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Research Papers: Natural and Mixed Convection

Laminar Natural Convection From an Isothermal Vertical Surface to Pseudoplastic and Dilatant Fluids

[+] Author and Article Information
Massimo Capobianchi

Professor
Mem. ASME
e-mail: capobianchi@gonzaga.edu

A. Aziz

Distinguished Research Professor
Life Fellow ASME
e-mail: aziz@gonzaga.edu
Department of Mechanical Engineering,
School of Engineering and Applied Science,
Gonzaga University,
502 E. Boone Avenue,
Spokane, WA 99258-0026

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received November 12, 2011; final manuscript received August 7, 2012; published online October 5, 2012. Assoc. Editor: Ali Ebadian.

J. Heat Transfer 134(12), 122502 (Oct 05, 2012) (9 pages) doi:10.1115/1.4007406 History: Received November 12, 2011; Revised August 07, 2012

This paper reports the results of a numerical study of natural convective heat transfer from a vertical isothermal surface to pseudoplastic and dilatant fluids. The analysis calculates the average Nusselt number in the laminar regime when the surface is exposed to an otherwise quiescent fluid. Because the solution utilizes constitutive equations that are valid over the entire shear rate range, the results map the behavior regardless of the shear rates that exist in the flow field. The Nusselt number is shown to approach Newtonian values when the shear rates throughout the flow field are predominantly either in the zero or in the high shear rate Newtonian regions of the flow curve. When they are principally in the power law regime, and if the fluid is also strongly non-Newtonian, then the Nusselt number approaches power law values. For all other cases, it is seen to attain intermediate values. Furthermore, a shear rate parameter is identified that determines the shear rate regime where the system is operating. The average Nusselt number is presented in both graphical and tabular forms over a broad range of system parameters.

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References

Ostrach, S., 1953, “An Analysis of Laminar Free-Convection Flow and Heat Transfer About a Flat Plate Parallel to the Direction of the Generating Body Force,” National Advisory Committee for Aeronautics, Report No. 1111, pp. 63–79.
Gebhart, B., Jaluria, Y., Mahajan, R. L., and Sammakia, B., 1988, Buoyancy-Induced Flows and Transport, Reference Edition, Hemisphere Publishing Corp., New York, pp. 1–132, 857–890, Chaps. 1–3, 16.
Shenoy, A. V., and Mashelkar, R. A., 1982, “Thermal Convection in Non-Newtonian Fluids,” Advances in Heat Transfer, 15, pp. 143–225. [CrossRef]
Irvine, T. F., Jr., and Capobianchi, M., 2005, “Non-Newtonian Fluids—Heat Transfer,” The CRC Handbook of Mechanical Engineering, 2nd ed., F.Kreith and Y.Goswami, eds., Boca Raton, pp. 4-269–4-278, Chap. 4.9.
Macosko, C. W., 1994, Rheology Principles, Measurements, and Applications, Wiley-VHC, Inc., New York, pp. 1–175, Part I, Chaps. 1–4.
Acrivos, A., 1960, “A Theoretical Analysis of Laminar Natural Convection Heat Transfer to Non-Newtonian Fluids,” Am. Inst. Chem. Eng. J., 6, pp. 584–590. [CrossRef]
Tien, C., 1967, “Laminar Natural Convection Heat Transfer From Vertical Plate to Power-Law Fluid”, Appl. Sci. Res., 17(3), pp. 233–248. [CrossRef]
Kawase, Y., and Ulbrecht, J. J., 1984, “Approximate Solution to the Natural Convection Heat Transfer From a Vertical Plate,” Int. Commun. Heat Mass Transfer, 11(2), pp. 143–155. [CrossRef]
Huang, M.-J., and Chen, C.-K., 1990, “Local Similarity Solutions of Free Convective Heat Transfer From a Vertical Plate to Non-Newtonian Power Law Fluids,” Int. J. Heat Mass Transfer, 33(1), pp. 119–125. [CrossRef]
Moulic, S. G., and Yao, L. S., 2009, “Non-Newtonian Natural Convection Along a Vertical Flat Plate With Uniform Surface Temperature,” ASME J. Heat Transfer, 131, p. 062501. [CrossRef]
Capobianchi, M., 2008, “Pressure Drop Predictions for Laminar Flows of Extended Modified Power Law Fluids in Rectangular Ducts,” Int. J. Heat Mass Transfer, 51, pp. 1393–1401. [CrossRef]
Dunleavy, J. E., Jr., and Middleman, S., 1966, “Correlation of Shear Behavior of Solutions of Polyisobutylene,” Trans. Soc. Rheol., 10(1), pp. 157–168. [CrossRef]
Capobianchi, M., and Irvine, T. F., Jr., 1992, “Predictions of Pressure Drop and Heat Transfer in Concentric Annular Ducts With Modified Power Law Fluids,” Heat Mass Transfer, 27(4), pp. 209–215. [CrossRef]
Cross, M. M., 1965, “Rheology of Non-Newtonian Fluids: A New Flow Equation for Pseudoplastic Systems,” J. Colloid Sci., 20, pp. 417–437. [CrossRef]
Capobianchi, M., and Aziz, A., 2012, “A Scale Analysis for Natural Convective Flows Over Vertical Surfaces,” Int. J. Therm. Sci., 54, pp. 82–88. [CrossRef]
Bejan, A., 1984, Convection Heat Transfer, John Wiley and Sons, Inc., New York, pp. 114–122.
Patankar, S. V., 1980, Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Company, New York, pp. 11–109, Chaps. 2–5.
Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P., 1992, Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed., Cambridge University Press, Cambridge, UK, pp. 129–130, Chap. 4.

Figures

Grahic Jump Location
Fig. 1

Flow curves of four related fluids. Curve A—a Newtonian fluid (log10(R*) = 0.00, n = 1.00) with a viscosity of 1000 Ns/m2. Curve B—the power law model with n = 0.50 and K = 5000 Nsn/m2. Curve C—a strongly non-Newtonian, pseudoplastic fluid with log10(R*) = −4.00, n = 0.50, K = 5000 Nsn/m2, and η0 = 1000 Ns/m2 plotted using the EMPL model, Eq. (1). Also shown are the shear rate regimes: I—zero shear rate Newtonian region; II—low shear rate transition region; III—power law region; IV—high shear rate transition region; and V—high shear rate Newtonian region. Curve D—a weakly non-Newtonian, pseudoplastic fluid with the same properties as those in Curve C except with log10(R*) = −1.00.

Grahic Jump Location
Fig. 2

Schematic of the geometry considered in this study, including the definitions of coordinate directions and associated velocity components. The system is of infinite extent in the direction normal to the figure (i.e., in the z-direction, unlabeled). “TBL” and “HBL” indicate the thermal and hydrodynamic boundary layers, respectively, and the schematic of the temperature profile arbitrarily assumes TW > T.

Grahic Jump Location
Fig. 3

NuL for pseudoplastic fluids, upper Pr0 range. The curves in heavy-weight lines are Newtonian fluid (n = 1.00) solutions. Curves in normal-weight lines are pseudoplastic fluid solutions for n = 0.50, 0.60, 0.70, 0.80, and 0.90, proceeding from the upper to the lower curve in each family. A—location where Pr* = Pr0; B—Newtonian fluid curve; C—family of curves for which log10(R*) = −3.00; D—midpoint of the valid log10(β*) range for the family of curves with log10(R*) = −4.00.

Grahic Jump Location
Fig. 4

NuL for pseudoplastic fluids, lower Pr0 range. The curves in heavy-weight lines are Newtonian fluid (n = 1.00) solutions. Curves in normal-weight lines are pseudoplastic fluid solutions for n = 0.50, 0.60, 0.70, 0.80, and 0.90, proceeding from the upper to the lower curve in each family. E—midpoint of the valid log10(β*) range where log10(Pr*) = 2.00 for the curve with log10(R*) = −2.00 and n = 0.50.

Grahic Jump Location
Fig. 5

NuL for dilatant fluids, upper Pr0 range. The curves in heavy-weight lines are Newtonian fluid (n = 1.00) solutions. Curves in normal-weight lines are dilatant fluid solutions for n = 1.50, 1.40, 1.30, 1.20, and 1.10, proceeding from the lower to the upper curve in each family. A—location where Pr* = Pr0; B—Newtonian fluid curve; C—family of curves for which log10(R*) = 3.00; D—midpoint of the valid log10(β*) range for the family of curves with log10(R*) = 4.00.

Grahic Jump Location
Fig. 6

NuL for dilatant fluids, lower Pr0 range. The curves in heavy-weight lines are Newtonian fluid (n = 1.00) solutions. Curves in normal-weight lines are dilatant fluid solutions for n = 1.50, 1.40, 1.30, 1.20, and 1.10, proceeding from the lower to the upper curve in each family.

Grahic Jump Location
Fig. 7

Dimensionless shear rate profiles at the midpoint of the plate, x+ = 0.500, and at the end of the plate, x+ = 1.000, along with the associated dimensionless apparent viscosity from the EMPL model, Eq. (31), and from the power law model. The parameters are those used in the calculation of Point D in Fig. 3 (i.e., n = 0.50, log10(R*) = −4.00, log10(β*) = 2.00, log10(Pr0) = 4.00, and log10(Bo) = 8.00).

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