- Email: [email protected]
A method to characterise heat transfer during high-pressure processing
Journal of Food Engineering 60 (2003) 131–135 www.elsevier.com/locate/jfoodeng
A method to characterise heat transfer during high-pressure processing T. Carroll *, P. Chen, A. Fletcher Fonterra Research Centre, Private Bag 11029, Dairy Farm Road, Palmerston North, New Zealand Received 6 August 2002; accepted 5 January 2003
Abstract An analytical solution to the model for conductive heat transfer in high-pressure processing of foods is developed. This solution can be used to extract thermal diffusivity, thermal expansivity and heat-transfer coefficient from an experimental cooling curve obtained during the hold at pressure of a high-pressure process cycle. The extracted heat transfer parameters can then be used to predict the temperature profiles in the process-vessel chamber. These profiles are the key to understanding (where appropriate) the contribution of thermal effects in high-pressure processing, for example in combined temperature–pressure processes. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: High pressure processing; Heat transfer; Adiabatic compression
1. Introduction The compression that occurs during high-pressure processing of food generates heat that increases the temperature within the pressure-vessel chamber. This temperature rise may complicate the interpretation of pressure-induced phenomena such as protein denaturation or microbial inactivation. (Ting, Balasubramaniam, & Raghubeer, 2002). In particular, if the pressure-vessel chamber is not maintained at a temperature below which thermal effects can be excluded, then combined thermal-pressure mechanisms must be considered in any interpretations. Thermal effects are a significant complication, as unlike pressure, temperature is not typically uniform across a vessel-pressure chamber. The combined pressure–temperature experienced by a given sample volume element varies with location within the chamber, the size of the chamber and the size of the sample (Carroll et al., submitted for publication). The effects of compressive heating on high-pressure processes are now being considered (Ting et al., 2002; Denys, Ludikhuyze, Van Loey, & Hendrickx, 2000; Otero, Molina-Garcia, & Sanz, 2000). Although the temperature is typically measured at one location within
*
Corresponding author. Fax: +64-63504663. E-mail address: [email protected] (T. Carroll).
0260-8774/03/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0260-8774(03)00026-8
the chamber, temperature varies both temporally and spatially during a pressure cycle. It follows that the temperature within a pressure vessel should be known at all locations, and at all times during a high-pressure process if thermal effects are to be captured. In essence, the heat-transfer coefficient of the pressure vessel and the thermal properties of the pressurising fluid must be determined. A meaningful comparison of results across various research groups using different vessels and pressurising media is not possible if heat-transfer parameters are not reported along with the typical basic parameters such as pressure and hold-time. This is particularly important for the development of combined pressure–heat processes for sterilisation. Furthermore, it is neither possible to develop combined pressure–temperature models of observed behaviour, nor to predict process performance at alternative scales (e.g. as part of process development) without first determining heattransfer parameters. A numerical method has been developed to model the temperature profiles caused by conductive heat transfer in high-pressure processes (Denys et al., 2000). In this method, the governing heat-transfer equation is solved numerically over a finite element mesh. Non-uniform element-based physical properties of the fluid can be included, where these are known. Such a numerical method is useful to predict the temperature profiles if the thermal properties of the fluid are known. However, the
132
T. Carroll et al. / Journal of Food Engineering 60 (2003) 131–135
method is less useful in reverse, namely to extract the thermal properties if the temperature is known. In this paper we develop an analytical solution for the temperature profile within a pressure-vessel chamber during a high-pressure process. This analytical solution is more general, more straightforward, and easier to apply than the available numerical solution. The analytical solution can be used to extract the thermal expansivity and thermal diffusivity of any pressuring fluid, and the heat-transfer coefficient for any vessel from an experimentally determined cooling curve obtained during a pressure-hold. These extracted heat-transfer parameters can then be used with the analytical solution to predict the temperature at all points within the chamber at all times during a high-pressure process. We have written software to perform this extraction and prediction from the analytical solution and we demonstrate the software by example.
2. Model of heat transfer in high-pressure processing A high-pressure food treatment process typically begins with a pressurisation step during which the pressure chamber is pumped up to the pressure set-point, followed by a hold at the pressure set-point for a predetermined time, and concludes with a depressurisation step (i.e. pressure is released). In theory the heat generated by compression is dissipated by a combination of conduction and convection within the pressurising fluid in the chamber and transfer across the chamber wall into the surroundings. Depending on whether an external intensifier or plunger press design is used, either a small volume of fluid is forced into the chamber, or a piston is displaced, to compensate for the loss of volume on compression during the pressurisation step. During the hold at pressure, the pressurising fluid and chamber volume are typically undisturbed. As a first approach, convection within the pressurising fluid is not considered, (and the resultant error is evaluated for a laboratory plunger press system). The temperature of a pressurising fluid in the chamber of a pressure vessel during such a high-pressure process is then given by the conductive heat-transfer equation r2 T
qcp oT T a oP ¼ K ot K ot
ð1Þ
subject to K^ n~ DT ¼ U ðT Tj Þ at the wall and T ¼ Tj everywhere initially (^ n is the unit normal vector). T and P are chamber temperature and pressure, respectively. K (W/m K), q (kg/m3 ), cp (J/kg K) and a (1/K) are the thermal conductivity, density, heat capacity and thermal expansion coefficient of the pressurising fluid, respectively. U (W/m2 K) is the heat-transfer coefficient from
the chamber inner wall to the jacket (referenced to the inner area) and Tj is the jacket temperature. In particular, during the hold at the pressure setpoint, oP =ot ¼ 0. In the case of a long cylindrical chamber (aspect-ratio < 1=3), the temperature profile is given by the solution to Eq. (1) as T ðr; tÞ Tj ¼ 2
1 X
e
N ¼1
Z
4jb2 Nt d2
b2N
b2N J0 ðbN rÞ 2 þ L J02 ðbN Þ
1
rf ðrÞJ0 ðbN rÞ dr
ð2Þ
0
where bN are roots of bN J1 ðbN Þ ¼ LJ0 ðbN Þ (adapted from Carslaw & Jaeger, 1959). j ¼ K=qcp (m2 /s) is the thermal diffusivity, L ¼ Ud=2K is the extra-chamber dimensionless heat-transfer coefficient (d (m) is the chamber diameter), and J0 , J1 are zero-order and firstorder cylindrical Bessel functions, respectively. f ðrÞ is the initial temperature profile during the hold. This general solution describes the cooling curve of a fluid in an infinitely long cylindrical container allowed to cool by a combination of conduction and convective interfacial heat-transfer. The temperature profile during the pressurisation is given by an analytical solution to Eq. (1) for the case where the pressurising fluid is heated at a constant rate. This case corresponds to a pressurisation at a constant rate over a pressure range across which physical properties are constant. The temperature profile is then given by 2 2T a oP d 2 4 1 r2 1 T ðr; tÞ Tj ¼ þ K ot 4 4L 8 3 1 4jb2 X Nt 1 J ðb rÞ 0 N 5 ð3Þ e d2 b2N 3 J1 ðbN Þ b 1þ N ¼1 N
L2
(adapted from Carslaw & Jaeger, 1959). Note that Eq. (3) also describes the temperature during the depressurisation if the temperature at the end of the hold is uniform. In particular, the temperature profiles at the end of pressurisation and at the beginning of the hold are the same. From Eq. (3) " T a oP d 2 2 f ðrÞ ¼ 1 r2 þ K ot 16 L # 1 4jb2 X Ns 1 J ðb rÞ 0 N 8L e d2 ð4Þ b2N ðb2N þ L2 Þ J0 ðbN Þ N ¼1 where s is come-up time to the pressure set-point (adapted from Carslaw & Jaeger, 1959). The temperature profile during the hold is then obtained by substituting Eq. (4) into Eq. (2) and inte-
T. Carroll et al. / Journal of Food Engineering 60 (2003) 131–135
grating. The radial integrals of the appropriate Bessel functions are given by Z 1 J1 ðbN Þ rJ0 ðbN rÞ dr ¼ ð5aÞ bN 0 Z 1 rð1 r2 ÞJ0 ðbN rÞ dr 0
¼ Z
2J2 ðbN Þ 4J1 ðbN Þ 2J0 ðbN Þ ¼ b2N b3N b2N
ð5bÞ
1
rJ0 ðaN rÞJ0 ðbN rÞ dr
0
¼
J12 ðbN Þ þ J02 ðbN Þ 2
if aN ¼ bN ; zero otherwise
ð5cÞ using integration by parts and Bessel function integral formulae (Abramovitz & Stegun, 1972). The solution is a cooling curve during the hold at the pressure set-point (t þ s > s) given by
4jb2 Ns 2 d 1 e 1 4jb2 Nt 2T a oP d 2 X J ðb rÞ 0 N T ðr; tÞ Tj ¼ e d2 b2N 3 K ot 4 N ¼1 J1 ðbN Þ bN 1 þ L2 ð6Þ and the heating curve during the pressurisation (t < s) is given by Eq. (3). In particular, at the centre of the chamber (r ¼ 0) the cooling curve during the hold is given by
4jb2 N 2 s d 1e 1 4jb2 Nt 2T a oP d 2 X 1 T ð0; tÞ Tj ¼ e d2 2 b 3 K ot 4 N ¼1 J1 ðbN Þ bN 1 þ LN2 ð7Þ and the heating curve at the centre of the chamber (r ¼ 0) is given by 2 2 2T a oP d 4 1 1 T ðr; tÞ Tj ¼ þ K ot 4 8 4L 3 1 4jb2 X Nt 1 1 5 ð8Þ e d2 b2 b3 1 þ N J1 ðbN Þ N ¼1 N
L2
The general cooling curve during the hold is given by Eq. (6), with r as the dimensionless thermocouple position (from r ¼ 0 at the centre to r ¼ 1 at the wall). This equation can be used to extract the thermal diffusivity ðj ¼ K=qcp Þ, dimensionless extra-chamber heattransfer coefficient ððUd=2KÞ ¼ ðU =qcp Þð1=jÞðd=2Þ ¼ LÞ and thermal expansivity ððT a=KÞðoP =otÞðd 2 =4Þ ¼ ðT a=qcp Þð1=jÞðoP =otÞðd 2 =4ÞÞ from an experimental cooling curve. These heat-transfer parameters can then be used in Eqs. (6) and (3) to predict the temperature profiles in the vessel chamber during the pressurisation and hold of a high-pressure process.
133
3. Extraction of heat-transfer properties from the analytical solution The most general approach to extract the three unknown thermal and heat-transfer parameters (thermal diffusivity, thermal expansivity and extra-chamber heattransfer coefficient) is to fit the experimental cooling curve to Eq. (6) directly. A simplex algorithm routine to minimise a least-squares-difference objective function (Press, Teukolsky, Vetterling, & Flannery, 1986) was adapted to perform this curve fit. The chamber diameter, thermocouple position and pressurisation time are required input parameters along with the experimental cooling curve. The thermal diffusivity, thermal expansivity and extra-chamber heat-transfer coefficient can then be calculated from the curve fit. Alternatively, the unknown thermal and heat-transfer properties can be obtained from the model of the cooling curve at sufficiently long times by the first-term truncation of the series given in Eq. (6). The temperature profile is then given by 2
4jb2 2 1s 6 2T a oP d T ðr; tÞ Tj ¼ 4 1 e d2 K ot 4 3 2
1 J ðb rÞ 4jb2 1 t 0 1 7 5e d 2 b b31 1 þ L12 J1 ðb1 Þ
ð9Þ
A plot of log-temperature against time (at long times) is a straight line of slope 4jb21 =d 2 and a log-intercept given by the logarithm of the parenthesised part of Eq. (9). Thus the slope and log-intercept can be used to calculate the thermal diffusivity and extra-chamber heattransfer coefficient. The maximum temperature (i.e. centre of the chamber at the beginning of the hold) can then be used to calculate the thermal expansivity from Eq. (3). This is a simple method for cases for which heat transfer out of the vessel chamber is rate-limiting. However, the method relies upon accurate temperature measurements at long times (typically ðjt=d 2 Þ P 0:04), when temperature differences are lowest, and the sensible heat accumulated in the vessel wall might be significant. Furthermore, the method loses accuracy when heat transfer within the vessel chamber is rate-limiting, as is typical for high-pressure processing equipment. A third method, if cooling curves can be measured at two different (radial) positions, is to use the long-time slope as defined above, and a temperature ratio rather than the extrapolated intercept (this ratio is obtained from Eq. (9) as the ratio of two zero-order Bessel functions each with a different argument as determined by the value of r). However this method requires a pair of thermocouple ports, and relies on accurate thermocouple positioning.
134
T. Carroll et al. / Journal of Food Engineering 60 (2003) 131–135
The accuracy of the calculated thermal and heattransfer properties depends upon the change in temperature and pressure over which measurements are taken, and therefore, some averaging of the measured property. However, if significant, variations in the measured property over the range of interest could be captured by measurements at several temperatures and pressures. The thermal expansivity and heat-transfer coefficient measurements both rely on the actual temperature difference, and therefore thermocouple position, whereas the thermal conductivity depends on the rate of temperature change, which less sensitive to thermocouple position. It is neither necessary nor possible to decouple the extra-chamber heat-transfer coefficient from the density and heat capacity of the pressurising fluid at high pressure, unless the latter are known. Water is the only fluid whose physical properties are well-characterised under conditions typical of high-pressure processes (NIST/ ASME Steam Properties), and as such, a vessel heattransfer coefficient could be calculated. However this coefficient would not be useful for alternative pressurising media whose product of density and heat capacity is unknown at high pressure, although it could be used to determine this product, and thereby the thermal properties. The temperature profile during decompression is modelled by Eq. (3) (with a negative pressurisation rate) if the temperature profile at the end of the hold is uniform. However this limit can only be achieved in small chambers with appropriately long hold-times. Otherwise, the analytical solution is substantially complicated. However, in practice, the depressurisation rate can often be made arbitrarily high to minimise thermal effects during depressurisation if necessary. A similar approach could form the basis of a method to determine the heat-transfer properties and temperature profiles of food products under pressure, in particular where the pressurising fluid and foodstuff are similar in composition (Ting et al., 2002) and there are no barriers to heat transfer at the sample-pressurising fluid interface (container, packaging etc). 4. Sample calculation of heat-transfer properties for water Pure water was pressurised to 600 MPa from an initial temperature of 51.3 °C at 5.81 MPa/s and held for 300 s in a Stansted Fluid Power Micro Food-lab plunger press. Pressure and temperature were logged every 1 s. Temperature control was achieved by circulating a 3%
ethylene glycol solution from a water bath through the vessel jacket. The chamber diameter was 17 mm, the pressurisation come-up time was 90 s from 35 to 575 MPa (and 20 s from 575 to 600 MPa) and the thermocouple was positioned in the centre of the chamber. These last three parameters were inputs to the curvefitting algorithm, along with the experimental temperature data file (i.e. cooling curve) for the cycle. The extracted thermal diffusivity and thermal expansivity for water are shown in Table 1. The corresponding values calculated from the NIST/ASME Steam Properties database are also shown as a test of the model and software. The calculated thermal diffusivity of the pressurising water is 20% higher than the NIST value, and the calculated thermal expansivity is 20–25% lower. Both calculated properties are consistent with an underprediction of heat loss by a conduction model, and the differences may reflect a convection contribution. The extra-chamber heat-transfer coefficient was greater than 1000 W/m 2 K, using the density and heat capacity from the NIST/ASME Steam Properties database. This value is indicative of the absence of a substantial heat-transfer barrier at the chamber wall. The theoretical cooling curve generated from the calculated heat-transfer properties is compared to the experimental cooling curve in Fig. 1.
62.0
Pressurisation rate reduced to avoid overshoot
60.0
58.0
56.0
Sensible heat in vessel wall
54.0
52.0
50.0 0
30
60
90
120
150
180
210
240
270
300
Fig. 1. Experimental and model cooling curves for 600 MPa/51 °C test cycle.
Table 1 Comparison of model prediction with NIST/ASME values for water as pressurising fluid
Model prediction NIST/ASME database a
qcp (J/m3 K) (600 MPa) (50–60 °C)
j (m2 /s) (600 MPa) (50–60 °C)
qcp (J/m3 K) (55 °C) (0–600 MPa)
ð4.39–4:36Þ 106
0:23 106 ð0.190–0:194Þ 106 a
ð4.11–4:37Þ 106
Extrapolated beyond recommended pressure range.
ðT a=qcp Þðd 2 =4Þ ðoP =otÞð1=T0 Tj Þ (m2 /s)
a (K1 ) (55 °C) (0–600 MPa)
1:173 106
0.00035 0.00047–0.00044
T. Carroll et al. / Journal of Food Engineering 60 (2003) 131–135 1.0 0.8
75s 90s 60s
(a)
Hold
30s
0.6
T(0,0) − Tj 0.4
(b)
0s 15s
45s
T ( r , t ) − Tj
30s
135
45s 60s
15s 90s 120s 180s
0.2 Pressurise
0.0 0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Fig. 2. Model-prediction of temperature profiles for 600 MPa/51 °C test cycle, calculated using thermal and heat-transfer properties extracted from experimental cooling curve. (a) Pressurise to 600 MPa in 90 s. (b) Hold at 600 MPa for 300 s.
The spatial and time-dependence of the temperature during any arbitrary cycle in any chamber can be predicted from Eqs. (6) and (3) once the heat-transfer properties are extracted from the cooling curve (or otherwise known). The predictions using the thermal properties extracted from the cooling curve in Fig. 1 are shown in Fig. 2. The 17 mm chamber pressurised to 600 MPa at 5.81 MPa/s is at the jacket temperature near the chamber wall, but reaches 59.5 °C at the centre at the end of the pressurisation. The chamber cools substantially during the pressurisation, as indicated by the temperature change over each time interval. Although the maximum temperature for this test cycle was 59.5 °C, the maximum possible temperature is predicted as 65.3 °C if no heat is lost (i.e. an adiabatic compression). In fact, in a larger chamber, it is possible to both achieve and maintain this maximum temperature during the hold, near the centre of the vessel chamber, in an ‘‘pseudo’’-adiabatic cycle. Conversely, in a smaller chamber, the pressurisation rate can be set to balance heat generated and heat lost, in a ‘‘pseudo’’-isothermal cycle. For the test cycle example, a pressurisation time of 360 s would guarantee a temperature rise of less than 3 °C. Thermal effects can be thus incorporated into the interpretation of pressure-treatment outcomes as necessary. Findings from different chambers/systems can be compared, impacts on scale-up can be quantified and realistic temperature–pressure models of targeted effects can be developed. For example, in a case where microbial inactivation under pressure is measured at elevated temperature, the time–temperature exposure of each sample volume element can be predicted for the cycle. Any additional contribution to inactivation could then be averaged out and subtracted from the measured inactivation for a genuine pressure contribution, or included in a refined pressure-thermal inactivation model. Furthermore, the time–temperature exposures could be predicted for alternative cycles, chambers, heat-transfer
coefficients etc. and differences could be accounted for or corrections made as appropriate.
5. Conclusions An analytical solution was developed to the model for conductive heat transfer in high-pressure processing. This solution can be used to extract the thermal diffusivity, thermal expansivity and heat-transfer coefficient from an experimental cooling curve during a highpressure process. The extracted heat-transfer parameters can then be used to predict temperature profiles in the vessel chamber. A user-friendly software package was written around a curve-fitting algorithm to perform the extraction and prediction for any pressure cycle run with any pressurising fluid in any vessel.
References Abramovitz, M., & Stegun, I. A. (Eds.). (1972). Handbook of mathematical functions (p. 484). New York: Dover Publications Inc. Carroll, T., Chen, P., Fletcher, A., & Pearce, L. P. Size does matter: The impact of heat transfer on microbial inactivation by highpressure processing. Innovative Food Science and Emerging Technologies, submitted for publication. Carslaw, H. S., & Jaeger, J. C. (1959). Conduction of heat in solids. London: Oxford University Press, pp. 201–205. Denys, S., Ludikhuyze, L. R., Van Loey, A. M., & Hendrickx, M. E. (2000). Modelling conductive heat transfer and process uniformity during batch high-pressure processing of foods. Biotech. Prog., 16, 92–101. Otero, L., Molina-Garcia, A. D., & Sanz, P. D. (2000). Thermal effects in foods during quasi-adiabatic pressure treatments. Innovative Food Science and Emerging Technologies, 1, 119–126. Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. (1986). Numerical recipes in Fortran 77. Cambridge University Press, pp. 402–406. Ting, E., Balasubramaniam, V. M., & Raghubeer, E. (2002). Determining thermal effects in high-pressure processing. Food Technology, 56, 31–35.
A method to characterise heat transfer during high-pressure processing T. Carroll *, P. Chen, A. Fletcher Fonterra Research Centre, Private Bag 11029, Dairy Farm Road, Palmerston North, New Zealand Received 6 August 2002; accepted 5 January 2003
Abstract An analytical solution to the model for conductive heat transfer in high-pressure processing of foods is developed. This solution can be used to extract thermal diffusivity, thermal expansivity and heat-transfer coefficient from an experimental cooling curve obtained during the hold at pressure of a high-pressure process cycle. The extracted heat transfer parameters can then be used to predict the temperature profiles in the process-vessel chamber. These profiles are the key to understanding (where appropriate) the contribution of thermal effects in high-pressure processing, for example in combined temperature–pressure processes. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: High pressure processing; Heat transfer; Adiabatic compression
1. Introduction The compression that occurs during high-pressure processing of food generates heat that increases the temperature within the pressure-vessel chamber. This temperature rise may complicate the interpretation of pressure-induced phenomena such as protein denaturation or microbial inactivation. (Ting, Balasubramaniam, & Raghubeer, 2002). In particular, if the pressure-vessel chamber is not maintained at a temperature below which thermal effects can be excluded, then combined thermal-pressure mechanisms must be considered in any interpretations. Thermal effects are a significant complication, as unlike pressure, temperature is not typically uniform across a vessel-pressure chamber. The combined pressure–temperature experienced by a given sample volume element varies with location within the chamber, the size of the chamber and the size of the sample (Carroll et al., submitted for publication). The effects of compressive heating on high-pressure processes are now being considered (Ting et al., 2002; Denys, Ludikhuyze, Van Loey, & Hendrickx, 2000; Otero, Molina-Garcia, & Sanz, 2000). Although the temperature is typically measured at one location within
*
Corresponding author. Fax: +64-63504663. E-mail address: [email protected] (T. Carroll).
0260-8774/03/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0260-8774(03)00026-8
the chamber, temperature varies both temporally and spatially during a pressure cycle. It follows that the temperature within a pressure vessel should be known at all locations, and at all times during a high-pressure process if thermal effects are to be captured. In essence, the heat-transfer coefficient of the pressure vessel and the thermal properties of the pressurising fluid must be determined. A meaningful comparison of results across various research groups using different vessels and pressurising media is not possible if heat-transfer parameters are not reported along with the typical basic parameters such as pressure and hold-time. This is particularly important for the development of combined pressure–heat processes for sterilisation. Furthermore, it is neither possible to develop combined pressure–temperature models of observed behaviour, nor to predict process performance at alternative scales (e.g. as part of process development) without first determining heattransfer parameters. A numerical method has been developed to model the temperature profiles caused by conductive heat transfer in high-pressure processes (Denys et al., 2000). In this method, the governing heat-transfer equation is solved numerically over a finite element mesh. Non-uniform element-based physical properties of the fluid can be included, where these are known. Such a numerical method is useful to predict the temperature profiles if the thermal properties of the fluid are known. However, the
132
T. Carroll et al. / Journal of Food Engineering 60 (2003) 131–135
method is less useful in reverse, namely to extract the thermal properties if the temperature is known. In this paper we develop an analytical solution for the temperature profile within a pressure-vessel chamber during a high-pressure process. This analytical solution is more general, more straightforward, and easier to apply than the available numerical solution. The analytical solution can be used to extract the thermal expansivity and thermal diffusivity of any pressuring fluid, and the heat-transfer coefficient for any vessel from an experimentally determined cooling curve obtained during a pressure-hold. These extracted heat-transfer parameters can then be used with the analytical solution to predict the temperature at all points within the chamber at all times during a high-pressure process. We have written software to perform this extraction and prediction from the analytical solution and we demonstrate the software by example.
2. Model of heat transfer in high-pressure processing A high-pressure food treatment process typically begins with a pressurisation step during which the pressure chamber is pumped up to the pressure set-point, followed by a hold at the pressure set-point for a predetermined time, and concludes with a depressurisation step (i.e. pressure is released). In theory the heat generated by compression is dissipated by a combination of conduction and convection within the pressurising fluid in the chamber and transfer across the chamber wall into the surroundings. Depending on whether an external intensifier or plunger press design is used, either a small volume of fluid is forced into the chamber, or a piston is displaced, to compensate for the loss of volume on compression during the pressurisation step. During the hold at pressure, the pressurising fluid and chamber volume are typically undisturbed. As a first approach, convection within the pressurising fluid is not considered, (and the resultant error is evaluated for a laboratory plunger press system). The temperature of a pressurising fluid in the chamber of a pressure vessel during such a high-pressure process is then given by the conductive heat-transfer equation r2 T
qcp oT T a oP ¼ K ot K ot
ð1Þ
subject to K^ n~ DT ¼ U ðT Tj Þ at the wall and T ¼ Tj everywhere initially (^ n is the unit normal vector). T and P are chamber temperature and pressure, respectively. K (W/m K), q (kg/m3 ), cp (J/kg K) and a (1/K) are the thermal conductivity, density, heat capacity and thermal expansion coefficient of the pressurising fluid, respectively. U (W/m2 K) is the heat-transfer coefficient from
the chamber inner wall to the jacket (referenced to the inner area) and Tj is the jacket temperature. In particular, during the hold at the pressure setpoint, oP =ot ¼ 0. In the case of a long cylindrical chamber (aspect-ratio < 1=3), the temperature profile is given by the solution to Eq. (1) as T ðr; tÞ Tj ¼ 2
1 X
e
N ¼1
Z
4jb2 Nt d2
b2N
b2N J0 ðbN rÞ 2 þ L J02 ðbN Þ
1
rf ðrÞJ0 ðbN rÞ dr
ð2Þ
0
where bN are roots of bN J1 ðbN Þ ¼ LJ0 ðbN Þ (adapted from Carslaw & Jaeger, 1959). j ¼ K=qcp (m2 /s) is the thermal diffusivity, L ¼ Ud=2K is the extra-chamber dimensionless heat-transfer coefficient (d (m) is the chamber diameter), and J0 , J1 are zero-order and firstorder cylindrical Bessel functions, respectively. f ðrÞ is the initial temperature profile during the hold. This general solution describes the cooling curve of a fluid in an infinitely long cylindrical container allowed to cool by a combination of conduction and convective interfacial heat-transfer. The temperature profile during the pressurisation is given by an analytical solution to Eq. (1) for the case where the pressurising fluid is heated at a constant rate. This case corresponds to a pressurisation at a constant rate over a pressure range across which physical properties are constant. The temperature profile is then given by 2 2T a oP d 2 4 1 r2 1 T ðr; tÞ Tj ¼ þ K ot 4 4L 8 3 1 4jb2 X Nt 1 J ðb rÞ 0 N 5 ð3Þ e d2 b2N 3 J1 ðbN Þ b 1þ N ¼1 N
L2
(adapted from Carslaw & Jaeger, 1959). Note that Eq. (3) also describes the temperature during the depressurisation if the temperature at the end of the hold is uniform. In particular, the temperature profiles at the end of pressurisation and at the beginning of the hold are the same. From Eq. (3) " T a oP d 2 2 f ðrÞ ¼ 1 r2 þ K ot 16 L # 1 4jb2 X Ns 1 J ðb rÞ 0 N 8L e d2 ð4Þ b2N ðb2N þ L2 Þ J0 ðbN Þ N ¼1 where s is come-up time to the pressure set-point (adapted from Carslaw & Jaeger, 1959). The temperature profile during the hold is then obtained by substituting Eq. (4) into Eq. (2) and inte-
T. Carroll et al. / Journal of Food Engineering 60 (2003) 131–135
grating. The radial integrals of the appropriate Bessel functions are given by Z 1 J1 ðbN Þ rJ0 ðbN rÞ dr ¼ ð5aÞ bN 0 Z 1 rð1 r2 ÞJ0 ðbN rÞ dr 0
¼ Z
2J2 ðbN Þ 4J1 ðbN Þ 2J0 ðbN Þ ¼ b2N b3N b2N
ð5bÞ
1
rJ0 ðaN rÞJ0 ðbN rÞ dr
0
¼
J12 ðbN Þ þ J02 ðbN Þ 2
if aN ¼ bN ; zero otherwise
ð5cÞ using integration by parts and Bessel function integral formulae (Abramovitz & Stegun, 1972). The solution is a cooling curve during the hold at the pressure set-point (t þ s > s) given by
4jb2 Ns 2 d 1 e 1 4jb2 Nt 2T a oP d 2 X J ðb rÞ 0 N T ðr; tÞ Tj ¼ e d2 b2N 3 K ot 4 N ¼1 J1 ðbN Þ bN 1 þ L2 ð6Þ and the heating curve during the pressurisation (t < s) is given by Eq. (3). In particular, at the centre of the chamber (r ¼ 0) the cooling curve during the hold is given by
4jb2 N 2 s d 1e 1 4jb2 Nt 2T a oP d 2 X 1 T ð0; tÞ Tj ¼ e d2 2 b 3 K ot 4 N ¼1 J1 ðbN Þ bN 1 þ LN2 ð7Þ and the heating curve at the centre of the chamber (r ¼ 0) is given by 2 2 2T a oP d 4 1 1 T ðr; tÞ Tj ¼ þ K ot 4 8 4L 3 1 4jb2 X Nt 1 1 5 ð8Þ e d2 b2 b3 1 þ N J1 ðbN Þ N ¼1 N
L2
The general cooling curve during the hold is given by Eq. (6), with r as the dimensionless thermocouple position (from r ¼ 0 at the centre to r ¼ 1 at the wall). This equation can be used to extract the thermal diffusivity ðj ¼ K=qcp Þ, dimensionless extra-chamber heattransfer coefficient ððUd=2KÞ ¼ ðU =qcp Þð1=jÞðd=2Þ ¼ LÞ and thermal expansivity ððT a=KÞðoP =otÞðd 2 =4Þ ¼ ðT a=qcp Þð1=jÞðoP =otÞðd 2 =4ÞÞ from an experimental cooling curve. These heat-transfer parameters can then be used in Eqs. (6) and (3) to predict the temperature profiles in the vessel chamber during the pressurisation and hold of a high-pressure process.
133
3. Extraction of heat-transfer properties from the analytical solution The most general approach to extract the three unknown thermal and heat-transfer parameters (thermal diffusivity, thermal expansivity and extra-chamber heattransfer coefficient) is to fit the experimental cooling curve to Eq. (6) directly. A simplex algorithm routine to minimise a least-squares-difference objective function (Press, Teukolsky, Vetterling, & Flannery, 1986) was adapted to perform this curve fit. The chamber diameter, thermocouple position and pressurisation time are required input parameters along with the experimental cooling curve. The thermal diffusivity, thermal expansivity and extra-chamber heat-transfer coefficient can then be calculated from the curve fit. Alternatively, the unknown thermal and heat-transfer properties can be obtained from the model of the cooling curve at sufficiently long times by the first-term truncation of the series given in Eq. (6). The temperature profile is then given by 2
4jb2 2 1s 6 2T a oP d T ðr; tÞ Tj ¼ 4 1 e d2 K ot 4 3 2
1 J ðb rÞ 4jb2 1 t 0 1 7 5e d 2 b b31 1 þ L12 J1 ðb1 Þ
ð9Þ
A plot of log-temperature against time (at long times) is a straight line of slope 4jb21 =d 2 and a log-intercept given by the logarithm of the parenthesised part of Eq. (9). Thus the slope and log-intercept can be used to calculate the thermal diffusivity and extra-chamber heattransfer coefficient. The maximum temperature (i.e. centre of the chamber at the beginning of the hold) can then be used to calculate the thermal expansivity from Eq. (3). This is a simple method for cases for which heat transfer out of the vessel chamber is rate-limiting. However, the method relies upon accurate temperature measurements at long times (typically ðjt=d 2 Þ P 0:04), when temperature differences are lowest, and the sensible heat accumulated in the vessel wall might be significant. Furthermore, the method loses accuracy when heat transfer within the vessel chamber is rate-limiting, as is typical for high-pressure processing equipment. A third method, if cooling curves can be measured at two different (radial) positions, is to use the long-time slope as defined above, and a temperature ratio rather than the extrapolated intercept (this ratio is obtained from Eq. (9) as the ratio of two zero-order Bessel functions each with a different argument as determined by the value of r). However this method requires a pair of thermocouple ports, and relies on accurate thermocouple positioning.
134
T. Carroll et al. / Journal of Food Engineering 60 (2003) 131–135
The accuracy of the calculated thermal and heattransfer properties depends upon the change in temperature and pressure over which measurements are taken, and therefore, some averaging of the measured property. However, if significant, variations in the measured property over the range of interest could be captured by measurements at several temperatures and pressures. The thermal expansivity and heat-transfer coefficient measurements both rely on the actual temperature difference, and therefore thermocouple position, whereas the thermal conductivity depends on the rate of temperature change, which less sensitive to thermocouple position. It is neither necessary nor possible to decouple the extra-chamber heat-transfer coefficient from the density and heat capacity of the pressurising fluid at high pressure, unless the latter are known. Water is the only fluid whose physical properties are well-characterised under conditions typical of high-pressure processes (NIST/ ASME Steam Properties), and as such, a vessel heattransfer coefficient could be calculated. However this coefficient would not be useful for alternative pressurising media whose product of density and heat capacity is unknown at high pressure, although it could be used to determine this product, and thereby the thermal properties. The temperature profile during decompression is modelled by Eq. (3) (with a negative pressurisation rate) if the temperature profile at the end of the hold is uniform. However this limit can only be achieved in small chambers with appropriately long hold-times. Otherwise, the analytical solution is substantially complicated. However, in practice, the depressurisation rate can often be made arbitrarily high to minimise thermal effects during depressurisation if necessary. A similar approach could form the basis of a method to determine the heat-transfer properties and temperature profiles of food products under pressure, in particular where the pressurising fluid and foodstuff are similar in composition (Ting et al., 2002) and there are no barriers to heat transfer at the sample-pressurising fluid interface (container, packaging etc). 4. Sample calculation of heat-transfer properties for water Pure water was pressurised to 600 MPa from an initial temperature of 51.3 °C at 5.81 MPa/s and held for 300 s in a Stansted Fluid Power Micro Food-lab plunger press. Pressure and temperature were logged every 1 s. Temperature control was achieved by circulating a 3%
ethylene glycol solution from a water bath through the vessel jacket. The chamber diameter was 17 mm, the pressurisation come-up time was 90 s from 35 to 575 MPa (and 20 s from 575 to 600 MPa) and the thermocouple was positioned in the centre of the chamber. These last three parameters were inputs to the curvefitting algorithm, along with the experimental temperature data file (i.e. cooling curve) for the cycle. The extracted thermal diffusivity and thermal expansivity for water are shown in Table 1. The corresponding values calculated from the NIST/ASME Steam Properties database are also shown as a test of the model and software. The calculated thermal diffusivity of the pressurising water is 20% higher than the NIST value, and the calculated thermal expansivity is 20–25% lower. Both calculated properties are consistent with an underprediction of heat loss by a conduction model, and the differences may reflect a convection contribution. The extra-chamber heat-transfer coefficient was greater than 1000 W/m 2 K, using the density and heat capacity from the NIST/ASME Steam Properties database. This value is indicative of the absence of a substantial heat-transfer barrier at the chamber wall. The theoretical cooling curve generated from the calculated heat-transfer properties is compared to the experimental cooling curve in Fig. 1.
62.0
Pressurisation rate reduced to avoid overshoot
60.0
58.0
56.0
Sensible heat in vessel wall
54.0
52.0
50.0 0
30
60
90
120
150
180
210
240
270
300
Fig. 1. Experimental and model cooling curves for 600 MPa/51 °C test cycle.
Table 1 Comparison of model prediction with NIST/ASME values for water as pressurising fluid
Model prediction NIST/ASME database a
qcp (J/m3 K) (600 MPa) (50–60 °C)
j (m2 /s) (600 MPa) (50–60 °C)
qcp (J/m3 K) (55 °C) (0–600 MPa)
ð4.39–4:36Þ 106
0:23 106 ð0.190–0:194Þ 106 a
ð4.11–4:37Þ 106
Extrapolated beyond recommended pressure range.
ðT a=qcp Þðd 2 =4Þ ðoP =otÞð1=T0 Tj Þ (m2 /s)
a (K1 ) (55 °C) (0–600 MPa)
1:173 106
0.00035 0.00047–0.00044
T. Carroll et al. / Journal of Food Engineering 60 (2003) 131–135 1.0 0.8
75s 90s 60s
(a)
Hold
30s
0.6
T(0,0) − Tj 0.4
(b)
0s 15s
45s
T ( r , t ) − Tj
30s
135
45s 60s
15s 90s 120s 180s
0.2 Pressurise
0.0 0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Fig. 2. Model-prediction of temperature profiles for 600 MPa/51 °C test cycle, calculated using thermal and heat-transfer properties extracted from experimental cooling curve. (a) Pressurise to 600 MPa in 90 s. (b) Hold at 600 MPa for 300 s.
The spatial and time-dependence of the temperature during any arbitrary cycle in any chamber can be predicted from Eqs. (6) and (3) once the heat-transfer properties are extracted from the cooling curve (or otherwise known). The predictions using the thermal properties extracted from the cooling curve in Fig. 1 are shown in Fig. 2. The 17 mm chamber pressurised to 600 MPa at 5.81 MPa/s is at the jacket temperature near the chamber wall, but reaches 59.5 °C at the centre at the end of the pressurisation. The chamber cools substantially during the pressurisation, as indicated by the temperature change over each time interval. Although the maximum temperature for this test cycle was 59.5 °C, the maximum possible temperature is predicted as 65.3 °C if no heat is lost (i.e. an adiabatic compression). In fact, in a larger chamber, it is possible to both achieve and maintain this maximum temperature during the hold, near the centre of the vessel chamber, in an ‘‘pseudo’’-adiabatic cycle. Conversely, in a smaller chamber, the pressurisation rate can be set to balance heat generated and heat lost, in a ‘‘pseudo’’-isothermal cycle. For the test cycle example, a pressurisation time of 360 s would guarantee a temperature rise of less than 3 °C. Thermal effects can be thus incorporated into the interpretation of pressure-treatment outcomes as necessary. Findings from different chambers/systems can be compared, impacts on scale-up can be quantified and realistic temperature–pressure models of targeted effects can be developed. For example, in a case where microbial inactivation under pressure is measured at elevated temperature, the time–temperature exposure of each sample volume element can be predicted for the cycle. Any additional contribution to inactivation could then be averaged out and subtracted from the measured inactivation for a genuine pressure contribution, or included in a refined pressure-thermal inactivation model. Furthermore, the time–temperature exposures could be predicted for alternative cycles, chambers, heat-transfer
coefficients etc. and differences could be accounted for or corrections made as appropriate.
5. Conclusions An analytical solution was developed to the model for conductive heat transfer in high-pressure processing. This solution can be used to extract the thermal diffusivity, thermal expansivity and heat-transfer coefficient from an experimental cooling curve during a highpressure process. The extracted heat-transfer parameters can then be used to predict temperature profiles in the vessel chamber. A user-friendly software package was written around a curve-fitting algorithm to perform the extraction and prediction for any pressure cycle run with any pressurising fluid in any vessel.
References Abramovitz, M., & Stegun, I. A. (Eds.). (1972). Handbook of mathematical functions (p. 484). New York: Dover Publications Inc. Carroll, T., Chen, P., Fletcher, A., & Pearce, L. P. Size does matter: The impact of heat transfer on microbial inactivation by highpressure processing. Innovative Food Science and Emerging Technologies, submitted for publication. Carslaw, H. S., & Jaeger, J. C. (1959). Conduction of heat in solids. London: Oxford University Press, pp. 201–205. Denys, S., Ludikhuyze, L. R., Van Loey, A. M., & Hendrickx, M. E. (2000). Modelling conductive heat transfer and process uniformity during batch high-pressure processing of foods. Biotech. Prog., 16, 92–101. Otero, L., Molina-Garcia, A. D., & Sanz, P. D. (2000). Thermal effects in foods during quasi-adiabatic pressure treatments. Innovative Food Science and Emerging Technologies, 1, 119–126. Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. (1986). Numerical recipes in Fortran 77. Cambridge University Press, pp. 402–406. Ting, E., Balasubramaniam, V. M., & Raghubeer, E. (2002). Determining thermal effects in high-pressure processing. Food Technology, 56, 31–35.