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

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.

Application of Particle Image Velocimetry and Computational Fluid Dynamics Methods for Analysis of Natural Convection over a Horizontal Heating Source

“…Natural convective flow occurs in fluids due to density gradients that are in turn developed due to temperature differences in the fluid. Accordingly, free or natural convection has been studied using similarity theory [3], integral theory [4], and by using experiments [5].…”

Magnetohydrodynamic free convection of non-Newtonian power-law fluids over a uniformly heated horizontal plate

“…The natural-convection flows over a semi-infinite horizontal plate subjected to variable heat flux or variable wall temperature have been presented in [5,6]. Guha and Samanta [7] performed an integral analysis to investigate natural-convection heat transfer over a semi-infinite horizontal plate subjected to either a variable wall temperature or variable heat flux.…”

Laminar natural convection of power‐law fluids over a horizontal heated flat plate

“…Samanta and Guha and Dholey utilized the similarity analysis to study the natural convection flow above a semi‐infinite horizontal plate with variable heat flux or variable wall temperature. Guha and Samanta also performed an integral analysis to solve the previously mentioned problem. Archana et al characterized a diffusive convective flow of nanoliquid induced by a horizontal plate under buoyancy force and thermal radiation.…”