ρT
∂s
∂t
þ u
∂s
∂x
¼ 0
or
Ds
Dt
¼ 0:
ð1:58Þ
This equation implies that the entropy of the fluid element or particle does not
change with time along the element’s path of motion and, as a result, the flow is
called isentropic. However, if discontinuities, such as, shock waves occur in the
flow, even in the case of an ideal fluid, there will be an increase in entropy across the
discontinuity and the equation Ds/Dt ¼ 0 no longer applies.
1.6 Entropy Change for an Ideal Gas
If the fluid is an ideal gas, we can calculate the entropy change (per unit mass) as
follows;
Tds ¼ de þ pdυ and pυ ¼ RT
Hence,
ds ¼
c V dT
T
þ
pdυ
T
,
where de ¼ c V dT, therefore,
ds ¼ c V
pdυ þ υdp
pυ
þ
Rdυ
υ
¼ c V þ R
ð
Þ
dυ
υ
þ c V
dp
p
,
where c P ¼ c V + R, hence,
ds ¼ c P
dυ
υ
þ c V
dp
p
,
therefore,
1.6 Entropy Change for an Ideal Gas
23