Freezing time of an infinite cylinder and sphereusing the method of lines

S.R. Ferreira a,*, L.O.A. Rojas b, D.F.S. Souza a, J.A. Oliveira a

a Department of Chemical Engineering, UFRN – University Federal of the Rio Grande do Norte, CampusUniversitário, 59078-970 Natal, RN, Brazilb Department of Chemical Engineering, UFPb – University Federal of the Paraíba, João Pessoa, PB, Brazil

A R T I C L E I N F O

Article history:

Received 26 July 2015

Received in revised form 17

November 2015

Accepted 22 January 2016

Available online 4 May 2016

A B S T R A C T

An energy balance model has been developed to simulate freezing processes for infinite cyl-

inders and spheres. The mathematical model (1) was numerically solved using the method

of lines. In this method, spatial derivatives are discretized by the finite difference method

and the resulting system of ordinary differential equations in time is integrated using an

appropriate solver. Freezing times obtained with the proposed model were compared to ex-

perimental data and results calculated by different published methods. The freezing times

predicted by the proposed model agreed well with the published experimental results and

predictions by other published methods. Model (1) gives a percentage error in the range

−4.61 ≤ E (%) ≤ 6.81, which includes the experimental data for the 123 spheres analyzed and

30 infinite cylinders, within range −2.96 ≤ E (%) ≤ 3.34.

© 2016 Elsevier Ltd and IIR. All rights reserved.

Keywords:

Freezing model

Freezing time

Method of lines

Numerical method

Temps de congélation d’un cylindre infini et d’une sphère enutilisant la méthode des lignes

Mots clés : Modèle de congélation ; Temps de congélation ; Méthode des lignes ; Méthode numérique

1. Introduction

The freezing process consists of freezing, frozen storage, andthawing, each of which must be properly conducted to obtainoptimum results when preserving food and living specimens

(Fennema et al., 1973, p. 151). Foods are frozen for severalreasons, the most common being to achieve effective preser-vation in the frozen state (Fennema et al., 1973, p. 504).

Foods and other biological materials are frozen to improvestorage ability, and subsequently thawed to permit consump-tion, use or processing. Some foods (e.g. ice cream and sherbet)

* Corresponding author. Department of Chemical Engineering, UFRN – University Federal of the Rio Grande do Norte, Campus Universitário,59078-970 Natal, RN, Brazil. Tel.: +55 84 32153753; Fax: +55 8432153770.

E-mail address: Ferreira@eq.ufrn.br (S.R. Ferreira).http://dx.doi.org/10.1016/j.ijrefrig.2016.01.0210140-7007/© 2016 Elsevier Ltd and IIR. All rights reserved.

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are consumed in the frozen or partially frozen state becauseof sensory attributes that state provides. Freezing may also beused to concentrate foods through the formation of ice whichcan be separated from the remaining food component. Freez-ing is also used to prepare foods and other biological materialsfor freeze drying (Schwartzberg, 1981).

The technological aspects of food freezing preservation havebeen reviewed by Fennema et al. (1973, pp. 504–50), whichinclude freezing foods and pre-freezing and freezing methodsand equipment for the major natural foods, frozen food storageand thawing.

Engineers concerned with freezing and thawing often wantto answer the following questions (Schwartzberg, 1981):

a) How much heat must be transferred to convert a givenamount of product from a given initial state to a desiredfinal state?

b) How rapidly can such heat be transferred?c) What effect will manipulation of the product and process

variables have on the rapidity of transfer?d) In the case of freeze concentration, what conditions are re-

quired to convert a desired portion of food water contentinto ice?

The answers to these questions are important in(Schwartzberg, 1981):

A) determining the size of equipment required to handlespecified freezing and thawing loads;

B) determining necessary holdup times for the materialbeing processed;

C) improving and optimizing the productivity andefficiency of freezing and thawing operations;and

D) determining how these processes might be manipu-lated to compensate for abnormal conditions andprocessing upsets.

According to Cleland et al. (1994), the prediction of freez-ing times for foods has been a popular area of research for someyears. The mathematical models to freezing time predictionhave been significantly improved over the last two decades andare now rarely a limiting factor of accurate design (Cleland andÖzilgen, 1998). Prediction of food freezing and thawing timeis necessary when designing and evaluating freezing equip-ment (Mannapperuma and Singh, 1989).The methods availablefor such predictions range from simple semi-theoretical equa-tions to computer programs based on finite difference (FDM),finite elements (FEM), finite volumes (FVM) and the methodof lines (MOL).

One of the most popular approaches to the numerical so-lution of PDE models is the MOL, which proceeds in twoseparate steps (Saucez et al., 2009):

Nomenclature

A parameter A = 0 for slab, A = 1 for infinitecylinder and A = 2 for sphere [–]

Cp apparent or effective specific heat [J kg−1 °C−1]dr step size for spatial variable [m]E calculated or experimental percentage error [%]h convective heat transfer coefficient [W m−2 °C−1]H specific enthalpy [J kg−1]i ith node or number of nodal points (–)k thermal conductivity [W m−1 °C−1]k(i) thermal conductivity calculated at temperature

of ith node T(i) [W m−1 °C−1]k0 thawed thermal conductivity [W m−1 °C−1]k1 freezing zone thermal conductivity [W m−1 °C−1]L half slab thickness [m]P distance of the center of an infinite cylinder or

sphere [m]R infinite cylinder or sphere radius [m]Rho infinite cylinder or sphere or slab density in

computational program [kg m−3]n number of nodes in r or x dimension [–]np number of divisions in the time t [–]t time [s]T temperature [°C]Ta ambient or cooling medium temperature [°C]Tc temperature in the center [°C]Tf initial freezing point (°C)T0 initial temperature [°C]r distance measured along the r axis [m]

x distance measured along the x axis [m]Xmax maximum deviation [%]Xmean mean deviation [%]Xmin minimum deviation [%]y temperature in computational program [°C]ya ambient or cooling medium temperature in

computational program [°C]yprime temperature derivative dT/dt in computational

program [°C s−1]w water mass fraction [–]

Greek symbolsΔr distance increment in r dimension [m]∂T/∂t temperature derivative [°C s−1]ρ density [kg m−3]σn-1 standard deviation [%]Ψ distance measured along the Ψ axis, where

Ψ = r for an infinite cylinder and sphere,and Ψ = x for a slab (m)

Subscriptsa ambient or cooling mediumc centerend end time of calculation in computational

programf freezingi ithn number of nodes in r or x dimension0 initial or thawed1 in freezing zone

38 i n t e rna t i ona l j o u rna l o f r e f r i g e r a t i on 6 8 ( 2 0 1 6 ) 3 7 – 4 9

• Approximation of the spatial derivatives using FDM or FEM.• Time integration of the resulting semi-discrete (discrete in

space, but continuous in time) equations using an appro-priate solver.

With accurate thermophysical property data (including varia-tion of apparent specific heat and thermal conductivity withtemperature), FEM and FDM are capable of predicting freez-ing times very accurately (Cleland and Özilgen, 1998). Acomprehensive study reported by Mannapperuma and Singh(1988, 1989) has illustrated suitably this aspect. The apparentor effective specific heat includes both specific heat and latentheat (Gulati and Datta, 2013). The effective (or apparent) spe-cific heat and enthalpy (H) of foods are related (Bird et al., 2002,p. 286; Schwartzberg, 1981).

Cleland (1990) reviewed in detail the numerical methods andtheir applications to freezing time prediction.This author statedthat accuracy of the results depends on appropriate choice ofscheme (particularly the use of centrally located thermophysicalproperty estimates), on correct selection of time and space stepsfor finite differences, and on the space discretization used forfinite elements (Cleland and Özilgen, 1998). Important numeri-cal analysis with simulation of food freezing processes has beenpresented by Saad and Scott (1997), reinforcing the conclu-sion obtained by Cleland and Özilgen (1998) about the needcentrally located for thermophysical properties.

The objective of the present study is to develop and evalu-ate a mathematical model for the prediction of freezing time.The numerical solution of a partial equation system result-ing from the problem was obtained by using the MOL. Tovalidate the present study’s simulations, some freezing timeexperimental data for infinite cylinders and spheres were se-lected from literature (Cleland, 1977; Ilicali and Saglam, 1987;Mannapperuma et al., 1994; Tocci and Mascheroni, 1994). Freez-ing times obtained with the proposed model were comparedto experimental data and results were calculated by differentmethods (Cleland, 1977; Mannapperuma and Singh, 1988, 1989;Mannapperuma et al., 1994; Tocci and Mascheroni, 1994; Wanget al., 2007).

2. Literature review

This section briefly reviews food freezing processes and theuse of different numerical methods for calculating freezing time.

2.1. Freezing process

A most important group of problems is that in which a sub-stance has a transformation point at which is changed fromone phase to another with emission or absorption of heat(Carslaw and Jaeger, 1980, p. 282). The most important prob-lems in many engineering applications are melting andsolidification, e.g. food freezing and thawing, ground freezingand metal casting (Sheen and Hayakawa, 1991). The exact so-lution of such a problem is difficult because of the movinginterface between the solid and liquid phase when latent heatis absorbed or released.Although there are many methods avail-able for solving these phase change or “moving boundary”

problems, they are usually only applicable to specific cases suchas in an infinite or semi-infinite domain with simple initial and/or boundary condition. When the material physical propertiesare functions of temperature, the analytical solutions are dif-ficult to obtain. Apart from the few exact solutions, all problemshave to be attacked by numerical methods (Carslaw and Jaeger,1980, p. 283).

Detailed discussion about food freezing processes, includ-ing analysis, design and simulation, is presented in variousarticles, books and theses (Cleland, 1977, 1990, 2003; Clelandand Özilgen, 1998; Cleland et al., 1994; Earle and Earle, 2004;Evans, 2008; Fennema et al., 1973; Gulati and Datta, 2013;Heldman, 2003; Kondjoyan, 2006; Mannapperuma and Singh,1988, 1989; Mannapperuma et al., 1994; Mascheroni, 2012; Pham,1985, 1995, 2008, 2014; Salvadori, 1994; Schwartzberg, 1981;Schwartzberg et al., 2007; Singh and Heldman, 2009; Sun, 2012;Tocci and Mascheroni, 1994; Valentas et al., 1997; Wang et al.,2007).

A key calculation in the design of a freezing process is thedetermination of freezing time (Cleland, 2003; Singh andHeldman, 2009). A detailed analysis of all factors influencingfreezing-time prediction has been presented by Heldman (1983).One of the first factors is the freezing-medium temperature,where lower magnitudes will decrease freezing time in a sig-nificant manner.The factor with the most significant influenceon freezing time is the convective heat-transfer coefficient, h(Singh and Heldman, 2009), and its estimation is a vast andcomplex subject (Kondjoyan, 2006; Pham, 2014, p. 6). This pa-rameter can be used to influence freezing time throughequipment design and should be analyzed carefully (Singh andHeldman, 2009). Among the most relevant properties are:product size, thermal conductivity, density, effective specificheat, initial freezing point, initial and final product tempera-tures and convective heat transfer coefficient (Gulati and Datta,2013; Pham, 2008, 2014; Singh and Heldman, 2009). Accordingto Kondjoyan (2006) accurate measurement of transfer coef-ficients is always difficult and time consuming especially undercomplex situations.

2.2. MOL and other numerical methods for freezing timeprediction

A review of prediction of freezing time and design of food freez-ers is presented by Cleland and Valentas (1997, pp. 71–124).

In general, the main advantages of numerical methods arerelated to ability to make predictions where (Cleland, 2003, p.397):

• other mechanisms distinct from the convection can occurat the surface;

• boundary conditions vary with position and time;• product is heterogeneous; and• product geometry is complex.

Detailed analyses of the use of the MOL and other numeri-cal methods are presented in various articles, books and theses(Brenan et al., 1996; Cleland, 1977, 1990; Cleland and Earle, 1977,1984; Cleland and Özilgen, 1998; Cleland et al., 1982; Cutlip andShacham, 1999; Gülkaç, 2005; Mannapperuma and Singh, 1988,1989; Mendoza et al., 2014; Northrop et al., 2013; Pregla, 2008;

39i n t e rna t i ona l j o u rna l o f r e f r i g e r a t i on 6 8 ( 2 0 1 6 ) 3 7 – 4 9

Sadiku, 2009; Schiesser and Griffiths, 2009; Schwartzberg, 1981;Schwartzberg et al., 2007; Shampine et al., 2003; Walas, 1991;Wouwer et al., 2014).

Sadiku and Obiozor (2000) and Zafarullah (1970) present in-troductions to and illustrate the use of the MOL. The MOL isregarded as a special finite difference method which is moreaccurate and effective with respect to computational time thanthe regular finite difference method (Sadiku, 2009; Sadiku andObiozor, 2000).

Besides, the MOL has the following properties which justifyits use: computational efficiency, numerical stability, reducedprogramming effort and reduced computational time. (Sadikuand Obiozor, 2000; Shakeri and Dehghan, 2008).

Schiesser and Silebi (2009) compare the MOL with PDE totaldiscretization based on the use of finite differences. In summary,they observed consistent numerical results for the two methods.

Wang et al. (2007) developed a one-dimensional unsteady-state mathematical model with variable physical propertyparameters to simulate the freezing time of individual foods.The numerical method used was a FDM with the Crank–Nicolson scheme.

According to Saad and Scott (1997) the accuracy of numeri-cal simulations of the freezing process has been discussed bymany researchers (Cleland and Earle, 1984; Cleland et al., 1982;Pham and Willix, 1990). Numerical results are usually ana-lyzed through comparison with experimental data.Disagreements between experimental results and numericalsimulations are generally attributed to experimental errors, nu-merical inaccuracies, and analytical biases (Saad and Scott,1997). Numerical inaccuracies are those associated with thenumerical solution technique, while analytical biases are thoseassociated with the formulation of the mathematical model.However, relatively little effort has been directed toward dif-ferentiating between the different types of disagreements and,therefore, there is no conclusive explanation for the primarycause of the disagreements. Saad and Scott (1997) performeda detailed numerical simulation of five experimental data fromLescano (1973) using the Crank–Nicolson and the two-step FDM.According to Saad and Scott (1997), the results of the two-step method again exhibited a better agreement with theexperimental data than the Crank–Nicolson results.

Pham (1985) suggested a simple method that results in afast and robust procedure and requires less computationaltime for a similar prediction accuracy level when comparedto other FDM. Three test problems were set up. The proposedmethod was compared to its “parent methods”, the explicitenthalpy method and the three-level temperature method(Pham, 1985).

In the effective specific heat methods, latent heat is mergedwith specific heat to produce a specific heat curve with a largepeak around the freezing point (Pham, 2014). The heat con-duction equation based on effective specific heat is numericallysolved based on properties function of the temperature, k(T),ρ(T) and Cp(T) (Pham, 1985). The heat conduction equation infunction of the enthalpy is the basis of enthalpy methods andis numerically solved on enthalpy (H). The properties are en-thalpy functions, e.g. k(H), ρ(H) and T(H). From the engineeringpoint of view, the real test of a food enthalpy model is whetherit will allow accurate prediction of freezing times and heat loads(Pham, 1996).

A number of fixed grid FEM were tested with problems in-volving heat conduction with phase change (Pham, 1995).According to Pham (2008) the FEM is probably the most popularmethod for modeling heat transfer and various other physi-cal phenomena. FEM equations are more time consuming toset up and solve than FDM equations (Pham, 2008). Accord-ing to Pham (2008) FDM is the easiest and fastest numericalmethod.

Computational fluid dynamics (CFD) models calculate thefluid flow and temperature around the products as well as insideit (Kondjoyan, 2006; Pham, 2008). In non-solid regions, the equa-tions of fluid flow must be solved to calculate fluid velocities.The great advantage of CFD is that they allow the heat trans-fer coefficients to be calculated rather than guessed ormeasured experimentally. So, in principle, the rate of coolingand freezing for any product in any situation can be pre-dicted without doing any experiment, provided the product’sproperties are known (Pham, 2008).

3. Materials and methods

The freezing process is modeled in Section 3.1 using the heattransfer equation and Section 3.2 discusses the discretiza-tion of this equation for an infinite cylinder.The discretizationfor a general equation valid for an infinite cylinder, sphere andslab are also presented.

Sections 3.3, 3.4 and 3.5 synthesize the method used, whichis presented in detail in the Results and Discussion section.

3.1. Heat transfer equation for the freezing problem in aninfinite cylinder

For purposes of illustration we will only discuss the case of theinfinite cylinder.Analogous equations can be written for spheresand slabs.

The partial differential equation for heat conduction in aninfinite cylinder can be represented by Cleland (1977) andCleland and Earle (1979):

ρ T Cp TTt r r

k T rTr

( ) ⋅ ( ) ⋅ ∂∂

= ⋅ ∂∂

( ) ⋅ ⋅ ∂∂

⎡⎣⎢

⎤⎦⎥

1(1)

Considering an infinite cylinder with radius R, cooled atambient temperature Ta, the initial and boundary conditionsare:

T T r R t= ≤ ≤ =0 0 0 (2)

− ( ) ∂∂

= = ≥k TTr

r t0 0 0 (3)

− ( ) ∂∂

= −( ) = ≥k TTr

h T T r R ta 0 (4)

The mathematical model Eq. (1) was solved with the initialand boundary conditions given in Eqs. (2) to (4) for an infinitecylinder.

Tylose gel was used as case study for evaluation of the modeland its thermophysical properties were considered in accor-

40 i n t e rna t i ona l j o u rna l o f r e f r i g e r a t i on 6 8 ( 2 0 1 6 ) 3 7 – 4 9

dance with the equations presented by Succar and Hayakawa(1983). The properties for tylose gel in thawed state arek0 = 0.55 W m−1 °C−1, ρ0 = 1006 kg m−3 and Cp0 = 3688 J kg−1 °C−1,and the initial freezing point is Tf = −0.6 °C (Cleland and Earle,1984).The thermophysical properties for tylose gel in the freez-ing zone, that is, for T < Tf, are (Succar and Hayakawa, 1983):

k T T1 1 573 0 00773 0 653≅ − +. . . (5)

ρ1 937 908 0 154 40 8≅ − +. . .T T (6)

Cp T11 6961418 122173≅ + −( ) . (7)

3.2. Model (1): discretization of the heat transferequation and solution by the MOL

The infinite cylinder with radius R is initially at a uniform tem-perature T0. The radius of the infinite cylinder is divided into(n − 1) sections with (n) node points, as shown in Fig. 1. Makingi = 1 in the center of the infinite cylinder and i = n on the in-finite cylinder surface in r = R, the initial condition of Eq. (2)can be written as:

T T i n ti for at= = … =0 1 0 (8)

Making a forward difference for first derivative forthe boundary condition at r = 0 and i = 1 in Eq. (3),− ( ) −( ) =k T T r1 02 1 , it is possible to find:

T T1 2( ) = ( ) (9)

Using a first-order backward finite difference for the firstderivative in Eq. (4) on a cylinder surface at i = n, results (Cutlipand Shacham, 1999; Schiesser, 1991):

∂ ( )∂

≅ ( ) − −( )T ir

T n T nr

1Δ

(10)

Substituting Eq. (10) in Eq. (4) is obtained the surfacetemperature T(n). The thermal conductivity k(n − 1/2) =(1/2){k[T(n)] + k[T(n − 1)]} is simplified as k(n − 1/2) ≅k[T(n − 1)] ≅ k[T(n)] resulting:

− −( ) ( ) − −( )⎡⎣⎢

⎤⎦⎥≅ ( ) −[ ]k n

T n T nr

h T n T1 21

Δ a (11)

Reordering Eq. (11) the cylinder surface temperature T(n),using the simplified form of thermal conductivity evaluationwith k(n − 1/2) ≅ k[T(n − 1)], is given by:

T nh rT k n T n

h r k n( ) ≅ + −( ) −( )

+ −( )Δ

Δa 1 1

1(12)

The second-order central approximation for the second de-rivative at point (i) in Eq. (1), using a simplified form k(i − 1/2) ≅ k(i + 1/2) ≅ k(i), resulting in a “centred” evaluation of k(i), isgiven by (Schiesser, 1991):

k TT ir

k TT i T i T i

r rk i T i

( ) ∂ ( )∂

≅ ( ) +( ) − ( ) + −( )⎡⎣⎢

⎤⎦⎥

= − −( )

2

2

1 2 1

1 2Δ Δ

(( ) − −( )[ ] + +( ) +( ) − ( )[ ]( )

≅ ( ) +( ) − ( ) +

T i k i T i T ir

k i T i T i T i

1 1 2 1

1 2

2Δ−−( )[ ]

( )1

2Δr(13)

Unsteady state heat transfer in an infinite cylinder, repre-sented by Eq. (1), can be rewritten using Eq. (13). Evaluating ρ(i)and Cp(i) at temperature T(i), and substituting Eq. (13) in Eq.(1) for 2 ≤ i ≤ n − 1, results:

∂ ( )∂

≅ ( ) +( ) − ( ) + −( )[ ]( ) ( )( )

T it

k i T i T i T ii Cp i r1 2 1

2ρ Δ (14)

Salvadori (1994) presents a similar discretization in spacefor a slab using total discretization in space and time, butCleland (1977, p. 108) and Cleland and Earle (1979, p. 962) used

Fig. 1 – Nodal illustration for an infinite cylinder or sphere radius.

41i n t e rna t i ona l j o u rna l o f r e f r i g e r a t i on 6 8 ( 2 0 1 6 ) 3 7 – 4 9

k(i) = (1/2)[k(i − 1/2) + k(i + 1/2)], while Wilson and Singh(1987, p. 151) used k(i − 1/2) = k{(1/2)[T(i) + T(i − 1)]} andk(i + 1/2) = k{(1/2)[T(i) + T(i + 1)]}.

The radius of the infinite cylinder is divided into (n-1) sec-tions with (n) node points, numbered from (1) to (n), and thespace grid is given by:

ΔrR

n=

− 1(15)

The problem then requires the solution of Eqs. (8), (9), (12),and (14), which results in (n-2) simultaneous ordinary differ-ential equations from Eq. (14) and two explicit algebraicequations, Eqs. (9) and (12), to obtain (n) temperatures at (n)node points. The preceding equation set can be integrated toany time (t) with one ODE solver.

A general one-dimensional heat conduction equation of aninfinite cylinder, where A = 1 and Ψ = r, or in a sphere, whereA = 2 and Ψ = r, or in a slab, where A = 0 and Ψ = x, is (Cleland,1977; Cleland and Earle, 1979):

ρψ ψ ψ ψ

T Cp TTt

k TT Ak T T( ) ( ) ∂

∂= ∂∂

( ) ∂∂

⎡⎣⎢

⎤⎦⎥+ ( ) ∂

∂(16)

The second-order central finite difference for the first de-rivative at point (i) with ρ(i) and Cp(i) in Eq. (16), is given by(Cutlip and Shacham, 1999; Schiesser, 1991):

k TT i

rk i

T i T ir

( ) ∂ ( )∂

≅ ( ) +( ) − −( )⎡⎣⎢

⎤⎦⎥

1 12Δ

(17)

The discretized form of Eq. (16), where P is the distance ofthe center of infinite cylinder or sphere P = (i − 1)*Δr for2 ≤ i ≤ n − 1 in a simplified form for thermal conductivity evalu-ation k(i), is:

∂ ( )∂

≅ ( ) +( ) − ( ) + −( )[ ]( ) ( )( )

+ ( ) +(T it

k i T i T i T ii Cp i r

Ak i T i1 2 1 12ρ Δ

)) − −( )[ ]( ) ( )( )

T ii Cp i r

12ρ Δ

(18)

3.3. Test and analysis of the developed computationalprogram

A numerical routine based on Runge–Kutta method devel-oped by Brankin et al. (1991) was implemented in Fortran 90to integrate the resulting system of EDOs. The program wasimplemented in Intel Visual Fortran using IMSL – Fortran Nu-merical Library (Imsl, 2006), using routine IVMRK based on thecodes contained in RKSUITE developed by Brankin et al. (1991).All the numerical runs were tested using a Compact PresarioCQ40 Notebook PC, Intel Core(TM) 2 Duo T6500 with a 2.1-GHz processor speed of 2.1 GHz, 4 GB RAM and a 64-bitoperational system.

Preliminary calculation was performed using the MOL andfood freezing experimental data from the literature to assessthe accuracy of the program developed.The results were com-pared to those of other methods in the literature, such as inCleland (1977), Pham (1985, 1995) and Saad and Scott (1997),and presented and discussed in Section 4.1 of Results andDiscussion.

3.4. Evaluation and selection of physical properties forthe calculation of freezing times

Preliminary calculations were performed to assess and selectthe thermal properties to be used in the freezing time calcu-lations; and the results of the calculations are presented anddiscussed in Section 4.2 of Results and Discussion.

3.5. Selection of the data set for the calculation of thefreezing times

Some infinite cylinder and sphere freezing times were se-lected from the literature for simulation in the developedprogram (Cleland, 1977; Ilicali and Saglam, 1987; Mannapperumaet al., 1994; Tocci and Mascheroni, 1994); and the results of thecalculations are discussed in Section 4.3 of Results andDiscussion.

Freezing times calculated by the proposed model were com-pared to experimental freezing times and those calculated bydifferent methods (Cleland, 1977; Mannapperuma and Singh,1988, 1989; Mannapperuma et al., 1994; Tocci and Mascheroni,1994; Wang et al., 2007).

Wang et al. (2007) developed a one-dimensional unsteady-state mathematical model with variable physical propertyparameters to simulate the freezing time of individual foods.According to Wang et al. (2007) the model’s numerical algo-rithm is finite-difference method with Crank–Nicolson scheme,which is programmed with Visual Basic 6.0.

Mannapperuma and Singh (1988, 1989) performed calcu-lations with the numerical method based on enthalpyformulation. In the enthalpy formulation, Fourier’s second lawis written as a function of enthalpy instead of temperature.

Mannapperuma et al. (1994) compared their experimentalfreezing time results for nine individual whole chickens to cal-culated results using the numerical solution proceduredeveloped by Mannapperuma and Singh (1988, 1989).

Cleland (1977) investigated various geometrical shapes andmeasured experimental freezing time over a wide range of con-ditions using a Karlsruhe test substance, Tylose-77, a 23%methylcellulose and 77% water gel, often used as an ana-logue in food freezing experiments (Cleland and Earle, 1977).

Tocci and Mascheroni (1994) obtained freezing times of beefmeat balls in a prototype of belt freezer and freezing times werepredicted by a numerical method, using three different setsof thermal properties of minced beef. Ilicali and Saglam (1987)obtained freezing times of golden delicious apple spheres.

4. Results and discussion

4.1. Test and analysis of the developed computationalprogram

Table 1 shows the experimental conditions for each tylose gelinfinite cylinder, the experimental freezing times obtained byCleland (1977), which were the time from cooling onset to theinstant that the center temperature reached Tc = −10 °C, andthe calculated freezing times (Cleland, 1977; Cleland and Earle,1979).

42 i n t e rna t i ona l j o u rna l o f r e f r i g e r a t i on 6 8 ( 2 0 1 6 ) 3 7 – 4 9

Tables 2 and 3 illustrate the computational program andtemperature and freezing time calculation subroutine for atylose infinite cylinder employing model (1), using Eqs. (9), (12),and (15) with the initial condition of Eq. (8), that is, T = T0 att = 0, Tf = −0.6 °C, as well as the previously presented k1, ρ1, andCp1 properties. The program was checked by comparing nu-merical calculation results to experimental values from Cleland(1977) for tylose gel with 77% water, a substance withthermophysical properties strongly dependent on tempera-ture and approximately the same as those of lean beef with74% water (Bonacina and Comini, 1973). Tables 2 and 3 presentthe input parameters necessary to calculate the first experi-ment in Table 1, CCl01, they are A = 1 for infinite cylinder, n = 4,tend = 32400 s = 9 h, np = 3240, R = 0.07675 m, y(1:n) = 28.4 °C,ya = −21 °C and h = 36.4 W m-2°C−1. The time needed to reachTc = −10 °C at the center point of the tylose gel infinite cylin-der was calculated, resulting in tcalc. = 29720.00 s = 8.26 h < tend =32400 s, with run time t = 1.2 s, experimental time texper = 8.36 hand percentage error in prediction of E = −1.25%.

The accuracy of the present numerical method was checkedout with five typical freezing temperature curves for tylose gelpresented by Cleland (1977, pp. 62, 86, 87, 88, 89) in one-dimensional geometry, a cylinder, two slabs, cylinder and sphere,respectively. Twenty-two selected experimental points in therange −10 < Tc(°C) < 10 were analyzed, giving Xmin = −7.88%,Xmean = −0.09%, Xmax = 7.19% and σn-1 = 3.83% using parameterA = 0 for the slab, A = 1 for the cylinder and A = 3 for the sphere.

The proposed method was compared to its “parentmethods”, the explicit enthalpy method and the three-level tem-perature method (Pham, 1985). Three test problems were setup by Pham (1985). Comparison was also made againstLongworth’s (1975) and Furzeland’s (1980) iterative implicit en-thalpy methods. This study compares our calculation resultsto those obtained by Pham (1985) for their problem numberthree.This problem has no analytical solution but gives an ex-perimental result of texper = 2300 s and error E = ±150 s, asmeasured by Cleland and Earle (1977). Pham (1985) used thecomputer processing (c.p.u.) time on a Digital Vax 1l/750 com-puter and obtained the following results by the explicit enthalpymethod tcalc = 2483 s and c.p.u. time t = 1.2 s, three-level tem-perature method tcalc = 2485 s and c.p.u. t = 10.8 s, Furzeland’s(1980) iterative implicit enthalpy method tcalc = 2485 s and c.p.u.t = 4.0 s, Longworth’s (1975) iterative implicit enthalpy methodtcalc = 2485 s and c.p.u. t = 4.0 s, and Pham’s (1985) tcalc = 2485 sand c.p.u. t = 0.4 s. Using model (1) the input parameters nec-essary to calculate are A = 0 for slab, n = 6, tend = 2400 s, np = 240,half slab thickness L = 0.0125 m, y(1:n) = 30 °C, ya = −40 °C andh = 51.9 W m−2 °C−1. The time needed to reach Tc = −10 °C at thecenter point of the tylose gel slab was calculated, resulting intcalc = 2260 s, with run time t = 1.0 s, experimental time by Clelandand Earle (1977) texper = 2300 s and E = −1,74%.

In a similar way, Pham (1995) tested a number of fixed gridFEM with problems involving heat conduction with phasechange.The methods were implemented in Lahey F77L Fortran

Table 1 – Experimental times by Cleland (1977) for tylose gel infinite cylinder to reach a center temperature of Tc = −10 °C(Cleland, 1977; Cleland and Earle, 1979).

Run R (m) h (W m−2 °C−1) T0 (°C) Ta (°C) texper (h) ± error (h)

CCl01 0.07675 36.4 28.4 −21 8.36 ± 0.46CCl02 0.05225 27.2 30 −20.3 6.14 ± 0.40CCl03 0.026 35.5 28 −19.9 2.16 ± 0.17CCl04 0.07675 36.4 21 −20 8.3 ± 0.47CCl05 0.05225 27.2 19 −20 5.82 ± 0.38CCl06 0.026 35.5 19.7 −20 1.98 ± 0.16CCl07 0.07675 36.4 4.8 −21.1 7.5 ± 0.39CCl08 0.05225 27.2 5.8 −20.5 5.3 ± 0.33CCl09 0.026 35.5 5.8 −20.3 1.84 ± 0.14CCl10 0.07675 36.4 27.2 −33.5 6.1 ± 0.24CCl11 0.05225 27.2 27.8 −33.1 4.1 ± 0.23CCl12 0.026 35.5 26.8 −33.4 1.28 ± 0.08CCl13 0.07675 36.4 16.2 −33.6 5.56 ± 0.23CCl14 0.05225 27.2 16.5 −33.4 3.64 ± 0.19CCl15 0.026 35.5 17.2 −33.2 1.18 ± 0.08CCl16 0.07675 36.4 10.8 −33.5 5.48 ± 0.23CCl17 0.05225 27.2 4.4 −34 3.44 ± 0.21CCl18 0.026 35.5 4.8 −33.2 1.08 ± 0.07CCl19 0.07675 36.4 26.7 −19.8 8.9 ± 0.46CCl20 0.07675 36.4 27.6 −19.8 9.36 ± 0.46CCl21 0.026 35.5 4.3 −33.5 1.06 ± 0.07CCl22 0.026 35.5 4 −33.7 1.04 ± 0.07CCl23 0.05225 27.2 20.1 −25.6 4.56 ± 0.27CCl24 0.05225 27.2 19.6 −25.8 4.74 ± 0.27CCl25 0.07675 21 13 −40.5 5.86 ± 0.37CCl26 0.05225 17.8 14.6 −39.6 4.14 ± 0.31CCl27 0.026 23.9 15.4 −40.2 1.34 ± 0.13CCl28 0.07675 21 4 −39.8 5.7 ± 0.31CCl29 0.05225 17.8 11.5 −39.9 3.96 ± 0.37CCl30 0.026 23.9 25.3 −30.2 2.1 ± 0.19

43i n t e rna t i ona l j o u rna l o f r e f r i g e r a t i on 6 8 ( 2 0 1 6 ) 3 7 – 4 9

for DOS with an 8 × 87 co-processor and single precision forall real variables. Pham (1995) set up three test problems. Testproblem (3) was for a 148-mm thick tylose gel slab initially at30 °C.The surface was suddenly brought to −30 °C and the timerequired for the center to reach −18 °C was calculated. Usingmodel (1) and a Compact Presario CQ40 Notebook PC, the timeneeded for the tylose gel slab center point to reach Tc = −18 °Cwas calculated, tcalc = 23240 s, with run time t = 1.2 s.

A minor modification was implemented to solve thisproblem with our program: the slab surface temperature wasset constant and equal to T(n) = −30 °C as a substitute for theconvective boundary condition of the initial program. Our cal-culated time, tcalc = 23240 s, was similar to those calculated withthe six methods presented by Pham (1995), tcalc = 23001 s,tcalc = 22998 s, tcalc = 22998 s, tcalc = 22998 s, tcalc = 23300 s andtcalc = 23384 s.

The accuracy of the present numerical method waschecked out with five sets of experimental data fromLescano (1973) for codfish slab, reproduced by Saad andScott (1997). As a first approximation, lean beef properties wereused in simulations, and evaluated with the equations pre-sented by Succar and Hayakawa (1983). Seventeen selectedexperimental points were analyzed in five of Lescano’s(1973) experiments in the range −25 < Tc(°C) < −5, withminimum deviation Xmin = −8.78%, mean deviation Xmean = 0.17%,maximum deviation Xmax = 8.41% and standard deviationσn-1 = 5.24% with parameter A = 0 for codfish slab, using citedlean beef properties. Evaluation using lean fish meat density,lean fish meat specific heat, and codfish thermal conductiv-ity (Succar and Hayakawa, 1983) gave Xmin = −8.14%,Xmean = 0.18%, Xmax = 7.96% and σn-1 = 4.87% with reasonableagreement.

Table 2 – Computer program for the model (1) for an infinite cylinder or sphere or slab.

44 i n t e rna t i ona l j o u rna l o f r e f r i g e r a t i on 6 8 ( 2 0 1 6 ) 3 7 – 4 9

4.2. Evaluation and selection of physical properties forthe calculation of freezing times

Some preliminary calculations were performed for selectionof the thermal properties to be used in the calculations withthe developed program and for comparison with experimen-tal freezing times found in Tocci and Mascheroni (1994). Thelean beef thermal properties presented by Succar and Hayakawa(1983) were used instead of minced beef thermal properties inmodel (1), resulting in calculated freezing times close to theexperimental data reported by Tocci and Mascheroni (1994)using cited properties.

A similar procedure was adopted to choose the physicalproperties to be used in the calculations with the experimen-tal data from Ilicali and Saglam (1987).The thermal conductivityfor golden delicious apple presented by Ramaswamy and Tung(1981) obtained using the authors’ Eq. (10) for T > Tf = −1.0 °C

and Eq. (12) for T ≤ Tf = −1.0 °C was used. The density usedwas based on experimental data from Ramaswamy and Tung(1981), and correlated and presented in the footnote of table6 in Succar and Hayakawa (1983). Golden delicious apple ex-perimental data from Ramaswamy and Tung (1981) containedw = 0.873. Specific heat for fruit and vegetable juices withw = 0.87 and Tf = −1.38 °C, which is also applicable to freshfruit and vegetables correlated by Succar and Hayakawa (1983),were used in simulation. The referred specific heat equationwas chosen only after comparison of two sets of enthalpy ex-perimental data from two different sources which had a goodagreed. The enthalpy of apple juice with w = 0.872 andTf = −1.44 °C is similar to enthalpy of apple sauce with w = 0.828and Tf = −1.67 °C from Dickerson (1968) based on 17 experi-mental data from Riedel (1951) in the range −40 ≤ T(°C) ≤ 4.44.Evidently, the specific heat data may be obtained from theenthalpy data.

Table 3 – Calculation subroutine for the model (1) for an infinite cylinder or sphere or slab.

45i n t e rna t i ona l j o u rna l o f r e f r i g e r a t i on 6 8 ( 2 0 1 6 ) 3 7 – 4 9

4.3. Calculation of the freezing times for the selected dataset

Table 4 compares calculated and experimental freezing times(Cleland, 1977; Cleland and Earle, 1979), and freezing times usingmodel (1). Cleland (1977) compared his experimental freezingtime data to values calculated using a specific heat-basedmethod and obtained for infinite cylinder Xmin = −10.8%,Xmean = −3.8%, Xmax = 2.4% and σn-1 = 3.1%. Most of the experi-ments (24 with cylinders and the same number with spheres)were carried out in a liquid immersion freezer because thesurface heat transfer coefficient variation in a liquid arounda freezing object is lower than in an air blast freezer (Clelandand Earle, 1979). The measurement and surface heat transfercoefficient control errors were much great than those in slabplate freezing (Cleland 1977). Model (1) gave Xmin = −2.96%,Xmean = 0.03%, Xmax = 3.34% and σn-1 = 2.03% for an infinite cyl-inder calculated based on Table 4.

Mannapperuma and Singh (1988, 1989) employed the en-thalpy formulation and obtained Xmin = −14.0%, Xmean = −4.6%,Xmax = 1.6% and σn-1 = 4.4% for an infinite cylinder. Wang et al.(2007) obtained Xmin = −4.29%, Xmean = 5.61%, Xmax = 12.77% andσn-1 = 3.67%, also for an infinite cylinder. Parameters of tylosegel used in numerical simulation are listed on table 2 of Wanget al. (2007). Table 5 summarizes the comparison of the ex-perimental and predicted freezing times in the present paper.

Similarly, the freezing time was calculated by Cleland (1977)and Mannapperuma and Singh (1988, 1989) and with model

(1) using the experimental data obtained by Cleland (1977) witha 30 tylose gel sphere until the center temperature reachedTc = −10 °C. The results are presented in Table 5.

Mannapperuma et al. (1994) employed the enthalpy fromRiedel (1957), thermal conductivity from Sweat et al. (1973) anddensity from Walters and May (1963). These physical proper-ties are similar to the tylose gel properties and were used incalculations with model (1). The results are presented inTable 5.

Tocci and Mascheroni (1994) compared their experimentalfreezing time results for 48 beef meatballs to calculated results.Wang et al. (2007) compared their results to only 20 selectedexperimental data. The 48 experimental data were comparedto the calculated results using model (1). Tocci and Mascheroni(1994) predicted freezing times using a numerical method withthree different sets of minced beef thermal properties simu-lating the properties of beef meatballs. In order to study theinfluence of physical properties on freezing time predictions,three sets of values from the literature were used to calcu-late the freezing time of minced beef as follows (Tocci andMascheroni, 1994). For the first set using the thermal data setfrom Sanz et al. (1989), for the second the data set from Clelandand Earle (1984) and for the third the data set from Clelandet al. (1986).

Experimental data published by Tocci and Mascheroni (1994)were compared with predicted results of numerical model ofWang et al. (2007). Data used in simulations of the numericalmodel are listed on table 6 of Wang et al. (2007).

Table 4 – Calculated and experimental freezing times of the tylose gel infinite cylinder (Cleland, 1977; Cleland and Earle,1979) and freezing times using the model (1).

Run texper. Cle. (h) Error in texper. Cle. (%) tnum. Cle. (h) Enum. Cle. (%) tmodel (1) (h) Emodel (1) (%) n (-)

CCl01 8.36 ±5.50 8.15 −2.5 8.26 −1.25 4CCl02 6.14 ±6.51 6.15 0.2 5.99 −2.39 5CCl03 2.16 ±7.87 2.14 −1.0 2.11 −2.47 6CCl04 8.3 ±5.66 8.22 −1.0 8.22 −0.93 4CCl05 5.82 ±6.53 5.86 0.7 5.67 −2.56 5CCl06 1.98 ±8.08 2.03 2.4 2.00 0.77 6CCl07 7.5 ±5.20 7.00 −6.7 7.37 −1.70 5CCl08 5.3 ±6.23 5.09 −4.0 5.31 0.12 6CCl09 1.84 ±7.61 1.79 −3.0 1.85 0.59 7CCl10 6.1 ±3.93 5.63 −7.6 6.26 2.69 6CCl11 4.1 ±5.61 3.83 −6.6 4.02 −1.95 6CCl12 1.28 ±6.25 1.27 −1.0 1.27 −1.05 6CCl13 5.56 ±4.14 5.17 −7.0 5.72 2.86 6CCl14 3.64 ±5.22 3.43 −5.9 3.65 0.25 7CCl15 1.18 ±6.78 1.19 0.6 1.18 0.13 6CCl16 5.48 ±4.20 4.89 −10.8 5.43 −0.83 6CCl17 3.44 ±6.10 3.23 −6.2 3.37 −2.14 7CCl18 1.08 ±6.48 1.03 −4.4 1.11 2.95 7CCl19 8.9 ±5.17 8.59 −3.5 8.66 −2.73 4CCl20 9.36 ±4.91 8.64 −7.9 9.60 2.58 5CCl21 1.06 ±6.60 1.03 −3.3 1.10 3.34 7CCl22 1.04 ±6.73 1.01 −3.2 1.01 −2.96 6CCl23 4.56 ±5.92 4.60 0.8 4.46 −2.28 5CCl24 4.74 ±5.70 4.54 −4.2 4.78 0.83 6CCl25 5.86 ±6.31 5.49 −6.5 5.96 1.79 6CCl26 4.14 ±7.49 3.94 −4.8 4.21 1.61 7CCl27 1.34 ±9.70 1.34 −0.6 1.36 1.68 7CCl28 5.7 ±5.44 5.41 −5.1 5.83 2.35 7CCl29 3.96 ±9.34 3.80 −4.2 4.05 2.35 7CCl30 2.1 ±9.05 1.98 −5.9 2.08 −0.78 9

46 i n t e rna t i ona l j o u rna l o f r e f r i g e r a t i on 6 8 ( 2 0 1 6 ) 3 7 – 4 9

The lean beef thermal properties presented by Succar andHayakawa (1983) were used instead of minced beef thermalproperties in model (1), resulting in calculated freezing timesclose to the experimental data reported by Tocci and Mascheroni(1994), as shown in Table 5.

The experimental freezing times for 20 golden deliciousapple spheres for Tc = −10 °C and from another 16 runs forTc = −18 °C obtained by Ilicali and Saglam (1987) were com-pared to those calculated using model (1). The calculatedfreezing times were close to the experimental data reportedby Ilicali and Saglam (1987); the statistical parameters are pre-sented in Table 5.

Model (1) gives a percentage error in the range −4.61 ≤ E(%) ≤ 6.81, which includes the experimental data for the 123spheres analyzed and 30 infinite cylinders, −2.96 ≤ E (%) ≤ 3.34.The MOL is easy to program, accurate and requires a shortercomputing time. The freezing times predicted by the pro-posed model agreed well with published experimental resultsand predictions by other published methods.

Stable oscillations were observed using the MOL in thepresent work, as for example, at surface temperature T(n), whentemperature T(n − 1) crossed the freezing zone of tylose, es-pecially with large convective heat transfer coefficients. Similarresults were observed by Mannapperuma and Singh (1989) usingthe explicit enthalpy method to simulate food freezing andthawing, but stable oscillations were observed when the surfacetemperature T(n) crossed the initial freezing point Tf. These os-cillations can be virtually eliminated by using time incrementssmaller than those dictated by the stability criteria(Mannapperuma and Singh, 1989).

Cleland and Earle (1979) modified the nodal configurationand the updating scheme to reduce oscillations (Mannapperumaand Singh, 1989) using the apparent heat capacity formula-tion. The FDM agreed well with a known analytical solutionfor constant thermal properties, but when it was applied toKarlsruhe test substance freezing oscillations, it led to a falsesolution (Cleland and Earle, 1979). These oscillations wereavoided by defining k(i) = (1/2){k[T(i + 1/2)] + k[T(i − 1/2)]}; however,this introduces a further slight error into the FDM when thethermal conductivity was not constant. In the freezing of bio-logical materials, this approximation results in a smalloverestimation of k(i) (Cleland, 1977; Cleland and Earle, 1979).

5. Conclusions

A mathematical model is proposed for the calculation of freez-ing time with tylose gel as a model food. The MOL, which iseasy to program, accurate and requires a generally shorter com-putational time, was used to solve the proposed model. Freezingtimes obtained with the proposed model were compared to ex-perimental data and results calculated by different publishedmethods.The freezing times predicted by the proposed modelagreed well with published experimental results and predic-tions obtained by other published methods. Model (1) gave apercentage error in the range −4.61 ≤ E (%) ≤ 6.81, which in-cludes the experimental data for the 123 spheres analyzed and30 infinite cylinders, −2.96 ≤ E (%) ≤ 3.34.

Tabl

e5

–S

um

mar

yof

com

par

ison

sbe

twee

nex

per

imen

tala

nd

pre

dic

ted

free

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mes

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ng

nu

mer

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met

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s.

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rce

ofd

ata

set

Met

hod

Th

erm

alp

rop

erty

Geo

met

ryT

c(°

C)

Ru

ns

Xm

in(%

)X

mea

n(%

)X

max

(%)

σ n-1

(%)

Cle

lan

d(1

977)

Cle

lan

d(1

977)

Man

nap

per

um

aan

dSi

ngh

(198

9)W

ang

etal

.(20

07)

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el(1

)

Cle

lan

d(1

977)

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um

aan

dSi

ngh

(198

9)W

ang

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.(20

07)

Succ

aran

dH

ayak

awa

(198

3)

Infi

nit

ecy

lin

der

−10

30−1

0.8

−14.

0−4

.29

−2.9

6

−3.8

−4.6 5.61

0.03

2.4

1.6

12.7

73.

34

3.1

4.4

3.67

2.03

Cle

lan

d(1

977)

Cle

lan

d(1

977)

Man

nap

per

um

aan

dSi

ngh

(198

9)M

odel

(1)

Cle

lan

d(1

977)

Man

nap

per

um

aan

dSi

ngh

(198

9)Su

ccar

and

Hay

akaw

a(1

983)

Sph

ere

−10

30−8

.70

−10.

0−3

.14

−4.5

−2.3 1.72

0.3

6.0

6.81

2.5

3.7

2.78

Man

nap

per

um

aet

al.(

1994

)M

ann

app

eru

ma

etal

.(19

94)

Mod

el(1

)R

ied

el(1

957)

,Sw

eat

etal

.(19

73),

Wal

ters

and

May

(196

3)Sp

her

e−1

89

−2.5

610

.28

19.0

55.

58

Succ

aran

dH

ayak

awa

(198

3)−0

.87

−0.0

60.

930.

63To

ccia

nd

Mas

cher

oni(

1994

)To

ccia

nd

Mas

cher

oni(

1994

)To

ccia

nd

Mas

cher

oni(

1994

)To

ccia

nd

Mas

cher

oni(

1994

)W

ang

etal

.(20

07)

Mod

el(1

)

San

zet

al.(

1989

)C

lela

nd

and

Earl

e(1

984)

Cle

lan

det

al.(

1986

)W

ang

etal

.(20

07)

Succ

aran

dH

ayak

awa

(198

3)

Sph

ere

−18

48 48 48 20 48

−13.

28−6

.03

−0.2

2−6

.27

−4.6

1

−0.9

26.

8413

.23

0.55

0.64

12.1

022

.19

29.1

99.

434.

72

5.65

6.31

6.59

4.16

2.50

Ilic

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Sagl

am(1

987)

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el(1

)R

amas

wam

yan

dTu

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−10

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.73

0.23

2.56

1.31

Idem

−18

16−0

.71

0.33

1.91

0.73

47i n t e rna t i ona l j o u rna l o f r e f r i g e r a t i on 6 8 ( 2 0 1 6 ) 3 7 – 4 9

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