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Research Papers: Forced Convection

Solution of the Graetz–Nusselt Problem for Liquid Flow Over Isothermal Parallel Ridges

[+] Author and Article Information
Georgios Karamanis

Department of Mechanical Engineering,
Tufts University,
Medford, MA 02155
e-mail: georgios.karamanis@tufts.edu

Marc Hodes

Department of Mechanical Engineering,
Tufts University,
Medford, MA 02155
e-mail: marc.hodes@tufts.edu

Toby Kirk

Department of Mathematics,
Imperial College London,
London SW7 2AZ, UK
e-mail: toby.kirk12@imperial.ac.uk

Demetrios T. Papageorgiou

Department of Mathematics,
Imperial College London,
London SW7 2AZ, UK
e-mail: d.papageorgiou@imperial.ac.uk

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received August 13, 2016; final manuscript received March 7, 2017; published online May 2, 2017. Assoc. Editor: Jim A. Liburdy.

J. Heat Transfer 139(9), 091702 (May 02, 2017) (12 pages) Paper No: HT-16-1508; 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 flow configurations are analyzed: one plate textured and the other one smooth; both plates textured and the ridges aligned; and both plates textured, but the ridges staggered by half a pitch. The liquid is assumed to be in the Cassie state on the textured surface(s), to which a mixed boundary condition of no-slip on the ridges and no-shear along flat menisci applies. 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 thermal energy equation is subjected to a mixed isothermal-ridge and adiabatic-meniscus boundary condition on the textured surface(s). 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 transverse coordinates, which is solved numerically. Expressions for the local Nusselt number and those averaged over the period of the ridges in the developing and fully developed regions are provided. Nusselt numbers averaged over the period and length of the domain are also provided. Our approach enables the aforementioned quantities to be computed in a small fraction of the time required by a general computational fluid dynamics (CFD) solver.

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Figures

Grahic Jump Location
Fig. 1

Liquid in the Cassie state and the composite interface

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Fig. 2

Schematic of the domain when one plate is textured and the other one is smooth

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Fig. 3

fRe versus ϕ for selected values of H/d when one plate is textured and the other one is smooth

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Fig. 4

Nufd versus ϕ for selected H/d when one plate is textured with isothermal ridges and the other one is smooth and adiabatic

Grahic Jump Location
Fig. 5

Nul,fd versus the normalized coordinate (x̃−1)/(d̃−1) along the ridge for H/d = 10 and selected values of ϕ when one plate is textured with isothermal ridges and the other one is smooth and adiabatic

Grahic Jump Location
Fig. 6

Nufd,ridge versus ϕ for selected H/d when one plate is textured with isothermal ridges and the other one is smooth and adiabatic

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Fig. 7

NuUIT and Nu¯UIT versus z* for selected H/d when one plate is textured with isothermal ridges and the other one is smooth and adiabatic for ϕ=0.01

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Fig. 8

NuUIT and Nu¯UIT versus z* for selected H/d when one plate is textured with isothermal ridges and the other one is smooth and adiabatic for ϕ=0.1

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Fig. 9

fRe versus ϕ for selected H/d when one plate is textured with isothermal ridges and the other one is adiabatic and either smooth (t–s) or textured with aligned ridges (al)

Grahic Jump Location
Fig. 10

Nufd versus ϕ for selected H/d when one plate is textured with isothermal ridges and the other one is adiabatic and either smooth (t–s) or textured with aligned ridges (al)

Grahic Jump Location
Fig. 11

Contour plot of w̃/w̃¯ when one plate is textured with isothermal ridges and the other one is adiabatic and smooth (t–s) for H/d = 4 and ϕ=0.3

Grahic Jump Location
Fig. 12

Contour plot of w̃/w̃¯ when one plate is textured with isothermal and the other one with adiabatic ridges and the ridges are aligned (al) for H/d = 4 and ϕ=0.3

Grahic Jump Location
Fig. 13

Nufd and Nufd,if versus ϕ for selected H/d when both plates are textured with aligned isothermal and isoflux ridges, respectively

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Fig. 14

Schematic of the domain when both plates are textured and the ridges are aligned

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Fig. 15

fRe versus ϕ for selected H/d when both plates are textured and the ridges are aligned

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Fig. 16

Nufd versus ϕ for selected H/d when the ridges of one plate are isothermal and the ridges of the other one are aligned and adiabatic

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Fig. 17

Nufd versus ϕ for selected H/d when the ridges of both plates are isothermal and aligned

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Fig. 18

Schematic of the domain when both plates are textured and the ridges are staggered by half a pitch

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Fig. 19

fRe versus ϕ for selected H/d when both plates are textured and the ridges are staggered by half a pitch

Grahic Jump Location
Fig. 20

Contour plot of w̃/w̃¯ when both plates are textured and the ridges are staggered by half a pitch for H/d = 4 and ϕ=0.3

Grahic Jump Location
Fig. 21

Nufd versus ϕ for selected H/d when the ridges of one plate are isothermal and the ridges of the other one are staggered by half a pitch and adiabatic

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Fig. 22

Nufd versus ϕ for selected H/d when the ridges of both plates are isothermal and staggered by half a pitch

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