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doi:10.1016/j.bu

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Building and Environment 40 (2005) 417–426

www.elsevier.com/locate/buildenv

Numerical prediction of ventilation patterns and thermalprocesses in ice rinks

O. Bellachea, M. Ouzzanea, N. Galanisb,�

aCANMET Energy Technology Centre Varennes QC Canada J3X 1S6bTHERMAUS, Faculte de genie, Universite de Sherbrooke, Sherbrooke QC, Canada J1K 2R1

Abstract

A numerical simulation of the heat and mass transfer phenomena in a ventilated ice rink has been carried out using the standard

k2e turbulence model. The complexity of the geometry has been taken into account by considering elements such as the plastic

protection panels which have a significant effect on the flow field. The results calculated for four different configurations show the

flow pattern, the isotherms and the lines of constant absolute and relative humidity. Values of the effective draft temperature are

presented for several positions in the zone occupied by the spectators. The heat losses through the walls and ceiling as well as the

latent, convective and radiative heat flux into the ice are presented and analyzed.

r 2004 Elsevier Ltd. All rights reserved.

Keywords: Numerical simulation; Turbulent mixed convection; Ice rinks; Ventilation; Heating; Thermal comfort; Humidity; Radiation;

Refrigeration load; Heat losses

1. Introduction

Canadian ice rinks use approximately 3500GWh ofelectricity annually and generate 5� 105 tons of gasescontributing to the greenhouse effect. The potential forimprovement in both areas is substantial. However, noconcerted effort has been undertaken to develop designprocedures and specify operational guidelines whichwould reduce energy consumption and gas emissionswhile maintaining the quality of the indoor air and ice aswell as the comfort of skaters and spectators. Thechallenge is great because of the diversity of size andconfiguration of the buildings, and the differences inheating and ventilation systems in use. This complexityand the coupling between air movement, heat transferthrough the building envelope, heat and mass transferbetween the air and ice surface, radiation exchangesbetween the surfaces, pollutant and water vapour

e front matter r 2004 Elsevier Ltd. All rights reserved.

ildenv.2004.08.004

ing author. Tel.: +1-819-821-7144; fax: +1-819-821-

ess: Nicolas.Galanis@USherbrooke.ca (N. Galanis).

dispersion in a large irregular domain, explain the lackof design and operation norms.The development of reliable CFD codes offers the

potential for systematic analysis of the velocity, tem-perature, pollutant and/or vapour concentration dis-tributions in ice rinks and other large buildings.Such numerical studies have been undertaken in 2Dand 3D configurations by Chen et al. [1], Yang [2],Nielsen et al. [3], Jones and Whittle [4]. However,these studies do not generally take into account theinteraction between convection, radiation, vapour diffu-sion and mass transfer between the ice surface and theair. They do not include the influence of thesephenomena on heat losses through the envelope.Furthermore, previous studies have not evaluated theeffect of these heat and mass transfer processes onthe refrigeration load. Finally, no CFD study hasevaluated systematically the effect of different ventila-tion and heating systems on energy consumption andthermal comfort.In view of this situation, the present project aims to

develop a model for ice rinks which takes into account

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Nomenclature

C absolute humidity (kg/kg)ðCe1;Ce2;Ce3;Cm;CdÞ turbulence model constantsDin characteristic length (m)g acceleration of gravity ðm=s2ÞGB buoyancy production of kG stress production of kH in height of the inlet (m)Lin width of the inlet (m)P static pressure (Pa)Prt turbulent Prandtl numberqi heat flux for each element i of the envelope

(W)qci convective flux between the air and inside

surface of the element i (W)qri net radiative heat flux (W)qli heat latent (W)T temperature ð

�CÞTc average (control) space dry bulb temperature

ð�CÞ

t time (s)U in inlet velocity (m/s)

V*

velocity vectoru,v components of velocity (m/s)Sf source termx horizontal coordinate (m)y vertical coordinate (m)bT coefficient of thermal expansion ðK�1Þ

bc coefficient of mass expansionGf diffusion coefficiente dissipation rate of turbulent kinetic energy

ðm2=s3Þk turbulent kinetic energy ðm2=s2Þm dynamic viscosity (kg/m s)n kinematic viscosity ðm2=sÞr density ðkg=m3Þ

st; se;sk turbulence model constantsf general field variable

O. Bellache et al. / Building and Environment 40 (2005) 417–426418

all of the above phenomena and predicts energyconsumption as well as ice and comfort conditions.For this purpose, we have calculated results for winterconditions and a particular ice rink in Montreal,Canada using the standard k2e model to simulate theflow field and taking into account radiation exchangesbetween the inside surfaces of the envelope and the ice.In an earlier paper [5], we have presented numericalpredictions for the velocity and temperature fields aswell as comfort conditions and heat fluxes through thebuilding envelope for four different combinations ofinlet air conditions (temperature and direction of theventilation air). In the present paper, we extendthe formulation by including mass transfer betweenthe ice surface and the air. Four different configurationsof the ventilation system are considered and results withand without vapour diffusion and condensation arepresented and compared in each case.

Fig. 1. (a,b) Cross section of the ice rink under consideration.

2. Modeling and calculation procedure

A schematic representation of a cross-section of theice rink is shown in Fig. 1. Since the length of thebuilding is considerable (64m) the flow field far fromthe end walls is considered to be two-dimensional (novelocities perpendicular to the shown cross-section). Asshown in Fig. 1, three rows of stands run the wholelength of the building on one side. A narrow corridorencircles the ice surface. Protective transparent barriersseparate the corridor and stands from the skatingsurface. Heating and ventilation for the spectators is

provided by blowing Qin ¼ 6:75m3=s of air througheight regularly spaced inlets (0:3m� 1m; each at aheight of 6m, above the ice surface). Air is extractedthrough a single outlet on the ceiling but for thepurposes of this study we have considered that there areeight smaller openings at the same Z coordinate as theinlets. Therefore, the flow field in the X–Y plane midwaythrough each set of inlets and outlets can be considered

ARTICLE IN PRESSO. Bellache et al. / Building and Environment 40 (2005) 417–426 419

to be two-dimensional (no velocity component in the Z

direction) and the numerical calculations in that planecan be considerably simplified.Fig. 1a with the air outlet to the right of the inlet

represents the actual configuration of the ventilationsystem. Fig. 1b with the air outlet to the left of the inletrepresents the configuration originally proposed in thearchitectural plans. In both cases the ventilation airenters through a 0.3m opening at x ¼ 27:25m; y ¼ 6mand leaves through a 1m wide opening in the ceilingðy ¼ 8mÞ; respectively, at x ¼ 30m in the first config-uration (Fig. 1a) and at x ¼ 17:63m in the secondconfiguration (Fig. 1b). The origin of the Cartesiancoordinate system is at the bottom left corner of thecross-section.Several turbulence models for the simulation of

indoor air motion have been compared during thepresent project, although their results are not presentedhere. It was found that the standard k2e model byLaunder and Spalding [6] performed quite well and wasvery stable. Chen [7] also has compared the performanceof five turbulence models in predicting natural convec-tion, forced convection, and mixed convection in rooms,and used experimental data for their validation. Hefound that the prediction of the mean velocity is moreaccurate than that of the turbulent velocity and statedthat the performance of the standard k–e model is good.Therefore, this model was selected for the presentinvestigation.The air is considered to have constant properties

except for its density in the expression of the bodyforces, which is assumed to vary linearly with bothtemperature and concentration (Boussinesq hypothesis).Its absorptivity is set equal to zero (totally transparentto radiation of all wavelengths). The flow is assumed tobe turbulent with mean values independent of time.The partial differential equations modeling the air

movement, heat transfer and mass diffusion can bewritten in the following general form:

@ðrfÞ@t

þ divðrV*f� GfrfÞ ¼ Sf: (1)

Table 1

Values of f; Gf and Sf

Equation f Gf

Mass 1 0

X-momentum U mþ mtY-momentum V mþ mtEnergy T m=Pr þ

Absolute humidity C ðmþ mTurbulent kinetic Energy k mt=skDissipation rate e mt=se

mt ¼ rCmCdk2=e; Prt ¼ 1

GB ¼ �gbðmt=PrtÞð@T=@xiÞ; G ¼ mtð@Ui=@xj þ @Uj=@xiÞ:ð@Ui=@xjÞ

Ce1 ¼ 1:44; Ce2 ¼ 1:92; Ce3 ¼ 1; Cm ¼ 0:5478; Cd ¼ 0:1643; sk ¼ 1; se ¼ 1

The appropriate values or expressions forf; Gf and Sf

are defined in Table 1.The boundary conditions for this problem are as

follows:

m

tÞ=

:3;

The ventilation air enters through the inlet at 30 �Cand 45% relative humidity with horizontal andvertical velocities equal to 2.435m/s and �1:406m=srespectively. The turbulence kinetic energy kin anddissipation rate ein are

kin ¼ ð0:05U inÞ2; ein ¼

ðCmCdÞ3=4k3=2in

0:1Din;

Din ¼ 2HL=ðH þ LÞ: ð2Þ

At the outlet the relative pressure is set equal to zero. A uniform temperature T ice ¼ �5:5 �C is imposedunder the ice at Y ¼ 0; for the calculation withvapour diffusion, the air layer in contact with the iceis assumed to be saturated. On all the other solid surfaces (walls, ceiling, stands)the velocity components, concentration gradient andturbulent kinetic energy are equal to zero. Thecorresponding temperature is related to that of theair within the flow domain and to the externaltemperature by an energy balance which takes intoaccount conduction through the solid, convection tothe inside and outside as well as net radiation fluxesbetween the inside surfaces. Thus, the heat flux qi foreach element i of the inside surfaces is equal to thefollowing algebraic sum [8]:

qi ¼ qci þ qri þ qli: (3)

The last term of Eq. (3) denotes the latent heatexchanged between the air and the ice.It should be noted that the seven partial differential

equations modeling the heat and mass transfer processesin the ice rinks are elliptic, non-linear and coupled sincethe air density depends on both the temperature and theabsolute humidity.

Sf

0

�@P=@x

�@P=@y � rgbTðT � T rÞ � rgbcðC � CrÞ

t=Prt ST

sc Sc

G � reþ GB

½eðCe1G � Ce2reÞ=k� þ Ce3GBðe=kÞ

sc ¼ 1:

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Fig. 2. Validation of the model and numerical code (a) Profile of

vertical component of velocity near the hot wall in section z ¼ 65mm;(b) Profile of vertical component of velocity near the hot wall in section

z ¼ 125mm:

O. Bellache et al. / Building and Environment 40 (2005) 417–426420

3. Numerical solution and validation

The simulations have been carried out with thenumerical code PHOENICS [9] which uses the finitevolume method and a staggered grid. The hybrid schemewas used for the numerical solution. The SIMPLEalgorithm [10] was used for the pressure correction andthe solution is obtained iteratively starting from anarbitrary initial condition. The calculation domain wassubdivided into zones corresponding to the existingphysical discontinuities appearing in Fig. 1. In thepresent case, there are 15 zones in the x direction and 11in the y direction. The number of grid points, or finitevolume, within each resulting subdivision of the domainhas been selected based on a preliminary analysis toensure that the results are independent of their number.The discretisation grid was non-uniform with highernode density near the inlet and the solid surfaces wheregradients are high. Numerical tests have also beencarried out to ensure that the results are independent ofthe number of iterations. Under-relaxation was oftennecessary to achieve convergence which was declaredwhen the cumulative residuals for each of the conserva-tion equations was less than 10�6:The experimental data by Cheesewright et al. [11] was

used to validate the predictions of the model. Thatexperiment was carried out using air in a squareenclosure 0:25m� 0:25m with adiabatic horizontalwalls and isothermal vertical walls. The temperaturedifference between the hot and cold walls was 43:4 �C:Fig. 2a shows the measured velocity vectors while thecorresponding results computed by the present modeland code are shown in Fig. 2b. The agreement betweenour predictions and the measured values is well withinthe uncertainty of the latter. Similarly, calculatedvelocities and temperatures for mixed convection of airin a rectangular enclosure with isothermal horizontalwalls and diabatic vertical walls were found to agreeclosely [5] with corresponding measurements by Bra-connier et al. [12]. Finally, the predicted air, ice and walltemperatures were successfully compared with corre-sponding values measured in the actual ice rink [5]. Inview of these successful validations, we consider that theselected model and calculation procedure are reliableand accurate. They can therefore be used for theparametric studies.

4. Results and discussion

The results presented here were obtained using morethan 100� 40 grid points and over 15� 103 iterations.The external temperature and the corresponding con-vection coefficient were considered constant and equalto �10 �C and 7W=m2K; respectively. The total errorfor mass, momentum and energy is, in all reported cases,

less than 1%. Four cases were studied for each of thetwo configurations shown in Fig. 1:

Case 1: the vertical duct providing the ventilation andheating air is supposed to constitute a physical barrier tothe airflow extending from Y ¼ 6m to the ceiling. Air isextracted through a single outlet on the right of the inlet(Fig. 1a).

Case 2: the inlet duct constitutes a physical barrier butthe outlet is situated on the left of the inlet (Fig. 1b).

Case 3: inlet without vertical duct and air is extractedthrough a single outlet on the right of the inlet as incase 1.

Case 4: inlet without vertical duct and the outlet issituated on the left of the inlet as in case 2.Calculations were performed twice for each of those

four cases. At first, vapour diffusion was neglected as in

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7.4 7.68.0 8.3 8.8 9.1 9.4 9.6

10.625.4

24.519.7

29.9

-5.5-

0.0058

0.01

0.0099

0.00740.0073

0.0072

0.0071

31

7579

828487899092

94

45

(a)

(b)

(c)

(d)

(e)

Fig. 3. Velocity vectors, streamlines, isotherms, absolute and relative

humidity for case 1.

O. Bellache et al. / Building and Environment 40 (2005) 417–426 421

our previous article [5]. Then the species conservationequation and the corresponding boundary conditionswere introduced in the model. A comparison of theresults calculated with these two formulations hasshown that the difference between correspondingvelocities and temperatures is rather small. This isattributed to the fact that, for the conditions underconsideration here, the range of absolute humidityvalues is small (0.01 kg/kg at the air inlet and0.0058 kg/kg just above the ice). Therefore, the corre-sponding variation of the humid air density is negligibleand the coupling between the species conservationequation and the mass, momentum, energy equationsfor the humid air is weak. In view of these observations,the description of the flow filed in Sections 4.1 and 4.2 isonly presented for the case with vapour diffusion. Onthe other hand, in Section 4.3, the calculated refrigera-tion load is presented for both formulations to illustrateits dependence on mass transfer between the ice andthe air.

4.1. Velocity, temperature and humidity distributions

Fig. 3 shows the principal characteristics of the flowfield for the first case. As illustrated by the velocityvectors in Fig. 3a and the streamlines in Fig. 3b, becauseof the relative proximity of the air inlet and outlet, thewarm ventilation air forms a rather narrow jet whichfollows a curved path from left to right between thesetwo openings. By entrainment, this jet creates twovortices. The first one is situated near the ceiling betweenthe inlet duct and the air outlet. It is very nearly circularand rotates in the anti-clockwise direction. The secondone, situated between the ventilation jet and the stands,is much weaker and rotates in the clockwise direction. Ithas the form of a boomerang, extending from near thetop right corner of the domain to approximately thebottom of the inlet duct. Near the right wall itsmovement is reinforced by the downward action of thebuoyancy force. The velocities in the right side corridoras well as between the protective transparent barrier andthe stands are negligible and therefore this area is poorlyventilated. The quality of the air herein risks to be poorif smoking is permitted. Similarly the velocities to theleft of the inlet duct, particularly those close to the icesurface, are very small. This numerical deduction wasconfirmed experimentally using a hot wire anemometerwhich did not register any significant air movement overthe ice for Yo1m: The isotherms in Fig. 3c indicate theexistence of very high temperature gradients normal tothe ice and the right wall, i.e. in the regions where thevelocities are small. On the other hand, a comparison ofair temperatures with those of the left wall shows thatthe temperature gradient normal to that wall is muchsmaller. For the ceiling, this gradient has intermediatevalues which increase from left to right up to the inlet

duct. Fig. 3d shows the absolute humidity in the ice rink.The absolute humidity is generally low on the left side ofthe inlet and high in the zone occupied by the spectators.Its variation is significant over the ice, while it does notvary much near the stands. This is consistent with theflow field since ventilation air does not enter in the leftside of the inlet. Finally, Fig. 3e shows that the relativehumidity is quite high above the ice but rather lowabove the stands. This is a direct consequence of theflow pattern which results in relatively high tempera-tures in the well-ventilated region above the stands andin low temperatures above the ice.The results of case 2 are shown in Fig. 4. The

ventilation air forms a fairly narrow jet which flowsfrom the inlet towards the stands, turns downwards andmoves leftwards towards the outlet. The model predictsa clockwise vortex situated above this jet between theinlet and exit. In the left half of the domain, the modelpredicts a weak anti-clockwise vortex with particularly

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29.9

20.120.4

20.821.1

21.6

22.6 24.1 27.8

27.8

-5.5.

0.0058

0.0099

0.0099

0.00910.0089

0.0088

0.0086

32 27

3639

4142

42

(a)

(b)

(c)

(d)

(e)

Fig. 4. Velocity vectors, streamlines, isotherms, absolute and relative

humidity for case 2.

-5.5

29

.9

17.416.0

15.314.914.614.4

25.525.1

0.00580.0078

0.0081

0.00810.0084

0.0094

45

5456

57585960606

1

30

31

(a)

(b)

(c)

(d)

(e)

Fig. 5. Velocity vectors, streamlines, isotherms, absolute and relative

humidity for case 3.

O. Bellache et al. / Building and Environment 40 (2005) 417–426422

small velocities near the ice. On the right of the inlet twovortices are generated: the one closest to the inlet is inthe anti-clockwise direction while the one closest to theright wall is in the clockwise direction. The latter isreinforced by the buoyancy force which acts downwardsnear the wall. Air temperatures over the ice are muchhigher than for the first case since the ventilation air is inthe present case flowing above the ice. Fig. 4d showsthat the absolute humidity on the left of the air inlet ishigher than in the first case since the outlet position hasbeen changed. On the other hand, to the right of theinlet and over the stands the absolute humidity variesvery little since there is little air renewal in that region.Fig. 4e shows that in this case the relative humidity ismuch lower than that shown in Fig. 3e, especially in theregion over the ice. This observation is consistent withthe fact that, in the present case (Fig. 4), the

corresponding temperatures are much higher as a resultof the path of the ventilation air.Fig. 5 shows the corresponding results for case 3 for

which the air inlet and outlet are in the same positions asfor case 1. The difference between these two cases is dueto the presence (case 1) or absence (case 3) of the verticalinlet duct which constitutes a physical barrier to theairflow. A comparison of Figs. 5a and b with 3a and bshows that the main characteristics of the velocity fieldare very similar, with one important exception: the twoanti-clockwise vortices situated above the ice and abovethe narrow ventilation jet in Fig. 3b have coalesced intoa single one in Fig. 5b. This communication of the left(above the ice) and right (above the stands) sides of theflow domain has very important effects on the tempera-ture and humidity distributions. Indeed, as illustrated inFigs. 5c and d the air temperature and its absolute

ARTICLE IN PRESSO. Bellache et al. / Building and Environment 40 (2005) 417–426 423

humidity above the ice are significantly higher for case 3than for case 1. This is due to the fact that in case 3 heatand vapour are transmitted from the right to the left sideacross the top of the domain. This transfer is notpossible in case 1 due to the presence of the vertical inletduct which obstructs mass and energy transfers betweenthe two sides. It is important to also note that, eventhough absolute humidity above the ice is higher in case3 than in case 1, the corresponding relative humidity islower in case 3. This is due to the fact that the increaseof absolute humidity which influences the partialpressure of the vapour is less important than theincrease in temperature which results in an increase ofthe saturation pressure.Fig. 6 shows the results for case 4 (positions of inlet

and outlet identical to case 2 but without the obstruction

-5.5

29.9

29.4

26.324.5

23.222.321.721.220.8

0.00580.0079

0.0081

0.00820.0085

0.0098

0.0098

24

2530

343537383940

41

(a)

(b)

(c)

(d)

(e)

Fig. 6. Velocity vectors, streamlines, isotherms, absolute and relative

humidity for case 4.

corresponding to the vertical inlet duct). There is nosignificant difference in the flow patterns of these twocases in the region above the ice. On the other hand,above the stands, the patterns in Figs. 6a and b are quitedifferent from those in Figs. 4a and b. The two verticallyelongated counter-rotating vortices of case 2 have beensucked into the low-pressure region above the narrowventilation jet and their long axis is now horizontal. Thetemperature and absolute humidity distributions arefairly similar. However, it should be noted that the lowerleft corner of the domain is warmer in case 4 while theabsolute humidity near the ice is higher in case 2.

4.2. Comfort conditions

Engineers and architects are not interested only in thedistribution of the basic variables such as velocity andtemperature, but also need information on derivedquantities to assess occupant comfort. For this purpose,we programmed the calculation of the effective drafttemperature (EDT) in the output routine of PHOE-NICS. The EDT represents the difference in tempera-ture between any point in the occupied zone and thecontrol condition and is calculated from the followingequation [13]:

EDT ¼ ðT � T cÞ � 8ðV � 0:15Þ: (4)

Positive values of this parameter indicate a feeling ofwarmth while negative values indicate a feeling ofcoolness. Ideally the EDT should be between �1:5 andþ1 �C and the air velocity should be less than 0.35m/s.Fig. 7 shows the values of EDT for each of the four

cases specified earlier at two different heights above eachof the three levels in the stands. Values denoted by theletter A are at 0.2m above the concrete stands andcorrespond to conditions near the ankles of thespectators. Values denoted by the letter N are at 1.2mabove the concrete stands and correspond to conditionsnear the neck of the seated spectators. The numbers 1, 2,3 associated with the letters A and N refer to the lowest,intermediate and highest levels of the stands (Fig. 1).These results indicate that all cases lead to uncomfor-tably warm conditions ðEDT41 �CÞ everywhere. This isdue to the relatively high temperature of the inlet air andto its initial inclination which results in fairly lowvelocities in the area occupied by spectators for bothoutlet positions under consideration.

4.3. Heat transfer calculation

Table 2 shows the heat losses through the walls andceiling of the building per unit length perpendicular toFig. 1 for the four cases under consideration and the twoadopted formulations (with and without mass diffu-sion). These values reflect the temperature distributionsof Figs. 3c, 4c, 5c and 6c since the outside temperature

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x

xxxxxxx* * * *

++++++

++ +

26 28 30 32 340

1

2

3

4

5

6

7

8

9

10

A1N1A2N2A3N3

x

*

+

Case 1

x

xxxxxxx *

* * *++++

++

+ +

+

26 28 30 32 340

1

2

3

4

5

6

7

8

9

10

A1N1A2N2A3N3

x

*

+

Case 2

x

xxxxxxx * * * *

++++++

+

++

26 28 30 32 340

1

2

3

4

5

6

7

8

9

10

A1N1A2N2A3N3

x

*

+

Case 3

xxxxxxx

x

* * * *

+++++++

+

+

X (m)X (m)

X (m)X (m)

ED

T (

°C)

ED

T (

°C)

ED

T (

°C)

ED

T (

°C)

26 28 30 32 340

1

2

3

4

5

6

7

8

9

10

A1N1A2N2A3N3

x

*

+

Case 4

Fig. 7. Calculated EDT profile at two heights above the stands (A and N are at 0.2m and 1.2m above the stands, respectively).

Table 2

Calculated heat losses (W/m)

Case no. Without mass transfer With mass transfer

1 608 610

2 838 837

3 801 803

4 877 902

O. Bellache et al. / Building and Environment 40 (2005) 417–426424

and heat transfer coefficient are the same for all cases.Therefore, since the indoor temperature distributionsare not influenced by mass diffusion, the heat losses withand without mass transfer are the same. They are lowestin case 1, since as shown in Fig. 3c most of the envelopeis in this case exposed to rather cool air, and highest incase 4 since in this case the air near the walls and ceilingis warmest. When the air outlet is to the left of the inlet

(cases 2 and 4) the heat losses are higher than when it isto the right of the inlet (cases 1 and 4) since in the lastcase a smaller part of the envelope is in contact withwarm air.Table 3 presents energy fluxes into the ice. Obviously,

for the formulation without mass transfer, the latentheat load is zero. On the other hand, for the formulationwith mass transfer this quantity is important andcorresponds to solidification of water vapour on theice. The rate of solidification is proportional tothe absolute humidity gradient perpendicular to the icesurface while the enthalpy of solidification depends onthe air temperature near the ice. Therefore, the latentheat load is lowest in case 1 since both the absolutehumidity and the air temperature above the ice areclosest to the corresponding values of the ice surface.Cases 2 and 4 with the air outlet to the left of the inletresult in higher latent heat loads than cases 1 and 3 since

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Table 3

Calculated energy fluxes into the ice (W/m2)

Latent heat load Convective heat load Radiation heat load Refrigeration load

Without mass transfer

Case 1 0 24 49 73

Case 2 0 44 83 127

Case 3 0 31 79 110

Case 4 0 50 79 129

With mass transfer

Case 1 36 25 48 109

Case 2 67 44 81 192

Case 3 52 32 79 163

Case 4 88 47 78 213

O. Bellache et al. / Building and Environment 40 (2005) 417–426 425

in the last two cases the air near the ice is cooler andcontains less vapour.Column four presents the convective heat flux

between the air and the ice. This flux is proportionalto the temperature difference between the air and the ice.It also depends on the local air velocity which influencesthe heat transfer coefficient. Therefore, although asstated earlier mass diffusion has little influence on thetemperature and velocity fields, the combined effect ofthese two variables results in an increase of almost 3%of the convection heat load for case 4 when massdiffusion is taken into consideration. The convectiveheat flux for cases 2 and 4 (with the air outlet to the leftof the inlet) is higher than the corresponding values forcases 1 and 3 (with the air outlet to the right of the inlet)since the air temperatures over the ice are considerablyhigher in cases 2 and 4 due to the different air flowpattern.Since the width of the domain is approximately four

times greater than its height, most of the radiationabsorbed by the ice originates at the ceiling. Therefore,the values in the fifth column of Table 3 are highestwhen the ceiling temperature is important. This ob-servation and the results in Figs. 3c, 4c, 5c and 6cexplain the small values of the radiation heat load forcase 1 and the fact that in the other three cases this fluxis almost the same. Consistently with previous remarks,the influence of mass diffusion on this flux is small.Finally, the summation of the three heat fluxes into

the ice is presented in the last column of Table 3. This isnot the total refrigeration load since these calculationshave not taken into account the contribution of iceresurfacing, system pump work, ground heat, andskaters. These values are considerably higher whenmass diffusion and vapour condensation on the icesurface are considered. The sum is minimum for case 1without mass diffusion for which only the right-handside of the building is maintained at high temperatures(see Fig. 3c). It is maximum for case 4 with massdiffusion for which the temperature of the air is

everywhere above 20 �C and the rate of condensationis also the highest.It is interesting to note that the magnitude of the

latent, convective and radiation heat loads presented inthe last four lines of Table 3 are very similar to thecorresponding data presented in ASHRAE Refrigera-tion Handbook (2002), [Chapter 34]. Specifically,according to this source the heat loads for indoors icerinks vary between 2.1 and 6:3m2=W while the latent,convective and radiation contributions are, respectively,15%, 13% and 26%. In terms of heat fluxes thecorresponding values are 71–24W=m2; 62–21W=m2

and 138–44W=m2 for, respectively, the latent, convec-tive and radiation loads. Thus, 11 of the 12 calculatedvalues lie within these measured values.

5. Conclusion

CFD calculations of air flow, heat transfer and massdiffusion in an indoor ice rink with warm air heatinghave been successfully carried out using the standardk2e model for turbulent mixed convection. The resultshave been obtained for two different positions of the airoutlet and indicate that the one initially envisaged in thearchitectural plans results in higher heat losses throughthe ceiling and walls as well as in a higher refrigerationload. However, neither of the two air outlet positionsgive satisfactory results as far as air renewal in the areaoccupied by the spectators is concerned, although thesituation is somewhat better with the outlet overthe stands. Finally, the calculated results confirm thatthe effects of vapour condensation and radiation on therefrigeration load are significant.

References

[1] Chen Q, Hoornstra TG, Vanderkoon J. Energy analysis of

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