Study of heat transfer conditions at the wall of an ingot heated by electron gun

A metal confined in a crucible is heated by an electron beam to produce metallic vapor. Only a small proportion of the electron beam energy is used to vaporize the metal. Another portion is lost because of backscattered electrons and thermal radiation of the ingot. The remaining energy is transmitted by convection and conduction to the water cooled crucible. The knowledge of this energy, and particularly its spatial distribution, is important. The magnitude of the heat flux can be analyzed to ensure component integrity. The spatial flux distribution provides information about the coolant system homogeneity, which is related to the position of the ingot in the crucible and the electron beam alignment. These data are useful for controlling operating conditions. In the present method, the heat flux at the crucible wall is identified by conventional measurements (thermocouples, flowmeter) and an inverse method. This approach makes it unnecessary to model complex mechanisms in the liquid metal pool, limiting the study to the heat transfers inside the crucible. Calculation of the flux at a given point of the crucible wall requires three essential data: coolant system temperature Te, crucible temperature Tmes at the measurement point Pmes and temperature sensitivity at this point to heat flux variations at the wall. This characteristic is also called the sensitivity coefficient. It is obtained by modeling heat transfers in the crucible (convective with coolant system, conductive in the crucible) with the boundary conditions Te = 0 K and φ 1 Wm-2. This approach is a simple application of the function specification method, which is a reference method in inverse problems. The problem is assumed to be linear and stationary because of the high thermal crucible conductivity and the measurement locations (close to the wall). If the sensitivity coefficient is known, the heat flux can be calculated by equation (1). Figure 2 shows an example of variations in Tmes and Te variation and the heat flux φ at the crucible wall as a function of time. This operation can be repeated for each, temperature measurement in the crucible. A heat flux mapping is therefore available whose integral is the picture of the exhausted power (Pcr1). It can be directly compared to the enthalpic balance, which is usually estimated (Pcr0). Figure 3 compares these experimental (Pcr0) and theoretical (Pcr1) values and reveals a wide discrepancy associated with the instrumentation indeterminacies (measurement error, error on thermocouple positioning, on water temperature) and with the sensitivity coefficient calculation (model error, heat transfer uncertainty). A corrective formula is proposed to correct temperature measurements as a function of experimental conditions, and particularly exhausted power (Tcorrected =Tmes + C.Pcr0). C is estimated with Pcr2(C) ≡ Pcr0 where Pcr2 = φcor.S (φcor: mean heat flux calculated by means of the inverse method and corrected temperature measurements, S crucible wall surface area). Corrective temperatures were measured in the crucible in parallel and perpendicular positions to isothermal areas. They confirm the corrective expression and show, in our study, that thermocouple location is a major source of indeterminacies. Five experiments with different operating conditions analyzed with this expression show C = 0,3 ± 0,02 K.kW-1. More accurate data are now available on heat flux at the ingot-crucible wall. This approach could be improved with better instrumentation that would limit measurement errors. (authors)