BOOKCOMP, Inc. — John Wiley & Sons / Page 835 / 2nd Proofs / Heat Transfer Handbook / Bejan SHELL-AND-TUBE HEAT EXCHANGER 835 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 [835], (39) Lines: 1791 to 1853 ——— 0.49445pt PgVar ——— Normal Page * PgEnds: Eject [835], (39) where r c = A bp A w (11.91a) ζ = N ss N cc (11.91b) where with X L as the longitudinal tube pitch and N ss taken as the number of sealing strip pairs, N cc = D s − 2l c X L C = 1.35 for Re s ≤ 100 1.25 for Re s > 100 (11.91c) Here A bp = L bc ( D s − D o + 0.5N P w P ) is the crossflow area for the bypass, where N P is the number of bypass divider lanes that are parallel to the crossflow stream B, w P is the width of the bypass divider lane (m), and L bc is the central baffle spacing. J S is the correction factor that accounts for variations in baffle spacing at the inlet and outlet sections as compared to the central baffle spacing: J S = N b − 1 + L ∗ i (1−n) − L ∗ o (1−n) N b − 1 + L ∗ i (1−n) + L ∗ o (1−n) (11.92) where N b is the number of baffles and L ∗ i = L bi L bc (11.93a) L ∗ o = L bo L bc (11.93b) n = 3 5 for turbulent flow 1 3 for laminar flow (11.93c) Here L bi is the baffle spacing at the inlet (m), L bo is the baffle spacing at the outlet (m), and L bc is the central baffle spacing (m) J R is the correction factor that accounts for the temperature gradient when the shell-side fluid is in laminar flow: BOOKCOMP, Inc. — John Wiley & Sons / Page 836 / 2nd Proofs / Heat Transfer Handbook / Bejan 836 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 [836], (40) Lines: 1853 to 1908 ——— 2.65828pt PgVar ——— Normal Page * PgEnds: Eject [836], (40) J R = 1 for Re s ≥ 100 10 N r,c 0.18 for Re s ≤ 20 (11.94) For 20 < Re s < 100, a linear interpolation should be performed between the two extreme values. In eq. (11.81), Re s is the shell-side Reynolds number and N r,c is the number of effective tube rows crossed through one crossflow section. 11.4.4 Pressure Loss Data Tube Side The pressure loss inside tubes of circular cross section in a shell-and- tube heat exchanger is the sum of the friction loss within the tubes and the turn losses between the passes of the exchanger. The friction loss inside the tubes is given by ∆P f = 4f ρu 2 2 L d i (Pa) (11.95a) where u is the linear velocity of the fluid in the tubes, or ∆P f = 4fG 2 2ρ L d (Pa) (11.95b) where G is the mass velocity of the fluid in the tubes. In eqs. (11.95), f is the friction factor. The fluid will undergo an additional pressure loss due to contractions and expan- sions that occur during fluid turnaround between tube passes. Kern (1950) and Kern and Kraus (1972) have proposed that this loss be given by one velocity head per turn: ∆P t = 4ρu 2 2 (Pa) (11.96) In an exchanger with a single pass, ∆P t = 4ρu 2 2 = 2ρu 2 (Pa) (11.97) and in an exchanger with n p passes, there will be n p − 1 turns. Hence ∆P t = 2(n p − 1)ρu 2 (11.98) Friction factors may be obtained from Fig. 11.12, which plots the friction factor as a function of the Reynolds number inside the tube and the relative roughness, /d i . The figure is due to Moody (1944), and it may be noted that when the flow is laminar, f = 16 Re (11.99) BOOKCOMP, Inc. — John Wiley & Sons / Page 837 / 2nd Proofs / Heat Transfer Handbook / Bejan 837 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 [837], (41) Lines: 1908 to 1918 ——— * 528.0pt PgVar ——— Normal Page * PgEnds: PageBreak [837], (41) Laminar flow Laminar flow Critical zone Transition zone Fully rough zone Smooth pipes 0.09 0.08 0.07 0.05 0.06 0.04 0.03 0.05 0.02 0.04 0.015 0.01 0.03 0.008 0.006 0.004 0.025 0.002 0.02 0.001 0.015 0.0008 0.0006 0.0004 0.0002 0.0001 0.01 0.000,05 0.000,01 e — d = 0.000,001 e — d = 0.000,005 0.009 0.008 0.1 f = 64 Re — Friction factor, f Relative roughness, — d e 10 3 10 4 10 5 10 6 10 7 10 8 222223333344444555556666688888 Reynolds number, Re = d e Re cr Figure 11.12 Moody chart for tube and annulus friction factors. (From Moody, 1944.) BOOKCOMP, Inc. — John Wiley & Sons / Page 838 / 2nd Proofs / Heat Transfer Handbook / Bejan 838 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 [838], (42) Lines: 1918 to 1969 ——— 5.58817pt PgVar ——— Normal Page PgEnds: T E X [838], (42) Many investigators have developed friction factor relationships as a function of the Reynolds number. The use of the friction factor give by eq. (11.77) in the Petukhov and Gnielinski correlations of eqs. (11.76) and (11.78) has been noted. Other func- tions can be fitted to the curves in Fig. 11.10. Two of them for smooth tubes are f = 0.046 Re 0.20 3 × 10 4 ≤ Re ≤ 10 6 (11.100) 0.079 Re 0.25 4 × 10 3 ≤ Re ≤ 10 5 (11.101) Shell Side Tinker (1951) also suggested a flow stream model for the determination of shell-side pressure loss. However, the lack of adequate data caused him to make rather gross simplifications in arriving at the effects to be attributed to the various flow streams. Willis and Johnston (1984) developed a simpler method which extends Tinker’s scheme to include end-space pressure losses and includes a simple method for nozzle pressure drop developed by Grant (1980). The flow streams in the Willis and Johnston method are shown in Fig. 11.13. For each of the streams, a coefficient n is defined so that n i = ∆p i ˙m i (i = b, c, s, t,w) (11.102) where the ∆p i ’s and the ˙m i ’s are the pressure drops and mass flow rates for the ith stream, respectively. The crossflow stream contains the actual crossflow path (path c) and the bypass path (path b). These paths merge into the window stream (path w), and continuity and compatability for these three paths give ˙m cr =˙m w (11.103a) where ˙m cr =˙m b +˙m c (11.103b) A c b w s t B Figure 11.13 Shell-side flow streams for the Willis and Johnston (1984) pressure-drop method. BOOKCOMP, Inc. — John Wiley & Sons / Page 839 / 2nd Proofs / Heat Transfer Handbook / Bejan SHELL-AND-TUBE HEAT EXCHANGER 839 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 [839], (43) Lines: 1969 to 2035 ——— 4.39621pt PgVar ——— Normal Page * PgEnds: Eject [839], (43) and the pressure loss between points A and B will be ∆p AB = ∆p cr + ∆p w (11.103c) because ∆p cr = ∆p b = ∆p c . It can be shown that ∆p cr = 1 n b 1/2 + 1 n c 1/2 2 ˙m 2 w (11.104) and in similar fashion, for the parallel combination of the shell-to-baffle leakage path (path s) and the tube-to-baffle leakage path (path t), ∆p l = 1 n s 1/2 + 1 n t 1/2 2 ˙m 2 l (11.105) where the leakage flow rate is ˙m l =˙m s +˙m t (11.106) With the total flow rate given by ˙m T =˙m s +˙m t +˙m w (11.107) a combination of eqs. (11.102) and (11.104)–(11.107) gives ˙m w = ˙m T 1 + n −1/2 c + n −1/2 b −2 + n w n −1/2 s + n −1/2 t −2 1/2 (11.108) and it is observed that the procedure depends on the values of the n i ’s. For n c , Butterworth (1979) has proposed that n c = C c F −b c (11.109a) with C c = 4a d 2 o d V (p −d o ) 3 ˙m T d o µA c −b δ ov 2ρd o A 2 c (11.109b) where for square or rotated square pitch, a = 0.061,b = 0.088, and F c = 1.00; and for equilateral triangular pitch, a = 0.45,b = 0.267, and F c = 0.50. In eq. (11.109b), D V = ap 2 − d 2 o d o BOOKCOMP, Inc. — John Wiley & Sons / Page 840 / 2nd Proofs / Heat Transfer Handbook / Bejan 840 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 [840], (44) Lines: 2035 to 2117 ——— 0.57655pt PgVar ——— Long Page * PgEnds: Eject [840], (44) with a = 1.273 for square and rotated square pitch and a = 1.103 for equilateral triangular pitch. In addition, δ ov = (1 − 0.02p b )D s is the height of the baffle overlap region, p b is the baffle cut, and A c = πD 2 o 2 − 2 D 2 o 2 θ 3 2 − sin θ 3 2 cos θ 3 2 L cb δ ov − N p zL cb where n p is the number of pass partitions and z is the path partition width in line with the flow. For n b , n b = 0.316(δ ov /X L )( ˙m T D e /µA bp ) −0.25 + 2N s 2ρA 2 bp (11.110) where A bp = (2w +N p z)L bc D e = 2A bp 2(w +L bc ) + N p (z + L bc ) For n s , n s = 4 [ 0.0035 + 0.264(2 ˙m T δ sb /µA sb ) ] −0.42 + (δ b /2δ sb ) + 2.03(δ b /2δ sb ) −0.177 2ρA 2 sb (11.111) where A sb = π(D s − δ sb )δ sb For n t , n t = 4 [ 0.0035 + 0.264(2 ˙m T δ tb /µA tb ) ] −0.42 + (δ b /2δ tb ) + 2.03(δ b /2δ tb ) −0.177 2ρA 2 tb (11.112) where A tb = nπ(d o − δ tb )δ tb For n w , n w = 1.90e 0.6856A w /A CL 2ρA 2 w (11.113) BOOKCOMP, Inc. — John Wiley & Sons / Page 841 / 2nd Proofs / Heat Transfer Handbook / Bejan SHELL-AND-TUBE HEAT EXCHANGER 841 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 [841], (45) Lines: 2117 to 2186 ——— 0.60612pt PgVar ——— Long Page * PgEnds: Eject [841], (45) where A w is the window flow area with n tw taken as the number of tubes in the window: A w = A w1 − πd 2 o n tw 4 A w1 = d s 4 θ 1 2 − sin θ 1 2 cos θ 1 2 and where for square and rotated square layouts, A CL = (D s − N CL d o )L bc and for equilateral triangular layouts, A CL = 2(N CL − 1)(p −D o ) + 2w Here, to the nearest integer, N CL = D o − d o P y with P y = p for square pitch, P y = 1.414p for rotated square pitch, and P y = 1.732p for equilateral triangular pitch. Equation (11.108) establishes the window mass flow as a function of the total mass flow, and a simple computation then determines the total baffle-space pressure loss via ∆p AB = ∆P cr + ∆p w (11.103c) where ∆p cr = n c ˙m c or ∆p cr = n b ˙m b ∆p w = n w ˙m w The total pressure loss contains components due to the baffle-space pressure loss established by the foregoing procedure, the end-space pressure loss, and the nozzle pressure loss. The end-space pressure loss is taken as ∆p e = N e ˙m 2 e + 1 2 n we ˙m 2 w (11.114) where n e = n cr D s + δ ov 2δ ov L bc L be 2 (11.115a) with BOOKCOMP, Inc. — John Wiley & Sons / Page 842 / 2nd Proofs / Heat Transfer Handbook / Bejan 842 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 [842], (46) Lines: 2186 to 2254 ——— -3.28177pt PgVar ——— Normal Page PgEnds: T E X [842], (46) n cr = 1 n c 1/2 + 1 n c 1/2 −2 (11.115b) and where n we = 1.9e 0.6856(A w L bc /A CL L) 2ρA 2 w (11.115c) The average end-space flow rate is ˙m e = ˙m t +˙m w 2 Grant (1980) gives the pressure drop in the inlet nozzle as ∆p n1 = G 2 1 A 1 ρ 1 A 2 A 1 A 2 − 1 (11.116) where G 1 is the entry mass velocity, G 1 = ρ 1 u 1 ,A 1 is the inlet nozzle area, and A 2 is the bundle entry area. For the outlet-nozzle pressure loss, ∆p n2 = G 2 2 2ρ 2 1 − A 3 A 4 2 + 1 c − 1 2 (11.117) where G 2 is the exit mass velocity, G 2 = ρ 2 u 2 ,A 3 is the outlet nozzle area, and A 4 is the bundle exit area. The recommended value of the contraction coefficient is c = 2 3 . The total shell-side pressure loss will be ∆P T = ∆p n1 + (F T + 1) ∆p e + (N b − 1) ∆p AB ∆p n2 (11.118) Equation (11.118) has assumed that the pressure losses in the end spaces at inlet and outlet are identical. The factor F T is the transitional correction factor and is based on the crossflow Reynolds number Re c = ˙m c d o µA c where for Re c < 300, the entire method is not valid; 0 ≤ Re c < 1000,F T = 3.646e −0.1934 ; and Re c ≥ 1000,F T = 1. 11.5 COMPACT HEAT EXCHANGERS 11.5.1 Introduction One variation of the fundamental compact exchanger element, the core, is shown in Fig. 11.14. The core consists of a pair of parallel plates with connecting metal BOOKCOMP, Inc. — John Wiley & Sons / Page 843 / 2nd Proofs / Heat Transfer Handbook / Bejan COMPACT HEAT EXCHANGERS 843 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 [843], (47) Lines: 2254 to 2266 ——— 0.097pt PgVar ——— Normal Page PgEnds: T E X [843], (47) Figure 11.14 Exploded view of a compact heat exchanger core: 1, plates; 2, side bars; 3, corrugated fins stamped from a continuous strip of metal. By spraying braze powder on the plates, the entire assembly of plates, fins, and bars can be thermally bonded in a single furnace operation. (From Kraus et al., 2001, with permission.) members that are bonded to the plates. The arrangement of plates and bonded mem- bers provides both a fluid-flow channel and prime and extended surface. It is observed that if a plane were drawn midway between the two plates, each half of the connecting metal members could be considered as longitudinal fins. Two or more identical cores can be connected by separation or splitter plates, and this arrangement is called a stack or sandwich. Heat can enter a stack through either or both end plates. However, the heat is removed from the successive separating plates and fins by a fluid flowing in parallel through the entire network with a single average convection heat transfer coefficient. For this reason, the stack may be treated as a finned passage rather than a fluid–fluid heat exchanger, and, of course, due consideration must be given to the fact that as more and more fins are placed in a core, the equivalent or hydraulic diameter of the core is lowered while the pressure loss is increased significantly. Next, consider a pair of cores arranged as components of a two-fluid exchanger in crossflow as shown in Fig. 11.15. Fluids enter alternate cores from separate headers at right angles to each other and leave through separate headers at opposite ends of the exchanger. The separation plate spacing need not be the same for both fluids, nor need the cores for both fluids contain the same numbers or kinds of fins. These are dictated by the allowable pressure drops for both fluids and the resulting heat transfer coefficients. When one coefficient is quite large compared with the other, it is entirely permissible to have no extended surface in the alternate cores through which the fluid with the higher coefficient travels. An exchanger built up with plates and fins as in Fig. 11.16 is a plate fin heat exchanger. The discussion of plate fin exchangers has concentrated thus far on geometries involving two or more fluids that enter the body of the compact heat exchanger by BOOKCOMP, Inc. — John Wiley & Sons / Page 844 / 2nd Proofs / Heat Transfer Handbook / Bejan 844 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 [844], (48) Lines: 2266 to 2283 ——— -4.933pt PgVar ——— Normal Page PgEnds: T E X [844], (48) Figure 11.15 Two-fluid compact heat exchanger with headers removed. (From Kraus et al., 2001, with permission.) means of headers. In many instances, one of the fluids may be merely air, which is used as a cooling medium on a once-through basis. Typical examples include the air-fin cooler and the radiators associated with various types of internal combustion engines. Similarly, there are examples in which the compact heat exchanger is a coil that is inserted into a duct, as in air-conditioning applications. A small selection of compact heat exchanger elements available is shown in Fig. 11.16. 11.5.2 Classification of Compact Heat Exchangers Compact heat exchangers may be classified by the kinds of compact elements that they employ. The compact elements usually fall into five classes: 1. Circular and flattened circular tubes. These are the simplest form of compact heat exchanger surface. The designation ST indicates flow inside straight tubes (ex- ample: ST-1), FT indicates flow inside straight flattened tubes (example: FT-1) and FTD indicates flow inside straight flattened dimpled tubes. Dimpling interrupts the boundary layer, which tends to increase the heat transfer coefficient without increas- ing the flow velocity. 2. Tubular surfaces. These are arrays of tubes of small diameter, from 0.9535 cm down to 0.635 cm, used in service where the ruggedness and cleanability of the conventional shell-and-tube exchanger are not required. Usually, tubesheets are com- paratively thin, and soldering or brazing a tube to a tubesheet provides an adequate seal against interleakage and differential thermal expansion. . δ tb )δ tb For n w , n w = 1.90e 0. 6856 A w /A CL 2ρA 2 w (11.113) BOOKCOMP, Inc. — John Wiley & Sons / Page 841 / 2nd Proofs / Heat Transfer Handbook / Bejan SHELL-AND-TUBE HEAT EXCHANGER 841 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 [841],. laminar flow: BOOKCOMP, Inc. — John Wiley & Sons / Page 836 / 2nd Proofs / Heat Transfer Handbook / Bejan 836 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 [836],. Moody, 1944.) BOOKCOMP, Inc. — John Wiley & Sons / Page 838 / 2nd Proofs / Heat Transfer Handbook / Bejan 838 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 [838],