# Numerical Analysis on Natural Convection Heat Transfer in a Single Circular Fin-Tube Heat Exchanger (Part 2): Correlations for Limiting Cases

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## Abstract

**:**

## 1. Introduction

_{o}/D → 1 and s/D are finite values for a single horizontal cylinder, where Nu is proportional to the power 0.25 of Ra, whereas when D

_{o}/D → ∞ and s/D << 1 for vertical parallel plates, Nu is proportional to the power unity of Ra. They studied 16 types of heat exchangers, with the diameter ratios (D

_{o}/D) ranging from 1.2 to 2.8, and the normalized gap of the fins (s/D) ranging from 1.2 to 2.6. The following correlation was finalized:

## 2. Heat Exchanger Model

_{o}are the diameters of the circular fin and tube, and ${P}_{f}$,$t$, and $s$ denote the pitch, thickness, and gap of the fins, respectively. This study expands the test cases from the original configurations of Kang and Chang [9] to 32 fin-tube combinations (see Table 2), and uses the commercial code ANSYS CFX 18 [10] for the numerical simulations. Unsteady and laminar flow conditions are assumed for the entire computation. The details of the numerical method are elucidated in Part 1 [1] of this paper.

## 3. Limiting Cases

#### 3.1. Lowest Case: D_{o}/D → 1, s/D = Finite Value (Single Horizontal Cylinder)

_{o}/D) of D12, D11, D10, … (the numbers following D mean ten times the value of D

_{o}/D; thus, 1.20, 1.07, 1.01, …) continue to decrease gradually to approach the limiting case of D

_{o}/D = 1.0. Then, 12 types of heat exchangers were numerically analyzed for the fin pitches P12, P17, P21, and P26.

#### 3.2. Hightest Case: D_{o}/D → ∞, s/D << 1 (Vertical Parallel Disks)

## 4. Classification Criteria for Types A and B

## 5. Correlation Expansion and Validity

#### 5.1. Expansion of Correlation

#### 5.2. Validity of Correlation

## 6. Conclusions

_{o}/D → 1, s/D = finite value) corresponds to a horizontal circular tube, while the upper limit (D

_{o}/D → ∞, s/D << 1) corresponds to vertical flat plates. The main idea of the empirical correlation proposed by Kang and Chang [9] was verified using extended parameters, as the experiment could not cover these conditions. The power of Ra (which is based on the gap of the fins) proportional to Nu was computed as 0.22 at a minimum (${D}_{o}/D=1.01$) and 0.78 at a maximum (${D}_{o}/D=10.0$). Although these values differ from the theoretical results of Kang and Chang’s correlation [9], they show the possibility of using the numerical analysis for prediction over a far wider range of parameters.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclatures

$C$ | proportional coefficient, $C$ is a function of the ${D}_{o}/D$ |

${c}_{p}$ | specific heat capacity [J/kg·K] |

$D$ | circular tube diameter [m] |

${D}_{o}$ | circular fin diameter [m] |

${\mathrm{Gr}}_{D}$ | Grashof number based on tube diameter |

$K$ | correction factor |

$k$ | thermal conductivity of air [Wm^{−1}K^{−1}] |

$L$ | characteristic length [m] |

${\mathrm{Nu}}_{D}$ | Nusselt number based on tube diameter |

${\mathrm{Nu}}_{L}$ | Nusselt number based on characteristic length |

${\mathrm{Nu}}_{s}$ | Nusselt number based on fin spacing |

$n$ | power, $n$ is the logarithmic function of ${D}_{o}/D$ |

${P}_{f}$ | fin pitch [m] |

$\mathrm{Pr}$ | Prandtl number |

${\mathrm{Ra}}_{D}$ | Rayleigh number based on tube diameter |

${\mathrm{Ra}}_{s}$ | Rayleigh number based on fin spacing |

$s$ | fin spacing [m] |

$t$ | fin thickness [m] |

$\mu $ | dynamic viscosity [kg/m·s] |

$\pi $ | ratio of the circumference of a circle to its diameter |

## References

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**Figure 1.**Primary regime between two extreme conditions for natural convection in the circular fin-tube configuration. (from Kang & Chang [9]).

**Figure 9.**Variation of Nusselt number with ${\mathrm{Ra}}_{s}$: Classification criteria for type A and B.

**Figure 14.**Comparison of the present correlation with experimental data: (

**a**) D18, (

**b**) D22, and (

**c**) D28.

${\mathbf{Ra}}_{\mathit{D}}$ | $\mathit{C}$ | $\mathit{n}$ |
---|---|---|

${10}^{-10}~{10}^{-2}$ | 0.675 | 0.058 |

${10}^{-2}~{10}^{2}$ | 1.020 | 0.148 |

${10}^{2}~{10}^{4}$ | 0.850 | 0.188 |

${10}^{4}~{10}^{7}$ | 0.480 | 0.250 |

${10}^{7}~{10}^{12}$ | 0.125 | 0.333 |

Case | D | D_{o} | P_{f} | t | D_{o}/D | s/D | Case | D | D_{o} | P_{f} | t | D_{o}/D | s/D | ||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

D10 | P12 | 15.88 | 16.1 | 2.89 | 1.0 | 1.01 | 0.119 | D22 | P12 | 15.88 | 34.9 | 2.89 | 1.0 | 2.20 | 0.119 |

P17 | 3.68 | 0.169 | P17 | 3.68 | 0.169 | ||||||||||

P21 | 4.26 | 0.205 | P21 | 4.26 | 0.205 | ||||||||||

P26 | 5.06 | 0.256 | P26 | 5.06 | 0.256 | ||||||||||

D11 | P12 | 17.1 | 2.89 | 1.07 | 0.119 | D28 | P12 | 44.5 | 2.89 | 2.80 | 0.119 | ||||

P17 | 3.68 | 0.169 | P17 | 3.68 | 0.169 | ||||||||||

P21 | 4.26 | 0.205 | P21 | 4.26 | 0.205 | ||||||||||

P26 | 5.06 | 0.256 | P26 | 5.06 | 0.256 | ||||||||||

D12 | P12 | 19.1 | 2.89 | 1.20 | 0.119 | D50 | P12 | 79.4 | 2.89 | 5.00 | 0.119 | ||||

P17 | 3.68 | 0.169 | P17 | 3.68 | 0.169 | ||||||||||

P21 | 4.26 | 0.205 | P21 | 4.26 | 0.205 | ||||||||||

P26 | 5.06 | 0.256 | P26 | 5.06 | 0.256 | ||||||||||

D18 | P12 | 27.8 | 2.89 | 1.75 | 0.119 | D100 | P12 | 158.8 | 2.89 | 10.0 | 0.119 | ||||

P17 | 3.68 | 0.169 | P17 | 3.68 | 0.169 | ||||||||||

P21 | 4.26 | 0.205 | P21 | 4.26 | 0.205 | ||||||||||

P26 | 5.06 | 0.256 | P26 | 5.06 | 0.256 |

Case | D | D_{o} | P_{f} | t | D_{o}/D | s/D | |
---|---|---|---|---|---|---|---|

D15 | P12 | 15.88 | 23.8 | 2.89 | 1.0 | 1.50 | 0.119 |

P17 | 3.68 | 0.169 | |||||

P21 | 4.26 | 0.205 | |||||

P26 | 5.06 | 0.256 |

**Table 4.**Error value between B type heat exchanger and Kang and Chang’s correlation [9].

Case | $\mathbf{\left(}\mathbf{1}\mathbf{-}\frac{{\mathbf{Nu}}_{\mathit{C}\mathit{F}\mathit{D}}}{{\mathbf{Nu}}_{\mathit{K}\mathbf{\&}\mathit{C}\mathit{c}\mathit{o}\mathit{r}\mathit{r}}}\mathbf{\right)}\mathbf{\times}\mathbf{100}\mathbf{,}\mathbf{\%}$ | Case | $\mathbf{\left(}\mathbf{1}\mathbf{-}\frac{{\mathbf{Nu}}_{\mathit{C}\mathit{F}\mathit{D}}}{{\mathbf{Nu}}_{\mathit{K}\mathbf{\&}\mathit{C}\mathit{c}\mathit{o}\mathit{r}\mathit{r}}}\mathbf{\right)}\mathbf{\times}\mathbf{100}\mathbf{,}\mathbf{\%}$ | ||||||
---|---|---|---|---|---|---|---|---|---|

Min. | Max. | Ave. | Min. | Max. | Ave. | ||||

D15 | P12 | 15.2 | 20.8 | 18.3 | D28 | P12 | 0.0 | 22.3 | 12.7 |

P17 | 6.3 | 13.9 | 11.6 | P17 | 0.2 | 17.0 | 6.5 | ||

P21 | 3.4 | 9.7 | 8.1 | P21 | 0.2 | 17.2 | 7.1 | ||

P26 | 7.7 | 14.2 | 12.4 | P26 | 0.3 | 21.2 | 14.4 | ||

D18 | P12 | 0.0 | 6.6 | 1.1 | D50 | P12 | 136.7 | 154.1 | 143.0 |

P17 | 2.9 | 16.0 | 12.0 | P17 | 143.7 | 165.0 | 150.8 | ||

P21 | 0.0 | 13.0 | 9.0 | P21 | 142.8 | 165.4 | 149.5 | ||

P26 | 4.6 | 15.1 | 11.6 | P26 | 138.3 | 156.6 | 143.2 | ||

D22 | P12 | 6.3 | 21.7 | 16.5 | D100 | P12 | 101.7 | 102.2 | 101.9 |

P17 | 0.0 | 17.3 | 11.4 | P17 | 102.6 | 103.7 | 103.1 | ||

P21 | 0.0 | 17.6 | 11.1 | P21 | 102.6 | 103.7 | 103.0 | ||

P26 | 0.0 | 19.4 | 13.9 | P26 | 102.1 | 103.2 | 102.5 |

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**MDPI and ACS Style**

Lee, J.H.; Son, Y.W.; Chang, S.-M.
Numerical Analysis on Natural Convection Heat Transfer in a Single Circular Fin-Tube Heat Exchanger (Part 2): Correlations for Limiting Cases. *Entropy* **2020**, *22*, 358.
https://doi.org/10.3390/e22030358

**AMA Style**

Lee JH, Son YW, Chang S-M.
Numerical Analysis on Natural Convection Heat Transfer in a Single Circular Fin-Tube Heat Exchanger (Part 2): Correlations for Limiting Cases. *Entropy*. 2020; 22(3):358.
https://doi.org/10.3390/e22030358

**Chicago/Turabian Style**

Lee, Jong Hwi, Young Woo Son, and Se-Myong Chang.
2020. "Numerical Analysis on Natural Convection Heat Transfer in a Single Circular Fin-Tube Heat Exchanger (Part 2): Correlations for Limiting Cases" *Entropy* 22, no. 3: 358.
https://doi.org/10.3390/e22030358