Heat Transfer Handbook part 70 docx

BOOKCOMP, Inc. — John Wiley & Sons / Page 684 / 2nd Proofs / Heat Transfer Handbook / Bejan 684 BOILING 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [684], (50) Lines: 1797 to 1806 ——— -4.65503pt PgVar ——— Long Page PgEnds: T E X [684], (50) 0.0 0.0 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1.0 1.0 0 0 50 1000 100 2000 150 3000 200 4000 250 5000 300 7000 6000 350 8000 400 Vapor quality Vapor quality Mass Velocity (kg/m . s) 2 Heat Transfer Coef. (W/m . K) 2 nT D q-butane sat = 60°C = 19.89mm = 15kW/m 2 nT D q-butane sat = 60°C = 19.89mm = 15kW/m 2 SW I A MF S G = 20kg/m s 2 G = 60kg/m s 2 G = 200kg/m s 2 Figure 9.16 Simulation of Kattan–Thome–Favrat model for pure n-butane at 60°C, showing flow pattern map and heat transfer coefficients. • Fully stratified flow for ˙m = 20 kg/m 2 · satallχ values, with a monotonic decrease in α tp with increasing χ • Stratified–wavy flow for ˙m = 60 kg/m 2 · satallχ values shown that give a moderate peak in α tp versus χ • Intermittent flow for ˙m = 200 kg/m 2 ·s with χ ≤ 0.4 that shows a moderate rise in α tp versus χ BOOKCOMP, Inc. — John Wiley & Sons / Page 685 / 2nd Proofs / Heat Transfer Handbook / Bejan FLOW BOILING IN HORIZONTAL TUBES 685 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [685], (51) Lines: 1806 to 1851 ——— 12.0pt PgVar ——— Long Page PgEnds: T E X [685], (51) • Annular flow for ˙m = 200 kg/m 2 · s with 0.4 < χ < 0.93 that results in an ever-steeper rise in α tp versus χ as the annular film thins out before the onset of dryout occurs at χ = 0.93 • Annular flow with partial dryout (modeled as stratified–wavy flow) for ˙m = 200 kg/m 2 · s with χ ≥ 0.93, where a sharp decline in α tp versus χ after the peak occurs Although not illustrated, α tp goes to its natural limit of α vapor at χ = 0. On the other hand, for all liquid flow, the convective boiling heat transfer coefficient α cb for liquid film flow does not go to the tubular value. Hence, α cb should be obtained with the Dittus–Boelter or Gnielinski correlation when χ = 0. Including more recent data for evaporation of ammonia for mass velocities as low as 16.3 kg/m 2 ·s and results for evaporation of refrigerant–oil mixtures, their flow boiling model is applicable over the following parameter ranges: • 1.12 ≤ p sat ≤ 8.9 bar • 0.0085 ≤ p r ≤ 0.225 • 16.3 ≤˙m ≤ 500 kg/m 2 · s • 0.01 ≤ χ ≤ 1.0 • 440 ≤ q ≤ 71,600 W/m 2 • 17.03 ≤ M ≤ 152.9 (but up to about 300 for refrigerant–oil mixtures) • 74 ≤ Re L ≤ 20,399 and 1300 ≤ Re G ≤ 376,804 • 1.85 ≤ Pr L ≤ 5.47 (but up to 134 for refrigerant–oil mixtures) • 0.00016 ≤ µ L ≤ 0.035N · s/m 2 (i.e., 0.16 to 35 cP) • Tube metals (copper, carbon steel, and stainless steel) For annular flows, the accuracy of this new method is similar to those of the Shah (1982), Jung et al. (1989), and Gungor-Winterton (1986, 1987) correlations; however, the latter methods do not provide a method to determine when annular flow conditions exist. For stratified–wavy flows, the Kattan–Thome–Favrat model has been shown to be twice as accurate as the best of these other methods, even though these other correlations have stratified flow threshold criteria and corresponding heat transfer correction factors. At χ > 0.85, typical of direct-expansion evaporator applications, the Kattan–Thome–Favrat model is three times more accurate than the best of these other methods, which have standard deviations of over ±80%. The Kattan–Thome–Favrat model is implemented as follows for a given tube internal diameter, specific design conditions, and fluid physical properties: 1. Determine the local flow pattern corresponding to the local design condition using the Kattan–Thome–Favrat flow pattern map (Section 9.7) together with the local heat flux, vapor quality, and mass velocity. 2. Calculate the local vapor void fraction ε with eq. (9.130). 3. Calculate the local liquid cross-sectional area A L with eq. (9.131). BOOKCOMP, Inc. — John Wiley & Sons / Page 686 / 2nd Proofs / Heat Transfer Handbook / Bejan 686 BOILING 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [686], (52) Lines: 1851 to 1884 ——— 10.0pt PgVar ——— Normal Page PgEnds: T E X [686], (52) 4. If the flow is annular or intermittent (intermittent flow is thermally modeled as if it were annular), determine the annular liquid film thickness δ from eq. (9.136) with θ dry set to 0. 5. If the flow is stratified–wavy (note that the flow pattern map classifies annular flow with partial dryout at the top of the tube as being stratified–wavy), eq. (9.132) is first utilized to calculate θ strat , then the values of ˙m high and ˙m low are determined with the flow pattern map at vapor quality χ. Next, θ dry is calculated with eq. (9.134) if χ ≤ χ max or with eq. (9.135) if χ > χ max , and then the annular liquid film thickness δ is determined from eq. (9.136) with this value of θ dry . 6. If the flow is fully stratified, use eq. (9.132) to calculate θ strat and then deter- mine the annular liquid film thickness δ from eq. (9.136) using the value of θ strat for θ dry . 7. Determine the convective boiling heat transfer coefficient α cb with eq. (9.128). 8. Calculate the vapor-phase heat transfer coefficient α vapor with eq. (9.129) if part of the wall is dry. 9. For a pure, single-component liquid or an azeotropic mixture, the nucleate pool boiling heat transfer coefficient α nb is determined with eq. (9.127) using the total local heat flux q. 10. Calculate the heat transfer coefficient on the wetted perimeter of the tube α wet with eq. (9.126) using the values of α nb and α cb . 11. Determine the local flow boiling coefficient α tp with eq. (9.125). For evaporation of zeotropic mixtures and refrigerant–oil mixtures, refer to Sec- tion 9.12. 9.9.3 Subcooled Boiling Heat Transfer Fully developed subcooled boiling is characterized by vapor formation at the heated wall in the form of single bubbles or as a bubbly layer parallel to the wall. These bubbles are swept into the subcooled area of the liquid flow by the variable shear stress on their boundary imposed by the turbulent flow velocity profile. The bubbles then condense in the subcooled core. To predict local heat transfer coefficients in the subcooled boiling regime, Gungor and Winterton (1986) have adapted their corre- lation by using separate temperature differences for driving the respective nucleate boiling and convective boiling processes so that the heat flux is calculated as a sum of their contributions as q = α L [ T w − T L (z) ] + Sα nb (T wall − T sat ) (9.137) This formula predicted their database with a mean error of ±25%. The methods pre- sented earlier for saturated forced-convective evaporation may be adapted to sub- cooled flow boiling in an analogous manner. BOOKCOMP, Inc. — John Wiley & Sons / Page 687 / 2nd Proofs / Heat Transfer Handbook / Bejan BOILING ON TUBE BUNDLES 687 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [687], (53) Lines: 1884 to 1896 ——— -0.073pt PgVar ——— Normal Page PgEnds: T E X [687], (53) 9.10 BOILING ON TUBE BUNDLES In the foregoing sections we addressed flow boiling when it occurs inside tubes. Boiling on the outside of horizontal tube bundles is another important process, typical of kettle and thermosyphon reboilers, waste heat boilers, fire tube steam generators, and flooded evaporators. 9.10.1 Heat Transfer Characteristics Figure 9.17 depicts a simplified tube bundle layout. Subcooled liquid enters the bun- dle from below and flows upward past the tubes. Until the wall temperature surpasses the saturation temperature of the liquid, single-phase convective heat transfer occurs. Once the wall temperature is above T sat , subcooled boiling may occur and the bubbly flow regime begins. Farther up the bundle, the bulk fluid temperature reaches the sat- uration temperature and saturated boiling begins. The rapid departure of sequential bubbles tends to form bubble jets from the top of tubes. With coalesce of these bub- bles into a larger size, sliding bubbles are formed as they pass between adjacent tubes, characterized by a thin evaporating film of liquid betweenthebubble and the wall. The flow becomes ever more chaotic and locally unstable and the chugging flow regime is Figure 9.17 Boiling on a horizontal tube bundle. (From Collier and Thome, 1994.) BOOKCOMP, Inc. — John Wiley & Sons / Page 688 / 2nd Proofs / Heat Transfer Handbook / Bejan 688 BOILING 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [688], (54) Lines: 1896 to 1925 ——— -4.58496pt PgVar ——— Short Page PgEnds: T E X [688], (54) encountered. At higher vapor qualities with increasing vapor shear on the liquid films on the tubes, the liquid becomes entrained in the vapor, which becomes the contin- uous phase. This is the spray flow regime and the tubes are wetted by the impact of droplets that may maintain a continuous liquid film on the tubes. Thus, the active heat transfer modes in bundle boiling are nucleate boiling, convective boiling, and thin- film evaporation. At some critical condition x cr illustrated in the diagram, dryout of the tubes may occur with a substantial decrease in the local heat transfer coefficient. Typically, for saturated inlet conditions the local flow boiling coefficient at the bottom of the tube is similar in value to that for nucleate pool boiling on a single tube (methods described in Section 9.5). As the local vapor quality rises from bottom to top in the bundle, the influence of convection becomes more and more important. At the top of the bundle, the heat transfer coefficient may become as high as three to four times that at the bottom. 9.10.2 Bundle Boiling Factor Bundle boiling coefficients can be analyzed by normalizing the bundle coefficient α b with the single tube nucleate pool boiling coefficient α st , where α st is either measured or calculated using a nucleate pool boiling correlation and α b may refer either to a local value within the bundle or to the mean value for the entire tube bundle. This ratio, known as the bundle boiling factor F b , indicates the relative enhancing effect of two-phase convection in the bundle compared to the pool boiling coefficient, such that F b = α b α st (9.138) The value of F b tends toward 1.0 at high heat fluxes and high reduced pressures because the nucleate boiling coefficient becomes dominant. For plain tubes and low finned tubes, local values of F b tend to range from about 1.0 at the bottom of the bundle, when the inlet flow is all liquid and thus the convective effect is minimal, up to as high as 3 or 4 near the top tube rows, where the convective contribution is very pronounced. Mean bundle values of F b , on the other hand, are normally in the range 1.5 to 2.0. 9.10.3 Bundle Design Methods The simplest thermal design method is to assume a value of the bundle boiling factor, such as F b = 1.5 as a conservative value. Then, after calculating the single-tube boiling heat transfer coefficient using one of the methods in Section 9.5, the mean bundle boiling coefficient is obtainable from eq. (9.138). Another simple approach has been proposed by Palen (1983), where the mean bundle boiling heat transfer coefficient α b is assumed to be a superposition of the contributions of boiling and natural convection as α b = α st F b F c + α nc (9.139) BOOKCOMP, Inc. — John Wiley & Sons / Page 689 / 2nd Proofs / Heat Transfer Handbook / Bejan POST-DRYOUT HEAT TRANSFER 689 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [689], (55) Lines: 1925 to 1947 ——— 1.67pt PgVar ——— Short Page PgEnds: T E X [689], (55) Here α st is the single-tube nucleate pool boiling coefficient, F b the bundle boiling factor, F c the mixture boiling correction factor (see Section 9.12), and α nc the natural convection heat transfer coefficient for the tube bundle. Palen recommends using α nc = 250 W/m 2 · K and F b = 1.5. For pure fluids and azeotropic mixtures, F c is equal to 1.0; for zeotropic mixtures, its value may vary from 0.1 to 1.0. Various Chen-type in-tube boiling correlations have been proposed for evaporation on the outside of plain tube bundles. In this approach, the liquid-only heat transfer co- efficient to the liquid phase in eq. (9.89) is calculated using a correlation for turbulent crossflow over a tube bundle rather than the Dittus–Boelter in-tube correlation, and the nucleate boiling coefficient is predicted with one of the methods in Section 9.5. New expressions for the boiling suppression factor S and two-phase multiplier F are then formulated, sometimes with the boiling suppression factor set to unity. So far, these methods have had only limited success in predicting local bundle boiling heat transfer coefficients since they are typically based on small databases composed of only one combination of tube diameter and tube pitch and one or two fluids, and hence are not applicable for general use. 9.11 POST-DRYOUT HEAT TRANSFER 9.11.1 Introduction Post-dryout heat transfer occurs during forced-flow evaporation when the heated surface becomes dry before complete evaporation. It refers to the heat transfer process downstream from the point at which the surface became dry and may occur at any vapor quality or even during subcooled flow boiling. Post-dryout heat transfer is also referred to as the liquid-deficient regime or as mist flow heat transfer; however, these terms do not describe the process when it occurs at low vapor quality. In general, the post-dryout heat transfer regime is entered from the wet wall regime by passing through one of the following transitions in the evaporation process: • Dryout of the liquid film. A liquid film (such as in an annular flow) may com- pletely evaporate, leaving only the entrained liquid droplets in the vapor to be evaporated. • Entrainment of the liquid film. For a high vapor shear stress on the liquid film, the liquid may be pulled from the surface and become entirely entrained in the vapor phase. • Critical heat flux. Imposing a large heat flux or wall superheat at the wall may create a continuous layer of vapor on the wall, starting from dryout under a single small bubble, a large Taylor bubble, or a dense packing of bubbles. For the transition from annular to mist flow, refer to Section 9.7 on two-phase flow maps. For further discussion on the mechanisms and prediction of the critical heat flux, refer to the reviews by Weisman (1992), Katto (1994, 1996), and Celata (1997). BOOKCOMP, Inc. — John Wiley & Sons / Page 690 / 2nd Proofs / Heat Transfer Handbook / Bejan 690 BOILING 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [690], (56) Lines: 1947 to 1969 ——— 0.0pt PgVar ——— Normal Page PgEnds: T E X [690], (56) The heated wall is not completely dry in the post-dryout heat transfer regime. En- trained droplets impinge on the surface and wet it locally before either evaporating or rebounding back into the vapor phase. Second, in horizontal channels, the upper portion of the heated periphery of the flow channel becomes dry while the bottom remains wetted by the flowing liquid, such that there is simultaneously flow boiling heat transfer on the bottom and post-dryout heat transfer around the top. This partial dryout of the perimeter of the heated channel may also occur when there is a signifi- cant variation in the peripheral heat flux, such as a boiler tube exposed to radiant heat from only one direction. The post-dryout regime may be reached during saturated boiling in a channel but may also be encountered during subcooled boiling at high heat fluxes. Only saturated boiling is discussed here. The post-dryout regime may be encountered in fossil-fuel boilers and fired heaters, on nuclear power plant fuel rod assemblies during a hypothetical loss-of-coolant accident, in direct-expansion evaporators and air-conditioning coils, and in cooling of various high-heat-fluxdevices. Heat transfer coefficients in the post-dryout regime are significantly lower than those for wet wall evaporation. In this chapter, first thermal nonequilibrium effects and heat transfer phenomena particular to post-dryout flow are described, and then methods for predicting heat transfer under post-dryout conditions inside channels with uniform boundary conditions are presented. 9.11.2 Thermal Nonequilibrium The wall superheat during wet wall evaporation remains relatively small, typically be- low 15 to 30 K. In the post-dryout regime, instead, the local wall temperature may be- come significantly higher than the saturation temperature, such that a departure from equilibrium occurs. The two limiting cases are illustrated in Fig. 9.18. For complete departure from equilibrium, heat is transferred only to the continuous vapor phase. If heat absorbed by the entrained droplets is insignificant, the vapor temperature T G (z) downstream from the point of dryout rises with the sensible heating of the vapor. Similarly, the wall temperature T w (z) rises like that of a single-phase convective flow, giving the temperature profile illustrated in Fig. 9.18a. Post-dryout evaporation tends toward the case of complete thermal nonequilibrium at low pressures and low mass flow rates at high vapor qualities. For complete thermodynamic equilibrium, illustrated in Fig. 9.18b, the rate of heat transfer to the entrained droplets is assumed to be so effective that the vapor tem- perature T G (z) remains at the saturation temperature as the droplets evaporate. The wall temperature T w (z) varies depending on the intensity of the droplet evaporation process. Evaporation tends toward thermal equilibrium at high reduced pressures and very high mass flow rates. A typical process path is illustrated in Fig. 9.19, where the local vapor temperature is lower than that occurring for complete nonequilibrium, but is still significantly above the local saturation temperature of the complete equilibrium case. Hence, for post-dryout heat transfer the temperature of the vapor is not known a priori but is part of the solution. Thermodynamic equilibrium means that all the heat absorbed by the fluid is utilized to evaporate the liquid, and hence the local equilibrium vapor quality BOOKCOMP, Inc. — John Wiley & Sons / Page 691 / 2nd Proofs / Heat Transfer Handbook / Bejan POST-DRYOUT HEAT TRANSFER 691 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [691], (57) Lines: 1969 to 1971 ——— 3.42099pt PgVar ——— Normal Page PgEnds: T E X [691], (57) Figure 9.18 Thermodynamic states in the post-dryout regime. (From Collier and Thome, 1994.) is χ e (z). If, instead, all the heat does into superheating the vapor after the onset of dryout, the vapor quality remains that at the dryout point χ DO (z). In between, the actual local vapor quality χ a (z) is somewhere between these two limits such that χ DO (z) < χ a (z) < χ e (z). Consider Fig. 9.19, which depicts post dryout in a vertical tube of internal diameter d i heated uniformly with a heat flux of q. Dryout occurs at a length z DO from the inlet, and it is assumed that thermodynamic equilibrium exists at the dryout point. If complete equilibrium is maintained after dryout, all the liquid will be evaporated when point z e is reached. However, in the actual situation, only a fraction (κ)ofthe surface heat flux is used to evaporate the remaining liquid in the post-dryout region while the remainder is used to superheat the bulk vapor. The liquid is thus evaporated completely only when a downstream distance of z a is reached. Assuming that the total heat flux q(z) from the tube wall to the fluid is comprised of the heat flux associated BOOKCOMP, Inc. — John Wiley & Sons / Page 692 / 2nd Proofs / Heat Transfer Handbook / Bejan 692 BOILING 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [692], (58) Lines: 1971 to 1992 ——— 0.54103pt PgVar ——— Long Page * PgEnds: Eject [692], (58) Dryout Post-dryout region Superheating region Uniform heat flux q z e z DO z a Temperature ( )T Vapor Quality ( )␹ Wall temp. Bulk vapor Based on thermodynamic equilibrium Saturation temp. Based on thermodynamic equilibrium Tz w () Tz G () Thermodynamic quality ( )␹ e z ␹ a ()z ␹ DO ␹()z Actual variation of vapor quality 1 Length ( )z Figure 9.19 Departure from thermodynamic equilibrium in the post-dryout regime. (From Collier and Thome, 1994.) with droplet evaporation q L (z) and the heat flux associated with vapor superheating q G (z), then q(z) = q L (z) + q G (z) (9.140) Furthermore, let κ = q L (z) q(z) (9.141) where κ is considered independent of tube length, so that the profiles of the actual bulk vapor temperature and actual vapor quality are linear. The vapor quality for z<z e is given by an energy balance BOOKCOMP, Inc. — John Wiley & Sons / Page 693 / 2nd Proofs / Heat Transfer Handbook / Bejan POST-DRYOUT HEAT TRANSFER 693 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [693], (59) Lines: 1992 to 2046 ——— 2.61432pt PgVar ——— Long Page PgEnds: T E X [693], (59) χ(z) − χ DO = 4 d i ˙mh LG (z − z DO ) (9.142) where h LG is the latent heat of vaporization, ˙m is the total mass velocity, and point z e is given by z e =  d i ˙mh LG 4q  1 − χ DO   + z DO (9.143) The variation in the actual vapor quality χ a (z) with length for z<z a is χ a (z) − χ DO = 4κq d i ˙mh LG (z − z DO ) (9.144) where z a is z a =  d i ˙mh LG 4κq (1 − χ DO )  + z DO (9.145) Combining eq. (9.142) with (9.145) yields κ = χ a (z) − χ DO χ(z) − χ DO = z a − z DO z e − z DO (9.146) The actual bulk vapor temperature T G,a (z) is thus T G,a (z) = T sat + 4(1 − κ)q(z − z DO ) ˙mc pG d i (9.147) for z<z a , while for z a >zit is T G,a (z) = T sat + 4q(z − z e ) ˙mc pG d i (9.148) The two limiting cases in Fig. 9.18 are obtained by setting κ = 0 and κ = 1, respectively, in the expressions above. In reality, κ is not independent of tube length and must be predicted from the actual process conditions. As illustrated in Fig. 9.19, small droplets may remain entrained in the vapor wall beyond the location of χ e (z), where one is tempted to believe that all the flow is superheated vapor. 9.11.3 Heat Transfer Mechanisms Post-dryout heat transfer may occur in the dispersed flow regime, in which the vapor phase becomes the continuous phase and all the liquid is entrained as dispersed droplets or as inverted annular flow, in which the vapor forms an annular film on the tube wall and the liquid is in the central core. The first typically occurs after dryout or entrainment of an annular film flow, while the second occurs when the critical heat flux is exceeded at low vapor quality or in a subcooled liquid. In inverted annular flow, . POST-DRYOUT HEAT TRANSFER 9.11.1 Introduction Post-dryout heat transfer occurs during forced-flow evaporation when the heated surface becomes dry before complete evaporation. It refers to the heat transfer. azeotropic mixture, the nucleate pool boiling heat transfer coefficient α nb is determined with eq. (9.127) using the total local heat flux q. 10. Calculate the heat transfer coefficient on the wetted perimeter. α nc (9.139) BOOKCOMP, Inc. — John Wiley & Sons / Page 689 / 2nd Proofs / Heat Transfer Handbook / Bejan POST-DRYOUT HEAT TRANSFER 689 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [689],