### Abstract

In this paper, we consider the heat transfer from a periodic array of isothermal pipes embedded in a rectangular slab. The upper surface of the slab is sustained at a constant temperature while the lower surface is insulated. The particular configuration is a classical heat conduction problem with a wide range of practical applications. We consider both the classical problem, i.e., estimating the shape factor of a given configuration, and the inverse problem, i.e., calculating the optimum shape that maximizes the heat transfer rate associated with a set of geometrical constraints. The way the present formulation differs from previous formulations is that: (i) the array of pipes does not have to be placed at the midsection of the slab and (ii) we have included an isoperimetric constraint (not changing in perimeter) through which we can control the deviation of the optimum shape from that of a circle. This is very important considering that most of the applications deal with buried pipes and a realistic shape is a practical necessity. The isoperimetric constraint is included through the isoperimetric quotient (IQ), which is the ratio between the area and the perimeter of a closed curve.

Original language | English |
---|---|

Article number | 094502 |

Journal | Journal of Heat Transfer |

Volume | 136 |

Issue number | 9 |

DOIs | |

Publication status | Published - 2014 |

Externally published | Yes |

### Fingerprint

### Keywords

- Generalized Schwarz-Christoffel transformation
- Heat conduction
- Isoperimetric quotient
- Laplace equation
- Shape factor
- Shape optimization
- Solid slab with periodic array of pipes/strips

### ASJC Scopus subject areas

- Mechanical Engineering
- Mechanics of Materials
- Materials Science(all)
- Condensed Matter Physics

### Cite this

*Journal of Heat Transfer*,

*136*(9), [094502]. https://doi.org/10.1115/1.4027780

**Shape optimization with isoperimetric constraints for isothermal pipes embedded in an insulated slab.** / Leontiou, Theodoros; Fyrillas, Marios M.

Research output: Contribution to journal › Article

*Journal of Heat Transfer*, vol. 136, no. 9, 094502. https://doi.org/10.1115/1.4027780

}

TY - JOUR

T1 - Shape optimization with isoperimetric constraints for isothermal pipes embedded in an insulated slab

AU - Leontiou, Theodoros

AU - Fyrillas, Marios M.

PY - 2014

Y1 - 2014

N2 - In this paper, we consider the heat transfer from a periodic array of isothermal pipes embedded in a rectangular slab. The upper surface of the slab is sustained at a constant temperature while the lower surface is insulated. The particular configuration is a classical heat conduction problem with a wide range of practical applications. We consider both the classical problem, i.e., estimating the shape factor of a given configuration, and the inverse problem, i.e., calculating the optimum shape that maximizes the heat transfer rate associated with a set of geometrical constraints. The way the present formulation differs from previous formulations is that: (i) the array of pipes does not have to be placed at the midsection of the slab and (ii) we have included an isoperimetric constraint (not changing in perimeter) through which we can control the deviation of the optimum shape from that of a circle. This is very important considering that most of the applications deal with buried pipes and a realistic shape is a practical necessity. The isoperimetric constraint is included through the isoperimetric quotient (IQ), which is the ratio between the area and the perimeter of a closed curve.

AB - In this paper, we consider the heat transfer from a periodic array of isothermal pipes embedded in a rectangular slab. The upper surface of the slab is sustained at a constant temperature while the lower surface is insulated. The particular configuration is a classical heat conduction problem with a wide range of practical applications. We consider both the classical problem, i.e., estimating the shape factor of a given configuration, and the inverse problem, i.e., calculating the optimum shape that maximizes the heat transfer rate associated with a set of geometrical constraints. The way the present formulation differs from previous formulations is that: (i) the array of pipes does not have to be placed at the midsection of the slab and (ii) we have included an isoperimetric constraint (not changing in perimeter) through which we can control the deviation of the optimum shape from that of a circle. This is very important considering that most of the applications deal with buried pipes and a realistic shape is a practical necessity. The isoperimetric constraint is included through the isoperimetric quotient (IQ), which is the ratio between the area and the perimeter of a closed curve.

KW - Generalized Schwarz-Christoffel transformation

KW - Heat conduction

KW - Isoperimetric quotient

KW - Laplace equation

KW - Shape factor

KW - Shape optimization

KW - Solid slab with periodic array of pipes/strips

UR - http://www.scopus.com/inward/record.url?scp=84902665712&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84902665712&partnerID=8YFLogxK

U2 - 10.1115/1.4027780

DO - 10.1115/1.4027780

M3 - Article

VL - 136

JO - Journal of Heat Transfer

JF - Journal of Heat Transfer

SN - 0022-1481

IS - 9

M1 - 094502

ER -