**1. Introduction**

Gas turbines are mainly used as prime movers for aircraft propulsion and natural gas power generation. Their thermal efficiency and output increase with their inlet temperature [1]. Current state-of-the-art turbine engines operate at inlet temperatures (1700 ◦C) well above the melting point of the material (1000 ◦C), and hence, the turbine blades are cooled by compressor bleed air (700 ◦C). Cooling air is supplied to the internal cooling passage of the turbine blade (Figure 1a), where ribs are installed to promote heat transfer. If the predicted temperature of the blade is out of 30 ◦C, the blade life can be halved, and therefore, local heat transfer by the rib and the blade temperature should be accurately predicted [2].

The effect of promoting heat transfer by using ribs has been studied under various conditions and by considering various variables over the past several decades. It has been reported that the channel blockage ratio, rib pitch (=*p*) [3], rib angle of attack [4], and rib shape [5,6] have a significant effect on performance. In the recirculating flow region behind the rib, heat transfer is not active. When the blockage ratio is increased, the pressure drop increases significantly, so the optimal values of blockage ratio and pitch exist [7]. It is recommended that the rib height (=*e*) be 10% of the channel height and the pitch be 10 times the rib height [8].

**Citation:** 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

Academic Editor: Artur Bartosik

Received: 31 July 2021 Accepted: 6 September 2021 Published: 10 September 2021

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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.

At a typical pitch of *p/e* = 10, since 90% of the channel wall is a bare surface, many studies have dealt with the heat transfer of the channel wall. Although rib heat transfer has received relatively little attention, on-rib heat transfer significantly contributes to the total heat transfer [6,9]. Most of the past studies have been performed under isothermal or iso-flux thermal boundary conditions [10]. In actual cooled turbine blades, solid conduction exists and is important for predicting the blade temperature [10,11]. In particular, heat transfer by the rib, where the conduction effect is concentrated, is not purely convective [12,13].

Iaccarino et al. [13] performed a Reynolds-averaged Navier–Stokes (RANS) simulation by adopting the *k*-*ε v*2*f* model and considering the conduction of the rib. They also imposed an iso-flux condition on the channel wall, despite conduction between the rib and the channel wall existing in actual cooled blades. Subsequently, studies conducted experiments on conjugate heat transfer including channel walls [14,15]. When conduction was considered, the average heat transfer coefficient decreased by 26% in the rib and 10% in the channel wall compared with the case of pure convection [15].

Using *k*-*ε v*2*f* improves [16], but in the ribbed channel problem, RANS underpredicts the heat transfer and does not accurately predict local peaks. Large eddy simulations (LESs) can solve these problems [17,18]. Furthermore, mechanisms for generating local peaks can be elucidated by observing instantaneous flow and thermal fields [18,19]. For studying the conjugate heat transfer of ribbed channels [14,15], Scholl et al. [20,21] performed a LES. The LES predicted that the heat transfer of the channel wall was in good agreement with the experiment, but the heat transfer coefficient at the front edge and back of the rib was higher than those observed in the experiment [20].

The difference observed between experimental and LES results at the edge of the rib indicates the possibility of an optical problem involving infrared camera images [22]. Additionally, the possibility of solid/fluid time disparity [23] caused by Scholl et al. [20,21] using the conduction–convection linkage as the heat transfer coefficient forward temperature back method was raised. Recently, Oh et al. [24] performed a fully coupled LES using the IBM (Immersed Boundary Method) for the same problem [13,14]. While the

thermal response of a solid and a fluid has been shown to affect temporal changes in the temperature, it does not significantly affect the time-averaged heat transfer [24].

In the above studies [14,15,20,21,24], the blockage ratio was 0.3 and the thickness of the channel wall was the same as the rib height. In actual turbine blades, the typical blockage ratio is about 0.1, and the wall thickness is usually 2 to 7 times the rib height [25,26]. Ahn et al. [27] performed a fully coupled LES of a ribbed channel with a blockage ratio of 0.1 by applying the IBM. They also set the channel wall thickness to be thrice the ribs to consider an appropriate length scale when defining the Biot number (Bi). They found that Bi was below 0.1, and the amount of reduction in the conjugate heat transfer compared with pure convection was predicted to be 3%.

The reasons for the effect of conduction being small (3%) are the blockage ratio and the thermal conductivity of the blade material, which is 500 to 600 times that of air [27]. On the basis of dimensional analysis, Cukurel and Arts [15] showed that the heat transfer characteristics of a ribbed channel with a conducting wall can be expressed as

$$\text{Nu} = f(\text{Re}, \text{K}^\*, \text{Bi}), \tag{1}$$

where the thermal conductivity ratio K\* (=*k*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].

The effect of the thermal conductivity ratio has been partially addressed in previous studies related to ribbed channels. Iaccarino et al. [13] compared cases with K\* = 0.1, 1, and 100, but their results were RANS results and they did not consider wall conduction. Oh et al. [24] only verified their code for laminar flow over a flat plate in the range of K\* = 0.01 to 1, and they did not report the effect of K\* in the ribbed channel. In the current study, we observed the effect of the thermal conductivity ratio on the turbulent heat transfer of the ribbed channel for K\* = 1, 10, 100, and 566.26.

In this study, a channel wall with a thickness thrice the rib height was considered in the calculation area and a fully coupled LES was performed using the IBM. The thermal conductivity ratio was analyzed for 1, 10, and 100 cases with 566, which is a typical value for a gas turbine blade, and the results were compared. On the basis of the time average temperature field and heat transfer distribution, it was examined whether the mechanism responsible for promoting heat transfer under pure convection conditions is valid even when the conductivity ratio is different. Furthermore, conjugate heat transfer characteristics for different K\* values were observed through the instantaneous flow and thermal fields. Finally, the thermal performance of a ribbed channel according to the thermal conductivity ratio and the Biot number is discussed.
