## 1 Introduction
To overcome the mathematical difficulty of the rigorous model by [PERSON], [PERSON] [5], in 1931, presented a simplified model called the \"pseudo-recuperator model\", where the thermal regenerator at cyclic equilibrium is modeled as a conventional recuperator where fluid and  
Figure 1: End-port (EP) glass furnaces design. solid temperatures are the time-mean temperatures in the real regenerator phases. Only 30 years later, [PERSON] [6] and [PERSON] [7] published a solution of the \"ideal regenerator model\" differential equation system. [PERSON] used the model to predict the regenerator performance under variable mass flow conditions [8] and to study the effects of the gas storage on the stack during the flow inversions [9]. In recent years, [PERSON] et al. [10] and [PERSON] et al. [11] developed simulation tools to investigate the energy performance of a regenerative glass furnace. [PERSON] et al. [12] developed a model based on the [PERSON]'s hypothesis that is useful to analyze the regenerator performance under different operating conditions. The numerical solution of the [PERSON]'s model is currently no longer a limit. In [13], [PERSON] et al. developed a numerical model of a glass furnace regenerator based on the [PERSON] assumptions. Nowadays, heat regenerators can be efficiently simulated with state-of-the-art computational fluid dynamic (CFD) 3D techniques. [PERSON] et al. [14] developed a CFD analysis to study the heat transfer in a glass furnace regenerator during the cold and the hot phases. In [15; 16], the authors described a similar CFD procedure where a specific model for the heat transfer and pressure drop phenomena into the stacking is detailed. A lower order numerical model for the regenerator--under steady operation--was set up and validated using the results from high fidelity CFD [17]. CFD models have also been developed for the Strategic Waste Gas Recirculation (WGR) system [18]. It is a thermal regenerator add-on module developed within the PrimeGlass project framework [19]. The WGR system, developed to reduce NOx emissions, is a ventilation system that feeds, at its base, the RC operating in the cold phase with a fraction of the waste gas flow rate extracted from the base of the RC operating in the hot phase. A preliminary numerical model developed for steady analysis [17] has been extended with a model for gas thermal radiation [20] in order to predict the additional thermal effects in the air phase in the presence of CO-CO\({}_{2}\)-H\({}_{2}\)O molecules (due to the exhaust gases using a WGR system). In this paper, a lower order numerical approach for the unsteady analysis of the thermal regenerator, with or without the WGR system, is presented. The model is firstly applied to predict the thermal performance of regenerators with different geometrical dimensions to demonstrate its applicability for design purposes. The model is then used to simulate the thermal effects of the WGR system. The present lower order model is developed for the design and development of the regenerative system to set up and optimize the plant with the WGR solution. The transient analysis option gives an increased opportunity to setup the WGR system and to optimize the switch time between hot and cold phases. In fact, the heat transfer process is altered by the presence of the exhaust gases into the combustion air due to the increase of radiation. The present model with transient analysis is a first step in the development of a digital twin for the entire furnace system.