Propagation of stochastic temperature fluctuations in refrigerated fruits: Propagation des fluctuations de température stochastiques dans des fruits réfrigérés

Propagation of stochastic temperature fluctuations in refrigeratedfruits

Bart M. Nicolaa,* , Bert Verlindena, Annelies Beuselinckb, Pal Jancsokb,Valery Quenonb, Nico Scheerlinckb, Pieter Verbovenb, Josse De Baerdemaekerb

aFlanders Centre for Postharvest Technology, Katholieke Universiteit Leuven, De Croylaan 42, B–3001 Heverlee, BelgiumbDepartment of Agro-Engineering and Economics, Katholieke Universiteit Leuven, Kardinaal Mercierlaan 92, B–3001 Heverlee, Belgium

Received 1 July 1996; received in revised form 4 May 1998; accepted 4 August 1998

Abstract

A steady state variance propagation algorithm is derived to investigate the effect of stochastic air temperature disturbances onthe variability of the temperature field inside refrigerated fruits during cold storage conditions. The disturbances are modelledby continuous-time autoregressive processes of variable order. The algorithm is based on the finite element formulation of theheat conduction equation and involves the numerical solution of matrix equations of the Lyapunov and Sylvester type. As anexample, the cold storage of pineapple has been considered. It is shown that disturbances of different order but with the samescale of fluctuation result in comparable centre temperature variances.q 1999 Elsevier Science Ltd and IIR. All rights reserved.

Keywords:Fruit; Refrigeration; Heat transfer; Temperature

Propagation des fluctuations de tempe´rature stochastiques dans desfruits refrigeres

Resume

Un derive d’algorithme de la variance de la propagation en re´gime permanent a e´te utilise pour etudier l’effet desperturbations de tempe´rature de l’air stochastiques a` l’interieur de fruits lors de leur entreposage frigorifique. Mode´lisationdes perturbations e´tait effectueé par des proce´des autoregressifs d’ordre variable en continu. L’algorithme est base´ sur laformulation par elements finis de l’e´quation de la conduction de la chaleur et comprend la re´solution nume´rique des e´quationsde matrice du type Lyapunov et Sylvester. L’exemple de l’entreposage de l’ananas a e´te utilise. Il a ete montreque lesperturbations d’ampleurs diffe´rentes mais avec la meˆme echelle de fluctuation donnent des variances de tempe´rature compar-ables au coeur des fruits.q 1999 Elsevier Science Ltd and IIR. All rights reserved.

Mots cles: Fruit; Refrigeration; Transfert de chaleur; Tempe´rature

Nomenclature

A companion matrixc heat capacity J kg21 K21

C finite element capacity matrixD diameter, mE mean value operatorf finite element thermal load vectorg vector functionG heat conduction domainh surface heat transfer coefficient, W m22 K21

International Journal of Refrigeration 22 (1999) 81–90

0140-7007/99/$ - see front matterq 1999 Elsevier Science Ltd and IIR. All rights reserved.PII: S0140-7007(98)00051-6

* Corresponding author: Tel:1 32-16-321456; Fax:1 32-16-321994; e-mail: [email protected]

h vector functionk thermal conductivity, W m21 K21

kair thermal conductivity of air, W m21 K21

K finite element tiffness matrixm order of autoregressive random processnnod number of nodesn outward normalNu Nusselt number0 zero matrixPr Prandtl numberRx,x correlation function, sRe Reynolds numbert time, st0 start time, sT temperature,8CT0 initial temperature,8CT∞ refrigeration temperature,8Cu nodal temperature vectoru0 nodal temperature vector at timet � t0V covariance matrixw white noise processx arbitrary stochastic process�x mean of random processx stochastic state vectory extended stochastic state vectorz position vectorDt time interval, s2G convection surfaceu scale of fluctuation, s or hr density, kg m23

s standard deviationj root of characteristic equation of AR process

1. Introduction

Many fruits and vegetables are injured by low tempera-tures in the 0–208C range [1–3]. Common symptomsinclude surface lesions (e.g. cucumber), browning of fleshtissue (e.g. black heart in pineapples) and development ofwater-soaked areas (e.g. cucumbers). The degree of chillinginjury depends on the temperature to which the produce isexposed, the duration of the exposure, and the sensitivity ofthe species to chilling temperatures [1]. While the actualinjury occurs during the storage, usually the symptomsdevelop only after removal of the commodity from thestorage facilities and exposure to higher temperatures.

The occurrence of chilling injury can be prevented bycareful monitoring and control of the temperature in thecold storage facilities. However, due to unpredictable distur-bances, such as the opening of doors or introduction of freshproduce, it is possible that the storage temperature tempora-rily drops into the region of increased chilling injury risk.This will affect the product centre temperature in such a waythat, because of the thermal inertia of the product, highfrequency fluctuations are smoothed. On the other hand,

low frequency fluctuations will be phase shifted but notattenuated. In the case of pineapples, the chilling injurystarts at the core of the product, and the frequency of thestorage temperature fluctuations will therefore affect thedevelopment of chilling injury to a large extent. A quanti-tative insight into the propagation of temperature fluctua-tions inside the fruits may guide towards the establishmentof temperature control performance criteria which minimizethe occurrence of chilling injury.

In the case of deterministic refrigeration temperature fluc-tuations, analytical and numerical methods have beensuggested by several authors (for an overview, see [4]).The finite element method thereby appears to be generallyapplicable as it is not restricted to regular product shapes.Zuritz and Singh [5] applied a finite element program topredict temperature distribution and time-temperaturehistory for frozen food products subjected to a stepwisechanging storage temperature. They found a good agree-ment between experimental and calculated product tempera-tures. However, they did not include room temperaturefluctuations in the computer model. In a subsequent articleZuritz et al. [6] considered periodic variations of the storagetemperature below the freezing point by analytical solutionof the Fourier equation with appropriate boundary condi-tions. A satisfactory agreement between measured andpredicted temperatures was observed. The authorsconcluded that the geometry of the package has a significantinfluence in the response of the product temperature to peri-odically fluctuating storage temperature. However, becausethe authors relied on an analytical solution, it is difficult toextend their results to more complicated geometries such asa pineapple. Further, in reality the storage temperaturedisturbances are stochastic and it is not clear how this affectsthe temperature of the product.

The objectives of this paper were:

(i) to establish a validated finite element model for heattransfer inside a complex shaped refrigerated fruit such asa pineapple;(ii) to develop a procedure to calculate the mean valueand the variability of the temperature in steady stateconditions at an arbitrary position inside a complexshaped refrigerated fruit corresponding to a stochasticallyvarying storage temperature; and(iii) to evaluate the effect of the scale of fluctuation of thestorage temperature on the temperature fluctuationsinside the fruit.

2. Materials and methods

2.1. Computation of heat conduction

In order to model the heat transfer during cold storage ofpineapple, the following assumptions were made:

• Heat transfer in the produce occurs through conduction

B.M. Nicola et al. / International Journal of Refrigeration 22 (1999) 81–9082

• As the temperature range considered in this article islimited, the thermophysical properties are considerednot to change as a function of temperature

• Latent heat transfer because of evaporation is negligible• Heat production because of respiration is negligible• The pineapple is considered as a single axisymmetric,

homogeneous and isotropic entity• The effect of bulk packaging is not taken into account.

Under these assumptions, the temperature fieldT atarbitrary space time coordinatesz and t in the productG enclosed by a boundary2G is governed by the Fourierequation [7]

k72T � rc2T2t

�1�

subjected to the initial and convection boundary conditions

T�z; t� � T0�z� at t � t0 �2�

k2

2nT�z; t� � h�T∞�t�2 T�z; t�� on2G �3�

The finite element method is widely used for the simula-tion of conductive heat transfer. In this method the conti-nuum is subdivided in (finite) elements of variable size andshape which are interconnected in a finite numbernnod ofnodal points. In every element the unknown temperature canbe approximated by a low order interpolating polynomial insuch a way that the temperature is uniquely defined in termsof the (approximate) temperaturesui(t) at the nodes. It ispossible to show [8] that the unknown nodal temperaturevectoru� �u1…unnod

�T may be found by solving the follow-ing linear differential system

C _u 1 Ku � f �4�

u � u0 at t � t0 �5�

with C the capacitance matrix andK the stiffness matrix,bothnnod × nnod matrices andf a nnod × 1 vector. The super-script T denotes the transpose of a vector or matrix.Cinvolvesr andc, K involvesk andh, f involvesT∞ andh.This system is usually discretized with a finite differencescheme.

All actual computations were done on a HP-9000 seriesC160 Unix workstation using a modified version of the heattransfer finite element code DOT [9,10].

2.2. Temperature measurements

A pineapple was purchased in a local retail shop. Theheight of the pineapple without leaves was 129 mm andthe major diameter was 96 mm. The pineapple was equili-brated at room temperature (21.68C) and put into a climateroom (Weiss Technik, Reiskirchen-Lindenstruth, Germany)at t � t0. The air temperature in the climate room was equalto 9.58C, and the relative humidity was set equal to 95% toavoid evaporative chilling effects. The air velocity wasmeasured with a hot film omnidirectional air velocity trans-ducer (type 8475, TSI, St. Paul, MN) and logged with a 20channel HP 3497A logger (Hewlett–Packard, USA). Thetime constant was equal to 50 ms, and the total acquisitiontime was 1 min. The velocity was measured at the fourcorner nodes of a rectangular grid of 16× 18 cm whichwas perpendicular to the direction of flow and upstream ofthe pineapple. The velocity was also measured at five addi-tional nodes of which four were on the edges of the rectan-gular grid and one was inside.

Five type T thermocouples were inserted at various posi-tions in the pineapple. The temperature was logged with thesame acquisition system as above. The thermocouples werecalibrated in an ice bath, and the accuracy of the temperaturereadings was equal to 0.58C. The thermocouple positionswere determined based on X-ray images which were takenfrom two perpendicular directions with a HOMX 161Microfocus X-ray system (IRT Corporation, San Diego,CA). The tube voltage and current were 70 kV and0.5 mA, respectively. The images were processed usingthe DVS-3000 real time image processing software (Hama-matsu Photonics K.K., Hamamatsu City, Japan). A rectan-gular iron frame was mounted in the pineapple as areference coordinate system. Iron reference marks at thesurface of the pineapple were used to establish a transforma-tion between real world coordinates and image coordinates.From a comparison of actual and measured positions of ironreference marks an overall accuracy of 1.5 mm was foundfor the rectangular cartesian co-ordinates. The thermocou-ples can be observed clearly in the X-ray image in Fig. 1.

The construction of the geometrical model for the finiteelement simulations was automated by means of a computervision system. The system consists of a JVC KY 55BE 3CCD color camera, a Matrox Comet frame grabber card, anillumination system and a Pentium based host computer.Two images were taken from perpendicular directions.

B.M. Nicola et al. / International Journal of Refrigeration 22 (1999) 81–90 83

Fig. 1. X-ray image of the pineapple with thermocouples (TC),reference coordinate system and reference marks.

Fig. 1. Image radiographique de l’ananas et des thermocouples(TC), le syste`me des coordonneés et des repe`res de reference.

The images were evaluated using an in house developeddigital image analysis program AGROPIC (Jancsok et al.,1997) [11], and an average cross section was calculated. Thelist of average boundary points was further used to generatean axisymmetric structured finite element grid consisting of180 quadrilateral elements with four nodes/element usingthe grid generator described in Ref. [12]. The grid is visua-lized in Fig. 2 using the ANSYS commercial finite elementpackage (Ansys Inc., Houston, TX). The grid was distortedslightly so that the thermocouple locations coincided withfinite element nodes. In this way the comparison betweenmeasured and simulated temperatures could be simplified.

The following values have been used for the thermophy-sical parametersr � 1010 kg m23 [1]; c� 3685 J kg21 K21

[13]; k � 0.549 W m21 K21 [14]. For all transient simula-tions the time step was set equal to 60 s.

The surface heat transfer coefficienth was calculatedfrom the Nusselt number using the following correlationformula (Incropera and de Witt, 1990) [7] for a sphere

hDkair� Nu� 2 1 �0:4Re1=2 1 0:06Re2=3�Pr0:4 m

ms

� �1=4

�6�

with D the diameter of the sphere,Rethe Reynolds number,Nu the Nusselt number,kair the thermal conductivity of air,Pr the Prandtl number,m the viscosity of air, andm s theviscosity of air at the surface temperature. For the computa-tions,D was set equal to the diameter of a sphere with thesame volume as an ellipsoid with major diameter equal tothe pineapple length (129 mm) and minor and intermediatediameter equal to the pineapple diameter (96 mm).

2.3. Autoregressive random processes

It is usually assumed that the storage temperature is

constant during cold storage of fruits. However, in practicedue to unpredictable disturbances, the refrigeration tempera-ture fluctuates in a random way and can hence be consideredas a stochastic process. It is assumed in this work that thecharacteristics of the stochastic process do not change intime, so that the process isstationary. Important probabil-istic characteristics of a stationary random processx withprobability density functionf(x,t) are its mean�x and covar-ianceVx,x

�x� E�x� WZ∞

2 ∞xf�x; t�dx

Vx;x�Dt� � E{ �x�t�2 �x��x�t 1 Dt�2 �x��}If Dt � 0 thenVx;x�0� W s2

x, the variance of the process.The covariance function describes how much the currentvalue of the random function will affect its future values.The correlation functionRx,x is found by normalization ofthe covariance function:

Rx;x�Dt� � Vx;x�Dt�=s2x �7�

Whereas for domestic and retail refrigerators a consider-able amount of data on temperature variability has beenpublished [15–20] there are few literature reports on thetemperature variability inside cold stores [10]. In thiswork it has been assumed that the refrigeration temperatureT∞ is an autoregressive random process [8–11]. Thisassumption is supported by the fact that the class of auto-regressive processes is a special case of the generic class ofphysically realizable stochastic processes. The lattercomprise most of the random processes seen in practice[21]. The validity of this assumption is currently beinginvestigated by the authors of this article. The effect ofspatial variation of the storage temperature on producetemperature has been considered elsewhere [22].

An autoregressive refrigeration temperature of orderm isdefined by the following stochastic differential equation

dm

dtmT∞�t�1 a1

dm21

dtm21 T∞�t�1 …1 amT∞�t� � w�t� �8�

wherea1,a2,…,am are constants, andm $ 1. w is a Gaussianstationary white noise process with covariance

Vw;w�Dt� � s2wd�Dt� �9�

whered is the Dirac delta. The (Gaussian) random variableinitial condition corresponding to the stochastic differentialequation, Eq. (8), is defined as:

E�T∞�t0�� T∞ �10�

E�T∞�t0�2 T∞�2 � s2T∞ �11�

The autoregressive process (8) can be written conveni-ently in the followingstate space form[23]:

ddt

x�t� � Ax�t�1 Bw�t� �12�


Fig. 2. Finite element grid.

Fig. 2. Grille d’elements finis.

where

x �

T∞

dT∞=dt

..

.

dm22T∞=dtm22

dm21T∞=dtm21

266666666664

377777777775�13�

B �

0

0

..

.

0

1

266666666664

377777777775�14�

A �

0 1 … 0

..

.]

..

.

0 0 … 1

2am 2am21… 2a1

26666664

37777775 �15�

The matrixA is called the companion matrix. The dimen-sion ofA is equal tom× m; B andx are of dimensionm× 1.

In order to describe the smoothness of a random processby means of a single measure, Vanmarcke [24] introducedthe conceptscale of fluctuationwhich is defined as

u �Z1 ∞

2 ∞RT∞ ;T∞ �Dt�dDt

This gives an indication of the time beyond which a futurevalue of a random process will no longer be affected by itscurrent value. In Table 1 the variance, the autocovariancefunction and the scale of fluctuation are given for AR(1)and AR(2) processes. For the latter, the characteristicpolynomial

jm 1 a1jm21 1 …1 am � 0 �16�

has two real or two complex conjugate roots, resultingin non-oscillating or oscillating correlation functions,respectively.

In Fig. 3 some correlation functions and correspondingrealizations of an AR(2)-process with different scales offluctuation are compared. Ifu ! 0 then the process approx-imates a white noise process. On the other hand, ifu ! 1 ∞then the values of the realization at arbitrary points arecompletely correlated. In this case the random processconcept is far too sophisticated to describe the physicalquantity since all the meaningful probabilistic features ofthe quantity can be captured by a simple random variablemodel.

The correlation functions and corresponding realizationsof different types of random processes are shown in Fig. 4.Clearly the scale of fluctuation is a measure of how frequent


Table 1Autocovariance function and scale of fluctuation of AR(1) andAR(2) processes

Tableau 1Fonction auto-covariance et ećhelle de fluctuation des proce´desAR(1) et AR(2)

AR(1) Vx;x�Dt� � s2xe2a1uDtu

u � 2/a1

AR(2)Real roots

Vx;x�Dt� � s2x�j2ej1uDtu 2 j1ej2uDtu��j2 2 j1��21

with j1 andj2 the roots ofj2 1 a1j 1 a2 � 0u � 2a1=a2

AR(2)Complex roots

Vx;x�Dt� � s2xe2

12a1uDtu�cos�puDtu�1

a12p sin�puDtu��with p� �a2 2 a2

1=4�12

u � 2a1=a2

Fig. 3. Correlation functions (a) and corresponding realizations (b)of a AR(2)-process with different scales of fluctuation.

Fig. 3. Fonctions de corre´lation (a) et realisations correspondantes(b) d’un procedeAR(2) avec diffe´rentes ećhelles de fluctuation.

the process wiggles around the mean-axis, irrespective ofthe order of the process.

2.4. Variance propagation algorithm

Because of the random refrigeration temperatureT∞, thetemperature field inside the fruit is a stochastically fluctuat-ing quantity which must be described in a probabilistic way.The Monte Carlo method is a common approach to achievethis. In this method, a series of simulations is initiated, andfor each simulation run a randomly fluctuating refrigerationtemperature is generated by the computer according to thepredefined stochastic properties. In the end, the mean valueand the variance of the centre temperature are estimated bymeans of classical statistical techniques. Other characteris-tics such as the probability density function of the tempera-ture field can also be estimated. However, to obtain a

reasonably accurate estimate of the probabilistic character-istics of the temperature field, typically 1000 or more simu-lations are required [10]. As in each simulation run a finiteelement problem must be solved, this implies an unaccept-ably high calculation time. An alternative algorithm whichis based on stochastic systems theory will therefore bederived.

Eqs. (4) and (12) can be stacked conveniently in thefollowing stochastic differential system

ddt

y�t� � g�y�t��1 hw�t�2 �w �17�

where

y �u

x

" #�18�

g�y�t�� C21f 2 C21Ku

Ax 1 b �w

" #�19�

h �0

B

" #�20�

A known result from stochastic systems theory is thefollowing steady state variance propagation algorithmforthe mean value and the covariance matrix ofy as describedby Eq. (17) [25]

g��y� � 0 �21�

2g2y

Vy;y�t�1 Vy;y�t� 2g2y

� �T

1hs2whT � 0 �22�

Substitution of Eqs. (18)–(20) in Eqs. (21) and (22) yieldsthe following system

K �u � �f �23�

C21 2f2T∞

VTu;T∞ 2 K �uVu;u

� �

1 Vu;T∞2fT

2T∞2 Vu; �u �u

TKT

!C2T� 0 �24�

2f2T∞

VT∞ ;x 2 KV u;x 1 CVu;xAT � 0 �25�

AVx;x 1 Vx;xAT 1 s2T∞ BBT � 0 �26�

In Eqs. (23)–(26)�f and2f =2T∞ are assembled using themean value�T∞.

Eqs. (24) and (26) are of the general form

DV 1 VDT 1 E � 0 �27�

with V, D, andE square matrices of equal dimension, and iscalled aLyapunovequation [26]. Eq. (25) is of the general


Fig. 4. Correlation functions (a) and corresponding realizations (b)of several types of autoregressive process with the same scale offluctuation.

Fig. 4. Fonctions de corre´lation (a) et realisations correspondantes(b) de plusieurs types de proce´de auto-regressif avec la meˆmeechelle de fluctuation.

form

DV 1 VE 1 F � 0

with D, E square matrices, andV and F matrices whichare in general not square. Eq. (2) is called aSylvesterequation [27]. Methods and code for the numerical solutionof Lyapunov and Sylvester matrix differential equationsare discussed in Ref. [28].

For validation purposes the results of the variance propa-gation algorithm were compared to those obtained by adirect Monte Carlo method. The samples were generatedby straightforward evaluation of Eq. (12) after temporaldiscretization by means of implicit Euler finite differencing.Gaussian white noise was generated by means of Nagroutine G05DDF. The time step for the transient MonteCarlo analysis was equal to 60 s. The total period consideredwas equal to 24 h.

3. Results and discussion

3.1. Heat transfer model

The average velocity of the air was equal to 0.88 m s21.The calculated surface heat transfer coefficient according toEq. (6) was equal to 11.3 W m22K21. Initial simulationswith this value ofh resulted in considerable discrepanciesbetween the calculated and measured temperatures. Thevalue ofh was subsequently adjusted so that a good matchbetween the temperature at the position with the slowestcooling rate could be obtained. The results forh �20 W m22K21 are shown in Fig. 5. Whereas there is agood match between measured and calculated temperaturesclose to the thermal centre of the pineapple, the resultsbecome worse closer to the surface of the pineapple. Thisis probably due to the fact that the pineapple is not reallyaxisymmetric, inaccuracies of the thermophysicalproperties, or the lack of inner structures (core) with

different thermophysical properties. The value ofh whichwas used is considerably higher than the value calculated bymeans of the correlation formulas. This is probably due tothe relatively large specific surface of the pineapple skinbecause of its special structure. Also, local turbulencesmay be present for the same reason.

In Fig. 6 the temperature variance in the thermal centre ofthe pineapple is shown as calculated with the variancepropagation algorithm and the Monte Carlo method (100and 1000 runs). The scale of fluctuation was set equal tou � 2h (ora1� 2.7× 1024) and the order of the AR processwas 1. For the Monte Carlo method the initial variance ofthe temperature was set equal to 08C2. It can be seen that forlarge times the variance approaches the steady state valuewhich was obtained with the variance propagationalgorithm. The time required for the variance propagationalgorithm was equal to 27.3 s, and for the Monte Carlomethod 232.6 s (100 runs) and 2121.4 s (1000 runs). Thevariance propagation algorithm is clearly much faster thanthe Monte Carlo method, even when only 100 runs arecarried out. Therefore the former was used uniquely forthe further analysis. Note that, because of the nature of theMonte Carlo method, a different set of random ambienttemperatures would lead to different results. A directcomparison of the Monte Carlo method and the variancepropagation algorithm is, hence, only possible in a statisticalsense. This is beyond the scope of this article; the reader isreferred to reference [10] for more details.

Samples of a stochastically varying storage temperatureand the corresponding surface and centre temperature in thepineapple are shown in Fig. 7 (u � 0.2 h) and Fig. 8 (u �2 h) . It is clear that foru � 0.2 h the fast fluctuations aresmoothed out as a consequence of the thermal inertia of thepineapple. Also, the surface temperature is more sensitive tostorage temperature fluctuations.

In Fig. 9 the influence of the scale of fluctuation on the


Fig. 5. Measured (lines) and simulated temperatures (symbols)inside pineapple at five different thermocouple (TC) positions.Legend:W: TC 1;X: TC 2; q: TC 3; 1 : TC 4;A: TC 5.

Fig. 5. Tempe´ratures mesureés (courbes) et simuleés (symboles) aucoeur de l’ananas pour cinq positions de thermocouple (TC) diffe´r-entes. Le´gende:W: TC 1; X: TC 2; q: TC 3; 1 : TC 4; A: TC 5.

Fig. 6. Temperature variance (K2) at the thermal centre of thepineapple. — — —: variance propagation algorithm;W: MonteCarlo with 100 runs;q: Monte Carlo with 1000 runs.

Fig. 6. Variance de tempe´rature (8C2) au coeur thermique del’ananas. — — —: algorithme de variance de la propagation;W: methode Monte Carlo avec 100 essais;q: methode de MonteCarlo avec 1000 essais.

centre temperature variance corresponding to autoregressivechilling temperature fluctuations of different order is shown.For the AR(2) process with real roots of Eq. (16),a1 anda2

where chosen such thatj2/j1 � 0.2; for the AR(2) processwith imaginary roots,a1 anda2 where chosen such that theimaginary part of the complex solution was equal to threetimes the real part. While for the AR(1) process and theAR(2) process with non-oscillating correlation functionthe variance is almost equal for equal scale of fluctuation,for the AR(2) process with oscillating correlation functionthe variance is slightly larger.

In Fig. 10 the mean and confidence interval of the centretemperature are shown as a function of the scale of fluctua-tion for AR(2) refrigeration temperature disturbances (realroots). Foru ! 1 ∞ the 95% confidence interval becomesequal to 10 28C, or equal to the confidence interval of therefrigeration temperature. In this case the random processdegenerates and becomes a random variable.

The above results are not unexpected, as it has beenshown [29] that for each homogeneous random process, aseries of cosine functions with random phase angles can beconstructed whose spectral density (the Fourier transform of

the correlation function) converges in the mean square senseto the original spectral density. Random processes with asmall scale of fluctuation are represented by cosine series inwhich the high frequencies dominate, and vice versa. Butrefrigeration temperature oscillations of high frequency areattenuated more than low frequency oscillations [30]. So,stochastic refrigeration temperature fluctuations character-ized by a small scale of fluctuation will be attenuated morethan those characterized by a large scale of fluctuation.Similar observations, but for deterministically fluctuatingsurface temperatures, have been made by Save and Niranjan[31]. They suggested a novel method for thermal surfacetreatment of foods based on these conclusions. More precisemeasures to describe the high frequency content of random


Fig. 7. Refrigeration, surface (—— ——) and centre temperature(- - -) in a pineapple;u � 2 h.

Fig. 7. Tempe´ratures de re´frigeration, a la surface (—— ——) etau coeur (- - -) d’un ananas;u � 2 h.

Fig. 8. Refrigeration, surface (—— ——) and centre temperature(- - -) in a pineapple;u � 0.2 h.

Fig. 8. Tempe´ratures de re´frigeration, a la surface (—— ——) etau coeur (- - -) d’un ananas;u � 0.2 h.

Fig. 9. Variance of the temperature in the thermal centre of thepineapple as a function of the scale of fluctuation for autoregressiveprocesses of different order. Legend:q: AR(1); A: AR(2), realroots;W: AR(2), complex roots.

Fig. 9. Variance de la tempe´rature au coeur de l’ananas en fonctionde l’echelle de fluctuation des proce´des auto-regressifs d’ordredifferent. Legende:q: AR(1); A: AR(2), vrai racines;W: AR(2),racines complexes.

Fig. 10. Mean (———) and 95% confidence margins (- - -) of thecore temperature in a pineapple as a function of the scale of fluctua-tion of a autoregressive refrigeration temperature of order 2 (realroots).

Fig. 10. Marges moyenne (———) et de 95% de confiance de latemperature au coeur de l’ananas en fonction de l’ećhelle de fluc-tuation d’une tempe´rature frigorifique autore´gressive d’ordre 2(vrai racines).

refrigeration temperature fluctuations are currently beinginvestigated by the authors.

4. Conclusions

A finite element model was established to describeconduction heat transfer in a pineapple. A good agreementwas found between measured and simulated temperaturesinside the pineapple during a cooling process. However, theobserved surface heat transfer coefficient was considerablylarger than the value calculated based on correlation formu-las. This is probably due to the special structure of the pine-apple surface which increases the active surface and maycause local turbulences.

Random temperature fluctuations can cause the centretemperature of chilled produce to decrease temporarilybelow the threshold level beyond which cold injury maydevelop. A variance propagation algorithm was derived tocalculate the steady state temperature variance at an arbi-trary position inside the pineapple. Simulations revealedthat, in the case of autoregressive refrigeration temperaturedisturbances, the centre temperature variance is dependenton the scale of fluctuation, and, to a much lesser extent, onthe order of the autoregressive processes. This indicates thatknowledge of the scale of fluctuation of the disturbancesmight be sufficient to predict the variability of the internaltemperature. As a consequence, stochastic estimation of thecoefficients of the autoregressive processes would be unne-cessary.

Eventually, simulations such as those presented in thisarticle should allow to specify acceptable thresholds forstorage temperature fluctuations. Further, these thresholdsmay vary depending on the actual produce which is to bestored. However, further research in this area is certainlynecessary.

Acknowledgements

The European Communities (FLAIR project AGRF–CT91–0047 and FAIR project FAIR–CT96–1192) aregratefully acknowledged for financial support. The authorswish to thank Dr. Martine Wevers of the Department of theDepartment of Metallurgy and Materials Engineering(MTM) of the Katholieke Universiteit Leuven for her assis-tance with X-ray images, and Dr. Gerard van Beek of theATO-DLO (Wageningen, Netherlands) for interestingdiscussions on surface heat transfer coefficients. AuthorBart Nicolaı is Postdoctoral Fellow with the Flemish Fundfor Scientific Research (F.W.O. Vlaanderen).

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Documents

Propagation of stochastic temperature fluctuations in refrigerated fruits: Propagation des fluctuations de température stochastiques dans des fruits réfrigérés