Jökull - 01.12.1966, Page 40

(ii) The specific kinetic energy of the mix-
ture is
E = 1/2(x + ÍT_l)vg2 (22)
IV^
(iii) For a workless expansion with no heat
transfer, continuity of stagnation enthalpy re-
quires that
h0 — (hf + Vf2/2g0J)
r + (Vg2 — Vf2)/2g0J
(23)
(iv) Regarding K as a constant mean value
of slip applicable to the whole expansion, equa-
tion (20) gives, after some manipulation
_ aVgdQm + QmdVg
“ 1 + (K(1-x)Qii,)/q7
where a is a factor of the form
(24)
1 Kgg d(log x)
a _ Of d(lQg Om)
d~ i + (K(i-x)0m)/ef
On the same basis, equation (22) yields
(25)
dE^íL^vv.
+ 1/2(1-4+)v -p- -de“
R2 do,n
(26)
(v) As stated above it is assumed that the
expansion can be described by the relation
(vi) Maximizing G for the critical flow case
gives on the basis of equation (24)
aVgd6m + emdVg = 0 (31)
Eliminating dVg by equation (29), and recogniz-
ing that dom is arbitrary we obtain
Vgc2 = Ss£- (32)
a
Since the right hand side of equation (32) is
a function of Vg we must recognise this in
solving for Vgc.
For the total critical mass flow, we find on
the basis of equations (20) and (32) that
o, = IVí^ = V^ <53)
where cp is a flow coefficient of the following
form
cp :
(s-t) (1 + K(1 -^) (34)
0f
where the factors s and t are respectively
s =
t =
d(logx) ~|
d(!og 0m) J
V% (! ■
K-
d(log x) /
d(log em) \
1 + K(l-x)
(35)
pvmn = Constant (27)
where n is deemed to be constant. Moreover
from equations (16) and (27) one obtains
d0„
( = x)
K2
VgdVg
+ Vz (l
from which results
1
K2
)Vg
2
dx
d0m
dpm
(28)
VgdVg = — bdpm (29)
where b is a factor of the form
— )V g2 —+
K2 d6m +
x +
(!~x)
en
o 2
ym
(30)
The above results will now be discussed on
the basis of the experimental data given by
James (1962). The theoretical results in equa-
tion (33) can be compared with the experiment-
ally derived equation (1), which gives the mea-
sured values of Gc for various values of h0
and p. Furthermore, if thermal equilibrium be-
tween the phases is assumed, the latter two
quantities can be used to clerive the values of
Qg, Qm and x. Hence, an experimental value
of the product cpn can be obtained on the basis
of equation (33). The two coefficients in this
product cannot be separated unless some values
are available for e and K. Both quantities are
unknown, but there are reasons to believe that
e is slightly less than unity, and the slip ratio
K is probably in the range 5 to 30 depending
on the dryness fraction and other variables. For
the purpose of calculation, e can be taken as
194 JÖKULL