Production order quantity equation terms

TABLE 28.2 Production Order Quantity Equation Terms... 

The terms in the production order quantity equation are the same as those in the EOQ equation with two exceptions (see Table 28.2). 

Each term on the right-hand side of the equation involves matrix products that contain v a specific number of times, either explicitly or implicitly (for the terms that involve AA). Recognizing that is a zeroth-order quantity, it is straightforward to make the associations... 

The rate of a process is expressed by the derivative of a concentration (square brackets) with respect to time, d[ ]/dt. If the concentration of a reaction product is used, this quantity is positive if a reactant is used, it is negative and a minus sign must be included. Also, each derivative d[ ]/dt should be divided by the coefficient of that component in the chemical equation which describes the reaction so that a single rate is described, whichever component in the reaction is used to monitor it. A rate law describes the rate of a reaction as the product of a constant k, called the rate constant, and various concentrations, each raised to specific powers. The power of an individual concentration term in a rate law is called the order with respect to that component, and the sum of the exponents of all concentration terms gives the overall order of the reaction. Thus in the rate law Rate = k[X] [Y], the reaction is first order in X, second order in Y, and third order overall. 

Table 4 the product has been calculated from eqns (7.24) and (7.25) using the additional data from Table 7. The direct third order term, has been neglected. The published value obtained from )i and jSj as given in the papers is given in the seventh coluttm. In rows 1 and 2 a dipole of 6.2D has been assumed (see text). In the final coluttm the published values that have been obtained from simplified equations such as (7.24) and (7.25) have been reduced by the ratio of the published / eqn (7.25) results of Teng and Garito. Values in brackets are unaltered, except in the case of Wortmann et al. where the published result has been multiplied by the missing 2a> internal field factor. TG Teng and Garito Bur Burland et al.f Boss Bosshard et Stah Stahlein et al. Wort Wortman et al.f° Pal Paley et al.- KS Kaatz and Shelton, Hyper Rayleigh measurement. The pi values are in atomic units and the value in the last column has been adjusted, where appropriate, by the same factors used for the macroscopic quantities in Table 4.

In order to assess the internal consistency of the emissions, as shown in Table 8, a calculation was made whereby the mean atmospheric input was equated to the world metal production emitted to the atmosphere plus natural emissions and other sources to the atmosphere. With the exceptions of Cu and Zn, the quantities of emissions balance rather well. There is no obvious reason why Cu is out of balance by nearly a factor of 2 (atmospheric input > sources). For Zn, with an imbalance of 1.7 for atmospheric input > sources, there is an obvious problem with other sources in that the impact of rubber tire wear. This source term will be addressed in the next section. However, even with this term, the right side of the equation would increase to a maximum emissions figure of 300,000 tyr (Table 8). It is possible that maximum Cu and Zn emissions to the atmosphere have been overestimated but there is no way to check this with the available data. 

The second-order tensors are characterized by three invariants, that is, it is possible to combine the nine components in three ways to get quantities that are independent of the coordinate systems and express some fundamental properties. For the siuface component there are only two such invariants. The first of these can be written as tr (xj, where tr is short for trace and the operation that sums the diagonal elements of the tensor. The second is l/2 [tr (xj] - XjiXj, where a double dot product has been introduced. Since the trace itself is an invariant, some authors drop this term from the second invariant. In addition, the second invariant of this symmetric siuface tensor is the same as the third invariant in three dimensions, which is the determinant of x (see the remark after Equation 7.E1.8). There is a very important second-order surface tensor in the form of... 

The quantity AG " in Equations (119)-(121) can be related qualitatively to the asymmetry of the transition state, and terms such as the degree of proton transfer have been widely used highly asymmetric transition states are often described as being reactant-like or product-like. These concepts could also be expressed quantitatively in terms of the bond lengths, bond orders, or force constants of the two bonds in a transition state However, the relation between these quantities is a matter for speculation, and AG° is the only one of them which can (in favourable cases) be... 

A note on index notation is in order for those unfamiliar with its use. Index notation is used to shorten the transport equations that are presented above. Each index (shown as a subscript) can represent one of the three Cartesian directions or one of the chemical components. The index that appears on both sides of the equation in all or most terms is the equation index. If it is x, the equation is a balance for quantities in the x-direction if it is 1, the equation is for chemical component 1. The indices that appear only on one side of the equation in occasional terms are running indices and take on each of the three values x, y, and z or each of the component values 1,2,3,... such that the term in which they appear will have several forms that are additive. In the case of component values, indices of product components may not be equal. To illustrate use of the... 

ROA and Raman intensity are proportional to the square of a tensor quantity, as expressed in Equation [1]. For Raman scattering only the square of the polarizability is needed, whereas ROA intensity arises from the product of the polarizability and an ROA tensor. The ROA tensor are approximately three orders of magnitude smaller than the polarizability, and hence an ROA spectrum is approximately three orders of magnitude smaller than its parent Raman spectrum. As noted above, the Greek subscripts of the tensor refer to the molecular axis system. However, for both Raman and ROA, linear combinations of products of tensors can be found that do not vary with the choice of the molecular coordinate frame. Such combinations are called invariants. All Raman intensities from samples of randomly oriented molecules can be expressed in terms of only three invariants, called the isotropic invariant, the symmetric... 

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