Influence of a cold crucible geometry parameters on electrical efficiency

Abstract

Cold crucible furnaces are widely used in multiple special applications. Unfortunately, the efficiency of this type of furnace is significantly lower than that of other induction units. The paper presents methodology and the analysis results of influence of the cold crucible components geometry on efficiency of transfer energy from an inductor to a charge. Distribution and the total value of the Joule heat generated in the charge inside an induction furnace with a cold crucible were determined for various modifications of construction of the typical furnace. The obtained results allowed for proposing a solution of the cold crucible with a lowered, in comparison to the crucible wall, bottom and a reduced number of segments.

Keywords

Cold crucible electrical efficiency numerical modelling

1. Introduction

A cold crucible furnace has low heating efficiency compared to other induction furnaces. It is a result of both low electrical efficiency of this device and escape of the generated heat from the charge to the intensively cooled crucible. The energy transfer from the inductor to the charge is mainly done through the eddy currents induced in fingers of the crucible. The currents induce the eddy currents in the heated charge. Due to such a mechanism of energy transport, the construction of a cold crucible furnace should minimize losses of energy in these components. A main task of the crucible bottom is to support and cool the charge in order to form a skull protecting liquid metal or another material against contamination. In terms of electrical efficiency of the furnace, the bottom causes losses. This component is an additional load and in a case of some crucible designs it causes leakage of currents from crucible fingers.

There are many proposals of the cold crucible designs. In two main groups of solutions, the cold crucible is implemented as a cut crucible [1, 2, 3, 4] or made from series of tubes [5]. Additionally, systems of shunts, rings and inserts made of material with high magnetic permeability are proposed in order to reduce energy losses in the crucible [6].

The paper [7] presents an analysis of the cold crucible efficiency for such parameters as the inductor current, the operating frequency, the height-diameter ratio of the crucible and the crucible filling level. In the paper [8], the authors proved that the efficiency increases with the thickness reduction of the wall. In that case, the crucible had no bottom. The paper [9] analyses an impact of the current frequency, the segment number and the wall thickness. The results for the wall thickness and the current frequency are consistent with the previous studies.

In this paper, the study is focused on the impact of the segment number of the electrical efficiency. At present, all the industrial, laboratory and theoretically analysed crucibles are built with many segments [10, 11, 12, 13]. This is consistent with the result of the works devoted to the segment number analysis [1, 9]. The mentioned studies indicate that the electrical efficiency increases with that number. This paper presents an attempt of undermining the dogma of the necessity to use many segments. It proposes a solution of the fingers and bottom system which enable to radically reduce the number of segments and improve the efficiency of the cold crucible.

2. Cold crucible model

The developed 3D model of the cold crucible takes into consideration the electromagnetic field and its effect which is the Joule heat generated in the charge. To simplify the model, it was assumed that the charge is in a solid state and the problem of a free surface dynamics does not occur [3, 4, 14]. The model was implemented with the use of the open source software GetDP [15]. The program using the finite element method enables to solve any system of partial differential equations.

The harmonic model of the electromagnetic field was based on formulation using the magnetic vector potential A and the electric scalar potential V:

$\displaystyle\nabla\times\left({\frac{1}{\mu}\nabla\times\underline{{{\bf A}}}% }\right)=\underline{{\bf J}}_{s}+\underline{{\bf J}}_{e}$ (1)

where: $\mu$ – magnetic permeability, A – magnetic vector potential, J ${}_{s}$ – source current density, J ${}_{e}$ – induced eddy currents density.

The eddy currents density was defined by the following equation:

$\displaystyle\underline{{\bf J}}_{e}=-\sigma\left(\text{j}\omega\underline{{% \bf A}}+\nabla\underline{{\bf V}}\right)$ (2)

where: $\sigma$ – electric conductivity, V – electric scalar potential.

The second component taking into account in the AV formulation is the current continuity equation, i.e. the Gauss law for the electric field after transformation:

$\nabla\cdot\underline{{\bf J}}_{e}=0$ (3)

The periodicity boundary conditions were applied to the sides of the computation domain in the shape of a wedge. In addition, a boundary condition of the zero magnetic potential was defined on the outer surfaces of the model.

On the basis of solution of the above equations the Joule heat was determined:

$\displaystyle q_{\textit{VJ}}=\frac{\left|\underline{{\bf J}}_{e}\right|^{2}}{\sigma}$ (4)

The electrical efficiency of the cold crucible was defined as:

$\displaystyle\eta=\frac{\int\limits_{V}q_{\textit{VJ}}{\bf d}V}{P_{a}}\cdot 100\%$ (5)

where: $V$ – volume of the charge, $P_{a}$ – supply active power.

3. Analysis

Table 1
Material properties, geometrical sizes, supply parameters employed in the analysis

Parameter name	Value
Magnetic permeability of all model components [H $\times$ m ${}^{-1}$ ]	12.56 $\times$ 10 ${}^{-7}$
Electric conductivity of titanium charge [S $\times$ m ${}^{-1}$ ]	2.4 $\times$ 10 ${}^{6}$
Electric conductivity of copper crucible and coil [S $\times$ m ${}^{-1}$ ]	5.5 $\times$ 10 ${}^{7}$
Coil current [A]	3000
Frequency [Hz]	10000
Number of crucible segments	16
Thickness of finger wall [mm]	3
Gap between segments [mm]	2

Figure 1.

General geometry and dimension of the analysed cold crucible system.

3.1 Construction of bottom-fingers system

The analysis included 5 different variants of the crucible bottom and fingers configurations. The calculations were carried out for a fixed number of segments, sizes of inductor, inner and outer radius of the crucible, material properties and supply parameters (see Table 1). For simplicity and acceleration of the multi-variant calculations, it was assumed that the charge had a fixed shape of the cylinder with the conductivity of titanium in solid state. The general geometry and dimensions of the analysed object are shown in Fig. 1. The considered variants of the crucible bottom system are shown in Fig. 2. The finger inner channels were closed by non-conductive walls (not shown in the figure).

Figure 2.

Crucible designs: a) separated bottom inside “the wall of fingers”, b) separated bottom below the wall (a gap between the bottom and the wall), c) separated bottom aligned with the lower wall edge, d) the solid bottom joined to the wall, e) cut bottom joined to the wall.

As shown in Fig. 3, the Joule heat distributions in the crucible shows that the lower zone of the charge is not heated effectively in all the variants. This is caused by the energy losses in the crucible bottom. This observation showed the way for searching a method of improving the efficiency of the crucible system.

Figure 3.

Distribution of the generated Joule heat in the charge for the considered cases presented in Fig. 2.

Figure 4.

Electrical efficiency of analysed crucible designs.

Figure 4 shows electrical efficiency of the analysed cold crucible designs. The worst efficiency was obtained for the system with the solid bottom joined to the fingers (the variant $d$ ). This is because of the eddy currents leakage from fingers to the bottom of the crucible (Fig. 5) and increased loss in this part of the system.

Figure 5.

Leakage of eddy currents to the bottom in the design $d$ .

In turn, the best results were obtained in the case of the separated bottom which top was aligned with the lower wall edge (the variant $c$ in Fig. 2). This construction of the cold crucible reduces losses in the bottom significantly. Figure 6 shows the electrical efficiency as a function of the overlapping degree of the wall to the bottom where the overlap for the design $c$ is equal to 0 mm and for the design $a$ is of 20 mm. Further lowering of the bottom position with respect to the wall was not considered because of a risk of leakage of liquid metal from the crucible.

Figure 6.

Effect of overlap of the wall to the bottom on the electrical efficiency.

Figure 7.

Effect of the number of segments for three degrees of the bottom-wall overlap on the electrical efficiency.

3.2 Number of the crucible segments

The analysis of influence of segments number on the electrical efficiency was based on the crucible design with the bottom separated from the wall (the designs $a$ and $c)$ . Figure 7 presents an effect of the number for three degrees of the bottom-wall overlap on the electrical efficiency. In general, a reduction of the segment number causes loss reduction in the walls of fingers connecting their inner and outer surfaces. However, in the case of non-zero value of the overlap, there are a pronounced optimum of the segments number. Further decreasing of the number of segments causes deterioration in the efficiency of the crucible. This is due to pulling eddy currents at the inner surface of a finger by the bottom (the proximity effect) and loss increase in its area (see Fig. 8). A smaller number of segments means the elongation of the path of current flow heating the bottom instead of the charge.

Figure 8.

Pulling eddy currents by the crucible bottom.

Previous studies [16] also indicated that with the decreasing number of segments, the efficiency decreased. However, this result was interpreted as an effect of the increasing crucible wall reluctance (with reduction of the number of slots in this wall), instead of an influence of the increasing losses in the crucible bottom.

In addition to the use of tubes as fingers of some crucible constructions, this phenomenon was probably the main reason for using multiple segments in the crucibles implemented so far.

Reduction of the proximity effect by the lowering of the bottom allows for a radical reduction of the number of segments resulting in the significant efficiency improvement.

Some papers [17] suggest that the arcing risk between the segments increases as the number of segments decreases. However, due to the electrical potential difference between segments in the analysed furnace does not exceed 1V and a vacuum chamber is used for melting titanium, the arcing risk seems to be very low.

4. Conclusion

The presented study showed that new configurations of the cold crucible furnace and the optimization of its geometric parameters can lead to significant improvement of the electrical efficiency in such a device. The proposed solution of the lowered, in comparison to the wall, bottom and the reduced number of segments allows the for the efficiency improvement with the use of simple means.

Of course, the obtained results are fully representative only for the first stage of the titanium melting process when the metal is in solid stated. However, they are useful in optimizing the whole process for the following reasons:

due to the high melting point of titanium (1668 ${}^{\circ}$ C) The main part of the process involves solid state,

the problem of losses caused by the bottom, which are analysed in the paper will be the same mechanism for liquid metal.

furthermore, meniscus shape will cause that more metal mass will be in the bottom area and thanks to this, the impact of the wall-bottom system on the crucible efficiency will be more visible.

the lower conductivity of molten titanium in comparison to the copper bottom will also cause more visible impact of the wall-bottom system.

Obviously, the solution can be combined with other previously developed solutions such as distribution controlling of the electromagnetic field using device components made of high magnetic permeability material [6] and the optimized geometry of the inductor [18].

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Influence of a cold crucible geometry parameters on electrical efficiency

Abstract

Keywords

1. Introduction

2. Cold crucible model

Table 1 Material properties, geometrical sizes, supply parameters employed in the analysis

References

Table 1
Material properties, geometrical sizes, supply parameters employed in the analysis