Modelo de madurez de capacidad integrada

Heat transfer is an energy-transfer phenomenon. Increased heat energy causes a product’s molecules to move rapidly. That is, the kinetic energy of a molecule increases as heat energy is absorbed. Heat is transferred when a fast moving molecule collides with a slow-moving molecule, causing the fast moving molecule to lose kinetic energy and the slow moving molecule to gain kinetic energy. From this it is apparent that heat transfer involves a molecular mechanism.

One manifestation of the level of heat energy in a group of molecules is temperature, a substance’s relative hotness or coldness. Several arbitrary

“yardsticks” or scales have been developed for measuring temperature, but the two most common ones are the Celsius and Fahrenheit scales. However, if the energy level of atoms and molecules are compared to zero kinetic energy (i.e., no motion of the particles and consequently zero on an absolute temperature scale) the temperature is expressed in Kelvin or degrees Rankine. The relationships among these temperature scales are shown in Table 3.1.

The amount of heat energy is also measured in arbitrary units. In the System International (SI) the joule is the unit of heat energy, whereas in the fps system the Btu (British thermal unit) is the unit of heat energy. A joule (J) is 0.239 cal, and a calorie is the amount of heat necessary to raise the temperature of 1 g of water 18C (from 14.5 to 15.58C). A Btu is the amount of heat necessary to raise 1 lb of water 18F (usually from 61 to 628F).

Heat capacity (C_p, the p indicating constant pressure) is a term used to define the amount of heat energy necessary to cause a given mass of a substance to change 18. In the SI system heat capacity is kJ/kg K. However, it is more convenient to define heat capacity in terms of calories. Then heat capacity is the calories required to raise 1 g of substance 18C; in the fps system, it is the British thermal units required to raise 1 lb of substance 18F.

T^ABLE3.1 Relationships Among Various Temperature Scales

Reference points for water

Boiling point (1 atm) 100 212 373 672

Freezing point (1 atm) 0 32 273 492

Zero kinetic energy 2273 2460 0 0

aC ¼ (5/9) (F2 32)

bK ¼ C þ 273

cR ¼ F þ 460

Since the definition of the calorie and of the Btu is based on water, the heat capacity for water is 1 cal/g8C (4.184 J/g K) or 1 Btu/lb8F. If the heat capacity of a substance is compared to that of water, the resulting ratio is called specific heat.

Specific heat is a dimensionless ratio that is numerically equal to the heat capacity of the substance.

Specific heat (sh) ¼C_p;substance

C_p;water ð1Þ

The heat capacities of a variety of common foods are given in Table 3.2 (Rahman, 1995). Note that heat capacity is given with energy in units of joules (or kJ).

One calorie is 4.187 J. Also, C_pusually has temperature unit K, and 1K ¼ 18C.

Since heat capacity is very dependent on water content and composition, some correlations for calculating heat capacity as a function of composition have been developed. Dickerson (1968) developed the following equation for heat capacity of meat products with moisture content greater than 26% and fruit juices with moisture content greater than 50%.

Cp ¼ 1:675 þ 0:025w

where w is water content (%). When the composition is known, heat capacity can be calculated from proximate composition of the food.

Cp ¼ 1:42mcþ 1:549mpþ 1:675mf þ 0:837maþ 4:187mm ð2Þ where m is mass fraction and the subscript are c, carbohydrate; p, proteins; f, fat;

a, ash; and m, water.

When heat energy is added to or taken from a substance, either the temperature or the physical state of the substance changes. Heat input which causes a rise in temperature is called sensible heat since the heat can be “sensed.”

Added heat that does not contribute to a temperature increase is called latent heat.

Latent heat is the heat that is used to bring about a change of state of a substance without a change in temperature. If the substance goes from a solid to a liquid, the latent heat is called heat of fusion. If the changes is from liquid to a gas, it is called heat of vaporization. For the reverse process of gas to liquid or liquid to solid the latent heat is usually called heat of condensation and heat of solidification (heat of crystallization), respectively. Energy involved in a change of state is added or emitted as a result of changes in various forces of attraction among molecules.

For many heating and cooling applications in food, a change in the physical state of water is the primary objective. For example, changing liquid water to vapor is the primary objective in dehydration and evaporation, and changing liquid water to a solid is the objective of freezing foods that are to be consumed or stored in a frozen state.

T^ABLE3.2 Heat Capacity of Food between 0 and 1008C

Material x^aw Heat capacity (kJ/kg K)

Fruit

Apples 0.75 – 0.85 3.724 – 4.017

Berries (fresh) 0.84 – 0.90 3.724 – 4.100

Berries (dried) 0.30 2.134

Fruits (fresh) 075 – 0.92 3.347 – 3.766

Fruits (dried) 0.30 2.092

Grape Marsh (fresh) — 3.703

Lemon Eureka (fresh) — 3.732

Orange Valencia (fresh) — 3.515

Orange Naval (fresh) — 3.661

Plums (Fresh) 0.75 to 0.78 3.514

Plum (dried) 0.28 to 0.35 2.218 – 2.469

Meat

Veal cutlet (fried) 0.58 3.096

Venison 0.70 3.389

Miscellaneous

Bread (white) 0.44–0.45 2.720 – 2.845

Bread (brown) 0.485 2.845

Curd (cottage cheese) 0.60–0.70 3.264

Dough — 1.883 – 2.176

Macaroni 0.125 – 0.135 1.841 – 1.883

(continued )

TABLE 3.2 Continued

Material x^aw Heat capacity (kJ/kg K)

Pearl barley — 2.803 – 2.845

Porridge (buckwheat) — 3.222 – 3.766

Raisins 0.245 1.966

Rice 0.105 – 0.135 1.757 – 1.841

Sugar — 1.25

White of egg 0.87 3.849

Yolk of egg 0.48 2.803

Soups

Dried (salted) 0.16 – 0.20 1.715 – 1.841

Fats

Butter 0.14 – 0.155 2.050 – 2.134

Margarine 0.9 – 0.15 1.757 – 2.092

Vegetable oils — 1.464 – 1.883

Beverages

Cacao — 1.841

Cream, 45 – 60%fat 0.57 – 0.73 3.054 – 3.264

Rich (cow) milk 0.875 3.849

Skim (cow) milk 0.91 3.975 – 4.017

Vegetables

Artichokes 0.90 3.891

Beet (dried) 0.044 2.008

Cabbage (dried) 0.054 2.176

Carrots (fresh) 0.86 – 0.90 3.807 – 3.933

Carrots (dried) 0.044 2.092

The latent heat of vaporization is dependent on the pressure and boiling point of the aqueous phase. For pure water at 1008C (2128F), the latent heat of vaporization is 2259 kJ/kg or 540 cal/g (971 Btu/lb). Values of latent heat of vaporization as a function of pressure and temperature can be found in most handbooks of the physical sciences (e.g., Handbook of Chemistry and Physics).

The latent heat of fusion for pure water at 08C (328F) is 335 kJ/kg or 80 cal/g (144 Btu/lb).

In many heat transfer processes it is necessary to know the total quantity of heat that must be transferred. Knowing this, the rate of the process can be estimated from process parameters of the heat transfer system. In assessing the total quantity of heat transferred, sensible heat, heat of vaporization, and heat of fusion all must be considered whenever appropriate. Sensible heat can be calculated from the relationship

Qs¼ MCpDT ð3Þ

where Qsis sensible heat transferred (kJ), M is mass of material (kg), Cpis heat capacity (kJ/kg K), and DT is temperature change of the product (8C). The heat of fusion or melting can be calculated from the relationship

Q_f ¼ M_flf ð4Þ

where Q_fis latent heat transfer (kJ), M_fis mass of material solidified or melted (kg), and l_f is heat of solidification or melting (kJ/kg). Likewise, the heat TABLE3.2 Continued

Material x^aw Heat capacity (kJ/kg K)

Parsley 0.65 – 0.95 3.180 – 4.058

Peas (dried) 0.14 1.841

Spinach 0.85 – 0.90 1.925

Spinich (dried) 0.059 4.017

Sweet potato (dried) 0.076 2.050

White cabbage (fresh) 0.90 – 0.92 3.891

White cabbage (boiled) 0.97 4.100

axw, mass fraction of water (wet basis).

Source: Rahman (1995).

removed or supplied for condensation or vaporization can be calculated from the relationship

Q_v¼ M_vlv ð5Þ

where Q_v is latent heat transfer (kJ), M_v is mass of material evaporated or condensed (kg), and l_vis heat of vaporization or condensation (kJ/kg).

The total heat required (or released) is then the sum of these three, i.e.,

Q total ¼ Q^sþ Qf þ Qv ð6Þ

Equation (6) can be used to calculate the total quantity of heat transferred but not the rate of heat transfer. To arrive at the rate of heat transfer the process must be expressed on a time basis. If it is known that the mass of material must undergo a change in sensible heat or a change of state in a given length of time, then the rate of heat transfer can be calculated. The rate of heat transfer is obtained by dividing Eq. (6) by the time involved. For example, if the process is to be completed in 1 h then Q_totalwould be in units of kJ/h. On a rate basis the change in sensible heat becomes

Q_s¼ WC_pDT ð7Þ

where W is mass flow rate (kg/h), Qsis heat transfer rate (kJ/h), and CpDT is as defined in Eq. (3). Likewise, Eqs. (4) and (5) can be expressed on a time basis as follows:

Q_f ¼ W_flf ð8Þ

where W_fis the mass of material solidified or melted per unit time (kg/h) and

Qv¼ Wvlv ð9Þ

where W_vis the mass of material evaporated or condensed per unit time (kg/h).

The total rate of heat transfer is then

Q total ¼ WC^pDT þ W_flf þ W_vlv ð10Þ

Heat transfer is a dynamic process wherein heat is transferred from a hotter body to a colder body, the rate being dependent on the temperature difference between the two bodies. A temperature difference is therefore necessary for heat transfer, and this temperature difference is called the driving force for heat transfer. Without a temperature difference there is no heat transfer. For example, water cannot be boiled at 1008C by heating with steam at 1008C.

In transferring heat energy from one body to another resistance may be encountered. This resistance to heat flow can occur within or at the surface of the material. Expressions for these resistances are given in Sec. II. B and II. C. Like all rate processes, the rate of heat transfer is directly proportional to driving force

and inversely proportional to resistance. Thus,

Rate of transfer ¼ driving force resistance

There are three basic mechanisms by which heat transfer can occur: conduction, convection, and radiation. Conduction heat transfer occurs when thermal energy is transferred from one molecule (or atom) to an adjacent molecule (or atom) without gross change in the relative positions of the molecules. Molecular movement is limited to oscillation about a fixed position, allowing any given molecule to make contact with only its immediate neighbors. In heat transfer by convection, the molecules are free to move about, resulting in mixing of warmer and cooler portions of the same material. Convection is thus restricted to flow of heat in fluids, either gases or liquids. Transfer of heat by radiation occurs by means of an electromagnetic radiation. Electromagnetic radiation passes unimpeded through space and is not converted to heat or any other form of energy until it collides with matter. Upon collision the radiation can be transmitted, absorbed, or reflected.

Only the absorbed energy can (but need not) appear as heat.

One final introductory concept should be discussed, namely the differences between steady-state and unsteady-state processes. The criterion of a steady state is that conditions at all points in the system do not change with time. If one of the conditions at any given point changes with time an unsteady state exists.

An example will help illustrate the difference. Consider a pipe through which water is flowing at a constant inlet temperature and a constant rate while surrounded by a jacket in which steam is circulating at a constant temperature.

Conditions in the water vary from one location to another throughout the length of the heat exchanger, but at any given point the conditions are constant with time.

This system is said to be in steady state. Obviously there is a flow of energy from the hot fluid (steam) to the cold fluid (water), but the equations which describe the conditions at any point in the exchanger are independent of time. Now consider a tank of cold water in which a steam coil is immersed. In this situation, the mean temperature of the water increases, and there is no point in the mass of water where conditions are constant with time. This system is said to be in unsteady state.

Generally, steady state is characteristic of continuous processes, whereas unsteady state is characteristic of batch or discontinuous operation. A steady-state system can be analyzed as unsteady state if the frame of reference is changed. For the example cited above for the tubular heat exchanger, if the frame of reference is a volume element of water flowing through the tube, then the temperature of the water changes with time as the volume element moves through the tube. This is the situation, for example, when the objective of heating is to inactivate microbial cells or spores and it is necessary to describe the time-temperature relationship. This will be demonstrated in Chapter 6 on thermal processing.

A. Conduction

The basic law of heat transfer by conduction is given by Fourier’s law. Fourier’s law states that the rate of heat transfer through a uniform material is directly proportional to the area for heat transfer, directly proportional to the temperature drop through the material, and inversely proportional to the thickness of the material. Stated mathematically, Fourier’s law is

dQ

dt ¼ 2kAdT

dx ð11Þ

where dQ/dt is rate of heat transfer (J/s ¼ W), A is area for heat transfer (m²), dT/dx is temperature drop per unit thickness (C/m), and k is a proportionality constant.

The principles of Fourier’s law are illustrated schematically in Fig. 3.1.

In Eq. (11) the negative sign is included on the right-hand side of the equation because the term dT/dx is negative and therefore must be multiplied by the minus sign to yield a positive heat transfer rate, dQ/dt. If the temperature gradient does not vary with time, then the rate of heat flow is constant and Eq. (11) can be written

q ¼2kAdT

dx ð12Þ

where q is the rate of heat flow (W). For constant conditions of A, L, and k, Eq. (11) can be separated and integrated, resulting in

q ¼ kADT

L ð13Þ

The proportionality constant k is called the thermal conductivity and it is a physical property of the material through which heat is transferred. The units of k may be

FIGURE3.1 Heat transfer by conduction.

derived from Eq. (11) as ðJ=s m8CÞ ¼ ðW=m8CÞ: Like most physical properties, k is dependent on temperature and the k in Eq. (13) is actually k between T₁and T₂. Thermal conductivity values of various materials are available in the literature [e.g., Rahman (1995)] and a few representative values are presented in Table 3.3.

The most notable feature of food products is their extremely low values of thermal conductivity compared to metals. This difference in thermal conductivity is due to differences in abundance of free electrons. In metals, electrons transmit most of the heat energy, whereas in foods, where water is the main constituent, the free electron concentration is low and the transfer mechanism involves primarily vibration of atoms and molecules. Therefore good electrical conductors are also good thermal conductors. Similarly, those materials which are good electrical insulators, such as air and glass, are also good thermal insulators (see Table 3.3).

Another striking feature in Table 3.3 is that the thermal conductivity of ice is nearly four times greater than that of water. This difference in thermal conductivity partly accounts for the difference in freezing and thawing rates of food tissues. During freezing, heat is conducted through an ice layer, a layer which has a higher conductivity than that of water and therefore offers comparatively little resistance to heat flow. During thawing, on the other hand, the frozen material becomes surrounded by a continually expanding layer of immobilized water and heat flow through this layer is comparatively difficult. In essence, the water is acting as an insulator compared to ice. Consequently the thawing process for foods that are not fluid in nature is inherently slower than the freezing process.

Sweat (1974, 1975) derived several equations to estimate thermal conductivity of food classes. Examples include the following:

T^ABLE3.3 Thermal Conductivity of Various Materials

Material Temperature (8C) Thermal conductivity (w/mK)

Ice 0 2.22

Vegetables (average) 0 – 27 0.38 – 0.50

Stainless steel 16.3

Copper 398

Pyrex glass 0.087

1. For fruits and vegetables with m. 60%, k ¼ 0.148 þ 0.0493w, where w is water content (%).

2. For meats at 0 , T , 60C and 60% , M , 80% (w.b), k ¼ 0.08 þ 0.0052w

3. For foods of known proximate composition, k ¼ 0.25 m_cþ 0.155 m_pþ 0.16 m_fþ 0.135 m_aþ 0.58 m_m

B. Convection

As pointed out earlier, convective heat transfer occurs as a result of bulk movement in a fluid. Obviously, heat transfer by conduction occurs simultaneously, but it is generally negligible compared to convective heat transfer. There are two types of convective heat transfer: natural (or free) convection and forced convection. In natural convection, bulk movement of the fluid occurs as a result of density gradients established during heating or cooling the fluid. With forced convection an external source of energy (such as a pump, stirrer, or fan) is used to move the fluid. In most processing applications with foods, forced-convection heat transfer is used since it is inherently much faster than free convection.

The rate of convective heat transfer is governed by Newton’s law of cooling. This law states that the rate of heat transfer by convection is directly proportional to the heat transfer area and the temperature difference between the hot and cold fluid. The quantitative relationship is

q ¼ hADT ð14Þ

where q is rate of heat transfer (W), A is heat transfer area (m²), DT is temperature difference (8C), and h is a proportionality constant (W/m²8C). The proportionality constant h is called the heat transfer coefficient.

The magnitude of h is dependent on properties of the fluid, nature of the surface, and velocity of flow past the heat-transfer surface. It can be regarded as the conductance of heat through a thin stagnant layer of fluid of thicknessdfand is often referred to as the film heat transfer coefficient. Thus, h ¼ k/df, where k is the thermal conductivity of the fluid. Some typical values of h are shown in Table 3.4.

A method for calculating individual film heat transfer coefficient must take into consideration properties of the fluid and the conditions of flow that affect heat transfer. Since there are so many factors affecting h, the most useful method for developing equations involves dimensional analysis. This method shows the relationship between variables in the form of dimensionless groups. Actual data are then used to establish the constants and exponents in a power function relating these dimensionless groups. Examples will be provided later in this chapter.

1. Overall Heat Transfer Coefficient

Most food heating applications involve indirect heating (i.e., heat is transmitted from a hot fluid, through a wall, into a cold fluid). Direct heating occurs when hot steam and cold fluids are mixed together.

The process of indirect heating is shown schematically in Fig. 3.2. In steady state, the rate of heat transfer across each physical boundary is constant. There is a heat transfer coefficient characterizing transfer of heat from the hot fluid to the wall, a thermal conductivity characterizing heat transfer through the wall, and finally a heat transfer coefficient for transfer of heat from the wall to the cold fluid. The resistance to heat transfer in each of these sections can be determined by rearranging Eqs. (13) and (14) as follows:

q ¼DT wall

ðL/kAÞ ¼ DT hot fluid ð1/hAÞhot fluid

¼ DT cold fluid ð1/hAÞcold fluid

ð15Þ T^ABLE3.4 Typical Film Heat Transfer Coefficients

Condition h (W/m²K)

Gases, natural convection 3 – 30

Gases, forced convection 10 – 100

Viscous liquids, forced convection 60 – 600

Water, forced convection 600 – 6000

Boiling water 1700 – 23000

Condensing steam 6000 – 18000

FIGURE3.2 Indirect heat transfer.

The term L/kA then represents the resistance of the wall to heat transfer, ð1/hAÞ_{hot fluid}is the resistance in the hot fluid, and (1/hA)_{cold fluid}is the resistance in the cold fluid.

It can be shown that the overall resistance to heat transfer is the sum of the individual resistances. Thus,

R total ¼ ð1/hAÞ hot fluid þ ðL/kAÞ wall þ ð1/hAÞ cold fluid ð16Þ The rate of heat transfer is then directly proportional to the overall driving force and inversely proportional to the total resistance

q ¼DT overall R total

¼ ðT12 T6Þ

ð1/hAÞhot fluid þ ðL/kAÞ wall þ ð1/hAÞ cold fluid

ð17Þ

Since the total resistance to heat transfer is inversely proportional to the overall conductance, Eq. (17) can be expressed as

q ¼DT overall

ð1/UAÞ ¼ UADToverall ð18Þ

where U is the overall heat transfer efficient (W/m²C). Equation (18) is similar to Eq. (14), except U is substituted for h. Equation (18) indicates that the rate of heat transfer is the product of three factors: overall heat-transfer coefficient, temperature drop, and area of heating surface.

For cylindrical geometries, as we saw earlier, the area is dependent on r and the heat transfer must be referred to a specific dimension, usually either the inside or outside radius of the tube. Consequently, if U is referred to the inside radius, then

In every heating application it is necessary to establish the magnitude of U so that the equipment can be properly sized. By knowing the rate of heat transfer desired, the overall heat transfer coefficient, and the driving force for heat

In every heating application it is necessary to establish the magnitude of U so that the equipment can be properly sized. By knowing the rate of heat transfer desired, the overall heat transfer coefficient, and the driving force for heat

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