Page 21 - Entrophy Analysis in Thermal Engineering Systems

Fundamental concepts 11

Eq. (1.19) provides a relation between the thermodynamic properties of the
system. An alternative expression may be obtained using the differential
form of Eq. (1.13) as

δQ
dΦ ¼ dS T s (1.20)
Thus, the combined first and second laws relation becomes
dU ¼ T s dS pdV T s dΦ (1.21)

Note that in Eq. (1.19) T is the system temperature whereas in Eq. (1.21) T s
denotes the surrounding temperature. Also, both Eqs. (1.19) and (1.21) are
valid for compressible fluids (gases) that exchange heat with the surroundings
and perform work due to the fluid pressure.
For an ideal gas whose state equation is given by
(1.22)
pV ¼ nRT
where n denotes the number of moles and R is the universal gas constant, the
internal energy is a function of temperature only; that is, dU¼nc v dT. In this
case, Eq. (1.19) can be expressed as

(1.23)
dT dV
¼ ds R
c v
T V
where c v denotes the specific heat at constant volume and s is the specific
entropy.
References
[1] http://www.fchart.com/ees/.
[2] https://www.irc.wisc.edu/properties/.
[3] R. Clausius, The Mechanical Theory of Heat, Translated by W. R. Brown, MacMillan
& Co., London, 1879.
[4] S. Carnot, R.H. Thurston (Ed.), Reflections on the Motive Power of Heat, second ed.,
Wiley, New York, 1897.
[5] B.F. Dodge, Chemical Engineering Thermodynamics, McGraw-Hill, New York, 1944.
[6] E.F. Obert, Thermodynamics, first ed., McGraw-Hill, New York, 1948.
[7] W. Nernst, Experimental and Theoretical Applications of Thermodynamics to Chem-
istry, Charles Scribner’s Sons, New York, 1907.
[8] E. Fermi, Thermodynamics, Dover Publications Inc., New York, 1956
[9] J.M. Smith, H.C. Van Ness, M.M. Abbott, M.T. Swihart, Introduction to Chemical
Engineering Thermodynamics, eighth ed., McGraw-Hill, New York, 2018.