Method of Krishna

The simplest solution to Eqs. 8.5.3 is obtained by assuming that the matrix can be 

The molar fluxes A, are then given by Eq. 8.5.3 with [jSlB] evaluated at some average composition (the arithmetic average of yg and is that used in practice) (Krishna, 1979d, 1981b) 

With the mole fraction gradients obtained by differentiating Eq. 8.5.4 we have 

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That is, the matrices [A] and [B][ ] are equal and not simply equivalent. Thus, the explicit method of Krishna could equally well be written in the form... 

The explicit method of Krishna (1979d, 1981b) is most successful if the are close together and, therefore (or for other reasons), the total flux is low. At high rates of mass transfer, the assumption of constant (or of [/3][B] ) is a poor one, particularly in... 

SOLUTION We shall use the method of Krishna and Standart described in Section 8.3 to compute the molar fluxes. Algorithms 8.3.1 or 8.3.2 may be used to determine the molar fluxes. Convergence is very rapid in this example. No more than two iterations are needed. Only the final results of the relevant computations are summarized below. 

To relate the multicomponent mass-transfer rates to binary mass-transfer coefficients, the method of Krishna and Standart [71] can be used. By this method, the diffusional fluxes for the liquid phase are calculated from... 

Measurement of adenyl cyclase activity was a laborious and imprecise procedure until the introduction of the method of Krishna et al. This method is based on the production of labeled cyclic AMP from ATP-a- P. Labeled product is separated from substrate by a combination of ion exchange chromatography and adsorption of substrate to nascent BaS04 precipitate. With appropriate equipment, it is possible to perform 300 adenyl cyclase assays in one day. A complication of the method arises from the fact that adenyl cyclase preparations invariably contain sufficient ATPase activity to deplete rapidly the substrate concentration, thereby shortening the period of time over which adenyl cyclase activity can be observed and precluding even the simplest kinetic analysis. This problem has generally been overcome by addition of an ATP-regenerating system to the adenyl cyclase assay medium . In addition, it has recently been shown that 5 -adenylyl-imidodiphosphate (AMP-PNP) is a substrate for adenyl cyclase but not for ATPase ... 

In plate column distillation the basic performance concept is that of plate efficiency, usually the Murphree type, although the Hausen and Standart variations have occasionally been recommended and used. The mass transfer process enters the plate efficiency concept through the relation between the flux, and hence mass transfer coefficients, and the point efficiency which, in turn, is related to the plate efficiency by flow behaviour on and between the plates. Since in multicomponent mixtures the mass transfer coefficients of the components are different, the same must apply to the point efficiencies and also to the plate efficiencies. Again, the method of Krishna and Standart can be used to predict plate efficiencies of individual components and it has been successfully tested for ternary mixtures. However, it has not as yet been accepted in general distillation practice. 

Adenylate cyclase assay was carried out into mice brain crude mitochondrial fraction by procedure based on the method of Krishna et al.(1968). 

Of the methods which have been published, that due to Krishna and Ghose has been recommended by Peyer and Gstirner for its simplicity and uniform results. Methods involving final extraction and evaporation of the base with chloroform should be avoided as ephedrine base reacts on standing with, or evaporation from, this solvent to form the hydrochloride. The B.P.C, 1954 adopted the method of Krishna and Ghose with small modifications, which make it more accurate and give results from 15 to 20 per cent higher. Their method has also been recommended as standard by the Analytical Methods Committee of the S,A,CA... 

For commercial Ephedra the British Pharmaceutical Codex, 1934, specifies a total alkaloidal content of not less than 1-25 per cent, when assayed by the method therein prescribed. The proportion of Z-ephedrine is generally about 70 per cent. Methods of assay for total alkaloids are described by Feng and Read and by Krishna and Chose, who discuss the various difficulties involved and comments on these and other methods have been made by various workers. Conditions affecting the results of such assays have also been discussed by T ang and Wang, and Brownlee has shown that chloroform is not a suitable medium for the assay since it converts ephedrine quickly and 0-ephedrine slowly to the hydrochloride. 

The linearized theory of Toor (1964a) and of Stewart and Prober (1964) is probably the most important method of solving multicomponent diffusion problems. Very often, the method provides the only practical means of obtaining useful analytical solutions of multicomponent diffusion problems. Additional applications of the method are developed in Chapters 8-10 and still more can be found in the literature [see Cussler (1976), Krishna and Standart (1979) and Taylor (1982c) for sources]. 

The solution to the matrix differential Eq. 10.4.6 can be found using the method of successive substitution (Appendix B.2). Here we follow closely the treatment by Taylor (1981b) (see, also Krishna, 1982). The solution to Eq. 10.4.6 can be written as... 

A number of investigators used the wetted-wall column data of Modine to test multicomponent mass transfer models (Krishna, 1979, 1981 Furno et al., 1986 Bandrowski and Kubaczka, 1991). Krishna (1979b, 1981a) tested the Krishna-Standart (1976) multicomponent film model and also the linearized theory of Toor (1964) and Stewart and Prober (1964). Furno et al. (1986) used the same data to evaluate the turbulent eddy diffusion model of Chapter 10 (see Example 11.5.3) as well as the explicit methods of Section 8.5. Bandrowski and Kubaczka (1991) evaluated a more complicated method based on the development in Section 8.3.5. The results shown here are from Furno et al. (1986). 

Repeat Example 8.8.1 (ternary distillation in a wetted wall column) using the method of calculating the fluxes developed in Section 8.3 but following the suggestion of Krishna and Standart as described in Section 8.8.3 for estimating the binary mass transfer coefficients. 

Revise the analysis of Example 11.5.3 and show how a method based on the film models of Chapter 8 could be used to compute the rates of mass transfer. Then use the Krishna-Standart method (of Sections 8.3 and 8.8.3) and compute the molar fluxes. Binary pair mass transfer coefficients may be estimated using the Chilton-Colburn analogy. 

Krishna, R. and Standart, G. L., A Multicomponent Film Model Incorporating an Exact Matrix Method of Solution to the Maxwell-Stefan Equations, AIChE J, 22, 383-389 (1976a). 

In recent years, considerable progress has been made to improve further upon this method for use with multicomponent mixtures. Detailed discussions of these methods may be found in Stephan [25], Hewitt et al. [193], and Webb [194]. The procedure of Sardesai et al. [205], which outlines the work of Krishna and Standart [206], is briefly described below. 

R. Krishna and G. L. Standart, A Multicomponent Film Model Incorporating a General Matrix Method of Solution to Maxwell-Stephan Equations, AIChE I, 22, pp. 383-389,1976. 

Feng and Read (104) found that the low yield of alkaloids obtained by previous workers was due to incomplete alkalinization of the herb before extraction with chloroform or ether. Hot extraction and the use of sodium hydrate to liberate the alkaloids has been found satisfactory. The ammonia-chloroform process has been critically studied and it was found that a large excess of ammonia was necessary to hberate the alkaloids. Feng (103) extracted Ephedra equisetina, first with 80% alcohol and finally with 0.2 % acetic acid. After working up the extracts, ephedrine was separated from f -ephedrine by crystallization of the hydrochlorides from 95% alcohol. -Ephedrine may be recovered from the mother liquors. Ghose and Krishna (114, 117, 118) described other methods of extraction and the preparation of alkaloid concentrates. They separated ephedrine from -ephedrine by extracting the dry hydrochlorides with chloroform, in which only the f -ephedrine salt is soluble. 

In the past decade or so, lipase-catalyzed esterifications and transesterifications in anhydrous media (e.g., organic solvents and supercritical fluids) have been an area of intensive research. In particular, the use of organic solvents, which normally allow a higher stability of enzymes than in water (Bock, Jimoh, Wozny, 1997), has been demonstrated. Reviews of the applications have been made by Hail Krishna and Karanth (2002) and Gandhi et al. (2000), dealing with fundamental and practical aspects of lipase catalysis. In particular, they concentrated on various immobilization strategies and factors (e.g., temperature, reaction medium, water activity) as weU as the methods of preparation (which affect and influence the stability of the lipases). 

Gaion, R. M., and Krishna, G., 1979a, Cytidylate cyclase The product isolated by the method of Cech and Ignarro is not cytidine 3, 5 -monophosphate, Biochem. Biophys. Res. Commun. 86 105. 

For the assay of ephedrine in the total alkaloids a colorimetric method based on the biuret reaction was used by Feng and Read and is described in detail by Feng. Krishna and Chose separated ephedrine and iji-ephedrine by treating the dry mixed hydrochlorides with dry chloroform in which the ephedrine salt is virtually insoluble and the -ephedrine salt soluble. ... 

Krishna Murthy H. M. (1996). The use of multiple wavelength anomalous diffraction in ab initio phase determination. Method Mol. Biol. 56,127-152. 

Krishna et al. reported a nitritometric method, using aminoanthraquinone dyes as indicators, for the determination of procaine, other 4-amino-benzenesulfonamides, and 4-aminobenzoic acid derivatives [77]. Using sodium dioctyl sulfosuccinate, Faicao and Vianna determined procaine hydrochloride [78]. 

Microwave extraction methods are now being developed [22-25]. Krishna-murti et al. [22] found that the microwave extraction of cadmium in a soil reference material gave results comparable to those obtained by conventional soil extraction methods. In another study, Kingston and Walter [23] compared... 

Diffusional interaction methods have also been applied successfully to packed columns. Gorak (204) found that the Krishna-Standart model (205) is relatively simple and sufficiently accurate to predict multicomponent composition profiles. Gorak s own variation of the diffusional interaction method was also reported to predict experimental data well, while use of HETP was reported to give poor data predictions. 

The hypervalent iodine reagents PIFA and PIDA have also been used in the synthesis of naturally occurring structures, primarily the amaryllidaceae alkaloids and related species. Work by White s group showed the feasibility of this method for the synthesis of 6a-epipretazettine and (-)-codeine [45, 46]. In the early 1990s, Rama Krishna and co-workers demonstrated that PIDA can promote the oxidative phenolic coupling of diaryl substrates 38a-e to deliver cyclohexadienones 39a-e, respectively, in consistent 30 % yields for all of the substrates examined (Scheme 10) [47]. 

If Eqs. (5-200) and (5-201) are combined, the multicomponent diffusion coefficient may be assessed in terms of binary diffusion coefficients [see Eq. (5-214)]. For gases, the values Dy of this equation are approximately equal to the binary diffusivities for the ij pairs. The Stefan-Maxwell diffusion coefficients may be negative, and the method may be applied to liquids, even for electrolyte diffusion [Kraaijeveld, Wesselingh, and Kuiken, Ind. Eng. Chem. Res., 33, 750 (1994)]. Approximate solutions have been developed by linearization [Toor, H.L., AlChE J., 10,448 and 460 (1964) Stewart and Prober, Ind. Eng. Chem. Fundam., 3,224 (1964)]. Those differ in details but yield about the same accuracy. More recently, efficient algorithms for solving the equations exactly have been developed (see Taylor and Krishna, Krishnamurthy and Taylor [Chem. Eng. J., 25, 47 (1982)], and Taylor and Webb [Comput Chem. Eng., 5, 61 (1981)]. 

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