Nicholson technique

Among the variety of methods which have been proposed for simulation of packed bed dynamics three techniques have been used with success (1) Crank-Nicholson technique [10], (2) transformation to integral equation [11], (3) orthogonal collocation on finite elements [12]. In the following computation, we have used the Crank-Nicholson method with the nonequidistant space steps in the Eigenberger and Butt version [10]. 

The time increment was corrected according to a number of iterations necessary for calculation of a new profile. A revision of the space finite difference mesh was performed after five time steps. Frequently, 30-60 mesh points were sufficient also for high values of B. The Eigenberger-Butt method lowers the computer time expenditure in comparison with the classical Crank-Nicholson technique by a factor 4-10. 

The Crank-Nicholson technique is a widely applied method for solving partial diiferential equations such as those for the radial dispersion model. However, it is implicit in approach and thus a little balky sometimes. Use this approach to develop an algorithm for the solution of the equations of Illustration 7.11. You will see that, if the method is developed properly, it will result in equations leading to a tridiagonal matrix similar to those treated in Illustration 6.4. 

The movement of the rubbery-solvent interface, S, was governed by the difference between the solvent penetration flux and the dissolution rate, derived earlier. An implicit Crank-Nicholson technique with a fixed grid was used to solve the model equations. A typical concentration profile of the polymer is shown in Fig, 24. Typical Case II behavior was observed. The respective positions of the interfaces R and S are shown in Fig. 25. Typical disentanglement-controlled dissolution was observed. Limited comparisons of the model predictions were made with experimental data for a PMMA-MIBK system. 

M. Foster, I. Grigorev, D. Lurie, V. Khramtsov, S. McCallum, I. Panagiotelis, J. Hutchison, A. Koptioug, and I. Nicholson, In vivo detection of a pH-sensitive nitroxide in the rat stomach by low-field ESR-based techniques. Magn. Reson. Med. 49, 558—567 (2003). 

M.M. Nicholson, Fundamentals and Techniques of Mathematics for Scientists, 1961, Longmans, Green and Co., London. 

Robinson, S., Nicholson, R. A., Pollard, A. M., and O Connor, T. P. (2003). An evaluation of nitrogen porosimetry as a technique for predicting taphonomic durability in animal bone. Journal of Archaeological Science 30 391 —403. 

Nicholson, H.M., and J.P. Field. 1949. Some experimental techniques for the investigation of the mechanism of flame stabilization in the wakes of bluff bodies. 3rd Symposium on Combustion, Flame and Explosion Phenomena Proceedings. Baltimore The Williams and Wilkins Co. 44-68. 

So far, relatively little attention has been given to the variational method of solving diffusion problems. Nevertheless, it is a technique which may become of more interest as the nature of problems becomes more complex. Indeed, the variational method is the basis of the finite element method of numerical calculations and so is, in many ways, an equal alternative to the more familiar Crank—Nicholson approach [505a, 505b]. The author hopes that the comments made in this chapter will indicate how useful and versatile this approach can be. 

Hrabetova S, Nicholson C. Biophysical properties of brain extracellular space explored with ion-selective microelectrodes, integrative optical imaging and related techniques. In Michael AC, Borland LM (Eds), Electrochemical Methods for Neuroscience. CRC Press/ Taylor Francis, Boca Raton, FL, 2007 167-204. 

Similarly to the above derivation, we can also use the technique to predict transient temperature fields. Again, as with finite elements and boundary elements, the time stepping is done using finite difference techniques. For a Crank-Nicholson transient energy equation formulation given by... 

Nicholson, A., Long, S.E., McEwen, I. and Sparkes, S.T. (1990) The development of a speciation technique for uranium in the environment. AERE R13436, AEA Environment and Technology, Harwell Laboratory, Oxon, UK, 22 pp. 

Applications of these techniques to drug metabolism are reviewed by Nicholson and Wilson (1989), Preece and Timbrell (1990), and Malet-Martino and Martino (1992). Their application to the study of plasma composition in cancer has been reviewed by Vion-Dury et al. (1993). 

Kross, B.C., H.F. Nicholson and L.K. Ogilvie (1996). Methods Development Study for Measuring Pesticide Exposure to Golf Course Workers using Video Imaging Techniques, Appl. Occup. Environ. Hyg., 11, 1346-1350. 

Capillary gas chromatography with optically active stationary phases became a well-established technique for stereochemical analysis after the pioneering work of Gil-Av and his associates in the mid-1960s (2). Thermal stability of the initially low-molecular-weight amino add and peptide derivatives was greatly improved after the introduction of chiral polysUox-anes by Frank, Nicholson, and Bayer (Chirasil-val [1], 1977) (3) and of... 

The effect of pressure on the rate of free radical propagation reactions has been studied in homopolymerizations only for styrene. Since aV is negative for this reaction, the apphcation of pressure increases the propagation rate. Nicholson and Norrish (12) list the propagation rate constant for styrene as 72.5 liters-mole- —sec.- at atmospheric pressure and 30° C. This increases to 206 liters-mole —sec. at 2000 atm. and 400 hters-mole —sec. at 3000 atm. From these data it is possible to calculate the value of AV for the propagation step to be —13.3 cc. per mole. Walling and Pellon (16) report a value of —11.5 cc. per mole for the same reaction measured by a different technique. 

Lindon JC, Holmes E, Nicholson JK. Metabonomics techniques and applications to pharmaceutical research development. Pharm Res 2006 23(6) 1075-88. 

Nicholson, B.L. Techniques in fish cell culture, In Techniques in the Life Sciences, Vol. Cl, edited by E. Kurstak, Limmerick, Elsevier, pp. C015/1-C015/16, 1985. 

EPD is an efiective technique to synthesize monolithic as well laminates thick film [1-8] and bulk samples [9-13]. Saikar and Nicholson [9,10] are the first to demonstrate the EPD can be successfully use for fabrication of laminated and functionally graded materials. There are few reports on brication of BaTiOj monolithic thick film by EPD [1,5,6]. Yamashita et al [7] is the only one used EPD to synthesize BaTiOj/SrTiO, laminated composite with varying BT/ST ratio and observed board Curie temperature response in the sample. 

P. Sarkar, X. Huang and P.S. Nicholson,"Zirconia-Alumina Functionally-Gradiented Composites by Electrophoretic Deposition Techniques", J. Am. Ceram. Soc. 76 (1993) 1055-1056. 

There is long standing theoretical interest in the question of how this affects the electronic DOS [5.18,28-33], Due to the inapplicability of Bloch s theorem, calculations are extremely difficult. Using Green s function techniques, Ballentine [5.28] could show for liquid metals that distinct deviations from the free-electron-like behaviour may occur whenever v(K) is significantly large at a peak of S(K) (see Fig. 5.2a, c). The width of the so-called pseudo gap may then correspond to 2 n(/C) as in crystalline matter, and the depth to the intensity of S(K), the structural weight. Theoretical considerations by Nicholson and Schwartz [5.30] as well as recent work by Fresard [5.32], Beck et al. [5.33], and Hafner et al. [5.18] could also show the structural effects on the DOS. 

Nicholson, M.D. and Fryer R. (1995) Techniques in the Marine Environmental Sciences - A Robust Method for Analysing Contaminant Trend Monitoring Data, International Council for the Exploration of the Sea. 

Nicholson, M.D., Fryer, R.J. and Larsen, J.R. (1998) ICES techniques in marine environmental sciences. No. 20, ICES, Copenhagen, Denmark. 

In this section, a model similar to that used by Cardoso and Luss (1969) is considered, with the exception that the assumption of solid isothermality is relaxed. A finite difference solution for the transient equations with symmetrical boundary conditions is presented using the Crank-Nicholson method (Lapidus, 1962). Another more efficient, method of solution is considered which is based on the orthogonal collocation technique first used by Villadsen and Stewart (1967), Finlayson (1972) and Villadsen and Michelsen (1978). Several assumptions for model reductions are investigated. 

Solution of Mathematical Model for Case 1. For the Case 1 solution iterative techniques were ruled unacceptable owing to the excessive time requirements of such methods. Several investigators (27, 28, 29, 30) working with similar noncoupled systems found that the Crank-Nicholson 6-point implicit differencing method (31) provided an excellent solution. For the solution of Equation (8) we decided to apply the Crank-Nicholson method to the second-order partials and corresponding explicit methods to the first-order partials. Nonlinear coefficients were treated in a special manner outlined by Reneau et al (5). 

Solution of Mathematical Model for Case 2 and Case 3. For the Case 2 and Case 3 solutions, the technique developed for the Case 1 solution, which was based on the Crank-Nicholson implicit method, was unacceptable when the time derivative was included. 

Holmes, E., J. K. Nicholson, A. W. Nicholls, J. C. Lindon, S. C. Connor, S. Policy, and J. Connelly. 1998. The identification of novel biomarkers of renal toxicity using automatic data reduction techniques and PCA of proton NMR spectra of urine. Chemometrics and Intelligent Laboratory Systems 44 245-255... 

To obtain field data to test and improve models of dry deposition of particles, improved techniques and/or new instrumentation appear to be critical (Nicholson, 1988). Particle-size-specific, eddy-flux measurements (see Section 19.5) of ambient particles appear promising (e.g., Neumann and den Hartog, 1985) but to date have suffered from poor statistics associated with few particles in specific size classes. Furthermore, similar studies at... 

Despite the first presentation of the network analyser in 1965, the large-band measurement techniques only appeared in 1974. After ameliorations on the accuracy and the development of Von Hippel s methods, the first data treatments were proposed. Weir [III] and Nicholson [112] used the reflection and transmission coefficients (S parameters) resulting when a test sample was inserted into a waveguide or a TEM transmission line as shown in Figure 8.7 From measurements, complex permittivity and permeability values were derived in the range from 50 MHz to 18 GHz. 

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