The CRC Handbook of Mechanical Engineering. Chapter 4. Heat and Mass Transfer (776127), страница 32
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The fluid pumping power à is approximately related to the core pressuredrop in the exchanger as (Shah, 1985).ì 1 m 4 L f Reï 2 g r2 Dhm˙ Dp ï cÃ=»í0.22.8rï 0.046 m 4 L m˙ï 2 g r2 D A1.8 D 0.2chh0îfor laminar flow(4.5.26)for turbulent flow(4.5.27)It is clear from Equations (4.2.26) and (4.2.27) that the fluid pumping power is strongly dependentupon the fluid density (à µ 1/r2) particularly for low-density fluids in laminar and turbulent flows, andupon the viscosity in laminar flow. In addition, the pressure drop itself can be an important considerationwhen blowers and pumps are used for the fluid flow since they are head limited. Also for condensingand evaporating fluids, the pressure drop affects the heat transfer rate.
Hence, the pressure drop determination in the exchanger is important.The pressure drop associated with a heat exchanger consists of (1) core pressure drop, and (2) thepressure drop associated with the fluid distribution devices such as inlet and outlet manifolds, headers,tanks, nozzles, ducting, and so on, which may include bends, valves, and fittings. This second Dpcomponent is determined from Idelchik (1994) and Miller (1990). The core pressure drop may consistof one or more of the following components depending upon the exchanger construction: (1) frictionlosses associated with fluid flow over heat transfer surface (this usually consists of skin friction, form(profile) drag, and internal contractions and expansions, if any); (2) the momentum effect (pressure dropor rise due to fluid density changes) in the core; (3) pressure drop associated with sudden contractionand expansion at the core inlet and outlet, and (4) the gravity effect due to the change in elevation© 1999 by CRC Press LLC4-140Section 4between the inlet and outlet of the exchanger.
The gravity effect is generally negligible for gases. Forvertical flow through the exchanger, the pressure drop or rise (“static head”) due to the elevation changeis given byDp = ±r m gLgc(4.5.28)Here the “+” sign denotes vertical upflow (i.e., pressure drop), the “–” sign denotes vertical downflow(i.e., pressure rise).
The first three components of the core pressure drop are now presented for platefin, tube-fin, plate, and regenerative heat exchangers.Plate-fin heat exchangers. For the plate-fin exchanger (Figure 4.5.2), all three components are considered in the core pressure drop evaluation as follows:ærör ùDp G 2 1 éL æ 1ö22=ê 1 - s + K c + f ri ç ÷ + 2ç i - 1÷ - 1 - s - K e i úpirh è r ø m2 gc pi ri êëro úûè roø()()(4.5.29)where f is the Fanning friction factor, Kc and Ke are flow contraction (entrance) and expansion (exit)pressure loss coefficients (see Figure 4.5.14), and s is a ratio of minimum free flow area to frontal area.Kc and Ke for four different entrance flow passage geometries are presented by Kays and London (1984).The entrance and exit losses are important at low values of s and L (short cores), high values of Re,and for gases; they are negligible for liquids.
The values of Kc and Ke apply to long tubes for whichflow is fully developed at the exit. For partially developed flows, Kc and Ke is higher than that for fullydeveloped flows. For interrupted surfaces, flow is never a fully developed boundary layer type. For highlyinterrupted fin geometries, the entrance and exit losses are generally small compared to the core pressuredrop and the flow is well mixed; hence, Kc and Ke for Re ® ¥ should represent a good approximation.The mean specific volume vm or (1/r)m in Equation (4.5.29) is given as follows. For liquids with anyflow arrangement, or for a perfect gas with C* = 1 and any flow arrangement (except for parallel flow),vi + v o 1 æ 1æ 1ö1ö= ç + ÷ç ÷ = vm =22 è ri r o øè rø m(4.5.30)where v is the specific volume in m3/kg. For a perfect gas with C* = 0 and any flow arrangement,æ 1öR˜Tlmç ÷ =è r ø m pave(4.5.31)Here R̃ is the gas constant in J/(kg K), pave = (pi + po)/2, and Tlm = Tconst + DTlm where Tconst is the meanaverage temperature of the fluid on the other side of the exchanger; the LMTD DTlm is defined in Table4.5.2.
The core frictional pressure drop in Equation 4.5.29 may be approximated asDp »4 fLG 2 æ 1 öç ÷2 gc Dh è r ø m(4.5.32)Tube-fin heat exchangers. The pressure drop inside a circular tube is computed using Equation (4.5.29)with proper values of f factors, and Kc and Ke from Figure (4.5.18) for circular tubes.For flat fins on an array of tubes (see Figure 4.5.5b), the components of the core pressure drop (suchas those in Equation 4.5.29) are the same with the following exception: the core friction and momentumeffect take place within the core with G = m˙ / A0 , where Ao is the minimum free-flow area within the© 1999 by CRC Press LLCHeat and Mass Transfer4-141FIGURE 4.5.14 Entrance and exit pressure loss coefficients: (a) circular tubes, (b) parallel plates, (C) squarepassages, and (d) triangular passages.
(From Kays, W.M. and London, A.L., Compact Heat Exchangers, 3rd ed.,McGraw-Hill, New York, 1984. With permission.) For each of these flow passages, shown in the inset, the fluidflows perpendicular to the plane of the paper into the flow passages.© 1999 by CRC Press LLC4-142Section 4core, and the entrance and exit losses occur at the leading and trailing edges of the core with the associatedflow area A0¢ such thatṁ = GAo = G ¢Ao¢or(4.5.33)G ¢s ¢ = Gswhere s¢ is the ratio of free-flow area to frontal area at the fin leading edges. The pressure drop for flownormal to a tube bank with flat fins is then given byDp G 2 1=pi2 gc pi rié L æ 1öærö ù G¢2 1 éri ù22ê f ri ç ÷ + 2ç i - 1÷ ú +úê 1 - s ¢ - Kc - 1 - s ¢ - Kero ûè roø úû 2 gc pi ri ëêë rh è r ø m() ()(4.5.34)For individually finned tubes as shown in Figure 4.5.5a, flow expansion and contraction take place alongeach tube row, and the magnitude is of the same order as that at the entrance and exit.
Hence, the entranceand exit losses are generally lumped into the core friction factor. Equation (4.5.29) then reduces forindividually finned tubes toæröùDp G 2 1 é L æ 1 ö=ê f ri ç ÷ + 2ç i - 1÷ úpi2 gc pi ri êë rh è r ø mè roø ûú(4.5.35)Regenerators.
For regenerator matrices having cylindrical passages, the pressure drop is computedusing Equation (4.5.29) with appropriate values of f, Kc, and Ke. For regenerator matrices made up ofany porous material (such as checkerwork, wire, mesh, spheres, copper wools, etc.), the pressure dropis calculated using Equation (4.5.35) in which the entrance and exit losses are included in the frictionfactor f.Plate heat exchangers. Pressure drop in a PHE consists of three components: (1) pressure dropassociated with the inlet and outlet manifolds and ports, (2) pressure drop within the core (plate passages),and (3) pressure drop due to the elevation change. The pressure drop in the manifolds and ports shouldbe kept as low as possible (generally < 10%, but it is found as high as 25 to 30% of higher in somedesigns).
Empirically, it is calculated as approximately 1.5 times the inlet velocity head per pass. Sincethe entrance and exit losses in the core (plate passages) cannot be determined experimentally, they areincluded in the friction factor for the given plate geometry. The pressure drop (rise) caused by theelevation change for liquids is given by Equation (4.5.28). Hence, the pressure drop on one fluid sidein a PHE is given byDp =1.5G 2 N p2 gc ri+æ 14 fLG 2 æ 1 ö1 ö G 2 r m gL- ÷±ç ÷ +ç2 gc De è r ø m è ro ri ø gcgc(4.5.36)where Np is the number of passes on the given fluid side and De is the equivalent diameter of flowpassages (usually twice the plate spacing).
Note that the third term on the right-hand side of the equalitysign of Equation (4.5.36) is for the momentum effect which is generally negligible in liquids.Heat Transfer and Flow Friction CorrelationsAccurate and reliable surface heat transfer and flow friction characteristics are a key input to theexchanger heat transfer and pressure drop analyses or to the rating and sizing problems (Shah, 1985).Some important analytical solutions and empirical correlations are presented next for selected exchangergeometries.The heat transfer rate in laminar duct flow is very sensitive to the thermal boundary condition. Hence,it is essential to identify carefully the thermal boundary condition in laminar flow. The heat transfer ratein turbulent duct flow is insensitive to the thermal boundary condition for most common fluids (Pr >© 1999 by CRC Press LLC4-143Heat and Mass TransferTABLE 4.5.6Solutions for Heat Transfer and Friction for Fully Developed Flow-Through Specified DuctsGeometry (L/Dh > 100)abNuH1NuH2NutfRejH1/f aNuH1/Nut3.0141.4742.39b12.6300.2691.263.1111.8922.4713.3330.2631.263.6083.0912.97614.2270.2861.214.0023.8623.34b15.0540.2991.204.1233.0173.39115.5480.2991.224.3644.3643.65716.0000.3071.195.3312.944.43918.2330.3291.206.4902.945.59720.5850.3551.168.2358.2357.54124.0000.3861.095.385—4.86124.0000.2531.11This heading is the same as NuH1 Pr–1/3/f Re with Pr = 0.7.Interpolated values.0.7); the exception is liquid metals (Pr < 0.03).
Hence, there is generally no need to identify the thermalboundary condition in turbulent flow for all fluids except liquid metals.Fully developed laminar flow analytical solutions for some duct shapes of interest in compact heatexchangers are presented in Table 4.5.6 for three important thermal boundary conditions denoted by thesubscripts H1, H2, and T (Shah and London, 1978; Shah and Bhatti, 1987). Here, H1 denotes constantaxial wall heat flux with constant peripheral wall temperature, H2 denotes constant axial and peripheralwall heat flux, and T denotes constant wall temperature.
The entrance effects, flow maldistribution, freeconvection, property variation, fouling, and surface roughness all affect fully developed analyticalsolutions. In order to account for these effects in real plate-fin plain fin geometries having fully developedflows, it is best to reduce the magnitude of the analytical Nu by at least 10% and to increase the valueof the analytical fRe by 10% for design purposes.The initiation of transition flow, the lower limit of the critical Reynolds number (Recrit), depends uponthe type of entrance (e.g., smooth vs. abrupt configuration at the exchanger flow passage entrance).
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