# Fully Coupled Large Eddy Simulation of Conjugate Heat Transfer in a Ribbed Channel with a 0.1 Blockage Ratio

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Methods

_{i}) was imposed to ensure a velocity of zero in the solid region. In the continuity equation (Equation (1)), mass source/sink (ms) was imposed to satisfy the conservation of mass in cells containing a solid–fluid interface. The amounts of momentum forcing and mass source/sink were determined using a procedure proposed by [26].

_{ij}is the subgrid stress, which is obtained from the strain rate tensor. Germano et al. [27] proposed a method of dynamically determining the proportionality coefficient between the strain rate and τ

_{ij}, and [28] improved upon this method. Tafti [23] confirmed that using the dynamic model better predicts heat transfer in the ribbed channel. Since the dynamic model predicts τ

_{ij}through scale similarity by setting a test filter around the grid, it is difficult for one to obtain good results in body-fitted coordinates including sharp corners. This simulation secured scale similarity by adopting a Cartesian grid while introducing IBM.

_{j}must be obtained. In this simulation, q

_{j}was dynamically determined from the scale similarity between the heat flux obtained from the grid filter and the test filter. When isothermal conditions on a solid wall were creating, as suggested by [29], a heat source/sink was imposed on the cell containing the solid. When considering conjugate heat transfer, the continuity of temperature and the heat flux must be satisfied at the solid–fluid interface.

_{i}) and mass (ms) were imposed inside the solid so that the velocity became zero, and Equation (3) became the heat diffusion equation. The difference in physical properties between the fluid and the solid was reflected by introducing the heat capacity ratio (C*) and the thermal conductivity ratio (K*) while maintaining the basis of the Prandtl number in Equation (3). A convection correction factor (τ) was introduced to maintain the second-order accuracy at the interface, and effective thermal conductivity was introduced to satisfy continuity of the heat flux at the interface. An explanation of how to determine each factor according to the interface configuration in the cell can be found in [25,30].

## 3. Results and Discussion

#### 3.1. Time-Averaged Flow Fields and Heat Transfer Coefficient

_{b}):

_{0}is the Nusselt number of the smooth channel wall obtained using the following Dittus–Boelter correlation:

#### 3.2. Turbulence Statistics and Instantaneous Thermal Fields

_{b}, and when it reaches 10, the bulk flow passes once through the computational domain. A mechanism that promotes heat transfer by entraining the cold core fluid at the shear layer and flowing it along the wall after reattachment is also effective for conducting walls. At t* = 2, 4, and 6, it can be observed that cold fluid flows in the shear layer.

#### 3.3. Thermal Performance and the Biot Number

_{f}) divided by the maximum possible heat transfer rate (q

_{max}). Here, since the rib plays the role of a fin. Fin efficiency can be obtained by using Equation (6) [34]:

_{c,b}is the fin cross-sectional area at the base. In the present study, A

_{c,b}per unit width is the bottom area e of the rib. Fin effectiveness is calculated to be ~8 in pure convection for both the data from the experiments by [15,16] and the present LES. At a blockage ratio of 0.1 (present LES), fin efficiency is 98.9%, so fin effectiveness is almost maintained even in conjugate heat transfer. However, at a blockage ratio of 0.3 (as in the experiments conducted by [15,16]), fin effectiveness decreases to 6.19 by conduction, but it is still a much larger value than 2, which is considered to be effective.

## 4. Conclusions

- (1)
- The heat transfer peak that occurs in front of the rib is not caused by unheated ribs but rather by impinging cold fluid. When the thermal properties of the gas turbine blade are applied, secondary heat transfer peaks occur in front of the ribs, even in cases of conjugate heat transfer.
- (2)
- In conjugate heat transfer, the average heat transfer rate and thermal performance were reduced by 3% compared with those during pure convection. On the channel wall, there appeared to be slight decreases in the variation of the local heat transfer in the windward face of the rib and on the top surface. Even for the conducting rib, high heat transfer was predicted at the upstream edge of the rib.
- (3)
- In the conjugate heat transfer, the overall distribution of the turbulent heat flux was similar to that in the isothermal heat transfer and was consistent with the local heat transfer distribution on the front and rear surfaces of the rib perpendicular to the main flow. The temperature fluctuation inside the solid was much smaller than that in the fluid region, and most of the fluctuation occurred in the rib.
- (4)
- When the thermal performance was evaluated using the rib as an extended surface, fin effectiveness and efficiency were 8.32 and 98.9%, respectively, under typical gas turbine operating conditions. Both indices are recommended values in fin design, meaning that the rib performs well as a fin.
- (5)
- Under typical gas turbine conditions, the Bi value calculated based on the internal heat transfer coefficient is 0.1 or less, and most of the temperature changes occur in the fluid region. In the solid interior, most of the temperature change occurs in the ribs, so when describing heat transfer only to the internal ribbed channel, the height of the ribs indicates the thermal resistance characteristics of Bi superior to the thickness of the channel.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

A_{c,b} | cross-sectional area at the base |

A_{rib} | rib surface area |

Bi | Biot number (=h d/k_{s}) |

Bi_{e} | Biot number based on rib height (=h e/k_{s}) |

C* | heat capacity ratio (=(ρc_{p})_{f}/(ρc_{p})_{s}) |

d | thickness of the channel wall |

D_{h} | hydraulic diameter of the channel |

e | rib height |

f | friction factor |

f_{i} | momentum forcing |

h | heat transfer coefficient |

H | channel height |

k_{f} | thermal conductivity of the fluid |

k_{s} | thermal conductivity of the solid |

K* | thermal conductivity ratio (=k_{s}/k_{f}) |

ms | mass source/sink |

Nu | Nusselt number (=h D_{h}/k_{f}) |

p | rib-to-rib pitch |

Pr | Prandtl number (=ν/α) |

q″ | heat flux |

q | heat transfer rate |

q_{f} | heat transfer rate through a fin |

Re | bulk Reynolds number (=U_{b} D_{h}/ν) |

t | time |

t* | dimensionless time (=t e/U_{b}) |

T | temperature |

T_{b} | bulk temperature |

T_{w} | wall temperature |

U_{b} | bulk velocity |

v′ | wall-normal velocity fluctuation |

W | channel width |

Greek symbols | |

α | thermal diffusivity |

ε_{f} | fin effectiveness |

η_{f} | fin efficiency |

ν | kinematic viscosity |

θ | dimensionless temperature (=(T − T_{b})/(T_{w} − T_{b})) |

Θ | time-averaged dimensionless temperature |

ω | index function between the solid and the fluid |

Subscripts | |

rms | root-mean-square value |

0 | fully developed value in a smooth pipe |

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**Figure 1.**Computational domain and grid system: (

**a**) schematic diagram of the internal cooling passage; (

**b**) the computational domain; and (

**c**) the grid system.

**Figure 2.**Time averaged flow and thermal fields for the isothermal wall; (

**a**) time averaged streamlines compared to particle image velocimetry (PIV) measurement data of Casarsa et al. [33]; (

**b**) the time averaged temperature field; and (

**c**) the Nusselt number ratio on the channel wall between ribs.

**Figure 3.**Time-averaged thermal fields; (

**a**) thermal fields in the xy plane; (

**b**) thermal fields with heat flux vectors around the rib; and (

**c**) experimental data reported by [15].

**Figure 4.**Local Nusselt number ratio variations on the channel wall between ribs. (See Table 1 for information related to the compared experimental data.).

**Figure 5.**Local Nusselt number ratio variations on the rib’s surface. (s is the coordinate defined along the rib’s surface. See Table 1 for information related to the compared data.).

**Figure 6.**Turbulence quantities; (

**a**) turbulent heat flux, (

**b**) turbulent stress, and (

**c**) temperature fluctuation.

**Figure 12.**Percentile change of the heat transfer coefficient by location, as caused by the thermal conduction of the rib.

**Figure 13.**Local Biot number variation on the rib’s surface: (

**a**) Biot number (Bi) and (

**b**) Biot number based on e (Bi

_{e}). (See Table 1 for information related to the compared data.).

Present Study | Liou et al. [24] | Cukurel et al. [15,16] | Scholl et al. [17,18] | |
---|---|---|---|---|

Method | LES | Hologram | IR Camera | LES |

K* | 566.36 | 1368.8 | 618.32 | 618.32 |

C* | 0.00031 | 0.00038 | 0.00031 | 0.00031 |

d/e | 3 | 0.75 | 1 | 1 |

Source | Reynolds Number | Method | e/H | p/e | Rib | W/H |
---|---|---|---|---|---|---|

Ahn et al. [8] | 30,000 | TLC | 0.1 | 10 | Isothermal | ∞ |

Cho et al. [3] | 30,000 | Naphthalene | 0.1 | 10 | Unheated | 2 |

Liou et al. [24] | 10,200 | Hologram | 0.1 | 10 | Heated | 4 |

Rau et al. [10] | 30,000 | TLC | 0.1 | 9 | Unheated | 1 |

Tafti [23] | 20,000 | LE | 0.1 | 10 | Iso-flux | 1 |

Cukurel et al. [15,16] | 40,000 | IR Camera | 0.3 | 10 | Conjugate | 1 |

Scholl et al. [17,18] | 40,000 | LES | 0.3 | 10 | Conjugate | 1 |

Casarsa et al. [33] | 40,000 | PIV | 0.3 | 10 | Conjugate | 1 |

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**MDPI and ACS Style**

Ahn, J.; Song, J.C.; Lee, J.S.
Fully Coupled Large Eddy Simulation of Conjugate Heat Transfer in a Ribbed Channel with a 0.1 Blockage Ratio. *Energies* **2021**, *14*, 2096.
https://doi.org/10.3390/en14082096

**AMA Style**

Ahn J, Song JC, Lee JS.
Fully Coupled Large Eddy Simulation of Conjugate Heat Transfer in a Ribbed Channel with a 0.1 Blockage Ratio. *Energies*. 2021; 14(8):2096.
https://doi.org/10.3390/en14082096

**Chicago/Turabian Style**

Ahn, Joon, Jeong Chul Song, and Joon Sik Lee.
2021. "Fully Coupled Large Eddy Simulation of Conjugate Heat Transfer in a Ribbed Channel with a 0.1 Blockage Ratio" *Energies* 14, no. 8: 2096.
https://doi.org/10.3390/en14082096