Page 441 - Advanced thermodynamics for engineers

18.2 LIQUEFACTION BY EXPANSION – METHOD (II) 431

at B then the downstream temperature, T 2 , will always be less than T 1 . It is possible to analyse whether
heating or cooling will occur by evaluating the sign of the derivative (vT/vp) h . This term is called the
Joule–Thomson Coefficient, m.
If m < 0, then there can be either heating or cooling depending on whether the downstream pressure
is between A and C, or to left of C. If m > 0, then the gas will be cooled on passing through the plug.
(i.e. the upstream state point defined by (p 1 ,T 1 ) is to the left of B on the isenthalpic line.)
This situation can be analysed in the following way. From the Second Law of Thermodynamics
dh ¼ Tds þ vdp: (18.4)
But for this process dh ¼ 0, and thus, from Eqn (18.4),

vs

0 ¼ T þ v: (18.5)
vp h
If it is assumed that entropy is a continuous function of pressure and temperature, i.e. s ¼ s (p,T)then
vs vs

ds ¼ dp þ dT (18.6)
vp T vT p
which can be rearranged to give
vs vs vs vT

¼ þ (18.7)
vp vp vT vp
h T p h
Hence, substituting this expression into Eqn (18.5) gives

vs vs vT
0 ¼ T þ þ v (18.8)
vp vT vp
T P h
Now, from thermodynamic relationships (Eqn (7.24)),

vs
T ¼ c p
vT
p
and, from the Maxwell relationships (Eqn (7.19(d)))

vs vv
¼ :
vp vT
T p
Thus

vv vT
0 ¼ T þ c p þ v (18.9)
vT vp
p h
which may be rearranged to give
" #
vT 1 vv

¼ T v ¼ m: (18.10)
vp c p vT
h p