BOOKCOMP, Inc. — John Wiley & Sons / Page 815 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT EXCHANGER ANALYSIS METHODS 815 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 [815], (19) Lines: 1029 to 1040 ——— -14.073pt PgVar ——— Short Page PgEnds: T E X [815], (19) 0 0 0 0 20 20 20 20 40 40 40 40 60 60 60 60 80 80 80 80 100 100 100 100 Effectiveness, , %⑀ Effectiveness, , %⑀ Effectiveness, , %⑀ Effectiveness, , %⑀ Counterflow exchanger performance hot fluid ( ) = . mc C ph h Crossflow exchanger with fluids unmixed. Parallel-flow exchanger performance hot fluid ( ) . mc ph Cross exchanger with one fluid mixed Cold fluid ( ) = . mc C pc c () . mc ph () . mc pc Cold fluid ( ) . mc pc Heat transfer surface Heat transfer surface ()a ()c ()b ()d 0.25 0.25 0.25 0.25 0.50 0.50 0.50 4 0.50 2 0.75 1.33 0.75 0.75 0.75 1.00 1.00 1.00 CC min max /=0 CC min max /=0 CC min max /=0 0 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 Number of Transfer Units, = /NUSC tu min Number of Transfer Units, = /NUSC tu min Number of Transfer Units, = /NUSC tu min Number of Transfer Units, = /NUSC tu min Mixed fluid Unmixed fluid Hot fluid Cold fluid C C mixed unmixed =0,ϱ C C mixed unmixed =1 Figure 11.5 Heat exchanger effectiveness as a function of N tu for 10 heat exchanger arrange- ments. (From Kakac¸, 1991, with permission.) BOOKCOMP, Inc. — John Wiley & Sons / Page 816 / 2nd Proofs / Heat Transfer Handbook / Bejan 816 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 [816], (20) Lines: 1040 to 1045 ——— -9.073pt PgVar ——— Normal Page PgEnds: T E X [816], (20) 0 0 0 0 40 20 20 20 60 40 40 40 70 60 60 60 80 80 80 80 90 100 100 100 Effectiveness, , %⑀ Effectiveness, ,%⑀ Effectiveness, , %⑀ Effectiveness, ,%⑀ Multipass cross-counter- flow exchanger / = 1 unmixed flow within passes CC min max () . mc ph Multipass counterflow exchanger performance (parallel-counterflow passes) Cold fluid () . mc pc Hot fluid ()e ()g ()f ()h Effect of number of shell passes for / = 1CC min max Exchanger performance effect of flow arrangement for / = 1CC min max Exchanger performance effect of /CC min max Counterflow / = 1.0CC min max CC min max / = 0.9 Crossflow both fluids unmixed / = 0.9CC min max CC min max / = 1.0 0 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 Number of Transfer Units, = /NUSC tu min Number of Transfer Units, = /NUSC tu min Number of Transfer Units, = /NUSC tu min Number of Transfer Units, = /NUSC tu min One pass Two passes Three passes Four passes Counterflow (= )n ϱ Counterflow Two-pass arrangement illustrated Crossflow one fluid mixed Crossflow fluids unmixed Parallel-counterflow one shell pass Parallel flow Counterflow ( = )n ϱ Four passes Three passes Two passes (1–2 exchanger) One pass (1–2 exchanger) Figure 11.5 (Continued) Heat exchanger effectiveness as a function of N tu for 10 heat exchanger arrangements. (From Kakac¸, 1991, with permission.) BOOKCOMP, Inc. — John Wiley & Sons / Page 817 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT EXCHANGER ANALYSIS METHODS 817 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 [817], (21) Lines: 1045 to 1086 ——— 0.13704pt PgVar ——— Normal Page * PgEnds: Eject [817], (21) Figure 11.5 (Continued) Heat exchanger effectiveness as a function of N tu for 10 heat exchanger arrangements. (From Kakac¸, 1991, with permission.) For C min = C shell and C max = C tube : For the effectiveness, use eq. (11.46a) with C ∗ replaced by 1/C ∗ ,N tu replaced by C ∗ N tu , and replaced by C ∗ . When C ∗ = 1, use eq. (11.46b), and when N tu −→ ∞ when C min = C shell , = 2 C ∗ + 2 + (C ∗2 + 4) 1/2 Graphs of the 10 arrangements just considered are shown in Fig. 11.5. 11.3.3 P–N tu,c Method In shell-and-tube heat exchangers, any possible confusion deriving from selection of the C min fluid is avoided through use of the P –N tu,c method. The method uses the cold-side capacity rate, so that P = C min C c = for C c = C min C ∗ for C c = C max (11.47) N tu,c = C min C c N tu = N tu for C c = C min N tu C ∗ for C c = C max (11.48) BOOKCOMP, Inc. — John Wiley & Sons / Page 818 / 2nd Proofs / Heat Transfer Handbook / Bejan 818 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 [818], (22) Lines: 1086 to 1101 ——— -0.24794pt PgVar ——— Normal Page * PgEnds: Eject [818], (22) 0.00.0 0.2 0.3 0.4 0.6 0.8 1 2 5 6 8 1034 N tu,c 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 3.0 4.0 5.0 6.0 8.0 10.0 2.5 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 R =0 F = 0.7 0.75 0.8 0.95 0.9 0.85 P= tt Tt 21 1 1 Ϫ Ϫ Shell fluid Tube fluid Figure 11.6 Temperature effectiveness P as a function of N tu ,c, and R for a 1-2 shell-and- tube heat exchanger with the shell fluid mixed. (From Kakac¸, 1991, with permission.) and it may be recalled that R = C c C h = T 1 − T 2 t 2 − t 1 (11.29b) The parameter P is the temperature effectiveness and is similar to the exchanger effectiveness . It is a function of N tu,c ,R, and the flow arrangement BOOKCOMP, Inc. — John Wiley & Sons / Page 819 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT EXCHANGER ANALYSIS METHODS 819 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 [819], (23) Lines: 1101 to 1166 ——— 1.16634pt PgVar ——— Normal Page * PgEnds: Eject [819], (23) P = f(N tu,c, R, flow arrangement) In the P –N tu,c method, the total heat flow from the hot fluid to the cold fluid will be q = PC c (T 1 − t 1 ) (11.49) and the P –N tu,c relationships can be derived from the –N tu relationships by replacing C ∗ , and N tu by R, P , and N tu,c . For example, for the counterflow exchanger, which has an –N tu representation of = 1 − e −N tu (1−C ∗ ) 1 − C ∗ e −N tu (1−C ∗ ) (11.37) the P –N tu,c representation is P = 1 − e −N tu,c (1−R) 1 − Re −N tu (1−R) Figure 11.6 is a chart of P plotted against N tu,c for the 1–2 shell-and-tube heat exchanger with the shell fluid mixed. 11.3.4 ψ–P Method The ψ–P method proposed by Mueller (1967) combines the variables of the LMTD and –N tu methods. Here ψ is introduced as the ratio of the true temperature dif- ference to the temperature head (the inlet temperature difference of the two fluids, T 1 − t 1 , ψ = θ m T 1 − t 1 = N tu = P N tu,c (11.50) The logarithmic mean temperature difference correction factor F = θ m LMTD c which can be written as F = N cf N tu,c (11.51) where N cf is the number of transfer units for the counterflow exchanger obtained by solving eq. (11.37), = 1 − e −N tu (1−C ∗ ) 1 − C ∗ e −N tu (1−C ∗ ) (11.37) for N tu : BOOKCOMP, Inc. — John Wiley & Sons / Page 820 / 2nd Proofs / Heat Transfer Handbook / Bejan 820 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 [820], (24) Lines: 1166 to 1192 ——— 6.80705pt PgVar ——— Normal Page PgEnds: T E X [820], (24) N cf = 1 1 − R ln 1 − RP 1 − P for R = 1 P 1 − P for R = 1 (11.52) Equations (11.30), (11.31), and (11.52) can be combined to yield ψ = FP (1 − R) ln [ (1 − RP )/(1 − P) ] (11.53) 0.00.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 P= tt Tt 21 1 1 Ϫ Ϫ 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.95 0.9 0.8 0.75 F = 0.5 3.0 2.5 2.0 1.8 1.6 1.4 1.2 1.0 1 NTU c 0.9 0.8 0.7 0.6 0.5 0.4 3.5 3.0 2.5 2.0 1.8 1.6 1.4 1.2 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 R =4 Figure 11.7 Mueller (1967) chart for Ψ as a function of P for a 1–2 shell-and-tube heat exchanger with the shell fluid mixed. (From Kakac¸, 1991, with permission.) BOOKCOMP, Inc. — John Wiley & Sons / Page 821 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT EXCHANGER ANALYSIS METHODS 821 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 [821], (25) Lines: 1192 to 1239 ——— 2.3401pt PgVar ——— Normal Page PgEnds: T E X [821], (25) so that q = US ψ(T 1 − t 1 ) (11.54) which shows that ψ = f(P,R,flow arrangement) The plot of ψ as a function of P is known as a Mueller chart (Mueller, 1967), and Fig. 11.7 shows such a chart for a 1–2 shell-and-tube heat exchanger with the shell fluid mixed. 11.3.5 Heat Transfer and Pressure Loss The relationships presented thus far refer to the principles of heat transfer and the conservation of energy among the streams that make up the heat exchangers. The energy analysis is completed by taking into account the pumping power needed to force the streams through the heat exchanger structure. Relations for pumping power or pressure loss calculations are presented in Section 11.4.4. Qualitatively speaking, in a heat exchanger with changing flow architecture, the heat exchanger performance and the pumping power performance compete with one another. For example, structural modifications such as the employment of extended surface (fins) that lead to heat transfer enhancement also cause an increase in pumping power. Trade-offs between these competing effects have been addressed extensively in thermal design (Bejan et al., 1996). For example, the confined thermodynamic irreversibility due to heat transfer and pumping power can be minimized by proper selection of the dimensions and aspect ratios of the flow passages (Bejan, 1997, 2000). 11.3.6 Summary of Working Relationships A summary of the pertinent relationships employed in the analysis of heat exchangers follows. • For the logarithmic mean temperature difference correction factor method (the LMTD method), q = US F θ m θ m = LMTD = ∆T 1 − ∆T 2 ln(∆T 1 /∆T 2 ) = ∆T 2 − ∆T 1 ln(∆T 2 /∆T 1 ) (11.24) where ∆T 1 = T 1 − t 2 ∆T 2 = T 2 − t 1 P = t 2 − t 1 T 1 − t 1 (11.29a) BOOKCOMP, Inc. — John Wiley & Sons / Page 822 / 2nd Proofs / Heat Transfer Handbook / Bejan 822 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 [822], (26) Lines: 1239 to 1287 ——— -2.79271pt PgVar ——— Normal Page PgEnds: T E X [822], (26) R = T 1 − T 2 t 2 − t 1 (11.29b) F = f(P,R,flow arrangement) • For the –N tu method, q = C min (T 1 − t 1 ) (11.37) = q q max = C h (T 2 − T 1 ) C min (T 1 − t 1 ) = C c (t 2 − t 1 ) C min (T 1 − t 1 ) (11.35) C ∗ = C min C max (0 ≤ C ∗ ≤ 1) (11.31) N tu ≡ US C min = 1 C min S UdS (11.33) = f(N tu, C ∗ , flow arrangement) • For the P –N tu,c method, q = PC c (T 1 − t 1 ) (11.49) N tu,c = US C c = C min C c N tu (11.48) P = f(N tu,c ,R, flow arrangement) • For the ψ–P method, q = US ψ(T 1 − t 2 ) (11.54) ψ = θ m T 1 − t 1 = N tu = P N tu,c (11.50) ψ = f(P,R, flow arrangement) 11.4 SHELL-AND-TUBE HEAT EXCHANGER 11.4.1 Construction Shell-and-tube heat exchangers are fabricated with round tubes mounted in cylin- drical shells with their axes coaxial with the shell axis. The differences between the many variations of this basic type of heat exchanger lie mainly in their construction features and the provisions made for handling differential thermal expansion between tubes and shell. BOOKCOMP, Inc. — John Wiley & Sons / Page 823 / 2nd Proofs / Heat Transfer Handbook / Bejan SHELL-AND-TUBE HEAT EXCHANGER 823 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 [823], (27) Lines: 1287 to 1300 ——— 0.757pt PgVar ——— Normal Page PgEnds: T E X [823], (27) A widely accepted standard is published by the Tubular Exchanger Manufacturers’ Association (TEMA). This standard is intended to supplement the ASME as well as other boiler and pressure vessel codes. The TEMA (1998) standard was prepared by a committee comprising representatives of 27 U.S. manufacturing companies, and their combined expertise and experience provide exchangers of high integrity at reasonable cost. TEMA provides a standard designation system that is summarized in Fig. 11.8. Six examples of the shell-and-tube heat exchanger arrangements are shown in Fig. 11.9. One pass shell Two pass shell with longitudinal baffle Split flow Shell Types Front End Stationary Head Types Channel and removable cover Bonnet (integral cover) Channel integral with tube- sheet and removable cover Channel integral with tube- sheet and removable cover Special high pressure closure Rear End Head Types Fixed tubesheet like “A” stationary head Fixed tubesheet like “B” stationary head Fixed tubesheet like “N” stationary head Outside packed floating head Floating head with backing device Pull through floating head U-Tube bundle Externally sealed floating tubesheet Double split flow Divided flow Kettle type reboiler Cross flow EL A F M B G N C H P N J S D K T X W U Removable tube bundle only Figure 11.8 TEMA standard designation system for shell-and-tube heat exchangers. (From Saunders, 1988, with permission.) BOOKCOMP, Inc. — John Wiley & Sons / Page 824 / 2nd Proofs / Heat Transfer Handbook / Bejan 824 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 [824], (28) Lines: 1300 to 1300 ——— -4.073pt PgVar ——— Normal Page PgEnds: T E X [824], (28) ()a ()b ()c ()d ()e ()f Figure 11.9 (a) Single-tube-pass baffled single-pass-shell shell-and-tube heat exchanger de- signed to give essentially counterflow conditions. The toroidal expansion joint in the center of the shell accommodates differential thermal expansion between the tubes and the shell. (b) U-tube single-pass-shell shell-and-tube heat exchanger. (c) Two-pass baffled single-pass- shell shell-and-tube heat exchanger. (d) Heat exchanger similar to that of (c) except for the floating head used to accommodate differential thermal expansion between the tubes and the shell. (e) Heat exchanger that is similar to the heat exchanger in (d) but with a different type of floating head. (f ) Single-tube-pass baffled single-pass-shell shell-and-tube heat exchanger with a packed joint floating head and double header sheets to assure that no fluid leaks from one fluid circuit into the other. (Courtesy of the Patterson-Kelley Co. and reproduced from Fraas, 1989, with permission.) . shell-and-tube heat exchanger with the shell fluid mixed. (From Kakac¸, 1991, with permission.) BOOKCOMP, Inc. — John Wiley & Sons / Page 821 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT EXCHANGER. a 1–2 shell-and-tube heat exchanger with the shell fluid mixed. 11.3.5 Heat Transfer and Pressure Loss The relationships presented thus far refer to the principles of heat transfer and the conservation. for shell-and-tube heat exchangers. (From Saunders, 1988, with permission.) BOOKCOMP, Inc. — John Wiley & Sons / Page 824 / 2nd Proofs / Heat Transfer Handbook / Bejan 824 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 [824],