16 Heat Transfer and Air Flow_Libro

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445

16 Heat Transfer and Air Flow

in a Domestic RefrigeratorOnrawee LaguerreUMR Gnie Industriel Alimentaire Cemagref-ENSIA-INAPG-INRA

Contents

16.1 Introduction ........................................................................................................................446

16.2 Literature Review o Natural Convection in Closed Cavity ...............................................447

16.2.1 Studies in Domestic Rerigerators ........................................................................447

16.2.2 Heat Transer and Airow Near a Vertical Plate ..................................................448

16.2.3 Heat Transer and Airow in Empty Closed Cavity .............................................450

16.2.4 Heat Transer and Airow in Cavity Completely or Partially

Filled with Porous Media ...................................................................................... 451

16.3 Cold Production System in Domestic Rerigerators/Freezers ............................................ 452

16.4 Temperatures and Heat Transer Modes in Domestic Rerigerators .................................. 453

16.4.1 Temperature in Rerigerating Compartment ........................................................ 453

16.4.2 Temperature in Freezer ......................................................................................... 453

16.4.3 Heat Transer Modes in Domestic Rerigerator .................................................... 453

16.5 Example o Heat Transer Analysis in a Typical Rerigerator............................................456

16.5.1 Empty Rerigerator Considered as being a Closed Rectangular Cavity ...............456

16.5.2 Empty Rerigerator Considered as a Combination o Vertical Plates .................. 45716.5.2.1 Estimation o the Convective Heat Transer Coefcient by

Natural Convection between the Evaporator and Air (hevap) ................ 457

16.5.2.2 Estimation o the Radiative Heat Transer Coefcient

between the Evaporator and the other Walls (hr) ................................. 458

16.5.2.3 Estimation o the Convective Heat Transer Coefcient

between Air and the Internal Walls o the

Rerigerator (hwi) ..................................................................................458

16.5.2.4 Estimation o the Overall Heat Transer Coefcient and

Rerigerating Capacity ......................................................................... 458

16.5.2.5 Estimation o the Thickness o the Thermal Boundary

Layer Near the Evaporator ...................................................................45916.6 Numerical Simulation in Domestic Rerigerator ................................................................460

16.6.1 Rerigerator Characteristics ..................................................................................460

16.6.2 Measurement o the Thermal Resistance o Rerigerator Insulation .................... 461

16.6.3 Temperature Measurement ...................................................................................462

16.6.4 Modeling ...............................................................................................................463

16.6.4.1 Main Assumptions and Boundary Conditions .....................................463

16.6.4.2 Mesh .....................................................................................................463

16.6.4.3 Discrete Ordinate (DO) Method or Radiation .................................... 463

16.6.5 Numerical Results (Taking into Account Radiation) ............................................465

16.6.5.1 Temperature Fields ...............................................................................465

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in : Mathematical Modelling of Food Processing, Mohammed M. Farid (ed.) ; CRC Press; Boca Raton, USA; 445-474; 2010


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446 Mathematical Modeling of Food Processing

16.6.5.2 Air Velocity Field .................................................................................465

16.6.5.3 Comparison with Numerical Simulation without Radiation ...............467

16.6.5.4 Comparison between the Measured and

Predicted Air Temperature .................................................................. 467

16.7 Conclusions ......................................................................................................................... 467Acknowledgment ........................................................................................................................... 472

Nomenclature ................................................................................................................................. 472

Reerences ......................................................................................................................................473

16.1 IntroduCtIon

Domestic rerigerators are widely used in industrialized countries. There are approximately 1 billion

domestic rerigerators worldwide1 and the demand in 2004, was 71.44 million units (including 11.2

in China, 10.7 in the United States, 4.43 in Japan, 3.36 in India, 3.14 in Brazil2). In developing coun-

tries, the production is rising steadily: total production rose 30% in 2000.3 In France, there are 1.7

rerigerators per household.

4

Epidemiological data rom Europe, North America, Australia and New Zealand indicate that

a substantial proportion o oodborne disease is attributed to improper ood preparation practices

in consumers homes.5 Data also illustrate that a large proportion o consumers lack knowledge o

adequate rerigeration temperatures. Surveys carried out in various countries on the temperature

and the microbial contamination in the rerigerating compartment under real use conditions show

an alarming situation.613 Compared with surveys on the rerigerating compartment, a ew surveys

have been carried out on temperatures in domestic reezers.14 Product temperature is a quality and

saety-determining actor. It is thereore necessary to ully understand the mechanism o heat trans-

er and airow.

Three types o domestic rerigerators are available in the market: static, brewed, and no-rost.

The static type (Figure 16.1a) is widely used in Europe. In this case, heat is transerred principally

by natural convection and airow is due to variations in air density. These variations are relatedprincipally to the temperature and humidity gradients. The vertical orce which results rom air

weight and buoyancy is ascendant i air is locally lighter than the average and descendant where the

opposite is true (hot/humid air is lighter than cold/dry air). There is a combination o heat transer

and airow inside rerigerators i.e., heat transer rom air to the evaporator (vertical plate), to the

other walls (cavity) and to products (various orms). Due to the principal o heat transer, tem-

perature heterogeneity is oten observed in this type o rerigerator. The position o the evaporator

(a) (b) (c)

FIgure 16.1 Three types o rerigerator: (a) static, (b) brewed, (c) no-rost.

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Heat Transfer and Air Flow in a Domestic Refrigerator 447

(horizontal/vertical, top/bottom o the compartment) determines the location o cold and warm

zones. The brewed type is a static rerigerator equipped with a an (Figure 16.1b). It allows air cir-

culation and the temperature decreases rapidly ater door opening. Air temperature is more homo-

geneous in this case than in the static type but the energy consumption is higher due to the an. In

a no-rost rerigerator (Figure 16.1c), a an (embedded in the back wall) pushes air to ow over theevaporator beore entering into the rerigerating compartment. Air temperature is more homogene-

ous compared to the two other rerigerator types. Disadvantages o no-rost type are noise, energy

consumption, drying on ood surace and high price.

It should be remembered that the rerigerator design in the United States and in Europe is quite

dierent: particularly, in the United States, the size o appliances is bigger and there are more rerig-

erators equipped with an.

Generally, the external dimensions o commercialized rerigerator are 60 60 cm (width depth)and the height varies between 90 cm and over 2 m. The wall thickness is approximately 4 cm and

the rerigerator is generally made o polystyrene (inner liner), polyurethane (insulating material)

and metal sheet (outer liner).

Knowledge o air temperature and velocity profles in a rerigerator is important or ood qualitycontrol. In act, i the consumer knows the position o warm and cold zones in the rerigerator, the

product can be placed correctly. Knowledge o thermal and hydrodynamic boundary layers near the

evaporator and the other walls is also important. I the product is too close to the evaporator wall,

reezing can occur, and i it is too close to the other walls, there may be health risks.

The objective o this chapter is to present the state o the art o knowledge on heat transer and

airow by natural convection in domestic rerigerators. Several subjects are dealt with: literature

review o natural convection in closed cavity, cold production system o domestic rerigerators,

temperatures and dierent heat transer modes in appliances. Finally, an example o heat transer

analysis and numerical simulation in a typical rerigerator will be shown.

16.2 LIterature revIew oF naturaL ConveCtIon In CLosed CavItyA literature review on natural convection in domestic rerigerators, near a vertical plate, in empty

closed cavities and in cavities flled with porous media will be presented. Some limits o the appli-

cation o these studies to our case (rerigerator loaded with a ood product) will also be given.

16.2.1 StudieSin domeStic RefRigeRatoRS

Several experimental studies were carried out on empty and loaded rerigerators.7,15 The objec-

tive was to analyze the eects o several parameters on the temperature in the rerigerating

compartment (thermostat setting, requency o door openings, flled volume, temperature and

humidity o ambient air). However, ew studies were carried out on airow measurement due

to the complexity o metrology techniques compared to temperature. Airow measurement in areezer compartment under real operating conditions was carried out by Lacerda et al.16 using PIV

(particles imagery velocimetry). It was observed that the ow feld was strongly inuenced by the

temperature variations due to the on and o operation cycles o compressor. This behavior

was attributed to natural convection and strong temperature dependency o air viscosity. Another

study on airow in a ventilated domestic reezing compartment was carried out by Lee et al.17 In

this study a comparison o velocity feld obtained by CFD simulation and by experiments (PIV

measurements) was undertaken. These authors observed that the ow was very complex: jet-like

ow around entrance ports, impinging and stagnation ow on the walls and a large recircula-

tion ow in cavity. To our knowledge, no study was carried out on air velocity measurement in

a rerigerating compartment. Moreover, airow being strongly inuenced by the aspect ratio

(height/width) o the cavity; the ow in a reezer is thereore dierent rom the one in a reriger-

ating compartment.

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To obtain useul inormation on natural convection in a domestic rerigerator, airow in some

well known confgurations will be presented: near a warm (or cold) vertical plate, empty cavity and

cavity flled with product. The temperature o the cold wall is constant or these three confgura-

tions in spite that this temperature uctuates due to the on and o compressor working cycles

in a real rerigerator.

16.2.2 Heat tRanSfeRand aiRflow neaRa VeRtical Plate

For a frst approach, literature on ow adjacent to a cold vertical plate placed in a warm environ-

ment (without other limiting walls) can be applied or a good understanding on how airow by

natural convection nears the rerigerators evaporator. I a tracer (e.g., smoke) is injected at one end

o the plate to visualize the ow, laminar ow is frstly observed near the wall and then turbulence

appears (Figure 16.2). The air velocity (u) is zero at the plate surace, then, it increases rapidly at

locations away rom the plate to attain a maximum value (um). Air velocity then decreases and

approaches zero, which is the velocity ar rom the plate (Figure 16.3). The zone o nonzero velocity

(u > um/100) is called the hydrodynamic boundary layer and its thickness () increases in the owdirection (x).The air temperature (T) increases rom the wall temperature (Tw) to the ambient temperature (T)

(Figure 16.4). The zone where the temperature diers rom ambient ((T T) > (Tw T)) is calledthe thermal boundary layer and its thickness (T) increases in the ow direction (x).

The equivalent boundary thermal layer thickness (T,eq) is also requently used in practice; it isdefned as T,eq = /h.

When Prandtl number (Pr) is near 1 such as in the case o air, the equivalent boundary layer

thickness (T,eq) is o the same order o magnitude as that o the thermal boundary layer ( T). Forexample, or laminar orced convection, T,eq = 2/3 T.

The ow regime in natural convection is characterized by the Rayleigh number (Ra) defned as:

Ra = g TL 3 (16.1)

Laminar

Turbulent

Xcritical

Tracer

Tw< T

x

y

T

FIgure. 16.2 Air ow by natural convection near the wall.

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In general, the critical Rayleigh number, which distinguishes the transition rom laminar to tur-bulent ows, is approximately 109 (depending on the geometry and boundary conditions.18

Heat transer phenomena depend on the ow regimes (laminar or turbulent). Khalia19 presents

a literature review o natural convection heat transer correlations or vertical or horizontal plates.

More than 40 articles are presented in this review. The experimental conditions are summarized:

dimension o the tested surace, uid type, temperature dierence between the plate and the uid,

and Rayleigh number range. A strong variation in the values o heat transer coefcient was ound

rom using these dierent correlations.

In general, the correlations are presented in the ollowing orm:

Nu Ra= a n. (16.2)

y

u

um

um

x

U=0

(x)

FIgure. 16.3 Hydrodynamic boundary layer and velocity profle in natural convection ow.

TTw

TTw

0 T y

T(x)

T

Tw

x

FIgure 16.4 Thermal boundary layer, temperature profle and dimensionless profle in natural convection.

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a and n are coefcients whose value depends on the ow regime. For example, or a vertical

plate, Incropera and Dewitt18 proposed:

a = 0.59 and n = 1/4 or laminar ow,

a = 0.10 and n = 1/3 or turbulent ow.

16.2.3 Heat tRanSfeRand aiRflowin emPty cloSed caVity

Several experimental studies were carried out to measure air temperature and/or velocity in closed

cavities.2024 Ostrach25, Catton26, and Yang27 carried out a literature review on this subject, which

included both the experimental and modeling results (2D and 3D). These authors emphasize the

importance o the aspect ratio o the cavity and the temperature dierence between walls on the

ow regimes.

When the bottom horizontal wall is cold, stable temperature stratifcation is observed in the

cavity (cold zone at the bottom and warm zone at the top), and there is no airow. When the upper

horizontal wall is cold, unstable ow is observed

25

due to gravity. The state o unstable equilibriumoccurs until a critical density gradient is exceeded. A spontaneous ow then results that eventually

becomes steady and cellular-like. When a vertical wall is cold, circular ow is observed along walls

and the air is almost stagnant at the center o the cavity; thermal stratifcation is also observed. This

case is similar to that o a domestic rerigerator, since the evaporator is oten inserted in the vertical

back wall.

There are ewer experimental studies on natural convection than on orced convection due to

experimental difculties in terms o metrology or low velocity and design o experimental devices

maintaining given wall conditions. In act, measurement is very sensitive to experimental and

boundary conditions. Henks and Hoogendoorn28 compared some experimental results obtained

with a standard case (Ra = 5 1010, cavity aspect rationH/L = 1 in 3D, adiabatic horizontal walls).Good agreement between results was ound, particularly about the temperature and velocity profles

within the boundary layers.Ramesh and Venkateshan29 used a dierential intererometer to visualize conditions in the bound-

ary layer along the wall (105 < Ra < 106). They ound that it is generally stable except in the corner.Mergui and Penot23 carried out a visualization o ow in an empty cavity using a laser tomography

(Ra = 1.7109); they observed the same phenomena as Ramesh and Venkateshan29.Deschamps et al.30 reported that in a domestic rerigerator, the Rayleigh number varies rom

108 and 109, and that ow is thereore, in the transition regime between laminar and turbulent

ow.

Heat exchange by radiation between internal walls o the cavity is as important as that achieved

with natural convection and this should be taken into account. Several authors3134 showed by experi-

mental and numerical approaches that these two heat transer modes occur simultaneously. Ramesh

and Venkateshan32

showed experimentally that or a square enclosure (vertical walls maintained at35 and 65C, adiabatic horizontal walls, Ra = 5 105), the heat transer by convection and radiationbetween high emissive vertical walls ( = 0.85) is twice o that o polished ones ( = 0.05). Balajiand Venkateshan31 proposed correlations established rom numerical simulations to express the

convection and radiation in a square cavity as a unction o , Ra, Tc/Th and a radiation convectioninteraction parameter

NT H

T TRC

wh

wh wc

=

4

( ).

These correlations show that the radiation eect increases when the wall emmisivity and/or

wall temperatures increase. Moreover, Li and Li34 reported that the radiation relative to convection

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increases as the size o enclosure increases. An estimation o convection and radiation heat transer in

a rerigerator was carried out in our previous study35, which confrms the importance o radiation.

16.2.4 Heat tRanSfeRand aiRflowin caVity comPletelyoR PaRtially

filledwitH PoRouS media

Several reviews on heat transer by natural convection in a cavity flled with porous media have been

carried out.3639 In the case o porous media, the Rayleigh number is defned as:

Rapp

g T H K =

(16.3)

When Rayleigh number is less than a critical value (Rac), the heat transer is dominated by conduc-

tion. When Rap > Rac, airow is observed, which leads to a heat transer dominated by convection.Oosthuizen39 reported a value o 40 or Rac in a rectangular cavity heated rom below.

Airow in a cavity flled with porous media is generally laminar. Circular ow, similar to that oan empty cavity, is observed in the boundary layer along the walls (Figure 16.5). Velocity is much

smaller at the center o the cavity.

Literature concerning heat transers in porous media and in packed beds4042 presents several

approaches taking into account heat transer by conduction, convection and radiation. Moreover,

these studies distinguish the one-temperature models, in which local equilibrium between product

and air is assumed, rom the two-temperature models, in which dierent temperatures represent

product and air statement not clear.

The literature on cavity flled with porous media cannot be applied directly to the case o loaded

domestic rerigerator principally due to the large variation in products dimension. For rerigerators,

the ratio between the dimension o product and cavity is about 0.10 (5 cm product width and 50

cm cavity width) while this ratio is 0.02 or porous media. There is notably a great inuence oproduct position on heat transer compared to the case o porous media. This was shown in ourprevious studies43 that demonstrate the inuence o these parameters on the heat transer at low air

velocity (< 0.2 m/s) in a stack o spheres.

Porous medium

Cold wall

Tc

Warm wall

Thg

y

x

0

0 L

Insulated wall

u

FIgure 16.5 Two-dimensional rectangular porous layer held between dierentially heated side walls.

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16.3 CoLd ProduCtIon system In domestICreFrIgerators/Freezers

The most common rerigerators and reezers have our major parts in their rerigeration systema

compressor, a condenser, an expansion valve and an evaporator (Figure 16.6). In the evaporatorsection, a rerigerant (commonly R600a and R134; still in use R12 and ammonia) is vaporized to

absorb heat added into the rerigerator due to heat transer across the rerigerators walls and infltra-

tion through the door and seals and during door opening. The rerigerant boils at 18 to 20C whenpressurized at 0.91 bar, so the evaporator temperature is maintained at or near that temperature ithe appliance is working correctly. In the next stage, an electric motor runs a small piston compres-

sor and the rerigerant is pressurized. This raises the temperature o the rerigerant and the resulting

superheated, high-pressure gas (it is still a gas at this point) is then condensed to a liquid in an air-

cooled condenser. In most rerigerators and reezers, the compressor is in the base and the condenser

coils are at the rear o the appliance. From the condenser, the liquid rerigerant ows through an

expansion valve (almost always a capillary tube), in which its pressure and temperature are reduced

and these conditions are maintained in the evaporator. The whole process operates continuously,

by transerring heat rom the evaporator section (inside the rerigerator) to the condenser section

(outside the rerigerator), by pumping rerigerant continuously through the system described above.

When the desired temperature is reached, the pump stops and so does heat transer.

The rerigerator/reezers may be equipped with one or two compressors. In the case o one com-

pressor, the operating cycle is both controlled by the air temperature in the rerigerating compart-

ment and in reezer. In the case o two compressors, each operating cycle is independently controlled

Condenser

Expansion

valve

Compressor

Evaporator

Termostat

FIgure 16.6 Cold production system in a domestic rerigerator/reezer.

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by the air temperature in the rerigerating compartment and in the reezer. The temperature in each

compartment is, thereore, better regulated but the price is higher.

16.4 temPeratures and Heat transFer modesIn domestIC reFrIgerators

The on and o operating cycle o the compressor leads to temperatures uctuations in the rerig-

erator. The air temperature inside the appliance is regulated by a thermostat. When this temperature,

measured at a given position, is higher than the maximum setting value, the compressor is on until

the minimum setting value is reached, and then it is switched o. The dierence between maxi-

mum and minimum settings is fxed by the manuacturer.

16.4.1 temPeRatuRein RefRigeRating comPaRtment

The temperatures in a rerigerating compartment o an empty one-door rerigerator equipped with

one compressor, measured using calibrated T-type thermocouples are presented here. The internaldimensions o this compartment were 50 50 cm (width depth) and the height was 90 cm. The ver-tical walls exchanged heat with the external ambience. The thickness o these walls was 4 cm and

the overall thermal conductivity was 0.027 W/(mC). The evaporator was ftted inside the vertical

back wall and it was 50 cm wide and 30 cm high.

The temperature variations are shown in Figure 16.7 or a thermostat setting at 6C and the

average ambient temperature at 20C. The on and o compressor work cycles lead to variations

in the evaporator wall temperature (Tevap) (Figure 16.7a). It can be seen that the temperature varies

within a range o+ 7C to 12C (average temperature 1.2C).The air temperature was measured at the top, middle, and bottom levels at the center o the

rerigerating compartment (Figure 16.7b). It can be seen that the temperature is heterogeneous in

the cavity: air at the bottom is cooler than that at the top. The mean air temperature calculated rom

25 measurements is also shown in Figure 16.7b and the average value o this temperature over 24 h

(Tai) is 6.3C (minimum value 3.8C and maximum value 8.3C).

The wall temperature variations are shown in Figure 16.7c, the mean value at the top level being

9.1C, at the middle level 5.4C and at the bottom level 5.7C. The average value o these three

temperatures (Twi) is 6.7C.

16.4.2 temPeRatuRein fReezeR

The air temperature uctuations in a domestic reezer equipped with one compressor are generally

more signifcant than those equipped with two compressors. An example o these uctuations is pre-

sented in Figure 16.8. Both rerigerators are two-door models, with a rerigerating compartment on

the top and a reezing compartment on the bottom. Air-temperature stratifcation can be observed inthe compartment in both cases. The air on the top shel was slightly higher than that on the middle

one (cold air is heavier). The characteristics o these two rerigerators are presented in Table 16.1.

The wall o the reezing compartment o these rerigerators was composed o an inner liner (1 mm

o polystyrene, = 0.15 W.m1.K1), oam (polyurethane, = 0.02 W.m1.K1, 5.8 cm or the onecompressor rerigerator and 6.3 cm or the two compressor rerigerator) and a metal outer sheet (0.7

mm, = 50 W.m1.K1). It was clearly shown, in this example, that the temperature was less stablein the reezing compartment o the one compressor rerigerator.

16.4.3 Heat tRanSfeR modeSin domeStic RefRigeRatoR

In an empty rerigerator, cold air near the evaporator ows downward and warm air near the

door and the other side walls ows upward (Figure 16.9). The heat exchanges inside the cavity

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0

2

4

6

8

10

12

60 1 2 3 4 5Time (h)

Temperature

(C)

Twall middle

Twall bottom

Twall top

0

2

4

6

8

10

12

6543210

Time (h)

Temperature

(C)

Tair middle

Tair bottom

Tair top

T

airmean

Tai= 6.3C

Time (h)

Temperature

(C)

Tevap= 1.2C

0 1 2 3 4 5 6

Compressor

off Compressor

on10

5

0

5

10

15

(a)

(b)

(c)

FIgure 16.7 Example o temperature variations in a rerigerating compartment o one compressor appli-ance or a thermostat setting at 6C: (a) evaporator wall, (b) air temperature, (c) wall temperature.

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are governed by natural convection between internal walls and air, radiation between evapora-

tor and the other walls and conduction within the walls.35 In the case o a rerigerator flled

with products, the products are cooled by natural convection, radiation between the surace o

the products and the internal walls o the rerigerator, and through conduction and radiationbetween products.

10.0(a)

(b)

15.0

20.0

25.0

30.0

35.00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

Temperature

(C)

Middle shelfTop shelf

Time (h)

10.0

15.0

20.0

25.0

30.0

35.00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

Temperature

(C)

Middle shelfTop shelf

Time (h)

FIgure 16.8 Example o air-temperature variations in the reezing compartment: (a) one compressorrerigerator, and (b) two compressor rerigerator.

tabLe 16.1Chciic ri

o-cp ri t-cp ri

External dimensions

(heightwidthdepth)

185 cm60 cm60 cm 195 cm60 cm60 cm

Internal dimensions o the reezing

compartment

62 cm48 cm38 cm 64 cm47 cm42 cm

Thermostat setting +4C (impossible to set the

temperature o reezing compartment)

+4C (rerigerating compartment)

and 18C (reezing compartment)

Power o compressor 120 W 160 W

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16.5 examPLe oF Heat transFer anaLysIs In a tyPICaL reFrIgerator

In order to study heat transer inside a rerigerator, natural convection theories covering the ollow-

ing cases can be applied:

Rectangular closed cavity representing heat transer inside the rerigerating compartment

Cold vertical plate placed in a warm ambience representing exchanges between the evapo-

rator and airRectangular closed cavity partially flled with porous media representing a loaded

rerigerator. This case is more complex and the study requires numerical simulations as

presented in Section 16.6.

16.5.1 emPty RefRigeRatoR conSideRedaSbeinga cloSed RectangulaR caVity

The simplest approach that can be used in order to approximate the transers inside an empty rerig-

erator is to consider it as a rectangular cavity.35 Circular air circulation is established, cool air close

to the evaporator moves downward and hot air in contact with the door moves upward.

In order to simpliy the study, the exchanges inside the cavity are initially, considered as a two

dimensional problem o heat transer between a cold vertical wall (Tevap) and a hot vertical wall (Twi).The vertical walls are assumed to have a homogeneous temperature and the horizontal walls are

adiabatic. This is a rough approximation because, in act, only part o the vertical wall o a rerigera-

tor is taken up by the evaporator and heat losses occur through at least three vertical walls.

Despite this, the order o magnitude o the Rayleigh number (Ra) can give some qualitative inor-

mation regarding the hydrodynamic and thermal boundary layers. The rerigerator described in

Section 16.4.1 will be used or the analysis. The aspect ratio (H/L) o this typical rerigerator is equal

to 1.8. The Rayleigh number can be based on height or width, but since H/L = 1.8, the values areo the same order o magnitude. For simplicitys sake, the walls and air temperatures are assumed

to be constant. The dierence between the inner wall temperature and the evaporator temperature

(T= Twi Tevap) is equal to 7.9C.The physical properties o air, such as conductivity (), thermal expansion coefcient (), di-

usivity () and kinetic viscosity (), are calculated at the reerence temperature (T) defned as the

Convective heat transferin the boundary layernear the evaporator

Cold wall(evaporator)

Door

External convectiveheat transfer

Conductive heat transferinside the wall

Convective heat transferin the boundary layer

near the door

Radiation heat transfer

Tair

.

Q

FIgure 16.9 Various heat exchange modes and airow inside a domestic rerigerator.

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average temperature o the evaporator and the other walls (T= 275.8 K). The Prandtl number orair is taken constant and equal to 0.72.

The Rayleigh number based on the width o the rerigerating compartment (L = 0.5 m) and thetemperature dierence between the evaporator and the other walls (T) is equal to 1.43 108.

Thus, the air ow inside the rerigerator is laminar (Ra < 109). This result is in agreement withthat o Deschamps et al.30 who showed that Ra varies between 108 and 109 in rerigerators.

As the Rayleigh number is higher than 103, a stationary core region can be expected, indicating

very low air velocities in the area where ood is stored.

For this range o Rayleigh number and aspect ratio, a vertical thermal gradient o approximately

(Twi Tevap)/2 is expected,44 that is 4C in our case.These expectations are qualitatively confrmed by wall temperature measurement inside the

rerigerator (Figure 16.7c). In particular, a wall temperature dierence between the top and bottom

levels o 3.4C was observed.

In order to estimate the heat transer coefcient in our rerigerator, the correlation proposed by

Catton,26 valid or 1


14/30


The Rayleigh number based on the height o evaporator (Hevap = 0.3 m) and the dierence betweenthe temperature o the evaporator and that o air is: Ra evap = 2.0107.

Since Ra < 109 , the correlation proposed by Incropera and Dewitt18 or laminar ow is used:

Nu = 0.59 Ra1/4 = 39.4, thus hevap = 3.28 W/(m2.C).

16.5.2.2 eii h rii H t Cfci h ep h h wll (hr)

Heat transer by radiation occurs between the evaporator and the other walls. An equivalent radia-

tive heat transer coefcient hrcan be defned. For parallel walls o emissivity near 1:

h T T T T r = + + 1 2 2 2( )( )evap wi evap wi (16.5)

= = Boltzmann constant W/(m .K )2 45 67 10 8.

In the case o the rerigerator, the emissivity o the internal suraces is approximately 1 = 2 = 0.9and we obtain hr= 3.85W/(m2.K).

One can note that the value o the radiative heat transer coefcient is o the same order o mag-

nitude as that o natural convection. Thereore, it should be taken into account when investigating

heat transer in rerigerators.

16.5.2.3 eii h Cci H t Cfci ai h Il wll h ri (hi)

The same approach is used or the side walls as or the evaporator. The Rayleigh number, based on

the temperature dierence between that o inner side wall and inner air and on the height o the

walls is 4.2

107 (Ra

( )1 0r s n I s nd

uux

+ r sT4 (16.7)

The walls are assumed as gray diuse: Iout rad out/= _Iin: intensity o incident radiation in

s direction (at

r position)

Iout: intensity o radiation leaving the surace (atr position)

n : normal vector

Ts

: surace temperature, K

: solid angle

53515_C016.indd 463 6/18/09 8:05:47 PM



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(a) (b) (c)

FIgure 16.15Mesh structure: (a) empty rerigerator, (b) rerigerator ftted with glass shelves, (c) rerigera-

tor loaded with the test product.

tabLe 16.4n Cll h sili

mh n Hih (136 c) Hl ih (26 c) dph (44 c) tl

Empty rerigerator 138 28 66 255,024

Rerigerator with shelves 222 28 66 410,256

Rerigerator with shelves and products 240 62 74 1,101,120

tabLe 16.3rli P i sili

rli Fc tp dicii

Pressure 0.8 Presto

Density 1

Gravity orces 1

Momentum 0.2 Second order upwind

Energy 1 Second order upwind

Radiation 1

Pressurevelocity Simple

53515_C016.indd 464 6/18/09 8:05:49 PM



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A sensitivity study o solid angle discretization was carried out beorehand in order to ensure that

the simulation results were not inuenced by the number o solid angle subdivisions.

16.6.5 numeRical ReSultS (takinginto account Radiation)

The results presented in this paragraph concern simulation, which takes into account heat transer

by convection between walls and air and by radiation between the internal walls o the rerigerat-

ing compartment. Two results will be presented: air temperature and velocity felds. A comparison

between numerical and experimental results is carried out only or temperature. In act, air velocity

in a real rerigerator is difcult to measure with precision in practice.

16.6.5.1 tp Fil

The temperature felds obtained rom simulations or the dierent cases studied are shown in

Figure 16.16. Considering only the main cavity (excluding the vegetable box), or all cases, thermal

stratifcation is observed with the cold zone at the bottom (~2C) o the rerigerating compartment

and the warm zone at the top (89C). In addition, a cold zone is also observed along the back

wall. This is related to cold air coming rom the evaporator. When the rerigerator is loaded with

products, the temperature o the product located near the evaporator is lower than that located

near the door. In the top hal o the compartment, the temperature is relatively homogeneous at a

given height (except in the boundary layers near the walls). The temperature o the vegetable box

is almost constant or all cases studied (~8C).

The temperature feld is slightly inuenced by the presence o obstacles: shelves and products. A

slightly lower temperature is observed at the bottom and a slightly higher one at the top compared

with the empty rerigerator case. This is due to the act that the shelves and/or the products slowed

down the air circulation in the central zone o the rerigerator. The presence o shelves and/or prod-

ucts also inuenced the main air circulation in the boundary layers situated along the evaporator

and the side walls. However, this inuence is weak because o the presence o air spaces between

the shelves and the vertical walls (1.2 cm between the back wall and the shelves), which acilitatesthe air ow. In our previous study, it was ound that the thickness o the boundary layer was less

than 2 cm.45

In addition to the overall thermal stratifcation in the cavity, stratifcation is also observed in

each gap between two shelves or between a shel and a product. It is to be emphasized that or the

rerigerator loaded with the test product, the symmetry plane is located in the gap between two

piles. This explains why the packages are invisible on this plane (Figure 16.16c). On the plane

situated at 8 cm rom a side wall which cuts the product pile (Figure 16.16d), a cold product zone

near the evaporator can be clearly distinguished. This is related to the blockage o cold air by

the product.

The average and maximum air temperatures in all cases are reported in Table 16.5. The air

temperatures increase with increasing numbers o obstacles.

16.6.5.2 ai vlci Fil

Figure 16.17 presents the air velocity felds on the symmetry plane (Figure 16.17ac) and on the

plane situated at 8 cm rom the side wall (Figure 16.17d) or the dierent cases studied. Considering

only the main cavity (excluding the vegetable box), or all cases, the main air circulation is observed

near the walls, and constitutes a recirculation loop. Air ows downward along the evaporator while

its velocity increases along the course to attain a maximum value at the bottom o the rerigerator

(umax 0.2 m s1). Air then ows upward along the door and the side walls o the rerigerator whileits velocity decreases progressively and becomes stagnant at the top o the rerigerator. This observa-

tion is in agreement with the air temperature feld shown in Figure 16.16, with cold air located at the

bottom o the cavity and warm air at the top. It can also be observed that there is a weak horizontal

air ow rom the door to the evaporator. However, the air velocity at the center o the cavity is very

53515_C016.indd 465 6/18/09 8:05:49 PM



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low (< 0.04 m s1). In the case o the rerigerator ftted with glass shelves, in addition to the mainair ow along the walls as mentioned previously, there are also small air loops between the shelves.

For the rerigerator loaded with products, air ows in the gaps between the shelves and the products(Figure 16.17d).

(a)

(c) (d)

(b)

Evaporator

Evaporator

Evaporator

Evaporator

8.20

7.667.11

6.57

6.02

5.48

4.94

4.39

3.85

3.31

2.76

2.22

1.671.13

0.59

0.04

0.50

9.00

8.41

7.81

7.22

6.63

6.03

5.44

4.84

4.25

3.66

3.06

2.47

1.88

1.28

0.69

0.09

0.50

9.06

8.46

7.87

7.27

6.67

6.07

5.48

4.88

4.28

3.68

3.09

2.49

1.89

1.29

0.70

0.10

0.50

8.98

8.39

7.79

7.20

6.61

6.02

5.43

4.83

4.24

3.65

3.06

2.46

1.87

1.28

0.69

0.09

0.50

T(C)T(C)

T(C) T(C)

FIgure 16.16 Predicted temperature felds (C): (a) on the symmetry plan o empty rerigerator, (b) on thesymmetry plan o rerigerator with glass shelves, (c) on the symmetry plan o rerigerator loaded with prod-

ucts, (d) on the plan situated at 8 cm rom the side wall o rerigerator loaded with products.

53515_C016.indd 466 6/18/09 8:05:51 PM



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It should be remembered that the containers attached to the door were not represented in our

study. In practice these containers are an obstacle to airow along the door and tend to reduce the

air velocity in this area.

Considering the vegetable box, one or two air recirculation loops were observed (Figure 16.17).

This is due to the presence o the glass shel (cold wall), which separates the vegetable box rom themain cavity, and the fve other walls which are warmer (heat loss through these walls).

16.6.5.3 Cpi ih nicl sili ih rii

Figure 16.18 presents the air temperature feld on the symmetry plane obtained by simulation with-

out taking into consideration radiation (between internal walls o the rerigerating compartment,

shelves and product surace). It was observed that overall the temperature feld is similar to that

present when radiation is taken into account (a cold zone at the bottom and a warm zone at the top).

However, stratifcation is more pronounced without radiation, and this leads to a higher temperature

at the top o the cavity. In act, or an empty rerigerator, the maximum temperature rises rom 8C

(with radiation) to 15C (without radiation). This temperature increase can be explained by the act

that, without radiation, there is no heat exchange between the warm top wall and the other colderwalls, particularly the evaporator wall. This contributes to a high air temperature at the top posi-

tion. When radiation is taken into account, the heat exchange between the top wall and the other

walls tends to reduce the top wall temperature and consequently reduces air temperature near this

wall. From a microbiological point o view, the growth rate is much higher at 15C than at 8C. It is

thereore necessary to take into consideration radiation in the simulation in order to better describe

the phenomena occurring in domestic rerigerators.

16.6.5.4 Cpi h m Pic ai tp

Figure 16.19 presents a comparison between the measured and predicted air temperature results

(with and without taking into account radiation). It can be seen that the simulation results with radia-

tion agreed with the experimental values to a greater extent, while simulation without radiation over-

estimated the air temperature, particularly at the top o the rerigerator. The peaks observed on the

temperature profle in the presence o shelves and/or products can be explained by the higher con-

ductivity o glass compared with air and by the cold air ow along the upper sides o the shelves.

The agreement between the experimental and simulation results is relatively poor in the case o

a loaded rerigerator, even though the radiative heat exchange between the product and the walls

was taken into account. This may be explained by the geometry complexity. Further grid refnement

could lead to a better agreement, but the computing time is already very excessive (about 8 days

using a cluster o our processors o 2 Gigaoctets (Go) o RAM).

16.7 ConCLusIons

Several studies illustrate that a large proportion o rerigerators operate at too high temperatures.These rerigerators are oten static types (without an) in which heat is transerred by natural

tabLe 16.5a mi ai tp h th sili

a tp i

h mi Ci (C)

mi tp

i h mi Ci (C)

a tp i

h vl b (C)

Empty rerigerator 3.8 8.2 7.4

Rerigerator with glass shelves 4.0 9.0 8.2

Rerigerator with glass shelves

and products

5.1 9.1 8.0

53515_C016.indd 467 6/18/09 8:05:51 PM



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(a)

(c) (d)

(b)

u (ms1)

u (ms1) u (ms1)

u (ms1)

0.21

0.20

0.190.18

0.17

0.16

0.15

0.14

Evaporator

Evaporator

Evaporator

Evaporator

0.13

0.12

0.11

0.09

0.08

0.07

0.060.05

0.04

0.03

0.02

0.01

0.00

0.23

0.22

0.21

0.20

0.19

0.17

0.16

0.15

0.14

0.13

0.12

0.10

0.09

0.08

0.07

0.06

0.05

0.03

0.02

0.01

0.00

0.23

0.22

0.21

0.20

0.19

0.17

0.16

0.15

0.14

0.13

0.12

0.10

0.09

0.08

0.07

0.06

0.05

0.03

0.02

0.01

0.00

0.18

0.17

0.16

0.15

0.14

0.13

0.13

0.12

0.11

0.10

0.09

0.08

0.07

0.06

0.050.04

0.04

0.03

0.02

0.01

0.00

FIgure 16.17 Path lines: (a) on the symmetry plan o the empty rerigerator, (b) on the symmetry plan othe rerigerator ftted with glass shelves, (c) on the symmetry plan o the rerigerator loaded with products, (d)

on the plan situated at 8 cm rom the side wall o rerigerator loaded with products.

53515_C016.indd 468 6/18/09 8:05:54 PM



25/30


T(C)

T(C) T(C)

T(C)

(a)

(c) (d)

(b)

Evaporator

Evaporator

Evaporator

Evaporator

15.35

14.36

13.37

12.38

11.39

10.40

9.41

8.42

7.42

6.43

5.44

4.45

3.462.47

1.48

0.49

0.50

13.82

12.93

12.03

11.1410.24

9.35

8.45

7.55

6.66

5.77

4.87

3.97

3.08

2.19

1.29

0.40

0.50

13.88

12.98

12.08

11.1810.29

9.39

8.49

7.59

6.69

5.79

4.89

3.99

3.10

2.20

1.30

0.40

0.50

15.08

14.1113.13

12.16

11.19

10.21

9.24

8.26

7.29

6.32

5.34

4.37

3.40

2.42

1.45

0.47

0.50

FIgure 16.18 Temperature feld (radiation not taken into account): (a) on the symmetry plan o the emptyrerigerator, (b) on the symmetry plan o the rerigerator ftted with glass shelves, (c) on the symmetry plan

o the rerigerator loaded with the test product, (d) on the plan situated at 8 cm rom the side wall o the

rerigerator loaded with products.

53515_C016.indd 469 6/18/09 8:05:55 PM



26/30


140

120

100

80

60

40

20

0

140

120

100

80

60

40

20

0

140

120

100

80

60

40

20

0

Height(cm

)

Height(cm

)

Height(cm

)

(a) (b)

(c)

0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8

0.00 0.20 0.40 0.60 0.80

21.5 cm 21.5 cm

21.5 cm

Symmetry plan Symmetry plan

Symmetry plan

Experimental

Numerical simulation with radiation

Numerical simulation without radiation

Experimental

Numerical simulation with radiation

Numerical simulation without radiation

Experimental

Numerical simulation with radiationNumerical simulation without radiation

(T Tevap)/(Tamb Tevap) (T Tevap)/(Tamb Tevap)

(T Tevap)/(Tamb Tevap)

FIgure 16.19 Comparison between experimental air temperatures and predicted values obtained by simu-lation with and without radiation: (a) empty rerigerator, (b) rerigerator ftted with glass shelves, (c) rerigera-

tor loaded with products.

53515_C016.indd 470 6/18/09 8:05:57 PM



27/30


convection. From a ood quality and saety point o view, it is necessary to ully understand the

mechanism o heat transer and airow in rerigerators.

A literature review on natural convection near a vertical plate, in empty and flled cavity was

presented in this chapter. Cold production systems and temperature evolutions in domestic rerig-

erators were shown. Several heat transer modes which may occur in rerigerators are presented:convection, conduction and radiation. It is to be emphasized that radiation may be the same order o

magnitude as natural convection.

The heat transer by natural convection in a typical domestic rerigerator was analyzed using

results recorded in the literature concerning:

Transer in a rectangular empty cavity

Transer between vertical plates and air

In spite o a simplifed hypothesis, the simple mathematical approach provided some useul

inormation on the rerigerator:

The model, which includes natural convection between the evaporator and air and between

the walls and air, radiation between the walls, and conduction inside the walls, gives a

good approximation o the required rerigerating capacity and air temperature. This model

slightly overestimates the wall temperatures.

Inside the boundary layer near the evaporator, there is a zone a ew millimeters wide

in which the temperature is below 0C. Thereore, placing ood in this zone can lead to

reezing.

Outside the boundary layers, the air is practically stagnant. This means that air circulation

is induced near the evaporator and the side walls. However, in the core region, where ood

is stored, there are low velocities, which do not ensure marked convective heat transer

between air and products.

The temperature dierence between the top and bottom levels o rerigerator can be esti-

mated as being a hal o the temperature gradient between the lateral walls and the evapo-

rator. In this type o rerigerator, sensitive ood should not be stored on the top shel.

CFD numerical simulation seems to be a powerul tool to study the complex 3D heat transer

and airow in rerigerator. Three confgurations were studied: an empty rerigerator, an empty

rerigerator ftted with glass shelves and a rerigerator loaded with products. When radiation was

taken into consideration in simulation, the predicted air temperatures were in good agreement with

the experimental values. However, when radiation was not taken into account, the temperature was

over-estimated, particularly at the top o the rerigerator. Radiation allows heat exchange, particu-

larly between the top wall and the cold wall (evaporator); consequently, it limits the stratifcation

phenomena.The obstacles (shelves and/or products) slow down the air circulation in the central zone o the

rerigerator and mildly inuence the main air circulation along the walls. This is confrmed by the

maximum values o air temperature: 8.2oC or an empty rerigerator without shelves and 9.1oC or a

rerigerator loaded with products.

Whatever the confguration studied (empty with/without shelves, loaded with products) or this

type o rerigerator, the air temperature at the top o the rerigerator is about 5C higher than the aver-

age air temperature, and thereore it is important to avoid placing sensitive products in this position.

The CFD simulation developed by our work can be urther used as a tool to study the inuence

on the temperature and velocity felds o operating conditions: evaporator temperature (parameter

related to the thermostat setting by the consumer), dimensions o the evaporator (parameter related

to design) and percentage o product-occupied volume in the rerigerating compartment.

53515_C016.indd 471 6/18/09 8:05:58 PM



28/30


aCknowLedgment

The author would like to thank Proessor Denis Flick (AgroParisTech, 16 rue Claude Bernard,

75231 Paris Cedex 05, France) or the review o this chapter.

nomenCLature

Cp Thermal capacity (J/kg/C)

h Heat transer coefcient (W/m2/C)

H Height (m)

I Intensity o radiation (W/m2) per unit solid angle

L Width or characteristic length (m)

NRC Radiation-convection interaction parameter

g Acceleration due to gravity (9.81 m/s2)

r Radius (m)

R Thermal resistance (C/W) or Radius (m)T Temperature (C or K)

T Temperature dierence between cold and warm walls, C or K

Q Rerigerating power (W)

u Air velocity in ow direction (m/s)

Greek symbol

Thermal diusivity (m2/s)

Thermal expansion coefcient (K1)

Thermal conductivity (W/m/C)

Kinetic viscosity (m2/s)

Boundary layer thickness (m) Density (kg/m3)

Emissivity o the wall

Solid angle

Radiative ux (W/m2)

Subscript

a Air

Far rom walls or bulk

ai Air inside the rerigerator

ae Air outside rerigeratorext External

evap Evaporator

Film

gl Overall

p Product

r Radiation

s Surace

T Thermal

w wall

wi Internal wall

we External wall

53515_C016.indd 472 6/18/09 8:05:58 PM



29/30


wc Cold wall

wh Hot wall

Dimensionless number

Pr Prandtl number =

Nu Nusselt number =hL

Ra Rayleigh number =g TL

3

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