ETE-11

CFD Simulations and Reduced Order Modeling of a Refrigerator Compartment Including Radiation Effects

54

Abstract—Considering the engineering problem of natural

convection in domestic refrigerator applications, this study aims

to simulate the fluid flow and temperature distribution in a single

commercial refrigerator compartment by using the

experimentally determined temperature values as the specified

constant wall temperature boundary conditions. The free

convection in refrigerator applications is evaluated as a 3-D,

turbulent, transient and coupled non-linear flow problem.

Radiation heat transfer mode is also included in the analysis.

According to the results, taking radiation effects into

consideration does not change the temperature distribution inside

the refrigerator significantly; however the heat rates are affected

drastically. The flow inside the compartment is further analyzed

with a reduced order modeling method called Proper Orthogonal

Decomposition (POD) and the energy contents of several spatial

and temporal modes that exist in the flow are examined. The

results show that approximately 95 % of all the flow energy can

be represented by only using one spatial mode.

Keywords—refrigerator compartment, CFD, POD, radiation

effects

I. INTRODUCTION

aintaining a preset low temperature by spending the least

amount of electricity is the most important characteristic

of a refrigerator for evaluating its performance. Optimizing its

design for performance requires a well understanding of the

natural convection inside it. Natural convection in enclosures

has been extensively studied both experimentally and

numerically. General reviews were focused on the importance

of scaling analysis and experiments to determine the flow

details. [1-3] The studies performed by Corcione [4], Markatos

and Pericleous [5], Davis [6] and Hyun and Lee [7] are

examples for 2D studies in the literature.

Experimental benchmark studies of low-level turbulence

natural convection in an air filled vertical cavity were

conducted by Tian and Karayiannis [8], Ampofo and

Karayiannis [9], Ampofo [10, 11] and Penot and N’Dame

[12]. A work different from the studies mentioned so far was a

preliminary attempt to study transient natural convection

phenomena in a two-dimensional cavity heated symmetrically

from both sides with a uniform heat flux [13]. There are also

several other studies related to the 2D simulations of cavities

in literature [14, 15, 16, 17, 18, 19].

Although 2D cavity model for a refrigerated space is good

enough when the dimensional conditions are satisfied [12], the

results may deviate from the experiments at the corners. On the

other hand, 3D modeling gives more realistic and accurate

results. One of the commonly used benchmark numerical

solutions for natural convection in a cubical cavity was

obtained by Wakashima and Sayitoh [20]. Transition to time-

periodicity of a natural convection flow in a 3D differentially

heated cavity was studied by Janssen et al. [21]. Fusegi et al.

[22] also worked on 3D natural convection of air in cubical

enclosures. There are also experimental studies in literature

related to the subject. [23- 27]. Other 3D analyses focused on

temperature and velocity distribution determination across the

enclosures caused by the heat source are also available in

literature [28, 29].

There are various studies related to natural convection in

enclosures; however refrigerator applications are limited. For

refrigerators, simulation includes steady-state simulation and

dynamic simulation. For steady-state simulation, the thermal

capacity of foam insulation is neglected. For dynamic

simulation, not only the refrigeration system, but also the

refrigerated space (cabinet) is considered to be dynamic, so the

simulation is complicated. Dynamic simulation of natural

convection bypass two-circuit cycle refrigerator for both the

component and system basis is performed by Ding et al. [30,

31]. Similarly, Salat et al. [32] investigated the turbulent

convection in a large air filled cavity by the help of direct

numerical simulation (DNS) and Large Eddy Simulation (LES)

methods. In a different study, the velocity and temperature

distributions in commercial refrigerated open display cabinets

are examined by applying finite element method. [33].

Laguerre and Flick [34] analyzed heat transfer by natural

convection in domestic unventilated refrigerators. Based on

[34], Laguerre et al. performed an experimental study of heat

CFD Simulations and Reduced Order Modeling

of a Refrigerator Compartment Including

Radiation Effects

O. Bayer1, R. Oskay

2, A. Paksoy

3, S. Aradag

4

1Middle East Technical University, Ankara/Turkey, [email protected]

2Middle East Technical University, Ankara/Turkey, [email protected]

3TOBB University of Economics and Technology, Ankara/Turkey, [email protected]

4TOBB University of Economics and Technology, Ankara/Turkey, [email protected]

M

mailto:[email protected]



O. Bayer, R Oskay, A. Paksoy and S. Aradag

55

transfer by natural convection in a cavity selecting the

application as a domestic refrigerator with the real dimensions

[35]. Air temperature profile in the boundary layers and in the

central zone of the empty refrigerator model was searched.

The effects of temperature and the surface area of the cold wall

were studied. After the experimental study [35], in turn,

Laguerre et al. performed the numerical simulation of air flow

and heat transfer [36] and experimental work of air flow [37].

The effect of radiation was investigated in [36] for a 3D

enclosure with the dimensions close to an actual refrigerator

and comparison of calculated air temperatures obtained from

the numerical analysis and the experimental values showed

good agreement when radiation was taken into account.

Proper Orthogonal Decomposition (POD) is a method used

to analyze time-dependent high-dimensional experimental or

computational processes by separating the system into its

space and time components, and to enable identification of the

most energetic modes in a sequence of snapshots from the

time-dependent system [38, 39]. The procedure was originally

developed in the context of pattern recognition, and it has been

used in various industrial and natural applications especially

for system identification and control [38, 40]. For instance, in

the studies performed by Paksoy et al. [41] and Apacoglu et al.

[42], the POD method is successfully used to analyze and

identify flow structures formed in the wake region of a 2D

circular cylinder for forced and unforced laminar fluid. In

another study carried out by Paksoy et al. [43], the classical

POD method is combined with the Fast Fourier Transform

(FFT) filtering technique to effectively observe the effects of

large-scale flow structures formed in the wake region of the

2D circular cylinder for forced and unforced turbulent fluid

flows.

Considering the engineering problem of natural convection

in domestic refrigerator applications, this study first aims to

simulate the fluid flow and temperature distribution in a single

commercial refrigerator compartment by using the

experimentally determined temperature values as the specified

constant wall temperature boundary conditions. The free

convection in refrigerator applications is evaluated as a 3-D,

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 [36] is also included

in the analysis. Another objective of the study is to further

analyze the flow inside the compartment with a reduced order

modeling method called Proper Orthogonal Decomposition

(POD) and examine the energy contents of several spatial and

temporal modes that exist in the flow.

II. METHODOLOGY

A. CFD Methodology

The computational domain is a single compartment of

21.5x47x62 cm height, depth and width respectively, which

represents a compartment of a real refrigerator of Arçelik

Refrigerator Company. The height, depth and width of the

refrigerator compartment are represented with the letters c, a

and b respectively in Figure 1. d shown in Figure 1 indicates

the distance of the evaporator at the back surface from the side

walls and it is 9.75 cm.

Figure 1: Schematic 3-D cavity model of a single compartment.

It is assumed that there is no mass flow across the

boundaries. For velocity, no slip boundary conditions are used

for the walls. The wall temperature values are obtained from

the experiments reported in 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 K and Tevap=270.06 K. The

initial conditions are 0 m/s for velocity and T=275 K for

temperature.

Realizable k-ε turbulence model is used. To include the

radiation effect in heat transfer, DO method is implemented.

ANSYS Fluent Computational Fluid Dynamics (CFD)

software package is used to perform the numerical analyses.

Segregated pressure-based solver with pressure implicit with

splitting of operators (PISO) algorithm is used. The strong

coupling between the flow and temperature fields and the

interaction between boundary layers and core flow make

computation stiff and the convergence difficult.

Characteristics of the 3-D cavity model used in the

numerical analysis and the total number of cells in the models

used for the simulations are tabulated in Table 1 and the mesh

is shown in Figure 2.

Figure 2: Mesh for single compartment.


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Table 1: Number of cells used for the simulations.

Single

Compartment

Mesh Number

Height Width Depth Total

37 75 66 183150

Celeron two core dual T7400 (2.3 GHz, 12 GB RAM)

computers are used in the simulations. Run time is about 24

hours for radiation omitted analysis, and it is about 72 hours

when the radiation is included.

B. Experimental Methodology for Boundary Conditions

It is necessary to perform experiments in order to form a

base for the boundary conditions of the numerical analysis. In

the experimental part, 4243 TMB model static (without

ventilation) household refrigerator (with outer dimensions

173x70x68 cm) in the research department of Arçelik

Çayırova factory is used and the temperatures of the walls and

specified points at different locations inside are measured.

Therefore it is made possible to substitute the values of the

temperature boundary conditions in the numerical analysis

with the experimental ones.

Temperature measurements are made only on one side of

the symmetry plane of the domain as shown in Figure 3.

Omega, T-type copper-constantan thermocouples with a

temperature measuring range of –250 °C to 350 °C and HP,

Agilent 34970A model data logger are used in measurements.

Temperatures of 54 points are measured. (9 points for all walls

and symmetry plane) The bottom wall of the compartment is

39 cm above the bottom of the whole refrigerator so one part

of the back wall is completely the evaporator region.

Temperature values are continuously measured and data is

recorded every ten seconds for three days. Average values are

used as boundary conditions.

The thermocouple locations are shown in Figure 4.

Figure 3: Experimental set-up for the single compartment.

Figure 4: Schematic view of the thermocouple locations.

C. POD Methodology

The POD approach based on the snapshot method is

originally developed by Sirovich [44], and it optimizes modes

based on energy.

In this study, CFD simulation results consisting of 900

snapshots are used as the data ensemble, where the snapshots

are equally spaced from each other, and they contain

temperature data with respect to the data of spatial y and z

coordinates of a single x-plane located at 0.1175 m. Each

snapshot is arranged to contain 5500 points in a matrix. All

matrices generating the snapshots ensemble have dimensions

of 125x44 where the y direction spatial domain changes within

-0.31 m and 0.31 m and the z direction spatial domain changes

within -0.1075 m and 0.1075 m with Δx=Δy=0.005 m. Further

mathematical procedure for the POD method is given in

Apacoglu et al [43].

III. RESULTS

A. Results of the Numerical Analysis

A time dependent natural convection analysis is performed

by including or omitting radiation. The corresponding

temperature and velocity profiles are determined at the

midplanes. The temperature and velocity profiles are

visualized at three different planes; x-z midplane which is the

symmetry plane orthogonal to the evaporator and front wall, a

plane parallel to symmetry plane and perpendicular to

evaporator at its one end and y-z midplane of the cavity.

In Figure 5, at t=150 seconds, the temperature profile in the

compartment is shown for the case with radiation. The profiles

are nearly the same for the cases, radiation included or

omitted. Moreover, on the symmetry plane of the problem (x-z

midplane) the onset of the flow is faster. Natural convection

characteristics are significant. Boundary layers developing on

the evaporator and bottom wall are observed.


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Figure 5: Temperature profile for the single compartment analysis,

t=150 s (with radiation).

When the results for several time instants are compared,

radiation only affects the maximum velocity value. Except the

maximum velocity value, the circulation loops formed,

boundary layers developed on the walls are the same for both

of the analyses, radiation included or neglected. However, the

time to reach steady state decreases when radiation is taken

into account as an additional heat transfer mechanism; i.e. it is

nearly 5 minutes and 3 minutes for the analysis without

radiation and with radiation respectively.

In Table 2, the total and radiative heat transfer rates

obtained from the numerical analyses of the single

compartment by applying radiation model or neglecting it are

tabulated. The radiative heat transfer is a significant portion of

the total heat transfer rates from or to the walls. For instance,

radiative heat transfer rate is about 55 % of the total heat

transfer rate for the evaporator.

Table 2: Radiative and total heat transfer rates (in Watts) for the

single compartment analysis, t=3600 s.

With Rad.

Model

Without Rad.

Model

Rad. Heat

Tr. Rate

Tot. Heat

Tr. Rate

Tot. Heat

Tr. Rate

Front Wall 0.67 1.15 0.49

Rear Wall 0.37 0.73 0.37

Top Wall 1.59 1.38 -0.14

Bottom Wall 0.22 2.27 2.05

Left Side Wall 0.68 1.08 0.41

Right Side Wall 0.68 1.08 0.41

Evaporator -4.18 -7.67 -3.59

Residual of the

Energy Balance 0.02 2.32E-06 -1.53E-04

B. Results of the Experiments

Sample temperature distributions at the locations of the

thermocouples positioned on the side wall of the compartment

are shown in Figure 6.

Figure 6: Temperature distribution on the side wall of the single

compartment.

Temperature distribution obtained for the reference lines on

symmetry plane in numerical analysis are compared with the

temperature values measured in the experiments. The reference

lines selected in numerical analysis shown in Figure 7 are the

vertical lines on which the thermocouples are located in

experimental work. In Figure 8, experimental temperature

values measured on three vertical lines away from the

evaporator on the symmetry plane are shown together with the

numerical simulations.

Figure 7: Configuration of reference lines on the symmetry plane

Experimental temperature values are in good agreement

with the numerical results especially on the upper half of the

symmetry plane. The frame used to locate the thermocouples

on the symmetry plane in the experimental study may be

disturbing the boundary layer at the lower part close to the

bottom wall and there is a conduction heat transfer through the

solid frame. Therefore, experimental temperature values at the

bottom level of all three vertical reference lines deviate from

the values obtained in numerical analysis but numerical results

still remain in the uncertainty range (error bar range is 1°C in

Figure 8) of the experimental temperature values.


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Figure 8: Comparison of temperature values on the symmetry plane

obtained from the numerical analysis and the experimental work

C. Results of Reduced Order Modeling

Application of the POD technique to the data ensemble

obtained from CFD simulations separates the flow structures

on the single x-plane located at 0.1175 m according to their

frequency content. In other words, it sorts the spatial modes

with respect to their energy content [45]. The energy content

distribution, in which it is observed that more than 99% of the

total energy can be represented with the most energetic four

POD modes, is shown in Table 3.

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

Mode Number Energy Content (%)

1 94.49

2 3.54

3 0.95

4 0.40

Total of four modes 99.38

Figure 9 shows the history of mode amplitudes

corresponding to the most energetic four POD modes. From

Figure 9, it can be concluded that after a certain snapshot

number (approximately 550) the system proceeds to the steady

state.

Figure 9: History of the mode amplitudes with respect to snapshot

number for x-plane located at 0.1175 m.

The most energetic two POD modes are shown in Figure 10

and they contain information about the temperature

distribution along the x-plane located at 0.1175 m.

Figure 10: The most energetic POD modes.

IV. DISCUSSION AND CONCLUSION

The free convection in refrigerator applications is

evaluated as a 3-D, turbulent, transient and coupled non-

linear flow problem. Radiation heat transfer mode is

included in the analysis. Experiments are performed for the

boundary conditions used in the analysis. According to the

results, taking radiation effects into consideration does not

change the temperature distribution inside the refrigerator

significantly; however the heat rates are affected drastically.

The flow inside the compartment is further analyzed with

Proper Orthogonal Decomposition (POD) The results show

that approximately 95 % of all the flow energy can be

represented by only using one spatial mode.

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ETE-11

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