The CRC Handbook of Mechanical Engineering. Chapter 4. Heat and Mass Transfer (776127), страница 67
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Since the heat transfer coefficient can varyconsiderably for different thermal boundary conditions, it is important that the boundary conditions bespecified correctly. Although the number of thermal boundary conditions is in principle infinite, severalclassical types have been identified and are in common use. They are usually identified in terms of theNusselt number, Nu = hcL/k, with a particular subscript. For example, for duct flow, the symbol NuT isused to specify the Nusselt number when the wall temperature is constant in both the flow and peripheraldirections. Other thermal boundary conditions are described in Table 4.9.1 for duct heat transfer andwill be used throughout this section.TABLE 4.9.11.2.3.4.ConstantConstantConstantConstantThermal Boundary Conditions for Duct Heat Transferwall temperature in both the flow and circumferential directionheat flux in the flow direction and constant temperature in the circumferential directionheat flux in the flow and circumferential directionsheat flux per unit volume in the wall with circumferential wall heat conductionNuTNuH1NuH2NuH4It should be noted that because of the symmetry in circular and parallel plate ducts, NuH1 and NuH2are identical and are referred to simply as NuH.
NuH4 with wall conduction is a more-complicated problemwhere the energy equations must be solved simultaneously in both the wall and the fluid. Such problemsare called conjugated. In the NuH4 situation, the designer has the flexibility of affecting the heat transferby varying either or both the characteristics of the duct wall or the convective fluid. In the heat transferrelations to be considered later, care will be taken to identify the proper thermal boundary conditionsusing the nomenclature in Table 4.9.1.Laminar Duct Heat Transfer — Purely Viscous, Time-Independent NonNewtonian FluidsAs discussed in Section 3.9, a convenient and comprehensive constitutive equation for pseudoplasticfluids (flow index, n < 1) is the modified power law equation:ma =momo1+(g˙ )1-nK(4.9.1)Equation (4.9.1) has the characteristic that at low shear rates, the equation approaches that for aNewtonian fluid while at large shear rates it describes a power law fluid.
In addition, solutions using© 1999 by CRC Press LLC4-280Section 4Equation (4.9.1) generate a shear rate parameter, b, which describes whether any particular system isin the Newtonian, transitional, or power law region. For duct flow, b is given bymb= oKæ u öçD ÷è Hø1- n(4.9.2)If log10 b > 2: Power law regionIf log10 b < –2: Newtonian regionIf –2 £ log10 b £ 2: Transition regionFor fully developed flow, the characteristic length is the hydraulic diameter, DH, and the fluid temperatureis the “bulk” temperature defined asTb =1Ac uò uTdAcAc(4.9.3)Figure 4.9.1 illustrates the values of NuT vs.
b for a circular duct with the flow index, n, as a parameter.It is seen from the figure that the effect of b on NuT is only moderate, but for some applications it maybe important to know at what value of b the system is operating. The situation is similar for boundarycondition NuH.Although Figure 4.9.1 shows the Nusselt number relation graphically, it is convenient to have simplecorrelation equations to represent the solutions for both boundary conditions.
For fully developed Nusseltnumbers with values of 0.5 £ n £ 1.0 and 10–4 £ b £ 104, Irvine et al. (1988) present the followingequation which represents both solutions with a maximum difference of 1.5%:Nu =Nu N (1 + b)Nu N b1+Nu P(4.9.4)The Newtonian Nusselt numbers are NuN = 3.6568 for NuT , and NuN = 4.3638 for NuH. In addition,Table 4.9.2 lists the power law Nusselt numbers, NuTP and NuHP , for log10 b = 4.Graetz solutions for the thermal entrance lengths are also available. They assume that the velocityprofile is fully developed at the duct entrance and present the duct lengths required for the Nusseltnumbers to reach within 1% of the fully developed values. Figure 4.9.2 shows these thermal entrancelengths for NuT thermal boundary condition.
The situation is similar for boundary condition NuH.A correlation equation for the thermal entrance lengths for both the NuT and NuH boundary conditionsby Irvine et al. (1988) represents the numerical solutions within 0.5% for 0.5 £ n £ 1.0 and –4 £ log10b £ 4. Table 4.9.3 lists the power law thermal entrance lengths which are needed to evaluate the followingcorrelation equation:+x ent,=b ,n+x ent,N (1 + b)1++x ent,N (b)(4.9.5)+x ent,P++where xent,is the modified power law dimensionless entrance length defined as xent,=b ,nb ,n++(xent,b,n /DH )/Pe, and xent, N and xent,P are the Newtonian and power law values, respectively.
The New++tonian dimensionless entrance lengths are xent,N = 0.03347 for NuT and xent, N = 0.04309 for NuH.© 1999 by CRC Press LLC4-281Heat and Mass TransferFIGURE 4.9.1 Variation of the fully developed circular duct Nusselt numbers, NuT , with the shear rate parameterb and n. (From Irvine, T.F., Jr. et al., in ASME Symposium on Fundamentals of Forced Convection Heat Transfer,ASME publ. HTD 101, 1988, 123–127. With permission.)TABLE 4.9.2 Power Law NuT and NuHSolutions for a Circular Duct (log10 b = 4)n1.0 (Newtonian)0.90.80.70.60.5NuTPNuHP3.65683.69343.73773.79213.86053.94944.36384.41094.46794.53854.62814.7456Source: Irvine, T.F., Jr.
et al., in ASME Symposiumon Fundamentals of Forced Convection HeatTransfer, ASME publ. HTD 101, 1988, 123–127.Only one noncircular geometry using the modified power law equation has been published in thearchival literature for laminar fully developed heat transfer (Capobianchi and Irvine, 1992). A correlationequation for NuH1 for annuli with constant heat flux at the inner wall and the outer wall insulated isn <1Nu H1 =1+ b1b+Nu H1, N Nu H1,P(4.9.6)Nusselt numbers for square ducts and power law fluids can be found in Chandrupatla and Sastri (1977)and, for isosceles triangular ducts, in Cheng (1984). Thermally developing and thermally developedlaminar heat transfer in rectangular channels has been studied by Hartnett and Kostic (1989).For other cross-sectional shapes, a power law approximate correlation has been proposed by Cheng(1984):© 1999 by CRC Press LLC4-282Section 4FIGURE 4.9.2 Thermal entrance lengths vs.
shear rate parameter b and n for NuT in circular ducts. (From Irvine,T.F., Jr. et al., in ASME Symposium on Fundamentals of Forced Convection Heat Transfer, ASME publ. HTD 101,1988, 123–127. With permission.)TABLE 4.9.3 Values of Circular Duct Thermal EntranceLengths for NuT and NuH for Use in Equation 4.9.5n1.0 (Newtonian)0.90.80.70.60.5+´ 102NuT , x ent,P+2NuH , x ent,P ´ 103.3473.3263.3063.2793.2503.2134.3094.2814.2484.2104.1664.114Source: Irvine, T.F., Jr., et al., in ASME Symposium on Fundamentalsof Forced Convection Heat Transfer, ASME publ. HTD 101, 1988,123–127.é (a + bn) ùNu P = Nu N êúë ( a + b) n û13(4.9.7)where a and b are the Kozicki geometric constants listed in Table 3.9.3 in the section on non-Newtonianflows. Equation (4.9.7) applies to any thermal boundary condition.
For circular ducts, Equation 4.9.7predicts the correct solution for both NuT and NuH.Turbulent Duct Flow for Purely Viscous Time-Independent Non-NewtonianFluidsIt is known that in turbulent flow, the type of thermal boundary conditions has much less effect than inlaminar flow. Therefore, turbulent flow heat transfer investigations are often reported without specifyingthe thermal boundary conditions. Yoo (1974) has presented an empirical correlation for turbulent heattransfer in circular ducts for purely viscous time-independent power law fluids.© 1999 by CRC Press LLC4-283Heat and Mass TransferStPra2 3 = 0.0152 Re a-0.155(4.9.8)Equation (4.9.8) describes all of the experimental data available in the literature at the time with a meandeviation of 2.3%.
Equation (4.9.8) is recommended in order to predict the turbulent fully developedheat transfer in the ranges 0.2 £ n £ 0.9 and 3000 £ Rea £ 90,000. The Reynolds number and Prandtlnumbers in Equation (4.9.8) are based on the apparent viscosity at the wall, ma, i.e.,Re a =Pra =ruDHma(4.9.9)m acp(4.9.10)kIn order to evaluate Equations (4.9.9) and (4.9.10) in terms of the rheological properties and operatingparameters, an expression must be obtained for ma in terms of these quantities. The value of ma is evaluatedby considering that ma is determined from fully developed laminar circular tube power law fluid flowfor which it can be shown that (Irvine and Karni, 1987)3n + 1öm a = Kæè 4n øn -1æ 8u öçD ÷è Høn -1(4.9.11)assuming that the quantities K, n, cp, and k are constant.
It is also of interest that the Prandtl numberis no longer a thermophysical property for power law fluids but depends upon the average velocity, u,and the hydraulic diameter, DH.Hartnett and Rao (1987) have investigated fully developed turbulent heat transfer for a rectangularduct with a 2:1 aspect ratio and propose the following equation which generally agreed with theirexperimental data within ±20%:Nu = (0.0081 + 0.0149n) Re 0a.8 Pra0.4(4.9.12)Viscoelastic FluidsAn important characteristic of viscoelastic fluids is their large hydrodynamic and thermal entrancelengths. Cho and Hartnett (1982) have reported hydrodynamic entrance lengths of up to 100 diametersand thermal entrance lengths up to 200 to 800 diameters depending upon the Reynolds and Prandtlnumbers. These can be compared with Newtonian fluids entrance lengths which are of the order of 10to 15 diameters.
Therefore, care must be used in applying fully developed relations to practical situations.Cho et al. (1980) reported heat transfer measurements in the thermal entrance region and recommendthe following empirical equation for saturated aqueous polymer solutions for 6000 £ Rea and x/DH valuesup to 450:J H = 0.13( x DH )-0.24Re -a0.45(4.9.13)where JH = St Pra2 3 and St - hc/rcp u.All of the reported fully developed turbulent flow heat transfer measurements have been plagued bysolute and solvent, thermal entrance, and degradation effects, and thus there is considerable scatter inthe results. Degradation effects can be reduced or eliminated by using large amounts of polymer (500© 1999 by CRC Press LLC4-284Section 4to 10,000 wppm) so that the solution becomes saturated. Cho and Hartnett (1982) attempted to eliminatethese effects by using a thermal entrance length of 430 diameters and saturated polymer solutions whichshould yield maximum heat transfer reductions.
Their experimental results for fully developed heattransfer were correlated for a Reynolds number range 3500 £ Rea £ 40,000 and concentration solutionsof 500 to 5000 wppm of polyacrylamide and polyethylene oxide byJ H = 0.03Re a-0.45(4.9.14)For viscoelastic fluids in fully developed (hydrodynamically and thermally) laminar flow in circularducts there is no apparent viscoelastic effect.
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