Post on 28-Oct-2014
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, byrzgr@gmail.com
2Middle East Technical University, Ankara/Turkey, roskay@metu.edu.tr
3TOBB University of Economics and Technology, Ankara/Turkey, apaksoy@etu.edu.tr
4TOBB University of Economics and Technology, Ankara/Turkey, saradag@etu.edu.tr
M
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.
CFD Simulations and Reduced Order Modeling of a Refrigerator Compartment Including Radiation Effects
56
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.
O. Bayer, R Oskay, A. Paksoy and S. Aradag
57
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.
CFD Simulations and Reduced Order Modeling of a Refrigerator Compartment Including Radiation Effects
58
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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