Poceedings20th NZ Geothemal Wokshop 1998 FLUID FLOW A FLASHING CYCLONE SEPARATOR Institute of Fundamental SciencesMathematics,Massey Univesity, Palmeston Noth,NZ SUMMARY In geothemal steamgatheingsystems fo electic powe geneation, ae used to povide a tubine woking fluid which is dy o slightly supeheated steam (wate vapou). This pape descibes the steam flow pattens and pessue distibution within a typical cyclone sepaato which is opeating at satuation (boiling o flashing) conditions. The mass and volume factions of the gas (steam) and liquid phases which must be sepaated ae fo a typical twophase flow example. The steam faction occupies most of the sepaato vessel volume; analysis of the motion of this steam phase allows depiction of the steam sufaces within the flow and an estimate of the pessue dop acoss the sepaato. Solutionsof a set of simplified consevation equations ae found by analytical and numeical methods. 1. INTRODUCTION Cyclone sepaatos ae used widely in geothemal technology to sepaatethe liquid and gas phases in twophase steams fom wellboes. While the main component in the flows is wate, thee may also be dissolved components such as chloides and cabonates as well as noncondensible gases such as cabon dioxide and hydogen sulphide. Tubine designs ae usually based on a dy o slightly supeheated fluid at inlet; the liquid must be emoved fom flows when the themodynamic conditions at the wellhead ae such that the flowing enthalpy is less than that of the gas phase. In this pape, some of the consideations which undelie the themodynamicdesign of a sepaation system ae outlined. These ae illustated by a typical example. The fluid dynamical equations which esult fiom physical consevation laws fo the flow within a sepaato ae set out; some simple analytic and numeical solutions ae found. Table 1: Themodynamic popeties of wate at p = 11 ba abs, T = = 184 "C. sp. volume, density, p sp. enthalpy, h dyn. viscosity, viscosity, v I I 0.001138 0.1772 878 5.64 781 2781 0.17e6 I I I 2.68e6 2.1 Mass, enegy and volume fluxes The flowing quality, o dyness, X, is the steam (gas phase) mass flow faction of the total mass flow M of the twophase fluid mixtue. The sepaatephase mass fluxes ae then given by (1 X)M 2. A TYPICAL EXAMPLE Mg XM A pipeline caies a twophase wate flow with a total mass flux of M = At the wellhead, the flowing mixtue specific enthalpy is measued to be The flow passes into a sepaato which has an inlet pessue of 10 ba gauge, o 11 ba abs. The coesponding sepaato inlet tempeatue is 11) = 184 "C. Data available fom steam tables give the following themodynamic popeties fo the liquid and gas (vapou, steam) phases of wate at the specified inlet conditions: The espective enthalpy flows: Hg Mghg with the total enegy flux H ae measued by the 221
The specific flowing enthalpy of the twophase mixtue = is expessed in tems of the espective liquid and gasphase values and as + Xhg Afte eaangement and substitution of the paamete values given in Table 1, the flowing quality of the example mixtue at the sepaato inlet is found to be: This means that the gas phase of the twophase flow caies 36 % of the mass,while 64 of the mass is caied in the liquid phase. Fo a total mass flow of M = the steam mass flowate is = 36 The steam flux (elative to a datum of 0 "C) is then given by With a themaltoelecticenegy convesion facto of 15 %, the equivalent electical enegy flux of the steam flow is theefoe 15 in this case. The liquid enegy flux (elativeto datum 0 "C)is The volume fluxes coesponding to each of the phase steams ae Vg with total volume flux In summay, while the liquid phase caies 64 % of the mass, it caies 33 of the themal enegy (elative to datum 0 and only about 1% of the volume. The volume flux of the steam, 99 of the total, dominates the flow in the sepaato. 2.2 Void faction in the pipeline Although not necessay fo the sepaato calculations which follow, it is inteesting to conside the twophase flow geomety in the inlet pipeline. Fo ou example, we use a pipeline of adius = 25 cm = 0.25 m,aea = 0.20 Commonly, the liquid and gas phases move in a pipeline in an annula flow egime, with the gas phase flowing in the cental egion suoundedby the sloweflowing liquid phase nea the pipe wall. The atio k = v of the gas speed to that of the liquid is slip atio. [If the fluid moves in a mist flow, the slip atio k = In ou example, the supeficial liquid and gas velocities ae, espectively, = = 0.35 and = = 32 Fom a map fo hoizontal et al., 1974) the egime is annula. The void faction a is the faction of the pipe cosssectional aea which is occupied by the gas phase. The sepaate phase mass fluxes ae then given by = XM = Elimination of M (and fom these equations gives an expession fo the void faction a in tems of the flowing quality X, the phase densities and the slip atio k, as follows: The aveage specific volume flux of the twophase flow, = is then = (1 + xvg The total volume flux fo ou example is, fom Equation (4) and Table 1, Using a coelation due to Haison (1975) fo the slip atio in annula flow, the expession in Equation (4) may be witten in the fom: + M = (100 = 6.45 The volumetic (the faction of the total volume flux due to the gas phase) is Fo ou example, using X = 0.36 and values fom Table 1, = 0.989. Fo ou example, with X = 0.36 and paametes as in Table 1, the slip atio is k = 10.3 and the void faction is a 0.89 o about 89 So, while the liquid phase caies 64 % of the mass and 1 of the volume flux, it occupies of the cosssectional flow aea, flowing in an annulus which occupies the outside 5 of the adius of the pipeline. 222
2.3 Flow in the pipeline The Reynolds numbe fo the inlet flow is This means that inetial effects dominate ove viscous effects, but that the flow is tubulent. The viscous tems in the NavieStokes equations may be neglected, but the tubulent Reynolds stesses may be impotant. 2.4 Flow in the sepaato Within the sepaato itself, diffeent length and velocity scales petain. dimensions fo a cyclone sepaato as used at Waiakei Geothemal Field ae shown in Figue 1. The oute case has a adius of about = 1m, while that of the steam outlet pipe is about = 0.3 m. The height of the steam flow egion is in the ange 4 6 m. The fluid entes tangentially fom the side; the liquid wate is flung to the oute suface by centifugal foce. The gas phase (steam) spials inwads and upwads to the top of the outlet tube; meanwhile, the liquid falls unde gavity to the bottom of the vessel, fom whee it is piped fo disposal o fo futhe flashing and sepaation at a lowe pessue. 2.5 Fluid esidence times With the assumption that thee is little phase change within the sepaato, the mass fluxes of the two phases emain nealy constant duing thei passage though the vessel. Assuming also that the density of the gas phase does not change too much, the volume flux of the steam is = X M 6.4 The volume of the steamfilled egion in the sepaatois, typically So the mean esidence time of the steam in the sepaato vessel is appoximately 2 seconds. Similaly, the volume flux of the liquid wate can be calculated to be about 0.073 and it occupies a volume of about 1.5 in the vessel fo a typical esidence time of 20 seconds. Thee is theefoe a contast in the esidence times of the two phases as they flow though the sepaato. 3. EQUATIONS OF MOTION Attention hee is focussed on the steam flow afte the liquid phase is emoved by centifugal action, In to take into account tubulent vaiations in the velocity, it is witten v = v + whee is the fluctuation to the mean flow Similaly the pessue is witten p = + The aveage values, indicated by an ae taken ove a long peiod of time: 1 and 22 7 whee the time scale is lage. Note that = 0. In the model of the steam flow pesented hee, it is assumed that: outlet liquid wate outlet Fig. 1: Typical layout and dimensions of a cyclone sepaato. [Dimensions: mm] the mean flow is steady 0); thee is cylindical symmety in cylindical coodinates (, 8, z) with z vetically upwads, the mean fluid velocity vecto = is of the functional fom z), while the mean pessue may be expessed = z); the inlet and outlet flows ae unifomly distibuted aound 0 the system is isothemal and the fluid is baotopic e. = The NavieStokes equations, which expess consevation of mass and momentum fo the flow in cylindical coodinates, with the above notation and assumptions ae (see, fo example, Hughes Gaylod, 1964): 223
Consevation of mass: a + = 0 Consevation of momentum: With negligible viscous dissipation, consevative body foces (gavity) and unifom density, assumption of a unifom velocity distibution at inlet implies that V x v = 0 fo flow within the sepaato. Then v whee the potential is a hamonic function (V v = 0 = 0)of the fom = B, with paametes A and B detemined by the bounday conditions. The adial and tangential velocity components ae then g + (12) whee g is the gavitational acceleation and the tems ae deived fom the components of the Reynolds stess tenso and ae given in tems of the components of as follows: whee A = and B = and whee the tangential inlet speed = at = is detemined by the cosssectional aea of the inlet pipe. The expessions (16) fo the velocity components satisfy Equations (9) and (11) identically. The components of the pessue gadient can be deduced (10)and (12)to be a la a = ) % a a = The tubulent stess tems may be evaluated in diffeent ways. The analysis of tubulent fluid flows is not an easy matte. Vaious coelations between the tubulent stess components and the mean velocity field have been poposed, but the closue of the poblem with appopiate bounday conditions emains difficult. 4. FLOW SOLUTIONS Simplifying assumptions may be made in ode to find appoximate solutions to the equations. To find whethe density vaiation with pessue is significant, it is fist assumed that the flow is in the (, diections only and the density is unifom; the coesponding pessue distibution is then calculated. 4.1 Radial flow appoximation = = mass flow fom oute adius to a line sink at inne adius distibuted ove height L; tubulent stesses 2 which give a Benoulli equation: + P = Po The pessue diffeence Ap between the oute adius = and the inne adius = at any elevation z is Fo the typical flow paametes given fo the example above, A = 32 and B = 0.25 With 5.6 = 1 m and = 0.3 m, the pessue diffeence Ap between the oute and inne adii of the cicula vessel is about 0.3 ba. The coesponding steam density vaiation though such a flow whee density vaiation due to pessue vaiation is taken account of, is then appoximately 0.15 a 3 change in value. This indicates that easonable esults should be pedicted by a model with unifom steam density. Note also that the pessue will dop adially inwads though the vessel due to the cyclonic flow and the centipetal acceleation induced by the otation. This will tend to "dy" the gas by effective supeheating of the steam as its pessue dops below the satuation value fo the tempeatue. 224
4.2 A Simplified 2D Model Chaacteistics of the mode1: Steady flow of a fluid of unifom density with otational symmety = 0) and with negligible viscous effects; neglect tubulent stesses; both the inlet and outlet flows ae unifomly distibuted thoughout 0 12n and fo a z c b and d L espectively; neglect the liquid wate flow; assume unifom flow in inlet piping (iotational flow). The fluid flows between the adii = and = and the uppe and lowe sufaces =0 and z =L. The bounday conditions, tanslated fom the flow conditions at inlet and outlet and that of zeo nomal flow at solid walls, ae, assuming that speeds ae unifom with z at the inflow and outflow sections: + [ + COS while the velocity components ae given by n=l Bo+ = and = Oczla n=l whee and ae the Bessel functions of the Fist and Second Kinds, of ode zeo and one espectively. The constant coefficients and (n = 0, 1,..., and ae evaluated by using the bounday conditions. A steamfunction fo the flow may also be defined, as follows. Reaangement of Equation (9) gives = on = 0, L: = 0. The steam flow, by Kelvin's theoem, emains iotational within the sepaato, x v = 0 and the flow velocity can then be expessed as the gadient of a scala potential function v = The equation fo consevation of mass fo the steady flow of a fluid of unifom density educes to v = 0. Substitution of v = then gives = 0, and is a hamonic function. In cylindical coodinates, otational symmety all functionally dependent on (, z) only, while satisfies the equation Afte witing and = whee = z), to identically satisfy Equation it takes only a small amount of calculation to show that n=l The constant coefficients ae the same as occu in the velocity potential A efeence value =0 is taken on z = 0. 225
As a vaiable which may be used to descibe the whole flow patten, the steamfunction satisfiesthe equation with the following conditions: SUMMARY A model fo flow of the steam phase in a flashing cyclone sepaato has been solved to give the flow and pessue distibution pattens. The calculated oveall pessue dop acoss the vessel agees well with design values. Futhe wok to investigate the flow and effects of the sepaatedwate is planned. ba Oczla aczcb ACKNOWLEDGEMENT I am gateful to Tom King fom Downe Enegy Sevices Limited, Waiakei, fo poviding infomation on typical sepaato vessel dimensions, and to Kevin Kooey fo his comments. REFERENCES on on z =0: on z = L: Haison, R.F (1975) Methods fo the analysis of geothemal twophase flow. ME thesis, Univesity of Auckland. Hughes, W.F. and E.W.(1964) Basic Equations of Engineeing Science. S chaum s Outline Seies, McGawHill. y=o Manhane, J.M., Gegoy, G. and Aziz, K. (1974) A flow patten map fo gasliquid flow in hoizontal pipes. Int. J. Multiphase Flow 1, 537554. Note that values of the steamfunction ae popotional to = the volume flux of the vapou. 4.3 A numeical solution Figue 2 shows a cosssectional view of a set of steam and isobaic sufaces found fom the numeical solution of Equation (21) using the bounday conditions set out above. The solution was confinned by the analytic fom given in The velocity components wee infeed fom (19) and the pessue distibution fom a Benoulli equation. The shape of the steam sufacesdepends only on vessel aspect atio and the inlet and outlet configuations; hee, the dimensions of the steam flow egion ae L = 4 m, = 1 m, = 0.3 m, while the lowe and uppe boundaies of the inlet and outlet ae given by a = 0.9 m, b = 1.10 m and d = 3.80 m; the vetical heights of the inlet and outlet ae equal, in this case, with b a = L d = 0.20 The calculated pessue within the vessel anges fiom the inlet value of 11.0 ba abs to the outlet value of 10.71 ba abs, a total dop of 0.29 ba. This is close to the estimate found fom the simple appoximation calculated in the fist pat of above, and also nea to the design pessue dop used commecially (Kevin Kooey, pesonal communication). The isobaic sufaces (sufaces of equal pessue) ae nealy vetical cylindes, which indicates the stong contol of the pessue by the cylindical motion. 3.5 2.5 1.5 0.5 0 0 0.5 1 0.5 1 Fig. 2: Cosssectional view of steam (left) and isobaic (ight) fo of a gas phase in a cyclone sepaato vessel. [Dimensions: m] 226