## 1 Introduction  
Submerged arc furnaces are critical pieces of metallurgical equipment used for ferroalloy smelting. Ferroalloy produced by submerged arc furnaces comprises more than 80% of their total production [1]. Electrical energy is delivered into the furnace through an electrode during the smelting process. The arc heat at the electrode end and the resistance heat between the material or slag to are used to melt and reduce the ore [2]. Therefore, the depth electrodes inserted into the charge directly affects a furnace's power consumption and melting temperature distribution [3]. The operator adjusts the depth of the electrode inserted into the charge by raising and lowering the electrode. However, the furnace temperature can reach up to 1350 degrees, and the electrodes are deeply buried in the charge. Therefore, measuring the depth of the electrodes into the charge is a significant challenge for many researchers. Since the last century, researchers have made many attempts to measure the electrode length. For example, workers have estimated the length of the electrode by touching the electrode to a metal bar during smelting. This method is exceptionally unsafe and inaccurate. From the published literature, we noticed that researchers tried to estimate the electrode length by the arc current, the electrode's weight, and the magnetic field's distribution around the electric furnace. A U.S. patent attempted to estimate the electrode length by establishing a functional relationship between the electrode current and the electrode depth in the furnace [4]. However, the electrode currents of each phase are coupled with each other. Changing one electrode current will affect the electrode currents of the two different phases. Therefore, the feasibility of this method is weak. The method using the weight of the electrode to estimate the electrode length is affected by the electrode clamping system, the addition of electrode materials, and the contact support of the charge, which makes the calculated result error too large [5]. Methods based on electric, magnetic, and temperature fields in the furnace to estimate the electrode length need to overcome the influence of electromagnetic interference, which is still in the theoretical research stage [6]. Researchers have used numerical simulation technology to analyze the relationship between the electrode length and the magnetic field radiation around the electric furnace [7; 8]. However, the experimental results are not ideal. The method using electromagnetic waves to detect the electrode length requires adding a hollow waveguide to the electrode [9]. The hollow waveguide will cause the combustible furnace gas to overflow, which is fatal to safe production. Fortunately, ultrasonic-guided wave technology has been successfully applied to measure concrete anchor rods [10; 11]. Ultrasonic-guided wave technology is not only used to measure the length of a bolt but also to evaluate its anchoring quality. These achievements mean that ultrasonic-guided wave technology has excellent potential in measuring consumable electrodes in submerged arc furnaces. Therefore, we used ultrasound waves reflected from the electrode tip to estimate the electrode length. In our previous work, we analyzed the effect of high temperature on the ultrasonic velocity in graphite [12]. However, the ultrasonic transmission loss in graphite is one of the foundations for establishing an ultrasonic measurement system, because the transmission loss directly affects the distance of the sound wave transmission and the intensity of the echo. The measurement standards of material transmission loss include the pulse-echo or spectral ratio method [13; 14]. However, experimental investigation of transmission loss is very time-consuming and costly. In addition, because obtaining transducers of various frequencies is challenging, only a few parameters can be studied. In contrast to experimental studies, numerical simulations allow studies with lower cost and time implications.