Mathematical modeling of the in-mold coating process

Fiber glass reinforced polyester parts compression molded from sheet molding compound (SMC) are prone to such surface inconsistencies as porosity and sinks. Even though it appears that some of these defects could be eliminated by techniques such as vacuum molding, the resulting surface, with current technology, is not yet consistently up to automotive standards for exterior body panels. In-mold coating (IMC) of SMC, is designed to fill porosity, reduce sinks, and furnish a primer-like coating, thus upgrading the part surface to automotive standards. As a consequence, IMC is generally an integral part of the molding cycle when producing compression molded SMC exterior automotive body panels. Most commonly, in-mold coating is injected after opening the press slightly so as to separate the mold cavity and the exterior surface of the part to make room for the coating. A second approach is to let the hydraulic pressure of the injected IMC open the mold. Here, we present a mathematical model of the process and show application in predicting injection pressures, fill times, and filling patterns. A comparison with experimental results is also presented. Cycle times required for IMC injection methods is also discussed.     

Mathematical modeling of gas dehydration using adsorption process

In this study, a mathematical model was developed to simulate an adsorption process for dehydration of a gas stream. In deriving the model, the following assumptions were made. Variation of the gas velocity along the bed length was accounted for and the bed pressure drop was calculated by the Ergun equation. Mass and heat transfer outside the solid particles were assumed to be convective and those inside the particle were assumed to be diffusive. The dual site Langmuir isotherm was employed in predicting adsorption equilibrium and the Peng–Robinson equation was used as the PVT relation. The resulting mathematical model was solved by the finite volume method and the results were verified against experimental data reported by Mohamadinejad et al. [2000. Separation Science and Technology 35,1] and Gorbach et al. [2004. Adsorption 10,1]. Good agreement was observed between the predictions of the model and the experimental data. The developed model was used to perform parametric study. Our results suggest that the break-through time decreases linearly with the square of particles diameter. It also decreases linearly with the inverse of particles tortuosity. A similar trend appears to exist for variations of the bed percent saturation. The bed pressure drop increases linearly with the inversed diameter of particles.     

Mathematical modeling of the freezing process of concrete and aggregates

A two dimensional finite element computer model for predicting frost penetration in saturated porous materials is presented. Based on available second order parabolic differential equations for predicting temperature and moisture content, and equations relating freezing temperature and water potential, the finite element model predicts temperature, water and ice content, frost penetration, and generated pore water pressures as a function of freezing time and given boundary conditions, Comparison of models results with available experimental data for mature cement pastes rendered reasonable agreement between the model and data.     

Mathematical modeling of the integrated microfiltration-desorption process at the microfiltration stage

A mathematical model of the integrated microfiltration-desorption process is proposed. Modeling of microfiltration using a stochastic approach is considered in detail.     

分子蒸馏过程数学建模及模拟验证

建立了分子蒸馏过程径向和轴向的二维数学模型,采用CFD方法对DBP-DBS混合物系的分离过程进行模拟计算,分析了蒸馏温度和进料速率等操作条件对蒸馏过程的影响。将模拟结果与文献中采用BGK模型计算得到的结果进行对比,两者具有较好的吻合性,验证了模型的合理性。     

Mathematical modeling and experimental study of conveyor belt continuous curing process

A mathematical model was developed to predict the temperature distribution within various layers of conveyor belts during the continuous curing process. The results predicted by the model are found to be in good agreement with the experimental results, hence justifying the capability of the model for simulation of the conveyor belt continuous process. This information was utilized to provide more insight into the curing process in terms of the state of cure (SOC) and/or the degree of conversion, which may, in turn, be utilized for the optimization of the curing process. © 2000 John Wiley & Sons, Inc. J Appl Polym Sci 77: 2448–2454, 2000     

减压间歇萃取精馏分离乙酸乙酯-乙醇的过程数学模型

减压间歇萃取精馏是一种用于分离近沸点混合物及共沸物分离的极具发展潜力的技术。本文分别以N,N-二甲基甲酰胺作为简单溶剂和由N,N-二甲基甲酰胺与二甲基亚砜组成的混合物作为混合溶剂,建立了减压间歇萃取精馏分离乙酸乙酯和乙醇共沸体物系的过程数学模型,利用追赶法求解了描述该过程的一组刚性较强的方程组,并通过实验对所建模型进行了验证。结果表明,模拟计算结果与实验结果吻合较好。另外,通过图示,过程模拟结果表明减压操作能够强化萃取精馏的分离过程。     

Mathematical modeling of the rheological behavior of thermoplastic slurry in the molding process of beryllium ceramics

The article presents the results of calculations of a mathematical model of the flow and heat exchange of a thermoplastic beryllium oxide slurry in a circular channel of a foundry installation. Taking into account the peculiarities of coagulation structure formation and the flow mechanism with boundary conditions, the article presents calculations of the speed of the viscoelastic flow of the slurry on the basis of two rheological models of Shvedov-Bingham and Herschel-Bulkley. The three-parameter equation is used to check the consistency of the experimental data of the non-isothermal slurry flow in comparison with the Shvedov-Bingham model. According to the results of calculations of various hydrodynamic conditions and thermal parameters, the fields of velocities and temperatures were obtained to determine the optimal modes of casting beryllium oxide slurry. The heat transfer parameters were determined in accordance with the physical properties of the cavity wall. The positions of the crystallization front along the circular channel are also shown depending on the thermal parameters. The results of the calculated data on the Shvedov-Bingham and Bulkley-Herschel rheological models showed sufficient adequacy with the experimental data.     

Mathematical modeling of the columnar vortex

The columnar laminar vortex has a two-zone structure in the case of a highly swirling flow. Various asymptotics for the laminar, wave, and turbulent flows in the Trefftz plane are discussed.     

Mathematical modeling and calculation of the spinning solution filtration process in the manufacture of man-made fibres

Conclusions A generalized filtration equation has been obtained, which permits one to carry out mathematical modeling of a broad class of filtration processes of spinning solutions of fibre-forming polymers; it has low values of the error in approximation.On the basis of the results of mathematical modeling, a procedure has been worked out for evaluating the complex mechanism of the filtration process, using which it is possible to carry out a comparative analysis of the experimental data and to take into account (quantatively) the effect of technological conditions on the filtration process.The extremeness of the behavior of the basic characteristics of filtration equipment as a function of change in filtration rate, which characterizes the operating regime of a filter, has been demonstrated.The use of the magnitude of specific costs as an optimicity criterion in designing and rational construction of filtration schemes is soundly based.Program provision has been devised for calculating and making an analysis of the characteristics of apparatus design in the stage of filtering spinning solutions of fibre-forming polymers using a computer for projected and operating man-made fibre manufacturing.Translated from Khimicheskie Volokna, No. 2, pp. 24–27, March–April, 1985.     

Mathematical modeling for simultaneous design of plants and supply chain in the batch process industry

Most supply chain design models have focused on the integration problem, where links among nodes must be settled in order to allow an efficient operation of the whole system. At this level, all the problem elements are modeled like black boxes, and the optimal solution determines the nodes allocation and their capacity, and links among nodes. In this work, a new approach is proposed where decisions about plant design are simultaneously made with operational and planning decisions on the supply chain. Thus, tradeoffs between the plant structure and the network design are assessed. The model considers unit duplications and the allocation of storage tanks for plant design. Using different sets of discrete sizes for batch units and tanks, a mixed integer linear programming model (MILP) is attained. The proposed formulation is compared with other non-integrated approaches in order to illustrate the advantages of the presented simultaneous approach.     

Mathematical and computational modeling of heat transfer and deformation in film blowing process

The blown‐film extrusion process was investigated both experimentally and theoretically. In experimental study, nonisothermal experiments were conducted using low‐density polyethylene. Rheological parameters were studied, considering the polymer melt as a power law fluid in nonisothermal conditions. Axial tension, bubble diameter, and film thickness at a variety of film extrusion conditions, that is, different flow rate, pressure difference across the film, and take‐up speeds were measured. In theoretical study, an analysis was employed to simulate the blown‐film extrusion process by setting up the force‐ and energy‐balance equations on the film bubble moving upward. Four nonlinear complex differential equations were integrated numerically, using an iterative backward shooting method and the fifth‐order Runge‐kutta technique. The program written, based on a mathematical model, predicts the bubble shape, temperature profile, and film thickness as a function of the distance along the machine axis. Furthermore, the model evaluates the elongational viscosity of LDPE in biaxial tension in terms of distance from die axis and take‐up speed. In this simulation, the total stress components in machine and the transverse directions were computed from the die exit up to the freeze line, the knowledge of which is necessary for evaluation of the elastic memory build up in heat‐shrinkable films. © 2002 Wiley Periodicals, Inc. J Appl Polym Sci 86: 2115–2123, 2002     

Mathematical simulation of wet spinning coagulation process: Dynamic modeling and numerical results

A dynamic model of polymer wet spinning coagulation process is proposed in this article. The model is based on the double diffusion phenomenon, phase separation process, continuity balance, and momentum balance of the entire coagulation process. The uniqueness of the model lies in its dynamic feature. The model can simulate the system's dynamic response to variations in system inputs/parameters. Steady‐state system solutions can also be produced as the long‐time solutions of the dynamic model; a settling time can be observed at the same time. This paper employs a computationally efficient method of lines numerical algorithm for solving the dynamic model. A simulation experiment on a selected non‐solvent‐solvent‐polymer ternary system is carried out to verify the model as well as the numerical method. The dynamic simulation results are analyzed and discussed. At the end of the article, h‐refinement and p‐refinement are used to confirm the spatial convergence of the numerical solutions. © 2016 American Institute of Chemical Engineers AIChE J, 62: 3432–3440, 2016     

Mathematical modeling of filtering process of two-phase suspensions in tubular filters under nonisothermal conditions

The hydrodynamic conditions in tubular filter cells operating under nonisothermal conditions are studied. The equations of mechanics of heterogeneous media are used to describe the separation process of two-phase suspensions, which are written and simplified in the cylindrical coordinate system taking into account characteristics of the flow. The challenge is solved semi-analytically. Using the methods of surfaces of equal consumptions and Slezkin, numerical calculations on the constructed mathematical model are presented for particular implementations of the separation process.     

Mathematical modeling of the dispersed phase dynamics

A vector equation of motion in the Lagrangian frame of reference is rearranged into a set of two scalar equations to determine the relative-velocity magnitude of a particle and its direction. The resulting system, written in terms of the Lamé coefficients, is solved jointly with continuum equations of motion in the Eulerian frame of reference. The method is exemplified by the flow of a heterogeneous mixture between permeable parallel walls and the flow of a heterogeneous non-Newtonian fluid over the surface of a sedimentation centrifuge rotor.Translated from Teoreticheskie Osnovy Khimicheskoi Tekhnologii, Vol. 39, No. 2, 2005, pp. 206–215.Original Russian Text Copyright © 2005 by Kholpanov, Ibyatov.     

Mathematical modeling of cyclic processes

The identification of cyclic chemical-engineering processes via high-degree linear differential equations is presented. Based on the elaborated approach, the identification of the cyclic process of the gaslift extraction of hydrocarbons from drowned wells of oil-and-gas condensate fields was performed. Analytical solutions of the mathematical model possessing stability and a good correlation with the experimental data are obtained.     

Mathematical modeling of biotechnological process of lactic acid production by batch fermentation: A review

The results of analyzing mathematical models of the process of batch fermentation for the production of lactic acid have been presented. Three groups of mathematical models have been considered. The first two groups are based exclusively on the notion of the specific growth rate, dependent or independent on substrate concentration. The third group uses the notions of the specific rate of substrate consumption and specific rate of product formation in additions to the notion of specific growth rate. The numeric estimates of constants are presented for the majority of models as well as comparison with the experimental studies. Mathematical models of the processes where the formation of side products is possible have been discussed, which may or may not be of value on their own. The possible practical applicability of the models has been evaluated based on the results of the analysis.     

Mathematical modeling of the kinetics of binder hardening

The paper intends to point out some methodological aspects to treat the difficult problem of mathematical modeling of concrete taking care of the main factors influencing the process. It will be demonstrated that the information coming from the physico-chemical phenomena like dissolution, nucleation, crystallisation etc. and the corresponding evolution of state, recognized by an adequate output of a measurement, can be reflected in a more or less realistic way. The investigation is done to have the necessary background to describe the changing properties of concrete under computer aided production control as well as to give a new insight in the outgrowth of variation in the composition of the system.     

Mathematical modeling study of the gasification of coke

Moscow. Translated from Fizika Goreniya i Vzryva, Vol. 24, No. 5, pp. 88–95, September–October, 1988.