Citation: Jaffer, A. Natural Convection Heat Transfer From an Isothermal Plate. Thermo 2023, 3, 148–175. https://doi.org/10.3390/ thermo3010010 Academic Editor: Johan Jacquemin Received: 22 November 2022 Revised: 16 January 2023 Accepted: 29 January 2023 Published: 3 February 2023 Copyright: © 2023 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Article Natural Convection Heat Transfer From an Isothermal Plate Aubrey Jaffer Independent Researcher, Waltham, MA 02452, USA; agj@alum.mit.edu Abstract: Using boundary-layer theory, natural convection heat transfer formulas that are accurate over a wide range of Rayleigh numbers (Ra) were developed in the 1970s and 1980s for vertical and downward-facing plates. A comprehensive formula for upward-facing plates remained unsolved because they do not form conventional boundary-layers. From the thermodynamic constraints on heat-engine efficiency, the novel approach presented here derives formulas for natural convection heat transfer from isothermal plates. The union of four peer-reviewed data-sets spanning 1 < Ra < 10 12 has 5.4% root-mean-squared relative error (RMSRE) from the new upward-facing heat transfer formula. Applied to downward-facing plates, this novel approach outperforms the Schulenberg (1985) formula’s 4.6% RMSRE with 3.8% on four peer-reviewed data-sets spanning 10 6 < Ra < 10 12 . The introduction of the harmonic mean as the characteristic length metric for vertical and downward- facing plates extends those rectangular plate formulas to other convex shapes, achieving 3.8% RMSRE on vertical disk convection from Hassani and Hollands (1987) and 3.2% from Kobus and Wedekind (1995). Keywords: natural convection; heat engine; Carnot efficiency 1. Introduction Natural convection is the flow caused by nonuniform density in a fluid. It is a fundamental process with applications from engineering to geophysics. When a stationary, immersed object changes temperature, nearby fluid can change density as it warms or cools. Under the influence of gravity, density changes cause fluid to flow. The rates of fluid flow and heat transfer from the object grow until reaching a plateau. This investigation seeks to predict the overall steady-state heat transfer rate from an external, flat, isothermal surface inclined at any angle in a Newtonian fluid. An “external” plate is one that fluid can flow around freely, especially horizontally. If enclosed, the enclosure must have dimensions much larger than the heated or cooled surface. Natural convection in an enclosure of size comparable to the heated or cooled surface can organize into cells of Rayleigh-Bénard convection, which is not treated here. The characteristic length L is the length scale of a physical system. For many heat transfer processes, it is the volume-to-surface-area or area-to-perimeter ratio of the heated or cooled object. There are several characteristic length metrics used for natural convection, some of which are valid only for convex objects. This investigation focuses on flat plates with convex perimeters. 1.1. Flow Topologies There are three topologies of convective flow from external, convex plates. For a horizontal plate with a heated upper face, streamline photographs in Fujii and Imura [1] show natural convection pulling fluid horizontally from above the plate’s perimeter into a rising central plume. Figure 1a, below, is a diagram of this upward-facing convection. Horizontal flow is nearly absent at the elevation of the dashed line. Kitamura, Mitsuishi, Suzuki, and Kimura [2] show top-views of plumes from heated rectangular plates with aspect ratios between 1:1 and 8:1. The plates with high aspect ratios have a plume over the plate’s mid-line parallel to the longer sides, but not as long. Thermo 2023, 3, 148–175. https://doi.org/10.3390/thermo3010010 https://www.mdpi.com/journal/thermo