単相電弧炉内の温度分布に関する理論的考察

高融点無機材料の研究 (第1報)

抄録

Thermal state in electric arc furnace in which many refractory materials are melted has not yet been clear enough to control melting process completely. This work deals theoretically with temperature distribution in the furnace using a simple model based upon experimental data.
The furnace shape is modeled as a spheroid having an isothermal surface with temperature θ_s. In the furnace, closely packed test specimen is heated by a heat source that locates on a straight line of segment limiting the both foci of the spheroid. Distribution of generated heat energy in the source is derived from a power density function, P(x′), that depends upon coordinate x′ set along the heat source and is normalized to supplied electric power P.
∫P(x′)dx′=P…(1)
Therefore a heat element generated in an elemental length of the heat source, Δx′, is P(x′) Δx′ which flows out through the specimen in all directions uniformly.
The furnace is assumed to be kept in thermally stationary state with heat transfer depending either on thermal conduction in the furnace or on thermal convection on its surface. If the heat source consists of only one element, temperature at any point in the furnace Δθ is virtually given in the following equation,
d²Δθ/dγ²+(2/γ)dΔθ/dγ=0…(2)
where γ is radius vector of any point from the element of the heat source. Heat flowing per second through a spherical surface having a center in the element of the heat source has to be tantamount to heat generated per second, then
4πγ²λ⋅dΔθ/dγ=P(x′)Δx′…(3)
where λ is thermal conductivity of the specimen in the furnace. And the temperature of outer shell of the furnace is virtually Δθ_s at any point having a radius vector R.
Δθ=Δθ_s at γ=R…(4)
Solving equation (2) under boundary conditions consisting of equations (3) and (4), virtual temperature is given as follows.
Δθ=Δθ_s+P(x′)Δx′/4πλ(1/γ-1/R)…(5)
As any point in the furnace receives heat energy from every element of the heat source, it is necessary for obtaining real temperature to superpose all virtual temperatures of that point. Using a rectangular coordinates, (x, y), in which the origin and the abscissa are common with those in the coordinate (x′), real temperature θ_ij at the point x_iy_j is obtained by integrating Δθ_ij over all range of the heat source, as follows:
θ_ij=θ_s+1/4πλ∫_-k^kP(x′){1/γ_ij-1/R_IJ}dx′
=θ_s+1/4πλ∫_-k^kP(x′){1/√y_j²+1/(x_i-x′)²-1/√y_J²+(x_I-x′)²}dx′…(6)
where k and -k are coordinates of both ends of the heat source and x_I and y_J are coordinates where isothermal curve (x²/a²+y²/b²=1, a and b are half length of major and minor axes of the isothermal