## 1 Introduction  
Distributed power generation is actively researched as a response to the instability of power supplies, such as a large-scale blackout, due to disaster. Distributed power generation directly produces and supplies power at demand, and stationary fuel cells are representative of distributed power generation [1; 2]. Stationary fuel cells for buildings have the advantages of using a relatively small installation space for producing the same amount of energy compared to other renewable energies and using the existing city gas infrastructure [3; 4]. Therefore, the demand for stationary fuel cell systems for buildings is increasing worldwide. Also, the electricity output, durability, system size, operation, installation, and operation costs of fuel cell systems are becoming competitive [5; 6]. In particular, the efficiency of a fuel processor and fuel cell combined system is mainly affected by the fuel processor efficiency [7; 8]. The flow rate of methane used for the reactants and burner fuel affects the reaction temperature in the steam reformer, resulting in a change in conversion and affecting the overall efficiency of the combined system [9]. In general, the hydrogen production rate increases as the temperature of the reactor increases, but the overall thermal efficiency may decrease due to the excessive input of burner fuel [10]. Therefore, the fuel processor needs to be operated at optimal temperature conditions and operate within the temperature range shown in Figure 1.  
The fuel processor is composed of a steam reformer, a water-gas shift reactor, a burner, a desulfurizer, and heat exchangers [11]. In a steam reformer, natural gas and steam react on nickel catalysts at 500 \({}^{\circ}\)C to 800 \({}^{\circ}\)C to produce hydrogen and carbon monoxide [12]. Since the reforming reaction is endothermic, the reactor must be designed to supply a sufficient amount of heat to keep the reaction temperature constant [13; 14]. For this reason, a burner is installed close to where the steam reforming reaction takes place to provide sufficient thermal energy. The heat from the exhaust gas, the reformed gas, and released from the steam reformer is recovered to heat the reactants, in particular, water [15]. Increasing the efficiency of fuel processors requires thermal energy optimization for the heat transfer between the exhaust gas, reformate, and reactants. The WGS (water-gas shift reactor) is classified into either an HTS (high-temperature shift reactor) or LTS (low-temperature shift reactor) depending on the operating temperature (shown in Figure 1). The WGS reactor shifts CO to CO\({}_{2}\) to prevent CO poisoning of the fuel cell. A water-gas shift reaction is an exothermic reaction so it is necessary to remove the reaction heat [16]. The LT-PEMFC (low temperature-proton exchange membrane fuel cell) requires the CO concentration to be less than 10 ppm [17; 18], whereas the HT-PEMFC (high temperature-proton exchange membrane fuel cell) has sufficient durability, even when the CO concentration is less than 1% [17]. As such, HT-PEMFCs may have greater tolerance to some impurities, such as carbon monoxide (CO) and sulfur compounds, than LT-PEMFCs. This is economically useful as it eliminates the purification process, similar to a preferential reactor. In addition, since the HT-PEMFC operates at a higher temperature (120 to 180 \({}^{\circ}\)C) than the LT-PEM (60 to 80 \({}^{\circ}\)C) fuel cell, the utilization of waste heat such as tri-generation is increased and the overall efficiency is increased [17; 18; 19; 20]. However, despite these benefits, combined systems of fuel processors and HT-PEMFCs can release more CO pollutants into the atmosphere. Therefore, CO reduction as well as heat recovery should be achieved at the burner by recycling the anode off-gas to the burner. One of the ways to increase the total efficiency of the fuel processor and fuel cell combined system is to recirculate the unreacted hydrogen discharged from the anode of the fuel cell to the burner as a fuel [21; 22; 23; 24; 25; 26].