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Applied Thermal Engineering 28 (2008) 1782–1790

Calculation of refrigeration loads by convection, radiationand condensation in ice rinks using a transient 3D zonal model

Ahmed Daoud a,*, Nicolas Galanis a, Omar Bellache b

a Departement de Genie Mecanique, Universite de Sherbrooke, Sherbrooke, Quebec, Canadab CANMET Energy Technology, Natural Resources Canada, Varennes, Quebec, Canada

Received 27 February 2007; accepted 12 November 2007Available online 22 November 2007

Abstract

This article presents the results of simulations using a zonal airflow model, a radiation model and a humidity model to predict the heatfluxes, the transient airflow patterns as well as the temperature and absolute humidity distributions in a 3D section of an indoor ice rinkin Montreal. The results of the simulations are in good agreement with published CFD (Computational Fluid Dynamics) results andexperimental measurements. The principal advantage of using the present model to simulate the thermal and airflow fields in an ice rinkis the computational time which is very short compared to that used by a CFD model. The effects of the ceiling emissivity and the tem-perature set point in the stands on heat transfer toward the ice were studied by conducting a parametric study.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Zonal models; Simulation; Airflow; Enclosure; Ice rink; Thermal load

1. Introduction

Canadian ice rinks use approximately 3500 GWh of elec-tricity annually and generate 5 � 105 tons of gases contrib-uting to the greenhouse effect. The potential forimprovement in both areas is substantial. Some efforts arebeginning to be undertaken to develop design proceduresand specify operational guidelines which would reduceenergy consumption and gas emissions while maintainingthe quality of the indoor air and ice as well as the comfortof skaters and spectators. The challenge is great becauseof the diversity of size and configuration of the buildings,and the differences in heating and ventilation systems inuse. This complexity and the coupling between air move-ment, heat transfer through the building envelope, heatand mass transfer between the air and ice surface, radiationexchanges between the surfaces, pollutant and water vapourdispersion in a large irregular domain, explain the lack ofdesign and operation norms. Numerical studies have been

1359-4311/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.applthermaleng.2007.11.011

* Corresponding author. Tel.: +1 819 8218000.E-mail address: ahmed.daoud@usherbrooke.ca (A. Daoud).

undertaken in 2D and 3D configurations by Jian and Chen[1], Yang et al. [2], and Jones and Whittle [3]. However,these studies did not take into account the interactionbetween convection, radiation, vapour diffusion and masstransfer between the ice surface and the air. They did notevaluate the effect of these heat and mass transfer processeson the refrigeration load and on heat losses through theenvelope. These phenomena were included in the recentnumerical studies by Bellache et al. [4–6] which are howeverlimited to 2D configuration. These studies, which used aCFD code to calculate temperature and velocity distribu-tions required considerable computer memory and CPUtime. On the other hand, Daoud and Galanis [7] proposeda different approach for the simulation of an ice rink by cou-pling a zonal model with a commercial code [8]. Some phe-nomena like long wave radiation exchange and vapourdispersion were not included in that simulation.

The present project is the first step in the development ofa 3D transient model for ice rinks based on the work ofDaoud and Galanis [7] which takes into account all of theabove phenomena and predicts refrigeration loads as wellas ice and comfort conditions with relatively modest

Nomenclature

A area (m2)cp specific heat at constant pressure (J/kg K)Cd discharge coefficienth height (m)G Gebhart coefficientP static pressure (Pa)Q power (W)T temperature (�C)x absolute humidity (kgmoisture/kgdryair)x,y,z cartesian coordinates

Greek symbols

q density of the air (kg/m3)e constant depending on flow direction (±1)

ei surface emissivityr Stephan Boltzmann constant

Subscripts

i cell i or surface i

i,j between surface or cell i and j

sat saturationice ice surfacevap vapoursl solidificationcond condensation

A. Daoud et al. / Applied Thermal Engineering 28 (2008) 1782–1790 1783

computer capabilities. For this purpose, we have calculatedresults for a typical year and a particular ice rink inMontreal, Canada using a modified version of a commer-cially available program [8] to simulate this building. Theflow field was calculated with a zonal model which was spe-cifically developed and integrated into the commercial pro-gram. The long wave radiation exchange between surfaceswas calculated using Gebhart’s model which was also addedto the commercial code for the present project. Watervapour transport and the condensation heat flux on theice surface were also taken into account. The comparisonsof these results with corresponding measurements and withvalues calculated by using CFD or standard components ofthe commercial program are presented and analysed.

2. Description and modeling of the ice rink

The studied building is based on the Camilien Houde icerink located in Montreal, Canada whose principal dimen-sions and characteristics are shown in Fig. 1. Its length is64 m in the direction perpendicular to the section depictedin Fig. 1. The physical objects (suspended ceiling over thestands; partitions between the ice, the corridors and thestands) are taken into account in the 3D transient modelingof the building.

Eight radiant heaters (8 � 22 kW) and seven air inlets(7 � 1.4 m3/s) situated above the stands and regularly dis-tributed along the length of the building are used to heatand ventilate the space occupied by the spectators. Theair exits through outlets on the east, north and south walls.The infrared heaters are controlled with a thermostat,whose set point is 15 �C, placed on the wall near theentrance of the ice rink. The ventilation functions simulta-neously with the heaters and resurfacing. Heated spaces,located behind and under the stands, are heated by a con-ventional hot water system.

Heat gains from lighting are 10 W/m2 above the ice and5 W/m2 above the stands. The contributions of heat and

moisture from the people in the building are not taken intoaccount in the conservation equations. The resurfacing ofthe ice is assimilated to the application of a layer of1 mm of water at 60 �C on the ice [9]. For the calculations,all of these inputs are modulated according to a schedulewhich differentiates between weekdays and weekends [5].

In this work we simulate a section of the ice rink whichincludes only one infrared heater, one ventilation inlet andone outlet (Fig. 2). For calculation purposes, this section ofthe ice rink is subdivided into 36 zones. Lengthwise it isdivided into two parts: the length of the first, whichincludes a radiant heater, is 0.56 m while that of the sec-ond, which includes an air inlet, is 0.76 m. Over the ice(Fig. 3) the height of the bottom zones is 0.2 m. The twomiddle zones have a height of 5 m and 2 m, respectively,and the top zones have a height of 2.16 m.

For the simulations, we used hourly average weatherdata (temperature of the air and solar radiation) for theMontreal area [10]. The boundary conditions consideredin the simulation are specified in Fig. 3. The time step usedfor these simulations is 360 s (0.1 h) which allows us to ana-lyse the functioning of the radiant heaters.

The following sections present the relations used tomodel the different coupled phenomena occurring in theice rink.

2.1. Energy transfer model

The energy transfer model which uses a commercialcode [8] has been previously described by Daoud and Gal-anis [7]. It is based on the following two relations:

– The energy conservation equation for each internal sur-face in contact with the air in the building, which is:

qcd ¼ qcv þ qrd þ qcond þ qrs ð1Þ

The conductive flux through the wall, qcd, is calculatedby commercial code [8] using the physical characteristics

Fig. 1. Schematic representation of the ice rink.

Radiant heater

Fan

Air exit

l=1.32m

L=25.9m

h=9.36m

Fig. 2. Position 3D representation of the simulated ice rink section of theradiant heater, the fan and the air exit.

y

x

Text

Text

0 °C 0 °C

21 °C

-5 °C

21 °C

Fig. 3. Zones and boundary condition in the simulated configuration.

1784 A. Daoud et al. / Applied Thermal Engineering 28 (2008) 1782–1790

of the wall while the convective flux, qcv, between thewall surface and the air in the building is evaluated usinga constant heat transfer coefficient (3 W/m2 K). The netradiation flux, qrd is evaluated by the method describedin Section 2.3 of the present paper. The last two terms,qcond and qrs, represent the condensation and resurfacingfluxes and are only considered in the case of the ice sur-face. The method used for the calculation of the formeris described in Section 2.5 while the resurfacing flux isevaluated from the relation suggested by ASHRAE [9]and the specifications given previously.

– The energy conservation equation for the air in zone i,which is:

ðMcpÞidT i

dt¼ ðqcvAÞi þ

Xj

_mj!icpT j þ _Qs ð2Þ

Since there are 36 zones, there are 36 such equationsrelating the 36 unknown zone temperatures to the massflow rates between adjacent zones. The last term on theright hand side of this equation is a source term which inthe present study is not taken into account.

2.2. Zonal airflow model

Fauconnier et al. [11], Bouia and Dalicieux [12] andfinally Wurtz [13] contributed to the development of zonalmodels for the prediction of air flows and temperaturefields in a building, based on the calculation of the pressurefield. In this zonal method, the studied space is divided intoa small number of control volumes, also called cells orzones, in which the temperature and the density of the airare supposed to be homogeneous, whereas the pressurevaries hydrostatically. The gas mixture is treated like a per-fect gas. The mass transfer between zones is expressedusing models based on the flows in conduits. The mass flowcrossing the surface common to both zones i and j dependson the difference in pressure between these two cells. Thus,in the case of a vertical interface

_mi;j ¼ ei;j � Cd �ffiffiffiffiffiffiffi2qi

p� Ai;jjP j � P ij0:5 ð3Þ

while in the case of a horizontal interface

_mi;j ¼ ei;j � Cd �ffiffiffiffiffiffiffi2qi

p� Ai;j ðP j � P iÞ �

1

2ðqighi þ qjghjÞ

��������0:5

ð4Þ

where Ai,j is the surface separating cells i and j with staticpressure Pi and Pj, respectively, while _mi;j is the flow of dry-air entering zone i from zone j. Finally, ei,j = ±1 accordingto the direction of flow.

The mass balance for the dryair in any zone i is:Xj

_mj!i ¼ 0 ð5Þ

where the summation is carried out over all zones adjoiningzone i.

A. Daoud et al. / Applied Thermal Engineering 28 (2008) 1782–1790 1785

The ventilation inlet airflow is introduced in the zonalmodel as a 2D thermal free jet. The governing equationsfor this type of jet are given by Ren [14].

By replacing the expressions of the mass flow rates in themass balance for each zone, we obtain a system of nonlin-ear equations whose unknowns are the corresponding pres-sures. This system is solved using Newton’s modifiedmethod. However, the solution of this system requires theaverage temperature in each zone in order to express thecorresponding density of the air in terms of its pressure.This unknown information is obtained by coupling thezonal model with the energy transfer model.

2.3. Radiation exchange model

The calculation of the net radiative exchange betweenthe inside surfaces of the building is based on the methodproposed by Gebhart [15]. According to this method, theradiation balance for a surface Ai is:

Qi ¼ AieirT 4i � A1e1rT 4

1G1�i þ � � � þ AjejrT 4j Gj�i

�þ � � � þ AN eNrT 4

N GN�i

�ð6Þ

where the first term on the right hand side represents thepower emitted by Ai while the other terms represent thepower absorbed by Ai coming from all other surfaces. So

Qi ¼ AieirT 4i �

XN

j¼1

AjejrT 4j Gj�i ð7Þ

The Gebhart factors Gi�j represent the sum of the poweremitted by i and absorbed directly by j plus the powerabsorbed after one or more reflections on all the other sur-faces. They are easily calculated from the view factors andthe surface properties by the resolution of the followingsystem:

Gi�j ¼ F i�jaj þXN

k¼1

F i�kqkGk�j ð8Þ

The Gebhart factors are also characterised by the two fol-lowing properties:

– Energy conservation:

XN

j¼1

Gi�j ¼ 1 ð9Þ

– Reciprocity relation:

eiAiGi�j ¼ ejAjGj�i ð10Þ

The principal advantage of this method is due to the factthat the Gebhart factors depend only on the physical char-acteristics and the relative position of the surfaces. Theycan therefore be calculated once, at the beginning of thesimulation, contrary to radiosity calculations which mustbe performed at each iteration.

The calculation of the Gebhart factors is done using Eq.(8). At any given time step Eq. (7) is then used to calculateeither the temperature of an adiabatic surface or the netradiation heat flux for a non-adiabatic surface.

2.4. Radiant heater model

The radiant heater is incorporated into the model as ahot surface. The temperature of this surface was evaluatedknowing the power of the radiant heater and assuming thatthis power is delivered by a gray surface

_Q ¼ eArT 4 ) T ¼

ffiffiffiffiffiffiffiffi_Q

eAr

4

sð11Þ

In the selected ice rink, eight radiant heaters are used toheat the stands with a total power of 22 kW. However, weare only studying a section of the ice rink which representsapproximately 2% of the total ice rink volume. Therefore_Q ¼ 3578:39 W and using Eq. (11) with e = 1, we obtaina heater temperature of 396 �C. Radiation from this sur-face is incorporated in the radiation exchange model andconvection towards the surrounding air is included in Eq.(2).

2.5. Humidity transport model

The absolute humidity of the air in zone i obeys Eq. (12)which expresses conservation of water vapour:

Mair;idxi

dt¼Xi�j

_mi�jðxj � xiÞ þ _W g;i ð12Þ

It indicates that the accumulation of water vapour is dueto net transport caused by air movement and to the contri-bution of any sources or sinks. Ventilation is the onlysource of humidity considered in this simulation. Specifi-cally, the air introduced above the stands is assumed tohave a temperature of 15 �C and the external absolutehumidity. On the other hand, the only considered sink isdue to condensation on the ice surface.

Eq. (12) applies to all 36 zones of the ice rink section.The solution of the corresponding system of 36 equationswith 36 unknowns gives the absolute humidity of each zoneat each time step.

To determine whether there is condensation on the ice,the following calculations are performed:

– The vapour pressure (Pvap,i) and the corresponding dewpoint (Tsat,i) in a zone i in contact with the ice are eval-uated from Eqs. (13) and (14) using the total pressure(Pi) and the absolute humidity (xi) in the zone:

P vap;i ¼xiP i

0:622þ xið13Þ

T sat;i ¼ 1:7741P 3vap;i � 12:682P 2

vap;i þ 35:92P vap;i � 17:814

ð14Þ

1786 A. Daoud et al. / Applied Thermal Engineering 28 (2008) 1782–1790

– Assuming that the air in contact with the ice has thesame temperature as the ice surface, its partial pressureand absolute humidity can be calculated using Eqs. (15)and (16)

P sat;ice;i ¼ P sat;25 �C exphslðT ice;i � T sat;25 �CÞ

R � T ice;iT sat;25 �C

� �ð15Þ

xice;i ¼0:622P sat;ice;i

P i � P sat;ice;ið16Þ

– Condensation on the ice surface occurs when its temper-ature is below the dew point of the air above it. In thiscase the mass transfer between the humid air and theice and the heat transfer due to condensation are calcu-lated using Eqs. (17) and (18)

_mcond;i ¼ hmAiðxi � xice;iÞ ð17Þ_qcond ¼ _mcond;iðcp;vapðT i � 0Þ þ hsl þ cp;iceð0� T ice;iÞÞ ð18Þ

The value of the mass transfer coefficient hm wasassumed to be a constant value equal to 10 kg/h m2.

2.6. Thermostat modeling

The radiant heater over the stands is controlled by athermostat with a set point of 15 �C and a hysteresis of±0.2 �C. That is, the heater is turned on when the air tem-perature falls below 14.8 �C and is turned off when itincreases above 15.2 �C. In the interval between these twoextreme values, the heater remains off if the air temperatureis falling towards 14.8 �C or on if the air temperature isincreasing towards 15.2 �C (see Fig. 4). The commercialcode [8] used for the simulations includes models for ther-mostats. However, these models create unwanted oscilla-tions during iterations within a time step because of theimportant addition of the energy input from the radiantheater when it is turned on or off. Thus, when the heatercomes on, the air temperature increases above 15.2 �C. It

1414.114.214.314.414.514.614.714.814.9

1515.115.215.315.415.515.615.715.815.9

1616.1

0.1 0.2 0.3 0.4 0.5 0.6Time

Tem

pera

ture

(°C

)

Thermostat responseAir temperature

Setpoint+hyst

Setpoint-hyst

Fig. 4. Sample of the th

is therefore turned off for the next iteration and the temper-ature falls below 14.8 �C. To avoid this cycling the adoptedthermostat model controls the radiant heater only at thebeginning of a time step. So that it remains on, or off, dur-ing all the iterations within that time step.

2.7. Coupling of the different models

The solution of the zonal airflow model, the radiationmodel and the humidity transport model were pro-grammed using MATLAB. This widely used programcan be coupled with the energy simulation software usingan existing interface. During the simulation, the energytransfer model takes the airflows calculated by the airflowzonal model and returns the air temperature of each zone(see Fig. 5). The energy model also supplies the surfacetemperatures to the radiation model which returns thenet radiation fluxes for each internal surface of the build-ing. The humidity transport model takes the airflow ratesbetween zones from the airflow zonal model and the tem-perature of the ice surface from the energy model to returnthe heat transfer due to condensation on ice which is intro-duced as a heat gain to the ice surface in the energy model.The coupling between all the models is based on the itera-tive onion method.

At the first time step, calculations of all models are car-ried out using arbitrary initial values. The convergence cri-terion for the airflow zonal model is 10�6 of variation onthe pressure value of each zone. The convergence criterionfor the energy simulation is 10�3. Time is incrementedwhen the outputs of each model vary less than 10�3. Inorder to ensure periodicity of the results, the simulationis carried out over 17 months by repeating the calculationsfor the January to May month at the beginning and the endof the simulation. Such a simulation with a time step of360 s (0.1 h) requires approximately 18 h on a computerequipped with Intel Core Duo CPU, and a RAM of 2

0.7 0.8 0.9 1 1.1 1.2 (hour)

0

1

Ther

mos

tat r

espo

nse

ermostat behaviour.

TRNSYSEnergy model

ZonalAirflow model

Radiation model

Humidity transport model

Ts,iTa,i

mTa,icond,iQ

mrd,iQ

Fig. 5. Coupling between the different models.

A. Daoud et al. / Applied Thermal Engineering 28 (2008) 1782–1790 1787

Go. This is considerably less than the time required for a2D 24-h transient CFD simulation of the same problem [6].

3. Validation

It is difficult to validate the results obtained with thismodel since the simulations are conducted using a typicalyear weather data. Another difficulty is to validate sepa-rately the heat fluxes due to radiation, convection and con-densation which are very difficult to measure separately.Nevertheless, to verify the accuracy of the results, the tem-perature profile above the ice obtained by the presentmodel is compared with those given by a 2D CFD simula-tion and measurements in the ice rink. Furthermore, mea-surements using four heat flux sensors under the ice sheetare used for the validation of the total heat transfer tothe ice. All these comparisons will be presented in the fol-lowing section.

4. Results and discussion

Fig. 6 shows the monthly average heat flux towards theice due to convection, radiation and condensation with a

Month1 2 3 4 5 6 7 8 9 10 11 12

0

10

20

30

40

50

60

70

80

90

100RadiationConvectionCondensation

Hea

t tra

nsfe

r (W

/m²)

Fig. 6. Monthly average heat flux due to convection, radiation andcondensation (Tthermostat = 15 �C, e = 0.5).

thermostat set point at 15 �C and a ceiling emissivity of0.5. We notice that radiation represents the major part ofthe heat flux received by the ice surface, followed by con-vection and finally, condensation. During the year themonthly average heat flux due to radiation varies from aminimum 39.85 W/m2 in January to a maximum 97.15W/m2 in July. The convection heat flux varies from5.80 W/m2 in February to 10.99 W/m2 in August. In thecase of radiation this variation is principally due to theincrease of the ceiling temperature which is affected bythe external temperature and solar radiation. In the caseof convection, the increase during the summer is due tothe corresponding increase of the temperature differencebetween the air (which is warmer) and the ice (which ismaintained at constant temperature). The condensationheat flux is null during December, January and Februarysince the outdoors absolute humidity is very low duringthose months. During the rest of the year, it varies between2.23 W/m2 in March and 11.87 W/m2 in November. Themonthly average heat flux due to resurfacing is constantand equal to 19.6 W/m2. It is therefore higher than thosedue to convection and condensation but lower than theone due to radiation. Because these results depend on localmeteorological conditions and the frequency of resurfacingoperations it is difficult to compare quantitatively the pres-ent results with those in other sources [6,9]. However, thesesources confirm the present prediction concerning theimportance of radiation.

Validation of the heat flux prediction is obtained bycomparing it with a measured value of the daily averageheat flux. The latter is obtained by integrating the readingsof four heat flux sensors installed under the ice sheet. Thecorresponding daily mean values for October 1, 2005 are:94.9 Wm2, 56.6 W/m2, 90.6 W/m2 and 113.0 W/m2. Dis-regarding the second sensor which gives significantly lowervalues than the other three, the mean value of the experi-mentally measured heat flux is 99.5 W/m2. On the otherhand, the model predicts an average total heat flux of108 W/m2 for October which is about 8% higher than themeasurements. In view of the fact that these values donot correspond to identical climate conditions, their agree-ment is judged to be acceptable.

Fig. 7 shows the average temperature profile in the fourregions above the ice during February with three differentthermostat set points (12 �C, 15 �C and 18 �C). The profileson the left of this figure are for the two regions far from thestands while those on the right are for the ones near thestands (cf. Fig. 3). The profiles in the top half of Fig. 7are for the regions which include the air inlet while thosein the bottom half are for the regions which include theradiant heater. We notice that the air temperature abovethe ice varies from a minimum of 1 �C to a maximum of18 �C which is normal in such buildings. Temperature pro-files near the stands are warmer than those far from themdue to the influence of heating. We also notice that theincrease of the thermostat set point causes an increase ofthe air temperatures above the ice, especially in the regions

-5 0 5 10 15 200

2

4

6

8

10

-5 0 5 10 15 200

2

4

6

8

Temperature (°C)

Hei

ght (

m)

-5 0 5 10 15 200

2

4

6

8

10

Temperature (°C)

Temperature (°C)Temperature (°C)

Hei

ght (

m)

Hei

ght (

m)

Hei

ght (

m)

-5 0 5 10 15 200

2

4

6

8

10

10

Thermostat = 12°CThermostat = 15°CThermostat = 18°C

Fig. 7. Temperature profiles above the ice.

1788 A. Daoud et al. / Applied Thermal Engineering 28 (2008) 1782–1790

near the stands. Far from the stands the influence of thethermostat set point is small. These results also show thatthe air temperatures in the regions which include the heaterand the fan inlet are not very different due to the mixing ofthe air over the ice.

Fig. 8 compares mean air temperature profiles above theice calculated with the present simulation code for Febru-ary with measurements recorded on the 24th of February2005 in the Montreal ice rink and with the CFD resultsby Bellache et al. [6]. It should be kept in mind that thepresent calculations were carried out with hourly meteoro-logical data for a typical year while the measurements and

-10 -5 0 5 10 15 200

1

2

3

4

5

6

7

8

9

Temperature (°C)

Hei

ght (

m)

CFDMesurementsPresent model

Fig. 8. Air temperature profile above the ice given by measurements, CFDand the present model.

CFD predictions correspond to a specific date and meteo-rological conditions. Despite this significant difference,Fig. 8 shows that the mean temperature profile predictedby the present model is in good qualitative agreement withthe available measurements, particularly near the ice wherethe air temperature is not influenced much by the outdoorsmeteorological conditions. In fact, for heights between 2 mand 5 m above the ice, the present predictions are closer tothe measured values than the CFD results.

Fig. 9 shows the effect of the ceiling emissivity on themonthly average flux due to net radiation transfer towardsthe ice for a thermostat set point equal to 12 �C. For a ceil-ing emissivity of 0.2, this heat flux varies from 29.07 W/m2

1 2 3 4 5 6 7 8 9 10 11 12

30

40

50

60

70

80

90

100

110

Month

ε = 0.2ε = 0.5ε = 0.9

Hea

t tra

nsfe

r (W

/m²)

Fig. 9. Effect of the ceiling surface emissivity on the monthly average heatflux due to long wave radiation (Tthermostat = 12 �C).

1 2 3 4 5 6 7 8 9 10 11 12

6

7

8

9

10

11

Month

Hea

t tra

nsfe

r (W

/m²)

Thermostat = 12°CThermostat = 15°CThermostat = 18°C

Fig. 11. Effect of the thermostat temperature set point on the monthlyaverage heat flux due to convection (e = 0.2).

A. Daoud et al. / Applied Thermal Engineering 28 (2008) 1782–1790 1789

in January to 82.56 W/m2 in July, which is an increase ofabout 184%. For a ceiling emissivity of 0.5, the correspond-ing values are 35.32 W/m2 for January and 97.08 W/m2 forJuly; which is an increase of about 175%. Finally, for a ceil-ing emissivity of 0.9, this heat flux increases from 41.84 W/m2 in January to 112.26 W/m2 in July, i.e. by about 169%.It should also be noticed that a ceiling emissivity of 0.2results in an average net radiation flux for August whichis significantly smaller than the corresponding ones forMay and October with a ceiling emissivity of 0.9. Theseresults are important since they show that a low ceilingemissivity can reduce significantly the load on the refriger-ation system of the ice rink.

Fig. 10 shows the effect of the thermostat set point onthe monthly average net radiation heat flux towards theice for a ceiling emissivity of 0.2. During this parametricstudy three thermostat set points (12 �C, 15 �C and18 �C) were considered. We notice from this figure thatdecreasing the thermostat set point reduces this heat flux,especially in winter and spring. This effect is not as signif-icant as the one due to the emissivity. In fact, in Februaryusing a thermostat set point of 12 �C instead of 18 �Creduces this heat flux from 35.59 W/m2 to 29.45 W/m2,i.e. by approximately 17.25%. During the summer, thethermostat set point has no effect whatsoever since ceilingtemperature (which is the principal contributor to the radi-ation flux towards the ice) is essentially established by theexternal meteorological conditions.

Fig. 11 shows the effect of the thermostat set point onthe monthly average convective heat flux towards the icefor a ceiling emissivity of 0.2. The results of this figure showthat a thermostat set point of 18 �C results in bigger con-vective heat fluxes towards the ice throughout the year.This is due to the fact that for this condition, the heatingcontinues functioning even during the summer andincreases the temperature of the air in contact with theice (see Fig. 7). In February, using a thermostat set point

1 2 3 4 5 6 7 8 9 10 11 12Month

30

40

50

60

70

80 Thermostat = 12°CThermostat = 15°CThermostat = 18°C

Hea

t tra

nsfe

r (W

/m²)

Fig. 10. Effect of the thermostat temperature set point on the monthlyaverage heat flux due to long wave radiation (e = 0.2).

of 12 �C instead of 15 �C decreases the convective towardsthe ice from 6.23 W/m2 to 5.09 W/m2. This corresponds toa reduction of 18.3%.

5. Conclusion

The present method of modeling the complex phenom-ena taking place in an ice rink constitutes an excellent alter-native to CFD calculations which require very substantialcomputational time and cannot be undertaken on conven-tional computers, especially for transient 3D cases. The cal-culated values of air temperatures above the ice and heatflux into the ice are in satisfactory agreement with corre-sponding measurements and CFD calculations. As far aswe can ascertain, this is the first time that the transientresponse of an ice rink to the changing meteorologicaland operational conditions has been simulated in 3D foran entire year.

This simulation of a 3D section of the ice rink consti-tutes a successful first step which will be followed by a tran-sient simulation of the complete ice rink with multipleventilation inlets and radiant heaters. The heat and humid-ity gains from spectators will also be considered. Due to itsrapidity the present model provides the opportunity to con-duct many parametric cases which can lead to correlationsof the refrigeration load with design and operationalparameters of a representative ice rink.

Acknowledgements

This study was financed by the Natural Sciences andEngineering Research Council (NSERC) of Canadathrough the Strategic Project Grant STPGP 306792 (Title:Development of design tools and of operation guidelinesfor the heating, ventilation, air conditioning and refrigera-tion systems of ice rinks).

1790 A. Daoud et al. / Applied Thermal Engineering 28 (2008) 1782–1790

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