## 1 Introduction  
A key step of steel wire production is the heat treatment of the wire. The most used method for this is patenting [1; 2; 3; 4]. This type of heat treatment is specially tailored to the requirements of the wire production and the carbon steel grades used in this process. The aim of the heat treatment is to achieve a very fine-lamellar pearlite microstructure, also known as sorbite [1; 3; 4]. The microstructure is characterised by good formability combined with high strength. In combination with the subsequent drawing process to reduce the wire diameter, this results in the high mechanical strength of steel wires. The patenting process basically consists of austenitization and subsequent cooling. The austenitization temperature depends primarily on the carbon content of the alloy and is carried out at temperatures between 850 and 1000 \({}^{\circ}\)C [3].  
Cooling typically takes place in lead baths that are heated to temperatures in the range of the pearlite formation (450-600 \({}^{\circ}\)C) to allow the material to pass through the so-called pearlite nose (see Figure 1). The faster the cooling to the desired transformation temperature is realised, the more complete the transformation to a pearlitic structure will be. The material then remains in the bath for an isothermal transformation. Both of these boundary conditions are met when cooling in molten lead baths, as lead provides high cooling rates and a stable bath temperature. Therefore, the use of lead as a cooling fluid is the state-of-the-art solution in many European plants. By adjusting the wire speed and thereby the duration in the bath, different wire dimensions and grades can be heat-treated simultaneously in a single wire production line to a high product quality, giving the industry the ability to react to market changes.  
At the same time, lead has toxic effects on the human body and the environment. For this reason, lead was added to the EU's REACH SVHC candidate list in 2018 [5]. The regulation (EC 1907/2006 of the European Parliament and of the Council concerning the Registration, Evaluation, Authorisation and Restriction of Chemicals [6]) sums up the properties of chemical substances and compounds made from them. In Germany, the use of lead is regulated by TRGS 505 (Technical Rules for Hazardous Substances) [7]. For example, the regulation prescribes that lead baths must be covered to reduce the escape/release of lead vapours, as well as regular blood tests for employees. Further regulations deal with the use of lead during production as well as with the distribution.  
As in any fluid, the heat flux transferred between the wire and the lead bath can be described by the heat transfer coefficient h and the temperature difference between wire and lead, according to [PERSON]'s law of cooling, as shown in Equation (1) [8].  
\[\dot{q}_{conv}^{\prime\prime}=\mathrm{h}\cdot(\mathrm{T_{s}}-\mathrm{T_{ \infty}}) \tag{1}\]  
where  
\(\dot{q}_{conv}^{\prime\prime}\)heat flux density in \(\mathrm{W/m^{2}}\);  
hheat transfer coefficient in \(\mathrm{W/(m^{2}K)}\);  
\(\mathrm{T_{s}}\)surface temperature in K;  
\(\mathrm{T_{\infty}}\)fluid temperature in K.  
The temperatures of both the fluid and the wire surface must be known, as well as the heat flux density during cooling, in order to determine the heat transfer coefficient. However, the heat flux density cannot be measured. Therefore, a different approach using the heat conduction in the wire is necessary to calculate the heat transfer coefficient.  
In the following section, an experimental setup for measuring the core temperature of wire samples of various sizes quenched in an industrial lead bath is presented. Additionally, the results of the measurements as well as an evaluation method developed to derive the heat transfer coefficient at the Department for Industrial Furnaces and Heat Engineering at RWTH Aachen University are shown.  
Figure 1: Schematic temperature curve during cooling in the patenting process.