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# Journal of Non-Equilibrium Thermodynamics

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# Geometry-Based Entropy Generation Minimization in Laminar Internal Convective Micro-Flow

Pallavi Rastogi
• Research Scholar, Department of Aerospace Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India
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• Professor, Department of Aerospace Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India
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Published Online: 2018-11-15 | DOI: https://doi.org/10.1515/jnet-2018-0036

## Abstract

In this theoretical study, fully developed forced convective laminar water flow is considered in circular micro-tubes, for the constant wall heat flux boundary condition. The change in entropy generation rate ($\mathrm{\Delta }{\stackrel{˙}{\mathit{S}}}_{\mathrm{gen}}$) for N micro-tubes (each of diameter ${\mathit{D}}_{\mathrm{N}}$) relative to a reference tube (of 1 mm diameter) was investigated towards the micro-scale, for different tube length (l). A given total heat flow rate is to be removed using a fixed total mass flow rate through N tubes. Hence, the wall heat flux for one of the N tubes decreases towards the micro-scale, which is “thermal under-loading”. For given l, $\mathrm{\Delta }{\stackrel{˙}{\mathit{S}}}_{\mathrm{gen}}$ due to fluid conduction decreases and $\mathrm{\Delta }{\stackrel{˙}{\mathit{S}}}_{\mathrm{gen}}$ due to fluid friction increases towards the micro-scale. There exists an optimum ${\mathit{D}}_{\mathrm{N}}$ ($={\mathit{D}}_{\mathrm{N}\mathrm{,}\mathrm{opt}}$) at which the change in sum-total ${\stackrel{˙}{\mathit{S}}}_{\mathrm{gen}}$ ($\mathrm{\Delta }{\stackrel{˙}{\mathit{S}}}_{\mathrm{gen}\mathrm{,}\mathrm{tot}}$) is minimum; where ${\mathit{D}}_{\mathrm{N}\mathrm{,}\mathrm{opt}}$ decreases with decreasing l. For given l, cooling capacity of the heat sink increases towards the micro-scale. A general criterion for minimization of $\mathrm{\Delta }{\stackrel{˙}{\mathit{S}}}_{\mathrm{gen}\mathrm{,}\mathrm{tot}}$ is obtained in terms of Reynolds number, Brinkman number, and dimensionless l.

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Revised: 2018-08-27

Accepted: 2018-09-27

Published Online: 2018-11-15

Funding Source: Ministry of Human Resource Development

Award identifier / Grant number: 154010008

Funding Source: Alexander von Humboldt-Stiftung

Award identifier / Grant number: 1104249

The authors thank the Ministry of Human Resource Development, Govt. of India, for the financial support to P. Rastogi (roll no. 154010008 at IIT-Bombay) for pursuing this research. The authors are grateful to the A. von Humboldt Foundation, Germany, for the rich exposure to research methodology through sponsorship no. 1104249 to S. P. Mahulikar.

Citation Information: Journal of Non-Equilibrium Thermodynamics, ISSN (Online) 1437-4358, ISSN (Print) 0340-0204,

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