Jökull - 01.12.1966, Page 37

w
G = "A = v°m (9)
in which gm, the mixture density is l/vm where
vm = (1 — x)vf + xvg, the properties being ap-
propriate to the observed critical pressure. The
velocity, V, is obtained with sufficient accuracy
by assuming isentropic equilibrium expansion
from the original liquid state in the hydro-
thermal flow to pressure pc. As the homo-
geneous theory fails to recognize the “slip”
which occurs between liquid and vapour at all
but the lowest qualities, it predicts values of
flow which are much less than those observed.
WPour core
1 lauio PrCfi/ULUS
i< \ , 1. ! | , ] £L£/ie*r
ap ) fo/z
i flRER fí r J ÍW! YS/S

Fiouj
Fig. 6. Annular Flow at Wellhead Exit.
acceleration pressure drop, and this could be
shown to occur for the value:
(iv) Method due to Fauske (1962)
Fauske postulated equilibriunr annular flow
and obtained, for the conservation of moment-
um, the expression
G2 ðvm dp
g0 dl
(10)
Observations from many sources suggested
that dp/dl assumed a finite maximum value
at the critical pressure. It follows that 3vm/31 is
also a maximum. Fauske argued that the slip
ratio K, being the only variable, maximized the
K
(11)
The effective specific volume for the two-
phase fiow was det'ined as
R„
(1 — x)2 vf
1 -R„
(12)
The void fraction, Rg, was defined as
-l
(13)
The critical flow then becomes
r -g0R
|^(l-x + Kx)xj 1 (3pS)h+ 1 (vg(l + 2Kx — 2x) +vf(2Kx —2K —2xK2 + K2) j | ( 3P / h
]l/2
where K = Kc, as given in equation (11).
(v) Method due to Cruver (1963)
Cruver recognized that there was uncertainty
concerning the thermodynamic stability of the
mixture near the pipe exit. The pressure gra-
dient dp/dl is known to be very steep in this
region, and the rapid rate of expansion coupl-
ed with the limited extent of the phase inter-
facial surface are conditions favourable to
supersaturation. A flow theory was devised in
which the effective specific volume was based
on a kinetic energy average value. This value
was so chosen that the slip ratio Kc = (v„ /vf)1/s,
the choice arising from tlie fact that the value
quoted maximises the fiow for a given energy
expenditure. The method was adaptable either
for equilibrium or for metastable flow. The
mass flow is given by
JÖKULL 191