Condensation mathematical model

Thermal conductivity increases with temperature. The insulating medium (the air or gas within the voids) becomes more excited as its temperature is raised, and this enhances convection within or between the voids, thus increasing heat flow. This increase in thermal conductivity is generally continuous for air-filled products and can be mathematically modeled (see Figure 11.3). Those insulants that employ inert gases as their insulating medium may show sharp changes in thermal conductivity, which may occur because of gas condensation. However, this tends to take place at sub-zero temperatures. 

Develop a mathematical model for the three-column train of distillation columns sketched below. The feed to the first column is 400 kg mol/h and contains four components (1, 2, 3, and 4), each at 25 mol %. Most of the lightest component is removed in the distillate of the first column, most of the next lightest in the second column distillate and the final column separates the final two heavy components. Assume constant relative volatilities throughout the system ai, CI2, and a3. The condensers are total condensers and the reboilers are partial. Trays, column bases, and reflux drums are perfectly mixed. Distillate flow rates are set by reflux drum... 

Derive a dynamic mathematical model of the flooded-condenser system. Calculate the transfer function relating steam flow rate to condensate flow rate. Using a PI controller with tj = 0.1 minute, calculate the closedloop time constant of the steam flow control loop when a closedloop damping coefTident of 0.3 is used. Compare this with the result found in (u). 

A chemical reactor is cooled by both jacket cooling water and condenser cooling water. A mathematical model of the system has yielded the following openloop transfer functions (time is in minutes) ... 

Prasarma et al. [185] were also able to observe an optimum thickness of DLs for fuel cells experimentally. They demonstrated that the thicker DLs experience severe flooding at intermediate current densities (i.e., ohmic region) due to low gas permeation and to possible condensation of water in the pores as the thickness of the DL increases. On the other hand, as the thickness of the DL decreases, the mass transport losses, contact resistance, and mechanical weakness increase significantly [113,185]. Through the use of mathematical modeling, different research groups have reported similar conclusions regarding the effect of DL thickness on fuel cell performance [186-189]. 

For the sake of brevity the reader is referred to Paper II for the details regarding the constitutive mathematical models of the method applied to measure the mass flow and stoichiometry of conversion gas as well as air factors for conversion and combustion system. Below is a condensed formulation of the mathematical models applied. Here a distinction is made between measurands and sought physical quantities of the method. 

Formulation of the mathematical model here adopts the usual assumptions of equimolar overflow, constant relative volatility, total condenser, and partial reboiler. Binary variables denote the existence of trays in the column, and their sum is the number of trays N. Continuous variables represent the liquid flow rates Li and compositions xj, vapor flow rates Vi and compositions yi, the reflux Ri and vapor boilup VBi, and the column diameter Di. The equations governing the model include material and component balances around each tray, thermodynamic relations between vapor and liquid phase compositions, and the column diameter calculation based on vapor flow rate. Additional logical constraints ensure that reflux and vapor boilup enter only on one tray and that the trays are arranged sequentially (so trays cannot be skipped). Also included are the product specifications. Under the assumptions made in this example, neither the temperature nor the pressure is an explicit variable, although they could easily be included if energy balances are required. A minimum and maximum number of trays can also be imposed on the problem. 

Barolo et al. (1998) developed a mathematical model of a pilot-plant MVC column. The model was validated using experimental data on a highly non-ideal mixture (ethanol-water). The pilot plant and some of the operating constraints are described in Table 4.13. The column is equipped with a steam-heated thermosiphon reboiler, and a water-cooled total condenser (with subcooling of the condensate). Electropneumatic valves are installed in the process and steam lines. All flows are measured on a volumetric basis the steam flow measurement is pressure- and temperature-compensated, so that a mass flow measurement is available indirectly. Temperature measurements from several trays along the column are also available. The plant is interfaced to a personal computer, which performs data acquisition and logging, control routine calculation, and direct valve manipulation. 

Ivleva, T. I., and Shkadinskii, K. G., Mathematical modeling of two-dimensional problems in filtration combustion of condensed systems. Proceedings of the Kinetics and Mechanisms of Physicochemical Processes, Chernogolovka, Russia, 74 (1981). 

Control of ceramic properties through control of parameters. Variation of the reaction conditions affects product morphologies and bulk properties. Variation of pH, temperature, concentration, and chemical control of the rates of hydrolysis and condensation dramatically affect the final product. The complexity of the reactions involved often precludes a complete mechanistic understanding but the use of computer simulation and mathematical models to predict the behavior of the precursors under different conditions allows an insight into how the reaction might proceed and thus how the nature of the final product can be controlled. 

First principle mathematical models These models solve the basic conservation equations for mass and momentum in their form as partial differential equations (PDEs) along with some method of turbulence closure and appropriate initial and boundary conditions. Such models have become more common with the steady increase in computing power and sophistication of numerical algorithms. However, there are many potential problems that must be addressed. In the verification process, the PDEs being solved must adequately represent the physics of the dispersion process especially for processes such as ground-to-cloud heat transfer, phase changes for condensed phases, and chemical reactions. Also, turbulence closure methods (and associated boundary and initial conditions) must be appropriate for the dis-... 

The huge extent of biological knowledge is also an obstacle since it does not allow for intuition-based reasoning due to its complexity. Mathematical modeling can help handle the complexity by condensing it into a model. 

Heat Transfer Correlations for External Condensation. Although the complexity of condensation heat transfer phenomena prevents a rigorous theoretical analysis, an external condensation for some simple situations and geometric configurations has been the subject of a mathematical modeling. The famous pioneering Nusselt theory of film condensation had led to a simple correlation for the determination of a heat transfer coefficient under conditions of gravity-controlled, laminar, wave-free condensation of a pure vapor on a vertical surface (either flat or tube). Modified versions of Nusselt s theory and further empirical studies have produced a list of many correlations, some of which are compiled in Table 17.23. 

Consider a distillation column with 20 trays, a reboiler, and a condenser. The feed is a two-component mixture. Then, as we have seen in Example 4.13, the mathematical model is composed of... 

A mathematical model of Silicaiite crystallization from clear solution was developed and solved numerically. Crystallization was considered to be solution-mediated, and to involve three categories of silica species and three steps activation, nucleation, and crystal growth. Nucleation was represented by a condensation reaction between the hydroxyl groups present on the surface of 10 nm amorphous silica particles and the soluble silica species, producing activated complexes, which transform to crystalline nuclei. Both nucleation and crystal growth were considered to be reaction controlled. The model simulated well the in situ experimental data. This suggests that the hypothesized nucleation mechanism can be used to qualitatively describe the nucleation event during clear solution Silicaiite crystallization. 

The macroscopic description of the adsorption on electrodes is characterised by the development of models based on classical thermodynamics and the electrostatic theory. Within the frames of these theories we can distinguish two approaches. The first approach, originated from Frumkin s work on the parallel condensers (PC) model,attempts to determine the dependence of upon the applied potential E based on the Gibbs adsorption equation. From the relationship = g( ), the surface tension y and the differential capacity C can be obtained as a function of E by simple mathematical transformations and they can be further compared with experimental data. The second approach denoted as STE (simple thermodynamic-electrostatic approach) has been developed in our laboratory, and it is based on the determination of analytical expressions for the chemical potentials of the constituents of the adsorbed layer. If these expressions are known, the equilibrium properties of the adsorbed layer are derived from the equilibrium equations among the chemical potentials. Note that the relationship = g( ), between and , is also needed for this approach to express the equilibrium properties in terms of either or E. Flere, this relationship is determined by means of the Gauss theorem of electrostatics. 

Section I provides an introduction to the basic processes, the technological schemes, and the components of the equipment employed in systems for the field preparation of oil, natural gas, and gas condensate. The emphasis is on the designs and the principles of operation of separators, absorbers, and cooling devices. Mathematical modeling of the processes in these devices is covered in subsequent sections of the book. 

---