BOOKCOMP, Inc. — John Wiley & Sons / Page 875 / 2nd Proofs / Heat Transfer Handbook / Bejan TRANSVERSE HIGH-FIN EXCHANGERS 875 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 [875], (79) Lines: 3275 to 3330 ——— -0.55058pt PgVar ——— Normal Page PgEnds: T E X [875], (79) d e = S b d a + S f (S f P f /2NL) S 1/2 with the diagonal pitch given by eq. (11.165). In eq. (11.173), the range of para- meters is 1000 < Re < 10,000 24.78 mm <P t < 49.55 mm 10.67 mm <d b < 26.01 mm 16.20 mm <d e < 34.00 mm 5.20 mm <b= d a − d b 2 < 9.70 mm 0.48 mm < P t − d b d b < 1.64 0.25 mm < δ f < 0.70 mm 4.34 < P t − d b P f − δ f + 1 < 25.2 2.28 mm <P f < 5.92 mm 0.45 < P t − d b P d − d r < 2.50 20.32 mm <P l < 52.40 mm Ganguli et al. (1985) proposed the following correlation for three or more rows of finned tubes: Nu = hd b k = 0.38Re 0.6 · Pr 1/3 S b S 0.15 (11.174) where Re = d b G µ The correlation of eq. (11.174) is valid for 1800 < Re < 100,000 2.30 mm <P f < 3.629 mm 11.176 mm <d b < 19.05 mm 27.432 mm <P t < 98.552 mm 5.842 mm <b= d a − d b 2 < 19.05 mm 1 < S S b < 50 0.254 mm < δ f < 0.559 mm Other correlations include those of Brauer (1964), Schulenberg (1965), Kuntysh and Iokhvedor (1971), and Mirkovic (1974). More recent correlations include thoseof Zhukauskas (1974), Weierman (1976), Hofmann (1976), Ehlmady and Biggs (1979), Biery (1981), Gianolio and Cuti (1981), Brandt and Wehle (1983), and Nir (1991). Many of them are cited by Kr ¨ oger (1998). 11.7.5 Pressure Loss Correlations for Staggered Tubes Some of the earlier correlations for the static pressure drop through bundles of circular finned tubes are those of Jameson (1945), Gunter and Shaw (1945), and Ward and BOOKCOMP, Inc. — John Wiley & Sons / Page 876 / 2nd Proofs / Heat Transfer Handbook / Bejan 876 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 [876], (80) Lines: 3330 to 3392 ——— 0.57925pt PgVar ——— Short Page * PgEnds: Eject [876], (80) Young (1959). A frequently used correlation is that of Robinson and Briggs (1966) for staggered tubes: ∆P = 18.03 G 2 ρ n r · Re −0.316 P t d b −0.927 P t P d 0.515 (11.175) for n r rows and where Re = d b G µ and where P d is given by eq. (11.165). Equation (11.175) is valid for 2000 < Re < 50,000 2.31 mm <P f < 2.82 mm 18.64 mm <d b < 40.89 mm 42.85 mm <P t < 114.3mm 39.68 mm <d a < 69.85 mm 37.11 mm <P l < 98.89 mm 10.52 mm <b= d a − d b 2 < 14.48 mm 1.8 < P t d b < 4.6 Vampola (1966) proposed the correlation ∆P = 0.7315 G 2 ρ n r · Re −0.245 P t − d b d b −0.90 × P t − d b P f − δ + 1 0.70 d e d b 0.90 (11.176) where the Reynolds number, equivalent diameter, and limits of applicability are identical to those following eq. (11.173). 11.7.6 Overall Heat Transfer Coefficient Because the air- and tube-side heat transfer coefficients, the bond and tube metal resistances, and the tube-side fouling factor all apply at very dissimilar surfaces, it is important that all of these resistances be corrected and summed properly. No provi- sion need be made for air-side fouling because the air-side heat transfer coefficient is low and becomes the controlling resistance. Usually, with the muff-type tube, the resistances are first referred to a hypothetical bare tube having outside diameter, d b . With diameter designations in Fig. 11.30, there are five inside resistances: 1. The inside film resistance: r io = 1 h i d b d i (11.177) BOOKCOMP, Inc. — John Wiley & Sons / Page 877 / 2nd Proofs / Heat Transfer Handbook / Bejan TRANSVERSE HIGH-FIN EXCHANGERS 877 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 [877], (81) Lines: 3392 to 3414 ——— 3.51714pt PgVar ——— Short Page * PgEnds: Eject [877], (81) Figure 11.30 Single fin in muff-type tubing. Notice that the diameter at the tips and base of the fin are designated as d a and d b , respectively. (From Kraus et al., 2001, with permission.) 2. The inside fouling resistance: r dio = r di d b d i (11.178) 3. The liner metal resistance is based on the mean liner diameter, and with the metal thickness δ l = d o − d i 2 the liner metal resistance is r mol = δ l k l 2d b d o + d i (11.179) BOOKCOMP, Inc. — John Wiley & Sons / Page 878 / 2nd Proofs / Heat Transfer Handbook / Bejan 878 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 [878], (82) Lines: 3414 to 3496 ——— -1.77864pt PgVar ——— Normal Page PgEnds: T E X [878], (82) 4. The bond resistance given by the tube manufacturer or calculated from the procedure of Section 11.2.3 is transferred appropriately via r Bo = r B d b d g (11.180) 5. The tube metal resistance is based on the mean tube diameter, and with the metal thickness δ t = d b − d g 2 the tube metal resistance is r mot = δ t k t 2d b d b + d g (11.181) The sum of these resistances is R io : R io = r io + r dio + r mol + r Bo + r mot and it is noted that R io is based on the equivalent bare outside tube surface. The gross outside surface to bare tube surface is S /πd b , so that the total resistance referred to the gross outside surface will be R is = R io S πd b (11.182) The air-side coefficient is h o and the fin efficiency is computed from eq. (11.9b). Then, with no provision for fouling, r oη = 1 h o η ov,o (11.183) where η ov,o is obtained from eq. (11.9b): η ov,o = 1 − S f S (1 − η f ) The overall heat transfer coefficient is then given by U o = 1 R is + r oηf (11.184) 11.8 PLATE AND FRAME HEAT EXCHANGER 11.8.1 Introduction An exploded view of the plate and frame heat exchanger, also referred to as a gas- keted plate heat exchanger, is shown in Fig. 11.31a. The terminology plate fin heat BOOKCOMP, Inc. — John Wiley & Sons / Page 879 / 2nd Proofs / Heat Transfer Handbook / Bejan PLATE AND FRAME HEAT EXCHANGER 879 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 [879], (83) Lines: 3496 to 3496 ——— * 63.927pt PgVar ——— Normal Page PgEnds: T E X [879], (83) Figure 11.31 (a) Exploded view of a typical plate and frame (gasketed-plate) heat exchanger and (b) flow pattern in a plate and frame (gasketed-plate) heat exchanger. (From Saunders, 1988, with permission.) BOOKCOMP, Inc. — John Wiley & Sons / Page 880 / 2nd Proofs / Heat Transfer Handbook / Bejan 880 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 [880], (84) Lines: 3496 to 3507 ——— 2.927pt PgVar ——— Normal Page PgEnds: T E X [880], (84) exchanger is also in current use but is avoided here because of the possibility of confusion with the plate fin surfaces in compact heat exchangers. The exchanger is composed of a series of corrugated plates that are formed by precision pressing with subsequent assembly into a mounting frame using full peripheral gaskets. Fig- ure 11.31b illustrates the general flow pattern and indicates that the spaces between the plates form alternate flow channels through which the hot and cold fluids may flow, in this case, in counterflow. Plate and frame heat exchangers have several advantages. They are relatively inexpensive and they are easy to dismantle and clean. The surface area enhancement due to the many corrugations means that a great deal of surface can be packed into a rather small volume. Moreover, plate and frame heat exchangers can accommodate a wide range of fluids. There are three main disadvantages to their employment. Because of the gasket, they are vulnerable to leakage and hence must be used at low pressures. The rather small equivalent diameter of the passages makes the pressure loss relatively high, and the plate and frame heat exchanger may require a substantial investment in the pumping system, which may make the exchanger costwise noncompetitive. Figure 11.32 Typical plates in plate and frame (gasketed-plate) heat exchanger (a) Inter- mating or washboard type and (b) Chevron or herringbone type. (From Saunders, 1988, with permission.) BOOKCOMP, Inc. — John Wiley & Sons / Page 881 / 2nd Proofs / Heat Transfer Handbook / Bejan PLATE AND FRAME HEAT EXCHANGER 881 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 [881], (85) Lines: 3507 to 3526 ——— 0.927pt PgVar ——— Normal Page PgEnds: T E X [881], (85) The two most widely employed corrugation types are the intermating or wash- board type and the chevron or herringbone type. Both of these are shown in Fig. 11.32. The corrugations strengthen the individual plates, increase the heat transfer surface area, and actually enhance the heat transfer mechanism. The outside plates of the assembly do not contribute the fluid-to-fluid heat transfer. Hence, the effective number of plates is the total number of plates minus two. This fact becomes less and less important as the number of plates becomes large. It may be noted that an odd number of plates must be used to assure an equal number of channels for the hot and cold fluids. Figure 11.31a indicates that the frame consists of a fixed head at one end and a movable head at the other. The fluids enter the device through ports located in one or both of the end plates. If both inlet and outlet ports for both fluids are located at the fixed-heat end, the unit may be opened without disturbing the external piping. A single traverse of either fluid from top to bottom (or indeed, bottom to top) is called a pass and single- or multipass flow is possible. Counterflow or co-current flow is achieved in what is called looped flow or 1/1 arrangement, shown in Fig. 11.33a and b. In Fig. 11.33a, termed the Z or zed arrangement, two ports are present on both the fixed and movable heads. In the U arrangement of Fig. 11.33b, all four Figure 11.33 Countercurrent single-pass flow (a)Z-arrangement and (b)U-arrangement. (From Saunders, 1988, with permission.) BOOKCOMP, Inc. — John Wiley & Sons / Page 882 / 2nd Proofs / Heat Transfer Handbook / Bejan 882 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 [882], (86) Lines: 3526 to 3557 ——— 0.98404pt PgVar ——— Normal Page * PgEnds: Eject [882], (86) Figure 11.34 Two-pass/two-pass flow. (From Saunders, 1988, with permission.) Figure 11.35 Two-pass/one-pass flow. (From Saunders, 1988, with permission.) ports are at the fixed-head end. The two pass/two pass flow or 2/2 arrangement is shown in Fig. 11.34, and the two pass/one pass flow or 2/1 arrangement is shown in Fig. 11.35. Observe that in Fig. 11.34, the arrangement is in true counterflow except for the center plate, where co-current flow exists. In Fig. 11.35, one half of the unit is in counterflow and the other half is in co-current flow. 11.8.2 Physical Data Figure 11.36a shows a sketch of a single plate for the chevron configuration. The chevron angle is designated by β, which can range from 25° to 65°. As shown in Fig. 11.36b, the mean flow channel gap is b, and it is seen that b is related to the plate pitch p p1 and plate thickness δ pl : b = p pl − δ pl (11.185) Because the corrugations increase the flat plate area, an enlargement factor Λ is employed: Λ = developed length projected length (11.186) where typically, 1.10 < Λ < 1.25. The cross-sectional area of one channel, A 1 ,isgivenby A 1 = bw (11.187) where w is the effective plate width shown in Fig. 11.36a. With the wetted perimeter of one channel, BOOKCOMP, Inc. — John Wiley & Sons / Page 883 / 2nd Proofs / Heat Transfer Handbook / Bejan PLATE AND FRAME HEAT EXCHANGER 883 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 [883], (87) Lines: 3557 to 3580 ——— 6.13701pt PgVar ——— Normal Page PgEnds: T E X [883], (87) P W 1 = 2(b + Λw) (11.188) the channel equivalent diameter will be d e = 4A 1 P W 1 = 4bw 2(b + Λw) Figure 11.36 Plate geometry for Chevron plates in plate and frame (gasketed-plate heat exchanger). (From Saunders, 1988, with permission.) BOOKCOMP, Inc. — John Wiley & Sons / Page 884 / 2nd Proofs / Heat Transfer Handbook / Bejan 884 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 [884], (88) Lines: 3580 to 3641 ——— 0.82622pt PgVar ——— Normal Page * PgEnds: Eject [884], (88) and because w b, d e = 2b Λ (11.189) If N p is high, the total surface area for heat flow may be based on the projected area: S = N P LW (11.190) where L is the length of each plate in the flow direction and W is its width. Because the hot- and cold-side surfaces are identical, the overall heat transfer coefficient will be given by U = 1 1/h c + 1/h c + R dc + R dh + δ pl /k p1 S m where R dc and R dh are the hot- and cold-side fouling resistances. For a thin plate of high thermal conductivity, U = 1 1/h c + 1/h c + R dc + R dh (11.191) If R dc = R dh = 0 (an unfouled or clean exchanger), U = h c h h h c + h h (11.192) In the case of plate and frame heat exchanger, the true temperature difference in q = US θ m depends on the flow arrangement. For true counterflow or co-current flow, eqs. (11.25) and (11.26) apply. For other arrangements, such as those shown in Figs. 11.34 through 11.36, the work of Shah and Focke (1988) should be consulted. 11.8.3 Heat Transfer and Pressure Loss The heat transfer and pressure loss in a plate and frame heat exchanger are based on a channel Reynolds number evaluated at the bulk temperature of the fluid given by eqs. (11.131): Re ch = d e G ch µ (11.131) Then the channel Nusselt number will be Nu ch = hd e k = j h kPr 1/3 φ 0.17 . plate heat exchanger, is shown in Fig. 11.31a. The terminology plate fin heat BOOKCOMP, Inc. — John Wiley & Sons / Page 879 / 2nd Proofs / Heat Transfer Handbook / Bejan PLATE AND FRAME HEAT. plates, increase the heat transfer surface area, and actually enhance the heat transfer mechanism. The outside plates of the assembly do not contribute the fluid-to-fluid heat transfer. Hence, the. frame (gasketed-plate heat exchanger). (From Saunders, 1988, with permission.) BOOKCOMP, Inc. — John Wiley & Sons / Page 884 / 2nd Proofs / Heat Transfer Handbook / Bejan 884 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 [884],