Multi-dimensional Mathematical Models Research Articles

Express analyzers of motor fuels: the problems of metrological support

The problems of identifying counterfeit motor fuels in real conditions are highlighted. It is noted that the use of express analyzers of motor fuels does not provide the necessary level of control over the accuracy and reliability of measurement results of the parameters of the fuel under study. The problems of metrological support of express analyzers of motor fuels based on near-infrared spectroscopy are considered. The method is based on the rejection of classical spectroscopic approaches in favor of complex multidimensional mathematical models, and the result of the analysis is a forecast; the usual metrological approaches do not work here, since quantitative analysis is replaced by some kind of qualitative analysis. The category of uncertainty is replaced by the category of reliability, which is much less developed. In this situation the usual accuracy control procedures are insufficient. For express analyzers based on the principles of dielmetry and near-infrared spectroscopy, the identification of counterfeit fuel turned out to be approximately the same: the addition of 8–10 % vol. of the counterfeit component practically does not change the measurement results. When the operations with a mathematical model is not allowed (it is the intellectual property of developers) some simple approaches to the assessment of metrological characteristics is proposed. This approach allows the metrologist or the user to verify the relevance of the program model for specific grades of motor fuel. A study of domestic express analyzers based on the principles of measuring the dielectric permittivity of fuels has been carried out. The ability of express analyzers of both types to identify counterfeit fuel is evaluated.

Enhancing Water Temperature Prediction in Stratified Reservoirs: A Process-Guided Deep Learning Approach

Data-driven models (DDMs) are extensively used in environmental modeling yet encounter obstacles stemming from limited training data and potential discrepancies with physical laws. To address this challenge, this study developed a process-guided deep learning (PGDL) model, integrating a long short-term memory (LSTM) neural network and a process-based model (PBM), CE-QUAL-W2 (W2), to predict water temperature in a stratified reservoir. The PGDL model incorporates an energy constraint term derived from W2′s thermal energy equilibrium into the LSTM’s cost function, alongside the mean square error term. Through this mechanism, PGDL optimizes parameters while penalizing deviations from the energy law, thereby ensuring adherence to crucial physical constraints. In comparison to LSTM’s root mean square error (RMSE) of 0.062 °C, PGDL exhibits a noteworthy 1.5-fold enhancement in water temperature prediction (RMSE of 0.042 °C), coupled with improved satisfaction in maintaining energy balance. Intriguingly, even with training on just 20% of field data, PGDL (RMSE of 0.078 °C) outperforms both LSTM (RMSE of 0.131 °C) and calibrated W2 (RMSE of 1.781 °C) following pre-training with 80% of the data generated by the uncalibrated W2 model. The successful integration of the PBM and DDM in the PGDL validates a novel technique that capitalizes on the strengths of multidimensional mathematical models and data-based deep learning models. Furthermore, the pre-training of PGDL with PBM data demonstrates a highly effective strategy for mitigating bias and variance arising from insufficient field measurement data.

Numerical back analysis method of three-dimensional in situ stress fields considering complex surface topography and variable collinearity

Numerical back analysis based on incomplete in situ stress measurements is widely used to estimate the three-dimensional (3D) in situ stress fields of large deep underground caverns. However, the variable collinearity caused by complex geological environments has rarely been considered. This paper proposes a numerical back analysis method for 3D in situ stress fields based on stepwise regression (BSSR). In BSSR, the collinearity between independent variables is associated with the surface topography, and insignificant variables caused by variable collinearity are eliminated through stepwise regression. BSSR was applied to the underground powerhouse of a hydropower station for validation, and the results showed that it can reliably estimate the 3D in situ stress field despite the complex geological environments. Multidimensional mathematical models of in situ stress fields were established with the improved predictive ability and clear physical meaning, which can quantify the contributions of six geological actions and three stress sources. The findings of this study can help with understanding the formation mechanism of in situ stress fields for large, deep underground caverns in complex geological environments and provide a useful reference for optimizing excavation schemes and support designs.

Multi-Dimensional Regression Models for Predicting the Wall Thickness Distribution of Corrugated Pipes.

Corrugated pipes offer both higher stiffness and higher flexibility while simultaneously requiring less material than rigid pipes. Production rates of corrugated pipes have therefore increased significantly in recent years. Due to rising commodity prices, pipe manufacturers have been driven to produce corrugated pipes of high quality with reduced material input. To the best of our knowledge, corrugated pipe geometry and wall thickness distribution significantly influence pipe properties. Essential factors in optimizing wall thickness distribution include adaptation of the mold block geometry and structure optimization. To achieve these goals, a conventional approach would typically require numerous iterations over various pipe geometries, several mold block geometries, and then fabrication of pipes to be tested experimentally—an approach which is very time-consuming and costly. To address this issue, we developed multi-dimensional mathematical models that predict the wall thickness distribution in corrugated pipes as functions of the mold geometry by using symbolic regression based on genetic programming (GP). First, the blow molding problem was transformed into a dimensionless representation. Then, a screening study was performed to identify the most significant influencing parameters, which were subsequently varied within wide ranges as a basis for a comprehensive, numerically driven parametric design study. The data set obtained was used as input for data-driven modeling to derive novel regression models for predicting wall thickness distribution. Finally, model accuracy was confirmed by means of an error analysis that evaluated various statistical metrics. With our models, wall thickness distribution can now be predicted and subsequently used for structural analysis, thus enabling digital mold block design and optimizing the wall thickness distribution.

Qualimetric vermouth authenticity model

Modern methods of physical and chemical analysis make it possible to determine the composition of wine with great accuracy, define its belonging to the extended group of the category, as well as to quantify and qualitatively evaluate the foreign impurities contained in it. However, these methods cannot give a complete assessment of wine authenticity, which is based on the complex interaction of the aroma and its components. Therefore, it is impossible to assess the quality of wines, especially special flavored wines, without auxiliary models for organoleptic analysis. A “blind” organoleptic analysis without these models does not allow the taster reliably determining the type and category of wines, as well as their authenticity. This paper presents the method of determining a sample of tasted vermouth as a category of special flavored wines by comparing it with the initial parameters of taste, aroma and bouquet of the “authentic vermouth”. These parameters are presented in the form of two models: schematic five-stage qualimetric and multidimensional mathematical models. All parameters are based on the previously obtained results of the study of unique parameters of taste compositions of the varieties of vermouth samples purchased from retail chains. The veracity of the developed models was tested using the generally recognized triangular diagram of the Vedel’s triangle.

NUMERICAL SIMULATION OF ATMOSPHERIC POLLUTION DURING AN EXTREME SITUATION AT A CHLORINE OVERFLOW STATION

Problem statement. The problem of forecasting the dynamics of the formation of zones of chemical pollution due to accidental emissions of chlorine at the chlorine overflow station is considered. There is a large village near this chlorine overflow station, so in case of an emergency there is a risk of toxic damage to people. From this point of view, the development of effective multidimensional mathematical models to predict the formation of zones of chemical contamination is of great relevance. The purpose of the article. Development of a numerical multi-parameter model and computer program for forecasting the dynamics of air pollution due to accidental leakage of chlorine at the chlorine station. Methodology. For mathematical modeling of the propagation of a chemically hazardous substance, a three-dimensional equation of mass transfer is used (the equation of G.I. Marchuk). This modeling equation takes into account different wind directions, changes in the vertical coefficient of atmospheric diffusion with height, the intensity of the release of a chemically hazardous substance, and the location of the emission source. For the numerical integration of the three-dimensional mass transfer equation, finite-difference splitting schemes are used. First, for the modeling equation of mass transfer, its physical splitting is carried out: the equations of transfer due to velocity and due to diffusion are considered separately. Also, a separate step is to change the concentration of a chemically hazardous substance due to the action of a pollution source. Next, a difference splitting scheme is constructed. At each step of the splitting, the value of the concentration of a chemically hazardous substance is determined according to an explicit scheme. Scientific novelty. A numerical model is proposed to calculate the dynamics of accidental air pollution due to the release of chemically hazardous substances. The mathematical model takes into account the physical factors that significantly affect the process of distribution of chemically hazardous substances in the atmosphere. Practical significance. Based on the developed model, a code is created that allows you to quickly calculate the process of accidental air pollution. The mathematical model can be used in developing an emergency response plan. Conclusions. The developed mathematical model and the computer code that implements it allow us to study the dynamics of the spread of chemically hazardous substances in the air. The developed computer program can be implemented on low and medium power computers. The results of a computational experiment are presented.

Developing features of the near-bottomhole zones in productive formations at fields with high gas saturation of formation oil

The article studies the formation features of the bottomhole zones in productive formations during operation of production wells in the north of the Perm Territory. Their distinctive feature is the high gas saturation of formation oil. The most widely used parameter in Russian and world practice – the skin factor was used as a criterion characterizing the state of the bottomhole zone. Analysis of scientific publications has shown that one of the main problems of applying the skin factor to assess the state of bottomhole zones is the ambiguity of interpretations of its physical meaning and the impossibility of identifying the prevailing factors that form its value. The paper proposes an approach to identifying such factors in the conditions of the fields under consideration, based on multivariate correlation-regression analysis. Choice of this tool is due to the complexity of the processes occurring in the “formation – bottomhole zone – well” system. When describing complex multifactorial processes, the chosen method demonstrates a high degree of reliability. For a large number of wells in the region, significant material was collected and summarized, including the results of determining the skin factor (1102 values) during hydrodynamic investigations, as well as data on the values ​​of various geological and technological indicators, which can probably be statistically related to the value of the skin factor. A series of multidimensional mathematical models has been built; the skin factor was used as a predicted parameter, and data on the values ​​of geological and technological indicators were used as independent indicators. Analysis of the constructed models is a key stage of this study. Set of parameters included in the multidimensional models, sequence of their inclusion and contribution to the total value of the achieved determination coefficient as the main indicator for the performance of the constructed models were studied. It has been established that the main factor influencing the state of the bottomhole zone is oil degassing. Significant differences in the formation features of the skin factor in the terrigenous and carbonate sediments at the fields under consideration have been determined.

Prediction of reservoir pressure and study of its behavior in the development of oil fields based on the construction of multilevel multidimensional probabilistic-statistical models

Determination of the current reservoir pressure in oil production wells selection zones is an urgent task of field development monitoring. The main method for its determination is hydrodynamic studies under unsteady conditions. At the same time, the process of restoring bottomhole pressure to the value of reservoir pressure often lasts a significant period of time, which leads to long downtime of the fund and significant shortfalls in oil production. In addition, it seems rather difficult to compare reservoir pressures with each other in the wells due to the different timing of the studies, since it is impossible to simultaneously stop the entire fund for measuring the reservoir pressure in the field. The article proposes a new method for determining the current reservoir pressure in the extraction zones, based on the construction of multidimensional mathematical models based on the data of geological and technological development indicators. As the initial data, the values of reservoir pressure, determined during processing of the materials of hydrodynamic studies of wells, as well as a set of geological and technological indicators, probably affecting its value, were used (initial reservoir pressure for each well, the duration of its operation at the time of study, liquid rate, bottomhole pressure, the initial permeability and the current collector in the drainage area, GOR accumulated values oil, and liquid water, and skin factor). In the course of the research, several variants of statistical modeling were used, in the process of which the regularities of the reservoir pressure behavior during the development of reserves were established, individual for the object of development. The obtained models are characterized by a high degree of reliability and make it possible to determine the desired value with an error of no more than 1.0 MPa.

Estimation of the reliability of determination of filtering parameters of productive formations using multi-dimensional regression analysis

Determination of reliable values of filtration parameters of productive strata is the most important task of monitoring the processes of developing reserves. One of the most effective methods for solving the problem is hydrodynamic testing of wells using the pressure recovery method, as well as modern methods - the pressure stabilization method and the method based on production analysis (Decline Analyze). This article is devoted to the assessment of the reliability of these three methods in determining the filtration parameters of terrigenous and carbonate productive deposits of oil fields in the Perm Krai. To solve the problem, multivariate regression analysis was used. A series of multidimensional mathematical models of well flow rates was built using filtration parameters determined for each of the methods. It is proposed to consider the filtration parameters included in the models with the maximum statistical estimates of performance as the most reliable. With regard to the fields under consideration, it was found that in terrigenous reservoirs, all three methods demonstrate stable results. In carbonate reservoirs, reliable values of filtration parameters are determined by processing pressure build-up curves. Pressure stabilization and production analysis methods show less robust results and require additional research in order to develop sound recommendations for their practical application. Keywords: permeability; skin factor; pressure stabilization curve; decline analyze; liquid flow rate; geological and technological parameters; oil deposit; carbonate deposits.

ПОСТРОЕНИЕ И АНАЛИЗ ПРОГНОЗИРУЮЩИХ МОДЕЛЕЙ ОПАСНЫХ ТЕХНИЧЕСКИХ СОСТОЯНИЙ МАШИНОСТРОИТЕЛЬНЫХ ИЗДЕЛИЙ

An approach to the design of adaptive empirical multidimensional mathematical predictive models based on the results of measurements used to assess the risk of transition of technically complex products to a dangerous technical condition during production (acceptance and control) tests is proposed. The issues of verification and evaluation of the speed of adaptation of the proposed models to changing conditions and test goals are considered. As an example, we consider the process of modeling and forecasting changes in the technical conditions of high-speed internal combustion engines with forced spark ignition.

Multi-Dimensional Mathematical Wear Models of Vibration Generated by Rolling Ball Bearings Made of AISI 52100 Bearing Steel

The paper features the development of multi-dimensional mathematical models used for evaluating the impact of selected factors on the vibration generated by 6304ZZ type rolling ball bearings from three manufacturers in the aspect of the wear process. The bearings were manufactured of AISI 52100 bearing steel. The analyzed factors included the inner and outer raceways’ roundness and waviness deviations, radial clearance and the total curvature ratio. The models were developed for vibration recorded in three frequency ranges: 50–300 Hz, 300–1800 Hz and 1800–10,000 Hz. The paper includes a specification of the principles of operation of innovative measuring systems intended for testing bearing vibration, raceway geometries and radial clearance. Furthermore, it features a specification of particular stages of the multi-dimensional mathematical models’ development and verification. Testing with the purpose of statistical evaluation of the analyzed factors is also presented. The test results and mathematical models indicate that the inner raceway’s waviness deviation had a dominant impact on the vibration examined in all frequencies. The roundness and waviness deviation of bearing raceways made of AISI 52100 steel propagates the bearing wear process.

Exact solutions of the generalized multidimensional mathematical physics models via sub-equation method

In this paper, our focus is on the multidimensional mathematical physics models. We employ the sub-equation method to obtain new exact solutions to the proposed strongly nonlinear time-fractional differential equations of conformable type. The models considered are generalized Benjamin equation, modified generalized multidimensional Kadomtsev–Petviashvili (KP) equations, modified generalized multidimensional KP–BBM equation and the variant Boussinesq system of equations. We also introduced a new modified generalized multidimensional KP type equation and its exact solutions. As the order of fractional derivative tends to one, the obtain exact solutions by the proposed method reduce to classical solutions. We successfully established varieties of soliton type solutions. The results obtained affirm that sub-equation method is an efficient and powerful technique for analytic solutions of nonlinear fractional partial differential equations.

Modeling the Impedance Behavior of Ionic Conductors (AgPO<sub>3</sub>)<sub>1-x</sub>(Ag<sub>2</sub>SO<sub>4</sub>)<sub>x</sub> Glass System Using Artificial Neural Network

The dielectric permittivity and conductivity of (AgPO₃)_1-x(Ag₂SO₄)_x compound was investigated at different concentrations of (Ag₂SO₄). The effect of concentration on AC conductivity and permittivity as well as temperature and frequency was investigated in order to model this behavior. Multidimensional mathematical models were as proposed to predict the impedance components <img src=image/15115401_01.gif> and the dielectric permittivity components <img src=image/15115401_02.gif> of the glass system as a function of temperatures, frequencies and concentrations. Artificial Neural Network (ANN) and nonlinear regression approaches were set as curve fitting techniques in order to construct models based on 1700 points of data. This model can be then used to predict these proprieties at any concentration and therefore helping the product designer to choose the proper mixing and temperature conditions. For ANN, 20, 50, and 100 nodes in a single hidden layer neural network were considered. Although data results of the two approaches showed a good alignment with experimental data, the ANN model with twenty nodes was able to predict the outputs with lower MSE values range from 0.008 to 0.012 for impedance and from 0.006 to 0.008 for dielectric losses than the regression technique. Moreover, R² values for the neural network were over 99% in both training and testing of impedance and dielectric permittivity while R² values for non-linear regression vary between 73.86% to 94.75%. The proposed ANN model can be of a great help to find the optimal dielectric permittivity and conductivity of (AgPO₃)_1-x(Ag₂SO₄)_x compound when dealing with a specific application.

MATHEMATICAL MODELLING OF INDICATIVE PROCESS PARAMETERS OF DUAL-FUEL ENGINES WITH CONVENTIONAL FUEL INJECTION SYSTEM

Modern engine research uses multi-dimensional Mathematical Models (MMs) that are applicable to multi-fuel engines. However, their use involves the availability of detailed technical data on the design and characteristics of the engine, which is not always possible. The use of a one-dimensional MM is more expedient for the prediction of engine parameters, but their application for this purpose has not yet been sufficiently investigated. This publication presents the results of numerical studies evaluating the application of a one-dimensional MM with bi-phase Vibe combustion laws for dual-fuel (DF) Diesel (D) and Natural Gas (NG) engine power parameters. The motor test results of a high-speed 4ČN79.5/95.5 Diesel Engine (DE) with a conventional fuel injection system were used as adequacy criteria. The engines were operated with D100 and DF D20/NG80, in high- (HLM), medium- (MLM), and low- (LLM) load modes, and the angle of Diesel-fuel Injection Timing (DIT) was changed from −1 to −13 °CA in the Before Top Dead Center (BTDC) range. Modelling of the single-phase Vibe combustion law has limited applicability for efficient use only in HLM (with an error of 7%). In the MLM and LLM regimes, owing to non-compliance with real bi-phasic combustion with a strongly extended NG diffusive second phase, the modelling error is 50%. Individual MM matching in MLM and LLM in a DF D20/NG80 experiment detected a burn time increase from between 45 and 50 °CA, to 110 and 200 °CA, respectively. Limited use of the one-dimensional MM in the evaluation of DF engine performance has been identified. When comparing a one-dimensional MM with experimental data, a bi-phase law of heat release characteristic should be considered for better coincidence. In addition, individual MM matching with an experiment on each engine load mode ensured acceptable accuracy in testing and optimising the parameters of the indicator process, including DIT.

ИССЛЕДОВАНИЕ И АНАЛИЗ МЕТОДОВ ОПРЕДЕЛЕНИЯ ЭФФЕКТИВНОСТИ ПРИМЕНЕНИЯ ТЕХНОЛОГИИ ПРОПАНТНОГО ГИДРОРАЗРЫВА ПЛАСТА

Актуальность. Большинство месторождений ООО «ЛУКОЙЛ-ПЕРМЬ» находятся на завершающей стадии разработки, и на них выполняется большое количество операций гидроразрыва пласта (ГРП). В условиях высокой выработки запасов, обводнения скважин и снижения пластового давления необходимо планировать проведение ГРП на тех скважинах, где может быть получен максимальный прирост дебитов нефти. Цель: разработка методики прогноза дебитов нефти с использованием вероятностно-статистических методов. Объекты: данные по использованию пропантного ГРП на карбонатном В3В4 объекте Батырбайского месторождения. Методы: анализ данных по оценке эффективности ГРП с помощью использования вероятностно-статистических методов. Результаты. С использованием вероятностно-статистического метода построены многомерные математические модели оценки эффективности ГРП, которые позволили определить приросты дебитов нефти. Одним из основных преимуществ вероятностно-статистического метода является то, что все используемые для прогноза показатели, имеющие различные размерности, были переведены в безразмерные величины, что упрощает сопоставительный анализ. Кроме этого, выполнено сравнение полученных приростов дебитов нефти с методом расчета дебита жидкости через удельный коэффициент продуктивности. Выполненный анализ показал, что разработанный вероятностно-статистический метод прогноза обеспечивает более точную сходимость с фактическими данными по приростам дебитов нефти, в отличие от результатов, полученных по методу удельного коэффициента продуктивности. Значения коэффициентов парной корреляции для вероятностно-статистического метода составили по объекту разработки В3В4 – 0,85, тогда как по методу удельной продуктивности, соответственно, – 0,15, что на статистическом уровне подтверждает более высокую эффективность предлагаемого метода.&#x0D;

QUALITATIVE AND QUANTITATIVE EVALUATION OF THE CHARACTERISTICS OF THE ISOMETRIC MUSCLE FORCE OF DIFFERENT MUSCLE GROUPS IN CADET JUDO ATHLETES: A GENDER-BASED MULTIDIMENSIONAL MODEL

The aim of this paper is to define the initial quantitative and qualitative multidimensional model for evaluating basic contractile characteristics of isometric muscle force in systematically trained and selected cadet judo athletes. In this research absolute values of the obtained results, and values derived by relativization of absolute values in relation to skeletal muscle mass (SMM) were considered. The basic method used in this research was laboratory testing. All data sampling was performed by the dynamometry method, using tensiometric probes. The research sample in this study consisted of 21 cadet judo athletes, of which 14 were male and 7 were female. All measurements were performed using standardized testing procedures on the following muscle groups: flexor muscles of the left (HGL) and the right hand (HGR), back (DL) and leg extensor muscles (LE) and ankle joint plantar flexor muscles (PF). Based on the obtained results, separate multidimensional mathematical models for the estimation of contractile potential and development level were defined for both basic characteristics of isometric muscle force: maximal isometric muscle force (Fmax) and maximal explosive isometric muscle force (RFDmax). A qualitative assessment of contractile potential for each of the tested muscle groups, i.e. variables, was enabled by defining standard values for 7 distinct preparedness levels for both basic isometric muscle force contractile characteristics of male and female cadet judo athletes.

Исследование процесса нефтеизвлечения в коллекторах различного типа пустотности с использованием многомерного статистического анализа

Oil recovery laws that take into account distribution of reservoirs with different void types within the same accumulation. Carbonate field data of development of Tournaisian-Famennian oil accumulation was used. For comparison purposes data of development of the field with similar oil properties but reservoir of clastic grain rock and pore type were used. One injector and neighbor producers were used as components of applied production scheme. The type of reservoir within one development object was determined by several studies including pressure build-up curve processed by Warren-Root method. At the first stage correlation coefficients between injection and production of neighbor well were calculated. Calculation was done for different time and with assumption, according to which correlation coefficient is a quantitative measure of interactions between two wells. It is determined that use of correlation coefficient for pore reservoirs is significantly differ to the character of its behavior for fracture reservoir type. Multidimensional mathematical models that characterize flooding and allow determining producer’s rate were obtained with considered void type. Linear discriminant functions are built with considered void type of reservoir. Analysis of those functions determined that replacement of oil by water in clastic and carbonates porous rocks and carbonate naturally fractured reservoirs follow different scenarios.

Mathematical Models Developed by Zone Method Considering Non-Gray Radiation Properties of Gas in Combustion Chamber

AbstractWater vapour and carbon dioxide as the main products in combustion chamber have strong non-gray radiation properties. Multidimensional mathematical models were developed by zone method considering the non-gray radiation properties of gas in combustion chamber. Edwards exponential wide band model (EBWM) was adopted to calculate the non-gray radiation properties of gas, and the three point Gauss-Legendre integral formula was used to calculate direct radiation exchange areas. Reflected radiation heat fluxes were obtained by Gauss elimination method, and energy balance equations were solved by main variable correction method. An example was given to validate the developed models, and further investigation of effects of flame distribution on heat transfer was carried on.

Solar adsorption refrigeration (SAR) system modeling

The solar adsorption refrigeration (SAR) system has economical and environmental aspects that motivate many researches to investigate its capability in cooling system design. In this study, multi-dimensional mathematical models have been generated to predict the coefficient of performance (COP) value of the SAR system as function of the evaporator, condenser, and generator temperatures. Fuzzy logic and regression analysis approaches were implemented to construct a mathematical model for this purpose from one-dimensional collected data that relates COP value separately to condensation, evaporation, and generation temperatures, respectively. The results of COP calculation from the two models were agreed quite well with the measured values. However, the fuzzy logic technique showed excellent accuracy than the regression model when compared to the calculated COP value, as its steps have the optimum nature in constructing the required model.

Optimal Control of the Secondary Drying Stage of Freeze Drying of Solutions in Vials Using Variational Calculus

: The problem of operating freeze drying of pharmaceutical products in vials loaded on trays of freeze dryer to obtain a desired final bound water content in minimum time is formulated as an optimal control problem. Two different types of freeze dryer designs were considered. In the type I freeze dryer design, upper and lower plate temperatures were controlled together, while in the type II freeze dryer design, upper and lower plate temperatures were controlled independently. The heat input to the material being dried and the drying chamber pressure were considered as control variables. Only the scorch temperature was considered as a constraint on the system state variables during the secondary drying stage, because all the free water content (frozen water) is removed from the solid matrix during the primary drying stage of freeze drying. Necessary conditions of optimality for the secondary drying stage of freeze drying process in vials were derived and presented by using rigorous multidimensional unsteady-state mathematical models. The theoretical approach presented in this work was applied in the freeze drying of skim milk. Significant reductions in drying times of the secondary drying stage of the freeze drying process in vials were observed and more uniform bound water and temperature distributions in the material being dried were obtained compared to the conventional operational policies.