22.06 Engineering of Nuclear Systems MIT Department of Nuclear Science and Engineering NOTES ON TWO‐PHASE FLOW, BOILING HEAT TRANSFER, AND BOILING CRISES IN PWRs AND BWRs Jacopo Buongiorno Associate Professor of Nuclear Science and Engineering JB / Fall 2010Definition of the basic two‐phase flow parameters In this document the subscripts “v” and “ℓ” indicate the vapor and liquid phase, respectively. The subscripts “g” and “f” indicate the vapor and liquid phase at saturation, respectively. AvVoid fraction:  Av  AM A v v vStatic quality: xst  Mv  M Av v  A  m v A v v v vFlow quality: x   m v  m vvAv v  v A  vSlip ratio: S  v vMixture density: m v  (1) Flow quality, void fraction and slip ratio are generally related as follows: 1  1 xv1  S  x Figures 1 and 2 show the void fraction vs flow quality for various values of the slip ratio and pressure, respectively, for steam/water mixtures. Figure 1. Effect of S on  vs x for water at 7 MPa. Figure 2. Effect of water pressure on  vs x for S=1. JB / Fall 2010In 22.06 we assume homogeneous flow, i.e. vv=vℓ or S=1. This assumption makes it relatively simple to treat two‐phase mixtures effectively as single‐phase fluids with variable properties. If S=1, it follows 1 x 1 ximmediately that xst=x and, after some trivial algebra,   .m v  Note that the assumption of homogeneous flow is very restrictive, and in fact accurate only under limited conditions (e.g. dispersed bubbly flow, mist flow). In general, a significant slip between the two phases is present, which requires the use of more realistic models in which vvvℓ. Such models will be discussed in 22.312 and 22.313. Two‐phase flow regimes With reference to upflow in vertical channel, one can (loosely) identify several flow regimes, or patterns, whose occurrence, for a given fluid, pressure and channel geometry, depends on the flow quality and flow rate. The main flow regimes are reported in Table 1 and shown in Figure 3. Note that what values of flow quality and flow rate are “low”, “intermediate” or “high” depend on the fluid and pressure. In horizontal flow, in addition to the above flow regimes, there can also be stratified flow, typical of low flow rates at which the two phases separate under the effect of gravity. Table 1. Qualitative classification of two‐phase flow regimes. Flow quality Flow rate Flow regime Low Low and intermediate Bubbly High Dispersed bubbly Intermediate Low and intermediate Plug/slug High Churn High High Annular High (post‐dryout) Mist JB / Fall 2010More quantitatively, one can de termine which flow regime is present in a particular situation of interest resorting to an empirical flow map (see Figure 4). Note that such maps depend on the fluid, pressure and channel geometry, i.e., there is no “universal” map for two‐phase flow regimes. However, there exist methods to generate a flow map for a particular fluid, pressure and geometry. These methods will be covered in 22.312 and 22.313. JB / Fall 2010 Typical configuration of (a) bubbly flow, (b) dispersed bubbly (i.e. fine bubblesdispersed in the continuous liquid phase), (c) plug/slug flow, (d) churn flow,(e) annular flow, (f) mist flow (i.e. fine droplets dispersed in the continuous vaporphase) and (g) stratified flow. Note: mist flow is possible only in a heated channel;stratified flow is possible only in a horizontal channel.a b cgd e fImage by MIT OpenCourseWare.Figure 4. A typical flow map obtained from data for low‐pressure air‐water mixtures and high‐pressure water‐steam mixtures in small (1‐3 cm ID) adiabatic tubes (adapted from Hewitt and Roberts 1969). How to calculate pressure changes in two‐phase flow channels For a straight channel of flow area A, wetted perimeter Pw, equivalent (hydraulic) diameter De (4A/Pw), inclination angle , and length L, connected to inlet and outlet plena, the total pressure drop (inlet plenum pressure minus outlet plenum pressure) can be obtained from the steady‐state momentum equation: P0  PL  Ptot  Pacc  Pfric  Pgrav Pform (1) j where the subscripts “0” and “L” refer to inlet and outlet plenum, respectively. The last term on the RHS of Eq. 1 represents the sum of all form losses in the channel, including thos e due to the inlet, the outlet and all other abrupt flow area changes in the channel (e.g. valves, orifices, spacer grids). Under the assumption of homogenous flow (S=1 or vv=vℓ): Pacc  G2[ 1  1] (2)m,L m,0 JB / Fall 2010 10510410310210110-1106105104103102101[xG]2/ρg (kg/s2-m)[(1-x)G]2/ρf (kg/s2-m)AnnularWispy-annularBubblyBubblesslugsSlugsChurnImage by MIT OpenCourseWare.L 1 G2 Pfric 0 fD 2 dz (3) e m LPgrav  mg cos dz (4)0 Note that the above equations are formally identical to the single‐phase case with the mixture density, m=v+(1‐)ℓ, used instead of the single‐phase density. The friction factor in Eq. 3 can be calculated from a single‐phase correlation and using the “liquid‐only” Reynolds number, Reℓo =GDe/ℓ. For each abrupt flow area change in the channel (again including the inlet and outlet), the associated pressure loss is the sum of an acceleration term and an irreversible loss term: Pform  Pform,acc  Pform,irr  1(G22  G12)  KG2 (5)2 2m m where the subscripts “1” and “2” refer to the location immediately upstream and downstream of the abrupt area change, respectively. Boiling heat transfer Boiling is the transition from liquid to vapor via formation (or nucleation) of bubbles. It typically requires heat addition. When the boiling process occurs at constant pressure (e.g. in the BWR fuel assemblies, PWR steam generators, and practically all other heat exchangers in industrial applications), the heat required to vaporize a unit mass of liquid is hfghg ‐hf, which can be found in the steam tables. Writing the (Young‐Laplace) equation for the mechanical equilibrium of a bubble surrounded by liquid, it can be shown that the vapor pressure within the bubble must be somewhat higher than the pressure of the surrounding liquid. It follows that the vapor (and liquid) temperature must be somewhat higher than the