4.1. INTRODUCTION AND MECHANISMS OF HEAT TRANSFER

4.1A. Introduction to Steady-State Heat Transfer

The transfer of energy in the form of heat occurs in many chemical and other types of processes. Heat transfer often occurs in combination with other separation processes, such as drying of lumber or foods, alcohol distillation, burning of fuel, and evaporation. The heat transfer occurs because of a temperature-difference driving force and heat flows from the high to the low-temperature region.

In Section 2.3 we derived an equation for a general property balance of momentum, thermal energy, or mass at unsteady state by writing Eq. (2.3-7). Writing a similar equation but specifically for heat transfer,

Equation 4.1-1


Assuming the rate of transfer of heat occurs only by conduction, we can rewrite Eq. (2.3-14), which is Fourier's law, as

Equation 4.1-2


Making an unsteady-state heat balance for the x direction only on the element of volume or control volume in Fig. 4.1-1 by using Eqs. (4.1-1) and (4.1-2), with the cross-sectional area being A m2,

Equation 4.1-3


Figure 4.1-1. Unsteady-state balance for heat transfer in control volume.


where is rate of heat generated per unit volume. Assuming no heat generation and also assuming steady-state heat transfer, where the rate of accumulation is zero, Eq. (4.1-3) becomes

Equation 4.1-4


This means the rate of heat input by conduction = the rate of heat output by conduction; or qx is a constant with time for steady-state heat transfer.

In this chapter we are concerned with a control volume where the rate of accumulation of heat is zero and we have steady-state heat transfer. The rate of heat transfer is then constant with time, and the temperatures at various points in the system do not change with time. To solve problems in steady-state heat transfer, various mechanistic expressions in the form of differential equations for the different modes of heat transfer such as Fourier's law are integrated. Expressions for the temperature profile and heat flux are then obtained in this chapter.

In Chapter 5 the conservation-of-energy equations (2.7-2) and (4.1-3) will be used again when the rate of accumulation is not zero and unsteady-state heat transfer occurs. The mechanistic expression for Fourier's law in the form of a partial differential equation will be used where the temperature at various points and the rate of heat transfer change with time. In Section 5.6 a general differential equation of energy change will be derived and integrated for various specific cases to determine the temperature profile and heat flux.

4.1B. Basic Mechanisms of Heat Transfer

Heat transfer may occur by any one or more of the three basic mechanisms of heat transfer: conduction, convection, and radiation.

1. Conduction

In conduction, heat can be conducted through solids, liquids, and gases. The heat is conducted by the transfer of the energy of motion between adjacent molecules. In a gas the "hotter" molecules, which have greater energy and motions, impart energy to the adjacent molecules at lower energy levels. This type of transfer is present to some extent in all solids, gases, or liquids in which a temperature gradient exists. In conduction, energy can also be transferred by "free" electrons, which is quite important in metallic solids. Examples of heat transfer mainly by conduction are heat transfer through walls of exchangers or a refrigerator, heat treatment of steel forgings, freezing of the ground during the winter, and so on.

2. Convection

The transfer of heat by convection implies the transfer of heat by bulk transport and mixing of macroscopic elements of warmer portions with cooler portions of a gas or liquid. It also often refers to the energy exchange between a solid surface and a fluid. A distinction must be made between forced-convection heat transfer, where a fluid is forced to flow past a solid surface by a pump, fan, or other mechanical means, and natural or free convection, where warmer or cooler fluid next to the solid surface causes a circulation because of a density difference resulting from the temperature differences in the fluid. Examples of heat transfer by convection are loss of heat from a car radiator where the air is being circulated by a fan, cooking of foods in a vessel being stirred, cooling of a hot cup of coffee by blowing over the surface, and so on.

3. Radiation

Radiation differs from heat transfer by conduction and convection in that no physical medium is needed for its propagation. Radiation is the transfer of energy through space by means of electromagnetic waves in much the same way as electromagnetic light waves transfer light. The same laws that govern the transfer of light govern the radiant transfer of heat. Solids and liquids tend to absorb the radiation being transferred through them, so that radiation is important primarily in transfer through space or gases. The most important example of radiation is the transport of heat to the earth from the sun. Other examples are cooking of food when passed below red-hot electric heaters, heating of fluids in coils of tubing inside a combustion furnace, and so on.

4.1C. Fourier's Law of Heat Conduction

As discussed in Section 2.3 for the general molecular transport equation, all three main types of rate-transfer processes—momentum transfer, heat transfer, and mass transfer—are characterized by the same general type of equation. The transfer of electric current can also be included in this category. This basic equation is as follows:

Equation 2.3-1


This equation states what we know intuitively: that in order to transfer a property such as heat or mass, we need a driving force to overcome a resistance.

The transfer of heat by conduction also follows this basic equation and is written as Fourier's law for heat conduction in fluids or solids:

Equation 4.1-2


where qx is the heat-transfer rate in the x direction in watts (W), A is the cross-sectional area normal to the direction of flow of heat in m2, T is temperature in K, x is distance in m, and k is the thermal conductivity in W/m · K in the SI system. The quantity qx/A is called the heat flux in W/m2. The quantity dT/dx is the temperature gradient in the x direction. The minus sign in Eq. (4.1-2) is required because if the heat flow is positive in a given direction, the temperature decreases in this direction.

The units in Eq. (4.1-2) may also be expressed in the cgs system, with qx in cal/s, A in cm2, k in cal/s · °C · cm, T in °C, and x in cm. In the English system, qx is in btu/h, A in ft2, T in °F, x in ft, k in btu/h · °F · ft, and qx/A in btu/h · ft2. From Appendix A.1, the conversion factors are, for thermal conductivity,

Equation 4.1-5


Equation 4.1-6


For heat flux and power,

Equation 4.1-7


Equation 4.1-8


Fourier's law, Eq. (4.1-2), can be integrated for the case of steady-state heat transfer through a flat wall of constant cross-sectional area A, where the inside temperature is T1 at point 1 and T2 at point 2, a distance of x2 - x1 m away. Rearranging Eq. (4.1-2),

Equation 4.1-9


Integrating, assuming that k is constant and does not vary with temperature and dropping the subscript x on qx for convenience,

Equation 4.1-10


EXAMPLE 4.1-1. Heat Loss Through an Insulating Wall

Calculate the heat loss per m2 of surface area for an insulating wall composed of 25.4-mm-thick fiber insulating board, where the inside temperature is 352.7 K and the outside temperature is 297.1 K.

Solution: From Appendix A.3, the thermal conductivity of fiber insulating board is 0.048 W/m · K. The thickness x2 - x1 = 0.0254 m. Substituting into Eq. (4.1-10),



4.1D. Thermal Conductivity

The defining equation for thermal conductivity is given as Eq. (4.1-2), and with this definition, experimental measurements have been made to determine the thermal conductivity of different materials. In Table 4.1-1 thermal conductivities are given for a few materials for the purpose of comparison. More-detailed data are given in Appendix A.3 for inorganic and organic materials and Appendix A.4 for food and biological materials. As seen in Table 4.1-1, gases have quite low values of thermal conductivity, liquids intermediate values, and solid metals very high values.

Table 4.1-1. Thermal Conductivities of Some Materials at 101.325 kPa (1 Atm) Pressure (k in W/m · K)
SubstanceTemp. (K)kRef.SubstanceTemp. (K)kRef.
Gases   Solids   
 Air2730.0242(K2) Ice2732.25(C1)
  3730.0316  Fire claybrick4731.00(P1)
 H22730.167(K2) Paper0.130(M1)
 n-Butane2730.0135(P2) Hard rubber2730.151(M1)
Liquids    Cork board3030.043(M1)
 Water2730.569(P1) Asbestos3110.168(M1)
  3660.680  Rock wool2660.029(K1)
 Benzene3030.159(P1) Steel29145.3(P1)
  3330.151   37345 
Biological materials and foods    Copper273 373388 377(P1)
 Olive oil293 3730.168 0.164(P1) Aluminum273202(P1)
 Lean beef2631.35(C1)    
 Skim milk2750.538(C1)    
 Applesauce2960.692(C1)    
 Salmon2770.502(C1)    
  2481.30     

1. Gases

In gases the mechanism of thermal conduction is relatively simple. The molecules are in continuous random motion, colliding with one another and exchanging energy and momentum. If a molecule moves from a high-temperature region to a region of lower temperature, it transports kinetic energy to this region and gives up this energy through collisions with lower-energy molecules. Since smaller molecules move faster, gases such as hydrogen should have higher thermal conductivities, as shown in Table 4.1-1.

Theories to predict thermal conductivities of gases are reasonably accurate and are given elsewhere (R1). The thermal conductivity increases approximately as the square root of the absolute temperature and is independent of pressure up to a few atmospheres. At very low pressures (vacuum), however, the thermal conductivity approaches zero.

2. Liquids

The physical mechanism of conduction of energy in liquids is somewhat similar to that of gases, where higher-energy molecules collide with lower-energy molecules. However, the molecules are packed so closely together that molecular force fields exert a strong effect on the energy exchange. Since an adequate molecular theory of liquids is not available, most correlations to predict the thermal conductivities are empirical. Reid et al. (R1) discuss these in detail. The thermal conductivity of liquids varies moderately with temperature and often can be expressed as a linear variation.

Equation 4.1-11


where a and b are empirical constants. Thermal conductivities of liquids are essentially independent of pressure.

Water has a high thermal conductivity compared to organic-type liquids such as benzene. As shown in Table 4.1-1, the thermal conductivities of most unfrozen foodstuffs, such as skim milk and applesauce, which contain large amounts of water have thermal conductivities near that of pure water.

3. Solids

The thermal conductivity of homogeneous solids varies quite widely, as may be seen for some typical values in Table 4.1-1. The metallic solids of copper and aluminum have very high thermal conductivities, while some insulating nonmetallic materials such as rock wool and corkboard have very low conductivities.

Heat or energy is conducted through solids by two mechanisms. In the first, which applies primarily to metallic solids, heat, like electricity, is conducted by free electrons which move through the metal lattice. In the second mechanism, present in all solids, heat is conducted by the transmission of energy of vibration between adjacent atoms.

Thermal conductivities of insulating materials such as rock wool approach that of air since the insulating materials contain large amounts of air trapped in void spaces. Superinsulations to insulate cryogenic materials such as liquid hydrogen are composed of multiple layers of highly reflective materials separated by evacuated insulating spacers. Values of thermal conductivity are considerably lower than for air alone.

Ice has a thermal conductivity much greater than water. Hence, the thermal conductivities of frozen foods such as lean beef and salmon given in Table 4.1-1 are much higher than for unfrozen foods.

4.1E. Convective Heat-Transfer Coefficient

It is well known that a hot piece of material will cool faster when air is blown or forced past the object. When the fluid outside the solid surface is in forced or natural convective motion, we express the rate of heat transfer from the solid to the fluid, or vice versa, by the following equation:

Equation 4.1-12


where q is the heat-transfer rate in W, A is the area in m2, Tw is the temperature of the solid surface in K, Tf is the average or bulk temperature of the fluid flowing past in K, and h is the convective heat-transfer coefficient in W/m2 · K. In English units, h is in btu/h · ft2 · °F.

The coefficient h is a function of the system geometry, fluid properties, flow velocity, and temperature difference. In many cases, empirical correlations are available to predict this coefficient, since it often cannot be predicted theoretically. Since we know that when a fluid flows past a surface there is a thin, almost stationary layer or film of fluid adjacent to the wall which presents most of the resistance to heat transfer, we often call the coefficient h a film coefficient.

In Table 4.1-2 some order-of-magnitude values of h for different convective mechanisms of free or natural convection, forced convection, boiling, and condensation are given. Water gives the highest values of the heat-transfer coefficients.

Table 4.1-2. Approximate Magnitude of Some Heat-Transfer Coefficients
 Range of Values of h
Mechanismbtu/h · ft2 · °FW/m2 · K
Condensing steam1000–50005700–28000
Condensing organics200–5001100–2800
Boiling liquids300–50001700–28000
Moving water50–3000280–17000
Moving hydrocarbons10–30055–1700
Still air0.5–42.8–23
Moving air2–1011.3–55

To convert the heat-transfer coefficient h from English to SI units,


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