Directional property ordinate

Paper made on a paper machine exhibits quite different properties in the x and y directions (the machine and cross machine directions), an example of which is a difference in stiffness which can be demonstrated by plotting the specific elastic stiffness in the x-y plane as a function of the machine direction and cross machine direction co-ordinates in the form of a polar diagram (Figure 4.7). 

An isotropic material has the same properties in all directions. Properties such as refractive index and Young s modulus are independent of direction, and if we wish to refer the properties to a set of rectangular cartesian co-ordinates, we can rotate the axes to be in any orientation without any preferoice. For an anisotropic material, where the properties differ with direction, it is usually convenient to choose coordinate systems which coincide with axes of S3rmmetry if this is possible. The material is then described by its properties referred to these principal directions, which affords considerable simplification. 

Each point corresponding to the ordinate axis is the value of the cumulative property of the cut. The Cg-EP properties of gasoline cuts or IP-EP properties of residue cuts are obtained directly from the curves, while properties of other cuts are calculated either directly for the properties that are additive by volume, weight or moles, or by using blending indices. 

The other important property of an electron that must be specified besides its spatial distribution is its spin. According to quantum-mechanical arguments, into which we need not go in detail, each electron has a spin which can take one of two values. It is convenient to include this description in the wave function by defining a and / so that at = 1 if the spin is in one direction and = 0 if it is in the other, ft is defined in a complementary manner. Thus the electron moving in an orbital ip(x, y, z) may be associated with two functions pix, y, z) . and fix, y, z)8 according to the direction of its spin. A function such as %p x, y, z)a whioh gives the probability distribution of the spin co-ordinate as well as that of its spatial co-ordinates is sometimes referred to as a spin orbital. 

Probably one of the commonest reactions encountered in the template synthesis of macrocycles is the formation of imine C=N bonds from amines and carbonyl compounds. We have seen in the preceding chapters that co-ordination to a metal ion may be used to control the reactivity of the amine, the carbonyl or the imine. If we now consider that the metal ion may also play a conformational role in arranging the reactants in the correct orientation for cyclisation, it is clear that a limitless range of ligands can be prepared by metal-directed reactions of dicarbonyls with diamines. The Tt-acceptor imine functionality is also attractive to the co-ordination chemist as it gives rise to strong-field ligands which may have novel properties. All of the above renders imine formation a particularly useful tool in the arsenal of preparative co-ordination chemists. Some typical examples of the templated formation of imine macrocycles are presented in Fig. 6-12. 

Ordinal relations Ordinal relations can vary from a partial order, where one or more elements have precedence over others, to a complete order, where all elements are ordered with respect to some property or properties. There are two separable issues in mapping order onto space. One is the devices used to indicate order, and the other is the direction of order. The direction of indicating order will be discussed after interval relations, as the same principles apply. 

To solve Equations (67)-(69), the Discrete Ordinate Method was applied (Duderstadt and Martin, 1979). From the solution of the RTE, the monochromatic radiation intensity at each point and each direction inside the reactor can be obtained. Considering constant optical properties of the catalyst and steady radiation supply by the emitting system, the radiation field can be considered independent of time. 

The terminal velocity of liquid drops is the same as solid spheres when the diameter is small. The drag coefficient versus Reynold s number can be recalculated to provide a diameter-free ordinate versus a velocity-free abscissa to facilitate direct solution, as shown in Fig. 15. With drops, a maximum velocity is attained, and this maximum has been correlated with a parameter based on physical properties of the system. 

High-temperature ionic solvents are known to contain relatively high total concentrations of cations (e.g. in the KCl-LiCl eutectic, the concentration of Li+ is approximately equal to 8.5 mol kg-1 of the melt). Usually, cation-anion complexes in molten salts are characterized by co-ordination numbers of the order of 4-6. This means that the maximal consumption of acidic cations does not exceed 0.4-0.6 mol kg-1 in diluted solutions with concentrations close to 0.1 mol kg-1. This estimate is considerably lesser than the initial concentration of acidic cations in the pure melt. In the case of the KCl-LiCl eutectic melt, this consumption is only of the order of 5-7%, and the value of NMe+ in equation (1.3.16) may be assumed to be constant. Therefore, for each ionic solvent of the second kind (kind II) the denominator in equation (1.3.16) is a constant which characterizes its acidic properties. We shall define p/L = -log /L to be the relative measure of acidic properties of a solvent and call it the oxobasicity index of ionic melt [37, 162, 181]. Since the direct determination of the absolute concentration of free oxide ions in molten salts is practically impossible, the reference melt should be chosen— for this melt, /L is assumed to be 1 and p/L = 0. The equimolar KCl-NaCl... 

In order to eliminate problems assoeiated with the differenees in surface properties and shapes of coimnercially available dehydrated vermicelli, Andre and Pauli (1978) ground the samples down to powders, then formed tablets from the powders using a hydraulic press. The 40 mm diameter by 10 mm thick tablets could be measured directly on the head of the colorimeter. Good correlations were found between colour co-ordinate values and the / -carotene content derived from the egg component of the pastas. Pastas made from different flours could also be differentiated in terms of colour values. 

Sintered ceramics made of lead-zirconium titanate (PZT Pb(Tii jZr,)03 x S 0.5) are usually used for phoioacoustic experiments [105, 106]. The unit cell of the lead-zirconium titanate has a perovskite structure. Below the Curie temperature (328 °C for the PZT-4 (Vemitron) used by us [24]), the cells are tetragonally deformed, i.e., positive and negative charges are shifted and electric dipole moments are produced. In analogy to ferromagnetism, domains with randomly distributed polarization direction are formed. By the application of an electric field, these can be orientated in a preferred direction, and the sintered polycrystalline ceramic is then remanently polarized. The properties of these anisotropic piezoelectric materials are described by various parameters which depend on the polarization and deformation direction. In the common terminology, the < ordinate system shown in Fig. 3 is obtained for the cylindrical piezoelectric crystals [24]. 

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