Heat transfer in flat-plate boundary layers: a correlation for laminar, transitional, and turbulent flow

The laminar and turbulent regimes of a boundary layer on a flat plate are often represented with separate correlations under the assumption of a distinct “transition Reynolds number.” Average heat coefficients are then calculated by integrating across the “transition point.” Experimental data do not show an abrupt transition, but rather an extended transition region in which turbulence develops. The transition region may be as long as the laminar region. Although this transitional behavior has been known for many decades, few correlations have incorporated it. One attempt was made by Stuart Churchill in 1976. Churchill, however, based his curve fit on some doubtful assumptions about the data sets. In this paper, we develop different approximations through a detailed consideration of multiple data sets for 0.7 ⩽ Pr ⩽ 257, 4,000 ⩽ Rex ⩽ 4,300,000, and varying levels of free stream turbulence for smooth, sharp-edged plates at zero pressure gradient. The result we obtain is in good agreement with the available measurements and applies smoothly over the full range of Reynolds number for either a uniform wall temperature or a uniform heat flux boundary condition. Fully turbulent air data are correlated to ±11%. Like Churchill’s result, this correlation should be matched to the estimated transition condition of any particular flow. We also review the laminar analytical solutions for a uniform wall heat flux, and we point out limitations of the classical Colburn analogy.

J.H. Lienhard V, “Heat transfer in flat-plate boundary layers: a correlation for laminar, transitional, and turbulent flow,” J. Heat Transfer, online 31 March 2020, 142(6):061805, June 2020. (doi: Open access) (presentation) (one-page summary) (DSpace)

Exterior conduction shape factors from interior conduction shape factors

Shape factors for steady heat conduction enable quick and highly simplified calculations of heat transfer rates within bodies having a combination of isothermal and adiabatic boundary conditions. Many shape factors have been tabulated, and most undergraduate heat transfer books cover their derivation and use. However, the analytical determination of shape factors for any but the simplest configurations can quickly come to involve complicated mathematics, and, for that reason, it is desirable to extend the available results as far as possible. In these two papers, we show that known shape factors for the interior of two-dimensional objects are identical to the corresponding shape factors for the exterior of those objects. Rigorous results are obtained using classical potential theory, and the concept of “induced heat flux” is introduced (Lienhard, 2023). Conformal mapping is also applied, with results demonstrated for several configurations of disks, squares, and rectangles (Lienhard, 2019). Both a geometrical and a mathematical argument are introduced to show that shape factors are invariant under conformal mapping. In addition, the “Yin-Yang” phenomenon for conduction shape factors is explained as a rotation of the unit disk prior to conformal mapping.

J.H. Lienhard V, “Steady two-dimensional conduction: simple and double layer potentials, corner singularities, and induced heat flux,” ASME Journal of Heat and Mass Transfer, online 29 September 2022, 145(3):031401, March 2023. (open access) (preprint)

J.H. Lienhard V, “Exterior shape factors from interior shape factors,” J. Heat Transfer, online 20 Feb. 2019, 141(6):061301, June 2019. (OPEN ACCESS) (preprint) (presentation)

Accurate linearization of non-gray radiation heat exchange

The radiation fractional function is the fraction of black body radiation below a given value of λT. Edwards and others have distinguished between the traditional, or “external,” radiation fractional function and an “internal” radiation fractional function. The latter is used for linearization of net radiation from a nongray surface when the temperature of an effectively black environment is not far from the surface’s temperature, without calculating a separate total absorptivity. This paper examines the analytical approximation involved in the internal fractional function, with results given in terms of the incomplete zeta function. A rigorous upper bound on the difference between the external and internal emissivity is obtained. Calculations using the internal emissivity are compared to exact calculations for several models and materials. A new approach to calculating the internal emissivity is developed, yielding vastly improved accuracy over a wide range of temperature differences. The internal fractional function should be used for evaluating radiation thermal resistances, in particular.

J.H. Lienhard V, “Linearization of Non-gray Radiation Exchange: The Internal Fractional Function Reconsidered,” J. Heat Transfer, online 3 Dec. 2018, 141(5):052701, May 2019. (OPEN ACCESS) (preprint) (presentation)