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Research Papers: Natural and Mixed Convection

Laminar Natural Convection Between a Vertical Surface With Uniform Heat Flux and Pseudoplastic and Dilatant Fluids

[+] Author and Article Information
Massimo Capobianchi

Professor
Mem. ASME
Department of Mechanical Engineering,
Gonzaga University,
502 E. Boone Avenue,
Spokane, WA 99258-0026
e-mail: capobianchi@gonzaga.edu

A. Aziz

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

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received October 3, 2013; final manuscript received May 20, 2014; published online June 24, 2014. Assoc. Editor: Terry Simon.

J. Heat Transfer 136(9), 092501 (Jun 24, 2014) (9 pages) Paper No: HT-13-1525; doi: 10.1115/1.4027781 History: Received October 03, 2013; Revised May 20, 2014

This paper reports the average Nusselt number for steady, laminar natural convection between a vertical surface and otherwise quiescent pseudoplastic and dilatant fluids under a constant and uniform surface heat flux boundary condition. Models for the fluids' apparent viscosity were utilized that are valid in all five regions of the flow curve. The results are thus applicable for whatever shear rates may exist within the flow field and a dimensionless shear rate parameter was identified that quantifies the shear rate region where the given system is operating. The data indicate that the average Nusselt numbers approach the corresponding Newtonian values when the shear rates are predominantly in either the zero or the infinite shear rate Newtonian regions. However, power law values are approached only when both of the following two conditions are met: (1) the shear rates are principally in the power law region and (2) the fluid's limiting zero and infinite shear rate Newtonian viscosities differ sufficiently, by approximately 4 orders of magnitude or more. For all other cases, the average Nusselt number was found to reside between the Newtonian and the power law asymptotes. Results are provided in both graphical and tabular form over a broad range of system parameters.

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References

Capobianchi, M., and Aziz, A., 2012, “Laminar Natural Convection From an Isothermal Vertical Surface to Pseudoplastic and Dilatant Fluids,” ASME J. Heat Transfer, 134(12), p. 122502. [CrossRef]
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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., CRC Press, Boca Raton, FL, pp. 4-269–4-278.
Gebhart, B., Jaluria, Y., Mahajan, R. L., and Sammakia, B., 1988, Buoyancy-Induced Flows and Transport, ref. ed., Hemisphere, New York, pp. 42–47, 54–64, 873–876.
Sparrow, E. M., and Gregg, J. L., 1956, “Laminar Free Convection From a Vertical Plate With Uniform Surface Heat Flux,” ASME J. Heat Transfer, 78, pp. 435–440.
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Chen, T. Y.-W., and Wollersheim, D. E., 1973, “Free Convection at a Vertical Plate With Uniform Flux Condition in Non-Newtonian Power-Law Fluids,” ASME J. Heat Transfer, 95(1), pp. 123–124. [CrossRef]
Dale, J. D., and Emery, A. F., 1972, “The Free Convection of Heat From a Vertical Plate to Several Non-Newtonian “Pseudoplastic” Fluids,” ASME J. Heat Transfer, 94(1), pp. 64–72. [CrossRef]
Huang, M. J., and Chen, C. K., 1990, “Local Similarity Solution 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]
Capobianchi, M., 2008, “Pressure Drop Predictions for Laminar Flows of Extended Modified Power Law Fluids in Rectangular Ducts,” Int. J. Heat Mass Transfer, 51(5–6), pp. 1393–1401. [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]
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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, pp. 209–215. [CrossRef]
Brewster, R. A., and Irvine, T. F., Jr., 1987, “Similitude Consideration in Laminar Flow of Modified Power Law Fluids in Circular Ducts,” Heat Mass Transfer, 21, pp. 83–86. [CrossRef]
Patankar, S. V., 1980, Numerical Heat Transfer and Fluid Flow, Hemisphere, New York, pp. 11–109.
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, Cambridge, UK, pp. 129–130.

Figures

Grahic Jump Location
Fig. 1

Shear rate regimes for pseudoplastic and dilatant fluids

Grahic Jump Location
Fig. 2

Schematic of the geometry of the current problem, 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. The heat flux, q·w", and the boundary layers are arbitrarily shown for convection from the surface.

Grahic Jump Location
Fig. 3

NuL results for pseudoplastic fluids. The heavy lines are the solutions for Newtonian fluids, n = 1. The thinner lines are the non-Newtonian results for n = 0.9, 0.8, 0.7, 0.6, and 0.5 ordered sequentially from inner to outer curve in each set.

Grahic Jump Location
Fig. 4

NuL results for pseudoplastic fluids, continued. See Fig. 3 caption for key.

Grahic Jump Location
Fig. 5

NuL results for dilatant fluids. The heavy lines are the solutions for Newtonian fluids, n = 1. The thinner lines are the non-Newtonian results for n = 1.1, 1.2, 1.3, 1.4, and 1.5 ordered sequentially from inner to outer curve in each set.

Grahic Jump Location
Fig. 6

NuL results for dilatant fluids, continued. See Fig. 5 caption for key.

Grahic Jump Location
Fig. 7

u+ profiles at x+= 1 for n = 0.5, 1.0 (Newtonian), and 1.5: |log10(R*)| = 4.00 for n = 0.5 and 1.5, and 0.00 for n = 1.0; log10(Bo) = 7.00; |log10(β*)| = 2.00 for n = 0.5 and 1.5, and 0.00 for n = 1.0; and log10(Pr0) = 1.00 for n = 0.5, −1.00 for n = 1, and −3.00 for n = 1.5 so that log10(Pr*) = −1.00 for all cases.

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