4.2.2. Pressure Oscillations

In rare cases, usually when reinitiating electric energy input after long power-off periods, the calculated pressure can start oscillating, causing the simulation to become very slow and eventually crash. This is caused by a feedback loop from the coupling of leak air ingress with the pressure

and temperature of the gas phase. The pressure is calculated using the ideal gas law according to Equations (2) and (3):

$$P = \frac{RT\sum \frac{m\_i}{M\_i}}{V},\tag{2}$$

$$\frac{dP}{dt} = \frac{RT\sum \frac{dm\_i}{dt}\frac{1}{M\_i}}{V} + \frac{\frac{dT}{dt}\sum \frac{m\_i}{M\_i}}{V} \,. \tag{3}$$

where *P* is the pressure, *t* is the time, *T* is the gas phase temperature, *R* is the gas constant, and *mi* and *Mi* are the masses and molar masses of species in the gas phase, respectively. *V* is the volume of the gas phase and is assumed to remain constant, whereas the mass of the gas phase depends on the density and composition of the gas.

The leak air intake is assumed to be a linear function of the pressure inside the furnace, as shown in Equation (4): .

$$
\dot{m}\_{\text{leak-air}} = a + b(P - \mathfrak{c}) \tag{4}
$$

where *a*, *b*, and *c* are empirical parameters fitted to match measured data and optimize stability. In some cases, an increased intake of leak air will cause the temperature to drop, with a resulting reduction in pressure that in turn will increase the leak air ingress and vice versa. This can lead to unstable behavior and pressure oscillations with large amplitudes. This problem was addressed by defining the leak air ingress as a differential variable according to Equation (5).

$$\frac{d\dot{m}\_{\text{leak-air}}}{dt} = 50(a + b(P - c) - \dot{m}\_{\text{leak-air}}) \,. \tag{5}$$

This allows the solver to detect steep changes in the leak air intake and adjust the time-step accordingly so that oscillations can be avoided. The difference in simulation results between Equations (4) and (5) is negligible.
