Energy Conversion and Management 69 (2013) 68–76

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Energy Conversion and Management

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CFD simulations and reduced order modeling of a refrigerator compartmentincluding radiation effects

Ozgur Bayer a, Ruknettin Oskay a, Akin Paksoy b, Selin Aradag b,⇑a Middle East Technical University, Department of Mechanical Engineering, Ankara 06531, Turkeyb TOBB University of Economics and Technology, Department of Mechanical Engineering, Sogutozu Cad. No. 43, Ankara 06560, Turkey

a r t i c l e i n f o a b s t r a c t

Article history:Received 31 January 2011Accepted 15 January 2013Available online 6 March 2013

Keywords:Refrigerator compartmentComputational Fluid DynamicsProper Orthogonal DecompositionRadiation effects

0196-8904/$ - see front matter � 2013 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.enconman.2013.01.024

⇑ Corresponding author. Tel.: +90 312 292 4267.E-mail address: saradag@etu.edu.tr (S. Aradag).

Considering the engineering problem of natural convection in domestic refrigerator applications, thisstudy aims to simulate the fluid flow and temperature distribution in a single commercial refrigeratorcompartment by using the experimentally determined temperature values as the specified constant walltemperature boundary conditions. The free convection in refrigerator applications is evaluated as a three-dimensional (3D), turbulent, transient and coupled non-linear flow problem. Radiation heat transfermode is also included in the analysis. According to the results, taking radiation effects into considerationdoes not change the temperature distribution inside the refrigerator significantly; however the heat ratesare affected drastically. The flow inside the compartment is further analyzed with a reduced order mod-eling method called Proper Orthogonal Decomposition (POD) and the energy contents of several spatialand temporal modes that exist in the flow are examined. The results show that approximately 95% of allthe flow energy can be represented by only using one spatial mode.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Maintaining a preset low temperature by spending the leastamount of electricity is the most important characteristic of arefrigerator for evaluating its performance. Optimizing its designfor performance requires a well understanding of the natural con-vection inside it. Natural convection in enclosures has been exten-sively studied both experimentally and numerically. The size of theresearch effort dedicated to this topic during the past three dec-ades reflects the fact that natural convection in enclosures is achallenging subject; and it is one of the simplest multiple-scale,coupled non-linear flow problems and provides a convenient toolfor the development of new numerical algorithms. General reviewswere focused on the importance of scaling analysis and experi-ments to determine flow details [1–3]. A study performed by Cor-cione [4] was steady laminar natural convection in air-filled, two-dimensional (2D) rectangular enclosures heated from below andcooled from above for a wide variety of thermal boundary condi-tions at the side walls. The influence of Rayleigh number uponthe flow patterns, temperature distributions and heat transfer rateswere analyzed and compared with a benchmark numerical solu-tion [5]. The studies performed by Markatos and Pericleous [5],de Vahl Davis [6], and Hyun and Lee [7] are examples for 2D stud-ies in the literature.

ll rights reserved.

Experimental benchmark studies of low-level turbulence natu-ral convection in an air filled vertical cavity were conducted byTian and Karayiannis [8], Ampofo and Karayiannis [9], Ampofo[10,11], and Penot and N’Dame [12]. A work different from thestudies mentioned so far was a preliminary attempt to study tran-sient natural convection phenomena in a 2D cavity heated sym-metrically from both sides with a uniform heat flux [13]. Thereare also several other studies related to the 2D simulations of cav-ities in literature [14–20].

Although 2D cavity model for a refrigerated space is good en-ough when the dimensional conditions are satisfied [12], the re-sults may deviate from the experiments at the corners. On theother hand, three-dimensional (3D) modeling gives more realisticand accurate results. One of the commonly used benchmarknumerical solutions for natural convection in a cubical cavitywas obtained by Wakashima and Saitoh [21]. In the study, 3D cav-ity has two differentially heated and isothermal vertical walls, andalso four adiabatic walls. The working fluid is air with Pr = 0.71. Inthe computations, the high accuracy finite differences of fourth-or-der were employed for the spatial discretization of governingequations and the boundary conditions. Transition to time-period-icity of a natural convection flow in a 3D differentially heated cav-ity was studied by Janssen et al. [22]. Fusegi et al. [23] also workedon 3D natural convection of air in cubical enclosures. There are alsoexperimental studies in literature related to the subject [24–28].Other 3D analyses focused on temperature and velocity distribu-tion determination across the enclosures caused by the heat sourceare also available in literature [29–31].

O. Bayer et al. / Energy Conversion and Management 69 (2013) 68–76 69

There are various studies available in the literature related tonatural convection in enclosures; however, refrigerator applica-tions are limited. For refrigerators, simulation includes steady-state simulation and dynamic simulation. For steady-state simula-tion, the thermal capacity of foam insulation is neglected. For dy-namic simulation, not only the refrigeration system, but also therefrigerated space (cabinet) is considered to be dynamic, so thesimulation is complicated. Dynamic simulation of natural convec-tion bypass two-circuit cycle refrigerator for both the componentand system basis was performed by Ding et al. [32,33]. Similarly,Salat et al. [34] investigated the turbulent convection in a largeair filled cavity with the help of direct numerical simulation(DNS) and Large Eddy Simulation (LES) methods. In some differentstudies, detailed numerical simulation of the performance ofhousehold refrigerator with experimental verification is investi-gated and the velocity and temperature distributions as well asthe cooling loads in commercial refrigerated open display cabinetsand freezers are examined [35–37].

Laguerre and Flick [38] analyzed heat transfer by natural con-vection in domestic unventilated refrigerators. The study consistedof natural convection theories covering some cases such as rectan-gular empty cavity, vertical plates and air and heat transfer aroundan isolated object. Based on [38], Laguerre et al. performed anexperimental study of heat transfer by natural convection in a cav-ity selecting the application as a domestic refrigerator with the realdimensions [39]. Air temperature profile in the boundary layersand in the central zone of the empty refrigerator model wassearched. The effects of temperature and the surface area of thecold wall were studied. After the experimental study [39], in turn,Laguerre et al. performed the numerical simulation of air flow andheat transfer [40], experimental work of air flow [41]. The effect ofradiation was investigated in reference [40] for a 3D enclosurewith the dimensions close to an actual refrigerator and comparisonof calculated air temperatures in numerical analysis and the exper-imental values showed good agreement when radiation was takeninto account.

Proper Orthogonal Decomposition (POD) is a method used toanalyze time-dependent high-dimensional experimental or com-putational processes by separating the system into its space andtime components, and to enable identification of the most ener-getic modes in a sequence of snapshots from the time-dependentsystem [42,43]. The procedure was originally developed in the con-text of pattern recognition, and it has been used in various indus-trial and natural applications especially for system identificationand control [42,44]. For instance, in the studies performed by Pak-soy et al. [45] and Apacoglu et al. [46], the POD method is success-fully used to analyze and identify flow structures formed in thewake region of a 2D circular cylinder for forced and unforced lam-inar fluid flows. In another study carried out by Paksoy et al. [47],the classical POD method is combined with the Fast Fourier Trans-form (FFT) filtering technique to separate unwanted interferencesof small bifurcations and to effectively observe direct effects oflarge-scale flow structures formed in the wake region of the 2D cir-cular cylinder for forced and unforced turbulent fluid flows.

2. Objective of the study

Considering the engineering problem of natural convection indomestic refrigerator applications, this study first aims to simulatethe fluid flow and temperature distribution in a single commercialrefrigerator compartment by using the experimentally determinedtemperature values as the specified constant wall temperatureboundary conditions. The free convection in refrigerator applica-tions is evaluated as a 3D, turbulent, transient and coupled non-linear flow problem. Radiation heat transfer mode which was

proved to be very important in the analysis by Laguerre et al.[40] is also included in the analysis. Another objective of the studyis to further analyze the flow inside the compartment with a re-duced order modeling method called Proper Orthogonal Decompo-sition (POD) and to examine the energy contents of several spatialand temporal modes that exist in the flow.

3. Methodology

3.1. Governing equations

With the assumptions of neglected viscous dissipation effectand work performed by pressure forces, constant thermophysicalproperties and Boussinesq approximation for free convection, con-tinuity, Navier–Stokes and energy equations in vector form can bewritten as,

@q@tþ qðr � uÞ ¼ 0 ð1Þ

qDuDt¼ �rP þ lr2uþ f ð2Þ

DTDt¼ ar2T; ð3Þ

where f is the body force term and defined as,

f ¼ 0 0 q0gðT � T0Þ½ � ð4Þ

where q0 and T0 are the reference density and temperature respec-tively, b is the volumetric thermal expansion coefficient. In theabove formulation, bold letters indicate vector quantities, and theterm D/Dt is the substantial derivative.

3.2. CFD methodology

The computational domain is a single compartment of21.5 � 47 � 62 cm height, depth and width respectively, whichrepresents a compartment of a real refrigerator of Arçelik Refriger-ator Company. The height, depth and width of the refrigeratorcompartment are represented with the letters c, a and b respec-tively in Fig. 1. d shown in Fig. 1 indicates the distance of the evap-orator at the back surface from the side walls, and it is 9.75 cm.

With or without taking radiation effect into consideration, anal-yses are performed for the same boundary conditions. It is as-sumed that there is no mass flow across the boundaries. Forvelocity, no slip boundary conditions are used for the walls. Thewall temperature values obtained from the experiments reportedin the next section as: Trear = 282.82 K, Tfront = 281.58 K,Tleft = 281.79 K, Tright = 281.79 K, Tbottom = 280.24 K, Ttop = 280.98 Kand Tevap = 270.06 K. The initial conditions are 0 m/s for velocityand T = 275 K for temperature.

Realizable k–e turbulence model which belongs to two-equa-tion models in the form of eddy viscosity model is used and kand e values are taken to be ‘‘1’’ and ‘‘1.2’’ respectively. To includethe radiation effect in heat transfer, DO method is implemented.ANSYS Fluent Computational Fluid Dynamics (CFD) software pack-age which is based on finite volume method is used to perform thenumerical analyses. Segregated pressure-based solver with pres-sure implicit with splitting of operators (PISOs) algorithm is used.Since the simulations are based on a natural convection problem, itis necessary to pack the mesh in regions of high gradient to resolvethe pressure variation adequately. For this reason the PressureStaggering option (PRESTO) scheme is selected. The strong cou-pling between flow and temperature fields and interaction be-tween boundary layers and core flow make computation verystiff and the convergence difficult. In addition, the flow in a cavity,

Fig. 1. Schematic 3D cavity model of a single compartment.

70 O. Bayer et al. / Energy Conversion and Management 69 (2013) 68–76

in particular for cooling of electronics or of refrigerator compart-ments, is likely to be transitional. This causes problems for mostof the Reynolds averaged Navier–Stokes (RANS) based turbulencemodels, which are calibrated in fully turbulent flow conditions.When the turbulence level in the core region of cavity is low forbuoyancy-driven flows, most of the models tend to relaminarizethe flow and, as a consequence, underpredict the near wall turbu-lence intensities and boundary layer thicknesses [30].

The numerical analysis was performed for all coarse meshed, fi-ner meshed and finest meshed control volumes. Total number ofcells, mesh faces, mesh edges and mesh nodes of the three controlvolumes are tabulated in Table 1. It was observed that, solutionsfor the finer meshed and the finest meshed volumes were thesame, therefore, the finer meshed domain with 183,150 volumeelements was used for the preliminary analysis.

The mesh is shown in Fig. 2. Absolute convergence criteria is setto be 10�3 for velocities, k�e values, 10�6 for energy and DO inten-

Table 1The mesh characteristics of models used for the simulations.

Numbers of mesh

Faces Edges Nodes Total

Coarse meshed cavity 9050 472 60,480 55,836Finer meshed cavity 20,334 712 193,496 183,150Finest meshed cavity 27,872 868 289,674 275,520

Fig. 2. Mesh for single compartment.

sity, and 5 � 10�3 for continuity. Celeron two core dual T7400(2.3 GHz, 12 GB RAM) computers are used in the simulations.Run time is about 24 h for radiation omitted analysis, and it isabout 72 h when the radiation is included.

3.3. Experimental methodology for boundary conditions

It is necessary to perform experiments in order to form a basisfor the boundary conditions of the numerical analysis. In theexperimental part, 4243 TMB model static (without ventilation)household refrigerator (with outer dimensions 173 � 70 � 68 cm)in the research department of Arçelik Çayırova factory is usedand the temperatures of the walls and specified points at differentlocations inside are measured. Therefore it is made possible to sub-stitute the values of the temperature boundary conditions in thenumerical analysis with the experimental ones.

Temperature measurements are made only on one side of thesymmetry plane of the domain as shown in Fig. 3. The coil seenin Fig. 3 is not actually a working condenser coil. It is there justfor fixing the thermocouples on the symmetry plane. Omega, T-type copper-constantan thermocouples with a temperature mea-suring range of –250 �C to 350 �C and HP, Agilent 34970A modeldata logger are used in measurements. Temperatures of 58 speci-

Fig. 3. Experimental set-up for the single compartment.

O. Bayer et al. / Energy Conversion and Management 69 (2013) 68–76 71

fied points (5 points each for top and bottom walls, 12 points eachfor the other walls and symmetry plane) are measured for the totalrefrigerator and 54 point measurements (9 points for all walls andsymmetry plane) are performed for single compartment. The bot-tom wall of the compartment is 39 cm above the bottom of thewhole refrigerator so one part of the back wall is completely theevaporator region. Temperature values are continuously measuredand data is recorded every 10 s through 3 days. Average values areused as boundary conditions.

Expected uncertainty including the measurement, switchingand transducer conversion errors in this experimental study arisesfrom the uncertainties in T-type thermocouples used and the datalogger integrated and these uncertainty values (standard limits oferror) are ±1 �C for both. The thermocouple locations are shownin Fig. 4.

3.4. Proper orthogonal decomposition methodology

The Proper Orthogonal Decomposition (POD) approach basedon the snapshot method is originally developed by Sirovich [48],and it optimizes modes based on energy. In this study, CFD simu-lation results consisting of 900 snapshots are used as the dataensemble, where the snapshots are equally spaced from eachother, and they contain temperature data with respect to the dataof spatial y and z coordinates of a single x-plane located at0.1175 m. Each snapshot is arranged to contain 5500 points in amatrix. All matrices generating the snapshots ensemble havedimensions of 125 � 44 where the y direction spatial domainchanges within �0.31 m and 0.31 m and the z direction spatial do-main changes within �0.1075 m and 0.1075 m withDx = Dy = 0.005 m. Further mathematical procedure for the PODmethod is given in Paksoy et al. [45] and Apacoglu et al. [46].

Fig. 4. Schematic view of the

4. Results

4.1. Results of the numerical analysis

A time dependent natural convection analysis is performedboth including and omitting radiation and corresponding temper-ature and velocity profiles are determined at the mid-planes. Thetemperature and velocity profiles are searched and visualized atthree different planes; x–z mid-plane which is the symmetry planeorthogonal to the evaporator and front wall, a plane parallel tosymmetry plane and perpendicular to evaporator at its one endand y–z midplane of the cavity. The results obtained for the y–zmid-plane are shifted below the domain in order to make thewhole picture clear.

In Figs. 5 and 6, at t = 150 s, the temperature profile in the com-partment is nearly the same for the cases, radiation included oromitted. Moreover, on the symmetry plane of the problem (x–zmidplane) the onset of the flow is faster. Natural convection char-acteristics are significant. Boundary layers developing on the evap-orator and bottom wall are observed. Although the maximumtemperature is the same in Figs. 5 and 6, maximum velocity valuein the domain is 0.119 m/s for the radiation included analysis andit is 0.124 m/s for the analysis where radiation is not taken into ac-count (Figs. 7 and 8).

When the results for several time instants are compared, radia-tion only affects the maximum velocity value. Except the maxi-mum velocity value, the circulation loops formed, boundarylayers developed on the walls are the same for both of the analyses,radiation included or neglected. However, the time to reach steadystate decreases when radiation is taken into account as an addi-tional heat transfer mechanism; i.e. it is nearly 5 min and 3 minfor the analysis without radiation and with radiation respectively.On the other hand, CPU time of the single compartment numericalanalysis with radiation is three times the computation time ofnumerical analysis for the same domain without radiation.

thermocouple locations.

Fig. 5. Temperature profile for the single compartment analysis, t = 150 s (with radiation).

Fig. 6. Temperature profile for the single compartment analysis, t = 150 s (without radiation).

72 O. Bayer et al. / Energy Conversion and Management 69 (2013) 68–76

The velocity boundary layers at the walls and the flow direc-tions can be compared in detail for the symmetry plane of the cav-ity (x–z midplane) at the time of 3600 s. As expected, at steadystate, the velocity distribution for both cases converges to the sameprofile inside the compartment. The trends in the study show sim-ilar results to the results reported by Moraga et al. [49], as well asthe results of Laguerre et al. [40].

Heat transfer analyses are performed for the single compart-ment models with or without radiation. Time dependent heat fluxand the effective heat transfer coefficient are determined. Totalheat transfer rates from the walls and change of the heat fluxand the effective heat transfer coefficient with spatial coordinatesare also examined for similar reference lines of the preliminarynumerical analysis. The effective heat transfer coefficient is theconvective heat transfer coefficient for the analysis that doesnot include radiation. While obtaining the heat flux values, area

weighted average of the flux at each face of the correspondingwall is calculated by the computer program and reference tem-perature is again the geometric center temperature of thedomain.

In Table 2, the total and radiative heat transfer rates obtainedfrom the numerical analyses of the single compartment by apply-ing radiation model or neglecting it are tabulated. It is shown that3600 s is a very good prediction of the time to reach steady state. InTable 2, it is also presented that including radiation in the analysisby applying radiation model makes the value of the residuals of theenergy balance of the system very close to zero which is the indi-cation of the convergence of the analysis. Moreover, it can bedetermined that radiative heat transfer is a significant portion ofthe total heat transfer rates from or to the walls. For instance, radi-ative heat transfer rate is about 55% of the total heat transfer ratefor the evaporator.

Fig. 7. Velocity profile for the single compartment analysis, t = 150 s (with radiation).

Fig. 8. Velocity profile for the single compartment analysis, t = 150 s (without radiation).

Table 2Radiative and total heat transfer rates (in Watts) for the single compartment analysis,t = 3600 s.

Location With rad. model Without rad.model

Rad. heat tr.rate

Tot. heat tr.rate

Tot. heat tr. rate

Front wall 0.67 1.15 0.49Rear wall 0.37 0.73 0.37Top wall 1.59 1.38 �0.14Bottom wall 0.22 2.27 2.05Left side wall 0.68 1.08 0.41Right side wall 0.68 1.08 0.41Evaporator �4.18 �7.67 �3.59Residual of the energy

balance0.02 2.32E-06 �1.53E-04

O. Bayer et al. / Energy Conversion and Management 69 (2013) 68–76 73

4.2. Results of the experiments

Sample temperature distributions at the locations of the ther-mocouples positioned on the side wall of the compartment areshown in Fig. 9. In Fig. 9, the temperature distribution on the sidewall of the single compartment is presented. When the graph isanalyzed, temperature values measured by thermocouples t12,t15 and t18 are higher than the others. These thermocouples areon the same vertical line, which is nearest to the front wall (door)of the refrigerator; therefore they may be affected from the ambi-ent room air. Temperature values measured by the thermocouples,t10, t13 and t16 are lower than the values measured by t12, t15and t18. These thermocouples are also positioned on a vertical lineclose to the relatively hot rear wall. On the other hand, t11, t14 andt17, which are located on the vertical line at the middle of the sidewall, measure the lowest temperatures. The fluctuations in the

Fig. 9. Temperature distribution on the side wall of the single compartment.

Fig. 10. Configuration of reference lines on the symmetry plane for the singlecompartment analysis.

Table 3Energy content for the most energetic four POD modes.

Mode number Energy content (%)

1 94.492 3.543 0.954 0.40Total of four modes 99.38

74 O. Bayer et al. / Energy Conversion and Management 69 (2013) 68–76

temperature measurements decrease from back to front, becausethermocouples close to the evaporator are to be affected by thecompressor. Although the experiments were performed in a tem-perature-controlled room, the room air temperature decreases

Fig. 11. Comparison of temperature values on the symmetry plane o

from 20.7 �C to 20.2 �C for the time interval considered in Fig. 9.In parallel to the change in the room air temperature, temperaturevalues measured by thermocouples on the compartment side wallare slightly decreasing. Fig. 9 implies that the compressor is onabout 15 min.

btained from the numerical analysis and the experimental work.

Fig. 12. History of the mode amplitudes with respect to snapshot number for x-plane located at 0.1175 m.

O. Bayer et al. / Energy Conversion and Management 69 (2013) 68–76 75

4.3. Comparison of temperature values obtained from the singlecompartment analysis and the experimental work

Temperature distribution obtained for the reference lines onsymmetry plane in numerical analysis are compared with the

Fig. 13. The most energe

temperature values measured in the experimental study of thesame compartment. The reference lines selected in numericalanalysis shown in Fig. 10 are the vertical lines on which the ther-mocouples are located in experimental work. In Fig. 11, experi-mental temperature values measured on three vertical linesaway from the evaporator on the symmetry plane with thenumerical simulations performed including radiation or omittingit are compared. Numerical analyses performed with or withoutradiation model give very close temperature values on the samevertical lines.

On the other hand, experimental temperature values do notoverlap the numerically obtained results; however they are in goodagreement especially on the upper half of the symmetry plane. Theframe used to locate the thermocouples on the symmetry plane inthe experimental study may disturb the boundary layer at the low-er part close to the bottom wall and there is a conduction heattransfer through the solid frame. Therefore, experimental temper-ature values at the bottom level of all three vertical reference linesdeviate from the values obtained in theoretical numerical analysisbut numerical results still remain in the uncertainty range (errorbar range is 1 �C in Fig. 11) of the experimental temperature values.

The results of the experiments presented herein show paralleltrends with the experimental study of Laguerre et al. [39,41] wherethe results of an experimental study of heat transfer by naturalconvection in a domestic refrigerator was investigated and temper-ature profiles in the vicinity of the walls of the compartment wereobtained.

tic four POD modes.

76 O. Bayer et al. / Energy Conversion and Management 69 (2013) 68–76

4.4. Results for reduced order modeling

Application of the POD technique to the data ensemble obtainedfrom CFD simulations separates the flow structures on the single x-plane located at 0.1175 m according to their frequency content. Inother words, it sorts the spatial modes with respect to their energycontent [49]. The energy content distribution, in which it is ob-served that more than 99% of the total energy can be representedwith the most energetic four POD modes, is shown in Table 3.Fig. 12 shows the history of mode amplitudes corresponding tothe most energetic four POD modes. From Fig. 12, it can be con-cluded that after a certain snapshot number (approximately 550)the system proceeds to the steady state. The most energetic twoPOD modes are shown in Fig. 13 and they contain informationabout the temperature distribution along the x-plane located at0.1175 m.

5. Discussion and conclusion

The free convection in refrigerator applications is evaluated as a3D, turbulent, transient and coupled non-linear flow problem.Radiation heat transfer mode is included in the analysis. Experi-ments are performed for the boundary conditions used in the anal-ysis. According to the results, taking radiation effects intoconsideration does not change the temperature distribution insidethe refrigerator significantly; however the heat rates are affecteddrastically. The flow inside the compartment is further analyzedwith Proper Orthogonal Decomposition (POD). The results showthat approximately 95% of all the flow energy can be representedby only using one spatial mode.

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