A Short Course by Reza Toossi, Ph.D., P.E. California State University, Long Beach. Heat Transfer Materials Storage, Transport, and Transformation Part II: Phase Change. Outline. Phase Change Materials Applications Properties Modeling Melting and Solidification Boiling and Condensation
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Reza Toossi, Ph.D., P.E.
California State University, Long Beach
Abhat, A., “Low temperature latent heat thermal energy storage: heat energy storage materials,” Solar Energy, 30 (1983) 313-332.
Velraj, R. , and Pasupathy, A., “PHASE CHANGE MATERIAL BASED THERMAL STORAGE FOR ENERGY CONSERVATION IN BUILDING ARCHITECTURE “Institute for Energy Studies, CEG, Anna University, Chennai - 600 025. INDIA.
Contact Melting (melting of a solid under its own weight)
Governing Equations (Neumann problem ):
Choose ∆t and ∆x to meet Neumann’s stability criterion
Determine the initial enthalpy at every node hjo (j = 1)
Calculate the enthalpy after the first time step at nodes (j = 2 ,..., N -1) by using equation (1).
Determine the temperature after the first time step at node (j = 1 ,..., N) by using equations (2) and (3).
Find a control volume in which the enthalpy falls between 0 and hsl , and determine the location of the solid-liquid interface by using equation (4).
Solve the phase-change problem at the next time step with the same procedure.
Unconditionally stable but is more complex because two unknown variables enthalpy and temperature are involved.
[See Alexiades , A ., and Solomon , A . D ., 1993 , Mathematical Modeling of Melting and Freezing Processes , Hemisphere , Washington , DC .]
Transform the energy equation into a nonlinear equation with a single variable h.
[See Cao , Y ., and Faghri , A ., 1989 , " A Numerical Analysis of Stefan Problem of Generalized Multi-Dimensional Phase-Change Structures Using the Enthalpy Transforming Model ," International Journal of Heat and Mass Transfer , Vol . 32 , pp . 1289-1298.]
Combination of the two methods [Cao , Y ., and Faghri , A ., 1990a , " A Numerical Analysis of Phase Change Problem including Natural Convection ," ASME Journal of Heat Transfer, Vol . 112 , pp . 812-815.]
Use finite volume approach by Patankar to solve the diffusion equation.
Jany , P ., and Bejan , 1988 , " Scaling Theory of Melting with Natural Convection in an Enclosure ," International Journal of Heat and Mass Transfer , Vol . 31 , pp . 1221-1235.
Rubner and Cohen, Nano Letters 6(6), 1213-1217 (2006)
Effect of substrate (Layered structure of an electric heater)
Garbero, et al., “Gas/surface heat transfer in spray deposition processes,” Intl. J. Heat and Fluid Flow, Vol. 27, Issue 1, Feb 2006, pp. 105-122
Two-layer model with enhanced wall function
Single round jet:
B Correlation number
D jet/nozzle diameter
d droplet diameter
K droplet splashing criterion
n number of droplets
number flux of droplets
Nu Nusselt number, hD/k
Nu0 Nusselt number in absence of particles
ω mass loading
σ surface tension
Comparison with parallel flow
Example: Substrate cooling of a plastic sheet
L = 20 cm, Ts = 95OC, Tf,∞= 20OC,
Uf,∞= 5 m/s for parallel flow; <uf> = 25 m/s in nozzle
Droplet deformation (spreading) during impact (dp = 200 μm, Up = 10 m/s).
Contours of total surface heat flux (seen from below)
Velocity vectors during the impact of three droplets: three-droplet
Garbero, Vanni, and Fritscling, “Gas/surface heat transfer in spray deposition processes,” Int’l J. Heat and Fluid Flow, Vol. 27, Issue 1. Feb 2006, pp. 105-122.
Cwb = 1/3
Park, K., and Watkins, A. P., “Comparison of wall spray impaction models with experimental data on drop velocities and sizes,” Int. J. Heat and Fluid Flow, Vol. 17, No. 4, August 1996.
Bai and Gosman (1995): Drop collision model
(Stick, Spread, Rebound, Rebound with breakup, Boiling-induced breakup, Random breakup, Splash)
Wang and Watkins (1990)
We < 80 We > 80
Splashing Criteria (Bussmann, 2000)
K = WeD Ohd-0.4
Kcrit = 649 + 3.76 ReD-0.63
Rebound, Rebound with breakup, Break-up, and Splash (Park and Watkins, 1996)
Embedding in Graphite Matrices