Browse
Keywords
Records with Keyword: Modelling
284. LAPSE:2023.4032
Modelling a Switching Process of IGBTs with Influence of Temperature Taken into Account
February 22, 2023 (v1)
Subject: Modelling and Simulations
Keywords: IGBT, Modelling, power semiconductor devices, SPICE, switching, thermal phenomena, transient analysis
In this article the problem of modelling a switching process of Insulated Gate Bipolar Transistors (IGBTs) in the SPICE software is considered. The new form of the considered transistor model is presented. The model includes controlled voltage and current sources, resistors and voltage sources. In the model, influence of temperature on dc and dynamic characteristics of the IGBT is taken into account. A detailed description of the dynamic part of this model is included in the article and some results of experimental verification are shown. Verification is performed for a transistor IRG4PC40UD by International Rectifier. The presented results of computations and measurements show clearly influence of temperature on on-time and off-time, and additionally switching energy losses are observed. Moreover, the results of investigations performed with the use of the new model are compared to the results of computations performed with classical models of the considered device given in the litera... [more]
285. LAPSE:2023.4011
Evaluating Magnetocaloric Effect in Magnetocaloric Materials: A Novel Approach Based on Indirect Measurements Using Artificial Neural Networks
February 22, 2023 (v1)
Subject: Materials
Keywords: artificial neural network, gadolinium, LaFe13 − x − yCoxSiy, magnetic refrigeration, magnetocaloric effect, Modelling
The thermodynamic characterisation of magnetocaloric materials is an essential task when evaluating the performance of a cooling process based on the magnetocaloric effect and its application in a magnetic refrigeration cycle. Several methods for the characterisation of magnetocaloric materials and their thermodynamic properties are available in the literature. These can be generally divided into theoretical and experimental methods. The experimental methods can be further divided into direct and indirect methods. In this paper, a new procedure based on an artificial neural network to predict the thermodynamic properties of magnetocaloric materials is reported. The results show that the procedure provides highly accurate predictions of both the isothermal entropy and the adiabatic temperature change for two different groups of magnetocaloric materials that were used to validate the procedure. In comparison with the commonly used techniques, such as the mean field theory or the interpol... [more]
286. LAPSE:2023.3952
A Predictive Model for Dry-Growth Icing on Composite Insulators under Natural Conditions
February 22, 2023 (v1)
Subject: Modelling and Simulations
Keywords: dry-growth icing, insulator, Modelling, natural conditions
Icing can adversely influence electric power system security. Two main issues are caused by icing: the overload of transmission lines, and the reduction in the insulation ability of the insulators. Most previous research has focused on the flashover characteristics of ice-covered insulators, but research on the icing process of the insulator is seriously lacking. Considering the effect of icing shape, the outer airflow field of an insulator was calculated and the local collision efficiencies of water droplets (β₁) were investigated according to the Lagrange algorithm. The simulation showed that the values of β₁ on the insulator edge and rod are much higher than on the insulator surface, and both were significantly influenced by the wind speed and median volume diameter (MVD) of the water droplets. Based on thermal balance equations, a dynamic dry-growth icing model was established. Using the natural icing conditions of Xuefeng Mountain (China) as an example, validation experiments were... [more]
287. LAPSE:2023.3871
Interaction and Coordination among Nuclear Power Plants, Power Grids and Their Protection Systems
February 22, 2023 (v1)
Subject: Modelling and Simulations
Keywords: coordination between units and grids, dynamic simulation, grid-related protection, Modelling, nuclear power plant (NPP), pressurized water reactor (PWR)
Nuclear power plants (NPPs) have recently undergone rapid development in China. To improve the performance of both NPPs and grids during adverse conditions, a precise understanding of the coordination between NPPs and grids is required. Therefore, a new mathematical model with reasonable accuracy and reduced computational complexity is developed. This model is applicable to the short, mid, and long-term dynamic simulation of large-scale power systems. The effectiveness of the model is verified by using an actual NPP full-scope simulator as a reference. Based on this model, the interaction and coordination between NPPs and grids under the conditions of over-frequency, under-frequency and under-voltage are analyzed, with special stress applied to the effect of protection systems on the safe operation of both NPPs and power grids. Finally, the coordinated control principles and schemes, together with the recommended protection system values, are proposed for both NPPs and grids. These res... [more]
288. LAPSE:2023.3842
Modeling and Experimental Validation of a Volumetric Expander Suitable for Waste Heat Recovery from an Automotive Internal Combustion Engine Using an Organic Rankine Cycle with Ethanol
February 22, 2023 (v1)
Subject: Energy Systems
Keywords: Ethanol, internal combustion engine (ICE), Modelling, organic Rankine cycle (ORC), swash-plate, thermal inertia, waste heat recovery (WHR)
Waste heat recovery (WHR) in exhaust gas flow of automotive engines has proved to be a useful path to increase the overall efficiency of internal combustion engines (ICE). Recovery potentials of up to 7% are shown in several works in the literature. However, most of them are theoretical estimations. Some present results from prototypes fed by steady flows generated in an auxiliary gas tank and not with actual engine exhaust gases. This paper deals with the modeling and experimental validation of an organic Rankine cycle (ORC) with a swash-plate expander integrated in a 2 L turbocharged petrol engine using ethanol as working fluid. A global simulation model of the ORC was developed with a maximum difference of 5%, validated with experimental results. Considering the swash-plate as the main limiting factor, an additional specific submodel was implemented to model the physical phenomena in this element. This model allows simulating the fluid dynamic behavior of the swash-plate expander us... [more]
289. LAPSE:2023.3779
Tool Chain for Deriving Consistent Storage Model Parameters for Optimization Models
February 22, 2023 (v1)
Subject: Numerical Methods and Statistics
Keywords: design of experiments, Energy Storage, Modelling, Optimization, regression analysis, scientific framework, Simulation, tool chain
Since existing energy system models often represent storage behavior in a simplified way, in this work, a tool chain for deriving consistent storage model parameters for optimization models is developed. The aim of our research work is to identify what are non-negligible influences on the the technical characteristics and dynamic behavior of the storage, to quantify the effect of these influences, and represent these effects in the model. This paper describes the developed tool chain and presents its application using an example. The tool chain consists of the steps “parameter screening”, “dynamic simulation”, “regression analysis” and “refining optimization model”. It is investigated which parameters have an influence on the storage system (here pumped hydroelectric energy storage (PHES)), how the storage behavior is modeled, which influencing factors have a measurable effect on the system, and how these findings can be integrated into optimization models. The main finding is that in... [more]
290. LAPSE:2023.3652
Model of an Air Transformer for Analyses of Wireless Power Transfer Systems
February 22, 2023 (v1)
Subject: Modelling and Simulations
Keywords: air transformer, inductive wireless power transfer, IPT, Modelling, SPICE, wireless power transfer, WPT
This article presents a new model of a dedicated air transformer for computer analyses of wireless power transfer systems. This model includes a form of subcircuit for SPICE. It takes into account the electric, magnetic and thermal properties of the modeled device. The form of the elaborated model is presented and the results of its experimental verification are shown. Some results from measurements and computations of an air transformer and a wireless power transfer system containing this transformer are shown and discussed. The structure of the tested system and the measuring setup used are also described. The results of measurements and computations illustrating the influence of the distance between the windings of the air transformer and the displacement between its windings on the output voltage of the power transfer system are presented and discussed. The influence of load resistance on the properties of the considered system is analyzed.
291. LAPSE:2023.3645
Simulation Analysis of a Single-Phase AC Circuit with a Nonlinear Load
February 22, 2023 (v1)
Subject: Modelling and Simulations
Keywords: equivalent circuit diagram, harmonics, IEEE 1459–2010, Modelling, nonlinear circuit, reactive power, Simulation
The IEEE 1459 standard states that: “There is a need to quantify correctly the distortions caused by the nonlinear and parametric loads …” The paper undertakes this problem. The article includes an analysis of selected steady-state properties of a single-phase AC circuit containing a series connection of a resistance, an inductance, and a nonlinear element described by a signum function. The circuit is supplied from a sinusoidal voltage source. The mathematical model of this circuit is described using dimensionless variables. Based on the harmonic balance, a description of the circuit for the fundamental harmonic and the circuit for the remaining harmonics was separated. On the basis of the relationship between the current and voltage waveforms, the solution for all harmonics, the harmonic content coefficients, and the elements of the equivalent scheme of the nonlinear load, were estimated symbolically. Based on the circuit equations, a circuit model and measurement systems were develo... [more]
292. LAPSE:2023.3534
The Influence of Distributed Generation on the Operation of the Power System, Based on the Example of PV Micro-Installations
February 22, 2023 (v1)
Subject: Modelling and Simulations
Keywords: distributed generation, Modelling, photovoltaic micro-installation, power supply quality, power system
This article describes the problems associated with distributed electrical power generation and the most frequent interruptions occurring in power grids. The most common methods of improving the quality of the power supply were analyzed and the possibilities offered by energy storages in this respect were considered. The operating parameters of an exemplary PV system connected to the power grid were analyzed. For this purpose, the model implemented in the Matlab/Simulink environment was used. Based on the conducted analysis and a review of the literature, conclusions were drawn and solutions were presented which could improve the quality and the reliability of power supply. The simulations conducted focused on the co-operation of individual photovoltaic, micro-installations, with rated powers of 12.2, 19.825, and 39.65 Kw in the power grid, which also corresponds to the co-operation of several, smaller micro-installations with low density.
293. LAPSE:2023.3381
Overview of Numerical Simulation of Solid-State Anaerobic Digestion Considering Hydrodynamic Behaviors, Phenomena of Transfer, Biochemical Kinetics and Statistical Approaches
February 22, 2023 (v1)
Subject: Numerical Methods and Statistics
Keywords: biogas, Computational Fluid Dynamics, degradation kinetics, diffusion, empirical models, Machine Learning, Modelling
Anaerobic digestion (AD) is a promising way to produce renewable energy. The solid-state anaerobic digestion (SSAD) with a dry matter content more than 15% in the reactors is seeing its increasing potential in biogas plant deployment. The relevant processes involve multiple of evolving chemical and physical phenomena that are not crucial to conventional liquid-state anaerobic digestion processes (LSAD). A good simulation of SSAD is of great importance to better control and operate the reactors. The modeling of SSAD reactors could be realized either by theoretical or statistical approaches. Both have been studied to a certain extent but are still not sound. This paper introduces the existing mathematical tools for SSAD simulation using theoretical, empirical and advanced statistical approaches and gives a critical review on each type of model. The issues of parameter identifiability, preference of modeling approaches, multiscale simulations, sensibility analysis, particularity of SSAD o... [more]
294. LAPSE:2023.3206
Reassessment of Thin-Layer Drying Models for Foods: A Critical Short Communication
February 22, 2023 (v1)
Subject: Food & Agricultural Processes
Keywords: drying curves, Modelling, regression, uncertainty
Modeling the thin-layer drying of foods is based on describing the moisture ratio versus time data by using a suitable mathematical model or models. Several models were proposed for this purpose and almost all studies were related to the application of these models to the data, a comparison and selecting the best-fitted model. A careful inspection of the existing drying data in literature revealed that there are only a limited number of curves and, therefore, the use of some models, especially the complex ones and the ones that require a transformation of the data, should be avoided. These were listed based on evidence with the use of both synthetic and published drying data. Moreover, the use of some models were encouraged, again based on evidence. Eventually, some suggestions were given to the researchers who plan to use mathematical models for their drying studies. These will help to reduce the time of the analyses and will also avoid the arbitrary usage of the models.
295. LAPSE:2023.3195
Temporal Evolution of Cooling by Natural Convection in an Enclosed Magma Chamber
February 22, 2023 (v1)
Subject: Modelling and Simulations
Keywords: fluid mechanics, heat flow, magma chamber processes, mechanics, Modelling, multiphase flows, numerical approximations and analysis
This research numerically studies the transient cooling of partially liquid magma by natural convection in an enclosed magma chamber. The mathematical model is based on the conservation laws for momentum, energy and mass for a non-Newtonian and incompressible fluid that may be modeled by the power law and the Oberbeck−Boussinesq equations (for basaltic magma) and solved with the finite volume method (FVM). The results of the programmed algorithm are compared with those in the literature for a non-Newtonian fluid with high apparent viscosity (10−200 Pa s) and Prandtl (Pr = 4 × 104) and Rayleigh (Ra = 1 × 106) numbers yielding a low relative error of 0.11. The times for cooling the center of the chamber from 1498 to 1448 K are 40 ky (kilo years), 37 and 28 ky for rectangular, hybrid and quasi-elliptical shapes, respectively. Results show that for the cases studied, natural convection moved the magma but had no influence on the isotherms; therefore the main mechanism of cooling is conduct... [more]
296. LAPSE:2023.3076
Research and Modelling the Ability of Waste from Water and Wastewater Treatment to Remove Phosphates from Water
February 21, 2023 (v1)
Subject: Modelling and Simulations
This research investigated the ability of two materials, which are waste generated during water treatment and wastewater treatment, to remove phosphates from water. The selected materials were quartz sand used in drinking water treatment plants (OQS) and incinerated (600 °C) sewage sludge (ISS). The materials were chosen for their composition: both contain aluminium, iron, and calcium. The experiments were carried out in the laboratory (in batch and in columns stand). Modelling of the sorption processes was performed on the basis of results from experiments in batches. The maximum adsorption capacity of the OQS was 1.14 mg/g obtained using the linearized Langmuir model and the maximum adsorption capacity of the ISS was 0.86 mg/g for the linearized Langmuir model (in batch). A pseudo-first-order model obtained using a nonlinear fit can accurately explain phosphate adsorption kinetics using both adsorbents: OQS and ISS. During the column filtration experiment, a higher sorption capacity... [more]
297. LAPSE:2023.3043
Quasi-One-Dimensional Flow Modeling for Flight Environment Simulation System of Altitude Ground Test Facilities
February 21, 2023 (v1)
Subject: Environment
Keywords: Flight Environment Simulation System, Modelling, numerical simulation, pipe, quasi-one-dimensional flow, regulating valve
The Flight Environment Simulation System (FESS) at Altitude Ground Test Facilities (AGTF) is used to test aircraft engines. The FESS model is the basis of research and verification of advanced control algorithms. To further improve the steady and dynamic accuracy of the FESS model, a modeling method based on quasi-one-dimensional flow is proposed. Firstly, based on the unified inlet/outlet boundary specifications, the component models of test equipment, such as the quasi-one-dimensional flow model of pipe, the regulating valve model considering the heat transfer process, the multi-inlet and multi-outlet volume model reflecting the mixing characteristics of air flow, and the air source model and engine model, were established. Secondly, according to the real structure and working mechanism of the FESS, the above component models were used to build the numerical simulation model of the FESS. The simulation results showed that the relative deviation of mass flow and pressure were less tha... [more]
298. LAPSE:2023.3038
Enabling Total Process Digital Twin in Sugar Refining through the Integration of Secondary Crystallization Influences
February 21, 2023 (v1)
Subject: Modelling and Simulations
Keywords: crystallization, digital twin, DoE, foods and drugs, Modelling, natural resources, quality by design, sugar
Crystallization is the main thermal process resulting in the formation of solid products and, therefore, is widely spread in all kinds of industries, from fine chemicals to foods and drugs. For these high-performance products, a quality by design (QbD) approach is applied to maintain high product purity and steady product parameters. In this QbD-context, especially demanded in the foods and drugs industry, the significance of models to deepen process understanding and moving toward automated operation is steadily rising. To reach these aspired goals, besides major process influences like crystallization temperature, other impacting parameters have to be evaluated and a model describing these influences is sought-after. In this work, the suitability of a population balance-based physico-chemical process model for the production of sugar is investigated. A model overview is given and the resulting model is compared to a statistical DoE scheme. The resulting process model is able to pictu... [more]
299. LAPSE:2023.2942
Modeling in Brewing—A Review
February 21, 2023 (v1)
Subject: Modelling and Simulations
Keywords: beer, general approach, Modelling, Optimization
Beer production has over a thousand-year tradition, but its development in the present continues with the introduction of new technological and technical solutions. The methods for modeling and optimization in beer production through an applied analytical approach have been discussed in the present paper. For this purpose, the parameters that are essential for the main processes in beer production have been considered—development of malt blends, guaranteeing the main brewing characteristics; obtaining wort through the processes of mashing, lautering and boiling of wort; fermentation and maturation of beer. Data on the mathematical dependences used to describe the different stages of beer production (one-factor experiments, modeling of mixtures, experiment planning, description of the kinetics of microbial growth, etc.) and their limits have been presented, and specific research results of various authors teams working in this field have been cited. The independent variables as well as... [more]
300. LAPSE:2023.2915
Design of Experiments for Modeling of Fermentation Process Characterization in Biological Drug Production
February 21, 2023 (v1)
Subject: Process Design
Keywords: design of experiments, Modelling, quality by design
Biological products are increasingly important, and therefore the industry has begun to adopt quality by design, as recommended by the ICH and the U.S. FDA. Smaller companies, however, have faced difficulties in employing full-scale experiments or the quality by design strategy. Thus, this study provides an alternative way to build a model from existing data with experimental software that does not require full-scale experiments. This empirical study hopes to provide a practical way to improve the efficiency of smaller biopharmaceutical companies and researchers. Moreover, the models provided here can be applied to process characterization in recombinant protein production.
301. LAPSE:2023.2840
Modeling and Experimental Studies on Carbon Dioxide Absorption with Sodium Hydroxide Solution in a Rotating Zigzag Bed
February 21, 2023 (v1)
Subject: Modelling and Simulations
The enhancement of mass transfer is very important in CO2 absorption, and a rotating zigzag bed (RZB) is a promising device to intensify the gas−liquid mass transfer efficiency. In this study, the mass transfer characteristics in an RZB in relation to the overall gas-phase volumetric mass-transfer coefficient (KGa) were investigated with a CO2−NaOH system. A mathematical model was established to illustrate the mechanism of the gas−liquid mass transfer with irreversible pseudo-first-order reaction in the RZB. The effects of various operating conditions on KGa were examined. Experimental results show that a rise in the liquid flow rate, inlet gas flow rate, rotational speed, absorbent temperature, and absorbent concentration was conducive to the mass transfer between gas and liquid in the RZB. It was found that the rotational speed had the largest impact on KGa in the RZB. The KGa predicted by the model agreed well with that by the experiments, with deviations generally within 10%. There... [more]
302. LAPSE:2023.2801
Analysis of the Time Step Influence in the Yearly Simulation of Integrated Seawater Multi-Effect Distillation and Parabolic trough Concentrating Solar Thermal Power Plants
February 21, 2023 (v1)
Subject: Modelling and Simulations
Keywords: CSP+D, Modelling, multi-effect distillation, parabolic trough, simulation tool, time step influence, yearly simulation
The joint demand for power and freshwater is continuously increasing due to population growth, the rise of economic activity, and climate change. Integrated concentrating solar thermal power and desalination (CSP+D) plants may provide a key solution for the pressing freshwater deficit and energy problems in many regions of the world. Simulation tools with an accurate prediction of the yearly electric energy and freshwater production are needed. This paper analyzed the influence of the time step in the annual simulation of a CSP+D plant composed of a seawater multi-effect distillation unit and a parabolic trough concentrating solar thermal power plant, considering the location of Tabernas (Spain). A dynamic simulation tool of this system was developed, implementing the models in Engineering Equation Solver. The annual electricity and water productions obtained for the study case considered were 154 GWh and 3.45 hm3, respectively, using 5 min time steps, and 94 GWh and 2.1 hm3, respectiv... [more]
303. LAPSE:2023.2780
Comparison of Knudsen Diffusion and the Dusty Gas Approach for the Modeling of the Freeze-Drying Process of Bulk Food Products
February 21, 2023 (v1)
Subject: Food & Agricultural Processes
Keywords: drying of frozen particles, dusty gas model, freeze-drying, improvement of mass transfer, internal porous structure, Modelling
Freeze-drying is generally used to achieve high quality products and preserve thermal sensitive components; however, it is also considered as a high energy and costly process. Modeling of the process can help to optimize the process to reduce these drawbacks. In this work, a mathematical model is presented to predict the heat and mass transfer behavior for freeze-drying of porous frozen food particles during freeze-drying to optimize the process. For the mass transfer, a comparison between Knudsen diffusion and the more complex dusty-gas approach is performed. Simulation results of a single particle are validated by experiments of single-layer drying to extend the usage of this model from a single particle to a particle bed. For the moisture transfer, adaption parameters are introduced and evaluated. A comparison shows a good agreement of the model with experimental results. The results furthermore suggest a strong correlation of the drying kinetics with pore size and particle porosity... [more]
304. LAPSE:2023.2696
Optimal Darwinian Selection of Microorganisms with Internal Storage
February 21, 2023 (v1)
Subject: Biosystems
Keywords: chemostat, Droop model, microalgae, Modelling, nonlinear control, optimal control, photobioreactor, Pontryagin’s principle, singular control
In this paper, we investigate the problem of species separation in minimal time. Droop model is considered to describe the evolution of two distinct populations of microorganisms that are in competition for the same resource in a photobioreactor. We focus on an optimal control problem (OCP) subject to a five-dimensional controlled system in which the control represents the dilution rate of the chemostat. The objective is to select the desired species in minimal-time and to synthesize an optimal feedback control. This is a very challenging issue, since we are are dealing with a ten-dimensional optimality system. We provide properties of optimal controls allowing the strain of interest to dominate the population. Our analysis is based on the Pontryagin Maximum Principle (PMP), along with a thorough study of singular arcs that is crucial in the synthesis of optimal controls. These theoretical results are also extensively illustrated and validated using a direct method in optimal control (... [more]
305. LAPSE:2023.2692
Investigating the Trade-Off between Design and Operational Flexibility in Continuous Manufacturing of Pharmaceutical Tablets: A Case Study of the Fluid Bed Dryer
February 21, 2023 (v1)
Subject: Process Design
Keywords: Modelling, operational envelopes, operational flexibility, pharmaceutical manufacture, uncertainty
Market globalisation, shortened patent lifetimes and the ongoing shift towards personalised medicines exert unprecedented pressure on the pharmaceutical industry. In the push for continuous pharmaceutical manufacturing, processes need to be shown to be agile and robust enough to handle variations with respect to product demands and operating conditions. In this paper we examine the use of operational envelopes to study the trade-off between the design and operational flexibility of the fluid bed dryer at the heart of a tablet manufacturing process. The operating flexibility of this unit is key to the flexibility of the full process and its supply chain. The methodology shows that for the fluid bed dryer case study there is significant effect on flexibility of the process at different drying times with the optimal obtained at 700 s. The flexibility is not affected by the change in volumetric flowrate, but only by the change in temperature. Here the method used a black box model to show... [more]
306. LAPSE:2023.2640
Review on Zigzag Air Classifier
February 21, 2023 (v1)
Subject: Process Design
Keywords: classifier, design, Modelling, separation, zigzag
The zigzag (ZZ) classifier is a sorting and classification device with a wide range of applications (e.g. recycling, food industry). Due to the possible variation of geometry and process settings, the apparatus is used for various windows of operation due to the specifications of the separation (e.g. cut sizes from 100 µm to several decimetres, compact and fluffy materials as well as foils). Since the ZZ classifier gains more and more interest in recycling applications, it is discussed in this paper, with regards to its design, mode of operation, influencing parameters and the research to date. Research on the ZZ-classifier has been ongoing on for more than 50 years and can be divided into mainly experimental studies and modelling approaches.
307. LAPSE:2023.2594
Need for a Next Generation of Chromatography Models—Academic Demands for Thermodynamic Consistency and Industrial Requirements in Everyday Project Work
February 21, 2023 (v1)
Subject: Modelling and Simulations
Keywords: adsorption thermodynamics, biochromatography, digital twins, Modelling, process chromatography
Process chromatography modelling for process development, design, and optimization as well as process control has been under development for decades. Still, the discussion of scientific potential and industrial applications needs is open to innovation. The discussion of next-generation modelling approaches starting from Langmuirian to steric mass action and multilayer or thermodynamic consistent real and ideal adsorption theory or colloidal particle adsorption approaches is continued.
308. LAPSE:2023.2537
Optimization of Biofertilizer Formulation for Phosphorus Solubilizing by Pseudomonas fluorescens Ur21 via Response Surface Methodology
February 21, 2023 (v1)
Subject: Process Design
Keywords: biofertilizer, central composite design, Modelling, phosphate solubilizing bacteria
This study aimed to analyze and quantify the effect of different ratios of vermicompost, phosphate rock, and sulfur on P solubilization and release by Pseudomonas fluorescens Ur21, and to identify optimal levels of those variables for an efficient biofertilizer. Twenty experiments were defined by surface response methodology based on a central composite design (CCD), and the effects of various quantities of vermicompost, phosphate rock, and sulfur (encoded by −1, 0, or +1) on P solubilization was explored. The results show that the CCD model had high efficiency for predicting P solubilization (R2 = 0.9035). The strongest effects of the included variables on the observed P solubilization were linear effects of sulfur and organic matter (vermicompost), a quadratic effect of phosphate rock, and an interactive effect of organic matter × phosphate rock. Statistical analysis of the coefficients in the CCD model revealed that vermicompost, vermicompost × phosphate rock, and phosphate rock × p... [more]