## 2 Theoretical Background
### Ultrasonic Non-Destructive Measurement Method for Consumable Electrodes  
Figure 2 shows a cross-sectional view of the consuming electrode in a medium-sized metallurgical electric furnace. The maximum electrode diameter reaches 2 m, the total length is close to 25 m, and the weight is 65 t. In the smelting process, the pre-made solid electrode material is put into the electrode casing installed on the submerged arc furnace. Then, the electrode is sintered using heat during the smelting process. Therefore, the electrode material in the electrode casing will be melted into a liquid from the original solid and then sintered into a solid. It means that from the top of the electrode to the end, the material in the electrode casing is sparse and uneven. This is detrimental to the propagation  
Figure 1: The electrode system of submerged arc furnaces with a graphite rod. of ultrasonic waves. Based on the abovementioned reasons, we proposed a method to form a uniform and continuous detection channel in the electrode by adding a graphite rod. The measurement principle of ultrasonic-guided wave technology is as follows. The ultrasonic transducer emits the ultrasound signal on top of the electrode. The emitted ultrasound signal will propagate to the end of the electrode through the graphite rod in the electrode. Because of interface reflection, the ultrasonic transducer will receive the ultrasonic signal reflected from the electrode end. The measurement system will evaluate the parameters of the electrode based on the reflected ultrasound signal. The ultrasonic measurement system can calculate the position of the electrode working end by measuring the flight time between the transmitted signal and the echo signal. Although the shape of the electrode tip in a submerged arc furnace is similar to an ellipse [26], the graphite rod's diameter is much smaller than that of the electrode's, so we can regard the graphite rod's end face in the molten pool as a plane. When the electrode cracks, part of the ultrasound signal will be reflected in the ultrasonic transducer by the crack in advance. Therefore, the measurement system can detect hidden cracks in the electrode based on the ultrasonic signal reflected in the ultrasonic transducer in advance. A metal tube should be added outside the graphite rod to avoid damaging the graphite rod when adding the electrode paste. The metal tube can protect the graphite rod. At the same time, adding a metal tube can form a clear interface between the metal tube and the graphite rod, which is conducive to reducing the diffusion and radiation of sound waves to the electrode in the graphite rod. In addition, the inner diameter of the metal tube should be greater than the outer diameter of the graphite rod so that the cylindrical surface of the graphite rod can be considered as a free boundary. When the ultrasonic signal propagates in the solid, some factors will attenuate the signal amplitude, such as the geometric dispersion caused by the columnar structure, the beam spread, the material's porosity, etc. [27]. The detection accuracy will be greatly reduced if the echo amplitude is too low. The electrode length in the medium-sized electric furnace is usually more than 10 m, which makes the non-destructive monitoring for consumable electrodes different from the general ultrasonic flaw detection. Obtaining sufficient echo amplitude is a significant challenge. However, a larger detection distance can be obtained by optimizing the ultrasonic transducer in the non-destructive detection of bolts [28]. However, the propagation characteristics of ultrasonic signals in graphite are different from those of structural steel. The most significant difference is that the ultrasonic velocity  
Figure 2: Cross-sectional view of the consuming electrode. in graphite increases instead of decreasing with increasing temperature [12]. Therefore, analyzing the acoustic characteristics of ultrasonic waves propagating in graphite rods is necessary.