Page 46 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

Introduction to Mechanics of Continua 31

3.1.1 Specific Heats. Enthalpy

The specific heat is defined as the amount of heat required to increase
by unity the temperature of a mass unit of the considered medium.
Correspondingly, the specific heat is

_ 4
dT”

Supposing that the temperature is a function of p and 5 = v, we have

ar = (& dp + or dv,
Op) Ov >

where the subscript denotes the fixed variable for partial differentiation.
Analogously, assuming that the specific internal energy e is also a func-
tion of p and v we have

Oe Oe
de = (5) a+ (5)

Referring to the case of fluids, as the work done by a unit mass

“against” the pressure forces is dw = pd (4) = pdv, the first law of
thermodynamics can be written

de = 6q — pdv,

where dqis the heat added to the unit mass. Because 7 is an integrating
factor for dq,in the sense that ds, = 9 , we get Tds, = de + pdv. Obvi-
ously, for reversible processes (ideal media) ds, = ds and the last relation

becomes T'ds = de + pdv, an equation which could be also deduced as a
consequence of Gibbs’ equations (for inviscid fluids).
Generally, for any fluids, by using the above expression for de and the
first law of thermodynamics, we have that

0 O
6g = (5) dp + (5) dv + pdv.
P/y Us»

Hence the specific heat is

a de
_ oq _ (32) dp + (35), a + pdv

dT oT oT
(4) dp + (Sr) p dv

From this expression it will be possible to define two “principal” spe-
cific heats: one Cp, for dp = 0 ( p = constant), called the specific heat at