Method Blasius

Blasius and coworkers have offered a somewhat different approach to systems of this general type. In the first of these, shown in Eq. (6.20), he utilizes a hydroxymethyl-substituted 15-crown-5 residue as the nucleophile. This essentially similar to the Mon-tanari method. The second approach is a variant also, but more different in the sense that covalent bond formation is effected by a Friedel-Crafts alkylation. In the reaction... 

An alternative copolymerization is illustrated by the method of Blasius. In this preparation, a phenol-formaldehyde (novolac) type system is formed. Monobenzo-18-crown-6, for example, is treated with a phenol (or alkylated aromatic like xylene) and formaldehyde in the presence of acid. As expected for this type of reaction, a highly crosslinked resin results. The method is illustrated in Eq. (6.23). It should also be noted that the additional aromatic can be left out and a crown-formaldehyde copolymer can be prepared in analogy to (6.22). ... 

The CSPs based on chiral crown ethers were prepared by immobilizing them on some suitable solid supports. Blasius et al. [33-35] synthesized a variety of achiral crown ethers based on ion exchangers by condensation, substitution, and polymerization reactions and were used in achiral liquid chromatography. Later, crown ethers were adsorbed on silica gel and were used to separate cations and anions [36-39]. Shinbo et al. [40] adsorbed hydrophobic CCE on silica gel and the developed CSP was used for the chiral resolution of amino acids. Kimura et al. [41-43] immobilized poly- and bis-CCEs on silica gel. Later, Iwachido et al. [44] allowed benzo-15-crown-5, benzo-18-crown-6 and benzo-21-crown-7 CCEs to react on silica gel. Of course, these types of CCE-based phases were used in liquid chromatography, but the column efficiency was very poor due to the limited choice of mobile phases. Therefore, an improvement in immobilization was realized and new methods of immobilization were developed. In this direction, CCEs were immobilized to silica gel by covalent bonds. 

The solution for f(q ) cannot be obtained analytically. Although the similarity transformation has reduced the set of PDEs, (10-64), to a single ODE, (10-75), the latter is still nonlinear. In fact, Blasius originally solved (10-75) by using a numerical method, but with the algebra carried out by hand Fortunately, today accurate numerical solutions can be obtained with a computer. The main difficulty in solving (10 75) numerically is that most methods for solving ODEs are set up for initial-value problems. 

Solve the Blasius problem using the shooting method in Mathcad. 

Figure 2.2 Solution of the Blasius problem by the shooting method.

Blasius steady-flow, laminar, flat-plate, boundary-layer solution is a numerical solution of his simplification of Prandtl s boundary-layer equations, which are a simplified, one-dimensional momentum balance and a mass balance. This type of solution is known in the boundary-layer literature as an exact solution. Exact solutions can be found for only a very limited number of cases. Therefore, approximate methods are available for making reasonable estimates of the behavior of laminar boundary layers (Prob. 11.8). 

For boundary layers on curved surfaces, the pressure will change with distance. This greatly complicates the solution of the boundary-layer equations compared with that on a flat plate (in which dPIdx was zero), and so very few exact solutions are known for such boundary layers. Some estimate of the behavior of such boundary layers is given by several methods. To illustrate, we apply them to the laminar boundary layer on a flat plate, where we can compare the results with Blasius exact solution. These methods begin by assuming a velocity profile of the form V tV where S is the boundary-layer thickness. 

Introduction and derivation of integral expression. In the solution for the laminar boundary layer on a fiat plate, the Blasius solution is quite restrictive, since it is for laminar flow over a flat plate. Other more complex systems cannot be solved by this method. An approximate method developed by von Karman can be used when the configuration is more complicated or the flow is turbulent. This is an approximate momentum integral analysis of the boundary layer using an empirical or assumed velocity distribution. 

As discussed in the analysis of the hydrodynamic boundary layer, the Blasius solution is accurate but limited in its scope. Other more complex systems cannot be solved by this method. The approximate integral analysis was used by von Karman to calculate the hydrodynamic boundary layer and was covered in Section 3.10. This approach can be used to analyze the thermal boundary layer. 

In a similar fashion, the integral momentum analysis method used for the turbulent hydrodynamic boundary layer in Section 3.10 can be used for the thermal boundary layer in turbulent flow. Again, the Blasius 7-power law is used for the temperature distribution. These give results that are quite similar to the experimental equations as given in Section 4.6. 

Copolymerization of polymeric crown ether with silica gel or other support materials is another way to incorporate crown ether into stationary phases. Blasius and coworkers thoroughly studied methods for polymerizing cyclic polyethers with various polymeric matrices and applied... 

For turbulent flows, the Bowen method, which is essentially a modification of the Bla-sius method, should be used. Bowen (1961) suggested the following modification to the Blasius equation ... 

Listing 11.30. Solution of Blasius third order boundary value equation by two methods. 

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