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A critical appraisal of some aspects of the analysis of the wet steam nozzle as used in total flow machines
0375 - 6505/85 $3.00 + 0.00 Pergamon Press Ltd. © 1985 CNR.
Geothermics, Vol. 14, No. 2/3, pp. 435 -447, 1985. Printed in Great Britain.
A CRITICAL APPRAISAL OF SOME ASPECTS OF THE ANALYSIS OF THE WET STEAM NOZZLE AS USED IN TOTAL FLOW MACHINES D. J. R Y L E Y University of Liverpool, P.O. Box 147, Liverpool L69 3BX, U.K. (Transmitted by the Government of the United Kingdom)
R.65 effort has been devoted to the design of the wet steam nozzle for total flow machines. This paper reviews critically some of the issues which arise in the analysis of the two-phase flow. Included are: mixing and atomization at nozzle entry, modelling the structure of a two-phase flow, sedimentation of droplets, the fracture of droplets in flight and thermodynamic phase equilibrium during expansion. It is suggested that future research should be directed towards increasing the size dispersion of the entry liquid and doing so with the minimum expenditure of energy. Abstract--Much
NOMENCLATURE ~t dynamic viscosity As specificinterfacial surface area density A projected area for drag force P 13 interfacial surface tension Co drag coefficient nozzle efficiency D distance between adjacent droplets qN d = 2r droplet diameter Reynolds number Re E energy We Weber number g gravitational constant Subscripts K slip factor 1, 2 initial, final p absolute pressure crit critical R lattice ratio f liquid phase S surface area g vapour phase s vertical distance, entropy mF mass flow basis T absolute temperature ml mass instantaneous basis tf droplet descent time s slip V velocity T terminal x dryness fraction INTRODUCTION G e o f l u i d delivered at the head o f a g e o t h e r m a l well t a p p i n g a n a q u i f e r will have a dryness f r a c t i o n d e p e n d i n g p r i n c i p a l l y u p o n the d o w n h o l e t e m p e r a t u r e at the well base. G e o t h e r m a l resources suitable for power p r o d u c t i o n require a m i n i m u m d o w n h o l e t e m p e r a t u r e o f a p p r o x i m a t e l y 100°C b u t d o w n h o l e t e m p e r a t u r e s m a y rise as high as - 3 5 0 ° C . F o r plants e m p l o y i n g the flash cycle, single or d o u b l e , which is the usual case, the mass o f v a p o u r available at the t u r b i n e stop valve will d e p e n d u p o n the t e m p e r a t u r e difference between that d o w n h o l e a n d that at the stop valve. F o r saturated water at 100°C, flashing to 0.5 b a r will yield a m i x t u r e o f q u a l i t y o f 0.03 only. F o r water at 200°C, flashing to 1.0 b a r will give a quality of 0.19. F o r a low salinity b r i n e these values w o u l d n o t differ significantly. I n b o t h cases the lowpressure s a t u r a t e d water is rejected, the fractions being respectively 0.97 a n d 0.81, a severe waste o f energy, n o t w i t h s t a n d i n g the recovery o f some energy in the long term if reinjection is employed. T h e total flow c o n c e p t a t t e m p t s to exploit the liquid heat in the live geofluid by e x p a n d i n g the two phases together as a wet mixture. T h e p r i m a r y c o n v e r s i o n o f e n t h a l p y to kinetic energy occurs within this type o f nozzle whether it is used in one o f the p r o p o s e d H e r o - t y p e reaction 435
D..I. R vksv
436
turbines (Austin and House, 1978), in an axial impulse turbine (Austin and Lundberg, 1978) or in a rotary separator (Cerini, 1978). Unlike the dry gas nozzle which is time-honoured and well understood, the wet steam nozzle presents many difficult problems which arise from the mutual interaction of the vapour and the dispersed liquid phase. A comprehensive analysis of the coupled phenomena occurring during expansion has been made by Elliott and Weinberg (1968) and used by other workers. The purpose of the present paper is to look critically at their model, to examine in detail some of the assumptions made and hopefully to identify the limitations of these assumptions and to clarify some of the basic problems encountered. M I X I N G A N D A T O M I Z A T I O N AT N O Z Z L E ENTRY It is customary to atomize the liquid fraction at the entry to or within the nozzle. Elliott and Weinberg (1968) employed 550 0.25 inch (6.35 mm) injector tubes each having three orifices which projected the liquid into the vapour at right angles to its flow. They did not state in detail the reason for atomization. Austin and his collaborators (Austin et al., 1977) atomized the liquid for the specific purpose of providing as nearly as possible a homogeneous flow within the specially designed impulse wheel served by the nozzle, thus avoiding centrifuging in the blade passages. Deposited water promotes losses in the passages due to increased friction. They estimated that a high degree of atomization (hopefully below d = 2 ~tm) was essential for homogeneous flow. They succeeded in obtaining a size-frequency distribution dominated by a large number of droplets less than d = 1 ~m but constituting only 20% of the mass of the total droplet population. The energy E required for isothermal atomization is the product of the interfacial surface tension, •, and the aggregate area of new surface created. For unit mass of liquid atomized to a monodispersion of droplets of diameter d, this is: E = A~ =
6o
d91
(1)
where A~ =
6 - is the specific surface. dpf
(2)
This quantity is very small. Thus to atomize 1 kg of bulk water at 60°C to a monodispersion with d = 50 ~m theoretically requires 8.4 J. The efficiency of a spinning disc atomizer a t o m i z i n g water to a s u r f a c e - v o l u m e d i a m e t e r (Sauter) of 2 0 - 1 2 0 ~tm was f o u n d experimentally by Ryley (1959) to be 0.08 - 0 . 2 0 % . At 0.1% atomizer efficiency this represents 8.3 k J / k g of input energy for atomization. Some other types of atomizer (Ryley, 1953) were also found to have very low efficiencies. It would be difficult to obtain estimates of the efficiency of geofluid nozzle entry atomization. The energy in the aggregate generated interfacial area could be assessed from the droplet diameter/frequency distribution but the energy withdrawn from the mixture to accomplish atomization and the resulting energy balance in the working fluid as a whole, would be difficult to measure. The atomizing efficiency is likely to be low. The fundamental problem with all atomizing devices is that the structural strength of drops rises with size reduction due to the increasing dominance of surface tension and it may be impossible with simple devices to concentrate the applied energy sufficiently to promote very highly dispersed liquid subdivision. The problem is intensified if the liquid masses are entrained. M O D E L L I N G T H E S T R U C T U R E OF A T W O - P H A S E FLOW As discussed above, it is usual to pass geofluid liquid/vapour mixture through devices at entry to the nozzle to secure dispersal and entrainment of the liquid phase. The flow is then
Analysis of Wet Steam Nozzle
437
modelled as a two-phase mixture in which the vapour is the continuous phase and the liquid is the dispersed phase. To make the nozzle flow amenable to analysis the liquid is assumed to be monodisperse in droplet size. No cognizance is taken of the space dispersion of the droplets. While not required for analysis it is convenient and instructive to model this space dispersion. The most useful model is to consider the droplets to be located at the intersection points on a rhombohedral lattice, all points being occupied. The dryness fraction may be expressed, for the case where slip is present, either as a mass flow dryness fraction, Xmv, or as an instantaneous mass dryness fraction, Xmt (Ryley, 1964), the former defined on the basis of the mixture traversing a stationary defined section in the flow and the latter defined on the basis of the mixture located instantaneously between two such defined sections. It is readily shown that 1 Xm~ = (3) 1 +
1
K
I
For a given pressure and slip ratio K = V~/V~, the average distance D between a droplet and its nearest neighbours may be characterized by the ratio, R, where = R=
d
E
(1 - xmv)Pg
-6
d3
117
(4)
13
where
1~3
is the slip correction factor. For a homogeneous flow K = 1,
unity and Xmv = Xm~. If slip Occurs but no other factor is changed, K > 1,
reduces to
< I and R is
diminished, i.e. the droplet volumetric population density increases to maintain Xmv constant as
required by mass conservat'on ,t is found that ,n most cases the term 0
6 4
andcan
be rejected. R then becomes independent of the droplet size and hence for all values of d, all other conditions being the same, the lattices are geometrically similar. Curves of R versus xmv are plotted from equation (4) and shown in Fig. 1 for the range of pressures likely to occur in geothermal work. The limit of validity for the model occurs when R = 1, corresponding to the closest possible three-dimensional packing of equal-sized spheres. In this case D = d, the spheres touch and the remaining void fraction is 0.26, corresponding to a low value of Xmv. Scale drawings are shown in Fig. 2 of droplet spacings in a rhombohedral plane for a range o f p and of XmF. Considering the third case in group (b), the introduction only of slip, K = Vg/Vf, would diminish the droplet velocity Vf, thus increasing the population density to preserve Xmv; D would then fall and R also. For K = 5, say, the slip correction factor has a value of - 0 . 6 and the modified lattice, with tighter packing, is shown. It is conceivable that if XmV is lOW at entry and the pressure p is high, a rapid expansion rate in a nozzle convergence might reduce R to unity causing a formal breakdown in the model and indicating the geometric limit of liquid-dispersed flow. It is evident from Fig. 2 that for low values of xmv and high values of p the high population density diminishes faith in the accuracy of this (or any) model to support analyses developed from them, especially for large droplets with their enhanced tendency to coalesce, to sediment and to flash explosively.
D. J. Ryley
438
Slip Correction ,1. . '2. .'3 . L. . 5 Lattice Ratio
Factor [~1~
6
7
8
1j t Slip Rotio, K
Slip1
Correction Factor
R=S/d
'9
I Pressure, p Bar . 1
'iN 7
6 5 4 3 5
2
N o 10 Slip
]
20
Limit of Volidity for Mode[
01 0
0'-1
0'.2
XmF
Fig. 1. Mass flow dryness fraction. Curves drawn for K = 1 h o m o g e n e o u s flow (lattice ratio on r h o m b o h e d r a l lattice).
-¢X O
~---~-~ -~ ¢~ 0-
o¢ O-
(ol
;
7---
/--
7-' Y-7-"
(b) Pressure
1 Bar
5 Bar (O) XmF=005
20 Bar (b) XmF = 0 2 0
Fig. 2. Droplet spacing in h o m o g e n e o u s flo,,s, and in slip flow.
Analysis o f Wet Steam Nozzle
439
S E D I M E N T A T I O N OF D R O P L E T S In the two-phase nozzle the intention is to keep the droplets entrained. Any droplet when deposited on the nozzle boundary loses its identity and converts its aggregate energies to friction reheat. Liquid, once deposited, forms films which will flow downstream but which tend to hug the wetted surface whatever its orientation. Re-entrainment does not readily occur except from pendant sheets and drops which can form at exposed edges. In the long c o n v e r g e n t - divergent nozzles, used for expanding two-phase mixtures of low initial quality, considerable deposition is to be expected. Its extent will depend, for given imposed terminal conditions, on the orientation of the nozzle axis and will be greatest for nozzles with a horizontal axis and least for those with a vertical axis. Sedimentation will be influenced by the following: (1) nature of the vapour flow, whether laminar, intermediate or turbulent, (2) extent of the thermal disequilibrium between the phases, (3) droplet size, (4) liquid and vapour densities, (5) droplet collision resulting in coalescence, fracture or bouncing, (6) dense droplet population effects, (7) aerodynamic fracture of droplets and (8) minor effects; Magnus effect, droplet oscillation, etc. Item (1) has been extensively studied. Item (2) promotes phase change and will be considered below. Items (5) and (7) have an extensive literature which may be unfamiliar to some readers and a number of references are included (Ryley and Wood, 1965- 1966; Ryley and BennettCowell, 1967; Ryley et al., 1970; Owe Berg et al., 1963; Brazier-Smith et al., 1972; Jaratne and Mason, 1964; Hanson et al., 1963; Taylor, 1949; Littaye, 1943; Hinze, 1949; Simpkins and Bales, 1972; Harper et al., 1972; Morrell, 1961). Clearly any analysis based on modelling which attempts to simulate all the coupled effects which might arise in the above list becomes unacceptably complex and would not be attempted. Nevertheless it is of value to have some " f e e l " for comparative rates of sedimentation in given circumstances. Such an estimate can be obtained as follows. The minimum time for a selected droplet to complete a free fall through the surrounding vapour can be found by assuming that the droplets fall through the available vertical distance s in a stagnant atmosphere having the appropriate conditions. The fall may be assumed to be unimpeded and the velocity everywhere equal to the terminal velocity Vx for the droplet's locality. If the surrounding vapour pressure does not change during the descent, the duration of the fall may be characterized by tf = s/VT. If the pressure does change the fall may be subdivided into finite length intervals and the corresponding time intervals summed. Droplets falling from rest in a stagnant atmosphere will accelerate within one or more of the regimes; laminar fall, intermediate fall and turbulent fall before attaining Vr in one of these regimes. These regimes are each characterized by a range of values for the terminal Reynolds number Rex defined as Rex -
VTdOf
(5)
Ixg
and given in column 2 of Table 1. All droplets of interest have diameters larger than those at which a Cunningham correction should be introduced. At terminal velocity gravitational and aerodynamic drag forces are equated: A
Co Pg VT~ =
nd 3 6 (p~ - pg)g
(6)
440
D. J. Ryley Table 1. Terminal Velocities 1
2
3
4
5
Fall regime
Rer
CD
F[
Factor B at termination of regimes
Laminar
0- 2
24
d~(pf-
VT
Rex
pg)g
3.67 (Rex = 2)
18p.g
(Stokes law) Intermediate
Turbulent
2-500
0.4 +
5 0 0 - 2 x 10'
40 ...... Rer
0.4VT +
40ttg
4 d (Ofpg)g " 3 pg
dog
0.44
d (Pl -
VT =
pg)g
= 0
6 . 2 4 (Rex = 2) 9174 (Re T = 500)
V2
0.44pg
8410 (Rex = 500)
Using the values of CD = CD(Rer) given in column 3, expressions for Vx may be obtained as given in column 4 (Table 1). The droplet diameter for the upper and lower terminal values of CD are given by d3 = B
pg(p;-- Pg)
where B is a constant, appropriate to the case, given in column 5 (Table 1) and its multiplier is a property constant for a given pressure. The use of simple functions Co -- CD(R~) in column 3 for the drag coefficient of a rigid sphere (Dalla Valle, 1948) confers sufficient accuracy for, and expedites calculations of, the present type. There is, however, a functional mis-match at the laminar/intermediate boundary as can be seen from the termina! velocity curves (Fig. 3).
v!
10-2,--
,
,
,
.....
i
~-
,
, w-~7---
,
i
i
/J ~
r , , , r,
,
r r ....
Bar
m/s 5
jz5 j
1
~
~"
10
VT
VT
m/s
m/s
j--
jl0 Bar
/
/ / /
J30 10-3
10-~
Turbulent
Laminar
o4
i
10-z
t/3
8
8 bq
10 2
10 -1
4
J
1 IO-S]~
1
I
5
10
. . . . . . . . . .
50
I
100
. . . . . . . . . . .
500
Fig. 3. T e r m i n a l velocities.
,
,
1000
d
~ . . . .
5000 ~.m
1
10-3
10 -2
Analysis o f Wet Steam Nozzle
441
The terminal velocities of all droplets below, say, d = 50 txm are small for any likely nozzle working pressure and the deposition rates will probably be acceptable. Larger drops, however, may sediment out of the flow and only a small number of these is needed to deposit a significant quantity of liquid. T H E FRACTURE OF D R O P L E T S IN F L I G H T There are three agencies that may promote the fracture of droplets in flight: (1) collision and coalescence followed by re-breakage, (2) distortion of shape, beyond recovery, by aerodynamic pressure forces and (3) explosive flashing arising from retarded cooling of the droplet core during cooling of the surrounding vapour.
(1) Coil&ion Limited experimental data is available for droplets in the diameter range 3 0 0 - 1 3 0 0 ~tm (Ryley and Wood, 1965 - 1966; Ryley and Bennett-Cowell, 1967; Ryley et al., 1970) but in the two-phase nozzle only very few droplets are likely to escape early fracture and remain in this size range. The author knows of no experimental data on smaller droplets but it seems likely that those for which d < 100 Ilm, say, are immune from fracture resulting from collision. (2) Aerodynamic fracture The critical Weber number W e , i, is usually employed to define the critical slip velocity Vs at which a droplet, diameter d , it, is fragmented by aerodynamic forces. Thus Wecrit -
Pg Vs2 dcrit 13
(8)
for which Recrit -
Pg Vs ~crit
(9)
~g
Wecr~t is a ratio: aerodynamic pressure forces/surface tensile forces and must be regarded as a
coarse criterion. A value of Wecrit is selected and equation (8) gives the diameter of the largest droplet that can survive in the prevailing flow conditions. The value chosen for Wecr~t is usually empirical and is more a "situational" value than one precisely related to a defined set of flow conditions. The criteria for droplet shattering are much more complex than equation (8) would suggest; for example, it has been shown (Hanson et al., 1963) that We,~t increases with and Recrit decreases with decreasing d for a given liquid, all other factors being equal. Wecrit is found experimentally and is related to the local geometry and the rapidity of exposure of the droplet at risk. The rapidity of exposure is difficult to quantify but must be related to d Vs/dt. Wecrit is also related to the length of time the droplet is exposed to intense slip, during which time it may flatten and form a film " b a g " supported at the opening by a toroid-type liquid ring. Even at this juncture if the slip velocity subsides gently the droplet may recover the spherical shape and survive. Elliott and Weinberg (1968) chose for their analytical procedure a value of Wecrit = 6 as obtained by Hanson et al. (1963) and Morrel (1961). Subsequent measurements by Elliott (1982b), on a two-phase steam water turbine, interpreted in the light of work by Alger (1978), suggested that the predicted drop size was too large by a factor of approximately two. The view is widely held that the droplet survival period when distorted is comparable with the periodic time (P.T.) of oscillation when spherical and is given by P.T.-
n 4
] g
pfd 3 o
(10)
D. J. Ryley
442
The time to complete the disruptive process has been studied by, among others, Taylor (1949), Littaye (1943) and Hinze (1949). More recently the response of a droplet to acceleration has been studied by Simpkins and Bales (1972) and its break-up process by Harper et al. (1972). (3) Flashing and explosive flashing* "Flashing" has been studied both by G y a r m a t h y (1962) and by Crowe and C o m f o r t (1978). Both assumed that heat transfer from core to surface takes place entirely by thermal conduction and this is probably correct. G y a r m a t h y (1962) assumed that a temperature excess of 5°C by the core over the interface was necessary to initiate explosive flashing but does not state the source of this information. Crowe and C o m f o r t (1978) recognise the retarding effect of the excess capillary pressure within the droplet but impose no arbitrary temperature interval. Wood (1960) pictures a " f l a s h i n g " droplet as subject to a short time delay before being " t o r n to pieces by internal explosion", a viewpoint widely held. Release of the vapour does not require fragmentation, it could occur by a perforation to the surface and might do so if the external pressure falls slowly. The author has seen no published experimental work on droplet flashing and such a study, albeit difficult, is overdue. For modelling, in order to make calculations manageable, it is convenient to assume that a droplet undergoing mechanical fracture divides into two droplets of equal size and hence at all locations within the flow the population is size-monodisperse. This is unlikely to be the case for the following reasons. If a single droplet is caused to divide into two droplets, radii rj and r2, where r~ = nr2, it is readily shown that the ratio: final surface area/original surface area is $2 (1 + n 2 ) t 1 )2/3 S~ = 1 + n3
(11)
Finding d(S2/Sl)/dn and equating to zero shows that $2/S~ is a m a x i m u m when n = 1, i.e. when r~ = r2 and the original droplet is divided into two equal droplets. The function [equation (11)] is shown in Fig. 4. A droplet breaking into two droplets will tend to break such that the final aggregate surface free energy is minimized so far as is permitted by the constraints attending the breakage. Hence breakage is more likely to occur in the interval AB than in interval BC. The principle of minimum creation of new surface applies however many fragments result from a single original liquid mass and is also operative when the disrupting forces arise from internal boiling. THERMODYNAMIC PHASE EQUILIBRIUM DURING TWO-PHASE EXPANSION~ Consider a low quality two-phase fluid entering a nozzle and suppose that the vapour phase comprises one system and the liquid phase another system. Suppose the systems to be independent and each to act as if the other were absent. Each system is thus confined within a conceptual system boundary. This model and the discussion to follow are unchanged in principle if the liquid is dispersed. The rate of expansion is neglected for the present and explosive flashing is assumed to be absent to simplify the discussion. Two extreme cases may be envisaged for the expansion: (a) expansion with both phases everywhere in equilibrium and (b) expansion with both phases everywhere in full disequilibrium. *There is some confusion in the literature about the term "flashing". Here, a distinction is made between "flashing" which is quiescent boiling at the external liquid/vapour interface and "explosive flashing" which is cavitation boiling within the liquid phase. tThis subject will be treated in detail in an analysis to be published by the author and 1. Owen.
443
Analysis of Wet Steam Nozzle 52
S~
1.26 1'2/.
1.22
C 52=(1+n2)(..1-'-~- )2/3 $1 1 + n~//"
/
1.20
oJ
1.18 o < U30
iT _
L~
1.16 1.1/. 1.12
B
"6 .~,
1 "10 1"08
c~
1.05 1.0/. 1.02 1"00 0
0.2
0.1.
0"6
0-8
10
r2 Radius Ratio r~Fig. 4. Division o f a d r o p l e t into two droplets.
(a) Full equilibrium The saturated vapour will condense continuously, yielding saturated liquid and saturated residual vapour at all points in the expansion. If the two operations proceed together such that both systems are always at the same temperature, the liquid fractions from the two sources need not be separately identified, nor need the two vapour fractions. Calculations treating the original liquid and the original vapour systems separately yield results identical with those obtained by treating the original mixture as homogeneous; the normal method of calculation. (b) Full disequilibrium The saturated vapour will expand wholly supersaturated until nucleation, which will occur if the expansion ratio exceeds - 2 . The saturated liquid will undergo no phase change and become increasingly superheated as expansion proceeds. If this case were to occur the kinetic energy delivered by the nozzle would arise only from the small vapour fraction undergoing an expansion disturbed in its flow by the necessity of following the interstices between the droplets. The effective nozzle efficiency would be very small. The original liquid would be conveyed as a "passenger". (c) Case intermediaie between Cases (a) and (b) We now partly relax the extreme conditions of Case (b) in respect of the original liquid phase and permit some flashing to take place from the drops. We now have an enthalpy drop component from the expansion of the system originally liquid which supplements the enthalpy drop of the original vapour and improves the nozzle efficiency. If now, in addition, we partly relax for Case (b) the extreme conditions previously imposed on the saturated vapour we shall have expansion with some relief of the undercooling. For given initial conditions and terminal pressures the difference in isentropic enthalpy drop between full equilibrium and full supersaturation for a vapour originally dry saturated is not great and only a small change in nozzle efficiency attends the relief of undercooling.
D. J. Ryley
444
(d) Representation on the T - S diagram Some of the above ideas are illustrated in Fig. 5 in which the phases are treated as independent systems. Thus, for equilibrium expansion from p, to p2 isentropic expansion of the liquid is defined by L f, and polytropic expansion by I, f > For a large droplet in which the core cools more slowly than the interface, less vapour is formed and the notional state point path is I, f2. The droplet core temperature exceeds the saturation temperature for the prevailing pressure p2 and the core state must lie on the liquid superheat line, say C,. Temperatures of shells of increasing radius define points along C,12 progressively approaching 12. Full equilibrium is never attained during the expansion process, but an expansion nearly in equilibrium has the vapour condition g, and the liquid condition C.~. Fully supersaturated expansion of the above would terminate at g2.
/
Pl
O. D
c2#'~ I2 -f2-f'
f3
g~O g 03
g # S Fig. 5. Thermal cquilibrium in two phase expansion.
(e) Further considerations Much is known about the expansion of a vapour originally dry saturated. Expansion will take place with almost full supersaturation until the onset of nucleation, which event is governed principally by the expansion rate parameter P =
-
i
dp
p
dt
(Gyarmathy, 1962). Nucleation
generates fog droplets, of critical size defined by the K e l v i n - H e r m h o l t z equation, the rapid growth of which affords almost Complete relief to the supersaturation, thus restoring the fluid to a condition of near-equilibrium. Further expansion is largely relieved by additional condensation on the now-available interfacial surfaces but in some cases the supersaturation may rise again sufficiently to promote a second nucleation. If some saturated liquid had been introduced into the original vapour the sequence of events just described would be nearly unaltered, as the dispersion of the associated liquid would provide insufficient condensation surface to assist significantly in the condensation process. This is because there is no known mechanical or aerodynamic method to subdivide bulk liquid into sub-micron sized droplets. We now consider the expansion of the fraction comprising the original liquid. Whatever may be its degree of dispersion vapour will be flashed from the surface, in quiescent fashion if the
Analysis of Wet Steam Nozzle
445
liquid is only slightly superheated, explosively if the superheat is large. In the nozzle used for a total flow machine the ingoing liquid fraction will be atomized so far as is practicable and the larger drops may suffer aerodynamic fracture in the early part of the nozzle. The rate of production of flashed vapour will depend upon the degree to which the liquid is dispersed, i.e. its mass is subdivided.
(f) Droplet heat and mass transfer Drops which form part of the original entry liquid and those nucleating from the expanding vapour behave in different ways. A drop originating from the liquid evaporates yielding flashed vapour which is transferred away from the drop, conveying with it saturated vapour enthalpy thus transporting heat to the surrounding vapour. This heat serves to depress the extent of the subcooling. If transfer is sufficiently rapid, i.e. if vapour is generated from a large aggregate area of interface, the supersaturation may never rise to the value required to promote nucleation. It follows that poor atomization of entry liquid a n d / o r inadequate aerodynamic fracture may deprive the expansion of relief for the rising supersaturation in addition to curtailing the enthalpy drop. The latter effect appears to be the more conducive to lowering the nozzle efficiency. (g) Rates of expansion and relaxation At any location within the nozzle there will be a large population of size-polydispersed droplets, each size with its distinctive thermal relaxation time. The overall status of the thermal equilibrium at that location will depend among other factors upon the ratio of the integrated mean effective thermal relaxation time to the transit time through the nozzle. (h) General It is well known that a nozzle expanding a mixture of high initial dryness fraction maintains a high efficiency despite the presence of significant undercooling. This has led to the widespread assumption that thermal disequilibrium can be neglected. Whilst this assumption is acceptable for high initial qualities it need not be so for low values of initial dryness fraction where the relaxation time for flashing from a large mass fraction of liquid may be significant. However, the few available published tests (Cerini, 1978; Alger, 1975; Elliott, 1982a) do in fact show quite high efficiencies for such cases. This suggests that the larger drops are fractured by explosive flashing and aerodynamic shattering, thus increasing the available heat/mass transfer surface and suppressing thermal disequilibrium. Agitation within the vapour, known to encourage nucleation, may also contribute to reversion. CONCLUSIONS The aim of the designer of nozzles for expanding low quality two-phase mixtures is to secure the highest possible nozzle efficiency tiN, i.e. to maximise the conversion of thermal energy at the nozzle entrance to kinetic energy at the nozzle exit by the suppression of boundary and stream frictional effects. Published experimental data on VlN appear to be few and are summarized in Table 2. In all cases the values predicted from the theory show good agreement. This is remarkable because it implies that the physical models used represent with significant accuracy the situation they purport to model. C o m m o n sense seems to reject this inasmuch that, even if it were possible to control the entry geofluid to provide monodispersion in droplet size and uniformity in space dispersion, rapid expansion early in the flow would be expected to re-impose chaos. Explosive flashing, coalescence, areodynamic fracture and sedimentation are all likely to concur if the droplets are relatively large. The agreement between predictions of tin and
446
D. J. R y / e y
Table 2. Efficiencies of two-phase nozzles (experimental) Terminal conditions Ref. Cerini (1978)
Inlet 5.5 bar, x~ = 5%
Exit 1 bar
Alger (1975)
24.1 bar, x, = 12.9%
0.3 bar
Elliott (1982)
29.3 bar, x, - 3%
0.46 bar
qN (%) 50-70 81
Notes qN = 8 0 - 9 0 % predicted for inlet conditions 13.8 27.6 bar
90 82
experimental confirmation sugggests that the droplet fracture both from aerodynamic shattering and explosive flashing dominates the flow process, reducing the statistical diameter o f t h e d r o p l e t p o p u l a t i o n . W e r e t h i s n o t so, d e p a r t u r e f r o m t h e r m a l e q u i l i b r i u m w o u l d o c c u r t o r e d u c e s i g n i f i c a n t l y t h e e x p e r i m e n t a l v a l u e o f rlN b e l o w its p r e d i c t e d v a l u e . T h i s o c c u r r e n c e m a y f u r t h e r e x p l a i n t h e 30°70 d i s c r e p a n c y t o w h i c h E l l i o t t a n d W e i n b e r g (1968) r e f e r in t h e pentultimate paragraph of their paper and which they correctly attribute to poor spatial d i s t r i b u t i o n o f l i q u i d a t inlet. T h e i d e a s i n t h i s p a p e r f u l l y s u p p o r t t h e a c c e p t e d v i e w t h a t t h e d r o p l e t size t o w h i c h t h e l i q u i d c a n b e a t o m i z e d is o f p a r a m o u n t i m p o r t a n c e . T h e m a i n t h r u s t o f f u t u r e r e s e a r c h o n t h e l o w quality two-phase nozzle should be towards increasing the dispersion of the entry liquid and d o i n g so w i t h t h e m i n i m u m e x p e n d i t u r e o f e n e r g y .
REFERENCES Alger, T. W. (1975) The performance of two-phase nozzles for total flow geothermal impulse turbines. Report UCRL-76417, Lawrence Livermore Laboratory, Livermore, CA 94550, U.S.A. Alger, T. W. (1978) Droplet phase characteristics in liquid-dominated steam-water nozzle flow. Report UCRL-52534, Lawrence Livermore Laboratory, Livermore, CA 94550, U.S.A. Austin, A. L. and House, P. A. (1978) New concepts for converting the energy in low- to medium-temperature liquids, with emphasis on geothermal applications. Report UCRL-52583, Lawrence Livermore Laboratory, Livermore, CA 94550, U.S.A. Austin, A. L. and Lundberg, A. W. (1978) A status report on the development of the total flow concept. Report UCRL-50046-77, Lawrence Livermore Laboratory, Livermore, CA 94550, U.S.A. Austin, A. L., Lundberg, A. W., Owen, L. B. and Tardiff, G. E. (1977) The LLL geothermal energy status report, January 1976-January 1977. Report UCRL-50046-76, Lawrence Livermore Laboratory, Livermore, CA 94550, U.S.A. Brazier-Smith, P. R., Jennings, S. G. and Latham, J. (1972) The interaction of falling water drops. Proc. R. Soc. A326, 393 408. Cerini, D. J. (1978) Demonstration of a rotary separator for two-phase brine and steam flows. U.S. Department of Energy Report T10-28519. Crowe, C. T. and Comfort III, W. J. (1978) Atomisation mechanisms in single-component, two-phase nozzle flows. First Int. Conf. on Liquid Atomisation and Spray Systems, Fuel Society of Japan, Tokyo. Dalla Valle, J. M. (1948) Micromeritics, p. 23. Pitman, New York. Elliott, D. G. (1982a) Tests of a two-stage, axial flow, two-phase turbine. U.S. Department of Energy, Report DOE/ER/10614-2. Elliott, D. G. (1982b)Theory and tests on two-phase turbines. U.S. Department of Energy, Report DOE/ER-10614-1. Elliott, D. G. and Weinberg, E. (1968) Acceleration of liquids in two-phase nozzles. NASA Technical Report 32-987. Prepared by Jet Propulsion Laboratory, California Inst. Technology, Pasadena, California, U.S.A. Gyarmathy, G. (1962) Grundlagen einer Theorie der Nassdampfturbine. Dissertation, ETH Zurich, Juris Verlag, 1962. [English translations: C.E.G.B. (London) Report, T-781 (1963) and USAF-FTO (Dayton, Ohio) Report TT-63-785.] Hanson, A. R., Domich, E. G. and Adams, H. S. (1963) Shock tube investigation of the break-up of drops by air blasts. Physics, Fluids 6, 1070- 1080. Harper, E. Y., Gruber, G. W. and I-Dee Chang (1972) On the break-up of accelerating liquid drops. J. FluM. Mech. 32, 565-591. Hinze, J. O. (1949) Critical speeds and sizes of liquid globules. AppL Sci. Res. A.I., No. 4, 273 288. Jaratne, O. W. and Mason, B. J. (1964) The coalescence and bouncing of water drops at an air/water interface. Proc. R. Soc. 280, 545-565.
Analysis of Wet Steam Nozzle
447
Littaye, G. (1943) Sur l'atomisation d'un jet liquide. C. R. held. S~anc. Acad. Sci. Paris 217, 340. Morrell, G. (1961) Critical conditions for drop and jet shattering. NASA TN D-677, National Aeronautics and Space Administration, Washington. Owe Berg, T. G., Fernish, G. C. and Gaukler, T. A. (1963) The mechanism of coalescence of liquid drops. J. atmos. Sci. 20, 153- 158. Ryley, D. J. (1959) Experimental determination of the atomising efficiency of a high speed spinning disk atomizer. Brit. J. appl. Phys. 10, 9 3 - 97. Ryley, D. J. (1964) Property definition in equilibrium wet steam. Int. J. mech. Sci. 6, 445-454. Ryley, D. J. and Bennett-Cowell, B. N. (1967)The collision behaviour of steam-borne water droplets. Int. J. mech. Sci. 9, 817-833. Ryley, D. J., Ralph, W. J. and Tubman, K. A. (1970) The collision behaviour of water drops within a low-pressure steam atmosphere. Int. J. mech. Sci. 12, 589-596. Ryley, D. J. and Wood, M. R. (1965- 1966) The collision in free flight of water droplets in atmospheres of air and steam. Proc. lnst. Mech. Engrs. 180, 2, 30. Simpkins, P. G. and Bales, E. L. (1972) Water drop response to sudden acceleration. J. Fluid Mech. 55, 629-639. Taylor, G. I. (1949) The shape and acceleration of a drop in a high-speed air stream. U.K. Ministry of Supply, Paper AC.10647/Phys. C64. Wood, B. (1960) Wetness in steam cycles. Proc. Inst.Mech.Engrs 174,491 -511.
Geothermics, Vol. 14, No. 2/3, pp. 435 -447, 1985. Printed in Great Britain.
A CRITICAL APPRAISAL OF SOME ASPECTS OF THE ANALYSIS OF THE WET STEAM NOZZLE AS USED IN TOTAL FLOW MACHINES D. J. R Y L E Y University of Liverpool, P.O. Box 147, Liverpool L69 3BX, U.K. (Transmitted by the Government of the United Kingdom)
R.65 effort has been devoted to the design of the wet steam nozzle for total flow machines. This paper reviews critically some of the issues which arise in the analysis of the two-phase flow. Included are: mixing and atomization at nozzle entry, modelling the structure of a two-phase flow, sedimentation of droplets, the fracture of droplets in flight and thermodynamic phase equilibrium during expansion. It is suggested that future research should be directed towards increasing the size dispersion of the entry liquid and doing so with the minimum expenditure of energy. Abstract--Much
NOMENCLATURE ~t dynamic viscosity As specificinterfacial surface area density A projected area for drag force P 13 interfacial surface tension Co drag coefficient nozzle efficiency D distance between adjacent droplets qN d = 2r droplet diameter Reynolds number Re E energy We Weber number g gravitational constant Subscripts K slip factor 1, 2 initial, final p absolute pressure crit critical R lattice ratio f liquid phase S surface area g vapour phase s vertical distance, entropy mF mass flow basis T absolute temperature ml mass instantaneous basis tf droplet descent time s slip V velocity T terminal x dryness fraction INTRODUCTION G e o f l u i d delivered at the head o f a g e o t h e r m a l well t a p p i n g a n a q u i f e r will have a dryness f r a c t i o n d e p e n d i n g p r i n c i p a l l y u p o n the d o w n h o l e t e m p e r a t u r e at the well base. G e o t h e r m a l resources suitable for power p r o d u c t i o n require a m i n i m u m d o w n h o l e t e m p e r a t u r e o f a p p r o x i m a t e l y 100°C b u t d o w n h o l e t e m p e r a t u r e s m a y rise as high as - 3 5 0 ° C . F o r plants e m p l o y i n g the flash cycle, single or d o u b l e , which is the usual case, the mass o f v a p o u r available at the t u r b i n e stop valve will d e p e n d u p o n the t e m p e r a t u r e difference between that d o w n h o l e a n d that at the stop valve. F o r saturated water at 100°C, flashing to 0.5 b a r will yield a m i x t u r e o f q u a l i t y o f 0.03 only. F o r water at 200°C, flashing to 1.0 b a r will give a quality of 0.19. F o r a low salinity b r i n e these values w o u l d n o t differ significantly. I n b o t h cases the lowpressure s a t u r a t e d water is rejected, the fractions being respectively 0.97 a n d 0.81, a severe waste o f energy, n o t w i t h s t a n d i n g the recovery o f some energy in the long term if reinjection is employed. T h e total flow c o n c e p t a t t e m p t s to exploit the liquid heat in the live geofluid by e x p a n d i n g the two phases together as a wet mixture. T h e p r i m a r y c o n v e r s i o n o f e n t h a l p y to kinetic energy occurs within this type o f nozzle whether it is used in one o f the p r o p o s e d H e r o - t y p e reaction 435
D..I. R vksv
436
turbines (Austin and House, 1978), in an axial impulse turbine (Austin and Lundberg, 1978) or in a rotary separator (Cerini, 1978). Unlike the dry gas nozzle which is time-honoured and well understood, the wet steam nozzle presents many difficult problems which arise from the mutual interaction of the vapour and the dispersed liquid phase. A comprehensive analysis of the coupled phenomena occurring during expansion has been made by Elliott and Weinberg (1968) and used by other workers. The purpose of the present paper is to look critically at their model, to examine in detail some of the assumptions made and hopefully to identify the limitations of these assumptions and to clarify some of the basic problems encountered. M I X I N G A N D A T O M I Z A T I O N AT N O Z Z L E ENTRY It is customary to atomize the liquid fraction at the entry to or within the nozzle. Elliott and Weinberg (1968) employed 550 0.25 inch (6.35 mm) injector tubes each having three orifices which projected the liquid into the vapour at right angles to its flow. They did not state in detail the reason for atomization. Austin and his collaborators (Austin et al., 1977) atomized the liquid for the specific purpose of providing as nearly as possible a homogeneous flow within the specially designed impulse wheel served by the nozzle, thus avoiding centrifuging in the blade passages. Deposited water promotes losses in the passages due to increased friction. They estimated that a high degree of atomization (hopefully below d = 2 ~tm) was essential for homogeneous flow. They succeeded in obtaining a size-frequency distribution dominated by a large number of droplets less than d = 1 ~m but constituting only 20% of the mass of the total droplet population. The energy E required for isothermal atomization is the product of the interfacial surface tension, •, and the aggregate area of new surface created. For unit mass of liquid atomized to a monodispersion of droplets of diameter d, this is: E = A~ =
6o
d91
(1)
where A~ =
6 - is the specific surface. dpf
(2)
This quantity is very small. Thus to atomize 1 kg of bulk water at 60°C to a monodispersion with d = 50 ~m theoretically requires 8.4 J. The efficiency of a spinning disc atomizer a t o m i z i n g water to a s u r f a c e - v o l u m e d i a m e t e r (Sauter) of 2 0 - 1 2 0 ~tm was f o u n d experimentally by Ryley (1959) to be 0.08 - 0 . 2 0 % . At 0.1% atomizer efficiency this represents 8.3 k J / k g of input energy for atomization. Some other types of atomizer (Ryley, 1953) were also found to have very low efficiencies. It would be difficult to obtain estimates of the efficiency of geofluid nozzle entry atomization. The energy in the aggregate generated interfacial area could be assessed from the droplet diameter/frequency distribution but the energy withdrawn from the mixture to accomplish atomization and the resulting energy balance in the working fluid as a whole, would be difficult to measure. The atomizing efficiency is likely to be low. The fundamental problem with all atomizing devices is that the structural strength of drops rises with size reduction due to the increasing dominance of surface tension and it may be impossible with simple devices to concentrate the applied energy sufficiently to promote very highly dispersed liquid subdivision. The problem is intensified if the liquid masses are entrained. M O D E L L I N G T H E S T R U C T U R E OF A T W O - P H A S E FLOW As discussed above, it is usual to pass geofluid liquid/vapour mixture through devices at entry to the nozzle to secure dispersal and entrainment of the liquid phase. The flow is then
Analysis of Wet Steam Nozzle
437
modelled as a two-phase mixture in which the vapour is the continuous phase and the liquid is the dispersed phase. To make the nozzle flow amenable to analysis the liquid is assumed to be monodisperse in droplet size. No cognizance is taken of the space dispersion of the droplets. While not required for analysis it is convenient and instructive to model this space dispersion. The most useful model is to consider the droplets to be located at the intersection points on a rhombohedral lattice, all points being occupied. The dryness fraction may be expressed, for the case where slip is present, either as a mass flow dryness fraction, Xmv, or as an instantaneous mass dryness fraction, Xmt (Ryley, 1964), the former defined on the basis of the mixture traversing a stationary defined section in the flow and the latter defined on the basis of the mixture located instantaneously between two such defined sections. It is readily shown that 1 Xm~ = (3) 1 +
1
K
I
For a given pressure and slip ratio K = V~/V~, the average distance D between a droplet and its nearest neighbours may be characterized by the ratio, R, where = R=
d
E
(1 - xmv)Pg
-6
d3
117
(4)
13
where
1~3
is the slip correction factor. For a homogeneous flow K = 1,
unity and Xmv = Xm~. If slip Occurs but no other factor is changed, K > 1,
reduces to
< I and R is
diminished, i.e. the droplet volumetric population density increases to maintain Xmv constant as
required by mass conservat'on ,t is found that ,n most cases the term 0
6 4
andcan
be rejected. R then becomes independent of the droplet size and hence for all values of d, all other conditions being the same, the lattices are geometrically similar. Curves of R versus xmv are plotted from equation (4) and shown in Fig. 1 for the range of pressures likely to occur in geothermal work. The limit of validity for the model occurs when R = 1, corresponding to the closest possible three-dimensional packing of equal-sized spheres. In this case D = d, the spheres touch and the remaining void fraction is 0.26, corresponding to a low value of Xmv. Scale drawings are shown in Fig. 2 of droplet spacings in a rhombohedral plane for a range o f p and of XmF. Considering the third case in group (b), the introduction only of slip, K = Vg/Vf, would diminish the droplet velocity Vf, thus increasing the population density to preserve Xmv; D would then fall and R also. For K = 5, say, the slip correction factor has a value of - 0 . 6 and the modified lattice, with tighter packing, is shown. It is conceivable that if XmV is lOW at entry and the pressure p is high, a rapid expansion rate in a nozzle convergence might reduce R to unity causing a formal breakdown in the model and indicating the geometric limit of liquid-dispersed flow. It is evident from Fig. 2 that for low values of xmv and high values of p the high population density diminishes faith in the accuracy of this (or any) model to support analyses developed from them, especially for large droplets with their enhanced tendency to coalesce, to sediment and to flash explosively.
D. J. Ryley
438
Slip Correction ,1. . '2. .'3 . L. . 5 Lattice Ratio
Factor [~1~
6
7
8
1j t Slip Rotio, K
Slip1
Correction Factor
R=S/d
'9
I Pressure, p Bar . 1
'iN 7
6 5 4 3 5
2
N o 10 Slip
]
20
Limit of Volidity for Mode[
01 0
0'-1
0'.2
XmF
Fig. 1. Mass flow dryness fraction. Curves drawn for K = 1 h o m o g e n e o u s flow (lattice ratio on r h o m b o h e d r a l lattice).
-¢X O
~---~-~ -~ ¢~ 0-
o¢ O-
(ol
;
7---
/--
7-' Y-7-"
(b) Pressure
1 Bar
5 Bar (O) XmF=005
20 Bar (b) XmF = 0 2 0
Fig. 2. Droplet spacing in h o m o g e n e o u s flo,,s, and in slip flow.
Analysis o f Wet Steam Nozzle
439
S E D I M E N T A T I O N OF D R O P L E T S In the two-phase nozzle the intention is to keep the droplets entrained. Any droplet when deposited on the nozzle boundary loses its identity and converts its aggregate energies to friction reheat. Liquid, once deposited, forms films which will flow downstream but which tend to hug the wetted surface whatever its orientation. Re-entrainment does not readily occur except from pendant sheets and drops which can form at exposed edges. In the long c o n v e r g e n t - divergent nozzles, used for expanding two-phase mixtures of low initial quality, considerable deposition is to be expected. Its extent will depend, for given imposed terminal conditions, on the orientation of the nozzle axis and will be greatest for nozzles with a horizontal axis and least for those with a vertical axis. Sedimentation will be influenced by the following: (1) nature of the vapour flow, whether laminar, intermediate or turbulent, (2) extent of the thermal disequilibrium between the phases, (3) droplet size, (4) liquid and vapour densities, (5) droplet collision resulting in coalescence, fracture or bouncing, (6) dense droplet population effects, (7) aerodynamic fracture of droplets and (8) minor effects; Magnus effect, droplet oscillation, etc. Item (1) has been extensively studied. Item (2) promotes phase change and will be considered below. Items (5) and (7) have an extensive literature which may be unfamiliar to some readers and a number of references are included (Ryley and Wood, 1965- 1966; Ryley and BennettCowell, 1967; Ryley et al., 1970; Owe Berg et al., 1963; Brazier-Smith et al., 1972; Jaratne and Mason, 1964; Hanson et al., 1963; Taylor, 1949; Littaye, 1943; Hinze, 1949; Simpkins and Bales, 1972; Harper et al., 1972; Morrell, 1961). Clearly any analysis based on modelling which attempts to simulate all the coupled effects which might arise in the above list becomes unacceptably complex and would not be attempted. Nevertheless it is of value to have some " f e e l " for comparative rates of sedimentation in given circumstances. Such an estimate can be obtained as follows. The minimum time for a selected droplet to complete a free fall through the surrounding vapour can be found by assuming that the droplets fall through the available vertical distance s in a stagnant atmosphere having the appropriate conditions. The fall may be assumed to be unimpeded and the velocity everywhere equal to the terminal velocity Vx for the droplet's locality. If the surrounding vapour pressure does not change during the descent, the duration of the fall may be characterized by tf = s/VT. If the pressure does change the fall may be subdivided into finite length intervals and the corresponding time intervals summed. Droplets falling from rest in a stagnant atmosphere will accelerate within one or more of the regimes; laminar fall, intermediate fall and turbulent fall before attaining Vr in one of these regimes. These regimes are each characterized by a range of values for the terminal Reynolds number Rex defined as Rex -
VTdOf
(5)
Ixg
and given in column 2 of Table 1. All droplets of interest have diameters larger than those at which a Cunningham correction should be introduced. At terminal velocity gravitational and aerodynamic drag forces are equated: A
Co Pg VT~ =
nd 3 6 (p~ - pg)g
(6)
440
D. J. Ryley Table 1. Terminal Velocities 1
2
3
4
5
Fall regime
Rer
CD
F[
Factor B at termination of regimes
Laminar
0- 2
24
d~(pf-
VT
Rex
pg)g
3.67 (Rex = 2)
18p.g
(Stokes law) Intermediate
Turbulent
2-500
0.4 +
5 0 0 - 2 x 10'
40 ...... Rer
0.4VT +
40ttg
4 d (Ofpg)g " 3 pg
dog
0.44
d (Pl -
VT =
pg)g
= 0
6 . 2 4 (Rex = 2) 9174 (Re T = 500)
V2
0.44pg
8410 (Rex = 500)
Using the values of CD = CD(Rer) given in column 3, expressions for Vx may be obtained as given in column 4 (Table 1). The droplet diameter for the upper and lower terminal values of CD are given by d3 = B
pg(p;-- Pg)
where B is a constant, appropriate to the case, given in column 5 (Table 1) and its multiplier is a property constant for a given pressure. The use of simple functions Co -- CD(R~) in column 3 for the drag coefficient of a rigid sphere (Dalla Valle, 1948) confers sufficient accuracy for, and expedites calculations of, the present type. There is, however, a functional mis-match at the laminar/intermediate boundary as can be seen from the termina! velocity curves (Fig. 3).
v!
10-2,--
,
,
,
.....
i
~-
,
, w-~7---
,
i
i
/J ~
r , , , r,
,
r r ....
Bar
m/s 5
jz5 j
1
~
~"
10
VT
VT
m/s
m/s
j--
jl0 Bar
/
/ / /
J30 10-3
10-~
Turbulent
Laminar
o4
i
10-z
t/3
8
8 bq
10 2
10 -1
4
J
1 IO-S]~
1
I
5
10
. . . . . . . . . .
50
I
100
. . . . . . . . . . .
500
Fig. 3. T e r m i n a l velocities.
,
,
1000
d
~ . . . .
5000 ~.m
1
10-3
10 -2
Analysis o f Wet Steam Nozzle
441
The terminal velocities of all droplets below, say, d = 50 txm are small for any likely nozzle working pressure and the deposition rates will probably be acceptable. Larger drops, however, may sediment out of the flow and only a small number of these is needed to deposit a significant quantity of liquid. T H E FRACTURE OF D R O P L E T S IN F L I G H T There are three agencies that may promote the fracture of droplets in flight: (1) collision and coalescence followed by re-breakage, (2) distortion of shape, beyond recovery, by aerodynamic pressure forces and (3) explosive flashing arising from retarded cooling of the droplet core during cooling of the surrounding vapour.
(1) Coil&ion Limited experimental data is available for droplets in the diameter range 3 0 0 - 1 3 0 0 ~tm (Ryley and Wood, 1965 - 1966; Ryley and Bennett-Cowell, 1967; Ryley et al., 1970) but in the two-phase nozzle only very few droplets are likely to escape early fracture and remain in this size range. The author knows of no experimental data on smaller droplets but it seems likely that those for which d < 100 Ilm, say, are immune from fracture resulting from collision. (2) Aerodynamic fracture The critical Weber number W e , i, is usually employed to define the critical slip velocity Vs at which a droplet, diameter d , it, is fragmented by aerodynamic forces. Thus Wecrit -
Pg Vs2 dcrit 13
(8)
for which Recrit -
Pg Vs ~crit
(9)
~g
Wecr~t is a ratio: aerodynamic pressure forces/surface tensile forces and must be regarded as a
coarse criterion. A value of Wecrit is selected and equation (8) gives the diameter of the largest droplet that can survive in the prevailing flow conditions. The value chosen for Wecr~t is usually empirical and is more a "situational" value than one precisely related to a defined set of flow conditions. The criteria for droplet shattering are much more complex than equation (8) would suggest; for example, it has been shown (Hanson et al., 1963) that We,~t increases with and Recrit decreases with decreasing d for a given liquid, all other factors being equal. Wecrit is found experimentally and is related to the local geometry and the rapidity of exposure of the droplet at risk. The rapidity of exposure is difficult to quantify but must be related to d Vs/dt. Wecrit is also related to the length of time the droplet is exposed to intense slip, during which time it may flatten and form a film " b a g " supported at the opening by a toroid-type liquid ring. Even at this juncture if the slip velocity subsides gently the droplet may recover the spherical shape and survive. Elliott and Weinberg (1968) chose for their analytical procedure a value of Wecrit = 6 as obtained by Hanson et al. (1963) and Morrel (1961). Subsequent measurements by Elliott (1982b), on a two-phase steam water turbine, interpreted in the light of work by Alger (1978), suggested that the predicted drop size was too large by a factor of approximately two. The view is widely held that the droplet survival period when distorted is comparable with the periodic time (P.T.) of oscillation when spherical and is given by P.T.-
n 4
] g
pfd 3 o
(10)
D. J. Ryley
442
The time to complete the disruptive process has been studied by, among others, Taylor (1949), Littaye (1943) and Hinze (1949). More recently the response of a droplet to acceleration has been studied by Simpkins and Bales (1972) and its break-up process by Harper et al. (1972). (3) Flashing and explosive flashing* "Flashing" has been studied both by G y a r m a t h y (1962) and by Crowe and C o m f o r t (1978). Both assumed that heat transfer from core to surface takes place entirely by thermal conduction and this is probably correct. G y a r m a t h y (1962) assumed that a temperature excess of 5°C by the core over the interface was necessary to initiate explosive flashing but does not state the source of this information. Crowe and C o m f o r t (1978) recognise the retarding effect of the excess capillary pressure within the droplet but impose no arbitrary temperature interval. Wood (1960) pictures a " f l a s h i n g " droplet as subject to a short time delay before being " t o r n to pieces by internal explosion", a viewpoint widely held. Release of the vapour does not require fragmentation, it could occur by a perforation to the surface and might do so if the external pressure falls slowly. The author has seen no published experimental work on droplet flashing and such a study, albeit difficult, is overdue. For modelling, in order to make calculations manageable, it is convenient to assume that a droplet undergoing mechanical fracture divides into two droplets of equal size and hence at all locations within the flow the population is size-monodisperse. This is unlikely to be the case for the following reasons. If a single droplet is caused to divide into two droplets, radii rj and r2, where r~ = nr2, it is readily shown that the ratio: final surface area/original surface area is $2 (1 + n 2 ) t 1 )2/3 S~ = 1 + n3
(11)
Finding d(S2/Sl)/dn and equating to zero shows that $2/S~ is a m a x i m u m when n = 1, i.e. when r~ = r2 and the original droplet is divided into two equal droplets. The function [equation (11)] is shown in Fig. 4. A droplet breaking into two droplets will tend to break such that the final aggregate surface free energy is minimized so far as is permitted by the constraints attending the breakage. Hence breakage is more likely to occur in the interval AB than in interval BC. The principle of minimum creation of new surface applies however many fragments result from a single original liquid mass and is also operative when the disrupting forces arise from internal boiling. THERMODYNAMIC PHASE EQUILIBRIUM DURING TWO-PHASE EXPANSION~ Consider a low quality two-phase fluid entering a nozzle and suppose that the vapour phase comprises one system and the liquid phase another system. Suppose the systems to be independent and each to act as if the other were absent. Each system is thus confined within a conceptual system boundary. This model and the discussion to follow are unchanged in principle if the liquid is dispersed. The rate of expansion is neglected for the present and explosive flashing is assumed to be absent to simplify the discussion. Two extreme cases may be envisaged for the expansion: (a) expansion with both phases everywhere in equilibrium and (b) expansion with both phases everywhere in full disequilibrium. *There is some confusion in the literature about the term "flashing". Here, a distinction is made between "flashing" which is quiescent boiling at the external liquid/vapour interface and "explosive flashing" which is cavitation boiling within the liquid phase. tThis subject will be treated in detail in an analysis to be published by the author and 1. Owen.
443
Analysis of Wet Steam Nozzle 52
S~
1.26 1'2/.
1.22
C 52=(1+n2)(..1-'-~- )2/3 $1 1 + n~//"
/
1.20
oJ
1.18 o < U30
iT _
L~
1.16 1.1/. 1.12
B
"6 .~,
1 "10 1"08
c~
1.05 1.0/. 1.02 1"00 0
0.2
0.1.
0"6
0-8
10
r2 Radius Ratio r~Fig. 4. Division o f a d r o p l e t into two droplets.
(a) Full equilibrium The saturated vapour will condense continuously, yielding saturated liquid and saturated residual vapour at all points in the expansion. If the two operations proceed together such that both systems are always at the same temperature, the liquid fractions from the two sources need not be separately identified, nor need the two vapour fractions. Calculations treating the original liquid and the original vapour systems separately yield results identical with those obtained by treating the original mixture as homogeneous; the normal method of calculation. (b) Full disequilibrium The saturated vapour will expand wholly supersaturated until nucleation, which will occur if the expansion ratio exceeds - 2 . The saturated liquid will undergo no phase change and become increasingly superheated as expansion proceeds. If this case were to occur the kinetic energy delivered by the nozzle would arise only from the small vapour fraction undergoing an expansion disturbed in its flow by the necessity of following the interstices between the droplets. The effective nozzle efficiency would be very small. The original liquid would be conveyed as a "passenger". (c) Case intermediaie between Cases (a) and (b) We now partly relax the extreme conditions of Case (b) in respect of the original liquid phase and permit some flashing to take place from the drops. We now have an enthalpy drop component from the expansion of the system originally liquid which supplements the enthalpy drop of the original vapour and improves the nozzle efficiency. If now, in addition, we partly relax for Case (b) the extreme conditions previously imposed on the saturated vapour we shall have expansion with some relief of the undercooling. For given initial conditions and terminal pressures the difference in isentropic enthalpy drop between full equilibrium and full supersaturation for a vapour originally dry saturated is not great and only a small change in nozzle efficiency attends the relief of undercooling.
D. J. Ryley
444
(d) Representation on the T - S diagram Some of the above ideas are illustrated in Fig. 5 in which the phases are treated as independent systems. Thus, for equilibrium expansion from p, to p2 isentropic expansion of the liquid is defined by L f, and polytropic expansion by I, f > For a large droplet in which the core cools more slowly than the interface, less vapour is formed and the notional state point path is I, f2. The droplet core temperature exceeds the saturation temperature for the prevailing pressure p2 and the core state must lie on the liquid superheat line, say C,. Temperatures of shells of increasing radius define points along C,12 progressively approaching 12. Full equilibrium is never attained during the expansion process, but an expansion nearly in equilibrium has the vapour condition g, and the liquid condition C.~. Fully supersaturated expansion of the above would terminate at g2.
/
Pl
O. D
c2#'~ I2 -f2-f'
f3
g~O g 03
g # S Fig. 5. Thermal cquilibrium in two phase expansion.
(e) Further considerations Much is known about the expansion of a vapour originally dry saturated. Expansion will take place with almost full supersaturation until the onset of nucleation, which event is governed principally by the expansion rate parameter P =
-
i
dp
p
dt
(Gyarmathy, 1962). Nucleation
generates fog droplets, of critical size defined by the K e l v i n - H e r m h o l t z equation, the rapid growth of which affords almost Complete relief to the supersaturation, thus restoring the fluid to a condition of near-equilibrium. Further expansion is largely relieved by additional condensation on the now-available interfacial surfaces but in some cases the supersaturation may rise again sufficiently to promote a second nucleation. If some saturated liquid had been introduced into the original vapour the sequence of events just described would be nearly unaltered, as the dispersion of the associated liquid would provide insufficient condensation surface to assist significantly in the condensation process. This is because there is no known mechanical or aerodynamic method to subdivide bulk liquid into sub-micron sized droplets. We now consider the expansion of the fraction comprising the original liquid. Whatever may be its degree of dispersion vapour will be flashed from the surface, in quiescent fashion if the
Analysis of Wet Steam Nozzle
445
liquid is only slightly superheated, explosively if the superheat is large. In the nozzle used for a total flow machine the ingoing liquid fraction will be atomized so far as is practicable and the larger drops may suffer aerodynamic fracture in the early part of the nozzle. The rate of production of flashed vapour will depend upon the degree to which the liquid is dispersed, i.e. its mass is subdivided.
(f) Droplet heat and mass transfer Drops which form part of the original entry liquid and those nucleating from the expanding vapour behave in different ways. A drop originating from the liquid evaporates yielding flashed vapour which is transferred away from the drop, conveying with it saturated vapour enthalpy thus transporting heat to the surrounding vapour. This heat serves to depress the extent of the subcooling. If transfer is sufficiently rapid, i.e. if vapour is generated from a large aggregate area of interface, the supersaturation may never rise to the value required to promote nucleation. It follows that poor atomization of entry liquid a n d / o r inadequate aerodynamic fracture may deprive the expansion of relief for the rising supersaturation in addition to curtailing the enthalpy drop. The latter effect appears to be the more conducive to lowering the nozzle efficiency. (g) Rates of expansion and relaxation At any location within the nozzle there will be a large population of size-polydispersed droplets, each size with its distinctive thermal relaxation time. The overall status of the thermal equilibrium at that location will depend among other factors upon the ratio of the integrated mean effective thermal relaxation time to the transit time through the nozzle. (h) General It is well known that a nozzle expanding a mixture of high initial dryness fraction maintains a high efficiency despite the presence of significant undercooling. This has led to the widespread assumption that thermal disequilibrium can be neglected. Whilst this assumption is acceptable for high initial qualities it need not be so for low values of initial dryness fraction where the relaxation time for flashing from a large mass fraction of liquid may be significant. However, the few available published tests (Cerini, 1978; Alger, 1975; Elliott, 1982a) do in fact show quite high efficiencies for such cases. This suggests that the larger drops are fractured by explosive flashing and aerodynamic shattering, thus increasing the available heat/mass transfer surface and suppressing thermal disequilibrium. Agitation within the vapour, known to encourage nucleation, may also contribute to reversion. CONCLUSIONS The aim of the designer of nozzles for expanding low quality two-phase mixtures is to secure the highest possible nozzle efficiency tiN, i.e. to maximise the conversion of thermal energy at the nozzle entrance to kinetic energy at the nozzle exit by the suppression of boundary and stream frictional effects. Published experimental data on VlN appear to be few and are summarized in Table 2. In all cases the values predicted from the theory show good agreement. This is remarkable because it implies that the physical models used represent with significant accuracy the situation they purport to model. C o m m o n sense seems to reject this inasmuch that, even if it were possible to control the entry geofluid to provide monodispersion in droplet size and uniformity in space dispersion, rapid expansion early in the flow would be expected to re-impose chaos. Explosive flashing, coalescence, areodynamic fracture and sedimentation are all likely to concur if the droplets are relatively large. The agreement between predictions of tin and
446
D. J. R y / e y
Table 2. Efficiencies of two-phase nozzles (experimental) Terminal conditions Ref. Cerini (1978)
Inlet 5.5 bar, x~ = 5%
Exit 1 bar
Alger (1975)
24.1 bar, x, = 12.9%
0.3 bar
Elliott (1982)
29.3 bar, x, - 3%
0.46 bar
qN (%) 50-70 81
Notes qN = 8 0 - 9 0 % predicted for inlet conditions 13.8 27.6 bar
90 82
experimental confirmation sugggests that the droplet fracture both from aerodynamic shattering and explosive flashing dominates the flow process, reducing the statistical diameter o f t h e d r o p l e t p o p u l a t i o n . W e r e t h i s n o t so, d e p a r t u r e f r o m t h e r m a l e q u i l i b r i u m w o u l d o c c u r t o r e d u c e s i g n i f i c a n t l y t h e e x p e r i m e n t a l v a l u e o f rlN b e l o w its p r e d i c t e d v a l u e . T h i s o c c u r r e n c e m a y f u r t h e r e x p l a i n t h e 30°70 d i s c r e p a n c y t o w h i c h E l l i o t t a n d W e i n b e r g (1968) r e f e r in t h e pentultimate paragraph of their paper and which they correctly attribute to poor spatial d i s t r i b u t i o n o f l i q u i d a t inlet. T h e i d e a s i n t h i s p a p e r f u l l y s u p p o r t t h e a c c e p t e d v i e w t h a t t h e d r o p l e t size t o w h i c h t h e l i q u i d c a n b e a t o m i z e d is o f p a r a m o u n t i m p o r t a n c e . T h e m a i n t h r u s t o f f u t u r e r e s e a r c h o n t h e l o w quality two-phase nozzle should be towards increasing the dispersion of the entry liquid and d o i n g so w i t h t h e m i n i m u m e x p e n d i t u r e o f e n e r g y .
REFERENCES Alger, T. W. (1975) The performance of two-phase nozzles for total flow geothermal impulse turbines. Report UCRL-76417, Lawrence Livermore Laboratory, Livermore, CA 94550, U.S.A. Alger, T. W. (1978) Droplet phase characteristics in liquid-dominated steam-water nozzle flow. Report UCRL-52534, Lawrence Livermore Laboratory, Livermore, CA 94550, U.S.A. Austin, A. L. and House, P. A. (1978) New concepts for converting the energy in low- to medium-temperature liquids, with emphasis on geothermal applications. Report UCRL-52583, Lawrence Livermore Laboratory, Livermore, CA 94550, U.S.A. Austin, A. L. and Lundberg, A. W. (1978) A status report on the development of the total flow concept. Report UCRL-50046-77, Lawrence Livermore Laboratory, Livermore, CA 94550, U.S.A. Austin, A. L., Lundberg, A. W., Owen, L. B. and Tardiff, G. E. (1977) The LLL geothermal energy status report, January 1976-January 1977. Report UCRL-50046-76, Lawrence Livermore Laboratory, Livermore, CA 94550, U.S.A. Brazier-Smith, P. R., Jennings, S. G. and Latham, J. (1972) The interaction of falling water drops. Proc. R. Soc. A326, 393 408. Cerini, D. J. (1978) Demonstration of a rotary separator for two-phase brine and steam flows. U.S. Department of Energy Report T10-28519. Crowe, C. T. and Comfort III, W. J. (1978) Atomisation mechanisms in single-component, two-phase nozzle flows. First Int. Conf. on Liquid Atomisation and Spray Systems, Fuel Society of Japan, Tokyo. Dalla Valle, J. M. (1948) Micromeritics, p. 23. Pitman, New York. Elliott, D. G. (1982a) Tests of a two-stage, axial flow, two-phase turbine. U.S. Department of Energy, Report DOE/ER/10614-2. Elliott, D. G. (1982b)Theory and tests on two-phase turbines. U.S. Department of Energy, Report DOE/ER-10614-1. Elliott, D. G. and Weinberg, E. (1968) Acceleration of liquids in two-phase nozzles. NASA Technical Report 32-987. Prepared by Jet Propulsion Laboratory, California Inst. Technology, Pasadena, California, U.S.A. Gyarmathy, G. (1962) Grundlagen einer Theorie der Nassdampfturbine. Dissertation, ETH Zurich, Juris Verlag, 1962. [English translations: C.E.G.B. (London) Report, T-781 (1963) and USAF-FTO (Dayton, Ohio) Report TT-63-785.] Hanson, A. R., Domich, E. G. and Adams, H. S. (1963) Shock tube investigation of the break-up of drops by air blasts. Physics, Fluids 6, 1070- 1080. Harper, E. Y., Gruber, G. W. and I-Dee Chang (1972) On the break-up of accelerating liquid drops. J. FluM. Mech. 32, 565-591. Hinze, J. O. (1949) Critical speeds and sizes of liquid globules. AppL Sci. Res. A.I., No. 4, 273 288. Jaratne, O. W. and Mason, B. J. (1964) The coalescence and bouncing of water drops at an air/water interface. Proc. R. Soc. 280, 545-565.
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Littaye, G. (1943) Sur l'atomisation d'un jet liquide. C. R. held. S~anc. Acad. Sci. Paris 217, 340. Morrell, G. (1961) Critical conditions for drop and jet shattering. NASA TN D-677, National Aeronautics and Space Administration, Washington. Owe Berg, T. G., Fernish, G. C. and Gaukler, T. A. (1963) The mechanism of coalescence of liquid drops. J. atmos. Sci. 20, 153- 158. Ryley, D. J. (1959) Experimental determination of the atomising efficiency of a high speed spinning disk atomizer. Brit. J. appl. Phys. 10, 9 3 - 97. Ryley, D. J. (1964) Property definition in equilibrium wet steam. Int. J. mech. Sci. 6, 445-454. Ryley, D. J. and Bennett-Cowell, B. N. (1967)The collision behaviour of steam-borne water droplets. Int. J. mech. Sci. 9, 817-833. Ryley, D. J., Ralph, W. J. and Tubman, K. A. (1970) The collision behaviour of water drops within a low-pressure steam atmosphere. Int. J. mech. Sci. 12, 589-596. Ryley, D. J. and Wood, M. R. (1965- 1966) The collision in free flight of water droplets in atmospheres of air and steam. Proc. lnst. Mech. Engrs. 180, 2, 30. Simpkins, P. G. and Bales, E. L. (1972) Water drop response to sudden acceleration. J. Fluid Mech. 55, 629-639. Taylor, G. I. (1949) The shape and acceleration of a drop in a high-speed air stream. U.K. Ministry of Supply, Paper AC.10647/Phys. C64. Wood, B. (1960) Wetness in steam cycles. Proc. Inst.Mech.Engrs 174,491 -511.