Compressible gas-liquid flow through pipeline restrictions

39

Compressible Restrictions

Gas- Liquid Flow through Pipeline

S. D. MORRIS Process Engineering Division, TP 250, Joint Communities, 21020 Ispru, VA (Italy)

(Received

February

Research

Centre

(Ispra),

Commission

of the European

6, 1991; in final form April 9, 1991)

Abstract A method is presented for estimating the mass flow rate-pressure drop characteristics of compressible gas-liquid mixtures flowing through pipeline restrictions. The method can be applied for all two-phase Mach numbers Matp G 1. Predictions of the method are compared with a number of experimental data sets for air-water flows through nozzles and air-water and air-oil flows through orifice plates. The agreement is typically within *So/o for nozzles and within _t 12% for orifice plates. Application of the method to flow through rupture discs and pressure relief valves is discussed. Introduction

for frictionless two-phase flow through a restriction such as a nozzle, orifice or valve may be written as

The flow of two-phase mixtures through restrictions such as nozzles, orifices and valves continues to be a topic of industrial interest. Some practical examples of problem areas are: - flow characteristics of rupture discs and pressure relief valves (PRVs) in emergency relief systems; - leak rates from ruptured vessels and pipes; - two-phase flow control through choke valves on oil production platforms; _ metering of two-phase flows. While a considerable amount of experimental and theoretical work has been done in relation to both choked and incompressible two-phase flow through restrictions, there has been relatively little effort directed at compressible subsonic flows. Apart from the numerical method reported by McNeil [ 11, there does not seem to be any fairly simple method available that is valid throughout the two-phase Mach number range 0 < Mat,, < 1. The present work addresses this problem for the case of non-flashing gas-liquid flows. It is important to note that the method presented here describes the mass flow rate-pressure drop characteristics for flow between the inlet and minimum area locations of a restriction; for estimations of overall pressure loss (which includes the downstream pressure recovery) one should refer to the work of McNeil [ 11.

dp -= dz where p is the pressure at axial location z, ti is the mass flux ( = M/A) and u, the ‘effective’ two-phase specific volume; a suitable expression for v, is described later. Equation (1) may be re-expressed in the form v, dp = -f d(fi2 v,‘)

(2)

and

integrated between inlet (p =pO) and throat locations to yield the following equation for the ideal mass flow rate: (p =p,)

Pt

Here Q, is the effective specific volume inlet conditions and j3 is the ratio d/D inlet diameters; for a non-circular flow p” is simply replaced by cr2, where cr is inlet area ratio. Introducing now the pressure ratio the two-phase discharge coefficient gives, for the actual mass flow rate,

evaluated at of throat-tocross-section, the throat-tor] =p/p, and C,, eqn. (3)

Analysis The differential 0255-2701/91/$3.50

form of the momentum Chem.

Eng.

Process.,

equation 30 (1991)

39-44

@J Elsevier Sequoia/Printed

in The Netherlands

40

Single-phase

gas flow

Equation (4) reduces to the standard textbook equation [2] for compressible single-phase flow when u,o = uy and

tends to underpredict experimentally measured mass flow rates, C, normally assumes values greater than unity; this indicates a deficiency in the model which, for non-flashing flows, is largely due to the assumption of equal phase velocities.

I/; %=vg= Y %o VP0 that is,

Two-phase flow with phase slip and liquid entrainment

ti

= C,A,

(5)

($)“‘Q”’

(6)

where

0, = [XV,+ k( 1 - x)uJ

and _& = ‘ll-*/; - 8”

Homogeneous

An ‘effective’ two-phase specific volume has previously been proposed which includes the effects of phase slip (differing phase velocities) and liquid entrainment. This takes the form [3]

x

(7) two-phase

{x+G

[I +(vg$Ly’;2): 1]}

where

flow

k = (u~/v,)“~

Two-phase flows through pipeline restrictions are generally characterized by differing phase velocities; therefore, the homogeneous flow model, which assumes equal phase velocities, is rather unrealistic in most cases. However, it is useful to consider this special case briefly, as it provides useful insight. For homogeneous flow, the effective two-phase specific volume is identical to the homogeneous specific volume, that is, v, = fIh = XUg+ ( 1 - X)UL

(8)

and U& = Vh”= XV@+ ( 1 - X)VLO

(9)

Assuming an incompressible liquid phase (vL = vLO) and the gas expansion law of eqn. (5), the effective specific volume ratio can be expressed in the form v,.u,o = Vh/VhO= 1 + EhO(V-I’) - 1)

(10)

where chO=

XV@’ XV@ +

( 1 - X)VLO

c&i,

(2J”($)“’

(12)

(16)

is the slip ratio of Chisholm [4]. Equations ( 15) and ( 16) have been used successfully in a number of previous publications dealing with choked flashing flows [5], choked non-flashing flows [6] and the determination of two-phase discharge coefficients [ 71. However, with regard to subsonic flow, the direct substitution of eqn. (15) in eqn. (4) results in an equation that has to be solved numerically. Since this is an undesirable feature of a practical engineering method an alternative and simpler approach is sought here. In particular, if an analytically integrable equation similar to that of eqn. (12) could be derived for the case of slip/entrainment flow, then this would greatly facilitate the problem and allow an explicit formula to be derived for the two-phase mass flow rate. This would in essence be a realistic two-phase counterpart to eqn. (6). To achieve this end, note that eqn. (16) for the slip ratio can be written as l/2

(11)

is the homogeneous flow void fraction. The righthand side of eqn. ( 10) can now be substituted in eqn. (4) and the necessary integration performed to yield the following mass flow equation for homogeneous flow: n;l=

(15)

(17) and, making

use of eqn. (lo),

k = k,( 1 + E,,/?)“~

it follows that (18)

where k, =

(vh”/uLO)“2

(19)

and

where

/IXV-I/“_l

Y, =(l

It is now possible to examine the pressure ratio dependence of eqn. ( 15) for the following two cases: (i) chOI + 1, that is, flows of high compressibility, where

Y*=[l

-&&(I

-P/rlt)+-&(l Y---l +&&-“1’I)]‘-/!I”

-yl,c;‘-I)“‘)

(13) (14)

The discharge coefficient C, in this case will differ from that for gas flow. Since the homogeneous flow

k z kO(EhOA)“*

(20)

(21)

41 (ii) E,,~/I<< 1, that is, flows of low compressibility, where k zkk,(l

+f~,,&)

(22)

Case (i) Substitution of eqn. (21) in eqn. (15) gives, after some algebra, the following equation for the effective two-phase specific volume ratio:

where a and b are constants which depend only on initial conditions. It also emerges from this analysis thata>O,b>Oandapb. Case (ii) A similar approach using eqn. (22) yields v,/v,~ = 1 +a,d

12% 1 +al+bA”’

(25)

where A = v,/v~. Therefore, if, corresponding to two pressure ratios ‘I, and q2, the parameters ii, A,, A,, A2 are found from eqns. (20), ( 18) and ( 15), then the simultaneous solutions for a and b are &“*(A, - 1) - &“*(A, - 1) 1 l/2 /2*1/2 &l/2 _ 2*1/2)

I

l

(

(26)

(30)

it can be shown that eqn. (29) becomes s,

z=

s

ds

-_n,(yl,-%_

1)W

(31)

0 (s2 + l);

The integral in this equation good aproximation by

(24)

where a, is also a constant depending on initial conditions (a, > 0). Comparing these results, it can be seen that the form of eqn. (23) also covers case (ii); therefore, it is reasonable to apply eqn. (23) for all pressure ratios. Thus, the change in effective specific volume can be expressed, through eqns. (23) and (20) entirely in terms of pressure ratio, thus facilitating the integration involved in eqn. (4). The constants a and b should be determined through simultaneous solution of eqn. (23) for known values of u,/u,~ at two pressure ratios (i.e. at two values of A). Equation (23) may be written more conveniently as

a=

Defining s*=rl-‘/;‘_

=f

[

3 tan-’

can be evaluated

(st) +

S’

s,‘+

1

to a

1

Then, combining eqns. (28), (31) and (33) with eqn. (4), the final working form of the two-phase mass flow equation is ti = C&l, (2$‘*@‘*

(34)

where

_ &I/; (Il,-‘iU_

1) ii* (+:_;)

(35)

Z, = AI’ - 8“

(36)

and A, = 1 + al, + bl,“’

(37)

and where Lt is found from eqn. (20) with q = qt.

Two-phase

discharge

coefficients

Nozzles and orljices

and (27) It is recommended that the pressure p, be chosen at or below the lowest pressure expected in the restriction and that p2 = (pO + p,)/2. Referring now to eqn. (4), the term involving the integral becomes

Methods

have already been proposed by Morris two-phase discharge coefficients of nozzles and orifices. Since these methods are derived from an analysis of choked flow data, and are quoted in ref. 7 in their choked flow form, it is appropriate to state them here in the subsonic form. For nozzles

[ 71 for estimating

C, = 0.75 + 0.25 v,r

s

where

A dq = (1 - a)( 1 - r/,)

‘I,

+

5

(38)

v,. = v,t 1% ( 1 _

v,(;’

-

‘l/Y) + bZ

is evaluated where

where

V tit =

(29)

xv,,

r?

(39) from eqn. (15) at throat -‘:‘+(l

-X)ULO

conditions

and (40)

For orifices C,, = ( 1.26 - 0.26p)C,,

(41)

where

which may be written

C&l = 6, C,, + ( 1 - c,)C,,

(421

more conveniently

as

tic= GA, ($‘[ - ($-‘”

(50)

and E,=

1 I

Z.‘hf

(I-xbLoq,,‘.

[

“2 -’

X

(43)

Making A4, is

use of eqn. (2.Q the working

VLO (>I VP0 is the void fraction at the orifice throat. The separate liquid- and gas-phase discharge coefficients C,, and C,, are given by [2, 81 C,, = 0.61375 + 0.13318/?2 - 0.26095p4 + 0.51 146ph

(44)

and c

= 1 -(l dg

-w)“2 (45)

2f tJt”>

where

f_&-L dL w

(46)

2GL2

=4qt2”(1

G2

2Y

&2=-

yl,2/y

1 _

v,(T -

I )/;)

y-l

(48)

for

-“2

(51) 1 where vc is the choking pressure ratio. Since a simple explicit equation for qc could not be derived from the present theory, the equation of McNeil and Morris [6] is recommended for use in eqn. (51) and also as a criterion to determine whether a given flow is choked or not. The discharge coefficient in eqn. (51) now relates to choking conditions and therefore the equations for C, given in ref. 7 must be used. Two-phase

-Y/&

equation

Mach number

This can be defined by Matp = $

Rupture discs and PRVs

Discharge coefficients for choked flow through rupture discs and PRVs have been discussed in ref. 7. Similar arguments are adopted here for subsonic flow. Friedel and Kissner [9] recommend that rupture discs be treated as equivalent orifices and provide equivalent orifice sizes for a number of rupture disc types. It is recommended here that these results are used together with the foregoing analysis for orifices. With regard to PRVs, the discharge coefficients are likely to have values somewhere between that for an orifice and that for a nozzle. For two-phase flow the difference between these values is less than for single-phase flow [7]. In the absence of information on single-phase discharge coefficients for PRVs it is recommended that a value of C, = 0.91 is assumed for two-phase flow [7]. If a single-phase C, is known, then eqn. (44) can be replaced by this value and the two-phase orifice C, evaluated as described. This value can then be compared with that for a nozzle (eqns. (38) -( 40)) and a mean value between the two adopted. Choked

flow

The choking (or maximum) obtained from the condition

A&=

C,A, [

-@j-“’

mass flow rate tic is

(49)

(52) c where A? and tic are determined from eqns. (34) and (5 l), respectively. The preceding analysis covers all cases where Matp < 1 and so also includes the incompressible flow regime. It is of interest to note the conclusion of McNeil [I] that two-phase flows through restrictions are essentially incompressible for Ma,r < 0.5. The analysis of the present work was also found to support this conclusion. Comparisons

with experimental

data

The above methods for subsonic and sonic (or choked) flow have been compared with a number of data sets; some details are given in Table 1. Most of these data lie in the compressible flow regime, although in some cases the flows are incompressible. Figure 1 shows mass flux comparisons for subsonic flow through nozzles. Of the 231 data points shown, 91% lie within f8% of measured values, while 73% lie within k 5%. Figure 2 shows a similar plot for choked flow through nozzles. Of the 90 data points, 82% lie within +6% of measured values. Figure 3 shows comparisons for subsonic flow through orifices. Here, of the 216 points plotted, 93% are within f 15% of measured values, while 78% are within + 12%. Lastly, Fig. 4 shows comparisons for choked flow through orifices. Of the 182 points in this category, 90% are within + 12% of measured values, while 77% are within f7%.

43 TABLE

1. Data

sets used in comparisons

Reference

Graham

[IO]

Watson

et al. [Ill

Muir and Eichhorn Wood and Dickson

[ 121 [ 131

“High and low viscosity;

Restriction type

Fluid

Throat diameter (mm)

Diameter ratio

Nozzle No22lc Nozzle l/4 circle orifice Orifice Orilice Orifice Nozzle Orifice Orifice Orifice

Air-water Air-water Air-water Air-water

15.875 25.4 34.925 25.4

0.3 125 0.5 0.6875 0.5

Air-water Air-water Air-water Air-water Air-oil” Air-oil” Air-oil”

9.525 15.875 25.4 10.133 15.85 9.55 25.41

0.1875 0.3125 0.5 0.2673 0.4182 0.252 0.6704

bhigh viscosity

only.

lo’

lo3 Measured 1. Subsonic

mass

flux

Measured

(kg/m’s)

nozzlr flow data: model/data

comparison.

Fig. 3. Subsonic

lo= Measured

Fig, 2. Choked

mc~ss flux

nozzle flow data: model/data

(kg/m’s)

comparison

flux

(kg/m’s)

comparison.

Lo’

IO3 Measured

Fig. 4. Choked

mass

orifice flow data: model/data

moss

flux

orifice flow data: model/data

(kg/m’s)

comparison.

44

These comparisons provide considerable for the method described above.

support

Conclusions

vht

VI.

VLO W X z

A method has been presented for estimating the mass flow rate-pressure drop characteristics of compressible gas-liquid flows through pipeline restrictions. The predictions of the method compare very well with measured values, typically within Jr8%1 for nozzles and within k 12% for orifice plates. The method is valid for two-phase Mach numbers Ma,, =$ 1 and can therefore also be applied to incompressible flows. The working equation for subsonic flow ( Matp < 1) is given by eqn. (34) with the discharge coefficient C’, determined from eqns. (38) -(40) for nozzles or from eqns. (41) -(48) for orifice plates. The constants a and b should be determined from eqns. (26) and (27) following the recommended procedure. With regard to choked flow (MatP = 1), eqn. (51) should be used together with the choking pressure ratio equation of McNeil and Morris [6] and the discharge coefficients reported previously in ref. 7. Guidelines have also been given on the application of these results to rupture discs and pressure relief valves.

P Y

&hO

4

3

b

Cd

C4s Cdl D

r” KN2 k

ko IFi

n;l, Matp Pi7 4 P PO

Pt s St ve

ve”

vet v,* %

VP0 vh vhO

diameter ratio (throat/inlet) ratio of specific heats homogeneous void fraction at inlet tions void fraction at throat conditions pressure ratio (throat/inlet) choking pressure ratio =pt/po, throat pressure ratio defined in eqn. (25) value of A at 2, value of A at AZ value of A at throat conditions parameter defined by eqn. (20) value of /z at downstream pressure value of /z at pressure (p. + p, )/2

I D. A. McNeil,

2

throat area, m2 constants constant discharge coefficient discharge coefficient for gas phase discharge coefficient for liquid phase pipe (or inlet) diameter, m throat diameter, m parameter given by eqn. (46) parameter given by eqn. (48) slip ratio slip ratio at inlet conditions mass flow rate, kg/s choking mass flow rate, kg/s two-phase Mach number mass flux, kg/m2 s throat mass flux, kg/m2 s pressure, N/m2 inlet pressure, N/m2 throat pressure, N/m2 parameter defined by eqn. (30) value of s at throat effective specific volume, m3/kg inlet value of v,, m3/kg throat value of v,, m3/kg ratio defined by eqn. (39) gas specific volume, m3/kg inlet value of vg, m3/kg homogeneous specific volume, m3/kg inlet value of vh, m3/kg

m

condi-

References

Nomenclature

a, aI

throat value of vh, m3/kg liquid specific volume, m3/kg inlet value of vL, m3/kg term defined by eqn. (47) mass flow quality axial distance in flow direction,

4 5

6

7

8 9 10

11

12

13

A momentum approach to pressure drop prediction in compressible two-phase flow through contraction/expansion type fittings, HTFS Research Symp., 1989, Nat. Eng. Lab., Harwell, Didcot, U.K., Paper No. NEL/ HTFS 119. R. P. BenedIct, Fundamentals of Pipe Flow, Wiley, 1980, ISBN o-471-03375-8. S. D. Morris, A simple model for estimating two-phase momentum flux, Proc. 1st Nat. Conf. Heat Transfer, Leeds, U.K., 1984. Vol. 2. pp. 773-784. D. Chisholm, Two-Phase Flow in Pipelines and Heat Exchangers. Godwin, London, 1983, ISBN O-71 14-574X-4. S. D. Morris, Choking conditions for flashing one-component flows in nozzles and valves-a simple estimation method, Proc. Symp. Preventing Major Chemical attd Related Process Accidents, 1988, Inst. Chem. Eng., London, pp. 281-299. D. A. McNeil and S. D. Morris, A simple explicit method for estimating gas-liquid choked flow conditions in pipclmc rcstrictions, Proc. 2nd Nat. Conf. Heat Trmsfer, Glasgow. U.K., 1988. Vol. 2, pp. 1243m 1256. S. D. Morris, Discharge coefficients for choked gas-liquid flow through nozzles and orifices and applications to safety devices, J. Loss Prevention Process Ind., 3 (1990) 303&310. D. A. Jobson, On the flow of a compressible fluid through orifices, Proc. Inst. Mech. Eng.. 169 (1955) 7677776. L. Friedel and H. M. Kissner, J. Loss Prevention Process Ind., 1 (1988) 31. E. J. Graham, The flow of air-water mixtures through nozzles, NE,!, Rep. No. 308, Nat. Eng. Lab., East Kilbride. Glasgow, U.K., 1967. G. G. Watson, V. E. Vaughan and M. W. McFarlane, Twophase pressure drop with a sharp-edged orifice, NEL Rep. No. 290, Nat. Eng. Lab., East Kilbride, Glasgow, LJ.K., 1967. J. F. Muir and R. Eichhorn, Compressible flow of an air-water mixture through a vertical, two-dimcnslonal. convergingdiverging nozzle, Proc. Heat Transfer und Fluid Mechanics Inst., IY63. Stanford Univ. Press, Stanford, CA, pp. 183-204. J. D. Wood and A. N. Dickson, Metering of oil-air mixtures with sharp-edged orifices, Dep. Mech. Eng. Rep., Heriot-Watt Univ., Kiccarton, Edinburgh, U.K.. 1973.