Pressure-based methods

The pressure-based method was introduced by Harlow and Welch [67] and Chorin [30] for the calculation of unsteady incompressible viscous flows (parabolic equations). In Chorines fractional step method, an incomplete form of the momentum equations is integrated at each time step to 3ueld an approximate velocity field, which will in general not be divergence free, then a correction is applied to that velocity field to produce a divergence free velocity field. The correction to the velocity field is an orthogonal projection in the sense that it projects the initial velocity field into the divergence free... 

Incompressible steady flows are commonly solved by pressure-based methods and methods based on the concept of artificial compressibility [183, 45]. The extension of pressure correction methods to steady flows, generally elliptic equations, has been performed by Patankar and Spalding [140] and Patankar [141]. The artificial compressibility method for calculating steady incompressible flows was proposed by Chorin [29]. In this method, an artificial compressibility term is introduced in the continuity equation, and the unsteady terms in the momentum equations are retained. Hence, the system of equations becomes hj perbolic and many of the methods developed for h3rperbolic systems can be applied. 

Many other methods for solving flow problems can be devised. It is impossible to describe all of them here. In this book, emphasis is placed on describing elements of particular pressure-based methods originally developed for incompressible flows. The basic methods are extended and used to simulate reactive flows. The standard algorithms used to solve multi-fluid models are extensions of particular pressure-based methods for single phase flows. 

In this section, particular pressure-based methods designed to solve the momentum equation are outlined. The numerical methods for solving the momentum component equations differ considerably from those designed to solve the generic scalar transport equation, because the velocity is a vector quantity. The special treatment of the terms in the momentum equations that are different from those in the generic transport equation is summarized. 

In this section, several strategies for determining the pressure in the incompressible flow limit is outlined for pressure-based methods. The extension of the pressure-correction approach to arbitrary Mach numbers is examined. 

Mathur SR, Murthy JY (1997) A pressure-based method for unstructured meshes. Numer Heat Trans Part B 31 195-215... 

CFX software focuses on one approach to solve the governing equations of motion (coupled algebraic multigrid), while the FLUENT product offers several solution approaches (density-segregated- and coupled-pressure-based methods). 

Principally the pressure based methods could also use conductive oven heating but would usually be used with a press to withstand the high pressure (usually between 5 and 8 bar).Hie mandatory use of closed mold arrangements leads to the most economic and feasible processes as follows ... 

In the pressure-based methods, the continuity is transformed to an equation for a pressure correction quantity. The transient term in the continuity vanishes for incompressible flows, as in this case the fluid properties are constant. The transient term is currently retained to emphasize that this method can be applied for compressible flows as well. In the latter case, the discrete form of the mass balance can stiU be interpreted as a constraint equation for pressure. 

The old, tedious, but quite reliable method is to measure the supply flow by the bag method. A tightly rolled plastic bag empty of air at the commencement of the test is pressed on the terminal with all the supply air passing into the bag. The filling time of the bag is measured and the flow rate calculated based on this information. The bag volume has to be determined in advance by a special measurement. Finally, the characteristic pressure difference method, menrumed above, can also be applied to supply terminals. 

Comparisons of the accuracy and efficiency for three numerical procedures, the direct summation, DC-FFT-based method and MLMI, are made in this section. The three methods were applied to calculating normal surface deformations at different levels of grids, under the load of a uniform pressure on a rectangle area 2a X 2fo, or a Hertzian pressure on a circle area in radius a. The calculations were performed on the same personal computer, the computational domain was set as -1.5a=Sx 1.5a and -1.5a=Sy 1.5a, and covered... 

General trends are focused on reduced-solvent extractions or adsorption-based methods — enviromnentaUy friendly solvents for both solid and liquid samples. In recent decades, advanced techniques like supercritical fluid extraction (SFE), ° pressurized liquid extraction (PLE)," microwave-assisted extraction (MAE), ultrasound-assisted extraction, countercurrent continued extraction (www.niroinc.com), solid... 

The principle of the pressure-jump method is based on the pressure dependence of the equilibrium constant, i.e.,... 

In the FA-based methods, a buffer gas (usually He) is passed through a reaction tube of about 1.0-meter length and 8-cm diameter at a velocity of about 10 cm/s. Typically, a pressure of about 0.5 torr is maintained along the entire length of the tube. Reactant ions and neutrals are added at specific points in the upstream portion... 

The effect on reactive compounds and potential alteration of mutagenic activity of the acid-base extraction procedure were major concerns. The TLC method was not recommended because of possible mutagen loss due to reactions on the plate. The Florisil and low-pressure chromatographic methods were considered to be research methods. 

Figure 2. (a) PSDs for 3.88 nm MCM-41 [22] calculated from nitrogen adsorption data at 77 K using i) the KJS approach with the BJH algorithm, ii) the hybrid DFT method, iii) the HK-based method with the relation between pore size and condensation pressure described by Eq. 2, and iv) the HK-based Saito-Foley method, (b) Comparison of PSDs for octyldimethylsilyl-bonded MCM-41 calculated from nitrogen adsorption using the KJS-calibrated BJH algorithm with t-curves for reference ODMS-modified silica and unmodified silica. 

Recently, the Horvath-Kawazoe (HK) method for slit-like pores [40] and its later modifications for cylindrical pores, such as the Saito-Foley (SF) method [41] have been applied in calculations of the mesopore size distributions. These methods are based on the condensation approximation (CA), that is on the assumption that as pressure is increased, the pores of a given size are completely empty until the condensation pressure corresponding to their size is reached and they become completely filled with the adsorbate. This is a poor approximation even in the micropore range [42], and is even worse for mesoporous solids, since it attributes adsorption on the pore surface to the presence of non-existent pores smaller than those actually present (see Fig. 2a) [43]. It is easy to verify that the area under the HK PSD peak corresponding to actually existing pores does not provide their correct volume, so the HK-based PSD is not only excessively broad, but also provides underestimated volume of the actual pores. This is a fundamental problem with the HK-based methods. An additional problem is that the HK method for slit-like pores provides better estimates of the pore size of MCM-41 with cylindrical pores than the SF method for cylindrical pores. This shows the lack of consistency [32,43]. Since the HK-based methods use CA, one can replace the HK or SF relations between the pore size and pore filling pressure by the properly calibrated ones, which would lead to dramatic improvement of accuracy of the pore size determination [43] (see Fig. 2a). However, this will not eliminate the problem of artificial tailing of PSDs, since the latter results from the very nature of HK-based methods. 

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