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Research Papers: Natural and Mixed Convection

# Closed-Form Analytical Solutions for Laminar Natural Convection on Horizontal Plates

[+] Author and Article Information
Abhijit Guha

Professor
e-mail: a.guha@mech.iitkgp.ernet.in

Subho Samanta

Research Student
Mechanical Engineering Department,
Indian Institute of Technology Kharagpur,
Kharagpur 721302, India

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received October 18, 2012; final manuscript received May 6, 2013; published online August 27, 2013. Assoc. Editor: Ali Ebadian.

J. Heat Transfer 135(10), 102501 (Aug 27, 2013) (9 pages) Paper No: HT-12-1580; doi: 10.1115/1.4024430 History: Received October 18, 2012; Revised May 06, 2013

## Abstract

A boundary layer based integral analysis has been performed to investigate laminar natural convection heat transfer characteristics for fluids with arbitrary Prandtl number over a semi-infinite horizontal plate subjected either to a variable wall temperature or variable heat flux. The wall temperature is assumed to vary in the form $T¯w(x¯)-T¯∞=ax¯n$ whereas the heat flux is assumed to vary according to $qw(x¯)=bx¯m$. Analytical closed-form solutions for local and average Nusselt number valid for arbitrary values of Prandtl number and nonuniform heating conditions are mathematically derived here. The effects of various values of Prandtl number and the index n or m on the heat transfer coefficients are presented. The results of the integral analysis compare well with that of previously published similarity theory, numerical computations and experiments. A study is presented on how the choice for velocity and temperature profiles affects the results of the integral theory. The theory has been generalized for arbitrary orders of the polynomials representing the velocity and temperature profiles. The subtle role of Prandtl number in determining the relative thicknesses of the velocity and temperature boundary layers for natural convection is elucidated and contrasted with that in forced convection. It is found that, in natural convection, the two boundary layers are of comparable thickness if Pr ≤ 1 or Pr ≈ 1. It is only when the Prandtl number is large (Pr > 1) that the velocity boundary layer is thicker than the thermal boundary layer.

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## Figures

Fig. 1

Local Nusselt number versus local Grashof number for an isothermal horizontal plate for Pr = 0.7: assessment of the present integral analysis (—— Present integral analysis (λ = 3, χ = 2), + Numerical solution [10], ⋄ Numerical solution [14], ▴ Integro-Differential analysis [22], ● Similarity solution [23], × Experimental correlation [24], □ Experimental correlation [25])

Fig. 2

Local Nusselt number versus modified local Grashof number for a horizontal plate with constant heat flux for Pr = 0.7: assessment of the present integral analysis (—— Present integral analysis (λ = 3, χ = 2 ), + Numerical solution [11], ▵ Integro-Differential analysis [22], • Similarity solution [23])

Fig. 3

Nondimensional velocity and temperature profiles for an isothermal horizontal plate for Pr = 0.01 (—— similarity solution [23]; – – – – third order velocity profile λ = 3 , second order temperature profile χ = 2 ; – · – · – eighth order velocity profile λ = 8, second order temperature profile χ = 2)

Fig. 4

Nondimensional velocity and temperature profiles for an isothermal horizontal plate for Pr = 0.7 (—— similarity solution [23]; – – – – third order velocity profile λ = 3 , second order temperature profile χ = 2 ; – · – · – eighth order velocity profile λ = 8 , second order temperature profile χ = 2)

Fig. 5

Nondimensional velocity and temperature profiles for an isothermal horizontal plate for Pr = 100 (—— similarity solution [23]; – – – – third order velocity profile λ = 3 , second order temperature profile χ = 2 ; – · – · – eighth order velocity profile λ = 8 , second order temperature profile χ = 2)

Fig. 6

Variation in nondimensional heat transfer coefficient with Prandtl number: assessment of the present integral analysis (——— Present integral analysis (λ = 3 , χ = 2 ), ▵ Numerical solution (isothermal plate) [10], ▴ Numerical solution (constant heat flux plate) [11], • Similarity solution (isothermal plate) [23], ○ Similarity solution (constant heat flux plate) [23])

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