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Use of a computer program for parameter sensitivity studies during thawing of foods
Journal of Food Engineering 52 (2002) 219–225 www.elsevier.com/locate/jfoodeng
Use of a computer program for parameter sensitivity studies during thawing of foods K. Hoke, M. Houska *, K. Kyhos, A. Landfeld Food Research Institute Prague, Radiova 7, 102 31 Prague 10, Czech Republic Received 9 December 2000; accepted 15 May 2001
Abstract Thawing is one of the important processes of the food industry. Its speed and conditions can influence the quality and initial concentration of the microbial population in raw materials used for further processing. Experimental prediction of the optimum conditions of the thawing process is very time consuming and expensive. Therefore, a mathematical model of the process can help in the design of efficient thawing apparatus and process conditions. This paper compares experimental data of different thawing procedures with numerically calculated predictions using the software package food product modeller (FPM). The model calculations agree very well with experimental data on three different thawing process arrangements with reasonable accuracy. The verified mathematical model for convective thawing was selected for sensitivity analysis of the process parameters. The time of thawing was calculated as a function of different parameters for three different points in the thawed food piece. Graphs of the relative change of thawing time as a function of the relative change of the process parameters have been constructed. From these graphs one can read that the air temperature, surface heat transfer coefficient (SHTC), thickness of the layer, water content and enthalpy are the parameters with the greatest influence on thawing time. On the other hand the thermal conductivity or initial freezing point does not influence the thawing time as much. 2002 Elsevier Science Ltd. All rights reserved. Keywords: Thawing; Mathematical modelling; Sensitivity analysis; Thawing time
1. Introduction Thawing is a thermodynamic process applied in many branches of the food industry. The process of thawing should be controlled. The surface temperature of different products should be kept under a predefined level, e.g., 10–15C. The time of thawing should be as short as possible to minimise the potential growth of microorganisms and spoilage of the raw material before the final processing. Also the water or juice losses and quality changes (colour, oxidation) can be minimised if the thawing is made in a special apparatus with controlled time-temperature profiles. One of the most frequently used designs is convective hot air thawing. This type of thawing is used because of low operating and investment costs. On the other hand the time of thawing is relatively long. The typical design is described in the producer’s documentation, e.g.,
*
Corresponding author. Tel.: +420-272-702-331; fax: +420-272-701983. E-mail address: [email protected] (M. Houska).
Morep (1999). Heated air is circulated in a closed chamber filled with the food, placed on trays to enable airflow between the layers. Control is executed on the inlet air temperature depending on measured temperatures of the food and temperatures of the air in the chamber. Another process is microwave thawing. This process can be used for large pieces of frozen products such as meat, fish blocks, etc. for increasing the food temperature from 18C up to 4C (tempering). The frequency mostly used in Europe is 2450 MHz with 915 MHz used in the USA (Decarean & Peterson, 1986; Rosenberg & Bogl (1987); Steele, 1987), this being more suitable for large food products. Steam thawing is also used (Kharabadzov, 1989). Ready frozen meals consisting of meat, vegetables and sauce were thawed in steam at a temperature of 98C. Samples were placed into dishes with height of 10–75 mm. If there was an air layer between the food and its packaging material, the thawing time was extended by 30% over cases where the meals were vacuum-packed. The quality of the thawed meals was very high and mass loss was minimised.
0260-8774/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 0 - 8 7 7 4 ( 0 1 ) 0 0 1 0 9 - 1
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K. Hoke et al. / Journal of Food Engineering 52 (2002) 219–225
Thawing can be also achieved by using a liquid. Water is mostly used as the heat-transmitting medium. Samples are submerged into a bath where the medium can have different temperatures or the sample can be sprayed with the warm medium, Merts and Lawson (1999a,b). The temperature of the medium can be selected carefully especially in the case of large food pieces. Spraying is frequently used to avoid freezing of the water on some parts of the surface since an ice layer can substantially lower the heat transfer. Merts and Lawson (1999a,b) refer also to controlled thawing by contact heating between two plates. Plates are heated by a circulating medium pumped into the cavities in the plates. This process is convenient for heat sensitive foods because the temperature of the plates can be carefully controlled. Our previous experimental work compares different thawing arrangements from the point of view of thawing time, (Hoke, Klima, Gree, & Houska, 2000). The processes were controlled so as to limit the surface temperature to a prescribed level. It was found that controlled vacuum-steam thawing is a very efficient method comparable with microwave thawing. The experimental prediction of the optimum conditions of thawing is very time consuming and expensive. If a mathematical model of the process can be developed, more efficient design and process control can be obtained. Mathematical modelling of thawing is complicated by the changes of physical properties during phase transition, heterogeneity of the food and mostly by the unknown surface heat transfer coefficient (SHTC) under the process conditions. This paper focuses on mathematical modelling of the process and its verification. Also a sensitivity analysis of the process parameters on thawing time for one thawing arrangement is presented.
2. Materials and methods The solution of differential equations describing the conduction of heat during thawing of a brick of frozen mashed potatoes was carried out using the software Food Product Modeller (FPM), 2001, MIRINZ Food Technology & Research, New Zealand. FPM allows simulating of the effects of processing regimes on food products. The underlying finite difference method of calculation can deal with the following shapes: infinite slab, infinite cylinder, sphere, infinite rod, rectangular prism (or brick), finite cylinder (or can). The calculation accepts input data to specify the thermal properties of the simulated material (heat capacity, thermal conductivity) and its boundary conditions (temperature, surface heat transfer coefficient). Thermal properties for over 70 food materials are included in the program. The data preparation procedure allows the user to specify these
properties in terms of the user problem domain (e.g., by specifying air velocity and packaging type rather than transfer coefficient). Boundary conditions may be varied over the period of process. We have chosen a brick geometry that was similar to the model food used for experimental testing of different thawing procedures. The calculation was carried out in 3D space. Only onequarter of the brick was analysed thanks to axial symmetry. Nodes were uniformly distributed in the brick ð20 20 20 ¼ 8000Þ. The length of time step was 5–6 s. Thawing experiments were carried out on a model food – mashed potatoes prepared by mixing a dried commercial product (450 g) with 2000 ml of water heated to the 70C. Basic chemical composition of the final product is given in Table 1. A flat cartoon tray having dimensions 120 100 45 mm was used for sample preparation. The hot prepared mashed potatoes were placed into the trays and vacuumed to remove the air bubbles. After cooling at ambient atmosphere the trays of potatoes were placed into the freezer and stored at 22C. The samples were removed from the trays before the experiments and holes were drilled into the frozen body for placement of thermocouple probes. Temperatures were measured at three points of the thawed sample body: in the centre (point 1), in the layer 10 mm under the surface above the centre (point 2) and in the layer 2 mm under the surface above the centre (point 3). The comparison with the mathematical model was made for the conditions of the following three selected experiments: • Thawing of the model food in a convective hot-air oven without air heating (air temperature 28C) with intensive air circulation – experiment no. 1. • Thawing of the model food in a convective hot-air oven at an air temperature of 70C with intensive air circulation – experiment no. 2. The surface temperature (measured at point 3) was regulated by stopping the heating and air circulation as soon as 15C was reached. As soon as the temperature was lowered to 11C, the heating and air circulation was switched on. • Thawing of the model food in vacuum cooker. The vacuum pump was switched on after closing the cooker and the absolute pressure was reduced to 10 kPa.
Table 1 Chemical composition of mashed potatoes Component
Concentration (%)
Water Protein Starch Fat Ash
82.7 1.2 15.2 0.4 0.5
K. Hoke et al. / Journal of Food Engineering 52 (2002) 219–225
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Table 2 Parameters for numerical modelling of thawing Parameter
Units
Experiment no. 1
Dimensions of the sample Layer thickness Ambient temperature Surface heat transfer coefficient Initial freezing point Density above freezing point Density below freezing point Initial temperature of the sample Thermal conductivity of the sample Enthalpy
(mm) (mm) (C) (W m2 K1 ) (C) (kg m3 ) (kg m3 ) (C) (W m1 K1 ) (MJ m3 )
120 100 40 40 28 32 )1 1050 960 )22
Input parameters for modelling the different thawing experiments are given in Table 2. The measured timetemperature dependence of the surrounding atmosphere during thawing (air, steam) was input into the software. The SHTC was predicted from preliminary experiments made on the same experimental set-up while heating an aluminium model of the sample of mashed potatoes, Klima (1999). The optimum SHTC was predicted by numerical modelling of the heating to give the best fit to the experimental heating curve. For the convective hotair oven and a surrounding air temperature of 70C the optimum value of SHTC was found in the range between 35 and 40 W m2 K1 . The optimum value of SHTC for the vacuum cooker with the fully opened steam valve at an absolute pressure 10 kPa was predicted at around 1000 W m2 K1 . The thermal conductivity and enthalpy of mashed potatoes, see Table 3,
Table 3 Predicted values of thermal conductivity and enthalpy using Costherm 3.11 for mashed potatoes Thermal conductivity (W m1 K1 )
Enthalpy (kJ kg1 )
)22.0 )20.0 )17.5 )15.0 )6.0 )3.5 )2.25 )1.5 )1.25 )1.1 )1.0 )0.5 7.5 15.0 22.5 30.0 40.0
2.0830 2.0581 2.0261 1.9921 1.7887 1.6063 1.3717 1.0399 0.8366 0.6675 0.5246 0.5253 0.5389 0.5510 0.5623 0.5729 0.5858
)287 )282 )275 )268 )224 )189 )146 )88.5 )54.5 )26.7 )3.71 )1.85 27.8 55.7 83.6 112 149
Experiment no. 3
120 100 40 120 100 40 40 40 See experiment See experiment 35 (8) 1000 )1 )1 1050 1050 960 960 )22 )22 Predicted with Costherm 3.11 Recalculated by using predicted values by Costherm 3.11
Then the steam valve was opened for the entire thawing process without regard to the surface temperature of the samples – experiment no. 3.
Temperature (C)
Experiment no. 2
were calculated by using the predictive software Costherm 3.11 (Allen, Liu, & Nesvadba, 1997) on the basis of the chemical composition and for the temperature range from 22C to 40C. The initial freezing point was predicted experimentally. Specific enthalpy calculated by the software Costherm 3.11 was recalculated with the help of the corresponding density values above and below the initial freezing point in MJ m3 for subsequent use in modelling software FPM. The software predicted the time-temperature histories at any point of the thawed model body. The points at which the experimental data readings were taken were selected for calculation.
3. Results and discussion The comparison of measured and calculated timetemperature histories for the conditions of experiment no. 1 at three points of the thawing body is shown in Fig. 1. Modelling the start-up of the experiment took account of manipulation time of 7 min (weighing of sample, introduction of probes, input of body into oven). The SHTC value of 10 W m2 K1 , air temperature of 20C, and 22C as the initial temperature of the body were used. This preliminary calculation meant that the initial temperatures (at time 0) were different at different positions in the body (see Fig. 1). This preliminary calculation ensured that the calculated timetemperature histories for thawing were more realistic. The surrounding air temperature of 28C and SHTC of 32 W m2 K1 were used for the thawing calculation. The similar comparison of measured and calculated time-temperature histories for the conditions of experiment no. 2 is shown in Fig. 2. It is apparent from this figure that the temperature at point 3 (2 mm under the surface) very quickly responds to changes in air temperature. The predicted temperature by FPM follows the experimental data fairly well. The numerical calculation has taken into account the changing conditions during thawing. The value 35 W m2 K1 of SHTC and a temperature of 70C were used in the first stage of
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Fig. 1. Thawing of mashed potatoes in a convective oven at 28C, comparison of experimental data with numerical calculation.
Fig. 2. Thawing of mashed potatoes in a convective oven at 70C, comparison of experimental data with numerical calculation.
thawing. An SHTC value of 8 W m2 K1 was used for modelling the situation with the fan and heater switched off. The variation in air temperature with time in the oven measured during the experiment was input into the calculation. An SHTC value of 35 W m2 K1 was used when the fan and heater were switched on. Modelling the start-up of the experiment took also account of manipulation time of 9 min (weighing of sample, introduction of probes, input of body into oven). The SHTC value of 10 W m2 K1 , air temperature of 20C and 22C as the initial temperature of the body were used. This preliminary calculation meant that the initial temperatures (at time 0) were different at different positions in the body (see Fig. 2). This preliminary calculation ensured that the calculated time-temperature histories for thawing were more realistic. Fig. 3 shows the comparison of measured and calculated time-temperature histories for the conditions of experiment no. 3. The comparison was done for three points in the food during thawing in the vacuum cooker. The temperature of the steam–air mixture changed from 22C (atmospheric air) to 17C (full vacuum) and then
up to 41C (full vacuum and full steam flow). The time necessary for closing the cooker was included in the modelling (about 7 min), see Fig. 3. The preliminary manipulation time (10.2 min) necessary for weighing, introducing the thermocouples and installing the food in the cooker was modelled by preliminary calculation. The temperature of the thawed body 22C (uniform), surrounding air temperature 20C and SHTC of 8 W m2 K1 were used. The temperature profile was calculated for zero time of thawing process, see Fig. 3. The same values of air temperature and SHTC used for the preliminary stage of the modelling were used for modelling the closing time of the vacuum cooker. The experimentally predicted time-temperature history of the inner atmosphere in the cooker was used in calculating temperatures during the vacuum and steaming stages in the cooker. An SHTC value of 1000 W m2 K1 was taken for modelling this stage of the thawing. The model calculations of the local time-temperature histories for the three different thawing arrangements agree with experimental data with reasonable accuracy
K. Hoke et al. / Journal of Food Engineering 52 (2002) 219–225
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Fig. 3. Thawing of mashed potatoes in a vacuum cooker, comparison of experimental data with numerical calculation.
(see Table 4). The use of inexact input parameters can cause differences between the experimental and calculated results (e.g. the thermal conductivity and enthalpy of the model material were predicted from chemical composition only). The density changes with temperature were ignored (constant values of densities predicted experimentally were used for temperatures below and above the initial freezing point, respectively). Also experimental errors in measured temperatures (0:2C for experiments no. 1 and 2; 0:5C for experiment no. 3) can be taken into account. The calculated thawing time for experiment no. 1 was 122 min (measured 110 min). The calculated thawing time for experiment no. 2 was 72 min (measured 58 min). The calculated thawing time for experiment no. 3 was 46 min (measured 36 min). In spite of these facts, the accuracy of modelling the thawing process seems to be good for the purpose of thawing time predictions and, therefore, the model can be regarded as verified. Therefore, the model could be used for sensitivity calculations of thawing time on changes in the input parameters (process parameters). The basic calculations were carried out using the conditions of experiment no. 1 (convective thawing in the hot-air oven at room temperature 28C). The thawing time for each point was defined as the time taken for this point to increase in temperature above the initial freezing point 1C. The input parameters were changed by a selected percentage below or above the mean values of the parameters of
Table 4 Standard deviations between measured and calculated temperatures
Point 1 Point 2 Point 3
Experiment no. 1 S.D. (C)
Experiment no. 2
Experiment no. 3
1.26 1.31 0.55
1.32 1.15 0.88
2.28 1.65 1.35
simplified experiment no. 1, see Tables 1–3 (no manipulation time, initial temperature of food 20C). Only one parameter was changed and the others were kept constant at nominal values. The response of the thawing time was calculated as a function of change of temperature of the surrounding air, SHTC, thickness of the thawed layer, physical properties and the composition of the thawed food (enthalpy, thermal conductivity, initial freezing point and water content). Calculations were conducted for three different points in the body (at the centre of the food – point 1, at one-quarter of the depth and above the centre – point 2 and at one-twentieth of the depth and above the centre – point 3). The input parameters were changed stepwise by 10% up to 50% from the nominal values. The water content was relatively changed from 6 to 18% and its change was compensated by a change in the starch content. Nominal calculated thawing times for points 1 (centre), 2 and 3 were 121.5, 87.1 and 42.1 min, respectively. The calculated thawing times for other than nominal combinations of input parameters were recalculated as a relative change of nominal thawing time. The results of the calculations are given in three graphs, Figs. 4–6. Fig. 4 shows the influence of change of individual input parameters on the relative change in thawing time at point 1 (centre of the body). Fig. 5 shows the influence of individual input parameters on the relative change in thawing time at point 2. Fig. 6 shows the influence of individual input parameters on the relative change in thawing time at point 3 (near the surface of the body). The thawing time for the arrangement studied is strongly influenced by air temperature and SHTC especially when they are decreasing. Their influence is most apparent at point 3 near the body surface. Their influence is comparatively lower at point 1 (centre of the body). The thickness of the body, enthalpy and water content are also strongly influential parameters in the model. Thermal conductivity has much less influence on thawing time at the body surface than at the body
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Fig. 4. Relative change in thawing time as a function of the size of different parameters, point 1 in the body, standard thawing time 121.55 min.
Fig. 5. Relative change in thawing time as a function of the size of different parameters, point 2 in the body, standard thawing time 87.1 min.
Fig. 6. Relative change in thawing time as a function of the size of different parameters, point 3 in the body, standard thawing time 42.1 min.
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the surrounding atmosphere. Accurate values of process parameters and physical properties of the thawing food should be input into the calculation. The most important parameters are the temperature of the surrounding atmosphere, the SHTC, the dimensions of the body and the temperature dependence of the specific enthalpy of the thawing food. The influence of thermal conductivity and initial freezing point on thawing time is weaker than the influence of the above-mentioned parameters. These results are derived from a sensitivity analysis of the input parameters on the thawing time prediction. The thawing process can be simulated if the proper values of these parameters are known.
Fig. 7. Calculated temperature profile in the centre of the mashed potato brick after 60 min of thawing in a convective oven at 28C.
centre. Lowering the thermal conductivity by 50% has a greater effect on thawing time than increasing it by 50% from the nominal value. (The change of thermal conductivity and enthalpy was made by multiplication of all numerical values for the whole temperature range in Table 3). Thawing time is almost independent of the changes in the initial freezing point. The results of the above analysis can be used also for assessing the accuracy of the model predictions when inaccurate process or physical parameters are sometimes input into the calculation. The mathematical modelling of the thawing process can be used for prediction of the time-temperature history for any point of the thawed body and at any time during thawing. Fig. 7 shows the temperature distribution at the centre of the thawing food (mashed potatoes) after 60 min of thawing in the convective hot-air oven at 28C. The effect of overheating of the sides and corners is apparent from this figure.
4. Conclusions The modelling software FPM can be used for prediction of temperature profiles and time dependencies during thawing with reasonable accuracy. The accuracy of calculations was tested by comparison of predicted time-temperature dependencies with experimental data measured at three different points using three different thawing arrangements with changing temperatures of
Acknowledgements This work was financed by Grant No. 101/99/1617, Grant agency of the Czech republic and Grant No. EP6260 of the Ministry of Agriculture of the Czech republic.
References Allen, A. R., Liu, M., & Nesvadba, P. (1997). Development of integrated software for the modelling of thermal processing of foods. In Proceedings of the conference, modelling of thermal properties and behaviour of foods during production, storage and distribution. Prague. Decarean, R. V., & Peterson, R. V. (1986). Microwave processing and engineering. London: VCH. Hoke, K., Klima, L., Gree, R., & Houska, M. (2000). Controlled thawing of foods. Czech Journal of Food Science, 18, 194–200 (in Czech). Kharabadzov, O. (1989). Thawing of ready meals by steam. Khranitelna–Promishlenost, 38(1), 21–23. Klima, L. (2000). Controlled thawing of foods. MS Thesis, Faculty of Mechanical Engineering, CTU Prague (in Czech). Food Product Modeller, version 2.00.988, 2000, MIRINZ, New Zealand: Food Technology & Research. Merts, I., & Lawson, C. R. (1999a). Thawing meat by water immersion and water spray. In 20th International congress of refrigeration. IIR/ IIF, Sydney, Paper No. 349. Merts, I., & Lawson, C. R. (1999b). Comparison of air, water and plate thawing methods for meat. In 20th International congress of refrigeration. IIR/IIF, Sydney, Paper No. 379. Morep (1999). Leaflet of Morep process systems limited. Halifax, West Yorkshire, England. Rosenberg, U., & Bogl, W. (1987). Microwave thawing, drying and baking in the food industry. Food Technology, 85–89. Steele, R. J. (1987). Microwaves in the food industry. CSIRO Division of Food Research, 47, 73–78.
Use of a computer program for parameter sensitivity studies during thawing of foods K. Hoke, M. Houska *, K. Kyhos, A. Landfeld Food Research Institute Prague, Radiova 7, 102 31 Prague 10, Czech Republic Received 9 December 2000; accepted 15 May 2001
Abstract Thawing is one of the important processes of the food industry. Its speed and conditions can influence the quality and initial concentration of the microbial population in raw materials used for further processing. Experimental prediction of the optimum conditions of the thawing process is very time consuming and expensive. Therefore, a mathematical model of the process can help in the design of efficient thawing apparatus and process conditions. This paper compares experimental data of different thawing procedures with numerically calculated predictions using the software package food product modeller (FPM). The model calculations agree very well with experimental data on three different thawing process arrangements with reasonable accuracy. The verified mathematical model for convective thawing was selected for sensitivity analysis of the process parameters. The time of thawing was calculated as a function of different parameters for three different points in the thawed food piece. Graphs of the relative change of thawing time as a function of the relative change of the process parameters have been constructed. From these graphs one can read that the air temperature, surface heat transfer coefficient (SHTC), thickness of the layer, water content and enthalpy are the parameters with the greatest influence on thawing time. On the other hand the thermal conductivity or initial freezing point does not influence the thawing time as much. 2002 Elsevier Science Ltd. All rights reserved. Keywords: Thawing; Mathematical modelling; Sensitivity analysis; Thawing time
1. Introduction Thawing is a thermodynamic process applied in many branches of the food industry. The process of thawing should be controlled. The surface temperature of different products should be kept under a predefined level, e.g., 10–15C. The time of thawing should be as short as possible to minimise the potential growth of microorganisms and spoilage of the raw material before the final processing. Also the water or juice losses and quality changes (colour, oxidation) can be minimised if the thawing is made in a special apparatus with controlled time-temperature profiles. One of the most frequently used designs is convective hot air thawing. This type of thawing is used because of low operating and investment costs. On the other hand the time of thawing is relatively long. The typical design is described in the producer’s documentation, e.g.,
*
Corresponding author. Tel.: +420-272-702-331; fax: +420-272-701983. E-mail address: [email protected] (M. Houska).
Morep (1999). Heated air is circulated in a closed chamber filled with the food, placed on trays to enable airflow between the layers. Control is executed on the inlet air temperature depending on measured temperatures of the food and temperatures of the air in the chamber. Another process is microwave thawing. This process can be used for large pieces of frozen products such as meat, fish blocks, etc. for increasing the food temperature from 18C up to 4C (tempering). The frequency mostly used in Europe is 2450 MHz with 915 MHz used in the USA (Decarean & Peterson, 1986; Rosenberg & Bogl (1987); Steele, 1987), this being more suitable for large food products. Steam thawing is also used (Kharabadzov, 1989). Ready frozen meals consisting of meat, vegetables and sauce were thawed in steam at a temperature of 98C. Samples were placed into dishes with height of 10–75 mm. If there was an air layer between the food and its packaging material, the thawing time was extended by 30% over cases where the meals were vacuum-packed. The quality of the thawed meals was very high and mass loss was minimised.
0260-8774/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 0 - 8 7 7 4 ( 0 1 ) 0 0 1 0 9 - 1
220
K. Hoke et al. / Journal of Food Engineering 52 (2002) 219–225
Thawing can be also achieved by using a liquid. Water is mostly used as the heat-transmitting medium. Samples are submerged into a bath where the medium can have different temperatures or the sample can be sprayed with the warm medium, Merts and Lawson (1999a,b). The temperature of the medium can be selected carefully especially in the case of large food pieces. Spraying is frequently used to avoid freezing of the water on some parts of the surface since an ice layer can substantially lower the heat transfer. Merts and Lawson (1999a,b) refer also to controlled thawing by contact heating between two plates. Plates are heated by a circulating medium pumped into the cavities in the plates. This process is convenient for heat sensitive foods because the temperature of the plates can be carefully controlled. Our previous experimental work compares different thawing arrangements from the point of view of thawing time, (Hoke, Klima, Gree, & Houska, 2000). The processes were controlled so as to limit the surface temperature to a prescribed level. It was found that controlled vacuum-steam thawing is a very efficient method comparable with microwave thawing. The experimental prediction of the optimum conditions of thawing is very time consuming and expensive. If a mathematical model of the process can be developed, more efficient design and process control can be obtained. Mathematical modelling of thawing is complicated by the changes of physical properties during phase transition, heterogeneity of the food and mostly by the unknown surface heat transfer coefficient (SHTC) under the process conditions. This paper focuses on mathematical modelling of the process and its verification. Also a sensitivity analysis of the process parameters on thawing time for one thawing arrangement is presented.
2. Materials and methods The solution of differential equations describing the conduction of heat during thawing of a brick of frozen mashed potatoes was carried out using the software Food Product Modeller (FPM), 2001, MIRINZ Food Technology & Research, New Zealand. FPM allows simulating of the effects of processing regimes on food products. The underlying finite difference method of calculation can deal with the following shapes: infinite slab, infinite cylinder, sphere, infinite rod, rectangular prism (or brick), finite cylinder (or can). The calculation accepts input data to specify the thermal properties of the simulated material (heat capacity, thermal conductivity) and its boundary conditions (temperature, surface heat transfer coefficient). Thermal properties for over 70 food materials are included in the program. The data preparation procedure allows the user to specify these
properties in terms of the user problem domain (e.g., by specifying air velocity and packaging type rather than transfer coefficient). Boundary conditions may be varied over the period of process. We have chosen a brick geometry that was similar to the model food used for experimental testing of different thawing procedures. The calculation was carried out in 3D space. Only onequarter of the brick was analysed thanks to axial symmetry. Nodes were uniformly distributed in the brick ð20 20 20 ¼ 8000Þ. The length of time step was 5–6 s. Thawing experiments were carried out on a model food – mashed potatoes prepared by mixing a dried commercial product (450 g) with 2000 ml of water heated to the 70C. Basic chemical composition of the final product is given in Table 1. A flat cartoon tray having dimensions 120 100 45 mm was used for sample preparation. The hot prepared mashed potatoes were placed into the trays and vacuumed to remove the air bubbles. After cooling at ambient atmosphere the trays of potatoes were placed into the freezer and stored at 22C. The samples were removed from the trays before the experiments and holes were drilled into the frozen body for placement of thermocouple probes. Temperatures were measured at three points of the thawed sample body: in the centre (point 1), in the layer 10 mm under the surface above the centre (point 2) and in the layer 2 mm under the surface above the centre (point 3). The comparison with the mathematical model was made for the conditions of the following three selected experiments: • Thawing of the model food in a convective hot-air oven without air heating (air temperature 28C) with intensive air circulation – experiment no. 1. • Thawing of the model food in a convective hot-air oven at an air temperature of 70C with intensive air circulation – experiment no. 2. The surface temperature (measured at point 3) was regulated by stopping the heating and air circulation as soon as 15C was reached. As soon as the temperature was lowered to 11C, the heating and air circulation was switched on. • Thawing of the model food in vacuum cooker. The vacuum pump was switched on after closing the cooker and the absolute pressure was reduced to 10 kPa.
Table 1 Chemical composition of mashed potatoes Component
Concentration (%)
Water Protein Starch Fat Ash
82.7 1.2 15.2 0.4 0.5
K. Hoke et al. / Journal of Food Engineering 52 (2002) 219–225
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Table 2 Parameters for numerical modelling of thawing Parameter
Units
Experiment no. 1
Dimensions of the sample Layer thickness Ambient temperature Surface heat transfer coefficient Initial freezing point Density above freezing point Density below freezing point Initial temperature of the sample Thermal conductivity of the sample Enthalpy
(mm) (mm) (C) (W m2 K1 ) (C) (kg m3 ) (kg m3 ) (C) (W m1 K1 ) (MJ m3 )
120 100 40 40 28 32 )1 1050 960 )22
Input parameters for modelling the different thawing experiments are given in Table 2. The measured timetemperature dependence of the surrounding atmosphere during thawing (air, steam) was input into the software. The SHTC was predicted from preliminary experiments made on the same experimental set-up while heating an aluminium model of the sample of mashed potatoes, Klima (1999). The optimum SHTC was predicted by numerical modelling of the heating to give the best fit to the experimental heating curve. For the convective hotair oven and a surrounding air temperature of 70C the optimum value of SHTC was found in the range between 35 and 40 W m2 K1 . The optimum value of SHTC for the vacuum cooker with the fully opened steam valve at an absolute pressure 10 kPa was predicted at around 1000 W m2 K1 . The thermal conductivity and enthalpy of mashed potatoes, see Table 3,
Table 3 Predicted values of thermal conductivity and enthalpy using Costherm 3.11 for mashed potatoes Thermal conductivity (W m1 K1 )
Enthalpy (kJ kg1 )
)22.0 )20.0 )17.5 )15.0 )6.0 )3.5 )2.25 )1.5 )1.25 )1.1 )1.0 )0.5 7.5 15.0 22.5 30.0 40.0
2.0830 2.0581 2.0261 1.9921 1.7887 1.6063 1.3717 1.0399 0.8366 0.6675 0.5246 0.5253 0.5389 0.5510 0.5623 0.5729 0.5858
)287 )282 )275 )268 )224 )189 )146 )88.5 )54.5 )26.7 )3.71 )1.85 27.8 55.7 83.6 112 149
Experiment no. 3
120 100 40 120 100 40 40 40 See experiment See experiment 35 (8) 1000 )1 )1 1050 1050 960 960 )22 )22 Predicted with Costherm 3.11 Recalculated by using predicted values by Costherm 3.11
Then the steam valve was opened for the entire thawing process without regard to the surface temperature of the samples – experiment no. 3.
Temperature (C)
Experiment no. 2
were calculated by using the predictive software Costherm 3.11 (Allen, Liu, & Nesvadba, 1997) on the basis of the chemical composition and for the temperature range from 22C to 40C. The initial freezing point was predicted experimentally. Specific enthalpy calculated by the software Costherm 3.11 was recalculated with the help of the corresponding density values above and below the initial freezing point in MJ m3 for subsequent use in modelling software FPM. The software predicted the time-temperature histories at any point of the thawed model body. The points at which the experimental data readings were taken were selected for calculation.
3. Results and discussion The comparison of measured and calculated timetemperature histories for the conditions of experiment no. 1 at three points of the thawing body is shown in Fig. 1. Modelling the start-up of the experiment took account of manipulation time of 7 min (weighing of sample, introduction of probes, input of body into oven). The SHTC value of 10 W m2 K1 , air temperature of 20C, and 22C as the initial temperature of the body were used. This preliminary calculation meant that the initial temperatures (at time 0) were different at different positions in the body (see Fig. 1). This preliminary calculation ensured that the calculated timetemperature histories for thawing were more realistic. The surrounding air temperature of 28C and SHTC of 32 W m2 K1 were used for the thawing calculation. The similar comparison of measured and calculated time-temperature histories for the conditions of experiment no. 2 is shown in Fig. 2. It is apparent from this figure that the temperature at point 3 (2 mm under the surface) very quickly responds to changes in air temperature. The predicted temperature by FPM follows the experimental data fairly well. The numerical calculation has taken into account the changing conditions during thawing. The value 35 W m2 K1 of SHTC and a temperature of 70C were used in the first stage of
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Fig. 1. Thawing of mashed potatoes in a convective oven at 28C, comparison of experimental data with numerical calculation.
Fig. 2. Thawing of mashed potatoes in a convective oven at 70C, comparison of experimental data with numerical calculation.
thawing. An SHTC value of 8 W m2 K1 was used for modelling the situation with the fan and heater switched off. The variation in air temperature with time in the oven measured during the experiment was input into the calculation. An SHTC value of 35 W m2 K1 was used when the fan and heater were switched on. Modelling the start-up of the experiment took also account of manipulation time of 9 min (weighing of sample, introduction of probes, input of body into oven). The SHTC value of 10 W m2 K1 , air temperature of 20C and 22C as the initial temperature of the body were used. This preliminary calculation meant that the initial temperatures (at time 0) were different at different positions in the body (see Fig. 2). This preliminary calculation ensured that the calculated time-temperature histories for thawing were more realistic. Fig. 3 shows the comparison of measured and calculated time-temperature histories for the conditions of experiment no. 3. The comparison was done for three points in the food during thawing in the vacuum cooker. The temperature of the steam–air mixture changed from 22C (atmospheric air) to 17C (full vacuum) and then
up to 41C (full vacuum and full steam flow). The time necessary for closing the cooker was included in the modelling (about 7 min), see Fig. 3. The preliminary manipulation time (10.2 min) necessary for weighing, introducing the thermocouples and installing the food in the cooker was modelled by preliminary calculation. The temperature of the thawed body 22C (uniform), surrounding air temperature 20C and SHTC of 8 W m2 K1 were used. The temperature profile was calculated for zero time of thawing process, see Fig. 3. The same values of air temperature and SHTC used for the preliminary stage of the modelling were used for modelling the closing time of the vacuum cooker. The experimentally predicted time-temperature history of the inner atmosphere in the cooker was used in calculating temperatures during the vacuum and steaming stages in the cooker. An SHTC value of 1000 W m2 K1 was taken for modelling this stage of the thawing. The model calculations of the local time-temperature histories for the three different thawing arrangements agree with experimental data with reasonable accuracy
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Fig. 3. Thawing of mashed potatoes in a vacuum cooker, comparison of experimental data with numerical calculation.
(see Table 4). The use of inexact input parameters can cause differences between the experimental and calculated results (e.g. the thermal conductivity and enthalpy of the model material were predicted from chemical composition only). The density changes with temperature were ignored (constant values of densities predicted experimentally were used for temperatures below and above the initial freezing point, respectively). Also experimental errors in measured temperatures (0:2C for experiments no. 1 and 2; 0:5C for experiment no. 3) can be taken into account. The calculated thawing time for experiment no. 1 was 122 min (measured 110 min). The calculated thawing time for experiment no. 2 was 72 min (measured 58 min). The calculated thawing time for experiment no. 3 was 46 min (measured 36 min). In spite of these facts, the accuracy of modelling the thawing process seems to be good for the purpose of thawing time predictions and, therefore, the model can be regarded as verified. Therefore, the model could be used for sensitivity calculations of thawing time on changes in the input parameters (process parameters). The basic calculations were carried out using the conditions of experiment no. 1 (convective thawing in the hot-air oven at room temperature 28C). The thawing time for each point was defined as the time taken for this point to increase in temperature above the initial freezing point 1C. The input parameters were changed by a selected percentage below or above the mean values of the parameters of
Table 4 Standard deviations between measured and calculated temperatures
Point 1 Point 2 Point 3
Experiment no. 1 S.D. (C)
Experiment no. 2
Experiment no. 3
1.26 1.31 0.55
1.32 1.15 0.88
2.28 1.65 1.35
simplified experiment no. 1, see Tables 1–3 (no manipulation time, initial temperature of food 20C). Only one parameter was changed and the others were kept constant at nominal values. The response of the thawing time was calculated as a function of change of temperature of the surrounding air, SHTC, thickness of the thawed layer, physical properties and the composition of the thawed food (enthalpy, thermal conductivity, initial freezing point and water content). Calculations were conducted for three different points in the body (at the centre of the food – point 1, at one-quarter of the depth and above the centre – point 2 and at one-twentieth of the depth and above the centre – point 3). The input parameters were changed stepwise by 10% up to 50% from the nominal values. The water content was relatively changed from 6 to 18% and its change was compensated by a change in the starch content. Nominal calculated thawing times for points 1 (centre), 2 and 3 were 121.5, 87.1 and 42.1 min, respectively. The calculated thawing times for other than nominal combinations of input parameters were recalculated as a relative change of nominal thawing time. The results of the calculations are given in three graphs, Figs. 4–6. Fig. 4 shows the influence of change of individual input parameters on the relative change in thawing time at point 1 (centre of the body). Fig. 5 shows the influence of individual input parameters on the relative change in thawing time at point 2. Fig. 6 shows the influence of individual input parameters on the relative change in thawing time at point 3 (near the surface of the body). The thawing time for the arrangement studied is strongly influenced by air temperature and SHTC especially when they are decreasing. Their influence is most apparent at point 3 near the body surface. Their influence is comparatively lower at point 1 (centre of the body). The thickness of the body, enthalpy and water content are also strongly influential parameters in the model. Thermal conductivity has much less influence on thawing time at the body surface than at the body
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Fig. 4. Relative change in thawing time as a function of the size of different parameters, point 1 in the body, standard thawing time 121.55 min.
Fig. 5. Relative change in thawing time as a function of the size of different parameters, point 2 in the body, standard thawing time 87.1 min.
Fig. 6. Relative change in thawing time as a function of the size of different parameters, point 3 in the body, standard thawing time 42.1 min.
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the surrounding atmosphere. Accurate values of process parameters and physical properties of the thawing food should be input into the calculation. The most important parameters are the temperature of the surrounding atmosphere, the SHTC, the dimensions of the body and the temperature dependence of the specific enthalpy of the thawing food. The influence of thermal conductivity and initial freezing point on thawing time is weaker than the influence of the above-mentioned parameters. These results are derived from a sensitivity analysis of the input parameters on the thawing time prediction. The thawing process can be simulated if the proper values of these parameters are known.
Fig. 7. Calculated temperature profile in the centre of the mashed potato brick after 60 min of thawing in a convective oven at 28C.
centre. Lowering the thermal conductivity by 50% has a greater effect on thawing time than increasing it by 50% from the nominal value. (The change of thermal conductivity and enthalpy was made by multiplication of all numerical values for the whole temperature range in Table 3). Thawing time is almost independent of the changes in the initial freezing point. The results of the above analysis can be used also for assessing the accuracy of the model predictions when inaccurate process or physical parameters are sometimes input into the calculation. The mathematical modelling of the thawing process can be used for prediction of the time-temperature history for any point of the thawed body and at any time during thawing. Fig. 7 shows the temperature distribution at the centre of the thawing food (mashed potatoes) after 60 min of thawing in the convective hot-air oven at 28C. The effect of overheating of the sides and corners is apparent from this figure.
4. Conclusions The modelling software FPM can be used for prediction of temperature profiles and time dependencies during thawing with reasonable accuracy. The accuracy of calculations was tested by comparison of predicted time-temperature dependencies with experimental data measured at three different points using three different thawing arrangements with changing temperatures of
Acknowledgements This work was financed by Grant No. 101/99/1617, Grant agency of the Czech republic and Grant No. EP6260 of the Ministry of Agriculture of the Czech republic.
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