# Dependence of Conjugate Heat Transfer in Ribbed Channel on Thermal Conductivity of Channel Wall: An LES Study

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## Abstract

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## 1. Introduction

_{s}/k

_{f}) is important for conjugate heat transfer. The blade material and coolant (air) are predetermined and therefore not actively considered. Recently, as 3D printing has been reviewed as a production technique for turbine blades [28,29], it has become possible to have a turbine blade with different thermal conductivities. Recently, as a new gas turbine cycle [30,31] has been studied as a countermeasure against global warming, cooling fluids with thermal conductivities different from that of air, such as carbon dioxide and water vapor, are being studied [32,33].

## 2. Numerical Methods

**x**, t) = −βx + p(

**x**, t),

**x**, t) = γx + θ(

**x**, t),

_{e}). The effective thermal conductivity was determined to satisfy the continuity of temperature and heat flux at the interface [34]. The parameter ω was a convection correction factor, and it was 0 in the cell containing the solid–fluid boundary and 1 in the remaining cells when conduction between the solid and the fluid was considered. In Equation (4), ξ was introduced to maintain second-order accuracy in the cell containing the interface [34]. The code was verified through the CHT problem involving a ribbed duct and a circular cylinder [34].

_{ij}and q

_{j}are the sub-grid scale turbulent stress and turbulent heat flux, respectively, and τ

_{ij}was determined as a dynamic sub-grid model by using scale similarity and setting a test filter around the grid [37,38]. The dynamic sub-grid model provided better results than the constant model for the ribbed channel problem [35]. Similar to τ

_{ij}, q

_{j}was determined dynamically; this approach yields better results in problems where the flow and heat transfer are dissimilar [39]. The simulation were performed for 10,000 time steps to reach a steady state. After that, additional 10,000 time steps (t U

_{b}/D

_{h}= 5) were carried out to obtain the statistics.

## 3. Results and Discussion

#### 3.1. Code Validation Study

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

#### 3.2. Time-Averaged Thermal Fields and Heat Transfer

_{0}is the Nusselt number without ribs, given by Equation (7). The Nusselt number presented in Figure 6 was defined on the basis of D

_{h}. In Figure 6a, the local heat transfer coefficient evidently increases (8 ≤ x/e ≤ 9) near the upstream corner for all thermal conductivity ratios as the cold core fluid collides with the rib. However, as the thermal conductivity ratio increases, the local heat transfer coefficient decreases noticeably and becomes spatially uniform. Quantitatively, the local heat transfer coefficient depends strongly on the thermal conductivity ratio, but its qualitative distribution does not significantly depend on the local thermal conductivity ratio. Except in the vicinity of the rib, heat conduction in the flow direction is not significant since the heat flux vector inside the solid is directed in the +y direction (Figure 4).

#### 3.3. Turbulent Heat Transfer

#### 3.4. Instantaneous Thermal Fields

#### 3.5. Thermal Performance and Biot Number

_{rib,conj}, it shows the same trend as the fin effectiveness. In actual gas turbine materials (K* = 566), the fin efficiency is close to 100%, but at K* = 100 it decreases to 78%. At K* = 10, the fin efficiency is less than 20%, and the fin does not perform its intended function properly.

_{0}is the heat transfer rate in the smooth channel for pure convection, and it is obtained from the Dittus–Boelter equation (Equation (7)). Figure 11b shows that the overall heat transfer rate decreases significantly as K* decreases. For K* = 566.26, there is no significant difference from the isothermal conditions, but for K* = 100, the overall heat transfer rate decreases by about 17%. At K* = 10, it is less than 1/3, and at K* = 1 it is considerably smaller than that in the isothermal smooth channel.

## 4. Conclusions

- When the thermal conductivity ratio was large (K* ≥ 100), the heat transfer characteristics were similar to those in isothermal conditions. In this case, the impingement of the cold core fluid into the rib and recirculation of the flow mainly affected the convective heat transfer. The heat flux from the solid wall was concentrated on the rib and directed toward both edges of the rib.
- When the thermal resistance of the solid increased (K* ≤ 10), the effect of cold core fluid impingement on the rib decreased, and the two vortices located at the corners played an important role in heat transfer. In this case, the temperature distribution of the solid wall that was affected by the two corner vortices determined the convective heat transfer. In particular, on the downstream face of the rib, a region with negative heat transfer appeared.
- For K* ≤ 10, the turbulent heat flux on the front face of the rib was concentrated at a corner, and the turbulent heat flux whose peak occurred near the channel wall disappeared.
- At K* = 100, the temperature fluctuation at the upstream edge of the rib reached 2%, and at K* = 1, the temperature fluctuation in the solid region was at a level similar to that in the fluid region.
- Below K* = 100, heat transfer enhancement was significantly reduced by conduction. Up to K* = 100, the rib promoted heat transfer, but below K* = 10, it did not promote heat transfer.
- Compared with the thermal resistance of the solid and fluid for the CHT of the ribbed channel, the Biot number that was defined on the basis of the thickness of the channel wall appropriately represented the heat transfer characteristics. In other words, for K* = 100 or higher, the Biot number at the channel wall was considerably smaller than 0.1, but at K* = 1, it was considerably larger than 1, which was consistent with the thermal performance of the rib.

## 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 [m^{2}] |

A_{rib} | rib surface area [m^{2}] |

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

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

d | thickness of the channel wall [m] |

D_{h} | hydraulic diameter of the channel [m] |

e | rib height [m] |

f | friction factor |

f_{i} | momentum forcing |

h | heat transfer coefficient [W/m^{2}K] |

H | channel height [m] |

k_{f} | thermal conductivity of the fluid [W/mK] |

k_{s} | thermal conductivity of the solid [W/mK] |

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

ms | mass source/sink |

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

p | rib-to-rib pitch [m] |

Pr | Prandtl number (=v/α) |

Q″ | heat flux {W/m^{2}} |

q | heat transfer rate [W] |

q_{f} | heat transfer rate through a fin [W] |

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

t | time [sec] |

T | temperature [K] |

T_{b} | bulk temperature [K] |

T_{w} | wall temperature [K] |

U_{b} | bulk velocity [m/s] |

V′ | wall-normal velocity fluctuation [m/s] |

W | channel width [m] |

Greek symbols | |

α | thermal diffusivity [m^{2}/s] |

β | mean pressure gradient [Pa/m] |

γ | mean temperature gradient [K/m] |

ε_{ϕ} | fin effectiveness |

η_{ϕ} | fin efficiency |

v | kinematic viscosity [m^{2}/s] |

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 |

Abbreviations | |

IBM | immersed boundary method |

LES | large eddy simulation |

RANS | Reynolds averaged Navier–Stokes simulation |

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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**) comparison of time-averaged streamlines with particle image velocimetry measurement data of Casarsa et al. [40], (

**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**) K* = 566.26, (

**b**) K* = 100.00, (

**c**) K* = 10.00, and (

**d**) K* = 1.00.

**Figure 4.**Heat flux vectors inside the solid wall: (

**a**) K* = 566.26, (

**b**) K* = 100.00, (

**c**) K* = 10.00, and (

**d**) K* = 1.00.

**Figure 5.**Local temperature distribution at the solid–fluid interface: (

**a**) the temperature along the interface between the ribs and (

**b**) the temperature on the rib.

**Figure 6.**Effect of thermal resistance on local convective heat transfer along the solid–fluid interface: (

**a**) heat transfer along the interface between the ribs and (

**b**) heat transfer on the rib.

**Figure 7.**Turbulent heat flux $\overline{{\theta}^{\prime}{v}^{\prime}}$ contours for (

**a**) K* = 566.26, (

**b**) K* = 100.00, (

**c**) K* = 10.00, and (

**d**) K* = 1.00.

**Figure 8.**Temperature fluctuations (θ

_{rms}) for (

**a**) K* = 566.26, (

**b**) K* = 100.00, (

**c**) K* = 10.00, and (

**d**) K* = 1.00.

**Figure 9.**Instantaneous thermal fields near the rib for (

**a**) K* = 566.26, (

**b**) K* = 100.00, (

**c**) K* = 10.00, and (

**d**) K* = 1.00.

**Figure 10.**Instantaneous local heat transfer: (

**a**) K* = 566.26; (

**b**) K* = 100.0; (

**c**) K* = 10.0; and (

**d**) K* = 1.0.

**Figure 11.**Thermal performance: (

**a**) fin performance of the rib and (

**b**) the total heat transfer rate.

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**MDPI and ACS Style**

Ahn, J.; Song, J.C.; Lee, J.S.
Dependence of Conjugate Heat Transfer in Ribbed Channel on Thermal Conductivity of Channel Wall: An LES Study. *Energies* **2021**, *14*, 5698.
https://doi.org/10.3390/en14185698

**AMA Style**

Ahn J, Song JC, Lee JS.
Dependence of Conjugate Heat Transfer in Ribbed Channel on Thermal Conductivity of Channel Wall: An LES Study. *Energies*. 2021; 14(18):5698.
https://doi.org/10.3390/en14185698

**Chicago/Turabian Style**

Ahn, Joon, Jeong Chul Song, and Joon Sik Lee.
2021. "Dependence of Conjugate Heat Transfer in Ribbed Channel on Thermal Conductivity of Channel Wall: An LES Study" *Energies* 14, no. 18: 5698.
https://doi.org/10.3390/en14185698