Frontal solution

Hood, P., 1976. Frontal solution program for unsymmetric matrices. Int. J. Numer. Methods Eng. 10, 379-399. 

The most frequently used modifications of the basic Gaussian elimination method in finite element analysis are the LU decomposition and frontal solution techniques. 

SOLUTION ALGORITHMS GAUSSIAN ELIMINATION METHOD 205 6.4.2 Frontal solution technique... 

Frontal solution requires very intricate bookkeeping for tracking coefficients and making sure that all of the stiffness equations have been assembled and fully reduced. The process time requirement in frontal solvers is hence larger than a straightforward band solver for equal size problems. 

Irons, B, M., 1970. A frontal solution for finite element analysis, hit. J. Numer. Methods Eng. 2, 5-32,... 

This type of chromatographic development will only be briefly described as it is rarely used and probably is of academic interest only. This method of development can only be effectively employed in a column distribution system. The sample is fed continuously onto the column, usually as a dilute solution in the mobile phase. This is in contrast to displacement development and elution development, where discrete samples are placed on the system and the separation is subsequently processed. Frontal analysis only separates part of the first compound in a relatively pure state, each subsequent component being mixed with those previously eluted. Consider a three component mixture, containing solutes (A), (B) and (C) as a dilute solution in the mobile phase that is fed continuously onto a column. The first component to elute, (A), will be that solute held least strongly in the stationary phase. Then the... 

At this stage the benzene is still just separated from the naphthalene. A 16 ml sample volume demonstrates the complete frontal analysis chromatogram of the mixture. The first and second steps containing pure benzene and benzene + naphthalene, respectively, the center peak containing all three solutes and the last two steps containing naphthalene plus anthracene and pure anthracene, respectively. 

This is an oversimplified treatment of the concentration effect that can occur on a thin layer plate when using mixed solvents. Nevertheless, despite the complex nature of the surface that is considered, the treatment is sufficiently representative to disclose that a concentration effect does, indeed, take place. The concentration effect arises from the frontal analysis of the mobile phase which not only provides unique and complex modes of solute interaction and, thus, enhanced selectivity, but also causes the solutes to be concentrated as they pass along the TLC plate. This concentration process will oppose the dilution that results from band dispersion and thus, provides greater sensitivity to the spots close to the solvent front. This concealed concentration process, often not recognized, is another property of TLC development that helps make it so practical and generally useful and often provides unexpected sensitivities. 

HETP of a TLC plate is taken as the ratio of the distance traveled by the spot to the plate efficiency. The same three processes cause spot dispersion in TLC as do cause band dispersion in GC and LC. Namely, they are multipath dispersion, longitudinal diffusion and resistance to mass transfer between the two phases. Due to the aforementioned solvent frontal analysis, however, neither the capacity ratio, the solute diffusivity or the solvent velocity are constant throughout the elution of the solute along the plate and thus the conventional dispersion equations used in GC and LC have no pertinence to the thin layer plate. 

The quality of the packed bed may also be determined by frontal analysis where the sample is applied until it reaches a plateau to give the residence time function and then the solution is momentarily switched to wash to give the washout function. The latter is used to calculate the plate height of the column... 

Figure 2. Transient pressure drop across the porous-medium micromodel of Figure 1 for foam pregenerated in an identical upstream medium. The foam frontal advance rate is 186 m/d. In the wet case, foam advanced into the downstream micromodel which was completely saturated with aqueous surfactant solution. In the dry case, the downstream micromodel contained only air.

Figure 3.2 Three major methods in chromatography. The commonest form of chromatography involves the introduction of a small volume of sample onto a column and is known as zonal chromatography. Movement down the column is effected by the mobile phase, which may be simply a solvent (A) allowing partition of the test molecules between the stationary and mobile phases. Alternatively, the mobile phase may be a solvent containing solute molecules (B), which actively displace test molecules from the stationary phase. A less frequently used method known as frontal separation (C) does not involve a separate mobile phase but a large volume of the sample is allowed to pass through the column and as the various components separate, concentration fronts develop and their movement can be monitored.

Plasma protein binding is also an important parameter in the pharmacokinetic field. Frontal analysis combined with capillary zone electrophoresis (CZE-FA) (67-69) is a powerful technique for high-throughput assay, because it is relatively rapid and easy to automate, in comparison with conventional methods such as dialysis, ultrafiltration, and ultracentrifugation. Recently, we introduced the EKC approach with ionic CDs to frontal analysis for anionic drugs that cannot be analyzed by conventional CZE-FA (70). In this approach, ionic CDs work as an EKC pseudostationary not for proteins but for small solutes. 

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