melting by natural convection
DESCRIPTION
Melting by Natural Convection. Solid initially at T s = uniform Exposed to surfaces at T > T s , resulting in growth of melt phase Important for a number of applications: Thermal energy storage using phase change materials - PowerPoint PPT PresentationTRANSCRIPT
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Melting by Natural Convection
• Solid initially at Ts = uniform
• Exposed to surfaces at T > Ts, resulting in growth of melt phase
• Important for a number of applications:– Thermal energy storage using phase change
materials– Materials processing: melting and solidification of
alloys, semiconductors– Nature: melting of ice on structures (roadways,
aircraft, autos, etc.)
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Melting by Natural Convection
• Solid initially at Ts = uniform
• At t = 0, left wall at Tw > Ts
– Ts = Tm
• Liquid phase appears and grows• Solid-liquid interface is now an unknown
– Coupled with heat flow problem– Interface influences and is influenced by heat flow
Liquid Solid, Ts
Tw
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Melting by Natural Convection
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Melting by Natural Convection
• Conduction regime
• Heat conducted across melt absorbed at interface
• s = location of solid-liquid interface
• hsf = enthalpy of solid-liquid phase change (latent heat of melting)
• ds/dt = interface velocity
)101.10(~dt
dsh
s
TTk sf
sw
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Melting by Natural Convection
• Non-dimensional form:
• Where dimensionless parameters are:FoSte
#2
FourierH
tFo
#Stefan
h
TTcSte
sf
sw
)102.10(~ 2/1H
s
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Melting by Natural Convection
• Note that melt thickness, s ~ t1/2
• Nusselt number can be written as
• Mixed regime:– Conduction and convection– Upper portion, z, wider than bottom due to
warmer fluid rising to top
– Region z lined by thermal B.L.’s, z
– Conduction in lower region (H-z)
)105.10(~~ 2/1s
HNu
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Melting by Natural Convection
• Mixed regime
• At bottom of z,
(boundary layer ~ melt thickness)
• Combining Eqs. (10.107, 10.106, and 10.102), we can get relation for size of z …
)107.10(~ 4/1zz Raz
swz
TTzgRa
3
)106.10(~ sz
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Melting by Natural Convection
• Height of z is:• Where we have re-defined:
• Thus:– Convection zone, z, moves downward as t2
– z grows faster than s– We can also show that:
– Constants K1, K2 ~ 1
ConvCond
KKNu )110.10(2/32
2/11
3
z
HRaRa z
)108.10(~ 2RaHz
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Melting by Natural Convection
• From Eq. (10.110), we can get two useful pieces of information:z ~ H when
• Quasisteady Convection regime• z extends over entire height, H• Nu controlled by convection only
)112.10(~~ 2/1min
4/1min
RaatRaNu
)111.10(~ 2/11
Ra
)113.10(~ 4/1RaNu
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Melting by Natural Convection
• Height-averaged melt interface x-location:
• Average melt location, sav extends over entire width, L, when
• Can only exists if:• Otherwise, mixed convection exists during
growth to sav ~ L
)115.10(~ 4/12
RaH
L
)114.10(~ 4/1 RaHsav
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Melting by Natural Convection
• Numerical simulations verify Bejan’s scaling• Fig. 10.25: Nu vs. for several Ra values
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Melting by Natural Convection
• Nu ~ for small (conduction regime)
• Numin at minRa (in mixed regime)
• Nu ~ Ra (convection regime)
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Melting by Natural Convection
• For large (– sav ~ L
– Scaling no longer appropriate– Nu decreases after “knee” point
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Melting by Natural Convection
• Fig. 10.26 re-plots data scaled to Ra-1/2,Ra1/4 or Ra-1/4