BOOKCOMP, Inc. — John Wiley & Sons / Page 1379 / 2nd Proofs / Heat Transfer Handbook / Bejan COLUMNS AND OTHER CONTACTORS 1379 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 [1379], (21) Lines: 649 to 684 ——— 0.81606pt PgVar ——— Normal Page PgEnds: T E X [1379], (21) Heat transfer results for liquid–liquid spray columns were correlated by Plass et al. (1979). For holdups greater than 5% (φ > 0.05), U V (Btu/hr-ft 2 -°F) = 4.5 × 10 4 (φ − 0.05)e −0.75( ˙m d / ˙m c ) + 600 (19.44) For columns where vaporization is taking place, the heat transfer can be calculated as follows (Jacobs and Boehm, 1980): St = U v V ˙m L C P,L = Q LMTD ·˙m L C P,L = 2(Ja V · Pr V ) 0.6 R −0.15 m (19.45) To determine this result, data from a variety of systems were analyzed, including some where boiling took place on a liquid surface. An even simpler result was given by Walter (1981). For all configurations with light hydrocarbons or refrigerants vaporizing in water, the following was recom- mended: U V = 894φ (19.46) The only complication in applying this relationship is that the φ is the value coming into the boiling section. Hence, if preheating and boiling are taking place in the same column, this would be the holdup leaving the preheater portion. More recently, Siquerios and Bonilla (1999) have evaluated the correlation above against experimental results they determined from vaporizing normal pentane in water. Although they found that the qualitative aspects of the performance matched well with that equation, their results were approximately 30% higher for the overall heat transfer coefficient. Direct contact boiling phenomena have been proposed foravariety of applications, including nuclear reactors. A proposal has been presented (Kinoshita andNishi,1994; Kinoshita et al., 1995) for an innovative steam generator system for fast breeder reactors that utilizes water and a molten metal. In experiments using Wood’s alloy (Bi–Pb–Sn–Cd) as the continuous phase fluid and water, volumetric heat transfer coefficients in the range 4 to 34 kW/m 3 ·K were demonstrated. Spray columns have been applied for waste heat recovery, and several of the papers cited in this chapter address this application. One clever approach is to use a falling cloud of solid particles in a stream of exhaust gas (Sagoo, 1981, 1982). No specific performance correlations were given, but the approach was deemed to be quite successful. The simplicity of the spray columns can lead to some undesirable aspects in overall performance. In many situations, the spray pattern may deteriorate at some distance from the nozzle. Spray columns are also prone to backmixing. Even in relatively organized flows, the wake following a droplet can impede heat transfer (Letan, 1988). A method was reported by Letan (1988)fordesigning liquid–liquid spray columns. Crucial aspects of this are the column diameter and the column length. Of course, the appropriate diameter of the column is that required to pass the necessary flows. As BOOKCOMP, Inc. — John Wiley & Sons / Page 1380 / 2nd Proofs / Heat Transfer Handbook / Bejan 1380 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 [1380], (22) Lines: 684 to 745 ——— 1.31425pt PgVar ——— Short Page PgEnds: T E X [1380], (22) the relative flow rate increases, the drag on the droplets increases. At some point, too much holdup of the dispersed fluid could occur. Flooding is the point where the flow rate of the dispersed fluid and its imposed drag on the droplets do not al- low the dispersed fluid to flow completely through the column. Considering these types of interactions, Letan presented a result to determine the proper diameter of the column: D column = 2 ˙ V C [(V d /V c )(1 − ¯ φ) + ¯ φ] πV T ¯ φ(1 − ¯ φ) ζ 1/2 (19.47) In this equation, the exponent ζ is given by the following empirical relationships: Re ≤ 0.2 ζ = 4.65 0.2 < Re < 1 ζ = 4.35Re −0.03 1 ≤ Re ≤ 500 ζ = 4.45Re −0.01 500 < Re ζ = 2.4 Here the Reynolds number is based on the terminal velocity of the droplets. The length of the column is determined to accomplish the necessary heat transfer. Letan (1988) has given a method that includes the effect of the wake of the droplets. This results in a fairly lengthy computational method, and the reference noted should be consulted to use this approach. An approximate approach has been recommended by Jacobs (1988a). This uses a formulation based on concepts used in closed heat exchangers to determine their length: Z = ˙m d ∆h d U V [(π/4)D 2 ] · LMTD (19.48) When the flow of the two fluids is in the same direction, several column operational characteristics are quite different. No longer is flooding a limitation to operation, as both fluids are being forced through the column co-currently. On the other hand, the residence time is generally less than in a counterflow tower, and the temperature differentials are less favorable. A recent comprehensive study of such a system was reported by Shiina (1997). In this work, R-113 was used for the dispersed phase injected through a jet, and water was used for the continuous phase. Work was performed in a 50-mm-inner-diameter column with a maximum length of 1.5 m. The injector level could be varied to show the effects of contact height on the results. A complicated set of correlations was presented that fit the general form St = U V V ˙m d C P,d = c 1 + c 2 (R m ) c 3 (Ja · Pr d,V ) c 4 Z D column c 5 D nozzle D column c 6 (19.49) The constants for this equation are shown in Table 19.1. BOOKCOMP, Inc. — John Wiley & Sons / Page 1381 / 2nd Proofs / Heat Transfer Handbook / Bejan COLUMNS AND OTHER CONTACTORS 1381 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 [1381], (23) Lines: 745 to 764 ——— 0.30038pt PgVar ——— Short Page PgEnds: T E X [1381], (23) TABLE 19.1 Empirical Correlation Constants for Eq. (19.49) Conditions a c 1 c 2 c 3 c 4 c 5 c 6 R m > 50, Ja < 11 0.37 −1.45 0.12 0.08 1.97 57.93 R m > 50, Ja > 11 0.37 0.12 0.12 0.08 4.73 2.27 R m < 50, Ja > 11 −0.29 0.11 0.39 0.08 1.30 13.95 R m < 50, Ja < 11 −0.29 −1.22 0.39 0.08 14.52 114.82 Source: Shiina (1997). a Range of applicability for table is 6.3 <R m < 380, 4.8 < Ja < 37. Differential Treatment If predictive information on the detailed behavior of dis- crete zones inside a spray column is desired, a differential approach can be used. In this formulation, the behavior of a typical bubble is predicted when it is interacting with various elevations of the continuous phase. Models of this type have been cast in terms of a one-dimensional equation set, with the height in the column being the independent parameter. There is no reason why two- and three-dimensional formu- lations cannot be used, but one-dimensional models generally yield good predictions for the overall performance of a spray column. The added complexity of higher-order models is normally not considered to be worthwhile. Several formulations have appeared. Among those found in the literature are for- mulations for liquid–liquid spray columns given by Jacobs and Golafshani (1989), Hutchins and Marschall (1989), and Summers and Crowe (1991), and three-phase exchanges in spray columns by Çoban and Boehm (1988), Tadrist et al. (1987, 1991), Ay et al. (1994), and Song et al. (1998). Brickman and Boehm (1994a,b) used a for- mulation similar to that of Çoban and Boehm (1988) to study design optimization of various configurations. A basic approach to the analysis of a three-phase spray column will now be out- lined. This draws on the work of Çoban and Boehm (1988) and Brickman and Boehm (1994a). First, consider the one-dimensional equations representing continuity, mo- mentum, and energy. These are written for the region influenced by each stream of bubbles as follows: dP dz =− ˙m d A dv d dz − ˙m c A dv c dz − [ρ c (1 − φ) + ρ d φ]g (19.50) dh d dz = A ˙m d Q d V (19.51) dh c dz = A ˙m c ηQ c V (19.52) Here the term η is used to denote the amount of heat that leaves (enters) the dispersed phase and is transferred to (from) the continuous phase. Heat loss to the surroundings from the column will render this parameter less than unity. BOOKCOMP, Inc. — John Wiley & Sons / Page 1382 / 2nd Proofs / Heat Transfer Handbook / Bejan 1382 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 [1382], (24) Lines: 764 to 808 ——— 3.38345pt PgVar ——— Normal Page PgEnds: T E X [1382], (24) A relationship for the droplet velocity can be written in terms the drag coefficient and the other parameters is modified slightly from the result given by Raina et al. (1984). v = 1.1547{[1 − (ρ d /ρ c )(R 0 /R) 3 ](2Rg/C D )} 1/2 R [(5/6)−(R/T c )] [(T 2 c + T 2 d )/2T d T c ] R (C P µ c /k c ) R 0 /1.6R (19.53) In this equation, the subscript 0 denotes beginning values, C D is the drag coefficient on a single droplet defined below, R is the droplet radius, and R is the average droplet radius from insertion to the point of interest in the column. This average droplet radius is given by the relationship R ≡ R 2 0 + R 2 2R 0 R (19.54) The drag coefficient on a single droplet can now be found (White, 1974): C D = 24 Re + 6 1 + Re 1/2 + 0.4 1 + (2µ c /3µ d ) 1 + (µ c /µ d ) (19.55) In general, heat transfer coefficient calculations must be made for both the interior and the exterior surfaces of the droplet. A variety of correlations can be used for this, as discussed earlier in the chapter. One distinction is if the droplet is all liquid, all vapor, or if there is evaporation or condensation taking place inside. For purposes of discussion here, it is assumed that the droplet is initially all liquid, and then at some point in the column vaporization begins. On the outside of the droplet, a variety of correlations are available, as noted previously. For example, eqs. (19.1), (19.2), or (19.3) could be applied, but a variety of other correlations appear in the literature [see, e.g., Sideman and Shabtai (1964)]. Any of these can be used to find the external heat transfer coefficient h e . Then a correlation for the interior-to-the-bubble heat transfer is determined. Again, a wide variety of correlations are available, depending on the presumed flow (or stationary) situation inside the bubble. Remember that the shape and size of the bubble have an influence on these aspects, as discussed by Grace (1983). One possible approach to this if the droplets are small is to assume that the internal heat transfer is solely by conduction when the droplet is all liquid. If and when evaporation begins, eq. (19.22) can be applied to find the h i . With both the external and internal heat transfer coefficients found, the overall heat transfer coefficient can be calculated: U = h e h i h e + h i (19.56) With this heat transfer coefficient determined, the heat transfer can be calculated. This is given by Q d = U(4πR 2 )N(T c − T d ) (19.57) BOOKCOMP, Inc. — John Wiley & Sons / Page 1383 / 2nd Proofs / Heat Transfer Handbook / Bejan COLUMNS AND OTHER CONTACTORS 1383 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 [1383], (25) Lines: 808 to 836 ——— 0.07602pt PgVar ——— Normal Page PgEnds: T E X [1383], (25) Here the term N denotes the number of bubble trains in a given spray column. Note that all of these factors just discussed are found at a given elevation z, and then the effect of height in the column is found from the mass, momentum, and energy balances shown in eqs. (19.50)–(19.52). If the droplet number density does not change throughout the column, the droplet radius follows the dispersed phase density in the following manner. R = R 0 ρ d0 ρ d 1/3 (19.58) This may not always be a good assumption, depending on possible breakup of drop- lets, agglomeration, and other factors. Droplet agglomeration or breakup could be included in the model. This could also include a variety of droplet sizes as may be the actual situation in a column. Both of these general aspects have been discussed by Song et al. (1998). They used probability functions to describe each of these aspects. A general design problem is one where the two fluids are flowing countercurrently and their corresponding mass flow rates and incoming temperatures are known. From a design standpoint, it is desired to estimate the outgoing temperatures of the two fluids as they might be inflenced by various physical parameters of the column (e.g., column diameter, column length, droplet diameter, operational pressure). Analysis next considers the differential zone. An example of this is shown in Fig. 19.8. Here N injection areas are considered for the dispersed fluid. It is assumed that these injectors are essentially equidistant from one another so that the volume of the active zone is made up of N more or less equal subvolumes that run the full length of the active portion of the column. Note that a particular arrangement (dispersed fluid in at the bottom that is vapor- ized by the continuous fluid as it travels up the column) has been assumed. This is done here only to give specific discussion points. The method can be applied to any general spray column combination of fluids, or particles flowing through a fluid. To proceed, thermodynamic and thermal–physical properties are needed. Many such routines have been developed over the years and are now found almost rou- tinely using thermodynamics and heat transfer texts or commercially available math- ematical solver software. A variety of numerical approaches can be used to solve this system of equations. Brickman and Boehm (1994a) used a fourth-order Runge–Kutta solution algorithm. At the bottom of the column, the dispersed phase temperature is known. This is the location where the solution begins. Unknown from the standpoint of the solution method are the velocity gradients and the temperature of continuous fluid. What is known is the incoming temperature of the continuous fluid at the top of the column. To handle this problem, a shooting type of method is used. The unknown temperature at the bottom is estimated, the solution carried out, and the continuous phase temperature calculated at the top is compared to the one given. If these temper- atures do not correspond to within a predetermined limit, a new value for the exiting continuous-fluid temperature is estimated and the calculation repeated. Using an approach similar to the one just described, Çoban and Boehm (1988) made comparisons to experimental results determined in large-scale direct contact BOOKCOMP, Inc. — John Wiley & Sons / Page 1384 / 2nd Proofs / Heat Transfer Handbook / Bejan 1384 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 [1384], (26) Lines: 836 to 839 ——— 0.097pt PgVar ——— Normal Page PgEnds: T E X [1384], (26) Figure 19.8 One-dimensional differential analysis volume that can be used to study perfor- mance of spray columns. spray towers used in energy conversion systems. Very good comparisons were found. Later Brickman and Boehm (1994a,b, 1995) applied this type of solution technique to optimize designs of direct contact heat exchangers of the spray column type. They showed the influence of a variety of parameters on the overall performance of the direct contact heat exchanger. It was noted that a wide range in overall performance could be effected by simple adjustments to the operational parameters. Melting and Solidification Applications One application of spray columns that has drawn interest over the years is that of solidification of a medium, typically for thermal storage or chemical separation. This can beaccomplishedinabatch mode, where the solidifying medium does not circulate outside the column. Alternatively, a means of moving a slurry can be used to circulate the partially solidified medium for BOOKCOMP, Inc. — John Wiley & Sons / Page 1385 / 2nd Proofs / Heat Transfer Handbook / Bejan COLUMNS AND OTHER CONTACTORS 1385 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 [1385], (27) Lines: 839 to 872 ——— 0.59639pt PgVar ——— Normal Page PgEnds: T E X [1385], (27) steady operation. The number of studies is sufficiently limited, and the working sub- stances and physical systems sufficiently numerous, that no broadly encompassing predictive technique has been developed. However, individual summaries of some the specific work are given here. Kim and Mersmann (1998) studied melt crystallization of n-dodecanol from a n- decanol/n-dodecanol mixture for separation purposes. In this set of experiments they used various coolants, including a gas, a liquid, and a vapor. Several semiempirical relationships were given to describe the data they found. Agitated columns were used for some of these studies, and the impeller diameter was used as a correlation variable in those cases. For a gas–liquid system, the following relationship fit the data very well: U V D 2 k melt = 1.79φWe −1.2 (19.59) Here the Weber number (We) is based on the droplet superficial velocity and the column diameter. For the situation when a liquid–liquid system is used, the authors separated the correlation into three ranges of holdup ratio: 0 < φ < 0.2: U V D 2 im k m = 1.06φWe −1.2 im (19.60a) 0.2 < φ < 0.8: U V D 2 im k m = 0.18We −1.2 im (19.60b) 0.8 < φ < 1: U V D 2 im k m = 0.34φWe −1.2 im (19.60c) In these correlations, the impeller characteristics are used. Moderately high temperature thermal storage (around 48°C) using direct contact processes was examined by Kiatsiriroat et al. (2000). In these studies, sodium thio- sulfate pentahydrate exchanged heat with a heat transfer oil. A direct contact storage unit made of acrylic and having a diameter of 0.12 m and a length of 0.7 m was used. The phase-change material remained in the unit. They found an empirical equation that fit the solidification data very well: St = 67.48 Ste −1.4033 · Pr −0.3508 (19.61) This also fit data they found for water–oil and water/R-12 systems. A direct contact technique applicable to cold storage investigated by Utaka et al. (1998) was related to earlier work they performed (Utaka et al., 1987). Their approach used a closed vessel partially filled with water where the hydration of HCFC142b took place. For this system, the critical decomposition temperature of the gas hydrate is 13.1°C, and the formation temperature of the gas hydrate depends on the system pressure. For their system, the pressure was controlled. Although no heat transfer correlation was recommended in this study of transient phenomena, they did show BOOKCOMP, Inc. — John Wiley & Sons / Page 1386 / 2nd Proofs / Heat Transfer Handbook / Bejan 1386 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 [1386], (28) Lines: 872 to 909 ——— 0.54407pt PgVar ——— Normal Page PgEnds: T E X [1386], (28) how the overall heat transfer coefficient varied with time (from about 4 kW/m 3 ·K down to an order of magnitude smaller) and showed how this was dominated by the heat transfer coefficient on the liquid side. A study of direct-contact freezing of tetradecane by a water–ethylene glycol so- lution in a spray column was reported by Inaba and Sato (1996). Both solidification rate versus temperature and some visual results were given. No correlation of heat transfer rate was noted. In a study related to the one just noted, latent heat transfer for cold storage using direct contact means has alsobeeninvestigated byInabaandSato (1997). In this work, tetradecane (paraffin) oil was used as the phase-change material, and cold water was used to accomplish the solidification. This phase change takes place at 5.8°C and liberates 229 kJ/kg of heat. Heat transfer characteristics were not given explicitly, but they did correlate volume fraction solidified as a function of the Stefan number based on the solid phase, the Reynolds number, and a dimensionless temperature ratio. A combined direct contact freezer and ice slurry district cooling system has been reported (Knodel, 1989). In this approach, the evaporator section of the refrigeration system was a direct contact heat transfer device. The ice slurry that resulted was then moved through a distribution system. No specific heat transfer performance was given. Thermal storage for solar heating was investigated by Fouda et al. (1980, 1984). A mixture of 68.2% Na 2 SO 4 and 31.8% water was used throughout the work as the latent heat storage material and the paraffinic solvent Varsol was used as the immisci- ble heat transfer fluid. For most runs a spray column was used, but a limited number were performed using a screen packing (see the discussion on packed columns be- low). Volumetric heat transfer data, based on the salt volume, were correlated with the relationship U V = c 1 ˜ V c 2 (19.62) In this equation, ˜ V is the superficial velocity of the heat transfer fluid based on the total column cross-sectional area. The correlation constants varied with the size of the column, the fluid combination used, and whether or not the column had packing in it. These are listed in Table 19.2. TABLE 19.2 Correlation Constants of Volumetric Heat Transfer Coefficients in Eq. (19.62) D ˜ V Range System (cm) Packing (cm/s) c 1 c 2 Sodium sulfate 27.8 No 0.13–0.73 31.4 1.28 27.8 Yes 0.10–0.39 18.6 0.78 12.7 No 0.61–1.48 25.2 1.16 Water–Varsol 27.8 No 0.07–0.20 34.6 1.26 12.7 No 0.61–0.91 29.8 0.43 Adapted from data presented by Fouda et al. (1984). BOOKCOMP, Inc. — John Wiley & Sons / Page 1387 / 2nd Proofs / Heat Transfer Handbook / Bejan COLUMNS AND OTHER CONTACTORS 1387 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 [1387], (29) Lines: 909 to 947 ——— 0.0pt PgVar ——— Normal Page PgEnds: T E X [1387], (29) Farid and Yacoub (1989) and Farid and Khalaf (1994) have reported studies on latent heat storage at temperatures in the approximate range 20 to 50°C. In the first of these studies a single column was used, and three hydrated salts were the focus of the work: Na 2 CO 2 · 10H 2 O, Na 2 SO 4 · 10H 2 O, and Na 2 HPO 4 · 12H 2 O. Volumetric heat transfer coefficients in the range 2 to 12 kW/m 3 ·K were demonstrated. The authors fit these data with a correlation like that shown eq. (19.62) for a variety of cases with values for the first constant in the range 23.1 ≤ c 1 ≤ 46.5 and the second constant in the range 0.70 ≤ c 2 ≤ 1.28. In the second study, two columns were used with Na 2 CO 2 ·10H 2 O and Na 2 S 2 O 3 ·5H 2 O. The volumetric heat transfer coefficients were not correlated, but they were in the same general range as noted in the earlier paper. In this second work, a thermal efficiency was defined as the ratio of the actual energy removed (or stored) from both columns to the energy theoretically available in both columns. Values for this parameter of 45.3 to 91% were demonstrated. 19.4.2 Baffled Columns Baffled columns are used as means of improving on the performance of spray columns. These devices rearrange flow that can be more favorable to heat transfer in both the dispersed and continuous phases. However, as internals are added to the col- umn, the capital cost increases: hence the improvements in performance need to outweigh the additional expense of installation. The flow and heat transfer situation in baffled-tray columns is more complicated to describe than that in spray columns. Figure 19.9 is a schematic showing some elements of this type of column. Important variables are the diameter of the column, the curtain area (shown in the figure), and the window area (column area minus the tray area). For this situation, Fair (1972, 1988) has given a heat transfer correlation for typical designs of baffles spaced about 0.6 m apart and window areas of 40 to 50% of the column area: U v (W/m 3 · K) = 585G 0.7 G G 0.4 L (19.63) This correlation fits data approximately for a range of systems, including air–water as well as hydrogen–light hydrocarbon/oil units. Baffled columns have also been the basis of increasing heat transfer to immersed tubes. The use of a slurry in a bubble column has been reported by Saxena and Chen (1994). They gave the following correlation for the heat transfer in this situation: St = 0.0389(Re · Fr · Pr 2 ) −0.25 [µ L (Pa · s)] −0.15 (19.64) In this equation, the liquid viscosity has the units shown. Sieve tray or perforated tray columns are a form of the baffled type. These can be used for applications of liquid–liquid and three-phase systems. In this arrange- ment, each of the baffles serves to catch the dispersed fluid and re-form the droplets. A typical tray interaction is shown in Fig. 19.10. At least two positive characteris- tics can result from this. One is that the droplet can be resized, which could be very BOOKCOMP, Inc. — John Wiley & Sons / Page 1388 / 2nd Proofs / Heat Transfer Handbook / Bejan 1388 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 [1388], (30) Lines: 947 to 947 ——— * 30.854pt PgVar ——— Normal Page PgEnds: T E X [1388], (30) Figure 19.9 Conceptual schematic of a baffled tray column with the liquid indicated. Figure 19.10 Fluid interactions at one unit in a perforated tray, baffled column direct-contact heat exchanger. . coefficients found, the overall heat transfer coefficient can be calculated: U = h e h i h e + h i (19.56) With this heat transfer coefficient determined, the heat transfer can be calculated. This is. unity. BOOKCOMP, Inc. — John Wiley & Sons / Page 1382 / 2nd Proofs / Heat Transfer Handbook / Bejan 1382 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 [1382],. As BOOKCOMP, Inc. — John Wiley & Sons / Page 1380 / 2nd Proofs / Heat Transfer Handbook / Bejan 1380 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 [1380],