The CRC Handbook of Mechanical Engineering. Chapter 4. Heat and Mass Transfer (776127), страница 30
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It simply consists of stainless steelplates and two end plates. The brazed unit can be mounted directly on piping without brackets andfoundations.© 1999 by CRC Press LLC4-128Section 4Printed Circuit Heat Exchangers. This exchanger, as shown in Figure 4.5.10, has only primary heattransfer surfaces as PHEs. Fine grooves are made in the plate by using the same techniques as thoseemployed for making printed electrical circuits. High surface area densities (650 to 1350 m2/m3 or 200to 400 ft2/ft3 for operating pressures of 500 to 100 bar respectively) are achievable. A variety of materialsincluding stainless steel, nickel, and titanium alloys can be used.
It has been successfully used withrelatively clean gases, liquids and phase-change fluids in chemical processing, fuel processing, wasteheat recovery, and refrigeration industries. Again, this exchanger is a new construction with limitedspecial applications currently.FIGURE 4.5.10 A section of a printed circuit heat exchanger. (Courtesy of Heatric Ltd., Dorset, U.K.)Exchanger Heat Transfer and Pressure Drop AnalysisIn this subsection, starting with the thermal circuit associated with a two-fluid exchanger, e-NTU, PNTU, and mean temperature difference (MTD) methods used for an exchanger analysis are presented,followed by the fin efficiency concept and various expressions. Finally, pressure drop expressions areoutlined for various single-phase exchangers.Two energy conservation differential equations for a two-fluid exchanger with any flow arrangementare (see Figure 4.5.11 for counterflow)dq = q ¢¢ dA = -Ch dTh = ±Cc dTc(4.5.1)where the ± sign depends upon whether dTc is increasing or decreasing with increasing dA or dx.
Thelocal overall rate equation isdq = q ¢¢ dA = U (Th - Tc ) local dA = U DT dA(4.5.2)Integration of Equations (4.5.1) and (4.5.2) across the exchanger surface area results in()(q = Ch Th,i - Th,o = Cc Tc,o - Tc,i)(4.5.3)andq = UA DTm = DTm Ro(4.5.4)where DTm is the true mean temperature difference (or MTD) that depends upon the exchanger flowarrangement and degree of fluid mixing within each fluid stream. The inverse of the overall thermalconductance UA is referred to as the overall thermal resistance Ro as follows (see Figure 4.5.12).© 1999 by CRC Press LLC4-129Heat and Mass TransferFIGURE 4.5.11 Nomenclature for heat exchanger variables.FIGURE 4.5.12 Thermal circuit for heat transfer in an exchanger.Ro = Rh + Rs,h + Rw + Rs,c + Rc(4.5.5)where the subscripts h, c, s, and w denote hot, cold, fouling (or scale), and wall, respectively. In termsof the overall and individual heat transfer coefficients, Equation (4.5.5) is represented as11111=++ Rw ++UA ( ho hA)hAhAhAhhh( o s )h( o s )c ( o )ch(4.5.6)where ho = the overall surface efficiency of an extended (fin) surface and is related to the fin efficiencyhf , fin surface area Af , and the total surface area A as follows:ho = 1 -AfA(1 - h )fThe wall thermal resistance Rw of Equation (4.5.5) is given by© 1999 by CRC Press LLC(4.5.7)4-130Section 4ìïïd A kï w wïï ln do d jRw = íï 2 pkw LNtïïéï 1 êï 2 pLNt êëî(for a flat wall)for a circular tube with a single-layer wallåj()ln d j +1 d j ùúúkw, jû(4.5.8)for a circular tube with a multiple-layer wallIf one of the resistances on the right-hand side of Equation (4.5.5) or (4.5.6) is significantly higherthan the other resistances, it is referred to as the controlling thermal resistance.
A reduction in thecontrolling thermal resistance will have much more impact in reducing the exchanger surface area (A)requirement compared with the reduction in A as a result of the reduction in other thermal resistances.UA of Equation (4.5.6) may be defined in terms of hot or cold fluid side surface area or wall conductionarea as(4.5.9)UA = U h Ah = Uc Ac = U w AwWhen Rw is negligible, Tw,h = Tw,c = Tw of Figure 4.5.12 is computed fromTw =) ( R + R )]T[(1 + [( R + R ) ( R + R )]Th + Rh + Rs,hs ,hhs ,ccc(4.5.10)s ,ccWhen Rs,h = Rs,c = 0, Equation (4.5.10) reduces toTw =Th Rh + Tc Rc ( ho hA) h Th + ( ho hA)c Tc=1 Rh + 1 Rc(ho hA)h + (ho hA)c(4.5.11)e-NTU, P-NTU, and MTD Methods.
If we consider the fluid outlet temperatures or heat transfer rate asdependent variables, they are related to independent variable/parameters of Figure 4.5.11 as follows.{}Th,o , Tc,o , or q = f Th,i , Tc,i , Cc , Ch , U , A, flow arrangement(4.5.12)Six independent and three dependent variables of Equation (4.5.12) for a given flow arrangement canbe transferred into two independent and one dependent dimensionless groups; three different methodsare presented in Table 4.5.2 based on the choice of three dimensionless groups.
The relationship amongthree dimensionless groups is derived by integrating Equations (4.5.1) and (4.5.2) across the surfacearea for a specified exchanger flow arrangement. Such expressions are presented later in Table 4.5.4 forthe industrially most important flow arrangements. Now we briefly describe the three methods.In the e-NTU method, the heat transfer rate from the hot fluid to the cold fluid in the exchanger isexpressed as(q = eCmin Th,i - Tc,i© 1999 by CRC Press LLC)(4.5.13)4-131Heat and Mass TransferTABLE 4.5.2MethodsGeneral Functional Relationships and Dimensionless Groups for e-NTU, P-NTU, and MTDe-NTU MethodP-NTU MethodaMTD Methodaq = eCmin(Th,i – Tc,i)q = P1C1|T1,i – T2,i|q = UAFDTlme = f(NTU, C*, flow arrangement)P1 = f(NTU1, R1, flow arrangement)F = f(P, R, flow arrangement)be=(Ch Th,i - Th,o() = C (T) C (TCmin Th,i - Tc,icminUA1NTU ==Cmin CminC* =abc ,oh ,i- Tc,iò U dAA( )( )˙ pmcCmin=Cmax˙ pmcmin))- Tc,iP1 =NTU 1 =R1 =T1,o - T1,iT2,i - T1,iUA T1,o - T1,i=DTmC1C1 T2,i - T2,o=C2T1,o - T1,iF=DTmDTlmLMTD = DTlm =DT1 - DT2ln( DT1 DT2 )DT1 = Th,i – Tc,o DT2 = Th,o – Tc,imaxAlthough P, R, and NTU1 are defined on fluid side 1, it must be emphasized that all the results of the P-NTU andMTD methods are valid if the definitions of P, NTU, and R are consistently based on Cc, Cs, Ch, or C.P and R are defined in the P-NTU method.Here the exchanger effectiveness e is an efficiency factor.
It is a ratio of the actual heat transfer ratefrom the hot fluid to the cold fluid in a given heat exchanger of any flow arrangement to the maximumpossible heat transfer rate qmax thermodynamically permitted. The qmax is obtained in a counterflow heatexchanger (recuperator) of infinite surface area operating with the fluid flow rates (heat capacity rates)and fluid inlet temperatures equal to those of an actual exchanger (constant fluid properties are idealized).As noted in Table 4.5.1, the exchanger effectiveness e is a function of NTU and C* in this method. Thenumber of transfer units NTU is a ratio of the overall conductance UA to the smaller heat capacity rateCmin.
NTU designates the dimensionless “heat transfer size” or “thermal size” of the exchanger. Otherinterpretations of NTU are given by Shah (1983). The heat capacity rate ratio C* is simply a ratio ofthe smaller to the larger heat capacity rate for the two fluid streams. Note that 0 £ e £ 1, 0 £ NTU £ ¥and 0 £ C* £ 1.The P-NTU method represents a variant of the e-NTU method. The e-NTU relationship is differentdepending upon whether the shell fluid is the Cmin or Cmax fluid in the (stream unsymmetric) flowarrangements commonly used for shell-and-tube exchangers. In order to avoid possible errors and toavoid keeping track of the Cmin fluid side, an alternative is to present the temperature effectiveness P asa function of NTU and R, where P, NTU, and R are defined consistently either for Fluid 1 side or Fluid2 side; in Table 4.5.2, they are defined for Fluid 1 side (regardless of whether that side is the hot or coldfluid side), and Fluid 1 side is clearly identified for each flow arrangement in Table 4.5.4; it is the shellside in a shell-and-tube exchanger.
Note thatq = P1C1 T1,i - T2,i = P2 C2 T2,i - T1,iP1 = P2 R2NTU1 = NTU 2 R2(4.5.14)P2 = P1 R1(4.5.15)NTU 2 = NTU1 R1(4.5.16)andR1 = 1 R2© 1999 by CRC Press LLC(4.5.17)4-132Section 4In the MTD method, the heat transfer rate from the hot fluid to the cold fluid in the exchanger is given byq = UA DTm = UAF DTlm(4.5.18)where DTm the log-mean temperature difference (LMTD), and F the LMTD correction factor, a ratio oftrue (actual) MTD to the LMTD, whereLMTD = DTlm =DT1 - DT2ln( DT1 DT2 )(4.5.19)Here DT1 and DT2 are defined asDT1 = Th,i - Tc,oDT2 = Th,o - Tc,ifor all flow arrangementsexcept for parallel flow(4.5.20)DT1 = Th,i - Tc,iDT2 = Th,o - Tc,ofor parallel flow(4.5.21)The LMTD represents a true MTD for a counterflow arrangement under the idealizations listed below.Thus, the LMTD correction factor F represents a degree of departure for the MTD from the counterflowLMTD; it does not represent the effectiveness of a heat exchanger.
It depends on two dimensionlessgroup P1 and R1 or P2 and R2 for a given flow arrangement.TABLE 4.5.3 Relationships between Dimensionless Groups of the P-NTU and LMTD Methodsand Those of the e-NTU MethodP1 =Cminìee=í *C1îeCR1 =C1 ìC *=íC2 î1 C *NTU 1 = NTUF=for C1 = Cminfor C1 = Cmaxfor C1 = Cminfor C1 = CmaxCmin ìNTU=í*C1îNTU Cfor C1 = Cminfor C1 = CmaxNTU cfé1 - C * e ù1e=ln êú **NTU=1eC1NTU(1 - e)NTU 1 - Cëû(F=)é 1 - RP1 ùP11ln êúNTU 1 (1 - R 1 ) êë 1 - P1 úû R 1 = 1 NTU 1 (1 - P1 )The relationship among the dimensionless groups of the e-NTU, P-NTU, and MTD methods arepresented in Table 4.5.3. The closed-form formulas for industrially important exchangers are presentedin terms of P1, NTU1, and R1 in Table 4.5.4.
These formulas are valid under idealizations which include:(1) steady-state conditions; (2) negligible heat losses to the surrounding; (3) no phase changes in thefluid streams flowing through the exchanger, or phase changes (condensation or boiling) occurring atconstant temperature and constant effective specific heat; (4) uniform velocity and temperature at theentrance of the heat exchanger on each fluid side; (5) the overall extended surface efficiency ho is uniformand constant; (6) constant individual and overall heat transfer coefficients; (7) uniformly distributed heattransfer area on each fluid side; (7) the number of baffles as large in shell-and-tube exchangers; (8) noflow maldistribution; and (9) negligible longitudinal heat conduction in the fluid and exchanger wall.© 1999 by CRC Press LLCHeat and Mass Transfer4-133The overall heat transfer coefficient can vary as a result of variations in local heat transfer coefficientsdue to two effects: (1) change in heat transfer coefficients in the exchanger as a result of changes in thefluid properties or radiation due to rise or drop of fluid temperatures and (2) change in heat transfercoefficients in the exchanger due to developing thermal boundary layers; it is referred to as the lengtheffect.
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