Jökull - 01.12.1966, Page 38

■pu
G -
[
g0K
2x + 1 / 3v„ \ / \ / 3x \
(1 + (K—1)(—--)) ^g(l + 2x(K-l))+vfK((l-2x)(K-l))j ( — j.
+ (l-x) (1 + x(K — 1)(1 + -iK))
3vf " V 3p / s
1
3p /s J
1/2
(15)
Critical flow ratio as predicted by thís model,
both for equilibrium and for metastable flow
are shown in Fig. 7, reproduced from Cruver
(1963).
Fig. 7. Critical Discharge (Cruver’s Method).
(Reproduced from Fig. 30, Cruver 1963).
COMPARATIVE RESULTS
Calculations employing the more sophisticat-
ed models required evaluation of the equations
(8), (14) and (15). The calculations are very
tedious and the interpretation to be placed
upon some of the derivations involved is ob-
scure. It is not profitable, therefore, at the
present juncture to evaluate a large number of
cases. To enable the foregoing to be appraised,
however, comparative figures have been cal-
culated for the Hveragerdi Well G-7 Run 8
quoted previously, and are as follows.
The value given in the last row of Table II
is probably reasonably correct and present
evidence, therefore, suggests that Methods (i)
and (v) due to James and Cruver respectively
are the most promising.
AN APPROXIMATE ANALYSIS
OF THE FLOW REGIME
AT THE WELL EXIT
It is well known that the critical exit velocity
of a perfect gas from a pipe can be obtained
by maximizing mass flow G = Vp subject to
the constraints of isentropic expansion and
conservation of energy. This simple approach
is not directly applicable to the much more
complicated case of a two-phase flow. No ideal
physical model, which can be handled by mathe-
matical means, can incorporate the complexities
of this type of flow. On the other hand, it
may be possible to obtain some rough quantita-
tive analysis by making a number of approxima-
tions which simplify the model. One such ap-
proach is offered below. It is based on the
above mentioned method of maximizing the
rnass flow at the exit under a number of con-
straints.
The present approach involves 4 basic as-
sumptions, (1) the slip ratio Iv remains constant
du-ring the expansion of the two-phase mixture,
(2) the pressure-volume relation for the expand-
ing mixture can be described by a polytropic
equation pvmn = C, where C is a constant and
n is at most a slowly varying function of p.
Moreover, (3) since there are no external heat
losses, the energy equation for the mixture can
be written
dE = — evdp, (16)
192 JÖKULL