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Chapter 1IntroductionThe engineering area frequently referred to as thermal science includes thermodynamics and heattransfer. The role of heat transfer is to supplement thermodynamic analyses, which consider onlysystems in equilibrium, with additional laws that allow prediction of time rates of energy transfer.These supplemental laws are based upon the three fundamental modes of heat transfer, namelyconduction, convection, and radiation.1.1 CONDUCTIONA temperature gradient within a homogeneous substance results in an energy transfer rate withinthe medium which can be calculated bywhere aT/dn is the temperature gradient in the directionnormal to the area A . The thermal conductivity k is anexperimental constant for the medium involved, and it maydepend upon other properties, such as temperature andpressure, as discussed in Section 1.4. The units of k areW/m.K or Btu/h.ft."F. (For units systems, see Section 1.5.)The minus sign in Fourier's law, ( I . ] ) , is required by thesecond law of thermodynamics: thermal energy transferresulting from a thermal gradient must be from a warmer toa colder region.If the temperature profile within the medium is linear(Fig. 1-1), it is permissible to replace the temperaturegradient (partial derivative) withAT- Ax--XFig. 1-1T2- Ti&-XISuch linearity always exists in a homogeneous medium of fixed k during steady state heat transfer.Steady state transfer occurs whenever the temperature at every point within the body, includingthe surfaces, is independent of time. If the temperature changes with time, energy is either being storedin or removed from the body. This storage rate iswhere the mass rn is the product of volume V and density p.1.2 CONVECTIONWhenever a solid body is exposed to a moving fluid having a temperature different from that ofthe body, energy is carried or convected from or to the body by the fluid.

INTRODUCTION2[CHAP. 1If the upstream temperature of the fluid is T, and the surface temperature of the solid is T,, theheat transfer per unit time is given byq hA(T, - T,)(1.4)which is known as Newton's law of cooling. This equation defines the convective heat transfer coefficienth as the constant of proportionality relating the heat transfer per unit time and unit area to the overalltemperature difference. The units of h are W/m2-Kor Btu/h.ft2."E It is important to keep in mind thatthe fundamental energy exchange at a solid-fluid boundary is by conduction, and that this energy isthen convected away by the fluid flow. By comparison of (2.2) and (1.4), we obtain, for y n,hA(Ts- T*) - kA(3where the subscript on the temperature gradient indicates evaluation in the fluid at the surface.1.3 RADIATIONThe third mode of heat transmission is due to electromagnetic wave propagation, which can occurin a total vacuum as well as in a medium. Experimental evidence indicates that radiant heat transferis proportional to the fourth power of the absolute temperature, whereas conduction and convectionare proportional to a linear temperature difference. The fundamental Stefan-Boltzmann law isq oAT4(1.6)where T is the absolute temperature. The constant o is independent of surface, medium, andtemperature; its value is 5.6697 X 10-' W/m2.K4or 0.1714 X 10-8 Btu/h.ft2-"R4.The ideal emitter, or blackbody, is one which gives off radiant energy according to (1.6). All othersurfaces emit somewhat less than this amount, and the thermal emission from many surfaces (graybodies) can be well represented bywhereE,q eo-AT4the emissivity of the surface, ranges from zero to one.(1.7)1.4 MATERIAL PROPERTIESThermal Conductivity of SolidsThermal conductivities of numerous pure metals and alloys are given in Tables B-1 (SI) and B-1(Engl.). The thermal conductivity of the solid phase of a metal of known composition is primarilydependent only upon temperature. In general, k for a pure metal decreases with temperature; alloyingelements tend to reverse this trend.The thermal conductivity of a metal can usually be represented over a wide range of temperaturebyk ko(l b 0 C O 2 )(1.8)where 8 T - Trctand k,, is the conductivity at the reference temperature TrePFor many engineeringapplications the range of temperature is relatively small, say a few hundred degrees, andk kO(l 60)(1.9)The thermal conductivity of a nonhomogeneous material is usually markedly dependent upon theapparent bulk density, which is the mass of the substance divided by the total volume occupied. Thistotal volume includes the void volume, such as air pockets within the overall boundaries of the pieceof material. The conductivity also varies with temperature. As a general rule, k for a nonhomogeneous

CHAP. 11INTRODUCTION3material increases both with increasing temperature and increasing apparent bulk density. Tables B-2(SI) and B-2 (Engl.) contain thermal conductivity data for some nonhomogeneous materials.Thermal Conductivity of LiquidsTables B-3 (SI) and B-3 (Engl.) list thermal conductivity data for some liquids of engineeringimportance. For these, k is usually temperature dependent but insensitive to pressure. The data of thistable are for saturation conditions, i.e., the pressure for a given fluid and given temperature is thecorresponding saturation value. Thermal conductivities of most liquids decrease with increasingtemperature. The exception is water, which exhibits increasing k up to about 150 C or 300 F anddecreasing k thereafter. Water has the highest thermal conductivity of all common liquids except theso-called liquid metals.Thermal Conductivity of GasesThe thermal conductivity of a gas increases with increasing temperature, but is essentiallyindependent of pressure for pressures close to atmospheric. Tables B-4 (SI) and B-4 (Engl.) presentk-data for several gases at atmospheric pressure. For high pressure (i.e., pressure of the order of thecritical pressure or greater), the effect of pressure may be significant.Two of the most important gases are air and steam. (No distinction is made between a gas and avapor in this chapter.) For air the atmospheric values listed in Table B-4 (SI) are suitable formost engineering purposes over the ranges: (i) 0 C d T G 1650 C and 1 atm p 6 100 atm; (ii)-75 "Cd T s 0 "C and 1atm 6 p s 10 atm.Thermal conductivity data for steam exhibit a strong pressure dependence. For approximatecalculations the atmospheric data of Tables B-4 (SI) and B-4 (Engl.) may be used together with FigureB-3. For other gases, one must resort to more extensive property tables.DensityDensity is defined as the mass per unit volume. All systems considered in this book will besufficiently large for statistical averages to be meaningful; that is, we will consider only a continuum,which is a region with a continuous distribution of matter. For systems with variable density we definedensity at a point (a specific location) as(1.10)where 6Vc is the smallest volume for which a continuum has meaning.Density data for most solids and liquids are only slightly temperature dependent and are negligiblyinfluenced by pressure up to 100 atm. Density data for solids and liquids are presented in Tables B-1,B-2, and B-3 in both SI and English Engineering units. The density of a gas, however, is stronglydependent upon the pressure as well as upon the temperature. In the absence of specific gas data theatmospheric density of Tables B-4 (SI) and B-4 (Engl.) may be modified by application of the idealgas law:(I.11)P PI(;)The specific volume is the reciprocal of the density,U -1P(1.12)

4INTRODUCTION[CHAP. 1and the specific gravity is the ratio of the density to that of pure water at a temperature of 4 C anda pressure of one atmosphere (760mmHg). Thuss -P(1.13)Pwwhere S is the specific gravity.Specific HeatThe specific heat of a substance is a measure of the variation of its stored energy with temperature.From thermo-dynamics the two important specific heats are:specific heat at constant volume:c,specific heat at constant pressure:c"r7U r7Tdh3-dT(1.14)( I . IS)Here U is the internal energy per unit mass and h is the enthalpy per unit mass. In general, U and hare functions of two variables: temperature and specific volume, and temperature and pressure,respectively. For substances which are incompressible, i.e., solids and liquids, c,, and c , are numericallyequal. For gases, however, the two specific heats are considerably different. The units of c, and cp areJ/kg.K or Btu/lb,."F.For solids, specific heat data are only weakly dependent upon temperature and even less affectedby pressure. It is usually acceptable to use the limited cI, data of Tables B-1 (SI), B-1 (Engl.), B-2 (SI),and B-2 (Engl.), over a fairly wide range of temperatures and pressures.Specific heats of liquids are even less pressure dependent than those of solids, but they aresomewhat temperature influenced. Data for some liquids are presented in Tables B-3 (SI) and B-3(Engl.).Gas specific heat data exhibit a strong temperature dependence. The pressure effect is slight exceptnear the critical state, and the pressure dependence diminishes with increasing temperature. For mostengineering calculations the data of Table B-4 (Engl.) (other than density) can be used for pressuresup to 200 psia, while Table B-4 (SI) is suitable for pressures up to 1.4 X 10' Pa.Thermal DiffusivityA useful combination of terms already considered is the thermal diffusivity a , defined bykcy-(1.1 6 )PCpIt is seen that a is the ratio of the thermal conductivity to the thermal capacity of the material. Its unitsare ft2/h or m2/s. Thermal energy diffuses rapidly through substances with high a and slowly throughthose with low a.Some of the tables of Appendix B, in both SI and English units, list thermal diffusivity data. Notethe strong dependence of a for gases upon both pressure and temperature; these data for gases areonly for atmospheric pressure, and they are only valid for the specified temperature.ViscosityThe simplest flow situation involving a real fluid, i.e., one that has a nonzero viscosity, is laminarflow along a flat wall (Fig. 1-2). In this model, fluid layers slide parallel to one another, the molecularlayer adjacent to the wall being stationary. The next layer out from the wall slides along this stationarylayer, and its motion is impeded or slowed because of the frictional shear between these layers.

CHAP. 115INTRODUCTIONContinuing outward, a distance is reached where theretardation of the fluid due to the presence of the wall isno longer evident.Consider plane P-P. The fluid layer immediatelybelow this plane has velocity U - 6u, and the fluid layerimmediately above has velocity U Su. Here U is thevalue of the velocity in the x-direction at the y-locationof the P-P plane. The difference in velocity betweenthese two adjacent fluid layers produces a shear stress 7.Newton postulated that this stress is directly proportionalto the velocity gradient normal to the plane:dur pf-dYFig. 1-2(1.I7)The coefficient of proportionality is called the coeficient of dynamic viscosity, or more simply thedynamic or absolute viscosity.Viscosity units As shown by (I.17), the units of pf are N d m 2 or lbf.s/ft2. In many applications it isconvenient to have the dynamic viscosity expressed in terms of a mass, rather than a force, unit. In thisbook, pmwill denote the mass-based viscosity coefficient. In the SI sys

reports, and has coauthored four books including one textbook and one Schaum’s Outline. His 30 years in academe have been focused on teaching and research in the thermal sciences (Thermal Dynamics, Fluid Mechanics, and Heat Transfer). Dr. Sissom is h