ciency than water electrolysis still remains, but the possibility appears to be remote.
SUBSCRIPTS
j
= = = = = = = =
I
=
n
= = =
CY
0 C
Nomenclature
heat capacity, calories per gram-mole ’ C. Gibbs free energy, calories per gram-mole = enthalpy, calories per gram-mole I = total number of reactions in a multistep process J = number of reactions having positive entropy changes in a multistep process KO, = Gibbs free energy change for Reaction 17 evaluated a t temperature of reaction, calories per gram-mole Ko2 = Gibbs free energy change for Reaction 18 evaluated a t temperature of reaction, calories per gram-mole K H 1 = Gibbs free energy change for Reaction 19 evaluated a t temperature of reaction, calories per gram-mole K H 2 = Gibbs free energy change for Reaction 20 evaluated a t temperature of reaction, calories per gram-mole L = number of reactions having negative entropy changes in a multistep process M = hydrogen flow rate, gram-moles per second N = number of elements that form unknoivn compound X P = pressure, atm. Q = heat generation rate, calories per second q = heat required by a reaction, calories per gram-mole s = entropy, calories per gram-mole ’ C. T = temperature, ’ C. T , = temperature of heat sink, ’ C. TH = temperature of heat source, ’ C. T , = standard state temperature = 25’ C. = temperature a t which Reactions 17 and 19 are operated, ’ C. T z = temperature a t which Reactions 18 and 20 are operated, ’ C. w = useful work, calories per gram-mole X = unknown element or compound XH2 = hydride of unknown element or compound X XO = oxide of unknoivn element or compound X
Cp
=
g h
=
GREEKLETTERS 7 = process thermal efficiency = efficiency of converting heat to work 5
f
g H
i
s
1,2
given phase given phase surroundings or heat sink of a process formation of a compound gaseous phase heat reservoir ith reaction in series of reactions individual reaction in series of reactions, each having positive entropy change individual reaction in series of reactions, each having negative entropy changes one of .V elements that form unknown compound X standard state specific states or conditions
literature Cited
(1) Chem. Eng. 61 (April 1954). (2) Chem. Eng. Progr. 61, No. 4 (April 1965). (3) Dodge, R. F., “Chemical Engineering Thermodynamics,” p. 498, McGraw-Hill, New York, 1944. (4) Funk, J. E., Keinstrom, R. M., “System Study of Hydrogen Generation by Thermal Energy,” Allison Division, General Motors, Rept. EDR 3714, Vol. 11, Suppl. A, 3.1-3.25 (June 1964). ( 5 ) Latimer, \V. M., “Oxidation Potentials,” 2nd ed., pp. 359-69, Prentice-Hall, New York, 1952. (6) Lindell, K. O., e t al. ,“Energy Depot Concept,” Society of Automotive Engineers, SP-263 (November 1964). (7) Petrol. R t f n e r 40 (April 1961’1. ( 8 ) Reinstrom, K. M., et al., “Ammonia Production Feasibility Studv.” Allison Division, General Motors,. Rept. . EDR 4200. 3,1-3,’31 (November 1965). ) Steinberg, M., et al., “Production of Industrial Chemicals by Chemonuclear Processes,” November 1964 meeting, American Nuclear Society, San Francisco, Calif. 0 ) LVenner, K. R., “Thermochemical Calculations,” pp. 114-95, McGraw-Hill, New York, 1941. 1 ) Zemansky, M. W.,“Heat and Thermodynamics,” p . 322 McGraw-Hill, New York: 1957. RECEIVED for review October 25, 1965 ACCEPTEDFebruary 3, 1966 LVork partially funded by the U. S. Army, Corps of Engineers, Division of Reactor Development (Nuclear Power Division), under Contract D.4-44-009-AMC-S55(X).
PARTIAL VAPORIZATION IN ORIFICES AND VALVES R. P. R O M I G I , R. R. R O T H F U S , A N D R. I. K E R M O D E
Carnegie Institute of Technology, Pittsburgh, Po.
The influence of a small amount of vapor generation on the flow o f pressurized water through four orifices and a control valve has been studied experimentally. Profiles of pressure recovery and temperature have been obtained at orifice diameter ratios from 0.253 to 0.608 and at various settings of the valve over the high range of Reynolds numbers. Pertinent expansion factors are defined and correlated. Pressure recovery is much reduced b y vapor generation, Pressure drops and temperature profiles are strongly dependent on the quality of the fluid ahead of the constriction.
they have appeared over a period of 30 years, there are only a few publishpd studies of the combined flow of vapor and liquid through constrictions such as orifices and valves. Stuart and Yarnell (8, 9 ) and Monroe (5) have investigated stable and metastable conditions associated with the flow of saturated liquids through orifices in series. Benjamin and Miller (4) and Bailey (2) have studied the flow of ALTHOUGH
1 Present address, Chemical Engineering Department, San Jose State College, San Jose, Calif.
342
I&EC
PROCESS DESIGN A N D DEVELOPMENT
saturated or nearly saturated water and of water-steam mixtures through orifices having a broad range of diameter ratios. Murdock (6) has measured and correlated orifice flow rates for a number of two-phase systems having a large volume ratio of vapor to liquid. From another standpoint, Numachi, Yamabe, and Oba (7) have studied the effect of cavitation on the behavior of an orifice and have described several degrees of cavitation in terms of an appropriate cavitation number. Ball ( 3 )has added some meaningful comments and data. Taken as a whole, the cited literature indicates that dis-
charge coefficients in the presence of vapor generation are about the same as those for liquid flow, provided the fluid ahead of the orifice is subcooled or, a t the most, saturated. F a r downstream from the constriction, there is evidence that the mixture is approximately saturated. The present work was undertaken to obtain a more comprehensive picture of how the adiabatic, spontaneous generation of a small a r o u n t of vapor affects the pattern of pressure drop and recovery in orifices and control valves. I n order to correlate the behavior of the constriction with geometry, operating variables, and fluid properties, it was necessary to establish the quality of the fluid a t various axial positions, and for this reason temperature profiles were measured. T h e vapor-liquid composition and the temperature proved to be of fundamental importance. Once these were determined, it was possible to move forward along analytical lines in a relatively simple way. Basic Relationships
Factors Related to Velocity Profile. To account for differences between the local velocity, u, and the average linear velocity, V, in a straight conduit of cross-sectional area A , the average momentum, 2, per unit mass of steadily flowing fluid, taken over the whole conduit, can be written in the form PA -
V
Jo
Conservation Equations. In the present work, attention is limited to steady flow in nonbranching, constricted ducts. A single inlet boundary, i, and a single outlet boundary, j, are therefore sufficient to specify the extent of a system having a finite length along the path of the flow. Since there is no accumulation to be considered, the mass balance on such a system is simply (4)
w = piArVf = pjAjVj T h e unidirectional momentum balance is
F.3
- mg ( 5 )
where Fs represents any additional net force due to the presence of the boundaries and mg is the force due to gravity, both properly resolved. If the flow can be considered adiabatic then in the absence of shaft work, the energy equation is simply
where the items within the parentheses are the internal energy, insertion work, kinetic energy, and potential energy in the order named, taken per unit mass of flowing fluid. T h e corresponding mechanical energy balance, so called, is
Jo
Similarly, the average kinetic energy, expressed as
E, per
unit mass can be
where the term ZFp is zero if the process is reversible and positive if there are irreversibilities. Equations for Orifice Meters. A standard reference on the behavior of head meters is the ASME report (7) in which the working equations for orifice meters are presented in the following form :
JO
I n dealing with steady, incompressible, turbulent flow in smooth tubes, the velocity profile over the main part of the stream is often approximated by the power rule
w
= AOKY
- 161)
d2gCp1(fi2
(8)
where
K
=
C/ d 1-
D4
(9)
(3) Exponent n is a weak function of the Reynolds number. T h e relationships among N R ~n,, am,and f f k are shown in Table I. Little error is introduced by considering the a’s to be unity. O n the other hand, when the flow is entirely laminar, f f k and omhave the values 0.50 and 0.75, respectively.
Table I.
Velocity, Momentum, and Kinetic Energy Factors for Smooth Tubes
Range of Reynolds iVo. 4 x 103 to 1 x 104 1 x 104 to 1 x 105 1 x 105 to 2 x 105 2 x 105 to 5 x 10; 5 x 105 to 1 x io6 Table II.
P 0,253 0.304 0.405 0.608
’/* 6 7 8 9 10
“rn
“k
0.974 0.978 0.984 0.987 0.990
0.928 0.945 0.956 0.964 0.970
Axial Position of Vena Contracta ( I ) A70. of Pipe Diameters from Plate Min. Av. Max. 0.41 0.83 1.22 0.44 0.80 1.14 0.47 0.74 0.99 0.42 0.56 0.70
and C is the discharge coefficient with the velocity of approach factor included. For standard orifices, the discharge coefficient, K , depends on the position of static pressure taps, the pipe size, the Reynolds number, and the diameter ratio. T h e expansion factor, Y,depends on the pressure level, the pressure drop across the meter, the ratio of heat capacities a t constant pressure and constant volume, the diameter ratio, and the position of the static pressure taps. The value of the expansion factor is unity if the flowing fluid has a constant density. The usual correlations of K and Y are based on data for flow of a single phase and do not consider vapor-liquid combinations. Since K can be established readily for wholly liquid flow, it is convenient to retain the form of Equation 8 when dealing with two-phase situations. The effect of vaporliquid composition on the flow rate is consequently reflected in the value of the expansion factor. This approach is used in the present work. Area of Downstream Jet. Behind an orifice, the fluid jet reaches its minimum cross-sectional area a t the so-called ”vena contracta,” whose position is affected very little by the Reynolds number or by the pipe size, provided the nominal pipe diameter VOL. 5
NO. 3
JULY 1 9 6 6
343
is 2 inches or more. The only important variable appears to be the diameter ratio, p, and Table I1 shows the positions of the vena contracta (7) for single-phase flow a t the ratios pertinent to the present work. When the fluid has a constant density and the process is taken to be entirely reversible, Equations 7 and 8 can be combined to yield
where A2 is the area of the jet a t the downstream tap and A , is the area of the opening in the orifice plate. Any change of potential energy is neglected. The principal problem is to establish the correct value of C Y ~ aZ t the downstream tap. Recovery of Static Pressure. I t is of practical interest to determine the “permanent” loss of static pressure across an orifice, since it is that quantity which influences the power required for maintaining the flow at a specified rate. If it is assumed that the downstream tap is situated a t the vena contratca, the simplified momentum equation for the expansion of the jet from the downstream tap to the transverse plane, n, where the jet completely fills the pipe is
I t is often convenient to introduce the specific volume, u, in place of the density, p, and to express the pressure recovery (pn - p2) as the fraction, @R, of the measured pressure drop (pl - p2) across the orifice taps. With these changes, combination of Equations 4, 8, IO, and 12 yields
For wholly liquid flow, the specific volume can be assumed constant and the fractional recovery in a specified type of orifice a t a particular Reynolds number depends on the diameter ratio, /3, and the velocity factor, a ~ .As shown in Table 111, the recovery is not much affected by the latter. I n fact, the computed area of the jet a t the vena contracta is more sensitive to the choice of a than is the fractional recovery of static pressure. The entries in the right-hand column of Table I11 corresponding to parabolic flow are merely for comparison and have no pertinent physical significance. Flow of Two Phases
Pressure Drop across Orifices. LIQUIDFLOW. A convenient way of establishing the behavior of a test orifice, T ,
Effect of Velocity Factors on Jet Area and Pressure Recovery in Wholly Liquid Flow 0.960 0.960 Olkl 1 .ooo 0.986 0.986 1 1.000 1.000 0.500 ak2 1.000 1.000 0.750 OlllZZ 1.000 Dis. Ratio, fi Ratio of J e t Areas, A2/A, 0.628 0.628 0.889 0.253 0.598 0.597 0.844 0.302 0.604 0.854 0.405 0.607 0.644 0.642 0.907 0.608 Fractional Pressure Recovery, @ R 0.253 0,077 0.077 0.073 0.302 0,105 0.104 0.098 0.180 0.180 0.405 0.169 0.384 0.383 0.354 0.608
Table 111.
is to place it in series with a metering orifice, M, installed in the same pipe. Then by virtue of Equation 8, w = Ao.wK.>rY.w ~ / ~ ~ , P I M-( @z).\r PI =
AOTKTYT 2/2gcPir(.bi - P z ) T
(14)
When the fluid is wholly in the liquid state, both expansion factors are simply unity. If in addition the liquid is a t the same temperature when it passes through each of the orifices, the specific volumes are equal for practical purposes and
Subscript L is simply a reminder that the fluid must be entirely liquid and a t the same temperature in both orifices. If both orifices have the same type of taps, the discharge coefficients are functions of the diameter ratio, p, and the Reynolds number of the pipe-namely,
Under the stated constraints, the Reynolds number in the pipe is the same a t the upstream taps of both orifices. When the Reynolds number is greater than about 50,000, the ratio of discharge coefficients is essentially constant and therefore the ratio (PI - ~ ~ ) T L / (-P PI ~ ) . ~ is I Lalso constant and independent of the Reynolds number. PARTIAL VAPORIZATION. T h e specific volume and enthalpy of a mixture containing saturated vapor, g, and saturated liquid, f, phases of a single component can be described in terms of the weight fraction, w, of vapor in the mixture-i.e., the “quality.” Thus u = (1
h = (1
+
- w)v,
- w ) h f + who
= u,
wug
= h,
+ w(u,
- u,)
+ ~ ( h g- h,)
=
UI
+
hf
+ wh,,
WUIU
(17)
=
(18)
LVhen a metering orifice, M , handling only liquid is placed in series with a test orifice, T , across which partial vaporization occurs, there are two possibilities to consider. O n the one hand, there may be no vapor a t the upstream tap of the orifice under test-Le., WI = 0-on the other hand, the mixture a t the upstream tap may have a quality-i.e., 01 >O. If there is only liquid ahead of the test orifice but a mixture of vapor and liquid behind it, a n appropriate expansion factor, YTO,can be defined such that
Combining this with Equation 15, and noting that since w1 = 0 the temperature a t the upstream taps of both orifices can be assumed to be the same, it follows immediately that yTo
=
[(PI - P P ) T L / ( f i l
- pZ)T]”2
(20)
In the same manner, if there is vapor present a t the upstream tap of the test orifice as well as behind it, another expansion simply factor, Y T , can be defined by Equation 19, with replacing YTo. In this case,
Y,,
=
KTL ~
KT
[
IT
1
PI - P ~ T 1’2 L
:l$(pl
- p2)T
(21)
Since the discharge coefficients depend on the Reynolds num344
I & E C PROCE.SS D E S I G N A N D DEVELOPMENT
bers in the pipe ahead of their respective orifices, the question of how to evaluate the viscosity of the two-phase mixture must be settled. I n the case of water, it appears that the two-phase Reynolds number is higher than that for entirely liquid flow, regardless of the way in which the vapor and liquid viscosities are weighted. At liquid-phase Reynolds numbers greater than 50,000, however, calculations show that the ratio K T J K T can be taken as uniLy with a n error of less than 2y0. With this approximation, Equations 17 and 21 yield the expansion factor in the form
Determination of Quality. T h e change of enthalpy across a constriction can be obtained by combining Equations 4 and 6 when the process is adiabatic. If in addition the change of potential energy is negligible, and if the system moves from a state of subcooled liquid to a state of partial vaporization with saturated phases, then for flow through an orifice it is readily shown that
In most ordinary cases, the kinetic energy term can be neglected without appreciable error and the quality a t the downstream tap is adequately represented by the first term on the right-hand side of the equation. T o the extent that the partial vaporization can be approximated by a set of equilibrium states, the quality can be determined through measurements of temperatures on both sides of the orifice. Cavitation Number. If the inception of cavitation at the downstream pressure tap of an orifice is to be specified, a reasonable cavitation number can be evolved to suit the purpose. I n general, the quantity of interest is the difference between the pressure, Pz,a t the downstream tap and the vapor pressure, Pz,corresponding to the temperature of the fluid a t the same position. Since no liquid is present u p to inception, the density is known and the pressure difference ( p ~- Pz) can be made dimensionless through division by the quantity ( p l V Z 2 / 2 g c ) . Noting that the vapor pressures a t the upstream and downstream taps are equal \\hen the flow is isothermal, combination of Equations 4 and 7 immediately yields the cavitation number
For practical purposes, the factor ( A 2 / A o ) L 2 / K 2can be considered independent of the Reynolds number in the range above S0,OOO and can therefore be taken as a function of the diameter ratio, P, alone. The critical value of the cavitation number-that is, the value a t which the inception of observable cavitation occursdepends greatly on the experimental means used to detect the phenomenon. This may be by visual or audio means or, in adiabatic systems, by measurement of local temperature changes within the fluid. I n any case, the reported values of the critical cavitation number are not apt to be very meaningful unless the means of detection is specified. Experimental Equipment and Procedure
T h e experimental unit was installed in the bypass of a closed hydraulic loop handling deionized, pressurized water. T h e loop consisted of 4-inch, schedule 40 steel pipe through which the water from a 500-gallon supply tank was circulated by
means of two pumps, each with a capacity of 500 gallons per minute and a head of 150 to 175 p.s.i., which could be used singly or in series. T h e loop was fitted with appropriate heat exchangers and accessories to maintain constant operating conditions a t specified temperature levels. T h e test section was constructed of 2l/'*-inch, schedule 40, steel pipe and contained the metering orifice, the test orifice (or valve), a 22-square foot, double-pass, shell-and-tube condenser, and a pump for returning the water to the main loop. T h e calming lengths, the specifications of the orifices, and the positions of pressure and temperature measurements are summarized in Table IV. Temperatures were measured by means of copper-constantan thermocouples enclosed in stainless steel tubing with a n outer diameter of 0.067 inch. T h e thermocouple beads extended from the sealed tubes and were exposed directly to the fluid. T h e probes could be adjusted to traverse the diameter of the pipe in the vertical direction. Pressures and pressure differences were measured by means of 40-inch mercury manometers or by means of 200-p.s.i. gages with increments of 0.2 p.s.i., which were calibrated by dead weight test gages. T o minimize the area of gas-liquid interface in the system and a t the same time allow for expansion and contraction of the water. a n auxiliary supply tank was installed. T h e interfacial area in this tank was only about 2 sq. feet. During operation the main supply tank was kept full and the interface was held in the auxiliary tank. Nitrogen pressure was maintained on the interface to prevent cavitation in the main pumps. Temperature profiles were measured in the pipe behind the test orifice a t Reynolds numbers between 144,000 and 658,000 based on conditions in the pipe a t the upstream tap and a t temperatures between 253' and 298' F. I n all, 97 temperature profiles were determined. Profiles of static pressure recovery and corresponding axial temperature profiles were measured a t Reynolds numbers between 39,000 and 604,000 based on liquid conditions in the pipe a t the upstream tap and a t temperatures between 153' and 297' F. One or both of these profiles were measured in 178 separate experiments. Pressure changes across the orifice taps were determined in 283 instances. Experiments similar to those performed on the orifices were repeated with a Fisher Governor Co. Type 657 AR, 2l/*-inch pneumatic control valve taking the place of the test orifice. T h e pressures immediately upstream and downstream from the valve were measured from the flange taps used in the preceding experiments. The downstream positions of temperature and pressure measurement were the same as before,
Table IV. Specifications of Experimental Equipment Orifice Dfameter, Pipe Diameter D u m . Ratio, D o , Inches D I , Inches DOlDi A. Standard Flange-Tap Orifices (Foxboro) in 2l/~-Inch, 0 0 1 1
625 750 000
500
Temp. or Press. Reading
Schedule 40, Steel Pipe 2 469 2 469 2 469 2 469 Direction from Test Orifice
0 0 0 0
253 304 405 608
Pipe Diameters from Plate of Test Orifice
Position of Pressure Taps and Thermocouples (Taken at upstream tap of metering orifice) To0 TO Upstream 22,40 P1 Upstream 0.40 PP,T P Downstream 0.40 Pa Downstream 1.93 T4 Downstream 3.71 p5 Downstream 7.78 T6 Downstream 11.82 p7 Downstream 15.90 p8 Downstream 47.0 T8 Downstream 49.4 B.
C. Calming Sections, Test Orifice Upstream 52.6 diameters, straight pipe Downstream 49.5 diameters, straight pipe
VOL. 5
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JULY 1 9 6 6
345
except that the distances shown in Table IV were increased by 2.43 pipe diameters in order to relate them accurately to the axial position of the valve stem. A metering orifice 1.000 inch in diameter (/3 = 0.405) was retained in series with the valve for the purpose of measuring the rate of flow. A total of six temperature profiles were measured behind the valve a t Reynolds numbers between 153,000 and 333,000 based on upstream conditions in the pipe and a t temperatures of 276' to 280' F . Seven pressure recovery profiles, 32 axial temperature profiles, and 61 pressure changes across the valve were determined a t Reynolds numbers between 94,000 and 466,000 based on liquid in the pipe a t the upstream tap and a t temperatures between 262' and 280' F. Experimental Results and Discussion
Orifice Meters. TEMPERATURE PROFILES.A typical group of temperature profiles across the pipe a t various distances behind the test orifice is shown in Figure l . The horizontal pipe is traversed in the vertical direction and positive values of the reduced radius, r / R , indicate positions above the center line. and the fluid temperaThe diameter of the orifice opening, Do, ture To,a t the upstream tap are also shown. When there is no generation of vapor (w, = w z = 0), the temperatures a t all planes are found to be constant across the stream and equal to To within experimental error. I n all cases, even when the upstream fluid has a small quality ( W L >0) the temperature ahead of the test orifice turns out to be equal to Toa t all points in the stream. When there is partial vaporization, however, the profiles taken behind the test orifice indicate that in most cases a sharply defined temperature jet emerges from the orifice opening and is dissipated as it moves down the pipe. T h e temperature data make it clear that the quality w 1 a t the upstream tap of the test orifice influences the downstream profile. First, the thermal j e t is wider and its edges are less sharp when the upstream quality is greater than zero. Second, the temperature a long distance downstream from the orifice plate is more uniform under this condition than when there is no vapor a t the upstream tap
o-X/D= 0.40 o-X/D= 3.7I A-X/D= 11.82
R U N Ill-P
- - - _ T- o
240-
The temperature profiles suggest that the position of maximum temperature moves somewhat toward the bottom of the pipe as the fluid passes farther and farther downstream from the orifice. This may mean that vapor-rich fluid tends to favor the upper part of the pipe, consistent with the upward buoyant force associated with the vapor. The profiles also suggest that most of the vaporization takes place a t the edges of the jet or in the dead-water region around it rather than within the jet itself. I n the absence of velocity profiles, it is not known a t what distance downstream from the orifice vaporization essentially ceases. If there were no further appreciable vaporization due to frictional pressure drop in the pipe, this would correspond to the position where the bulk average temperature of the fluid no longer changes with axial distance from the orifice. Calculation of the bulk average temperature, of course, requires information about the velocity distribution and this is not obtainable from the present study. The basis for calculating the qualities wn a t various doivnstream positions has to be selected rather arbitrarily, since it would be most reasonable to base them on the bulk average temperature if both temperature and velocity profiles were available in the j e t and pipe. In the absence of velocity data, the maximum temperature in the j e t a t the downstream tap is used as the basis for computing the apparent quality, w2. T h e sharpness of the temperature gradient a t the edge of the jet suggests that the jet passes through a region relatively rich in vapor. Within the jet itself the temperature is very uniform and the average is not far from the maximum. At any other downstream plane, n, the minimum temperature in the cross section is chosen as the basis for con. Since most of the temperature profiles are somewhat asymmetric, the maximum point may be associated with a variety of local velocities. On the other hand, much of the profile is ordinarily not far from the minimum and the latter affords a reproducible basis for computing the apparent quality, w,. These choices of temperature yield values of w z a little lower than the true average and values of con a little higher than the true average. The differences are small, but the amount of vapor generated in the region behind the orifice is somewhat overestimated by this procedure. O n the other hand, the qualities thus defined are associated with readily measured local temperatures and are more immediately useful than the true average ones when velocity data are not available. PRESSURE DROPACROSS ORIFICES, LIQUIDFLOW. A typical comparison of the pressure drops across the metering orifice and test orifice is shown in Figure 2. The data for entirely liquid flow confirm that the slope is constant and in accordance with Equation 15. T h e values of the discharge coefficient, K , can be calculated directly and the experimental results are compared with ASME predictions in Table V. Both the plates and flanges are standard, commercial products in every case. The experimental coefficients agree closely with the standard values except for the orifice with a diameter ratio of 0.253. In other respects, this orifice behaves satisfactorily, so it is included without further distinction.
Table V.
tt--Do----Fi I
I
I
I
I
I
I
I
I
l
I
I
I
I
l
I
I
I
Discharge Coefficients
(Flange-tap orifices in Z1/1-inch, Schedule 40, steel pipe at N R =~ ~
I
50,000)
I
Diam. Ratio,
Figure 1. Typical temperature profiles at various distances behind orifice NReL
= w1
346
I&EC
= 178,000
0.304
=0
PROCESS DESIGN A N D DEVELOPMENT
P
K.4S11E
0.253 0.304 0.405 0.608
0.598 0.601 0.610 0.656
KEXPTL 0.629 0.599 0.610 0,663
200
I
I
l
l
l
1
1
I
A- p = 0 . 2 5 3
*-HIGH PRESSURE
U-p=0.304 o-p=0.405
-p
=0.608
-Y, = 0.94
Fz: a
- 0.01(;J 2
t 0.2 0.!! "I L
Y
W
Iv)
W
I& I
>-
I 50'
'
I
I
'
I
I
I
'
I
I
2,O 30 40 50 60 CM. O F Hg METERING ORIFICE (/3=0.405) Figure 2. Comparison of pressure drops across test orifice and metering orifice for liquid flow and zero quality at upstream tap
1
l'ol.O
3.0
5.0
7.0
9.0
VI/VlL
Figure 3.
Expansion factor for flange-tap orifice w1
>0
wi = 0
PRESSURE DROPACROSS ORIFICES, w 1 = 0. Figure 2 also contains typical data for the generation of vapor with pure liquid a t the upstream tap. I n agreement with other investigators (4, 8, 9 ) it is found there is little effect of partial vaporization on the measured pressure drop. T h e small changes can be correlated in terms of the expansion factor Yo,defined in Equation 20. I t turns out that Yo can be represented very satisfactorily as a linear function of the diameter ratio, pnamely, Yo= 0.968 0.112p (25)
+
T h e average values of the experimental expansion factor a t each diameter ratio are compared with Equation 25 in Table VI. The agreement is excellent a t all four diameter ratios. PRESSURE DROP IN ORIFICES, w 1 >O. T h e temperature profiles show that the temperature To, ahead of the metering orifice and the temperature To ahead of the test orifice are uniform across the pipe. T h e quality w 1 a t the upstream tap of the test orifice can therefore be calculated immediately by means of Equation 23. Once the pressure drop is measured, the expansion factor Y1 can be obtained from Equation 21. T h e expansion factor is found to be dependent mainly on the condition of the upstream fluid and can be correlated as a
Table VI.
Expansion Factor, Y$, for Partial Vaporization with Zero Quality at Upstream Tap
Diam. Ratio,
P 0.253 0.304 0.405 0,608
YO _________ Eq. 25 Exptl. 0.996 0.997 1.002 1.003 1.013 1.011 1.036 1.035
function of the specific volume ratio, ( u ~ / u ~ ~ u)1 .is the actual specific volume a t the upstream tap of the test orifice and u l L is the specific volume of pure liquid a t the upstream temperature, To. As shown in Figure 3, the expansion factors can be represented adequately by the empirical equation
The data of Figure 3 are taken a t temperatures between 254' and 287' F. and the pipe Reynolds number AVRe,L based on liquid properties a t the upstream tap ranges from 159,000 to 406,000. I t is reasonable to expect some effects of temperature and diameter ratio on the expansion factor, but the data imply that they are of secondary importance relative to the influence of the specific volume ratio. The expansion factors of Figure 3 correspond to experimental values of ( u l / u l L ) between 1.5 and 8 or, in other words, to upstream qualities amounting to no more than a few weight per cent vapor. T h e expansion factors are greater than unity, even though values less than unity are ordinarily obtained a t the other end of the composition range where the mixture is mainly vapor. Since vapor-in-liquid flow is far removed from mist flow in the hydrodynamic sense, the difference in expansion factors is to be expected, and the results of the present work are strongly supported by the data of Benjamin and Miller (4). STATICPRESSURE RECOVERY BEHIND ORIFICES, LIQUID FLOW. Typical data o n pressure recovery are shown in Figure 4. For purely liquid flow, the same recoveries are obtained a t 7.78 (not shown on the graph) and a t 15.90 pipe diameters from the orifice plate. The recovery is therefore fully established a t 15.90 diameters and the experimental data can be compared with the prediction of Equation 13. By virtue of Table 111, the fractional pressure recovery is not much inVOL. 5
NO. 3
JULY 1966
347
2'01 I80
b
AA
A 0
A
I
A
A
I
I I
AA
1 I I
0I O '/
e
A
I
3?.[1
I
'
3I 6I OF ORIFICE AP
'
I
9
Figure 4. Typical recovery of static pressure behind orifice with and without generation of vapor
4 = 0.253
liquid flow, as might be expected. T h e recovery falls very sharply with increased vapor, however, and the behavior shown in Figure 4 is typical of any case in which the presence of vapor can be detected with a reasonable degree of certainty by the technique of temperature measurement. Although the data on the graph d o not go beyond 15.9 pipe diameters from the orifice, measurements have been obtained a t points as far as 47.0 diameters downstream from the plate. The maximum recovery never exceeds 30y0 of the corresponding value for purely liquid flow. Vapor formation increases the axial distance needed for completion of the recovery, even though the total recovery amounts to only a few per cent. When the fluid is entirely liquid, the recovery profile is essentially completed within 8 pipe diameters. O n the other hand, when vaporization occurs, something between 20 and 50 pipe diameters is needed. T h e present data indicate that for practical purposes a length of 25 pipe diameters can be assumed satisfactorily. The fractional recovery appears to increase with Reynolds number a t a fixed specific volume ratio, u,/u1 (where n is the plane of observation), and to decrease with increased u,/uI. I t is small in any case and the differences are not ordinarily important. I t is possible to obtain negative fractional recoveries if the vaporization causes the vena contracta to move away from the downstream tap of the orifice and such behavior is observed in some of the experimental data. AREAOF DOWNSTREAM JET. Once pressure recoveries are measured, Equation 13 affords a means of estimating the area of the jet a t the vena contracta. Equation 11, based on the energy equation, has already been used to predict the area ratios, A2/Ao, shown in Table 111. I t is of interest to compare these results with the predictions of Equation 13, which is based on the momentum equation. When the fluid is entirely liquid and all of the cu-factors are taken to be unity, Equation 13 becomes
fluenced by the choice of velocity factors. If all of the CYfactors are taken to be unity, adequate results are obtained and calculations are simplified. T h e ASME report (7) indicates that the fractional pressure recovery can be approximated by the value of p2 for p
