Capitulo 11 - Thermal Properties

© 2003 by CRC Press LLC

309

11

Thermal Properties of Food

11.1 Thermal Conductivity

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. . . . . .

k XWATERm= +0 148 0 493. .

TX69299 ch01 frame.book Page 309 Wednesday, September 4, 2002 2:13 PM


310

Unit Operations in Food Engineering

In the case of milk, Fernández–Martín (1982) gives a second order poly-nomial expression with respect to temperature.

(11.3)

in which the parameters

A

,

B

, and

C

are a function of the fat and nonfatcontent of milk.

An equation that allows one to obtain the thermal conductivity of cream(Gromov, 1974) is:

(11.4)

in which thermal conductivity is expressed in kcal/(h ·m ·°C), and

f

is the fatcontent between 10 and 60%;

ρ

is the density of the sample at the corre-sponding temperature and composition expressed in kg/m

3

; while

T

is thetemperature in °C in the 30 to 70°C range.

Also, Fernández–Martín and Montes (1977) gave an expression for cream:

(11.5)

where the thermal conductivity is expressed in cal/(s·cm·°C) and temperaturein °C in the 0 to 80°C range. Also,

f

is the fat percentage between 0.1 and 40%,while

X

VG

is the volumetric fraction of the fat phase for values lower than 0.52.If the food composition is known, it is possible to find its thermal conduc-

tivity from the equation:

(11.6)

in which

k

i

is the thermal conductivity of the component

i

, and

X

Vi

is thevolumetric fraction of this component.

The volumetric fraction of the component

i

is given by the expression:

(11.7)

in which

X

mi

is the mass fraction of the component

i

and

ρ

i

is its density.

foods, while the conductivity of water and ice as a function of temperature

k A BT CT= + + 2

k

f

T=

− −( )[ ] ×

− −( )−411 6 4 26 10 10

1 0 0041 30

6. .

.ρ

k = + T T + T XGV12 63 0 051 0 000175 1 0 843 0 0019 102 4. . . . .−[ ] − ( )[ ]⋅ −

k k Xi i

V

i

= ( )∑

X

X

XiV

im

i

im

ii

=

∑

ρ

ρ


Table 11.1 presents the thermal conductivity values of some foods.

is given in Table 11.3.

Table 11.2 shows the thermal conductivity of the main pure components of


Thermal Properties of Food

311

11.2 Specific Heat

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):

(11.8)

TABLE 11.1

Thermal Conductivity of Some Foods

ProductWater Temperature Thermal Conductivity

Content (%) (°C) (J/s ·m ·°C)

OilOlive 15 0.189

— 100 0.163Soybean 13.2 7–10 0.069Vegetable and animal — 4–187 0.169

Sugars — 29–62 0.087–0.22Cod 83 2.8 0.544Meats

PorkPerpendicular to the fibers 75.1 6 0.488

60 0.54Parallel to the fibers 75.9 4 0.443

61 0.489Fatty meat — 25 0.152

LambPerpendicular to the fibers 71.8 5 0.45

61 0.478Parallel to the fibers 71.0 5 0.415

61 0.422Veal

Perpendicular to the fibers 75 6 0.47662 0.489

Parallel to the fibers 75 5 0.44160 0.452

BeefFreeze-dried

1000 mm Hg — 0 0.0650.001 mm Hg — 0 0.035

LeanPerpendicular to the fibers 78.9 7 0.476

78.9 62 0.485Parallel to the fibers 78.7 8 0.431

78.7 61 0.447Fatty — 24–38 0.19

Strawberries — –14–25 0.675Peas — 3–17 0.312

ˆ . .C XP WATER

m= +0 837 3 349



312

Unit Operations in Food Engineering

in which ˆ

C

P

is expressed in kJ/(kg ·°C) and

X

mWATER

is the mass fraction of thewater in food.

An equation given by Charm (1971) is:

(11.9)

TABLE 11.2

Equations for Calculating Thermal Properties

ThermalProperty Component

Equation as aFunction of Temperature

k

(W/m ·°C) Carbohydrate

k

= 0.20141 + 1.3874

×

10

–3

T

– 4.3312

×

10

–6

T

2

Ash

k

= 0.32962 + 1.4011

×

10

–3

T

– 2.9069

×

10

–6

T

2

Fiber

k

= 0.18331 + 1.2497

×

10

–3

T

– 3.1683

×

10

–6

T

2

Fat

k

= 0.18071 + 2.7604

×

10

–3

T

– 1.7749

×

10

–7

T

2

Protein

k

= 0.17881 + 1.1958

×

10

–3

T

– 2.7178

×

10

–6

T

2

α

·10

6

(m

2

/s) Carbohydrate

α

= 8.0842

×

10

–2

+ 5.3052

×

10

–4

T

– 2.3218

×

10

–6

T

2

Ash

α

= 1.2461

×

10

–1

+ 3.7321

×

10

–4

T

– 1.2244

×

10

–6

T

2

Fiber

α

= 7.3976

×

10

–2

+ 5.1902

×

10

–4

T

– 2.2202

×

10

–6

T

2

Fat

α

= 9.8777

×

10

–2

+ 1.2569

×

10

–4

T

– 3.8286

×

10

–8

T

2

Protein

α

= 6.8714

×

10

–2

+ 4.7578

×

10

–4

T

– 1.4646

×

10

–6

T

2

ρ

(kg/m

3

) Carbohydrate

ρ

= 1.5991

×

10

3

– 0.31046

T

Ash

ρ

= 2.4238

×

10

3

– 0.28063

T

Fiber

ρ

= 1.3115

×

10

3

– 0.36589

T

Fat

ρ

= 9.2559

×

10

2

– 0.41757

T

Protein

ρ

= 1.3299

×

10

3

– 0.51840

T

CP (kJ/kg ·°C) Carbohydrate CP = 1.5488 + 1.9625 × 10–3 T – 5.9399 × 10–6 T 2

Ash CP = 1.0926 + 1.8896 × 10–3 T – 3.6817 × 10–6 T 2

Fiber CP = 1.8459 + 1.8306 × 10–3 T – 4.6509 × 10–6 T 2

Fat CP = 1.9842 + 1.4733 × 10–3 T – 4.8008 × 10–6 T 2

Protein CP = 2.0082 + 1.2089 × 10–3 T – 1.3129 × 10–6 T 2

Source: Choi and Okos (1986b).

TABLE 11.3

Equations to Calculate Thermal Properties of Water and Ice

Temperature Functionsa

Water kA = 0.57109 + 1.7625 × 10–3 T – 6.7036 × 10–6 T 2 (W/m ·°C)αA = [0.13168 + 6.2477 × 10–4 T – 2.4022 × 10–6 T 2].10–6 (m2/s)ρA = 997.18 + 3.1439 × 10–3 T – 3.7574 × 10–3 T 2 (kg/m3)CPA1 = 4.0817 – 5.3062 × 10–3 T + 9.9516 × 10–4 T 2 (kJ/kg ·°C)CPA2 = 4.1762 – 9.0864 × 10–5 T + 5.4731 × 10–6 T 2 (kJ/kg ·°C)

Ice kH = 2.2196 – 6.2489 × 10–3 T + 1.0154 × 10–4 T 2 (W/m ·°C)αH = [1.1756 – 6.0833 × 10–3 T + 9.5037 × 10–5 T 2] × 10–6 (m2/s)ρH = 916.89 – 0.13071 T (kg/m3)CPH = 2.0623 + 6.0769 × 10 –3 T (kJ/kg ·°C)

a CPA1 = For a temperature range between –40 and 0°C.CPA2 = For a temperature range between 0 and 150°C.

Source: Choi and Okos (1986b).

ˆ . . .C X X XP GF

msm

WATERm= + +2 309 1 256 4 187



Thermal Properties of Food 313

in which XmGF and Xm

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):

ˆ . .C X T XP WATER

mTSm= + +( )0 238 0 0027

ˆ . . .C X T XP WATER

mWATERm= + −( ) −( )4 187 16 8 3 242 1

ˆ . . .C T CP = + × −( )−4 18 6 839 10 0 05035

ˆ ˆC C XP Pi im

i

= ( )∑


The specific heat values of different foods are listed in Table 11.4. Table 11.2

components as a function of temperature, while in Table 11.3, equations that


314 Unit Operations in Food Engineering

(11.14)

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:

(11.16)

TABLE 11.4

Specific Heat for Some Foods

ProductWater Specific heat

(%) (kJ/kg.K)

Meats Bacon 49.9 2.01Beef

Lean beef 71.7 3.433Roast beef 60.0 3.056Hamburger 68.3 3.520Veal 68.0 3.223

Prawns 66.2 3.014Eggs

Yolk 49.0 2.810Milk

Pasteurized, whole 87.0 3.852Skim 90.5 3.977–4.019

Butter 15.5 2.051–2.135Apples (raw) 84.4 3.726–4.019Cucumbers 96.1 4.103Potatoes 79.8 3.517

75.0 3.517Fish 80.0 3.600

Fresh 76.0 3.600Cheese (fresh) 65.0 3.265Sardines 57.4 3.014Carrots (fresh) 88.2 3.810–3.935

Source: Reidy, G.A., M.S. thesis, Michigan StateUniversity, 1968.

ρ =

−+

ss

2 12

64 20 206

16 0185.

..

ρ = + ( ) − × −0 82780 0 34708 0 01 5 479 10 4. . exp . . X T

ρρ

=− × −

WATER

0 992417 3 7391 10 3. . X




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

ρ = − × + × + ×− − −0 98998 5 050 10 5 1709 10 0 0308 104 3 5. . . .T C C2

ρ = − × + × + ×− − −1 0113 5 4764 10 3 713 10 1 744 104 3 5 2. . . .T C C

ρ = + − + × + ×− −1002 4 61 0 460 7 001 10 9 175 103 2 5 3. . . .C T T T

ρ =+( )

0 524484330 872

170 435

2

. exp.

,C

ρ = + +1000 4 092 0 03136 2. . C C

ρ = + +a bX c Xo o2


coefficients of this equation for different temperatures are given in Table 11.6.



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:

ρ = − −

− + × − ×( )− −

1040 51 0 2655 0 01

0 967 0 969 10 0 478 10

2

2 4 2

. . .

. . .

T T

T T f

ρ = − × − × + ×( )− − −1 0435 1 17 10 0 52 10 1 6 105 3 8. . . .X X TG G

ρ

ρ

=

∑

1

Xim

ii

α

ρ=

k

CPˆ


Tables 11.2 and 11.3 show the expressions that allow calculation of the



(11.27)

where α is the thermal diffusivity in m2/s, XmWATER is the water mass fraction,

and T is the temperature in Kelvin.On the other hand, Dickerson (1969) presented an expression in which the

food’s thermal diffusivity is a function only of the water content and itsthermal diffusivity:

(11.28)

Choi and Okos (1986b) expressed thermal diffusivity as a function of thecomponents, similar to other thermal properties:

(11.29)

where αi is the thermal diffusivity of the component i and XVi is the volu-

metric fraction of such component.

ities of pure components:

TABLE 11.5

Thermal Diffusivity for Some Foods

ProductWater Temperaturea Thermal Diffusivity

(%) (°C) × 105 (m2/s)

Fruits, VegetablesAvocado (pulp) — 24 (0) 1.24

Seed — 24 (0) 1.29Whole — 41 (0) 1.54

Sweet potato — 35 1.06— 55 1.39— 70 1.91

Cherries (pulp) — 30 (0) 1.32Squash — 47 (0) 1.71Strawberries (pulp) 92 5 1.27Beans (purée) — 26–122 1.80Peas (purée) — 26–128 1.82String beans (cooked) — 4–122 1.68Limes — 40 (0) 1.07Apples 85 0–30 1.37

Applesauce 37 5 1.0537 65 1.1280 5 1.2280 65 1.40— 26–129 1.67

α = × + ×− −5 7363 10 2 8 108 10. .X TWATERm

α α= × −( ) +−8 8 10 18. X XWATERm

WATER WATERm

α α= ( )∑ i iV

iX


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-



Peach — 27 (4) 1.39Turnip — 48 (0) 1.34Potato

Pulp — 25 1.70Mashed (cooked) 78 5 1.23

Banana (pulp) 76 5 1.1876 65 1.42

Grapefruit(pulp) 88.8 — 1.27(albedo) 72.2 — 1.09

Beet — 14 (60) 1.26Tomato (pulp) — 4.26 1.48

Fishes and MeatsCod 81 5 1.22

81 65 1.42Hipogloso 76 40–65 1.47Salted meat 65 5 1.32

65 65 1.18Ham (smoked) 64 5 1.18

64 40–65 1.38Beef

Loinb 66 40–65 1.23Round 71 40–65 1.33Tongue 68 40–65 1.32

Water — 30 1.48— 65 1.60

Ice — 0 11.82

a The first temperature is the initial one, and that in parentheses is the oneof the surroundings.

b Data are applicable if the juices exuded during storage remain in foods.

Source: Singh, R.P., Food Technol., 36(2): 87–91.

TABLE 11.6

Values of the Parameters of Equation 11.22

T Skim Milk Whole Milk (c = 0)(°C) a b × 103 c × 105 a b × 103

5 1.0000 3.616 1.827 1.0010 2.5520 0.9982 3.519 1.782 1.0080 2.0935 0.9941 3.504 1.664 1.0137 1.6650 0.9881 3.568 1.366 0.9953 2.1160 0.9806 3.601 1.308

TABLE 11.5 (continued)

Thermal Diffusivity for Some Foods

ProductWater Temperaturea Thermal Diffusivity

(%) (°C) × 105 (m2/s)




Problems

11.1

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:

Density: obtained from Equation 11.25:

Specific heat: obtained from Equation 11.13:

Component�i ki CPi �i × 107

(kg/m3) (W/m ·°C) (kJ/kg ·°C) (m2/s)

Water 994.91 0.6110 4.1773 1.458Carbohydrate 1591.34 0.2334 1.5942 0.927Protein 1316.94 0.2070 2.0376 0.797Fat 915.15 0.2496 2.0180 1.019Ash 2416.78 0.3628 1.1375 1.332

Component Xmi Xv

i

Water 0.77 0.8398Carbohydrate 0.19 0.1296Protein 0.03 0.0247Fat 0.002 0.0024Ash 0.008 0.0036

k = ( ) = ⋅°( ) = × °( )∑ −k Xi i

v 0 55 5 5 10 4. W m C . kJ s.m. C

ρ

ρ

=

∑=

11085

Xim

ii

kg m3

ˆ ˆ .C C XP Pi i

m

i= ( )∑ = ⋅°( )3 594 kJ kg C




Thermal diffusivity: obtained from Equation 11.29:

It can also be calculated by Equation 11.26:

Result:ρ = 1085 kg/m3

k = 0.50 W/(m ·°C)CP = 3.94 kJ/(kg ·°C)α = 1.7 × 10–7 m2/s

α α= ( )∑ = × −

i iV

iX 1 37 10 7. m s2

α

ρ= = × −k

CPˆ .1 41 10 7 m s2


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Capitulo 11 - Thermal Properties