Use of the finite element method in modelling the heat transfer inside anartificial skating rink

COMAN GELU, UZUNEANU KRISZTINA, PANAIT TANASE, DRAGAN MARICELDepartment of Thermal Systems & Environmental Engineering

"Dunărea de Jos" University of Galaţi111 Domneasca St., Galati 800201

ROMANIAgelu.coman@ugal.ro, kristina.uzuneanu@ugal.ro

Abstract: The paper presents the modeling of heat transfer and ice formation around the pipes of a skating rinkby the finite element method. In case of skating rinks equipped with pipe registers, the temperature field duringthe ice formation process can be modeled by analytical methods. Thus the paper advances a method forcalculating the temperature and distribution of heat around the track pipes The heat transfer in the skating rinktrack is nonstationary and phase changing.

Key-Words: solidification, ice, skating rink, heat transfer, phase changing.

1 IntroductionThe problem of melting and solidification ofsubstances, regarded in terms of determining thetemperature field in solid and liquid phases andsolid – liquid interface propagation , has arousedconsiderable (theoretical and practical) interest ,because the conductive transfer processesaccompanied by the phenomenon of phase transitionare present in many applications, such assolidification of ingots, controlled solidification ofalloys to obtain certain metallographic structures,food refrugeration , soil freezing and defreezing,the ablation phenomenon under aerodynamicheating, thermal storage with phase change, etc..Mathematical modeling of this phenomenon andespecially mathematical solution to the problemare made difficult by the fact that general solutionsrelate to three-dimensional nonstationarytemperature fields in bodies whose physicalproperties are temperature dependent [1,2]

2 Problem FormulationNumerical modeling was performed using Fluent6.3 software for two cases of skating trackconstruction, namely: track with water immersedpipeline and track with pipes buried in sand. Forsymmetry purpose the study was conducted using aflat discretization network, bordered left and right

by two vertical lines passing through the middistance between two consecutive tubes, a straighthorizontal line representing the water (or ice)surface and another one that represents the trackfoundation board (Fig 1.).With the water submerged pipe geometry , thecalculation range is identically bounded, specifyingthat the entire area around the pipe was consideredof liquid type .The grid with the water submerged pipe geometrycontains 13563 nodes and 12952 quadrilateralelements, and in the second case ,with the pipeburied in the sand, 11243 nodes and 10896elements.Due to the multiblock structured grid, withquadrilateral elements, the grid contains a relativelysmall number of elements, which will positivelyinfluence the calculation time, solution convergenceand results accuracy.The Fluent program has an impressive library ofsolid and fluid materials and their properties [6].Numerical modeling of heat transfer with phasechange made with this software, physicallycorresponds to the real solidification processaround the cylindrical surfaces. Thus, thethermophysical properties of materials are no longerconstant but vary with temperature and the phasechange is not isothermal .

Recent Advances in Fluid Mechanics and Heat & Mass Transfer

ISBN: 978-1-61804-026-8 277

The mathematical model chosen is based onpressure based, since heat transfer modeling is donein incompressible fluids. Formulating themathematical model is of default type which makessolver stable and convergent, although the numberof equations and complexity of calculationsincreases. Moreover, the default scheme, unlike theexplicit one, is recommended for heat transferprocesses with phase change, by providing freedomof choice of time step within a wide range of values

The type of solver adopted was Solidification andmellting.This type of solver is intended to solve the problemsof heat transfer with phase change. For a betteranalysis of the thermal field in certain areas of thecomputing range a number of surfaces have beenintroduced to monitor the evolution of temperaturein the course of the process (Figure 3).At the same time to follow the surface temperatureevolution was monitored , namely, water surfacetemperature, track foundation board surfacetemperature and pipe surface temperature.Equations underlying the mathematical model [3]and the initial and limit conditions are:

0V

(1)

uukx

puVu

(2)

g)(vvky

pvVv

m

(3)

LVtLh

ckhVh

(4)

2

30

1KK

where:τ - time, - density, p - pressure, V - velocityvector, u, v - velocity components on x and y axes,μ - dynamic viscosity, β - liquid fraction, L - latent

Trackfoundation

boardsurface

Sandsurface

Watersurface

Fig.3 The surfaces of the analyzed domain

Pipesurface

60

10

50

25

28

x [mm]

y [mm]

Fig.2 The grid of the analyzed domainFig.1 The geometry of the analyzed domain

Recent Advances in Fluid Mechanics and Heat & Mass Transfer

ISBN: 978-1-61804-026-8 278

heat, K - permeability, K0 - Kozeny-Carmanconstant.

fl

fsTTTCTTTC

h

T

refTpref dTchh

where: Tf – solidification temperature,Cs,Cl – specific heat for the solid and liquid phases ,h – sensitive enthalpy, Tref – reference temperature.href – reference enthalpy, cp – specific heat atconstant pressure

3 Numerical resultsWith numerical simulation of ice formation aroundthe pipe submerged in water, a number of 25 210iterations were made at a time step τ = 10 seconds.The total process running time was 7 hours,considering the process conclusion at the ice freesurface temperature of -5,55 0CFor the pipe is buried in sand a number of 18.000iterations at a time step of 10 s were carried out .The time taken for solidification process was 5hours, considering the process complete at an icefree surface temperature of -4,62 0C.Initialized parameters in the mathematical modelwere:water temperature on the skating track tapa = 100Cair temperature at the water free surface taer = 100Ctemperature of the refrigerating agent inside the pipetagent = -100Cmass flow fate fo the refrugerating agent

s/kg185.0m

convection coefficient in the pipe agent = 550W/m2Kair Convection coefficient on water surfaceaer= 15W/m2Kheat flow density o the track foundation boardq= 10W/m2.Fig. 4 and 5 illustrate the time variation of thewater surface temperature in the two cases analyzed:tube submerged in the water and pipe buried insand. When immersed in water (Figure 4) there is asharp decrease in temperature in the nodes above thepipe and a slower decrease in the points locatedmidway between the pipes. Under isothermal 0 0Cafter τ = 300min, there is an almost constantdistribution in nodes.

When the pipe is buried in the sand (Fig.5) ther isa more rapid decrease in temperature compatedwith the pipe submerged in water. Isothermal 0 0Creaches the water surface after approximately 90minutes. Temperature distribution in the watersurface nodes is almost uniform under isothermal0 0C, indicating a high quality of the ice.

t[0C]

x[m]

Fig.4 Temperature variation on the watersurface at pipe submerged in water

Fig.5 Temperature variation on the watersurface at pipe buried in the sand

t[0C]

x[m]

Recent Advances in Fluid Mechanics and Heat & Mass Transfer

ISBN: 978-1-61804-026-8 279

Fig.6 Temperature variation on the pipe surfaceafter τ =180 min

t[0C] t[0C]

x[m] x[m]

Fig.7 Temperature variation on the water-sandseparation surface

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

-0.03 -0.02 -0.01 0.00 0.01 0.02 0.03

1

2

x[m]

t[0C]

Fig.9 Temperature variation on the track foundationboard surface after τ = 180 min; 1- the pipe submergedin water; 2- the pipe buried in sand

-6

-5

-4

-3

-2

-1

0

1

2

-0.040 -0.030 -0.020 -0.010 0.000 0.010 0.020 0.030 0.040

1

2

x[m]

t[0C]

Fig.8 Temperature variation on the watersurface after τ = 180 min; 1- the pipe submergedin water; 2- the pipe buried in sand

Recent Advances in Fluid Mechanics and Heat & Mass Transfer

ISBN: 978-1-61804-026-8 280

Fig.6 presents the pipe surface temperaturevariation at τ = 180 min. Different temperaturedistribution is observed, namely in lateral directionand under the pipe temperature falls rapidlycompared with the area over the domain where theflow decreases slowly due to heat given by theexternal air.Unlike pipe submerged in water, when the pipe isburied in the sand there is a new and specific surfacearea of separation between water and sand layer. InFig 7. temperature variation is presented on thewater-sand separation surface at different times.Note that the 0C isotherm reaches the surface afterabout 30 minutes.Fig.8 provides for the water surface temperaturevariation in the two cases analyzed at τ = 180minutes. It may be noted that while the minimumtemperature when the pipe is immersed in water ist = 0.58 0C, if pipe is buried in the sand theminimum temperature is t = -4,72 0C. Moreover,changes in water surface temperature in the secondcase is much smaller which results in a high qualityof the ice . Another important aspect is thesolidification time which is much shorter in case ofthe pipe buried in sand.Figure 9 illustrates the difference of temperaturevariation on the foundation board under the pipe inthe two cases analyzed at the time τ = 180 minutes.For the same time interval , the temperature for pipeburied in sand decreases greatly compared with theother case.

4 ConclusionCompared with the submerged pipes, there is amuch faster decrease of surface temperature and amore uniform distribution at air-water contact.At theend of the process minimum temperatures on thesurfaces studied took the following values:- temperature of the inner surface of pipe = -8.4 0C- temperature on the bottom board surface of thetrack t = - 8,32 0C-temperature on sand-water contact surface

t =- 5,15 0C- ice surface temperature t = -4,62 0CFrom the data analyzed it follows that in terms ofheat transfer and ice quality , the sand buried pipedesign is recommended due to the followingaspects:- Complete water solidification time over the rangeconcerned is reduced- Water surface temperature distribution is uniformresulting in a high quality ice.- Solidification front is spreading quickly andevenly. - More uniform temperature distribution onthe inner surface of the pipe, so there is an increasein the heat transfer from the refrigerant to theskating rink

References:[1] Lewis R.W., Roberts P.M. - Finite elementsimulation of solidification problems. App.Scientific Research, Vol. 44, pp 61-92,1987.[2] Sasaguchi K., Viskanta R. - Phase-change heattransfer during melting and resolidification of meltaround cylindrical heat source(s)/Sinks, Jl. ofEnergy ResourcesTechnology, vol.111, 1989[3] Shih Y,Chou H, Numerical study ofsolidification around staggered cylinders in a fixedspace, Numerical Heat Transfer, No.40, 1997,pp1343-1354[4] Vick B, Nelson DJ, Yu X, Model of an ice-on-pipe brine thermal storage component, ASHRAETrans, No.102,1996, pp 45-54.[5] Vick B, Nelson DJ, Yu X, Freezing and meltingwith multiple phase fronts along the outside of atube, J Heat Transfer, No.120 (1998), pp. 422-429[6] Zhang R. Zhang C.,Jiang J., Direct Solution of2D Heat Transfer Problems in Frequency Domainwith Dynamic Boundary Conditions Proceedings ofthe 7th WSEAS International Conference onSimulation, Modelling and Optimization, Beijing,China, September 15-17, 2007.[7] FLUENT Inc, Fluent user’s guide, 2003.

Recent Advances in Fluid Mechanics and Heat & Mass Transfer

ISBN: 978-1-61804-026-8 281