An Approximate Analytic Solution to a Non-Linear ODE for Air Jet Velocity Decay through Tree Crops Using Piecewise Linear Emulations and Rectangle Functions

Mathematical Modelling of the Entrainment Ratio of High Performance Supersonic Industrial Ejectors

For many years now, manufacturers have been producing supersonic ejectors with a high entrainment ratio for the chemical, oil, and food industries. In the present work, mathematical modelling of the entrainment ratio of such industrial ejectors is carried out, in which a variation of the diffuser efficiency is also assumed to be a function of the Mach number of the motive gas. To determine this unknown relationship, the mathematical modelling was overturned by inserting the entrainment ratios of ten different high-performance industrial ejectors, as identified through an experimental investigation. The mathematical modelling, completed through the use of the relationship between the diffuser efficiency and the Mach number of the motive gas, was applied to sixty-eight ejectors, built and tested experimentally over the last twenty years as part of research aimed at the development of thermal ejector refrigeration systems (ERSs), to obtain the entrainment ratios proposed by the manufacturers (industrial entrainment ratios). A comparison of the experimental entrainment ratios with respect to the industrial ones demonstrated that the former were always lower, ranging from a minimum of −17% to a maximum of −82%. These results indicate that the lab-built ejectors for ERS prototypes can be improved. Therefore, in the future, researchers should apply numerical analysis iteratively, starting from a given geometry of the ejector, and modifying it until the numerical analysis provides the industrial value of the entrainment ratio.

Conveyor-Belt Dryers with Tangential Flow for Food Drying: Development of Drying ODEs Useful to Design and Process Adjustment

The mathematical investigation presented in this paper concerns the conveyor-belt dryer with tangential flow operating in co-current. This dryer is bigger than the continuous through-circulation conveyor dryer but has the advantage of better preserving the organoleptic and nutritional qualities of the dried product. In a previous work a mathematical modeling of the conveyor-belt dryer with tangential flow was carried out to offer guidelines for its optimized design. The last of those design guidelines indicated the need for an optimized adjustment of the dryer to ensure the constant maintenance of the final moisture content of the product. The fast and very precise measurement of the moisture content as the first step in the feedback chain was therefore necessary. Considering the difficulty of this type of measurement, two specific ordinary differential equations (ODEs) were obtained with the mathematical investigation of this work. Their solution became a relationship between the final moisture content of the product, the outlet air temperature, and other quantities that could be easily implemented in an automatic dryer control system. Therefore, the fast and accurate and much less expensive measurement of the temperature of the air leaving the dryer, owing to the relationship found, replaces the measurement of moisture content for the adjustment system. The experimental verification of the relationship highlighted the need to introduce a modification by which the relationship was finally validated.