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In the heat transfer by conduction processes under steady state, the flow ofheat transmitted (
Q
) through a solid is directly proportional to the transmis-sion area (
A
) and to the increase of temperature (
∆
T
), and is inversely pro-portional to the thickness of the solid (
e
). The proportionality constant iscalled thermal conductivity:
Heat conduction under steady state has been used in different experimentsto calculate the thermal conductivity of food, although experiments underunsteady state can also be used. Either way, mathematical relationships aresought that allow calculation of the thermal conductivity of a given food asa function of temperature and composition.
An equation that allows calculation of the thermal conductivity of sugarsolutions, fruit juices, and milk is (Riedel, 1949):
(11.1)
in which
k
is expressed in J/(s ·m ·°C);
T
in °C, and
X
mWATER
is the mass fractionof water. This equation is valid for a temperature range between 0 and 180°C.
Sweat (1974) gives the following equation for different fruits and vegetables:
(11.2)
valid for water contents higher than 60%, although it cannot be used withlow-density foods or with foods that have pores (e.g., apples).
Q k
A Te
=∆
k T T XWATERm= + −( ) +( ) × −326 8 1 0412 0 00337 0 44 0 54 1 73 102 3. . . . . .
The specific heat is defined as the energy needed to increase by 1°C thetemperature of one unit mass. For foods with a high water content abovethe freezing point, the following equation can be used (Siebel, 1982):
s are the mass fractions of fat and solids, respectively.For milk at temperatures higher than the final point of fusion of milk fat,
the following expression can be used (Fernández–Martín, 1972a):
(11.10)
in which the specific heat is expressed in kcal/(kg ·°C), temperature T in °Cin a range from 40 to 80°C, and Xm
WATER and XmTS are the mass fractions of
water and total solids, respectively.Gromov (1979) gives the next equation for cream:
(11.11)
expressing the specific heat in J/(kg.K), temperature T in Kelvin, for the 272to 353 K range, and the fat content between 9 and 40%.
Manohar et al. (1991) gave the following equation for tamarind juices:
(11.12)
in which the specific heat is expressed in kJ/(kg K) if the temperature isgiven in Kelvin, and C is the soluble solids content expressed in °Brix.
Choi and Okos (1986b) proposed an equation for the case in which thecomposition of the product is known:
(11.13)
where CPi is the specific heat of the component i, while Xmi is the mass fraction
of the component i.
also presents expressions for the calculation of the specific heat of pure
allow calculation of the specific heat of water and ice as a function of tem-perature are given.
11.3 Density
Density is defined as the relation between the mass of a given sample andits volume. Different expressions for the calculation of food density can befound in the literature. Thus, for fruit juices, density can be expressed as afunction of the refraction index according to the expression (Riedel, 1949):
where ρ is the density expressed in kg/m3 and s is the refraction index.Some equations express density as a function of temperature and soluble
solids content. For clarified apple juices, Constenla et al. (1989) presentedthe following:
(11.15)
in which density is expressed in g/cm3, X is the concentration in °Brix, andT is the absolute temperature. This expression can be applied in the 20 to80°C temperature range and in the 12 to 68.5°Brix range. These same authorsexpresed the density of these juices as a function of °Brix and density ofwater:
However, Aguado and Ibarz (1988) gave different expressions for clarifiedapple juices in the 5 to 70°C temperature range and in the 10 to 71°Brixconcentration range. One of these expressions is:
(11.17)
where density is expressed in g/cm3, C in °Brix, and T in °C.Ibarz and Miguelsanz (1989) reported a similar equation for clarified pear
juice in the 5 to 70°C temperature range and in the 10 to 71°Brix concentrationrange:
(11.18)
Alvarado and Romero (1989) presented the following expression for dif-ferent juices, for temperatures from 20 to 40°C and for concentrations from5 to 30°Brix:
(11.19)
in which density is expressed in kg/m3, C in °Brix, and T in °C.For sucrose solutions with concentrations between 6 and 65°Brix and a
temperature of 20°C, Kimball (1986) reported the equation:
(11.20)
in which density is expressed in g/cm3 and C in °Brix.Manohar et al. (1991) presented a second order polynomial equation as a
function of the total soluble solids content for tamarind juices:
(11.21)
in which density is obtained in kg/m3 and the concentration C is expressedin °Brix.
Rambke and Konrad (1970) reported a second order polynomial equationfor milk as a function of the dry mass percentage:
(11.22)
where ρ is expressed in g/cm3 and Xo is the dry mass percentage. The
For temperatures higher than the boiling point, the equation of Berstschet al. (1982) can be used:
(11.23)
where ρ is expressed in kg/m3; T is temperature in °C for the range from 65to 140°C; and f is the fat content for values between 0.02 and 15.5%.
Andrianov et al. (1968) reported the following equation for cream in the40 to 80°C range and fat content between 30 and 83%:
(11.24)
in which density is expressed in g/cm3, temperature is in °C, and the fatcontent XG is the mass fraction.
Choi and Okos (1986b) suggested an expression as a function of the densityof the components of the product:
(11.25)
in which Xmi is the mass fraction of the component i and ρi its density.
densities of the pure components as a function of temperature.
11.4 Thermal Diffusivity
A widely used property in calculations of heat transfer by conduction is thethermal diffusivity, defined according to the expression:
(11.26)
The value of the thermal diffusivity of a given food can be calculated if thethermal conductivity, density, and specific heat are known. However, somemathematical expressions allow calculation of the thermal diffusivity accord-ing to water content. Thus, Martens (1980) reported the following equation:
Table 11.5 presents thermal diffusivity values for some foods. Tables 11.2and 11.3 show the expressions that allow calculation of the thermal diffusiv-
Determine the density, thermal conductivity, specific heat, and thermal dif-fusivity, at 25°C, of a food product that has been chemically analyzed, andwhose weight composition is: 77% water, 19% carbohydrate, 3% protein,0.2% fat, and 0.8% ash.
The method of Choi and Okos is used; therefore, the thermal propertiesof each component at 25°C are previously calculated. The following tablecontains the results obtained.
The volumetric fraction of each component is calculated by means ofEquation 11.7. The mass and volumetric fractions of each component arepresented next.
Thermal conductivity: obtained from Equation 11.6: