## 1 Introduction  
The improvement of the energy conversion efficiency in the process of energy consumption is of great significance and has attracted increasing attention in the global energy shortage scenario [1]. As an indispensable device in the coal-fired power plant, rotary air preheaters (RAPH) are widely applied to recover the waste heat from flue gas and heat combustion air owing to its compactness and superior performance [2, 3]. The combustion air heated by RAPH can not only enhance the combustion, but also is necessary for drying and transportation of the pulverized coal [4]. The efficient operation of RAPH can improve the overall efficiency of the boiler by 5-10% [5]. However, an inherent defect of the operating RAPH is known as leakage, which would seriously hinder the efficient and safe operation of RAPH. The leakage in the RAPH includes the direct and entrained parts. As shown in Figure 0(a), there are six general leakage paths in the RAPH. Under certain pressure differences, the leak streams 1-2 and 3-4 take place in the gap between the stationary and rotating structures, which are called the direct leakage. The leak streams 5-6 represent the leakage carried by the rotor void during rotor rotation, calledentrained leakage [6; 7]. The air leakage of RAPH will cause a series of problems, such as the decrease in heat transfer efficiency, increase in fan power consumption, and aggravation of low-temperature corrosion [8]. To reduce the indirect leakage, the rotational speed of RAPH is usually set as low as possible under the premise of efficient operation. By contrast, owing to the mushroom deformation of the rotor and the fouling of the matrix, both of the leaking area and the pressure difference would be enlarged [9; 10; 11; 12], which would result in a large direct leakage accounting for almost 80% of the whole [13]. To reduce the direct leakage, a number of methods have been proposed, among which multi-seal sealing system (Figure 1b) has more advantages and is most widely employed. Much work has been done on the leakage distributions and their effects on the efficiency of the RAPH. [PERSON] et al. [14] proposed a method to measure the gap size of direct air leakage by installing probes on each sector plate. Several studies have been conducted on air leakage by Skiekpo, including the methods for the gap monitoring [15], the evaluation of leakage distribution [16], the effect of leakage distribution on the thermal performance [17], as well as the effect of the reduction in sealing gap size on RAPH [6]. They found that radial gap has a significant effect on leakage and the reduction in radial gap size would improve the efficiency of the power plant. [PERSON] et al. [18] showed that various leakages could cause significant decrease in the heat transfer efficiency of the RAPH, and the decline in effectiveness depends on the leakage type. [PERSON] et al. [5] investigated the effect of air leakage on the flue gas composition and the outlet temperature of the RAPH. Their results showed that the cold end leakage could significantly reduce the flue gas temperature, whereas the hot end leakage has less effect on the flue gas temperature. [PERSON] et al. [19] established a simplified flow path model for numerical calculation of direct air leakage. It is found that the results calculated by empirical formula are lower than the simulated values due to the ignorance of the gradual change of the gap size along the radial direction. [PERSON] et al. [20] proposed an online application approach to monitor the gap size and estimate the direct leakage of the air preheater. [PERSON] et al. [21] established a model by coupling the air leakage and the thermodynamics to investigate the leakage matter. The results show that the total leakage of quad-sectional air preheater is less than that of tri-sectional by 33.1%. [PERSON] et al.