1

EVALUATION OF THE THERMOPHYSICAL PROPERTIES OF

TYLOSE GEL UNDER PRESSURE IN THE PHASE CHANGE

DOMAIN

L. Oteroa, A. Ousegui b, B. Guignona, A. Le Bailb, P. D. Sanz a

aInstituto del Frío, C.S.I.C, c/ José Antonio Nováis, 10. E-28040 Madrid, Spain.

bUMR GEPEA (UA CNRS 6144-SPI). ENITIAA. Rue de la Géraudière BP 82225,

F-44322 Nantes Cedex 03, France.

ABSTRACT

The accurate knowledge of the thermophysical properties of foods under pressure

is essential to optimize high-pressure processes in the food industry. In this paper,

two generic models are proposed based on the food composition: the ‘additive

model under pressure’ and the ‘shifting approach’. The first one employs

thermophysical properties of ice and water under pressure calculated as accurately

as possible using sophisticated routines. The ‘shifting approach’ offers a less

accurate, but considerably faster alternative for modeling purposes. This obtains

thermophysical properties of foods under pressure shifting data at atmospheric

Corresponding author. Tel. : +34 91 5445607; fax: +34 91 5493627.

E-mail address: psanz@if.csic.es (P.D. Sanz)

brought to you by COREView metadata, citation and similar papers at core.ac.uk

provided by Digital.CSIC

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conditions according to the freezing point depression at the prevalent pressure.

These models were applied to the case of tylose, a methylcellulose gel that is often

used as a model food system. At atmospheric conditions, both models gave similar

results for all the studied properties. Under high pressure, the ‘shifting approach’

undervalued the density and overestimated the specific heat values. However, both

methods gave similar results of thermal conductivity, apparent specific heat and

enthalpy.

Keywords: high-pressure, thermophysical properties, food, modeling, freezing,

thawing

INTRODUCTION

High-pressure processing is an expanding technique that is being successfully

introduced in the food industry mainly due to the consumer demand for fresh-like

food products with minimal degradation of nutritional and organoleptic properties

(San Martín, Barbosa-Cánovas & Swanson, 2002). Among all the innovative non

thermal processes, those based on the effect of pressure on the phase transitions

of water have a special interest. High-pressure freezing, high-pressure thawing and

preservation at subzero temperatures without phase transition (Kalichevsky, Knorr

& Lillford, 1995; Le Bail, Chevalier, Mussa & Ghoul, 2002; Sanz, 2004) open new

perspectives in food preservation, promise high product safety and quality,

reductions in process times and offer the potential for an environmentally friendly

exploitation.

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However, the implementation of such a technology in the food industry requires the

knowledge of many properties in order to design efficient processes that will permit

to achieve the expected quality and the stability of the processed products.

Furthermore, it is essential to optimize the industrial equipment and, of course, to

control the operating costs. Thus, heat transfer phenomena during high-pressure

processing must be simulated in order to design equipment, estimate process

times or predict the impact of a treatment on the quality of a food, for example. An

appropriate modeling implies a precise knowledge of the thermophysical properties

of the materials involved. In the case of conductive heat transfer, thermophysical

properties of interest include the density, the thermal conductivity and the specific

heat. When the processes to model involve a change of phase, the ice fraction, the

enthalpy and the initial freezing point are also important. The techniques usually

employed to determine these food properties at atmospheric conditions are time

consuming. When determinations must be made at high pressure, the practical

problems increase significantly (Otero & Sanz, 2003). Some experimental

determinations have been made under pressure: density, thermal expansion

coefficient and thermal conductivity in tomato paste and apple sauce (Denys &

Hendrickx, 1999; Denys, Ludikhuyze, Van Loey & Hendrickx, 2000a; Denys, Van

Loey & Hendrickx, 2000b;) and also, latent heat of tylose (Denys, Van Loey &

Hendrickx, 2000c) and MgSO4 aqueous solutions (Chourot, Le Bail & Chevalier,

2000); but the practical problems hamper extensive measurements.

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At atmospheric conditions, numerous models, considering food as a two-phase

and two-component system of water and dry matter, have been proposed and

reviewed by different authors (Mannapperuma & Sight, 1989; Lind, 1991; Saad &

Scott, 1996; Fikiin & Fikiin, 1999). Having in mind the negligible influence exerted

on the food thermophysical properties by the differences in the thermophysical

characteristics of the various elements that compose the dry matter

(carbohydrates, proteins, fats ...), the interpretation of the food materials as a two-

component system of water (solid and liquid phase) and dry matter is completely

justified (Fikiin et al., 1999). This kind of generic models, able to predict the

thermophysical properties of foods from their composition, results much more

practical than the direct experimental measurements taking into account the almost

infinite recipes and compositions of foods.

Below the initial freezing point, foods represent a dynamic complex of three

fractions: dry substance, liquid water and ice. At a given pressure, these fractions

are continuously changing their quantitative ratios with temperature since water in

a food product is frozen over a range of temperatures rather than at a single one.

This phenomenon is attributed to the continuous concentration in insoluble solids

and solutes that occurs in the unfrozen product fraction as ice is formed. The

increased concentration depresses the freezing point of the unfrozen product

fraction and involves the removal of latent heat over a range of temperatures.

Since the product properties are closely related to the state of water in the product,

it is evident that, at a given pressure, they vary with temperature and more

specifically with the extent to which the phase change of water has occurred

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(Heldman, 1982). Therefore, in order to model the physical properties of food

products in the phase change domain, the knowledge of the initial freezing point

and the ice fraction is essential. Thus, once the ice content is known, thermal

properties of food at atmospheric conditions are relatively easy to model using

classical additive models (Heldman, 1982; Mannapperuma et al., 1989; Fikiin et al.,

1999) as long as the properties of the components of the food (liquid water, ice and

dry matter) are known.

At high pressures, different approaches have been made in the literature to predict

thermophysical properties of foods. Miles and Morley (1978) used thermodynamic

equations to estimate the effect of hydrostatic pressure on some of the thermal and

physical properties of frozen foods. Denys, Van Loey, Hendrickx and Tobback

(1997) shifted the atmospheric pressure conductivity data of tylose according to the

freezing point depression associated with the applied pressure. They also adjusted

the apparent specific heat curve, reducing its height or width to simulate the

reduction of the latent heat of pure water at high pressures (Denys et al., 2000c). In

a similar way, Schlüter, George, Heinz & Knorr (1999) fitted cumulative and density

Weibull distributions to thermal conductivity and apparent specific heat data for

potato at atmospheric conditions. Then, they modified these distributions according

to the freezing point depression induced by pressure, maintaining the scale and

shape parameters. All these methods need as a starting point a set of known

thermophysical data of the implied food at atmospheric conditions to predict the

corresponding values under high pressure.

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The first objective of this paper is to obtain a generic model, able to predict

thermophysical properties of foods under pressure from their composition in a

similar way as the classic additive models described at atmospheric conditions. To

do that, it is essential to know the thermophysical properties of liquid water and ice

under pressure. These properties are available in the existing literature (e. g.,

Tanishita, Nagashima & Murai, 1971; Ter Minassian, Pruzan & Soulard, 1981;

Sato, 1989; Chizhov & Nagornov, 1991). The International Association for the

Properties of Water and Steam (IAPWS) adopted, in 1995, a new formulation for

water thermodynamic properties ‘for general and scientific use’ (IAPWS, 1996).

Nevertheless, this formulation is only valid in the stable fluid region of water, from

the melting-pressure curve to 1000ºC at pressures up to 1000 MPa and so, it does

not include data for supercooled water. These data are of special importance when

modeling processes supporting a phase change. Otero, Molina-García and Sanz

(2002) proposed a review about some interrelated thermophysical properties of

liquid water and ice I on the pressure and temperature range of interest for high-

pressure food processing, including supercooled water. The corresponding

calculation routines are publicly available through http:/www.

if.csic.es/programas/ifiform.htm.

But the implementation of this kind of calculation routines in a heat transfer model

is time-consuming and simple models are needed to reduce the computation time.

The second objective of this paper is therefore to offer and to evaluate an easy-to-

use and fast alternative to the ‘additive model under pressure’ described above.

This alternative is based on a shifting of the thermophysical properties obtained at

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atmospheric pressure towards the high pressure domain (‘shifting approach’)

according to the freezing point depression at the prevalent pressure. Such an

approach has already been used by Denys et al. (1997).

MATERIAL AND METHODS

SAMPLE

Tylose (77% water content), a hydroxymethylcellulose gel, was employed to test

and validate the models. Tylose was chosen because it is a food analogue

material, usually employed to model freezing and thawing processes, with known

thermal properties at atmospheric pressure (Riedel, 1960; Cleland & Earle, 1984).

ADDITIVE MODEL UNDER PRESSURE

The ‘additive model under pressure’ is a generic model, able to predict properties

of foods from their composition and the corresponding thermophysical properties of

ice and water calculated at the desired pressure/temperature conditions as

accurately as possible.

Initial freezing point

The initial freezing point of foods as a function of pressure can be approximately

obtained from the melting curve of pure water considering that the freezing point

depression observed at atmospheric pressure remains constant under high

pressure. This fact has experimentally observed in 4.3% NaCl aqueous solutions

(Chourot, Cornier, Legrand & Le Bail, 1996), in tomato paste (Denys, 2000b) or in

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potato (Schlüter & Knorr, 2002), for example. Accurate equations to obtain the

pressure dependent freezing point of pure water are described in Wagner, Saul &

Pru (1994). Also simpler expressions can be obtained from the experimental data

measured by Bridgman (1912) by linear regression.

In this paper, equation (1) is proposed to calculate the initial freezing point of foods,

tif(P), as a function of pressure (P) expressed in MPa. This equation has been

obtained combining a linear regression from Bridgman’s data (1912) for pure water

(Chourot, Boillereaux, Havet & Le Bail, 1997) together with the initial freezing point

of the food at atmospheric conditions (tif(Patm)):

200015500721920 PPPtPt atmifif ..)()( (1)

Ice fraction

Water contained in foods can be schematically classified as ‘free’ water or as

‘bounded’ water that is bounded to the hydrophilic sites of the various components

(carbohydrates, proteins, ...). When calculating the ice content as a function of

temperature and pressure, it must be taken into account that only free water can

undergo state transitions such as ice crystallization; this statement is valid if one

considers that the so called ‘free’ water is able to be completely frozen. Bound

water can be approximately obtained from tables in the literature for particular

foods (Pham, 1987; Pham, 1996) or more accurately if enthalpy data are known

(Pham, 1987), from the moisture sorption isotherm data (Karmas & Chen, 1975) or

the state of water data by NMR (Kerr, Kauten, Ozilgen, McCarthy & Reid, 1996).

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Miles (1974) obtained a simple expression, based on the equation of the freezing

point depression, to calculate the ice fraction (Xi) as a function of temperature,

taking into account the bound water fraction (Xbw). Equation (2) shows this

expression adapted to take into account the prevalent pressure:

)(

)()(1)(),(

0

0

PTT

PTPTXXTPX

if

bwtwi (2)

Traditionally, this equation has been only used at atmospheric conditions; but, in

this paper, we have employed it at high pressure substituting the appropriate

values. Both, the freezing point of pure water as a function of pressure (T0(P)) and

the initial freezing point of the food (Tif(P)), in Kelvin degrees, can be calculated

from equation (1), considering the corresponding values at atmospheric pressure

(t0(Patm) = 0ºC for pure water) and changing units (Celsius to Kelvin degrees).

Density

The density of food materials is directly dependent on the mass fraction of water

and the solid ingredients existing in the food matter. In this paper, a weight additive

model has been applied, based on the mass fraction of water, ice and the solid

particles, along with the appropriate density for each of the components. Equation

(3) and (4) permit to obtain the density above and below the initial freezing point

respectively.

s

s

w

tw XTP

XTP

1

),(

1

),(

1 (3)

s

s

i

i

w

ufw XTP

TPXTP

TPXTP

1

),(

1),(

),(

1),(

),(

1 (4)

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In these equations, is the density (kg/m3) of the product. Moreover, the unfrozen

water fraction (Xufw) can be expressed as:

),(),( TPXXTPX itwufw (5)

In this ‘additive model under pressure’, the densities of liquid water ( w) and ice ( i)

have been obtained as functions of pressure and temperature according to Otero

et al. (2002). The intrinsic density of the dry matter ( s) was calculated applying

equation (3) to known density data of unfrozen tylose at atmospheric conditions as

suggested by Saad et al. (1996). As a simplification, in this paper s has been

assumed to be independent of pressure and of temperature.

Specific heat

The specific heat model is based on the summation of the specific heat of the

different food components weighted by their respective mass ratio. Equation (6)

and (7) were used to evaluate the specific heat above and below the initial freezing

point respectively.

sswtw CpXTPCpXTPCp ·),(),( (6)

ssiiwufw CpXTPCpTPXTPCpTPXTPCp ·),(),(),(),(),( (7)

In these quations, Cp is the specific heat (J/kg·K) of the product. In this model,

specific heats of liquid water (Cpw) and ice (Cpi) have been calculated as functions

11

of pressure and temperature according to Otero et al. (2002). Since these

calculation routines are only valid for temperatures higher than –40ºC, the specific

heats of ice and water below this temperature were considered constant and equal

to the corresponding value at –40ºC. The intrinsic specific heat of the dry matter,

Cps, was obtained applying equation (6) to known specific heat data of unfrozen

tylose at atmospheric conditions as suggested by Saad et al. (1996). As a

simplification, in this paper, Cps has been assumed to be constant (not pressure,

nor temperature dependent).

Conductivity

The thermal conductivity of a frozen food is probably the most complex property to

predict. Three main factors are implied: pressure, temperature and the structure of

the product. The size, shape and distribution of the ice crystals affect this structure

and longer depend on the freezing method. In general terms, low freezing rates

produce large dendritic ice crystals unlike high rates that induce small crystals.

Moreover, high-pressure shift freezing implies instantaneous nucleation throughout

the sample and the resulting ice crystals are granular in shape with no specific

orientation. This resulting ice structure by no way resembles that of a product

frozen at low rates at atmospheric pressure or by high-pressure assisted freezing

(Le Bail et al., 2002). Thus, determining an accurate conductivity value in a frozen

product implies not only knowing the ice fraction at a concrete temperature and

pressure but also the size, shape and distribution of the ice crystals formed. There

exist in the literature different models to predict the thermal conductivity of a food.

The simpler models (series and parallel models) only take into account the volume

fraction of each component (ice, water and dry matter) and its intrinsic thermal

conductivity value. They give the lower and the upper limits of the thermal

conductivity, whilst the real value should be between these two extremes. A more

accurate prediction implies taking into account the morphology of the ice crystals.

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The Maxwell’s model represents a heterogeneous system as a dispersion of

spherical particles (ice crystals) in a continuous phase. The De Vries’s model, an

extension of Maxwell’s equation, is a semi-empirical model that introduces shape

factors to take into account the nonsphericity of the ice crystals (Cogné, Andrieu,

Laurent, Besson & Nocquet, 2003). The suitability of each model depends on the

freezing method and the freezing rate and must be experimentally evaluated. So,

taking into account the general scope of this paper, we have employed the parallel

model to study the dependency of the conductivity on the pressure and

temperature conditions.

Equations (8) and (9) were used to evaluate the thermal conductivity above and

below the initial freezing point respectively:

sswtw kTPTPkTPTPk ),(),(),(),( (8)

ssiiwufw kTPTpkTPTPkTPTPk ),(),(),(),(),(),( (9)

where,

),(

),(),(

TP

TPXTP

w

twtw (10)

s

ss

TPXTP

),(),( (11)

),(

),(),(),(

TP

TPTPXTP

i

ii (12)

),(),(1),(),(),( TPTPTPTPTP siitwufw (13)

In this ‘additive model under pressure’, the thermal conductivity of liquid water has

been calculated according to Tanishita et al. (1971). Since Tanishita’s model is

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only valid for temperatures between 0 and 700ºC and pressures between 1 and 50

MPa, values out of this range were obtained by extrapolation. At high pressures,

there are not conductivity data for ice and dry matter available. So, in this paper,

the thermal conductivity of ice (ki) has been assumed to be only temperature

dependent:

2170.2100402.7107811.9101559.4)( 32537 ttttki (14)

Equation (14) was obtained by lineal regression from data in the literature

(Heldman, 1992).

Moreover, the intrinsic thermal conductivity of the product solids fraction (ks) has

been considered as not pressure, nor temperature dependent.

Apparent specific heat and enthalpy

At atmospheric conditions, the apparent specific heat of foods has been

traditionally calculated considering the contribution of the apparent heat capacity of

each pure component and incorporating the latent heat of fusion as shown below

(Bartlett, 1944; Miles, 1991; Cogné et al., 2003).

In this paper, that kind of model has been adapted to be used under high-pressure

conditions considering pressure-temperature dependent properties:

dT

TPdXPTPLTPCpTPXTPC i

ifj

j

japp

),())(,(),(),(),( (15)

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Also, the pure ice latent heat of fusion, L(P,Tif(P)) has been calculated according

to:

))()(()()())(,( 00 PTPTPCpPLPTPL ifif (16)

2

0 388.0369.399295.333549)( PPPL (17)

))(,())(,()( 00 PTPCpPTPCpPCp iw (18)

Moreover, the temperature rate of ice formation was derived from equation (2):

2

0

0

)(

)()()(

PTT

PTPTXX

dT

dX ifbwtwi (19)

Enthalpy was mathematically determined by:

dTTPCHT

Tapp

Pref

),()(

(20)

where, the reference temperature Tref (P), at which H = 0, was considered as:

40)()( 0 PTPTref (21)

SHIFTING APPROACH

In this paper, the ‘shifting approach’ has been developed as an easy-to-use and

fast alternative (always thinking about modeling purposes) to calculate the

thermophysical properties of food under pressure.

So, at atmospheric conditions, thermophysical properties of tylose were obtained

using the additive models previously described (equations (3) and (4) for density,

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(6) and (7) for specific heat, (8) and (9) for thermal conductivity and (15) for

apparent specific heat); but the intrinsic properties of liquid water and ice were

obtained using much simpler equations than the calculation routines by Otero et al.

(2002):

)/(4.1000105.5105.9

102104102),(

3323

344658

mkgtt

ttttPatmw (22)

)/(48.9161003.17108),( 3224 mkgtttPatmi (23)

)º/(4.42537732.61048.36

103.1410810104),(

22

33445568

CkgJtt

tttttPCp atmw (24)

)º/(211578.7),( CkgJttPCp atmi (25)

57109.0107625.1107036.6),( 326 tttPk atmw (26)

The thermal conductivity of ice was calculated from equation (14).

Equations (22) - (25), used to calculate the density and the specific heat of water

and ice at atmospheric conditions, were obtained by linear regression from data

previously calculated according to the routines by Otero et al. (2002). High order

polynomials were needed to properly reproduce the anomalous behavior of

supercooled water.

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At high pressure, properties were estimated using the ‘shifting approach’ proposed

by Denys et al. (1997) instead of the additive models. Thus, the density, the

specific heat and the thermal conductivity under pressure were modeled by shifting

the corresponding data at atmospheric pressure according to the freezing point

depression at the prevalent pressure calculated using equation (1). In the other

hand, an appropriate modeling of the apparent specific heat also implies taking into

account the reduction of the latent heat of pure water under pressure (Bridgman,

1912). So, it was necessary to reduce the area (height or width) of the apparent

specific heat peak together with shifting atmospheric data. In this paper, we have

reduced the height of the specific heat peak to a certain percentage, in the

temperature interval where the main part of the transition occurs (an interval

between the corresponding initial freezing point under pressure and 10ºC below

this point). The applied reduction for a given pressure was obtained from the

following expression optimized by Denys (2000) for tylose:

PreductionOptimal 1100220.10296.3(%) (27)

RESULTS

Initial freezing point

The reported initial freezing point of the tylose that was used in this study (77%

water content) is tif = -0.6ºC at atmospheric conditions (Riedel, 1960).

The solid line in Figure 1 shows the melting curve of tylose obtained according to

equation (1) together with the melting curve of pure water (bold line). Also initial

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freezing points of tylose under pressure have been calculated using the more

accurate equations of Wagner et al. (1994) for pure water (dashed curve). Both

curves are overlapped in Figure 1, giving an evidence that equation (1) is accurate

enough for modeling purposes. This equation, obtained from Bridgman’s data for

pure water (1912) and the initial freezing point of the food at atmospheric

conditions, is preferable when implementing a heat transfer model because it is

much faster than calculation routines proposed by Wagner et al. (1994).

Ice content

Pham (1987) estimated, from enthalpy-temperature data, that 9.7% of the water in

tylose (77% total water content and tif = -0.6ºC) was in the bounded state. Taking

into account this bounded water fraction, the ice content as a function of pressure

and temperature has been calculated using equation (2) and is represented in

Figure 2. It clearly shows how the ice content increases very quickly from 0% at the

initial freezing point (as a function of pressure) to approximately 75 percent when

the temperature drops. Then, the ice fraction increases gradually up to about 86%

when the temperature is 40ºC below the initial freezing point. The higher the

pressure, the lower must be the temperature to obtain a given ice fraction due to

the initial freezing point depression. For example, 75% of the frre water will be

frozen at -4.6ºC at atmospheric pressure whereas such a percentage is obtained at

–25.2°C at 200 MPa.

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Density

There is a great scarcity of data on the density of tylose at atmospheric conditions.

After an extensive bibliographic review, a value of 1006 kg/m3 for tylose (77%

water content) in the thawed state was found (Succar & Hayakawa, 1983).

Substituting this value in equation (3) at the initial freezing point at atmospheric

conditions, the intrinsic density of the dry matter obtained was s = 1026.3 kg/m3, a

value in close agreement with that proposed by Devres (1991) for dry matter

(1026.0 kg/m3).

Succar et al. (1983) proposed an empirical formula for predicting the density of

tylose at freezing temperatures and atmospheric pressure. These authors obtained

their formula from semi experimental values calculated using equations (3) and (4)

together with published Mollier charts. But they did not take into account the

bounded water content of the product in their calculus. The importance of

considering the bounded water fraction is reflected in Figure 3. Crosses show the

results by Succar et al. (1983). In the other hand, the dashed curve shows the

density of tylose at atmospheric pressure calculated according to the ’additive

model under pressure’ without taking into account the bounded water content. Both

curves fitted well. But, when the bounded water is considered in the calculus (solid

line) the results are rather different, since the ice content for a given temperature is

lower and so, the density of the product higher.

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Figure 4 compares the density of tylose as a function of temperature for selected

pressures (0.1, 100 and 200 MPa) taking into account the bounded water fraction.

Solid lines show results obtained with the ‘additive model under pressure’ and

dashed curves those calculated according to the ‘shifting approach’. At

atmospheric conditions, both curves, solid and dashed, are almost overlapped,

showing the good agreement between the easy-to-use equations (22) and (23) and

the calculation routines by Otero et al. (2002) in the entire range of temperatures

shown.

At higher pressures, the dashed lines obtained by shifting the atmospheric

pressure density data according to the freezing point depression under pressure

did not render satisfactory results. Density values of tylose under pressure are

underestimated when this approach is applied since it does not take into account

the increment registered in liquid water and ice densities under pressure.

Figure 5 shows the density of tylose as a function of pressure and temperature

calculated according to the ‘additive model under pressure’. Tylose density always

increases with pressure. Also, in the thawed state, density increases when

temperature decreases. Once, the initial freezing temperature is reached, there is a

sharp decrease in density values since ice density is much lower than the

corresponding to liquid water. Thus, the lower the temperature, the higher the ice

content and therefore the lower the density.

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Specific heat

The intrinsic specific heat of the product solids fraction (Cps) has been obtained

applying equation (6) to known data of the specific heat of the unfrozen product at

atmospheric pressure (Succar et al., 1983). The obtained value was Cps = 1924.07

J/kg·K.

Solid lines in Figure 6 show the temperature dependent specific heat data

calculated for different pressures (0.1, 100 and 200 MPa) according to the ‘additive

model under pressure’. At low temperatures, the anomalous shape of the curves is

due to the well-known anomalous behavior of supercooled water (Angel, Shuppert

& Tucker, 1973; Ter Minassian et al., 1981). Dashed curves were obtained using

the ‘shifting approach’. At atmospheric conditions, solid and dashed curves fitted

well, showing the good agreement between equation (24) and (25) and the

routines by Otero et al. (2002). Nevertheless, at high pressure, the ‘shifting

approach’ overvalues the tylose Cp data since the drop in pure water specific heat

values with pressure was not taken into account.

Figure 7 shows the specific heat of tylose as a function of pressure and

temperature calculated using the ‘additive model under pressure’. Tylose specific

heat always decreases with pressure as water does (Otero et al., 2002). Also, in

the thawed state, the specific heat increases with temperature. Once, the initial

freezing temperature is reached, there is a sharp decrease in Cp values since the

specific heat of ice is much lower than the corresponding to liquid water. Figure 7

also shows the effects of the anomalous behavior of supercooled water. The high

21

values of Cpw at low pressure and temperatures produce the corresponding

anomalous ascending peak in the surface of Cp data of tylose shown. But, when

pressure increases, this anomalous behavior disappears as occurs for pure water.

Thermal conductivity

Figure 8 shows the temperature dependent conductivity data of tylose at

atmospheric conditions calculated by different approximations. The solid curve

shows the results obtained according to the ‘additive model’ using equations by

Tanishita et al. (1971) to calculate the conductivity values of liquid water. At

temperatures below 0ºC, values for liquid water were obtained by extrapolation.

Equation (14) was used to obtain the thermal conductivity of ice and the

conductivity of the solids fractions was considered as ks = 0.26 W/m·K as Devres

(1991) suggested. The dashed curve was obtained using the easy-to-use equation

(26) to calculate the thermal conductivity of liquid water (‘shifting approach’)

instead of Tanishita’s model.

Cleland et al. (1984) gave some values of tylose conductivity at atmospheric

pressure between 40ºC and –40ºC obtained from compositional factors (diamonds

in Figure 8). Also, Succar et al. (1983) developed an empirical formula for

predicting thermal conductivity of tylose from Cleland’s data. Results calculated

using this formula are shown in Figure 8 by crosses. All the data fit together rather

well except at –40ºC, where Cleland’s value is considerably lower.

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Figure 9 compares the conductivity of tylose as a function of temperature for

different pressures (0.1 MPa, 100 MPa and 200 MPa). The solid lines show results

obtained by the ‘additive model under pressure’ and the dashed curves those using

the ‘shifting approach’. In this case, results by one or the other method are much

closer than for other thermal properties giving an explanation of the good results

obtained by Denys et al. (1997) when used the ‘shifting approach’ in potato.

Figure 10 shows the conductivity of tylose as a function of pressure and

temperature calculated using the ‘additive model under pressure’. In the thawed

state, conductivity maintains rather stable and variations with temperature or

pressure are small. But, once the initial freezing temperature is reached, there is a

sharp increase in conductivity since ice conductivity is much higher than the

corresponding to liquid water. It is worth noting that, in the temperature/pressure

range studied, increasing pressure produces little variations in this property.

Apparent specific heat and enthalpy

Figure 11 shows the apparent specific heat of tylose at atmospheric conditions

calculated according to the ‘additive model’ (solid curve) and the ‘approach’

(dashed curve). Both curves are overlapped in the Figure. Diamonds represent

some values derived by Cleland et al. (1984) from experimental data (Riedel,

1960). Also, Succar et al. (1983) developed an empirical formula to predict the

apparent specific heat of tylose at atmospheric conditions from these same data.

Crosses show the results calculated using this empirical formula. Figure 11 shows

the good agreement found among all the data at atmospheric conditions.

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Figure 12 compares the apparent specific heat of tylose (Figure 12a) and the

enthalpy (Figure 12b) as functions of temperature for different pressures (0.1, 100

and 200 MPa) calculated according to the ‘additive model under pressure’ (solid

curve) and the ‘shifting approach’ (dashed curve). The agreement between both

methods is really good in the entire range of pressure studied.

Figure 13 shows the apparent specific heat (Figure 13a) and the enthalpy (Figure

13b) of tylose as functions of pressure and temperature using the ‘additive model

under pressure’. In the thawed state, the apparent specific heat maintains rather

stable being variations with the temperature or pressure small. When the initial

freezing temperature is reached the contribution of the latent heat of fusion

becomes patent as a sharp increment. The maximum height of the peak decreases

with pressure as occurs with latent heat values.

CONCLUSIONS

The ‘additive model under pressure’ and the ‘shifting approach’ presented in this

paper are useful methods to model the thermophysical properties of foods under

pressure from their composition. The ‘additive model under pressure’ employs

sophisticated calculation routines to obtain the thermophysical properties of liquid

water and ice under pressure and, therefore, gives the more accurate results. But,

the implementation of this kind of routines in a heat transfer model to simulate real

24

processes is a hard and time-consuming task. The ‘shifting approach’ offers an

easy-to-use and fast alternative to the ‘additive model under pressure’. The

polynomials employed to obtain thermal properties of liquid water and ice at

atmospheric conditions and the subsequent shifting of the data to obtain the

thermophysical properties of foods under pressure are calculation routines that can

be easily implemented and quickly solved when modeling high-pressure

processes. The ‘shifting approach’ underestimated the density and overestimated

the specific heat values of tylose under pressure. But, the results obtained for the

conductivity, the apparent specific heat and the enthalpy of tylose fitted well those

calculated according to the ‘additive model under pressure’. Taking into account

that these last properties are the really important data when modeling heat transfer

phenomena, the ‘shifting approach’ can be considered as a useful alternative to the

more tedious, but also more accurate, ‘additive model under pressure’.

ACKNOWLEDGEMENTS

This research has been carried out with financial support from the Spanish ‘Plan

Nacional de I+D+i (2004-2006) MCYT’ through the AGL 2003-06862-C02-00

project and the Commission of the European Communities, specific RTD

programme ‘Quality of Life and Management of Living Resources’, QLK1- CT-

2002- D2230 project, ‘Low temperature-pressure processing of foods: Safety and

quality aspects, process parameters and consumer acceptance’. It does not

25

necessarily reflect the Commission’s views and in no way anticipates the

Commission’s future policy in this area.

L. Otero thanks the CSIC of Spain for her contract in the context of the I3P

Program, partially supported by the European Social Fund.

26

NOMENCLATURE

Capp: Apparent specific heat (J/kg·K)

Cp: Specific heat (J/kg·K)

H: Enthalpy (J/kg)

k: Thermal conductivity (W/m·K)

L: Latent heat of fusion of pure water (J/kg)

P: Pressure (MPa)

T: Temperature (K)

t: Temperature (ºC)

X: Mass fraction

Greek

: Volume fraction

: Density (kg/m3)

Subscripts

atm: At atmospheric conditions

bw: Bound water

i: Ice

if: At the initial freezing point of the product

ref: Reference

s: Solids fraction, dry matter

27

tw: Total water

ufw: Unfrozen water

w: Water

0: At the initial freezing point of pure water

28

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36

FIGURE CAPTIONS

Figure 1: Melting curve of tylose calculated according to equation 1 ( ) or

using the more accurate equations by Wagner et al. (1994) for pure

water (---).

Figure 2: Ice content (%) in tylose as a function of pressure and temperature.

Figure 3: Density of tylose at atmospheric conditions as a function of

temperature.

Values obtained according to the ‘additive model’ taking into account

the bound water content ( ) or not (---) and using the empirical

formula by Succar and Hayakawa, 1983 (xx).

Figure 4: Density of tylose as a function of temperature for different pressures

(0.1, 100 and 200 MPa) calculated according to the ‘additive model

under pressure’ ( ) or using the ‘shifting approach’ (---).

Figure 5: Density of tylose as a function of pressure and temperature

calculated according to the ‘additive model under pressure’.

Figure 6: Specific heat of tylose as a function of temperature for different

pressures (0.1, 100 and 200 MPa) calculated according to the

37

‘additive model under pressure’ ( ) or using the ‘shifting approach’ (-

--).

Figure 7: Specific heat of tylose as a function of pressure and temperature

calculated according to the ‘additive model under pressure’.

Figure 8: Thermal conductivity of tylose at atmospheric conditions as a function

of temperature.

Values obtained according to the ‘additive model’ ( ) using

equations by Tanhishita, Nagashima and Murai (1971) or according

to the ‘approach’ (---) using equations (14) and (26).

(xx): Values obtained using the empirical formula by Succar and

Hayakawa (1983).

( ): Values from Cleland and Earle (1984).

Figure 9: Thermal conductivity of tylose as a function of temperature for

different pressures (0.1, 100 and 200 MPa) calculated according to

the ‘additive model under pressure’ ( ) or using the ‘shifting

approach’ (---).

Figure 10: Thermal conductivity of tylose as a function of pressure and

temperature calculated according to the ‘additive model under

pressure’.

38

Figure 11: Apparent specific heat of tylose at atmospheric conditions as a

function of temperature.

Values obtained according to the ‘additive model’ ( ) or using the

‘shifting approach’ (---).

(xx): Values obtained using the empirical formula by Succar and

Hayakawa, (1983).

( ): Values derived by Cleland and Earle (1984) from experimental

data (Riedel, 1960).

Figure 12: (a) Apparent specific heat and (b) enthalpy of tylose as functions of

temperature for different pressures (0.1, 100 and 200 MPa)

calculated according to the ‘additive model under pressure’ ( ) or

using the ‘shifting approach’ (---).

Figure 13: (a) Apparent specific heat and (b) enthalpy of tylose calculated as

functions of pressure and temperature according to the ‘additive

model under pressure’.

39

FIGURE 1

40

FIGURE 2

41

FIGURE 3

42

FIGURE 4

43

FIGURE 5

44

FIGURE 6

45

FIGURE 7

46

FIGURE 8

47

FIGURE 9

48

FIGURE 10

49

FIGURE 11

50

FIGURE 12

51

FIGURE 13