# Similarity Analysis for Time Series-Based 2D Temperature Measurement of Engine Exhaust Gas in TDLAT
## 2 Theory of TDLAS  
Tunable diode laser absorption spectroscopy (TDLAS) is a measurement technique commonly used in a variety of applications recently as well as temperature and density of molecules in gas [15; 16]. When a laser beam permeates an absorption medium, its intensity will be attenuated by the absorbing gas species. This phenomenon is measured by TDLAS system that continuously scans laser wavelengths. The principle of TDLAS is theoretically expressed in terms of absorption by Beer-Lambert law that describes the relationship between the incident intensity \(I_{v0}\) and transmitted intensity \(I_{v}\) as follows:  
\[\frac{I_{v}}{I_{v0}}=exp\{-A_{v}\}\ =exp\{-\int_{0}^{L}\alpha_{v}(\ s)ds\}, \tag{1}\]  
where \(A_{v}\) denotes the spectral absorbance of representative wavenumber \(\
u\). \(L\) denotes the optical path length, and \(\alpha_{v}\) is the local mass-weighted absorption coefficient at position s along the laser beam, from 0 to \(L\). This quantity is given by the following expression:  
\[\alpha_{v}=\sum_{i}\frac{x_{i}p}{kT}\sum_{j}S_{i,j}(T)f_{v}\Big{(}v;v_{j},\ \Delta v_{G},\Delta v_{L}\ \Big{)}, \tag{2}\]  
where, \(k\) [J/K] denotes [PERSON]'s constant, \(f_{v}\) is the Voigt profile, with Lorentzian and Gaussian half width at half maximums (HWHMs), \(\Delta v_{L}\) and \(\Delta v_{G}\)[17; 18; 19]. \(p\) denotes pressure, assumed to be uniform and \(x_{i}\) denotes mole-fraction of absorbing species \(i\), where, \(x_{i}\), \(p\), and \(T\) are evaluated locally. \(S_{i,j}(T)\) denotes temperature dependent line strength of absorption transition \(j\) and can be calculated by a function of temperature as follows:  
\[S_{i,j}(T)=S_{i,j}(T_{0})\frac{Q(T_{0})}{Q(T)}\frac{T}{T_{0}}exp\{-\frac{hcE^ {\prime\prime}_{i}}{k}\left(\frac{1}{T}-\frac{1}{T_{0}}\right)\}\left[1-exp \left(\frac{-hcv_{0,i}}{kT}\right)\right]\left[1-exp\left(\frac{-hcv_{0,i}}{ kT_{0}}\right)\right]^{-1}, \tag{3}\]  
where, \(h\) (J-s) and \(c\) (cm/s) denote [PERSON]'s constant and the speed of light, \(v_{0}\) (cm\({}^{-1}\)) denotes the line-center wavenumber, \(E_{i}\)\" (cm\({}^{-1}\)) denotes lower-state energy of the transition, \(T_{0}\) (K) is the reference temperature (296 K), and \(Q(T)\) denotes the partition function of molecular absorption at a particular temperature. It is usually expressed as a 4 th order polynomial function [20]. Although it is difficult to accurately describe the chemical formula of gasoline, iso octane (C\({}_{8}\)H\({}_{18}\)) was selected because it is the hydrocarbon component with similar thermal properties to the actual composition. The combustion reaction of iso octane is as follows:  
\[C_{8}H_{18}+\ 12.05(O_{2}+3.76N_{2})\rightarrow\ 8 CO_{2}+\ 9H_{2}O\ +\ 12.05 \text{-}(3.76)N_{2}. \tag{4}\]  
Among the combustion products, nitrogen (N\({}_{2}\)) is an inert gas and hence, there is no chemical or absorption reaction. Therefore, water-vapor (H\({}_{2}\)O) was chosen for exhaust gas diagnosis because of its high mole fraction and high absorption strength.  
Figure 1 shows the theoretical H\({}_{2}\)O absorption spectra calculated using the HITRAN2012 molecular spectroscopic database [21]. The patterns of the absorption spectra can be observed as the temperature changes. In this study, three peaks lines (7202.255 cm\({}^{-1}\), 7202.909 cm\({}^{-1}\), and 7203.891 cm\({}^{-1}\)) with high line strengths and no interference with other molecules were chosen for calculating reconstruction temperature.