^{1}

Different high-performance fins are mathematically analyzed in this work. Initially, three types are considered: (i) exponential, (ii) parabolic, and (iii) triangular fins. Analytical solutions are obtained. Accordingly, the effective thermal efficiency and the effective volumetric heat dissipation rate are calculated. The analytical results were validated against numerical solutions. It is found that the triangular fin has the maximum effective thermal length. In addition, the exponential pin fin is found to have the largest effective thermal efficiency. However, the effective efficiency for the straight one is the maximum when its effective thermal length based on profile area is greater than 1.4. Furthermore, the exponential straight fin is found to have effective volumetric heat dissipation that can be 440% and 580% above the parabolic and triangular straight fins, respectively. In contrast, the exponential pin fin is found to possess effective volumetric heat dissipation that can be 120% and 132% above the parabolic and triangular pin fins, respectively. Finally, new high performance fins are mathematically generated that can have effective volumetric heat dissipation of 24% and 12% above those of exponential pin and straight fins, respectively.

Fins are widely used in industry, especially in heat exchanger and refrigeration industries [

Later on, Schmidt [

The fin thermal efficiency,

In this work, high-performance fins with effective thermal lengths are mathematically analyzed. Three types are initially considered: (i) exponential, (ii) parabolic, and (iii) triangular fins. Analytical forms for the excess temperature are obtained. As such, the fin effective thermal efficiency and the effective volumetric heat dissipation are calculated both analytically and numerically. Comparisons between the performances of each fin are performed. Finally, ultrahigh performance fin geometries are extrapolated from the derived solutions.

In this work, Murray [

Consider a rectangular fin having a thickness

Schematic diagram for the straight fin and the system coordinate.

The application of the energy equation [

By solving (

Equations (

Equations (

Consider a pin fin having a radius

Schematic diagram for the pin fin and the system coordinate.

The application of the energy equation [

By solving (

Equations (

Equations (

The variation of the high-performance fin profile with its

The variation of the high-performance fin profile with its

Equations (

Comparisons between the numerical and the analytical results.

Fin type | Thermal | Numerical | Analytical | Percent difference |
---|---|---|---|---|

Straight-triangular | 0.4703 | 0.4704 | 0.02% | |

Straight-parabolic | 0.5457 | 0.5459 | 0.04% | |

Straight-exponential | 0.4414 | 0.4419 | 0.1% | |

Pin-triangular | 0.7753 | 0.7756 | 0.04% | |

Pin-parabolic | 0.9174 | 0.9178 | 0.04% | |

Pin-exponential | 0.5763 | 0.5764 | 0.02% |

The results generated by solving (_{1}–g_{12} for the different studied cases are listed in Table

Correlation constants, see (

Coefficient | Triangular | Triangular | Exponential | Triangular | Triangular | Exponential |
---|---|---|---|---|---|---|

_{1} | 0.69428 | 598.742 | 3.8343 | 1.46608 | 7.27154 | 0.931451 |

_{2} | −1.98283 | −7.1912 | −0.894435 | 0.117199 | 0.311742 | −0.514116 |

_{ 3} | 0.198954 | 12.3343 | 7.0634 | 4.2461 | −0.546715 | 4.46038 |

_{ 4} | −1.76853 | −2.80184 | −0.274186 | −1.98447 | 4.10486 | 0.513971 |

_{ 5} | 0.167474 | 2.12319 | 0.208524 | −0.578418 | 0.40409 | −5.04956 |

_{ 6} | 1.06061 | 0.746752 | −0.131276 | 0.116989 | 4.33523 | 0.523092 |

_{7} | 0.886024 | 588.714 | 0.704077 | 0.973387 | −0.389977 | 0.119439 |

_{8} | −2.9514 | −8.19517 | −1.7562 | −1.29981 | 2.18417 | −0.367153 |

_{9} | 0.141659 | 3.82157 | 3.17823 | 4.63058 | 4.673 | 0.105864 |

_{10} | 0.623454 | −1.09171 | −1.10017 | −2.99168 | −0.699218 | −1.22631 |

_{11} | 0.0680181 | −0.0615233 | 2.84166 | 0.33941 | 1.77476 | 0.000821297 |

_{12} | 1.04416 | 3.2981 | −0.273702 | 0.116343 | −0.695973 | 0.483324 |

Figures

Effects of the fin dimensionless length

Effects of the fin dimensionless length

Figure

Effect of the fin effective thermal length based on profile area on the fin effective thermal efficiency

Effect of the fin minimum effective thermal

Exponential fins have the largest effective heat dissipation per unit volume,

Effect of the fin minimum effective thermal length

Effect of the fin minimum effective thermal length

New high-performance straight fins profiles,

New high-performance pin fins profiles,

Heat transfer through high-performance fins was mathematically analyzed under conditions that lead to useful thermal lengths. Three fin types were considered: parabolic, triangular, and exponential straight or pin fins. Analytical solutions were obtained. The effective thermal length was obtained for each case. Accordingly, the effective thermal efficiency and the effective heat dissipation per unit volume were calculated. The analytical results were compared against numerical solutions and excellent agreements were found. The following remarks were concluded:

Triangular fins have always-larger effective thermal lengths than parabolic fins.

Exponential pin fins possess the largest effective thermal efficiencies.

The exponential straight fin possesses the maximum effective thermal efficiency when its effective thermal length based on profile area is greater than 1.4.

The triangular straight fin has the maximum effective thermal efficiency when its effective thermal length based on profile area is smaller than 0.95.

Exponential straight fins were found to possess effective volumetric heat dissipation that can be 440% and 580% above parabolic and triangular straight fins.

Exponential pin fins were found to possess effective volumetric heat dissipation that can be 120% and 132% above parabolic and triangular pin fins.

The derived analytical solutions were used to generate new high-performance fins that possess volumetric heat dissipation 24% and 12% above those of exponential pin and straight fins, respectively.

Effective fin profile area

Exponential functions indices

Fin thickness

Tin thickness at its base

Convection heat transfer coefficient

Modified Bessel functions of the first kind of order

Modified Bessel functions of the second kind of order

Fin thermal conductivity

Fin length

Effective fin length

Fin thermal index

Fin heat transfer rate per unit width

Maximum fin heat transfer rate per unit width

Fin radius

Fin radius at its base

Fin temperature

Fin base temperature

Free stream temperature of the adjoining fluid

Fin volume

Dimensionless exponential fin parameter

Coordinate axis along the fin centreline.

Fin effective dimensionless volumetric heat dissipation

Fin thermal efficiency

Fin effective thermal efficiency.

The support of this work by King Abdulaziz City for Science and Technology (KACST) under Project no. 8-ENE192-3 is acknowledged.