1 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 –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 T s = uniform At t = 0, left wall at T w > T s –T s = T m 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 LiquidSolid, T s TwTw

3 Melting by Natural Convection

4 Conduction regime Heat conducted across melt absorbed at interface s = location of solid-liquid interface h sf = 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 ~ t 1/2 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)

7 Melting by Natural Convection Mixed regime At bottom of z, (boundary layer ~ melt thickness) Combining Eqs. (10.107, , and ), 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 t 2 –z grows faster than s –We can also show that: –Constants K 1, K 2 ~ 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, s av extends over entire width, L, when Can only exists if: Otherwise, mixed convection exists during growth to s av ~ L

11 Melting by Natural Convection Numerical simulations verify Bejan’s scaling Fig : Nu vs.  for several Ra values

12 Melting by Natural Convection Nu ~   for small  (conduction regime) Nu min at  min  Ra  (in mixed regime) Nu ~ Ra  (convection regime)

13 Melting by Natural Convection For large  (    –s av ~ L –Scaling no longer appropriate –Nu decreases after “knee” point

14 Melting by Natural Convection Fig re-plots data scaled to Ra -1/2,Ra 1/4 or Ra -1/4