Solution of the Graetz-Nusselt Problem for Liquid Flow Over Isothermal Parallel Ridge

[+] Author and Article Information
Georgios Karamanis

Department of Mechanical Engineering Tufts University Medford, MA 02155

Marc Hodes

Department of Mechanical Engineering Tufts University Medford, MA 02155

Toby Kirk

Department of Mathematics Imperial College London London, UK

Demetrios T. Papageorgiou

Department of Mathematics Imperial College London London, UK

1Corresponding author.

ASME doi:10.1115/1.4036281 History: Received August 13, 2016; Revised March 07, 2017


We consider convective heat transfer for laminar flow of liquid between parallel plates that are textured with isothermal ridges oriented parallel to the flow. Three different configurations are analyzed: one plate textured and the other one smooth; both plates textured and the ridges aligned symmetrically; and both plates textured but the ridges staggered by half a pitch. Heat is exchanged with the liquid either through the ridges of one plate with the other plate adiabatic, or through the ridges of both plates. The liquid is assumed to be in the Cassie state on the textured surfaces, to which a mixed boundary condition of no-slip on the ridges and no-shear along flat menisci applies. The thermal energy equation is subjected to a mixed isothermal-ridge and adiabatic-meniscus boundary condition on the textured surfaces of the non-adiabatic plates. Axial conduction is neglected and the inlet temperature profile is arbitrary. We solve for the three-dimensional developing temperature profile assuming a hydrodynamically-developed flow, i.e., we consider the Graetz-Nusselt problem. Using the method of separation of variables, the thermal problem is essentially reduced to a two-dimensional eigenvalue problem in the spanwise coordinates, which must be solved numerically. Expressions are found for the local, average and fully developed Nusselt number in terms of the eigenfunctions, eigenvalues and inlet temperature profile. The present analysis allows the aforementioned quantities to be computed in a small fraction of the time required by a general computational fluid dynamics solver.

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