## 1 Introduction  
The carrot (_Daucus carota_ L.) is a popular root vegetable which is commonly seen as orange in color. From a nutrition point of view, carrots are a good source of alpha- and beta-carotene, vitamin K, and vitamin B6. Carrots are planted worldwide, with a production of around 44 million tons in 2019 according to the data from Food and Agriculture Organization (FAO). Traditionally, the taproot of carrot is freshly eaten in daily meals. Nowadays, in addition to their fresh consumption, carrots have been dried to prolong storage time and save transport cost. Dried carrot is used in the production of instant soups, instant sauces, ingredients for stuffings, casseroles, stews, and ped food [1]. Carrot is often dried at high temperatures in tunnel, tray, and belt dryer systems, with hot air serving as the drying agent. The heat and mass transfer kinetics of carrot in the course of hot-air drying has been investigated in several previous studies. Traditionally, the moisture diffusivity of carrots during the hot-air-drying process has been described by the Arrhenius equation and the modified Arrhenius equation [2; 3; 4; 5; 6; 7]. The difference in core and annular moisture diffusivity in carrot was also investigated [8]. The impact of drying conditions on the product quality aspects (color, chemical composition: \(\beta\)-carotene, antioxidants, vitamin B1, B2, polyphenols, etc.) was experimentally investigated [9; 10; 11]. Recently, to improve product quality and to reduce energy consumption, carrot has been dried in a low-temperature air environment [12; 13; 14]. However, since the temperature difference between the drying agent and the product is small, the drying process at low temperatures is often slow. External energy sources (such as microwave, infrared, and ultrasound (US)) have, therefore, been used to assist the drying process by enhancing water vaporization [12; 14; 15]. The US-assisted drying process has experimentally been proven to shorten drying time and improve product quality. This implies that US-assisted drying techniques have the potential to save energy and improve the economic benefit of food processing products. In the laboratory-scale dryer, it has been demonstrated that the integration of US pre-treatment in conjunction with microwave and hot-air drying effectively reduces the specific energy consumption required to remove one kilogram of water. The reduction falls within the range of 10% to 20% for various products, such as pistachio kernel, basil, and raspberries [16; 17; 18]. Furthermore, the introduction of US pre-treatment can enhance overall energy efficiency by a margin, ranging from 16% to 29%, as reported in [19]. Thus, for the industrial sector to effectively implement US-assisted drying techniques, a comprehensive understanding of drying kinetic in US-assisted drying is essential [20]. In addition to using US sources to assist the drying process directly, US pre-treatment has also been progressively applied in recent years and has demonstrated the potential to significantly reduce the drying time of carrots as reported, for example, in the work of [PERSON] et al. [21], [PERSON] et al. [22], and [PERSON] et al. [23]. This positive impact of US pre-treatment on drying behavior is attributed to the cavitation due to powerful waves in liquid solutions which is dependent on the properties of the liquid, the presence of air, and the acoustic power of the ultrasonic system. The use of US causes cavitation by producing micro gas bubbles within a liquid. These bubbles, when they burst, release powerful shock waves and free radicals across the cell membrane, both of which contribute to the inactivation of microorganisms. Despite this understanding, however, the mathematical description of the drying processes remains at the empirical level, where models such as Page and Newton models have been used. Furthermore, the enhancement of US pre-treatment on moisture diffusivity has not been considered.