BOOKCOMP, Inc. — John Wiley & Sons / Page 825 / 2nd Proofs / Heat Transfer Handbook / Bejan SHELL-AND-TUBE HEAT EXCHANGER 825 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 [825], (29) Lines: 1300 to 1341 ——— 3.71513pt PgVar ——— Normal Page PgEnds: T E X [825], (29) Figure 11.9a shows a single-pass tube side and a baffled single-pass shell side. A toroidal expansion joint in the center of the shell accommodates the differential thermal expansion between the tubes and the shell. Figure 11.9b employs U-tubes within the baffled single-pass shell. In this case, account must be taken of the fact that approximately half of the tube-side surface is in counterflow and the other half of the tube-side surface is in co-current flow with the shell-side fluid. The unit shown in Fig. 11.9c has a flow pattern that is similar to the unit shown in Fig. 11.9b. However, the construction is more complex, to facilitate inspection of the inside of the tubes, the cleaning of the tubes mechanically, and the replacement of defective tubes. While the configuration of Fig. 11.9d does not provide for differential thermal expansion between the tubes and the shell, the floating head allows for thermal expansion between the tubes and the shell and for large temperature differences between the fluids. In Fig. 11.9e, leakage from one fluid stream into the other through the packed joints of the floating head goes directly to the exterior of the shell, where it can be detected readily and without contamination of the otherstream.Figure11.9f indicates a further variation of the type of exchanger shown in Fig. 11.9e. 11.4.2 Physical Data Tube Side For a tube bundle containing n t tubes of length L with outside and inside diameters of d o and d i , respectively, the flow area will be A f = πn t d 2 i 4n p (11.55) where n p is the number of passes. The surface area for the n t tubes is S o = πd o Ln t (11.56) The equivalent diameter to be used in establishing the heat transfer coefficient is d e = d i (11.57) and the length of the tube to be used in pressure loss computations is L T : L T = L + 2δ sh (11.58) where δ sh is the thickness of the tube sheet. Because the tube side is concerned only with the number of tubes and tube passes and not with the tube layout, eqs. (11.55) through (11.58) pertain to tubes on square (), rotated square (♦), and equilateral triangular () pitch. The number of tubes within a given shell can be obtained from a list of tube counts such as those provided by Kern (1950) and Saunders (1988). Shell Side Saunders (1988) has provided the pertinent physical data for the shell side with segmented baffles. A concise pictorial summary of the various shell-side BOOKCOMP, Inc. — John Wiley & Sons / Page 826 / 2nd Proofs / Heat Transfer Handbook / Bejan 826 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 [826], (30) Lines: 1341 to 1415 ——— 8.29523pt PgVar ——— Normal Page PgEnds: T E X [826], (30) parameters is shown in Fig. 11.10. With the nomenclature contained in Fig. 11.10, several flow areas can be identified. The crossflow area is A c = C 1 + (D s − C 1 − d o )(p − d o ) yp (11.59) where the tube pitch factor y varies with the tube arrangement: y = 1.000 for equilateral triangular , 30° pitch 0.866 for equilateral triangular , 60° pitch 1.000 for square pitch 0.707 for square rotated ♦45° pitch The tube bundle bypass area is A bp = C 1 L bc (11.60) where L bc is the central baffle spacing The shell-to-baffle leakage area for one baffle is A sb = 1 2 (π + θ 1 )D s δ sb (11.61) where δ sb is the shell-to-baffle spacing: δ sb = D s − D b 2 θ 1 = arcsin 1 − 2l c D s (rad) The fraction of the total number of tubes in one window is F w = θ 3 − sin θ 3 2π (11.62) where θ 3 = 2 arccos D s − 2l c D s − C 1 (rad) The tube-to-baffle leakage area for one baffle is A tb = πd o (1 − F w )n t δ tb 2 (11.63) where δ tb is the tube to baffle spacing. BOOKCOMP, Inc. — John Wiley & Sons / Page 827 / 2nd Proofs / Heat Transfer Handbook / Bejan SHELL-AND-TUBE HEAT EXCHANGER 827 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 [827], (31) Lines: 1415 to 1415 ——— -1.073pt PgVar ——— Normal Page PgEnds: T E X [827], (31) ( /2) 3 1 C 1 /2 C 1 /2 d 0 ( /2) 2 Baffle cut = (100)( / )ID cs Baffle cut ratio = ( / )ID cs I c Baffle edge Baffle edge Number rows ( ) Crossed in one Cross-flow space n c Outer tube limit D o p p p p (30°) (90°) ()a ()b ()c ()e ()d (60°) (45°)( / 1.706)pd o Baffle diameter ␦ sb /2 ␦ sb /2 ␦ tb /2 ␦ tb /2 Shell inside diameter Shell-baffle leakage area ( )—shaded portionA sb Tube o.d. Baffle-hole diameter “Window baffle” Conventional baffle Bundle by-pass area shownA bp Figure 11.10 Shell-side terminology and areas for shells with segmental baffles: (a) end view showing tube layout and baffles; (b) shell-baffle leakage area; (c) conventional baffle arrangement; (d) tube–baffle leakage area; (e) window baffle. (From Saunders, 1988, with permission.) BOOKCOMP, Inc. — John Wiley & Sons / Page 828 / 2nd Proofs / Heat Transfer Handbook / Bejan 828 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 [828], (32) Lines: 1415 to 1494 ——— 2.20824pt PgVar ——— Normal Page * PgEnds: Eject [828], (32) The free area for fluid flow in one window section will be A w = A wg − A wt where A wg is the gross window area: A wg = D 2 s 8 (θ 2 − sin θ 2 ) where θ 2 = arccos 1 − 2l c D s and the area occupied by the tubes in one window is A wt = π 4 n tw d 2 o where the number of tubes in the window is n tw = F w n t Hence A w = D 2 s 8 (θ 2 − sin θ 2 ) − π 4 n tw d 2 o (11.64) The number of tubes crossed in one crossflow space is n c = D s − 2l c qp (11.65) where the factor q varies with the tube pitch: q = 0.866 for equilateral triangle , 30° pitch 1.000 for equilateral triangle , 60° pitch 1.000 for square pitch 0.707 for square rotated ♦45° pitch The effective number of tubes crossed in one window is n cw = 0.8l c qp (11.66) The equivalent diameter of one window to be used in establishing the heat transfer coefficient is D e = 4A w πd o n tw + D s θ 2 /2 (11.67) BOOKCOMP, Inc. — John Wiley & Sons / Page 829 / 2nd Proofs / Heat Transfer Handbook / Bejan SHELL-AND-TUBE HEAT EXCHANGER 829 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 [829], (33) Lines: 1494 to 1542 ——— 2.92703pt PgVar ——— Normal Page * PgEnds: Eject [829], (33) 11.4.3 Heat Transfer Data The establishment of the heat transfer coefficient on the tube side and the shell side of a shell-and-tube exchanger is of fundamental importance to the design and analysis of the shell-and-tube heat exchanger. Tube Side Investigations that pertain to heat transfer and friction data within tubes have been reported by Pohlhausen (1921), DeLorenzo and Anderson (1945), Deissler (1951), McAdams (1954), Hausen (1959, 1974), Stefan (1959), Barnes and Jackson (1961), Dalle Donne and Bowditch (1963), Yang (1962), Petukhov and Popov (1963), Perkins and W ¨ orsœ-Schmidt (1965), W ¨ orsœ-Schmidt (1966), Test (1968), Webb (1971), Oskay and Kakac¸ (1973), Shah and London (1978), Rogers (1980), Kays and Crawford (1993), Gnielinski (1976), Kakac¸ et al. (1985, 1987), Shah and Bhatti (1987), and Kakac¸ and Yener (1994). These are summarizedbyKraus et al. (2001). Some heat transfer correlations depend on a viscosity correction, φ n = µ µ w n (11.68) where µ and µ w are the dynamic viscosities at the bulk and wall temperature, respec- tively, and where n is an exponent depending on whether the process is one of heating or one of cooling. The heat transfer correlations that follow are subdivided into three listings: for laminar flow, for transition flow, and for turbulent flow. In all of these, unless other- wise indicated, all fluid properties are based on the bulk temperature: T b = T 1 + T 2 2 and t b = t 1 + t 2 2 Laminar Flow: Re ≤ 2300 For situations in which the thermal and velocity pro- files are fully developed, the Nusselt number depends only on the thermal boundary conditions. For circular tubes with Pr ≥ 0.60 and Re ·Pr · L/d > 0.05, the Nusselt numbers have been shown to be Nu = 3.657 (11.69) for constant-temperature conditions and Nu = 4.364 (11.70) for constant-heat-flux conditions. Here Re d is the Reynolds number based on the tube diameter d, and L is the tube length. In many cases, the Graetz number, which is the product of the Reynolds and Prandtl numbers and the diameter-to-length ratio: BOOKCOMP, Inc. — John Wiley & Sons / Page 830 / 2nd Proofs / Heat Transfer Handbook / Bejan 830 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 [830], (34) Lines: 1542 to 1601 ——— 2.3482pt PgVar ——— Normal Page * PgEnds: Eject [830], (34) Gz ≡ Re · Pr d L is employed. At the entrance of a tube, the Nusselt number is infinite and decreases asymptoti- cally to the value for fully developed flow as the flow progresses along the length of the tube. The Sieder–Tate (1936) equation gives a good correlation for both liquids and gases in the region where the thermal and velocity profiles are both developing: Nu = hd k = 1.86 Re · Pr d L 1/3 φ 0.14 (11.71) for T w constant and within the following ranges: 0.48 ≤ Pr ≤ 16,700 0.0044 ≤ φ ≤ 9.75 Re · Pr d L 1/3 φ 0.14 ≥ 2 The limitations should be observed carefully as the Sieder–Tate equation yields a zero heat transfer coefficient for extremely long tubes. The correlation of Hausen (1943) is good for both liquids and gases at constant wall temperature: Nu = hd k = 3.66 + 0.668Re · Pr(d/L) 1 + 0.40 [ Re · Pr(d/L) ] 2/3 (11.72) The heat transfer coefficient obtained from this correlation is the average value for the entire length of the tube, and it may be observed that when the tube is sufficiently long, the Nusselt number approaches the constant value of 3.657. Transition: 2300 ≤ Re ≤ 10,000 For transition flow for both liquids and gases, the Hausen (1943) correlation for both liquids and gases may be employed: Nu = hd k = 0.116 Re 2/3 − 125 Pr 1/3 φ 0.14 1 + d L 2/3 (11.73) Turbulent Flow: Re ≥ 10,000 For both liquids and gases, Dittus and Boelter (1930) recommend Nu = hd k = 0.023Re 0.80 · Pr n (11.74) where n = 0.30 for cooling and n = 0.40 for heating. Sieder and Tate (1936) removed the dependence on heating and cooling by setting the exponent on Pr to 1/3 and adding a viscosity correction: Nu = hd k = 0.023Re 0.80 · Pr 1/3 φ 0.14 (11.75) BOOKCOMP, Inc. — John Wiley & Sons / Page 831 / 2nd Proofs / Heat Transfer Handbook / Bejan SHELL-AND-TUBE HEAT EXCHANGER 831 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 [831], (35) Lines: 1601 to 1675 ——— 0.44637pt PgVar ——— Normal Page PgEnds: T E X [831], (35) This correlation is valid for liquids and gases for L/d > 60, Pr > 0.60, and moderate T w − T b . The correlation of Petukhov (1970), Nu = hd k = (f/8)Re · Pr 1.07 + 12.7(f/8) 1/2 (Pr 2/3 − 1) (11.76) where f = 1 (1.82 log 10 Re − 1.64) 2 (11.77) is valid for 1 ≤ Pr ≤ 2000 and 10 4 ≤ Re ≤ 5 × 10 5 . Bejan (1995) has suggested that the most accurate correlation is that of Gnielinski (1976), who provided a modification of the Petukhov (1970) correlation of eq. (11.76) with f given by eq. (11.64): Nu = hd k = (f/8)(Re − 1000)Pr 1.00 + 12.7(f/8) 1/2 (Pr 2/3 − 1) (11.78) in order to extend the range to 1.0 < Pr < 10 6 and 2300 < Re < 5 × 10 6 . Two simpler alternatives to eq. (11.78) have been suggested by Gnielinski (1976): Nu = hd k = 0.0214(Re 0.80 − 100)Pr 0.40 (11.79) for 0.50 < Pr < 1.50 and 10 4 < Re < 5 × 10 6 , and Nu = hd k = 0.012(Re 0.87 − 280)Pr 0.40 (11.80) for 0.50 < Pr < 500 and 3 × 10 3 < Re < 10 6 . Equation (11.76) can be modified to account for variable properties: Nu = hd k = (f/8)Re · Pr 1.07 + 12.7(f/8) 1/2 (Pr 2/3 − 1) φ n (11.81) where n = 0.11 for heating and n = 0.25 for cooling and where f is given by eq. (11.64). In addition to the restrictions on L/d and Re cited with eq. (11.64), 1 ≤ φ ≤ 40 and 0.5 ≤ Pr ≤ 140. Sleicher and Rouse (1975) give Nu = hd k = 5 + 0.015Re m · Pr n (11.82) where m = 0.88 − 0.24 Pr + 4.00 BOOKCOMP, Inc. — John Wiley & Sons / Page 832 / 2nd Proofs / Heat Transfer Handbook / Bejan 832 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 [832], (36) Lines: 1675 to 1691 ——— 0.45103pt PgVar ——— Normal Page PgEnds: T E X [832], (36) n = 1 3 + 0.5e −0.60Pr for 1 ≤ Pr ≤ 1000 and 10 4 ≤ Re ≤ 10 6 . Shell Side It is not practical to manufacture a shell-and-tube heat exchanger in which fluid flow between the baffles and the shell is prevented by welding each segmented baffle to the inside of the shell. It is also not practical to try and prevent fluid flow in the annular space between each tube and a baffle by fitting the annular space around each tube with a tightly fitting sleeve and or by trying to assure that the tubes completely fill the shell in a uniform manner such that there are no gaps between the tube bundle and the shell. Yet an exchanger with a shell side fabricated with these features would yield the idealized flow pattern shown in Fig. 11.11a. Notice that in this ideal flow pattern, there is no bypassing of the tube bundle within a baffle space and no leakage of the shell-side fluid between adjacent baffle spaces. Figure 11.11 Tinker (1951) model for the shell-side flow streams: (a) idealized model; (b, c) model proposed by Tinker. (From Saunders, 1988, with permission.) BOOKCOMP, Inc. — John Wiley & Sons / Page 833 / 2nd Proofs / Heat Transfer Handbook / Bejan SHELL-AND-TUBE HEAT EXCHANGER 833 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 [833], (37) Lines: 1691 to 1727 ——— 0.12007pt PgVar ——— Normal Page * PgEnds: Eject [833], (37) Because of the need to have certain tube bundles removable and because of the cost, shell-and-tube heat exchangers always possess gaps between the baffles and the shell and between the baffles and the tubes. Moreover, there may also be gaps between the tube bundle and the shell, and these gaps will be due to impingement baffles and/or pass partitions. Tinker (1951) proposed the shell-side fluid flow model shown in Fig. 11.11b as the departure from the ideal. The stream designations A, B, C, and E were proposed by Tinker (1951), and later, stream F , shown in Fig. 11.8c, was added. Tinker also provided a method for determination of the individual flow stream components from which the overall heat transfer coefficient and pressure loss could be determined. While the Tinker (1951) approach was fundamentally sound, experimental data were sparce and unreliable, and, of course, there were no computers in the 1950s. Although the Tinker (1951) method was later simplified by Tinker (1958) and simplified further by Devore (1962) and Fraas (1989), many heat exchanger designers continued to rely on the methods provided by Donohue (1949), Kern (1950), and Gilmour (1952–54), which assumed that all the shell-side fluid flowed across the tube bundle in crossflow without leakage, as shown in Fig. 11.11a. A correction factor was later applied to heat transfer coefficients obtained from these methods to account for all the leakage streams. Bell (1963) published a method based on extensive research at the University of Delaware, and this produced the name Bell–Delaware method. This method accounts for the various leakage streams and involves relatively straightforward calculations. Details of the method, complete with supporting curves, have been presented by Bell (1988) and Taborek (1983). In the Bell–Delaware method, an ideal heat transfer coefficient h id is determined for pure crossflow using the entire shell-side fluid flow stream at (or near) the center of the shell. It is computed from the correlations of Zhukauskas (1987) outlined in Chapter 6 and repeated here as eqs. (11.83) and (11.84): For in-line tube bundles with the number of tube rows n r ≥ 16: Nu d = 0.90Re 0.4 d Pr 0.36 Υ 0.25 for 1 ≤ Re d < 100 0.52Re 0.5 d Pr 0.36 Υ 0.25 for 100 ≤ Re d < 1000 0.27Re 0.63 d Pr 0.36 Υ 0.25 for 1000 ≤ Re d < 2 × 10 5 0.033Re 0.8 d Pr 0.36 Υ 0.25 for 2 × 10 5 ≤ Re d < 2 × 10 6 (11.83) and for staggered tube bundles with the number of tube rows n r ≥ 16: Nu d = 1.04Re 0.4 d Pr 0.36 Υ 0.25 for 1 ≤ Re d < 500 0.71Re 0.5 d Pr 0.36 Υ 0.25 for 500 ≤ Re d < 1000 0.35 0.2 Re 0.63 d Pr 0.36 Υ 0.25 for 1000 ≤ Re d < 2 × 10 5 0.031 0.2 Re 0.8 d Pr 0.36 Υ 0.25 for 2 × 10 5 ≤ Re d < 2 × 10 6 (11.84) BOOKCOMP, Inc. — John Wiley & Sons / Page 834 / 2nd Proofs / Heat Transfer Handbook / Bejan 834 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 [834], (38) Lines: 1727 to 1791 ——— 1.34021pt PgVar ——— Normal Page * PgEnds: Eject [834], (38) In eqs. (11.83) and (11.84), Υ = Pr Pr w and = X T X L The ideal heat transfer coefficient is then corrected using the product of five correction factors to provide the shell-side heat transfer coefficient h s : h s = J C J L J B J S J R h id (11.85) The numerical values of the correction factors were determined by Bell (1963) and with a subsequent curve-fitting procedure due to Taborek (1998). They are now considered in detail. J C is the correction factor for the baffle cut and spacing and is the average for the entire exchanger. It is expressed as a fraction of the number of tubes in crossflow J C = 0.55 + 0.72F C (11.86) where with ϕ = D s − 2l c D o F C = 1 π [ π + ϕ sin(arccos ϕ) − 2 arccos ϕ ] (11.87) In eqs. (11.86) and (11.87), D s is the shell inside diameter (m), D o is the diameter at the outer tube limit (m), and l c is the distance from the baffle tip to the shell inside diameter (m). J L is the correction factor for baffle leakage effects, including both the tube-to- baffle and the baffle-to-shell effects (the A and E streams in Fig. 11.11b and c): J L = 0.44(1 − r a ) + [ 1 − 0.044(1 − r a ) ] e −2.2r b (11.88) where r a = A sb A sb + A tb (11.89a) r b = A sb + A tb A w (11.89b) J B is the correction factor for bundle and partition bypass effects (the C and F streams in Fig. 11.11b and c): J B = 1forζ ≥ 1 2 e −Cr c [1−2ζ 1/3 ] for ζ < 1 2 (11.90) . establishing the heat transfer coefficient is D e = 4A w πd o n tw + D s θ 2 /2 (11.67) BOOKCOMP, Inc. — John Wiley & Sons / Page 829 / 2nd Proofs / Heat Transfer Handbook / Bejan SHELL-AND-TUBE HEAT. 10 5 0.031 0.2 Re 0.8 d Pr 0.36 Υ 0.25 for 2 × 10 5 ≤ Re d < 2 × 10 6 (11 .84) BOOKCOMP, Inc. — John Wiley & Sons / Page 834 / 2nd Proofs / Heat Transfer Handbook / Bejan 834 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 [834],. (11.83) and (11 .84) , Υ = Pr Pr w and = X T X L The ideal heat transfer coefficient is then corrected using the product of five correction factors to provide the shell-side heat transfer coefficient