CONTRIBUTION TO THE THEORY OF TWO-PHASE, ONE-COMPONENT CRITICAL FLOW

Because the mathemat ica l models r equ i red for theore t ica l analyses of the p rob lem of two-phase flow a re quite complicated, the more successful approaches have been empir ica l and s e m i e m p i r i c a l . The major i ty of theore t ica l analyses [e.g., Harvey and Foust(41)] have assumed that the phases a r e mixed sufficiently to be cons idered as a homogeneous fluid. The work of Linning(66) is an example of a s e m i e m p i r i c a l approach; Linning s e t u p "one-dimensional" models for froth, s trat if ied, andannular flow configurations, determining unknown p a r a m e t e r s exper imental ly . Lockhar t and Martinelli(71) p resen ted a modera te ly successful cor re la t ion for the two-component problem. Mart inel l i and Nelson( ' ^ adapted the twocomponent cor re la t ion of Lockhar t and Martinelli(71) to s ingle-component sys tems by means of a simple cor rec t ion to account for changes in the axial component of momentum. The potential advantage of this approach over that of Harvey and Foust is the re la t ive simplici ty of the computations. However, Mart inel l i and Nelson did not p o s s e s s sufficient exper imenta l data to verify their approach. One of the most recent works on the predict ion of p r e s s u r e drop in two-phase , s ingle-component fluid flow is that of Isbin et al.(48) They conducted extensive exper imenta l work with s t eam-wate r mix tures at qual i t ies ranging from 3 to 98 per cent and at p r e s s u r e s ranging from 25 to 1415 psia . They at tempted to co r re l a t e their data by use of the Mart inel l i cor re la t ion , but found se r ious f low-rate and p r e s s u r e effects. The use of a homogeneous fr ict ion-factor model also proved unsuccessful . Therefore , they co r re l a t ed their data in a r e s t r i c t i ve manner which took into account the f low-rate and p r e s s u r e dependencies. In spite of the la rge amount of l i t e r a tu re in this a r e a , the information is still inadequate for obtaining accura te and re l iable design p rocedures for two-phase , s ingle-component flow sys t ems . 2.3. P a s t Work on Cr i t ica l Flow Much more is known about c r i t i ca l flow of a s ingle-phase fluid than is known about the two-phase c r i t i ca l flow of fluids. Many text books, such as that of HalH^O) a n d that of Shapiro,(89) a r e available which give the theory of s ingle-phase flow. Most of the r e s e a r c h which has been done in the genera l r e a l m of two-phase flow has concerned ei ther two-phase , two-component sys tems or two-phase , s ingle-component sys tems very close to sa turat ion condit ions. Most of the work involving the f i r s t has been done in the UnitedStates whereas Br i t i sh inves t igators have taken the initiative with the l a t t e r , by studying the behavior of sa tura ted water or mix tures of liquid water and s team resul t ing from some kind of an expansion or flashing p r o c e s s . The field being of such a complexity, it was not surpr i s ing to find a var ie ty of possible simplifying assumptions being made by different inves t iga to r s in an at tempt to bring theory close to fact. To date, however , no single design method has yet been proposed that will prove sat isfactory, except for a few limiting cases . The f i rs t r ea l a t tempts at a ser ious study of two-phase flow were probably those of Sauvage(85) i n 1892. According to Isbin,(46) Rateau(80) in 1902 showed the exis tence of c r i t i ca l flow in the flow of boiling water through nozzles . He obtained this c r i t i ca l condition by dropping the back p r e s s u r e until the d ischarge reached a maximum. He also developed a method of calculating the quantity of sa tura ted water d ischarged through a nozzle based upon isent ropic expansion. Mellanby and K e r r ( ' 5 ) in 1922 studied the flow of wet s t eam through nozzles . All four nozzles used were of the simple convergent or convergent para l le l type. Two of the nozzles were used in conjunction with a sea rch tube so that the t e s t s included p r e s s u r e m e a s u r e m e n t s as well as flow determinat ions . Only flow tes t s were c a r r i e d out on the other two, as these were not fitted with a sea rch tube. In all c a s e s , flow determinat ions were made with varying initial superheats at a supply p r e s sure of about 75 psia . The flow curves that these inves t igators have p r e sented for these four nozzles showed cer ta in var ia t ions between themselves but they all agreed on the two points: (1) that the flow, at and near the initially dry s ta te , was excess ive when compared with the theore t ica l as obtained on the assumption of stable expansion, and (2) that the form of the flow curve over a smal l range of superheat beyond the initially dry condition was not in agreement with the theore t ica l assumption of complete super saturat ion on any rat ional bas i s of expansion l o s s e s . Equations were developed in o rder to es t imate the "fractional revers ion" (fraction liquid actually p resen t / f r ac t ion liquid with no super saturat ion) existing in the nozzles . The fractional r eve r s ion was found to be l e s s for the nozzle with cent ra l search tube than for the ful l-bore jet. The difference seems r a the r unaccountable. F r o m their calcula t ions , it was noticed that the re was no indication of r eve r s ion before a value of roughly 1 \ per cent equivalent wetness was reached. The work of Stanton(94) i n 1926 gave valuable insight into the highspeed flow of a i r through sharp-edge or i f ices , convergent nozz les , and convergent-divergent nozzles . This invest igator obtained both axial and radia l p r e s s u r e profi les by using both search and Pi tot tubes . The most important conclusions derived in this work a re as follows: (1) In each of the three cha rac t e r i s t i c types of or if ices which may be used for the d i s charge of gases from a vesse l at constant p r e s s u r e into a r ece ive r at a p r e s s u r e appreciably below the c r i t i ca l value (0.527 t imes the u p s t r e a m stagnation p r e s s u r e ) , the a r ea of the je t diminishes to a minimum value, at which the velocity is that of sound under the conditions exist ing, and then i n c r e a s e s . This minimum section in the case of a free jet is not constant in a r e a or position re la t ive to the plane or throat of the or i f ice , but depends on the total ra t io of expansion. (2) In a jet in which the expansion takes place within solid boundar ies , i .e . , a diverging nozzle , the minimum section may for all p rac t ica l purposes be regarded as coincident with the throat of the nozzle for all ra t ios of expansion. (3) The flow of the fluid up to the minimum section is adiabatic in cha rac t e r . Goodenoughw^) i n 1927 said the fact that the speed of the water drops is only a small fraction of the speed of the s team has a decided effect on the discharge of wet s team, for the total friction is compr ised of not only the interact ion between the fluid and wal l s , but also of the internal interact ion between s team and water pa r t i c l e s . Goodenough further showed by calculation that super saturat ion would give a net i nc rease in isentropic flow for a given p r e s s u r e drop. He also pointed out that the surface energy of the droplets formed in such flows could be quite l a rge . Another compli cation is that any entrained drops (usually l a rge r than those formed by condensation) probably do not cool fast enough during the expansion to maintain the rmal equi l ibr ium with the s team. He points out that all the above could occur simultaneously in a flow p r o c e s s . Kit tredge and Dougherty(58) i n 1934 p resen ted an express ion for the calculation of c r i t ica l d ischarge capacity for s t eam t raps based on the initial and d ischarge p r e s s u r e s , and the specific volume of the mixture evaluated at the d ischarge p r e s s u r e . Their calculated values p resen ted in graphical form a re said to be 100 to 200 per cent g r e a t e r than exper imenta l values through a nozzle . In 1934, Yellott(105) investigated drop size and super saturat ion by studying the flow of l o w p r e s s u r e s team through nozzles s imi la r to those used in tu rb ines . This invest igator exper imental ly verif ied the existence of super sa turat ion with the aid of light r a y s , applications of the laws of opt ics , and a sea rch tube for nozz les . He found that the condensation occu r r ed between the 3 and 4 per cent mois tu re l ines on a Mollier d iagram if an isent ropic expansion is assumed. Rettaliatta '^-U in 1936 extended Yel lot t ' s work and intended p r i mar i ly to de termine the effect of wall roughness on the flow of s team in nozzles . It was found that wall roughness , causing a re ta rda t ion of s team flow, made the point of initial condensation occur at a higher p r e s s u r e and far ther downst ream than in the smooth-walled nozzle. This caused the Wilson line to occur at the 3.2 per cent mois tu re line on the Mollier diag r a m for the rough-walled nozzle , instead of at the 3.7 per cent mois tu re line as was found for the smooth nozzle. Ret tal ia t ta c la imed that this was the resu l t of the inc reased time of expansion in the case of the rough nozzle. In 1937, urged by a need for more information on c r i t i ca l flow for the design of cascading drain p ipes , Bottomley.i^O) experimenting with flow of boiling water through or i f ices , found that his actual flows based on exper imenta l data were 5 t imes g rea t e r than the predic ted values based on computation by the s ingle-phase equi l ibr ium theory. His explanation is that the effects of surface tensions lower the p r e s s u r e at which the propagation takes place. Because of surface tension, the condition of flow at the throat is that of unstable equilibrium. Bottomley further cites one experi