Journal of Food Engineering 149 (2015) 78–86

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Journal of Food Engineering

journal homepage: www.elsevier .com/locate / j foodeng

Simplified heat transfer modeling in a cold room filled with foodproducts

http://dx.doi.org/10.1016/j.jfoodeng.2014.09.0230260-8774/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +33 1 40 96 61 21; fax: +33 1 40 96 60 75.E-mail address: onrawee.laguerre@irstea.fr (O. Laguerre).

O. Laguerre a,⇑, S. Duret a,b,d, H.M. Hoang a, L. Guillier d, D. Flick b,c

a Irstea, UR GPAN, 1 rue Pierre-Gilles de Gennes, 92761 Antony, Franceb AgroParisTech, UMR 1145-GENIAL, F-91300 Massy, Francec INRA, UMR 1145-GENIAL, F-91300 Massy, Franced Université Paris-Est, Anses, French Agency for Food, Environmental and Occupational Health and Safety, Food Safety Laboratory, 23 avenue du Général de Gaulle, F-94706 Maisons-Alfort Cedex, France

a r t i c l e i n f o

Article history:Received 25 March 2014Received in revised form 4 September 2014Accepted 10 September 2014Available online 2 October 2014

Keywords:Cold roomExperimentSimplified modelTemperature

a b s t r a c t

High product and air temperatures and moistures are often observed in certain positions of a cold roomleading to deterioration of food quality and safety. To reduce food losses, it is necessary to understandheat and mass transfers. However, these transfers are complex phenomena because of the presence ofthe product (airflow obstacle, heat of respiration. . .) and the coupling between the transfers and airflow.Temperature and velocity measurements were carried out in a ventilated cold room filled with four applepallets. Because of the room small dimensions (3.4 � 3.4 � 2.5 m), the cold supply air headed directly tothe rear part of the room. Therefore, the rear pallets were submitted to lower air temperatures comparedto the front ones. This leads to more rapid product cooling rate and lower product final temperature atthe rear. A simplified model was developed. It describes the evolution of product and air temperaturesat different zones in the cold room. Good agreement between the predicted and experimental resultswas found for both product cooling rate and final product temperature.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

The cooling process and storage conditions have to be properlycarried out in order to obtain high quality and safety for food prod-ucts. The International Institute of Refrigeration has reported thefood losses due to the lack of refrigeration worldwide (IIR, 2009).It was estimated that, if developing countries could acquire thesame level of refrigerated equipment as that in industrializedcountries, over 200 million tonnes of perishable foods would bepreserved, this being roughly 14% of the current consumption inthese countries (Table 1). In Europe, food losses of 280 kg/yearper capita were reported, 65% are related to improper handlingduring production to distribution and 35% in consumer behavior(FAO, 2011). The good control of temperature in refrigeratingequipment is, thus, a key factor which can be achieved usingappropriate techniques.

The understanding of the airflow, heat and mass transfer in acold room is the main objective of this study. Concretely, a simpli-fied heat transfer model in a ventilated cold room filled with foodproducts of a complex configuration (stack of food in pallets) was

developed. This model which is based on the zonal approach canbe used to predict the product cooling rate (transient state) andthe product final temperature after long storage period (stationarystate). The model validation was carried out by comparing theexperimental and calculated product temperatures measured atdifferent positions in the cold room. By combining it with a weightloss model, the product weight evolution during storage in the coldroom can be predicted.

This simplified cold room model will be combined with theones already developed by our team for refrigerated vehicle(Hoang et al., 2012), display cabinet (Laguerre et al., 2012) anddomestic refrigerator (Laguerre and Flick, 2010) to describe theproduct time–temperature history along the cold chain untilconsumption.

2. Literature review

Uniform storage conditions in cold stores are difficult to attainin practice. Several studies have shown temperature and moistureheterogeneity, with non-uniform airflow in cold rooms (Miradeand Daudin, 2006; Chourasia and Goswami, 2007; Kolodziejczykand Butrymowicz, 2011). This uneven distribution of airflow isrelated to the presence of the product and the cooling equipment

Nomenclature

aw water activity dimensionlessBi Biot number Bi ¼ hc � Rp=k dimensionlessC product heat capacity, J kg�1 K�1

Ca air heat capacity, J kg�1 K�1

h overall heat transfer coefficient, W m�2 K�1

Cw water vapor concentration at product skin,kg water=m3 humid air

Cw,1 water vapor concentration of surrounding air,kg water=m3 humid air

hc convective heat transfer coefficient, W m�2 K�1

K thermal insulation coefficient (transmittance) of coldroom, W m�2 K�1

kta moisture transfer coefficient of the apple skin, m s�1

m product weight, kgMH2O molecular weight of water, kg mol�1 (=0.018)_m air mass flow rate, kg s�1

P pressure, Paq heat of respiration, WRp product radius, mR universal gas constant, J K�1 mol�1

RH relative humidity of air, %S surface area of cold room, m2

T temperature, �C or KV air velocity magnitude, m s�1

a, c, b air distribution coefficient dimensionlessk product thermal conductivity, W m�1 K�1

Subscriptsa airave averagec cores surfacei initialf finalth thermostatfl front loadrl rear loadfw front wallrw rear wall

O. Laguerre et al. / Journal of Food Engineering 149 (2015) 78–86 79

(Ho et al., 2010). Variation of heat transfer coefficient between theair and the product at different positions in the cold room was alsoobserved by Flick et al. (1999) and Mirade (2007), leading to differ-ent product cooling rates. The heat transfer phenomena involvedduring product cooling and storage are: conduction withinproducts, convection (between cold air and product surface) andradiation (between product surface and cold room walls) (Huand Sun, 2000). Simultaneously, moisture evaporation from theproduct surface can be significant which causes product weightloss. In the case of beef carcasses for example, the weight loss isaround 1.5% for 1 day storage and 2.3% for 2 days storage, whichrepresents around 20 times the cost of the process function(Gigiel and Collett, 1989). Both the heat and moisture transfersare influenced by flow characteristics (such as cooling air temper-ature and velocity), air properties (viscosity, density, conductivityand specific heat), product properties, shape, dimension andarrangement of the load.

The comprehension of the heat/mass transfer and airflow incold stores is a complex task because of several interdependentfactors acting simultaneously (Smale et al., 2006). Failure to under-stand the phenomena taking place in the equipment results inexcessive weight loss, reduced shelf life or deterioration in productquality (James, 1996). This deterioration rate is more significant inthe case of high product respiration rate at warm zone or chillinginjury at cold zone (James, 1996).

Table 1Physical properties of ‘‘Jonagold’’ apple variety (van der Sluis et al., 2012).

Average weight (kg) 0.245Average diameter (mm) 77.5Water content (% of mass fraction) 85.3Density (kg m�3) 898Bulk density (kg m�3) 320Thermal conductivity at 10 �C (W m�1 K�1) 0.463 (Lisowa et al.,

2002)Specific heat at 10 �C (J kg�1 K�1) 3780Respiratory heat at 10 �C (kJ day�1 per ton

apples)Early ripening 3520–5230Late ripening 1760–2680

Computational Fluid Dynamic (CFD) models (Delele et al., 2008;Hoang et al., 2000; Nahor et al., 2005; Mirade and Daudin, 2006)were developed to predict temperature and velocity in refrigerat-ing equipment of the cold chain. There are 2 main advantages ofthe CFD approach. Firstly, it enables the knowledge of local param-eters such as temperature, humidity and velocity allowing theunderstanding of phenomena. Secondly, it enables the predictionof the influence of operating conditions and the equipment designwithout doing experience which is expensive or impossible toundertaken. Although CFD is a powerful simulation tool, its useis sometimes limited because of the high number of cells neededwhich leads to high calculation time. As a complementary to CFDapproach and to reduce the calculation time, simplified modelswere developed for cold room (Wang and Touber, 1990), domesticrefrigerator (Laguerre and Flick, 2010), display cabinet (Laguerreet al., 2012) and refrigerated vehicle (Hoang et al., 2012). Thesesimplified models are more appropriated to predict the time–tem-perature history of a high number of product items in a cold chaincomposed of several equipments.

3. Experimental study

Experiments were carried out in a cold room loaded by productto better understand the airflow and heat transfer. The obtaineddata were, then, used to compare with the ones calculated by thesimplified model.

3.1. Description of cold room

The cold room was 3.4 m long, 3.4 m wide and 2.5 m high (totalvolume 29 m3) in which 4 pallets were placed. Each pallet (1.2 mlong, 1 m wide and 1.75 m high) was composed of 64 bins (0.5 mlong, 0.3 m wide and 0.2 m high) and each bin was filled with 34apples (Jonagold variety). The total weight of the apples was about2560 kg. It is to be emphasized that in Europe, the ‘‘Jonagold’’ vari-ety held the 4th place in volume in 2009. The physical properties ofthis apple variety are presented in Table 1.

The cooling unit, located at the ceiling of the room, includedtwo axial fans of 30 cm of diameter rotating at 1320 RPM. The flow

Fig. 1. Dimensions (in meter) and position of the pallets in the cold room (a) – top view (b) – side view.

Fig. 2. Heat transfer during apple cooling (a) – conduction and convection (b) –temperature profile.

80 O. Laguerre et al. / Journal of Food Engineering 149 (2015) 78–86

rate of the cooling unit was 2450 m3 h�1 corresponding to the sup-plied air velocity of 4.8 m s�1 (measured data). Position of pallets:rear (pallet A and B), front (pallet C and D) and dimensions of thecold room are presented in Fig. 1. At first, the apples were placedfor 3 days at a room temperature of 20 �C to homogenize the tem-perature in pallets and for instrumentation. Then, the pallets weretransferred in a cold room previously set at 4 �C and stayed for4 months during which the measurements were carried out.

After 4 months storage at 4 �C, the product was warmed up bysetting the thermostat temperature of the cold room at 16 �C andthe measurements were carried out in the same manner.

3.2. Weight loss measurements

The weight loss is an indicator of quality of apple in our study.Indeed, the weight loss is related to water evaporation from theproduct surface which leads to the product shrinkage and texturemodification.

The weight of 6 apples was measured at the center of the palletsB and D using a digital balance (Sartorius, CPA 34001 P, accuracy0.1 g). The product weight was recorded at 0, 5, 10, 20, 37, 60and 87 days. This manipulation was performed inside the closeddoor cold room to avoid ambient perturbation. The experimentwas finished after 87 days because the product was toodeteriorated.

3.3. Temperature measurement

Temperature was measured (using T-type calibrated thermo-couples) at 82 points in the cold room and at 1 point outside.The product and neighboring air temperatures were measured ateach pallet center. For the pallets B (rear) and D (front), six morepositions were measured (3 at the top, 3 at the bottom).

When an apple (initial temperature 20 �C) was placed in thecold room (thermostat setting temperature 3 �C), the product tem-perature decreased and approached the ambient temperature. Thisis related to the heat transfer by conduction inside the apple (applethermal conductivity k) and by convection between the productsurface and external air (convective heat transfer hc), Fig. 2.

The Biot number (Bi) of the apple is about 2.2 (for hc about10 W m�2 K�1, our measured data). In the case of spherical productsuch as apple and for 0.1 < Bi < 10, the average product tempera-ture (Tavg) is located at 3/4Rp (van der Sman, 2003). The experimen-tal average temperature can be estimated using the measured core(Tc) and surface temperature (Ts) by the Eq. (1) assuming a

parabolic temperature profile in the apple. This average tempera-ture was used to compare with the one predicted by the simplifiedmodel.

Tavg ¼ Tc þ ðTs � TcÞ34

� �2

ð1Þ

3.4. Velocity measurement

Velocity measurements were performed to describe the airflowin the entire cold room in terms of magnitude and direction (theidentification of the turbulence intensity was out of our objective).A hot wire anemometer (TESTO 435-4) with a velocity range of 0–20 m s�1 was used. Its calibrated accuracy assured by the supplieris 5% of the read value or �0:03 m s�1 which restricts the accuracyat the low boundary of the range. In spite of the good balancebetween cost, accuracy and convenience of hot wire anemometer(Melikov et al., 2007), the velocity measurements remain a toughtask at low velocity (<0.2 m s�1).

Measurements were carried on 24 points at three differentheights (0.25 m, 0.9 m and 1.75 m) in the cold room. For eachpoint, velocity was measured in the three directions x, y, and z

Top view

Side view

Fig. 3. Experimental air velocity field (m s�1) and air direction on the symmetry plane in the cold room for the supply air velocity of 4.8 m s�1 and thermostat settingtemperature of 4 �C.

Ta1

II I

IIIIV

TS

Ta2

(β(1-γ)+γ)(1+α)ṁ

Tth Ta6

Ta5

(1-β)(1-γ)(1+α)ṁ

Ta3

β(1-γ)(1+α)ṁ

γ(1+α)ṁ

(1-γ)(1+α)ṁ

Ta4

αṁṁ

(1+α)ṁṁ

KSf KSrText

Tfl, mfl, qfl Trl, mrl, qrl

rearfront

hflSfl

hrlSrl

Air mixing point

Heat exchanged by convection orconduction

Heat exchange zones between airand load

Heat exchange zones betweeninternal and external air (heatlosses through the walls)

I and II

III and IV

Fig. 4. Simplified airflow and heat transfer model in the cold room (side view).

O. Laguerre et al. / Journal of Food Engineering 149 (2015) 78–86 81

(see Fig. 1a) by changing the orientation of the sensor, to determinethe airflow direction. The velocity magnitude (V) was then calcu-

lated using: V ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiV2

x meas þ V2y meas þ V2

z meas

q.

3.5. Measurement of the thermal insulation of cold room (K coefficient)

Measurement of the thermal insulation was carried out insidethe ‘‘switch off’’ empty cold room in which a heating coil wasplaced. To ensure a homogeneous air temperature inside the coldroom, a small fan was installed near the heating coil. The heatingpower was adjusted in such a manner that the average internalair temperature Tint was maintained at 40 �C while the externalambient temperature Text was 18 �C. In this manner, at steady statethe supplied heat is equal to the heat loss to the external airthrough the cold room walls. The power supplied to the heatingcoil and fan were recorded when the steady state was attained(after 10 h) and the average values were calculated over 3.5 h(Q = 1380 W). Thus, the thermal insulation of cold room can be cal-culated (K = Q/(Tint � Text)/S) where S is the total surface of the coldroom wall (57 m2). The K value (1.10 W m�2 K�1) was used in theheat balance equations of the simplified model presented below.

4. Simplified heat transfer model

The knowledge of air velocity distribution obtained from exper-iment was used to develop a simplified heat transfer model. Fig. 3shows, for example, the values of air velocity and direction on the

symmetry plane for the supply air velocity of 4.8 m s�1 and ther-mostat setting temperature of 3 �C. On the axis in front of the fans,the air velocity decreases because air spreads out over the entirespace between the ceiling of the cold room and the top of the pal-lets. A part of air coming out from the front pallet is entrained bythe blown air; the other part flows to the return air duct locatedbehind the fans position. It was also observed experimentally thata part of air flows throughout the pallets (in the small spaces insidethe stack of food) and the other part flows around the pallets (in airspace between two pallets, between cold room walls and pallets,under pallets).

The experimental results lead to develop a simplified modelwhich considers 2 food product zones in the cold room (Fig. 4) witha circular air flow pattern.

The model network (diagram of Fig. 4) was proposed accordingfollowing rules:

– Ability to predict the final (steady state) product tempera-ture with an indication of spatial variability.

– Ability to predict cooling rate (e.g. half cooling time) of theproduct with an indication of spatial variability.

– The most important fluid flow and heat transfer phenomenaare taken into account.

– The number of parameters to be adjusted is minimized.

Each product zone (front and rear load) is composed of 2 pallets.This consideration is true for all balance equations. The simplifiedmodel is not able to predict in detail the local temperature

82 O. Laguerre et al. / Journal of Food Engineering 149 (2015) 78–86

evolution. The rear and front pallet temperature cannot be consid-ered as the minimum and maximum values. More details on localexperimental temperature evolution can be found in Duret et al.(2014).

The effect of air entrainment in front of the fans is characterizedby the coefficient of air distribution a. The distribution betweenthe air flowing inside and around the rear pallets is characterizedby the coefficient c which depends on the ratio of internal andexternal friction losses. A mix of air which previously passesthrough and around the rear pallets then flows through the frontpallets. This mix was characterized by the coefficient b. The otherpart flows around the front pallet.

4.1. Heat balance equations

In order to simplify the model, the thermal inertia of the air wasneglected compared to that of the load. Heat balance equationswere developed for the two loads and different zones inside thecold room (Fig. 4). The rectangles I and II represent the heatexchange zones between air and loads, rectangles III and IV theheat losses through the walls. The mixing zones of different air-flows above and between the loads and at the return air were alsoconsidered.

4.1.1. Heat balance for the air above the front load (Ta1)The air above the front load [temperature Ta1; mass flow rate

ð1þ aÞ _m] is a mix of the air from the supply [Ta7, mass flow rate_m] and a part of the air from the return air [thermostat setting tem-

perature Tth; mass flow rate a _m].

_mTa7 þ a _mTth ¼ ð1þ aÞ _mTa1

�Ta7 þ ð1þ aÞTa1 ¼ aTth ð2Þ

4.1.2. Heat exchange between the air and the rear load (rectangle I)Convective exchange (heat transfer hrl) takes place between the

air (temperature Ta1) and the rear load (temperature Trl). After thisexchange, the air temperature is Ta2.

cð1þ aÞ _mCadTa ¼ hrlðTrl � TaÞdSrl ) ðTa2 � TrlÞ ¼ d1ðTa1 � TrlÞ

d1Ta1 � Ta2 ¼ �ð1� d1ÞTrl ð3Þ

where d1 ¼ exp hrlSrlcð1þaÞ _mCa

� �.

The same procedure was applied to the front load (rectangle II).

d2Ta3 � Ta6 ¼ �ð1� d2ÞTfl ð4Þ

where d21 ¼ exp hflSfl

ðbð1�cÞþcÞð1þaÞ _mCa

� �.

It is to be emphasized that the heat transfer coefficient (hrl or hfl)is an overall value taking into account the conduction inside theproduct and convection between the product surface and air(Fig. 2). This coefficient can be estimated as follow:

h ¼ Rp=4kþ 1

hc

� ��1

The convective heat transfer coefficient (hc) was measured in steadystate regime using an instrumented brass sphere equipped with athermocouple (type K) and a heating wire (Ben Amara et al.,2004). Then this sphere was introduced at various positions insidethe stack of apples.

4.1.3. Temperature evolution in the rear loadThe variation of the internal energy of the rear load is the sum

of the heat generated by the product respiration and the heatexchanged by convection with air. Transient state is considered.

mrlCrldTrl

dt¼ mrlqrl þ cð1þ aÞ _mCaðTa1 � Ta2Þ

dTrl

dt¼ qrl

Crlþ cð1þ aÞ _mCa

mrlCrlðTa1 � Ta2Þ ð5Þ

The same procedure was applied to the front load.

dTfl

dt¼

qfl

Cflþ ðbð1� cÞ þ cÞð1þ aÞ _mCa

mflCflðTa3 � Ta6Þ ð6Þ

4.1.4. Heat losses through the rear cold room wall (rectangle III)The air flowing around the rear load [temperature Ta1, mass

flow rate ð1� cÞð1þ aÞ _m] exchanges heat with the cold room walls(rectangle III). After this exchange, the air temperature is Ta4. Thisheat loss is located at the bottom of diagram in Fig. 4 in spite thatin reality it occurs over all the walls.

ð1� cÞð1þ aÞ _mCadTa ¼ KðText � TaÞdS) ðTa4 � TextÞ¼ d3ðTa1 � TextÞ

Ta4 � d3Ta1 ¼ ð1� d3ÞText ð7Þ

where d3 ¼ exp � KSrwð1�cÞð1þaÞ _mCa

� �The same procedure was applied to

the front cold room wall (rectangle IV).

Ta5 � d4Ta4 ¼ ð1� d4ÞText ð8Þ

where d4 ¼ exp � KSfw

ð1�bÞð1�cÞð1þaÞ _mCa

� �4.1.5. Heat balance for the air between the front and rear loads (Ta3)

The air entering to the front load [temperature Ta3; mass flowrate ðbð1� cÞ þ cÞð1þ aÞ _m] is a mix of the air exiting the rear load[temperature Ta2; mass flow rate cð1þ aÞ _m] and the air turningaround the rear load [Ta4, mass flow rate bð1� cÞð1þ aÞ _m].

ðbð1� cÞ þ cÞð1þ aÞ _mTa3 ¼ cð1þ aÞ _mTa2 þ bð1� cÞð1þ aÞ _mTa4

cTa2 þ bð1� cÞTa4 � ðbð1� cÞ þ cÞTa3 ¼ 0 ð9Þ

4.1.6. Heat balance in the return air (Tth)It is assumed that the returned air has exactly the thermostat

setting temperature (Tth). A part of the returned air flows towardthe cooling unit (mass flow rate _m) and another part is entrainedby the supply jet [mass flow rate ð1þ aÞ _m]. This air is a mix ofthe air from the front pallet [temperature Ta6, mass flow rateðbð1� cÞ þ cÞð1þ aÞ _m] and the air passing round the load [temper-ature Ta5; mass flow rate ð1� bÞð1� cÞð1þ aÞ _m].

ð1þaÞ _mTth¼ð1�bÞð1�cÞð1þaÞ _mTa5þðbð1�cÞþcÞð1þaÞ _mTa6

Tth ¼ ð1� bÞð1� cÞTa5 þ ðbð1� cÞ þ cÞTa6 ð10Þ

4.2. Matrix form of the equations

For transient state, the equations developed previously can besummarized in matrix form.

dTl!

dt¼ A Ta

�!þ B~q ð11Þ

C Ta�! ¼ D

!Text þ E

!Tth � F Tl

! ð12Þ

Tl!¼

Tfl

Trl

� ð13Þ

O. Laguerre et al. / Journal of Food Engineering 149 (2015) 78–86 83

T!

a ¼

Ta1

Ta2

Ta3

Ta4

Ta5

Ta6

Ta7

266666666666664

377777777777775

ð14Þ

where A;B;C; D!; E!; F are matrices or vectors involving several

parameters which depend on the air flow rate, the heat transfercoefficients, the product physical properties and the cold roomcharacteristics. They are defined as follows:

A ¼0 0 1=sfl 0 0 �1=sfl 0

1=srl �1=srl 0 0 0 0 0

� ð15Þ

B ¼1=Cfl 0

0 1=Crl

� ð16Þ

~q ¼qfl

qrl

� ð17Þ

where sfl ¼mflCfl

ðbð1�cÞþcÞð1þaÞ _mCaand srl ¼ mrlCrl

cð1þaÞ _mCa.

C¼

d1 �1 0 0 0 0 00 0 d2 0 0 �1 0�d3 0 0 1 0 0 0

0 0 0 �d4 1 0 0ð1þaÞ 0 0 0 0 0 �1

0 c �ðbð1�cÞþcÞ bð1�cÞ 0 0 00 0 0 0 ð1�bÞð1�cÞ ðbð1�cÞþcÞ 0

0BBBBBBBBBBB@

1CCCCCCCCCCCAð18Þ

D!¼

00

1� d3

1� d4

000

0BBBBBBBBBBB@

1CCCCCCCCCCCA

ð19Þ

E!¼

0000a01

0BBBBBBBBBBB@

1CCCCCCCCCCCA

ð20Þ

F ¼

0 1� d1

1� d2 00 00 00 00 00 0

0BBBBBBBBBBB@

1CCCCCCCCCCCA

ð21Þ

The Eqs. (11) and (12), leading to the Eq. (22), were solved usingMatlab software (vR2012a; The MathWorks Inc., Natick, MA, USA,4th order Runge–Kutta).

dTl�!dtþ AC�1F� �

~Tl ¼ AC�1D� �

Text þ AC�1~E� �

Tth þ B~q ð22Þ

4.3. Estimation of air distribution coefficients

The coefficients a, c and b represent, respectively, the propor-tion of air entrained by the supply jet, the proportion of the airflowing through the rear pallets, and the proportion of the air flow-ing through the front pallet which previously exchanged heat withthe cold room walls. Due to the complex airflow, the experimentalevaluation of these coefficients is difficult and no data were foundin the literature. In consequence, an Approximate Bayesian Com-putation (particle filtering method, Turner and Van Zandt, 2012)was carried out to estimate these coefficients according to thecooling experiment. The following steps were achieved (Fig. 5).

1. The distribution of these 3 coefficients a, c and b was fittedby Beta law. This law is characterized by 2 parameters. Inour study, non-informative distribution Beta(0.5, 0.5) arechosen as initial distributions. A value of coefficients wassampled from the Beta distribution.

2. The sampled values were applied to the simplified model ofcold room to calculate the load temperatures and the halfcooling times.

3. If the 4 criteria (objective function) were attained, the‘‘accepted’’ values of a, c and b were saved. These 4 criteriaare final front load temperature (4.9 �C ± 0.2 �C), half-cool-ing time of the front load (5.23 h ± 0.5 h), final temperatureof the rear load (4.4 �C ± 0.2 �C) and half-cooling time of therear load (4.08 h ± 0.5 h). These values were obtained byexperiments and will be presented in Section 5. The steps2 and 3 were repeated until 30 values of ‘‘accepted’’ a, cand b were obtained.

4. New Beta distributions of a, c and b were fitted from the 30‘‘accepted’’ values.

5. The steps 2–4 were repeated 500 times. For the iterationi + 1 and at step 2, the values of a, c and b in iteration i wereused.

The final (mean) values of a, c and b are obtained in the itera-tion 500.

It is to be emphasized that the half-cooling time (t1/2) is definedas the time required to reduce the temperature of the apple half-way between its initial temperature (Ti) and its final one (Tf, atthe end of the cooling process).

4.4. Weight loss model

The weight loss of food products is driven by the difference inwater vapor concentration between the apple skin (Cw) and thesurrounding air (Cw,1). A simple model was proposed byGwanpua et al. (2012):

dmdt¼ ktaAðCw � Cw;1Þ ð23Þ

kta (moisture transfer coefficient, 0.1825 m h�1, Gwanpua et al.,2012) represents the reverse of the resistance of water vapor migra-tion of the apple skin. This internal resistance is much higher thanthe one of the surrounding air.

Cw is related to the saturated vapor pressure (Pa) at the loadsurface temperature (Tsfl or Tsrl). For the front load, for example,Cw can be calculated as follow:

Cw ¼MH2OPsatðTsflÞaw

RTsflð24Þ

The water activity aw is considered constant and equal to 1 duringthe experiment.

It is to be noticed that the front load surface temperature can becalculated by the following equation:

Fig. 5. Steps for the identification of coefficients of air distribution (a, c and b).

84 O. Laguerre et al. / Journal of Food Engineering 149 (2015) 78–86

Tsfl ¼4kRp

Tavg:fl þ hcTa3þTa6

24kRpþ hc

!ð25Þ

A similar equation can be used for the rear load.The partial pressure of the water in air is considered homoge-

nous in the cold room and equal to the saturated vapor pressureof the air at the blowing temperature (Ta7). The following expres-sion can be developed for the surrounding air Cw,1:

Cw;1 ¼MH2OPsatðTa7Þ

RTa7ð26Þ

5. Result and discussion

The adjustment of air distribution coefficients led to a = 0.80,c = 0.10 and b = 0.12. The value of a is relatively high comparedto the entrainment coefficient of wall jets. This is certainly duethe fact that air is rotating at the outlet of the fans which increasesthe entrainment effect.

5.1. Product cooling and warming

The load temperatures calculated from the simplified modelwere compared with the average experimental final temperatures.This average experimental final temperature was obtained usingthe temperature measured at the core and the surface (see Eq.(1)) and the product located at 3 heights (7 positions, see Fig. 1b)in each pallet was considered. The final temperatures (at 30 h ofcooling) and the half cooling time are compared in Table 2.

Table 2Half cooling/warming time and final product temperature in the front and rear pallets; co

Studied condition Thermostat setting temperature (�C) Load position Initial loa

Product cooling 4 Front 20.0Rear 20.0

Product warming 16 Front 4.0Rear 4.0

For thermostat setting temperature of 4 �C (product cooling),the half-cooling time of the front pallet is higher because of thehigher temperature of the air at this position. Indeed, the air waswarmed up due to the heat exchange with the rear products beforeflowing through the front pallet. As a consequence, higher half-cooling time is observed in the front pallets.

For the thermostat setting temperature of 16 �C (product warm-ing), a similar result was obtained. The half-warming time and thefinal temperature of the front pallet are higher than that of therear. The comparison between these 2 results is in good agreementfor both cooling and warming conditions.

The comparison of the evolution of product average tempera-ture obtained from measurement and simplified model is pre-sented in Fig. 6 for the product cooling (thermostat settingtemperature of 4 �C). As cited previously, the experimental averagetemperature shown in this figure was calculated from themeasurements at 3 heights and at 7 positions in each pallet at dif-ferent time. The cooling rate of the front pallets predicted by thesimplified model is in agreement with the experimental (Fig. 6a).For the rear pallets (Fig. 6b), the cooling rate is well predicted forthe 5 first hours, then the temperature is underestimated. Themaximum difference between the experimental and calculatedtemperature is 2 �C for every studied position. The simplifiedmodel allows predicting food products temperature evolution withacceptable accuracy in a short calculation time.

5.2. Product weight loss

Fig. 7 shows the comparison between the experimental and cal-culated weight losses of apples (in percentage) at the middle of the

mparison between experiment and simplified model.

d temperature (�C) Half cooling/warming time(h)

Final temperature at 30 h(�C)

Exp Model Exp Model

5.2 5.4 4.9 4.84.1 4.3 4.4 4.45.5 5.0 15.5 15.74.7 4.1 15.2 15.6

Fig. 6. Comparison between the experimental average product temperature and the calculated value of simplified model (a) front pallets (b) rear pallets. The experimentalsupply air temperature is also shown for the thermostat setting temperature of 4 �C.

0 20 40 60 80 1000

0.5

1

1.5

2

2.5

Time (Days)

Wei

ght l

oss

(%)

Rear expRear simFront expFront sim

Fig. 7. Measured and calculated weight losses of apples at the middle of the frontand rear pallets for the thermostat setting temperature of 4 �C.

O. Laguerre et al. / Journal of Food Engineering 149 (2015) 78–86 85

front and rear pallets. The experimental and calculated results arein good agreement: weight losses of the front load are higher thanthe one at the rear. This can mainly be explained by the fact thatthe front pallet is exposed to higher temperature as mentionedpreviously. Air on the front load is also slightly more humidbecause of the weight loss in the rear load. This can lead to expectless weight loss at the front load. However, the effect of air humid-ity is less significant than the one of air temperature. The maxi-mum weight loss of about 1.8% was observed experimentally atthe front pallet while the predicted value is 2.0% after 87 days pres-ervation in the cold room. In spite of the model simplification, theweight loss prediction is acceptable.

5.3. Sensitivity study

A sensitivity study of several model input parameters was car-ried out. It was found that the thermostat setting temperature andthe airflow rate have the most influence on the predicted load tem-peratures at steady state. For example, if the thermostat is set at8 �C instead of 4 �C, the load temperatures increase also about4 �C. If the air mass flow rate is set at 0.15 kg s�1 instead of0.30 kg s�1, the load temperature heterogeneity is more notice-able: the difference between the rear and the front load tempera-tures is almost doubled.

6. Conclusion

Temperature and weight losses measurements were carried outin a ventilated cold room filled with four apple pallets. A simplifiedmodel was developed using the knowledge obtained from velocitymeasurements. This model considers 2 zones: front and rear and asimplified air circulation in the cold room were proposed. It allowsthe prediction of product cooling rate (transient state), the productfinal temperature (steady state) at different positions in the coldroom and the product weight loss. Good agreement between theexperimental and predicted values was obtained. This model couldbe extended to other food product preserved in a cold room. In thiscase, the air distribution coefficients (a, b and c) may be differentand must be determined again.

The main advantage of this model is the short CPU computingtime (<1 s) compared to that of CFD for the same configuration(>24 h for about 8 � 105 cells). This aspect enables the rapid eval-uation of the influence of input parameters such as external airtemperature, wall insulation, thermostat setting and product heatof respiration on the load temperatures. Also, this simplified modelis appropriated for the study of time–temperature evolution of ahigh number of product items (>104) along the cold chain. Thismodel will be implemented together with others models alreadydeveloped (refrigerated truck, display cabinet and domestic refrig-erator) allowing monitoring the product time–temperature historyalong the cold chain. This model allows the prediction of the vari-ability of load temperature. In fact, there are few studies whichtake into account several links in a cold chain at a time such as thatproposed by our work. Quality and microbiological evolution canalso be coupled with this approach and the developed model canthen be used as a risk evaluation tool. It can help the operatorsto manage the logistic chain to avoid excessive food.

Acknowledgement

The research leading to this result has received funding fromRégion Ile de France and European Community’s Seventh Frame-work Programme (FP7/2007-2013) under the Grant AgreementNo. 245288.

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