The CRC Handbook of Mechanical Engineering. Chapter 4. Heat and Mass Transfer (776127), страница 68
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Thus, the heat transfer relations are the same as those fortime-independent fluids such as power law or modified power law fluids. The same situation holds forthermal entrance region heat transfer (Graetz problem). Relations for laminar Nusselt numbers in thermalentrance regions are presented by Cho and Hartnett (1982).Free Convection Flows and Heat TransferFree convection information available in the heat transfer literature up to the present time is concentratedon heat transfer to power law fluids for vertical plates and parallel plate channels.
For free convectionflows, however, the velocities and thus the shear rates are low and care must be taken that the flow fora particular fluid is in the power law shear rate region before using power law solutions or correlations.Comprehensive review articles on free convection with non-Newtonian fluids have been presented byShenoy and Mashelkar (1982) and Irvine and Karni (1987).For a single vertical plate with a modified power law fluid and a thermal boundary condition NuT , inlaminar flow, the following relation is recommended by Shenoy and Mashelkar (1982):1 / (2 n+2 )n / ( 3n +1)NuTL = T (n) GrTLPrTL(4.9.15)where NuTL is the average Nusselt number andGrTL =PrTL =rc p æ K öç ÷k è rør2 Ln+2ga(Ts - T¥ )K2[2-n](4.9.16)2 ( n +1)[( 3n -3) ( 2 n + 2 )]L( n-1) ( 2 n+2 ) ga(Ts - T¥ )(4.9.17)where a is the isobaric thermal expansion coefficient.In the range 0.5 £ n £ 1, T(n) can be approximated byT (n) = 0.1636n + 0.5139(4.9.18)The characteristic dimension in the Nusselt and Grashof numbers is the plate height, L.For thermal boundary conditions NuH, the following relation is also recommended by Shenoy andMashelkar (1982).
Since the heat flux, qw is specified in this case, the local plate temperature at any x(measured from the bottom of the plate) can be obtained from the local Nusselt number NuHx. The heattransfer coefficient is defined in terms of the difference between the wall and free-stream temperatures.[( 3n + 2 ) ( n + 4 ) nNu Hx = 0.619 GrHxPrHxwhere© 1999 by CRC Press LLC]0.213(4.9.19)4-285Heat and Mass TransferGrHx =PrHx =rc p æ K öç ÷K è rør2 x 4 æ ga qw ö÷çk2 è k ø5 (n+4)2-n(4.9.20)æ ga q w öx ( 2 n-2 ) ( n+ 4 ) ç÷è k ø( 3n -3) ( n + 4 )(4.9.21)Vertical Parallel PlatesFor power law fluids and laminar flow, Figure 4.9.3 presents the graphical results of a numerical solution.Of interest are the average Nusselt number NuTb and the dimensionless average flow velocity betweenthe plates, Uo+ . These are shown on the left and right ordinates respectively in Figure 4.9.3 (Irvine etal., 1982).
The characteristic dimension in the Nusselt and Grashof numbers is the plate spacing, b. Thedimensionless quantities used in Figure 4.9.3 are defined as follows:NuTb =hc bkUo+ =buoLu *éù1 (2-n)úrc p êvkêúPrg =k ê æ L ö (1-n ) ( 2-n ) ( 2 n-2 ) ( 2-n ) úbêè øúë bûGrg =ga(Ts - T¥ ) b ( n+2 ) ( 2-n )Lv K2 ( 2-n ) æ öè bøn (2-n)u* =vK =Krv1K ( 2-n ) b (1-2 n ) ( 2-n )L(1-n ) ( 2-n )For vertical parallel plates for the average Nusselt number, Nu Hb , and the between plate average velocity,Schneider and Irvine (1984) have presented graphical results similar to Figure 4.9.3.Lee (1992) has presented a numerical solution for laminar flow of a modified power law fluid betweenvertical plates.
Lee has also calculated thermal entrance regions and shown that if a parallel plate systemis actually operating in the transition region and if the power law solution is used, both the total heattransfer and the velocity between plates can differ by over an order of magnitude. It is important toconsider the shear rate parameter in order to determine which free convection solution to use.Sphere and Horizontal Cylinder — Power Law FluidsFor flow over a sphere, the correlation for power law fluids by Amato and Tien (1976) isNuTr = CZ D(4.9.22)1 (2 n+2 )Z = GrTrPrTrn (3n+1)(4.9.23)whereand© 1999 by CRC Press LLCC = 0.996 ± 0.120, D = 0.682for Z < 10C = 0.489 ± 0.005, D = 1.10for 10 £ Z £ 404-286Section 4FIGURE 4.9.3 Free convection average Nusselt number, Nub and dimensionless average velocity U o+ betweenvertical plates for a power law fluid vs.
generalized Raleigh number for the NuT boundary condition. (From Irvine,T.F., Jr. et al., ASME Paper 82-WA/HT-69, 1982. With permission.)where the characteristic dimension in all dimensionless variables is the sphere radius, r, and GrTr andPrTr are defined in Equations (4.9.16) and (4.9.17).For pseudoplastic fluids flowing over a cylinder, an experimental correlation proposed by Gentry andWorllersheim (1974) for the average Nusselt number, NuTD , isNuTD =hc D0.2= 1.19(GrTD PrTD )k(4.9.24)where GrTD and PrTD are defined as in Equations (4.9.16) and (4.9.17) with the cylinder diameter, D,being used instead of L.ReferencesAcrivos, A. 1960.
A theoretical analysis of laminar natural convection heat transfer to non-Newtonianfluids, AIChE J., 6, 584–590.Amato, W.S. and Tien, C. 1976. Free convection heat transfer from isothermal spheres in polymersolutions, Int. J. Heat Mass Transfer, 19, 1257–1266.Capobianchi, M. and Irvine, T.F., Jr.
1992. Predictions of pressure drop and heat transfer in concentricannular ducts with modified power law fluids, Wärme Stoffübertragung, 27,209–215.© 1999 by CRC Press LLCHeat and Mass Transfer4-287Chandrupatla, A.R. and Sastri, V.M. 1977. Laminar forced convection heat transfer of a non-Newtonianfluid in a square duct, Int. J. Heat Mass Transfer, 20, 1315–1324.Cheng, J.A.
1984. Laminar Forced Convection Heat Transfer of Power Law Fluids in Isosceles TriangularDucts, Ph.D. Thesis, Mechanical Engineering Department, State University of New York at StonyBrook.Cho, Y.I. and Hartnett, J.P. 1982. Non-Newtonian fluids in circular pipe flow, Adv. Heat Transfer, 15,59–141.Cho, Y.I., Ng, K.S., and Hartnett, J.P.
1980. Viscoelastic fluids in turbulent pipe flow — a new heattransfer correlation, Lett. Heat Mass Transfer, 7, 347.Gentry, C.C. and Wollersheim, D.E. 1974. Local free convection to non-Newtonian fluids from ahorizontal isothermal cylinder, ASME J. Heat Transfer, 96, 3–8.Hartnett, J.P. and Kostic, M. 1989. Heat transfer to Newtonian and non-Newtonian fluids in rectangularducts, Adv. Heat Transfer, 19, 247–356.Hartnett, J.P. and Rao, B.K.
1987. Heat transfer and pressure drop for purely viscous non-Newtonianfluids in turbulent flow through rectangular passages, Wärme Stoffüberetragung, 21, 261.Irvine, T.F., Jr. and Karni, J. 1987. Non-Newtonian flow and heat transfer, in Handbook of Single PhaseConvective Heat Transfer, John Wiley & Sons, New York, 20-1–20-57.Irvine, T.F., Jr., Wu, K.C., and Schneider, W.J. 1982. Vertical Channel Free Convection to a Power LawFluid, ASME Paper 82-WA/HT-69.Irvine, T.F., Jr., Kim, S.C., and Gui, F.L. 1988. Graetz problem solutions for a modified power law fluid,in ASME Symposium on Fundamentals of Forced Convection Heat Transfer, ASME publ. HTD101, pp.
123–127.Lee, S.R. 1992, A Computational Analysis of Natural Convection in a Vertical Channel with a ModifiedPower Law Fluid, Ph.D. Thesis, Mechanical Engineering Department, State University of NewYork at Stony Brook.Schneider, W.J. and Irvine, T.F., Jr. 1984. Vertical Channel Free Convection for a Power Law Fluid withConstant Heat Flux, ASME Paper 84-HT-16.Shenoy, A.V. and Mashelkar, R.A. 1982 Thermal convection in non-Newtonian fluids, Adv. Heat Transfer,15, 143–225.Yoo, S.S.
1974. Heat Transfer and Friction Factors for Non-Newtonian Fluids in Turbulent Pipe Flow,Ph.D. Thesis, University of Illinois at Chicago Circle.Further InformationOther sources which may be consulted for more detailed information are Cho and Hartnett (1982),Shenoy and Mashelkar (1982), Irvine and Karni (1987), and Hartnett and Kostic (1989).© 1999 by CRC Press LLC.
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