A presentation on the heat transfer with specific topic forced convection.

1. Heat Transfer
Forced Convection

2. Outline
• Boundary layer, its types & its thickness
• Skin surface co-efficient
• Blasius exact equation
• Von Karman momentum equation
• Dimensional relationship of forced convection with Reynolds's number and Prandts
number
• Boiling heat transfer
• Condensation heat transfer

3. Boundary Layer Thickness
• The boundary layer thickness 𝛿 , is the distance across a
boundary layer from the wall to a point where the flow velocity
has essential reached the ‘free stream’ velocity, 𝑢0.
• The distance is defined normal to the wall. it is customarily
defined as the point y where:
• At a point, on the wall 𝑥, For laminar boundary layers over a
flat plate, the blasius solution to the flow governing equations
gives:
• For turbulent boundary layers over a flat plate, the boundary
layer thickness is given by:
where, Re 5
1
37.0 x



0
Re
xu

uyu 0
99.0)( 
0
0.5
u
vx


4. Here, 𝛿 is the overall thickness of the boundary layer.
Re is the Reynolds number.
𝜌 is density.
𝑢0 is the free stream velocity.
𝑥 is the distance downstream from start of the boundary layer.
𝜇 is the dynamic viscosity.
• Actually the retardation of fluid is due to shear (viscous) stress acting in opposite
opposite direction of flow.
• With increase in 𝑥, the flow will be unstable as more fluid will be retarded. So the
So the flow will be converted from laminar to transition and finally turbulent.
turbulent.
• Critical value of Reynolds's number, at which boundary layer changes from
laminar to turbulent, depends upon surface roughness, pressure gradient, plate
plate curvature and intensity of turbulence in free stream flow.
• For Laminar Flow, Re < 3 ∗ 105
• For Turbulent Flow, Re > 5 ∗ 105
• For Transition form, 3 ∗ 105
< Re < 5 ∗ 105
• For pipe flow, the development of boundary layer is similar to flow over a flat
flat plate, but in this case, the thickness of boundary layer is limited to the pipe
pipe radius.

6. Skin Surface Co-efficient
• Skin friction coefficient is also known as “friction coefficient” or “drag
coefficient”.
• Skin friction drag is a component of parasitic drag that occurs
differently depending on the type of flow over the lifting body (Laminar
or Turbulent).
• Just like any other form of drag, the coefficient of skin friction drag is
calculated with various equation and measurements depending on the
flow.
• Laminar flow is when layers of the fluids move smoothly past each other
in parallel lines.
• As the fluid flows over an object, it applies frictional forces on the
surface of the object which works to impede forward movement of the
object. In other words, create skin friction drag.
• Turbulent Flow has a fluctuating and irregular pattern of flow which is
attributed to the formation of vortices.
• This results in a thinner boundary layer. which relative to laminar flow,
depreciates the magnitude of friction force as the fluid flows over the
object. This suggests that the total parasitic drag observed in turbulent
flow is minimally impacted by skin friction drag.

7. Calculation
• The calculating of skin friction drag is heavily based on the Reynolds
number of the body. For reference, Reynolds number(Re) is calculated
with:
Where,
V = Velocity of flow.
l = Length of the body that the flow travels across
v = Kinematic viscosity of fluid.
• From this equation, the Reynolds number is known. So, the coefficient of
the skin friction drag can be calculated.
V
vl
Re

8. • For Laminar flow,
, Also known as the blasius friction law.
• For Turbulent flow,
, Also known as the Schlichting empirical formula.
• The total force on the body caused by skin friction drag in units of force
can be calculated with:
Where,
is the total surface area that is in contact with the fluid.
Re
328.1
fC
  58.2
Relog
445.0
fC
wettedf S
v
CF
2
2


wettedS

9. BLASIUS EXACT SOLUTIN FOR LAMINAR
BOUNDARY LAYER FLOWS
0





y
u
v
x
u
u
2
2
y
uv
y
u
v
x
u
u








The velocity distribution in the boundary layer can be obtained by solving the equation
of the motion for hydrodynamic boundary layer equation
The continuity equation :
As u > v , therefore the way may be write as 2
2
y
u
x
u
u





 








v
Also as u α U and , along a plate length L, therefore ,we have
L
U
x
u



2
2


 U
L
U

U
vx
U
vL
U
L





10. From experiments it has been observed that velocity profiles at different locations along the
plate geometrically similar , i.e., they differs only by a stretching factor the y-direction ,this
implies that the dimensionless velocity can be expressed at any location x as a function
of the dimensionless distance from the wallU
u

y







 
y
f
U
u
Substituting the value of δ
 f
vx
U
yf
U
u










vx
U
y
The vertical component of the velocity occurs in the boundary layer equation of motion ,it
is essential to define a stream function ψ such that ,
 

f
U
vx
U








)( fvxU

11. the continuous stream function Ψ is the mathematical postulation such that its partial
differential with respect to x gives the velocity in Y-direction and its partial differential
with respect to y gives the velocity in x-direction
y
u




x
v










d
df
U
vx
U
d
df
U
vx
Uu
vx
U
y
y
f
U
vx
U
yy
u
































 )(
2
2
2
2
2
.
2
1
.





d
fd
x
U
x
u
vx
U
y
xd
fd
U
xd
df
U
d
df
x
U
x
u

























Here f is abbreviated as f(n)

12. 2
2
..


 d
fd
vx
U
U
y
f
U
d
df
y
U
y
u























3
32
2
2
.
d
fd
vx
U
y
u



 








































































x
f
xxv
U
y
f
x
U
v
Uv
x
f
vx
U
y
x
f
x
U
v
Uv
x
f
x
f
x
U
v
Uv
x
x
f
x
f
x
U
v
Uv
fx
xU
v
Uf
U
vx
xx
vv
2
1
2
1
..
2
..
2
..
)(..
)(.)(






Now,
Similarly,
Again,

13. 




















f
d
df
x
Uv
v
f
d
df
x
Uv
v
f
vx
U
y
d
df
x
Uvv





2
1
2
1
2
1
3
32
2
2
2
2
2
2
3
32
2
2
2
2
.
2
1
..
2
1
..
2
1
.
2
.










d
fd
x
U
f
d
df
d
fd
U
xd
fd
d
df
U
x
d
fd
vx
U
v
d
fd
vx
U
Uf
d
df
x
Uv
d
fd
x
U
d
df
U














Inserting the values of in the hydrodynamic
equation ,we get
v
x
u
y
u
x
u
u ,,,, 2
2







14. 0'''''2
02
.
2
1
2
1
2
2
3
3
3
32
2
2
2
3
32
2
2
2










fff
d
fd
f
d
fd
d
fd
x
U
f
d
fd
U
x
d
fd
x
U
f
d
df
d
df
d
fd
U
x








15. Which is the ordinary differential equation for f, the num of the prime on f denoted the
number of successive derivation of f(n) with respect to y
Physical boundary conditions
0
0

0,0  uy
0,0  vy
Uuy  ,
)1(
)2(
)3(

16. 1. The single curve 2 shows the variation of the normal velocity . It is to be note
that at the outer edge of the boundary layer where ,this does not go to zero
but approaches the value
U
v

Ux
v
Uv 865.0
(2) The graph 1 enables us to calculate the parameters:
(i) boundary layer thickness, δ :
The boundary thickness δ is taken to be the distance from the plate surface to a point
at which the velocity is within 1% of the asymptotic limit, i.e. it occurs at
η=5.0 therefore, the value of at the edge of the boundary layer ( y = δ) is given by
x
Ux
v
x
vx
U
vx
U
y
Re
5
5
5




99.0
U
u

17. (ii) Skin friction coefficient ;
the skin friction coefficient( ) is define as the shear stress at the plate of the
dynamic head caused by the free stream velocity. Thus the local skin friction
coefficient
the average value of the skin friction coefficient can be determined by
integrating the local skin friction coefficient
from x=0 to x=L and dividing the integrated result by the plate length
fC
fC
0)(
2
1 2
U
x
fxC
Re
664.0

__
fC
xCf
L
fC
Re
328.1__


18. Von Karman Integral Momentum Equation
• It is difficult to obtain the exact solution of hydrodynamic boundary layer
even for as simple geometry as flat plate.
• A substitute procedure entailing adequate accuracy has been developed
which is known as “ Approximate Integral Method” and this is based upon a
boundary layer momentum equation derived by Von Karman.
• Flow flowing over a thin plate with free steam velocity equal to v.
• Consider a small length dx of the plate at a distance x from the leading
edge.
• Consider unit width of plate perpendicular to the direction of flow.

20.  Let,
ABCD be a small element of a boundary Layer.
 Mass rate of fluid entering through AD =
 Mass rate of fluid leaving through BC =
 Mass Flow rate of fluid entering the control volume through the
surface ,
DC = Mass rate of fluid BC – Mass flow rate of fluid through AD
=
 Momentum rate of fluid entering the control volume in x-direction through
AD =



0
.. dyu
dxdyu
dx
d
dyu ]..[..
00
 


 


000
..]..[.. dyudxdyu
dx
d
dyu
dxdyu
dx
d
dyu ]..[..
0
2
0
2
 



21.  Momentum rate of fluid leaving the control in x-direction through
BC =
 Fluid is entering through DC with a uniform velocity V.
 Momentum rate of fluid entering the control volume through DC in
x-direction.
=
=
Vdxdyu
dx
d
*]..[
0



dxdyu
dx
d
dyu ]..[..
0
2
0
2
 





0
]...[ dxdyvu
dx
d

22.  Rate of change of momentum of control volume = momentum rate
of fluid through BC – momentum rate of fluid through AD –
momentum rate of fluid through DC.
=
=
=
 Drag force
dxdyvu
dx
d
dyudxdyu
dx
d
dyu ]...[..]..[..
00
2
0
2
0
2
 


dxdyvudyu
dx
d
].....[
00
2
 


dxdyvuu
dx
d
]).([
0
2
 


dxFb *0


23.  Total external force in the direction of change of momentum,
=
 As per momentum principle the rate of change of momentum on
the control volume ABCD must be equal to the total force on the
control volume in the same direction.
)*( 0 dx
dxdyvuu
dx
d
dx ]).([)*(
0
2
0  


dxdyvuu
dx
d
]).([
0
2
0  


dxdyuvu
dx
d
]).([
0
2
 



24. ])([
0
2
2
2
dy
v
u
v
u
v
dx
d
 


])1([
0
2
0 dy
v
u
v
u
dx
d
v  


])1([
0
2
0
dy
v
u
v
u
dx
d
v
 



dx
d
v



 2
0

25.  Where,
= Momentum Thickness
 Above equation is known as Von Karman momentum
equation for boundary layer flow.
dy
v
u
v
u
)1(
0
 

 

26. Relation between Nussle No. ,Reynolds No. , Prandtl No. .
k
hl
Nu 

vl
Re Cp
Using system , the dimensions of various quantities can be obtained as
under .
Quantity Symbol dimension
Heat transfer co-
efficient
H
Fluid density
Length l
Fluid velocity
Fluid viscosity
Specific heat
Thermal
conductivity
k

v
Cp
13 
MT
 TLM
3
ML
1
LT
11 
TML
122 
TL
13 
MLT

27. • Rayleigh’s method :
•
• Therefore …..(1)
Putting dimensions of all quantities
= [1]
equating exponents of , we get :
For M : 1 = b + c + e
L : 0 = a + b -3c + d – e + 2f
T : -3 = -3b – d – e – 2f
: -1 = -b - f
Here , we have four equations and six unknowns . So we have to select two components in
terms of which all other exponents are obtained.
[ here we have selected V and because both are coming in two separate groups Reynolds
and prandtl numbers respectively.]
),,,,,( pCVklfh 
           f
p
dfddfd
CVklCh

 11
1
13 
MT
 TLM

pC
           fedcba
TLTMLLTMLMLTL 122111313 


28. So expressing a,b,c,e in terms of d,f
We get, a = d - 1 , b = 1 – f
c = d e = f - d
Putting these all values in equation (1)
combining similar exponents
OR
           f
p
dfddfd
CVklCh

 11
1
f
p
d
k
CVl
l
k
Ch 





















1







k
Cvl
f
k
hl pn



,

29. • Buckingham’s -theorem :
Here, total No. of quantities(n) = 7 and total No. of fundamental quantities(m)=4
Therefore No. of - constant = n - m = 7 – 4 =3
For obtaining -groups, we would select length ‘l’ (geometric), velocity ‘V’ (flow
characteristics ), density ‘ ‘ (fluid property) and thermal conductivity ‘k’ (thermal property)
as repeating variables
- terms :
Equating components of
for M : 0 =
L : 0 =
T : 0 =
: 0 =




1 hkvl dcba 1111
1  
,,, TLM
111  dc
11  d
33 11  db
 11  d
         1313310000 1111 
  MTMLTMLLTlTLM
dcba

30. • Equating above equating we get ,
• Thus
• - term :
putting dimensions,
Equating components
For M : 0 =
L : 0 =
T : 0 =
: 0 =
3
k
hl
hkVl  1001
1 
2  2222
2
dcba
kvl
         1113310000 2222 
 TMLMLTMLLTlTLM
dcba

122  dc
13 2222  dcba
13 22  db
 2d

31. • Equating above equation we get ,
• Thus ,
- term :
Now equating exponents
For M : 0 =
L : 0 =
T : 0 =
: 0 =
Equating above equation we get ,
Thus
0,1,1,1 2222  dcba
Vl
kVl


   0111
2
3 p
dcba
Ckvl 3333
3  
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33. Boiling Heat Transfer
• INTRODUCTION:-
• Boiling can said to be a liquid to vapor phase
change process similar to evaporation, but there
are quite difference between both.
• Here, boiling occurs at the solid to liquid
interface when liquid is brought into contact with
a surface maintained at a temp which is above the
saturation temperature.
• It can be characterized by rapid formation and
growth of vapor bubbles at contact surface of
solid-liquid interface.

34. • The process of boiling contains large no. of variable properties such as density,
friction, thermal conductivity surface tension, energy absorbed by unit mass of
liquid vaporized at specific pressure & temperature etc. kind of thermal properties
& also rapid formation & growth of vapor bubbles plays very important role in such
heat transfer, which makes it complex to examine.
• Boiling is a forced convection heat transfer process.
• As we know that the basic of the forced convection is Newton's law of cooling,
Hence; we have, boiling heat flux
• The unit of boiling heat flux is,
Tq
TTq
excessboiling
satsboiling
h
h


.
).(
m
W 2

35. • Boiling heat transfer takes place because of difference between the vapor bubbles
inner temperature and outer liquids temperature.
• When inner temperature of bubble is high, boiling heat transfer takes place from
bubble to surrounding liquid and as the temperature of bubble reduces it comes to
the free surface and collapses. When outer liquid temperature is high than bubbles
inner temperature, heat transfer takes place from liquid to bubble and causes in
growth of bubble.
• Boiling can be classified by two types,
– Pool boiling,
– Flow boiling.
• These types of boiling depends upon presence of bulk fluid motion.
• If bulk fluid motion is absent than it is pool boiling and id not, then its flow boiling.

36. • One can say that pool boiling is natural convection in which
vapor bubbles can be only considered under buoyancy and
other forces are neglected. i.e. heating of water on stove in
pan.
• Flow boiling can be considered as forced convection in
which liquid is moving and so as bubbles and causing the
heat transfer process.
• Pool boiling & flow boiling can be further classified in sub
cooled boiling and saturated boiling.
• When temperature of main body is below saturation
temperature it is sub cooled boiling and if it is equal to the
saturation temperature then it is saturated liquid.

37. • POOL BOILING:-
• In pool boiling, stationary liquid is not forced to flow by a mover such as pumps.
• Any motion is due to natural convection.
• For example, consider boiling of water in pan on top of a stove. The water is
initially at temperature about 15 degree Celsius temperature, far below the
saturation temperature about 100 degree Celsius at atm. At early stages one will not
notice anything except some bubbles. These bubbles are caused by the release of air
molecules dissolved in liquid water. As the vapor temperature rises there are chunks
of water rolling up & down can be seen by natural convection, followed by the
vapor bubbles forming at the bottom surface. These bubbles get detached from the
surface & starts rising, ten collapses in cooler water at above ( Sub cooled boiling ).
• The bubbles formation increases as the water temperature rises further, and
becomes faster at saturation temperature( Saturated boiling ).

38. • Boiling regimes & boiling curve:-
• As per the researches boiling is most familiar type of heat transfer in every days
life.
• S. Nukiyama was the first who have done complete examination in boiling. He
used electrically heated nichrome & platinum wires immersed in liquids in his
experiment & noticed change in different forms of boiling, depending on the value
of excess temperature.
• Mainly, as observed; there are 4 different boiling regimes,
– Natural convection boiling
– Nucleate boiling
– Transition boiling
– Film boiling
– Critical heat flux & burnout point.

39. o Natural convection boiling:-
 Pure substances starts boiling in specific pressure when it
reaches to certain temperature. There are number of
bubbles generates up to certain temperature hence
temperature slightly increases & liquid evaporates when
it rises to free surface. Heat transfer takes place by
natural convection. In boiling curve it ends at 5 degree
Celsius temperature.
o Nucleate boiling:-
 Bubbles starts to generate at point a at various heating
sites, hence the point a is called onset of nucleate boiling.
bubble forming and growth increases as we move to
point C.

41.  There are two regions in nucleate boiling,
i. A to B at 5 to 10 degree Celsius:- bubbles dissipated in liquid shortly after they
separate from free surface. The vacant places caused by bubbles is now filed with
vicinity of heater surface liquid and this process repeats. Stirring & agitation
caused by entrainment of liquid to heater surface is responsible for increase in heat
transfer co-efficient & heat flux.
ii. B to C at 10 to 30 degree Celsius:- because of rise in temperature the rate of
formation & growth of bubbles increases in this region. They together forms a
continuous column kind of structure of vapor in liquid. The bubbles moves faster
to free surfaces and collapses and because of that the vapor releases. In this region,
large heat flux is obtained.
 At large values of temperature differences, the rate of evaporation at heater surface
reaches such high values that a large fraction of the heater surface is covered by
bubbles making it difficult for liquid to reach to heater surface.

42.  Heat flux increases with increase in temperature difference and reaches to
maximum at point C.
 On point C, heat flux is maximum/critical.
From Newton’s law of cooling, heat transfer co-efficient at point C is,
o Transition boiling:-
 As temperature increases, heat flux decreases after point C. This is because of large
heater surface is covered by vapor film acts as insulation of low thermal
conductivity (K).
 Transition boiling can be explained as sum of Nucleate boiling and insulation film.
 Nucleate boiling at point C is completely converted in insulation film at point C.
 This region is called Unstable film boiling region.
K
W
h
mT
q
excess 


2
4
6
max
1010 3.3
30

43. o Film boiling:-
 At point D, where the heater surface is completely covered by vapor film, this
region starts.
 Value of heat flux is minimum at point D, it is called Leiden Frost point.
• A typical boiling process doesn’t follow boiling curve beyond point C.
• S. Nukiyama observed, if power more than maximum heat flux is applied on
nichrome wire immersed in water, it suddenly reaches to melting temperature and
burnout happens. And no such burnout takes place in case of platinum wire.
• When he gradually reduced power, he obtained cooling curve up to heat flux
minimum.
• Practically, boiling process does not follow transition boiling untill power applied is
reduced suddenly.

44. • Burnout Phenomenon:-
• In order to move more beyond point C, where
maximum heat flux occurs, we must increase
heater surface temperature.
• Liquid cannot receive this energy at excess
temperature just beyond point C. hence heater
surface stops to absorb more energy causing
heater surface to rise & liquid can receive even
less energy than that hence heater surface
temperature further rises.
• Hence, heater surface temperature reaches at
point E, where no temperature rises & heat can
be transferred to liquid steadily.
• After any attempt to increase heat flux makes
curve to jump from point C to point E &
burnout occurs.

45. • Enhancement in Pool boiling:-
• Heat transfer in Nucleate boiling depends upon nucleation sites on heater surface.
• From experimental results, it is known that roughness & dirt also helps in
increasing the heat transfer and works as nucleation sites.

46. • Flow Boiling:-
• In this case, liquid is forced to move by external source such as pump as it
undergoes a phase change process.
• Flow boiling can be said as sum of convection and pool boiling.
• Flow boiling can be classified in two types,
– External, over the heater surface
– Internal, inside the heated tube
• External flow boiling is similar to pool boiling, but only involves motion of liquid,
which increases nucleate boiling heat flux as well as total flux.
• Internal flow boiling referred as two phase flow and much complicated, because no
free surface for vapor to escape. Hence, vapor and liquid caused to flow together.

47. Condensation Heat Transfer
• Condensation occurs when the temperature of vapor is reduced below saturation
temperature. This is usually done by bringing the vapor in to contact with a solid
surface whose temperature is below saturation temperature of vapor.
• Condensation also takes place in liquid as well as gas.
• There are two distinct forms of condensation,
– Film condensation
– Drop wise condensation
• In film condensation, the condensate wets the surface & forms a liquid film on the
surface that slides down under the influence of gravity.
• Thickness of liquid film increases in flow direction as more vapor condensates.
• As thickness of liquid layer increases, this ‘liquid wall’ resists the heat transfer
between solid and vapor. The heat from vapor required to pass through the liquid
layer to complete heat transfer.

48. • In drop wise condensation, the condensed vapor forms
droplets on the surface instead of continuous film, &
surface is covered by countless droplets of varying
diameters.
• The droplets slides down when they reach a certain size,
cleaning the surface and exposing it to vapor.
• As a result, heat transfer rates that are 10 times larger
than those associated with film condensation can be
achieved with drop wise condensation. Therefore, it is
referred mostly in condensation applications.
• But there is a disadvantage or drawback of drop wise
condensation is that, till now no one has achieved it for
a long time in experiments.