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Since heat transfer is one of the most common unit operations in food processing, in this section we will examine several cases of steady-state heat transfer which have wide application in processing food. Recall from the earlier description of modes of heat transfer that steady-state heat transfer means that the temperature is constant (i.e., independent of time) at any given location in the system but may vary from location to location.

A. Conduction Heat Transfer in Series

Consider a situation in which a solid wall is composed of several layers of material of different thickness each with its unique thermal conductivity (Fig. 3.4). This situation would exist, for example, in the wall of a cold storage room. There, in steady state, the rate of heat transfer across the wall may be considered using Fourier’s law of heat transfer by conduction.

Q ¼ kiAdT_i dxi

 

i¼1;2;3

ð26Þ

Since steady state exists, then the heat transfer through each material must be constant and equal. That is, Q₁¼ Q₂¼ Q₃.

For materials 1, 2, and 3, Eq. (26) can be rewritten as DT1 ¼QAx1

k₁ DT2¼QAx2

k₂ DT3¼QAx3

k₃

or

This simple analysis illustrates the important principle of thermal resistances in series. That is, the total thermal resistance is the sum of the individual resistances.

This also leads to the analogy with electrical systems in which rate ¼ driving force/resistance (I ¼ E/R). This analysis can be used to assess the value of insulating walls of processing equipment.

The previous discussion applies to objects which are rectilinear in which two dimensions are much larger than the third dimension. In the case of F^IGURE3.4 Composite rectilinear wall.

cylindrical objects such as pipes, the analysis is more complex. Consider the tubular system in Fig. 3.5. Fourier’s law for heat conduction in cylindrical coordinates is

qr ¼ 2k12AdT dr

where q_r is the heat transfer in the radial direction and k₁₂ is the thermal conductivity of material between r₁and r_2.The area for heat transfer is A ¼ 2prL and boundary conditions across the inner most layers are

T ¼ T1 at r ¼ r1

T ¼ T2 at r ¼ r2

Then

qr ¼ 2k12ð2prLÞdT

dr and qr

2pL Z r2

r₁

dr r ¼

Z T2

T₁

2 k12dT

Integration for constant k₁₂yields q_r

2pLlnr₂ r1

¼ 2k12ðT22 T1Þ

and

qr ¼ 2pLk₁₂

ln rð 2/r1ÞðT12 T2Þ ð28Þ

FIGURE3.5 Cylindrical geometry for heat transfer.

Similarly, for the outer layer r₂to r₃,

q_r¼ 2pLk₂₃

ln rð 3/r2ÞðT₂2 T3Þ ð29Þ

In a manner similar to that used for rectilinear thermal resistances in series, Eq.(28) and (29) can be solved for DT and summed yielding:

ðT12 T3Þ ¼ qr Since the area for heat transfer is dependent on r, define a logarithmic mean area such that

Substituting Eq. (31) into Eq. (30) for each layer yields ðT₁2 T3Þ ¼ q_r r22 r1

B. Convection Heat Transfer Coefficient

As mentioned earlier, the convective heat transfer coefficient h may be thought of as a heat transfer resistance caused by a film of immobile liquid of thicknessdand thermal conductivity k_f. Since it is not practical to measuredin most convective heat transfer applications, empirical equations have been developed from experi-mental data using dimensional analysis. Dimensional analysis consists of identi-fying all the variables or parameters in a system that influence the value of another variable or parameter (e.g., h), combining those variables into dimensionless numbers [dimensionless numbers consist of a combination of multiplying and dividing variables so that the dimensions (mass, distance, time, and energy)

exactly cancel one another], and developing correlations which enable one to estimate the value of a parameter in a particular system.

The following section illustrates dimensionless correlations that can be applied to free and forced convection and are applicable for Newtonian fluids only (i.e., constant viscosity independent of shear rate).

C. Free or Natural Convection

Earlier it was pointed out that free or natural convection occurs when bulk movement of fluid occurs solely due to density gradients which are established when heating or cooling a fluid. These situations occur in food processing systems in which there are thermal gradients in the fluid (liquid, gas, or vapor).

Examples include vegetables in brine in containers subjected to nonagitated thermal processing (still retorts), freezing or cooling ambient fruits and vegetables in still air (not a recommended practice), and heat gain or loss from sides of buildings and cold stores (warehouses) in still air.

The dimensionless numbers which are important include the Nusselt number, Grashof number, and Prandl number. The Nusselt number (Nu) is the ratio of convective to conductive heat transfer in the fluid and is given by

Nu ¼hD kf

¼ convective heat transfer

conductive heat transfer ð33Þ

where h is the convective heat transfer coefficient (W/m²8C), D is the characteristic dimension of the system (m), and k_fis the thermal conductivity of the fluid (W/m8C).

The Grashof number (Gr) represents the physical properties of the fluid and the temperature difference between the surface and the bulk fluid, one of the most important variables in the system.

Gr ¼D³r²gBDT m^{2 ¼}

buoyancy forces

viscous force ð34Þ

where D is the characteristic dimension (m),ris the fluid density (kg/m³), g is the acceleration due to gravity (9.80665 m/s²), B is the coefficient of thermal expansion (1/T) (K²¹), DT is temperature of the wall minus bulk fluid temperature (T_w2 Tb) (C), andm is viscosity (Pas)(kg/ms). The values of all properties are estimated at the average temperature [T_f¼ (T_wþ T_b)/2].

The Prandl number (Pr) represents the relative magnitude of the thermal boundary layer to the momentum boundary layer.

Pr ¼mC_p

k ¼ viscous effect

thermal diffusion effect ð35Þ

wheremis fluid viscosity (Pas) (kg/ms), C_pis heat capacity (kJ/kg8C), and k is thermal conductivity of the fluid (W/m8C).

The empirical correlations generally take the following form

Nu ¼ aðGrPrÞ^m ð36Þ

and a and m are functions of the geometry and magnitude of GrPr (sometimes identified as the Rayleigh number, Ra). Values for a and m for various conditions are illustrated in Table 3.7.

D. Forced Convection

In forced convection, the fluid is moved by the application of forces upon the fluid. Conventionally the forces are created by a pump, fan or stirrer. This is the most common type of convection in food processing since not only are the heating and cooling media moved (e.g., air, water, brine, refrigerant) but also the product is moved (e.g., milk, ice cream mix, vegetables or fruit in brine or syrup).

It is easy to imagine that forced convection will result in more rapid heat transfer than free convection because additional energy is used to cause bulk T^ABLE3.7 Constants a and m for Free Convection

Configuration GrPr a m

Vertical Plates and cylinders Length. 1 m

Laminar ,10⁴ 1.36 1/5

Laminar 10⁴, GrPr , 10⁹ 0.55 1/4

Turbulent .10⁹ 0.13 1/3

Spheres and

horizontal cylinders Diameter,0.2 m

Laminar 10³, GrPr ,10⁹ 0.53 1/4

Turbulent .10⁹ 0.13 1/3

Horizontal plates Heated plate facing up

(Or cold plate facing down)

Laminar 10⁵, GrPr , 2 £ 10⁷ 0.54 1/4

Turbulent 2£ 10⁷, GrPr , 3 £ 10¹⁰ 0.14 1/3 Heated plate facing down

(Or cold plate facing up)

Laminar 3£ 10⁵, GrPr , 3 £ 10¹⁰ 0.27 1/4

Source: Perry and Green (1997).

mixing. Furthermore, the value of the convective coefficient should be a function of not only the physical properties of the fluid and the relative thermal and momentum boundary layers (i.e., the Prandl number) but also of the nature of the fluid flow [i.e., laminar or turbulent flow, represented by the Reynolds number (Re), the kinetic forces/viscous forces].

Correlations for forced convection generally take the form Nu ¼ a Reⁿ Pr^m mb All properties are evaluated at the average temperature [i.e., (T_wþ T_b/2)]. If T_b varies from inlet to outlet, then the average bulk temperature is used for T_b. Table 3.8 gives several equations for various geometric and flow conditions.

Handbooks on chemical and mechanical engineering provide many more correlations for specific operating conditions [see, for example, Perry and Green (1997)].

IV. UNSTEADY-STATE HEAT TRANSFER