Unidimensional case

Viscosity is the measure of the resistance that a fluid offers to an applied shearing force. A gas whose velocity in the direction Oy is a function of the spatial coordinates z, is considered. NEWTON s law states that the flux density of momentum in the direction Oz perpendicular to Oy is proportional to the velocity gradient  

The SI unit of viscosity is the Pascal s. A fluid has a viscosity of 1 Pa s if a force of 1 N is required to move a plane of 1 m at a velocity of 1 m s with respect to a plane surface 1 m away and parallel to it. 

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Solvent transport in organic polymer matrices is usually depicted as a two-step mechanism. The first step is the dissolution of the solvent in the superficial polymer layer. This process, which can be considered almost instantaneous in the case of water, creates a concentration gradient. The second step is the diffusion of the solvent in the direction of the concentration gradient. This process may be described by a differential mass balance (often called Fick s second law), which, in the unidimensional case, may be written as... 

The Poisson equation has, in the unidimensional case considered here, the form... 

Initially, efforts were made to find expressions for the deviation of currents at the UMDE from that at a so-called planar electrode, which is the unidimensional case (called a shrouded plane by Oldham [425]). One can regard this as a disk (or any shape) at the bottom of a deep well, so that the system can be reduced to one dimension. Here the Cottrell equation defines the current for a potential step, as in Chap. 2 (2.37) and (2.44). At a flush UMDE, the current deviates from the Cottrell value very soon after the potential jump. Lingane [365] suggested that to a good approximation and for a range of small values of time t, the current iuMDE at a UMDE could be expressed as... 

Exactly the same consideration can be applied to ordinary diffusion, easiest in the unidimensional case. As was shown in 1936 by Christiansen (23), this leads to the following expression for s the diffusion current in a force field. 

The local charge density for this unidimensional case is linear so this flux of electron holes leaking through the predominantly ionic conductor is ... 

Thus, estimating the properties of filled polymers, an investigator rests on a conventional system of simple indices, which are measured in unidimensional (desirably-viscometric) fields under the conditions of stationary flow or low periodic deformations. This, of course, does not exclude the possibility of going over to more complicated conditions of deformation, but as in other cases, it may turn out to be difficult to handle the results of such measurements. 

The physical situation in a fluidized bed reactor is obviously too complicated to be modeled by an ideal plug flow reactor or an ideal stirred tank reactor although, under certain conditions, either of these ideal models may provide a fair representation of the behavior of a fluidized bed reactor. In other cases, the behavior of the system can be characterized as plug flow modified by longitudinal dispersion, and the unidimensional pseudo homogeneous model (Section 12.7.2.1) can be employed to describe the fluidized bed reactor. As an alternative, a cascade of CSTR s (Section 11.1.3.2) may be used to model the fluidized bed reactor. Unfortunately, none of these models provides an adequate representation of reaction behavior in fluidized beds, particularly when there is appreciable bubble formation within the bed. This situation arises mainly because a knowledge of the residence time distribution of the gas in the bed is insuf-... 

Two halide-centered cubane silver cluster complexes, containing discrete units, [Ag8X Se2P(OEt)2 6][PF6], or unidimensional chains, Ag8X[Se2-P(OEt)2]6X ra (X = Br and Cl) have been reported. This is an interesting case where the counter ions dictate the molecular/supramolecular architecture.440... 

In the case where transitions involve low-symmetry phases, general equation 2.50 may be recast into the unidimensional form... 

If [TO4] groups are linked to form chains, multiplicity is the number of linked chains the observed multiplicities in this case are m = 2,3, 4, and 5. Analogously, multiplicity is defined as the number of mutually linked sheets. If several [TO4] groups are linked in a chain, periodicity (p) is the number of groups that defines the structural motive (i.e., after p groups, the chain is obtained by simple translation). Dimensionality of [TO4] groups is the degree of condensation of a crystal structure. The condensation of an infinite number of tetrahedra may lead to infinite unidimensional chains d = 1), bidimensional sheets d = 2), and tridimensional lattices d = 3). The dimensionality of an isolated [TO4] tetrahedron is zero d = 0). 

One of the key parameters in reactive processing is the distribution of residence times and temperatures for all particles in the liquid, because their reactivity depends on temperature, and the degree of conversion is determined by the dwell time inside the mold. Since the fountain effect changes residence time distribution from that in the hypothetical case of steady unidimensional flow, this factor becomes especially important in chemical (reactive) processing, more so than in standard injection molding of thermoplastic materials.284... 

Most thermosetting materials are polymerized in heated molds. Figure 9.4 shows a schematic diagram of the mold L is the part thickness, which is assumed to be much less than the other two dimensions. Therefore, the system may be modeled as a case of unidimensional heat transfer with simultaneous heat generation. 

Dunia R, Qin S. A unified geometric approach to process and sensor fault identification and reconstruction the unidimensional fault case. Computers and Chemical Engineering 1998, 22, 927-943. 

Other convective systems have been used in electroanalytical chemistry. The oldest one is the dropping mercury electrode [74,257]. Convection here arises by virtue of the expansion of the growing mercury drop, and the transport equation is pleasantly simple and unidimensional for the simplified case,... 

Titanium-containing pure-silica ZSM-48 (e.g., [71, 72]), a unidimensional medium pore zeolite, and titano-aluminosilicates with the structure of zeolite Beta [72-74] are materials which are currently scrutinized in catalytic oxidation reactions [75]. In the latter case, however, residual acidity created by framework aluminum leads to undesired side reactions. Since, so far, the direct synthesis of Al-free pure titaniumsilicate Beta was not successful, van Bekkum et al. [76] developed a special post-synthesis modification technique. The three-step procedure... 

Each relativistic wavefunction for the electron of H (and hydrogen-like atoms) has four components, as already mentioned for the simple case of a particle in a unidimensional box. For example, for the 2s orbital (level 2S1/2) we have (see, for example, ref. 34) ... 

In this case, the second-order hyperpolarizability y can be considered as a unidimensional parameter defined along the axis x of the molecule following the single component Yxxxx- Equation 3 can be simpUfied into the relationship Eq. 5, when applying the three-level model in which the lowest excited state 11) and the two-photon excited state 2) are considered to be the most significant... 

A scenario referred to as a sub-Chandrasekhar-mass supernova envisions a C-O WD capped with a helium layer accreted by a companion, and which explodes as the result of a hydrodynamical burning before having reached the Chandrasekhar limit. This type of explosions may exhibit properties which do not match easily the observed properties of typical SNIa events. It cannot be excluded, however, that they are responsible for some special types of events, depending in particular on the He accretion rate and on the CO-sub-Chandrasekhar WD (SCWD) initial mass (e.g. [85]). Unidimensional simulations of He cataclysmics characterized by suitably selected values of these quantities reach the conclusion that the accreted He-rich layer can detonate. Most commonly, this explosion is predicted to be accompanied with the C-detonation of the CO-SC WD. In some specific cases, however, this explosive burning might not develop, so that a remnant would be left following the He detonation. Multidimensional calculations cast doubt on the nature, and even occurrence, of the C-detonation in CO-SC WD (e.g. [86]). 

The large number of TIs, and the fact that many of them are highly correlated, confounds the development of predictive models. Therefore, we attempted to reduce the number of TIs to a smaller set of relatively independent variables. Variable clustering " was used to divide the TIs into disjoint subsets (clusters) that are essentially unidimensional. These clusters form new variables which are the first principal component derived from the members of the cluster. From each cluster of indexes, a single index was selected. The index chosen was the one most correlated with the cluster variable. In some cases, a member of a cluster showed poor group membership relative to the other members of the cluster, i.e., the correlation of an index with the cluster variable was much lower than the other members. Any variable showing poor cluster membership was selected for further studies as well. A correlation of a TI with the cluster variable less than 0.7 was used as the definition of poor cluster membership. 

Figure 2, is unidimensional with channels circumscribed by rings consisting of 18-mem-bered tetragonal (T) atoms which possess free diameters of approximately 1.2 nm. Since the pore diameter is extremely large, VPI-5 adsorbs tri-isopropyl benzene readily, whereas the adsorbed amount of triisopropyl benzene is quite small in the case of NaY zeolite Furthermore, the pore diameter of VPI-5 is large enough that iron-phthalocyanines can be synthesized in the pores. However, the thermal stability of this VPI-5 is rather poor and it is easily transformed to AlPO-8 above 423... 

It turned out that the mechanical dispersion may be formally described by the same Pick s laws if the diffusion coefficient is replaced with mechanical dispersion coefficient This is a proportionality coefficient between value of a deflection of component i migration rate from the average seepage velocity on the one hand, and the component concentration gradient between mixed waters, on the other. In a case of unidimensional and bidimensional fluxes this correlation by analogy with the first Pick s law has the following format... 

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