Two-Dimensional Approaches

The quasi-one-dimensional model used in the previous sections for analysis of various characteristics of fiow in a heated capillary assumes a uniform distribution of the hydrodynamical and thermal parameters in the cross-section of micro-channel. In the frame of this model, the general characteristics of the fiow with a distinct interface, such as position of the meniscus, rate evaporation and mean velocities of the liquid and its vapor, etc., can be determined for given drag and intensity of heat transfer between working fluid and wall, as well as vapor and wall. In accordance with that, the governing system of equations has to include not only the mass, momentum and energy equations but also some additional correlations that determine 

In contrast with the one-dimensional model, the two-dimensional model allows to determine the actual parameter distribution in flow fields of the working fluid and its vapor. It also allows one to calculate the drag and heat transfer coefficients by the solution of a fundamental system of equations, which describes the flow of viscous fluid in a heated capillary. 

The general features of two-dimensional flow with evaporating liquid-vapor meniscus in a capillary slot were studied by Khrustalev and Faghri (1996). Following this work we present the main results mentioned in their research. The model of flow in a narrow slot is presented in Fig. 10.16. Within a capillary slot two characteristic regions can be selected, where two-dimensional or quasi-one-dimensional flow occurs. Two-dimensional flow is realized in the major part of the liquid domain, whereas the quasi-one-dimensional flow is observed in the micro-film region, located near the wall. 

At the assumption that in a narrow slot the gravitational forces are negligible as compared to surface tension, the governing equations describing flow of an incompressible liquid in the central part of flow read as 

10 Laminar Flow in a Heated Capillary with a Distinct Interface 

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The most reliable technique to find the global optimum by means of common methods is the transition from the quasi-two-dimensional approach (Fig. 5.3b,c) to a complete two-dimensional one. It consists of a certain number of experiments as shown in Fig. 5.4. 

Contrary to traditional one-dimensional models, two-dimensional models are required for taking into consideration the effects of the radial distribution of the most influential thermophysical properties. As can be seen from the above literature survey, not many studies have adopted the two-dimensional approach for simulation the drying process in a vertical tube. In addition, their predictions were not validated with experimental data and in some cases only the momentum transfer was taken into account. 

Various techniques exist and more are under development for separating components along two independent coordinate axes instead of one. The principal motivation for this development is that separation power is greatly multiplied by the addition of a second separation coordinate. The two-dimensional approach accordingly has great promise for future use [8]. 

Glasser et al. (1987) and Hildebrandt et al. (1990) demonstrated this two-dimensional approach on a number of small reactor network problems, with better results than previously reported. Moreover, Omtveit and Lien (1993) were able to consider higher-dimensional problems as well through projections in concentration space that allow a complete two-dimensional represention. These projections were accomplished through the principle of reaction invariants (Fjeld et al., 1974) and the imposition of system specific constraints. 

The system of equations is discretized in space by a finite voJume approach, while for the time integration an implicit Euler method is used. Particle and flow model are solved consecutively, which implies that conditions in the bed change slowly compared to the integration step. To reduce the required computation time the flow model is solved here only for one dimension even if the software library TOSCA provides also classes for a two dimensional approach. 

Turning to pathway c), the concerted-reaction mechanism, we have formulated two approaches to predicting the rate constant, a double-adiabatic and a two-dimensional approach (4,5). In the double-adiabatic theory, the electron is considered to be coupled to two nuclear modes, a solvent (orientational polarization) mode that is treated classically in view of its low... 

The modeling study was based on a modification of the applicable Fourier relationship, Eq. (1). The modification includes the addition of another heat source from the Joule heating by the eurrent. The rate of heat generation from this source is aE, where cr is the electrical conductivity and E is the applied field. The modeling was initially done on the synthesis of SiC, using a two-dimensional approach and accounting for heat loss by radiation, to bring the results closer to... 

S.-A. Chen, Polymer miscibility in organic solvents and in plasticizers - a two-dimensional approach, J. Appl. Polym. Sci., 15 (1971) 1247-1266. 

In Section 6.11, Fischer-Tropsch synthesis in a multitube reactor was used as an example to illustrate the differences between the one- and two-dimensional approaches for the simulation of cooled fixed bed reactors. For o-xylene oxidation to phthalic anhydride, only the two-dimensional reactor model is considered. 

Introducing correction factors into Eq. [11] implies that molecular topology needs to be taken into account when partition coefficients are calculated. These correction factors have led to the development of more elaborate fragmental systems, as described in the section on two-dimensional approaches. 

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