Conduction|Convection|Radiation
Symmetry criteria for equality of interior and exterior shape factors with exact solutions
In this paper, we show that specific symmetries must be present for interior conduction shape factors to equal exterior shape factors. We deduce a simple criterion which, when satisfied, ensures the equality of interior and exterior shape factors in general. Our criterion notably relies on a beautiful and little-known symmetry method due to Hersch which we introduce in a tutorial manner. In addition, we derive a new formula for the shape factor of objects meeting our N-fold symmetry criterion, encompassing exact solutions for regular polygons and more complex shapes. A number of exact shape factors are tabulated.
K.I. McKee and J.H. Lienhard V, “Symmetry criteria for the equality of interior and exterior shape factors with exact solutions,” ASME J. Heat Mass Transfer, online 4 July 2024, 146(11):111401, Nov. 2024. (open access) (Graphical Abstract) (arXiv:2403.19030) (high rez preprint)
Heat diffusion during thin-film composite membrane formation
hin-film composite (TFC) membranes, the backbone of modern reverse osmosis and nanofiltration, combine the high separation performance of a thin selective layer with the robust mechanical support. Previous studies have shown that heat released during interfacial polymerization (IP) can have a significant impact on the physical and chemical structure of the selective layer. In this study, we develop a multilayer transient heat conduction model to analyze how the thermal properties of the materials used in TFC fabrication impact interfacial temperature, focusing on support-free (SFIP), conventional (CIP), and interlayer-modulated IP (IMIP). Using a combination of analytic solutions and computational models, we demonstrate that the thermal effusivities of fluid and material layers can have a significant effect on the temporal evolution of interfacial temperature during IP. In CIP, we show that the presence of a polymeric support adjacent to the reaction interface yields a 20% to 60% increase in interfacial temperature rise, lasting for ∼ 0.1 s. Furthermore, we demonstrate that inorganic or metallic interlayers, which have high thermal effusivities, can lead to short-lived order-of-magnitude reductions in interfacial temperature rise. Finally, we provide analytical approximations for transient heat conduction through multilayered systems, enabling rapid evaluation of the thermal impact of novel membrane support and interlayer materials and structures on interfacial temperature during TFC fabrication. Quantifying how the thermal properties of solvents, support layers, and interlayers affect interfacial temperature during IP is critical for the rational design of new TFC membranes.
A. Deshmukh, J.H. Lienhard, and M. Elimelech, “Heat Diffusion During Thin-Film Composite Membrane Formation,”J. Membrane Science,696:122493, March 2024. Editor’s Choice Article for March 2024. There are still fun things to do with classical heat conduction!
On the Nusselt number for thermally developed flow between isothermal parallel plates with dissipation
This paper revisits the effect of dissipation and flow work on the Nusselt number for laminar flow between parallel plates. The aim is to refute a recently published claim that the entire literature is in error. I show that the commonly reported value of the thermally developed Nusselt number, 7.541, is quite acceptable for commonly encountered situations. In particular, for ideal gases, the wall heat flux is predicted exactly without accounting for dissipation and flow work because they cancel one another. For liquids, I derive the channel length within which dissipation makes a negligible contribution to heat flux. This length will often span the entire range within which the bulk temperature changes in response to a wall temperature change. The residual bulk temperature rise from dissipation can amount to only millikelvin. The Nusselt number following a change in wall temperature should be calculated after separating the temperature rise and heat flux caused by dissipation. Failure to do so gives a Nusselt number that can be zero, negative, or singular. The effects of dissipation on flux and temperature can be added to the ordinary Graetz solution if they are not negligible. The present results show that the Seban–Shimazaki criterion for thermally developed flow is misleading when dissipation is considered. Instead, the flow may be called thermally developed when the Graetz series is well approximated by its first term.
J. H. Lienhard, “On the Nusselt number for thermally developed flow between isothermal parallel plates with dissipation,” ASME J. Heat Mass Transfer, online 17 June 2025, 147(11):111801, Nov. 2025, (open access) (presentation) (DSPACE)
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)
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)