A distribution

The capital cost of most aqueous waste treatment operations is proportional to the total flow of wastewater, and the operating cost increases with decreasing concentration for a given mass of contaminant to be removed. Thus, if two streams require different treatment operations, it makes no sense to mix them and treat both streams in both treatment operations. This will increase both capital and operating costs. Rather, the streams should be segregated and treated separately in a distributed effluent treatment system. Indeed, effective primary treatment might mean that some streams do not need biological treatment at all. 

One has seen that the number of individual components in a hydrocarbon cut increases rapidly with its boiling point. It is thereby out of the question to resolve such a cut to its individual components instead of the analysis by family given by mass spectrometry, one may prefer a distribution by type of carbon. This can be done by infrared absorption spectrometry which also has other applications in the petroleum industry. Another distribution is possible which describes a cut in tei ns of a set of structural patterns using nuclear magnetic resonance of hydrogen (or carbon) this can thus describe the average molecule in the fraction under study. 

Each of the input parameters reguires an estimate of the range of values, which can itself be represented by a distribution, or expectation curve. Ideally, the expectation curves for the input parameters are combined together statistically. 

A random number (between 0 and 1) is picked, and the associated value of gross reservoir thickness (T) is read from within the range described by the above distribution. The value of T close to the mean will be randomly sampled more frequently than those values away from the mean. The same process is repeated (using a different random number) for the net-to-gross ratio (N/G). The two values are multiplied to obtain one value of net sand thickness. This is repeated some 1,000-10,000 times, with each outcome being equally likely. The outcomes are used to generate a distribution of values of net sand thickness. This can be performed simultaneously for more than two variables. 

Once oil and gas have been processed the products have to be evacuated from the site. Stabilised crude is normally stored in tank farms at a distribution terminal which may involve an extended journey by pipeline. At a distribution terminal, crude is stored prior to further pipeline distribution or loading for shipment by sea (Figure 10.28). 

As the first well is being drilled the template is connected to the host facility with flowlines, umbilicals and risers. A Chemical Injection Umbilical will also typically be laid to the template or subsea facility and connected to a distribution manifold. 

The computed CWT leads to complex coefficients. Therefore total information provided by the transform needs a double representation (modulus and phase). However, as the representation in the time-frequency plane of the phase of the CWT is generally quite difficult to interpret, we shall focus on the modulus of the CWT. Furthermore, it is known that the square modulus of the transform, CWT(s(t)) I corresponds to a distribution of the energy of s(t) in the time frequency plane [4], This property enhances the interpretability of the analysis. Indeed, each pattern formed in the representation can be understood as a part of the signal s total energy. This representation is called "scalogram". 

The transducer is modelled by a distribution of particle velocity source over the radiating surface Tr as... 

The field radiated into the coupling medium by such a distribution of sources may be obtained by means of the well-known Rayleigh integral. The field at the considered point r is computed by a simple integral over the whole radiating surface of the contributions of each elementary source acting as a hemispherical point source. 

The first term on the right is the common inverse cube law, the second is taken to be the empirically more important form for moderate film thickness (and also conforms to the polarization model, Section XVII-7C), and the last term allows for structural perturbation in the adsorbed film relative to bulk liquid adsorbate. In effect, the vapor pressure of a thin multilayer film is taken to be P and to relax toward P as the film thickens. The equation has been useful in relating adsorption isotherms to contact angle behavior (see Section X-7). Roy and Halsey [73] have used a similar equation earlier, Halsey [74] allowed for surface heterogeneity by assuming a distribution of Uq values in Eq. XVII-79. Dubinin s equation (Eq. XVII-75) has been mentioned another variant has been used by Bonnetain and co-workers [7S]. 

The induction energy is inlierently non-additive. In fact, the non-additivity is displayed elegantly in a distributed polarizability approach [28]. Non-additive induction energies have been found to stabilize what appear to be highly improbable crystal structures of the alkalme earth halides [57]. 

Such an ensemble of systems can be geometrically represented by a distribution of representative points m the F space (classically a continuous distribution). It is described by an ensemble density fiinction p(p, q, t) such that pip, q, t)S Q is the number of representative points which at time t are within the infinitesimal phase volume element df p df q (denoted by d - D) around the point (p, q) in the F space. 

Only at extremely high electric fields are the water molecules fiilly aligned at the electrode surface. For electric fields of the size normally encountered, a distribution of dipole directions is found, whose half-widtli is strongly dependent on whether specific adsorption of ions takes place. In tlie absence of such adsorption the distribution fiinction steadily narrows, but in the presence of adsorption the distribution may show little change from that found at the PZC an example is shown in figure A2.4.10 [30]. 

One can introduce a distributed micleation rate J(R, f)dR for nucleating clusters of radius between R and R + dR. Its integral overi is the total micleation rate J(0- Equation (A3,3.1031 can be viewed as a radius-dependent droplet energy which has a maximum at = R. If one assumes j R, t) to be a Gaussian fiinction, then... 

For a sample at diennal equilibrium there is a distribution of speeds which depends on the mass of the molecules and on the temperature according to the Boltzmaim distribution. This results in a line shape of the form... 

Here the ijk coordinate system represents the laboratory reference frame the primed coordinate system i j k corresponds to coordinates in the molecular system. The quantities Tj, are the matrices describing the coordinate transfomiation between the molecular and laboratory systems. In this relationship, we have neglected local-field effects and expressed the in a fomi equivalent to simnning the molecular response over all the molecules in a unit surface area (with surface density N. (For simplicity, we have omitted any contribution to not attributable to the dipolar response of the molecules. In many cases, however, it is important to measure and account for the background nonlinear response not arising from the dipolar contributions from the molecules of interest.) In equation B 1.5.44, we allow for a distribution of molecular orientations and have denoted by () the corresponding ensemble average ... 

It has been shown that spherical particles with a distribution of sizes produce diffraction patterns that are indistingiushable from those produced by triaxial ellipsoids. It is therefore possible to assume a shape and detemiine a size distribution, or to assume a size distribution and detemiine a shape, but not both simultaneously. 

A distributiony (v ) of NM) test particles (cm of species A in a beam collisionally interacts with a distribution of N () field particles of species B. Collisions with B will scatter A out of the beam at the... 

Single molecules also have promise as probes for local stmcture when doped into materials tliat are tliemselves nonfluorescent. Rlrodamine dyes in botli silicate and polymer tliin films exliibit a distribution of fluorescence maxima indicative of considerable heterogeneity in local environments, particularly for the silicate material [159]. A bimodal distribution of fluorescence intensities observed for single molecules of crystal violet in a PMMA film has been suggested to result from high and low viscosity local sites witliin tire polymer tliat give rise to slow and fast internal conversion, respectively [160]. 

Obtaining high-quality nanocry stalline samples is the most important task faced by experimentalists working in tire field of nanoscience. In tire ideal sample, every cluster is crystalline, witli a specific size and shape, and all clusters are identical. Wlrile such unifonnity can be expected from a molecular sample, nanocrystal samples rarely attain tliis level of perfection more typically, tliey consist of a collection of clusters witli a distribution of sizes, shapes and stmctures. In order to evaluate size-dependent properties quantitatively, it is important tliat tire variations between different clusters in a nanocrystal sample be minimized, or, at tire very least, tliat tire range and nature of tire variations be well understood. 

Therefore, the absorjDtion line is massively inlromogeneously broadened at low temperature. An inliomogeneous lineshape can be used to detennine the static or quasistatic frequency spread of oscillators due to a distribution of environments, but it provides no dynamical infonnation whatsoever [94, 95]. As T is increased to 300 K, the absorjDtion linewidth decreases and increases. At 300 K, the lineshape is nearly homogeneously broadened and dominated by vibrational dephasing, because fast dephasing wipes out effects of inliomogeneous environments, a well known phenomenon tenned motional narrowing [951. 

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