BOOKCOMP, Inc. — John Wiley & Sons / Page 865 / 2nd Proofs / Heat Transfer Handbook / Bejan LONGITUDINAL FINNED DOUBLE-PIPE EXCHANGERS 865 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 [865], (69) Lines: 2905 to 2966 ——— 3.87216pt PgVar ——— Normal Page PgEnds: T E X [865], (69) For the annulus, the tube diameter is replaced with the equivalent diameter for pres- sure loss: ∆P f = 8n hp f ρV 2 2 L d e = 4n hp f ρV 2 L d e (11.156) The turn loss for both inner pipe and annulus with N hp hairpins is ∆P t = 2(2n hp − 1)ρV 2 (11.157) 11.6.6 Wall Temperature and Further Remarks It may be noted that the wall resistance presents the lowest resistance to the flow of heat between the hot and cold fluids. Hence, an excellent approximation to the wall temperature may be obtained via the computation of the product of R is and the heat flux. Then, if the hot fluid is carried within the inner tube, the wall temperature will be T w = T b − R is q S (11.158) where R is = R io S πd o (11.149) R io = r io + r do + r mo In the event, that the cold fluid is carried in the inner tube, the wall temperature will be T w = t b + R is q S (11.159) 11.6.7 Series–Parallel Arrangements When two streams are arranged for counterflow, the LMTD represents the maximum thermal potential for heat transfer that can be obtained. Often, on the industrial scale, a single process service may entail the use of more than a single long hairpin. It then follows that it is desirable to connect the hairpins in series on both the annulus and inner pipe sides, as in Fig. 11.26. In this configuration, the temperature potential remains the LMTD for counterflow. In some services, there may be a large quantity of one fluid undergoing a small temperature change and a small quantity of another fluid undergoing a large temper- ature change. It may not be possible to circulate the large volume of fluid through the required number of hairpins with the pressure drop available. Under these circum- stances, the larger volume of fluid may be manifolded in the series–parallel arrange- ment shown in Fig. 11.27. The inner pipe fluid has been split between the exchangers BOOKCOMP, Inc. — John Wiley & Sons / Page 866 / 2nd Proofs / Heat Transfer Handbook / Bejan 866 HEAT EXCHANGERS 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 [866], (70) Lines: 2966 to 2971 ——— 0.34799pt PgVar ——— Normal Page * PgEnds: Eject [866], (70) Figure 11.26 Double-pipe heat exchangers in series. (From Kraus et al., 2001, with per- mission.) T 1 T 2 t 2 t 1 II I Figure 11.27 Double-pipe heat exchangers in series–parallel. (From Kraus et al., 2001, with permission.) designated I and II. Both of these exchangers are in counterflow relative to each other but not in the same sense as in Fig. 11.26. In Fig. 11.27, the T ’s refer to the series streams and the t’s refer to the parallel streams. Departures from true counterflow and true co-current (parallel) flow can be han- dled by the logarithmic mean temperature difference correction factor F . Kern (1950) presents a derivation for the factor γ to be used in a modification of the heat transfer rate equation BOOKCOMP, Inc. — John Wiley & Sons / Page 867 / 2nd Proofs / Heat Transfer Handbook / Bejan LONGITUDINAL FINNED DOUBLE-PIPE EXCHANGERS 867 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 [867], (71) Lines: 2971 to 3023 ——— 2.19043pt PgVar ——— Normal Page * PgEnds: Eject [867], (71) q = U o Sγ(T 1 − t 1 ) (11.160) where T 1 − t 1 represents the total temperature potential, the difference in the fluid stream inlet temperatures, in the exchanger configuration. After a laborious and de- tailed derivation, Kern (1950) gives, for one series hot fluid and n parallel cold fluid streams, γ = 1 − T 2 − t 1 T 1 − t 1 Z − 1 nZ ln Z − 1 Z T 1 − t 1 T 2 − t 1 1/n + 1 Z (11.161) where Z = T 1 − T 2 n(t 2 − t 1 ) For one series cold fluid and n parallel hot fluid streams, Kern (1950) gives γ = 1 − T 1 − t2 T 1 − t 1 1 − Z nz ln (1 − Z) T 1 − t 1 T 1 − t 2 1/n + Z (11.162) TABLE 11.4 Dimensions of Multitube Double-Pipe Exchangers a Nom. Pipe Pipe No. No. Tube Tube Fin Dia. Thick. OD of of OD Thick. Height (in.) (mm) (mm) Tubes Fins (mm) (mm) (mm) 4 6.02 114.3 7 16 19.02 2.11 5.33 6.02 114.3 7 20 22.2 2.11 5.33 6 7.11 168.3 19 16 19.02 2.11 5.33 7.11 168.3 14 16 19.02 2.11 5.33 7.11 168.3 7 20 25.4 2.77 12.7 8 8.18 219.1 19 16 19.02 2.11 8.64 8.18 219.1 19 20 22.2 2.11 7.11 8.18 219.1 19 20 25.4 2.77 5.33 8.18 219.1 19 16 19.02 2.11 7.11 8.18 219.1 19 20 22.2 2.11 5.33 Source: After Saunders (1988). a Fin thicknesses are identical to those listed in Table 11.3. The dimensions shown here are for low-pressure units. BOOKCOMP, Inc. — John Wiley & Sons / Page 868 / 2nd Proofs / Heat Transfer Handbook / Bejan 868 HEAT EXCHANGERS 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 [868], (72) Lines: 3023 to 3060 ——— 2.382pt PgVar ——— Long Page PgEnds: T E X [868], (72) where Z = n(T 1 − T 2 ) t 2 − t 1 11.6.8 Multiple Finned Double-Pipe Exchangers There are numerous applications for longitudinal fin pipes and tubes. Closest to the double-pipe exchanger is the hairpin with multiple longitudinal-fin pipes. A vari- ety of pipes and tubes are available with longitudinal fins whose numbers, heights, thicknesses, and materials differ. Data for some of these configurations are shown in Table 11.4. The procedure for the design and analysis of the multiple-tube exchanger differs little for that used for the single-tube exchanger. 11.7 TRANSVERSE HIGH-FIN EXCHANGERS 11.7.1 Introduction Pipes, tubes, and cast tubular sections with external transverse high fins have been used extensively for heating, cooling, and dehumidifying air and other gases. The fins are preferably called transverse rather than radial because they need not be circular, as the latter term implies, and are often helical. The air-fin cooler is a device in which hot-process fluids, usually liquids, flow inside extended surface tubes and atmospheric air is circulated outside the tubes by forced or induced draft over the extended surface. Unlike liquids, gases are compressible, and it is usually necessary to allocate very small pressure drops for their circulation through industrial equipment or the cost of the compression work may entail a substantial operating charge. Except for hydrogen and helium, which have relatively high thermal conductivities, the low thermal conductivities of gases coupled with small allowable pressure drops tend toward low-external-convection heat transfer coefficients. In the discussion of longitudinal high-fin tubes in Section 11.6.1, it was noted that a steel fin 1.27 cm high and 0.0889 cm thick could be used advantageously with a fluid producing a heat transfer coefficient as high as 250 W/m 2 ·K. Aluminum and copper have thermal conductivities much higher than steel, 200 and 380 versus 45 W/m ·K. It would appear that thin high fins made of aluminum or copper would have excellent fin efficiencies when exposed to various heating and cooling applications of air and other gases at or near atmospheric pressure. In air-fin cooler services, the allowable pressure drop is measured in centimeters or inches of water and air can be circulated over a few rows of high-fin tubes with large transverse fin surfaces and, at the same time, require a very small pressure drop. Transverse high-fin tubular elements are found in such diverse places as economizers of steam power boilers, cooling towers, air-conditioning coils, indirect-fired heaters, waste-heat recovery systems for gas turbines and catalytic reactors, gas-cooled nu- clear reactors, convectors for home heating, and air-fin coolers. BOOKCOMP, Inc. — John Wiley & Sons / Page 869 / 2nd Proofs / Heat Transfer Handbook / Bejan TRANSVERSE HIGH-FIN EXCHANGERS 869 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 [869], (73) Lines: 3060 to 3074 ——— 0.0pt PgVar ——— Long Page PgEnds: T E X [869], (73) In the services cited involving high temperatures, hot gases flow over the fins and water or steam flows inside the tubes. The extended surface element usually consists of a chromium steel tube whose chromium content is increased with higher anticipated service temperatures. A ribbon, similar in composition to the tube, is helically wound and continuously welded to the tube. The higher and thicker the fins, the fewer the maximum number of fins per centimeter of tube which can be arc- welded because the fin spacing must also accommodate the welding electrode. High- temperature high-fin tubes on a closer spacing are fabricated by electrical resistance welding of the fins to the tube. High-fin tubes can also be extruded directly from the tube-wall metal, as in the case of integral low-fin tubing. However, it becomes increasingly difficult to extrude a high fin from ferrous alloys as hard as those required for high-temperature services, which are often amenable to work hardening while the fin is being formed. Whether fins are attached by arc welding or resistance welding, the fin-to-tube attachment for all practical design considerations introduces a neglible bond or contact resistance. High-fin tubes are used in increasing numbers in devices such as the air-fin cooler, in which a hot fluid flows within the tubes, and atmospheric air, serving as the cooling medium, is circulated over the fins by fans. Several high-fin tubes for air-fin cooler service are shown in Fig. 11.28. Type a can be made by inserting the tubes through sheet metal strips with stamped or drilled holes and then expanding the tubes slightly to cause pressure at the tube-to-strip contacts. The tubes and strips may then be brazed. If the tubes are only expanded into the plates to produce an “interference fit,” some bond or contact resistance must be anticipated. For practical purposes, when the tubes and strips are brazed together, the joint may be considered a metallurgical bond and the bond resistance can be neglected. In Fig. 11.28, tubes b through e are made by winding a metal ribbon in tension around the tube. These types are not metallurgically bonded and rely entirely upon the tension in the ribbon to provide good contact. Type f combines tension winding with brazing, and for the combination of a steel tube and an aluminum fin, the common tin–lead solder is not compatible and a zinc solder is used. Type g employs a tube as a liner, and high fins are extruded from aluminum, which, like copper, is a metal that can be manipulated to a considerable fin height. Types d,e, and f employ aluminum for the fins and are arranged to protect the tube from the weather because air-fin coolers are installed outdoors. Type g, sometimes called a muff-type high-fin tube, has its contact resistance between the inside of the integral finned tube and the liner or plain tube. Type h has a mechanical bond which can closely match a metallurgical bond for contact resistance. Type i, an elliptical tube with rectangular fins, may employ galvanized steel fins. When tube ends are circular, they are rolled into headers. Consider a typical air-fin cooler application with a hot fluid inside the tubes. In many instances, carbon steel meets the corrosion-resistance requirements of the tube-side fluid. From the standpoint of high thermal conductivity and cost, aluminum ribbon is very suitable for tension-wound fins. However, aluminum has twice the thermal coefficient of expansion of steel, and the higher the operating temperature of the fluid inside the tubes, the greater the tendency of the fins to elongate away from their room-temperature tension-wound contact with the tube, and the greater is BOOKCOMP, Inc. — John Wiley & Sons / Page 870 / 2nd Proofs / Heat Transfer Handbook / Bejan 870 HEAT EXCHANGERS 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 [870], (74) Lines: 3074 to 3085 ——— 0.097pt PgVar ——— Normal Page PgEnds: T E X [870], (74) Figure 11.28 Various types of high-fin tubing. (From Kraus et al., 2001, with permission.) the bond or contact resistance. In one variation of type d in Fig. 11.28, the ribbon is wound with J rather than L feet, with the J ’s pressing against each other and the tube at room temperature. As the feet become heated during operation, they expand against each other. 11.7.2 Bond or Contact Resistance of High-Fin Tubes The bond resistance of several types of interference-fit high-fin tubes shown in Fig. 11.28 has been studied by Gardner and Carnavos (1960), Shlykov and Ganin (1964), and Yovanovich (1981). Gardner and Carnavos pointed out that in its most general sense, the term interference fit implies the absence of a metallurgical bond, as opposed to the extrusion of a fin from a tube wall, and the welding, soldering, or brazing of a fin to the tube. The interference fit is produced by mechanically developing contact pressure through elastic deformation either by winding a ribbon under tension about a tube, as in types b through e in Fig. 11.28, or by expanding a tube against the fins as in type a, or a combination of pressing the root tube against the liner or the liner against the root tube, as in type g. BOOKCOMP, Inc. — John Wiley & Sons / Page 871 / 2nd Proofs / Heat Transfer Handbook / Bejan TRANSVERSE HIGH-FIN EXCHANGERS 871 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 [871], (75) Lines: 3085 to 3127 ——— 2.61522pt PgVar ——— Normal Page PgEnds: T E X [871], (75) 11.7.3 Fin Efficiency Approximation The fin efficiency of the radial fin of rectangular profile was given in Chapter 3: η = 2r b m r 2 a − r 2 b I 1 (mr a )K 1 (mr b ) − K 1 (mr a )I 1 (mr b ) I 0 (mr b )K 1 (mr a ) + I 1 (mr a )I o (mr b ) where m = 2h kδ 1/2 ρ = r b r a φ = (r a − r b ) 3/2 2h kA p 1/2 R a = 1 1 − ρ R b = ρ 1 − ρ The modified Bessel functions in the radial fin efficiency expressions are obtained from tables or from software, and their employment to obtain the efficiency involves a somewhat laborious procedure. An alternative has been provided by McQuiston and Tree (1972), who suggest the approximation η = tanh mψ mψ (11.163) where m = 2h kδ 1/2 ψ = r b 1 − ρ ρ 1 + 0.35 ln 1 ρ (11.164) where ρ is the radius ratio, ρ = r b r a 11.7.4 Air-Fin Coolers The air-fin cooler consists of one or more horizontal rows of tubes constituting a section through which air is circulated upward by mechanical draft. The fan that moves the air may be above the section providing an induced draft or it may be below the section, providing a forced draft. In the induced-draft air-fin cooler, the heated air is thrown upward to a good height by its high exit velocity. A relatively small amount of the heated air is sucked back to reenter the air intake below the section and thereby cut down the temperature difference available between the ambient air and the process fluid. In a forced-draft unit, the air leaves at a low velocity at a point not far from the high entrance velocity BOOKCOMP, Inc. — John Wiley & Sons / Page 872 / 2nd Proofs / Heat Transfer Handbook / Bejan 872 HEAT EXCHANGERS 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 [872], (76) Lines: 3127 to 3138 ——— 0.927pt PgVar ——— Normal Page PgEnds: T E X [872], (76) of the air to the fan below the section. Hot air is more apt to be sucked back into the fan intake, causing recirculation. Following a trend in cooling towers that started some years ago, induced-draft units now appear to be preferred. Usually, the section has cross bracing and baffles to increase rigidity and reduce vibration. The design and analysis of air-fin coolers differs only in a few respects from the longitudinal fin exchangers in Section 11.6. The principal difference is in the air side, where air competes with other fluids as a coolant. Because air is incompressible and liquids are not, only a small pressure drop can be expended for air circulation across the finned tubes, lest the cost of air-compression work become prohibitive. In most applications, the allowable air-side pressure drop is only about 1.25 cm ( 1 2 in.) of water. The air passes over the finned tubing in crossflow, and this merely requires the use of the proper heat transfer and flow friction data. The temperature excursion of the air usually cannot be computed at the start of the calculations because the air volume, and hence the air temperature rise, are dependent on the air pressure drop and flow area of the cooler. Most widely used are the integral-fin muff-type tube (Fig. 11.28g), the L-footed tension-wound tube (Fig. 11.28g), and the grooved and peened tension-wound tube (Fig. 11.28h). These tubes usually employ nine or eleven fins per inch. Numerous other tubes are manufactured in accordance with the types shown in Fig. 11.28b, c, and f . Other tubes have serrated or discontinuous fins. The latter tubes are fabricated to their own standards by manufacturers of air-fin coolers. Physical Data As indicated in Fig. 11.29, tubes may be arranged in either triangu- lar or in-line arrangements. Observe that the pitch in these arrangements is designated by P t ,P l ,orP d , where P t = transverse pitch (m), P l = longitudinal pitch (m), and p t p t p d p d p l p l Flow Flow ()a ()b Figure 11.29 Tube arrangements: (a) triangular; (b) in-line. (From Kraus et al., 2001, with permission.) BOOKCOMP, Inc. — John Wiley & Sons / Page 873 / 2nd Proofs / Heat Transfer Handbook / Bejan TRANSVERSE HIGH-FIN EXCHANGERS 873 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 [873], (77) Lines: 3138 to 3202 ——— 2.7445pt PgVar ——— Normal Page * PgEnds: Eject [873], (77) P d = diagonal pitch (m). The diagonal pitch is related to the transverse and longitu- dinal pitch by P d = P t 2 2 + P 2 l 1/2 (11.165) and in the case of an equilateral triangular arrangement, P d = P t When there are n tubes in a row and n r rows, the total number of tubes will be n t = n n r (11.166) Let z be the clear space between the tubes, which are L meters long. The fins are b meters high: b = d a − d b 2 where d a and d b are, respectively, the outer and inner diameters of the fin. The fins are δ f thick and the minimum flow area A = A min will depend on the transverse pitch P t .For P t > 2P d − d b − 2zδ f z + δ f A = A min = n L P t − d b − 2zδ f z + δ f (11.167) and for P t < 2P d − d b − 2zδ f z + δ f A = A min = 2n L P d − d b − 2zδ f z + δ f (11.168) The surface area of the tube (between the fins) will be S b = πn t Ld b z z + δ f (11.169) and the surface of the fins, which accounts for the heat transfer from the tips of the fins, will be S f = πn t L z + δ f 1 2 d 2 a − d 2 b + d a δ f (11.170) BOOKCOMP, Inc. — John Wiley & Sons / Page 874 / 2nd Proofs / Heat Transfer Handbook / Bejan 874 HEAT EXCHANGERS 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 [874], (78) Lines: 3202 to 3275 ——— -1.66566pt PgVar ——— Normal Page PgEnds: T E X [874], (78) This makes the total surface S = S b + S f (11.171a) the finned surface per total surface, S f S = S f S b + S f (11.171b) and the surface per unit length per tube, S = S NL (11.171c) Heat Transfer Correlations Early investigations that pertain to heat transfer and friction data in tube bundles containing high-fin tubes have been reported by Jameson (1945), Kutateladze and Borishaniskii (1966), and Schmidt (1963). The correlation of Briggs and Young (1963) is based on a wide range of data. Their general equation for tube banks containing six rows of tubes on equilateral triangular pitch is Nu = hd b k = 0.134Re 0.681 · Pr 1/3 2(P f − δ f ) d a − d b 0.20 P f − δ f δf 0.1134 (11.172) where Re = d b G µ and where the range of parameters is 1000 < Re < 18,000 0.33 mm < δ f < 20.02 mm 11.13 mm <d b < 40.89 mm 1.30 mm <P f < 4.06 mm 11.42 mm <b= d a − d b 2 < 16.57 mm 24.99 mm <P t < 111 mm Vampola (1966) proposed a correlation based on an extensive study of different finned tubes. For more than three tube rows, Nu = hd e k = 0.251Re 0.67 P t − d b d b −0.20 × P t − d b P f − δ f + 1 −0.20 P t − d b P d − d b 0.40 (11.173) where Re = d e G µ . factor γ to be used in a modification of the heat transfer rate equation BOOKCOMP, Inc. — John Wiley & Sons / Page 867 / 2nd Proofs / Heat Transfer Handbook / Bejan LONGITUDINAL FINNED DOUBLE-PIPE. the exchangers BOOKCOMP, Inc. — John Wiley & Sons / Page 866 / 2nd Proofs / Heat Transfer Handbook / Bejan 866 HEAT EXCHANGERS 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 [866],. the greater is BOOKCOMP, Inc. — John Wiley & Sons / Page 870 / 2nd Proofs / Heat Transfer Handbook / Bejan 870 HEAT EXCHANGERS 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 [870],