Water represents the major component of most food systems. During thermal or high-pressure processing, physical and chemical properties of water are changed. The p-T diagram represents an obvious presentation of isoproperty lines and their pressure and temperature dependencies. In this work, 15 different properties of pure water are shown as isoproperty lines in the pressure-temperature landscape. By using functional relationships from the "International Association for the Properties of Water and Steam” and databases from the "National Institute of Standards and Technology,” highest accuracy is guaranteed. Applying the generated graphs, a compact overview is given and a wide range of thermal and high-pressure processes can easily be compared. The different pressure and temperature dependencies of all properties showed the complexity of medium conditions during thermal and high-pressure processing. An extended understanding of pressure-temperature dependencies will improve process concepts as well as industrial applications at high temperature and high isostatic pressur

Knorr, D.

Mathys, A.

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ORIGINAL ARTICLEThe Properties of Water in the Pressure–TemperatureLandscapeA. Mathys & D. KnorrReceived: 7 August 2008 /Accepted: 11 February 2009 /Published online: 12 March 2009# Springer Science + Business Media, LLC 2009Abstract Water represents the major component of mostfood systems. During thermal or high-pressure processing,physical and chemical properties of water are changed. Thep–T diagram represents an obvious presentation of iso-property lines and their pressure and temperature depen-dencies. In this work, 15 different properties of pure waterare shown as isoproperty lines in the pressure–temperaturelandscape. By using functional relationships from the“International Association for the Properties of Water andSteam” and databases from the “National Institute ofStandards and Technology,” highest accuracy is guaranteed.Applying the generated graphs, a compact overview is givenand a wide range of thermal and high-pressure processes caneasily be compared. The different pressure and temperaturedependencies of all properties showed the complexity ofmedium conditions during thermal and high-pressure process-ing. An extended understanding of pressure–temperaturedependencies will improve process concepts as well asindustrial applications at high temperature and high isostaticpressure.Keywords Water . p–T diagram . High pressure .Thermal treatmentIntroductionWater is essential to all known forms of life. It representsthe major component of most food systems and is typicallyused as pressure transmitting medium. In food processing,the application of high isostatic pressure as emergingtechnology has steadily increased during the past 10 years,and in 2008, 125 industrial installations existed worldwidewith volumes ranging from 35 to 420 L and an annualproduction volume of more than 200,000 tons (TonelloSamson, C., NC Hyperbaric, Spain, 2008, personal com-munication). The fast-growing user community in foodtechnology needs concise overviews about process param-eters in the p–T domain. Data for the pressure andtemperature dependencies of water are available in complextables and functional relationships. This work presents thepressure and temperature dependencies of 15 fundamentalparameters as most obvious presentation form with iso-property lines in the p–T landscape. These data enable anadequate comparison of different thermal or high-pressureprocesses with chemical or biological changes in the p–Tdomain. Some biological research studies worked up to1,200–1,400 MPa,1,2 and innovative laboratory-scale equip-ment for basic studies is developed up to 1,400 MPa (byusing an extra booster system) and 140 °C.3 Hence,thermodynamic, material, and transport properties at theseextreme conditions need to be estimated.Material and MethodsExtensive data and formulations of the main thermody-namic properties of water are available from the “Interna-tional Association for the Properties of Water and Steam”(IAPWS) and in the database from the “National Institute ofFood Biophysics (2009) 4:77–82DOI 10.1007/s11483-009-9103-1A. Mathys (*) :D. KnorrDepartment of Food Biotechnology and Food Process Engineering,Berlin University of Technology,Koenigin-Luise-Str. 22,14195 Berlin, Germanye-mail: alexander.mathys@rdls.nestle.comPresent Address:A. MathysNestlé Research Centre Lausanne,Vers-Chez-Les-Blanc,1000 Lausanne 26, SwitzerlandStandards and Technology” (NIST). The functional rela-tionships for the pressure and temperature dependencies ofdensity, ρ, internal energy, u, enthalpy, h, Helmholtz freeenergy, f, Gibbs free energy, g, entropy, s, and speed ofsound, c, are accurate up to 1,000 MPa and 1,000 °C.4Material properties in the pressure and temperature land-scape can be obtained by derivation of Eqs. 1, 2, and 3.According to the first fundamental theorem of thermo-dynamics and for a constant pressure, isobaric heatcapacity, cp, is usually defined as:cp  @h@T p¼ T @s@T p: ð1ÞThe isobaric coefficient of thermal expansion can bedefined as5:ap   1v@v@T p¼  1r@r@T pð2Þwith the specific volume v.Isothermal compressibility βT is defined by Eq. 35:bT  1v@v@p T¼  1r@r@p T: ð3ÞThese properties are seen to be the three possible secondderivatives of the Gibbs free energy with respect to temper-ature and pressure.The functional relationship for the pressure and temper-ature dependencies of the relative static permittivity, ɛ, isgiven up to 1,000 MPa and 800 °C.6By using the second law of thermodynamics, which derivesa relation for enthalpy, entropy, and temperature (Eq. 1), theadiabatic–isentropic heating (Eq. 4) of a system can beobtained by combining Eq. 1 with the basic equations for thecompressibility of a system (Eq. 3) at constant pressure@T@p S¼ bTTcpr: ð4ÞThe adiabatic heating of water at different initialtemperatures was obtained by an iteration loop (MathCAD2001i Professional, MathSoft Engineering & Education,Inc., USA).By using the regression equation fromMarshall and Frank,7the negative logarithm of the ion product Kw [(mol kg−1)2] ofwater substance (Eq. 5) can be calculated with an absolutedeviation of ±0.01 units at T<200 °C and vapor pressure(max. possible relative error 0.08%) as well as within±0.03 units at pressures up to 1,000 MPa when thetemperature is below 250 °C (max. possible relative error0.33%), log10 KWð Þ ¼  log10 H3Oþ½   OH½ ð Þ ð5Þwith the oxonium [H3O+] and hydroxide [OH−] ionconcentrations.Functional relationships for the transport properties thermalconductivity, λ, and dynamic viscosity, η, are given up to400 MPa/125 °C8 and 500 MPa/150 °C8 respectively(IAPWS). Using the NIST databases 10 or 23 (NISTSTEAM or NIST REFPROP, US Department of Commerce,USA), pressure ranges up to 1,000 MPa can be obtained.In this work, data were extrapolated up to 1,400 MPa,which presents the current limit in food technology.2,3,9Using a Boolean logic as complete system for logicaloperations with the function “equal to operator”(operator =[bold] in MathCAD 2001i Professional, MathSoft Engineer-ing & Education, Inc.), it was possible to search the pressureor temperature conditions for a fixed water property value.The fixed water property value was connected to thefunctional relationship by the “equal to operator” function[e.g., fixed water property value=function(p,T)]. This proce-dure enabled the generation of isoproperty lines in thepressure and temperature landscape. The thermal conductiv-ity, λ, and dynamic viscosity, η, at higher pressure levels wasmodeled using a linear interpolation function of theisothermal pressure dependence. The specific volumetricwork, w(p,T), in pure water was calculated with Eq. 6,ZBAdw ¼ZBApvbTdp ð6Þwith the volumetric work dw from A to B.The data for the isothermal specific volumetric workwere also modeled with a linear interpolation function.Using the mentioned Boolean operation (MathCAD 2001iProfessional, MathSoft Engineering & Education, Inc.), itwas possible to search the pressure at constant temperaturefor a fixed λ, η, and w value. The presentation ofisoproperty lines in the p–T landscape was arranged bythe Origin 7SR1 software (OriginLab Corporation, USA).Results and DiscussionBridgman5,10 was the first to determine the phase diagram ofwater as function of temperature and pressure. The phasetransition lines with adiabatic lines due to compression ofwater and its different ice modifications, according to Bridg-man, are shown in Figure 1a. At present, 12 different crystalstructures plus two amorphous states are known. At thetransition from the liquid to the solid state, Ice I represents aspecialty since only this ice modification shows a positivevolume change ΔV. Lower initial temperatures have signif-icantly lower (approximately 2 K/100 MPa) temperatureincreases due to compression than higher starting temper-atures (up to 5 K/100 MPa over 80 °C). A temperature rise is78 Food Biophysics (2009) 4:77–82accompanied by a dissipation of heat within and through thepressure vessel, which is dependent on the vessel size, rate ofcompression, heat transfer parameters, as well as initial andboundary conditions.In Figure 1b, the density, ρ, of water with adiabatic lines dueto compression in the p–T landscape is shown. At high pres-sures, the compressibility decreases and there is a volume contrac-tion of 10%, 17%, or 23%at 400, 800, or 1,400MPa, respectively.Fig. 1 a Phase diagram5,10 withdifferent ice modifications and bdensity, ρ (kg m−3), with adia-batic lines due to compression(dashed line) of water in the p–TlandscapeFig. 2 Fundamental thermody-namic parameters with adiabaticlines due to compression(dashed line) of water in the p–Tlandscape with internal energy, u(a), enthalpy, h (b), Helmholtzfree energy, f (c), Gibbs freeenergy, g (d), entropy, s (e), andthe speed of sound, c (f)Food Biophysics (2009) 4:77–82 79Figure 2 presents the p–T landscape of fundamentalthermodynamic parameters of water with (a) internal energy,u, (b) enthalpy, h, (c) Helmholtz free energy, f, (d) Gibbs freeenergy, g, (e) entropy, s, and (f) the speed of sound, c. Thefour thermodynamic potentials, internal energy, enthalpy,Helmholtz free energy, and Gibbs free energy, are allpressure- and temperature-dependent (Figure 2a–d) and havethe dimensions of energy. They describe the amount ofpotential energy in a thermodynamic system when it issubjected to certain constraints. The internal energy is theenergy needed to create a system. In Figure 2a, all isothermsshow a decreasing inner energy, u, with pressure because ofvolume contraction.The enthalpy is the energy needed to create a system plusthe work needed to make room for it. Adiabatic lines inFigure 2b show a higher increase of the enthalpy, h, becauseof the heat of compression in comparison to the isotherms.All thermodynamic properties of water in this commu-nication are based on the IAPWS-95 formulation,4 which isin the form of a fundamental equation explicit in theHelmholtz free energy, f. The Helmholtz free energy is theenergy needed to create a system minus the energy you canget from the environment. Pressure is steadily increasingand temperature is steadily decreasing the Helmholtz freeenergy, f (Figure 2c).The Gibbs free energy is the total energy needed tocreate a system and the work needed to make room for itminus the energy you can get from the environment. InFigure 2d, the strong pressure and low temperaturedependence of the specific Gibbs free energy, g, is obvious.The Gibbs function of free energy, dG(T,p), is thefundamental equation for a system where pressure andtemperature are the independent variables.The difference in the free energy ΔG (T, p) is used toexplain and model the elliptical shape of transitionphenomena, e.g., protein denaturation11 or inactivation ofmicroorganisms.12The entropy is a measure of the disorder of a system. Theentropy s is increased when ice is melted because of theincrease in the disaggregation of the molecules of the body ofFig. 3 Material properties withadiabatic lines due to compres-sion (dashed line) of water inthe p–T landscape with isobaricheat capacity, cp (a), relativestatic permittivity, ɛ (b), thermalexpansion coefficient, αp (c),and isothermal compressibility,βT (d)Fig. 4 Dissociation equilibrium shift [negative logarithm of the ionproduct KW (mol kg−1)2] in pure water under different p–T conditionswith adiabatic lines due to compression (dashed line; according toMarshall and Franck7)1480 Food Biophysics (2009) 4:77–82ice. At higher temperatures up to 140 °C, the specific entropys is also steadily increased. Iso-entropy and adiabatic linesare equal (Figure 2e) because adiabatic processes arereversible. According to the second fundamental theoremof thermodynamic, ds during reversible processes is zero.The speed of sound, c, describes the velocity of thevibration that travels through an elastic medium, in thiscase water, as a wave (Figure 2f). It shows strong pressureand very low temperature dependencies (Figure 2f).Figure 3 presents the p–T landscape of materialproperties of water with (a) isobaric heat capacity, cp, (b)relative static permittivity, ɛ, (c) isobaric thermal expansioncoefficient, αp, and (d) isothermal compressibility, βT.The heat capacity, cp, shows a strong decrease whenincreasing the pressure, but features only a very small changefrom 500 MPa between 30 °C and 60 °C (Figure 3a). Hence,high-pressure processes at moderate temperatures for pas-teurization and high-pressure thermal processes at temper-atures over 100 °C for sterilization show different pressuredependencies of the heat capacity, cp.The relative static permittivity, ɛ, has importance in thecalculation of ionic reactions13 and shows noticeablechanges with pressure and temperature (Figure 3b).The thermal expansion coefficient, αp, and the coeffi-cient of compressibility, βT, are most important in thecalculation of adiabatic heating as a result of compression.βT as an intrinsic physical property of the material exhibitsa high variability in gases, liquids, and solids and shows anextensive decrease with pressure at all temperatures.The dissociation equilibrium shift in water showed alsolarge variations with pressure and temperature (Figure 4).Pure water is neutral ([H3O+] = [OH−]), and the negativelogarithm of KW has to be divided by two (according toEq. 5) to obtain the pH or pOH value. During the shift ofthe dissociation equilibrium, the pH value would bechanged with all reaction partners, but no change of theconcentration difference on one site occurs during thereaction. Under pressure, there are increased oxonium andhydroxide concentrations, but still “neutral” conditions.Consequently, the pH shift alone cannot exactly describethe dissociation equilibrium shift, and thus, the pKa shiftshould be used.14An important aspect for basic research studies underpressure is that commonly used buffer solutions showeddifferent shifts of the dissociation equilibrium. The diver-sity of pKa shifts in buffer solutions resulted mainly fromthe different dissociation reactions and, consequently,different standard molar enthalpies (ΔrH0) or reactionvolumes (ΔV).14Figure 5 shows the p–T landscape of transport propertieswith (a) thermal conductivity, λ, and (b) dynamic viscosity,η. The thermal conductivity, λ, has a strong pressuredependence and improves the heat transfer in and to theproduct during pressure build-up as well as dwell time. Atmolecular processes, the dynamic viscosity, η (Figure 5b),has an important role and shows strong temperaturedependence.On the basis of the first law of thermodynamics, aftercompression of 1 kg water up to 1,400 MPa, a maximumvolumetric work of 130 kJ kg−1 at 140 °C is performed onthe system according to Eq. 6.In Figure 6, the specific volumetric work, w, in pure water(p,T) is presented using previous results of this communica-tion with the isothermal compressibility, βT(p,T) (Figure 3d)and the specific volume v(p,T) [v(p,T)=ρ−1(p,T)] (Figure 1b).Fig. 5 Transport properties withadiabatic lines due to compres-sion (dashed line) of water inthe p–T landscape with thermalconductivity, λ (a) and dynamicviscosity, η (b)Fig. 6 Specific volumetric work w (kJ kg−1) with adiabatic lines dueto compression (dashed line) of water in the p–T landscape accordingto Eq. 6Food Biophysics (2009) 4:77–82 81The high pressure and low temperature dependence resultedfrom the functional relationships of the isothermal compress-ibility, βT(p,T), and the specific volume, v(p,T).ConclusionThe generated p–T diagrams give a concise overview aboutthe pressure and temperature dependencies of pure water.Complete different pressure and temperature dependenciesof the investigated properties can be observed. This differ-ences show the complexity of medium properties duringhigh-pressure processing with or without higher temper-atures. The modeled data are also interesting for otherthermal processes (e.g., pasteurization and sterilization byheat) or technologies (e.g., microwave or ohmic heating).Currently, there are only limited data for food or othermatrices available. Experimental investigations and theformulation of functional relationships, for example byregression methods, have to be performed for such kinds ofmedia to clarify the different behavior at process conditions.NotationG Gibbs free energy (kJ kg−1)ΔrH0 Standard molar enthalpy (kJ mol−1)IAPWS International Association for the Properties ofWater and SteamKW Ion product of water [(mol kg−1)2]NIST National Institute of Standards and TechnologyT Temperature (°C or K)ΔV Reaction volume at atmospheric pressure(cm3 mol−1)c Speed of sound (m s−1)cp Isobaric heat capacity (kJ kg−1 K−1)d Derivatef Specific Helmholtz energy (kJ kg−1)g Specific Gibbs free energy (kJ kg−1)h Specific enthalpy (kJ kg−1)p Pressure (MPa)pH Decadic logarithm of the activity ofhydroxonium ions [-]pKa Decadic logarithm of acid dissociation constant[-]pOH Decadic logarithm of the activity of hydroxideions [-]s Specific entropy (kJ kg−1 K−1)u Specific internal energy (kJ kg−1)v Specific volume (m3 kg−1)w Volumetric work (kJ kg−1)αp Isobaric expansion coefficient (K−1)βT Isothermal compressibility (MPa−1)Δ Increment or gradientɛ Relative static permittivity [-]η Dynamic viscosity (μPa s or mPa s)λ Thermal conductivity (mW m−1 K−1 or W m−1K−1)ρ Density (kg m−3)∂ Partial derivativeReferences1. W.P. Larson, T.B. Hartzell, H.S. Diehl, J. Infect. Dis. 22, 271–279(1918)2. D. Margosch, M.A. Ehrmann, R. Buckow et al., Appl. Environ.Microbiol. 72, 3476–3481 (2006). doi:10.1128/AEM.72.5.3476-3481.20063. A. Mathys, K. Reineke, V. Heinz, D. Knorr, High. Press. Res.(2009) (in press). doi:10.1080/089579508025264694. W. Wagner, A. Pruss, J. Phys. Chem. Ref. Data 31, 387–535(2002). doi:10.1063/1.14618295. P.W. Bridgman, Proc. Am Acad. Arts Sci. 47, 439–558 (1912)6. D.P. Fernandez, A.R.H. Goodwin, E.W. Lemmon et al., J. Phys.Chem. Ref. Data 26, 1125–1166 (1997)7. W.L. Marshall, E.U. Franck, J. Phys. Chem. Ref. Data 10, 295–304 (1981)8. J. Kestin, J.V. Sengers, B. Kamgarparsi, J.M.H.L. Sengers, J.Phys. Chem. Ref. Data 13, 175–183 (1984)9. A. Ardia, PhD thesis, Berlin University of Technology, 200410. P.W. Bridgman, Proc. Am. Acad. Arts Sci. 47, 441–558 (1911)11. L. Smeller, BBA-Protein Struct. M. 1595, 11–29 (2002)12. V. Heinz, D. Knorr, in Ultra high pressure treatments of foods, ed.by M.E.G. Hendrickx, D. Knorr (Kluwer, New York, 2002)13. P. Debye, E. Hueckel, Phys. Z. 24, 185–206 (1923)14. A. Mathys, R. Kallmeyer, V. Heinz, D. Knorr, Food Control 19,1165–1173 (2008). doi:10.1016/j.foodcont.2008.01.00382 Food Biophysics (2009) 4:77–82

The Properties of Water in the Pressure-Temperature Landscape

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The Properties of Water in the Pressure-Temperature Landscape

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