In many applications, convection heat transfer is coupled with conduction and radiation heat transfer, which generate temperature gradients along the walls and may greatly affect natural convection heat transfer. The main objective of this study is to calculate the heat-transfer characteristics for natural convection from a non-isothermal vertical at plate into a supercritical fluid. The influence of the non-uniformity of wall temperature on the heat transfer by natural convection along a vertical plate, having a linearly distributed temperature (characterized by the slope S) is also investigated. The thermal expansion coefficient is considered as a function of the temperature, the pressure, the van der Waals constants and the compressibility factor. The trends of the curves obtained with this equation and with values from tables of thermodynamic properties were similar and diverged at a critical point. These features confirmed the validity of this equation. Then, the governing systems of partial differential equations are solved numerically using the finite difference method. The local Nusselt number was then calculated and plotted as a function of the local Rayleigh number. It was observed that a positive slope of temperature distribution increases the heat transfer rate and a negative slope decreases it.

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