Melting by Natural Convection

1
Melting by Natural Convection

• Solid initially at Ts uniform
• Exposed to surfaces at T gt 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.)

2
Melting by Natural Convection

• Solid initially at Ts uniform
• At t 0, left wall at Tw gt 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

3
Melting by Natural Convection
4
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

5
Melting by Natural Convection

• Non-dimensional form
• Where dimensionless parameters are

6
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, dz
• Conduction in lower region (H-z)

7
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

8
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

9
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

10
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

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Melting by Natural Convection

• Numerical simulations verify Bejans scaling
• Fig. 10.25 Nu vs. q for several Ra values

12
Melting by Natural Convection

• Nu q-1/2 for small q (conduction regime)
• Numin at qmin Ra-1/2 (in mixed regime)
• Nu Ra1/4 (convection regime)

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Melting by Natural Convection

• For large q (q gt q2)
• 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