## 4 Description of the Application Problem and Employed Multi-Fidelity Models
### Conceptual Modeling
#### 5.1.2 Definition of Target and Intermediate Variables  
The method developed in this work is applied to obtain a surrogate for the condensation mass flow rate that 3D CFD simulations (HF model) would predict. Therefore, this is the target variable \(z_{t}\) sought:  
\[z_{t}=\dot{m}_{cond,3D} \tag{11}\]  
Furthermore, the method suggests (see Table 1) that additional intermediate variables could be modeled through surrogates. The underlying idea is that the distribution of such variables across the design space could be more adequate to build surrogate models than that of the target variable. Table 4 presents the target and intermediate variables metamodeled in this work. \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Parameters** & **Variables** & **Units** & **Low, Medium and High Values** \\ \hline Inlet air mass flow & \(\dot{m}_{air}\) & [kg/h] & \(50,175,300\) \\ Inlet temperature & \(\dot{T}_{air}\) & [\({}^{\circ}\)C] & \(-10,0,10\) \\ Inlet relative humidity & \(R_{Hair}\) & [\%] & \(50,75,100\) \\ Inlet turbulent intensity & \(\dot{I}_{air}\) & [\%] & \(1,5.5,10\) \\ EGR temperature & \(T_{egr}\) & [\({}^{\circ}\)C] & \(40,50,60\) \\ EGR rate & \(\dot{EGR}_{rate}\) & [\%] & \(10,25,40\) \\ EGR specific humidity & \(\dot{w}_{EGR}\) & [g/kg] & \(30,50,70\) \\ EGR turbulent intensity & \(\dot{I}_{EGR}\) & [\%] & \(1,5.5,10\) \\ Outlet pressure & \(P_{out}\) & [bar] & \(0.8,0.9,1\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: HF model parameters and levels considered for the different designs of experiments. An intermediate variable can be defined within the domain of the HF model (\(z_{m}\)), as long as it involves the target variable and other input parameters. This is the case of the condensation humidity, which normalizes the target condensation mass flow rate by the inlet mass flow rate of ambient dry air:  
\[\left(z_{m=w_{cond,3D}}\right):w_{cond,3D}=\frac{\dot{m}_{cond,3D} \cdot(1+w_{air})}{\dot{m}_{air}}\cdot 1000 \tag{12}\] \[\left(f_{m=w_{cond,3D}}\right):\dot{m}_{cond,3D}=\frac{\dot{m}_{ air}}{1+w_{air}}\cdot\frac{w_{cond,3D}}{1000} \tag{13}\]  
Notice that the equation for the function \(f_{m}\) is also included before, as it will be required to obtain later \(\dot{m}_{cond,3D}\) on the basis of the surrogate model for \(w_{cond,3D}\) and the parameters \(w_{air}\) and \(\dot{m}_{air}\) corresponding to the working point studied. But aside from the HF intermediate variables, this work aims to show how to build and assess multi-fidelity surrogate models for space design exploration. In this way, [PERSON] et al. [1] identified three types of variables for MF models based on scaling functions: multiplicative, additive, and hybrid variables. For instance, a multiplicative variable can be defined as just the ratio between the HF and LF predictions for condensation mass flow rate:  
\[(z_{m=\varphi}^{MF}):\varphi=\frac{\dot{m}_{cond,3D}}{\dot{m}_{ cond,0D}} \tag{14}\] \[(f_{m=\varphi}^{MF}):\dot{m}_{cond,3D}=\dot{m}_{cond,0D}\cdot\varphi \tag{15}\]  
An MF additive variable is also built by calculating the difference between the condensation mass flow rates obtained by the 0D (LF) and 3D CFD (HF) models:  
\[(z_{m=\Delta}^{MF}):\Delta=\dot{m}_{cond,0D}-\dot{m}_{cond,3D} \tag{16}\] \[(f_{m=\Delta}^{MF}):\dot{m}_{cond,3D}=\dot{m}_{cond,0D}-\Delta \tag{17}\]  
Finally, a hybrid variable may combine multiplications and additions of the LF and HF variables in its definition. In this work, a so-called junction efficiency \(\eta\) is defined:  
\[(z_{m=\eta}^{MF}):\eta=\frac{\dot{m}_{cond,0D}-\dot{m}_{cond,3D}}{\dot{m}_{ cond,0D}}\cdot 100 \tag{18}\]  
\[(f_{m=\eta}^{MF}):\dot{m}_{cond,3D}=\dot{m}_{cond,0D}\cdot(1-\eta/100), \tag{19}\]  
So that it is useful not only as an MF intermediate variable for design space exploration but also as a figure of merit to assess the junction design [74; 75]. Indeed, since \(\dot{m}_{cond,0D}\) is an upper bound for \(\dot{m}_{cond,3D}\), the junction efficiency defined in 18 ranges from 0% (maximum condensation) to 100% (avoids all condensation).