*3.5. Thermal Performance and Biot Number*

Since ribs can be considered to be extended surfaces or fins, the performance of fins was analyzed. Fin performance is evaluated by examining the fin effectiveness and fin efficiency [41]. The fin effectiveness is defined as follows:

$$
\varepsilon\_f = \frac{q\_{\text{rib,conj}}}{h\_{\text{conv}} A\_{c,b} (T\_w - T\_b)}. \tag{8}
$$

Fins are judged to be effective when their effectiveness is 2 or more [41], and this condition is satisfied when K\* is 100 or higher (see Figure 11a).

The fin efficiency is defined as follows [41]:

$$\eta\_f = \frac{q\_f}{q\_{\text{max}}} = \frac{q\_{\text{rib\,\,conj}}}{h\_{\text{conv}} A\_{\text{rib}} (T\_w - T\_b)}.\tag{9}$$

Since numerator is identical to *qrib,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.

The total heat transfer rate (*q*) is shown in Figure 11b; *q*<sup>0</sup> 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.

**Figure 11.** Thermal performance: (**a**) fin performance of the rib and (**b**) the total heat transfer rate.

Thermal performance considering both heat transfer and pressure drop is defined by Equation (10), where *f* stands for frication factor [10].

$$\text{Thermal performance} = \frac{q/q\_0}{\left(f/f\_0\right)^{1/3}}.\tag{10}$$

Thermal performance is proportional to the total heat transfer rate as the flow field (i.e., friction factor) does not change even when the conductivity changes. For the heat transfer enhancement to be greater than the pressure drop penalty, the value of thermal performance should exceed 1, but this criterion is satisfied only when K\* = 100 or higher. If K\* = 10 or lower, the rib is not effective in promoting heat transfer.

Figure 12 shows the variation of the local Biot number with the conductivity ratio. The Biot number on the coolant side, which is within the scope of this study, is defined by Equation (11), using the thickness of the solid wall (*d*) as the characteristic length [42]:

$$\text{Bi} = \frac{h\_{\text{conj}}d}{k\_{\text{s}}} = \frac{\text{Nu}}{K^\*} \frac{d}{D\_{\text{h}}}.\tag{11}$$

In a cooled gas turbine blade, Bi is typically around 0.3 [43]. If the Biot number is less than 0.1, the fluid domain accounts for most of the thermal resistance, similar to the isothermal case [43]. Ahn et al. [27] reported that under typical gas turbine blade conditions, the Biot number is less than 0.1 (black circles in Figure 12) and reflects heat transfer characteristics close to pure convection.

At K\* = 100 (red squares), Bi exceeds 0.1 at 3 < *x/e* < 6 on the channel wall, making the conduction thermal resistance non-negligible (Figure 12a), and at the rib surface, Bi goes up to 0.6 at the upstream edge (Figure 12b). At K\* = 10 (blue triangles), Bi is close to 1, and the thermal resistance of the solid is similar to that of the fluid. About half of the temperature drop is expected in the solid region. The data for K\* = 10 in Figure 5 shows that *θ* is around 0.4, confirming that the Biot number is an appropriate indicator. At K\* = 1 (green diamonds), Bi exceeds 1 in most regions, resulting in a higher temperature drop in solids than in fluids, consistent with the results presented in Figures 3–5.

**Figure 12.** Local Biot number variations: (**a**) on the channel wall and (**b**) on the rib.
