Page 139 - Chemical Process Equipment - Selection and Design

6.8. LIQUID-GAS FLOW IN PIPELINES 111

one is obtained as follows. Begin with Although the key equations are transcendental, they are
readily solvable with hand calculators, particularly those with
root-solving provisions. Several charts to ease the solutions before
(6.78) the age of calculators have been devised: M.B. Powley, Can. J.
Chem. Eng., 241-245 (Dec. 1958); C.E. Lapple, reproduced in
(6.79) Perry’s Chemical Engineers’ Handbook, McGraw-Hill, New York,
1973, p. 5.27; 0. Levenspiel, reproduced in Peny’s Chemical
Engineers’ Handbook, McGraw-Hill, New York, 1984, p. 5.31;
from which
Hougen, Watson, and Ragatz, Themodyiurmics, Wiley, New York,
1959, pp. 710-711.
(6.80) In all compressible fluid pressure drop calculations it is usually
justifiable to evaluate the friction factor at the inlet conditions and
to assume it constant. The variation because of the effect of
and the integral is temperature change on the viscosity and hence on the Reynolds
number, at the usual high Reynolds numbers, is rarely appreciable.
(6.81)
NONlDEAL GASES
Also
Without the assumption of gas ideality, Eq. (6.71) is
dV
VdP = d(PV) - (PV) 7 (6.82)

Substitutions into Eq. (6.71) result in
‘In the isothermal case, any appropriate PVT equation of state may
be used to eliminate either P or V from this equation and thus
d(PV) - PV- + - VdV +- fG2 dL = 0. (6.83) permit integration. Since most of the useful equations of state are
dV G2
gc 2gcD pressure-explicit, it is simpler to eliminate P. Take the example of
one of the simplest of the non-ideal equations, that of van der
Further substitutions from Eqs. (6.80) and (6.81) and multiplying Waals
through by 2kgc,lG2V2 result in
RT
dV
2-- dV [ ;; -I- (k - l)Vi] p + (k - 1) - - dL = 0. p=--- a (6.91)
dv kf
-- ZkgPV
+
V-b V2’
V I V D
(6.84) of which the differential is
Integrating from VI to V2 and L = 0 to L gives
dp= -~ RT + ”> dV.
( (V-b)’ V3
Substituting into Eq. (6.90),
(6.85)
or (6.93)

Although integration is possible in closed form, it may be more
convenient to perform the integration numerically. With more
accurate and necessarily more complicated equations of state,
numerical integration will be mandatory. Example 6.13 empioys the
In terms of the inlet Mach number, van der Waals equation of steam, although this is not a particularly
suitable one; the results show a substantial difference between the
~
~
MI = u l / = GVl/aT/M,, ~ (6.87) ideal and the nonideal pressure drops. At the inlet condition, the
compressibility factor of steam is I = PV/RT = 0.88, a substantial
the result becomes deviation from ideality.

6.8. LIQUID-GAS FLOW IN PIPELINES
When everything else is specified, Eqs. (6.86) or (6.88) may be In flow of mixtures of the two phases in pipelines, the liquid tends
solved for the exit specific volume V,. Then Pz may be found from to wet the wall and the gas to concentrate in the center of the
Eq. (6.81) or in the rearrangement channel, but various degrees of dispersion of each phase in the
other may exist, depending on operating conditions, particularly the
(6.89) individual flow rates. The main patterns of flow that have been
recognized are indicated on Figures 6.7(a) and (b). The ranges of
conditions over which individual patterns exist are represented on
from which the outlet temperature likewise may be found. maps like those of Figures 6.7(c) and (d). Since the concept of a