# Heat Transfer Enhancement in a Novel Annular Tube with Outer Straight and Inner Twisted Oval Tubes

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## Abstract

**:**

## 1. Introduction

## 2. Physical Model and Numerical Method

_{0}) and short-axis (b

_{0}) of the outer straight oval tube are 41 mm and 30.75 mm, respectively. The inner oval tube has a constant long-axis (a

_{i}) of 24 mm, and the studied three short-axes (b

_{i}) are 9.6 mm, 12 mm, and 14.4 mm. The twist pitches (P) considered are 240 mm, 360 mm, and 480 mm. The twist ratio (s) and the aspect ratio (e) of the inner oval tube are determined as s = P/a

_{i}and e = b

_{i}/a

_{i}, respectively. The dimensionless length is defined as X = x/P. In this study, the twist ratios are equal to s = 10, 15, 20 and the aspect ratios are equal to e = 0.4, 0.5, 0.6.

_{t}and λ

_{t}are the turbulent dynamic viscosity and thermal conductivity, respectively. μ

_{t}and λ

_{t}are equal to zero for laminar flow.

_{k}is the production of turbulent kinetic energy caused by the mean velocity gradients. The turbulent dynamic viscosity μ

_{t}and turbulent thermal conductivity λ

_{t}are given by μ

_{t}= ρC

_{μ}k

^{2}/ε and λ

_{t}= C

_{p}μ

_{t}/Pr

_{t}. The value of the turbulent Prandtl number Pr

_{t}is set as 0.85, and the values of C

_{1ε}, C

_{2ε}, and C

_{μ}are equal to 1.42, 1.68, and 0.0845, respectively.

_{P}are calculated as

_{local}on the inner tube wall is calculated as follows:

_{local}on the inner tube wall is calculated as follows:

- (a)
- A uniform velocity u
_{in}and constant temperature (T_{in}= 293K) are adopted at the inlet. - (b)
- The turbulence intensity (I) of the inlet is obtained by I = 0.16Re
^{−1/8}. - (c)
- All of the tube walls are no-slip.
- (d)
- A constant temperature (T
_{w}= 363K) and adiabatic condition are adopted at the inner wall and outer wall, respectively. - (e)
- Outlet: an outflow boundary condition is applied, i.e., $\frac{\partial u}{\partial x}=\frac{\partial v}{\partial x}=\frac{\partial w}{\partial x}=\frac{\partial T}{\partial x}=\frac{\partial k}{\partial x}=\frac{\partial \epsilon}{\partial x}=0$.

^{−8}and 10

^{−6}, respectively.

^{+}is very important for the calculation of the turbulence model. The values of y

^{+}and Re determine the first spacing of the grid boundary layer. The value of y

^{+}≈ 1 is adopted, and the growth factor of the near-wall grid is 1.2 for all the models.

## 3. Results

#### 3.1. Secondary Flow in the Annulus

#### 3.2. Temperature in the Annular Space

#### 3.3. Effects of s and e on Nu and f

#### 3.4. Effects of s and e on JF

#### 3.5. Correlations

## 4. Conclusions

- (1)
- The fluid mixing in the annulus is obviously improved by the inner twisted oval tube.
- (2)
- Nu and f increase as both the aspect ratio and twist ratio decrease. The largest relative increments of Nu and f are 35% and 13% between different aspect ratios, and 26% and 18% between different twist ratios.
- (3)
- The inner twisted oval tube yields 116% and 46% increases in Nu and f, respectively, compared with the inner straight tube.
- (4)
- The thermal performance enhancement is more significant in the laminar regime. The largest JF = 1.9 is obtained for aspect ratio 0.4 and twist ratio 10 at Re = 3000, which is the largest Re in the laminar regime.
- (5)
- The deviations for Nu and f of the correlations are within ±5% and ±4%, respectively.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

a_{o} | long-axis length of outer tube, m |

a_{i} | long-axis length of inner tube, m |

A | cross-sectional area, m^{2} |

b_{o} | short-axis length of outer tube, m |

b_{i} | short-axis length of inner tube, m |

D_{h} | hydraulic diameter, m |

e | aspect ratio |

f | friction factor |

G_{k} | turbulent kinetic energy due to mean velocity gradient, J/kg |

h_{i} | heat transfer coefficient, W/(m^{2} K) |

JF | thermal performance factor |

k | turbulent kinetic energy, J/kg |

L | length of the annular domain, m |

L_{P} | wetted perimeter, m |

Nu | Nusselt number |

Δp | pressure loss, Pa |

P | length of twisted pitch, m |

q | wall heat flux, W/m^{2} |

Re | Reynolds number |

s | twist ratio |

S | fin surface area, m^{2} |

T | temperature, K |

T_{s} | bulk temperature, K |

u, v, w | components of velocity vector, m/s |

X | non-dimensional distance |

x, y, z | coordinates, m |

Greek symbols | |

ε | turbulent dissipation rate, J/(kg s) |

λ | thermal conductivity, W/(m K) |

μ | dynamic viscosity, kg/(m s) |

ρ | density, kg/m^{3} |

Subscripts | |

0 | straight inner and outer oval tubes |

in | inlet |

t | turbulent |

w | wall surface |

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**Figure 1.**Physical model, (

**a**) annular tube, (

**b**) cross section, and (

**c**) portion I and portion II of inner oval tube.

**Figure 7.**Temperature fields for cross sections of the third twisted pitch for e = 0.4, s = 10 at Re = 3000.

Calculation Model | Laminar, Re = 1000 | RNG k–ε, Re = 11,000 | ||
---|---|---|---|---|

fRe | Nu | fRe | Nu | |

Experiment [24] | 22.7 | 4.41 | 87.4 | 43.31 |

Present | 21.45 | 4.01 | 97.39 | 38.67 |

Relative error | 5.5% | 9.1% | 11.4% | 10.7% |

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## Share and Cite

**MDPI and ACS Style**

Luo, C.; Song, K.; Tagawa, T.; Liu, T.
Heat Transfer Enhancement in a Novel Annular Tube with Outer Straight and Inner Twisted Oval Tubes. *Symmetry* **2020**, *12*, 1213.
https://doi.org/10.3390/sym12081213

**AMA Style**

Luo C, Song K, Tagawa T, Liu T.
Heat Transfer Enhancement in a Novel Annular Tube with Outer Straight and Inner Twisted Oval Tubes. *Symmetry*. 2020; 12(8):1213.
https://doi.org/10.3390/sym12081213

**Chicago/Turabian Style**

Luo, Chao, KeWei Song, Toshio Tagawa, and TengFei Liu.
2020. "Heat Transfer Enhancement in a Novel Annular Tube with Outer Straight and Inner Twisted Oval Tubes" *Symmetry* 12, no. 8: 1213.
https://doi.org/10.3390/sym12081213