I.CHEM.E. SYMPOSIUM SERIES NO. 110

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1 CHOKING CONDITIONS FOR FLASHING ONE-COMPONENT FLOWS IN NOZZLES AND VALVES - A SIMPLE ESTIMATION METHOD S. D. Morris* The present work addresses the problem of choked two-phase flow in nozzles and valves and describes a reasonably simple method for estimation of the pertinent flow parameters (pressure ratio and mass discharge rate) where account is taken of the main physical effects, namely thermal nonequi1ibriurn, phase slip and nozzle length. Predictions of the method are compared with a number of data sets and good accuracy is demonstrated. INTRODUCTION Although there are many methods available for calculating choked (or critical) flashing flows in pipeline restrictions such as nozzles and orifices, these methods are often iterative in nature and their use requires the aid of a computer. From the point of view of a one-off or occasional calculation therefore, the effort required to code up such a method may be difficult to justify. Furthermore, there does not seem to be a method reported in the literature that handles the main physical effects together, namely those of thermal nonequi1ibriurn, phase slip and nozzle length. There is then a need for a simplified estimation method that does take account of these effects. The present work addresses this need and recommends a method which includes these main physical effects yet retains an ease of use suitable for hand calculations. It is important to note, however, that the model involves a few physical property parameters. Those given in the present work are applicable only to steam-water mixtures. In order to apply the model to other fluids, these parameterswould need to be derived from relevant physical property data. General equations are given in the Appendix for this purpose. Energy Technology Unit, Department of Mechanical Engineering Heriot-Watt University, Riccarton, Edinburgh EH14 4AS 281

2 ANALYSIS By following principles similar to those of isentropic gas-dynamic theory, it can be shown that the two-phase critical mass flowrate M (kg/s) through a nozzle or orifice of throat area A can be determined from Here, C. is the discharge coefficient and v is an 'effective 1 two-phase specific volume (v = xv + (1-x)v for homogeneous flow). The derivative is required to be evaluated at throat conditions. However, the choking (or critical) pressure p. at the throat is not normally known a priori but must be determined by a numerical procedure that iterates on p until a value is found that satisfies equation (2.1) and the momentum equation simultaneously. For purposes of a quick-estimate method, it is useful to investigate whether or not the choking pressure p.can be found from some simple correlation based on experimental data and requiring knowledge only of upstream (or stagnation) conditions. Such an approach offers considerable simplification since, knowing p., equation (2.1) can be evaluated directly thus dispensing with the need for iterative numerical procedures. Choking Pressure Ratio A number of data sets of choking pressure ratio for steam-water flow through converging-diverging nozzles have been assembled for the present work and are shown in Fig. 1. Choking pressure ratios are plotted against corresponding nozzle inlet qualities x 0. Inlet pressures cover the range bar abs. and so inlet density ratios are in the range In the range x > 0.003, the data seem to be In quite reasonable o agreement. For qualities x < there is only one set of data and therefore any conclusions drawn regarding behaviour in this range must be regarded as tentative for the present; further experimental confimation is clearly necessary. However, physical reasoning can offer some explanation of the decreasing behaviour of with decreasing x in the range x < If, due to thermal nonequi 1ibrium effects (manifested as delayed flashing), the choking plane occurs not at the minimum flow area but at some larger cross-sectional area downstream in the diverging section of the nozzle, then a higher mass discharge rate will result.this higher flow rate will then cause a reduction in throat pressure below that value that would pertain were choking to occur at the throat. On the basis of this argument, the pressure ratios for x < are not, strictly speaking, 282

3 chok i nq pressure ratios since the choking plane is located in the diverging section and not at the throat where the pressure is measured. However, it is impossible from the experimental information available to determine the actual location of the choking plane and the choking pressure there, while the measured throat pressure offers a practical basis for calculations. It is therefore proposed that the data in Fig. 1 be used as a basis for determining throat pressures during choked flow through coverging - diverging nozzles. A best-fit curve has been hand-drawn through the data and equations for this curve are given in Fig.1. Hence, given inlet quality x and pressure p, the throat (or choking) pressure p can be determined directly from where is read directly from F1g.1 or calculated from the equations supplied. With regard to subcooled inlet conditions, the data of Sozzi and Sutherland (1975) for nozzle 1 have been re-worked into the form shown in Fig. 2 where choking pressure ratio ri Is plotted against inlet s u b c o o l i n g A n equation is also supplied for this curve. It is of interest to note that the data upon which this curve is based exhibited no variation with pressure over the range bar abs. For fluids other than steam-water mixtures for which there are no available choking pressure ratio data, it Is suggested that Figs. 1 and 2 should offer reasonable estimates. Choking Mass Flowrate In order to express equation (2.1) in a working form, a choice must be made for the 'effective 1 two-phase specific volume v. For present purposes, the expression of Morris (1984) is used, this having been shown to give good agreement with momentum flux data; this expression is given by 1s the slip (or velocity) ratio of Chisholm (1983). Equ. (2.3) has also been shown by Morris (1985) to give good predictions of two-phase pressure drop across valves and orifice plates when used in the form of a two-phase multipller 1.e. In addition, in order to account for nonequi1ibrium effects, nonequi1ibrium quality at the nozzle throat Is defined by 283

4 where x is the equilibrium throat quality based on isentropic expansion along the saturation line, i.e. and where termed the relaxation number, is given by and In this latter expression, r is the vapour expansion exponent which, for steam, is given in Fig.3. The quantity v is the effective two-phase eo specific volume (equations (2.3) and (2.4)) evaluated at the nozzle Inlet where x = x. o This quantity may be read from Fig. 4 where the ratio v /v is plotted against x over a range of density ratios. eo HO o Lastly, the parameter N_, termed the flashing number, is given by the expression where A and A may be read from F1g.5 or evaluated from the equations given thereon. A full derivation of equations (2.7) - (2.9) Is given in the Appendix; It Is also shown here that equation (2.1) may be cast into the following working equation: Here, S is a phase slip parameter which can be read from F1g.6 or n evaluated from the equations given in the Appendix. It remains now to account for nozzle length effects. Nozzle Length Effects There appears to be only a limited amount of data available in the literature from which nozzle length effects can be estimated. In particular, the steam-water choked flow data of Sozzi and Sutherland (1975) for round and sharp entry tubes (12.5mm ID) offer some scope for 284

5 analysing length effects since the tube lengths varied from 44.5mm to mm and fluid inlet conditions ranged from subcooled liquid to saturated two-phase mixtures. As will be shown from the data comparisons of the next section, the proposed model performs well for nozzle lengths less than 44.5mm and it was felt that the model could be used as a basis for analysing nozzles/tubes of longer length. Since longer nozzle lengths afford the two-phase mixture more time to relax towards equilibrium before the choke location is reached, it is likely that the relaxation number N is strongly affected by nozzle length. Based on this argument then, a modified relaxation number is defined by where LF is termed the length factor. In analysing the data of Sozzi and Sutherland (1975) for tubes of varying length, N was used in place of N R R in equation (2.5) and the measured discharge rates were input to the model thus allowing the length factor LF to be backed-out of the calculations. It was found that LF was adequately described by the equation where L is the nozzle length (entry to exit) expressed in mm. For L < 44.5mm it was found through analysis of the data sets of several authors that there was little noticeable effect on discharge rates, i.e. LF = 1 for L < 44.5mm. COMPARISONS WITH DATA First of all, Table 1 shows a comparison of model predictions with the converging-diverging nozzle data of Carofano and McManus (1969). The high inlet qualities in these tests would suggest an equilibrium flow and the predicted choking mass fluxes m ( = M /A) are seen to underpredict the measured values by the average underprediction over the parameter range being 13.6%. Only one value of measured choking pressure ratio is given and the agreement here is seen to be good. Further comparisons of model predictions with a number of data sets are shown in F1gs In all cases, including the previous comparison with the data of Carofano and McManus (1969), the discharge coefficient C. was d taken as unity. Figs. 7-9 show the performance of the model under subcooled, saturated and low-quality inlet conditions when compared with the data of Sozzi and Sutherland (1975) and Morrison (1977). The flows that are initially subcooled are probably metastable at least as far as the nozzle throat while the saturated and low-quality flows are highly nonequl1ibrium. In either case, the model appears to perform rather well. However, comparisons with other data that are not given in this paper would Indicate that the model should not be used to predict choked flow through nozzles greater than 200mm in length when inlet subcoolings are greater than a few degrees. 285

6 The data of Neusen (1962) for choked flow through converging-diverging nozzles are shown in Figs 10 and 11. These data cover fairly wide ranges of inlet quality and pressure and the proposed model is seen to give good agreement here also. Fig. 12 compares predictions with the data of Friedrich (1960). These data also cover large ranges of quality and pressure. There seems to be a tendency for the model to overpredict for sharp-edged inlet geometries, a feature also noticeable in Fig. 13. Although the discharge coefficient for these cases was taken as unity, there seems to be scope for optimising model performance through a better choice of C. The sensitivity of the model to nozzle length is one feature also of the comparisons shown in Fig. 14 with the various nozzle data of Dryndrozhik (1975). Good agreement is demonstrated here over the whole range of data. In particular, nozzle C8/cyl30 has a 30mm extension from the throat and the data indicate that this causes a reduction in choking mass flux. This reduction is reflected by the model predictions at least over the lower-quality range. Lastly, Fig. 15 compares predictions with the low-pressure, high-quality data of Deich et al (1969). The good agreement shown here is due mainly to the slip effects incorporated in the model rather than to nonequi1ibruim or length effects. APPLICATION TO VALVES There is a great paucity of available experimental data relating to choked two-phase flow through valves. With regard to pressure relief valves (PRVs) the work of Sallet et al (1981), Sallet (1984) and Sallet and Somers (1985) provide some welcome insight into two-phase flow aspects but much more experimental effort is required in order to produce the data necessary for reliable design methods. Of note too is the report by HSE (1987) which provides an extensive review of work on PRVs and some related aspects of venting. Until more data is available for choked two-phase flow through valves in general, one must make use of available nozzle data and methods with an appropriate choice of discharge coefficient C for the valve of Interest. d It seems normal practice to use the'cold water' discharge coefficient of the valve which Is sometimes given by the manufacturer, or through if the loss coefficient K (or *K-factor') is given, or through if the (valve sizing coefficient) is given. Note that, In this last expression, is In the standard units of US p in and A is the line cross-sectional area in 286

7 Only one experimental test for a valve could be traced against which the predictions of the proposed method could be compared. This was the saturated water PRV test reported by Sal let and Somers (1985). In this test, the saturated water at inlet to the PRV was at a pressure of 6.89 bar abs and the measured discharge rate (M ) was kg/s. Cold water tests showed the discharge coefficient to be 0.85 and the PRV orifice size was given as mm. The proposed model was applied to this problem and yielded a choking mass flowrate of 3.51 kg/s, some 14% higher than the measured flowrate. The various calculational steps followed in achieving this result are given as a worked example in the next section by way of demonstrating the use of the proposed model. Lastly, it is of Interest to note that had the homogeneous equilibrium model been used to estimate the discharge rate for the above PRV, a value of 0.91 kg/s would be obtained. This is clearly a serious underpredlction and would have led to gross oversizing of the valve. On the other hand the Modified Frozen Flow Model of Sal let and Somers (1985) gave a flowrate of 3.52 kg/s, in close agreement with the model proposed in the present work. Estimating the flowrate based on al1-1iquid flow (Bernoulli's equation) gives a value of 6.14 kg/s, a serious overprediction which would have grossly undersized the valve for such service. WORKED EXAMPLE Saturated water at a pressure of 6.89 bar abs. discharges from a vessel through a PRV with discharge coefficient 0.85 and orifice area 220.6mm. Use the proposed model to estimate the maximum mass discharge rate. Step 1. Find choking pressure ratio t) and throat pressure p 287

8 CONCLUSIONS A model has been proposed for the estimation of choking mass flowrates through nozzles and valves. Comparison of predictions with a number of data sets covering wide ranges of inlet conditions demonstrated good agreement. Although the model includes the effects of thermal nonequi1ibrium, phase slip and nozzle length, it is still simple enough to use as a hand-calculation method. Equations are provided for the relevant parameters enabling the method to be coded-up on a computer if necessary and to be extended for use with fluids other than steam-water. There is a clear need for additional experimental data on flashing choked flows through valves. Only one data point was found in the literature and, while the proposed model demonstrated good agreement, a wider data base for valves is essential in order that this and other models may be assessed and a reliable design procedure recommended. NOTATION 288

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10 DEICH, M.E., DANILIN, V.S. TSIKLAURI, G.V. and SHANIN, U.K. (1969). "Investigation of the flow of wet steam in ax 1 symmetric de Laval nozzles over a wide range of moisture content ". High Temperature, Vol. 7, p DRYNDROZHIK, E.I. (1975). "Critical flow regime of a low-quality steam-liquid mixture in convergent nozzles". Fluid Mechanics - Soviet Research, Vol, 4, No.1, Jan-Feb., pp FRIEDRICH, H. (1960). "Flow through single-stage nozzles with different thermodynamic states". Energie, Vol 12, p.3. HENRY, R.E. and FAUSKE, H.K. (1971). "The two-phase critical flow of one-component mixtures in nozzles, orifices and short tubes". Trans. ASME, J. Heat Transfer, May, Vol 93, No. 2, pp HSE (1987), "Pressure relief valves and related aspects of venting". Health and Safety Executive, Research and Laboratory Services Division, Harpur Hill, Buxton, Derbyshire U.K. KEV0RK0V, L.R., LUTOVIN0V, S.Z. and TIKHONENKO, L.K. (1977). "Influence of the scale factors on the critical discharge of saturated water from straight tubes with a sharp inlet edge". Teploenergetika, Vol. 24, No. 7, pp Also Thermal Engineering, Vol 24, No. 7, pp MORRIS,S.D. (1984). "A simple model for estimating two-phase momentum flux". Presented at 1st UK National Conf. on Heat Transfer, University,f Leeds, Leeds, UK, 3-5 July. Vol.2, pp MORRIS,S.D. (1985). "Two-phase pressure drop across valves and orifice plates". Presented at the European Two-Phase Flow Group Meeting, Marchwood Engineering Laboratories, Marchwood, Southamption, U.K., 4-7 June. (Paper E2). Copies from author. MORRISON, A.F. (1977). "Blowdown flow in the BWR BDHT test apparatus". GEAP-21656, NRC-2. Report prepared by General Electric Co. for US Nuc. Reg. Comm. & EPRI under contract No. AT(49-24) NEUSEN, K.F. (1962) "Optimizing of flow parameters for the expansion of very low-quality steam". Report No. UCRL-6152, University of California Radiation Laboratory, Calif., U.S.A. SALLET, D.W., NASTOLL, W., KNIGHT, R.W., PALMER, M.E. and SINGH, A. (1981). "An experimental investigation of the internal pressure and flow fields in a safety valve". ASME Paper No. 81-WA/NE-19 (presented at ASME Winter Annual Meeting, Washington DC, Nov , 1981). SALLET, D.W. (1984). "Thermal hydraulics of valves for nuclear applications". Nuclear Science and Engineering, Vol. 88, pp SALLET, D.W. and SOMERS, G.W. (1985). "Flow capacity and response of safety valves to saturated water flow". Paper No. 55e, presented at the 19th Annual Loss Prevention Symposium (DIERS-1), AICHE Spring National Meeting, Houston, Texas, March

11 SOZZI, G.L. and SUTHERLAND. W.A. (1975). "Critical flow of saturated and subcooled water at high pressure". Ijn Non-Equi1ibruim Two Phase Flows, ASME Winter Annual Meeting, Houston, Texas, 30 Nov. - 5 Dec, pp (eds. R.I. Lahey and G.B. Wall is). APPENDIX - Derivation of Equations (2.7) - (2.10) The term in equation (2.1) must be evaluated at throat conditions. in the form Using equations (2.3) and (2.4), this term can be expressed 291

12 The isentropic vapour expansion term In equ. (A.1) is replaced by expansion along the saturation line as this 1s more likely to be representative of actual flow conditions, particularly with low-quality flows with significant flashing at the throat. That is, (9v /9p) is G S replaced by where, for steam, T Is given In terms of reduced throat pressure In Fig.3. The flashing term in equ. (A.1) may be evaluated using equ. (2.5). Since for isentropic expansion along the saturation line and where it has been assumed that Also, since it follows by differentiation that If a flashing number N. 1s now defined by then this may be written as The above parameters A, A are evaluated from physical properties data and, for saturated steam-water mixtures, are given in Fig.5 as functions of reduced throat pressure. Given the throat pressure then, N_ can be calculated from equ. (A.10). 292

13 These developments allow equ. written as (2.1) for the choking mass flowrate to be and it remains to find some way of determining the relaxation number N. Since S (plotted in Fig. 6) and T are partly functions of x, and x itself depends on N, some approximation is now required in order to achieve a tractable result. Examinations of equs. (A.2) and (A.3) shows that as 1. Furthermore, as it can be shown t h a t a n d so, in this limit, equ.(a.12) becomes where N denotes the value of N for very small x. Now, equ. (A. 13) Ro R n should give mass flowrates comparable to those based on the assumption of Bernoul1i-type flow, i.e. where v is eo the 'effective' two-phase specific volume 'frozen' at inlet conditions. Combining equs.(a.13) and (A.14) yields the following expression for At the other extreme as equilibrium flow prevails and this requires that N = 1 (see equ. (2.5)). Also, as x» 1, it can be shown n that Hence, by way of simplification, it is assumed that = 1 for all conditions and that N be determined from the n R following expression which is a 'smoothing function' between end values: 293

14 The final recommended form of the choking mass flowrate equation is then 294

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Mass flow determination in flashing openings

Int. Jnl. of Multiphysics Volume 3 Number 4 009 40 Mass flow determination in flashing openings Geanette Polanco Universidad Simón Bolívar Arne Holdø Narvik University College George Munday Coventry University