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It is known that the temperature of crystallization during the synthesis of zeolite is one of the most important process parameters. However, during the research work on the synthesis of zeolite 13X and the introduction of this material into regular industrial production, it was noticed that the heating rate of the starting reaction suspension can have an equally important influence. This influence can be so pronounced that a difference of just a few minutes in reaching the crystallization temperature can make a significant difference in product quality, affect the presence of other phases in the crystal, or even determine the direction of zeolite crystallization. Therefore, the aim of this work was to show the influence of the heating rate on the quality of the obtained 13X zeolite powders. The obtained samples were analysed in terms of crystallinity (by X-ray diffraction), chemical composition, granulometry and specific surface area (by Brunauer-Emmett-Teller analysis), and regarding water and CO2 adsorption capacities. Additionally, scanning electron microscopy analysis of the samples showed the morphological characteristics of different 13X zeolite powders. The analysis results of the obtained powders confirmed the influence of the heating rate and helped to define the optimal synthesis parameters i.e. the initial temperature and heating time, that resulted in stable product quality.

This paper presents research on hydrodynamics and mass transfer in a packed absorption column. Experimental data on dry column pressure drop, flooding point, and efficiency of absorption of CO2 in water is obtained on a lab-scale absorption column packed with Raschig rings. Auxiliary parts of equipment together with chemical analyses provide simple monitoring and collecting the data. All obtained data were used to test different mathematical models for a given problem, i.e. for determination of the dry column pressure drop, flooding point and the overall gas transfer unit height. For dry column pressure drop, models developed primarily for packed columns described the data the best, with the Billet model generating a 6.54 % mean error, followed by Mackowiak and Stichlmair models. In flooding point calculations, empirical models were tested and models of Lobo, Leva and Takahshi gave the best results. Mass transfer (absorption) experiments gave expected results, since absorption efficiency increased with the increase in the liquid/gas flow rate ratio, i.e. with approaching the flooding point. The Onda’s model was used to calculate partial mass transfer coefficients in liquid and gas phases based on which the height of the overall gas transfer unit was estimated and subsequently compared with the experimental data. Deviation of calculated and experimental results for the height of the overall gas transfer unit is in the expected range of 0-20 %, with mean value of 15.5 %. In conclusion, the available models for determination of the investigated hydrodynamics and mass transfer parameters in packed absorption columns gave adequate results in comparison to the experimental values.

The paper shows an example of performed optimization of sizes in terms of welding costs in a characteristic loaded welded joint. Hence, in the first stage, the variables and constant parameters are defined, and mathematical shape of the optimization function is determined. The following stage of the procedure defines and places the most important constraint functions that limit the design of structures, that the technologist and the designer should take into account. Subsequently, a mathematical optimization model of the problem is derived, that is efficiently solved by a proposed method of geometric programming. Further, a mathematically based thorough optimization algorithm is developed of the proposed method, with a main set of equations defining the problem that are valid under certain conditions. Thus, the primary task of optimization is reduced to the dual task through a corresponding function, which is easier to solve than the primary task of the optimized objective function. The main reason for this is a derived set of linear equations. Apparently, a correlation is used between the optimal primary vector that minimizes the objective function and the dual vector that maximizes the dual function. The method is illustrated on a computational practical example with a different number of constraint functions. It is shown that for the case of a lower level of complexity, a solution is reached through an appropriate maximization of the dual function by mathematical analysis and differential calculus.

In a semi-ideal gas, entropy changes cannot be determined through the medium specific heat capacity in a manner as determined by the change of internal energy and enthalpy, i.e. the amount of heat exchanged. Taking this into account, the authors conducted two models through which it is possible to determine the change in the specific entropy of a semi-ideal gas for arbitrary temperature interval using the spread sheet method, using the mean values of the appropriate functions. The idea is to replace integration, which occurs here in evitably, with mean values of the previous functions. The models are derived based on the functional dependence of the actual specific heat capacity on the temperature. The theorem used is that of the mean value of a function as well as the mathematical properties of the definite integral. The mean value of a fractional function is determined via its integrand while the logarithmic functions were performed by applying a suitable transformation of the differential calculus. The relations derived relation, using the computer program, have enabled the design of appropriate thermodynamic tables through which it is possible to determine the change in entropy of arbitrary state changes in an efficient and rational manner, without the use of calculus or finished forms. In this way, the change in the entropy of a semi-ideal gas is determined for an arbitrary temperature interval using the method which is analogous to that applied in determining the change of internal energy and enthalpy or the amount of heat exchanged, which was the goal of the work. Verification of the proposed method for both the above functions was performed for a a few characteristic semi-ideal gases where change c(T) is significant, for the three adopted temperature intervals, for the characteristic change of state. This was compared to the results of the classical integral and the proposed method through the prepared tables. In certain or special cases, it is possible to apply the presented method also in determining the change in entropy of the real gas. Apart from that, the paper shows that the change in entropy for the observed characteristic case can be represented or graphically determined using the planimetric method of diagrams with suitably selected coordinates.

Starting from the definition of the average specific heat capacity for chosen temperature range, the analytic dependence between the real and the mean specific heat capacities is obtained using differential and integral calculation. The obtained relation in differential form for the defined temperature range allows for the problem to be solved directly, without any special restrictions on its use. Using the obtained relation, a general model in the form of a polynomial of arbitrary degree in the function of temperature was derived, which has more suitable and faster practical application and is more general in character than the existing model. New graphical method for solving the problem is obtained based on differential geometry and using the derived equation. This may also have practical significance since many problems in thermodynamics are solved analytically and graphically. This result was used in order to obtain the amount of specific heat exchanged using an analytical model or a planimetric method. In addition, this graphical solution was used for the construction of the diagram showing the dependence between the specific heat exchanged and temperature. This diagram also gives a simple graphical procedure for the calculation of the real and the average specific heat capacity for arbitrary temperature or temperature interval. The confirmation for all graphic constructions is obtained using the differential properties between thermodynamic units. In order for the graphical solutions presented to be applicable in practice, suitable ratio coefficients have been determined for all cases. Verification of the model presented, as well as the possibilities of its application, were given using several characteristic examples of semi-ideal and real gas. Apart from linear and non-linear functions in the form of polynomials, the exponential function of the dependence between specific heat capacities and temperature was also analysed in this process.

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