momentum heat mass transfer mhmt12 heat transfer at melting, condensation and boiling (pool,...
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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.