Momentum Heat Mass TransferMHMT12

Heat transfer at melting, condensation and boiling (pool, convective)

Rudolf Žitný, Ústav procesní a zpracovatelské techniky ČVUT FS 2010

Heat transfer at phase changes

sourceDt

D

Whalley P.B.: Boiling, Condensation and Gas-Liquid flow. Oxford Sci.Pub. 1987

Phase changes T-s chartMHMT12

Gas

Liquid

L+G

L+S

S

Solid

Tcr critical temperature

TTP triple point temperature

SG sublimation

LG evaporation

GL condensation

s

there is only gas above the critical temperature

heating solids

heating liquid

melting

evaporation

superheated steam

isobar p=const

T boiling point

temperature at pressure p

s [kJ/kg.K] entropy

T [K]

The temperature-entropy chart enables to calculate for example the heat Q [J/kg] necessary for evaporation (Q=Ts), condensation, or melting of 1kg of substance (heat Q is the area under isobar in the case of heating at constant pressure, see the red curve)

HEAT transfer melting

Duchamp

MHMT12

Melting, freezing, baking are thermal processes characterised by moving interface between two phases – liquid and solid.

Description of the interface motion is so called Stefan problem.

HEAT transfer meltingMHMT12

Tm melting point

temperature

TS

(t)

z

(t+dt)

Stefan problem in 1D: Let us assume that a semi-infinite space is a solid at a melting point temperature Tm. Since the time zero the temperature of surface is increased to Ts.

Enthalpy balance (assuming linear temperature profile in liquefied layer) during time interval dt

dtTT

dhTTc msLSLmspL

))(

2

1(

2 ( )4

1( ( ) )2

s mm

pL s m SL L

T T ta t

c T T h

Solution:

21

( )

Lm

SL

pL s m

aa

h

c T T

where and aL is temperature diffusivity of liquid.

HEAT transfer condensation

Dropwise condensation

Film condensation

Duchamp

MHMT12

Steam at temperature of saturation Tsat

Twall < Tsat

z

T

HEAT transfer film condensation

Film condensation (Nusselt)

2 2

( ) GL

T T gq x dm dx d

h

2

2

3 2( ) ( )

2

u y yu y

3 u

g

gravity Viscous force at wall

Transversal parabolic velocity profile and balance of forces

Transversal linear temperature profile, heat and mass fluxes

2 3

3

gm u

2 3 24

4GL GL

T g T gdx d x

h h

Thickness of film determines the heat transfer coefficient

2 3

4( )4GLh g

xTx

Tw

Ts=Tw+T

x

dx

dmMass flow rate of condensed steam

Gravity acting in the flow direction

increases

The following analysis holds only for laminar films (Re<1800). It is usually sufficient, because majority of practical cases are laminar.

MHMT12

Heat transfer – BoilingMHMT12

Duchamp

z

T

Tsat Tw

Tsat overheating

Steam bubbles are generated at bottom only it Tsat exceeds a critical limits (pressure of steam inside the bubbles must overcome surface tension and hydrostatic pressure)

temperature of liquid

Tsat increases with hydrostatic pressure

Heat transfer – BoilingMHMT12

Nukyama curve (q-TSAT) see A.Bejan, A.Kraus: Heat transfer handbook. Willey 2003

Boiling crisis of the first kind

convection

natural

4/5satTq

3

bubble regime

satq T

4

radiation

satq T

Heat transfer – Pool BoilingMHMT12

m

LSCNu PrRe

1 3/2

,D

NuL

b

,Du

ReL

LbL

.a

PrL

L

Nucleate (pool) boiling Rohsenow (1952)

Exponent m is 0,7 for all liquids with the exception of water (m=0). The coefficient CLS depends upon the combination surface-liquid (tables see Özisik (1985)) and for

the most common combination steel-water CLS=0,013.

Db is the Laplace constant characterizing diameter of bubble ( )bL G

Dg

)(

12

12)(

3

GLGL g

DD

gD

All parameters are related to liquid L

uL - velocity of liquid surface

LLG L

qu

h

Interpretation of Db follows from the equilibrium of surface stress and buoyancy forces

Rohsenow W.M., Trans.ASME, Vol.74,pp.969-975 (1952)

Heat transfer – Flow BoilingMHMT12

Flow boiling in vertical pipes is characterized by gradual changes of flow regime and the vapor quality x increase along the pipe

, ,L SAT

LG

h hx

h

Enthalpy of liquid at saturation

temperature

Vapor quality x<0 means subcooled liquid, vapor quality x=0 liquid at the beginning of evaporation, x=1 state when all liquid is evaporated and x>1 superheated steam.

Heat transfer by forced convection (e.g.Dittus Boelter)

Nucleate boiling (bubbles), e.g.

Rohsenow correlation

Slug flow

Annular flow (rising film)

Vapor quality is related to the Martinelli’s parameter (ratio of pressure drops corresponding to liquid and vapor)

0,10,50,9( / ) 1

.( / )

GL L

G L G

p z xX

p z x

Vapor quality and Martinelli’s parameter are used in most correlations for convective boiling heat transfer.

EXAMMHMT12

Heat transfer at phase changes

What is important (at least for exam)MHMT12

Film Condensation

2 3

4( )4GLh g

xTx

Pool boiling

m

LSCNu PrRe

1 3/2

Reynolds and Nusselt numbers use Laplace constant (diameter of a bubble) as a characteristic dimension

( )L G

Dg