Freezing Time Predictions for Different IF inal Product Temperatures

A Research Note
Freezing Time Predictions
for Different IFinal Product Temperatures
A. C. CLELAND and R. L. EARLE
ABSTRACT
A modification to an empirical food freezing time prediction formula is
proposed that allows the formula to be used for a range of final product
temperatures. Over a large data set (275 runs) the percentage difference
between experimentally measured times and predictions has a mean of
0.2% and a standard deviation of 6.8% for the improved formula compared
to - 2.4% and 8.5%. respectively, prior to modification. Ninety percent
of the predicted freezing times by the new method were within 2 11% of
the experimentally measured times and +- 9% of predictions by an accurate
finite difference scheme. This performance compares favorably with other
published freezing time prediction methods. The same type of modification
for varying final product temperature may be suitable for other empirical
formulae.
INTRODUCTION
ONE OF THE DESIRED consequences arising from the use of a
recently proposed systematic methodology for assessmentof freezing
time prediction
methods for foods (Cleland
and Earle, 1984) was
that areas in which improvements to prediction methods could be
made would emerge. One area highlighted was that prediction
formulae based on curve-fitting of experimental data collected in
experiments terminated at one final product temperature performed poorly in comparison when applied to situations with different final product temperatures. This was because there is no
term for varying Tn, in such prediction formulae. The formulae
of Cleland and Earle (1982) and Hung and Thompson (1983) were
found to be limited in this way, whereas those of Pham (1984a,
b) and de Michelis and Calvelo (1983) were not. This note proposes a modification
to the Cleland and Earle formula of the form:
t, = (tf to T,,) .
1 -
E
1.65 Ste
k,
In(E-1
(1)
where T,r is - lO”C, and T, must be less than T,.. A further
condition is that Tn, must be below the temperature range where
the bulk of the latent heat release takes place. The modification
may also be applicable to other empirical formulae such as that
of Hung and Thompson.
The form of Eq. (1) was derived by consideration of finite
difference results in Cleland and Earle (1984). Plots of In [(T,,
- T,)/(T,r - T,)] versus time were constructed using temperature data after the main phase change region. As would be expected from heat conduction theory for constant thermal properties
these showed straight lines. The slopes of the straight lines were
linearly related to the Stefan number, and were found to be inversely proportional to the frozen material thermal conductivity.
The factor of 1.65 was the fitted constant required. This constant
applies for all materials. The variation of it between materials,
and between different Biot numbers was sufficiently small (less
than 10%) for a single average value to be used.
RESULTS
THE SPECIFIC FORM of the new Cleland and Earle formula
uses T,r = - 10°C (because - 10°C was the final product center
temperature in the data set from the earlier work) and is:
Authors
Massey
C/eland and Earle are affiliated with the Dept. of Biotechnology,
Univ., Palmerston North, New Zealand.
1230-JOURNAL
OF FOOD SCIENCE-
Volume 49 (1984J
where P = 0.5 [1.026 + 0.5808 Pk +
Ste (0.2296 Pk + O.lOSO)]
R = 0.125 [1.202 + Ste (3.41 Pk + 0.7336)]
Pk = CL(Ti - Tr)/AH,o
Ste = C,(Tf - T,)/AHlo
(3)
(4)
(5)
(6)
and EHTD is defined in the manner of Cleland and Earle (1982).
As a calculation example the problem in Heldman (1983) was
chosen. A 2 cm diameter strawberry is frozen from 10°C initial
temperature to a final temperature of - 25°C. The cooling medium
is at - 35”C, and the surface heat transfer coefficient is 70 W/
m2”C. Thermal data were taken from Table 3 of Cleland and Earle
(1984), using the procedures described in that paper to ‘evaluate
frozen phase properties:
CL = 4.09 X lo6 J/m3’C
AHlo = 318 x lo6 J/m3
Tf = -O.S”C
C, = 2.15 x lo6 J/m3”C
k, = 2.08 W/m”C
Application of Eq. (2) to (7) led to the following parameters:
T, = -35°C
T fin = - 25°C
Ti = 10°C
EHTD = 3
Pk = 0.135
Ste = 0.233
P = 0.568
R = 0.185
(Tfin - T,)/( - 10 - T,) = 0.400; tf = 10.13 X 1.169 min =
11.85 min.
This freezing time to - 25°C of 11.85 min is close to the finite
difference prediction by Scheme A in Cleland and Earle (1984)
which was 11.55 min. The freezing time to - 10°C is 10.13 min
by this procedure. These results appear plausible but testing is
required over a range of conditions so new calculations were carried out for the composite data set in Cleland and Earle (1984).
The consequential changes to Tables 6 and 8 of that paper are
shown in Tables 1 and 2 respectively.
It was also considered worthwhile to attempt to apply the analogous procedure to modify the method of Hung and Thompson
(1983) using Tref = - 18°C.
AH18
tf = U(T, - T,)(EHTD)
(8)
where P = 0.7306 - 1.083 Pk + Ste
(15.4OU - 15.43 + 0.01329 Ste/Bi)
R = 0.2079 - 0.2656 U Ste
Pk = CL(Ti - Tr)lAHI,
Ste = C&T, - T,)/AHI,
,
u = 1 + 0.5C~ (Ti - Tj2 - 0.5Cs (Tr + 18)2
AHIS 0, - T,)
and Bi = hD/k,
(8)
(9)
(10)
(11)
(12)
(13)
This procedure can only apply for values of T, below - 18°C.
Table l-Summary
of comparisons
between experimental
and predicted
freezing times for five prediction
method9
Methods
Finite
differences
Cleland
and Earle
de Michelis
and Calvelo
Hung and
Thompson
Pham
Data source
Mean
sd.
Mean
s.d.
Mean
sd.
Mean
s.d.
Mean
s.d.
Hung and Thompson
23 Tylose slabs
9 potato slabs
9 carp slabs
9 ground beef slabs
9 lean beef slabs
-1.3
-3.9
3.1
6.8
2.8
8.7
4.9
12.9
16.6
11.6
-3.3
-7.3
-1.6
2.6
-4.3
7.0
3.3
9.9
13.4
7.4
-3.8
-5.1
-2.4
-1.8
-8.0
7.7
3.5
11.1
13.9
8.9
1.1
2.9
2.8
2.5
0.8
2.9
4.0
4.1
4.8
4.0
-4.3
-4.2
6.1
10.1
7.9
14.1
8.4
18.9
18.9
14.7
C/e/and and Ear/e
43 Tylose slabs
6 potato slabs
6 lean beef slabs
30 Tylose cylinders
30 Tylose spheres
72 Tylose bricks
0.0
-0.5
4.8
-1.8
-0.3
-3.8
5.3
5.1
4.7
5.2
3.3
5.8
1.6
-0.5
2.1
-1.0
1.6
-0.9
2.8
4.0
3.6
5.2
5.7
3.9
3.7
4.1
0.8
2.3
-1.3
4.4
2.3
4.8
5.1
5.1
4.7
8.0
11.3
10.8
1.8
3.5
0.8
11.9
5.7
11.8
7.9
8.6
9.5
-4.1
-2.6
5.6
-6.5
-0.5
-26.1
6.2
3.5
4.4
8.2
10.3
9.6
-0.1
8.6
7.4
6.3
8.2
8.2
10.0
-0.9
4.1
4.9
6.6
1.8
3.8
9.2
11.5
-2.2
-11.3
10.5
10.8
were calculated
by the modified
de Michelis and Calvelo
5 lean beef slabs
24 lean beef bricks,
rods, finite cylinders
1.8
2.6
a All data expressed as percentage differences from experimental data. Results for the Cleland and Earle. and Hung and Thompson
* Ignores 4 experiments
methods
forms.
in which T, Z T,Bt
Table 2-Summary
of percentage differences between (A) experimental
under test and predicted freezing times from finite difference@
data and predicted
freezing times, and (6) predicted
freezing times from the methods
Methods
Data source
A// 275 experiments; (A) comparison
Mean
Standard deviation
Maximum
Minimum
Range enclosing 90% of data
Correlation (r) with finite differences
to experimental
207 experiments to - 10°C; (A) comparison
Mean
Standard deviation
Maximum
Minimum
Range
enclosing
90%
Hung and
Thompson
de Michelis
and Calvelo
0.2
6.8
23
-21
-10 to 11
0.746
0.3
6.7
18
-24
-10t09
0.762
3.2’
9.3
33
-18
-11 to 20
0.480
-8.7
15.1
38
-48
-35to15
0.619
3.5*
10.1
33
0.865
1.3
5.4
15
-13
-9t08
0.738
- 11.8
13.9
16
-48
-38to8
0.501
data
-1.1
9.4
22
-21
-13to15
0.838
-2.8
9.2
18
-24
-17to
13
0.929
“;
-5t09
0.552
0.5
5.1
14
3.0
8.2
31
to experimental
of data
data
0.5
6.0
23
-21
-9
68 experiments to - 18°C; (A) comparison
Mean
Standard deviation
Maximum
Minimum
Range enclosing 90% of data
Correlation (r) with finite differences
(B) comparison
Pham
data
Correlation (r) with finite differences
All 275 experiments;
Mean
Standard deviation
Maximum
Cleland
and Earle
to experimental
to 10
12to21
0.522
2.3
6.2
0.8
14.9
38
-28
17to28
0.857
to finite differences
-0.5
5.3
17
Minimum
-18
-15
Range enclosing 90% of data
-9t08
-9t08
207 experiments to - 10°C; (6) comparison
Mean
Standard deviation
Maximum
Minimum
Range enclosing 90% of data
to finite differences
68 experiments to - 18%; (B) comparison
Mean
Standard deviation
to finite differences
-2.7
6.1
13
Maximum
Minimum
-18
1.3
5.0
18
-21
-7t09
Range enclosing 90% of data
a Results for the Cleland and Earle, and Hung and Thompson
* Ignores 4 experiments in which T, > T,d
2.1
4.3
14
-14
-5t08
-4.4
4.3
4
-14
-15
-12to8
-1210
formulae
were calculated
by the modified
For the strawberry
example the following data were calculated
using Eq. (8) to (13) and Table 3 of Cleland and Earle (1984):
Bi = 0.673
AH,, = 378 X lo6 J/m3
Sii = 0.196
Pk = 0.114
U = 0.992
P = 0.578
R = 0.156
(T,, - T,)/( - 18 - T,) = 0.588
tr = 11.97 x 1.083 min = 12.96 min.
The freezing time to - 18°C is 11.97 min and the time to - 25°C
-19
-8t017
-8.4
12.0
15
-50
-3oto9
4.4'
8.7
31
-21
-9 to 19
-11.0
12.1
15
-49
-31 to 6
0.8
-0.8
7.8
13
-28
-11 to9
9.3
19
-24
-18to
1
14
forms.
is 12.96 min. Results from application of Eq. (7) to (13) to the
composite data set in Cleland and Earle (1984) are shown in Tables 1 and 2.
DISCUSSION
THE PREDICTION ACCURACY for the Cleland and Earle formula shown in Tables 1 and 2 is now as good as for any other
method for a final center temperature of - 18°C and over the
Volume 49 (19841-JOURNAL
OF FOOD SCIENCE-
1231
FREEZING TIME PREDICTIONS
...
whole data set, thus vindicating the use of Eq. (1). Correlation
with finite difference predictions is also much improved. The improved formula thus meets the criteria’in Cleland and Earle (1984).
The proposed modification improves the performance of the Hung
and Thompson formula at - 10°C. Correlation with finite difference predictions is now virtually identical at both final center
temperatures in the data set. However, because of variations in
data from different sources (Cleland and Earle, 1984) agreement
with experiments is not even across the whole data set. The similarity in the agreement with finite differences at the two Tfi,
values suggests that the modification may suit the Hung and
Thompson formula across a range of final center temperatures.
CONCLUSIONS
A MODIFICATION
to the Cleland and Earle freezing time pre. diction formula is shown to make it as accurate as any other
method over the data set studied for final product temperatures
that differ from - 10°C. The prediction accuracy of the Hung and
Thompson formula is also improved, by a corresponding modification, for final temperatures other than - 18°C.
SYMBOLS
Bi
-
CL
-
cs
-
D
EHTD
h
H
-
Biot number hD/k,
volumetric specific heat capacity of unfrozen material (J/m3”C)
volumetric specific heat capacity of frozen material
(J/m3”C)
characteristic dimension (full thickness) (m)
equivalent heat transfer dimensionality
surface heat transfer coefficient (W/m2”C)
enthalpy (J/m3)
E. COLI IN SEAFOODS
USING FLUOROGENIC
ASSAYS
4). For the M-Endo-MUG assay, a total confirmatory
rate
of 90.0% was obtained for the fluorescent colonies (56 out
of 62 fluorescent colonies). A false-positive rate of 10.0%
was observed (Table 6).
One of the problems commonly
encountered during
seafood analysis is the presence of stressed or injured microbial cells. Stressed cells have the ability to recuperate and
grow on nonselective media, but not on selective media
(Clark and Ordal, 1969; Ray and Speck, 1973). The ability
of Lauryl Tryptose
Broth-MUG
(LTB-MUG)
to detect
heat-stressed cells was examined by Feng and Hartman
(1981, 1982). The LTB-MUG
system (fluorescence) was
superior to the VRB-2 method in detecting heat-injured
cells. This is expected, since LTB is a relatively nonselective
medium. Perhaps, this is one of the reasons why the LBMUG assay in this study resulted in higher recoveries of E.
cali from the seafood samples tested.
The three assays described in this study are rapid (24
hr), sensitive, and save labor and material cost. Any of the
three fluoragenic assays can be routinely used by the seafood industry to discriminate seafood samples with high
total coliforms and/or E. coli (fecal contamination)
present.
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1232~JOURNAL
OF FOOD SCIENCE-Volume
49 (1984j
AHlo
-
L
ks
-
r
t
T,
Tf
Tfi,
-
Ti
T ref
-
U
-
enthalpy change in product between Tf and - 10°C
(J/m3)
enthalpy change in product between Ti and - 18°C
(J/m3)
thermal conductivity of frozen material (W/m”C)
latent heat of freezing (derived from AHI or AH,s
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correlation coefficient
freezing time (set, min or hr)
cooling medium temperature (“C)
temperature at which freezing commences (“C)
final product center temperature at end of freezing
process (“C)
initial product temperature (“C)
final product temperature used as a reference in empirical freezing formulae (“C)
ratio of temperature driving forces defined in eq. (12)
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