A process heat balance combined with the calculated heat distribution in the ESR or VAR process is critical to linking the model boundary conditions to the process operating parameters.
The ESR heat balance consists of three components comprised of the electrode, slag, and ingot domains. At steady state, the slag heat balance is assumed to be equal to zero. This is achieved by adjusting the furnace voltage to close the heat balance while satisfying Ohm’s law with the slag resistance model. This permits calculation of heat flows from the slag to the electrode (to calculate the melt rate) and the slag to the ingot (as the top boundary condition).
A graphical heat balance can be useful in illustrating the differences between melt scenarios. The figures below compare the early and late stages for two ESR melts with different slag practices.
The figure above illustrates an ESR melt with a slag practice in which slag is added to maintain a constant slag volume. Due to the crucible taper, the slag depth gradually increases throughout the melt despite the slag loss due to slag skin formation. The increasing lateral slag heat losses are offset by the reduced annular slag losses as a result of the increasing fill ratio, such that the specific power consumption decreases.
The case above for which no slag is added exhibits a lower specific power consumption toward the end of melting due to the lower heat loss associated with a reduced slag height and the lower annular losses associated with an increasing fill ratio.
The VAR heat balance also comprises individual balances for the electrode, arc, and ingot domains. Heat flows within the arc region of the VAR process are assumed to occur by radiation through a transparent arc, with the distribution among various surfaces being determined using an electrical analogy for radiative heat transfer. The solution to this network incorporating surface emissivities and view factors yields the heat flow to the electrode (to calculate the melt rate) and the heat flow to the top of the ingot (as the top boundary condition).
This information combined with other ingot boundary conditions permits calculation of the temperature field within the ingot from which solidification parameters and heat distributions can be calculated. Again, a graphical heat balance can be used to compare different melt scenarios to better understand the remelt process.
The figure above for an inverted electrode, constant melt rate scenario shows the decreasing specific power consumption due to opposing tapers on the electrode and crucible which produces an increasing fill ratio (decreasing annulus) leading to reduced heat losses toward the end of melting.
By contrast, an upright electrode melt in which electrode and crucible tapers match so as to yield a decreasing fill ratio (increasing annulus) leads to increased heat losses and an increasing specific power consumption toward the end of melting.