T J ScanlonDepartment of Mechanical EngineeringUniversity of StrathclydeGlasgow, Scotland
Abstract
This paper presents a numerical (CFD) model of vertical ice cylinders melting inwater. Incorporated within the model are mechanisms to allow for the densityinversion of water and a step-wise variation in the specific heat capacity at thesolid-liquid interface to account for the phase change. The natural convectionmelting process appears to be reasonably well captured by the numerical model incomparison with experimental data.
1 Introduction
The analysis of convective heat transfer with solid-liquid phase change isimportant in a broad range of scientific and engineering fields. These may befound in the solidification and melting phenomena commonly encountered inmetallurgical processes, latent heat thermal energy storage, oceanography andnuclear reactor safety. Such a case involving heat transfer with phase changecoupled with a moving solid-fluid boundary is often referred to as a Stefanproblem.
In this paper a numerical (Computational Fluid Dynamics - CFD) andexperimental (Shadowgraph) analysis has been carried out in order to betterunderstand and predict the thermodynamic and transport processes involved in theStefan problem by considering melting of a vertical ice cylinder in a surroundingwater environment. A literature review has highlighted that general work in thisfield has been relatively extensive, however, little or no studies appear to havebeen carried out on vertical ice cylinders particularly with regard to numericalsimulation of natural convection heat transfer involving a moving boundary withcoupled phase change. The problem is also an interesting one as it
covers the range over which the density extremum in water occurs (maximumdensity at approximately 4°C) which influences the natural convection process.
Experimental investigations have been carried out on vertical ice surfacesmelting into water. In these studies, water at temperatures in the vicinity of thedensity extremum was considered. Bendell and Gebhart [1] used thermocouples toindicate whether the natural convection flow was up or down. In the experimentalwork of Van P. Carey and Gebhart [2], flow patterns were visualised by seedingthe water and illuminating the flow field with a sheet of laser light. Wilson andVyas [3] applied the thymol-blue technique in order to visualise the velocityprofile occurring in the natural convection boundary layer.
Experimental studies for melting problems with other geometries have beenconsidered. Oosthuizen and Xu [4] presented evidence that the flow around ahorizontal melting ice cylinder is three-dimensional in nature. Gebhart, Bendelland Shaukatullah [5] were the first to analyse natural convection flows adjacent tohorizontal surfaces in cold water. Gebhart and Wang [6] melted short vertical icecylinders into cold fresh water in order to visualise the melting and resultingconvective motions.
Fukasako and Yamada [7] have presented an extensive summary of the workcarried out on water freezing and ice melting problems. Subject areas such asanalytical and numerical methods for freezing and melting problems, freezing ofwater with and without convective flow and atmospheric and marine icings havebeen considered.
In relation to the numerical side of melting problems, various problemgeometries and solution methods have been considered. Wilson and Lee [8]presented a finite difference analysis for the leading portion of a vertical ice sheetas it melted into fresh water. Wang [9] conducted a numerical study into thebuoyancy-induced flows next to a vertical wall of ice melting in porous media thatwas saturated with water. In the paper of Van P. Carey, Gebhart, and Mollendorf[10] numerical results for laminar, buoyancy induced flow adjacent to a verticalisothermal surface in cold, pure or saline water were presented.
In [11], Sparrow, Patankar and Ramadhyani used a finite difference scheme toanalyse the melt region created by a heated vertical tube embedded in a solid,which was at fusion temperature. Ng, Gong, and Mujumdar [12] used astreamline-upwind / Petrov-Galerkin finite element method to simulate themelting of a phase change material in a cylindrical, horizontal annulus heatedisothermally from the inside wall. Wu and Lacroix [13] used a stream function-vorticity-temperature formulation to track the position of the solid-liquid interfacefor an ice cylinder melting.
It is apparent that various approaches have been taken by the preceding authorsin their attempts to numerically simulate the melting process however none haveso far attempted a solution involving the primitive variables of pressure andvelocity. This paper is an attempt to consider this type of analysis.
2 Experimental data
The equipment used to obtain qualitative experimental data for the ice-meltingproblem is shown below in Figure 1.
Figure 1: Shadowgraph equipment used during the melting process
A homogeneous ice cylinder was a necessary pre-requisite for the shadowgraphexperiment, because any air bubbles released into the water during melting couldpotentially affect the flow patterns and the melting process itself. An ice cylinderof 50 mm diameter and length 200 mm with a reduced amount of trapped air wasproduced by vacuum pumping distilled water in an acrylic mould, whilesubjecting the mould and its contents to high frequency shocks from an ultrasonicbath. The mould and its contents were then sealed from the atmosphere, andfrozen in a chest freezer at -20°C. The ice was then allowed to thaw at roomtemperature, before the vacuum pumping process was repeated. The water in themould was then re-frozen in the freezer.
2.1 Experimental procedure
Experiments were performed in order to obtain qualitative data about the meltingprocess. A strong light source was used to illuminate the envelope of the icecylinder. The light was reflected across a parabolic mirror to produce a collimatedlight beam as shown in Figure 1. The ice cylinder was filmed melting with a CCDvideo camera for average initial water temperatures of 7°C and 17°C. Still imagesof the cylinder at different stages during the transient were captured from thedigitised film, and the ice-water interfaces were superimposed on top of eachother to show how the melting progressed with time.
During the melting process, the temperature history of the inside wall of theglass barrel was recorded using thermocouples at various within the ice cylinderand on the Perspex barrel itself. The temperature histories on the barrel were thenused as approximate boundary conditions for the CFD simulation of the meltingproblem.
A numerical analysis of the ice cylinder configuration shown in has been carriedout. The following equations have been solved using the commercially availableCFD code PHOENICS [14]:
Continuity equation:
h V.pw = 0 (1)dt
Momentum equation:
^- + V. (pan) = -Vp + V. (jiVii) + pg (2)dt
Temperature energy equation:
. , .(puc T\ = V.(kVT) (3)9f \ /
The PHOENICS code has been applied to solve the above equations usingaxisymmetric generalised curvilinear co-ordinates. The pressure-velocity couplingis accounted for using a method based on the SIMPLEST [15] algorithm. As suchlow velocities were encountered (typically 1.0 cm/s) and cell Peclet numbers wereof the order of 1.0, numerical diffusion was not considered to be significant thusthe HYBRID [15] scheme was adopted for convective discretisation of alltransported variables. For temporal discretisation a fully implicit scheme wasemployed.
A computational grid of 120 X by 150 Y was employed with time steps in theorder of 6 seconds. This led to calculation times typically of 40 hours on aPentium III PC and the solution was declared converged within each time stepwhen the global sum of the mass, momentum and thermal residuals were 0.001times their initial values.
Eqns (2) and (3) have been modified in order to account for the followingfeatures of the flow:
1) The density inversion within water between 0.01 °C to 17 °C within whichthe density follows a non-linear path, attaining its maximum value atapproximately 4°C. This density-temperature relationship has been formulated byGebhart and Mollendorf [16] according to:
where Pm is the maximum density (p ,„ = 999.972 kg/nf) and w = 9.2793 X 10"̂
(°C)~S Tn = 4.0293 °C and q = 1.894816 are constant values proposed by Gebhartand Mollendorf
2) The transport properties k and p within the ice and k and |Ll within the
water were set as constant values based on a mean temperature between the iceand water.
3) The energy and momentum equations are solved in such a manner that if thetemperature falls below 0.01 °C within any iteration within any numerical timestep the convection terms within these equations are switched off thus allowingonly transient, diffusion and source terms to influence this 'solidified" region.
4) harmonic averaging of exchange coefficients was employed as detailed inFigure 2 below:
Ax
-o
SolidInterface
Liquid
Figure 2 : Harmonic averaging of thermal conductivity at solid-liquid interface
and /, =Ax
such that for equal cell sizes
5) a 'mushy' zone is identified where matter is considered to be a mixture ofsolid and liquid with the respective thermodynamic and transport properties(except Cp) assumed to be the arithmetic mean of the two phases.
6) The 'mushy' region is confined to lie within a particular temperaturebandwidth A7\ which typically has a range -0.5 °C < A7 < 0.01 °C. It is withinthis bandwidth that phase change is assumed to occur and thus the variation inCp(mush) contained within the transient term of the energy equation may be written
as ) = A/7 , / AT where A/i .. is the latent heat of fusion (ice to water), as
The following sample of results compares the variations in the ice melting patternswith time for the experimental work against the numerical simulation (temperaturecontours) for experiment 2 (initial 7^ = 17°C):
Figure 4 : Experimental and numerical ice patterns, time - 1 min.
<igure 5 : Experimental and numerical ice patterns, time = 5 min.
Figure 6 : Experimental and numerical ice patterns, time = 11 mm.
Figure 7 : Experimental and numerical ice patterns, time ~ 15 ins.
Figure 8 : Experimental and numerical ice interface positions with time.
Figures 4 to 8 show that the numerical model has qualitatively captured themelting process by incorporating the phase change mechanism and the densityinversion within its coding. The natural convection melting process is wellcaptured at the top of the ice cylinder where there is evidence of a 'necking'formation within the solid. The smoothness of the predicted ice interface does notappear to be entirely regular, however, sensitivity studies for the temperaturebandwidth over which the phase change is assumed to occur (see Figure 3) areongoing and may yield a solution which is closer to reality.
5. CONCLUSIONS
An experimental and numerical investigation of natural convection ice melting hasbeen carried out. The results show that this complex transient phenomenon can bequalitatively captured using computational fluid dynamics..
Future work will include programming the influence of all transport propertiesas a function of temperature in the numerical model with grid adaptation also apossibility and the use of Particle Image Velocimetry (PIV) for the experimentalwork
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