## 1 Introduction
### The EGT as a Parameter Useful for Indirectly Measuring the Engine Performances
Hence, exploiting the Taylor series, it is possible to approximate the torque as follows [35]:  
\[M=f(n,\Delta T)=b_{0}+b_{1}\cdot\Delta T+b_{2}\cdot\Delta T^{2}+b_{3}\cdot n \cdot\Delta T+b_{4}\cdot n^{2} \tag{1}\]  
(where: \(\Delta T\) is the difference between the external air temperature and the EGT, \(n\) is the engine rotational speed in rpm, \(b_{0}\) to \(b_{4}\) are constants to be determined from some performance tests). The above-reported equation shows that the torque is a function of a temperature difference (hence: a temperature) and the engine speed, which are parameters that can be measured in an easier way than a direct measurement of \(M\). Many studies on engines fueled with traditional and innovative fuels used instead artificial neural networks (ANNs) to predict performance and pollutant emissions [36; 37; 38; 39]; however in those works EGT is used to train or check the results and no one uses it to individuate an explicit relation to \(M\). In an earlier work [12], it was proposed a methodology to calculate \(M\) starting from EGT and engine speed using ANNs, and it was found that this method can predict very well the torque (\(\mathbb{R}^{2}\cong 0.998\)) with a quite short training time (\(\leq\)5 s) and a reasonable amount of data to be collected for the training and validation phase (about 219 trials). However, independently from the method employed, the deduction of the torque starting from the EGT strictly relies on the accuracy of the temperature measurement. This experimental implementation can be complicated considering that the EGT, which it is pretended to be used to deduce the torque, is detected in a generic point downstream of the exhaust pipeline, with a thermally-conductive interposed material (i.e., with a thermocouple not in direct contact with the exhaust gas stream). Indeed, due to the many heat-exchange phenomena, which take place along the exhaust gas path (involving all the three heat exchanges modes), depending also on the speed of the exhaust gas (hence by its flow rate), there is no linear proportionality between the detected temperature variations (increments/decrements) and the associated torque variations (increments/decrements) also associated to the time response delays between the torque output and the temperature measurement due to the thermal inertias. As it will be illustrated in this study, the use of a numerical simulation program, such as computational fluid dynamics (CFD) software using the finite volume method (FVM), constitutes a valid and viable tool to preliminary relate the temperature detected at some points the exhaust line (\(T_{i}\), with \(i=1\) to \(n\), where \(n\) is the number of TCs positioned on the exhaust line), in correspondence with a reduced set of engine speeds (hence, with the associated mass flow rates of the exhaust gas), with the real exhaust gas temperature \(T\) in the same cases, thus increasing the accuracy of the general model to predict the torque \(M\):  
\[\left\{ \begin{array}{l}M=a\cdot T_{ex}+b\Rightarrow M=a\cdot T+b\\ T=f(T_{i},\text{CFD}-\text{FVM model parameters, incl. }n)\end{array} \right. \tag{2}\]  
In this case, even if using an overall model that is linear (the first equation of the two reported above), it is still possible to handle successfully all the non-linearities of such a system using the CFD-FVM in the setup phase before implementing the equations in a possible monitoring system. Given the importance of EGT, many aspects must be considered to collect the initial data properly and then tune a CFD-FVM model useful to further correct the measurement errors. Proper EGT acquisition relies on a preliminary evaluation of the most adequate instrumentation to be used.