Explicit, rigorous solutions to two-dimensional heat transfer: Two-component media and optimization of cooling fins

A. R. Kacimov, Yu V. Obnosov

Research output: Contribution to journalArticle

18 Citations (Scopus)

Abstract

New analytical solutions to the problem of steady heat conduction from the wall with longitudinal fins to the environment are derived. Within the two media two temperature fields are harmonic functions with rigorous conjugation of temperature and normal flux along the interface between the two components. First, for high values of the ratio e = k1/k2, with k1 and k2 being thermal conductivities of the grooved wall and environment, respectively, we derive the optimal fin contour providing extreme heat flux (total heat dissipation) from the fin surface at prescribed fin cross-sectional area. This optimizer is found in the class of arbitrary curves and both necessary and sufficient extremum conditions are satisfied. The extreme line coincides with the contour of constant hydraulic gradient calculated by Polubarinova-Kochina for a seepage flow under a concrete dam. At arbitrary e the same isoperimetric problem is solved in the class of elliptic fins assuming fin spacing large enough to consider an isolated profile. Two non-trivial local extrema exist depending on e. For arbitrary e the case of long rectangular fins with arbitrary direction of the outer field is studied. Streamline refraction illustrates non-trivial fluxes near the finger tips and roots.

Original languageEnglish
Pages (from-to)1191-1196
Number of pages6
JournalInternational Journal of Heat and Mass Transfer
Volume40
Issue number5
Publication statusPublished - Mar 1997

Fingerprint

cooling fins
Fins (heat exchange)
fins
heat transfer
Fluxes
Heat transfer
Cooling
Harmonic functions
Concrete dams
optimization
Seepage
Refraction
Heat losses
Heat conduction
Heat flux
Enthalpy
Thermal conductivity
Temperature distribution
Hydraulics
range (extremes)

ASJC Scopus subject areas

  • Fluid Flow and Transfer Processes
  • Energy(all)
  • Mechanical Engineering

Cite this

@article{06c9ef59578c47d7b52306e37f554d52,
title = "Explicit, rigorous solutions to two-dimensional heat transfer: Two-component media and optimization of cooling fins",
abstract = "New analytical solutions to the problem of steady heat conduction from the wall with longitudinal fins to the environment are derived. Within the two media two temperature fields are harmonic functions with rigorous conjugation of temperature and normal flux along the interface between the two components. First, for high values of the ratio e = k1/k2, with k1 and k2 being thermal conductivities of the grooved wall and environment, respectively, we derive the optimal fin contour providing extreme heat flux (total heat dissipation) from the fin surface at prescribed fin cross-sectional area. This optimizer is found in the class of arbitrary curves and both necessary and sufficient extremum conditions are satisfied. The extreme line coincides with the contour of constant hydraulic gradient calculated by Polubarinova-Kochina for a seepage flow under a concrete dam. At arbitrary e the same isoperimetric problem is solved in the class of elliptic fins assuming fin spacing large enough to consider an isolated profile. Two non-trivial local extrema exist depending on e. For arbitrary e the case of long rectangular fins with arbitrary direction of the outer field is studied. Streamline refraction illustrates non-trivial fluxes near the finger tips and roots.",
author = "Kacimov, {A. R.} and Obnosov, {Yu V.}",
year = "1997",
month = "3",
language = "English",
volume = "40",
pages = "1191--1196",
journal = "International Journal of Heat and Mass Transfer",
issn = "0017-9310",
publisher = "Elsevier Limited",
number = "5",

}

TY - JOUR

T1 - Explicit, rigorous solutions to two-dimensional heat transfer

T2 - Two-component media and optimization of cooling fins

AU - Kacimov, A. R.

AU - Obnosov, Yu V.

PY - 1997/3

Y1 - 1997/3

N2 - New analytical solutions to the problem of steady heat conduction from the wall with longitudinal fins to the environment are derived. Within the two media two temperature fields are harmonic functions with rigorous conjugation of temperature and normal flux along the interface between the two components. First, for high values of the ratio e = k1/k2, with k1 and k2 being thermal conductivities of the grooved wall and environment, respectively, we derive the optimal fin contour providing extreme heat flux (total heat dissipation) from the fin surface at prescribed fin cross-sectional area. This optimizer is found in the class of arbitrary curves and both necessary and sufficient extremum conditions are satisfied. The extreme line coincides with the contour of constant hydraulic gradient calculated by Polubarinova-Kochina for a seepage flow under a concrete dam. At arbitrary e the same isoperimetric problem is solved in the class of elliptic fins assuming fin spacing large enough to consider an isolated profile. Two non-trivial local extrema exist depending on e. For arbitrary e the case of long rectangular fins with arbitrary direction of the outer field is studied. Streamline refraction illustrates non-trivial fluxes near the finger tips and roots.

AB - New analytical solutions to the problem of steady heat conduction from the wall with longitudinal fins to the environment are derived. Within the two media two temperature fields are harmonic functions with rigorous conjugation of temperature and normal flux along the interface between the two components. First, for high values of the ratio e = k1/k2, with k1 and k2 being thermal conductivities of the grooved wall and environment, respectively, we derive the optimal fin contour providing extreme heat flux (total heat dissipation) from the fin surface at prescribed fin cross-sectional area. This optimizer is found in the class of arbitrary curves and both necessary and sufficient extremum conditions are satisfied. The extreme line coincides with the contour of constant hydraulic gradient calculated by Polubarinova-Kochina for a seepage flow under a concrete dam. At arbitrary e the same isoperimetric problem is solved in the class of elliptic fins assuming fin spacing large enough to consider an isolated profile. Two non-trivial local extrema exist depending on e. For arbitrary e the case of long rectangular fins with arbitrary direction of the outer field is studied. Streamline refraction illustrates non-trivial fluxes near the finger tips and roots.

UR - http://www.scopus.com/inward/record.url?scp=0031106490&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0031106490&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0031106490

VL - 40

SP - 1191

EP - 1196

JO - International Journal of Heat and Mass Transfer

JF - International Journal of Heat and Mass Transfer

SN - 0017-9310

IS - 5

ER -