu 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
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

Thermal Development of Internal Flows.ppt

More Related Content

Similar to Thermal Development of Internal Flows.ppt (20)

Recently uploaded (20)

Thermal Development of Internal Flows.ppt