Loss coefficient general

We now turn to the case of discontinuity coefficients k x), q x) and f x) and assume without loss of generality that k, q and / have discontinuities of the first kind only at a single point a = G (0,1), so that... 

The variational condition determining the coefficients cJt is cubic in the unknowns, but iterative techniques permit these coefficients to be determined by repeated use of matrix diagonalization methods. Under most conditions it is possible to choose an iterative process facilitating convergence there is much RHF experience, and inordinate difficulties are not usually experienced. Because of the occupancy assumptions, it is possible without loss of generality to take the RHF spatial orbitals as orthogonal, and this is an important feature simplifying the calculations. 

In the general case, the second-quantized operator linear combination of irreducible tensorial products of electron creation operators. The combination must be selected so that a classification of states according to additional quantum numbers be provided for. Without loss of generality, all the numerical coefficients in the linear combinations can be considered real. Then, from (14.14), we can introduce the operators... 

The flow in pipe fittings, e.g. bends and valves, is generally too complex to determine theoretically. For turbulent flow, these minor losses are approximately equal to the square of the flow velocity. Thus, we define a loss coefficient, K,... 

Low-loss materials can be obtained with relative permittivities exceeding 500 but accompanied by high negative temperature coefficients, generally exceeding... 

Here, rate equations are given in terms of concentrations C , in units of moles per unit volume. This is the most convenient choice for liquid-phase reactions. For gas-phase reactions, partial pressures p, suggest themselves as an alternative and can be substituted without loss of generality. However, the dimensions and numerical values of the rate coefficients change accordingly. 

For the calculation of stationary mutant distributions we restrict attention to a uniform error rate per digit (1 — ) and assume equal degradation rate coefficients Dy = D2= =D = D. Since the addition of a constant to all diagonal elements of a matrix just shifts the spectrum of eigenvalues and has no influence on the eigenvectors, we need only consider the case D = 0 without loss of generality. Then the elements of the matrix W are determined by the replication rate coefficients (as in Section III.2) and are of the form... 

The symmetry projection of the wavefunction is equivalent to a particular orbital transformation among the occupied orbitals of the wavefunction. If the CSF expansion is full within these sets of symmetry-related orbitals, no new CSFs will be generated by this orbital transformation. This type of wavefunction could have been computed directly in terms of symmetry orbitals with no loss of generality. (In fact, the CSF expansion expressed in terms of symmetry orbitals will usually result in fewer expansion terms because the symmetry blocking of the individual CSFs allows those of the incorrect symmetries to be deleted from the expansion.) However, if the CSF expansion is not full within these orbital sets, it is possible that the symmetry transformation of the orbitals will generate new CSF expansion terms. The coefficients of these new CSF expansion terms are determined by the old expansion coefficients and the symmetry transformation coefficients. For example, consider the case of two H2 molecules, described in terms of localized orbitals, separated by a reflection plane. Assume that the localized description of the two H2 molecules is of the form... 

Let us suppose that a system contains initial masses kmol of chemical A, Mb kmol of chemical B and Me kmol of chemical C. Let the reaction defined by equation (13.6) begin, so that after a time interval, St, an additional SMc kmol of chemical C have been formed. There is no loss of generality if we assume that SMc is the mathematical product of the stoichiometric coefficient, c, and an increase in mass SMk (kmol) defined by ... 

An implicit assumption made in deriving Eqs. (16.14) and (16.16) was that scattering could be neglected. In general, however, the loss coefficient must account for all losses, hence... 

Edwards et al. studied the frictional head loss for different pipe fittings for flow of Newtonian and non-Newtonian liquids in laminar flow condition [14]. They proposed generalized correlation with loss coefficient and Reynolds number (Re = VpD/p for Newtonian liquid and Re = Re for non-Newtonian liquid) of individual fittings. 

For a description of the remaining symbols, the reader is referred to section Nomenclature at the end of this chapter. Admitting that physical properties, transfer coefficients, porosity, velocity, etc. can be assumed to be constant and uniform, or without loss of generality, represented by the same notation for scaling purposes, the order of magnitude of each term in the model equations is represented by the associated timescale. We will take advantage of this formulation in our analysis (Section 3.3). Some common dimensionless parameters appear when ratios between these timescales are taken, as shown in the following sections. 

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