## 1 Introduction  
Industrial drying is a typical energy-intensive operation, and the energy consumed by the operation accounts for about 10-20% of the national industrial energy consumption per year in some countries [1; 2; 3]. Although there are various innovative drying technologies utilized around the world, such as vacuum [4], microwave [5], infrared [6] and pulse combustion [7] drying technologies, hot air drying has irreplaceable advantages (high efficiency and economy) in industrial drying production. The heat exchanger is the key component for transferring the energy in the industrial hot air drying system, and it is also the most important component affecting the energy consumption of the whole system. Considering the environmental effect and the limited amount of natural resources to produce energy, it is of great economic and social significance to optimize the heat exchanger. The heat exchanger is an important piece of heat transfer equipment which carries out the heat transfer task between two or more fluids and is widely used in refrigeration [8], food [9] and engineering [10] fields. For example, [PERSON] et al. conducted an experimental study on the energetic performance and cooling capacity of a liquid-line heat exchanger in a CO\({}_{2}\) subcritical refrigeration plant with a gas cooler, and the results showed that the internal heat exchanger does not improve the performance of the subcritical cycle, but it can improve the energetic performance if it is used inside a cascade refrigeration system [11]. [PERSON] et al. proposed a novel rhombus-shaped heat surface to enhance the heat exchange performance of a shell and tube heat exchanger, and the results showed that the \(Nu\) increases with the increase in the Re; thus, the relationship between the \(Nu\) and the Re as well as the friction factor were also investigated [12]. In order to optimize the heat exchange performance of an industrial-scale heat exchanger in the cement production industry, [PERSON] et al. [13] conducted an industrial-scale experiment to investigate the relationship between the heat exchange performance and the flow resistance for different pipe arrangements. They found that the surface heat transfer coefficient of an in-line cross flow heat exchanger varies in the range of 47.9-65.6%, which is higher than that of an in-line heat exchanger at the same mass flow rate, while the pressure drop of each row of tube bundles for the in-line cross flow is 1.6-2.5 times higher than that of in-line tube bundles. Though there are many studies focusing on the heat exchange performance and fluid dynamics of industrial heat exchangers in refrigeration, engineering and other fields, few works have reported on their use in the agricultural field. The fluid dynamics and heat exchange of heat exchangers have been considered as part of the typical interdisciplinary technology field involving fluid mechanics, heat transfer, engineering thermodynamics and other disciplines [14]. Numerical simulation is an important technical means in engineering applications which can solve some complex operations (e.g., the dynamic flow process, mechanical process and heat transfer process) that cannot be completed in the actual system [15]. In recent years, the methodology has been widely used in engineering applications; for example, [PERSON] et al. [16] numerically simulated the flow and heat transfer characteristics in the fully developed periodic section of the flow across the dense tube bundles with different tube spacings in the Re ranges of 2000-40,000. The results showed that the maximum heat transfer coefficient of tube bundles with small spacings is 3 times of that of tube bundles with large spacings, indicating that the tube bundles with small spacings may be the best pipe arrangement for the achieving the maximum heat exchange performance. [PERSON] et al.