## 2 Materials and Methods
### Experimental Tests at the Engine Test Stand (Step 1)
The TCs were applied in three different locations of the experimental exhaust system (see Figure 4 left) with different fixing systems:  
* The first TC (TC1) has a grommet that allowed it to be fixed on the bolt of the bended connector holding the exhaust duct on the engine; the tip is close to the grommet external circumference, and, hence, when fixed, it is in contact with the bolt surface and measures the temperature of it (right); this TC is 29.6 mm apart from the exhaust duct axis, in a position that is 144 mm distant (equivalent to about 6.3 duct diameters) from the exhaust valve position;  
Figure 4: **(left)** Exhaust duct layout and sensors positioning; (right) close-up view of the TC1 with indicated the position of the tip. Figure 3: Experimental setup (the dotted rectangle includes the components that are part of the tested engine); black lines represent mechanical/physical connections (shaft, pipes, screw bolts); green lines represent sensor electrical connections for data logging. * The second TC (TC2) has a clamp, which allowed to place the TC with its tip in contact with the exhaust pipe external surface; the clamp was tightened on the exhaust pipe in a position close to the third TC, 70 mm (3.0 diameters) from the straight tube junction with the connector, 190 mm (8.3 diameters) from the engine head junction, 334 mm (14.5 diameters) from the exhaust valve position;
* The third one (TC3) is a TC probe whose tip was inserted into a thermowell screwed in a specific seat on the exhaust pipe, 50 mm (2.2 diameters) from the straight tube junction with the connector, 170 mm (7.4 diameters) from the engine head junction, 314 mm (13.7 diameters) from the exhaust valve position. Therefore, the tip of this TC is spatially inside the exhaust pipe (approximatively at a depth corresponding to the center of the pipe section), and the external surface of the thermowell is in direct contact with the flow of exhaust gases. In other future applications of the exposed methodology, it is possible to place the TCs in other points, which will be specific to the exhaust line present in the inquired machine (hence, even very different from the one used here). As will be explained later, an experimenter should only take care to individuate the exact positions of the TCs' tips and place them in the CFD-FVM model to proceed with the model calibration procedure that will be explained in the following paragraphs. For this case study, a total of 4 trials were performed, each one characterized by different conditions of engine load and speed imposed by the dyno, hence determining 4 different operative points for the engine (named from P\({}_{0}\) to P\({}_{3}\)). As mentioned before, only two of these operative points are effectively needed for this procedure (in this case, P\({}_{0}\) and P\({}_{3}\) were chosen), the other two (in this case P\({}_{1}\) and P\({}_{2}\)) have been used as verification points and, therefore, in future applications of this procedure they may not be carried out. The load and speed conditions were kept constant for a minimum of 500 s per trial, so as the detected temperatures (\(T_{1}\) for TC1, \(T_{2}\) for TC2, \(T_{3}\) for TC3) could reach a constant value, and this value was kept for at least 50 s (meaning that the system was in a steady-state condition in this last part of the observation period). The room (ambient) temperature was stable around 27 \({}^{\circ}\)C during all the trials. During each trial, many parameters of interest were collected at the same time: the temperatures from the three TCs, the rotational speed, the torque, the fuel weight, and the value of lambda. The data acquisition interval was set at 10 s (i.e., the sampling frequency was 0.1 Hz) by using a proper program developed within the LABVIEW software environment (by National Instruments Corporation, Austin, Texas, USA [53]) and running on the datalogger. The same program also allowed the calculation of the mobile average on the last 5 acquired values for each measured quantity.