# NUMERICAL STUDY FOR A THREE DIMENSIONAL LAMINAR NATURAL CONVECTION HEAT TRANSFER FROM AN ISOTHERMAL HEATED HORIZONTAL AND INCLINED SQUARE PLATE AND WITH A CIRCULAR HOLE

## Abstract

A theoretical study for a three-dimensional natural convection heat transfer from an isothermal horizontal , vertical and inclined heated square flat plates (with and without circular hole) has been done in the present work. The study involved the numerical solution of the transient Navier-Stokes and energy equations by using finite deference method (F.D.M.). The complete Navier-Stokes
equation are transformed and expressed in terms of vorticity-vector potential. The Energy and Vorticity equations were solved by using an Alternating Direction Implicit (ADI) method because they are transient equations of parabolic portion, and the Vector potential is solved by using an equations Successive Over-Relaxation (S.O.R) method because it is from elliptic portion. The
numerical solution is capable of calculating the Vector potential, three components of Vorticity and temperature field of the calculation domain. The numerical results were obtained in rang of Grashof number (103≤Gr≤5x104 ) with Prandtl number of (0.72) for square flat plate and the other consist a circle hole with ratio 0.6 and 0.8 diameter of the hole to main square side length. The numerical results showed that the main process of heat transfer is conduction for Grashof number less than 103 and convection for Grashof number larger than 103 and the results of local Nusselt number show fairly large dependence on inclination angle. For horizontal plate facing upward and downward, average Nusselt number is proportional to one-fifth power of Rayleigh number, and there is a significant difference in heat transfer rates between the upward and downward cases. For horizontal plate with circle hole facing upward for Grashof number 104 , the effect of core portion caused a limited increment in the heat transfer rate, where as for the facing downward case, the effect was larger and the maximum value of heat transfer rates is be for square flat plate with circle hole by ratio 0.6 for all inclination angles. With the increase of Grashof number to 5x104 heat transfer rates decrease except the square horizontal flat plate with circle hole by ratio 0.6 . The average Nusselt number increases with the increase of inclination of plates facing upward to reach to the higher average Nusselt number at vertical position then decrease with increase of inclination of plates. And the maximum value of average Nusselt number is depended on the ratio of diameter of the hole to main square side length, showed that the maximum temperature gradient occurs at the external edge of the horizontal plate (with and without circle hole) facing upward and at the lower external edge in inclined case. The numerical results was made through comparison with a previous numerical and experimental work, the agreement was good.

## Article Details

How to Cite
“NUMERICAL STUDY FOR A THREE DIMENSIONAL LAMINAR NATURAL CONVECTION HEAT TRANSFER FROM AN ISOTHERMAL HEATED HORIZONTAL AND INCLINED SQUARE PLATE AND WITH A CIRCULAR HOLE” (2024) Journal of Engineering, 13(02), pp. 1474–1484. doi:10.31026/j.eng.2007.02.11.
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Articles

## How to Cite

“NUMERICAL STUDY FOR A THREE DIMENSIONAL LAMINAR NATURAL CONVECTION HEAT TRANSFER FROM AN ISOTHERMAL HEATED HORIZONTAL AND INCLINED SQUARE PLATE AND WITH A CIRCULAR HOLE” (2024) Journal of Engineering, 13(02), pp. 1474–1484. doi:10.31026/j.eng.2007.02.11.

## References

Ahmad W.Mustafa, 2001, "Numerical and Experimental Study of Natural Convection Heat Transfer from Isothermal Horizontal Disks and Rings". M. Sc. Thesis Univ. Technology.

Ayad K.Hassan , 2003,"Prediction of Three Dimensional Natural Convection from Heated Disks and Rings at Constant Temperature ". J. Eng. & Technology. Vol. 22, No.5, PP. 229-248 .

C.F.Gerald 1970. "Applied Numerical Analysis" Addison-Wesley Publishing Company.

Chuen-Yen Chow, Jone Wilely & Sons, 1979,"An Introduction to Computational Fluid Mechanics.” F.Geoola&A.R.H.Cornish.1982"Numerical Simulation of Free Connective Heat Transfer From a Sphere" Int. J.Heat&Mass Transfer. Vol. 25, No. 11, PP. 1677-1687.

Francis J.Suriano & Kwang-Tzu Yang ,1968 ,"Laminar Free Convection About Vertical And Horizontal Plates At Small And Moderate Grashof Numbers". Int. J.Heat & Mass Transfer.Vol. 11, PP.473-490.

K. Aziz & J.D.Hallums, 1967,"Numerical Solution of The Three-Dimensional Equations of Motion for Laminar Natural Convection ". The physics of fluids .Vol. 10, No. 2 , PP.314-325.

K.E.Torrance ,1985,"Numerical Method In Heat Transfer". Handbook of Heat Transfer Fundamentals, McGraw-Hill, 2nd Edition.

Luciano Pera & Benjamin Gebhart , 1973,"Natural Convection Boundary Layer Flow Over Horizontal And Slightly Inclined Surfaces".Int.J.Heat & Mass Transfer. Vol. 16, PP. 1131-1145.

L.R.Cairnie & A.J.Harrison,1982,"Natural Convection Adjacent To a Vertical Isothermal Hot Plate with a High Surface-To-Ambient Temperature Difference ".Int. J.Heat & Mass Transfer. Vol. 25, No. 7, PP. 925-934.

R.F.Boehm&D.Kamyab. May,1977,"Established Stripwise Laminar Natural Convection on Horizontal Surfaces".Transactions of the Asme. Vol. 99, PP. 294-299.

R.J.Goldstein & Kei-Shun Lau,1983,"Laminar Natural Convection From A Horizontal Plate And The Influence of Plate-Edge Extensions ". J.Fluid Mech. Vol.129 , PP.55-75.

Rafah Aziz , 2002,"Instructional System To Study Free Convection Heat Transfer from Isothermal Horizontal Square Flat Surfaces ".M. Sc.Thesis Univ. Technology.