Heat Exchangers CHAPTER 11 ALLAN D. KRAUS

Transcription

1 CHAPTER Heat Exchangers ALLAN D. KRAUS University of Akron Akron, Ohio. Introduction. Governing relationships.. Introduction.. Exchanger surface area.. Overall heat transfer coefficient.. Logarithmic mean temperature difference. Heat exchanger analysis methods.. Logarithmic mean temperature difference correction factor method.. ɛ N tu method Specific ɛ N tu relationships.. P N tu,c method.. ψ P method.. Heat transfer and pressure loss.. Summary ofworking relationships. Shell-and-tube heat exchanger.. Construction.. Physical data Tube side Shell side.. Heat transfer data Tube side Shell side.. Pressure loss data Tube side Shell side. Compact heat exchangers.. Introduction.. Classification ofcompact heat exchangers.. Geometrical factors and physical data.. Heat transfer and flow friction data Heat transfer data Flow friction data. Longitudinal finned double-pipe exchangers.. Introduction [Fi [ * No * [

2 HEAT EXCHANGERS Physical data for annuli Extruded fins Welded U-fins.. Overall heat transfer coefficient revisited.. Heat transfer coefficients in pipes and annuli.. Pressure loss in pipes and annuli.. Wall temperature and further remarks.. Series parallel arrangements.. Multiple finned double-pipe exchangers. Transverse high-fin exchangers.. Introduction.. Bond or contact resistance ofhigh-fin tubes.. Fin efficiency approximation.. Air-fin coolers Physical data Heat transfer correlations.. Pressure loss correlations for staggered tubes.. Overall heat transfer coefficient. Plate and frame heat exchanger.. Introduction.. Physical data.. Heat transfer and pressure loss. Regenerators.. Introduction.. Heat capacity and related parameters Governing differential equations.. ɛ N tu method.. Heat transfer and pressure loss Heat transfer coefficients Pressure loss.0 Fouling.0. Fouling mechanisms.0. Fouling factors Nomenclature References. INTRODUCTION A heat exchanger can be defined as any device that transfers heat from one fluid to another or from or to a fluid and the environment. Whereas in direct contact heat exchangers, there is no intervening surface between fluids, in indirect contact heat exchangers, the customary definition pertains to a device that is employed in the transfer of heat between two fluids or between a surface and a fluid. Heat exchangers may be classified (Shah,, or Mayinger, ) according to () transfer processes, [ 0. Nor [

3 GOVERNING RELATIONSHIPS () number offluids, () construction, () heat transfer mechanisms, () surface compactness, () flow arrangement, () number offluid passes, and () type ofsurface. Recuperators are direct-transfer heat exchangers in which heat transfer occurs between two fluid streams at different temperature levels in a space that is separated by a thin solid wall (a parting sheet or tube wall). Heat is transferred by convection from the hot (hotter) fluid to the wall surface and by convection from the wall surface to the cold (cooler) fluid. The recuperator is a surface heat exchanger. Regenerators are heat exchangers in which a hot fluid and a cold fluid flow alternately through the same surface at prescribed time intervals. The surface of the regenerator receives heat by convection from the hot fluid and then releases it by convection to the cold fluid. The process is transient; that is, the temperature ofthe surface (and of the fluids themselves) varies with time during the heating and cooling of the common surface. The regenerator is a also surface heat exchanger. In direct-contact heat exchangers, heat is transferred by partial or complete mixing ofthe hot and cold fluid streams. Hot and cold fluids that enter this type ofexchanger separately leave together as a single mixed stream. The temptation to refer to the direct-contact heat exchanger as a mixer should be resisted. Direct contact is discussed in Chapter. In the present chapter we discuss the shell-and-tube heat exchanger, the compact heat exchanger, the longitudinal high-fin exchanger, the transverse high-fin exchanger including the air-fin cooler, the plate-and-frame heat exchanger, the regenerator, and fouling.. GOVERNING RELATIONSHIPS.. Introduction Assume that there are two process streams in a heat exchanger, a hot stream flowing with a capacity rate C h = ṁ h C ph and a cooler (or cold stream) flowing with a capacity rate C c = ṁ c c ph. Then, conservation ofenergy demands that the heat transferred between the streams be described by the enthalpy balance q = C h (T T ) = C c (t t ) (.) where the subscripts and refer to the inlet and outlet of the exchanger and where the T s and t s are employed to indicate hot- and cold-fluid temperatures, respectively. Equation (.) represents an ideal that must hold in the absence oflosses, and while it describes the heat that will be transferred (the duty ofthe heat exchanger) for the case of prescribed flow and temperature conditions, it does not provide an indication ofthe size ofthe heat exchanger necessary to perform this duty. The size ofthe exchanger derives from a statement ofthe rate equation: q = UηSθ m = U h η ov,h S h θ m = U c η ov,c S c θ m (.) where S h and S c are the surface areas on the hot and cold sides of the exchanger, U h and U c are the overall heat transfer coefficients referred to the hot and cold sides of [. No [

4 00 HEAT EXCHANGERS the exchanger and θ m is some driving temperature difference. The quantities η ov,h and η ov,c are the respective overall fin efficiencies and in the case of an unfinned exchanger, η ov,h = η ov,c =. The entire heat exchange process can be represented by q = U h η ov,h S h θ m = U c η ov,c S c θ m = C h (T T ) = C c (t t ) (.) which is merely a combination ofeqs. (.) and (.)... Exchanger Surface Area Consider the unfinned tube oflength L shown in Fig..a and observe that because ofthe tube wall thickness δ w, the inner diameter will be smaller than the outer diameter and the surface areas will be different: S i = πd i L S o = πd o L (.a) (.b) In the case ofthe finned tube, shown with one fin on the inside and outside ofthe tube wall in Fig..b, the fin surface areas will be S fi = n i b i L S fo = n o b o L (.a) (.b) where n i and n o are the number offins on the inside and outside ofthe tube wall, respectively, and it is presumed that no heat is transferred through the tip of either of the inner or outer fins. In this case, the prime or base surface areas The total surface will then be or S bi = (πd i n i δ fi )L S bo = (πd o n o δ fo )L S i = S bi + S fi = (πd i n i δ fi + n i b i )L S i = [ πd i + n i (b i δ fi ) ] L S o = [ πd o + n o (b o δ fo ) ] L The ratio of the finned surface to the total surface will be S fi n i b i L = [ S i πdi + n i (b i δ fi ) ] L = n i b i πd i + n i (b i δ fi ) (.a) (.b) (.a) (.b) (.a) [00 0. Sho [00

5 GOVERNING RELATIONSHIPS S fo n o b o L = [ S o πdo + n o (b o δ fi ) ] L = n o b o πd o + n o (b o δ fo ) (.b) The overall surface efficiencies η ov,h and η ov,c are based on the base surface operating at an efficiency of unity and the finned surface operating at fin efficiencies of η fi and η fo. Hence or and in a similar manner, η ov,i S i = S bi + η fi S fi = S i S fi + η fi S fi η ov,i = S fi S i ( ηfi ) η ov,o = S fo S o ( ηfo ) (.a) (.b) Figure. End view of(a) a bare tube and (b) a small central angle ofa tube with both internal and external fins oflength, L. [0. Sho [0

6 0 HEAT EXCHANGERS Notice that when there is no finned surface, S fi = S fo = 0 and eqs. (.) reduce to η ov,i = η ov,o = and that with little effort, the subscripts in eqs. (.) through (.) can be changed to reflect the hot and cold fluids... Overall Heat Transfer Coefficient In a heat exchanger containing hot and cold streams, the heat must flow, in turn, from the hot fluid to the cold fluid through as many as five thermal resistances:. Hot-side convective layer resistance: R h = h h η ov,h S h (K/W) (.0). Hot-side fouling resistance due to an accumulation offoreign (and undesirable) material on the hot-fluid exchanger surface: R dh = h dh η ov,h S h (K/W) (.) Fouling is discussed in a subsequent section.. Resistance ofthe exchanger material, which has a finite thermal conductivity and which may take on a value that is a function ofthe type ofexchanger: R m = δ w k m S m (K/W) plane walls ln(d o )(d i ) πk m Ln t (K/W) circular tubes (.) where δ m is the thickness ofthe metal, S m the surface area of the metal, and n t the number oftubes.. Cold-side fouling resistance: R dc =. Cold-side convective layer resistance: R h = h dc η ov,c S c (K/W) (.) h c η ov,c S c (K/W) (.) The resistances listed in eqs. (.0) (.) are in series and the total resistance can be represented by [0 0.0 Nor * [0

7 GOVERNING RELATIONSHIPS US = h h η ov,h S h + h dh η ov,h S h + R m + h dc η ov,c S c + h c η ov,c S c (.) where, for the moment, U and S on the left side of eq. (.) are not assigned any subscript. Equation (.) is perfectly general and may be put into specific terms depending on the selection of the reference surface, whether or not fouling is present and whether or not the metal resistance needs to be considered. Ifeq. (.) is solved for U, the result is U = S h h η ov,h S h + S h dh η ov,h S h + SR m + S h dc η ov,c S c + S h c η ov,c S c (.) and ifthe thickness ofthe metal is small and thermal conductivity ofthe metal is high, the metal resistance becomes negligible and U = S h h η ov,h S h + Several forms of eq. (.) are: S h dh η ov,h S h + For a hot-side reference with fouling, U h = h h η ov,h + h dh η ov,h + For a cold-side reference with fouling, U c = h h η ov,h S c S h + S h dc η ov,c S c + S h c η ov,c S c h dc η ov,c S h S c + h c η ov,c S h S c h dh η ov,h S c S h + For a hot-side reference without fouling, U h = h h η ov,h + h c η ov,c S h S c For a cold-side reference without fouling, U c = h h η ov,h S c S h + h c η ov,c + h dc η ov,c h c η ov,c (.) (.) (.) (.0) (.) [0. No * [0

8 0 HEAT EXCHANGERS For an unfinned exchanger where η ov,h = η ov,c = and a hot-side reference without fouling, U h = h h + h c S h S c For an unfinned exchanger and a cold-side reference without fouling, U c = h h S h S c + h c (.) (.).. Logarithmic Mean Temperature Difference For the four basic simple arrangements indicated in Fig.., θ m in eqs. (.) and (.) is the logarithmic mean temperature difference, which can be written as T T t T s t θ m = LMTD = T T ln( T / T ) = T T ln( T / T ) () a () c T T T t t L L t T t s () b () d T t T L L (.) T T T T t t t t Figure. Four basic arrangements for which the logarithmic mean temperature difference may be determined from eq. (.): (a) counterflow; (b) co-current or parallel flow; (c) constant-temperature source and rising-temperature receiver; (d) constant-temperature receiver and falling-temperature source. [0. Nor [0

9 HEAT EXCHANGER ANALYSIS METHODS For the counterflow exchanger where the fluids flow in opposite directions through the exchanger (Fig..a), LMTD = (T t ) (T t ) (.) ln [(T T )/(T t )] For the co-current or parallel flow exchanger where the fluids flow in the same direction through the exchanger (Fig..b), LMTD = (T t ) (T t ) (.) ln [(T t )/(T t )] For an exchanger that has a constant-temperature source, T s = T = T, and a rising-temperature receiver (Fig..c), LMTD = t t ln [(T s t )/(T s t )] (.) For an exchanger that has a constant-temperature receiver, t s = t = t, and a falling-temperature source (Fig..d), LMTD = T T ln [(T t s )/(T t s )] (.) These simple expressions for the logarithmic mean temperature difference cannot be employed for arrangements other than those shown in Fig... The procedure for the case of crossflow and multipass exchangers is given in the next section.. HEAT EXCHANGER ANALYSIS METHODS.. Logarithmic Mean Temperature Difference Correction Factor Method The logarithmic mean temperature difference developed in Section.. is not applicable to multipass or crossflow heat exchangers. The temperature parameter θ m in eqs. (.) and (.) is the true or effective mean temperature difference and is related to the logarithmic mean temperature difference: and the functions θ m = LMTD = T T ln( T / T ) = T T ln( T / T ) P = t t T t (.) (.a) [0. No * [0

10 R = HEAT EXCHANGERS defined as the cold-side effectiveness and R = T T t t = C c C h (.b) defined as a capacity rate ratio. The true or effective mean temperature difference in multipass or crossflow exchangers, θ m, will be related to the counterflow logarithmic mean temperature difference via where the correction factor θ m = F(LMTD c ) F θ m LMTD c (.0) is a function of P,R, and the flow arrangement. The quest for the logarithmic mean temperature difference correction factor apparently began in the early 0s (Nagle, ; Underwood, ; Fischer, ; Bowman et al., 0). The correction factors are available in chart form, such as indicated in Figs.. and.. Correction Factor, F Temperature Efficiency, P P= t t T t R= T T t t Figure. Logarithmic mean temperature difference correction factor for the shell-andtube heat exchanger with one shell pass and two tube passes. (From Kakaç,, with permission.) [0 0. Nor [0

11 R = 0.0 HEAT EXCHANGER ANALYSIS METHODS Correction Factor, F Temperature Efficiency, P P= t t T t R= T T t t Figure. Logarithmic mean temperature difference correction factor for the shell-andtube heat exchanger with two shell passes and four tube passes. (From Kakaç,, with permission.).. ɛ N tu Method The parameter P in the logarithmic mean temperature difference correction factor method requires three temperatures for its computation. The inlet temperature of both the hot and cold streams is usually a given, but when the cold-side outlet temperature is not known, a trial-and-error method is required to evaluate P. The trial-and-error procedure may be avoided in the ɛ N tu method, and because ofits suitability for computer-aided design, the ɛ N tu method is gaining in popularity. Kays and London () have shown that the heat exchanger transfer equations may be written in dimensionless form, which results in three dimensionless groups:. Capacity rate ratio: C = C min (0 C ) (.) C max Notice that this differs from the capacity rate ratio R used in the determination ofthe logarithmic mean temperature difference correction factor. Here, the capacity ratio C is always less than unity.. Exchanger heat transfer effectiveness: ɛ = q q max (0 ɛ ) (.) [0 - No * [0

12 0 HEAT EXCHANGERS This is the ratio ofthe actual heat transferred to the maximum heat that could be transferred if the exchanger were a counterflow exchanger.. Number of transfer units: N tu US = UdS (.) C min C min The number oftransfer units is a measure ofthe size ofthe exchanger. The actual heat transfer is given by the enthalpy balance of eq. (.). Observe that if and if C h >C c then (T T )<(t t ) C c >C h then (t t )<(T T ) the fluid that might experience the maximum temperature change, T t,isthe fluid that has the minimum capacity rate. Thus, the maximum possible heat transfer can be expressed as or S q max = C c (T t ) (C c <C h ) (.a) q max = C h (T t ) (C h <C c ) (.b) and either ofthese can be obtained with the counterflow exchanger. Therefore, the exchanger effectiveness can be written as ɛ = q = C h(t T ) q max C min (T t ) = C c(t t ) C min (T t ) (.) Observe that the value of ɛ will range between zero and unity and that for a given ɛ and q max, the actual heat transfer in the exchanger will be Because q = ɛc min (T t ) (.) ɛ = f(c,n tu, flow arrangement) each exchanger arrangement has its own effectiveness relationship. The formal introduction ofthe ɛ N tu method ofheat exchanger analysis was apparently by London and Seban (, 0). However, Sekulic et al. () point out that solutions for the single-pass crossflow exchanger were originally obtained by Nusselt (, 0), Mason (), and Baclic and Heggs (). Additional solutions for single-pass [0 0. Nor [0

13 HEAT EXCHANGER ANALYSIS METHODS unmixed unmixed crossflow can be attributed to Hausen (), Baclic (), and Li (). Stevens et al. () provided data for one-, two-, and three-pass exchangers, and Baclic (0) reported on an analysis of different two-pass crossflow arrangements. Solutions for shell-and-tube heat exchanger arrangements are also numerous. In addition to the Nagle (), Underwood (), Fischer (), and Bowman et al. (0) studies already cited, Gardner (, ), Schindler and Bates (0), Jaw (), and Crozier and Samuels () all provided analyses ofdifferent arrangements. Kraus and Kern () solved the problem ofthe n (n even) exchanger by direct integration, and this was later modified by Baclic (). Baclic () also gave general explicit solutions for the n (n odd) exchanger. More complex situations with shell-and-tube heat exchangers with unbalanced passes and/or without axial flow nonuniformities have been the subject of papers by Roetzel and Spang () and Spang (). Specific ɛ N tu Relationships Specific ɛ N tu relationships along with their limiting values for 0 flow arrangements, summarized from the work of Kakaç et al. (), follow:. For counterflow, When C =, and when N tu, ɛ = e N tu( C ) C e N tu( C ) ɛ =. For co-current (parallel) flow, When C =, and when N tu, N tu ɛ = + N tu for all C (.) ɛ = e N tu(+c ) + C (.) ɛ = ( e N tu ) ɛ = + C [0-0 No * [0

14 0 HEAT EXCHANGERS For crossflow with both fluids unmixed, When C =, ɛ = e (+C )N tu [ I0 ( Ntu C /) + C n/ I ( Ntu C /)] and when N tu, C C C n/ ( I n Ntu C /) (.) n= ɛ = [I 0 (N tu ) + I (N tu )] e N tu ɛ = for all C. For crossflow with one fluid mixed and one fluid unmixed. With C min mixed and C max unmixed: When C =, and when N tu, With C max mixed and C min unmixed: When C =, and when N tu, ɛ = e { [ e( C Ntu) /C ]} ɛ = e ( e N tu ) ɛ = e /C (.0a) ɛ = C { e C ( e N tu ) } (.0b) ɛ = e ( e N tu ) ɛ = e C C. For crossflow with both fluids mixed, ɛ = C + e N tu e C N tu N tu (.) [0 0. Sho * [0

15 HEAT EXCHANGER ANALYSIS METHODS When C =, and when N tu, ɛ = e N tu ɛ = + C + N tu. For the shell-and-tube exchanger with the shell fluid mixed [Tubular Exchanger Manufacturers Association (TEMA) E-shell], where When C =, where and when N tu, ɛ = ( + C ) + ( + C ) / coth(γ/) ɛ = Γ = N tu ( + C ) / ɛ = + / cothγ/ Γ = / N tu ( + C ) + ( + C ) / (.). For the shell-and-tube exchanger with the shell fluid unmixed (TEMA E- shell). For C min = C tube and C max = C shell : When C =, ɛ = C C + [ C + e N tu(c +/) ] C e N tu(c /) ɛ = + e N tu + N tu (.a) [ 0. Sho * [

16 HEAT EXCHANGERS and when N tu, For C min = C shell and C max = C tube : When C =, and when N tu, for C 0. ɛ = + C f or C < 0. ɛ = C C C ( + C ) ɛ = ɛ = [ + C e N tu(+c /) ] C e N tu(+c /) [ + e N tu / ] e N tu/ + C (.b). For the shell-and-tube exchanger with the shell fluid mixed (TEMA E- shell). For C min = C tube and C max = C shell : where ɛ = When C =, where ( + C ) + ( + C ) / coth(γ/) + tanh(n tu /) ɛ = and when N tu with C min = C tube, ɛ = For C min = C shell and C max = C tube : Γ = N tu ( + C ) / + / coth(γ/) + tanh(n tu /) Γ = / N tu ( + C ) + ( + C ) / + (.a) [. Sho * [

17 HEAT EXCHANGER ANALYSIS METHODS where ɛ = When C =, where and when N tu, ( + C ) + ( + C ) / coth(γ/) + C tanh(c N tu /) ɛ = ɛ = Γ = N tu ( + C ) / + / coth(γ/) + tanh(n tu /) Γ = / N tu ( + C ) + ( + C ) / + C (.b). For the split-flow shell-and-tube exchanger with the shell fluid mixed (TEMA G-shell). For C min = C tube and C max = C shell : where When C =, ɛ = When N tu and C >, and when C, ( + G + C G) + (C + )De α e α ( + G + C G) + C ( D) + C De α (.a) D = e α C + G = e β C α = N tu(c + ) and β = N tu(c ) e Ntu/ e N tu/ ɛ = e N tu/ e N tu/ e N tu/ ɛ = C + C + C + ɛ = (.b) [ 0. Sho [

18 HEAT EXCHANGERS For C min = C shell and C max = C tube : For the effectiveness, use eq. (.a) with C replaced by /C,N tu replaced by C N tu, and ɛ replaced by C ɛ. When C =, use eq. (.b). When N tu with C <, and with C, ɛ = C + C + C + ɛ = C 0. For the divided-flow shell-and-tube exchanger with the shell fluid mixed (TEMA J-shell). For C min = C tube and C max = C shell : where When C =, where ɛ = + C Φ (.a) Φ = + γ + Φ + ( Φ)e C Ntu(γ )/ γγφ Φ ( Φ) + γ( Φ ) Φ = e γc N tu γ = ( + C ) / C ɛ = + Φ (.b) Φ = + γ + Φ + ( Φ)e Ntu(γ )/ γγφ Φ ( Φ) + γ( Φ ) Φ = e γn tu γ = / and when N tu, ɛ = + C + ( + C ) / [. Sho * [

19 HEAT EXCHANGER ANALYSIS METHODS Effectiveness,, % Effectiveness,, % Number of Transfer Units, Ntu = US/ Cmin Counterflow exchanger performance. hot fluid ( mcph ) = Ch. Cold fluid ( mcpc ) = Cc Heat transfer surface () a () c Cmin/ Cmax = Crossflow exchanger with fluids unmixed.. ( mc pc ) Cold fluid Cmin/ Cmax = ( mc ph ) Hot fluid 0 0 Number of Transfer Units, Ntu = US/ Cmin Effectiveness,, % Effectiveness,, % Number of Transfer Units, Ntu = US/ Cmin Parallel-flow exchanger performance. hot fluid ( mc p ) h Heat transfer surface () b () d Cmin/ Cmax =0 Cmixed =0, Cunmixed. Cold fluid ( mc pc ) Cross exchanger with one fluid mixed Mixed fluid Unmixed fluid C C mixed = unmixed Number of Transfer Units, Ntu = US/ Cmin Figure. Heat exchanger effectiveness as a function of N tu for 0 heat exchanger arrangements. (From Kakaç,, with permission.) [ - Sho [

20 HEAT EXCHANGERS Effectiveness,, % Effectiveness,,% Number of Transfer Units, Ntu = US/ Cmin ( mc ph ) Hot fluid () e () g Multipass cross-counterflow exchanger Cmin/ Cmax = unmixed flow within passes Two-pass arrangement illustrated. ( mc pc ) Cold fluid Four passes Counterflow ( n = ) Three passes One pass Multipass counterflow exchanger performance (parallel-counterflow passes) Effect of number of shell passes for Cmin/ Cmax = Counterflow ( n = ) Two passes Four passes Three passes Two passes ( exchanger) One pass ( exchanger) 0 0 Number of Transfer Units, Ntu = US/ Cmin Effectiveness,, % Effectiveness,,% Number of Transfer Units, Ntu = US/ Cmin ( f ) Counterflow () h Exchanger performance effect of flow arrangement for Cmin/ Cmax = Crossflow one fluid mixed Parallel flow Exchanger performance effect of Cmin/ Cmax Counterflow Cmin/ Cmax =.0 Cmin/ Cmax = 0. Crossflow fluids unmixed Parallel-counterflow one shell pass Crossflow both fluids unmixed Cmin/ Cmax = 0. Cmin/ Cmax = Number of Transfer Units, Ntu = US/ Cmin Figure. (Continued) Heat exchanger effectiveness as a function of N tu for 0 heat exchanger arrangements. (From Kakaç,, with permission.) [ -. Nor [

21 HEAT EXCHANGER ANALYSIS METHODS Figure. (Continued) Heat exchanger effectiveness as a function of N tu for 0 heat exchanger arrangements. (From Kakaç,, with permission.) For C min = C shell and C max = C tube : For the effectiveness, use eq. (.a) with C replaced by /C,N tu replaced by C N tu, and ɛ replaced by C ɛ. When C =, use eq. (.b), and when N tu when C min = C shell, ɛ = C + + (C + ) / Graphs ofthe 0 arrangements just considered are shown in Fig..... 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 ofthe P N tu,c method. The method uses the cold-side capacity rate, so that P = C { min ɛ for Cc = C min ɛ = C c ɛc for C c = C max (.) N tu,c = C { min Ntu for C N tu = c = C min C c N tu C for C c = C max (.) [ 0. No * [

22 HEAT EXCHANGERS P= t t T t Shell fluid F = 0. Tube fluid N tu,c Figure. Temperature effectiveness P as a function of N tu,c, and R for a - shell-andtube heat exchanger with the shell fluid mixed. (From Kakaç,, with permission.) and it may be recalled that R = C c C h = T T t t R = (.b) The parameter P is the temperature effectiveness and is similar to the exchanger effectiveness ɛ. It is a function of N tu,c,r, and the flow arrangement [ -0. Nor * [

23 HEAT EXCHANGER ANALYSIS METHODS 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 t ) (.) and the P N tu,c relationships can be derived from the ɛ N tu relationships by replacing C, ɛ and N tu by R,P, and N tu,c. For example, for the counterflow exchanger, which has an ɛ N tu representation of ɛ = e N tu( C ) (.) C e N tu( C ) the P N tu,c representation is P = e N tu,c( R) Re N tu( R) Figure. is a chart of P plotted against N tu,c for the shell-and-tube heat exchanger with the shell fluid mixed... ψ P Method The ψ P method proposed by Mueller () combines the variables ofthe LMTD and ɛ N tu methods. Here ψ is introduced as the ratio ofthe true temperature difference to the temperature head (the inlet temperature difference of the two fluids, T t, ψ = θ m T t = ɛ N tu = The logarithmic mean temperature difference correction factor which can be written as θ m F = LMTD c P N tu,c (.0) F = N cf N tu,c (.) where N cf is the number of transfer units for the counterflow exchanger obtained by solving eq. (.), for N tu : ɛ = e N tu( C ) C e N tu( C ) (.) [. No * [

24 R = HEAT EXCHANGERS N cf = RP ln R P P P for R = for R = Equations (.0), (.), and (.) can be combined to yield ψ =.0 0. FP ( R) ln [( RP )/( P)] F = 0. (.) (.) P= t t T t NTU c Figure. Mueller () chart for Ψ as a function of P for a shell-and-tube heat exchanger with the shell fluid mixed. (From Kakaç,, with permission.) [0. Nor [0

25 HEAT EXCHANGER ANALYSIS METHODS so that which shows that q = USψ(T t ) (.) ψ = f(p,r,flow arrangement) The plot of ψ as a function of P is known as a Mueller chart (Mueller, ), and Fig.. shows such a chart for a shell-and-tube heat exchanger with the shell fluid mixed... Heat Transfer and Pressure Loss The relationships presented thus far refer to the principles of heat transfer and the conservation ofenergy 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... 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 ofextended 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., ). For example, the confined thermodynamic irreversibility due to heat transfer and pumping power can be minimized by proper selection ofthe dimensions and aspect ratios ofthe flow passages (Bejan,, 000)... Summary of Working Relationships A summary ofthe pertinent relationships employed in the analysis ofheat exchangers follows. For the logarithmic mean temperature difference correction factor method (the LMTD method), where q = USF θ m θ m = LMTD = T T ln( T / T ) = T T ln( T / T ) T = T t T = T t P = t t T t (.) (.a) [. No [

26 HEAT EXCHANGERS For the ɛ N tu method, R = T T t t F = f(p,r,flow arrangement) (.b) q = ɛc min (T t ) (.) ɛ = For the P N tu,c method, For the ψ P method, q q max = C h(t T ) C min (T t ) = C c(t t ) C min (T t ) (.) C = C min (0 C ) (.) C max N tu US = UdS (.) C min C min S ɛ = f(n tu, C, flow arrangement) q = PC c (T t ) (.) N tu,c = US C c = C min C c N tu (.) P = f(n tu,c,r, flow arrangement) q = USψ(T t ) (.) ψ = θ m T t = ɛ N tu = ψ = f(p,r, flow arrangement). SHELL-AND-TUBE HEAT EXCHANGER.. Construction P N tu,c (.0) Shell-and-tube heat exchangers are fabricated with round tubes mounted in cylindrical shells with their axes coaxial with the shell axis. The differences between the many variations ofthis basic type ofheat exchanger lie mainly in their construction features and the provisions made for handling differential thermal expansion between tubes and shell. [ -. Nor [

27 SHELL-AND-TUBE HEAT EXCHANGER 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 () standard was prepared by a committee comprising representatives of U.S. manufacturing companies, and their combined expertise and experience provide exchangers ofhigh integrity at reasonable cost. TEMA provides a standard designation system that is summarized in Fig... Six examples ofthe shell-and-tube heat exchanger arrangements are shown in Fig... A Front End Stationary Head Types Channel and removable cover Bonnet (integral cover) Channel integral with tubesheet and removable cover Channel integral with tubesheet and removable cover Special high pressure closure E F Shell Types One pass shell Two pass shell with longitudinal baffle B G N C N D Removable tube bundle only H J K X Split flow Double split flow Divided flow Kettle type reboiler Cross flow L M P S T U W 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 Figure. TEMA standard designation system for shell-and-tube heat exchangers. (From Saunders,, with permission.) [ 0. No [

28 HEAT EXCHANGERS () a () b () c () d () e () f Figure. (a) Single-tube-pass baffled single-pass-shell shell-and-tube heat exchanger designed to give essentially counterflow conditions. The toroidal expansion joint in the center ofthe 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-passshell 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 offloating 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 ofthe Patterson-Kelley Co. and reproduced from Fraas,, with permission.) [ -. Nor [

29 SHELL-AND-TUBE HEAT EXCHANGER Figure.a shows a single-pass tube side and a baffled single-pass shell side. A toroidal expansion joint in the center ofthe shell accommodates the differential thermal expansion between the tubes and the shell. Figure.b employs U-tubes within the baffled single-pass shell. In this case, account must be taken of the fact that approximately halfofthe tube-side surface is in counterflow and the other halfofthe tube-side surface is in co-current flow with the shell-side fluid. The unit shown in Fig..c has a flow pattern that is similar to the unit shown in Fig..b. However, the construction is more complex, to facilitate inspection ofthe inside ofthe tubes, the cleaning ofthe tubes mechanically, and the replacement ofdefective tubes. While the configuration offig..d 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..e, leakage from one fluid stream into the other through the packed joints ofthe floating head goes directly to the exterior ofthe shell, where it can be detected readily and without contamination ofthe other stream. Figure.f indicates a further variation ofthe type ofexchanger shown in Fig..e... Physical Data Tube Side For a tube bundle containing n t tubes oflength L with outside and inside diameters of d o and d i, respectively, the flow area will be A f = πn td i n p (.) where n p is the number ofpasses. The surface area for the n t tubes is S o = πd o Ln t (.) The equivalent diameter to be used in establishing the heat transfer coefficient is d e = d i (.) and the length ofthe tube to be used in pressure loss computations is L T : L T = L + δ sh (.) where δ sh is the thickness ofthe tube sheet. Because the tube side is concerned only with the number oftubes and tube passes and not with the tube layout, eqs. (.) through (.) pertain to tubes on square ( ), rotated square ( ), and equilateral triangular ( ) pitch. The number oftubes within a given shell can be obtained from a list of tube counts such as those provided by Kern (0) and Saunders (). Shell Side Saunders () has provided the pertinent physical data for the shell side with segmented baffles. A concise pictorial summary of the various shell-side [. No [

30 HEAT EXCHANGERS parameters is shown in Fig..0. With the nomenclature contained in Fig..0, several flow areas can be identified. The crossflow area is A c = C + (D s C d o )(p d o ) yp where the tube pitch factor y varies with the tube arrangement:.000 for equilateral triangular, 0 pitch 0. for equilateral triangular, 0 pitch y =.000 for square pitch 0.0 for square rotated pitch The tube bundle bypass area is where L bc is the central baffle spacing The shell-to-baffle leakage area for one baffle is where δ sb is the shell-to-baffle spacing: (.) A bp = C L bc (.0) A sb = (π + θ )D s δ sb (.) δ sb = D s D b ( θ = arcsin l ) c D s (rad) The fraction ofthe total number oftubes in one window is where F w = θ sin θ π θ = arccos D s l c D s C The tube-to-baffle leakage area for one baffle is A tb = πd o( F w )n t δ tb where δ tb is the tube to baffle spacing. (rad) (.) (.) [. Nor [

31 SHELL-AND-TUBE HEAT EXCHANGER Baffle cut = (00)( I c / D s ) Baffle cut ratio = ( I / D) Baffle edge Baffle edge d 0 c s Ic n c Number rows ( ) Crossed in one Cross-flow space ( /) ( /) Outer tube limit D C / o C / p (0 ) (0 ) Baffle diameter sb / sb / () d p () a Shell inside diameter Shell-baffle leakage area ( A sb )shaded portion tb / Tube o.d. tb / () b () c Baffle-hole diameter p p (0 ) ( pd / o.0) ( ) Conventional baffle Window baffle Bundle by-pass area shown Figure.0 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,, with permission.) A bp () e [ - No [

32 HEAT EXCHANGERS The free area for fluid flow in one window section will be where A wg is the gross window area: where A w = A wg A wt A wg = D s (θ sin θ ) ( ) lc θ = arccos D s and the area occupied by the tubes in one window is A wt = π n twd o where the number oftubes in the window is Hence n tw = F w n t A w = D s (θ sin θ ) π n twdo (.) The number oftubes crossed in one crossflow space is n c = D s l c qp where the factor q varies with the tube pitch: 0. for equilateral triangle, 0 pitch.000 for equilateral triangle, 0 pitch q =.000 for square pitch 0.0 for square rotated pitch The effective number of tubes crossed in one window is (.) n cw = 0.l c (.) qp The equivalent diameter ofone window to be used in establishing the heat transfer coefficient is D e = A w πd o n tw + D s θ / (.) [. Nor * [

33 SHELL-AND-TUBE HEAT EXCHANGER Heat Transfer Data The establishment ofthe heat transfer coefficient on the tube side and the shell side of a shell-and-tube exchanger is offundamental importance to the design and analysis ofthe shell-and-tube heat exchanger. Tube Side Investigations that pertain to heat transfer and friction data within tubes have been reported by Pohlhausen (), DeLorenzo and Anderson (), Deissler (), McAdams (), Hausen (, ), Stefan (), Barnes and Jackson (), Dalle Donne and Bowditch (), Yang (), Petukhov and Popov (), Perkins and Wörsœ-Schmidt (), Wörsœ-Schmidt (), Test (), Webb (), Oskay and Kakaç (), Shah and London (), Rogers (0), Kays and Crawford (), Gnielinski (), Kakaç et al. (, ), Shah and Bhatti (), and Kakaç and Yener (). These are summarized by Kraus et al. (00). Some heat transfer correlations depend on a viscosity correction, ( ) µ n φ n = (.) where µ and µ w are the dynamic viscosities at the bulk and wall temperature, respectively, and where n is an exponent depending on whether the process is one ofheating or one ofcooling. 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 otherwise indicated, all fluid properties are based on the bulk temperature: T b = T + T µ w and t b = t + t Laminar Flow: Re 00 For situations in which the thermal and velocity profiles are fully developed, the Nusselt number depends only on the thermal boundary conditions. For circular tubes with Pr 0.0 and Re Pr L/d > 0.0, the Nusselt numbers have been shown to be for constant-temperature conditions and Nu =. (.) Nu =. (.0) 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 ofthe Reynolds and Prandtl numbers and the diameter-to-length ratio: [. No * [

34 0 HEAT EXCHANGERS Gz Re Pr d L is employed. At the entrance ofa tube, the Nusselt number is infinite and decreases asymptotically to the value for fully developed flow as the flow progresses along the length of the tube. The Sieder Tate () 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 =. Re Pr L) d / φ 0. (.) for T w constant and within the following ranges: 0. Pr, φ. ( Re Pr d L) / φ 0. The limitations should be observed carefully as the Sieder Tate equation yields a zero heat transfer coefficient for extremely long tubes. The correlation ofhausen () is good for both liquids and gases at constant wall temperature: Nu = hd k =. + 0.Re Pr(d/L) [Re Pr(d/L)] / (.) The heat transfer coefficient obtained from this correlation is the average value for the entire length ofthe tube, and it may be observed that when the tube is sufficiently long, the Nusselt number approaches the constant value of.. Transition: 00 Re 0,000 For transition flow for both liquids and gases, the Hausen () correlation for both liquids and gases may be employed: Nu = hd k = 0. ( Re / ) [ ( ) ] d / Pr / φ 0. + (.) L Turbulent Flow: Re 0,000 (0) recommend For both liquids and gases, Dittus and Boelter Nu = hd k = 0.0Re0.0 Pr n (.) where n = 0.0 for cooling and n = 0.0 for heating. Sieder and Tate () removed the dependence on heating and cooling by setting the exponent on Pr to / and adding a viscosity correction: Nu = hd k = 0.0Re0.0 Pr / φ 0. (.) [0. Nor * [0

35 SHELL-AND-TUBE HEAT EXCHANGER This correlation is valid for liquids and gases for L/d > 0, Pr > 0.0, and moderate T w T b. The correlation ofpetukhov (0), Nu = hd k = (f/)re Pr.0 +.(f/) / (Pr / (.) ) where f = (. log 0 Re.) (.) is valid for Pr 000 and 0 Re 0. Bejan () has suggested that the most accurate correlation is that ofgnielinski (), who provided a modification ofthe Petukhov (0) correlation ofeq. (.) with f given by eq. (.): Nu = hd k = (f/)(re 000)Pr.00 +.(f/) / (Pr / (.) ) in order to extend the range to.0 < Pr < 0 and 00 < Re < 0. Two simpler alternatives to eq. (.) have been suggested by Gnielinski (): Nu = hd k = 0.0(Re0.0 00)Pr 0.0 (.) for 0.0 < Pr <.0 and 0 < Re < 0, and Nu = hd k = 0.0(Re0. 0)Pr 0.0 (.0) for 0.0 < Pr < 00 and 0 < Re < 0. Equation (.) can be modified to account for variable properties: Nu = hd k = (f/)re Pr.0 +.(f/) / (Pr / ) φn (.) where n = 0. for heating and n = 0. for cooling and where f is given by eq. (.). In addition to the restrictions on L/d and Re cited with eq. (.), φ 0 and 0. Pr 0. Sleicher and Rouse () give Nu = hd k = + 0.0Rem Pr n (.) where m = Pr +.00 [ 0. No [

36 HEAT EXCHANGERS for Pr 000 and 0 Re 0. n = + 0.e 0.0Pr 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..a. Notice that in this ideal flow pattern, there is no bypassing ofthe tube bundle within a baffle space and no leakage ofthe shell-side fluid between adjacent baffle spaces. Figure. Tinker () model for the shell-side flow streams: (a) idealized model; (b, c) model proposed by Tinker. (From Saunders,, with permission.) [ 0. Nor [

37 SHELL-AND-TUBE HEAT EXCHANGER Because ofthe need to have certain tube bundles removable and because ofthe 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 () proposed the shell-side fluid flow model shown in Fig..b as the departure from the ideal. The stream designations A, B, C, and E were proposed by Tinker (), and later, stream F, shown in Fig..c, 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 () approach was fundamentally sound, experimental data were sparce and unreliable, and, ofcourse, there were no computers in the 0s. Although the Tinker () method was later simplified by Tinker () and simplified further by Devore () and Fraas (), many heat exchanger designers continued to rely on the methods provided by Donohue (), Kern (0), and Gilmour ( ), which assumed that all the shell-side fluid flowed across the tube bundle in crossflow without leakage, as shown in Fig..a. A correction factor was later applied to heat transfer coefficients obtained from these methods to account for all the leakage streams. Bell () 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 ofthe method, complete with supporting curves, have been presented by Bell () and Taborek (). 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 ofthe shell. It is computed from the correlations ofzhukauskas () outlined in Chapter and repeated here as eqs. (.) and (.): For in-line tube bundles with the number oftube rows n r : Nu d = 0.0Re 0. d Pr0. Υ 0. for Re d < 00 0.Re 0. d Pr0. Υ 0. for 00 Re d < Re 0. d Pr 0. Υ 0. for 000 Re d < 0 0.0Re 0. d Pr0. Υ 0. for 0 Re d < 0 and for staggered tube bundles with the number of tube rows n r : Nu d =.0Re 0. d Pr0. Υ 0. for Re d < 00 0.Re 0. d Pr0. Υ 0. for 00 Re d < ϱ 0. Re 0. d Pr 0. Υ 0. for 000 Re d < 0 0.0ϱ 0. Re 0. d Pr0. Υ 0. for 0 Re d < 0 (.) (.) [ 0. No * [

38 HEAT EXCHANGERS In eqs. (.) and (.), Υ = Pr Pr w and ϱ = 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 (.) The numerical values ofthe correction factors were determined by Bell () and with a subsequent curve-fitting procedure due to Taborek (). 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 ofthe number oftubes in crossflow where with ϕ = D s l c D o J C = F C (.) F C = [π + ϕ sin(arccos ϕ) arccos ϕ] (.) π In eqs. (.) and (.), 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-tobaffle and the baffle-to-shell effects (the A and E streams in Fig..b and c): where J L = 0.( r a ) + [ 0.0( r a )] e.r b (.) A sb r a = A sb + A tb r b = A sb + A tb A w (.a) (.b) J B is the correction factor for bundle and partition bypass effects (the C and F streams in Fig..b and c): J B = { f or ζ e Cr c[ ζ / ] for ζ < (.0) [. Nor * [

39 SHELL-AND-TUBE HEAT EXCHANGER where r c = A bp A w ζ = N ss N cc (.a) (.b) where with X L as the longitudinal tube pitch and N ss taken as the number ofsealing strip pairs, Here N cc = D s l c X L {. for Res 00 C =. for Re s > 00 A bp = L bc (D s D o + 0.N P w P ) (.c) is the crossflow area for the bypass, where N P is the number ofbypass divider lanes that are parallel to the crossflow stream B, w P is the width ofthe 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: where N b is the number ofbaffles and J S = N b + ( ) L ( n) ( ) i L ( n) o N b + ( ) L ( n) ( ) i + L ( n) (.) o L i = L bi L bc L o = L bo n = L bc { for turbulent flow for laminar flow (.a) (.b) (.c) 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: [ 0. No * [

40 HEAT EXCHANGERS for Re s 00 ( ) J R = 0 0. for Re s 0 N r,c (.) For 0 < Re s < 00, a linear interpolation should be performed between the two extreme values. In eq. (.), Re s is the shell-side Reynolds number and N r,c is the number ofeffective tube rows crossed through one crossflow section... Pressure Loss Data Tube Side The pressure loss inside tubes ofcircular cross section in a shell-andtube heat exchanger is the sum ofthe friction loss within the tubes and the turn losses between the passes ofthe exchanger. The friction loss inside the tubes is given by P f = f ρu where u is the linear velocity ofthe fluid in the tubes, or P f = fg ρ L d i (Pa) (.a) L d (Pa) (.b) where G is the mass velocity ofthe fluid in the tubes. In eqs. (.), f is the friction factor. The fluid will undergo an additional pressure loss due to contractions and expansions that occur during fluid turnaround between tube passes. Kern (0) and Kern and Kraus () have proposed that this loss be given by one velocity head per turn: P t = ρu In an exchanger with a single pass, (Pa) (.) P t = ρu = ρu (Pa) (.) and in an exchanger with n p passes, there will be n p turns. Hence P t = (n p )ρu (.) Friction factors may be obtained from Fig.., which plots the friction factor as a function of the Reynolds number inside the tube and the relative roughness, ɛ/d i. The figure is due to Moody (), and it may be noted that when the flow is laminar, f = (.) Re [. Nor * [

41 Critical 0.0 Laminar zone flow Transition zone Fully rough zone Friction factor, f 0.0 Laminar flow Smooth pipes f = Re Re cr , , de Relative roughness, Reynolds number, Re = d e [ * No * e d = 0.000,00 e d = 0.000,00 Figure. Moody chart for tube and annulus friction factors. (From Moody,.) [

42 HEAT EXCHANGERS Many investigators have developed friction factor relationships as a function of the Reynolds number. The use ofthe friction factor give by eq. (.) in the Petukhov and Gnielinski correlations ofeqs. (.) and (.) has been noted. Other functions can be fitted to the curves in Fig..0. Two ofthem for smooth tubes are 0.0 ( 0 Re 0.0 Re 0 ) (.00) f = 0.0 ( 0 Re 0. Re 0 ) (.0) Shell Side Tinker () also suggested a flow stream model for the determination ofshell-side pressure loss. However, the lack ofadequate data caused him to make rather gross simplifications in arriving at the effects to be attributed to the various flow streams. Willis and Johnston () 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 (0). The flow streams in the Willis and Johnston method are shown in Fig... For each ofthe streams, a coefficient n is defined so that n i = p i ṁ i (i = b, c, s, t, w) (.0) where the p i s and the ṁ 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 where Figure. method. c A ṁ cr = ṁ w ṁ cr = ṁ b +ṁ c b w s t (.0a) (.0b) Shell-side flow streams for the Willis and Johnston () pressure-drop B [. Nor [

43 SHELL-AND-TUBE HEAT EXCHANGER and the pressure loss between points A and B will be p AB = p cr + p w because p cr = p b = p c. It can be shown that p cr = [ ( n b (.0c) ) / ( ) ] / + ṁ w (.0) n c 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 = where the leakage flow rate is With the total flow rate given by [ ( n s ) / ( ) ] / + ṁ l (.0) n t ṁ l = ṁ s +ṁ t (.0) ṁ T = ṁ s +ṁ t +ṁ w (.0) a combination ofeqs. (.0) and (.0) (.0) gives ṁ T ṁ w = [( / n c + n / ) ] / b + nw + ( / n s + n / ) and it is observed that the procedure depends on the values ofthe n i s. For n c, Butterworth () has proposed that with n c = C c F b c do C c = a d ) V (ṁt d b o δ ov (p d o ) µa c ρd o A c t (.0) (.0a) (.0b) where for square or rotated square pitch, a = 0.0,b = 0.0, and F c =.00; and for equilateral triangular pitch, a = 0.,b = 0., and F c = 0.0. In eq. (.0b), D V = ap d o d o [. No * [

44 0 HEAT EXCHANGERS with a =. for square and rotated square pitch and a =.0 for equilateral triangular pitch. In addition, δ ov = ( 0.0p b )D s is the height ofthe baffle overlap region, p b is the baffle cut, and A c = πd o D o ( θ sin θ cos θ ) Lcb N p zl cb δ ov where n p is the number ofpass partitions and z is the path partition width in line with the flow. For n b, where For n s, n b = 0.(δ ov/x L )(ṁ T D e /µa bp) 0. + N s ρa bp A bp = (w + N p z)l bc D e = A bp (w + L bc ) + N p (z + L bc ) (.0) n s = [ (ṁ T δ sb /µa sb )] 0. + (δ b /δ sb ) +.0(δ b /δ sb ) 0. ρa sb (.) where For n t, A sb = π(d s δ sb )δ sb n t = [ (ṁ T δ tb /µa tb )] 0. + (δ b /δ tb ) +.0(δ b /δ tb ) 0. ρa tb (.) where For n w, A tb = nπ(d o δ tb )δ tb n w =.0e0.A w/a CL ρa w (.) [0 0. Lon * [0

45 SHELL-AND-TUBE HEAT EXCHANGER where A w is the window flow area with n tw taken as the number oftubes in the window: A w = A w πd o n tw A w = d s ( θ sin θ cos θ and where for square and rotated square layouts, and for equilateral triangular layouts, Here, to the nearest integer, A CL = (D s N CL d o )L bc A CL = (N CL )(p D o ) + w N CL = D o d o P y with P y = p for square pitch, P y =.p for rotated square pitch, and P y =.p for equilateral triangular pitch. Equation (.0) 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 where p AB = P cr + p w p cr = n c ṁ c or p cr = n b ṁ b p w = n w ṁ w ) (.0c) 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 where with p e = N e ṁ e + n weṁ w (.) n e = n cr D s + δ ov δ ov ( Lbc L be ) (.a) [ 0. Lon * [

46 HEAT EXCHANGERS and where n cr = The average end-space flow rate is [ ( n c ) / ( ) ] / + (.b) n c n we =.e0.(a wl bc /A CL L) ρa w ṁ e = ṁt +ṁ w Grant (0) gives the pressure drop in the inlet nozzle as p n = G A ρ A ( ) A A (.c) (.) where G is the entry mass velocity, G = ρ u,a is the inlet nozzle area, and A is the bundle entry area. For the outlet-nozzle pressure loss, [ ( ) ( ) ] p n = G A + ρ A c (.) where G is the exit mass velocity, G = ρ u,a is the outlet nozzle area, and A is the bundle exit area. The recommended value ofthe contraction coefficient is c =. The total shell-side pressure loss will be P T = p n + (F T + ) p e + (N b ) p AB p n (.) Equation (.) 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 = ṁcd o µa c where for Re c < 00, the entire method is not valid; 0 Re c < 000,F T =.e 0. ; and Re c 000,F T =.. COMPACT HEAT EXCHANGERS.. Introduction One variation ofthe fundamental compact exchanger element, the core, is shown in Fig... The core consists ofa pair ofparallel plates with connecting metal [ -. Nor [

47 COMPACT HEAT EXCHANGERS Figure. Exploded view ofa compact heat exchanger core:, plates;, side bars;, corrugated fins stamped from a continuous strip of metal. By spraying braze powder on the plates, the entire assembly ofplates, fins, and bars can be thermally bonded in a single furnace operation. (From Kraus et al., 00, with permission.) members that are bonded to the plates. The arrangement ofplates and bonded members provides both a fluid-flow channel and prime and extended surface. It is observed that ifa plane were drawn midway between the two plates, each halfofthe 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, ofcourse, due consideration must be given to the fact that as more and more fins are placed in a core, the equivalent or hydraulic diameter ofthe core is lowered while the pressure loss is increased significantly. Next, consider a pair ofcores arranged as components ofa two-fluid exchanger in crossflow as shown in Fig... 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.. is a plate fin heat exchanger. The discussion ofplate fin exchangers has concentrated thus far on geometries involving two or more fluids that enter the body ofthe compact heat exchanger by [ 0.0 No [

48 HEAT EXCHANGERS Figure. Two-fluid compact heat exchanger with headers removed. (From Kraus et al., 00, with permission.) means ofheaders. In many instances, one ofthe 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 ofinternal 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..... Classification of Compact Heat Exchangers Compact heat exchangers may be classified by the kinds ofcompact elements that they employ. The compact elements usually fall into five classes:. Circular and flattened circular tubes. These are the simplest form of compact heat exchanger surface. The designation ST indicates flow inside straight tubes (example: ST-), FT indicates flow inside straight flattened tubes (example: FT-) and FTD indicates flow inside straight flattened dimpled tubes. Dimpling interrupts the boundary layer, which tends to increase the heat transfer coefficient without increasing the flow velocity.. Tubular surfaces. These are arrays oftubes ofsmall diameter, from 0. cm down to 0. cm, used in service where the ruggedness and cleanability ofthe conventional shell-and-tube exchanger are not required. Usually, tubesheets are comparatively thin, and soldering or brazing a tube to a tubesheet provides an adequate seal against interleakage and differential thermal expansion. [ -. Nor [

49 COMPACT HEAT EXCHANGERS Header Air fin Tube Tube plates Side bar Tube ( a) Round tube and fin Spacer bar ( d) Bar and plate Header Figure. Louvered air fins Heating fins ( g) Formed plate fin Air fin Tube Tube plates Side bar Turbulator strip Reinforcement rod Header Louvered air fins Louvered air fins ( b) Flat tube and fin Header bar ( e) Bar and plate Tube Corrugated air fins ( h ) Formed plate fin Louvered air fins Tube Header Tube plates Header ( c) Tube and center Turbulator dimples Side bar Header bar ( f ) Bar and plate ( i) Dimpled strut tube Louvered air fins Turbulator strip Some compact heat exchanger elements. (Courtesy ofharrison Radiator Division.). Surfaces with flow normal to banks of smooth tubes. Unlike the radial low fin tubes, smooth round tubes are expanded into fins that can accept a number oftube rows, as shown in Fig..a. Holes may be stamped in the fin with a drawn hub or foot to improve contact resistance or as a spacer between successive fins, as shown, or brazed directly to the fin with or without a hub. Other types reduce the flow resistance outside the tubes by using flattened tubes and brazing, as indicated in Fig..b and c. Flat tubing is made from strips similar to the manufacture of welded circular tubing but is much thinner and is joined by soldering or brazing rather than welding. [ -0 No [

50 HEAT EXCHANGERS The designation considers staggered (S) and in-line (I) arrangements oftubes and identifies transverse and longitudinal pitch ratios. The suffix (s) indicates data correlation from steady-state tests. All other data were correlated from a transient technique. Examples include the surface S.0-.(s), which is a staggered arrangement with data obtained via steady-state tests with transverse pitch-to-diameter ratio of.0 and longitudinal pitch-to-diameter ratio of.. The surface I.-. has an inline arrangement with data obtained from transient tests with both transverse and longitudinal pitch-to-diameter ratios of... Plate fin surfaces. These are shown in Figs..d through i. (a) The plain fin is characterized by long uninterrupted flow passages and is designated by a numeral that indicates the number offins per inch. The suffix T is appended when the passages are ofdefinite triangular shape. Examples are the surfaces.,.0, and.t. (b) The louvered fin is characterized by fins that are cut and bent into the flow stream at frequent intervals and is designated by a fraction which indicates the length ofthe fin in the flow direction (inches) followed by a numeral that indicates the number offins per inch. For example, the designation.0 indicates.0 -in.-long fins per inch. (c) The strip fin is designated in the same manner as the louvered fin. The suffixes (D) and (T) indicate double and triple stacks. The strip fins are frequently referred to as offset fins because they are offset at frequent intervals and the exchanger is essentially a series ofplate fins with alternate lengths offset. (d) The wavy fin is characterized by a continuous curvature. The change in flow direction introduced by the waves in the surface tends to interrupt the boundary layer, as in the case oflouvered and strip fins. Wavy fin designations are always followed by the letter W. For example, the. W is a wavy fin with. fins per inch and a -in. wave. (e) The pin fin surface is constructed from small-diameter wires. This surface yields very high heat transfer coefficients because the effective flow length is very small. The designation ofthe pin fin surfaces is nondescriptive. (f) The perforated fin surface has holes cut in the fins to provide boundary layer interruption. These fins are designated by the number offins per inch followed by the letter P.. Finned-tube surfaces. Circular tubes with spiral radial fins are designated by the letters CF followed by one or two numerals. The first numeral designates the number offins per inch, and the second (ifone is used) refers to the nominal tube size. With circular tubes with continuous fins, no letter prefix is employed and the [. Sho [

51 COMPACT HEAT EXCHANGERS two numerals have the same meaning as those used for circular tubes with spiral radial fins. For finned flat tubes, no letter prefix is used; the first numeral indicates the fins per inch and the second numeral indicates the largest tube dimension. When CF does not appear in the designation ofthe circular tube with spiral radial fins, the surface may be presumed to have continuous fins.. Matrix surfaces. These are surfaces that are used in rotating, regenerative equipment such as combustion flue gas air preheaters for conventional fossil furnaces. In this application, metal is deployed for its ability to absorb heat with minimal fluid friction while exposed to hot flue gas and to give up this heat to incoming cold combustion air when it is rotated into the incoming cold airstream. No designation is employed... Geometrical Factors and Physical Data Compact heat exchanger surfaces are described in the literature by geometric factors that have been standardized largely through the extensive work ofkays and London (). These factors and the relationships between them are essential for application of the basic heat transfer and flow friction data to a particular design problem. They are listed and defined in Table.. Physical data for a number of compact heat exchanger surfaces are given in Table.. Relationships between the geometric factors in Table. will now be established. Consider an exchanger composed of n layers ofone type ofplate fin surface and n layers ofa second type, as shown in Fig... The separation plate thickness is established by the pressure differential to which it is exposed or through designer discretion. Retaining the subscripts and for the respective types of surface, the overall exchanger height H is H = n (b + a) + n (b + a) (.) where b and b are separation distances between the plates for the two kinds of surface. With the width W and depth D selected, the overall volume V is V = WDH (.0) In Fig.., the length L is along the depth ofthe exchanger (L = D) and the length L is along the width (L = W). The frontal areas are also established. Again referring to Fig.., we have A fr, = HW A fr, = HD (.a) (.b) If the entire exchanger consisted of a single exchanger surface, surface or surface, the total surface area would be the product ofthe ratio oftotal surface to total [.0 Sho [

52 HEAT EXCHANGERS TABLE. Factor and Symbol A A fr a b d e L P p r h S S f V α β δ f η f η ov σ Compact Heat Exchanger Geometric Factors Descriptive Comments Free flow area on one side ofthe exchanger. To distinguish hot and cold sides, the free flow areas are frequently designated by A h and A c. Frontal area on one side ofthe exchanger. This is merely the product ofthe overall exchanger width and height or depth and height. Separation plate thickness. This applies only to plate fin surfaces and its value is at the designer s discretion. Separation plate spacing. This dimension is an approximation ofthe fin height. Applies to plate fin surfaces only. Equivalent diameter used to correlate flow friction and heat transfer; four times the hydraulic radius, r h. Flow length on one side ofthe exchanger. Note that this factor always concerns the flow length ofa single side ofthe exchanger, although two sides may be present, and that the ambiguity is avoided with the overall exchanger dimensions, which are designated width, depth, and height. It is therefore reasonable to have the overall exchanger depth be the length on one side of the exchanger and the overall width the length on the other side. Perimeter ofthe passage. Porosity, the ratio ofthe exchanger void volume to the total exchanger volume. Applies to matrix surfaces only. Hydraulic radius; the ratio ofthe passage flow area to its wetted perimeter. Heat transfer surface on one side of the exchanger. Subscripts are often appended to distinguish between hot- and cold-side surfaces. Surface of the fins, only, on one side of the exchanger. Applies to finned surfaces only. Total exchanger volume. This applies to both sides ofthe heat exchanger and is merely the product ofthe overall heat exchanger width, depth, and height. Ratio ofthe total surface area on one side ofthe exchanger to the total volume on both sides ofthe exchanger. Applies to tubular, plate fin surfaces, and crossed-rod matrices only. Ratio ofthe total surface area to the total volume on one side ofthe exchanger. The surface alone is S. The total volume includes the overall exchanger dimensions. Applies to plate fin surfaces only. Fin thickness. Fin efficiency. Overall passage efficiency. Ratio of the free flow area to the frontal area on one side of the exchanger. [. Nor [

53 COMPACT HEAT EXCHANGERS TABLE. Surface Geometry of Some Plate Fin Surfaces Plain Plate Fins..0..T b( 0 m) Fins per inch..0.. d e ( 0 m) δ f ( 0 m) β(m /m ) S f /S Louvered Fins b( 0 m).... Fins per inch d e ( 0 m) δ f ( 0 m) β(m /m ) S f /S Strip Fins D -.D -0.0 b( 0 m).... Fins per inch d e ( 0 m).... δ f ( 0 m) β(m /m ) S f /S Wavy and Pin Fins.- W.- W AP- PF- b( 0 m) Fins per inch or fin pattern.. In-line In-line d e ( 0 m) δ f or pin diameter ( 0 m) β(m /m ) S f /S volume β(m /m ) and the total volume V. However, where there are two surfaces, it is necessary to employ the factor α, which is the ratio ofthe total surface on one side to the total surface on both sides ofthe exchanger. By taking simple proportions, α = α = b b + b + a β b b + b + a β (.a) (.b) [ 0.0 No * [

54 0 HEAT EXCHANGERS and the total surfaces will be S = α V S = α V (.a) (.b) The hydraulic radius is defined as the flow area divided by the wetted perimeter ofthe passage: r h = A P = AL S and the ratio ofthe flow area to the frontal area is designated as σ: (.) σ = A A fr (.) For all but matrix surfaces, because eq. (.) leads to A = Sr h /L, Thus, the flow areas are given by σ = Sr h A fr L = Sr h V = αr h (.) A = σ A fr, A = σ A fr, (.a) (.b).. Heat Transfer and Flow Friction Data Heat Transfer Data Heat transfer data for compact heat exchangers are correlated on an individual surface basis using a Colburn type of representation. This representation plots the heat transfer factor j h : j h = St Pr / = h ( cp µ ) / (.) Gc p k as a function of the Reynolds number, which is obtained by employing the equivalent diameter d e = r h : Re = r hg µ = d eg (.) µ The Stanton number St is the ratio ofthe Nusselt number Nu to the product ofthe Reynolds and Prandtl numbers, with the specific heat c taken as the value for constant pressure, [0. Sho [0

55 ( COMPACT HEAT EXCHANGERS L/ rh = Fin pitch =. per in. Plate spacing, b = 0.00 in. Fin length flow direction =. in. Flow passage hydraulic diameter, r h = 0.00 ft. Fin metal thickness = 0.00 in., stainless steel Total heat transfer area/volume between plates, =. ft /ft Fin area/total area = L/ rh = 0. Fin pitch =.0 per in. Plate spacing, b = 0. in. Flow passage hydraulic diameter, r h = 0.00 ft. Fin metal thickness = 0.00 in., aluminum Total heat transfer area/volume between plates, = ft /ft Fin area/total area = 0.0 f, dimensionless / c hg h =, dimensionless k ( c j () a () c L/ rh = 0. Fin pitch =. per in. Plate spacing, b = 0.0 in. Flow passage hydraulic diameter, r h = 0.00 ft. Fin metal thickness = 0.00 in., aluminum Total transfer area/volume between plate, = ft /ft Fin area/total area = 0. L/ rh = () b Fin pitch =. per in. Plate spacing, b = 0.0 in. Flow passage hydraulic diameter, r h = 0.00 ft. Fin metal thickness = 0.00 in., aluminum Total heat transfer area/volume between plates, = ft /ft Fin area/total area = dg Re = e, dimensionless () a () b () d () c () d.t..0. Figure. Heat transfer and flow friction characteristics of some plain plate fin compact heat exchanger surfaces. (From Kays and London,.) [ * 0 Sho [

56 ( HEAT EXCHANGERS Fin pitch =. per in. Plate spacing, b = 0.0 in. Louver spacing = 0. in. Fin gap = 0.0 in. Louver gap = 0.0 in. Flow passage hydraulic diameter, r h = 0.00 ft. Fin metal thickness = 0.00 in., aluminum Total heat transfer area/volume between plates, = ft /ft Fin area/total area = () a 0. / Fin pitch =.0 per in. Plate spacing, b = 0.0 in. Louver spacing = 0. in. Fin gap = 0.0 in. Louver gap = 0.0 in. Flow passage hydraulic diameter, r h = 0.00 ft. Fin metal thickness = 0.00 in., aluminum Total transfer area/volume between plates, = ft /ft Fin area/total area = 0.0 () c Fin pitch =. per in. Plate spacing, b = 0.0 in. Louver spacing = 0. in. Fin gap = 0.0 in. Louver gap = 0.0 in. Flow passage hydraulic diameter, r h = 0.00 ft. Fin metal thickness = 0.00 in., aluminum Total heat transfer area/volume between plates, = ft /ft Fin area/total area = f, dimensionless / c ( hg h =, dimensionless k c j /-..0 () b.0. Fin pitch =.0 per in. Plate spacing, b = 0.0 in. Louver spacing = 0.0 in. Fin gap = 0.0 in. Louver gap = 0.0 in. Flow passage hydraulic diameter, r h = 0.00 ft. Fin metal thickness = 0.00 in., aluminum Total heat transfer area/volume between plates, = ft /ft Fin area/total area = 0.0 () d dg Re = e, dimensionless () a () b () c /-. /-.0 /-.0 Figure. Heat transfer and flow friction characteristics of some louvered fin compact heat exchanger surfaces. (From Kays and London,.) () d [ -. Nor [

57 ( COMPACT HEAT EXCHANGERS Fin pitch = 0.0 per in. Plate spacing, b = 0.0 in. Splitter symmetrically located Fin length flow direction = 0. in. Flow passage hydraulic diameter, r h = 0.00 ft. Fin metal thickness = 0.00 in., aluminum Splitter metal thickness = 0.00 in. Total heat transfer area/volume between plates, = ft /ft Fin area (including splitter)/total area = 0. () a...0 Fin pitch =.00 per in. Plate spacing, b = 0. in. Splitter symmetrically located Fin length flow direction = 0. in. Flow passage hydraulic diameter, r h = 0.00 ft. Fin metal thickness = 0.00 in., aluminum Splitter metal thickness = 0.00 in. Total heat transfer area/volume between plates, = 0 ft /ft Fin area (including splitter)/total area = 0. f, dimensionless / c h =, dimensionless k ( h Gc j () c Fin pitch =. per in. Plate spacing, b = 0.0 in. Splitter symmetrically located Fin length flow direction = 0. in. Flow passage hydraulic diameter, r h = ft. Fin metal thickness = 0.00 in., nickel Splitter metal thickness = 0.00 in. Total heat transfer area/volume between plates, = 0 ft /ft Fin area (including splitter)/total area = 0. () b..0 Fin pitch =. per in. Plate spacing, b = 0. in. Fin length = 0. in. Flow passage hydraulic diameter, r h = 0.00 ft Fin metal thickness = 0.00 in., aluminum Total heat transfer area/volume between plates, = ft /ft Fin area/total area = 0.0 Note: The fin surface area on the leading and trailing edges of the fins have not been included in area computations. () d dg Re = e, dimensionless () a /-0.0( D) () b () c () d /-.( D) /-.00( D /-. ) Figure. Heat transfer and flow friction characteristics of some strip fin compact heat exchanger surfaces. (From Kays and London,.) [ -0 No [

58 ( HEAT EXCHANGERS Pin diameter = 0.0 in., aluminum Pin pitch parallel to flow = 0.0 in. Pin pitch perpendicular to flow = 0.0 in. Plate spacing, b = 0.0 in. Flow passage hydraulic diameter, r h = 0.00 ft Total heat transfer area/volume between plates, = ft /ft Fin area/total area = 0. () a Fin pitch =. per in. Plate spacing, b = 0. in. Flow passage hydraulic diameter, r h = 0.00 ft Fin metal thickness = 0.00 in., aluminum Total heat transfer area/volume between plates, = ft /ft Fin area/total area = 0. Note: Hydraulic diameter based on free-flow area normal to mean flow direction. f, dimensionless / c h =, dimensionless k ( h Gc j () c () a.0 /.0 Approx.. Fin pitch =. per in. Plate spacing, b = 0. in. Flow passage hydraulic diameter, r h = 0.00 ft Fin metal thickness = 0.00 in., aluminum Total heat transfer area/volume between plates, = ft /ft Fin area/total area = 0. Note: Hydraulic diameter based on free-flow area normal to mean flow direction. () b...0 dia. 0. Min free flow area Pin diameter = 0.0 in., copper Pin pitch parallel to flow = 0. in. Pin pitch perpendicular to flow = 0. in. Plate spacing, b = 0. in. Flow passage hydraulic diameter, r h = 0.0 ft Total heat transfer area/volume between plates, = ft /ft Fin area/total area = 0. () c () d dg Re = e, dimensionless () b PF-.-/W.-/W AP- Figure.0 Heat transfer and flow friction characteristics of some wavy and pin fin compact heat exchanger surfaces. (From Kays and London,.) () d [ - Nor [

59 COMPACT HEAT EXCHANGERS Entrance and exit loss coefficients K and K, dimensionless c e K c Laminar Re = 000 Re = 000 Re = 000 Re = 0,000 Re = Re = Re = 0,000 Re = 000 Re = 000 Re = 000 Laminar Ratio of free flow to frontal area on one side,, dimensionless Figure. Entrance and exit loss coefficients for flow through plate fin exchanger cores. (From Kays and London, ). St = Nu Re Pr = hd e /k (d e G/µ)(cµ/k) = h Gc K e (.0) The fluid properties in eqs. (.) and (.0) are evaluated at the bulk temperature T b = (T + T ) t b = (t + t ) (.a) (.b) [ 0. No [

60 HEAT EXCHANGERS Entrance and exit loss coefficients K and K, dimensionless c e K c Laminar Re = 000 Re = 000 Re = 000 Re = 0,000 Re = Re = Re = 0,000 Re = 000 Re = 000 Re = 000 Laminar Ratio of free flow to frontal area on one side,, dimensionless Figure. Entrance and exit loss coefficients for flow through rectangular passages. (From Kays and London,.) Flow Friction Data Kays and London () suggest that the pressure drop P in a compact heat exchanger be computed from the equation where P P = G v g c P (Φ + Φ + Φ Φ ) (.) Φ = + K c σ ( ) ν Φ = ν K e [. Nor [

61 LONGITUDINAL FINNED DOUBLE-PIPE EXCHANGERS Φ = f S ν m A ν Φ = ( σ K e ) ν ν 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. (.). These terms denote, respectively, the entrance or contraction loss as the fluid approaches the exchanger 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 () have presented heat transfer and flow friction data for approximately 0 surfaces described by the foregoing. Some typical examples are shown in Figs.. through.0. Entrance and exit loss coefficients for plate fin cores and rectangular passages are plotted in Figs.. and.. Because the pitch ofthe fins is small, the height ofthe 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/ mb/ where b/ is halfofthe separation plate distance.. LONGITUDINAL FINNED DOUBLE-PIPE EXCHANGERS.. Introduction (.) The double-pipe exchanger consists ofa pair ofconcentric tubes or pipes. One process stream flows through the inner pipe, and the other flows, either in counter- or cocurrent (parallel) flow in the annular region between the two pipes. The inner pipe may be bare or it may contain as many as longitudinal fins equally spaced around its periphery. Consider the plain double-pipe exchanger shown in Fig... It usually consists of two pairs ofconcentric 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. Ifthere 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 ofconcentric pipes are used to form a hairpin because ofthe convenience the hairpin affords for manifolding streams and the natural loop it can provide [. No [

62 HEAT EXCHANGERS Return bend Figure. Gland Return head Gland Tee Gland Plain double-pipe exchanger. (From Kraus et al., 00, 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 ofthe 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 appreciable inequity appears in the composite thermal resistance ofa pair offluids 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 resistance is the sum ofthe 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 ofboth fluids are low, as discussed subsequently, there may still be an advantage in the use ofthe finned inner pipe. Fins are usually 0.0 cm thick (0.0 in. and 0 BWG). A steel fin with a thermal conductivity of0 W/m K and a height of. cm ( in.) on exposure to a composite resistance of0.00 m K/W (corresponding to a film coefficient of 0 W/m K) has a fin efficiency ofabout 0.. Exposed to a composite resistance of0.00 m K/W, the efficiency drops to about 0.. Hence, the high fin has its limitations, although metals ofhigher thermal conductivity extend the range ofapplication. Fin surface is inexpensive compared with prime surface, but its usefulness diminishes significantly below a composite resistance of0.00 m 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 ofexchanger. Moreover, inner pipes are available with fins on the inside as well as the outside ofthe 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 ofthe fins about the pipe is shown clearly in Fig... They form a radial array ofchannels, with each channel composed oftwo fins. Channels may be attached by continuously spot-welding them to the outside ofthe inner pipe or by other brazing or welding procedures. It should be noted that contact between the [ 0.0 Nor [

63 LONGITUDINAL FINNED DOUBLE-PIPE EXCHANGERS Figure. Configurations to be used for the determination of the flow area, wetted perimeter, and surfaces for the annular region in the double-pipe heat exchanger: (a) extruded fins; (b) welded U-fins; (c) detail oftwo welded U-fins. (From Kraus et al., 00, with permission.) channels and the other pipe should be continuous over the entire channel length but need not be very wide. In another method ofattaching longitudinal fins, grooves are plowed in the outside diameter ofthe 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 0 to 00 cm. The common double-pipe exchanger units available are summarized in Table.. TABLE. Dimensions of Double-Pipe Exchangers a Outer Outer Inner Inner Nominal Pipe Pipe Max. Pipe Pipe Fin Diameter Thickness OD No. ofod Thickness Height (in.) (mm) (mm) Fins (mm) (mm) (mm) Source: After Saunders (). a One outer tube one inner tube: standard units. The fin thickness for extruded or soldered fins is 0. mm for fins up to. mm high and 0. mm high for greater heights. Fin thickness for welded fins is 0. mm for fin heights up to. mm. The dimensions shown here are for low-pressure units. [ 0. No [

64 0 HEAT EXCHANGERS Physical Data for Annuli Extruded Fins For the finned annular region between the inner and outer pipes shown in Fig..a, the cross-sectional area for n t identically finned inner pipes each having n f extruded fins will be A = π ( π ) D i d o n f b f δ f n t (.) There are two wetted perimeters. One of them is for heat transfer: P Wh = [ πd o + n f (b f δ f ) ] n t (.) where the tips ofthe fins are presumed adiabatic. The other is for pressure loss: or P Wf = πd i + P Wh P Wf = πd i + [ πd o + n f (b f δ f ) ] n t (.) The equivalent diameter for heat transfer will be d e = A [ ] = (π/)d i (π/)d o n f b f δ f nt (.) P Wh (πd o n f δ f + n f b f )n t and the equivalent diameter for pressure drop will be d e = A [ ] = (π/)d i (π/)d o n f b f δ f nt (.) P Wf πd i + (πd o n f δ f + n f b f )n t 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, Then, per unit length, with S b = πd o n f δ f S f = b f n f the surface area on the annulus side of the inner pipe per unit length is S = S = πd o + (b f δ f )n f (.a) (.b) (.c) [0 0. Sho [0

65 LONGITUDINAL FINNED DOUBLE-PIPE EXCHANGERS Welded U-Fins The configuration for the annular region that accommodates welded U-fins is shown in Fig..b, and detail for a pair of the fins is shown in Fig..c. Observe that z is the fin root width and thickness, which is usually taken as δ f. The free area for flow for n f fins and n t inner tubes with δ f = z/ is A = π [ π ( D i d o + n f δ f b f + z )] n t (.0) If d o z, the wetted perimeter for heat flow will be the circumference of the inner tube less the thicknesses ofthe n f fins plus twice the heights ofthe n f fins. P Wh = [ πd o + n f (b f δ f ) ] n t (.) Here, too, the tips are presumed to be adiabatic. The wetted perimeter for pressure loss is or P Wf = πd i + P Wh P Wf = πd i + [ πd o + n f (b f δ f ) ] n t (.) Then the two equivalent diameters are, for heat transfer, d e = A [ = (π/)d i (π/)d o + n f δ f (b f + z/) ] n t [ P Wh πdo + n f (b f δ f ) ] (.) n t and for pressure drop, d e = (π/)d i [ (π/)d o + n f δ f (b f + z/) ] n t πd i + [ πd o + n f (b f δ f ) ] n t (.) 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. (.)... Overall Heat Transfer Coefficient Revisited Kern (0), Kern and Kraus (), and Kraus et al. (00) 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 ofseveral thermal resistances that are identified in Fig... These resistances are giveninm K/W. After both inside and outside heat transfer coefficients, h i and h o, have been determined and after both fouling resistances, r di and r do, have been specified (either [ -0 Sho [

66 HEAT EXCHANGERS UD, (on s ) R s i, R i,od Fin r o w r di (at 0) r i (at 0) t c Dirt r do t fw r, t o r i,od r di,od f t f r o ro, tf r do r o T c Dirt Tube Figure. Location of thermal resistances for a fouled longitudinal fin double-pipe exchanger. 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 unfinned tube.. With h io = h i (d i /d o ), form the inside film resistance: r io = (m K/W) (.) h io. The inner pipe fouling resistance r di must be referred to the outer tube surface. Hence, r dio = d o r di (m K/W) (.) d i. The pipe wall resistance referred to the outside of the inner pipe is r mo = d o ln(d o /d i ) (m K/W) k m Dirt [ -. Nor * [

67 LONGITUDINAL FINNED DOUBLE-PIPE EXCHANGERS However, when the diameter ratio d i /d o 0.,r mo can be computed with an error ofless than % from the arithmetic mean diameter: r mo = d o d i k m πd o π(d o + d i ) = d o(d o d i ) k m (d o + d i ) (m K/W) (.). At this point, the sum ofthe internal resistances referred to the outside ofthe inner pipe is Rio = r io + r dio + r mo (m K/W) and this resistance must be referred to the gross outside surface of each inner pipe per meter: Thus, S = πd o + (b δ)n f (.) Ris = R io S. The annulus heat transfer coefficient is h o, so that πd o (m K/W) (.) r o = h o (m K/W) (.0). The annulus fouling resistance r do must be combined with h o to obtain the value ofthe annulus coefficient working on the fin and prime outer surface. Let this resistance be designated as r o, so that r o = h o + r do = r o + r do The fin efficiency will be given by where h o = r o + r do η fo = tanh mb mb ( h m = o k m δ f ) / (m K/W) Then, with the weighted fin efficiency defined by eq. (.b), (.) [. No * [

68 HEAT EXCHANGERS or η ov,o = S fo S o ( η fo ) η ov,o = η fos fo + S bo S fo + S bo (.b) 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 so that the resistance is or r oη = h oη h oη = h o η o (m K/W) (.). The overall resistance is the sum ofeqs. (.) and (.). Thus, U o = R is + r oη U o = Ris + r oη (.) The overall heat transfer coefficient given by eq. (.) is the coefficient to be used in the rate equation: q = U o S o θ m (.).. 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..... Pressure Loss in Pipes and Annuli The friction relationships ofeqs. (.) and the turn loss relationship ofeq. (.) also pertain to the double-pipe heat exchanger. However, when hairpins are considered, the total friction loss in the inner pipe will be P f = n hpf ρv L d = n hpf ρv L (.) d i [. Nor * [

69 LONGITUDINAL FINNED DOUBLE-PIPE EXCHANGERS For the annulus, the tube diameter is replaced with the equivalent diameter for pressure loss: P f = n hpf ρv L = n hp f ρv L (.) d e d e The turn loss for both inner pipe and annulus with N hp hairpins is P t = (n hp )ρv (.).. Wall Temperature and Further Remarks It may be noted that the wall resistance presents the lowest resistance to the flow of heat between the hot and cold fluids. Hence, an excellent approximation to the wall temperature may be obtained via the computation ofthe product of R is and the heat flux. Then, ifthe hot fluid is carried within the inner tube, the wall temperature will be T w = T b q R is (.) S where Ris = S R io (.) πd o Rio = r io + r do + r mo In the event, that the cold fluid is carried in the inner tube, the wall temperature will be T w = t b + q R is (.) S.. Series Parallel Arrangements When two streams are arranged for counterflow, the LMTD represents the maximum thermal potential for heat transfer that can be obtained. Often, on the industrial scale, a single process service may entail the use ofmore than a single long hairpin. It then follows that it is desirable to connect the hairpins in series on both the annulus and inner pipe sides, as in Fig... In this configuration, the temperature potential remains the LMTD for counterflow. In some services, there may be a large quantity ofone fluid undergoing a small temperature change and a small quantity ofanother fluid undergoing a large temperature change. It may not be possible to circulate the large volume offluid through the required number ofhairpins with the pressure drop available. Under these circumstances, the larger volume offluid may be manifolded in the series parallel arrangement shown in Fig... The inner pipe fluid has been split between the exchangers [. No [

70 HEAT EXCHANGERS Double-pipe heat exchangers in series. (From Kraus et al., 00, with per- Figure. mission.) Figure. permission.) II I Double-pipe heat exchangers in series parallel. (From Kraus et al., 00, with designated I and II. Both ofthese exchangers are in counterflow relative to each other but not in the same sense as in Fig... In Fig.., the T s refer to the series streams and the t s refer to the parallel streams. Departures from true counterflow and true co-current (parallel) flow can be handled by the logarithmic mean temperature difference correction factor F. Kern (0) presents a derivation for the factor γ to be used in a modification of the heat transfer rate equation T T t t [ 0. Nor * [

71 LONGITUDINAL FINNED DOUBLE-PIPE EXCHANGERS q = U o Sγ(T t ) (.0) where T t represents the total temperature potential, the difference in the fluid stream inlet temperatures, in the exchanger configuration. After a laborious and detailed derivation, Kern (0) gives, for one series hot fluid and n parallel cold fluid streams, ( T ) t Z where γ = ln [ Z Z T t nz ( ) ] (.) T t /n + T t Z Z = T T n(t t ) For one series cold fluid and n parallel hot fluid streams, Kern (0) gives ( T ) t Z T t nz γ = [ ( ) ] (.) T t /n ln ( Z) + Z T t TABLE. Dimensions of Multitube Double-Pipe Exchangers a Nom. Pipe Pipe No. No. Tube Tube Fin Dia. Thick. OD ofofod Thick. Height (in.) (mm) (mm) Tubes Fins (mm) (mm) (mm) Source: After Saunders (). a Fin thicknesses are identical to those listed in Table.. The dimensions shown here are for low-pressure units. [. No * [

72 HEAT EXCHANGERS where Z = n(t T ) t t.. Multiple Finned Double-Pipe Exchangers There are numerous applications for longitudinal fin pipes and tubes. Closest to the double-pipe exchanger is the hairpin with multiple longitudinal-fin pipes. A variety ofpipes and tubes are available with longitudinal fins whose numbers, heights, thicknesses, and materials differ. Data for some of these configurations are shown in Table.. The procedure for the design and analysis of the multiple-tube exchanger differs little for that used for the single-tube exchanger.. TRANSVERSE HIGH-FIN EXCHANGERS.. Introduction Pipes, tubes, and cast tubular sections with external transverse high fins have been used extensively for heating, cooling, and dehumidifying air and other gases. The fins are preferably called transverse rather than radial because they need not be circular, as the latter term implies, and are often helical. The air-fin cooler is a device in which hot-process fluids, usually liquids, flow inside extended surface tubes and atmospheric air is circulated outside the tubes by forced or induced draft over the extended surface. Unlike liquids, gases are compressible, and it is usually necessary to allocate very small pressure drops for their circulation through industrial equipment or the cost ofthe compression work may entail a substantial operating charge. Except for hydrogen and helium, which have relatively high thermal conductivities, the low thermal conductivities ofgases coupled with small allowable pressure drops tend toward low-external-convection heat transfer coefficients. In the discussion oflongitudinal high-fin tubes in Section.., it was noted that a steel fin. cm high and 0.0 cm thick could be used advantageously with a fluid producing a heat transfer coefficient as high as 0 W/m K. Aluminum and copper have thermal conductivities much higher than steel, 00 and 0 versus W/m K. It would appear that thin high fins made ofaluminum or copper would have excellent fin efficiencies when exposed to various heating and cooling applications ofair and other gases at or near atmospheric pressure. In air-fin cooler services, the allowable pressure drop is measured in centimeters or inches ofwater and air can be circulated over a few rows ofhigh-fin tubes with large transverse fin surfaces and, at the same time, require a very small pressure drop. Transverse high-fin tubular elements are found in such diverse places as economizers ofsteam power boilers, cooling towers, air-conditioning coils, indirect-fired heaters, waste-heat recovery systems for gas turbines and catalytic reactors, gas-cooled nuclear reactors, convectors for home heating, and air-fin coolers. [. Lon [

73 TRANSVERSE HIGH-FIN EXCHANGERS In the services cited involving high temperatures, hot gases flow over the fins and water or steam flows inside the tubes. The extended surface element usually consists ofa chromium steel tube whose chromium content is increased with higher anticipated service temperatures. A ribbon, similar in composition to the tube, is helically wound and continuously welded to the tube. The higher and thicker the fins, the fewer the maximum number offins per centimeter oftube which can be arcwelded because the fin spacing must also accommodate the welding electrode. Hightemperature high-fin tubes on a closer spacing are fabricated by electrical resistance welding ofthe fins to the tube. High-fin tubes can also be extruded directly from the tube-wall metal, as in the case ofintegral low-fin tubing. However, it becomes increasingly difficult to extrude a high fin from ferrous alloys as hard as those required for high-temperature services, which are often amenable to work hardening while the fin is being formed. Whether fins are attached by arc welding or resistance welding, the fin-to-tube attachment for all practical design considerations introduces a neglible bond or contact resistance. High-fin tubes are used in increasing numbers in devices such as the air-fin cooler, in which a hot fluid flows within the tubes, and atmospheric air, serving as the cooling medium, is circulated over the fins by fans. Several high-fin tubes for air-fin cooler service are shown in Fig... Type a can be made by inserting the tubes through sheet metal strips with stamped or drilled holes and then expanding the tubes slightly to cause pressure at the tube-to-strip contacts. The tubes and strips may then be brazed. Ifthe tubes are only expanded into the plates to produce an interference fit, some bond or contact resistance must be anticipated. For practical purposes, when the tubes and strips are brazed together, the joint may be considered a metallurgical bond and the bond resistance can be neglected. In Fig.., tubes b through e are made by winding a metal ribbon in tension around the tube. These types are not metallurgically bonded and rely entirely upon the tension in the ribbon to provide good contact. Type f combines tension winding with brazing, and for the combination of a steel tube and an aluminum fin, the common tin lead solder is not compatible and a zinc solder is used. Type g employs a tube as a liner, and high fins are extruded from aluminum, which, like copper, is a metal that can be manipulated to a considerable fin height. Types d,e, and f employ aluminum for the fins and are arranged to protect the tube from the weather because air-fin coolers are installed outdoors. Type g, sometimes called a muff-type high-fin tube, has its contact resistance between the inside ofthe integral finned tube and the liner or plain tube. Type h has a mechanical bond which can closely match a metallurgical bond for contact resistance. Type i, an elliptical tube with rectangular fins, may employ galvanized steel fins. When tube ends are circular, they are rolled into headers. Consider a typical air-fin cooler application with a hot fluid inside the tubes. In many instances, carbon steel meets the corrosion-resistance requirements ofthe tube-side fluid. From the standpoint ofhigh thermal conductivity and cost, aluminum ribbon is very suitable for tension-wound fins. However, aluminum has twice the thermal coefficient ofexpansion ofsteel, and the higher the operating temperature ofthe fluid inside the tubes, the greater the tendency ofthe fins to elongate away from their room-temperature tension-wound contact with the tube, and the greater is [ 0.0 Lon [

74 0 HEAT EXCHANGERS Figure. Various types ofhigh-fin tubing. (From Kraus et al., 00, with permission.) the bond or contact resistance. In one variation oftype d in Fig.., the ribbon is wound with J rather than L feet, with the J s pressing against each other and the tube at room temperature. As the feet become heated during operation, they expand against each other... Bond or Contact Resistance of High-Fin Tubes The bond resistance ofseveral types ofinterference-fit high-fin tubes shown in Fig.. has been studied by Gardner and Carnavos (0), Shlykov and Ganin (), and Yovanovich (). Gardner and Carnavos pointed out that in its most general sense, the term interference fit implies the absence ofa metallurgical bond, as opposed to the extrusion ofa fin from a tube wall, and the welding, soldering, or brazing ofa fin to the tube. The interference fit is produced by mechanically developing contact pressure through elastic deformation either by winding a ribbon under tension about a tube, as in types b through e in Fig.., or by expanding a tube against the fins as in type a, or a combination ofpressing the root tube against the liner or the liner against the root tube, as in type g. [0 0.0 Nor [0

75 TRANSVERSE HIGH-FIN EXCHANGERS Fin Efficiency Approximation The fin efficiency ofthe radial fin ofrectangular profile was given in Chapter : r b η = m ( ra ) I (mr a )K (mr b ) K (mr a )I (mr b ) r b I 0 (mr b )K (mr a ) + I (mr a )I o (mr b ) where m = ( ) h / ρ = r ( ) b h / φ = (r a r b ) / kδ r a ka p R a = ρ R b = ρ ρ The modified Bessel functions in the radial fin efficiency expressions are obtained from tables or from software, and their employment to obtain the efficiency involves a somewhat laborious procedure. An alternative has been provided by McQuiston and Tree (), who suggest the approximation where where ρ is the radius ratio,.. Air-Fin Coolers m = η = ( ) h / kδ ψ = r b ρ ρ tanh mψ mψ ( + 0. ln ) ρ ρ = r b r a (.) (.) The air-fin cooler consists ofone or more horizontal rows oftubes constituting a section through which air is circulated upward by mechanical draft. The fan that moves the air may be above the section providing an induced draft or it may be below the section, providing a forced draft. In the induced-draft air-fin cooler, the heated air is thrown upward to a good height by its high exit velocity. A relatively small amount ofthe heated air is sucked back to reenter the air intake below the section and thereby cut down the temperature difference available between the ambient air and the process fluid. In a forced-draft unit, the air leaves at a low velocity at a point not far from the high entrance velocity [. No [

76 HEAT EXCHANGERS ofthe air to the fan below the section. Hot air is more apt to be sucked back into the fan intake, causing recirculation. Following a trend in cooling towers that started some years ago, induced-draft units now appear to be preferred. Usually, the section has cross bracing and baffles to increase rigidity and reduce vibration. The design and analysis of air-fin coolers differs only in a few respects from the longitudinal fin exchangers in Section.. The principal difference is in the air side, where air competes with other fluids as a coolant. Because air is incompressible and liquids are not, only a small pressure drop can be expended for air circulation across the finned tubes, lest the cost ofair-compression work become prohibitive. In most applications, the allowable air-side pressure drop is only about. cm ( in.) of water. The air passes over the finned tubing in crossflow, and this merely requires the use of the proper heat transfer and flow friction data. The temperature excursion ofthe air usually cannot be computed at the start ofthe calculations because the air volume, and hence the air temperature rise, are dependent on the air pressure drop and flow area ofthe cooler. Most widely used are the integral-fin muff-type tube (Fig..g), the L-footed tension-wound tube (Fig..g), and the grooved and peened tension-wound tube (Fig..h). These tubes usually employ nine or eleven fins per inch. Numerous other tubes are manufactured in accordance with the types shown in Fig..b, c, and f. Other tubes have serrated or discontinuous fins. The latter tubes are fabricated to their own standards by manufacturers of air-fin coolers. Physical Data As indicated in Fig.., tubes may be arranged in either triangular or in-line arrangements. Observe that the pitch in these arrangements is designated by P t,p l,orp d, where P t = transverse pitch (m), P l = longitudinal pitch (m), and p t Flow Figure. permission.) p l p d Flow () a () b Tube arrangements: (a) triangular; (b) in-line. (From Kraus et al., 00, with p t p l p d [ 0. Nor [

77 TRANSVERSE HIGH-FIN EXCHANGERS P d = diagonal pitch (m). The diagonal pitch is related to the transverse and longitudinal pitch by [ (Pt ) ] / P d = + Pl (.) and in the case ofan equilateral triangular arrangement, P d = P t When there are n tubes in a row and n r rows, the total number oftubes will be n t = n n r (.) Let z be the clear space between the tubes, which are L meters long. The fins are b meters high: b = d a d b where d a and d b are, respectively, the outer and inner diameters ofthe fin. The fins are δ f thick and the minimum flow area A = A min will depend on the transverse pitch P t.for and for P t > P d d b zδ f z + δ f ( A = A min = n L P t d b zδ ) f z + δ f P t < P d d b zδ f z + δ f ( A = A min = n L P d d b zδ ) f z + δ f The surface area of the tube (between the fins) will be (.) (.) S b = πn tld b z z + δ f (.) and the surface of the fins, which accounts for the heat transfer from the tips of the fins, will be S f = πn [ ] tl ( d z + δ f a db ) + da δ f (.0) [. No * [

78 HEAT EXCHANGERS This makes the total surface S = S b + S f the finned surface per total surface, S f S = S f S b + S f and the surface per unit length per tube, S = S NL (.a) (.b) (.c) Heat Transfer Correlations Early investigations that pertain to heat transfer and friction data in tube bundles containing high-fin tubes have been reported by Jameson (), Kutateladze and Borishaniskii (), and Schmidt (). The correlation ofbriggs and Young () is based on a wide range ofdata. Their general equation for tube banks containing six rows of tubes on equilateral triangular pitch is where Nu = hd b k = 0.Re0. Pr / and where the range ofparameters is [ (Pf δ f ) d a d b Re = d bg µ ] 0.0 ( Pf δ f δf ) 0. (.) 000 < Re <, mm < δ f < 0.0 mm. mm <d b < 0. mm.0 mm <P f <.0 mm. mm <b= d a d b <. mm. mm <P t < mm Vampola () proposed a correlation based on an extensive study ofdifferent finned tubes. For more than three tube rows, where Nu = hd e k = 0.Re0. ( Pt d b d b ) 0.0 ( ) Pt d 0.0 ( ) b Pt d 0.0 b + (.) P f δ f P d d b Re = d eg µ [ -. Nor [

79 TRANSVERSE HIGH-FIN EXCHANGERS d e = S / bd a + S f (S f P f /NL) S with the diagonal pitch given by eq. (.). In eq. (.), the range ofparameters is 000 < Re < 0,000. mm <P t <. mm 0. mm <d b <.0 mm.0 mm <d e <.00 mm.0 mm <b= d a d b <.0 mm 0. mm < P t d b d b <. 0. mm < δ f < 0.0 mm. < P t d b P f δ f + <.. mm <P f <. mm 0. < P t d b P d d r <.0 0. mm <P l <.0 mm Ganguli et al. () proposed the following correlation for three or more rows of finned tubes: Nu = hd ( ) 0. b k = 0.Re0. Pr / Sb (.) S where Re = d bg µ The correlation ofeq. (.) is valid for 00 < Re < 00,000.0 mm <P f <. mm. mm <d b <.0 mm. mm <P t <. mm. mm <b= d a d b <.0 mm < S < 0 S b 0. mm < δ f < 0. mm Other correlations include those ofbrauer (), Schulenberg (), Kuntysh and Iokhvedor (), and Mirkovic (). More recent correlations include those of Zhukauskas (), Weierman (), Hofmann (), Ehlmady and Biggs (), Biery (), Gianolio and Cuti (), Brandt and Wehle (), and Nir (). Many ofthem are cited by Kröger ()... Pressure Loss Correlations for Staggered Tubes Some ofthe earlier correlations for the static pressure drop through bundles ofcircular finned tubes are those ofjameson (), Gunter and Shaw (), and Ward and [ -0 No [

80 HEAT EXCHANGERS Young (). A frequently used correlation is that of Robinson and Briggs () for staggered tubes: for n r rows and where P =.0 G ρ n r Re 0. Re = d bg µ ( Pt d b ) 0. ( ) 0. Pt (.) and where P d is given by eq. (.). Equation (.) is valid for 000 < Re < 0,000. mm <P f <. mm. mm <d b < 0. mm. mm <P t <.mm. mm <d a <. mm. mm <P l <. mm 0. mm <b= d a d b Vampola () proposed the correlation P d <. mm. < P t d b <. P = 0. G ρ n r Re 0. ( ) Pt d 0.0 ( b P f δ + de ( Pt d b d b d b ) 0.0 ) 0.0 (.) where the Reynolds number, equivalent diameter, and limits ofapplicability are identical to those following eq. (.)... 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 ofthese resistances be corrected and summed properly. No provision 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..0, there are five inside resistances:. The inside film resistance: r io = h i d b d i (.) [ 0. Sho * [

81 TRANSVERSE HIGH-FIN EXCHANGERS Figure.0 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., 00, with permission.). The inside fouling resistance: r dio = r di d b d i (.). The liner metal resistance is based on the mean liner diameter, and with the metal thickness the liner metal resistance is δ l = d o d i r mol = δ l d b (.) k l d o + d i [. Sho * [

82 HEAT EXCHANGERS The bond resistance given by the tube manufacturer or calculated from the procedure ofsection.. is transferred appropriately via r Bo = r B d b d g (.0). The tube metal resistance is based on the mean tube diameter, and with the metal thickness the tube metal resistance is δ t = d b d g r mot = δ t d b (.) k t d b + d g The sum ofthese resistances is R io : Rio = 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 Ris = S R io (.) πd b The air-side coefficient is h o and the fin efficiency is computed from eq. (.b). Then, with no provision for fouling, r oη = where η ov,o is obtained from eq. (.b): η ov,o = S f S ( η f ) The overall heat transfer coefficient is then given by U o =. PLATE AND FRAME HEAT EXCHANGER.. Introduction h o η ov,o (.) Ris + r oηf (.) An exploded view ofthe plate and frame heat exchanger, also referred to as a gasketed plate heat exchanger, is shown in Fig..a. The terminology plate fin heat [ -. Nor [

83 PLATE AND FRAME HEAT EXCHANGER Figure. (a) Exploded view ofa typical plate and frame (gasketed-plate) heat exchanger and (b) flow pattern in a plate and frame (gasketed-plate) heat exchanger. (From Saunders,, with permission.) [ * No [

84 0 HEAT EXCHANGERS exchanger is also in current use but is avoided here because ofthe possibility of confusion with the plate fin surfaces in compact heat exchangers. The exchanger is composed ofa series ofcorrugated plates that are formed by precision pressing with subsequent assembly into a mounting frame using full peripheral gaskets. Figure.b 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 ofsurface can be packed into a rather small volume. Moreover, plate and frame heat exchangers can accommodate a wide range offluids. There are three main disadvantages to their employment. Because ofthe gasket, they are vulnerable to leakage and hence must be used at low pressures. The rather small equivalent diameter ofthe 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. Typical plates in plate and frame (gasketed-plate) heat exchanger (a) Intermating or washboard type and (b) Chevron or herringbone type. (From Saunders,, with permission.) [0. Nor [0

85 PLATE AND FRAME HEAT EXCHANGER The two most widely employed corrugation types are the intermating or washboard type and the chevron or herringbone type. Both ofthese are shown in Fig... The corrugations strengthen the individual plates, increase the heat transfer surface area, and actually enhance the heat transfer mechanism. The outside plates ofthe assembly do not contribute the fluid-to-fluid heat transfer. Hence, the effective number ofplates is the total number ofplates minus two. This fact becomes less and less important as the number of plates becomes large. It may be noted that an odd number ofplates must be used to assure an equal number of channels for the hot and cold fluids. Figure.a indicates that the frame consists ofa fixed head at one end and a movable head at the other. The fluids enter the device through ports located in one or both ofthe end plates. Ifboth 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 ofeither 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 / arrangement, shown in Fig..a and b. In Fig..a, termed the Z or zed arrangement, two ports are present on both the fixed and movable heads. In the U arrangement offig..b, all four Figure. Countercurrent single-pass flow (a) Z-arrangement and (b) U-arrangement. (From Saunders,, with permission.) [ 0. No [

86 HEAT EXCHANGERS Figure. Figure. Two-pass/two-pass flow. (From Saunders,, with permission.) Two-pass/one-pass flow. (From Saunders,, with permission.) ports are at the fixed-head end. The two pass/two pass flow or / arrangement is shown in Fig.., and the two pass/one pass flow or / arrangement is shown in Fig... Observe that in Fig.., the arrangement is in true counterflow except for the center plate, where co-current flow exists. In Fig.., one halfofthe unit is in counterflow and the other halfis in co-current flow... Physical Data Figure.a shows a sketch ofa single plate for the chevron configuration. The chevron angle is designated by β, which can range from to. As shown in Fig..b, the mean flow channel gap is b, and it is seen that b is related to the plate pitch p p and plate thickness δ pl : b = p pl δ pl (.) Because the corrugations increase the flat plate area, an enlargement factor Λ is employed: Λ = developed length projected length where typically,.0 < Λ <.. The cross-sectional area ofone channel, A,isgivenby (.) A = bw (.) where w is the effective plate width shown in Fig..a. With the wetted perimeter ofone channel, [ 0. Nor * [

87 PLATE AND FRAME HEAT EXCHANGER the channel equivalent diameter will be d e = A bw = P W (b + Λw) P W = (b + Λw) (.) Figure. Plate geometry for Chevron plates in plate and frame (gasketed-plate heat exchanger). (From Saunders,, with permission.) [. No [

88 HEAT EXCHANGERS and because w b, d e = b (.) Λ If N p is high, the total surface area for heat flow may be based on the projected area: S = N P LW (.0) where L is the length ofeach 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 = /h c + /h c + R dc + R dh + δ pl /k p 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 = If R dc = R dh = 0 (an unfouled or clean exchanger), /h c + /h c + R dc + R dh (.) U = h ch h h c + h h (.) In the case of plate and frame heat exchanger, the true temperature difference in q = USθ m depends on the flow arrangement. For true counterflow or co-current flow, eqs. (.) and (.) apply. For other arrangements, such as those shown in Figs.. through., the work ofshah and Focke () should be consulted... 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 ofthe fluid given by eqs. (.): Then the channel Nusselt number will be Re ch = d eg ch µ Nu ch = hd e k = j hkpr / φ 0. (.) [ 0. Nor * [

89 PLATE AND FRAME HEAT EXCHANGER where j h is a heat transfer parameter and φ = µ/µ w : j h = C h Re y ch (.) where C h and y have been determined by Kumar (). Both ofthese values are listed in Table. as a function of the chevron angle β and the Reynolds number. Thus h = j hkpr / φ 0. (.) d e The channel pressure loss is the sum ofthe friction loss p ch and the port pressure loss p port, which accounts for the entrance and exit losses: In eq. (.), P = p ch + p port (.) p ch = f chn p L p G ρd e φ 0. (.) where N p is the number ofpasses, L p is indicated in Fig..a, and where f ch = K p Re z (.) TABLE. Kumar s () Constants for Single-Phase Heat Transfer and Pressure Loss in Plate and Frame Heat Exchangers Chevron Reynolds Reynolds Angle (deg) Number C h y Number K p z < > > < < > > < < > > < < > > < < > > Source: After Saunders (). [. No [

90 HEAT EXCHANGERS According to Kumar (), the port pressure loss may be taken as p port =.N pg so that the exchanger pressure loss, as required by eq. (.), will be ρ P = f chn p L p G ρd e φ 0. +.N pg ρ (.) The Kumar () constants depend on the chevron angle and the channel Reynolds number. They are displayed in Table... REGENERATORS.. Introduction The regenerator represents a class ofheat exchangers in which heat is alternately stored and removed from a surface. This heat transfer surface is usually referred to as the matrix ofthe regenerator. For continuous operation, the matrix must be moved into and out ofthe fixed hot and cold fluid streams. In this case, the regenerator is called a rotary regenerator. If, on the other hand, the hot and cold fluid streams are switched into and out ofthe matrix, the regenerator is referred to as a fixed matrix regenerator. In both cases the regenerator suffers from leakage and fluid entrainment problems, which must be considered during the design process. An example ofa rotary regenerator is shown in Fig... This is the Lungstrom air preheater used in power plants to warm the incoming combustion air using the exhaust or flue gases from the steam generator... Heat Capacity and Related Parameters A sketch ofthe end view ofa rotary regenerator is shown in Fig..a. The shaded lines represent the radial shield between the hot sector represented by the angle φ h and the cold sector represented by the angle φ c. Ifthe space occupied by the radial shield is represented by the angle φ sh, the total periphery in terms ofthese angles will be φ t = φ h + φ c + φ sh = π and because the magnitude of φ sh is much less than either φ h or φ c, φ sh may be neglected, leaving φ t = φ h + φ c = π (.) The entire assembly shown in Fig..a rotates in the counterclockwise direction at N r (rev/s). The hot fluid with capacity rate C h = ṁ h c ph enters the hot sector (φ h ) [ -0. Nor [

91 REGENERATORS Figure. A Lungstrom air preheater. (From Fraas,, with permission.) at T and flows out of the plane ofthe paper at T. The cold fluid with capacity rate C c = ṁ c c pc enters the cold sector (φ c )att and flows into the plane ofthe paper and leaves at t. Details ofa single passage ofthe hot and cold sides ofthe matrix are indicated, respectively, in Fig..b and c. In Fig..b, which represents the hot side ofthe matrix, the fluid flows from left to right, so that T(x = 0) = T and T(x = L) = T The matrix surface S h is presumed to be at T mh. For the cold fluid shown in Fig..c, the fluid flows from right to left, making t(x = 0) = t and t(x = L) = t and the matrix surface S c is presumed to be at T mc. [ 0. No [