Heat TransferForced Convection

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

Boundary Layer Thickness• The boundary layer thickness , is the distance across a

boundary layer from the wall to a point where the flow velocity has essential reached the ‘free stream’ velocity, .

• The distance is defined normal to the wall. it is customarily defined as the point y where:

• At a point, on the wall 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 51

37.0 x

0

Re xu

uyu0

99.0)(

0

0.5uvx

Here, is the overall thickness of the boundary layer. Re is the Reynolds number. is density. is the free stream velocity. is the distance downstream from start of the boundary layer. is the dynamic viscosity.• Actually the retardation of fluid is due to shear (viscous) stress acting

in opposite direction of flow.• With increase in the flow will be unstable as more fluid will be

retarded. So the flow will be converted from laminar to transition and finally turbulent.

• Critical value of Reynolds's number, at which boundary layer changes from laminar to turbulent, depends upon surface roughness, pressure gradient, plate curvature and intensity of turbulence in free stream flow.

• For Laminar Flow, Re < • For Turbulent Flow, Re > • For Transition form, Re < • For pipe flow, the development of boundary layer is similar to flow

over a flat plate, but in this case, the thickness of boundary layer is limited to the pipe radius.

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.

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.

Vvl

Re

• 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.

Re328.1

fC

58.2Relog445.0

fC

wettedf SvCF2

2

wettedS

BLASIUS EXACT SOLUTIN FOR LAMINAR BOUNDARY LAYER FLOWS

0

yuv

xuu

2

2

yuv

yuv

xuu

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

yu

xuu

v

Also as u α U and                    , along a plate length L, therefore ,we have L

Uxu

2

2

U

LU

Uvx

UvL

UL

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 wall

Uu

y

yf

Uu

                                          Substituting the value of   δ 

fvxUyf

Uu

vxUy

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 ,

fUvx

U

)( fvxU

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

yu

xv

ddfU

vxU

ddf

UvxUu

vxUy

yf

UvxU

yyu

)(

2

2

2

2

2

.21.

dfd

xU

xu

vxUy

xdfdU

xddfU

ddf

xU

xu

Here f is abbreviated as f(n)

2

2

..

d

fdvxUU

yfU

ddf

yU

yu

3

32

2

2

.dfd

vxU

yu

xf

xxvUyfx

UvUv

xf

vxUy

xfx

UvUv

xf

xfx

UvUv

xx

fxfx

UvUv

fxxU

vUfUvx

xxvv

21

21..

2..

2..

)(..

)(.)(

Now,

Similarly,

Again,

fddf

xUvv

fddf

xUvv

fvxUy

ddf

xUvv

21

21

21

3

32

2

22

2

22

3

32

2

2

2

2

.21..

21

..21.

2.

dfd

xUf

ddf

dfdU

xdfd

ddfU

x

dfd

vxUv

dfd

vxUUf

ddf

xUv

dfd

xU

ddfU

Inserting the values of in the hydrodynamic equation ,we get

vxu

yu

xuu ,,,, 2

2

0'''''2

02

.21

21

2

2

3

3

3

32

2

22

3

32

2

22

fffdfdf

dfd

dfd

xUf

dfdU

x

dfd

xUf

ddf

ddf

dfdU

x

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 uy0,0 vy

Uuy ,

)1(

)2(

)3(

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

Uv

UxvUv 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

xUxv

x

vxU

vxUy

Re55

5

99.0Uu

(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

fCfC

0)(21 2U

xfxC Re

664.0

__

fC

xCf

L

fCRe328.1__

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.

 

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

dxdyudxddyu ]..[..

00

000

..]..[.. dyudxdyudxddyu

dxdyudxddyu ]..[..

0

2

0

2

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.

=

=

Vdxdyudxd *]..[

0

dxdyudxddyu ]..[..

0

2

0

2

0

]...[ dxdyvudxd

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

dxdyvudxddyudxdyu

dxddyu ]...[..]..[..

00

2

0

2

0

2

dxdyvudyudxd ].....[

00

2

dxdyvuudxd ]).([

0

2

dxFb *0

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

dxdyvuudxddx ]).([)*(

0

20

dxdyvuudxd ]).([

0

20

dxdyuvudxd ]).([

0

2

])([0

2

22 dy

vu

vuv

dxd

])1([0

20 dy

vu

vu

dxdv

])1([0

20 dy

vu

vu

dxd

v

dxd

v

20

Where,

= Momentum Thickness

Above equation is known as Von Karman momentum equation for boundary layer flow.

dyvu

vu )1(

0

Relation between Nussle No. ,Reynolds No. , Prandtl No. . k

hlNu vl

Re C p

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

vC p

13 MT

TLM

3ML

1LT11 TML122 TL

13 MLT

• 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 - fHere , 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

fpdfddfd CVklCh 11

1

13 MT

TLM

pC

fedcba TLTMLLTMLMLTL 122111313

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

fpdfddfd CVklCh 11

1

fp

d

kCVl

lkCh

1

kCvlf

khl pn

,

• 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 11111

,,, TLM

111 dc

11 d

33 11 db 11 d

1313310000 1111 MTMLTMLLTlTLMdcba

• Equating above equating we get ,

• Thus

• - term : putting dimensions,

Equating components

For M : 0 = L : 0 = T : 0 = : 0 =

3

khlhkVl 1001

1

2 22222

dcba kvl

1113310000 2222 TMLMLTMLLTlTLMdcba

122 dc13 2222 dcba

13 22 db 2d

• 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

VlkVl

0111

2

3 pdcba Ckvl 3333

3

12213310000 3333 TLMLTMLLTLTLMdcba

33 dc 23 3333 dcba

23 33 db

13 d

1,1,1,1 3333 dcba

pp CkVlCkVl 1111

3

equating the dimension of we get

which is the dimension of viscosity , so we can use in place of therefore (or prandtl No.)

Now considering OR OR

OR

vl

1113 TMLLLTML

Vl

kC p

3

0,, 321

0,,

kC

vlkhl p

PrRe,Nu

kC

vlkhl p

,

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.

• 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,

TqTTqexcessboiling

satsboiling

h

h

.

).(

mW

2

• 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.

• 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.

• 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 ).

• 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.

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.

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.

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.

KWhmT

qexcess

2

46

max 1010 3.330

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.

• 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.

• 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.

• 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.

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.

• 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.