coupled conduction, convection, radiation heat transfer with simultaneous mass transfer in ice rinks

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Coupled Conduction, Convection,Radiation Heat Transfer withSimultaneous Mass Transfer in Ice RinksO. Bellache a , M. Ouzzane a & N. Galanis ba CANMET Energy Technology Center , Varennes, QC, Canadab THERMAUS, Faculté de génie, Université de Sherbrooke ,Sherbrooke, QC, CanadaPublished online: 02 Sep 2006.

To cite this article: O. Bellache , M. Ouzzane & N. Galanis (2005) Coupled Conduction, Convection,Radiation Heat Transfer with Simultaneous Mass Transfer in Ice Rinks, Numerical Heat Transfer, PartA: Applications: An International Journal of Computation and Methodology, 48:3, 219-238, DOI:10.1080/10407780590945588

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COUPLED CONDUCTION, CONVECTION, RADIATIONHEAT TRANSFER WITH SIMULTANEOUS MASSTRANSFER IN ICE RINKS

O. Bellache and M. OuzzaneCANMET Energy Technology Center, Varennes QC, Canada

N. GalanisTHERMAUS, Faculte de genie, Universite de Sherbrooke,Sherbrooke QC, Canada

This article presents numerical predictions of velocity, temperature, and absolute humidity

distributions in an indoor ice rink with ventilation and heating. The computational fluid

dynamics (CFD) simulation includes the effects of radiation between all inside surfaces

of the building envelope, turbulent mixed convection, and vapor diffusion, as well as conduc-

tion through the walls and condensation on the ice. The net radiative fluxes for each element

of the envelope’s inside surfaces are calculated with a modified version of Gebhart’s method.

This modification reduces the calculation time and the memory required to store the radi-

ation view factors for the discretized elements of the inside surfaces of the envelope. The

predicted temperatures show very good agreement with measured data. The CFD code also

calculates the heat fluxes toward the ice due to convection from the air, to condensation of

vapor, and to radiation from the walls and ceiling. It is shown that a low emissivity ceiling

reduces the sum of these fluxes and the risk of vapor condensation on the ceiling.

INTRODUCTION

Air flow velocities as well as temperature and absolute humidity distributionswithin buildings are essential for the calculation of heating and cooling loads as wellas for the design of efficient HVAC systems. However, their prediction is difficult asa result of complicated flow patterns, the coexistence of zones in which forced con-vection predominates with others in which natural convection is more important,different turbulence structures near and far from the walls, as well as the interactionbetween the indoor and outdoor environments. During the last few decades, predic-tion methods for indoor air flow have increasingly relied on numerical analysis basedon different turbulent models and the time-averaged form of the Navier–Stokesequations. To this date, however, no study has undertaken the simultaneous deter-mination of all these variables for complex geometries such as ice rinks, in which

Received 18 March 2004; accepted 30 December 2004.

Address correspondence to Nicolas Galanis, THERMAUS, Faculte de genie, Universite de

Sherbrooke, 2500, Boul. Universite, Sherbrooke, Quebec J1K 2R1, Canada. E-mail: nicolas.galanis@

usherbrooke.ca

219

Numerical Heat Transfer, Part A, 48: 219–238, 2005

Copyright # Taylor & Francis Inc.

ISSN: 1040-7782 print=1521-0634 online

DOI: 10.1080/10407780590945588

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mechanical ventilation, mass transfer, and all modes of heat transfer play a signifi-cant role.

Nielsen [1] was among the first to apply computational fluid dynamics (CFD)methods for indoor air flow studies. He solved the steady-state two-dimensionalequations for the conservation of mass, momentum, temperature, turbulence energy,and dissipation rate for a rectangular ventilated room with adiabatic surfaces exceptfor the bottom one, where a uniform heat flux was prescribed. The reported velocityand temperature distributions were deemed to be in ‘‘reasonable agreement’’ withexperimental data [2]. This study quantified the effects of buoyancy and their depen-dence on the room and inlet slot geometries.

Davidson [3] performed a similar numerical study using a 3-D transient formu-lation for a ventilated room with an internal heat source and adiabatic walls. Thepredicted flow pattern was shown to agree reasonably well with experiments usingwater. However, the agreement between calculated and measured temperature pro-files was not always satisfactory.

All these studies ignored the interaction between the indoor and outdoor envir-onments and neglected heat transfer by radiation between the interior surfaces of theroom. Chen Qingyan [4] and Chen and Van der Kooi [5] proposed a methodology

NOMENCLATURE

ce1; ce2; ce3; cm; cd turbulence model constants

C absolute humidity (=kg

water=kg dry air)

df distance from the surface to

the first grid point in the

adjacent fluid

Din characteristic length (m)

F view factor

g acceleration of gravity

(m=s2)

G stress production of k

GB buoyancy production of k

Gij Gebhart’s absorption factor

for surface element j

hig specific enthalpy of

sublimation (J=kg)

Hin height of the air inlet (m)

Lin width of the air inlet (m)_mm ventilation mass flow rate

(kg=s)

P static pressure (Pa)

Prt turbulent Prandtl number

qcd conduction heat flux in solid

envelope (W)

qcv convective flux between the

air and the inside surface of

the envelope (W)

q‘ latent heat flux toward the

ice (W)

qr net radiative heat flux (W)

S area of surface element i

(m2)

S/ source term

t time (s)

T temperature (�C)Td dew-point temperature (�C)u, v velocity components (m=s)

Uin, Vin inlet velocity components

(m=s)

V velocity vector

X horizontal coordinate (m)

Y vertical coordinate (m)

b expansion coefficient (K�1)

b� coefficient of mass expansion

C/ diffusion coefficient

ei emissivity of surface element

i

e dissipation rate of turbulent

kinetic energy (m2=s3)

k turbulent kinetic energy

(m2=s2)

k thermal conductivity

(W=m�K)

m; mt molecular, turbulent

viscosity (kg=m�s)q density (kg=m3)

rt; re; rj turbulence model constants

/ general field variable

220 O. BELLACHE ET AL.

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for the solution of combined problems of energy analysis and indoor flow whichtakes into account these effects. They used time-independent air flow patterns calcu-lated by a CFD code with the k–e turbulence model for typical situations. They thensolved the algebraic room energy balance equation with a load calculation codewhich used transient temperature differences between the air near the inside surfacesand the middle point of the room. This temperature difference was evaluated fromthe solution of the transient differential energy equation obtained by the CFD codewith the predetermined time-independent velocity distribution. Iterations betweenthese two equations were necessary to achieve convergent results.

Li [6] extended the application of CFD codes in the field of HVAC by includ-ing heat conduction through walls and radiation transfer between their internal sur-faces. This formulation was then applied [7] to study the radiative effects on flow andtemperature fields in a room ventilated by displacement. The calculated results wereshown to be in good agreement with measured data. This approach is conceptuallysimpler than the one proposed by Chen [4], since it eliminates the need to interfacethe load calculation and CFD codes. However, the inclusion of radiation signifi-cantly increases the size of the coefficient matrix of the discretized conservationequations. To prevent this, Li et al. [7] applied a decoupling technique using a coar-ser grid for radiation calculations which were not performed at every iteration.

Recently, Singh and Sharif [8] studied two-dimensional mixed-convection cool-ing of a rectangular enclosure with isothermal or adiabatic thermal conditions appliedat the interface between the fluid and solid boundaries. Cold fluid is blown into thecavity from an inlet in one side wall and exits through an outlet in the opposite sidewall. They assumed that the flow is laminar, did not include radiation exchanges, anddid not take into account the presence and effects of humidity gradients.

In the present study, a formulation similar to the one developed by Li [6, 7] isused to calculate the velocity, temperature, and absolute humidity distribution in anindoor ice rink with ventilation and heating. The geometric configuration of this icerink is more complex than the simple parallelipipedic form adopted in all previousstudies, due to the presence of spectator stands and the transparent protective bar-riers which separate them from the ice surface. Another particularity of indoor rinksis due to the importance of radiative exchanges between the ice and the ceiling, whichcontributes significantly to the refrigeration load [9]. The model includes the effectsof radiation between all inside surfaces of the building envelope, turbulent mixedconvection, and vapor diffusion, as well as conduction through the walls and con-densation on the ice. A new approach is proposed to simplify radiation calculations.The CFD code used to calculate the flow field also evaluates the heat fluxes towardthe ice due to convection from the air, to vapor condensation, and to radiation fromthe walls and ceiling. The combination of a complex geometry, thermal boundaryconditions applied outside the envelope, and turbulent mixed convection due toboth thermal and concentration gradients distinguish the present study from allprevious ones.

MODELING OF THE PROBLEM

A schematic representation of a cross section of the ice rink under considera-tions is shown in Figure 1. Three rows of stands run the whole length of the building

SIMULTANEOUS HEAT AND MASS TRANSFER IN ICE RINKS 221

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on one side, and a narrow corridor encircles the ice surface. Protective transparentbarriers separate the corridor and stands from the ice surface. Heating and venti-lation for the spectators is provided by blowing 6.75 m3=s of air through eight regu-larly spaced inlets (0:3m� 1m, each at a height of 6m above the ice surface). Air isextracted through a single outlet on the ceiling, but for the purposes of this study wehave considered that there are eight smaller openings at the same Z coordinates asthe inlets. Since the length of the building is considerable (64m), the flow field inthe X–Y plane midway through a set of inlets and outlets situated near the mid-planeat Z ¼ 32m can be considered to be two-dimensional (no velocity component andno heat–mass transfer in the Z direction). The numerical calculations in such a planecan be considerably simplified.

Several turbulence models for the simulation of indoor air motion have beencompared during the present project, although their results are not presented here.It was found that the standard k–e model by Launder and Spalding [10] performedquite well and was very stable. Chen [11] has also compared the performance of fiveturbulence models in predicting natural convection, forced convection, and mixedconvection in rooms and used experimental data for his validation. He found thatthe prediction of the mean velocity is more accurate than that of the turbulentvelocity and stated that the performance of the standard k–emodel is good. Therefore,this model was selected for the present investigation.

The partial differential equations modeling the air motion, heat transfer, andvapor diffusion for steady-state conditions can be written in the following generalform:

qðq/Þqt

þ div qV/� C/r/� �

¼ S/ ð1aÞ

The appropriate values or expressions for /, C/, and S/ in the region occupied by airare defined in Table 1. It should be noted that in this region these seven partial dif-ferential equations are elliptic, nonlinear, and coupled, since the air density is

Figure 1. Cross section of the ice rink under consideration.


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considered to vary with both the temperature and the absolute humidity:

q ¼ q0 þ bðT � T0Þ þ b�ðC � C0Þ ð1bÞOn the other hand, within the solids in the calculation domain (ceiling, walls,

ice, stands, protective barriers), the velocity is zero. For the purposes of this study wealso assume that these solids are impermeable to vapor and do not include any heatsources. Therefore their temperature is the only unknown quantity and can be calcu-lated from the two-dimensional conduction equation.

The boundary conditions for this problem are as follows.

The ventilation air enters through the inlet at 30�C and 45% relative humidity withhorizontal and vertical velocities equal to 2.435 and �1.406m=s, respectively.The turbulence kinetic energy kin and dissipation rate ein are [12]

kin ¼ ð0:05UinÞ2 ein ¼ ðCmCdÞ0:75k1:5in

0:1DinDin ¼ 2HL

H þ Lð2Þ

At the outlet, the relative pressure is set equal to zero.A uniform temperature, Tice ¼ �5:5�C, is imposed under the ice at Y ¼ 0; for the

calculation with vapor diffusion, the air layer in contact with the ice is assumedto be saturated.

On all the other solid surfaces (walls, ceiling, stands), the velocity components, con-centration gradient, and turbulent kinetic energy are equal to zero. The corre-sponding temperature is related to that of the air within the flow domain andto the external temperature by an energy balance which takes into account con-duction through the solid, convection to the inside and outside air, as well as netradiation fluxes between the inside surfaces. Thus, the conductive heat flux qcdfor each element of the inside surfaces is equal to the following algebraic sum [7]:

qcd ¼ qcv þ qr þ q‘ ð3Þ

The last term, which denotes the latent heat due to vapor condensation, is taken intoaccount only for the ice surface.

Table 1 Values of /, C/ and S/

Equation / C/ S/

Mass 1 0 0

X momentum U mþ mt �qP=qxY momentum V mþ mt � qP=qy� qgbðT � T0Þ � qgb�ðC � C0ÞEnergy T m=Prþ mt=Prt ST

Absolute humidity C ðmþ mtÞ=rc Sc

Turbulent kinetic energy k mt=rk G � qeþ GB

Dissipation rate e mt=re ½eðce1G � ce2qeÞ=k� þ ce3GBðe=kÞ

mt ¼ qcmcdk2=e; Prt ¼ 1

GB ¼ �gbðmt=PrtÞðqT=qxiÞ; G ¼ mtðqUi=qxj þ qUj=qxiÞ � ðqUi=qxjÞce1 ¼ 1:44; ce2 ¼ 1:92; ce3 ¼ 1:44; cm ¼ 0:5478; cd ¼ 0:1643; rk ¼ 1; re ¼ 1:3; rc ¼ 1


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Gebhart’s method [13] is used to calculate the net radiative heat flux qr betweensurface j and all the N surfaces of the ice rink envelope. In terms of the Gebhart fac-tors Gij, defined as the fraction of radiant energy that leaves surface i and is absorbedby surface j, this net radiative heat flux for surface j is

qrj ¼ rXNi¼1

eiGijT4i Si � rejT

4j Sj ð4Þ

The values of the Gebhart factors are calculated from

AGj ¼ Bj ð5Þ

where

GTj � G1j;G2j; . . . ; GNj

� �

BTj � �F1jej ; �F2jej; . . . ;�FNjej

� �

and

A �

F11 q1 � 1ð Þ F12 q2 � � � F1N qNF21 q1 ðF22 q2 � 1Þ F2N qN� � � � � �� FN1 qN FN2 q2 � � � FNN qN � 1ð Þ

26666664

37777775

The calculation of qrj requires the emissivities of all N elements of the ice rink envel-ope and the N2 radiation view factors Fij between these surfaces.

CALCULATION PROCEDURE AND VALIDATION

The numerical solution has been carried out using the finite-volume method, astaggered grid, and the hybrid scheme for discretization of the convection term. TheSIMPLE algorithm [14] was used for the pressure correction, and the solution wasobtained iteratively using the numerical code PHOENICS [12]. To simplify the prob-lem, steady-state conditions are assumed for the purpose of this article.

The calculation domain was first subdivided into zones corresponding to theexisting physical discontinuities appearing in Figure 1. In the present case thereare 15 zones in the X direction and 11 in the Y direction. The number of grid points,or finite volumes, within each resulting 165 subdivisions of the domain has been var-ied to ensure that the results are independent of their number. The discretization gridwas nonuniform, with higher node density near the air inlet and the solid surfaces,where gradients are higher. Numerical tests have been carried out with 80� 35,100� 50, and 200� 100 grid points in the X and Y directions, respectively, to ensurethat the results were grid-independent. Table 2 shows that the results for the two


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denser grids are essentially identical. All the results reported in this article were there-fore calculated with 100� 50 grid points. Underrelaxation was often necessary toachieve convergence, which is declared when the cumulative residuals for each ofthe conservation equations are less than 10�6.

The relatively large number of grid points results in a large number of surfaceelements on the inside of the building envelope. Thus, for the 100� 50 grid there areapproximately 250 such surface elements. The evaluation of the corresponding 2502

radiation view factors necessary for the calculation of the net radiative fluxes is parti-cularly tedious and time-consuming, due to the irregular geometry of Figure 1.Therefore an approximate, easily programmable method was adopted for thecalculation of the radiative fluxes between the surface elements. During each iter-ation, this approximate method considers the 10 surfaces identified in Figure 2.The corresponding 102 view factors are determined fairly easily, while the Gebhart

Table 2 Grid refinement study

No. of grid points (X, Y) 80� 35 100� 50 200� 100

X ¼ 18m, Y ¼ 1m

T(�C) 9.7 9.6 9.6

V(m=s) 0.17 0.20 0.20

X ¼ 18m, Y ¼ 5m

T(�C) 10.3 10.3 10.3

V(m=s) 0.042 0.058 0.062

X ¼ 32m, Y ¼ 5m

T(�C) 26.6 26.3 26.2

V(m=s) 0.18 0.17 0.16

Heat losses through the envelope (W=m) 561 594 595

Conductive heat flux in the ice (W=m) 2,716 2,751 2,750

Radiation heat flux into the ice (W=m) 1,332 1,318 1,318

Lack of closure in energy balancePðEout � EinÞ ðW=mÞ 29 7 6PðEout�EinÞEin

0.01 0.003 0.002

Lack of closure in mass balancePð _mmout� _mminÞ

_mmin0 0 0

Figure 2. Simplified geometry for radiation calculations.


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factors are calculated from Eq. (5) assuming that all elements of any one surfacehave the same emissivity.

The temperature Tj of each of these 10 surfaces is set equal to the arithmeticmean of those of its constituent elements, and the net radiative flux for each ofthe 10 surfaces is evaluated from Eq. (4) with N ¼ 10. Finally, in order to applythe boundary condition expressed by Eq. (3) to each individual surface element,the net radiative flux for element k of surface j is calculated from

qr ¼X10i¼1

SieirT4i Gi;kj � SkjejrT

4kj ð6Þ

where

Sj ¼XMk¼1

Skj and Gi;kj ¼ GijSkj

Sj

Here i and j indicate the number of surfaces in Figure 3 (1 � i � 10 and 1 � j � 10),while k is the number of elements on surface j (1 � k � M).

Essentially, this approximation consists of distributing the energy leaving sur-face i and absorbed by surface j (equal to Gij) among the M elements of surface jproportionally to their respective area.

The conductive and convective fluxes in Eq. (3) for element k of surface j(walls) are

qcd ¼ Skj Ts;kj � Text

� � ewkw

þ 1

hext

� ��1

ð7Þ

For the ice, qcd was calculated using the ice conductivity and a two-nodeapproximation of the temperature gradient in the ice.

qcv ¼kf Skj Tint � Ts;kj

� �df

ð8Þ

Finally, the latent heat due to vapor condensation on the ice is calculated fromand distributed uniformly among all the corresponding surface elements.

q‘ ¼ _mmin Cin � Coutð Þhig ð9Þ

The experimental data of Kim [15] were used to validate the model and the cal-culation procedure. That experiment was carried out using air in a square enclosurewith thick walls. The outside surface of the left wall was 52�C warmer than the out-side surface of the right wall, while the outside surfaces of the horizontal walls wereadiabatic. Figure 3 shows a comparison of measured and calculated nondimensionalair temperatures. When natural convection, conduction in the walls, and radiationare considered, the agreement is quite good. On the other hand, if radiationis neglected, the calculated results are very different from the measured data.Validation of the velocity predictions has also been obtained by comparison with


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Figure 3. Measured and numerically predicted steady-state isotherms with and without radiation;

Ra� ¼ 1.01� 106, Pr ¼ 0.71, TH ¼ 47�C, TC ¼ �5�C, ew ¼ 0.98.


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corresponding turbulent measurements and has been presented elsewhere [16]. Thesecomparisons illustrate the importance of radiation and show that the adopted modeland calculation procedure are quite accurate.

Comparisons of wall temperatures in ice rinks are presented in the next sectionof this article and show equally good agreement between measured and calculatedvalues. We therefore conclude that the model and numerical code used for its sol-ution are valid and can be used for the purposes of this study.

RESULTS AND DISCUSSION

The results presented were calculated by considering a constant outside tem-perature of �10�C, and a constant value of the corresponding convection coefficientof 7W=m2 �C. This value of the heat transfer coefficient was calculated from theaverage winter wind velocity in Montreal [9]. The physical properties of the wallsand the ice are the following:

thermal resistance of the walls per unit area: 0.23 �C �m2=Wthermal resistance of the ice per unit area: 0.09 �C �m2=Wemissivity of the walls: 0.9emissivity of the ice: 0.97

Results are presented for two cases: ceiling with low (ec ¼ 0:25) and high(ec ¼ 0:9) emissivity. By comparing these two cases it is possible to evaluate the effectof radiation on the velocity and temperature fields as well as on the heat fluxes acrossthe envelope.

Flow Field

Typical profiles of the horizontal and vertical velocity components predictedby the previously described model are shown in Figures 4 and 5, respectively. Thesefigures and many other similar comparisons show that the velocity distribution is es-sentially the same for the two ceiling emissivities under consideration. This result isdue to the fact that the air has been assumed to be totally transparent and its velocityis therefore not directly influenced by a change in the radiation fluxes. The indirectinfluence, due to a change of the buoyancy forces caused by radiation-inducedchanges of the envelope’s temperature, is obviously very small. Thus, we will describethe hydrodynamic fields for only one case (Figure 6). The warm ventilation air formsa rather narrow jet which follows an anticlockwise curved path between the air inletand outlet. This narrow focused jet is the result of the uniform, fairly large velocity(2.8 m=s) imposed at the rather small air inlet (DY ¼ 0:3m) and the relative proxim-ity of these two openings. By entrainment, this jet creates two vortices. The first oneis situated near the ceiling between the inlet duct and the air outlet. It is very nearlycircular and rotates in the anticlockwise direction. The second one, situated betweenthe ventilation jet and the stands, is much weaker and rotates in the clockwise direc-tion. Near the right wall its movement is reinforced by the downward action of thebuoyancy forces. The velocities in the right side corridor as well as between the pro-tective transparent barrier and the stands are negligible, and therefore this area is


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poorly ventilated. The quality of the air herein risks being poor if smoking is permit-ted. Similarly, the velocities to the left of the inlet duct, particularly those over the icesurface, are very small. This numerical deduction was confirmed experimentallyusing a hot-wire anemometer which did not register any significant air movementover the ice for Y<1m.

Vertical Temperature Profiles

Figures 7 and 8 show the temperature profiles over the ice and in the spectators’zone, respectively. Contrary to the velocity field, the temperatures are considerablyinfluenced by the ceiling emissivity. Air temperatures corresponding to ec ¼ 0:25 areeverywhere higher than those for ec ¼ 0:9. This is due to the importance of the radi-ative heat loss by the ceiling with high emissivity. Figure 7 shows that on the left ofthe inlet (X � 27:25m), the vertical gradient of the temperature is very importantnear the ceiling and the ice. Elsewhere in this region the temperature is essentiallyuniform and increases with X because of the proximity of the spectators’ zone, whichis heated by the warm supply air. Indeed, as shown in Figure 8, the temperature onthe right of the inlet (X > 27:25m) is considerably higher than over the ice and variessignificantly with both X and Y due to the important air movement in this region.

Figure 4. Profiles of horizontal component of velocity at different positions X.


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However, it does not exceed 26�C in the area really occupied by the spectators, i.e.,within the first 2m above the stands.

Vertical Absolute Humidity Profiles

Figure 9 shows the profiles of the absolute humidity in the ice rink. It isgenerally lower on the left side of the inlet and higher in the zone occupied by the

Figure 6. Velocity vectors (maximum velocity ¼ 2.8m=s at the inlet).

Figure 5. Profiles of vertical component of velocity at different heights Y.


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spectators. This is consistent with the flow field, since ventilation air does notpenetrate the left region of the inlet. Its variation with height is generally small.The effect of ceiling emissivity on this variable is also quite small.

Effect of Ceiling Emissivity on the Surface Temperature

The effect of ceiling emissivity on the temperature of the ice, ceiling, and wallsurfaces is shown in Figure 10. It must first be noted that, for ec ¼ 0:9, which isclose to the actual ceiling emissivity, the numerically predicted surface tempera-tures for the ice (Figure 10a) and walls (Figures 10c and 10d) are in close agree-ment with the corresponding measured values. This agreement is further proofof the validity of the model and solution procedure adopted in the present study.In particular, it validates the simplifications used in the calculation of the radiativefluxes.

Figure 10a shows that, for both values of ec, the ice surface temperature is uni-form except very near the edges of the ice sheet. On the other hand, the temperatureof the ceiling surface (Figure 10b) increases with X, since the region over the specta-tors’ stands is heated by the incoming air supply. A decrease of the ceiling emissivity

Figure 7. Temperature profiles at different positions X over the ice surface.


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results in an increase of the ceiling temperature and a decrease of the ice temperature.Both these effects are a result of the decrease of the ceiling’s net radiative flux, whichis 926W=m for ec ¼ 0:9 and only 514W=m for ec ¼ 0:25.

Figure 10b illustrates an important effect of a low ceiling emissivity by compar-ing the temperature of its surface with the dew point of the nearby air. The latter iscalculated from the previously presented results (see Appendix B) and is higher thanthe ceiling temperature for X < 13m when ec ¼ 0:9. Under such conditions, conden-sation will take place on the ceiling. This is undesirable, since the liquid can migrateinto the insulation or reduce the quality of the skating surface if water droplets fallfrom the ceiling. As shown by the results in Figure 10b, these adverse effects can betotally avoided by lowering the ceiling emissivity to 0.25.

Figure 10c shows that the ceiling emissivity has an important effect on the tem-perature of the left side wall. The temperature of this wall increases when ec decreasesbecause it is influenced by the higher ceiling temperature through convection (seeFigure 6) and radiation.

Finally, Figure 10d shows that the ceiling emissivity has a small effect on thetemperature of the right side wall. There are two reasons for this. First, the radiationview factors between this wall and the horizontal surfaces are much smaller than thecorresponding value between the ceiling and the ice. Second, the air and surface tem-peratures in the region above the spectators’ stands are primarily dependent on the

Figure 8. Temperature profiles at different positions X in the spectators zone.


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temperature of the warm supply air. Thus the effects of radiation are quite small forthis surface.

Energy Fluxes through the Building Envelope

Table 3 shows the effect of the ceiling emissivity on the energy fluxes throughthe envelope. The second row of this table shows that the heat losses through thewalls and ceiling of the building increase by approximately 15% when the ceilingemissivity decreases from 0.9 to 0.25. This change is due primarily to the correspond-ing increase of the ceiling temperature (see Figure 10b), since the outside temperatureand heat transfer coefficient are the same for both values of ec.

The third row of Table 3 presents the latent heat load which corresponds tosolidification of water vapor on the ice. The rate of solidification is proportionalto the absolute humidity gradient perpendicular to the ice surface (see Figure 9),while the enthalpy of solidification depends on the air temperature near the ice(see Figure 7). Since these two quantities do not change significantly with ec, thevalues of the latent heat load are of the same order of magnitude for the two valuesof the ceiling emissivity. The fourth row of Table 3 presents the convective heat fluxbetween the air and the ice. This flux is proportional to the temperature differencebetween the air and the ice. It also depends on the local air velocity, which influences

Figure 9. Absolute humidity profiles at different positions X.


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the heat transfer coefficient. Therefore, as the radiative exchanges have an appreci-able influence on the temperature of the ice surface (see Figure 10a), the reduction ofthe ceiling emissivity results in a increase of almost 29% of the convective heat loadbetween the air and the ice.

The most significant result of the reduction of ec is a decrease of approximately54% of the net radiation flux into the ice. This important reduction is almost totallydue to the corresponding decrease of the radiation flux emitted by the ceiling.Indeed, because the width of the domain is approximately four times greater thanits height, most of the radiation absorbed by the ice originates at the ceiling.

Table 3 Effect of ceiling emissivity on heat transfer (W=m)

Heat flux component ec ¼ 0.9 ec ¼ 0.25

Heat losses through walls and ceiling 594 682

Latent heat load 930 884

Convective heat load 503 641

Radiation heat load 1,318 603

Conductive heat flux in the ice 2,751 2,128

Figure 10. Effects of emissivity on the surface temperatures.


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Finally, the summation of the three heat fluxes into the ice is presented in thelast row of Table 3. It is noted that the reduction of the ceiling emissivity from 0.9 to0.25 results in a decrease of approximately 23% of this value. This sum is not thetotal refrigeration load, since the present calculations have not taken into accountthe contribution of ice resurfacing, system pump work, ground heat, and skaters.However, it is a significant part of the refrigeration load, and the calculatedreduction has significant economic implications. It is interesting to note that therelative magnitude of the latent, convective, and radiation heat loads presented inthe first column of Table 3 are very similar to the corresponding typical proportionspresented in Chapter 34 of the ASHRAE Refrigeration Handbook [9].

CONCLUSION

Turbulent flow with simultaneous mass diffusion inside a large ventilated icerink coupled to conduction through its envelope and radiation between the insidesurfaces of the latter has been solved using a CFD code. The standard k–e modeland a modified version of Gebhart’s method were used to evaluate the flow fieldand radiative fluxes, respectively.

The results have been calculated for fixed outdoor conditions, a fixed tempera-ture at the bottom of the ice sheet, and two different ceiling emissivities. They includea detailed description of the hydrodynamic field, temperature profiles in the air andalong the inside surfaces of the envelope, as well as profiles of the absolute humidity.The results indicate the air velocities and temperatures as well as absolute humiditydistributions are not influenced by the value of the ceiling emissivity. On the otherhand, a reduction of the ceiling emissivity results in a decrease of the ice surfacetemperature and an increase of the ceiling surface temperature. The risk of vaporcondensation on the ceiling is therefore considerably reduced.

Finally, the CFD code also calculates the most important contribution to therefrigeration load, namely, the heat fluxes toward the ice due to convection fromthe air, to condensation of vapor, and to radiation from the ceiling and the walls.The results show that a decrease of the ceiling emissivity from 0.9 to 0.25 reducesthe sum of these three fluxes by about 25%.

REFERENCES

1. P. V. Nielsen, Flow in Air Conditioned Rooms (English translation of Ph.D. thesis, Tech-nical University of Denmark, 1974), Danfos A=S, Nordborg, Denmark, 1976.

2. P. V. Nielsen, A. Restivo, and J. H. Whitelow, Buoyancy Affected Flows in VentilatedRooms, Numer. Heat Transfer, vol. 2, pp. 115–127, 1979.

3. L. Davidson, Ventilation by Displacement in a Three-Dimensional Room: A NumericalStudy, Building and Environment, vol. 24, no. 4, pp. 363–372, 1989.

4. Q. Chen, Indoor Airflow, Air Quality and Energy Consumption of Buildings, Ph.D.thesis, Delft University of Technology, Delft, The Netherlands, 1988.

5. Q. Chen and J. Van der Kooi, Accuracy—A Program for Combined Problems of EnergyAnalysis, Indoor Airflow and Air Quality, ASHRAE Trans., vol. 94, part. 2, pp. 196–214,1988.


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6. Y. Li, Simulation of Flow and Heat Transfer in Ventilated Rooms, Trans. Dept.Mech.=Appl. Comput. Fluid Dynam., R. Inst. Technol. S-10044, Stockholm, Sweden, 1992.

7. Y. Li, L. Fuchs, and M. Sandberg, Numerical Prediction of Airflow and Heat RadiationInteraction in a Room with Displacement Ventilation, Energy and Buildings, vol. 20, no. 1,pp. 27–43, 1993.

8. S. Singh and M. A. R. Sharif, Mixed Convective Cooling of a Rectangular Cavity withInlet and Exit Openings on Differentially Heated Side Walls, Numer. Heat Transfer A,vol. 44, pp. 233–253, 2003.

9. 2002 ASHRAE Handbook of Refrigeration, chap. 34, American Society of Heating,Refrigeration and Air-Conditionings Engineers, Atlanta, GA, USA, 2002.

10. B. E. Launder and D. B. Spalding, The Numerical Computation of Turbulent Flows,Comput. Meth. Appl. Mech. Eng., vol. 3, pp. 269–289, 1974.

11. Q. Chen, Comparison of Different k–e Models for Indoor Air Flow Computations,Numer. Heat Transfer B, vol. 28, pp. 353–369, 1995.

12. CHAM, Phoenics 3.4, CHAM, London, UK, 2001.13. R. Siegel and J. R. Howell, Thermal Radiation Heat Transfer, Hemisphere, Washington,

DC, 1981.14. S.V.Patankar,NumericalHeatTransfer and Fluid Flow,Hemisphere,Washington,DC, 1980.15. D. M. Kim and R. Viskanta, Effect of Wall Conduction and Radiation on Natural

Convection in a Rectangular Enclosure, Numer. Heat Transfer, vol. 7, pp. 449–470, 1984.16. O. Bellache, N. Galanis, M. Ouzzane, and R. Sunye, Modelisation des ecoulements et

transferts thermiques dans un arena ventile, VIe colloque inter universitaire franco-quebe-cois, thermique des systemes, Quebec, Canada, 2003.

APPENDIX A: CALCULATION OF NET RADIATIVE HEAT TRANSFER

Gebhart’s method is briefly presented here. Additional discussion can be foundin [12]. The main advantage of this method is that it yields the fraction of energyemitted by a surface that is absorbed at another surface after reaching the absorbingsurface by all possible paths.

Consider an enclosure with N diffuse-gray surfaces. The net energy flux for sur-face Sk is the difference between the energy it absorbs and that it emits. The emittedflux is SkekrT4

k . Let Gjk be the fraction of the emission from surface Sj that reachesSk and is absorbed. This includes all the paths for reaching Sk, that is, the directpath, paths by means of one reflection, and paths by means of multiple reflections.Thus SjejrT4

j Gjk is the amount of energy emitted by Sj that is absorbed by Sk.A heat balance on Sk gives

qrk ¼ S1e1rT41G1k þ S2e2rT

42G2k þ � � � þ SNeNrT

4NGNk

� �� SkekrT

4k ðA:1Þ

or

qrk ¼XNj¼1

SjejrT4j Gjk � SkekrT

4k ðA:2Þ

The coefficients Gkk would generally not be zero, since even for a plane or con-vex surface some of the emission from a surface will be returned to itself by reflectionfrom other surfaces. Equation (A.2) can be written for each surface; this will relate


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each of the qr to the surface temperatures in the enclosure. The G factors must nextbe determined.

The quantity Gjk is the fraction of energy emitted by Sj that reaches Sk and isabsorbed. The total emitted energy from Sj is SjejrT4

j . The portion traveling by adirect path to Sk and absorbed is SjejrT4

j Fj�kek, where for a gray surface e is equalto the absorbtivity. All other radiation from Sj arriving at Sk will first undergo onereflection. The emission from Sj that arrives at a typical surface Sn and is thenreflected is SjejrT4

j Fj�nqn.The fraction Gnk then reaches Sk and is absorbed. Thus all the energy absorbed

at Sk originating by emission from Sj is

ðSjejrT4j Fj�1q1G1kþSjejrT

4j Fj�2q2G2kþ��þSjejrT

4j Fj�kqkGkk

þ��þSjejrT4j Fj�NqNGNkÞ

Dividing this energy by the total emission from Sj gives the fraction

Gjk ¼ Fj�kek þ Fj�1q1G1k þ Fj�2 q2 G2k þ � � � þ Fj�k qkGkk þ � � � þ Fj�NqNGNk

ðA:3Þ

By letting j take on all values from 1 to N, the following set of equations is obtained:

G1k ¼ F1�kek þ F1�1q1G1k þ F1�2q2G2k þ � � � þ F1�kqkGkk

þ � � � þ F1�NqNGNk

G2k ¼ F2�kek þ F2�1q1G1k þ F2�2q2G2k þ � � � þ F2�kqkGkk

þ � � � þ F2�NqNGNk

..

.

GNk ¼ FN�kek þ FN�1 q1G1k þ FN�2q2G2k þ � � � þ FN�kqkGkk

þ � � � þ FN�NqNGNk

ðA:4Þ

That gives the following matrix form [cf. Eq. (5) in the main text]:

½qiFji � dji�½Gik� ¼ �ekFjk

� �ðA:5Þ

Finally, energy conservation implies that

XNk¼1

Gjk ¼ 1 1 < j < N ðA:6Þ

APPENDIX B: DEW-POINT TEMPERATURE Td

The dew-point temperature Td of moist air with absolute humidity C and press-ure P can be calculated directly with one of the following equations [8]:

For the dew-point temperature range 0–93�C,

Td ¼ C14 þ C15aþ C16a2 þ C17a

3 þ C18ðPwÞ0:1984 ðB:1Þ


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For temperatures below 0�C,

Td ¼ 6:09þ 12:608aþ 0:4959a2 ðB:2Þ

where

a ¼ lnðPwÞ Pw ¼ ðP � CÞ=ð0:62198þ CÞ

C14 ¼ 6:54 C15 ¼ 14:526 C16 ¼ 0:7389 C17 ¼ 0:09486 C18 ¼ 0:4569


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coupled conduction, convection, radiation heat transfer with simultaneous mass transfer in ice rinks

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