Thermodynamics

4–5 INTERNAL ENERGY, ENTHALPY, ANDSPECIFIC HEATS OF SOLIDS AND LIQUIDSA substance whose specific volume (or density) is constant is called anincompressible substance. The specific volumes of solids and liquidsessentially remain constant during a process (Fig. 4–33). Therefore, liquidsand solids can be approximated as incompressible substances without sacrificingmuch in accuracy. The constant-volume assumption should be takento imply that the energy associated with the volume change is negligiblecompared with other forms of energy. Otherwise, this assumption would beridiculous for studying the thermal stresses in solids (caused by volumechange with temperature) or analyzing liquid-in-glass thermometers.It can be mathematically shown that (see Chap. 12) the constant-volumeand constant-pressure specific heats are identical for incompressible substances(Fig. 4–34). Therefore, for solids and liquids, the subscripts on c pand c v can be dropped, and both specific heats can be represented by a singlesymbol c. That is,c p c v c(4–32)This result could also be deduced from the physical definitions of constantvolumeand constant-pressure specific heats. Specific heat values for severalcommon liquids and solids are given in Table A–3.Internal Energy ChangesLike those of ideal gases, the specific heats of incompressible substancesdepend on temperature only. Thus, the partial differentials in the definingequation of c v can be replaced by ordinary differentials, which yielddu c v ˛˛dT c 1T˛2 dT(4–33)The change in internal energy between states 1 and 2 is then obtained byintegration:2¢u u 2 u 1 c 1T2 dT1kJ>kg21(4–34)The variation of specific heat c with temperature should be known beforethis integration can be carried out. For small temperature intervals, a c valueat the average temperature can be used and treated as a constant, yielding¢u c avg 1T 2 T 1 21kJ>kg2(4–35)Chapter 4 | 189INTERACTIVETUTORIALSEE TUTORIAL CH. 4, SEC. 5 ON THE DVD.LIQUIDv l = constantSOLIDv s = constantFIGURE 4–33The specific volumes ofincompressible substances remainconstant during a process.IRON25°Cc = c v = c p= 0.45 kJ/kg . °CFIGURE 4–34The c v and c p values of incompressiblesubstances are identical and aredenoted by c.Enthalpy ChangesUsing the definition of enthalpy h u Pv and noting that v constant,the differential form of the enthalpy change of incompressible substances canbe determined by differentiation to beIntegrating,→0dh du v dP P dv du v dP¢h ¢u v ¢P c avg ¢T v ¢P1kJ>kg2(4–36)(4–37)