BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI
Publicat de
Universitatea Tehnică „Gheorghe Asachi” din Iaşi
Tomul LVIII (LXII), Fasc. 4, 2012
SecŃia
AUTOMATICĂ şi CALCULATOARE
ESTIMATION OF FOOD PRODUCT FREEZING TIME
BY
MOHAMED FRIHAT1*, BASSAM AL-ZGOUL1, JEHAD RADAEDEH1,
MAEN AL-RASHDAN1 and AHMAD H. AL-FRAIHAT2
1
Al-Huson Applied University College, Al-Balqa Applied University, Jordan
2
Ajloun University College, Al-Balqa Applied University, Ajloun, Jordan
Received: December 3, 2012
Accepted for publication: December 18, 2012
Abstract. The freezing process is important to guarantee the quality of the
frozen food. The objective of the investigation described here was to outline the
factors associated with assuring accurate prediction of food freezing rates with
minimum dependence on experimental inputs. Due to the complexity of the
freezing process as utilized throughout the food industry, it is impossible to
thoroughly investigate and establish the most efficient freezing times for all
situations. For this purpose, computer simulation becomes the most efficient tool
for comparison of freezing processes. Knowledge of initial moisture content as
well as unfrozen water content as a function of temperature allow the prediction of
frozen product density, thermal conductivity and apparent specific heat. The most
important input parameter to the freezing rate prediction is the surface heat transfer
coefficient. The local temperature history data from the acrylic transducer and
ground beef product were used to compute local surface heat transfer coefficients.
Key words: food product, freezing, refrigeration, estimation, cooling.
2010 Mathematics Subject Classification: 81T80.
*
Corresponding author; e-mail: [email protected]
40
Mohamed Frihat et al.
1. Introduction
For air-blast convective cooling and freezing operations to be costeffective, refrigeration equipment should fit the specific requirements of the
particular cooling or freezing application. The design of such refrigeration
equipment requires estimation of cooling and freezing times of foods. This
freezing time represents the length of time for which the product is exposed to
the freezing medium and represents a critical factor in determining the
efficiency of energy utilization for the process. Cooling and freezing food is
complex process. Before freezing, sensible heat must be removed from food to
decrease its temperature to the initial freezing point of the food. This initial
freezing point is somewhat lower than the freezing point of pure water because
of dissolved substances in the moisture within the food. At the initial freezing
point, a portion of water within the food crystallizes and the remaining solution
becomes more concentrated, reducing the freezing point of the unfrozen portion
of the food further. As the temperature decreases, ice crystal formation
increases the concentration of the solutes in solution and depresses the freezing
point further. Thus, the ice and water fractions in the frozen food, and
consequently the foods thermo-physical properties, depend on temperature.
Because most foods are irregularly shaped and have temperature dependent
thermo-physical properties, exact analytical solutions for their cooling and
freezing times cannot be derived.
Due to the complexity of the freezing process as utilized throughout the
food industry, it is impossible to thoroughly investigate and establish the most
efficient freezing times for all situations. For this purpose, computer simulation
becomes the most efficient tool for comparison of freezing processes. In order
for computer simulation of freezing processes to gain maximum potential, the
mathematical models describing the process must be accurate and accurate input
parameters must be available. Most research has focused on developing
empirical freezing time prediction methods that use simplifying assumption.
The objective of this research described here was to outline the factors
associated with assuring accurate prediction of food freezing rates with minimum
dependence on experimental inputs. More specifically, the purpose of the
investigation was to analyze the approaches required to assure accurate prediction
of surface heat transfer coefficients for food products during the freezing process.
2. Theoretical Considerations
The estimation of freezing time for a food product is dependent on a
variety of factors including product properties. The property predictions are of
importance due to the inability to accurately measure most properties of frozen
product, especially thermal properties. Because water is the predominant
Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 4, 2012
41
constituent in most foods, water content significantly influences the thermophysical properties of foods.
When carrying out engineering calculations, the following equations
can be used to find the thermal properties of frozen foods as follow.
Taking into account that the moisture content of food products is
between 0.50 and 0.96, the thermal and physical properties of food products
could be expressed as a function of moisture content.
The apparent specific heat can be computed as:
C pa = 1465.4 + 1482.7 ⋅ (W p − W ) ,
conductivity of food products is expressed as:
K p = 0.58 + 1.917 ⋅ (W p − W ) ,
and density of food products can be calculated as:
ρ p = 1005 + 208.3(W p − W ) ,
where: W p is the initial moisture content of food product, [%], W − the lowest
initial moisture content in the food products (W = 50%).
To achieve an accurate prediction of thermal properties of frozen foods,
which depend strongly on the fraction of ice in the food, the mass fraction of
water that has crystallized must be determined. Below the initial freezing point,
the mass fraction of water that has crystallized in a food is a function of
temperature. As illustrated by (Chen, 1985) this relationship can be predicted
with accuracy using the following expressions:
xice =
xs RT02 (t f − t )
M s L0t f t
,
(1)
where: xs is the mass fraction of solids in food, M s − the relative molecular
mass of solids, [kg/kmol], R − the universal gas constant (R = 8.314 J/mol K),
T0 − the freezing point of water ( T0 = 273.2 K), L0 − the latent heat of fusion
of water at 273.2 K ( L0 = 333.6 kJ/kg), t f − the initial freezing point of food
[ºC], t − the food temperature [ºC].
The relative molecular mass of the soluble solids in the food may be
estimated as follows:
Ms =
xs RT02
,
− ( xwo − xb ) L0 t f
(2)
42
Mohamed Frihat et al.
where: xwo is the mass fraction of water in the unfrozen food, xb − the mass
fraction of bound water in the food (Schwartzberg, 1976).
Bound water is the portion of water in a food that is bound to solids in
the food, and thus is unavailable for freezing. The mass fraction of bound water
may be estimated as follows:
xb = 0.4 x p ,
(3)
where: x p is the mass fraction of protein in the food.
Substituting eq. (3) into eq. (2) yields a simple way to predict the ice
fraction (Miles, 1974).
 tf 
xice = ( xwo − xb ) 1 −  .
t 

(4)
Because eq. (4) underestimates the ice fraction at temperatures near the
initial freezing point and overestimates the ice fraction at lower temperatures,
(Tchigeov, 1979) proposed an empirical relation to estimate the mass fraction of
ice:
xice =
1.105 xwo
.
0.7138
1+
ln ( t f − t + 1)
(5)
Fikiin (1998) notes that eq. (5) applies to a wide variety of foods and
provides satisfactory accuracy. Modeling the density of foods and beverages
requires knowledge of the food porosity, as well as the mass fraction and
density of the food components. The density of foods and beverages can be
calculated accordingly:
ρ=
(1 − ε )
∑ xi
ρi
,
(6)
where: ε is the porosity, xi − the mass fraction of the food constituents, ρi −
the density of food constituents.
One of the most important product properties is the specific heat, or, in
the case of frozen food, the apparent specific heat. Below food's freezing point,
the sensible heat from temperature change and the latent heat from fusion of
water must be considered. Because latent heat is not released at constant
temperature, but rather over a range of temperatures, an apparent specific heat
must be used to account for both the sensible and latent heat effects. A slightly
simpler apparent specific heat model, which is similar in form to that of
Schwartzberg (2006), was developed by Chen (1985).
Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 4, 2012
ca = 1.55 + 1.26 xs +
xs RT02
M st 2
.
43
(7)
In order to account for the influence of frozen water fraction as a
function of temperature during freezing of product, the change in heat content
of various product components must be considered. An accurate prediction of
product enthalpy can be obtained from the definition of constant-pressure
 ∂H 
specific heat c p = 
 .
 ∂T  P
Mathematical models for enthalpy may be obtained by integrating
expression of specific heat with respect to temperature. By integration eq. (7)
between the reference temperature tr and the food temperature t, (Chen, 1985)
obtained the following expression for enthalpy below the initial freezing point:

x RT 2 
H = ( t − tr )  1.55 + 1.26 xs + s 0 
M s tr t 

(8)
Substituting eq. (2) for the relative molecular mass of the soluble solids
simplifies Chen's method as follows:
( xwo − xb ) L0t f 

H = ( t − tr ) 1.55 + 1.26 xs −
.
tr t


(9)
A food's thermal conductivity depends on factors such as composition,
structure, and temperature. For an isotropic, two-component system composed
of continuous and discontinuous phases, in which thermal conductivity is
independent of direction of heat flow (Kopelman, 1966) developed the
following expression for thermal conductivity k:
 1 − L2

k = kc 
,
2
1 − L (1 − L ) 
(10)
where: kc is the thermal conductivity of the continuous phase, L − the volume
fraction of the discontinuous phase.
In eq. (10), thermal conductivity of the continuous phase is assumed to
be much larger than that of the discontinuous phase. However, if the opposite if
true, the following expression is used to calculate the thermal conductivity of
the isotropic mixture:
 1− M

k = kc 
,
1 − M (1 − L ) 
(11)
44
Mohamed Frihat et al.
where: kd is the thermal conductivity of the discontinuous phase and
M = L2 (1 − kd kc ) .
3. Experimental Procedures
The influence of the surface heat transfer coefficient (h) on freezing
times has been clearly demonstrated by (Hsieh, 1998). At low values of the
surface heat transfer coefficient (~30 W/m2·K) small variations in the
coefficient cause significant changes in the freezing time. It follows that small
inaccuracies in the prediction of the surface heat transfer coefficient may have a
significant influence on the accuracy of the freezing time prediction.
The details of the experimental design and procedure have been
described by (Charvarria, 2000). A flat plate transducer constructed with acrylic
was used to measure surface heat transfer coefficients at temperature and air
flow conditions typical for food product freezing. The transducer was used to
record temperature history curves for air speeds ranging from 2 to 14.2 m/s at
air temperatures from −27 to −15ºC and for transducer thicknesses of 0.96 and 2
cm. Cooling and freezing experiments were conducted using ground beef in the
same shape and location as the acrylic transducer.
The purpose of the experimental design was the measurement of
transient temperature during cooling and freezing processes for a range of
thermal and air velocity conditions. The initial and boundary conditions were
chosen to illustrate those accounted for in the mathematical models of onedimensional heat transfer during cooling and freezing.
4. Results and Discussions
Local Nusselt Number , Nu
The solver computer program Eureka was used to obtain coefficient
estimates for each experimental condition. The experimental results obtained
during cooling and freezing of ground beef are presented in Fig. 1.
air velocity 3.7 m/s
air velocity 7.5 m/s
air velocity 10.2 m/s
air velocity 11.4
1200
1000
800
600
400
200
0
10000
210000
410000
610000
Reynolds Number , Re
Fig. 1 − Local and average Nusselt number versus Reynolds number-food freezing result.
Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 4, 2012
45
The relationship between local Nusselt (Nu) number and Reynolds (Re)
number is:
Nu x = 20.041Rex0.298
(12)
The experimental results from the acrylic transducer are presented in
Fig. 2 where the correlation is expressed by:
Nu x = 214.87 Ln( Rex ) − 1899.2
(13)
1000
air velocity 3.7 m/s
air velcity 7.2 m/s
500
air velocity 9.8 m/s
air velocity 14.5 m/s
700000
600000
500000
400000
300000
200000
100000
0
0
Local Nusselt Number , Nu
The local temperature history data from the acrylic transducer and
ground beef product were used to compute local surface heat transfer
coefficients.
The influence of the air velocity and axial location on the surface heat
transfer coefficient is illustrated by the relationship of Reynolds number (Re)
and local Nusselt number (Nu).
The results obtained from both the acrylic transducer and the
temperature history in ground beef illustrate the influence of axial location in
the direction of air flow on the magnitude of the surface heat transfer
coefficient. It is important, therefore, to recognize that direction of air flow, as
well as air speed and geometry of product, must be accounted for in
measurement or prediction of surface heat transfer coefficient to be used in food
product freezing rate predictions.
Reynolds Number , Re
Fig. 2 − Local Nusselt number versus Reynolds number-acrylic
transducer cooling result.
Since correlations of the type presented by eqs. (12) and (13) are
uniquely associated with a geometry and flow conditions represented in the
experimental arrangement, consideration must be given to more flexible
approach. The use of eqs. (12) and (13) must be conducted with recognition that
46
Mohamed Frihat et al.
the appropriate location must be selected to coincide with the location of
reference for freezing time determination.
Air Temperature -26 C, Surface Heat Transfer Coefficient
115.5 w/m2 k, Air Velocity 12 m/s
Dimensionless
Temperature
12
10
Predicted
8
Experimental
6
4
2
0
96 92 82 71 61 41 31 21 10
Time , (min)
Fig. 3 − Influence of surface heat transfer coefficient obtained from acrylic
transducer cooling on ground beef freezing curve prediction.
The application of the transducer approach to obtaining a surface heat
transfer coefficient for use in a food product freezing prediction is illustrated in
Fig. 3. These results compare the experimental results for cooling and freezing
of ground beef with a predicted temperature history curve for the same product
using the surface heat transfer coefficient obtained from an acrylic transducer.
In most freezing processes, product geometry will vary significantly and air
flow patterns may not be well described by an easily described set of conditions. In
order to deal with the variety of circumstances existing during commercial food
product freezing, the acrylic transducer appears to be an acceptable alternative. This
approach provides an opportunity to select the geometry closest to that of the
product and to measure the surface heat transfer coefficient under air flow
conditions most similar to those existing under actual conditions.
5. Conclusions
The transducer approach to the measurement of surface heat transfer
coefficients during cooling and freezing of food product offers accurate
alternative to models for the prediction of these coefficients. There is a
possibility to develop a mathematical model to predict the temperatures on the
process of freezing food using only a few experimental inputs. The effect of
location along the product surface and the direction of movement of the air
stream on the surface heat transfer coefficient is very clear. Transducer cooling
and food products freezing temperature history curves can be used to determine
experimental values of surface heat transfer coefficients.
Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 4, 2012
47
REFERENCES
Charvarria V.M., Experimental Determination of Surface Heat Transfer Coefficient
Under Food Freezing Conditions. M.S. Thesis, 2000.
Chen C.S., Thermodynamic Analysis of the Freezing and Thawing of Foods: Enthalpy
and Apparent Specific Heat. Journal of Food Science, 50, 4, 1158−1162, 1985.
Fikiin K.A., Ice Content Prediction Methods During Food Freezing: A Survey of the
Eastern European Literature. Journal of Food Engineering, 38, 3, 331−339,
1998.
Hsieh R.-C., Influence of Food Product Properties on the Freezing Time. M.S. Thesis,
1998.
Kopelman I.J., Transient Heat Transfer and Thermal Properties in Food Systems. Ph.
D. Diss., Michigan State University. Department of Food Science, 1966.
Miles C.A., Meat Freezing-why and how?, Proceedings of the Meat Research Institute
Symposium, 3, 15.1−15.7, 1974.
Schwartzberg H.G., Effective Heat Capacities for the Freezing and Thawing of Food.
Journal of Food Science, 41, 1, 152−156, 2006.
Tchigeov G., Thermophysical Processes in Food Refrigeration Technology. Food
Industry, Moscow, 1979.
ESTIMAREA DURATEI DE CONGELARE A
PRODUSELOR ALIMENTARE
(Rezumat)
În procesul de congelare a produselor alimentare, este important a garanta
calitatea produselor. Obiectivul investigaŃiilor prezentate în lucrarea de faŃă este de a
scoate în evidenŃă factorii care influenŃează precizia de predicŃie a ratelor de congelare
în condiŃiile unei dependenŃe minime de intrările experimentale. Datorită complexităŃii
procesului de congelare utilizat pe scară largă în industria alimentară, este imposibil de
investigat în detaliu şi de stabilit duratele de îngheŃare pentru toate situaŃiile posibile.
Din acest motiv, simularea pe calculator reprezintă cea mai eficientă metodă de
comparaŃie între diverse procese de congelare. Cunoaşterea gradului de umiditate iniŃial
ca şi a dependenŃei conŃinutului de apă neîngheŃată în funcŃie de temperatură permit
estimarea densităŃii produsului congelat, a conductivităŃii termice şi a căldurii specifice
aparente. Cel mai important parametru de intrare pentru estimarea ratei de congelare îl
reprezintă coeficientul de transfer superficial al căldurii. Temperatura locală a
produsului, achiziŃionată cu ajutorului unui transducer acrilic, a fost utilizată pentru a
calcula coeficienŃii de transfer superficial al căldurii.