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Thawning time prediction for simple shaped foods using a generalized graphical method
Thawing time prediction for simple shaped foods using a generalized graphical method V. O. Salvadori and R. H. Mascheroni Centro de Investigacion y Desarrollo en Criotecnologia de Alimentos (CIDCA), Facultad de Ciencias Exactas, 47 y 116 (1900) La Plata, Argentina Received 2 February 1988; revised 10 January 1989
A graphical method is proposed for the estimation of thawing times of foods with a high water content. It has been developed from the predictions of a numerical model which solves the heat balance for a food undergoing thawing. The method is valid for foods with different shapes (slab, cylinder and sphere) and covers a wide range of working conditions for industrial thawers. It also enables the prediction to be made for any given end temperature in the thermal centre of the food. Thawing times predicted by this method have been compared with published experimental data, giving an average error in the predicted values of only 6.5%. (Keywords:thawing;thawingtime; foodproducts)
Prevision du temps de decongelation d'aliments de forme simple A l'aide d'une m&hode graphique generalisee On propose une mkthode graphique pour l'estimation des temps de d~congklation des aliments it forte teneur en eau. Cette mbthode est obtenue it partir d'un modble numbrique donnant le bilan thermique pour un produit en dkcongblation. On peut appliquer cette m~thode it des aliments de differentes formes (plaque, cylindre et sphkre). Elle inclut diverses conditions de d~congklation et permet de faire des pr~visions pour diff~rentes temperatures finales au centre thermique des aliments. En outre, on a comparb les temps calculus avec les temps experimentaux publids, et on a trouvb une difference moyenne de 6.5 % seulement.
(Mots cles: decongelation;temps de decongelation; aliments)
Food thawing processes are becoming increasingly important because of the continuous increase in the production of frozen food products. These are thawed for consumption or for further processing during the production of prepared meals. However, there is a lack of experimental data obtained under normalized conditions as well as simplified methods for the calculation of thawing times. Regarding food thawing, only recently have systematized experiments been performed, covering the most common food shapes, both for uni- and multidimensional heat conduction ~,2. With respect to thawing time predictions, several authors have developed numerical models valid for different geometries with uni- or multi-dimensional heat conduction 1'3-1°. These methods give an accurate prediction because they take into account the variations in food thermal properties during the thawing process. They require the use of computers with sufficient memory and reduce important process time, mainly in multidimensional heat conduction. On the other hand, simplified methods have been developed, which are generally modifications of Plank's equation or regressions obtained from experimental data l'3,s'1M3. Normally they appear in the form of algebraic equations, are valid for defined ranges of working conditions (Ti, Th, h, L) and only aim to predict the time necessary to 014(~7007/89/040232-05503.00 © 1989 Butterworth & Co (Publishers)Ltd and IIR 232 Int. J. Refrig. 1989 Vol 12 July
obtain a specified final temperature, T~, generally 0°C, in the thermal centre of the food. As has been adequately proved, mainly in the related area of food freezing, there is a limit in the precision attainable with both methods, due to the lack of precise information about food thermal properties and heat transfer coefficients in industrial equipment 14. In these situations a simplified graphical method of the type proposed in this work provides the same order of accuracy with far fewer calculations. Besides, it enables the prediction of thawing times for any desired final temperature. This is very important in reprocessing of previously frozen foods which are heated to temperatures slightly lower than T~r before being cut or minced.
Mathematical formulation In a previous work Salvadori et al. 15 used a finitedifferences nmnerical method to obtain generalized graphs of T~ versus t during the freezing of food slabs, infinite cylinders or spheres. Freezing times calculated employing those curves gave a very good correlation to experimental results, measured for the three geometries for a wide variety of high water content foods and test substances. In this work a similar methodology was used to obtain the proposed method.
Thawing time prediction for simple shaped foods. V. O. Salvadori and R. H. Mascheroni
Nomenclature Bi c Cpo Fo h L
n p
Biot number (11 1 ko-t) Constant in the definition of X Heat capacity of unfrozen food (J kg- ~ °C-~) Fourier number (t % L-=) Heat transfer coefficient (W m - ~ °C- ~) Food dimension (half thickness of slab, radius of cylinder or sphere) (m) Constant in the definition of X Constant in the definition of X
Table 1 Extremes range of processing conditions used in the definition of the method, and values of empirical constants used in the calculation of X Tableau 1 Domaines extremes de conditions de traitement utilisbes dans la dbfinition de la m&hode et valeurs des constantes empiriques utilisbes dans le calcul de X Range of thawing parameters Geometry
Th
Slab Infinite cylinder Sphere
5-35 5~45 5-45
Ti -10to -31 - 10 to - 35 - 1 0 to - 3 5
Value of constants in X definition
Bi
n
p
1 151 1-44 1~4
0.74 0.03 0.74 0.05 0.715 0.03
t tt T Tc T~ Th T~ X %
Time (s) Thawing time (s) Temperature (°C) Thermal centre temperature (°C) Beginning of phase change temperature (°C) Heating medium temperature (°C) Initial temperature (°C) Characteristic variable (dimensionless) Thermal diffusivity of unfrozen food (m 2 s - l ) Percentage error (%)
values of c, n and p which give the lower average deviation between this curve and the limits of variation of all the individual generalized graphs. Figures 3 and 4 are similar to Figure 2, for infinite cylinder and sphere, respectively. The values of c, n and p for each geometry are listed in Table 1. Results and discussion
c 0.45 0.47 0.45
Table 2 presents a comparison of thawing times obtained in the present work against published experimental times
:~J i , t:~ i l t.~ tt :;t .
o
To predict temperature profiles during the thawing of foods of different shapes a proven numerical method, which was developed in explicit finite-differences, was used t 6. The numerical model considers the variation with temperature of the thermophysical properties. From the time temperature graph for the thermal centre of the food, which coincides with the geometrical centre, thawing time can be calculated for any final temperature. For the three geometries thermal properties of boneless beef were used, considering heat transfer perpendicular to fibres for slab, and minced beef for cylinder and sphere. For slabs additional runs were made using thermal properties of a test gel. In all cases values for the properties were taken from previous work 6,~v. A wide range of thawing conditions were tested, using the most extreme values of Ti, Th and Bi found in industrial equipment. These conditions are presented in Table 1. The basis of the method is to replace the independent variable time t for a new dimensionless variable X which takes into account the combined influence of time, thawing conditions, food size and properties of T~. More details can be found in Salvadori et al. ~5. For thawing, X is defined according to Equation (1):
w rr
-1
I<
~-5 o_
le.."
.
.. ,...f . . ,.;~ /,
lg
w b-
?"/t
-9
i;,ilj i
-13
! i
-17
110
0
]
210
n
310
X (DIMENSIONLESS)
Figure 1 Temperatures during thawing of a slab of meat versus independent variable X, for a wide range of thawing conditions. - - , Proposed method Figure 1 Evolution de la temperature au cours de la decong~lation d'une plaque de viande en fonction de la variable ind~pendante X, pour une grande skrie de conditions de decongelation: , M~thode proposed
/i!]/ .
.
f
; ~,li/ t~ f
t tt
re
X-
F°[(Tcr- Th)/Ycr]n [(Ti - Tcr)/Tcr]P(1/Bi -4-c)
(3-
(1)
In numerical calculations Tcr= - 1°C was considered. c, n and p are empirical constants whose values were chosen to give the best fit between the proposed test curve and the generalized ones. This is illustrated in Figures 1 and 2. Figure 1 shows curves of Tc versus X for slabs of beef being thawed under different working conditions. Figure 2 is a more detailed plot of the same curves in the final thawing zone. The test curve was obtained using the
-4
t
0
5
10
15
20
25
30
X (DIMENSLONLESS)
Figure 2
Reduced curves for the temperature of the centre of a slab in the zone of interest for thawing time predictions. - - , Proposed method Figure 2 Courbes r~duites de la tempOrature centrale d'une plaque plane dans la zone d'int~r~t pour les previsions du temps de decongelation: - - , Mbthode proposee
Rev. Int. Froid 1989 Vo112 Juillet
233
Thawing time prediction for simple shaped foods." V. O. Salvadori and R. H. Mascheroni S
,7
!
*_ 2
[
#
iJ / it •
..... 4
5
7
6
8
, =, _----e._~y-'--~.-' --~." g
-"
i:
;i ~.-
a.
10 11 12 13 X (DIMENSIONLESS)
-4
Figure 3 R e d u c e d c u r v e s for the t e m p e r a t u r e of the centre of a cylinder in the z o n e of interest for t h a w i n g t i m e predictions. - - , Proposed method F i g u r e 3 Courbes reduites de la temperature au centre d'un c ylindre dans
la zone d'int~r~t pour les pr~visions du temps de dbcongelation: , M~thode proposeb
Table 2
illi
D
.'.~;~" 2
z~'" 3
i 4
I . I'° 5
I 6
I 7
I,1 I!
[ I 8 9 X (DIMENSIONLESS)
Figure 4 R e d u c e d curves for the t e m p e r a t u r e of the centre of a sphere in the zone of interest for t h a w i n g time predictions. - - , Proposed method F i g u r e 4 Courbes reduites de la temperature au centre d'une sphbre dans
la zone d'intbr~t pour les previsions du temps de d~cong~lation." , M~thode proposee
C o m p a r i s o n of published t h a w i n g times with those predicted by this w o r k for slabs
Tableau 2
Comparaison des temps de decongelation publi~s avec ceux prbvus d'aprbs cette btude pour une plaque plane
Material Tylose 1 ko = 0.55 ~o = 1.48 x 1 0 - 7
L (cm)
h (W m - a oC - 1)
Th (°C)
Ti (°C)
q exp
Bi
(h)
tt calc (h)
e (%)
1.3 1.3 1.3 2.625 2.625 2.625 2.625 2.625 2.625 2.625 2.625 2.625 3.85 3.85 3.85 5.0 5.25 5.25 5.25
50.4 78.1 78.1 24.5 29.5 50.4 50.4 50.4 50.4 50.4 78.1 78.1 37.3 78.1 78.1 24.5 37.3 78.1 172.7
1.19 1.85 1.85 1.17 1.41 2.41 2.41 2.41 2,41 2,41 3,73 3.73 2,61 5.47 5.47 2.23 3.56 7.46 16.48
4.6 5.2 12.4 5.2 12.8 5.1 13.4 13.4 13.4 13.4 5.2 13.3 5.2 5.0 12.9 12.8 5.2 5.2 13.4
- 26.3 - 12.3 - 27.7 - 10.7 - 20.9 - 25.0 - 23.6 - 23.6 - 20.2 - 24.1 - 28.9 - 22.5 - 30.2 - 14.2 -21.0 - 9.4 - 10.4 - 28.8 - 23.8
3.49 2.51 1.42 13.20 6.26 9.41 4.60 4.50 4.68 4.65 7.58 3.91 18.49 14.39 7.25 16.92 29.33 23.62 10.61
3.33 2.31 1.34 12.36 6.21 8.52 4.51 4.51 4.31 4.34 7.02 3.75 17.55 13.36 7.25 17.01 27.67 22.84 10.64
-
24.5 23.0 22.0 22.0
20.37 20.37 20.33 20.28
- 16.85 - 11.43 --7.61 -- 7.80
-
-
4.67 7.88 5.54 6.36 0.80 9.33 2.06 0.22 7.96 6.68 7.36 3.99 5.08 7.32 0.00 0.53 5.65 3.29 0.31
Beef, boneless 11 ko = 0.464 % = 1.34 x 1 0 - 7
5.0 5.0 5.0 5.0
27.0 27.0 27.0 27,0
2.91 2.91 2.91 2.91
10.0 10.0 10.0 10.0
-30.0 - 30.0 --28.0 -- 26.0
M a c k e r e l , fillets 18 ko = 0.463 c~o = 1.33 x 10 7
3.0 3.25 3.25 3.3 3.35
300.0 300.0 300.0 300.0 300.0
19.44 21.06 21.06 21.38 21.71
19.5 19.1 19.1 18.3 19.8
- 30.6 - 28.3 -30.8 - 28.9 - 30.6
3.43 3.25 3.47 3.72 3.28
2.95 3.47 3.48 3.69 3.60
- 14.09 6.86 0.35 - 0.80 9.64
A g a r gel 11 ko = 0.564 c~o = 1.34 x 10- 7
3.0 3.0 3.0 3.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0
150.0 150.0 150.0 300.0 150.0 150.0 600.0 600.0 300.0 600.0 150.0
7.98 7.98 7.98 15.96 10.64 10.64 42.55 42.55 21.28 42.55 10.64
10.0 10.0 15.0 18.0 10.0 10.0 18.0 15.0 15.0 20.0 20.0
-
5.5 5.0 4.0 3.17 9.25 8.67 5.5 6.0 6.0 4.83 5.5
5.21 5.19 3.90 3.47 8.74 8.70 5.08 5.73 5.97 4.73 5.42
- 5.24 3.79 - 2.45 9.52 - 5.48 0.32 - 7.66 - 4.56 -0.44 - 2.16 - 1.49
S a r d i n e 19 ko = 0.464 % = 1.73 x 1 0 - 7
3.25
600.0
42.02
15.0
- 20.0
3.0
2.98
- 0.73
C o d 13 ko = 0.470 ~o = 1.26 x 10-7
5.0
14.0
1.49
15.0
-18.0
20.0
22.82
14.12
Beef, m i n c e d ix ko = 0.4635 c%= 1.33 x 10 - 7
3.0 4.0 4.0
150.0 150.0 300.0
9.71 12.94 25.89
15.0 10.0 10.0
- 15.8 - 19.6 - 19.4
3.82 8.61 7.98
4.18 1.31 --3.30
234
Int. J. Refrig. 1989 Vol 12 July
16.0 14.0 11.0 10.5 15.0 12.8 15.0 12.0 10.0 16.0 15.0
3.67 8.5 8.25
Thawing time prediction for simple shaped foods." V. O. Salvadori and R. H. Mascheroni for slabs. For each product its thermal properties in thawed state, the experimental thawing conditions and the bibliographic citation from which data were taken are given. In a similar way Tables 3 and 4 compare predicted and experimental data for cylinders and spheres. All
calculated. It was defined according to Equation (2): =
thawing times were calculated for a final temperature Tc = O ° C . In order proposed
In
all
procedure, to obtain
an estimation
methodology,
the
of the precision
percentage
error
(predicted
tt-
experimental
tt) 1 0 0
(2)
experimental tt
cases
tt
was
a practical
obtained guide
following
for using
the
the method
same is:
of the e
was
1.
the
product
is defined
and
its thermal
properties
in
T a b l e 3 C o m p a r i s o n o f p u b f i s h e d t h a w i n g times w i t h t h o s e p r e d i c t e d b y this w o r k for a s p h e r e T a b l e a u 3 Comparaison des temps de d~cong~lation publi~s avec ceux prkvus d'apr~s cette ~tude pour une sphbre
Material Tylose 1 ko=0.55 eo = 1.48 x 1 0 - 7
L (cm)
h (W m - 2 oC
6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4
41.9 41.9 51.6 51.6 51.6 51.6 51.6 74.8 74.8 74.8 246.2 246.2 246.2 246.2
-
1)
Bi
Th (°C)
Ti (°C)
tt e x p (h)
tt c a l c (h)
e (%)
4.88 4.88 6.0 6.0 6.0 6.0 6.0 8.7 8.7 8.7 28.65 28.65 28.65 28.65
5.5 12.0 43.6 5.1 14.5 22.0 43.3 5.3 11.9 18.3 44.0 13.0 21.1 43.3
- 17.1 -33.0 - 20.3 - 15.5 - 20.3 - 18.8 - 27.9 - 26.5 - 23.1 - 18.8 - 15.1 - 19.7 - 25.3 - 9.7
12.82 7.63 3.13 12.26 5.81 4.31 2.76 11.33 6.39 4.78 2.66 5.42 3.92 2.26
12.13 7.54 2.90 11.91 6.17 4.64 2.94 10.85 6.47 4.82 2.24 5.21 3.79 2.23
- 5.41 - 1.16 - 7.45 -2.84 6.15 7.65 6.52 -4.28 1.24 0.81 - 15.71 - 3.86 - 3.33 - 1.11
T a b l e 4 C o m p a r i s o n o f p u b l i s h e d t h a w i n g times w i t h t h o s e p r e d i c t e d b y this w o r k for a n infinite c y l i n d e r T a b l e a u 4 Comparaison des temps de dbeongelation publi~s avec ceux prevus d'apr~s certe ~tude pour un cylindre
Material Tylose 1 ko = 0.55 ~ o = 1 . 4 8 x 10
L (cm)
h (W m - 2 ° C - 1)
Bi
Th (°C)
Ti (°C)
tt e x p (h)
tt c a l c (h)
E (%)
2.55 2.55 2.55 2.55 2.55 2.55 2.55 2.55 2.55 2.55 2.55 5.15 5.15 5.15 5.15 5.15 5.15 5.15 5.15 5.15 5.15 5.30 5.30 5.30 7.80 7.80 7.80 7.80 7.80 7.80 7.80 7.80 7.80 7.80
19.0 19.0 19.0 19.0 27.9 27.9 27.9 27.9 46.5 46.5 46.5 19.5 19.5 19.5 25.1 25.1 25.1 25.1 37.4 37.4 37.4 113.0 113.0 113.0 23.5 23.5 23.5 43.5 43.5 43.5 43.5 90.7 90.7 90.7
0.88 0.88 0.88 0.88 1.29 1.29 1.29 1.29 2.16 2.16 2.16 1.83 1.83 1.83 2.35 2.35 2.35 2.35 3.5 3.5 3.5 10.89 10.89 10.89 3.38 3.38 3.38 6.17 6.17 6.17 6.17 12.86 12.86 12.86
5.8 9.6 14.6 43.9 5.3 13.2 18.9 40.3 5.1 8.5 44.0 5.8 18.3 43.9 5.3 13.2 18.7 40.3 5.3 13.0 43.3 5.1 8.5 43.3 5.1 21.1 43.3 5.3 8.2 11.9 40.3 5.1 13.0 43.2
-
7.15 5.03 3.76 1.64 5.96 3.0 2.49 1.30 4.22 2.93 0.87 18.20 7.54 4.33 16.73 8.84 6.63 3.79 13.89 7.25 3.26 11.36 7.94 2.47 34.41 12.43 7.34 25.70 19.21 15.40 6.09 23.74 12.30 5.31
6.48 4.88 3.58 1.68 5.56 2.98 2.37 1.32 4.04 2.93 0.92 16.79 7.81 4.32 16.39 8.62 6.76 3.85 13.06 7.36 3.26 11.11 7.86 2.43 33.61 12.75 7.47 25.52 19.90 15.51 6.47 23.41 12.19 5.17
-9.39 - 3.01 - 4.73 2.44 - 6.79 - 0.69 - 5.02 1.53 - 4.38 0.00 6.05 -7.79 3.64 - 0.33 - 2.04 -2.44 1.97 1.56 - 5.98 1.48 0.00 - 2.23 - 1.07 - 1.64 - 2.32 2.58 1.72 - 0.68 3.57 0.70 6.21 - 1.37 -0.89 - 2.66
12.1 28.2 18.2 28.1 28.0 18.5 26.5 11.8 10.0 11.9 10.6 13.1 14.9 26.9 31.2 14.4 14.1 10.6 10.6 14.5 30.5 28.8 20.2 10.7 28.4 20.6 14.0 14.9 26.9 27.4 21.2 27.9 13.6 11.9
Rev. Int. Froid 1989 Vo112 Juillet
235
Thawing time prediction for simple shaped foods." V. O. Salvadori and R. H. Mascheroni t h a w e d state are o b t a i n e d from the literature or by measurement; 2. a desired final t e m p e r a t u r e is selected for the t h e r m a l centre; 3. w o r k i n g c o n d i t i o n s are defined (Ti, T h and Bi); 4. the X value is o b t a i n e d from the curve c o r r e s p o n d i n g to the p r o d u c t shape; 5. t h a w i n g t i m e is c a l c u l a t e d from E q u a t i o n (3), which is o b t a i n e d r e a r r a n g i n g E q u a t i o n (1):
3 4
5 6 7
XL~(1/Bi + c)[(W~- To~)/W.]~ t, ~o[(L,- T~)/T.] °
(3) 8
Conclusions
A n a l y s i n g the results o b t a i n e d , the following conclusions can be d r a w n . 1. This is an accurate m e t h o d with similar precision to m u c h m o r e involved a p p r o x i m a t i o n s or even n u m e r i c a l models, with an average e r r o r of - 6.52 %, a n d a range of - 1 6 . 8 5 to + 1 4 . 1 2 % . 2. This m e t h o d is valid for p r o d u c t s with very different p r o p e r t i e s a n d c o m p o s i t i o n , like boneless or minced meats, w h o l e o r filleted fish, tylose, a g a r gel, etc. and in g e n e r a / f o r a n y high water c o n t e n t material. 3. The simplicity of a p p l i c a t i o n of this m e t h o d m a k e s it very useful for industrial calculations, where detailed information about thermal properties and/or computer p r o g r a m s are not usually available, and the m a i n need is to get the process time for a given set of t h a w i n g conditions.
9 10
11
12 13 14 15 16 17
References
1 2
2.36
Cleland, D. J., Cleland, A. C., Earle, R. L., Byrne, S. J. Prediction of thawing times for foods of simple shape lnt J Refi'ig (1986) 9 220-228 Cleland, D. J., Cleland, A. C., Earle, R. L., Byrne, S. J. Experimental data for freezing and thawing of multidimensional objects lnt J Refrig (1987) 10 22-31
Int. J. Refrig. 1989 Vo112 July
18
19
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 Cleland, D. J., Cleland, A. C., Earle, R. L, Byrne, S. J. Prediction of freezing and thawing times for multidimensional shapes by numerical methods lnt J Refrig (1987) 10 32-39 Creed, P. G., James, S. J. Predicting thawing times of frozen boneless beef blocks lnt J Refrig (1981) 4 355-358 Flares, E. S., Mascherani, R. H. Theoretical and experimental study of thawing of frozen food blocks by aspersion with water Lat Am J Heat Mass Transf (1983) 7 263-279 Flares, E. S., Maseheroni, R. H. Descongelacion de bloques de carne par aspersibn con agua, Prediccion de perfiles de temperatura y tiempos de descongelacion Mecimica Computacional (1985) 1 196-208 Maseheroni, R. H. The utilization of numerical methods for the solution of the heat balance during the thawing of meat blocks under industrial conditions Lat Am J Heat Mass Transf (1982) 6 13-29 Pham, Q. T. A fast, unconditionally stable finite difference scheme for heat conduction problems with phase change lnt J Heat Mass Transf (1985) 28 2079-2084 Sehwartzberg, H. G., Rosenau, J. R., Haight, J. R. The prediction of freezing and thawing temperature vs time behaviour through the use of effective heat capacity equation I1R Commissions CI and C2 Karlsruhe FRG (1977) 311-318 Flares, E. S., Bazan, H. C., De Miehelis, A., Maseheroni, R. H. Thawing time of blocks of boneless or minced meats. Measure and prediction for different types of equipments. Int d Food Sci Technol (1988) submitted for publication Scott, K. R., Hayakawa, K. Simplified prediction of freezing and thawing times of foods ASHRAE Trans (1977) 83 Part 1 541-548 Vanichseni,S. Thawing of frozen lamb shoulders M I R I N Z REPORTNo 233 Hamilton, New Zealand (1971) Cleland, A. C., Earle, R. L., Cleland, Do J, The effect of freezing rate on the accuracy of numerical freezing time predictions lnr d Refrig (1982) 5 294-301 Salvadori, V. O., Reynoso, R. O., De Miehelis, A., Maseheroni, R. H. Freezing time predictions for regular shaped foods: a simplified graphical method lnt d Refrig (1987) 10 357-361 James, S., Bailey, C., Ono, S. Determination of freezing and thawing times in the centre of blocks of meat by measurement of surface temperature J Food Technol (1976) 11 505-513 Sanz, P. D., Dominguez, M., Maseheroni, R. H. Thermophysical properties of meat products. General bibliography and experimental data Trans ASAE (1987) 30 283-289 Flechtenmaeher, W. Measurement of heat transfer in thawing of fish fillet block. Part II: Thawing of mackerel fillets in running water lntern Zeits Lebens Technol Ferfahr (1983) 34 362, 364, 366-368 Crepey, J. R., Beeel, P. Etude experimentale de certaines methodes de decongelation appliqu~es au than eta la sardine IIR Commissions C2, D1 y D2 Budapest, Hungary (1978)
A graphical method is proposed for the estimation of thawing times of foods with a high water content. It has been developed from the predictions of a numerical model which solves the heat balance for a food undergoing thawing. The method is valid for foods with different shapes (slab, cylinder and sphere) and covers a wide range of working conditions for industrial thawers. It also enables the prediction to be made for any given end temperature in the thermal centre of the food. Thawing times predicted by this method have been compared with published experimental data, giving an average error in the predicted values of only 6.5%. (Keywords:thawing;thawingtime; foodproducts)
Prevision du temps de decongelation d'aliments de forme simple A l'aide d'une m&hode graphique generalisee On propose une mkthode graphique pour l'estimation des temps de d~congklation des aliments it forte teneur en eau. Cette mbthode est obtenue it partir d'un modble numbrique donnant le bilan thermique pour un produit en dkcongblation. On peut appliquer cette m~thode it des aliments de differentes formes (plaque, cylindre et sphkre). Elle inclut diverses conditions de d~congklation et permet de faire des pr~visions pour diff~rentes temperatures finales au centre thermique des aliments. En outre, on a comparb les temps calculus avec les temps experimentaux publids, et on a trouvb une difference moyenne de 6.5 % seulement.
(Mots cles: decongelation;temps de decongelation; aliments)
Food thawing processes are becoming increasingly important because of the continuous increase in the production of frozen food products. These are thawed for consumption or for further processing during the production of prepared meals. However, there is a lack of experimental data obtained under normalized conditions as well as simplified methods for the calculation of thawing times. Regarding food thawing, only recently have systematized experiments been performed, covering the most common food shapes, both for uni- and multidimensional heat conduction ~,2. With respect to thawing time predictions, several authors have developed numerical models valid for different geometries with uni- or multi-dimensional heat conduction 1'3-1°. These methods give an accurate prediction because they take into account the variations in food thermal properties during the thawing process. They require the use of computers with sufficient memory and reduce important process time, mainly in multidimensional heat conduction. On the other hand, simplified methods have been developed, which are generally modifications of Plank's equation or regressions obtained from experimental data l'3,s'1M3. Normally they appear in the form of algebraic equations, are valid for defined ranges of working conditions (Ti, Th, h, L) and only aim to predict the time necessary to 014(~7007/89/040232-05503.00 © 1989 Butterworth & Co (Publishers)Ltd and IIR 232 Int. J. Refrig. 1989 Vol 12 July
obtain a specified final temperature, T~, generally 0°C, in the thermal centre of the food. As has been adequately proved, mainly in the related area of food freezing, there is a limit in the precision attainable with both methods, due to the lack of precise information about food thermal properties and heat transfer coefficients in industrial equipment 14. In these situations a simplified graphical method of the type proposed in this work provides the same order of accuracy with far fewer calculations. Besides, it enables the prediction of thawing times for any desired final temperature. This is very important in reprocessing of previously frozen foods which are heated to temperatures slightly lower than T~r before being cut or minced.
Mathematical formulation In a previous work Salvadori et al. 15 used a finitedifferences nmnerical method to obtain generalized graphs of T~ versus t during the freezing of food slabs, infinite cylinders or spheres. Freezing times calculated employing those curves gave a very good correlation to experimental results, measured for the three geometries for a wide variety of high water content foods and test substances. In this work a similar methodology was used to obtain the proposed method.
Thawing time prediction for simple shaped foods. V. O. Salvadori and R. H. Mascheroni
Nomenclature Bi c Cpo Fo h L
n p
Biot number (11 1 ko-t) Constant in the definition of X Heat capacity of unfrozen food (J kg- ~ °C-~) Fourier number (t % L-=) Heat transfer coefficient (W m - ~ °C- ~) Food dimension (half thickness of slab, radius of cylinder or sphere) (m) Constant in the definition of X Constant in the definition of X
Table 1 Extremes range of processing conditions used in the definition of the method, and values of empirical constants used in the calculation of X Tableau 1 Domaines extremes de conditions de traitement utilisbes dans la dbfinition de la m&hode et valeurs des constantes empiriques utilisbes dans le calcul de X Range of thawing parameters Geometry
Th
Slab Infinite cylinder Sphere
5-35 5~45 5-45
Ti -10to -31 - 10 to - 35 - 1 0 to - 3 5
Value of constants in X definition
Bi
n
p
1 151 1-44 1~4
0.74 0.03 0.74 0.05 0.715 0.03
t tt T Tc T~ Th T~ X %
Time (s) Thawing time (s) Temperature (°C) Thermal centre temperature (°C) Beginning of phase change temperature (°C) Heating medium temperature (°C) Initial temperature (°C) Characteristic variable (dimensionless) Thermal diffusivity of unfrozen food (m 2 s - l ) Percentage error (%)
values of c, n and p which give the lower average deviation between this curve and the limits of variation of all the individual generalized graphs. Figures 3 and 4 are similar to Figure 2, for infinite cylinder and sphere, respectively. The values of c, n and p for each geometry are listed in Table 1. Results and discussion
c 0.45 0.47 0.45
Table 2 presents a comparison of thawing times obtained in the present work against published experimental times
:~J i , t:~ i l t.~ tt :;t .
o
To predict temperature profiles during the thawing of foods of different shapes a proven numerical method, which was developed in explicit finite-differences, was used t 6. The numerical model considers the variation with temperature of the thermophysical properties. From the time temperature graph for the thermal centre of the food, which coincides with the geometrical centre, thawing time can be calculated for any final temperature. For the three geometries thermal properties of boneless beef were used, considering heat transfer perpendicular to fibres for slab, and minced beef for cylinder and sphere. For slabs additional runs were made using thermal properties of a test gel. In all cases values for the properties were taken from previous work 6,~v. A wide range of thawing conditions were tested, using the most extreme values of Ti, Th and Bi found in industrial equipment. These conditions are presented in Table 1. The basis of the method is to replace the independent variable time t for a new dimensionless variable X which takes into account the combined influence of time, thawing conditions, food size and properties of T~. More details can be found in Salvadori et al. ~5. For thawing, X is defined according to Equation (1):
w rr
-1
I<
~-5 o_
le.."
.
.. ,...f . . ,.;~ /,
lg
w b-
?"/t
-9
i;,ilj i
-13
! i
-17
110
0
]
210
n
310
X (DIMENSIONLESS)
Figure 1 Temperatures during thawing of a slab of meat versus independent variable X, for a wide range of thawing conditions. - - , Proposed method Figure 1 Evolution de la temperature au cours de la decong~lation d'une plaque de viande en fonction de la variable ind~pendante X, pour une grande skrie de conditions de decongelation: , M~thode proposed
/i!]/ .
.
f
; ~,li/ t~ f
t tt
re
X-
F°[(Tcr- Th)/Ycr]n [(Ti - Tcr)/Tcr]P(1/Bi -4-c)
(3-
(1)
In numerical calculations Tcr= - 1°C was considered. c, n and p are empirical constants whose values were chosen to give the best fit between the proposed test curve and the generalized ones. This is illustrated in Figures 1 and 2. Figure 1 shows curves of Tc versus X for slabs of beef being thawed under different working conditions. Figure 2 is a more detailed plot of the same curves in the final thawing zone. The test curve was obtained using the
-4
t
0
5
10
15
20
25
30
X (DIMENSLONLESS)
Figure 2
Reduced curves for the temperature of the centre of a slab in the zone of interest for thawing time predictions. - - , Proposed method Figure 2 Courbes r~duites de la tempOrature centrale d'une plaque plane dans la zone d'int~r~t pour les previsions du temps de decongelation: - - , Mbthode proposee
Rev. Int. Froid 1989 Vo112 Juillet
233
Thawing time prediction for simple shaped foods." V. O. Salvadori and R. H. Mascheroni S
,7
!
*_ 2
[
#
iJ / it •
..... 4
5
7
6
8
, =, _----e._~y-'--~.-' --~." g
-"
i:
;i ~.-
a.
10 11 12 13 X (DIMENSIONLESS)
-4
Figure 3 R e d u c e d c u r v e s for the t e m p e r a t u r e of the centre of a cylinder in the z o n e of interest for t h a w i n g t i m e predictions. - - , Proposed method F i g u r e 3 Courbes reduites de la temperature au centre d'un c ylindre dans
la zone d'int~r~t pour les pr~visions du temps de dbcongelation: , M~thode proposeb
Table 2
illi
D
.'.~;~" 2
z~'" 3
i 4
I . I'° 5
I 6
I 7
I,1 I!
[ I 8 9 X (DIMENSIONLESS)
Figure 4 R e d u c e d curves for the t e m p e r a t u r e of the centre of a sphere in the zone of interest for t h a w i n g time predictions. - - , Proposed method F i g u r e 4 Courbes reduites de la temperature au centre d'une sphbre dans
la zone d'intbr~t pour les previsions du temps de d~cong~lation." , M~thode proposee
C o m p a r i s o n of published t h a w i n g times with those predicted by this w o r k for slabs
Tableau 2
Comparaison des temps de decongelation publi~s avec ceux prbvus d'aprbs cette btude pour une plaque plane
Material Tylose 1 ko = 0.55 ~o = 1.48 x 1 0 - 7
L (cm)
h (W m - a oC - 1)
Th (°C)
Ti (°C)
q exp
Bi
(h)
tt calc (h)
e (%)
1.3 1.3 1.3 2.625 2.625 2.625 2.625 2.625 2.625 2.625 2.625 2.625 3.85 3.85 3.85 5.0 5.25 5.25 5.25
50.4 78.1 78.1 24.5 29.5 50.4 50.4 50.4 50.4 50.4 78.1 78.1 37.3 78.1 78.1 24.5 37.3 78.1 172.7
1.19 1.85 1.85 1.17 1.41 2.41 2.41 2.41 2,41 2,41 3,73 3.73 2,61 5.47 5.47 2.23 3.56 7.46 16.48
4.6 5.2 12.4 5.2 12.8 5.1 13.4 13.4 13.4 13.4 5.2 13.3 5.2 5.0 12.9 12.8 5.2 5.2 13.4
- 26.3 - 12.3 - 27.7 - 10.7 - 20.9 - 25.0 - 23.6 - 23.6 - 20.2 - 24.1 - 28.9 - 22.5 - 30.2 - 14.2 -21.0 - 9.4 - 10.4 - 28.8 - 23.8
3.49 2.51 1.42 13.20 6.26 9.41 4.60 4.50 4.68 4.65 7.58 3.91 18.49 14.39 7.25 16.92 29.33 23.62 10.61
3.33 2.31 1.34 12.36 6.21 8.52 4.51 4.51 4.31 4.34 7.02 3.75 17.55 13.36 7.25 17.01 27.67 22.84 10.64
-
24.5 23.0 22.0 22.0
20.37 20.37 20.33 20.28
- 16.85 - 11.43 --7.61 -- 7.80
-
-
4.67 7.88 5.54 6.36 0.80 9.33 2.06 0.22 7.96 6.68 7.36 3.99 5.08 7.32 0.00 0.53 5.65 3.29 0.31
Beef, boneless 11 ko = 0.464 % = 1.34 x 1 0 - 7
5.0 5.0 5.0 5.0
27.0 27.0 27.0 27,0
2.91 2.91 2.91 2.91
10.0 10.0 10.0 10.0
-30.0 - 30.0 --28.0 -- 26.0
M a c k e r e l , fillets 18 ko = 0.463 c~o = 1.33 x 10 7
3.0 3.25 3.25 3.3 3.35
300.0 300.0 300.0 300.0 300.0
19.44 21.06 21.06 21.38 21.71
19.5 19.1 19.1 18.3 19.8
- 30.6 - 28.3 -30.8 - 28.9 - 30.6
3.43 3.25 3.47 3.72 3.28
2.95 3.47 3.48 3.69 3.60
- 14.09 6.86 0.35 - 0.80 9.64
A g a r gel 11 ko = 0.564 c~o = 1.34 x 10- 7
3.0 3.0 3.0 3.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0
150.0 150.0 150.0 300.0 150.0 150.0 600.0 600.0 300.0 600.0 150.0
7.98 7.98 7.98 15.96 10.64 10.64 42.55 42.55 21.28 42.55 10.64
10.0 10.0 15.0 18.0 10.0 10.0 18.0 15.0 15.0 20.0 20.0
-
5.5 5.0 4.0 3.17 9.25 8.67 5.5 6.0 6.0 4.83 5.5
5.21 5.19 3.90 3.47 8.74 8.70 5.08 5.73 5.97 4.73 5.42
- 5.24 3.79 - 2.45 9.52 - 5.48 0.32 - 7.66 - 4.56 -0.44 - 2.16 - 1.49
S a r d i n e 19 ko = 0.464 % = 1.73 x 1 0 - 7
3.25
600.0
42.02
15.0
- 20.0
3.0
2.98
- 0.73
C o d 13 ko = 0.470 ~o = 1.26 x 10-7
5.0
14.0
1.49
15.0
-18.0
20.0
22.82
14.12
Beef, m i n c e d ix ko = 0.4635 c%= 1.33 x 10 - 7
3.0 4.0 4.0
150.0 150.0 300.0
9.71 12.94 25.89
15.0 10.0 10.0
- 15.8 - 19.6 - 19.4
3.82 8.61 7.98
4.18 1.31 --3.30
234
Int. J. Refrig. 1989 Vol 12 July
16.0 14.0 11.0 10.5 15.0 12.8 15.0 12.0 10.0 16.0 15.0
3.67 8.5 8.25
Thawing time prediction for simple shaped foods." V. O. Salvadori and R. H. Mascheroni for slabs. For each product its thermal properties in thawed state, the experimental thawing conditions and the bibliographic citation from which data were taken are given. In a similar way Tables 3 and 4 compare predicted and experimental data for cylinders and spheres. All
calculated. It was defined according to Equation (2): =
thawing times were calculated for a final temperature Tc = O ° C . In order proposed
In
all
procedure, to obtain
an estimation
methodology,
the
of the precision
percentage
error
(predicted
tt-
experimental
tt) 1 0 0
(2)
experimental tt
cases
tt
was
a practical
obtained guide
following
for using
the
the method
same is:
of the e
was
1.
the
product
is defined
and
its thermal
properties
in
T a b l e 3 C o m p a r i s o n o f p u b f i s h e d t h a w i n g times w i t h t h o s e p r e d i c t e d b y this w o r k for a s p h e r e T a b l e a u 3 Comparaison des temps de d~cong~lation publi~s avec ceux prkvus d'apr~s cette ~tude pour une sphbre
Material Tylose 1 ko=0.55 eo = 1.48 x 1 0 - 7
L (cm)
h (W m - 2 oC
6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4
41.9 41.9 51.6 51.6 51.6 51.6 51.6 74.8 74.8 74.8 246.2 246.2 246.2 246.2
-
1)
Bi
Th (°C)
Ti (°C)
tt e x p (h)
tt c a l c (h)
e (%)
4.88 4.88 6.0 6.0 6.0 6.0 6.0 8.7 8.7 8.7 28.65 28.65 28.65 28.65
5.5 12.0 43.6 5.1 14.5 22.0 43.3 5.3 11.9 18.3 44.0 13.0 21.1 43.3
- 17.1 -33.0 - 20.3 - 15.5 - 20.3 - 18.8 - 27.9 - 26.5 - 23.1 - 18.8 - 15.1 - 19.7 - 25.3 - 9.7
12.82 7.63 3.13 12.26 5.81 4.31 2.76 11.33 6.39 4.78 2.66 5.42 3.92 2.26
12.13 7.54 2.90 11.91 6.17 4.64 2.94 10.85 6.47 4.82 2.24 5.21 3.79 2.23
- 5.41 - 1.16 - 7.45 -2.84 6.15 7.65 6.52 -4.28 1.24 0.81 - 15.71 - 3.86 - 3.33 - 1.11
T a b l e 4 C o m p a r i s o n o f p u b l i s h e d t h a w i n g times w i t h t h o s e p r e d i c t e d b y this w o r k for a n infinite c y l i n d e r T a b l e a u 4 Comparaison des temps de dbeongelation publi~s avec ceux prevus d'apr~s certe ~tude pour un cylindre
Material Tylose 1 ko = 0.55 ~ o = 1 . 4 8 x 10
L (cm)
h (W m - 2 ° C - 1)
Bi
Th (°C)
Ti (°C)
tt e x p (h)
tt c a l c (h)
E (%)
2.55 2.55 2.55 2.55 2.55 2.55 2.55 2.55 2.55 2.55 2.55 5.15 5.15 5.15 5.15 5.15 5.15 5.15 5.15 5.15 5.15 5.30 5.30 5.30 7.80 7.80 7.80 7.80 7.80 7.80 7.80 7.80 7.80 7.80
19.0 19.0 19.0 19.0 27.9 27.9 27.9 27.9 46.5 46.5 46.5 19.5 19.5 19.5 25.1 25.1 25.1 25.1 37.4 37.4 37.4 113.0 113.0 113.0 23.5 23.5 23.5 43.5 43.5 43.5 43.5 90.7 90.7 90.7
0.88 0.88 0.88 0.88 1.29 1.29 1.29 1.29 2.16 2.16 2.16 1.83 1.83 1.83 2.35 2.35 2.35 2.35 3.5 3.5 3.5 10.89 10.89 10.89 3.38 3.38 3.38 6.17 6.17 6.17 6.17 12.86 12.86 12.86
5.8 9.6 14.6 43.9 5.3 13.2 18.9 40.3 5.1 8.5 44.0 5.8 18.3 43.9 5.3 13.2 18.7 40.3 5.3 13.0 43.3 5.1 8.5 43.3 5.1 21.1 43.3 5.3 8.2 11.9 40.3 5.1 13.0 43.2
-
7.15 5.03 3.76 1.64 5.96 3.0 2.49 1.30 4.22 2.93 0.87 18.20 7.54 4.33 16.73 8.84 6.63 3.79 13.89 7.25 3.26 11.36 7.94 2.47 34.41 12.43 7.34 25.70 19.21 15.40 6.09 23.74 12.30 5.31
6.48 4.88 3.58 1.68 5.56 2.98 2.37 1.32 4.04 2.93 0.92 16.79 7.81 4.32 16.39 8.62 6.76 3.85 13.06 7.36 3.26 11.11 7.86 2.43 33.61 12.75 7.47 25.52 19.90 15.51 6.47 23.41 12.19 5.17
-9.39 - 3.01 - 4.73 2.44 - 6.79 - 0.69 - 5.02 1.53 - 4.38 0.00 6.05 -7.79 3.64 - 0.33 - 2.04 -2.44 1.97 1.56 - 5.98 1.48 0.00 - 2.23 - 1.07 - 1.64 - 2.32 2.58 1.72 - 0.68 3.57 0.70 6.21 - 1.37 -0.89 - 2.66
12.1 28.2 18.2 28.1 28.0 18.5 26.5 11.8 10.0 11.9 10.6 13.1 14.9 26.9 31.2 14.4 14.1 10.6 10.6 14.5 30.5 28.8 20.2 10.7 28.4 20.6 14.0 14.9 26.9 27.4 21.2 27.9 13.6 11.9
Rev. Int. Froid 1989 Vo112 Juillet
235
Thawing time prediction for simple shaped foods." V. O. Salvadori and R. H. Mascheroni t h a w e d state are o b t a i n e d from the literature or by measurement; 2. a desired final t e m p e r a t u r e is selected for the t h e r m a l centre; 3. w o r k i n g c o n d i t i o n s are defined (Ti, T h and Bi); 4. the X value is o b t a i n e d from the curve c o r r e s p o n d i n g to the p r o d u c t shape; 5. t h a w i n g t i m e is c a l c u l a t e d from E q u a t i o n (3), which is o b t a i n e d r e a r r a n g i n g E q u a t i o n (1):
3 4
5 6 7
XL~(1/Bi + c)[(W~- To~)/W.]~ t, ~o[(L,- T~)/T.] °
(3) 8
Conclusions
A n a l y s i n g the results o b t a i n e d , the following conclusions can be d r a w n . 1. This is an accurate m e t h o d with similar precision to m u c h m o r e involved a p p r o x i m a t i o n s or even n u m e r i c a l models, with an average e r r o r of - 6.52 %, a n d a range of - 1 6 . 8 5 to + 1 4 . 1 2 % . 2. This m e t h o d is valid for p r o d u c t s with very different p r o p e r t i e s a n d c o m p o s i t i o n , like boneless or minced meats, w h o l e o r filleted fish, tylose, a g a r gel, etc. and in g e n e r a / f o r a n y high water c o n t e n t material. 3. The simplicity of a p p l i c a t i o n of this m e t h o d m a k e s it very useful for industrial calculations, where detailed information about thermal properties and/or computer p r o g r a m s are not usually available, and the m a i n need is to get the process time for a given set of t h a w i n g conditions.
9 10
11
12 13 14 15 16 17
References
1 2
2.36
Cleland, D. J., Cleland, A. C., Earle, R. L., Byrne, S. J. Prediction of thawing times for foods of simple shape lnt J Refi'ig (1986) 9 220-228 Cleland, D. J., Cleland, A. C., Earle, R. L., Byrne, S. J. Experimental data for freezing and thawing of multidimensional objects lnt J Refrig (1987) 10 22-31
Int. J. Refrig. 1989 Vo112 July
18
19
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 Cleland, D. J., Cleland, A. C., Earle, R. L, Byrne, S. J. Prediction of freezing and thawing times for multidimensional shapes by numerical methods lnt J Refrig (1987) 10 32-39 Creed, P. G., James, S. J. Predicting thawing times of frozen boneless beef blocks lnt J Refrig (1981) 4 355-358 Flares, E. S., Mascherani, R. H. Theoretical and experimental study of thawing of frozen food blocks by aspersion with water Lat Am J Heat Mass Transf (1983) 7 263-279 Flares, E. S., Maseheroni, R. H. Descongelacion de bloques de carne par aspersibn con agua, Prediccion de perfiles de temperatura y tiempos de descongelacion Mecimica Computacional (1985) 1 196-208 Maseheroni, R. H. The utilization of numerical methods for the solution of the heat balance during the thawing of meat blocks under industrial conditions Lat Am J Heat Mass Transf (1982) 6 13-29 Pham, Q. T. A fast, unconditionally stable finite difference scheme for heat conduction problems with phase change lnt J Heat Mass Transf (1985) 28 2079-2084 Sehwartzberg, H. G., Rosenau, J. R., Haight, J. R. The prediction of freezing and thawing temperature vs time behaviour through the use of effective heat capacity equation I1R Commissions CI and C2 Karlsruhe FRG (1977) 311-318 Flares, E. S., Bazan, H. C., De Miehelis, A., Maseheroni, R. H. Thawing time of blocks of boneless or minced meats. Measure and prediction for different types of equipments. Int d Food Sci Technol (1988) submitted for publication Scott, K. R., Hayakawa, K. Simplified prediction of freezing and thawing times of foods ASHRAE Trans (1977) 83 Part 1 541-548 Vanichseni,S. Thawing of frozen lamb shoulders M I R I N Z REPORTNo 233 Hamilton, New Zealand (1971) Cleland, A. C., Earle, R. L., Cleland, Do J, The effect of freezing rate on the accuracy of numerical freezing time predictions lnr d Refrig (1982) 5 294-301 Salvadori, V. O., Reynoso, R. O., De Miehelis, A., Maseheroni, R. H. Freezing time predictions for regular shaped foods: a simplified graphical method lnt d Refrig (1987) 10 357-361 James, S., Bailey, C., Ono, S. Determination of freezing and thawing times in the centre of blocks of meat by measurement of surface temperature J Food Technol (1976) 11 505-513 Sanz, P. D., Dominguez, M., Maseheroni, R. H. Thermophysical properties of meat products. General bibliography and experimental data Trans ASAE (1987) 30 283-289 Flechtenmaeher, W. Measurement of heat transfer in thawing of fish fillet block. Part II: Thawing of mackerel fillets in running water lntern Zeits Lebens Technol Ferfahr (1983) 34 362, 364, 366-368 Crepey, J. R., Beeel, P. Etude experimentale de certaines methodes de decongelation appliqu~es au than eta la sardine IIR Commissions C2, D1 y D2 Budapest, Hungary (1978)