BOOKCOMP, Inc. — John Wiley & Sons / Page 1369 / 2nd Proofs / Heat Transfer Handbook / Bejan EVAPORATION AND CONDENSATION 1369 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 [1369], (11) Lines: 351 to 371 ——— 1.13803pt PgVar ——— Normal Page PgEnds: T E X [1369], (11) Phenomena Related to LOCA Interest in the condensation of a vapor in a liquid has been motivated by concern about loss-of-coolant accidents (LOCA) in light-water nuclear reactors. In this situation, unwanted steam generated in the core is forced through a pool of water. These phenomena can be quite complicated because of the various forms of unsteady steam flow that can exist. The heat transfer coefficients and the liquid temperatures can experience wide-ranging time fluctuations here. Aya and Nariai (1991) have summarized the various modes of condensation that can occur in this situation. They have indicated the importance of pool water subcool- ing. Note that these types of phenomena occur normally near standard atmospheric pressure. They gave an approximate map that shows the differentiation between the various types of behavior. This is shown in Fig. 19.3 and is discussed below. If a steam jet is directed into a liquid water pool at lower mass flux rates (i.e., the lower-left-hand region of Fig. 19.3), a phenomenon denoted as chugging can occur. For this situation, the results of Young et al. (1974) can be used: h = 6.5ρ L C P,L v 0.6 S ν L d 0.4 (19.19) Aya and Nariai (1991) noted that this result did not include the effect of subcooling, which they felt was very important. At higher mass flux rates (say, greater than about 25 kg/s · m 2 ) than those asso- ciated with eq. (19.19), another type of oscillation can occur. The frequency of this 010203040 10 20 30 40 50 60 70 80 90 100 Steam Mass Flow Flux (kg/s m ) 2 Water Temperature (°C) Bubbling region Chugging region Condensation oscillation region Transition region Figure 19.3 Map of condensation phenomena that can develop in loss of coolant accidents. (After Aya and Nariai, 1991.) BOOKCOMP, Inc. — John Wiley & Sons / Page 1370 / 2nd Proofs / Heat Transfer Handbook / Bejan 1370 DIRECT CONTACT HEAT TRANSFER 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 [1370], (12) Lines: 371 to 391 ——— 0.81305pt PgVar ——— Short Page PgEnds: T E X [1370], (12) one is closely related to that of the periodic growth and reduction of a steam bubble attached to the jet exit. This phenomenon is referred to in the literature simply as condensation oscillation. The region where this phenomenon takes place is shown on the right-hand side of Fig. 19.3. A correlation that represents the data quite well has been given by Fukuda (1982): ¯ h = 43.78 k L d dG S ρ L ν L 0.9 C P,L ∆T h fg (19.20) Aya and Nariai (1991) also outlined the magnitudes of the various types of con- densation phenomena when steam is condensed in pool water. These are shown in Fig. 19.4. Although the values are affected by numerous variables, including time, and are difficult to illustrate exactly, general trends can be shown. The highest of these modes is the film coefficient on the vapor side (not the overall value). This is denoted as “steam-side interfacial” in the figure. Moving down in Fig. 19.4, the next variation is shown for the “chugging” region. Along the line indicated, Steam side interfacial Chugging Condensation oscillation Jet in steam flow Jet in vessel Liquid drop Stratified Laminar jet 10 7 10 6 10 5 10 4 10 3 10 2 10 50 100 Subcooling (K) Heat transfer coefficient (W/m . K) 2 Figure 19.4 Various regimes of condensation of steam in water according to Aya and Nariai (1991). BOOKCOMP, Inc. — John Wiley & Sons / Page 1371 / 2nd Proofs / Heat Transfer Handbook / Bejan EVAPORATION AND CONDENSATION 1371 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 [1371], (13) Lines: 391 to 413 ——— 0.65005pt PgVar ——— Short Page PgEnds: T E X [1371], (13) the higher values are for the individual transient processes, while the lower values denote the average values. “Condensation oscillation” is affected by both the pool subcooling as well as the steam mass flow flux. Higher values of the latter are found in the upper portion of the shaded region. Heat transfer coefficients for this regime can be predicted using eq. (19.20). Near the middle levels, the “jet in steam flow” and the “jet in vessel” typical ranges are shown. Values in the former category are affected by the steam mass flux at higher levels of subcooling. The water pool water can be stratified in certain situations. When this is the case, the heat transfer coefficients are then decreased. This variation is shown by the range denoted as “stratified” in Fig. 19.4. Here the heat transfer is influenced by many variables, including the mass flux and the degree of stratification. Ranges of the heat transfer coefficients for laminar jets and droplets (Sideman and Moalem-Maron, 1982) are also shown for comparison purposes. The jet values are influenced by the diameter of the jet as well as mass flux rate. The droplet variations shown assume a small contact time. Longer contact times would greatly reduce the magnitudes of h. More recently, Ju et al. (2000) reported a study on details of the condensation process. They used holographic interferometry to determine the heat transfer to con- densing bubbles associated with the application of core makeup tanks. 19.3.3 Evaporation of a Liquid by a Surrounding Vapor, Gas or Liquid Droplet Evaporation in a Vapor or Gas A problem of great importance in applications is the spray cooling of a hot gas. In this situation, liquid droplets are evaporated by the warmer gas, cooling the latter. Unlike condensing systems, the presence of the gas does not impede the process. Several workers have focused on this problem, motivated by a variety of applications. Using a stochastic modeling approach, Carey and Hawks (1995) analyzed small droplets evaporating in their own superheated vapor. For larger microdroplets, they found the following relationship: hd k = 2 ln(Ja v − 1) Ja v (19.21) where the Jakob number is given by Ja = C pv (T ∞ − T R )/h fg . A more complicated result is given for microdrops. Tong and Sirignano (1984) analyzed the evaporation of multicomponent droplets in a hot gas. They used a simplified model for this problem that has application to evaporation of fuel in combustion systems. No correlation was presented in this work, but time-varying results were shown for some specific cases. Droplet Evaporation in a Liquid Çoban and Boehm (1989) modified the results given by several others for the evaporation of immiscible droplets in a continuous liquid. Their particular application was for organic liquids evaporating in water. In the model, the extremely high heat transfer coefficients associated with the evaporating BOOKCOMP, Inc. — John Wiley & Sons / Page 1372 / 2nd Proofs / Heat Transfer Handbook / Bejan 1372 DIRECT CONTACT HEAT TRANSFER 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 [1372], (14) Lines: 413 to 463 ——— 3.0052pt PgVar ——— Short Page * PgEnds: Eject [1372], (14) portion of the droplet were combined in appropriate ways with the poorer heat transfer through the vaporized portion of the droplet. This result is hd k L = (Re c · Pr c ) 1/3 + 5k c k L (1 −) (19.22) where ≡ 0.466(π − β + 0.5 sin 2β) 2/3 and β is vapor half-opening angle of the droplet, assuming that the vapor portion of the evaporating droplet accumulates in the top part of the droplet. When the flow of the two fluids is in the same direction, operational aspects change considerably. This is discussed further in Section 19.4.1. Work has also been performed for a three-phase exchanger where the dispersed phase is injected into a stagnant continuous phase (Smith et al., 1982). Specific results were found for cyclopentane injected into a vessel of water, and these results were compared to a numerical approach that used the drift flux model. Only short transient runs were possible for this situation. Both a preagglomeration state and a postagglomeration stage were considered. In both cases, the thermal resistance of the dispersed phase was assumed to be negligible, and single-droplet velocity was assumed to be of the form v = v 0 r c 1 (19.23) In this equation and the two below, the 0 subscript denotes the initial value of the droplet. Further, the single droplet heat transfer was calculated as hd k c = c 2 · Re c 3 C · Pr 1/3 C (19.24) The value of c 3 is known to be in the range 0.7 to 1.0. For the preagglomeration stages, the heat transfer of the bubble as a function of the travel distance in the stagnant water was expressed by h(z) = 2ψφ 0 h d0 d 0 (1 + ψz) 3/ψ − 1 ψBz (19.25) where B ≡ 2h 0 ∆T ν 0 d 0 h fg ρ d,L − ρ d,V ρ d,L ρ d,V and ψ ≡ (1 − c 1 )(1 + c 3 ) + 1 and with c 1 , c 2 , and c 3 as defined by comparison to eqs. (19.23) and (19.24). For the postagglomeration correlation, Smith et al. (1982) found the following result: BOOKCOMP, Inc. — John Wiley & Sons / Page 1373 / 2nd Proofs / Heat Transfer Handbook / Bejan COLUMNS AND OTHER CONTACTORS 1373 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 [1373], (15) Lines: 463 to 492 ——— -0.37991pt PgVar ——— Short Page * PgEnds: Eject [1373], (15) h(z)d 0 Bz 2h d0 = φ 0 [(1 + ψ 0 Bz a ) 3/ψ 0 − 1] + φ max 1 − φ max × 1 + ψ a Bz a + 6(1 − φ max ) 1−c 3 ψ a B(z − z a ) 1/2ψ a − (1 + ψ a Bz a ) 1/2ψ a (19.26) In this equation the subscript a is used to denote the onset of agglomeration. The same order of magnitude of predicted heat transfer coefficients for evaporation were noted as those for results reported in the literature. Studies of direct-contact evaporation of R114 injected into stagnant water near saturation were reported by Celata et al. (1995). Photographic means were used to quantify the rate at which the refrigerant evaporated and the amount of the liquid remaining at after the droplet. More recently they investigated a related problem, but with set amounts of subcooling (Celata et al., 1999). 19.4 COLUMNS AND OTHER CONTACTORS 19.4.1 Spray Columns Columns, including spray columns and columns with internals, can be used for a variety of direct contact heat transfer processes. Columns with internals can include tray columns and columns with packing. Packings can range from spheres to more complicated geometries called structured packings. The latter are preferred from the perspective of minimizing cost and maximizing performance. Columns allow intimate contact between a dispersed phase and a continuous phase. In condensation situations, the liquid (sink) phase is usually continuous. A review of a variety of process heating devices has been given by Jacobs (1988b). Three analysis approaches have been used. One draws on previous design practice for conventional, indirect heat exchangers, another uses concepts from mass transfer device analysis, and the third uses more detailed analyses of the behavior of a typical droplet in the device and accumulates the effects for all, to find overall performance. Both the first and third approaches are discussed here in some detail. First consider the design approach analogous to that used for closed heat exchang- ers. No matter which of the many types of heat transfer phenomena are present (e.g., desuperheating, condensation, and subcooling could all be taking place in a given device), the overall heat transfer is calculated using the LMTD approach. If the area between the two separate fluids is known, the calculation can be made: Q = U A A i · LMTD = U A A i ∆T 1 − ∆T 2 ln(∆T 1 /∆T 2 ) (19.27) BOOKCOMP, Inc. — John Wiley & Sons / Page 1374 / 2nd Proofs / Heat Transfer Handbook / Bejan 1374 DIRECT CONTACT HEAT TRANSFER 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 [1374], (16) Lines: 492 to 533 ——— 8.0261pt PgVar ——— Normal Page PgEnds: T E X [1374], (16) In this equation, U A denotes the overall heat transfer coefficient based upon the fluid interface area A i . The subscripts 1 and 2 denote the temperature differences between the two fluids at each of the ends of the contacting device. Several problems exist with this approach. For one, the area-based heat transfer coefficient on the fundamental droplet level may be difficult to estimate. Second, the area of that interface can also be difficult to determine. Third, even if the area is known, it is not unusual for it to vary considerably throughout the contactor. A more easily evaluated approach to this that has been used in many studies reported in the literature is to replace the area-based heat transfer characteristics with volume-based values. This is given as Q = U V V · LMTD = U V V ∆T 1 − ∆T 2 ln ( ∆T 1 /∆T 2 ) (19.28) In this equation, V denotes the total contactor internal volume and U V is the volumet- ric heat transfer coefficient based on that volume. Although the latter can suffer from imprecision about details of the heat transfer variations as indicated in the area-based approach, the assumption made here is that the result is a composite for the overall situation. Another way of dealing with the elusive concept of the basis for the overall heat transfer coefficient is to use a term for the surface area of the droplets per unit volume of the column. In this manner, the heat transfer coefficient can be written as U A a. Here the U A is the traditional surface heat transfer coefficient, but it is now based on the area of the droplets, while a is the area of the droplets per unit volume of the heat exchanger. Hence U A a = U V (19.29) This does not simplify the problems of determining the area-to-volume information explained above; it is simply another notation used in the literature. Reminiscent of mass transfer operations, where columns have traditionally been used most frequently, the transfer unit technique is often applied. In this approach, a volumetric heat transfer coefficient, usually determined from empirical data, is used. The concept of a stage comes into play. This is illustrated in Fig. 19.5. One situation that occurs frequently is that of partial condensation of a superheated vapor by a cool liquid. In this case, the volumetric heat transfer coefficient can be written as (Fair, 1972; Sideman and Moalem-Maron, 1982) U V = 1 1/h V,L + (1/βh V,G )(Q G /Q T ) (19.30) In this equation, h V,L and h V,G represent the heat transfer coefficients on the liquid side and the gas (vapor) side in direct contact, respectively; and Q T and Q G , respec- tively, are the total heat transfer and the amount of heat transfer involved in removing the superheat. The subscript G is used to differentiate it from locations where actual condensation is occurring. Finally, the term β is used for the following: BOOKCOMP, Inc. — John Wiley & Sons / Page 1375 / 2nd Proofs / Heat Transfer Handbook / Bejan COLUMNS AND OTHER CONTACTORS 1375 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 [1375], (17) Lines: 533 to 567 ——— 0.9722pt PgVar ——— Normal Page PgEnds: T E X [1375], (17) Figure 19.5 Temperature versus distance plot of heat exchanger with stages outlined. The first stage is at the left, the second is indicated explicitly, and a portion of a third exists at the right. β ≡ ˆ NC P,G /h V,G V 1 − e − ˆ NC P,G /h V,G V (19.31) Here the term ˆ N is used to denote the mass transfer rate. Note that when the contactor only includes condensation (no desuperheating) and the liquid (sink) is the same compound as the vapor, the vapor–liquid portion is not present, and the vapor-side heat transfer coefficient in the overall computation can be neglected. For this case, the following results: U V = h L,V (19.32) The overall height of a heat transfer unit (Z i,T ) can be written as Z i,T = Z i,G + Z i,L G G C P,G G L C P,L Q L Q G (19.33) This is a characteristic of the specific heat transfer equipment and process. Finally, the number of transfer units (NTUs) can be found: NTU g = T g2 T g1 dT g T g − T ≈ ln T − T g1 T − T g2 (19.34) From this the total column height can be found: Z T = V T A = NTU g · Z i,T (19.35) In this equation, A denotes the cross-sectional area of the empty column. Global Treatments Spray columns offer the simplest contactor configuration. In this arrangement, a column is outfitted with inlets for each of the continuous BOOKCOMP, Inc. — John Wiley & Sons / Page 1376 / 2nd Proofs / Heat Transfer Handbook / Bejan 1376 DIRECT CONTACT HEAT TRANSFER 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 [1376], (18) Lines: 567 to 574 ——— 0.927pt PgVar ——— Normal Page PgEnds: T E X [1376], (18) and dispersed fluids and their corresponding outlets. The lighter fluid is admitted at the bottom of the column and flows upward in a down-coming heavier fluid, due to the influence of gravity. The lighter fluid can serve as either the dispersed or the continuous component, although most applications find it as the former. Special arrangements (often called disengagement zones) are incorporated in both the top and bottom of the column to allow the two fluids to be separated. A simple schematic of one of these types of columns is shown in Fig. 19.6. Although the details of the actual introduction of the two fluids is not shown in this figure, the general arrangement is quite simple. One of the more important variables that affect heat transfer in a spray column is the holdup (denoted here by the symbol φ). This is defined as the amount of the Figure 19.6 Spray column when the dispersed phase enters from the bottom. Other than as a means of introducing the two fluids, the column does not contain any internals. BOOKCOMP, Inc. — John Wiley & Sons / Page 1377 / 2nd Proofs / Heat Transfer Handbook / Bejan COLUMNS AND OTHER CONTACTORS 1377 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 [1377], (19) Lines: 574 to 627 ——— 6.72122pt PgVar ——— Normal Page * PgEnds: Eject [1377], (19) dispersed fluid in the continuous fluid at any time. In a general heat transfer device, the holdup can vary with distance up the column. The holdup is often used to correlate both heat transfer and mass transfer. Other variables of interest in discussions of spray columns are the superficial velocities ( ˜ V ) of each of the phases and the slip velocity. The superficial velocity is the actual velocity that the dispersed or continuous phase is moving, and this depends on the holdup. For the dispersed phase, this is ˜ V d = v d φ (19.36) while the continuous phase velocity is given as ˜ V c = v c 1 − φ (19.37) Finally, the slip velocity ˜ V slip is given as the difference of the two component velocities shown above. As shown by Letan (1988), the operational relation of the system, in terms of the terminal velocity v T ,is v T (1 − φ) γ−1 = ˜ V d φ + ˜ V c 1 − φ (19.38) The flooding holdup can then be found. The flooding situation is found at the following condition (Letan, 1988): ∂ ˜ V c ∂φ ˜ V d = 0 (19.39) With this condition, the flooding limit on holdup can be found: φ f = (γ + 1) 2 + 4γ(1/R − 1) 1/2 − (γ + 1) 2γ(1/R − 1) (19.40) where R ≡ ˜ V d / ˜ V c and γ is as defined in eq. (19.38). At the limit where the con- tinuous phase velocity is zero (R becomes unlimited), the maximum holdup can be determined: φ f,max = 1 γ (19.41) Finally, the pressure drop in the continuous phase in a vertical column is found as follows: ∆P ∆Z = (ρ c − ρ d )gφ (19.42) BOOKCOMP, Inc. — John Wiley & Sons / Page 1378 / 2nd Proofs / Heat Transfer Handbook / Bejan 1378 DIRECT CONTACT HEAT TRANSFER 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 [1378], (20) Lines: 627 to 649 ——— 1.927pt PgVar ——— Normal Page PgEnds: T E X [1378], (20) As will be noted again in the discussion of columns that incorporate internals (baffles, packings, etc.), the spray column offers some benefits and shortcomings. Of the benefits, by far the most important is the relative simplicity and resulting low cost of these devices. Also, the heat transfer performance can be very high. One of the shortcomings is that there could be a fair amount of backmixing, which would hinder overall performance. Examples of possible temperature versus distance traces that might be encountered are given in Fig. 19.7. Means of estimating the heat transfer in these devices have been addressed in the literature. For example, for gas–liquid systems, Fair (1988) has given the following correlation: U V,G (W/m 3 · K) = 867G 0.82 G G 0.47 L Z −0.38 T (19.43) In this equation, Z T denotes the height of a single spray zone. Fair indicates that this equation is only for the gas side of the heat transfer process, but because of circulation inside the droplets, there is relatively little resistance to heat transfer in the liquid phase. Figure 19.7 Various types of performance of a spray column that can result from mixing phenomena. (After Letan, 1988.) . the heat transfer coefficients on the liquid side and the gas (vapor) side in direct contact, respectively; and Q T and Q G , respec- tively, are the total heat transfer and the amount of heat transfer. evaporating BOOKCOMP, Inc. — John Wiley & Sons / Page 1372 / 2nd Proofs / Heat Transfer Handbook / Bejan 1372 DIRECT CONTACT HEAT TRANSFER 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 [1372],. ∆T 2 ln(∆T 1 /∆T 2 ) (19.27) BOOKCOMP, Inc. — John Wiley & Sons / Page 1374 / 2nd Proofs / Heat Transfer Handbook / Bejan 1374 DIRECT CONTACT HEAT TRANSFER 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 [1374],