BOOKCOMP, Inc. — John Wiley & Sons / Page 664 / 2nd Proofs / Heat Transfer Handbook / Bejan 664 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 [664], (30) Lines: 1026 to 1052 ——— -0.03pt PgVar ——— Normal Page PgEnds: T E X [664], (30) upward. One explanation for the instability is that the gravity and shear forces acting on the thin film of liquid of Taylor bubbles become similar in magnitude, such that the flow direction of the film oscillates between upward and downward. This flow pattern is a transition regime between the slug flow and annular flow regimes. In small-diameter tubes, churn flow may not develop such that the flow passes directly from slug flow to annular flow. • Annular flow. Here the bulk of the liquid flows as a thin film on the wall with the gas as the continuous phase flowing up the center of the tube, forming a liquid annulus with a gas core whose interface is disturbed by both large-magnitude waves and chaotic ripples. Liquid may be entrained in the high-velocity gas core as small droplets; the liquid fraction entrained may be similar to that in the film. This flow regime is quite stable and is often desirable for system operation and pipe flow. • Wispy annular flow. When the flow rate isincreased further, the entrained droplets congregate to form large lumps or wisps of liquid in the central vapor core with a very disturbed annular liquid film. • Mist flow. When the flow rate is increased even further, the annular film becomes very thin, such that the shear of the gas core on the interface is able to entrain all the liquid as droplets in the continuous gas phase (i.e., the inverse of the bubbly flow regime). The wall is intermittently wetted locally by impinging droplets. The droplets in the mist may be too small to be seen without special lighting and/or magnification. Flow patterns in horizontal two-phase flows are influenced by the effect of gravity, which acts to stratify the liquid to the bottom and the gas to the top of the channel. Flow patterns encountered in co-current flow of gas and liquid in a horizontal tube are shown in Fig. 9.10. The commonly identifiable flow patterns are: • Bubbly flow. The bubbles are dispersed in the continuous liquid with a higher concentration in the upper half of the tube because of buoyancy effects. However, at high mass velocities, the bubbles tend to be dispersed uniformly in the tube as shear forces become dominant. • Stratified flow. At low liquid and gas velocities, there is complete separation of the two phases, with the gas in the top and the liquid in the bottom, separated by an undisturbed horizontal interface. • Stratified–wavy flow. With increasing gas velocity, waves form on the liquid– gas interface traveling in the direction of the flow. The amplitude of the waves depends on the relative velocity of the two phases, but their crests do not reach the top of the tube. The waves have a tendency to wrap up around the sides of the tube, leaving thin films of liquid on the wall after passage of the wave. • Intermittent flow. Further increasing the gas velocity, the waves grow in magni- tude until they reach the top of the tube. Thus, large amplitude waves wash the top of the tube intermittently, while slower-moving smaller-amplitude waves are often evident in between. The large-amplitude waves contain a large amount of BOOKCOMP, Inc. — John Wiley & Sons / Page 665 / 2nd Proofs / Heat Transfer Handbook / Bejan TWO-PHASE FLOW PATTERNS 665 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 [665], (31) Lines: 1052 to 1064 ——— 0.097pt PgVar ——— Normal Page PgEnds: T E X [665], (31) Figure 9.10 Flow pattens in horizontal flow. liquid and often have entrained bubbles. The top wall is often wet continuously from the thin liquid film left behind by each large-amplitude wave. • Annular flow. Similar to vertical upflow, at large gas flow rates the liquid forms a continuous annular film around the perimeter of the tube, which tends to be noticably thicker at the bottom than the top. The interface of the film is typically disturbed by small-amplitude waves, and droplets may be dispersed in the gas core. At high gas fractions, the top of the tube eventually becomes dry first, with the flow reverting to the stratified–wavy flow regime. • Mist flow. Similar to that occurring in vertical flow, all the liquid may become entrained as small droplets in the high-velocity continuous gas phase. Intermittent flow is actually a composite of the plug and slug flow regimes. These subcategories may be described as follows: BOOKCOMP, Inc. — John Wiley & Sons / Page 666 / 2nd Proofs / Heat Transfer Handbook / Bejan 666 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 [666], (32) Lines: 1064 to 1099 ——— -1.71793pt PgVar ——— Normal Page PgEnds: T E X [666], (32) • Plug flow. This pattern is characterized by liquid plugs that are separated by elongated gas bubbles. The diameters of the bubbles are smaller than the tube, such that the liquid phase is continuous along the bottom of the tube. Plug flow may also be referred to as elongated bubble flow. • Slug flow. At higher gas velocities, bubbles are entrained in the liquid slugs, and the elongated bubbles become similar in size to the channel height. The liquid slugs can also be characterized as large-amplitude waves. 9.7.2 Flow Pattern Maps for Vertical Flows For vertical upflow, Fig. 9.13 (shown later) shows the typical regimes that would be encountered from inlet to outlet of an evaporator tube. It is beyond our scope in this chapter to present flow pattern maps for vertical flows. A flow pattern map is a diagram utilized to delineate the transitions between the flow patterns, typically plotted on log-log axes using dimensionless parameters to represent the liquid and gas velocities. Hewitt and Roberts (1969) and Fair (1960) are widely quoted flow pattern maps for vertical upflows. 9.7.3 Flow Pattern Maps for Horizontal Flows For evaporation, Fig. 9.14 (shown later) depicts the typical sectional views of the flow structure. For condensation, the flow regimes are similar with the exception that the top tube wall is not dry in stratified types of flow, but instead, is coated with a thin condensing film of condensate. The most widely quoted flow pattern maps for predicting the transition between two-phase flow regimes for adiabatic flow in horizontal tubes are those of Baker (1954) and Taitel and Dukler (1976), whose descriptions are available in numerous books and publications. Their transition curves should be considered as zones similar to that between laminar and turbulent flow. For small-diameter tubes typical of heat exchangers, Kattan et al. (1998a) pro- posed a modification of the Steiner (1993) map, which itself is a modified Taitel– Dukler map, and included a method for predicting the onset of dryout at the top of the tube in evaporating annular flows. This flow pattern map is presented here as it is used in Section 9.9 for predicting local flow boiling coefficients based on the local flow pattern. The flow regime transition boundaries of the Kattan–Thome–Favrat flow pattern map are depicted in Fig. 9.11 (bubbly flow is at very high mass velocities and is not shown). This map provides the transition boundaries on a linear–linear graph with mass velocity plotted versus gas or vapor fraction for the particular fluid and flow channel, which is much easier to use than the log-log format of other maps. The transition boundary curve between annular and intermittent flows to stratified– wavy flow is ˙m wavy = 16A 3 Gd gd i ρ L ρ G χ 2 π 2 1 −(2h Ld − 1) 2 0.5 π 2 25h 2 Ld (1 −χ) −F 1 (q) We Fr −F 2 (q) L + 1 0.5 + 50 (9.70) BOOKCOMP, Inc. — John Wiley & Sons / Page 667 / 2nd Proofs / Heat Transfer Handbook / Bejan TWO-PHASE FLOW PATTERNS 667 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 [667], (33) Lines: 1099 to 1125 ——— 4.07436pt PgVar ——— Normal Page * PgEnds: Eject [667], (33) 0.0 0.2 0.4 0.6 0.8 1.0 0 100 200 500 300 600 400 700 Vapor quality Mass Velocity (kg/m . s) 2 R-134a Dia = 12mm = 10°CT sat Stratified wavy Intermittent Annular Mist flow Stratified [Strat] [IA] [Mist] [Wavy] q=0 kW/m 2 q=15 kW/m 2 Figure 9.11 Kattan–Thome–Favrat flow pattern map illustrating flow regime transition boundaries. The high-vapor-quality portion of this curve depends on the ratio of the Froude number (Fr L ) to the Weber number (We L ), where Fr L is the ratio of the inertia to the surface tension forces and We L is the ratio of inertia to gravity forces. The mass velocity threshold for the transition from annular flow to mist flow is ˙m mist = 7680A 2 Gd gd i ρ L ρ G χ 2 π 2 ξ Ph Fr We L 0.5 (9.71) Evaluating the expression above for the minimum mass velocity of the mist flow transition gives the value of χ min , which for χ > χ min is ˙m mist =˙m min (9.72) The transition between stratified–wavy flow and fully stratified flow is given by the expression ˙m strat = (226.3) 2 A Ld A 2 Gd ρ G (ρ L − ρ G )µ L g χ 2 (1 −χ)π 3 1/3 (9.73) The transition threshold into bubbly flow is ˙m bubbly = 256A Gd A 2 Ld d 1.25 i ρ L (ρ L − ρ G )g 0.3164(1 −χ) 1.75 π 2 P id µ 0.25 L 1/1.75 (9.74) BOOKCOMP, Inc. — John Wiley & Sons / Page 668 / 2nd Proofs / Heat Transfer Handbook / Bejan 668 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 [668], (34) Lines: 1125 to 1182 ——— 2.59824pt PgVar ——— Normal Page PgEnds: T E X [668], (34) In the equations above, the ratio of We to Fr is We Fr L = gd 2 i ρ L σ (9.75) and the friction factor is ξ Ph = 1.138 +2 log π 1.5A Ld −2 (9.76) The nondimensional empirical exponents F 1 (q) and F 2 (q) in the ˙m wavy boundary equation include the effect of heat flux on the onset of dryout of the annular film: the transition of annular flow into annular flow with partial dryout, the latter classified as stratified–wavy flow by the map. They are F 1 (q) = 646.0 q q DNB 2 + 64.8 q q DNB (9.77a) F 2 (q) = 18.8 q q DNB + 1.023 (9.77b) The Kutateladze (1948) correlation for the heat flux of departure from nucleate boil- ing, q DNB , is used to normalize the local heat flux: q DNB = 0.131ρ 1/2 G h LG g(ρ L − ρ G )σ 1/4 (9.78) The vertical boundary between intermittent flow and annular flow is assumed to occur at a fixed value of the Martinelli parameter X tt , equal to 0.34, where X tt is defined as X tt = 1 −χ χ 0.875 ρ G ρ L 0.5 µ L µ G 0.125 (9.79) Solving for χ, the threshold line of the intermittent-to-annular flow transition at χ IA is χ IA = 0.2914 ρ G ρ L −1/1.75 µ L µ G −1/7 + 1 −1 (9.80) Figure 9.12 defines the geometrical dimensions of the flow, where P L is the wetted perimeter of the tube, P G the dry perimeter in contact with only vapor, h the height of the completely stratified liquid layer, and P i the length of the phase interface. Similarly, A L and A G are the corresponding cross-sectional areas. Normalizing with the tube internal diameter d i , six dimensionless variables are obtained: h Ld = h d i P Ld = P L d i P Gd = P G d i P id = P i d i BOOKCOMP, Inc. — John Wiley & Sons / Page 669 / 2nd Proofs / Heat Transfer Handbook / Bejan TWO-PHASE FLOW PATTERNS 669 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 [669], (35) Lines: 1182 to 1230 ——— 0.17027pt PgVar ——— Normal Page PgEnds: T E X [669], (35) A Ld = A L d 2 i A Gd = A G d 2 i (9.81) For h Ld ≤ 0.5: P Ld = 8(h Ld ) 0.5 − 2 [ h Ld (1 −h Ld ) ] 0.5 3 P Gd = π −P Ld A Ld = 12 [ h Ld (1 −h Ld ) ] 0.5 + 8(h Ld ) 0.5 h Ld 15 A Gd = π 4 − A Ld (9.82) For h Ld > 0.5: P Gd = 8(1 −h Ld ) 0.5 − 2 [ h Ld (1 −h Ld ) ] 0.5 3 P Ld = π −P Gd A Gd = 12 [ h Ld (1 −h Ld ) ] 0.5 + 8(1 −h Ld ) 0.5 (1 −h Ld ) 15 A Ld = π 4 − A Gd (9.83) For 0 ≤ h Ld ≤ 1: P id = 2 [ h Ld (1 −h Ld ) ] 0.5 (9.84) Since h is unknown, an iterative method utilizing the following equation is necessary to calculate the reference liquid level h Ld : X 2 tt = P Gd + P id π 1/4 π 2 64A 2 Gd P Gd + P id A Gd + P id A Ld π P Ld 1/4 64A 3 Ld π 2 P Ld (9.85) Once the reference liquid level h Ld is known, the dimensionless variables are calcu- lated from eqs. (9.82)–(9.84) and the transition curves for the new flow pattern map are determined with eqs. (9.70)–(9.80). Figure 9.12 Cross-sectional and peripheral fractions in a circular tube. BOOKCOMP, Inc. — John Wiley & Sons / Page 670 / 2nd Proofs / Heat Transfer Handbook / Bejan 670 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 [670], (36) Lines: 1230 to 1265 ——— 4.58908pt PgVar ——— Normal Page PgEnds: T E X [670], (36) This map was developed from a database for five refrigerants: two single-com- ponent fluids (R-134a and R-123), two near-azeotropic mixtures (R-402A and R- 404A), and one azeotropic mixture (R-502). The test conditions covered the following range of variables: mass flow rates from 100 to 500 kg/m 2 · s, vapor qualities from 4 to 100%, heat fluxes from 440 to 36,500 W/m 2 , saturation pressures from 0.112 to 0.888 MPa, Weber numbers from 1.1 to 234.5, and liquid Froude numbers from 0.037 to 1.36. The Kattan–Thome–Favrat flow pattern map correctly identified 96.2% of these flow pattern data. Z ¨ urcher et al. (1997) obtained additional two-phase flow pattern observations for the zeotropic refrigerant mixture R-407C at an inlet saturation pressure of 0.645 MPa, and the map accurately identified these new flow pattern data. Z ¨ urcher et al. (1999) also obtained two-phase flow pattern data for ammonia with a 14–mm bore sight glass for mass velocities from 20 to 140 kg/m 2 ·s, vapor qualities from 1 to 99% and heat fluxes from 5000 to 58,000 W/m 2 , all taken at a saturation temperature of 4°C and saturation pressure of 0.497 MPa. Thus, the mass velocity range in the database was extended from 100 kg/m 2 ·sdownto20kg/m 2 ·s. In particular, it was observed that the transition curve ˙m strat was too low and eq. (9.73) was corrected empirically by adding +20χ as follows: ˙m strat = (226.3) 2 A Ld A 2 Gd ρ G (ρ L − ρ G )µ L g χ 2 (1 −χ)π 3 1/3 + 20χ (9.86) where ˙m strat is in kg/m 2 ·s. The transition from stratified–wavy flow to annular flow at high vapor qualities was, instead, observed to be too high, and hence an additional empirical term with an exponential factor modifying the boundary at high vapor qualities was added to eq. (9.70) to take this into account as ˙m wavy(new) =˙m wavy − 75e −(χ 2 −0.97) 2 /χ(1−χ) (9.87) where the mass velocity is in kg/m 2 · s. The movement of these boundaries has an effect on the dry angle calculation θ dry in the Kattan et al. (1998c) flow boiling heat transfer model and shifts the onset of dryout to slightly higher vapor qualities, which is in agreement with the ammonia heat transfer test data. To utilize this map, the following parameters are required: vapor quality (χ), mass velocity ( ˙m), tube internal diameter (d i ), heat flux (q), liquid density (ρ L ), vapor density (ρ G ), liquid dynamic viscosity (µ L ), vapor dynamic viscosity (µ G ), surface tension (σ), and latent heat of vaporization (h LG ), all in SI units. The local flow pattern is identified by the following procedure: 1. Solve eq. (9.85) iteratively with eqs. (9.79), (9.82), (9.83), and (9.84). 2. Evaluate eq. (9.81). 3. Evaluate eqs. (9.75)–(9.78). 4. Evaluate eqs. (9.70), (9.71) or (9.72), (9.73), (9.74), and (9.80). 5. Compare these values to the given values of χ and ˙m to identify the flow pattern. BOOKCOMP, Inc. — John Wiley & Sons / Page 671 / 2nd Proofs / Heat Transfer Handbook / Bejan FLOW BOILING IN VERTICAL TUBES 671 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 [671], (37) Lines: 1265 to 1276 ——— 0.25099pt PgVar ——— Normal Page PgEnds: T E X [671], (37) Note that eq. (9.87) should be used in place of eqs. (9.70) and (9.86) should be used in place of eq. (9.80) to utilize the most updated version. The map is thus specific to the fluid properties, flow conditions (heat flux), and tube internal diameter input into the equations. The map can be programmed into any computer language, evaluating the transition curves in incremental steps of 0.01 in vapor quality to obtain a tabular set of threshold boundary points, which can then displayed as a complete map with ˙m versus χ as coordinates. 9.8 FLOW BOILING IN VERTICAL TUBES Convective evaporation in a vertical tube is depicted in Fig. 9.13. At the inlet, the liquid enters subcooled. As the liquid heats up, the wall temperature rises until it Figure 9.13 Flow patterns during evaporation in a vertical tube with a uniform heat flux. (From Collier and Thome, 1994.) BOOKCOMP, Inc. — John Wiley & Sons / Page 672 / 2nd Proofs / Heat Transfer Handbook / Bejan 672 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 [672], (38) Lines: 1276 to 1315 ——— 4.24005pt PgVar ——— Normal Page PgEnds: T E X [672], (38) surpasses the saturation temperature of the liquid, at which point subcooled boiling begins, where bubbles nucleate and grow in the thermal boundary layer and condense in the subcooled core. Farther up the tube, the liquid bulk reaches its saturation temperature and the convective boiling process passes through the bubbly flow, churn flow, and finally, into the annular flow regime. At the dryout point, the annular liquid film is depleted completely, either by evaporation or by entrainment into the vapor core, and the wall temperature rises significantly in order to dissipate the applied heat flux. The process proceeds in the post-dryout (mist or drop flow) regime. First, saturated wet wall convective boiling is described, and post-dryout heat transfer is then described in Section 9.11. In the forced-convective evaporation regime the heat transfer coefficient is less dependent on heat flux than in nucleate pool boiling, while its dependence on the local vapor quality appears as a new and important parameter. Both the nucleate and convective heat transfer mechanisms must be taken into account to predict heat transfer data in the convective boiling regime. Their relative importance varies from dominance of nucleate boiling at low vapor qualities and high heat fluxes to the dominance of convection at relatively high vapor qualities and low heat fluxes. 9.8.1 Chen Correlation The Chen (1963) correlation was the first to attain widespread acclaim and has served as the starting point for the development of most other flow boiling correlations since. Superposition of the nucleate boiling and convective boiling mechanisms is assumed, such that the local two-phase flow boiling coefficient α tp is obtained as α tp = α nb + α cb (9.88) where α nb is the nucleate boiling contribution and α cb is the convective contribution, which Chen presented as α tp = α FZ S +α L F (9.89) The Forster and Zuber (1955) correlation is used to calculate the pool boiling heat transfer coefficient α FZ as α FZ = 0.00122 λ 0.79 L c 0.45 pL ρ 0.49 L σ 0.5 µ 0.29 L h 0.24 LG ρ 0.24 G ∆T 0.24 sat ∆p 0.75 sat (9.90) while for liquid-only heat transfer, coefficient α L is obtained with the Dittus–Boelter correlation: α L = 0.023Re 0.8 L · Pr 0.4 L λ L d i (9.91) The liquid Reynolds number Re L above is based on the fraction of liquid flowing alone in the channel [i.e., using ˙m(1 − χ) to calculate Re L ]. In the Forster–Zuber BOOKCOMP, Inc. — John Wiley & Sons / Page 673 / 2nd Proofs / Heat Transfer Handbook / Bejan FLOW BOILING IN VERTICAL TUBES 673 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 [673], (39) Lines: 1315 to 1350 ——— 1.66414pt PgVar ——— Normal Page PgEnds: T E X [673], (39) equation, the wall superheat ∆T sat and pressure difference ∆p sat are the active pa- rameters: the pressure difference refers to that obtained with the fluid’s vapor pres- sure curve evaluated at the wall and saturation temperatures, respectively, and ∆p sat is in N/m 2 . Thus, this approach leads to an iterative calculation when the heat flux is specified since the wall temperature is not known beforehand. Forced flow creates a sharper temperature gradient at the wall relative to that in nucleate pool boiling, which has an adverse effect on bubble nucleation. Thus, nucle- ation is partially suppressed, which Chen accounted for by introducing a nucleation suppression factor S. The convective boiling contribution α cb is a product of α L times a two-phase multiplier F , which enhances this heat transfer mode. The suppression factor S, two-phase multiplier F , Martinelli parameter X tt , and two-phase Reynolds number Re tp used in his method are calculated as follows: S = 1 1 +0.00000253Re 1.17 tp (9.92) F = 1 X tt + 0.213 0.736 (9.93) X tt = 1 −χ χ 0.9 ρ G ρ L 0.5 µ L µ G 0.1 (9.94) Re tp = Re L · F 1.25 (9.95) Note, however, that when 1/X tt ≤ 0.1,F is set equal to 1.0. The Chen correlation is applicable over the entire evaporation range in which the heated wall remains wet. 9.8.2 Shah Correlation Shah (1982) proposed a method for implementing his chart method. Similar to Chen, he included two distinct mechanisms: nucleate boiling and convective boiling. How- ever, instead of adding these two contributions, his method chooses the larger of the nucleate boiling coefficient α nb and the convective boiling coefficient α cb .Inhis method, the first step is to calculate the dimensionless parameter N , which for vertical tubes at all values of the liquid Froude number Fr L is given as N = C 0 (9.96) where C 0 is determined from C 0 = 1 −χ χ 0.8 ρ G ρ L 0.5 (9.97) For N>1.0, the values of α nb and α cb are calculated from the following expressions and the larger value is chosen as the local heat transfer coefficient α tp . The value of the liquid-only convective heat transfer coefficient α L used in these expressions . 668 / 2nd Proofs / Heat Transfer Handbook / Bejan 668 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 [ 668] ,. Kattan et al. (1998c) flow boiling heat transfer model and shifts the onset of dryout to slightly higher vapor qualities, which is in agreement with the ammonia heat transfer test data. To utilize. new and important parameter. Both the nucleate and convective heat transfer mechanisms must be taken into account to predict heat transfer data in the convective boiling regime. Their relative