# Numerical Study of Fin-and-Tube Heat Exchanger in Low-Pressure Environment: Air-Side Heat Transfer and Frictional Performance, Entropy Generation Analysis, and Model Development

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model Description

#### 2.1. Physical Model

^{3}and 237.2 W/m·K, respectively. The air pressure ranges from 1 kPa to normal atmosphere pressure, and the inlet airflow velocity ranges from 0.5 to 6 m/s.

#### 2.2. Governing Equations and Boundary Conditions

- Continuity equation:$$\frac{\partial {u}_{i}}{\partial {x}_{i}}=0$$
- Momentum conservation equation:$$\frac{\partial}{\partial {x}_{i}}({u}_{i}{u}_{k})=\frac{\mu}{\rho}\frac{\partial}{\partial {x}_{i}}\left(\frac{\partial {u}_{k}}{\partial {x}_{i}}\right)-\frac{1}{\rho}\frac{\partial p}{\partial {x}_{k}}$$
- Energy conservation equation:$$\frac{\partial}{\partial {x}_{i}}({u}_{i}T)=\frac{k}{\rho {c}_{p}}\frac{\partial}{\partial {x}_{i}}\left(\frac{\partial T}{\partial {x}_{i}}\right)$$

#### 2.3. Numerical Methods and Grid Independence Validation

^{−6}and 10

^{−4}for continuity.The meshing model independence was verified, the grid number ranges from 72,566 to 282,594, and the calculation was conducted at the inlet air velocity of 3 m/s, and the data are shown in Figure 4. With the increasing quantity of grids, the Colburn j-factor and friction factor f rise rapidly at first and become stable. The number of structured grid increase from 161,538 to 282,594, and the difference in averaged calculation results is below 0.2%. Thus, the mesh with 161,538 cells was finally adopted to the simulations.

#### 2.4. Parameter Definitions

_{c}is the collar tube diameter.

#### 2.5. Model of Entropy Analysis

_{1}and T

_{2}are the inlet and outlet temperature; p

_{1}and p

_{2}are the inlet and outlet pressure; R is gas constant. Equation (7) presents the entropy generation on the side of low-temperature heat source:

_{T}and ∆S

_{P}are the entropy generation generated by heat transfer temperature difference and pressure penalty, respectively.

## 3. Model Validation

## 4. Results and Discussions

#### 4.1. Heat Transfer and Frictional Performance Analysis

^{2}K at the air pressure of 25 kPa. In contrast, h ranges from 30.13 to 218.64 W/m

^{2}K at atmospheric pressure of 101 kPa. Furthermore, it can be observed that the higher the airflow inlet velocity, the stronger the increase of h. For the case of inlet air velocity of 3 m/s, as the pressure decreases from 101 kPa to 0 kPa, the convective heat transfer coefficient on the air-side reduced by 32.4%, 67.25%, and 92.8% on average when the air pressure is 60 kPa, 25 kPa, and 5 kPa, respectively. It is indicated that the air-side convective heat transfer of fin-and-tube is dramatically deteriorated due to the thin air under low pressure. The density of air is positively related with the air pressure. Nevertheless, the specific heat capacity is not changed. Thus, the heat transfer capacity reduces when the airflow across the fin with the surrounding pressure decreases.

#### 4.2. Entropy Generation Analysis

_{T}and entropy generation by pressure drop ΔS

_{P}. ΔS

_{T}and ΔS

_{P}at various environment pressure and air velocity are described in Figure 14. It can be found that both of them keep increasing with the uprising velocity and environment pressure of airflow. When the air pressure reduces from 101~5 kPa, ΔS

_{T}decreases by 92.52%, 92.44%, and 91.62% and reduction of ΔS

_{P}up to 50.58%, 73.86%, and 79.45% at the air velocities of 1, 3 and 5 m/s, separately. It is also attributed to the variation in thermal properties. In addition, keeping air pressure at 101 kPa, the entropy generation increased by 242.59% when the air speed ranges from 1 to 5 m/s, this due to the increment of airflow velocity leading to larger heat transfer temperature difference and flow resistance, resulting in the increase in heat exchange capacity and entropy generation. It can also be found that ΔS

_{T}is much higher than ΔS

_{P}under a certain working condition. For the cases discussed in this work, ΔS

_{T}accounts for above 99% of the total entropy generation while the entropy generated by pressure drop is really small. For instance, ΔS

_{P}accounting for 0.00011%, 0.00055% and 0.0013% at inlet air velocities of 1, 3 and 5 m/s, separately. Hence, it is concluded that during the design of a practical air-and-tube heat exchanger, the temperature difference between air and working fluid should be reduced to further reduce the entropy generation. Furthermore, considering the irreversible losses, air higher inlet air velocities are not recommended during the air-and-tube heat exchangers’ design and operation.

## 5. Model Development

_{0}is the normal pressure of 101 kPa.

_{D}

_{c}≤ 10,000. The error for 97.5% of results is within 15% and the absolute mean deviations of the correlation calculation results are 7.48% and 9.42%, respectively. This indicates that Equations (19) and (20) can describe precisely the heat transfer and flow mechanism in plain finned tube exchangers at low even extreme pressure. Meanwhile, when compered with relevant studies [28,29] in low pressure, the influences of tube rows and diameter are also considered in the present correlations. Thus, these correlations can be used for practical designs in practical engineering scenarios.

## 6. Conclusions

- (1)
- The heat transfer and pressure drop behavior in the air-side of the exchanger has changed dramatically in the negative pressure environment. The temperature gradient around tubes decreases with the reduction in the air pressure. Moreover, the pressure gradient around tubes is larger at higher air pressure.
- (2)
- The convective h and pressure drop reduced significantly when compared with ones at usual atmosphere pressure. At air pressure of 25 kPa, the h reduced by an average of 67.92% and pressure drop decreased by 53.45% on average when compared with that at 101 kPa.
- (3)
- The entropy generation of the air-side increases with the increase in air pressure and airflow velocity. The entropy generation increases about 205.8% by increasing the air pressure from 25 kPa to 101 kPa.
- (4)
- The entropy generation by temperature difference ΔS
_{T}accounts for the vast majority proportion of the overall entropy generation compared with that by pressure drop ΔS_{P}. The temperature difference between the air and the refrigerant should be reduced to further reduce the entropy generation. - (5)
- The models of j and f at the plain fin air-side in environment with negative pressure are developed with a mean absolute error of 7.48% and 9.42%, respectively, which shows high accuracy.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

A_{c} | minimum flow cross-sectional area (m^{2}) |

A_{f} | fin surface area (m^{2}) |

A_{fr} | frontal area (m^{2}) |

A_{0} | total surface area (m^{2}) |

D_{c} | tube outer diameter (m) |

D_{h} | hydraulic diameter, 4A_{c} * L/A_{0} (m) |

F_{p} | fin pitch (m) |

f | friction factor |

h | heat transfer coefficient (W m^{−2} K^{−1}) h = Q/(A_{f}*ΔT), or the height of the delta winglet (m) |

j | Colburn factor |

L | fin length along the main flow direction (m) |

l | chord length of the large winglet (m) |

m | massflow rate (kg/s) |

Nu | Nusselt number |

p | pressure (kPa) |

Δp | pressure drop (kPa) |

P_{l} | longitudinal tube pitch (m) |

P_{t} | transverse tube pitch (m) |

P_{r} | Prandtl number |

Re | Reynolds number based on tube collar diameter |

Q | heat transfer rate (W) |

p | Pressure (kPa) |

ΔS | entropy generation (J K^{−1}) |

ΔS_{T} | entropy generation by temperature difference |

ΔS_{P} | entropy generation by pressure drop |

T | temperature (K) |

T_{w} | wall temperature (K) |

ΔT | Temperature difference (K) |

u, v, w | velocity components in x-, y-, z- directions (m s^{−1}) |

u_{m} | mean velocity at the minimum flow cross-sectional area (m s^{−1}) |

U | velocity vector (m s^{−1}) |

x, y, z | Cartesian coordinates (m) |

Z | heat transfer power per unit temperature and per unit volume (W m^{−3} K^{−1}) |

Greek Symbols | |

δ_{f} | fin thickness (m) |

μ | dynamic viscosity (kg m^{−1} s^{−1}) |

ρ | density (kg m^{−3}) |

λ | thermal conductivity (W m^{−1} K^{−1}) |

## References

- Sadeghianjahromi, A.; Wang, C.-C. Heat transfer enhancement in fin-and-tube heat exchangers–A review on different mechanisms. Renew. Sustain. Energy Rev.
**2021**, 137, 110470. [Google Scholar] [CrossRef] - Ahmed, S.A.E.S.; Mesalhy, O.M.; Abdelatief, M.A. Flow and heat transfer enhancement in tube heat exchangers. Heat Mass Transf.
**2015**, 51, 1607–1630. [Google Scholar] [CrossRef] - Lu, C.-W.; Huang, J.-M.; Nien, W.C.; Wang, C.-C. A numerical investigation of the geometric effects on the performance of plate finned-tube heat exchanger. Energy Convers. Manag.
**2011**, 52, 1638–1643. [Google Scholar] [CrossRef] - Wongwises, S.; Chokeman, Y. Effect of fin pitch and number of tube rows on the air side performance of herringbone wavy fin and tube heat exchangers. Energy Convers. Manag.
**2005**, 46, 2216–2231. [Google Scholar] [CrossRef] - Pongsoi, P.; Pikulkajorn, S.; Wang, C.-C.; Wongwises, S. Effect of number of tube rows on the air-side performance of crimped spiral fin-and-tube heat exchanger with a multipass parallel and counter cross-flow configuration. Int. J. Heat Mass Transf.
**2012**, 55, 1403–1411. [Google Scholar] [CrossRef] - Sharifi, K.; Sabeti, M.; Rafiei, M.; Mohammadi, A.H.; Shirazi, L. Computational fluid dynamics (CFD) technique to study the effects of helical wire inserts on heat transfer and pressure drop in a double pipe heat exchanger. Appl. Therm. Eng.
**2018**, 128, 898–910. [Google Scholar] [CrossRef] - Wen, M.-Y.; Ho, C.-Y. Heat-transfer enhancement in fin-and-tube heat exchanger with improved fin design. Appl. Therm. Eng.
**2009**, 29, 1050–1057. [Google Scholar] [CrossRef] - Fathi, S.; Yazdi, M.E.; Adamian, A. Numerical investigation of heat transfer enhancement in heat sinks using multiple rows vortex generators. J. Theor. App. Mech.-Pol.
**2020**, 58, 97–108. [Google Scholar] [CrossRef] - Han, H.; Wang, S.; Sun, L.; Li, Y.; Wang, S. Numerical study of thermal and flow characteristics for a fin-and-tube heat exchanger with arc winglet type vortex generators. Int. J. Refrig.
**2019**, 98, 61–69. [Google Scholar] [CrossRef] - Liu, A.; Wang, G.; Wang, D.; Peng, X.; Yuan, H. Study on the thermal and hydraulic performance of fin-and-tube heat exchanger based on topology optimization. Appl. Therm. Eng.
**2021**, 197, 117380. [Google Scholar] [CrossRef] - Che, M.; Elbel, S. Experimental quantification of air-side row-by-row heat transfer coefficients on fin-and-tube heat exchangers. Int. J. Refrig.
**2021**, 131, 657–665. [Google Scholar] [CrossRef] - He, Y.L.; Han, H.; Tao, W.Q.; Zhang, Y.W. Numerical study of heat-transfer enhancement by punched winglet-type vortex generator arrays in fin-and-tube heat exchangers. Int. J. Heat Mass Transf.
**2012**, 55, 5449–5458. [Google Scholar] [CrossRef] - Erek, A.; Özerdem, B.; Bilir, L.; İlken, Z. Effect of geometrical parameters on heat transfer and pressure drop characteristics of plate fin and tube heat exchangers. Appl. Therm. Eng.
**2005**, 25, 2421–2431. [Google Scholar] [CrossRef] [Green Version] - Cobian-Iñiguez, J.; Wu, A.; Dugast, F.; Pacheco-Vega, A. Numerically-based parametric analysis of plain fin and tube compact heat exchangers. Appl. Therm. Eng.
**2015**, 86, 1–13. [Google Scholar] [CrossRef] [Green Version] - Aslam Bhutta, M.M.; Hayat, N.; Bashir, M.H.; Khan, A.R.; Ahmad, K.N.; Khan, S. CFD applications in various heat exchangers design: A review. Appl. Therm. Eng.
**2012**, 32, 1–12. [Google Scholar] [CrossRef] - Välikangas, T.; Folkersma, M.; Dal Maso, M.; Keskitalo, T.; Peltonen, P.; Vuorinen, V. Parametric CFD study for finding the optimal tube arrangement of a fin-and-tube heat exchanger with plain fins in a marine environment. Appl. Therm. Eng.
**2022**, 200, 117642. [Google Scholar] [CrossRef] - Yaïci, W.; Ghorab, M.; Entchev, E. 3D CFD study of the effect of inlet air flow maldistribution on plate-fin-tube heat exchanger design and thermal–hydraulic performance. Int. J. Heat Mass Transf.
**2016**, 101, 527–541. [Google Scholar] [CrossRef] - Alnakeeb, M.A.; Saad, M.A.; Hassab, M.A. Numerical investigation of thermal and hydraulic performance of fin and flat tube heat exchanger with various aspect ratios. Alex. Eng. J.
**2021**, 60, 4255–4265. [Google Scholar] [CrossRef] - Liu, X.; Wang, M.; Liu, H.; Chen, W.; Qian, S. Numerical analysis on heat transfer enhancement of wavy fin-tube heat exchangers for air-conditioning applications. Appl. Therm. Eng.
**2021**, 199, 117597. [Google Scholar] [CrossRef] - Kalantari, H.; Ghoreishi-Madiseh, S.A.; Kurnia, J.C.; Sasmito, A.P. An analytical correlation for conjugate heat transfer in fin and tube heat exchangers. Int. J. Therm. Sci.
**2021**, 164, 106915. [Google Scholar] [CrossRef] - Lindqvist, K.; Skaugen, G.; Meyer, O.H.H. Plate fin-and-tube heat exchanger computational fluid dynamics model. Appl. Therm. Eng.
**2021**, 189, 116669. [Google Scholar] [CrossRef] - Xie, G.; Wang, Q.; Sunden, B. Parametric study and multiple correlations on air-side heat transfer and friction characteristics of fin-and-tube heat exchangers with large number of large-diameter tube rows. Appl. Therm. Eng.
**2009**, 29, 1–16. [Google Scholar] [CrossRef] [Green Version] - Tang, L.H.; Zeng, M.; Wang, Q.W. Experimental and numerical investigation on air-side performance of fin-and-tube heat exchangers with various fin patterns. Exp. Therm. Fluid Sci.
**2009**, 33, 818–827. [Google Scholar] [CrossRef] - Wang, C.-C.; Chi, K.-Y.; Chang, C.-J. Heat transfer and friction characteristics of plain fin-and-tube heat exchangers, part II: Correlation. Int. J. Heat Mass Transf.
**2000**, 43, 2693–2700. [Google Scholar] [CrossRef] - Wang, C.-C.; Chi, K.-Y. Heat trabsfer and friction characteristics of plain fin-and-tube heat exchangers, part 1 new experimental data. Int. J. Heat Mass Transf.
**2000**, 43, 2681–2691. [Google Scholar] [CrossRef] - Jia, R.-z.; Wang, Y.-c.; Guo, J.; Yu, Z.-y.; Kang, H.-f. Research on the heat transfer and flow characteristics of fin-tube exchanger under low pressure environment. Appl. Therm. Eng.
**2017**, 112, 1163–1171. [Google Scholar] [CrossRef] - Jia, R.-z.; Wang, Y.-c.; Zhang, X.-l.; Wan, R. Research on the heat and mass transfer characteristics of fin-tube exchanger under low pressure environment. Appl. Therm. Eng.
**2017**, 115, 692–701. [Google Scholar] [CrossRef] - Li, Y.-W.; Wang, Y.-C.; Wan, R.; Liu, Z.; Wu, H.-W.; Mei, X.-Q. Study on the air-side flow and heat transfer characteristics of corrugated fin under low-pressure environment. Exp. Heat Transf.
**2020**, 1–17, Ahead-of-print. [Google Scholar] [CrossRef] - Wan, R.; Wang, Y.; Kavtaradze, R.; Ji, H.; He, X. Research on the air-side thermal hydraulic performance of louvered fin and flat tube heat exchangers under low-pressure environment. Exp. Heat Transf.
**2019**, 33, 81–99. [Google Scholar] [CrossRef] - Bejan, A. Entropy generation minimization: The new thermodynamics of finite-size devices and finite-time processes. J. Appl. Phys.
**1996**, 79, 1191–1218. [Google Scholar] [CrossRef] [Green Version] - Bejan, A. The Concept of Irreversibility in Heat Exchanger Design: Counterflow Heat Exchangers for Gas-to-Gas Applications. J. Heat Transf.
**1977**, 99, 374–380. [Google Scholar] [CrossRef] - Liu, J.; Jiang, Y.; Wang, B.; He, S. Assessment and optimization assistance of entropy generation to air-side comprehensive performance of fin-and-flat tube heat exchanger. Int. J. Therm. Sci.
**2019**, 138, 61–74. [Google Scholar] [CrossRef] - Pu, L.; Qi, D.; Xu, L.; Li, Y. Optimization on the performance of ground heat exchangers for GSHP using Kriging model based on MOGA. Appl. Therm. Eng.
**2017**, 118, 480–489. [Google Scholar] [CrossRef] - Sahiti, N.; Krasniqi, F.; Fejzullahu, X.; Bunjaku, J.; Muriqi, A. Entropy generation minimization of a double-pipe pin fin heat exchanger. Appl. Therm. Eng.
**2008**, 28, 2337–2344. [Google Scholar] [CrossRef] - Tian, L.; He, Y.; Tao, Y.; Tao, W. A comparative study on the air-side performance of wavy fin-and-tube heat exchanger with punched delta winglets in staggered and in-line arrangements. Int. J. Therm. Sci.
**2009**, 48, 1765–1776. [Google Scholar] [CrossRef] - Bhuiyan, A.A.; Amin, M.R.; Islam, A.K.M.S. Three-dimensional performance analysis of plain fin tube heat exchangers in transitional regime. Appl. Therm. Eng.
**2013**, 50, 445–454. [Google Scholar] [CrossRef] - Modi, A.J.; Rathod, M.K. Comparative study of heat transfer enhancement and pressure drop for fin-and-circular tube compact heat exchangers with sinusoidal wavy and elliptical curved rectangular winglet vortex generator. Int. J. Heat Mass Transf.
**2019**, 141, 310–326. [Google Scholar] [CrossRef] - Gholami, A.; Wahid, M.A.; Mohammed, H.A. Thermal-hydraulic performance of fin-and-oval tube compact heat exchangers with innovative design of corrugated fin patterns. Int. J. Heat Mass Transf.
**2017**, 106, 573–592. [Google Scholar] [CrossRef] - Bhuiyan, A.A.; Islam, A.K.M.S. Thermal and hydraulic performance of finned-tube heat exchangers under different flow ranges: A review on modeling and experiment. Int. J. Heat Mass Transf.
**2016**, 101, 38–59. [Google Scholar] [CrossRef]

**Figure 5.**Comparison between calculated and experimental datas of Wang et al. [25]. (

**a**) Colburn j-factor; (

**b**) Friction factor f.

**Figure 6.**Velocity fields on the middle xy-plane with different pressure conditions. (

**a**) 1 kPa (Re = 56); (

**b**) 5 kPa (Re = 280); (

**c**) 15 kPa (Re = 846); (

**d**) 25 kPa (Re = 1413); (

**e**) 45 kPa (Re = 2546); (

**f**) 101 kPa (Re = 5600).

**Figure 7.**Temperature distribution on the middle xy-plane with different pressure conditions. (

**a**) 1 kPa (Re = 56); (

**b**) 5 kPa (Re = 280); (

**c**) 15 kPa (Re = 846); (

**d**) 25 kPa (Re = 1413); (

**e**) 45 kPa (Re = 2546); (

**f**) 101 kPa (Re = 5600).

**Figure 8.**Air–side pressure field at different pressure conditions. (

**a**) 1 kPa; (

**b**) 5 kPa; (

**c**) 15 kPa; (

**d**) 25 kPa; (

**e**) 45 kPa; (

**f**) 101 kPa.

**Figure 15.**Correlation calculation error of h. (

**a**) h

_{0}; (

**b**) h (0 kPa < p < 25 kPa); (

**c**) h (25 kPa ≤ p < 101 kPa).

**Figure 17.**Correlation calculation error of friction factor f. (

**a**) Re < 500; (

**b**) (500 ≤ Re < 11,136).

**Figure 18.**Verification of fitted correlations of j and f factor. (

**a**) Colburn j-factor; (

**b**) Friction factor.

Parameter | Size or Value |
---|---|

Tube diameter (D_{c}) | 9.52 mm |

Transverse tube spacing (P_{t}) | 25.4 mm |

Longitudinal tube spacing (P_{l}) | 22 mm |

Fin pitch (F_{p}) | 1.23 mm |

Fin thickness (δ_{f}) | 0.1 mm |

Frontal velocity (v_{in}) | 0.5~6 m/s |

Wall temperature (T_{w}) | 203.15 K |

Tube bank number (N) | 3 |

Thermal conductivity of the fin (λ) | 236 W m^{−1} K^{−1} |

Inlet temperature of air (T_{in}) | 213.15 K |

Air pressure (p) | 1~101 kPa |

P_{t} (mm) | P_{l} (mm) | σ (mm) | D_{c} (mm) | N | F_{p} (mm) | P_{t} (mm) | |
---|---|---|---|---|---|---|---|

1 | 25 | 22 | 0.1 | 9.52 | 3 | 1.23 | 1 |

2 | 9.52 | 3 | 2.23 | 2 | |||

3 | 9.52 | 3 | 2.5 | 3 | |||

4 | 9.52 | 3 | 2.8 | 4 | |||

5 | 9.52 | 3 | 3.23 | 5 | |||

6 | 9.52 | 3 | 3.6 | 6 | |||

7 | 5 | 3 | 2.5 | 7 | |||

8 | 7 | 3 | 2.5 | 8 | |||

9 | 7.94 | 3 | 2.5 | 9 | |||

10 | 10.23 | 3 | 2.5 | 10 | |||

11 | 9.52 | 4 | 2.5 | 11 | |||

12 | 9.52 | 6 | 2.5 | 12 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhang, L.; Wang, J.; Liu, R.; Li, G.; Han, X.; Zhang, Z.; Zhao, J.; Dai, B.
Numerical Study of Fin-and-Tube Heat Exchanger in Low-Pressure Environment: Air-Side Heat Transfer and Frictional Performance, Entropy Generation Analysis, and Model Development. *Entropy* **2022**, *24*, 887.
https://doi.org/10.3390/e24070887

**AMA Style**

Zhang L, Wang J, Liu R, Li G, Han X, Zhang Z, Zhao J, Dai B.
Numerical Study of Fin-and-Tube Heat Exchanger in Low-Pressure Environment: Air-Side Heat Transfer and Frictional Performance, Entropy Generation Analysis, and Model Development. *Entropy*. 2022; 24(7):887.
https://doi.org/10.3390/e24070887

**Chicago/Turabian Style**

Zhang, Lei, Junwei Wang, Ran Liu, Guohua Li, Xiao Han, Zhiqiang Zhang, Jiayi Zhao, and Baomin Dai.
2022. "Numerical Study of Fin-and-Tube Heat Exchanger in Low-Pressure Environment: Air-Side Heat Transfer and Frictional Performance, Entropy Generation Analysis, and Model Development" *Entropy* 24, no. 7: 887.
https://doi.org/10.3390/e24070887