VM104

VM104
Liquid-Solid Phase Change

Overview

Reference:J. A. Dantzig, "Modeling Liquid-Solid Phase Changes with Melt Convection", Int. Journal Numerical Methods in Engineering, Vol. 28, 1989, pg. 1774, problem I.
Analysis Type(s):Thermal Analysis (with phase change) (ANTYPE = 4)
Element Type(s):2D Thermal Solid Elements (PLANE55)
Input Listing:vm104.dat

Test Case

A layer of liquid (of depth a), covering an insulated surface and initially at its freezing temperature To, is suddenly subjected to a free surface temperature Ts (less than To). Determine the time, tf, taken for the liquid to solidify completely, and the temperature distribution in the solid phase at time t1 seconds.

Figure 146: Liquid-Solid Phase Change Problem Sketch

Liquid-Solid Phase Change Problem Sketch

Material PropertiesGeometric PropertiesLoading
ρ = 1000 kg/m3
Cp = 4200.0 J/kg-°C
k = 0.60 W/m-°C
Lf = Latent heat of fusion
= 42000. J/kg
a = 0.01 m
b = 0.001 m
To = 0°C (t = 0)
Ts = -5°C (x = 0; t > 0)
t1 = 500 sec

Analysis Assumptions and Modeling Notes

The problem is formulated in two dimensions, with all faces insulated, except the face representing the liquid surface. The latent heat effect (accompanying change in phase from liquid to solid), is approximated by specifying a rapid variation in enthalpy (material property ENTH), over the "mushy" zone in a temperature range of ΔT (taken as 1.0°C). The enthalpy (H) variation is computed from the equation H = ρc dT. An adjusted specific heat of Lf/ΔT = 42000.0 J/kg-°C, is used in the freezing zone, resulting in a slope discontinuity, as shown in Figure 148: Temperature Distribution at Time = 501 Seconds. For the given case, a time step of 3.0 seconds is found to be adequate to give more than one time step through the freezing zone. Automatic time stepping is used.

POST1 is used to obtain the temperature distribution through the two phases and POST26 is used to display the temperature history of points in each of the phases.

Figure 147: Enthalpy vs. Temperature

Enthalpy vs. Temperature

Results Comparison

Target[1]Mechanical APDLRatio
t f, seconds810.0between (787 to 797) [2]-
At t = 500 seconds
T , °C (x = 0.002m)-3.64-3.711.019
T , °C (x = 0.004m)-2.32-2.461.059

  1. From Equations 20-31, in J. A. Dantzig, "Modeling Liquid-Solid Phase Changes with Melt Convection".

  2. Corresponds to the time interval at which Node 2 temperature crosses the ΔT = 1°C freezing zone.

Figure 148: Temperature Distribution at Time = 501 Seconds

Temperature Distribution at Time = 501 Seconds

Figure 149: Temperature History of Solidification

Temperature History of Solidification