## 1 Introduction  
Regenerative heat exchanger for heat recovery in glass production processes is a type of heat exchanger where the thermal energy is intermittently stored by a solid medium usually made by a matrix of high thermal capacity and high surface to volume ratio refractories bricks. In the charging phase, a hot flow releases heat to the medium; in the discharge phase, a cold flow receives heat from the medium. In 1816, [PERSON] invented the first thermal regenerator as a part of his well known engine [1]. Since the Industrial Revolution, thermal regenerators have been used for heat recovery from waste gases discharged by industrial processes. In 1857, Sir [PERSON] developed a regenerative furnace [2] where a continuous heat recovery was accomplished by a couple of regenerator chambers (RC) made of refractory material. The Siemens regenerator furnace layout is still largely used in industrial sectors where high temperature waste heat is produced, as in the case of glass industry. Figure 1 shows the layout of a typical end-port (EP) glass furnace, a regenerative plant commonly used in the container glass production industry. The raw material is heated and molten in a 1 to 1.5 m deep rectangular shaped tank. The glass pool is mainly heated by a flame in the combustion chamber (CC). Natural gas or heavy oil are used as fuel. The combustion process typically takes over the 90% of the energy input. Additional heat is frequently introduced by means of electrodes immersed into the glass with the purpose of increasing the convective flows into the glass bath. The raw material is introduced in the so-called \"dog house\" placed in the first part of the glass pool. The glass melts in the central part of the tank, and then it flows into a throat to be distributed to the channels that feed the forming machines. The glass bath is heated up to temperatures that can reach 1550 \({}^{\circ}\)C, and the exhaust gas leaves the CC at very high temperatures, over 1450 \({}^{\circ}\)C, corresponding to about 80-85% of the total energy consumption of the furnace. For this reason, the heat recovery of the waste gases is essential. Usually, heat regenerators are preferred over heat recuperators due to their higher conversion efficiency that significantly compensates the higher installation costs. In fact, a well-designed heat regenerator can easily recover 60% of the heat available from the exhaust gas, while a recuperator, due to the metallic materials involved, can recover up to 40% only. The EP furnace has the typical regenerative furnace layout for a medium size plant. These two ports connect the CC to a couple of RCs--two large brick structures that contain perforated or cruciform brick stacks made of refractory material. The stack geometry produces internal ducts shaped to optimize the fluid to solid heat transfer. The plant design is symmetrical. The combustion is fed alternatively from the preheated air by the left or the right RC. In the first phase, the air is preheated, flowing from the bottom to the top of the left RC, and it flows in the CC by the left port. The air is mixed with the fuel injected by the burners located under the port, creating a flame that covers about two thirds of the tank length. The exhaust gases come across the right ports in the right RC. Exhaust gases flow from the top to the bottom and release heat to the solid medium. A continuous combustion process and heat recovery is guaranteed by switching the cold and the hot flows into the two RCs. Since 1930, there has been a significant scientific effort in the development of numerical models for the design of heat regenerators [3]. [PERSON] [4] published a mathematical model called the \"ideal regenerator model\" that uses differential equations to rigorously describe the heat transfer process in thermal regenerators.