BOILING HEAT TRANSFER AND TWO-PHASE FLOW. In a discussion of two-phase flow, several fundamental quantities need defining: (1) A (2) A (3) A

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1 BOILING HEAT TRANSFER AND TWO-PHASE FLOW When a heated surface exceeds the uration temperature of the surroundin olant, boilin on the surface bemes possible. This is true whether the bulk fluid temperature is at or below the local uration temperature. If the bulk fluid temperature is below the uration temperature, boilin is referred to as "local" or "suboled" boilin. If the bulk fluid temperature is equal to the uration temperature, then "bulk" boilin is said to occur. Bubbles formed on the heated surface depart the surface and are transported by the bulk fluid, such that a ndition of two-phase flow is said to exist. Dependin on the deree of subolin and the lenth of the heated channel, the bubbles may or may not ndense and llapse prior to exitin the channel. In suboled boilin this process results in further heatin of the fluid toward the uration temperature. In urated or bulk boilin, bubbles can be transported alon the entire lenth of the heated channel without llapsin. Fundamental Concepts In a discussion of two-phase flow, several fundamental quantities need definin: Phase Velocity v k = velocity of phase k, where k may represent the liquid ( k ) or vapor (k ) phase Volume Fraction The volume fraction of phase k in a two-phase mixture is k Vk () V where V k is the volume occupied by phase k, and V is the total two-phase volume. If the volume in question nsists of the cross sectional area of a flow channel times a lenth sement z, the volume fraction can be nsidered an area fraction k A (2) A where A x is the total cross section flow area. The vapor volume fraction is often referred to as the void fraction () even thouh the vapor volume is filled with a low density as and no true void exist. Note: Volumetric Flux (Superficial Velocity) The volumetric flux of phase k is defined to be the volumetric flow rate of phase k divided by the total flow area, i.e. k x Slip Ratio j k vk Ak kvk (3) A x The slip ratio is defined as the ratio of the vapor and liquid phase velocities. v S v (4) 43

2 If the liquid and vapor velocities are equal, then the slip ratio is one and the flow is said to be Homoeneous. Quality Three qualities are of particular interest in our analysis of two-phase systems, these include: Equilibrium Quality (x e ) x e h h h f f (5) The equilibrium quality rresponds to the flow fraction of vapor only if thermodynamic equilibrium exists between the phases. Since the equilibrium quality is defined in terms of the fluid enthalpy, it can have values reater than one and less than zero. Under these nditions the equilibrium quality can be thouht of as a measure of the deree of subolin or superheat of the fluid, but can not be used to determine the fluid state. Flow Quality (x) The flow quality is the true flow fraction of the vapor phase and is always between zero and one, reardless of the fluid's state. We define the flow quality as m x m m (6) m m The mass flow rate of the liquid phase is Similarly for the vapor phase, m v A v A x (7) m v A v A (8) x such that v v v m x m m (9) Note: The equilibrium quality has physical meanin only when between zero and one, even thouh values can be calculated outside of this rane. The flow quality represents the true flow fraction of the vapor phase and can only have values between zero and one. Under uration nditions, the equilibrium and flow quality are equivalent. Static Quality ( x s ) The static quality is the mass fraction of the vapor phase, and like the flow quality is always between zero and one. We define the static quality as 44

3 x s M M (0) M M M The mass of the liquid phase is M V V () Similarly for the vapor phase, M V V (2) such that M x s (3) M M Note: The static quality and the flow quality are equivalent under Homoeneous Flow nditions. Void-Quality Relationships Mass flux is defined as the mass flow rate divided by the cross sectional flow area of the channel. In a two-phase system, the total system mass flow rate is the sum of the liquid and vapor mass flow rates such that From the definition of flow quality, v v G m m A x (4) Gx v v (5) and Dividin these two equations and rearranin ives for the slip ratio G( x) v ( ) v (6) x S x (7) or solvin for void fraction x S x (8) Equation 8 is often referred to as the Fundamental Void-Quality-Slip relation. In eneral, the quality and void fraction increase ntinuously alon the channel, which implies the slip ratio also varies ntinuously alon the channel. 45

4 The distribution of vapor in a boilin system affects both the heat transfer and the flow properties of the fluid. A number of flow patterns or flow reimes have been observed experimentally by viewin flow of liquid-vapor mixtures throuh transparent tubes. While the number and characteristics of specific flow reimes are somewhat subjective, four principal flow reimes are almost universally accepted. These patterns are illustrated in Fiure and include Bubbly Flow (a & b), Slu Flow (c), Churn or Churn-Turbulent Flow (d), and Annular Flow (e). (a) (b) (c) (d) (e) Bubbly Bubbly Slu Churn Annular Fiure : Typical Flow Reimes (From Thermohydraulics of Two-Phase Systems for Industrial Desin and Nuclear Enineerin, by J. M. Delhaye, M. Giot and M. L. Riethmuller) These flow reimes may be enerally characterized as Bubbly Flow: Slu Flow: Churn Flow: Annular Flow: Individual dispersed bubbles transported in a ntinuous liquid phase. Lare bullet shaped bubbles separated by liquid plus. The vapor flows in a somewhat chaotic manner throuh the liquid, with the vapor enerally ncentrated in the center of the channel, and the liquid displaced toward the channel walls. The vapor forms a ntinuous re, with a liquid film flowin alon the channel walls. To predict the existence of a particular flow reime, or the transition from one flow reime to another, requires that the visually observed flow patterns be quantified in terms of measurable (or mputed) quantities. This is normally acmplished throuh the use of flow reime maps. Two typical flow reime maps are iven below. The Hewitt and Roberts map is valid for both air-water and steam-water systems. The Govier and Aziz map was obtained for air-water flows in inch diameter tubes. 46

5 Fiure 2: Hewitt and Roberts Flow Reime Map (from Delhaye, Giot and Riethmuller) Fiure 3: Govier and Aziz Flow Reime Map for Air-Water (from Lahey and Moody) 47

6 Flow Boilin Reimes A number of different heat transfer mechanisms are possible when a fluid is heated in a boilin channel. These different heat transfer mechanisms are often represented by use of a boilin curve. A boilin curve for low quality, hih heat flux systems is illustrated below. c* ln q" c d e c' a b G < G 2 < G 3 G 3 G G 2 ln (Twall - T) Fiure 4: Flow Boilin Curve for Low Quality, Hih Heat Flux Systems In Fiure 4, the heat transfer rate is plotted as a function of the wall superheat (the difference between the wall temperature and the fluid uration temperature). The curve is divided into 5 reions (a e), each of which rresponds to a different heat transfer mechanism. These different heat transfer reions are discussed briefly below. Reion a: The minimum criteria for boilin, is that the temperature of the heated surface exceed the local uration temperature, i.e. some deree of wall superheat is required for boilin to occur. In reion a, wall superheat is insufficient to support bubble formation and rowth. Heat transfer is by sinle-phase forced nvection and is a stron function of fluid velocity (mass flux) and temperature. Reion b: Bubbles bein formin at nucleation sites on the heated surface. These nucleation sites are enerally associated with pits or crevices on the heated surfaces in which non dissolved ases or vapor can accumulate allowin bubble formation. As the bubbles row and depart the surface they carry latent heat, as well as enerate increased turbulence and mixin which increases the heat transfer rate. Boilin under these nditions is referred to as nucleate boilin. In reion b, heat transfer is a mplicated mixture of sinle-phase forced nvection and nucleate boilin. As a result, this reion is often called the mixed boilin or partial nucleate boilin reion. In eneral, as the wall temperature increases, the fraction of the wall surface subject to nucleate boilin increases, until bubble formation occupies the entire heated surface. Reion c: In reion c, bubble density increases rapidly with increasin wall superheat. Heat transfer is dominated by local nditions in the vicinity of the wall enerated by bubble rowth and departure. These bubbles transport lare amounts of latent heat from the surface at the fluid uration temperature and reatly increase fluid turbulence and mixin in the vicinity of the wall. As a result, heat transfer bemes independent of bulk fluid nditions such as flow velocity and temperature. Heat transfer is said to be by fully developed nucleate boilin and is 48

7 characterized by substantial increases in heat transfer rate for moderate increases in wall temperature. The bubble density at the wall however can not be increased indefinitely. At point c*, the bubble density bemes sufficiently hih to impede liquid flow back to the surface. Bubbles tend to alesce, formin insulatin vapor patches that reduce the heat transfer rate. Point c* is called the point of Critical Heat Flux (CHF), and the critical heat flux mechanism is Departure from Nucleate Boilin or DNB. Reion d: Further increases in wall superheat result in increasinly reater portions of the heated surface vered by insulatin vapor patches. The reduction in effective heat transfer area more than mpenes for the increase in wall temperature to reduce the overall heat transfer rate. This reion is referred to as the partial film or transition film boilin reion. Reion e: A ntinuous vapor film mpletely blankets the heated surface. Heat transfer is by nduction and nvection throuh the superheated vapor layer with evaporation at the liquid/vapor interface. Wall temperatures can beme sufficiently hih, such that radiative heat transfer bemes important. This reion is called the stable film boilin reime. Steady state operation beyond the point of critical heat flux is only possible for wall temperature ntrolled systems, where the heat input to the surface can be adjusted to maintain a iven wall temperature. In reactor systems, it is power and therefore heat flux which is ntrolled. In a heat flux ntrolled system, an increase in the heat flux beyond the critical point results in Departure from Nucleate Boilin with an associated increase in the wall temperature. This increase in the wall temperature causes more of the heated surface to be blanketed by vapor, further increasin the wall temperature. The wall temperature would then follow a transient path, jumpin from c* to c where steady state operation at the new heat flux would be possible, assumin the hihly elevated wall temperatures were within the material limits of the system. In reactors, these elevated wall temperatures can easily lead to fuel failure. Departure from Nucleate Boilin is the dominant critical heat flux mechanism in Pressurized Water Reactors. In low heat flux, hih quality systems typical of Boilin Water Reactor operation, thermal-hydraulic nditions within the re allow for the transition to annular flow. Vapor velocities and interfacial turbulence are sufficiently hih to suppress nucleation in the thin liquid film adjacent to the heated surface. Heat is transferred by nduction and nvection throuh the liquid film with evaporation at the liquid/vapor interface. This heat transfer mechanism is referred to as forced nvection vaporization and is characterized by extremely hih heat transfer efficients. In fact, heat transfer efficients can be so hih, that increased heat transfer rates can be achieved with decreasin wall temperatures. The boilin curve under these nditions is illustrated in Fiure 5. 49

8 d d* d' ln q" a b c e f G < G 2 < G 3 G 3 G G 2 ln (Twall - T) Fiure 5: Flow Boilin Curve for Hih Quality, Low Heat Flux Systems Reions a - c are identical to those in Fiure 4, with Reion d the forced nvection vaporization reion. At point d*, the liquid film bemes so thin that dry patches can appear on the heated surface lowerin the heat transfer rate. The critical heat flux under these nditions is not due to DNB but results from the mplete evaporation or dryout of the liquid film flowin alon the heated surface. The critical heat flux mechanism is then said to be dryout dominated. Further increases in wall temperature beyond the point of critical heat flux results in dryout of increasinly reater portions of the heated surface and a rrespondin reduction in the heat transfer rate, eventually leadin to mplete evaporation of the liquid film. As in the DNB dominated system, steady state operation beyond the point of critical heat flux is only possible in wall temperature ntrolled systems. Heat transfer in reion f is due primarily to sinle-phase forced nvection to a super heated vapor, mbined with evaporation of entrained liquid droplets. Radiation can also beme important at hih wall temperatures. Temperature excursions followin dryout are typically less severe than those followin DNB, as some sinle-phase forced nvection olin is available from the vapor re. In reactor systems however, wall temperatures can still reach levels such that fuel failure is likely. Fiure 6 illustrates the heat transfer and flow reimes which miht be expected in a dryout ntrolled channel. The point at which critical heat flux is reached, reardless of the mechanism, has also been referred to as the boilin crisis. 50

9 Fiure 6: Heat Transfer and Flow Reimes in a Boilin Channel (from Convective Boilin and Condenion, by J. G. Collier) 5

10 Heat Transfer in Boilin Channels As illustrated in Fiure 6, a number of different heat transfer reimes can occur simultaneously within a boilin channel. This is further illustrated in Fiure 7 below. Fully Developed Nucleate Boilin z = z B Mixed Boilin z = z n Sinle Phase Forced Convection z = 0 Fiure 7: Boilin Reimes in a Heated Channel The position z n is called the Incipient Boilin Point or the Nucleation Point and is the position where the wall superheat is sufficient to support bubble rowth. The position zb denotes the onset of fully developed nucleate boilin. Prior to boilin z [0,zn ], heat transfer is by sinle phase forced nvection. The heat flux is linear with wall temperature and iven by Newton s Law of olin q ( z) h ( T ( z) T ( z)) (9) where the nvective heat transfer efficient h c is enerally rrelated in the form c 0.8 m Nu C Re Pr. (20) Correlations of this type include the Dittus-Boelter Correlation for flow in nduits and annuli, and the Weisman rrelation for flow parallel to rod bundles. At the onset of boilin, the heat flux bemes nonlinear with respect to wall temperature. In the fully developed nucleate boilin reime ( z z ), heat flux is usually rrelated in the form B 52

11 where: q( z) Btu / hr - ft 2 T F q() z m T () z T 0 6 (2) or solvin for the clad surface temperature T () z T m m q() z 0 6 (22) Two popular rrelations of this type valid for both local and bulk boilin nditions in water are the Jens-Lottes rrelation exp( 4P 900) 4 60 m 4 and the Thom rrelation exp( 2P 260) 2 72 m 2 where P is pressure in psia. Assumin the Jens-Lottes rrelation, the outer clad temperature is T ( z) T. 897q( z) 4 exp( P 900) (23) A number of models exist for mputin the wall temperature in the mixed boilin reion ( z [zn,zb] ). One such model for suboled boilin is that by Berles and Rohsenow / 2 2 ( ) ( ) ( ) ( ) q NB z qnb zn q z q FC z (24) ( ) ( ) q FC z q NB z where: q FC (z) h [ T ( z) T ( z)] is the heat flux associated with sinle phase forced nvection c q NB (z) T m 0 6 [ T ( z) ] is the heat flux associated with fully developed nucleate boilin The Incipient Boilin Point is enerally rrelated in terms of a critical wall superheat. A mmonly used rrelation of this type is / P q ( z ) 5.6P [ T ( z ) T ] (25) n n where q z ) is the local heat flux at the Incipient Boilin Point (Btu/hr-ft 2 ) and P is pressure (psia). ( n In order to apply Equations 9, 22 and 24 to an arbitrarily heated channel where both sinle-phase forced nvection and nucleate boilin may occur, the transition points from sinle-phase forced nvection to mixed boilin and from mixed boilin to fully developed nucleate boilin must be determined. These transition points are usually taken to 53

12 insure the wall temperature is ntinuous at the transition point. The followin procedure can be used to determine the transition points and mpute wall temperature in a boilin channel. In the sinle phase forced nvection reion, the wall temperature at any point is iven by q z T () z T () z ( ) (26) h where h c is the sinle-phase forced nvection heat transfer efficient and T () z is the local fluid temperature. In a sinle channel with no mixin, the local fluid temperature can be obtained from the simple enery balance c p 0 T ( z) T (0) q( z) Ddz mc z (27) At the Incipient Boilin Point, we require the wall temperature in Equation 25 to isfy Equation 26. The Incipient Boilin Point is then the solution of 2.30 / P.56 q( zn ) q ( zn ) 5.6P T ( zn ) T (28) hc where p 0 T ( zn ) T (0) q( z) Ddz mc zn (29) such that for a iven mass flow rate, inlet nditions and heat flux profile, the only unknown in Equation 28 is z n. In eneral, Equation 28 is transcendental and must be solved iteratively. Once z n is known, where NB n 6 n m q ( z ) 0 [ T ( z ) T ] (30) T q( zn ) ( zn ) T ( zn ) (3) h such that q NB ( z n ) is a nstant. The wall temperature at any location in the mixed boilin reion is the solution of Equation 24, q( z) q FC ( ) q z q NB FC c ( z) q ( ) NB zn ( z) q NB ( z) / 2 2 where the only unknown at any location is T (z). 54

13 At the transition point between mixed and fully developed nucleate boilin, the wall temperature isfies q ( z ) 0 [ T such that at the transition point, Equation 24 reduces to B 6 ( z B ) T ] m q NB ( z B ) where / 2 2 ( ) ( ) ( ) ( ) q z B qnb zn q z B qfc zb (32) ( ) ( ) q FC zb q zb q z ) h [ T ( z ) T ( z )] (33) FC ( B c B B T m q ( z m B ) ( zb ) T 6 (34) 0 and p 0 zb T ( zb ) T (0) q( z) Ddz mc (35) For a iven heat flux profile and channel operatin nditions, Equations 32, 33, 34 and 35 can be reduced to a sinle nonlinear equation in the boilin transition point z B. No adequate criteria has been established to determine the transition from nucleate boilin to forced nvection vaporization. However, a sinle rrelation that is valid for both nucleate boilin and forced nvection vaporization has been developed by Chen for urated boilin nditions and extended to include suboled boilin by others. Acrdin to the Chen rrelation, the heat flux can be related to the wall (or clad) temperature by q( z) h [ T ( z) T ( z)] h2 [ T ( z) T ] (36) lo w w where h lo is a liquid only nvective heat transfer efficient and h 2 is a nucleate boilin heat transfer efficient. The liquid only nvective heat transfer efficient is similar to the Dittus-Boelter rrelation where the Reynolds number is mputed based on the liquid mass flux h lo G( x) D C e p k k D e F (37) The Reynolds number factor F is an experimentally determined rrection factor and is defined to be the ratio of the true two-phase Reynolds number to the sinle-phase, liquid only Reynolds number, i.e. F Re Re lo : Re lo G x D e The Reynolds number factor is illustrated in Fiure 8 below and can be expressed analytically in terms of the turbulent Martinelli Parameter tt as 55

14 tt F tt tt (38) where tt x x 09. f f (39) The nucleate boilin efficient has the form h kf Cpf f c f hf hf J Tv f 075. T T S w 099. (40) and is in terms of an experimentally determined nucleate boilin suppression factor S. The suppression factor is a measure of the true super heat in the liquid film and is defined as Tave S Tw T The suppression factor is illustrated in Fiure 9 and can be expressed analytically as (4) Re 2 S tan (42) 56

15 Fiure 8: Reynolds Number Factor Fiure 9: Suppression Factor 57

16 Example: The hot channel in a PWR operates over a sinificant fraction of its lenth under nucleate boilin. Assumin that boilin occurs at the position of maximum heat flux, determine the clad temperature and the boilin heat transfer efficient at this point. Clad Temperature Problem Data Core Averaed Heat Flux 89,800 Btu/hr-ft 2 Power Peakin Factor 2.5 System Pressure 2250 psia Assumin the Jens-Lottes rrelation for the boilin heat transfer efficient, the clad temperature at any point alon the boilin lenth is T ( z) T. 897q( z) exp( P 900) 4. If we denote the maximum heat flux by q 0, then the clad temperature at this point is iven by T T. 897q exp( P 900) At a system pressure of 2250 psia, the fluid uration temperature is F. The maximum heat flux in the channel is obtained from Boilin Heat Transfer Coefficient q0 qmax Fq q ( 2. 5)( 89, 800) 474, 500 Btu Hr - ft 2 T ( 474, 500) 4 exp( ) F Writin the heat flux in terms of Newton's Law of Coolin, q( z) q( z) hc ( T ( z) T ) hc T ( z) T At the position of maximum heat flux, the boilin heat transfer efficient is then 474, h c 4, 89 Btu Hr-ft -F Note the manitude of the boilin heat transfer efficient relative to a nvective heat transfer efficient