Computation of FREE CONVECTION P M V Subbarao Associate Professor Mechanical Engineering Department IIT Delhi Quantification of Free …….

Computation of FREE CONVECTION

P M V Subbarao

Associate Professor

Mechanical Engineering Department

IIT Delhi

Quantification of Free …….

Governing Equations

• Now, we can see buoyancy effects replace pressure gradient in the momentum equation.

2

2

)( y

uvTTg

y

uv

x

uu

0

y

v

x

u

2

2

2

y

u

cy

T

y

Tv

x

Tu

p

Strongly coupled and must be solved simultaneously

•The buoyancy effects are confined to the momentum equation, so the mass and energy equations are the same.

Dimensionless Similarity Parameter

• The x-momentum and energy equations are

velocityreferencearbitrary an is u

and length, sticcharacteri a is L where

T

TT

0

*

00

sTT

u

vvand

u

uu

L

yyand

L

xx

2*

*2*

20

*

**

*

**

Re

1 T

)(

y

u

u

LTTg

y

uv

x

uu

L

s

PrRe

1 2*

*2

*

**

*

**

y

T

y

Tv

x

Tu

L

Dimensionless Similarity Parameter

• Define new dimensionless parameter,

1Re2

L

LGr1

Re2

L

LGr

2

32

020

)()(

LTTgLu

u

LTTgGr ss

L

forced natural

•Grashof number in natural convection is analogous to the Reynolds number in forced convection.

•Grashof number indicates the ratio of the buoyancy force to the viscous force.

•Higher Gr number means increased natural convection flow

TTs

u(x,y)

y

g

sT

T

x

v

u

Laminar Free Convection on Vertical Surface

• As y : u = 0, T = T

• As y 0 : u = 0, T = Ts

• With little or no external driving flow, Re 0 and forced convection effects can be safely neglects

Pr),( LL GrfNu

1Re2

L

LGr

Analytical similarity solution for the local Nusselt number in laminar free convection

(Pr)4

4/1

fGr

k

hxNu L

x

4/1Pr238.1Pr 1.2210.609

Pr 75.0Pr

f

(Pr)43

4

4/1

fGr

k

LhNu L

L

Average Nusselt # =

Where

Effects of Turbulence

• Just like in forced convection flow, hydrodynamic instabilities may result in the flow.

• For example, illustrated for a heated vertical surface:

• Define the Rayleigh number for relativemagnitude of buoyancy and viscous forces

TTs

3

,,

)(

Pr

xTTg

GrRa

s

cxcx

Effects of Turbulence

• Transition to turbulent flow greatly effects heat transfer rate.

Empirical Correlations

Typical correlations for heat transfer coefficient developed from experimental data are expressed as:

3 Pr

LTTg GrRa s

LL

nLL CRa

k

LhNu

3/1

4/1

n

n For Turbulent

For Laminar

Vertical Plate at constant Ts

LNuLog10

LRaLog10

•Alternative applicable to entire Rayleigh number range (for constant Ts)

2

27/816/9

6/1

Pr)/492.0(1

387.0825.0

L

LRa

Nu

Vertical Cylinders

•Use same correlations for vertical flat plate if:

4/1

35 ~

LGrL

D

Free Convection from Inclined Plate

Cold plate or Hot fluid

Hot plate or Cold fluid

Horizontal Plate

Cold Plate (Ts < T)

Hot Plate (Ts > T)

Active Upper SurfaceActive Lower Surface

Empirical Correlations : Horizontal Plate

•Define the characteristic length, L asP

AL s

•Upper surface of heated plate, or Lower surface of cooled plate :

1173/1

744/1

1010 15.0

1010 54.0

LLL

LLL

RaRaNu

RaRaNu

•Lower surface of heated plate, or Upper surface of cooled plate :

1054/1 1010 27.0 LLL RaRaNu

Note: Use fluid properties at the film temperature2

TTT s

f

Empirical Correlations : Long Horizontal Cylinder

•Very common geometry (pipes, wires)

•For isothermal cylinder surface, use general form equation for computing Nusselt #

nDD CRa

k

DhNu

RaD C n

0.333 0.125 10 - 10

0.250 0.480 10 - 10

0.188 0.850 10 - 10

0.148 1.02 10 - 10

0.058 0.675 10 - 10

127

74

42

22

210

Constants for general Nusselt number Equation

free convection turbulent heat transfer in an enclosure

• Turbulent flow in an enclosed cavity or box is a model for many flows of practical interest:

• Heating of a room.• Flow in a double glazing Window.• Spreading of fire and fire generated gases in an

building.

Velocity Vectors on A Central Vertical Plane

Isotherms on A Central Vertical Plane

Nusselt Number Correlations

Small Window Large Window

)( satss TThq

Natural Convection in A Pool of Saturated Liquid

Tsat

Onset of Convection Tsurface

Further Behavior of Saturated Liquid

Increasin

g T

Natural Convection

Onset of Boiling

Isolated Bubble Regime

High Overshoots !!!

Wall Superheat (T=Ts – Tsat)

Heat Flux

Overshoot

A BA: Onset of Natural convection

B: Onset of Nucleate Boiling

BOILING HEAT TRANSFER

P M V Subbarao

Associate Professor

Mechanical Engineering Department

IIT Delhi

A Basic means of Power Generation……A science which made Einstein Very Happy!!!

Boiling

• In a steam power plant convective heat transfer is used to remove heat from a heat transfer surface.

• The  liquid  used  for  cooling  is  usually  in  a  compressed  state,  (that   is,  a  subcooled  fluid)  at pressures higher than the normal saturation pressure for  the given temperature.

• Under certain conditions some type of boiling can take place.

• It is  an important  process  in nuclear  field  when  discussing convection heat transfer.

• More  than  one  type  of  boiling  can  take  place  within  a  

nuclear facility.

Nuclear Power Plant

Steam Boiler

Classification of Boiling

• Microscopic classification or Boiling Science basis:

• Nucleated Boiling

• Bulk Boiling

• Film Boiling

• Macroscopic Classification or Boiling Technology basis:

• Flow Boiling

• Pool Boiling

Nucleate Boiling

• The most common type of local boiling encountered in nuclear facilities is nucleate boiling.

• In nucleate boiling, steam bubbles form at the heat transfer surface and then break away and are carried into the main stream of the fluid.  

• Such movement enhances heat transfer because the heat generated at the surface is carried directly into the fluid stream.   

• In the main fluid stream, the bubbles collapse because the bulk temperature of the fluid is not as high as the heat transfer surface  temperature  where  the  bubbles  were  created.   

• This  heat  transfer  process  is  sometimes desirable  because  the  energy  created  at  the  heat  transfer  surface  is  quickly  and  efficiently "carried" away.

Bulk Boiling

• As  system  temperature  increases  or  system  pressure drops,  the  bulk  fluid  can  reach  saturation conditions.  

• At this point, the bubbles entering the coolant channel will not collapse.  

• The bubbles will tend to join together and form bigger steam bubbles.  

• This phenomenon is referred to as bulk boiling.

• Bulk  boiling  can  provide  adequate  heat  transfer  provided  that  the  steam  bubbles  are carried  away  from  the  heat  transfer  surface  and  the  

surface  is  continually  wetted  with  liquid water.   

• When this cannot occur film boiling results.

Film Boiling

• When the pressure of a system drops or the flow decreases, the bubbles cannot escape as quickly from  the  heat  transfer  surface.    

• Likewise,  if  the  temperature  of  the  heat  transfer  surface  is increased, more bubbles are created.  

• As the temperature continues to increase, more bubbles are formed  than  can  be  efficiently  carried  away.   

• The  bubbles  grow  and  group  together,  covering small  areas  of  the  heat  transfer  surface  with  a  film  of  steam.    

• This  is  known  as  partial  film boiling.    • Since  steam  has  a  lower  convective  heat  transfer  coefficient  than

 water,  the  steam patches on the heat transfer surface act to insulate the surface making heat transfer more difficult.

• As  the  area  of  the  heat  transfer  surface  covered  with  steam  increases,  the  temperature  of  the surface  increases  dramatically,  while  the  heat  flux  from  the  surface  decreases.   

• This  unstable situation continues until the affected surface is covered by a stable blanket of steam, preventing contact between the heat transfer surface and the liquid in the center of the flow channel.   

• The condition after the stable steam blanket has formed is referred to as film boiling.

• The process of going from nucleate boiling to film boiling is graphically represented in Figure.   

• The figure illustrates the effect of boiling on the relationship between the heat flux and the temperature difference between the heat transfer surface and the fluid passing it.

Boiling Curve