Research oriented on identification of operating states variations with the application of mathematical models of thermal processes has been developed in the field of energy processes diagnostics. Simple models, characterised by short calculation time, are necessary for thermal diagnostics needs. Such models can be obtained using empirical modelling methods. Good results brings the construction of analytical model with auxiliary empirical built-in functions. The paper presents a mathematical model of a steam-water cycle containing mass and energy balances and semiempirical models of steam expansion line in turbine as well as heat transfer in exchangers. A model of steam expansion line in a turbine is worked out with the application of a steam flow capacity equation and an internal efficiency of process equation for each group of stages for the analysed turbine. A model of a heat exchanger contains energy balance and the relation describing heat transfer in an exchanger, proposed by Beckman. Estimation of empirical equations coefficients was realised with the application of special and reliable measurements. Estimation criterion was a weighted relative sum of the remainder squares. There are exemplary calculations results presented in the final part of paper.

The present study modelled the effects of operational parameters on the performance of the Falcon concentrator. For this purpose, the Falcon L40 concentrator was tested in narrow particle-size fractions (−600 + 425 μm, −425 + 300 μm, −300 + 212 μm, −212 + 150 μm, −150 + 106 μm, and −106 + 75 μm) at different washing water pressures and artificial gravity forces generated by a spinning bowl. The test samples were prepared artificially, comprising 2% magnetite (Fe3O4) and 98% calcite (CaCO3) by weight. The recovery and grade values of the 60 experimental conditions were investigated and compared for different operational parameters, including particle-size distributions, water pressures, and artificial gravity forces. Two empirical models were developed using non-linear regression analysis to indicate the effects of the operating parameter of the Falcon concentrator on its recovery and grade values. The operational parameters were found to impact the separation performance considerably. Therefore, the Falcon concentrator should operate under optimum conditions, which can be easily predicted using these models, to achieve improved recovery and grade values.

Rainfall is one of the main components of the hydrologic cycle; thus, the availability of accurate rainfall data is fundamental for designing and operating water resources systems and infrastructure. This study aims to develop an empirical model of rainfall intensity ( I_t,p) as a function of its probability ( p) and duration ( t). In 1999–2020, data on the hourly duration of rainfall were collected from automatic rainfall recorder (ARR) gauges. The empirical model has been developed using a statistical approach based on duration ( t) and probability ( p), and subsequently they have been validated with those obtained from ARR data. The resulting model demonstrates good performance compared with other empirical formulas (Sherman and Ishiguro) as indicated by the percent bias ( PBIAS) values (2.35–3.17), ratio of the RMSE (root mean square error) between simulated and observed values to the standard deviation of the observations ( RSR, 0.028–0.031), Nash–Sutcliffe efficiency ( NSE, 0.905–0.996), and index of agreement (d, 0.96–0.98) which classified in the rating of “very good” in model performance. The reliability of the estimated intensity based on the empirical model shows a tendency to decrease as duration ( t) increases, and a good accuracy mainly for the rainfall intensity for shorter periods (1-, 2-, and 3-hours), whereas low accuracy for long rainfall periods. The study found that the empirical model exhibits a reliable estimate for rainfall intensity with small recurrence intervals ( Tr) 2-, 5-, 10-, and a 20-year interval and for a shorter duration ( t). Validation results confirm that the rainfall intensity model shows good performance; thus, it could be used as a reliable instrument to estimate rainfall intensity in the study area.