BOOKCOMP, Inc. — John Wiley & Sons / Page 855 / 2nd Proofs / Heat Transfer Handbook / Bejan COMPACT HEAT EXCHANGERS 855 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 [855], (59) Lines: 2526 to 2550 ——— 0.88106pt PgVar ——— Normal Page PgEnds: T E X [855], (59) 0 0.2 0.4 0.6 0.8 1.0 Ratio of free flow to frontal area on one side, , dimensionless Ϫ0.8 Ϫ1.0 Ϫ0.7 Ϫ0.9 Ϫ0.6 Ϫ0.5 Ϫ0.4 Ϫ0.3 Ϫ0.2 Ϫ0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 Entrance and exit loss coefficients and , dimensionlessKK ce Re = ϱ Re = 2000 Re = 10,000 Re = 3000 Re = 5000 Re = 5000 Re = 3000 Re = 10,000 Re = 2000 Re = ϱ K e K c Laminar Laminar Figure 11.21 Entrance and exit loss coefficients for flow through plate fin exchanger cores. (From Kays and London, 1984). St = Nu Re · Pr = hd e /k (d e G/µ)(cµ/k) = h Gc (11.130) The fluid properties in eqs. (11.129) and (11.130) are evaluated at the bulk tem- perature T b = 1 2 (T 1 + T 2 ) (11.131a) t b = 1 2 (t 1 + t 2 ) (11.131b) BOOKCOMP, Inc. — John Wiley & Sons / Page 856 / 2nd Proofs / Heat Transfer Handbook / Bejan 856 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 [856], (60) Lines: 2550 to 2568 ——— 1.60608pt PgVar ——— Normal Page PgEnds: T E X [856], (60) 0 0.2 0.4 0.6 0.8 1.0 Ratio of free flow to frontal area on one side, , dimensionless Ϫ0.8 Ϫ0.7 Ϫ0.6 Ϫ0.5 Ϫ0.4 Ϫ0.3 Ϫ0.2 Ϫ0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 Entrance and exit loss coefficients and , dimensionlessKK ce Re = ϱ Re = 2000 Re = 10,000 Re = 3000 Re = 5000 Re = 5000 Re = 3000 Re = 10,000 Re = 2000 Re = ϱ K e K c Laminar Laminar Figure 11.22 Entrance and exit losscoefficients for flow through rectangular passages. (From Kays and London, 1984.) Flow Friction Data Kays and London (1984) suggest that the pressure drop ∆P in a compact heat exchanger be computed from the equation ∆P P 1 = G 2 v 1 2g c P 1 (Φ 1 + Φ 2 + Φ 3 − Φ 4 ) (11.132) where Φ 1 = 1 + K c − σ 2 Φ 2 = 2 ν 2 ν 1 − 1 BOOKCOMP, Inc. — John Wiley & Sons / Page 857 / 2nd Proofs / Heat Transfer Handbook / Bejan LONGITUDINAL FINNED DOUBLE-PIPE EXCHANGERS 857 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 [857], (61) Lines: 2568 to 2602 ——— 4.90512pt PgVar ——— Normal Page PgEnds: T E X [857], (61) Φ 3 = f S A ν m ν 1 Φ 4 = 1 − σ 2 − K e ν 2 ν 1 Friction factors are correlated on an individual surface basis and are usually plotted as a function of the Reynolds number. The entrance and exit loss coefficients differ for the various types of passages and are plotted as functions of the parameter σ and the Reynolds number. Four terms may be noted within the parentheses in eq. (11.132). These terms de- note, respectively, the entrance or contraction loss as the fluid approaches the ex- changer at line velocity and changes to the exchanger entrance velocity, acceleration loss, or acceleration gain as the fluid expands or contracts during its passage through the exchanger, flow friction loss, and exit loss. Kays and London (1984) have presented heat transfer and flow friction data for approximately 120 surfaces described by the foregoing. Some typical examples are shown in Figs. 11.17 through 11.20. Entrance and exit loss coefficients for plate fin cores and rectangular passages are plotted in Figs. 11.21 and 11.22. Because the pitch of the fins is small, the height of the fin is approximately equal to b, the distance between the separation plates. The fin efficiency for the parallel-plate heat exchanger may be taken as η f = tanhmb/2 mb/2 (11.133) where b/2 is half of the separation plate distance. 11.6 LONGITUDINAL FINNED DOUBLE-PIPE EXCHANGERS 11.6.1 Introduction The double-pipe exchanger consists of a pair of concentric tubesor pipes. One process stream flows through the inner pipe, and the other flows, either in counter- or co- current (parallel) flow in the annular region between the two pipes. The inner pipe may be bare or it may contain as many as 48 longitudinal fins equally spaced around its periphery. Consider the plain double-pipe exchanger shown in Fig. 11.23. It usually consists of two pairs of concentric pipes with a return bend and a return head made leaktight by packing glands. The packing glands and bends returning outside rather than inside the return head are used only where the annulus has low fluid pressure. If there is no problem with differential thermal expansion, the glands may be omitted and the outer and inner pipes may be welded together to provide a leaktight construction. Two pairs of concentric pipes are used to form a hairpin because of the conve- nience the hairpin affords for manifolding streams and the natural loop it can provide BOOKCOMP, Inc. — John Wiley & Sons / Page 858 / 2nd Proofs / Heat Transfer Handbook / Bejan 858 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 [858], (62) Lines: 2602 to 2613 ——— 0.097pt PgVar ——— Normal Page PgEnds: T E X [858], (62) Return bend Gland Gland Tee Gland Return head Figure 11.23 Plain double-pipe exchanger. (From Kraus et al., 2001, with permission.) for differential thermal expansion between the inner and outer pipes. The hairpin brings all inlets and outlets close together at one end, which is particularly important when multiple hairpins are connected in batteries. Moreover, the hairpins need not have the same length. An additional merit of the double-pipe heat exchanger is the ease in which it usually can be disassembled for inspection and cleaning or reused in another service whenever a process becomes obsolete. The longitudinal fin double-pipe exchanger is used advantageously where an ap- preciable inequity appears in the composite thermal resistance of a pair of fluids in a plain double-pipe exchanger. Because heat transfer equipment is usually purchased on the basis of its performance in the fouled condition, the composite thermal resis- tance is the sum of the convective film resistances and the fouling resistances. The advantage of the finned annulus lies in its ability to offset the effects of poorer heat transfer in one fluid by exposing more surface to it than the other. Indeed, even if the composite resistances of both fluids are low, as discussed subsequently, there may still be an advantage in the use of the finned inner pipe. Fins are usually 0.089 cm thick (0.035 in. and 20 BWG). A steel fin with a thermal conductivity of 50 W/m ·K and a height of 1.27 cm ( 1 2 in.) on exposure to a composite resistance of 0.004 m 2 · K/W (corresponding to a film coefficient of 250 W/m 2 · K) has a fin efficiency of about 0.65. Exposed to a composite resistance of 0.002 m 2 · K/W, the efficiency drops to about 0.5. Hence, the high fin has its limitations, although metals of higher thermal conductivity extend the range of application. Fin surface is inexpensive compared with prime surface, but its usefulness diminishes significantly below a composite resistance of 0.002 m 2 · K/W. For the case where both composite resistances are very large, any improvement in the surface exposed to the higher resistance may save considerable linear meters of exchanger. Moreover, inner pipes are available with fins on the inside as well as the outside of the pipe, and the inner pipes are also available with continuous twisted longitudinal fins, which cause some mixing in the annulus. As a class, however, these show a small increase in heat transfer coefficient for a large expenditure of pressure loss, and for viscous fluids, the mixing and its effects decay rapidly. The disposition of the fins about the pipe is shown clearly in Fig. 11.24. They form a radial array of channels, with each channel composed of two fins. Channels may be attached by continuously spot-welding them to the outside of the inner pipe or by other brazing or welding procedures. It should be noted that contact between the BOOKCOMP, Inc. — John Wiley & Sons / Page 859 / 2nd Proofs / Heat Transfer Handbook / Bejan LONGITUDINAL FINNED DOUBLE-PIPE EXCHANGERS 859 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 [859], (63) Lines: 2613 to 2643 ——— 0.47134pt PgVar ——— Normal Page PgEnds: T E X [859], (63) Figure 11.24 Configurations to be used for the determinationof the flow area, wetted perime- ter, and surfaces for the annular region in the double-pipe heat exchanger: (a) extruded fins; (b) welded U-fins; (c) detail of two welded U-fins. (From Kraus et al., 2001, with permission.) channels and the other pipe should be continuous over the entire channel length but need not be very wide. In another method of attaching longitudinal fins, grooves are plowed in the outside diameter of the inner pipe. Metal ribbon is then inserted into the grooves as fins and the plowed-up metal is peened back to form a tight bond between fins and the inner pipe. In the laminar or transition flow regimes, fins are sometimes offset every 30 to 100 cm. The common double-pipe exchanger units available are summarized in Table 11.3. TABLE 11.3 Dimensions of Double-Pipe Exchangers a Outer Outer Inner Inner Nominal Pipe Pipe Max. Pipe Pipe Fin Diameter Thickness OD No. of OD Thickness Height (in.) (mm) (mm) Fins (mm) (mm) (mm) 2 3.91 60.3 20 25.4 2.77 11.1 3 5.49 88.9 20 25.4 2.77 23.8 3 5.49 88.9 36 48.3 3.68 12.7 3 1 2 5.74 101.6 36 48.3 3.68 19.05 3 1 2 5.74 101.6 40 60.3 3.91 12.7 4 6.02 114.3 36 48.3 3.68 25.4 4 6.02 114.3 40 60.3 3.91 10.05 4 6.02 114.3 48 73.0 5.16 12.7 Source: After Saunders (1988). a One outer tube–one inner tube: standard units. The fin thickness for extruded or soldered fins is 0.5 mm for fins up to 12.7 mm high and 0.8 mm high for greater heights. Fin thickness for welded fins is 0.89 mm for fin heights up to 25.4 mm. The dimensions shown here are for low-pressure units. BOOKCOMP, Inc. — John Wiley & Sons / Page 860 / 2nd Proofs / Heat Transfer Handbook / Bejan 860 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 [860], (64) Lines: 2643 to 2712 ——— 0.61221pt PgVar ——— Short Page PgEnds: T E X [860], (64) 11.6.2 Physical Data for Annuli Extruded Fins For the finned annular region between the inner and outer pipes shown in Fig. 11.24a, the cross-sectional area for n t identically finned inner pipes each having n f extruded fins will be A = π 4 D 2 i − π 4 d 2 o − n f b f δ f n t (11.134) There are two wetted perimeters. One of them is for heat transfer: P Wh = πd o + n f (2b f − δ f ) n t (11.135) where the tips of the fins are presumed adiabatic. The other is for pressure loss: P Wf = πD i + P Wh or P Wf = πD i + πd o + n f (2b f − δ f ) n t (11.136) The equivalent diameter for heat transfer will be d e = 4A P Wh = (π/4)D 2 i − (π/4)d 2 o − n f b f δ f n t (πd o − n f δ f + 2n f b f )n t (11.137) and the equivalent diameter for pressure drop will be d e = 4A P Wf = (π/4)D 2 i − (π/4)d 2 o − n f b f δ f n t πD i + (πd o − n f δ f + 2n f b f )n t (11.138) The surface area per unit length per tube will be S = S = S b + S f where S b is the unfinned surface of the inner tube per unit length, S b = πd o − n f δ f (11.139a) Then, per unit length, with S f = 2b f n f (11.139b) the surface area on the annulus side of the inner pipe per unit length is S = S = πd o + (2b f − δ f )n f (11.139c) BOOKCOMP, Inc. — John Wiley & Sons / Page 861 / 2nd Proofs / Heat Transfer Handbook / Bejan LONGITUDINAL FINNED DOUBLE-PIPE EXCHANGERS 861 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 [861], (65) Lines: 2712 to 2763 ——— -0.66777pt PgVar ——— Short Page PgEnds: T E X [861], (65) Welded U-Fins The configuration for the annular region that accommodates welded U-fins is shown in Fig. 11.24b, and detail for a pair of the fins is shown in Fig. 11.24c. Observe that z is the fin root width and thickness, which is usually taken as 2δ f . The free area for flow for n f fins and n t inner tubes with δ f = z/2is A = π 4 D 2 i − π 4 d 2 o + n f δ f b f + z 2 n t (11.140) If d o z, the wetted perimeter for heat flow will be the circumference of the inner tube less the thicknesses of the n f fins plus twice the heights of the n f fins. P Wh = πd o + n f (2b f − δ f ) n t (11.141) Here, too, the tips are presumed to be adiabatic. The wetted perimeter for pressure loss is P Wf = πD i + P Wh or P Wf = πD i + πd o + n f (2b f − δ f ) n t (11.142) Then the two equivalent diameters are, for heat transfer, d e = 4A P Wh = (π/4)D 2 i − (π/4)d 2 o + n f δ f (b f + z/2) n t πd o + n f (2b f − δ f ) n t (11.143) and for pressure drop, d e = (π/4)D 2 i − (π/4)d 2 o + n f δ f (b f + z/2) n t πD i + πd o + n f (2b f − δ f ) n t (11.144) The surface areas, S b ,S f ,S, and the surface area per unit length S will be the same as those for the extruded fin configuration and are given by eqs. (11.139). 11.6.3 Overall Heat Transfer Coefficient Revisited Kern (1950), Kern and Kraus (1972), and Kraus et al. (2001) all report on a method originally proposed by Kern for evaluation of the overall heat transfer coefficient when it has a component that involves fouling in the presence of fins. The equation for an overall heat transfer coefficient is a complicated expression because of the annulus fouling and the fin efficiency. It can be developed from a series summation of several thermal resistances that are identified in Fig. 11.25. These resistances are giveninm 2 · K/W. After both inside and outside heat transfer coefficients, h i and h o , have been de- termined and after both fouling resistances, r di and r do , have been specified (either BOOKCOMP, Inc. — John Wiley & Sons / Page 862 / 2nd Proofs / Heat Transfer Handbook / Bejan 862 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 [862], (66) Lines: 2763 to 2788 ——— -4.34386pt PgVar ——— Normal Page * PgEnds: Eject [862], (66) Fin Dirt rtЈЈ of , rЉ o w rtЈЈ of , r o T c t f t fw r o r do r do r do r i,OD ⌺R i,OD ⌺ЉRs i , Us D , (on )Љ r i (at 10) r di (at 10) r di,OD Dirt Dirt Tube t c Figure 11.25 Location of thermal resistances for a fouled longitudinal fin double-pipe ex- changer. The thermal resistances are based on gross fin and outer pipe surface, and the tip of the fin is considered adiabatic. one or both can be zero), the steps can be arranged in systematic order. The detailed procedure that follows is based on a finned annular passage and an internally un- finned tube. 1. With h io = h i (d i /d o ), form the inside film resistance: r io = 1 h io (m 2 · K/W) (11.145) 2. The inner pipe fouling resistance r di must be referred to the outer tube surface. Hence, r dio = d o d i r di (m 2 · K/W) (11.146) 3. The pipe wall resistance referred to the outside of the inner pipe is r mo = d o ln(d o /d i ) 2k m (m 2 · K/W) BOOKCOMP, Inc. — John Wiley & Sons / Page 863 / 2nd Proofs / Heat Transfer Handbook / Bejan LONGITUDINAL FINNED DOUBLE-PIPE EXCHANGERS 863 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 [863], (67) Lines: 2788 to 2846 ——— 1.85434pt PgVar ——— Normal Page * PgEnds: Eject [863], (67) However, when the diameter ratio d i /d o ≥ 0.75,r mo can be computed with an error of less than 1% from the arithmetic mean diameter: r mo = d o − d i 2k m 2πd o π(d o + d i ) = d o (d o − d i ) k m (d o + d i ) (m 2 · K/W) (11.147) 4. At this point, the sum of the internal resistances referred to the outside of the inner pipe is R io = r io + r dio + r mo (m 2 · K/W) and this resistance must be referred to the gross outside surface of each inner pipe per meter: S = πd o + (2b − δ)n f (11.148) Thus, R is = R io S πd o (m 2 · K/W) (11.149) 5. The annulus heat transfer coefficient is h o , so that r o = 1 h o (m 2 · K/W) (11.150) 6. The annulus fouling resistance r do must be combined with h o to obtain the value of the annulus coefficient working on the fin and prime outer surface. Let this resistance be designated as r o , so that r o = 1 h o + r do = r o + r do (m 2 · K/W) h o = 1 r o + r do The fin efficiency will be given by η fo = tanh mb mb (11.151) where m = 2h o k m δ f 1/2 Then, with the weighted fin efficiency defined by eq. (11.9b), BOOKCOMP, Inc. — John Wiley & Sons / Page 864 / 2nd Proofs / Heat Transfer Handbook / Bejan 864 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 [864], (68) Lines: 2846 to 2905 ——— 2.51837pt PgVar ——— Normal Page * PgEnds: Eject [864], (68) η ov,o = 1 − S fo S o (1 − η fo ) (11.9b) or η ov,o = η fo S fo + S bo S fo + S bo the value of the heat transfer coefficient to the finned and prime surface corrected for the weighted fin efficiency and based on the outside surface of the inner pipe will be h oη = h o η o so that the resistance is r oη = 1 h oη (m 2 · K/W) (11.152) 7. The overall resistance is the sum of eqs. (11.149) and (11.152). Thus, 1 U o = R is + r oη or U o = 1 R is + r oη (11.153) The overall heat transfer coefficient given by eq. (11.153) is the coefficient to be used in the rate equation: q = U o S o θ m (11.154) 11.6.4 Heat Transfer Coefficients in Pipes and Annuli Heat transfer coefficients for both the inside of the tubes and the annular region containing the fins have been presented in Section 11.4.3. 11.6.5 Pressure Loss in Pipes and Annuli The friction relationships of eqs. (11.95) and the turn loss relationship of eq. (11.96) also pertain to the double-pipe heat exchanger. However, when hairpins are consid- ered, the total friction loss in the inner pipe will be ∆P f = 8n hp f ρV 2 2 L d = 4n hp f ρV 2 L d i (11.155) . overall heat transfer coefficient given by eq. (11.153) is the coefficient to be used in the rate equation: q = U o S o θ m (11.154) 11.6.4 Heat Transfer Coefficients in Pipes and Annuli Heat transfer. t 2 ) (11.131b) BOOKCOMP, Inc. — John Wiley & Sons / Page 856 / 2nd Proofs / Heat Transfer Handbook / Bejan 856 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 [856],. it can provide BOOKCOMP, Inc. — John Wiley & Sons / Page 858 / 2nd Proofs / Heat Transfer Handbook / Bejan 858 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 [858],