## 2 Materials and Methods
### Studied Materials
#### 2.3.2 Inconel 600  
Due to the high temperature sensitivity of the material properties of the nickel alloy Inconel 600, that is, according to ISO 9950 standard (\(D\) = 12.5 mm; \(H\) = 60 mm), a mandatory material that is used in the determination of the quenching power of quenchant liquids, the temperature change in thermal conductivity, and the specific heat capacity need to be taken into consideration in the present study. According to the data on thermal conductivities with respect to the temperature given in Hannoschock [21], one may obey the polynomial distribution of thermal conductivity as follows:  
\[\lambda_{s}(T)=AT^{3}+BT^{2}+CT+D \tag{10}\]  
where the constants are \(A\) = 1.34 \(\times\) 10\({}^{-10}\) W/(m K\({}^{4}\)); \(B\) = 2.75 \(\times\) 10\({}^{-6}\) W/(m K\({}^{3}\)); \(C\) = 1.24 \(\times\) 10\({}^{-2}\) W/(m K\({}^{2}\)); and \(D\) = 1.09 \(\times\) 10\({}^{1}\) W/(m K). Integrating Equation (10) in the temperature limits \(T_{1}\) and \(T_{2}\), and division by the temperature difference \(\Delta T\) = \(T_{2}-T_{1}\) yields an average thermal conductivity within the desired temperature range that reads  
\[\lambda_{m}=[\lambda_{s}]_{T_{1}}^{T_{2}}=\frac{1}{\Delta T}\int_{T_{1}}^{T_{2 }}\lambda_{s}(T)dT=\frac{\frac{A}{4}\left(T_{2}^{4}-T_{1}^{4}\right)+\frac{B} {3}\left(T_{2}^{3}-T_{1}^{3}\right)+\frac{C}{5}\left(T_{2}^{2}-T_{1}^{2} \right)+D(T_{2}-T_{1})}{T_{2}-T_{1}} \tag{11}\]  
This value is used in the lumped heat conduction model applied herein and reads 21 W/(m K) for the temperature range 959.3 to 1123.1 K, that is, from 686.2 to 849.9 \({}^{\circ}\)C.  
Furthermore, in the direct estimation of the heat transfer coefficient using the lumped heat conduction model, it is necessary to provide the average specific heat capacity of the solid material. Due to its strong temperature dependence, a polynomial distribution of the specific heat capacity is found and reads  
\[c_{s}(T)=AT^{4}+BT^{3}+CT^{2}+DT+E \tag{12}\]where the constants are \(A\) = \(-\)1.42 \(\times\) 10\({}^{-9}\) J/(kg K\({}^{5}\)); \(B\) = 4.13 \(\times\) 10\({}^{-6}\) J/(kg K\({}^{4}\)); \(C\) = \(-\)4.25 \(\times\) 10\({}^{-3}\) J/(kg K\({}^{3}\)); \(D\) = 2.02 J/(kg K\({}^{2}\)); and \(E\) = 1.14 \(\times\) 10\({}^{2}\) J/(kg K). The average specific heat capacity is thus  
\[c_{s,m}=\frac{1}{\Delta T}\int_{T_{1}}^{T_{2}}c_{s}(T)dT=\frac{\frac{A}{4} \left(T_{2}^{4}-T_{1}^{4}\right)+\frac{B}{3}\left(T_{2}^{3}-T_{1}^{3}\right)+ \frac{C}{2}\left(T_{2}^{2}-T_{1}^{2}\right)+D(T_{2}-\ T_{1})}{T_{2}-T_{1}} \tag{13}\]  
In this case, the obtained average specific heat capacity for Inconel 600 is 610.3 J/(kg K) in the temperature range \(T\) \(\in\) [849.90, 686.18] \({}^{\circ}\)C = [1123.05, 959.33] K.  
Assuming the heat transfer coefficient of value 500 W/(m\({}^{2}\) K), and by applying the average thermal conductivity calculated using Equation (11), one may obtain the value of the Biot number equal to 0.149, which is on the edge of the aforementioned criterion 0.1 but still in the frame of thermodynamically thin material. Therefore, for the sake of simplicity, the lumped heat conduction model will be applied herein in the determination of the heat transfer coefficient for the selected time of the quenching process. It is, however, known that since it is derived under the assumption that there are no temperature gradients with respect to space in a solid body, the lumped heat conduction model may be applied in the center of the solid, where the temperature gradients are the smallest in value. The material density is taken as a constant value, \(\rho_{\rm s}\) = 8470 kg/m\({}^{3}\). The thermal effusivity is depicted in Figure 4 using the temperature-dependent thermal conductivity and specific heat capacity according to Equations (10) and (12). The relatively high values of the thermal effusivity indicate the standard boiling modes would occur during the cooling of a high-temperature object made of this material.