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Adiabatic Heat Modelling for Pressure Build-up During High-pressure Treatment in Liquid-food Processing
0960–3085/04/$30.00+0.00 # 2004 Institution of Chemical Engineers Trans IChemE, Part C, March 2004 Food and Bioproducts Processing, 82(C1): 89–95
www.ingentaselect.com=titles=09603085.htm
ADIABATIC HEAT MODELLING FOR PRESSURE BUILD-UP DURING HIGH-PRESSURE TREATMENT IN LIQUID-FOOD PROCESSING A. ARDIA, D. KNORR and V. HEINZ* Department of Food Biotechnology and Food Process Engineering, Berlin University of Technology, Berlin, Germany
P
ressure build-up is always accompanied by an increase of temperature in the product due to the transformation of energy during compression. Pure water and sucrose solutions were pressurized up to 600 MPa starting from different initial temperatures. The thermal history at the centre of the product was recorded and then related to the change in the thermo-physical properties of the product as a function of pressure, temperature and solid content. These thermo-physical properties were incorporated into a model equation and the predictions were compared with NIST (National Institute of Standards and Technology, Gaithersburg) database formulations—no significant deviations between the results and model predictions were found. Comparing the predicted results for sucrose solutions with the adiabating heating of a real product like orange juice also resulted in no significant deviations. The temperature distribution was then modelled all over the sample and used to predict the inactivation of relevant microorganisms in orange juice. Keywords: adiabatic heating; high pressure; pressure build-up.
INTRODUCTION
for isothermal situations using the functional relation V ¼ f ( p) for pure water given by the NIST formulation (IAPWS, 1996). In Figure 1 the specific work of compression Wcompr is presented for pure water up to 800 MPa. Following the first and the second laws of thermodynamics and by rearrangement of the Maxwell equations (Perry, 1984), the temperature change (i.e. heating during compression and cooling during decompression) can be described as a function of thermo-physical properties of the compressible product. According to the relevant Maxwell equation: @T @V ¼ (2) @p S @S p
High hydrostatic pressure treatment is one of the more recent technologies which represent an alternative to thermal processing for food preservation. Although no large-scale commercial process has been developed yet, there are attempts to use hydrostatic pressure for decontamination processes, since it is possible to kill bacterial cells, including spores, while retaining most of the typical characteristics of the fresh product (Bender et al., 1982). The benefits of high pressure technology can be maximized when the adiabatic heat of compression which occurs during the pressure build-up is considered, since microbial inactivation by high pressure is improved at higher temperature levels. All materials change temperature during physical compression, depending on their compressibility and specific heat (Ting et al., 2002). For water, Wcompr is the specific work of compression which is given by: ð (1) Wcompr ¼ V dp
where T, p, V, and S denote the temperature, pressure, volume and entropy, respectively. The right-hand side of equation (2) can be re-written as: @V @V @T ¼ (3) @S p @T p @S p
It is less than 55 kJ when 1 kg is compressed up to 800 MPa. This value has been obtained by integrating of equation (1)
Making use of the definition of the thermal expansion coefficient b, i.e. 1 @V b¼ (4) V @T p
*Correspondence to: Dr V. Heinz, Department of Food Biotechnology and Food Process Engineering, Berlin University of Technology, Ko¨nigin-Luiseshr.22, D-14195, Berlin, Germany. E-mail: [email protected]
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Figure 1. Work of compression for water expressed in kJ kg1, as function of applied pressure level.
where V is the specific volume, equation (3) can be modified as: @V @T b @T ¼bV ¼ (5) @S p @S p r @S p From the second law of thermodynamics, enthalpy is given by: @H @S ¼T (6) @T p @T p where the left-hand side is defined as the heat capacity at constant pressure: @H Cp ¼ (7) @T p By combining equations (5), (6) and (7), a general expression for the adiabatic temperature increase upon compression is obtained: @T b T (8) ¼ @p S r Cp Since momentum transport in liquids and solids happens practically without delay, each volume element of the product is characterized by the same pressure level and, in adiabatic situations, by the same temperature. When there is a heat flux across the boundary of the system, the transient temperature field which occurs in the product must be taken into account. The maximum temperature is reached after compression. Thermal equilibration occurs in the product when held under pressure, due to the temperature difference between the warmest point (the centre of the product) and the coldest point of the system (the metal high-pressure vessel).
Immediately after the pressure is released, the product returns to its initial temperature, or even to a lower value. The high cooling capacity is of considerable interest in the production of high quality foods (Meyer et al., 2000). A sterilizing end-temperature can be achieved through instantaneous adiabatic heating. Very high pressure levels (>800 MPa) combined with appropriate mild temperatures (<60 C) before compression can lead to the inactivation of highly resistant bacterial spores. Thus, coupling pressure and temperature can form the basis of a new approach to food sterilization with significant improvement in product quality. The aim of this work is, therefore, to develop a mathematical model which can describe the adiabatic heating occurring during pressure build-up in pure water and sucrose solutions. Starting with water, whose thermophysical properties are known, the mathematical formulation is extended to the model food systems by simple mixing rules and then applied to predict adiabatic heating in orange juice. Since reliable experimental data on the pressure dependence of thermophysical properties of foods are scarcely available (Otero and Sanz, 2003), this approach can only be considered as an empirical method to estimate the extent of adiabatic heating, given the composition of the food. In combination with numerical modelling of the transient temperature field and the p–T relation of microbial inactivation, a valuable tool for process design and process control can be created.
MATERIAL AND METHODS High-pressure Equipment The high-pressure equipment (Model U111, Unipress, Warsaw, PL) consisted of one pressure vessel, completely immersed in a thermostatic bath filled with silicon oil, and connected to a pressure intensifier through capillary tubes. This design allowed compression at close to isothermal conditions. The chamber was equipped with a K-type thermocouple and a pressure sensor to monitor the temperature and pressure history of the sample during the treatment cycle. A hydraulic pump (Mannesman Rexroth Polska Ltd, Warszawa, Poland) produced a pressure of 70 MPa in the low pressure part of the intensifier while, due to the section reduction in the intensifier (multiplying factor 11), it was possible to reach 700 MPa on the high pressure side. In all experiments, the compression rate was set constant to reach a pressure level of 600 MPa in 24 seconds. Making use of a multimeter (Keithley, Multimeter 7001) and a computer acquisition program, the pressure and temperature evolutions were recorded at a rate of 3.5 scans per second during the pressure build-up.
Pressure Treatments The samples were taken in special containers (Nunc Cryo Tubes Nr. 375299, Nunc A=S, Roskilde, DK) filled to a volume of 1.8 ml and equilibrated to the starting temperature (5 to 90 C). The temperature at the centre of the sample was measured using a pressure resistant
Trans IChemE, Part C, Food and Bioproducts Processing, 2004, 82(C1): 89–95
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shielded K-type thermocouple (1 mm diameter and 0.45 s response time, Unipress, Warsaw, Pl). Sugar solutions were prepared by mixing pure water and sucrose in 4 different weight percentages of solid content: 10%, 20%, 40% and 60%. Since Cp, b and r change with pressure and temperature, a wide range of the independent variables ( p ¼ 0–600 MPa; T ¼ 5–90 C) were investigated. Mixing Rules In order to predict adiabatic heating of sucrose solutions having different concentrations, the following mixing rules were used to estimate the thermo-physical properties needed to evaluate (@ T= @ p)S (equation (8)). Basically, the equations for estimating physical properties of mixtures of pure substances at ambient pressure were used (Lewis et al., 1987). [W ] and [S] denote the amount of water and solid, expressed in percentage, respectively: rmixture ¼
1 ([W ]=rwater ) þ ([S]=rsolid )
Cpmixture ¼ [W ] Cpwater þ [S] Cpsolid
(9) (10)
The density and specific heat of solid sucrose are reported as a function of temperature (Bubnik et al., 1995; Van der Poel et al., 1998): 1587:9 [kg=m3 ] 1 þ 0:000107 (T 15) ¼ x (1622 þ 7:125 T ) [kJ=kg C]
rsolid ¼
(11)
Cpsolid
(12)
Cpwater (T , 0:1 MPa) Cpwater (T , p)0:75
The numerical routine was written in MathCAD by implementing the NIST formulations for regressive calculation of the thermal expansion coefficient b, the density r and the specific heat Cp. Integration of equation (14) was performed by the Rhomberg method for 0.1 MPa pressure increments. Numerical Modelling
The empirical correction factor x is given by: x¼
Figure 2. Comparison between experimental and theoretical data given by NIST, for distilled water at different starting temperatures.
(13)
where Cpwater (T , 0:1 MPa) represents the specific heat of water at the actual temperature and initial pressure, while Cpwater (T , p)0:75 denotes the specific heat at the same temperature and under pressure. It is assumed that mixing rules are not affected by the pressure. In comparison with pure water, it is assumed no significant deviation in the thermal expansion coefficient for mixed solutions in response to changes in pressure. Regression Analysis In order to validate the experimental set-up, the increase in temperature of distilled water was plotted against pressure (see Figure 2), and compared with data deduced from NIST databse. The NIST formulation consists of a fundamental equation for the Helmholtz energy per unit mass (kg), as a function of temperature and density. As shown in Figure 2, no significant deviations were found between the experimental data and the theoretical results, which proves the suitability of experimental method. The main model equation (equation (14)) was then used as a function of the thermo-physical properties of water, which were themselves function of pressure and temperature. ð p1 b T dp (14) DT ¼ p0 r Cp
In addition to the heat of compression calculated by equation (14) valid for adiabatic situations, the superposition of a heat flow across the boundary has to be taken into account in real high pressure systems. The developing transient temperature field strongly depends on geometry and boundary heat transfer, and on the thermophysical and transport properties of the compressed materials. A finite difference code based on heat conduction in radial coordinates was written which includes dT=dp as a time dependent heat source or sink during the compression or decompression phase, respectively. Since the use of equation (14) requires b, r and Cp for the actual temperature and pressure, these values have to be calculated at each spatial node and for every time step of the numerical routine. It is well known that the inactivation rate of microorganisms is related to pressure and temperature. For transient situations, like those encountered in high pressure vessels, fluctuations in microbial reduction during the treatment should be anticipated. To account for this behaviour, a model for microbial inactivation has been included in the finite difference scheme, yielding the degree of inactivation at any radial position. The time dependent inactivation was mathematically described by a n-th order decay reaction: dN ¼ k N n dt upon integration N ¼ (1 þ k N0(n1) t (n 1))1=1n N0
Trans IChemE, Part C, Food and Bioproducts Processing, 2004, 82(C1): 89–95
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The influence of pressure and temperature on the rate constant k was represented by the well-known Eyring equation: Ea DV þ ln(k) ¼ p (16) RT RT where the activation volume DV* is the characteristic parameter for the pressure dependence of the rate constant; Ea is the activation energy; and R is the universal gas constant (8.314 J mol1 K1). RESULTS AND DISCUSSION In Figure 3 the comparison between the results obtained from the predictive model and the theoretical data for pure water given by NIST is shown. The comparison shows no significant deviation between the results over a wide range of starting temperatures and pressures, from 5 to 90 C and from 0.1 to 600 MPa. In Figures 4–7 experimental and predicted results obtained by applying the mathematical model for sucrose solutions (equations (9)–(14)) at different concentrations are shown, for different values of initial temperature. The predictions were extrapolated to a maximum pressure of 1400 MPa. It is evident that good continuity was found for extrapolation of the results from 600 to 1400 MPa, even though the data are not experimentally verified in the extrapolated pressure range. No significant deviations were detected in the range from 0.1 to 600 MPa. Hence, the mathematical model was applied to predict the thermal behaviour for a real product. Orange juice (Hohes C, Eckes-Granini, Deutschland GmbH, pH 3.8) with a sugar content of 9% was pressurized up to 600 MPa starting from different initial temperatures. The model was then applied, simulating the adiabating heat of compression in orange juice equivalent to sucrose solution of 9% solid content.
Figure 3. Comparison between theoretical results given by NIST and results obtained from the predictive model for water for different starting temperatures.
Figure 4. Comparison between experimental results and model predictions for sugar solutions at 10% of solid content at different starting temperatures. The model extrapolated up to 1400 MPa was not experimentally verified.
In Figure 8 the comparison between the experimental and predicted results is shown over a range of pressures between 0.1 and 600 MPa and initial temperatures between 5 and 90 C. No significant deviations are evident. To demonstrate the effect of transient temperature conditions, two different microorganisms have been chosen. Previous experimental studies on microbial inactivation showed that reaction orders n ¼ 1 and n ¼ 1.1, well reproduced the inactivation kinetics of Lactobacillus rhamnosus and Alicyclobacillus acidoterrestris, respectively (Ardia et al., 2003).
Figure 5. Comparison between experimental results and model predictions for sugar solutions at 20% of solid content at different starting temperatures. The modelled data extrapolated up to 1400 MPa was not experimentally verified.
Trans IChemE, Part C, Food and Bioproducts Processing, 2004, 82(C1): 89–95
ADIABATIC HEAT MODELLING FOR PRESSURE BUILD-UP
Figure 6. Comparison between experimental results and model predictions for sugar solutions at 40% of solid content at different starting temperatures. The model extrapolated up to 1400 MPa was not experimentally verified.
A PET container having a diameter of 80 mm, a length of 100 mm and a thickness of 0.5 mm was chosen for the calculations. Pure water was assumed as the pressure transmitting medium. Figure 10 shows the prediction of adiabatic heating and the inactivation kinetics of Lactobacillus rhamnosus (Figure 10a) and Alicyclobacillus acidoterrestris spores (Figure 10b) in orange juice, at two different locations in the sample (see Figure 9). The simulation of Lactobacillus rhamnosus was performed for pressure ranging between 0.1 and 600 MPa
Figure 7. Comparison between experimental results and model predictions for sugar solutions at 60% of solid content at different starting temperatures. The modelled data extrapolated up to 1400 MPa was not experimentally verified.
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Figure 8. Comparison between experimental results for orange juice and results obtained from the predictive model (continuous line) for 9% sucrose at different starting temperatures.
starting at room temperature (20 C), while for Alicyclobacillus acidoterrestris spores at a higher operative pressure of 800 MPa, and a higher starting temperature of 50 C, were chosen because of its stronger pressure and temperature resistance.
Figure 9. Schematic view of the geometry used for numerical simulation indicating the three points where adiabatic heating has been predicted: A) centre of the sample; B) inner layer of the sample container; C) sample container (PET).
Trans IChemE, Part C, Food and Bioproducts Processing, 2004, 82(C1): 89–95
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Figure 10. Simulation of adiabating heat and inactivation kinetics of Lactobacillus rhamnosus (a) and Alicyclobacillus acidoterrestris spores (b) at the critical points of the sample: the centre (A), the inner layer of the product sample (B) and the PET container (C).
Simulating a pressure treatment on orange juice up to 600 MPa starting from room temperature, a difference of 2 C between the centre (A) and the bulk of the sample (B) was observed, which predicted a difference of almost 1 Log-cycle in the inactivation of Lactobacillus rhamnosus. Similar results were detected for Alicyclobacillus acidoterrestris. A pressure treatment up to 800 MPa starting from an initial temperature of 50 C produced a temperature increase of almost 30–32 C at the centre of the juice. Adiabatic heating of the product at inner side-wall of the PET bottle was characterized by lower temperatures compared to the centre, because of the heat flux to the vessel which was assumed to maintain a temperature of 50 C. A difference of 3–4 C between the two crucial points resulted in a strong difference in the spores inactivation:
soon after pressure release, a difference of approximately 6 Log-cycles between the centre and the inner side-wall of the bottle was predicted. CONCLUSIONS Simple mixing rules considerations have been used to model the temperature increase which occurs in the product during pressure build-up. It was possible to predict temperature evolution in sucrose solutions at different concentrations as the first step, before simulating adiabatic heating and temperature distribution in real food products. By the use of simple mixing rules it has been shown that the NIST formulation for b, r and Cp of pure water can be used to accurately predict the temperature increase in
Trans IChemE, Part C, Food and Bioproducts Processing, 2004, 82(C1): 89–95
ADIABATIC HEAT MODELLING FOR PRESSURE BUILD-UP response to pressure levels up to 600 MPa. Since the NIST formulation provides data for pure water up to much higher pressures, it is suggested that the model equations can be extrapolated up to 1400 MPa, which is still a relevant pressure level for food processing. The model can be used to optimize pressure and temperature processing conditions needed for pasteurization and sterilization processes. Knowledge about the end temperature as a function of the processing pressure will avoid thermal over-processing and yet result in safer products with better quality. Of course, the temperature gradient between the critical points of the system has to be taken into account. As shown in Figure10b, the strong difference in microbial inactivation between the centre and the boundary of the product is related to the temperature gradient developed during adiabatic heating. Several options can be considered to minimize this effect: use of an insulating layer along the inner wall of the vessel; heating the vessel to an appropriate temperature which minimizes the gradient; use of a pressure transmitting medium with higher thermal conductivity in combination with mild heating of the vessel. Once the temperature gradient is controlled, the pressure and temperature conditions can be optimised in order to achieve the desired inactivation levels and guarantee the safety margins that are required for pasteurized and sterilized products.
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REFERENCES Ardia, A., Heinz, V. and Knorr, D., 2003, Kinetic studies on combined high-pressure and temperature on the inactivation of Alicyclobacillus acidoterrestris spores in orange juice, App Biotech Food Science and Policy, (in press). Bender, G.R. and Marquis, R.E., 1982, Sensitivity of various salt forms of bacillus megaterium spores to the germinating action of hydrostatic pressure, J Microbiol, 28: 643–649. Bubnik, Z., Kadlec, P., Urban, D. and Bruhns, M., 1995, Sugar Technologists Manual, Bartens, A. (ed) (KG, Berlin, Germany), p 116. Lewis, M.J., 1987, Physical Properties of Foods and Food Processing Systems (Ellis Horwood Ltd., Chichester, England), p 54, 222. Meyer, R.S., Cooper, K.L., Knorr, D., Lelieveld, H.L.M., 2000, High-pressure sterilization of foods, Foodtechnology, 54(11): 67–72. NIST (National Institute of Standards and Technology) database, U.S. Commerce Department’s Technology Administration. Otero, L. and Sanz, P.D., 2003, Modelling heat transfer in high pressure food processing: A review. Innovative Food Sci and Emerging Technol, 4: 121–134. Perry, R.H., 1984, Perry’s Chemical Engineers’ Handbook, 6th edition (McGraw-Hill Book Co), p 54. Ting, E., Balasubramanian, V.M. and Taghubeer, E., 2002, Determining thermal effects in high-pressure processing. Food Technology Feature, 56(2): 31–35. Van der Poel, P.W., Schiweck, H. and Schwartz, T., 1998, Zuckertechnologie Bartens, A. (ed) (KG, Berlin, Germany), p 88. The manuscript was received 24 June 2003 and accepted for publication after revision 11 February 2004.
Trans IChemE, Part C, Food and Bioproducts Processing, 2004, 82(C1): 89–95
www.ingentaselect.com=titles=09603085.htm
ADIABATIC HEAT MODELLING FOR PRESSURE BUILD-UP DURING HIGH-PRESSURE TREATMENT IN LIQUID-FOOD PROCESSING A. ARDIA, D. KNORR and V. HEINZ* Department of Food Biotechnology and Food Process Engineering, Berlin University of Technology, Berlin, Germany
P
ressure build-up is always accompanied by an increase of temperature in the product due to the transformation of energy during compression. Pure water and sucrose solutions were pressurized up to 600 MPa starting from different initial temperatures. The thermal history at the centre of the product was recorded and then related to the change in the thermo-physical properties of the product as a function of pressure, temperature and solid content. These thermo-physical properties were incorporated into a model equation and the predictions were compared with NIST (National Institute of Standards and Technology, Gaithersburg) database formulations—no significant deviations between the results and model predictions were found. Comparing the predicted results for sucrose solutions with the adiabating heating of a real product like orange juice also resulted in no significant deviations. The temperature distribution was then modelled all over the sample and used to predict the inactivation of relevant microorganisms in orange juice. Keywords: adiabatic heating; high pressure; pressure build-up.
INTRODUCTION
for isothermal situations using the functional relation V ¼ f ( p) for pure water given by the NIST formulation (IAPWS, 1996). In Figure 1 the specific work of compression Wcompr is presented for pure water up to 800 MPa. Following the first and the second laws of thermodynamics and by rearrangement of the Maxwell equations (Perry, 1984), the temperature change (i.e. heating during compression and cooling during decompression) can be described as a function of thermo-physical properties of the compressible product. According to the relevant Maxwell equation: @T @V ¼ (2) @p S @S p
High hydrostatic pressure treatment is one of the more recent technologies which represent an alternative to thermal processing for food preservation. Although no large-scale commercial process has been developed yet, there are attempts to use hydrostatic pressure for decontamination processes, since it is possible to kill bacterial cells, including spores, while retaining most of the typical characteristics of the fresh product (Bender et al., 1982). The benefits of high pressure technology can be maximized when the adiabatic heat of compression which occurs during the pressure build-up is considered, since microbial inactivation by high pressure is improved at higher temperature levels. All materials change temperature during physical compression, depending on their compressibility and specific heat (Ting et al., 2002). For water, Wcompr is the specific work of compression which is given by: ð (1) Wcompr ¼ V dp
where T, p, V, and S denote the temperature, pressure, volume and entropy, respectively. The right-hand side of equation (2) can be re-written as: @V @V @T ¼ (3) @S p @T p @S p
It is less than 55 kJ when 1 kg is compressed up to 800 MPa. This value has been obtained by integrating of equation (1)
Making use of the definition of the thermal expansion coefficient b, i.e. 1 @V b¼ (4) V @T p
*Correspondence to: Dr V. Heinz, Department of Food Biotechnology and Food Process Engineering, Berlin University of Technology, Ko¨nigin-Luiseshr.22, D-14195, Berlin, Germany. E-mail: [email protected]
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Figure 1. Work of compression for water expressed in kJ kg1, as function of applied pressure level.
where V is the specific volume, equation (3) can be modified as: @V @T b @T ¼bV ¼ (5) @S p @S p r @S p From the second law of thermodynamics, enthalpy is given by: @H @S ¼T (6) @T p @T p where the left-hand side is defined as the heat capacity at constant pressure: @H Cp ¼ (7) @T p By combining equations (5), (6) and (7), a general expression for the adiabatic temperature increase upon compression is obtained: @T b T (8) ¼ @p S r Cp Since momentum transport in liquids and solids happens practically without delay, each volume element of the product is characterized by the same pressure level and, in adiabatic situations, by the same temperature. When there is a heat flux across the boundary of the system, the transient temperature field which occurs in the product must be taken into account. The maximum temperature is reached after compression. Thermal equilibration occurs in the product when held under pressure, due to the temperature difference between the warmest point (the centre of the product) and the coldest point of the system (the metal high-pressure vessel).
Immediately after the pressure is released, the product returns to its initial temperature, or even to a lower value. The high cooling capacity is of considerable interest in the production of high quality foods (Meyer et al., 2000). A sterilizing end-temperature can be achieved through instantaneous adiabatic heating. Very high pressure levels (>800 MPa) combined with appropriate mild temperatures (<60 C) before compression can lead to the inactivation of highly resistant bacterial spores. Thus, coupling pressure and temperature can form the basis of a new approach to food sterilization with significant improvement in product quality. The aim of this work is, therefore, to develop a mathematical model which can describe the adiabatic heating occurring during pressure build-up in pure water and sucrose solutions. Starting with water, whose thermophysical properties are known, the mathematical formulation is extended to the model food systems by simple mixing rules and then applied to predict adiabatic heating in orange juice. Since reliable experimental data on the pressure dependence of thermophysical properties of foods are scarcely available (Otero and Sanz, 2003), this approach can only be considered as an empirical method to estimate the extent of adiabatic heating, given the composition of the food. In combination with numerical modelling of the transient temperature field and the p–T relation of microbial inactivation, a valuable tool for process design and process control can be created.
MATERIAL AND METHODS High-pressure Equipment The high-pressure equipment (Model U111, Unipress, Warsaw, PL) consisted of one pressure vessel, completely immersed in a thermostatic bath filled with silicon oil, and connected to a pressure intensifier through capillary tubes. This design allowed compression at close to isothermal conditions. The chamber was equipped with a K-type thermocouple and a pressure sensor to monitor the temperature and pressure history of the sample during the treatment cycle. A hydraulic pump (Mannesman Rexroth Polska Ltd, Warszawa, Poland) produced a pressure of 70 MPa in the low pressure part of the intensifier while, due to the section reduction in the intensifier (multiplying factor 11), it was possible to reach 700 MPa on the high pressure side. In all experiments, the compression rate was set constant to reach a pressure level of 600 MPa in 24 seconds. Making use of a multimeter (Keithley, Multimeter 7001) and a computer acquisition program, the pressure and temperature evolutions were recorded at a rate of 3.5 scans per second during the pressure build-up.
Pressure Treatments The samples were taken in special containers (Nunc Cryo Tubes Nr. 375299, Nunc A=S, Roskilde, DK) filled to a volume of 1.8 ml and equilibrated to the starting temperature (5 to 90 C). The temperature at the centre of the sample was measured using a pressure resistant
Trans IChemE, Part C, Food and Bioproducts Processing, 2004, 82(C1): 89–95
ADIABATIC HEAT MODELLING FOR PRESSURE BUILD-UP
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shielded K-type thermocouple (1 mm diameter and 0.45 s response time, Unipress, Warsaw, Pl). Sugar solutions were prepared by mixing pure water and sucrose in 4 different weight percentages of solid content: 10%, 20%, 40% and 60%. Since Cp, b and r change with pressure and temperature, a wide range of the independent variables ( p ¼ 0–600 MPa; T ¼ 5–90 C) were investigated. Mixing Rules In order to predict adiabatic heating of sucrose solutions having different concentrations, the following mixing rules were used to estimate the thermo-physical properties needed to evaluate (@ T= @ p)S (equation (8)). Basically, the equations for estimating physical properties of mixtures of pure substances at ambient pressure were used (Lewis et al., 1987). [W ] and [S] denote the amount of water and solid, expressed in percentage, respectively: rmixture ¼
1 ([W ]=rwater ) þ ([S]=rsolid )
Cpmixture ¼ [W ] Cpwater þ [S] Cpsolid
(9) (10)
The density and specific heat of solid sucrose are reported as a function of temperature (Bubnik et al., 1995; Van der Poel et al., 1998): 1587:9 [kg=m3 ] 1 þ 0:000107 (T 15) ¼ x (1622 þ 7:125 T ) [kJ=kg C]
rsolid ¼
(11)
Cpsolid
(12)
Cpwater (T , 0:1 MPa) Cpwater (T , p)0:75
The numerical routine was written in MathCAD by implementing the NIST formulations for regressive calculation of the thermal expansion coefficient b, the density r and the specific heat Cp. Integration of equation (14) was performed by the Rhomberg method for 0.1 MPa pressure increments. Numerical Modelling
The empirical correction factor x is given by: x¼
Figure 2. Comparison between experimental and theoretical data given by NIST, for distilled water at different starting temperatures.
(13)
where Cpwater (T , 0:1 MPa) represents the specific heat of water at the actual temperature and initial pressure, while Cpwater (T , p)0:75 denotes the specific heat at the same temperature and under pressure. It is assumed that mixing rules are not affected by the pressure. In comparison with pure water, it is assumed no significant deviation in the thermal expansion coefficient for mixed solutions in response to changes in pressure. Regression Analysis In order to validate the experimental set-up, the increase in temperature of distilled water was plotted against pressure (see Figure 2), and compared with data deduced from NIST databse. The NIST formulation consists of a fundamental equation for the Helmholtz energy per unit mass (kg), as a function of temperature and density. As shown in Figure 2, no significant deviations were found between the experimental data and the theoretical results, which proves the suitability of experimental method. The main model equation (equation (14)) was then used as a function of the thermo-physical properties of water, which were themselves function of pressure and temperature. ð p1 b T dp (14) DT ¼ p0 r Cp
In addition to the heat of compression calculated by equation (14) valid for adiabatic situations, the superposition of a heat flow across the boundary has to be taken into account in real high pressure systems. The developing transient temperature field strongly depends on geometry and boundary heat transfer, and on the thermophysical and transport properties of the compressed materials. A finite difference code based on heat conduction in radial coordinates was written which includes dT=dp as a time dependent heat source or sink during the compression or decompression phase, respectively. Since the use of equation (14) requires b, r and Cp for the actual temperature and pressure, these values have to be calculated at each spatial node and for every time step of the numerical routine. It is well known that the inactivation rate of microorganisms is related to pressure and temperature. For transient situations, like those encountered in high pressure vessels, fluctuations in microbial reduction during the treatment should be anticipated. To account for this behaviour, a model for microbial inactivation has been included in the finite difference scheme, yielding the degree of inactivation at any radial position. The time dependent inactivation was mathematically described by a n-th order decay reaction: dN ¼ k N n dt upon integration N ¼ (1 þ k N0(n1) t (n 1))1=1n N0
Trans IChemE, Part C, Food and Bioproducts Processing, 2004, 82(C1): 89–95
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The influence of pressure and temperature on the rate constant k was represented by the well-known Eyring equation: Ea DV þ ln(k) ¼ p (16) RT RT where the activation volume DV* is the characteristic parameter for the pressure dependence of the rate constant; Ea is the activation energy; and R is the universal gas constant (8.314 J mol1 K1). RESULTS AND DISCUSSION In Figure 3 the comparison between the results obtained from the predictive model and the theoretical data for pure water given by NIST is shown. The comparison shows no significant deviation between the results over a wide range of starting temperatures and pressures, from 5 to 90 C and from 0.1 to 600 MPa. In Figures 4–7 experimental and predicted results obtained by applying the mathematical model for sucrose solutions (equations (9)–(14)) at different concentrations are shown, for different values of initial temperature. The predictions were extrapolated to a maximum pressure of 1400 MPa. It is evident that good continuity was found for extrapolation of the results from 600 to 1400 MPa, even though the data are not experimentally verified in the extrapolated pressure range. No significant deviations were detected in the range from 0.1 to 600 MPa. Hence, the mathematical model was applied to predict the thermal behaviour for a real product. Orange juice (Hohes C, Eckes-Granini, Deutschland GmbH, pH 3.8) with a sugar content of 9% was pressurized up to 600 MPa starting from different initial temperatures. The model was then applied, simulating the adiabating heat of compression in orange juice equivalent to sucrose solution of 9% solid content.
Figure 3. Comparison between theoretical results given by NIST and results obtained from the predictive model for water for different starting temperatures.
Figure 4. Comparison between experimental results and model predictions for sugar solutions at 10% of solid content at different starting temperatures. The model extrapolated up to 1400 MPa was not experimentally verified.
In Figure 8 the comparison between the experimental and predicted results is shown over a range of pressures between 0.1 and 600 MPa and initial temperatures between 5 and 90 C. No significant deviations are evident. To demonstrate the effect of transient temperature conditions, two different microorganisms have been chosen. Previous experimental studies on microbial inactivation showed that reaction orders n ¼ 1 and n ¼ 1.1, well reproduced the inactivation kinetics of Lactobacillus rhamnosus and Alicyclobacillus acidoterrestris, respectively (Ardia et al., 2003).
Figure 5. Comparison between experimental results and model predictions for sugar solutions at 20% of solid content at different starting temperatures. The modelled data extrapolated up to 1400 MPa was not experimentally verified.
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ADIABATIC HEAT MODELLING FOR PRESSURE BUILD-UP
Figure 6. Comparison between experimental results and model predictions for sugar solutions at 40% of solid content at different starting temperatures. The model extrapolated up to 1400 MPa was not experimentally verified.
A PET container having a diameter of 80 mm, a length of 100 mm and a thickness of 0.5 mm was chosen for the calculations. Pure water was assumed as the pressure transmitting medium. Figure 10 shows the prediction of adiabatic heating and the inactivation kinetics of Lactobacillus rhamnosus (Figure 10a) and Alicyclobacillus acidoterrestris spores (Figure 10b) in orange juice, at two different locations in the sample (see Figure 9). The simulation of Lactobacillus rhamnosus was performed for pressure ranging between 0.1 and 600 MPa
Figure 7. Comparison between experimental results and model predictions for sugar solutions at 60% of solid content at different starting temperatures. The modelled data extrapolated up to 1400 MPa was not experimentally verified.
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Figure 8. Comparison between experimental results for orange juice and results obtained from the predictive model (continuous line) for 9% sucrose at different starting temperatures.
starting at room temperature (20 C), while for Alicyclobacillus acidoterrestris spores at a higher operative pressure of 800 MPa, and a higher starting temperature of 50 C, were chosen because of its stronger pressure and temperature resistance.
Figure 9. Schematic view of the geometry used for numerical simulation indicating the three points where adiabatic heating has been predicted: A) centre of the sample; B) inner layer of the sample container; C) sample container (PET).
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ARDIA et al.
Figure 10. Simulation of adiabating heat and inactivation kinetics of Lactobacillus rhamnosus (a) and Alicyclobacillus acidoterrestris spores (b) at the critical points of the sample: the centre (A), the inner layer of the product sample (B) and the PET container (C).
Simulating a pressure treatment on orange juice up to 600 MPa starting from room temperature, a difference of 2 C between the centre (A) and the bulk of the sample (B) was observed, which predicted a difference of almost 1 Log-cycle in the inactivation of Lactobacillus rhamnosus. Similar results were detected for Alicyclobacillus acidoterrestris. A pressure treatment up to 800 MPa starting from an initial temperature of 50 C produced a temperature increase of almost 30–32 C at the centre of the juice. Adiabatic heating of the product at inner side-wall of the PET bottle was characterized by lower temperatures compared to the centre, because of the heat flux to the vessel which was assumed to maintain a temperature of 50 C. A difference of 3–4 C between the two crucial points resulted in a strong difference in the spores inactivation:
soon after pressure release, a difference of approximately 6 Log-cycles between the centre and the inner side-wall of the bottle was predicted. CONCLUSIONS Simple mixing rules considerations have been used to model the temperature increase which occurs in the product during pressure build-up. It was possible to predict temperature evolution in sucrose solutions at different concentrations as the first step, before simulating adiabatic heating and temperature distribution in real food products. By the use of simple mixing rules it has been shown that the NIST formulation for b, r and Cp of pure water can be used to accurately predict the temperature increase in
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ADIABATIC HEAT MODELLING FOR PRESSURE BUILD-UP response to pressure levels up to 600 MPa. Since the NIST formulation provides data for pure water up to much higher pressures, it is suggested that the model equations can be extrapolated up to 1400 MPa, which is still a relevant pressure level for food processing. The model can be used to optimize pressure and temperature processing conditions needed for pasteurization and sterilization processes. Knowledge about the end temperature as a function of the processing pressure will avoid thermal over-processing and yet result in safer products with better quality. Of course, the temperature gradient between the critical points of the system has to be taken into account. As shown in Figure10b, the strong difference in microbial inactivation between the centre and the boundary of the product is related to the temperature gradient developed during adiabatic heating. Several options can be considered to minimize this effect: use of an insulating layer along the inner wall of the vessel; heating the vessel to an appropriate temperature which minimizes the gradient; use of a pressure transmitting medium with higher thermal conductivity in combination with mild heating of the vessel. Once the temperature gradient is controlled, the pressure and temperature conditions can be optimised in order to achieve the desired inactivation levels and guarantee the safety margins that are required for pasteurized and sterilized products.
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REFERENCES Ardia, A., Heinz, V. and Knorr, D., 2003, Kinetic studies on combined high-pressure and temperature on the inactivation of Alicyclobacillus acidoterrestris spores in orange juice, App Biotech Food Science and Policy, (in press). Bender, G.R. and Marquis, R.E., 1982, Sensitivity of various salt forms of bacillus megaterium spores to the germinating action of hydrostatic pressure, J Microbiol, 28: 643–649. Bubnik, Z., Kadlec, P., Urban, D. and Bruhns, M., 1995, Sugar Technologists Manual, Bartens, A. (ed) (KG, Berlin, Germany), p 116. Lewis, M.J., 1987, Physical Properties of Foods and Food Processing Systems (Ellis Horwood Ltd., Chichester, England), p 54, 222. Meyer, R.S., Cooper, K.L., Knorr, D., Lelieveld, H.L.M., 2000, High-pressure sterilization of foods, Foodtechnology, 54(11): 67–72. NIST (National Institute of Standards and Technology) database, U.S. Commerce Department’s Technology Administration. Otero, L. and Sanz, P.D., 2003, Modelling heat transfer in high pressure food processing: A review. Innovative Food Sci and Emerging Technol, 4: 121–134. Perry, R.H., 1984, Perry’s Chemical Engineers’ Handbook, 6th edition (McGraw-Hill Book Co), p 54. Ting, E., Balasubramanian, V.M. and Taghubeer, E., 2002, Determining thermal effects in high-pressure processing. Food Technology Feature, 56(2): 31–35. Van der Poel, P.W., Schiweck, H. and Schwartz, T., 1998, Zuckertechnologie Bartens, A. (ed) (KG, Berlin, Germany), p 88. The manuscript was received 24 June 2003 and accepted for publication after revision 11 February 2004.
Trans IChemE, Part C, Food and Bioproducts Processing, 2004, 82(C1): 89–95