Holland’s equation

Existence and Uniqueness of Solutions to Holland s Equations fora Case of Multicolumn... 

In writing the Lagrangean density of quantum mechanics in the modulus-phase representation, Eq. (140), one notices a striking similarity between this Lagrangean density and that of potential fluid dynamics (fluid dynamics without vorticity) as represented in the work of Seliger and Whitham [325]. We recall briefly some parts of their work that are relevant, and then discuss the connections with quantum mechanics. The connection between fluid dynamics and quantum mechanics of an electron was already discussed by Madelung [326] and in Holland s book [324]. However, the discussion by Madelung refers to the equations only and does not address the variational formalism which we discuss here. 

The Pasquill-Gifford dispersion parameters and Holland s plume rise equations. 

A proposed source is to emit 72 g/s of a toxic pollutant from a stack 30 meters liigh with a diameter of 1.5 meters. Tlic effluent gases are emitted at a temperature of 250 F (394K) with an exit velocity of 13 m/s. Using Holland s plume rise equation, obtain the plume rise as a function of wind speed for stability classes B and D. Assume that the design atmospheric pressure is 970 mbar and that the design ambient air temperature is 20 C (293K). 

Comparative evaluation of the predictive properties of equations 3.85 and 3.86 is given in table 3.6. Column I lists values obtained by simple summation of the entropies of the constituent oxides. This method (sometimes observed in the literature) should be avoided, because it is a source of significant errors. The lower part of table 3.6 lists the adopted exchange reactions and the Sj terms of Holland s model. 

At 2000 K and 1 atm, Hollander s state-specific rate constant becomes k. = 1.46 x 1010 exp(-AE/kT) s-1, where AE is the energy required for ionization. For each n-manifold state the fraction ionized by collisions is determined, as well as the fraction transferred to nearby n-manifold states in steps of An = 1. Then the fractions ionized from these nearby n-manifold states are calculated. In this way a total overall ionization rate is evaluated for each photo-excited d state. The total ionization rate always exceeds the state-specific rate, since some of the Na atoms transferred by collisions to the nearby n-manifold states are subsequently ionized. Table I summarizes the values used for the state-specific cross sections and the derived overall ionization and quenching rate constants for each n-manifold state. The required optical transition, ionization, and quenching rates can now be incorporated in the rate equation model. Figure 2 compares the results of the model calculation with the experimental values. 

Many more plume rise equations may be found in the literature. The Environmental Protection Agency (EPA) is mandated to use Brigg s equations to calculate plume rise. In past years, industry has often chosen to use the Holland or Davidson-Bryant equation. The Holland equation is... 

R. Holland, Finite-difference solution ofMaxwell s equations in generalized nonorthog-onal coordinates, IEEE Trans. Nucl. Sci., vol. NS-30, no. 6, pp. 4589-4591, Dec. 1983. 

The development of mathemafical models is described in several of the general references [Giiiochon et al., Rhee et al., Riithven, Riithven et al., Suzuki, Tien, Wankat, and Yang]. See also Finlayson [Numerical Methods for Problems with Moving Front.s, Ravenna Park, Washington, 1992 Holland and Liapis, Computer Methods for Solving Dynamic Separation Problems, McGraw-Hill, New York, 1982 Villadsen and Michelsen, Solution of Differential Equation Models by... 

This equation must be solved for yn +l. The Newton-Raphson method can be used, and if convergence is not achieved within a few iterations, the time step can be reduced and the step repeated. In actuality, the higher-order backward-difference Gear methods are used in DASSL [Ascher, U. M., and L. R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, SIAM, Philadelphia (1998) and Brenan, K. E., S. L. Campbell, and L. R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, North Holland Elsevier (1989)]. 

There is no direct coupling between lb = + 2 and lb = - 2 because of the selection rule (4.128). The values of a, and a2 are obtained by using the equations of the previous sections and are given in general in terms of qu, q22, quin view of the special form of the rotation-vibration matrices, it is convenient to introduce a transformed basis (Wang s basis, 1929 see also Herman et al., 1991 Holland et al., 1992), defined as... 

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes (North-Holland, Amsterdam, 1981). 

Produced from the manipulation of the Jacobian are the changes in the variables, i,e., the Ax vector. The variables for the next trial are calculated from x + = x + s Ax (i.e,, . + T = xlk + sk Ajc1jA, etc,). The s scalar is generated to ensure that the norm of functions improve between trial k + 1 and trial k. Usually, s = 1 but may have to be smaller on some trials. The Newton-Raphson method assumes that the curves of the independent functions are close to linear and the slopes will point toward the answers. The MESH equations can be far from linear and the full predicted steps, Ax, can take the next trial well off the curves. The s scaler helps give an improved step search or prevente overstepping the solution. Holland (8) and Broyden (119) present formulas for getting s. ... 

Brenan. K. E., S. L. Campbell, and L. R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. North-Holland, New York (1989),... 

For large aspect ratios (///L S 12), this equation (Hollands et al., 1976] correlates experimental data extremely well for till angles up to 70°,... 

Equation 13.32 is used to calculate an updated Kj for each stage based on current o s and X,/s. The stage temperatures are then calculated from some function of temperature correlation of (Holland, 1975). 

In the original treatment of the separated light components in a conventional column at minimum reflux by Holland,4 the material balances were written around the top of the column and each plate down to and including plate s — 1. (a) Formulate these balances and (b) show that the set of equations found in part (a) is equivalent to the set given by Eq. (11-12). 

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