Page 155 - Advanced Thermodynamics for Engineers, Second Edition

142 CHAPTER 7 GENERAL THERMODYNAMIC RELATIONSHIPS

The chemical potential terms will be omitted in the following analysis, although similar equations
to those below can be derived by taking them into account.
It can be seen from Eqns (7.2) and (7.2a) that the specific internal energy can be represented by a
three-dimensional surface based on the independent variables of entropy and specific volume. If this
surface is continuous then the following relationships can be derived based on the mathematical
properties of the surface. The restriction of a continuous surface means that it is ‘smooth’. It can be
seen from Fig. 2.5 that the p–v–T surface for water is continuous over most of the surface, but there are
discontinuities at the saturated liquid and saturated vapour lines shown in Fig. 2.6. Hence, the
following relationships apply over the major regions of the surface, but not across the boundaries. For
a continuous surface,

z ¼ zðx; yÞ where z is a continuous function:
Then,
vz vz

dz ¼ dx þ dy: (7.3)
vx y vy x
Let

vz vz
M ¼ and N ¼ : (7.4)
vx vy
y x
Then,
dz ¼ Mdx þ Ndy: (7.5)
2 2
For continuous functions the derivatives v z and v z are equal, and hence vM ¼ vN .
vxvy vyvx vy x vx y
Consider also the expressions obtained when z ¼ z(x,y) and x and y are themselves related to additional
variables u and v, such that x ¼ x(u,v) and y ¼ y(u,v). Then,

vz vz vx vz vy
¼ þ (7.6)
vu vx vu vy vu
v y v x v
Let z ¼ v and u ¼ x, then x ¼ x(z) and

vz vx
¼ 0; and ¼ 1: (7.7)
vu vu
v v
Hence

vz vz vy
0 ¼ þ (7.8)
vx vy vx
y x z
giving
vz vy vx

¼ 1: (7.9)
vx vz vy
y x z