# Laser-Ultrasound-Based Grain Size Gauge for the Hot Strip Mill
## 3 Results and Discussion  
### Grain-Size Calibration Curve  
After the thermal treatment and EBSD/MPGR analysis of the samples as described in the methods section, the ultrasonic response of the relevant part of the GLUS measurements were extracted, i.e., the part of the annealing cycle wherein the samples had attained a stable-enough grain size. The attenuation from this relevant part of all of the GLUS measurements was then fed into a weighted least square analysis to solve for a function \(\Gamma(T)\) that translates the measured attenuation to a grain size. Each calculated attenuation in the LUS measurement was weighted by the signal-to-noise ratio of the echoes used for the attenuation calculation, the width of the frequency span, and the R\({}^{2}\) of the 3 rd-order polynomial fit of \(b^{\prime}\) i.e., the least square of Equation (3). A few different transfer functions i.e., versions of Equation (5), were tested, and the simplest and best solution to the data at hand was the following equation, which describes the average grain size \(\overline{D}\) as a function of spectral attenuation \(b^{\prime}\) and temperature \(T\):  
\[\overline{D}\big{(}T,b^{\prime}\big{)}=\frac{b^{\prime}}{c_{1}+c_{2}Z_{A}(T)}. \tag{6}\]  
The function \(Z_{A}(T)\) is the Zener A-parameter, which is a temperature-dependent variable describing the amount of crystal anisotropy:  
\[Z_{A}(T)=\frac{2C_{44}}{C_{11}-C_{12}}, \tag{7}\]  
where \(C_{ij}\) refers to elastic constants in Voigt notation that vary with temperature and is usually enhanced at elevated temperatures [28]. The material anisotropy effect on attenuation is well known [19] and has been suggested for laser ultrasonics earlier by, e.g., [23; 29]. The need for an offset parameter \(c_{1}\) is attributed to the many assumptions and simplifications made and partly to the measurement system, i.e., the total frequency response of the optical and digital components and the postprocessing of the data. However, the constant \(c_{1}\) is relatively small in comparison to \(c_{2}Z_{A}\). Below, in Figure 5, the average grain size calculated with the measured frequency-dependent attenuation together with Equation (6) is compared to the measured grain size by the EBSD and MPGR techniques. Figure 4: (**a**) Illustration of the hot strip mill in Borlänge, Sweden. The different components are denoted from the reheating furnaces on the left to the downcoilers on the right. The position of the new LUS grain size gauge is marked by the yellow arrow just before the runout table. (**b**) Image of the LUS grain size gauge, produced by EMG Automation GmbH. For an ideal model, the plotted points in the above figure would be positioned exactly on the diagonal 1:1 line. The graph is based on 52 different LUS measurements containing a total of ~23,000 A-scans. For each A-scan, the spectral attenuation was calculated for each of the available P-echoes that could be automatically located. These \(b^{\prime}\) values were then averaged to one single value and served as input for the weighted least square fit. When the final grain-size calibration curve is compared to the microscopic grain size, c.f., Figure 5, the residual, R\({}^{2}\), is just over 0.8. Some of the points have significantly larger error bars, which is mainly attributed to the GLUS measurements that, for those samples, were performed in transmission and which can result in significant oxide growth during the measurement. This oxide scale can have the disadvantage of the reflected intensity of the detection laser being lower. Furthermore, it is also believed to lead to the laser being screened from the actual sample surface that deteriorates the frequency content in the measurement. One should also note that the statistical grain-size distribution will also affect the perceived attenuation as predicted in [19], which is not taken into consideration herein; hence, some small variation in the vertical axis is expected due to this.