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Prediction of thawing times for foods of simple shape
Prediction of thawing times for foods of simple shape D. J. Cleland, A. C. Cieland, R. L. Earle and S. J. Byrne* Massey University, Palmerston North, New Zealand Received 1 January 1986
A set of 104 experimental measurements of thawing time were made over a wide range of conditions for slab, infinite cylinder and sphere shapes of a food analogue material. These results were used to assess existing thawing time prediction methods. Versions of both the finite difference and the finite element numerical methods that accounted for continuously temperature-variable thermal properties gave accurate predictions. No previously published simple prediction formula was found that was both sufficiently accurate and expressed in a form suitable for it to be adopted as a general thawing time prediction method. Four accurate, but simple, empirical formulae based on Plank's equation were developed. These formulae predicted thawing times that were both highly correlated with those predicted by the numerical methods and agreed with the experimental data to within + 10%o at the 95~ level of confidence. The agreement was more limited by uncertainties in the experimental and thermal property data than by inaccuracy in the prediction formulae. Significantly more accurate simple formulae are unlikely to be developed unless more accurate experimental data are available. (Keywords:foodproducts;thawing;thawingtimes)
Pr vision des temps de d cong lation des aliments de forme simple Une s&ie de 104 mesures exp&imentales du temps de dkcongblation a &tk effectube pour un grand nombre de conditions de formes : plaque, cylindre infini et sphere, d'un mat&iau analogue h un aliment. On a utilisb ces r~sultats pour ~valuer les m&hodes existantes de prbvision du temps de dbcongblation. Des versions des m&hodes num&iques aux diff&ences finies et blkments finis, tenant compte des propribtks thermiques variant continuellement en fonction de la temp&ature, ont donnb des prbvisions prkcises. On n'a pas trouvk parmi les formules de pr~vision simple publibes des formules suffisamment pr~cises et exprim~es sous une forme permettant de les adopter comme m~thode 9bn&ale de pr~vision du temps de d~congblation. Quatre formules empiriques, prkcises, mais simples, ont btb &ablies d'aprOs l'bquation de Plank, Ces formules prbvoient des temps de d~conoblation fortement correl~s avec ceux pr~vus par les mbthodes num&iques et concordant avec les r~suhats exp&imentaux fi -t-10% prds avec un taux de confiance de 95%. La-concordance btait plus limitbe par les incertitudes des rbsultats exp&imentaux et des propri&bs thermiques que par l'inexactitude des formules de prbvision. Des formules simples nettement plus prbcises ont peu de chance d'btre raises au point, fi moins qu'on ne dispose de rbsultats exp&imentaux plus prkcis. (Mots cl~s: aliments; decongblation; temps de d~congblation)
Recently, with greater quantities of goods being frozen and particularly with greater emphasis on further processing of frozen products, thawing has become an important industrial process. Accurate prediction of thawing times for a wide range of conditions and geometries would enable design, operation and control of thawing equipment to be optimized so that product quality is maintained while processing costs are kept low. This paper considers the problem of predicting thawing times for the simple slab, infinite cylinder and sphere shapes. Physically, thawing of slabs, infinite cylinders and spheres of food is governed by the partial differential equation for one-dimensional heat conduction in a solid, subject to the appropriate initial and boundary conditions:
where a = 0, 1 or 2 for slabs, infinite cylinders and spheres, * Present address: Auckland University, Auckland, New Zealand 0140-7007/86/040220~9503.00 ~') 1986 Butterworth & Co (Publishers) Ltd and IIR
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Int. J. Refrig. 1986 Vol 9 July
respectively, and the thermal properties, k and C, are temperature variable. Typically, foods are frozen to temperatures substantially below their thawing temperature to prevent chemical and biological deterioration. The frozen storage temperature defines the process initial condition. The third kind of boundary conditions (convective heat transfer to the food surface) is both the most physically realistic and the most common boundary condition for freezing and thawing of food 1 so only it need be considered: h(T~ -
T) =
k ~T
(2)
~r r=D/2
To complete the definition of thawing an endpoint for the process must be defined. For any food, thawing must be complete once the thermodynamic centre temperature reaches 0°C, yet it is desirable for chemical and biological reasons that the mass average temperature is as low as possible at the end of the process. A process endpoint of 0°C at the thermodynamic centre is therefore the most appropriate.
Thawing times of simple shaped foods: D. J. Cleland et al. Nomenclature A Area (m 2) Bi Biot Number (= hd/kl) C Volumetric specific heat capacity (j m-3 oC -1 ) corr FDM Correlation coefficient compared with the percentage differences for the finite difference method D Thickness or diameter (m) Fo Fourier Number (= kltt/GD 2) h Surface heat transfer coefficient (Wm-2oc -1 ) AH Change in enthalpy from - 10°C to 0°C (J m - 3) k Thermal conductivity (W m- 1 °C- t) L Latent heat (J m-3) max Maximum percentage difference rain Minimum percentage difference Pk Plank Number (= Cs(Tf-Tin)/AH) r Distance from the centre, radius (m) Mathematical solution of Equations (1) and (2) enables a method for thawing time prediction to be derived. There are no exact solutions because of the non-linearity that arises from both the temperature variable thermal properties of foods and the nature of the boundary condition. Approximate solutions fall into two broad groups - those requiring computer calculations (numerical methods) and those where the computations can be carried out by hand (simple formulae). Both numerical methods and simple formulae will be used to predict thawing times as computing facilities are not always available or convenient to use. Numerical methods (finite differences and finite elements) can take into account the temperature variable thermal properties, k and C, and consequently model the physical behaviour of food materials very closely for both freezing and thawing2. If they are correctly formulated and implemented these methods should give accurate prediction of thawing times 3. For prediction of freezing times for slabs, infinite cylinders and spheres both the finite difference method 4-6 and the finite element method T'8 have been formulated, implemented as computer programs and tested against experimental data. They were shown to predict freezing times to within ___10~o. These programs were available and were subsequently used for thawing time prediction. A simplifying assumption commonly made to facilitate analytical solution of phase change problems is that all the latent heat is added or removed at a unique temperature ( ~ and that thermal properties undergo a step change in value at this temperature. Although this assumption of a unique thawing temperature is true in some situations such as the melting of pure liquids, it is not true for most foods where the latent heat is added over a range of temperatures below Tr. The possibility of deriving prediction methods by using this assumption has been explored extensively as indicated by the bibliographies given in various reviews 4,7,9 14. However, the resulting methods are only approximate for predicting thawing times of foods. Many attempts have been made either to modify these existing analytical solutions to reduce their limitations and give them less dependence on assumptions made in
SD
Standard deviation of percentage differences Stefan Number (=CI(Ta- Tf)/AH) Time (s) Temperature (°C) Volume (m 3)
Ste t T V Subscripts a ave f fave fin i in 1 s t
Ambient thawing medium Average final Freezing or thawing Average for freezing or thawing Final centre ith case Initial Unfrozen Frozen Thawing
their theoretical development, or to develop simple formulae on an entirely empirical basis 4'7' 15,16. Although most of these modifications have been derived for freezing problems, the same general approaches can also be applied to thawing. However, the simplistic approach of treating thawing as the reverse of freezing has not been proved to be accurate. A search of the literature showed that those methods specifically derived for thawing are nearly all product specific. Commonly these simple formulae are based on the well known Plank's equation 17 for slabs, infinite cylinders and spheres: t 2V
a-- f\0"5
+0.125~-
(3)
or in dimensionless form:
AD Fo 2 T
0.5 0.125 Bi Ste i Ste
(4)
Plank assumed a unique phase change temperature, constant thermal properties and that sensible heat effects above and below the phase change temperature are negligible compared with the latent heat. Equations (3) and (4) suggest that when these assumptions are valid, the ratio of thawing times for slabs, infinite cylinders and spheres under identical conditions and with the same characteristic dimension is 6:3:2. This ratio arises from the relationship between volume and surface area for each shape. However, this ratio can be shown to apply only when the surface is isothermal with respect to position 7. In thawing of biological materials the above conditions are not all met, so the 6:3: 2 ratio is not necessarily correct. Assuming the ratio to be constant allows a unified approach, in which a single prediction method is equally applicable for all three simple shapes, to be used. However, there is a different distribution of volume with respect to displacement from the geometric centre for the three shapes, which could lead to deviations from the theoretical ratio. Previous studies for freezing 18-2° suggested that within the tolerance of the data they used, there was no error in assuming that the ratio was constant at 6:3:2.
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Thawing times of simple shaped foods. D. J. Cleland et al. An investigation of the validity of the ratio for thawing has been carried out using numerical method predictions for the three simple shapes. It showed that on average the 6:3:2 ratio holds for a wide range of conditions representative of typical thawing operations 7. There were discernible effects related to the variations in the Biot number (Bi), Stefan number (Ste), and Plank number (Pk), but it was concluded that these were sufficiently small compared with other probable sources of uncertainty in thawing time predictions that they could be ignored. Therefore the prediction of thawing times for slabs, infinite cylinders and spheres can use a single unified approach. Ultimately, verification of the accuracy of any method to predict thawing times must be made by comparison with accurate experimental data 3. The most useful data set would be large and diverse in order to differentiate between errors due to imprecise control and measurement of thawing conditions (experimental error), error due to uncertainty in thermal property data (data error) and inaccuracy arising from the prediction method itself (prediction error) 15. No such data set for thawing of foods could be found in the literature, which probably explains why no comprehensive assessment of existing prediction
1 T h e r m a l p r o p e r t y d a t a used in calculations with simple formulae T a b l e a u 1 Propriktbs thermiques utilisbes dans les calculs avec des Table
formules simples Property
Tylose
M i n c e d lean beef
ks (W m - 1 oC - 1) kl (W m - 1 ° C - 1) Cs ( M J m - a ° C - 1) CI ( M J m - 3 ° C - 1) A H (MJ m - a ) Tr (°C) 7~ave (°C)
1.65 0.55 1.90 3.71 226 - 0.6 -2.1
1.55 0.50 1.90 3.65 230 - 1.0 -2.5
2 Tableau
Table
methods has been made. The aims of the present work are therefore: 1. to collect accurate thawing data for the simple slab, infinite cylinder and sphere shapes; 2. to assess the accuracy of existing thawing time prediction methods for these shapes; and, if necessary, 3. to develop new and improved thawing time prediction methods. Experimental Full details of the experimental procedures can be found elsewhere7. The techniques and equipment used were very similar to those used by Cleland and Earle 1'6'zl for experimental freezing of slabs, infinite cylinders and spheres. Tylose (Karlsruhe test substance), a defined 23~o methyl--cellulose gel, was used almost exclusively for the experiments because it is cheap and convenient to use, its thermal properties are well known, and thawing of Tylose closely models thawing of many real food materials 3'22. A small number of confirmatory runs were made with an actual food material (minced lean beef). For both materials the thermal property data given by Cleland and Earle 3 were used (Tables 1 and 2). All temperature measurements were made with copper/constantan thermocouples connected to a 12 point recording potentiometer, operating on a 100 second print cycle, that was calibrated to within 0.3°C in the range - 50 to 50°C. Before thawing, the objects of Tylose were kept in constant-temperature cold stores. In spite of insulating material being used to prevent temperature change during handling of the objects immediately before thawing, the initial temperatures within the objects varied by up to 2.0°C about the mean values. Water was used as the thawing medium. A well mixed 1 m 3 immersion tank with both heating and refrigeration elements, actuated by on/off control, gave a large stable ambient heat source.
Thexrnal p r o p e r t y d a t a used in calculations with n u m e r i c a l m e t h o d s
2 Propri~t~s thermiques utilisdes dans les calculs avec des formules numdriques M i n c e d lean beef
Tylose T (°C)
C ( M J m - 3 ° C - 1)
T (°C)
k (W m - 1 ° C - ~)
T (°C)
C ( M J m - 3 oC
40.0 30.0 20.0 18.0 16.0 14.0 12.0 - 10.0 - 9.0 - 8.0 - 7.0 - 6.0 -5.0 -4.0 -3.0 - 2.5 - 2.0 - 1.5 1.3 0.6 -0.5 0.0 40.0
1.88 1.92 1.95 2.00 2.20 2.30 2.80 3.70 4.20 5.00 5.90 7.20 11.0 17.0 25.0 33.0 45.0 70.0 100 100 19.9 3.71 3.71
-
1.67 1.67 1.66 1.64 1.63 1.61 1.60 1.58 1.56 1.52 1.46 1.35 1.28 1.18 1.04 0.82 0.66 0.55 0.49 0.49 0.56 0.62
-
1.89 1.91 2.02 2.70 3.58 4.55 5.26 7.71 10.4 15.7 31.2 53.1 73.7 255 255 3.66 3.66
-
-
222
40.0 30.0 20.0 15.0 10.0 -9.0 - 8.0 - 7.0 - 6.0 - 5.0 - 4.0 - 3.0 -2.5 -2.0 - 1.5 - 1.0 - 0.8 - 0.7 - 0.6 0.0 20.0 40.0
Int. J. Refrig. 1986 Vol 9 July
40.0 30.0 25.0 20.0 15.0 10.0 - 8.0 - 6.0 - 5.0 - 4.0 - 3.0 - 2.0 - 1.5 - 1.3 - 1.2 - 1.0 40.0
T
-
1)
(oc) - 40.0 - 24.0 12.0 -8.0 -4.0 -2.0 -1.0 40.0
k (Wm-I 1.58
1.53 1.48 1.40 1.28
0.98 0.48 0.49
oc-1)
Thawing times of simple shaped foods: D. J. Cleland et al. Variations in water temperature over the course of an experiment were not detectable with the recording equipment used. Taking recorder calibration error into account, the overall error in temperature measurement and control was estimated to be less than +0.5°C. One-dimensional heat transfer in an infinite slab was approximated by a finite-sized slab insulated on all but two faces in an arrangement similar to a plate freezer. Water from the immersion tank was circulated through two 0.4m x 0.4 m plates at a sufficiently high rate that there was no observed change in water temperature across the plates. Good thermal contact between the slab surface and the plates was provided by a 100 kg weight put on the top plate. Slabs of 0.025-0.1 m (___0.5mm) in thickness were used. The slabs were pre-frozen in a plate freezer to ensure a constant thickness, a fiat surface and hence good thermal contact with the plates. Each slab was a 0.22 m diameter of Tylose surrounded by 0.09 m of polystyrene insulation around the edges to ensure only onedimensional heat transfer. Seven thermocouples were embedded at various places in each slab to measure the centre and surface temperatures. The two surfaces of each slab were wrapped with tinfoil and brown paper to prevent moisture losses and to give structural strength. Varying numbers of rubber sheets were inserted between the plates and the slabs to provide thermal resistance during thawing. The resulting surface heat transfer coefficients (h) were measured by the methods given by Cleland and Earle 23 using separate experiments at temperatures greater than 0°C. From variations in the estimated h values and in four repeated thawing experiments, the overall experimental error was estimated to be + 5.3~ at the 95~o level of confidence, indicating a high degree of repeatability. In the infinite cylinder and sphere experiments the objects were immersed directly in the water tank and oscillated through 300 ° every 30 s. This ensured that heat transfer was uniform across the whole object surface. The cylinders were made by filling 0.45 m lengths of 0.050.15 m diameter steel or PVC plastic pipes with Tylose. Centre and surface thermocouples were introduced through isothermal regions, and polystyrene foam 'caps' were put on the ends of the cylinders to reduce longitudinal heat flow to a minimum, thereby closely approximating an infinite cylinder. The different pipe wall materials and thicknesses gave some variation in the surface heat transfer coefficient, but greater variation was achieved by gluing sheets of rubber onto the outside of each cylinder for some of the experiments. This also ensured that the range ofh values included those normally encountered in air thawing. As for the slab experiments, separate heating experiments above 0°C were used to estimate h for each combination of wall type and rubber covering. A problem was encountered owing to imperfect contact of the Tylose to the inside surface of the cylinder walls during thawing as slight shrinking of the Tylose and expansion of the pipe wall material occurred. The air voids created were found to be randomly distributed and ill defined in size so it was impossible to evaluate their individual effect on heat transfer directly. The best way to account for them was to use an average value of h estimated by the heat penetration method 24 based on measured thermodynamic centre temperatures. The effect of any surface variation in the heat transfer conditions was effectively averaged by this procedure. The experimental
error, estimated to be 6.1~, for the cylinder experiments was larger than for the slabs, This reflects the increased uncertainty in the control and measurement of the surface heat transfer coefficient. Hollow metal balls were used to model the spherical geometry. Three sizes, approximately 0.05, 0.10 and 0o13m in diameter were used. Each sphere was constructed from two hemispheres. These were filled with Tylose, thermocouples were inserted and then the two halves were welded or soldered together. To seal each sphere, especially around the thermocouple leads, and to give a variation in the surface heat transfer coefficient, the spheres were coated with layers of silicone rubber after construction. The same problem with thermal contact that was observed for the infinite cylinders occurred for the spheres, so some small random variation of h over the surface was expected. Again the heat penetration method 24 using the centre temperature profiles from independent heating and cooling experiments without phase change were used to estimate h for each combination of sphere and rubber coating thickness. More variation in the measured h values occurred than for the slabs and infinite cylinders. Consequently, the overall 95~o confidence bound for the experimental error was estimated to be slightly higher at _ 8.0~o.
Results and discussion
Thawing data for slabs, infinite cylinders and spheres For thawing of a homogeneous material under constant environmental conditions to a final thermodynamic centre temperature of 0°C, the time taken is affected by six factors - shape, thermal properties, size (D), surface heat transfer coefficient (h), thawing medium temperature (Ta), and initial temperature (Ti,). For each shape an experimental design was set up to cover a wide range of conditions that are commonly found in food thawing situations. The heating medium temperature was varied from 5 to 45°C, the initial temperature of the material to be thawed was varied from - 10 to - 33°C and the surface heat transfer coefficient was varied from 13 to 246 W m - 2 oC -1. Because D and h could not always be set to pre-selected levels, the design was not orthogonal. Therefore it was decided not to attempt to control T~and T~, exactly to the pre-selected levels; instead, provided a value close to the desired levels was obtained it was considered satisfactory and care was then taken to control the conditions as close as possible to these constant values. Tables 3, 4 and 5 show the experimental conditions and measured thawing times for the thermodynamic centre temperature to reach 0°C for Tylose slabs, infinite cylinders and spheres, respectively.
Assessment of existing thawing time prediction methods The first part of Table 6 compares the thawing times predicted by the finite difference and finite element numerical methods with the experimental thawing times. The percentage differences were calculated from: ~o difference =
predicted t i m e - experimental time 100 xexperimental time 1
(5)
Rev. Int. Froid 1986 Vol 9 Juillet
223
Thawing times of simple shaped foods. D. J. Cleland et al. Table 3 Experimental data for thawing of slabs of Tylose Tableau 3 Rbsultats expbrimentaux de dbcongklation de plaques de
Tylose Run
D (m)
h (W m 2 °C- t)
"/a (C)
7in (:'C)
tt (h)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
0.0260 0.0260 0.0525 0.0260 0.0525 0.0525 0.0260 0.0525 0.1000 0.1020 0.0770 0.0525 0.0770 0.0525 0.0260 0.1000 0.0525 0.0770 0.1000 0.0260 0.0525 0.0525 0.0525 0.0525 0.0525 0.1050 0.0770 0.0280 0.0525 0.1050 0.0770 0.0770 0.1000 0.0770 0.1050
13.2 24.5 13.2 24.5 13.2 13.2 50.4 24.5 13.2 13.2 18.2 29.5 37.3 50.4 78.1 24.5 50.4 37.3 24.5 78.1 50.4 50.4 50.4 50.4 78.1 37.3 50.4 172.7 78.1 78.1 78.1 78.1 78.1 78.1 172.7
12.8 5.2 5.2 45.9 46.1 12.8 4.6 5.2 45.9 43.0 46.2 12.8 5.2 5.1 5.2 12.8 46.0 46.1 45.8 12.4 13.4 13.4 13.4 13.4 5.2 5.2 45.7 43.0 13.3 5.2 5.0 45.7 46.2 12.9 13.4
- 20.9 - 29.4 - 11.4 - 8.3 -28.6 -20.5 - 26.3 - 10.7 -8.3 - 13.7 -26.8 - 20.9 - 30.2 -25.0 - 12.3 -9.4 - 10.6 - 24.7 - 32.5 - 27.7 - 20.2 - 24.1 - 23.6 - 23.6 -28.9 - 10.4 - 13.5 - 31.0 -22.5 -28.8 - 14.2 -9.4 -28.2 -21.0 -23.8
4.69 5.63 19.32 0.90 3.89 10.33 3.49 13.20 7.82 8.83 5.11 6.26 18.49 9.41 2.51 16.92 1.64 3.72 6.75 1.42 4.68 4.65 4.60 4.50 7.58 29.33 2.99 0.43 3.91 23.62 14.39 2.61 4.59 7.25 10.61
It should be noted that the finite element method used was that defined by Equations (5)-(7) of Cleland et al. 8 and not the simplified formulation (Equations (5a)-{7a)). The simplified formulation was developed by assuming no variation of the thermal properties with position within each element, in order to reduce the computation time. For one-dimensional heat transfer, computer computation time is not a limitation, so the most accurate finite element method was used. Applying the criterion of Cleland and Earle 3 to comparisons of the numerical method predictions with the experimental results showed no trends in the data that might suggest major systematic experimental or thermal property data error. The 95% confidence bounds of the numerical predictions are consistent with the estimated experimental error bounds as a whole and for each shape individually, suggesting that only minimal uncertainty has arisen in the application of the numerical methods. This was expected because relatively fine space and time intervals were used in the calculations. Predictions significantly better than those achieved by these numerical methods cannot be expected if the same experimental and thermal data are used, even if a more exact or sophisticated form of the numerical methods was used. As was expected, for the simple regular shapes the finite element method gave almost identical results to the finite difference method. Thawing times for the slab, infinite cylinder and sphere data were also calculated by existing simple prediction methods that considered the third kind of boundary
224
Int. J. Refrig. 1986 Vol 9 July
condition 7. Many existing methods had been developed specifically for freezing time prediction. Where possible the analogous form of the method for thawing was developed and used. Table 7 summarizes the prediction accuracy for the best methods as well as several poorer, but well known, methods. Plank's equation ~ gave a mean prediction error close to zero, but the spread of predictions was large and the correlation with the finite difference method results was poor. For thawing the under-prediction caused by ignoring sensible heat effects tends to compensate for over-prediction arising from assuming a unique phase change temperature and constant thermal properties. However, because the amount of each effect depends on the combination of environmental conditions, the amount of under- and over-prediction is variable and leads to the high standard deviation of predictions and the low correlation with the finite difference method results (which take these effects into account in a physically correct manner). The results for the method of Goodman 25 are typical of a large group of methods that are similar to Plank's equation, except that they take account of the sensible heat in the unfrozen phase. Although the mean difference is higher than for Planks's equation, the spread of predictions is reduced because some of the sensible heat effects are taken into account. However, the predictions are still poor because of the other limitations. There are a number of methods that modify analytical methods by multiplicative factors to account for all the Table 4 Experimental data for thawing of infinite cylinders of Tylose Tableau 4 R~suhats expbrimentaux de dbcongblation de cylindres infinis
de Tylose D
h
Run
(m)
(W m -2 C -
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34
0.158 0,158 0.158 0.156 0.156 0.156 0.156 0.156 0.156 0.156 0.106 0.106 0.106 0.103 0.103 0,103 0.103 0,103 0.103 0.103 0.103 0.103 0.103 0.051 0.051 0.051 0.051 0.051 0.051 0.051 0.051 0.051 0.051 0.051
23.5 23.5 23.5 90.7 90.7 90.7 43.5 43.5 43.5 43.5 113.0 113.0 113.0 37.4 37.4 37.4 25.1 25.1 25.1 25.1 19.5 19.5 19.5 46.5 46.5 46.5 27.9 27.9 27.9 27.9 19.0 19.0 19.0 19.0
~)
Ta
~,,
tt
("C)
(C)
(h)
43.3 21.1 5.1 43.2 13.0 5.1 40.3 11.9 8.2 5.3 43.3 8.5 5.1 43.3 13.0 5.3 40.3 18.7 13.2 5.3 43.9 18.3 5.8 44.0 8.5 5.1 40.3 18.9 13.2 5.3 43.9 14.6 9.6 5.8
- 14.0 -20.6 -28.4 - I1.9 - 13.6 -27.9 -21.2 - 27.4 26.9 - 14.9 - 10.7 -20.2 -28.8 -30.5 - 14.5 - 10.6 - 10.6 - 14.1 - 14.4 -31.2 -26.9 - 14.9 - 13.1 - 10.6 11.9 - 10.0 - 11.8 - 26.5 - 18.5 - 28.0 -28.1 - 18.2 - 28.2 - 12.1
7.34 12,43 34,41 5,31 12,30 23.74 6.09 15.40 19.21 25.70 2.47 7.94 11.36 3.26 7.25 13.89 3.79 6.63 8.84 16.73 4.33 7.54 18.20 0.87 2.93 4.22 1.30 2.49 3.00 5.96 1.64 3.76 5.03 7.15
Thawing times of simple shaped foods." D. J. Cleland Table 5 Experimental data for thawing of spheres of Tylose Tableau 5 Rbsultats expkrimentaux de dbcongblation de sphOres de Tylose Run
D (m)
h (W m - 2 ° C - 1)
Ta (°C)
7~n (°C)
tt (h)
1 2 3 4 5 6 7 8 9 i0 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
o. 128 0.128 0.128 0.128 0.128 0.128 0.128 0.128 0.128 0.128 0.128 0.128 0.128 0.128 0.112 0.112 0.112 0.112 0.112 0.112 0.112 0.112 0.i 12 0.056 0.056 0.056 0.056 0.056 0.056 0.056 0.056 0.056 0.056 0.056 0.056
246.2 246.2 246.2 246.2 74.8 74.8 74.8 51.6 51.6 51.6 51.6 51.6 41.9 41.9 76.0 76.0 76.0 59.4 59.4 59.4 45.7 45.7 45.7 137.2 137.2 137.2 87.0 87.0 87.0 87.0 87.0 57.5 57.5 57.5 57.5
43.3 21.1 13.0 44.0 18.3 11.9 5.3 43.3 22.0 14.5 5.1 43.6 12.0 5.5 43.9 8.0 5.3 43.6 14.5 5.0 43.6 8.9 5.5 18.3 11.9 5.3 43.6 22.2 14.5 7.0 5.0 43.6 22.3 12.1 5.5
- 9.7 -25.3 - 19.7 - 15.1 - 18.8 -23.1 -26.5 -27.9 - 18.8 -20.3 - 15.5 -20.3 -33.0 -17.1 - 14.5 -13.9 -28.0 -27.1 - 16.6 -18.2 -30.2 -32.9 - 17.4 - 16.9 - 13.4 - 23.5 - 14.2 - 20.3 - 14.8 - 22.8 -24.7 - 22.8 - 14.9 -21.7 - 16.3
2.26 3.92 5.42 2.66 4.78 6.39 11.33 2.76 4.31 5.81 12.26 3.13 7.63 12.82 1.89 5.96 7.87 2.02 4.05 9.02 2.43 7.18 9.67 0.86 1.15 2.08 0.58 0.87 1.11 1.88 2.33 0.67 1.06 1.58 2.67
et al.
s e n s i b l e h e a t e f f e c t s . T h e y all t e n d t o s u b s t a n t i a l l y o v e r predict thawing times and the predictions tend to have high standard deviations. The results shown for the method of Nagaoka et al. 26 a r e t y p i c a l o f t h i s g r o u p o f methods. The results using the approach of Mellor and S e p p i n g s 27 a r e t y p i c a l o f a g r o u p o f m e t h o d s b a s e d o n P l a n k ' s e q u a t i o n 17 t h a t u s e m e a n t h a w i n g t e m p e r a t u r e s and/or average thermal conductivities as well as sensible h e a t m u l t i p l y i n g f a c t o r s t o t r y t o t a k e a c c o u n t o f all t h e a s s u m p t i o n s m a d e b y P l a n k . S u c h m e t h o d s still d o n o t t a k e a c c o u n t o f all t h e l i m i t a t i o n s i n a p h y s i c a l l y r e a l i s t i c m a n n e r s o t h e s p r e a d o f t h e p r e d i c t i o n s is s t i l l l a r g e . Other methods break the phase change process into three stages and attempt to approximate the heat transfer in each stage with simple formulae. The method of P h a m 19 w a s t h e b e s t o f t h e s e m e t h o d s e x a m i n e d . T h e
Table 7 S u m m a r y of percentage differences between experimental thawing times for Tylose slabs, infinite cylinders and spheres, and thawing times calculated by existing simple prediction formulae Tableau 7 Rkcapitulation des diffkrences de pourcentaoe entre les temps de dbcongblation expbrimentaux pour des plaques, des cylindres infinis et des spheres de Tylose et les temps de dbcongblation calculbs suivant les formules de prbvision simples existantes Mean (%)
Method
Plank 17 6.0 G o o d m a n 25 13.3 Nagaoka et al. 26 79.1 Mellor and Seppings 27 - 4 . 0 Pham 19 14.7 H u n g and T h o m p s o n 2s 92.2 Calvelo 3° 0.0 Creed and James 29 0.2
SD (%)
Min (%)
Max (%)
Corr FDM
21.4 17.6 15.2 14.4 7.2 26.6 6.0 9.1
-28.0 - 15.3 45.4 -36.4 0.6 14.7 - 12.1 - 20.0
50.5 53.7 122.6 30.7 35.0 134.2 17.1 21.9
0.08 0.14 0.33 0.28 0.54 0.23 0.66 0.52
Table 6 S u m m a r y of percentage differences between experimental thawing times for Tylose slabs, infinite cylinders and spheres, and thawing times calculated by various methods Tableau 6 Rbcapitulation des diffbrences de pourcentage entre les temps de dbcongblation expbrimentaux pour des plaques, des cylindres infinis et des sphbres de Tylose et temps de d~cong~lation calculbs suivant diverses mbthodes
Method
Data
Mean (%)
SD (%)
Finite difference method
All 3 shapes Slab Cylinder Sphere All 3 shapes Slab Cylinder Sphere
-1.3 - 0.8 - 2.2 - 0.9 - 0.9 - 2.0 - 1.7 1.2
4.0 3.5 2.4 5.5 4.4 3.6 2.5 5.8
All 3 shapes Slab Cylinder Sphere All 3 shapes Slab Cylinder Sphere All 3 shapes Slab Cylinder Sphere All 3 shapes Slab Cylinder Sphere
0.1 0.4 - 1.6 1.5 -0.2 - 0.2 - 1.8 1.3 0.0 1.7 - 1.1 - 0.5 0.3 0.1 - 1.3 2.1
5.7 6.0 3.2 6.9 4.9 4.5 3.3 6.0 5.4 6.0 3.1 6.2 5.6 5.2 3.9 6.9
Finite element method
Eqn (6)
Eqn (7)
Eqn (8)
Eqn (9)
Min (%)
Max (%)
Corr FDM
-9.5 - 6.3 - 6.8 - 9.5 - 9.2 - 7.0 - 6.0 - 9.2
9.1 6.8 3.6 9.1 11.4 6.5 4.6 11.4
1.00 1.00 1.00 1.00 0.94 0.99 0.97 0.99
-10.2 - 7.2 - 7.7 - 10.2 -9.2 - 7.0 - 9.2 - 9.0 -12.6 - 7.0 - 7.1 - 12.6 - 10.2 - 8.8 - 8.8 - 10.2
15.2 15.2 5.2 14.6 12.0 10.2 5.9 12.0 14.3 14.3 4.4 10.7 10.3 11.1 5.9 14.8
0.72 0.71 0.66 0.76 0.80 0.72 0.61 0.89 0.80 0.77 0.64 0.90 0.76 0.72 0.69 0.79
Rev. Int. Froid 1986 Vol 9 Juillet
225
Thawing times of simple shaped foods." D. J. Cleland et al. spread of predictions is low because the physical conditions are modelled more realistically. However, because the averaging techniques used to retain simplicity are not always accurate this method tends to over-predict the thawing times. The empirical formulae developed for freezing time prediction all tended to over-predict thawing times, since the same empirical correction factors are not appropriate for thawing. All these methods gave similar prediction accuracy to that shown for the method of Hung and Thompson 28. Two empirical methods specifically developed for thawing time prediction gave the best prediction accuracy. Creed and James 29 and Calvelo 3° both developed empirical formulae for thawing based primarily on fit to numerical method predictions for thawing of beef. Neither takes account of differences in thermal properties (kl, k, CI, C, AH and Tf) between food materials. The error introduced by this may not be large for predominantly aqueous materials but these methods will be less accurate than if thermal property variations had been considered. The results using the Calvelo formula are particularly accurate as beef and Tylose have very similar thermal properties. The error bounds are -11.8% to 11.8~o at the 95~o level of confidence. The correlation of the percentage differences with the finite difference method predictions is high, suggesting that all major sources of thawing time variation for Tylose are accurately accounted for by this formula. In summary, of the existing prediction methods tested only that of Calvelo 3° gave predictions of both reasonable accuracy and reliability (mean percentage difference close to zero and a low standard deviation of the percentage differences). However, this method has the disadvantage of not taking into account thermal property variations between foodstuffs. Therefore it seemed worthwhile to seek a new prediction method.
Development of improved thawing time prediction methods Four approaches to developing a simple thawing time prediction method proved superior to all others considered. First, the basic form of Calvelo 3° was used, but the formula was expressed in dimensionless terms that included the thermal properties. Analysis of the experimental data by multiple non-linear regression gave:
AD Fo~=
{ 0.5 0.125~ 1"°248 712pko.o610 1.4291~+~-e ) Ste °'2 (6)
By substituting the mean thermal properties for Tylose into Equation (6) the empirical constants are changed to values very similar to those suggested by Calvelo 3°. Zaritsky et al. 31 used a similar approach to predict freezing times but it was not particularly successful 7. Second, weighted multiple linear regression was used to modify Plank's equation in a similar manner to that used successfully for freezing time prediction by Cleland and Earle18.
AD P R F° 2V - Bi Ste ~-Ste
226
Int. J. Refrig. 1986 Vol 9 July
(7)
where: P = 0.5(0.7754 + 2.2828Ste Pk) R = 0.125(0.4271 + 2.1220Ste - 1.4847Ste 2 ) Further terms were statistically significant in this model but did not lead to worthwhile reduction in the standard deviation of the percentage differences between the experimental and predicted thawing times and so were not included. Third, a three-stage calculation approach similar to that of Pham ~9'2° was developed:
AD
~ AH~D ( +hD~
(8)
where: AH 1 = Cs(Tf.ve- Tin) AL
= Z-(T~n
kl
= ks
+ Travo)/2
AH 2 = L AT2 = Ta- Tr.vo k2
= 0.25ks + 0.75kl
AH 3 = Cl(Tave- Zfave) AT3 = Z - (Tave+ ~ave)/2 k3
= kl
'/'ave
= Tfin -- (Tfin -
Ta)/(2 -{-
4/ni)
~vo = T~- 1.5 The choice of weighting for k2 in the phase change period, although physically reasonable, has no basis apart from convenience. To retain the analytical basis of Plank's equation the values of the mean thawing temperature (Tfaw) and the average final temperature (Tave) were calculated by the methods suggested by Pham 19. However, the method should still be considered as an empirical, rather than an analytical, modification of Plank's equation because of the arbitrary choice of the weighting factor. Last, a direct fitting of a correction to Plank's equation using an approach proposed by Pham 32 for freezing time prediction gave:
AD
+ ~
)(
08941
0'0244 4-0.6192 Pk )
The prediction accuracy of these four prediction formulae assessed against the slab, infinite cylinder and sphere data are summarized in the second part of Table 6. All four approaches gave accurate predictions and the high correlations with the finite difference method results indicate that the predictions take account of all major sources of variation in the thawing times. All the formulae are simple and have some physical basis although their derivations were not analytical.
Thawing times of simple shaped foods: D. J. Cleland et al. Further refinement of these methods may be possible but it is unlikely that significant increases in prediction accuracy could be achieved. If further terms are added there is the danger that they could be fitting a systematic component of the experimental error. For all four formulae the spread of predictions for each of the three shapes was slightly higher than for the numerical methods. The spread was also larger than that which could be achieved by fitting each formula for each shape separately. This is because the unified approach using the 6:3:2 ratio is not completely valid. However the small improvement in accuracy of shape-specific formulae is outweighed by the simplicity of the unified approach. The possibility must still exist that an approach different from any of those investigated could lead to a significantly more accurate formula. This is not considered likely as the major contributors to the difference between experimental and predicted values for all four formulae are the random experimental and data errors. None of the four approaches can be recommended as being significantly more accurate or simple than the others for the three basic shapes. However, testing against data other than that used to derive the formulae, especially for materials other than Tylose, would be helpful to confirm this and fully assess these methods. A paucity of suitable, accurate experimental thawing data in the literature prevented comprehensive checking. Comparison of predictions with experimental thawing times for the six runs performed with minced lean beef (Table 8) confirmed the accuracy of the four proposed methods (Table 9). The slightly greater offset of the mean prediction error from zero and the slightly larger standard deviations have probably arisen from the increased uncertainty in the thermal property data for minced lean beef (compared with that for Tylose) and the small sample size, rather than from the inherent inaccuracy in the prediction methods for non-Tylose data. Each of the four formulae perform better on some parts of the data than the other three approaches but there was nothing systematic in these observations for the full data set. Therefore all four simple approaches can be used with equal confidence. Engineers may have preferences amongst them based on their ease of understanding of the concepts but it is not really important which is used. Should accuracy be particularly important, users may wish to make predictions by more than one method and take an average. This procedure also helps guard against the possibility of calculation error, as predictions disagreeing by more than a few percent have almost certainly arisen from user error. Because Equations (6)-(9) were derived from Tylose thawing data only, their use should be restricted in Table 8
E x p e r i m e n t a l d a t a for t h a w i n g o f slabs o f m i n c e d lean beef
Rbsultats expbrimentaux pour la d~cong~lation de plaques de boeuf maigre hach~ Tableau 8
Run
D (m)
h (W m - 2 °C - 1)
Ta (°C)
Tin (°C)
tt (h)
1 2 3 4 5 6
0.024 0.024 0.047 0.047 0.075 0.075
13.2 50.4 18.2 78.1 24.5 172.7
5.8 26.9 8.0 15.8 43.2 9.3
- 19.1 - 15.6 -27.1 - 24.4 - 29.0 - 16.9
8.58 0.84 12.07 2.85 5.21 8.94
Table 9
S u m m a r y of p e r c e n t a g e differences b e t w e e n e x p e r i m e n t a l a n d p r e d i c t e d t h a w i n g t i m e s for m i n c e d lean beef s l a b s T a b l e a u 9 R~capitulation des diff&ences de pourcentage entre les temps
de d~congblation expbrimentaux et calculbs pour des plaques de boeuf maigre hachb Mean (%)
SD (%)
Min (%)
Max (%)
Corr
Method F i n i t e difference method E q n (6) E q n (7) E q n (8) E q n (9) C a l v e l o 3°
- 4.5 -7.2 - 7.0 - 6.4 - 5.4 - 7.0
5.9 8.5 6.9 7.9 6.0 6.5
-
2.5 3.6 1.1 3.6 2.5 2.2
1.00 0.81 0.91 0.92 0.96 0.95
11.9 15.4 16.0 19.4 10.6 13.4
FDM
practice to those foods with thermal properties similar to those of Tylose until testing for a wider range of materials has been published. However, a wide range of medium to high moisture foods meet this restriction so it is not unduly limiting. The ranges of the thawing conditions used in the experimental data are a further set of restrictions to the applicability of Equations (6)-(9), outside which the predictions may be unreliable. These ranges are: 0.6
Conclusion Numerical methods (finite differences and finite elements) that accounted for thermal properties continuously variable with temperature accurately predicted thawing times for slabs, infinite cylinders and spheres. No previously published simple prediction formula was found that was both sufficiently accurate and expressed in a form suitable for it to be adopted as a general thawing time prediction method. Four improved formulae that gave comparable results with the numerical methods were developed. Each of these formulae represented a different approach to modifying the well known Plank's equation. All the approaches were limited more by the accuracy of the data from which they were derived than by the inherent inaccuracy of the approach used, and are of similar accuracy and simplicity. It seems unlikely that significantly better methods can be developed that retain the advantage of simplicity. Further testing of these prediction formulae against independently measured thawing data for materials other than Tylose would be helpful.
Acknowledgements The authors acknowledge the financial support of the Meat Industry Research Institute of New Zealand for this research. In addition, D. J. Cleland acknowledges the support of the New Zealand Meat Producers Board in t he form of a Sir Walter Mulholland Fellowship.
Rev. Int. Froid 1986 Vol 9 Juillet
227
Thawing times of simple shaped foods. D. J. Cleland et al. References 1
2
3 4 5 6 7
8 9 10 11 12 13 14 15
228
Cleland, A. C., Earle, R. L. A comparison of analytical and numerical methods for predicting the freezing times of foods J Food Sci (1977) 42 1390-1394 Comini, G., dei Guidiee, S., Lewis, R. W., Zienkiewiez, O. C. Finite element solution of non-linear heat conduction problems with special reference to phase change lnt J Num Meth Engng (19741 8 613 624 Cleland, A. C., Earle, R. L. Assessment of freezing time prediction methods J Food Sci (1984) 49 1034~1042 Cleland, A. C. Heat transfer during freezing of foods and prediction of freezing times PhD Thesis Massey University, New Zealand (1977) Cleland, A. C., Earle, R. L. The third kind of boundary condition in numerical freezing calculations lnt J Heat Mass Transjer (1977) 20 1029 1034 Cleland, A. C., Earle, R. L. A comparison of methods for predicting the freezing times of cylindrical and spherical foodstuffs J Food Sci (1979) 44 958~63 Cleland, D. J. Prediction of freezing and thawing times for foods PhD Thesis Massey University, New Zealand (1985) Cleland, D. J., Cleland, A. C., Earle, R. L., Byrne, S. J. Prediction of rates of freezing, thawing and cooling in solids of arbitrary shape using the finite element method Int J Refrig (1984) 7 6-13 Bakal, A., Hayakawa, K. Heat transfer during freezing and thawing of food Adv Food Res (1973) 20 217-256 Bankoff, S. G. Heat conduction or diffusion with change of phase Adv Chem Engng (1964) 5 75-150 Hayakawa, K. Estimation of heat transfer during freezing or defrosting of food Bull IIR Annexe-I (1977) 293-301 Muelbauer, J. C., Sunderland, J. E. Heat conduction wit h freezing or melting Appl Mech Rev 18 951-959 Ockendon, J. R., Hodgkins, W. R. (Eds) Moving Boundary Problems in Heat Flow and Diffi~sion Clarendon Press, Oxford, UK (1975) Wilson, D. G., Solomon, A. D., Boggs, P. T. (Eds) Moving Boundary Problems Academic Press, London, UK (1978) Cleland, A. C. A review of methods for predicting the duration of freezing processes, Paper presented at the 4th ICEF conference, Edmonton (1985)
Int. J. Refrig. 1986 Vol 9 July
16 17 18 19 20 21 22 23
24 25 26 27 28 29 30 3l 32
Kinder, E., Lamb, J. The prediction of freezing times of foodstufl~ Meat Freezing How and Why? MRI Syrup No. 3. ILl 17.6 Plank, R. Die gefrierdauer von eisblocken Zeitschri/i fi," die gesamte Kalte-lndustrie (1913) 20(6) 109 114 Cleland, A. C., Earle, R. L. Freezing time prediction for foods a simplified procedure lnt J Refrig (1982) 5 134-140 Pham, Q. T. An extension to Plank's equation for predicting freezing times of foodstuffs of simple shape Int J Re/i'iq (1984) 7 377 383 Pham, Q. T. A simplified equation for predicting freezing time J Food Technol in press Cleland, A. C., Earle, R. L. A comparison of freezing calculations including modification to take into account initial superheat Bull IlR Am~exe-I (1976) 369--376 Riedel, L. Eine Prufsubstanz fiir Gefrierversuche Kahetechnik (1960) 12 222-225 Cleland, A. C., Earle, R. L. A new method for prediction of surface heat transfer coefficients in freezing Bull fiR Annexe-1 (1976) 361 368 Charm, S. E. A method for experimentally evaluating heat transfer coefficients in freezers and thermal conductivities of frozen foods Food Technol (1963) 17(10) 93 96 Goodman, T. R. The heat balance integral and its application to problems involving a change of phase Trans ASME (1958) 80 335 342 Nagaoka, J., Takaji, S., Hohani, S. Experiments on t he freezing of fish in an air blast freezer Proc 9th lnt Congr Re/i'i9 (1955) 4 105 110 Melior, J. D., Seppings, A. H. Thermophysical data for designing a refrigerated food chain Bull IIR Annexe-I (1976) 349-359 Hung, Y. C., Thompson, D. R. Freezing time prediction for slab shape foodstuffs by an improved analytical method J Food Sci (1983) 48 555-560 Creed, P. G., James, S. J. Predicting thawing times of frozen boneless beef blocks lnt J Refrig (1981) 4 355 358 Calvelo, A. Recent studies on meat freezing in Developments in Meat Science 2 (Ed. R. Lawrie) Applied Science Publishers Ltd, London, UK (1981) 125 158 Zaritsky, N. E., Anon, M. C., Calvelo, A. Rate of freezing effect on the colour of frozen beef liver Meat Sci (1982) 7 299 312 Pham, Q. T. Personal communication (1983)
A set of 104 experimental measurements of thawing time were made over a wide range of conditions for slab, infinite cylinder and sphere shapes of a food analogue material. These results were used to assess existing thawing time prediction methods. Versions of both the finite difference and the finite element numerical methods that accounted for continuously temperature-variable thermal properties gave accurate predictions. No previously published simple prediction formula was found that was both sufficiently accurate and expressed in a form suitable for it to be adopted as a general thawing time prediction method. Four accurate, but simple, empirical formulae based on Plank's equation were developed. These formulae predicted thawing times that were both highly correlated with those predicted by the numerical methods and agreed with the experimental data to within + 10%o at the 95~ level of confidence. The agreement was more limited by uncertainties in the experimental and thermal property data than by inaccuracy in the prediction formulae. Significantly more accurate simple formulae are unlikely to be developed unless more accurate experimental data are available. (Keywords:foodproducts;thawing;thawingtimes)
Pr vision des temps de d cong lation des aliments de forme simple Une s&ie de 104 mesures exp&imentales du temps de dkcongblation a &tk effectube pour un grand nombre de conditions de formes : plaque, cylindre infini et sphere, d'un mat&iau analogue h un aliment. On a utilisb ces r~sultats pour ~valuer les m&hodes existantes de prbvision du temps de dbcongblation. Des versions des m&hodes num&iques aux diff&ences finies et blkments finis, tenant compte des propribtks thermiques variant continuellement en fonction de la temp&ature, ont donnb des prbvisions prkcises. On n'a pas trouvk parmi les formules de pr~vision simple publibes des formules suffisamment pr~cises et exprim~es sous une forme permettant de les adopter comme m~thode 9bn&ale de pr~vision du temps de d~congblation. Quatre formules empiriques, prkcises, mais simples, ont btb &ablies d'aprOs l'bquation de Plank, Ces formules prbvoient des temps de d~conoblation fortement correl~s avec ceux pr~vus par les mbthodes num&iques et concordant avec les r~suhats exp&imentaux fi -t-10% prds avec un taux de confiance de 95%. La-concordance btait plus limitbe par les incertitudes des rbsultats exp&imentaux et des propri&bs thermiques que par l'inexactitude des formules de prbvision. Des formules simples nettement plus prbcises ont peu de chance d'btre raises au point, fi moins qu'on ne dispose de rbsultats exp&imentaux plus prkcis. (Mots cl~s: aliments; decongblation; temps de d~congblation)
Recently, with greater quantities of goods being frozen and particularly with greater emphasis on further processing of frozen products, thawing has become an important industrial process. Accurate prediction of thawing times for a wide range of conditions and geometries would enable design, operation and control of thawing equipment to be optimized so that product quality is maintained while processing costs are kept low. This paper considers the problem of predicting thawing times for the simple slab, infinite cylinder and sphere shapes. Physically, thawing of slabs, infinite cylinders and spheres of food is governed by the partial differential equation for one-dimensional heat conduction in a solid, subject to the appropriate initial and boundary conditions:
where a = 0, 1 or 2 for slabs, infinite cylinders and spheres, * Present address: Auckland University, Auckland, New Zealand 0140-7007/86/040220~9503.00 ~') 1986 Butterworth & Co (Publishers) Ltd and IIR
220
Int. J. Refrig. 1986 Vol 9 July
respectively, and the thermal properties, k and C, are temperature variable. Typically, foods are frozen to temperatures substantially below their thawing temperature to prevent chemical and biological deterioration. The frozen storage temperature defines the process initial condition. The third kind of boundary conditions (convective heat transfer to the food surface) is both the most physically realistic and the most common boundary condition for freezing and thawing of food 1 so only it need be considered: h(T~ -
T) =
k ~T
(2)
~r r=D/2
To complete the definition of thawing an endpoint for the process must be defined. For any food, thawing must be complete once the thermodynamic centre temperature reaches 0°C, yet it is desirable for chemical and biological reasons that the mass average temperature is as low as possible at the end of the process. A process endpoint of 0°C at the thermodynamic centre is therefore the most appropriate.
Thawing times of simple shaped foods: D. J. Cleland et al. Nomenclature A Area (m 2) Bi Biot Number (= hd/kl) C Volumetric specific heat capacity (j m-3 oC -1 ) corr FDM Correlation coefficient compared with the percentage differences for the finite difference method D Thickness or diameter (m) Fo Fourier Number (= kltt/GD 2) h Surface heat transfer coefficient (Wm-2oc -1 ) AH Change in enthalpy from - 10°C to 0°C (J m - 3) k Thermal conductivity (W m- 1 °C- t) L Latent heat (J m-3) max Maximum percentage difference rain Minimum percentage difference Pk Plank Number (= Cs(Tf-Tin)/AH) r Distance from the centre, radius (m) Mathematical solution of Equations (1) and (2) enables a method for thawing time prediction to be derived. There are no exact solutions because of the non-linearity that arises from both the temperature variable thermal properties of foods and the nature of the boundary condition. Approximate solutions fall into two broad groups - those requiring computer calculations (numerical methods) and those where the computations can be carried out by hand (simple formulae). Both numerical methods and simple formulae will be used to predict thawing times as computing facilities are not always available or convenient to use. Numerical methods (finite differences and finite elements) can take into account the temperature variable thermal properties, k and C, and consequently model the physical behaviour of food materials very closely for both freezing and thawing2. If they are correctly formulated and implemented these methods should give accurate prediction of thawing times 3. For prediction of freezing times for slabs, infinite cylinders and spheres both the finite difference method 4-6 and the finite element method T'8 have been formulated, implemented as computer programs and tested against experimental data. They were shown to predict freezing times to within ___10~o. These programs were available and were subsequently used for thawing time prediction. A simplifying assumption commonly made to facilitate analytical solution of phase change problems is that all the latent heat is added or removed at a unique temperature ( ~ and that thermal properties undergo a step change in value at this temperature. Although this assumption of a unique thawing temperature is true in some situations such as the melting of pure liquids, it is not true for most foods where the latent heat is added over a range of temperatures below Tr. The possibility of deriving prediction methods by using this assumption has been explored extensively as indicated by the bibliographies given in various reviews 4,7,9 14. However, the resulting methods are only approximate for predicting thawing times of foods. Many attempts have been made either to modify these existing analytical solutions to reduce their limitations and give them less dependence on assumptions made in
SD
Standard deviation of percentage differences Stefan Number (=CI(Ta- Tf)/AH) Time (s) Temperature (°C) Volume (m 3)
Ste t T V Subscripts a ave f fave fin i in 1 s t
Ambient thawing medium Average final Freezing or thawing Average for freezing or thawing Final centre ith case Initial Unfrozen Frozen Thawing
their theoretical development, or to develop simple formulae on an entirely empirical basis 4'7' 15,16. Although most of these modifications have been derived for freezing problems, the same general approaches can also be applied to thawing. However, the simplistic approach of treating thawing as the reverse of freezing has not been proved to be accurate. A search of the literature showed that those methods specifically derived for thawing are nearly all product specific. Commonly these simple formulae are based on the well known Plank's equation 17 for slabs, infinite cylinders and spheres: t 2V
a-- f\0"5
+0.125~-
(3)
or in dimensionless form:
AD Fo 2 T
0.5 0.125 Bi Ste i Ste
(4)
Plank assumed a unique phase change temperature, constant thermal properties and that sensible heat effects above and below the phase change temperature are negligible compared with the latent heat. Equations (3) and (4) suggest that when these assumptions are valid, the ratio of thawing times for slabs, infinite cylinders and spheres under identical conditions and with the same characteristic dimension is 6:3:2. This ratio arises from the relationship between volume and surface area for each shape. However, this ratio can be shown to apply only when the surface is isothermal with respect to position 7. In thawing of biological materials the above conditions are not all met, so the 6:3: 2 ratio is not necessarily correct. Assuming the ratio to be constant allows a unified approach, in which a single prediction method is equally applicable for all three simple shapes, to be used. However, there is a different distribution of volume with respect to displacement from the geometric centre for the three shapes, which could lead to deviations from the theoretical ratio. Previous studies for freezing 18-2° suggested that within the tolerance of the data they used, there was no error in assuming that the ratio was constant at 6:3:2.
Rev. Int. Froid 1986 Vol 9 Juillet
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Thawing times of simple shaped foods. D. J. Cleland et al. An investigation of the validity of the ratio for thawing has been carried out using numerical method predictions for the three simple shapes. It showed that on average the 6:3:2 ratio holds for a wide range of conditions representative of typical thawing operations 7. There were discernible effects related to the variations in the Biot number (Bi), Stefan number (Ste), and Plank number (Pk), but it was concluded that these were sufficiently small compared with other probable sources of uncertainty in thawing time predictions that they could be ignored. Therefore the prediction of thawing times for slabs, infinite cylinders and spheres can use a single unified approach. Ultimately, verification of the accuracy of any method to predict thawing times must be made by comparison with accurate experimental data 3. The most useful data set would be large and diverse in order to differentiate between errors due to imprecise control and measurement of thawing conditions (experimental error), error due to uncertainty in thermal property data (data error) and inaccuracy arising from the prediction method itself (prediction error) 15. No such data set for thawing of foods could be found in the literature, which probably explains why no comprehensive assessment of existing prediction
1 T h e r m a l p r o p e r t y d a t a used in calculations with simple formulae T a b l e a u 1 Propriktbs thermiques utilisbes dans les calculs avec des Table
formules simples Property
Tylose
M i n c e d lean beef
ks (W m - 1 oC - 1) kl (W m - 1 ° C - 1) Cs ( M J m - a ° C - 1) CI ( M J m - 3 ° C - 1) A H (MJ m - a ) Tr (°C) 7~ave (°C)
1.65 0.55 1.90 3.71 226 - 0.6 -2.1
1.55 0.50 1.90 3.65 230 - 1.0 -2.5
2 Tableau
Table
methods has been made. The aims of the present work are therefore: 1. to collect accurate thawing data for the simple slab, infinite cylinder and sphere shapes; 2. to assess the accuracy of existing thawing time prediction methods for these shapes; and, if necessary, 3. to develop new and improved thawing time prediction methods. Experimental Full details of the experimental procedures can be found elsewhere7. The techniques and equipment used were very similar to those used by Cleland and Earle 1'6'zl for experimental freezing of slabs, infinite cylinders and spheres. Tylose (Karlsruhe test substance), a defined 23~o methyl--cellulose gel, was used almost exclusively for the experiments because it is cheap and convenient to use, its thermal properties are well known, and thawing of Tylose closely models thawing of many real food materials 3'22. A small number of confirmatory runs were made with an actual food material (minced lean beef). For both materials the thermal property data given by Cleland and Earle 3 were used (Tables 1 and 2). All temperature measurements were made with copper/constantan thermocouples connected to a 12 point recording potentiometer, operating on a 100 second print cycle, that was calibrated to within 0.3°C in the range - 50 to 50°C. Before thawing, the objects of Tylose were kept in constant-temperature cold stores. In spite of insulating material being used to prevent temperature change during handling of the objects immediately before thawing, the initial temperatures within the objects varied by up to 2.0°C about the mean values. Water was used as the thawing medium. A well mixed 1 m 3 immersion tank with both heating and refrigeration elements, actuated by on/off control, gave a large stable ambient heat source.
Thexrnal p r o p e r t y d a t a used in calculations with n u m e r i c a l m e t h o d s
2 Propri~t~s thermiques utilisdes dans les calculs avec des formules numdriques M i n c e d lean beef
Tylose T (°C)
C ( M J m - 3 ° C - 1)
T (°C)
k (W m - 1 ° C - ~)
T (°C)
C ( M J m - 3 oC
40.0 30.0 20.0 18.0 16.0 14.0 12.0 - 10.0 - 9.0 - 8.0 - 7.0 - 6.0 -5.0 -4.0 -3.0 - 2.5 - 2.0 - 1.5 1.3 0.6 -0.5 0.0 40.0
1.88 1.92 1.95 2.00 2.20 2.30 2.80 3.70 4.20 5.00 5.90 7.20 11.0 17.0 25.0 33.0 45.0 70.0 100 100 19.9 3.71 3.71
-
1.67 1.67 1.66 1.64 1.63 1.61 1.60 1.58 1.56 1.52 1.46 1.35 1.28 1.18 1.04 0.82 0.66 0.55 0.49 0.49 0.56 0.62
-
1.89 1.91 2.02 2.70 3.58 4.55 5.26 7.71 10.4 15.7 31.2 53.1 73.7 255 255 3.66 3.66
-
-
222
40.0 30.0 20.0 15.0 10.0 -9.0 - 8.0 - 7.0 - 6.0 - 5.0 - 4.0 - 3.0 -2.5 -2.0 - 1.5 - 1.0 - 0.8 - 0.7 - 0.6 0.0 20.0 40.0
Int. J. Refrig. 1986 Vol 9 July
40.0 30.0 25.0 20.0 15.0 10.0 - 8.0 - 6.0 - 5.0 - 4.0 - 3.0 - 2.0 - 1.5 - 1.3 - 1.2 - 1.0 40.0
T
-
1)
(oc) - 40.0 - 24.0 12.0 -8.0 -4.0 -2.0 -1.0 40.0
k (Wm-I 1.58
1.53 1.48 1.40 1.28
0.98 0.48 0.49
oc-1)
Thawing times of simple shaped foods: D. J. Cleland et al. Variations in water temperature over the course of an experiment were not detectable with the recording equipment used. Taking recorder calibration error into account, the overall error in temperature measurement and control was estimated to be less than +0.5°C. One-dimensional heat transfer in an infinite slab was approximated by a finite-sized slab insulated on all but two faces in an arrangement similar to a plate freezer. Water from the immersion tank was circulated through two 0.4m x 0.4 m plates at a sufficiently high rate that there was no observed change in water temperature across the plates. Good thermal contact between the slab surface and the plates was provided by a 100 kg weight put on the top plate. Slabs of 0.025-0.1 m (___0.5mm) in thickness were used. The slabs were pre-frozen in a plate freezer to ensure a constant thickness, a fiat surface and hence good thermal contact with the plates. Each slab was a 0.22 m diameter of Tylose surrounded by 0.09 m of polystyrene insulation around the edges to ensure only onedimensional heat transfer. Seven thermocouples were embedded at various places in each slab to measure the centre and surface temperatures. The two surfaces of each slab were wrapped with tinfoil and brown paper to prevent moisture losses and to give structural strength. Varying numbers of rubber sheets were inserted between the plates and the slabs to provide thermal resistance during thawing. The resulting surface heat transfer coefficients (h) were measured by the methods given by Cleland and Earle 23 using separate experiments at temperatures greater than 0°C. From variations in the estimated h values and in four repeated thawing experiments, the overall experimental error was estimated to be + 5.3~ at the 95~o level of confidence, indicating a high degree of repeatability. In the infinite cylinder and sphere experiments the objects were immersed directly in the water tank and oscillated through 300 ° every 30 s. This ensured that heat transfer was uniform across the whole object surface. The cylinders were made by filling 0.45 m lengths of 0.050.15 m diameter steel or PVC plastic pipes with Tylose. Centre and surface thermocouples were introduced through isothermal regions, and polystyrene foam 'caps' were put on the ends of the cylinders to reduce longitudinal heat flow to a minimum, thereby closely approximating an infinite cylinder. The different pipe wall materials and thicknesses gave some variation in the surface heat transfer coefficient, but greater variation was achieved by gluing sheets of rubber onto the outside of each cylinder for some of the experiments. This also ensured that the range ofh values included those normally encountered in air thawing. As for the slab experiments, separate heating experiments above 0°C were used to estimate h for each combination of wall type and rubber covering. A problem was encountered owing to imperfect contact of the Tylose to the inside surface of the cylinder walls during thawing as slight shrinking of the Tylose and expansion of the pipe wall material occurred. The air voids created were found to be randomly distributed and ill defined in size so it was impossible to evaluate their individual effect on heat transfer directly. The best way to account for them was to use an average value of h estimated by the heat penetration method 24 based on measured thermodynamic centre temperatures. The effect of any surface variation in the heat transfer conditions was effectively averaged by this procedure. The experimental
error, estimated to be 6.1~, for the cylinder experiments was larger than for the slabs, This reflects the increased uncertainty in the control and measurement of the surface heat transfer coefficient. Hollow metal balls were used to model the spherical geometry. Three sizes, approximately 0.05, 0.10 and 0o13m in diameter were used. Each sphere was constructed from two hemispheres. These were filled with Tylose, thermocouples were inserted and then the two halves were welded or soldered together. To seal each sphere, especially around the thermocouple leads, and to give a variation in the surface heat transfer coefficient, the spheres were coated with layers of silicone rubber after construction. The same problem with thermal contact that was observed for the infinite cylinders occurred for the spheres, so some small random variation of h over the surface was expected. Again the heat penetration method 24 using the centre temperature profiles from independent heating and cooling experiments without phase change were used to estimate h for each combination of sphere and rubber coating thickness. More variation in the measured h values occurred than for the slabs and infinite cylinders. Consequently, the overall 95~o confidence bound for the experimental error was estimated to be slightly higher at _ 8.0~o.
Results and discussion
Thawing data for slabs, infinite cylinders and spheres For thawing of a homogeneous material under constant environmental conditions to a final thermodynamic centre temperature of 0°C, the time taken is affected by six factors - shape, thermal properties, size (D), surface heat transfer coefficient (h), thawing medium temperature (Ta), and initial temperature (Ti,). For each shape an experimental design was set up to cover a wide range of conditions that are commonly found in food thawing situations. The heating medium temperature was varied from 5 to 45°C, the initial temperature of the material to be thawed was varied from - 10 to - 33°C and the surface heat transfer coefficient was varied from 13 to 246 W m - 2 oC -1. Because D and h could not always be set to pre-selected levels, the design was not orthogonal. Therefore it was decided not to attempt to control T~and T~, exactly to the pre-selected levels; instead, provided a value close to the desired levels was obtained it was considered satisfactory and care was then taken to control the conditions as close as possible to these constant values. Tables 3, 4 and 5 show the experimental conditions and measured thawing times for the thermodynamic centre temperature to reach 0°C for Tylose slabs, infinite cylinders and spheres, respectively.
Assessment of existing thawing time prediction methods The first part of Table 6 compares the thawing times predicted by the finite difference and finite element numerical methods with the experimental thawing times. The percentage differences were calculated from: ~o difference =
predicted t i m e - experimental time 100 xexperimental time 1
(5)
Rev. Int. Froid 1986 Vol 9 Juillet
223
Thawing times of simple shaped foods. D. J. Cleland et al. Table 3 Experimental data for thawing of slabs of Tylose Tableau 3 Rbsultats expbrimentaux de dbcongklation de plaques de
Tylose Run
D (m)
h (W m 2 °C- t)
"/a (C)
7in (:'C)
tt (h)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
0.0260 0.0260 0.0525 0.0260 0.0525 0.0525 0.0260 0.0525 0.1000 0.1020 0.0770 0.0525 0.0770 0.0525 0.0260 0.1000 0.0525 0.0770 0.1000 0.0260 0.0525 0.0525 0.0525 0.0525 0.0525 0.1050 0.0770 0.0280 0.0525 0.1050 0.0770 0.0770 0.1000 0.0770 0.1050
13.2 24.5 13.2 24.5 13.2 13.2 50.4 24.5 13.2 13.2 18.2 29.5 37.3 50.4 78.1 24.5 50.4 37.3 24.5 78.1 50.4 50.4 50.4 50.4 78.1 37.3 50.4 172.7 78.1 78.1 78.1 78.1 78.1 78.1 172.7
12.8 5.2 5.2 45.9 46.1 12.8 4.6 5.2 45.9 43.0 46.2 12.8 5.2 5.1 5.2 12.8 46.0 46.1 45.8 12.4 13.4 13.4 13.4 13.4 5.2 5.2 45.7 43.0 13.3 5.2 5.0 45.7 46.2 12.9 13.4
- 20.9 - 29.4 - 11.4 - 8.3 -28.6 -20.5 - 26.3 - 10.7 -8.3 - 13.7 -26.8 - 20.9 - 30.2 -25.0 - 12.3 -9.4 - 10.6 - 24.7 - 32.5 - 27.7 - 20.2 - 24.1 - 23.6 - 23.6 -28.9 - 10.4 - 13.5 - 31.0 -22.5 -28.8 - 14.2 -9.4 -28.2 -21.0 -23.8
4.69 5.63 19.32 0.90 3.89 10.33 3.49 13.20 7.82 8.83 5.11 6.26 18.49 9.41 2.51 16.92 1.64 3.72 6.75 1.42 4.68 4.65 4.60 4.50 7.58 29.33 2.99 0.43 3.91 23.62 14.39 2.61 4.59 7.25 10.61
It should be noted that the finite element method used was that defined by Equations (5)-(7) of Cleland et al. 8 and not the simplified formulation (Equations (5a)-{7a)). The simplified formulation was developed by assuming no variation of the thermal properties with position within each element, in order to reduce the computation time. For one-dimensional heat transfer, computer computation time is not a limitation, so the most accurate finite element method was used. Applying the criterion of Cleland and Earle 3 to comparisons of the numerical method predictions with the experimental results showed no trends in the data that might suggest major systematic experimental or thermal property data error. The 95% confidence bounds of the numerical predictions are consistent with the estimated experimental error bounds as a whole and for each shape individually, suggesting that only minimal uncertainty has arisen in the application of the numerical methods. This was expected because relatively fine space and time intervals were used in the calculations. Predictions significantly better than those achieved by these numerical methods cannot be expected if the same experimental and thermal data are used, even if a more exact or sophisticated form of the numerical methods was used. As was expected, for the simple regular shapes the finite element method gave almost identical results to the finite difference method. Thawing times for the slab, infinite cylinder and sphere data were also calculated by existing simple prediction methods that considered the third kind of boundary
224
Int. J. Refrig. 1986 Vol 9 July
condition 7. Many existing methods had been developed specifically for freezing time prediction. Where possible the analogous form of the method for thawing was developed and used. Table 7 summarizes the prediction accuracy for the best methods as well as several poorer, but well known, methods. Plank's equation ~ gave a mean prediction error close to zero, but the spread of predictions was large and the correlation with the finite difference method results was poor. For thawing the under-prediction caused by ignoring sensible heat effects tends to compensate for over-prediction arising from assuming a unique phase change temperature and constant thermal properties. However, because the amount of each effect depends on the combination of environmental conditions, the amount of under- and over-prediction is variable and leads to the high standard deviation of predictions and the low correlation with the finite difference method results (which take these effects into account in a physically correct manner). The results for the method of Goodman 25 are typical of a large group of methods that are similar to Plank's equation, except that they take account of the sensible heat in the unfrozen phase. Although the mean difference is higher than for Planks's equation, the spread of predictions is reduced because some of the sensible heat effects are taken into account. However, the predictions are still poor because of the other limitations. There are a number of methods that modify analytical methods by multiplicative factors to account for all the Table 4 Experimental data for thawing of infinite cylinders of Tylose Tableau 4 R~suhats expbrimentaux de dbcongblation de cylindres infinis
de Tylose D
h
Run
(m)
(W m -2 C -
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34
0.158 0,158 0.158 0.156 0.156 0.156 0.156 0.156 0.156 0.156 0.106 0.106 0.106 0.103 0.103 0,103 0.103 0,103 0.103 0.103 0.103 0.103 0.103 0.051 0.051 0.051 0.051 0.051 0.051 0.051 0.051 0.051 0.051 0.051
23.5 23.5 23.5 90.7 90.7 90.7 43.5 43.5 43.5 43.5 113.0 113.0 113.0 37.4 37.4 37.4 25.1 25.1 25.1 25.1 19.5 19.5 19.5 46.5 46.5 46.5 27.9 27.9 27.9 27.9 19.0 19.0 19.0 19.0
~)
Ta
~,,
tt
("C)
(C)
(h)
43.3 21.1 5.1 43.2 13.0 5.1 40.3 11.9 8.2 5.3 43.3 8.5 5.1 43.3 13.0 5.3 40.3 18.7 13.2 5.3 43.9 18.3 5.8 44.0 8.5 5.1 40.3 18.9 13.2 5.3 43.9 14.6 9.6 5.8
- 14.0 -20.6 -28.4 - I1.9 - 13.6 -27.9 -21.2 - 27.4 26.9 - 14.9 - 10.7 -20.2 -28.8 -30.5 - 14.5 - 10.6 - 10.6 - 14.1 - 14.4 -31.2 -26.9 - 14.9 - 13.1 - 10.6 11.9 - 10.0 - 11.8 - 26.5 - 18.5 - 28.0 -28.1 - 18.2 - 28.2 - 12.1
7.34 12,43 34,41 5,31 12,30 23.74 6.09 15.40 19.21 25.70 2.47 7.94 11.36 3.26 7.25 13.89 3.79 6.63 8.84 16.73 4.33 7.54 18.20 0.87 2.93 4.22 1.30 2.49 3.00 5.96 1.64 3.76 5.03 7.15
Thawing times of simple shaped foods." D. J. Cleland Table 5 Experimental data for thawing of spheres of Tylose Tableau 5 Rbsultats expkrimentaux de dbcongblation de sphOres de Tylose Run
D (m)
h (W m - 2 ° C - 1)
Ta (°C)
7~n (°C)
tt (h)
1 2 3 4 5 6 7 8 9 i0 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
o. 128 0.128 0.128 0.128 0.128 0.128 0.128 0.128 0.128 0.128 0.128 0.128 0.128 0.128 0.112 0.112 0.112 0.112 0.112 0.112 0.112 0.112 0.i 12 0.056 0.056 0.056 0.056 0.056 0.056 0.056 0.056 0.056 0.056 0.056 0.056
246.2 246.2 246.2 246.2 74.8 74.8 74.8 51.6 51.6 51.6 51.6 51.6 41.9 41.9 76.0 76.0 76.0 59.4 59.4 59.4 45.7 45.7 45.7 137.2 137.2 137.2 87.0 87.0 87.0 87.0 87.0 57.5 57.5 57.5 57.5
43.3 21.1 13.0 44.0 18.3 11.9 5.3 43.3 22.0 14.5 5.1 43.6 12.0 5.5 43.9 8.0 5.3 43.6 14.5 5.0 43.6 8.9 5.5 18.3 11.9 5.3 43.6 22.2 14.5 7.0 5.0 43.6 22.3 12.1 5.5
- 9.7 -25.3 - 19.7 - 15.1 - 18.8 -23.1 -26.5 -27.9 - 18.8 -20.3 - 15.5 -20.3 -33.0 -17.1 - 14.5 -13.9 -28.0 -27.1 - 16.6 -18.2 -30.2 -32.9 - 17.4 - 16.9 - 13.4 - 23.5 - 14.2 - 20.3 - 14.8 - 22.8 -24.7 - 22.8 - 14.9 -21.7 - 16.3
2.26 3.92 5.42 2.66 4.78 6.39 11.33 2.76 4.31 5.81 12.26 3.13 7.63 12.82 1.89 5.96 7.87 2.02 4.05 9.02 2.43 7.18 9.67 0.86 1.15 2.08 0.58 0.87 1.11 1.88 2.33 0.67 1.06 1.58 2.67
et al.
s e n s i b l e h e a t e f f e c t s . T h e y all t e n d t o s u b s t a n t i a l l y o v e r predict thawing times and the predictions tend to have high standard deviations. The results shown for the method of Nagaoka et al. 26 a r e t y p i c a l o f t h i s g r o u p o f methods. The results using the approach of Mellor and S e p p i n g s 27 a r e t y p i c a l o f a g r o u p o f m e t h o d s b a s e d o n P l a n k ' s e q u a t i o n 17 t h a t u s e m e a n t h a w i n g t e m p e r a t u r e s and/or average thermal conductivities as well as sensible h e a t m u l t i p l y i n g f a c t o r s t o t r y t o t a k e a c c o u n t o f all t h e a s s u m p t i o n s m a d e b y P l a n k . S u c h m e t h o d s still d o n o t t a k e a c c o u n t o f all t h e l i m i t a t i o n s i n a p h y s i c a l l y r e a l i s t i c m a n n e r s o t h e s p r e a d o f t h e p r e d i c t i o n s is s t i l l l a r g e . Other methods break the phase change process into three stages and attempt to approximate the heat transfer in each stage with simple formulae. The method of P h a m 19 w a s t h e b e s t o f t h e s e m e t h o d s e x a m i n e d . T h e
Table 7 S u m m a r y of percentage differences between experimental thawing times for Tylose slabs, infinite cylinders and spheres, and thawing times calculated by existing simple prediction formulae Tableau 7 Rkcapitulation des diffkrences de pourcentaoe entre les temps de dbcongblation expbrimentaux pour des plaques, des cylindres infinis et des spheres de Tylose et les temps de dbcongblation calculbs suivant les formules de prbvision simples existantes Mean (%)
Method
Plank 17 6.0 G o o d m a n 25 13.3 Nagaoka et al. 26 79.1 Mellor and Seppings 27 - 4 . 0 Pham 19 14.7 H u n g and T h o m p s o n 2s 92.2 Calvelo 3° 0.0 Creed and James 29 0.2
SD (%)
Min (%)
Max (%)
Corr FDM
21.4 17.6 15.2 14.4 7.2 26.6 6.0 9.1
-28.0 - 15.3 45.4 -36.4 0.6 14.7 - 12.1 - 20.0
50.5 53.7 122.6 30.7 35.0 134.2 17.1 21.9
0.08 0.14 0.33 0.28 0.54 0.23 0.66 0.52
Table 6 S u m m a r y of percentage differences between experimental thawing times for Tylose slabs, infinite cylinders and spheres, and thawing times calculated by various methods Tableau 6 Rbcapitulation des diffbrences de pourcentage entre les temps de dbcongblation expbrimentaux pour des plaques, des cylindres infinis et des sphbres de Tylose et temps de d~cong~lation calculbs suivant diverses mbthodes
Method
Data
Mean (%)
SD (%)
Finite difference method
All 3 shapes Slab Cylinder Sphere All 3 shapes Slab Cylinder Sphere
-1.3 - 0.8 - 2.2 - 0.9 - 0.9 - 2.0 - 1.7 1.2
4.0 3.5 2.4 5.5 4.4 3.6 2.5 5.8
All 3 shapes Slab Cylinder Sphere All 3 shapes Slab Cylinder Sphere All 3 shapes Slab Cylinder Sphere All 3 shapes Slab Cylinder Sphere
0.1 0.4 - 1.6 1.5 -0.2 - 0.2 - 1.8 1.3 0.0 1.7 - 1.1 - 0.5 0.3 0.1 - 1.3 2.1
5.7 6.0 3.2 6.9 4.9 4.5 3.3 6.0 5.4 6.0 3.1 6.2 5.6 5.2 3.9 6.9
Finite element method
Eqn (6)
Eqn (7)
Eqn (8)
Eqn (9)
Min (%)
Max (%)
Corr FDM
-9.5 - 6.3 - 6.8 - 9.5 - 9.2 - 7.0 - 6.0 - 9.2
9.1 6.8 3.6 9.1 11.4 6.5 4.6 11.4
1.00 1.00 1.00 1.00 0.94 0.99 0.97 0.99
-10.2 - 7.2 - 7.7 - 10.2 -9.2 - 7.0 - 9.2 - 9.0 -12.6 - 7.0 - 7.1 - 12.6 - 10.2 - 8.8 - 8.8 - 10.2
15.2 15.2 5.2 14.6 12.0 10.2 5.9 12.0 14.3 14.3 4.4 10.7 10.3 11.1 5.9 14.8
0.72 0.71 0.66 0.76 0.80 0.72 0.61 0.89 0.80 0.77 0.64 0.90 0.76 0.72 0.69 0.79
Rev. Int. Froid 1986 Vol 9 Juillet
225
Thawing times of simple shaped foods." D. J. Cleland et al. spread of predictions is low because the physical conditions are modelled more realistically. However, because the averaging techniques used to retain simplicity are not always accurate this method tends to over-predict the thawing times. The empirical formulae developed for freezing time prediction all tended to over-predict thawing times, since the same empirical correction factors are not appropriate for thawing. All these methods gave similar prediction accuracy to that shown for the method of Hung and Thompson 28. Two empirical methods specifically developed for thawing time prediction gave the best prediction accuracy. Creed and James 29 and Calvelo 3° both developed empirical formulae for thawing based primarily on fit to numerical method predictions for thawing of beef. Neither takes account of differences in thermal properties (kl, k, CI, C, AH and Tf) between food materials. The error introduced by this may not be large for predominantly aqueous materials but these methods will be less accurate than if thermal property variations had been considered. The results using the Calvelo formula are particularly accurate as beef and Tylose have very similar thermal properties. The error bounds are -11.8% to 11.8~o at the 95~o level of confidence. The correlation of the percentage differences with the finite difference method predictions is high, suggesting that all major sources of thawing time variation for Tylose are accurately accounted for by this formula. In summary, of the existing prediction methods tested only that of Calvelo 3° gave predictions of both reasonable accuracy and reliability (mean percentage difference close to zero and a low standard deviation of the percentage differences). However, this method has the disadvantage of not taking into account thermal property variations between foodstuffs. Therefore it seemed worthwhile to seek a new prediction method.
Development of improved thawing time prediction methods Four approaches to developing a simple thawing time prediction method proved superior to all others considered. First, the basic form of Calvelo 3° was used, but the formula was expressed in dimensionless terms that included the thermal properties. Analysis of the experimental data by multiple non-linear regression gave:
AD Fo~=
{ 0.5 0.125~ 1"°248 712pko.o610 1.4291~+~-e ) Ste °'2 (6)
By substituting the mean thermal properties for Tylose into Equation (6) the empirical constants are changed to values very similar to those suggested by Calvelo 3°. Zaritsky et al. 31 used a similar approach to predict freezing times but it was not particularly successful 7. Second, weighted multiple linear regression was used to modify Plank's equation in a similar manner to that used successfully for freezing time prediction by Cleland and Earle18.
AD P R F° 2V - Bi Ste ~-Ste
226
Int. J. Refrig. 1986 Vol 9 July
(7)
where: P = 0.5(0.7754 + 2.2828Ste Pk) R = 0.125(0.4271 + 2.1220Ste - 1.4847Ste 2 ) Further terms were statistically significant in this model but did not lead to worthwhile reduction in the standard deviation of the percentage differences between the experimental and predicted thawing times and so were not included. Third, a three-stage calculation approach similar to that of Pham ~9'2° was developed:
AD
~ AH~D ( +hD~
(8)
where: AH 1 = Cs(Tf.ve- Tin) AL
= Z-(T~n
kl
= ks
+ Travo)/2
AH 2 = L AT2 = Ta- Tr.vo k2
= 0.25ks + 0.75kl
AH 3 = Cl(Tave- Zfave) AT3 = Z - (Tave+ ~ave)/2 k3
= kl
'/'ave
= Tfin -- (Tfin -
Ta)/(2 -{-
4/ni)
~vo = T~- 1.5 The choice of weighting for k2 in the phase change period, although physically reasonable, has no basis apart from convenience. To retain the analytical basis of Plank's equation the values of the mean thawing temperature (Tfaw) and the average final temperature (Tave) were calculated by the methods suggested by Pham 19. However, the method should still be considered as an empirical, rather than an analytical, modification of Plank's equation because of the arbitrary choice of the weighting factor. Last, a direct fitting of a correction to Plank's equation using an approach proposed by Pham 32 for freezing time prediction gave:
AD
+ ~
)(
08941
0'0244 4-0.6192 Pk )
The prediction accuracy of these four prediction formulae assessed against the slab, infinite cylinder and sphere data are summarized in the second part of Table 6. All four approaches gave accurate predictions and the high correlations with the finite difference method results indicate that the predictions take account of all major sources of variation in the thawing times. All the formulae are simple and have some physical basis although their derivations were not analytical.
Thawing times of simple shaped foods: D. J. Cleland et al. Further refinement of these methods may be possible but it is unlikely that significant increases in prediction accuracy could be achieved. If further terms are added there is the danger that they could be fitting a systematic component of the experimental error. For all four formulae the spread of predictions for each of the three shapes was slightly higher than for the numerical methods. The spread was also larger than that which could be achieved by fitting each formula for each shape separately. This is because the unified approach using the 6:3:2 ratio is not completely valid. However the small improvement in accuracy of shape-specific formulae is outweighed by the simplicity of the unified approach. The possibility must still exist that an approach different from any of those investigated could lead to a significantly more accurate formula. This is not considered likely as the major contributors to the difference between experimental and predicted values for all four formulae are the random experimental and data errors. None of the four approaches can be recommended as being significantly more accurate or simple than the others for the three basic shapes. However, testing against data other than that used to derive the formulae, especially for materials other than Tylose, would be helpful to confirm this and fully assess these methods. A paucity of suitable, accurate experimental thawing data in the literature prevented comprehensive checking. Comparison of predictions with experimental thawing times for the six runs performed with minced lean beef (Table 8) confirmed the accuracy of the four proposed methods (Table 9). The slightly greater offset of the mean prediction error from zero and the slightly larger standard deviations have probably arisen from the increased uncertainty in the thermal property data for minced lean beef (compared with that for Tylose) and the small sample size, rather than from the inherent inaccuracy in the prediction methods for non-Tylose data. Each of the four formulae perform better on some parts of the data than the other three approaches but there was nothing systematic in these observations for the full data set. Therefore all four simple approaches can be used with equal confidence. Engineers may have preferences amongst them based on their ease of understanding of the concepts but it is not really important which is used. Should accuracy be particularly important, users may wish to make predictions by more than one method and take an average. This procedure also helps guard against the possibility of calculation error, as predictions disagreeing by more than a few percent have almost certainly arisen from user error. Because Equations (6)-(9) were derived from Tylose thawing data only, their use should be restricted in Table 8
E x p e r i m e n t a l d a t a for t h a w i n g o f slabs o f m i n c e d lean beef
Rbsultats expbrimentaux pour la d~cong~lation de plaques de boeuf maigre hach~ Tableau 8
Run
D (m)
h (W m - 2 °C - 1)
Ta (°C)
Tin (°C)
tt (h)
1 2 3 4 5 6
0.024 0.024 0.047 0.047 0.075 0.075
13.2 50.4 18.2 78.1 24.5 172.7
5.8 26.9 8.0 15.8 43.2 9.3
- 19.1 - 15.6 -27.1 - 24.4 - 29.0 - 16.9
8.58 0.84 12.07 2.85 5.21 8.94
Table 9
S u m m a r y of p e r c e n t a g e differences b e t w e e n e x p e r i m e n t a l a n d p r e d i c t e d t h a w i n g t i m e s for m i n c e d lean beef s l a b s T a b l e a u 9 R~capitulation des diff&ences de pourcentage entre les temps
de d~congblation expbrimentaux et calculbs pour des plaques de boeuf maigre hachb Mean (%)
SD (%)
Min (%)
Max (%)
Corr
Method F i n i t e difference method E q n (6) E q n (7) E q n (8) E q n (9) C a l v e l o 3°
- 4.5 -7.2 - 7.0 - 6.4 - 5.4 - 7.0
5.9 8.5 6.9 7.9 6.0 6.5
-
2.5 3.6 1.1 3.6 2.5 2.2
1.00 0.81 0.91 0.92 0.96 0.95
11.9 15.4 16.0 19.4 10.6 13.4
FDM
practice to those foods with thermal properties similar to those of Tylose until testing for a wider range of materials has been published. However, a wide range of medium to high moisture foods meet this restriction so it is not unduly limiting. The ranges of the thawing conditions used in the experimental data are a further set of restrictions to the applicability of Equations (6)-(9), outside which the predictions may be unreliable. These ranges are: 0.6
Conclusion Numerical methods (finite differences and finite elements) that accounted for thermal properties continuously variable with temperature accurately predicted thawing times for slabs, infinite cylinders and spheres. No previously published simple prediction formula was found that was both sufficiently accurate and expressed in a form suitable for it to be adopted as a general thawing time prediction method. Four improved formulae that gave comparable results with the numerical methods were developed. Each of these formulae represented a different approach to modifying the well known Plank's equation. All the approaches were limited more by the accuracy of the data from which they were derived than by the inherent inaccuracy of the approach used, and are of similar accuracy and simplicity. It seems unlikely that significantly better methods can be developed that retain the advantage of simplicity. Further testing of these prediction formulae against independently measured thawing data for materials other than Tylose would be helpful.
Acknowledgements The authors acknowledge the financial support of the Meat Industry Research Institute of New Zealand for this research. In addition, D. J. Cleland acknowledges the support of the New Zealand Meat Producers Board in t he form of a Sir Walter Mulholland Fellowship.
Rev. Int. Froid 1986 Vol 9 Juillet
227
Thawing times of simple shaped foods. D. J. Cleland et al. References 1
2
3 4 5 6 7
8 9 10 11 12 13 14 15
228
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