1. Momentum Heat Mass Transfer
MHMT12
Heat transfer at phase changes
Heat transfer at melting, condensation and boiling (pool, convective)
Whalley P.B.: Boiling, Condensation and Gas-Liquid flow. Oxford Sci.Pub. 1987
Rudolf Žitný, Ústav procesní a zpracovatelské techniky ČVUT FS 2010

2. Phase changes T-s chart
MHMT12
The temperature-entropy chart enables to calculate for example the heat Q [J/kg] necessary for evaporation (Q=Ts), 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)
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]

3. HEAT transfer melting 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.
Duchamp

4. HEAT transfer melting MHMT12
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.
Tm melting point temperature TS
(t) z
(t+dt)
Enthalpy balance (assuming linear temperature profile in liquefied layer) during time interval dt
Solution:
where
and aL is temperature diffusivity of liquid.

5. HEAT transfer condensation
MHMT12
Dropwise condensation
Film condensation
Steam at temperature of saturation Tsat T
Twall < Tsat z
Duchamp

6. HEAT transfer film condensation
MHMT12
Film condensation (Nusselt)
The following analysis holds only for laminar films (Re<1800). It is usually sufficient, because majority of practical cases are laminar.
Transversal parabolic velocity profile and balance of forces Tw
Ts=Tw+T x dx 
Mass flow rate of condensed steam
gravity
Viscous force at wall
Transversal linear temperature profile, heat and mass fluxes
Thickness of film determines the heat transfer coefficient
Gravity acting in the flow direction increases 

7. Heat transfer – Boiling
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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) z T
Tsat Tw
Tsat overheating
temperature of liquid
Tsat increases with hydrostatic pressure
Duchamp

8. Heat transfer – Boiling
MHMT12
Nukyama curve (q-TSAT) see A.Bejan, A.Kraus: Heat transfer handbook. Willey 2003
Boiling crisis of the first kind

9. Heat transfer – Pool Boiling
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All parameters are related to liquid L
Nucleate (pool) boiling Rohsenow (1952)
uL - velocity of liquid surface
Rohsenow W.M., Trans.ASME, Vol.74,pp (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
Interpretation of Db follows from the equilibrium of surface stress  and buoyancy forces

10. Heat transfer – Flow Boiling
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Flow boiling in vertical pipes is characterized by gradual changes of flow regime and the vapor quality x increase along the pipe
Enthalpy of liquid at saturation temperature
Annular flow (rising film)
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.
Slug flow
Vapor quality is related to the Martinelli’s parameter (ratio of pressure drops corresponding to liquid and vapor)
Nucleate boiling (bubbles), e.g. Rohsenow correlation
Vapor quality and Martinelli’s parameter are used in most correlations for convective boiling heat transfer.
Heat transfer by forced convection (e.g.Dittus Boelter)

11. EXAM
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Heat transfer at phase changes

12. What is important (at least for exam)
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Film Condensation
Pool boiling
Reynolds and Nusselt numbers use Laplace constant (diameter of a bubble) as a characteristic dimension