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Calculation of Steam Volume Fraction in Subcooled Boiling

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Calculation of Steam Volume Fraction in Subcooled Boiling AE-28G UDC 535.248.2 CD 00 CN UJ < Calculation of Steam Volume Fraction in Subcooled Boiling S. Z. Rouhani AKTIEBOLAGET ATOMENERGI STOCKHOLM, SWEDEN 1967 AE -286 CALCULATION OF STEAM VOLUME FRACTION IN SUBCOOLED BOILING by S. Z . Rouhani SUMMARY An analysis of subcooled boiling is presented. It is assumed that he at is removed by vapor generation, heating of the liquid that replaces the detached bubbles, and to some extent by single phase heat transfer. Two regions of subcooled boiling are considered and a criterion is provided for obtaining the limiting value of subcooling between the two regions. Condensation of vapor in the subcooled liquid is analysed and the relative velocity of vapor with respect to the liquid is neglected in these regions. The theoretical arguments result in some equations for the calculation of steam volume frac­ tion and true liquid subcooling. Printed and distributed in June 1 967. LIST OF CONTENTS Page Introduction 3 Literature Survey 3 Theory 4 Assumptions and the Derivation of the Basic Equations 4 Remarks on the Liquid Subcooling 6 The integrated Equation of Subcooled Void 7 Estimation of the Rate of Bubble Condensation 9 Effects of Void and Channel Geometry on cp and cp^ 1 0 Effect of Subcooling on cp^ and cp^ 1 0 Effects of Mass Velocity, Heat Flux, and Physical Properties on cpj and cp,, 12 The Limiting Value of Subcooling in the First Region 1 3 Resume of Calculation Procedure 14 Calculations of a in the First Region 14 Calculation of O' in the Second Region 1 4 Steam Quality in Subcooled Boiling 1 5 Comparison with Experimental Data 1 5 Conclusion 15 Nomenclature 18 Greek Letters 20 References 22 Figure Captions 23 - 3 - INTRODUCTION The intensity of subcooled boiling depends on many factors, among which the degree of subcooling, surface heat flux, mass veloc­ ity, and physical properties of the liquid and its vapor. This report is intended to give a new method of calculating the steam volume fraction in the region of subcooled boiling. The method is based on a new way of considering the different mechanisms of heat removal in this region. LITERATURE SURVEY The subject of subcooled boiling has been treated by many inves­ tigators. However, the works have mostly been in connection with studies of heat transfer and surface temperature and only a few authors have concentrated on the question of determining the steam volume fractions in the region of subcooled boiling. Among the published works one may refer to the following. Griffith, Clark, and Rohsenow [l ] made photographic studies of subcooled boiling at high pressures on a rather short strip of metal in a rectangular channel. A fundamental result of their work was the observation of two separate regions in subcooled boiling. They sug­ gested a calculation method which was based on neglecting the steam generation in the first region but including it in the heat balance in the second region. For the first region they suggested that heat was re­ moved partly by the single phase heat transfer mechanism and partly by condensation. For the second region they assumed a constant con­ densing film coefficient. In order to find the transition point they drew upon some observation by Rohsenow [2 ] and assumed that at that point 80 per cent of the surface heat flux would go to steam generation. Maurer [3 ] made use of data from different sources and gave a method of void calculation which seems to be only suitable in cases of very high liquid velocities (several meters per second), and high pres­ sures (over 60 ata). For the subcooled void he used the arguments and derivations of Griffith et al. with a modification in the calculation of wall superheat. - 4 - Bowring [4] gave a detailed method of calculation of void which was partly based on the forementioned works and included the assump­ tion that the bubble condensation was negligible in the second region. But, in calculating subcooled void according to his model it was neces­ sary to use an assumed and relatively high value of slip ratio. Costello [5] treated the problem of determining the time-averaged vapor bubble coating on the heater surface only for the case of high subcoolings. Delayre and Lavigne [6] gave a very brief treatment of the prob­ lem of calculating voids in local boiling with the assumption of a conden­ sation coefficient without further specifications. THEORY Assumptions and the Derivation of the Basic Equations In the process of subcooled boiling the supplied heat through the surface is removed mainly by the following three mechanisms 1) single-phase heat transfer which will be effective as long as the heated surface is not covered with bubbles. 2) Steam generation. 3) Heating of that mass of water which replaces the detached bubbles. Vapor condensation is a major factor in the heat exchange between the generated steam and subcooled liquid. This effect will be considered later. These mechanisms have been previously listed by some authors, among the possible forms of heat transfer in subcooled boiling [7 ]. An observation of the final results will show that the relative effective­ ness of each of the suggested mechanisms vary as mass velocity, pres ­ sure, and subcooling are varied. One may assume that in a heated channel local boiling starts at a point where q/A - h(t -1^) > o. In local boiling at high subcoolings the single phase heat transfer will be still effective but augmented by the other mechanisms. In this region a part of the heated surface will be covered with bubbles and hence be unaccesible to the liquid flow. - 5 - This is equivalent to a reduction in the single phase heat transfer co­ efficient. In the absence of an accurate method of estimating the ef­ fective h , we assume sb h valid for t > t (0 sb w s Based on this argument the non-boiling fraction of heat flux will be . = hsn r-r-T- (tw-V = hRn- At (2) nb w o As subcooling further decreases vapor clotting [5] takes place viz. , the bubbles cover the heated surface and eliminate single phase heat transfer. At this subcooling the steam volume fraction begins to rise rapidly with decreasing subcoolings. The heat balance on the heated surface is m qj; = h • At + m • X + —— C p. At (3) A s p p t ' ‘ g \ The amount of heat which goes to steam generation per unit time within an element of height of the channel, dz, will be (^;) - hAt dQ, = m \ P, dz Pg'^SP^AtPg" Ph^z b s h (4 ) where mg is substituted from eq. (3). The amount of heat removed from the steam bubbles by con­ densation is an unknown factor. The available information on the col­ lapse of the individual bubbles [8,9, 10] is not easily applicable to an unknown population of bubbles in a boiling system. Therefore, one may try to express the rate of condensation in terms of some macro­ scopic parameters of the system. Here we assume that the amount of heat which will be removed from steam bubbles by condensation per unit time within an element of height, dz, will be dQ = k (t - t.) dz (5 ) c cx s -v - 6 - where k is a condensation constant with the same dimensions as c that of the thermal conductivity. This constant will be analysed in more detail later. Considering the flow of steam into and out of a small section of the channel, ' dz, one should get steam heat flow out - steam heat flow in = dQ^ - dQ^ (6) Neglecting the relative velocity of vapor with respect to the liquid in the subcooled region: m V ~ Yl ~ = V (7) g A [cvp + (1 - of) p . ] C g ' -V and eq. (6) becomes A [ (V + dV) (a + da) - Va ] p X = dQ. - dQ (8) c ' g be But from eq. (7) (° l - P g) da m dV A~ (9) c [apg+ (1 - a) p^]2 Elimination of dQ^, dQ^, and dV between cqs. (4), (5), (8) and (9) yields : mX dof - (J) - h At p X P - k At dz (10) ~T Li - (i-i/p)of32 g h c . Remarks on the Liquid Subcooling In subcoolcd boiling a part of the supplied heat goes with the 1« iin bubbles. Tt is clear th.il the average liquid subcooling is dif b rent 11 •»i n I hr average bulk subcooling which is normal Iv obtained from a simple heat balance, viz. z o At, a At. (H) b in m • C P - 7 - This would represent the liquid subcooling only if the steam bub ­ bles had zero velocity. The assumption of equal velocities of vapor and liquid yields the steam quality in the region of subcooled boiling; a Xs “ a + (l - a) p (12) The heat carried by steam bubbles will be m X a x m X = (13) Q. a + 0 -<*) P And hence a closer approximation of the average liquid subcooling will be . z At. (1 4 - a) Att in rh- C P or a • X At (14- b) % AV [ar+ (1 -a) p] C The last term becomes increasingly significant as pressure and steam fraction are increased. THE INTEGRATED EQUATION OF SUBCOOLED VOID The more correct value of At^ from eq. (l 4-a) yields m C dO dz =--------- 2----- [d(At^) - -—-] (15) (q/A) mC But, the second term in the brackets may be neglected and hence m C dz = - d(At„) (16) ^ (q/A) Rewriting eq. (10) with dz substituted from eq. (l 6) we get Cp^ q/A-hAt^ dQf 2 ” q/A ' X • p ^ + C^p^A t^ Ll -a(l-l/p)] ‘ g P k At, p • x • p ) d (At-t) (17) h g _ 8 - In the case of constant values of heat flux and phy­sic ai prope 1 uc s this equation may be integrated between the inlet (a = 0) and any other point within the first region (this will be defined later).
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