## 1 Introduction
discovered through their studies that the diffusivity of coal particles is minimally affected by increasing particle size beyond a certain threshold (~6 mm) [15; 25]. Similarly, [PERSON] and [PERSON] demonstrated experimentally that a size limit exists for the rate of gas emission from coal particles [26]. Within the limit, the gas discharging rate decreases as the particle size increases, a phenomenon that is not apparent when the coal particle size exceeds the size limit. To investigate this, the gas diffusion amount was measured at different times for gas-bearing coal particles classified according to their size using sieving techniques. [PERSON] et al. claimed that gas transport along cracks or cleats becomes the critical factor while the inter-cleat diffusion distances remain basically constant in larger particles [27]. Conditions being equal, the finer the coal sample, the greater the initial release rate [19]. Once the coal particle size is reduced, rapid gas desorption can then take place [5]. [PERSON] et al. indicated in their work that the size effect of the gas desorbing rate from coal particles is due to the fact that the smaller the constituent coal grains, the shorter the route for absorbed gas to escape from inside the coal, which accordingly implies a faster gas desorbing rate and a severer outburst proneness [28]. Interpreting desorption rate experiments requires a combined gas desorption model on the coal particle scale. Generally, gas desorption amount from coal follows the rule of the square root of desorption time, which is derived from the unipore diffusion model [29; 30; 31; 32]. The unipore diffusion model accurately describes gas diffusing behavior in porous media with a unipore structure. However, it fails to produce an accurate fit for gas diffusion in coal due to the complexity of the coal structure. As a result, several models have been proposed by assuming a dual porosity structure [33; 34; 35]. Subsequently, diffusion models have been developed to more accurately represent gas transport in coal by further prioritizing pore structure assumptions in coal [23; 36]. In diffusion models, the observed temporal decrease in the diffusion coefficient is addressed through a proposed time-dependent model derived using variable separation [37; 38]. The multi-scale dynamic diffusion model introduces the decay coefficient as a key parameter to describe gas diffusion decay in coal. This concept is extended to larger-scale coal cores, encompassing the entire gas diffusion process [39; 40]. Additionally, a microscopic analysis is conducted to examine the multi-scale dynamic diffusion mechanism of gas in coal [41]. Sophisticated gas diffusion models will help researchers calculate the gas diffusing rate accurately in coal particles of different sizes [42]. At the same time, the experimental studies show that the effective diffusion coefficient and gas desorption rate increase as the particle size decreases [19; 26; 43; 44]. [PERSON] et al. quantitatively studied the gas desorption rate based on the gas emission model [19]. [PERSON] et al. derived that the average desorption speed of gas-bearing coal is inversely proportional to particle size by assuming a constant diffusion coefficient [45]. Despite significant research efforts, there is still a lack of theoretical understanding regarding the relationship between effective diffusion coefficients and coal grain size based on diffusion models. The crushed coal particles in the outburst process have different particle distributions. The most commonly used models for particle size distribution currently include the Gates-Gaudin-Schuhmann (GGS) model, the Rosin-Rammler (RR) model, the Lognormal model, the Normal model, et al. [46; 47]. The smaller the coal particles are crushed, the greater the increase in desorption rate. However, previous studies disregard the analysis of how the distribution of coal particle fragmentation affects the desorption rate.