Approximations equations

Marmur [12] has presented a guide to the appropriate choice of approximate solution to the Poisson-Boltzmann equation (Eq. V-5) for planar surfaces in an asymmetrical electrolyte. The solution to the Poisson-Boltzmann equation around a spherical charged particle is very important to colloid science. Explicit solutions cannot be obtained but there are extensive tabulations, known as the LOW tables [13]. For small values of o, an approximate equation is [9, 14]... 

Redhead [89] gives the approximate equation EjRTm - ln(A7) //3)- 3.64. Check the usefulness of this equation by comparing with the answers to Problems 5 and 6. 

When, for a one-component system, one of the two phases in equilibrium is a sufficiently dilute gas, i.e. is at a pressure well below 1 atm, one can obtain a very usefiil approximate equation from equation (A2.1.52). The molar volume of the gas is at least two orders of magnitude larger than that of the liquid or solid, and is very nearly an ideal gas. Then one can write... 

Equation (A2.1.53) is frequently called the Clausius-Clapeyronequation, although this name is sometimes applied to equation (A2.1.52). Apparently Clapeyron first proposed equation (A2.1.52) in 1834, but it was derived properly from thennodynamics decades later by Clausius, who also obtained tlie approximate equation (A2.1.53).)... 

Hence, in order to contract extended BO approximated equations for an N-state coupled BO system that takes into account the non-adiabatic coupling terms, we have to solve N uncoupled differential equations, all related to the electronic ground state but with different eigenvalues of the non-adiabatic coupling matrix. These uncoupled equations can yield meaningful physical... 

The ordinary BO approximate equations failed to predict the proper symmetry allowed transitions in the quasi-JT model whereas the extended BO equation either by including a vector potential in the system Hamiltonian or by multiplying a phase factor onto the basis set can reproduce the so-called exact results obtained by the two-surface diabatic calculation. Thus, the calculated hansition probabilities in the quasi-JT model using the extended BO equations clearly demonshate the GP effect. The multiplication of a phase factor with the adiabatic nuclear wave function is an approximate treatment when the position of the conical intersection does not coincide with the origin of the coordinate axis, as shown by the results of [60]. Moreover, even if the total energy of the system is far below the conical intersection point, transition probabilities in the JT model clearly indicate the importance of the extended BO equation and its necessity. 

The equations developed in Chapters 11 and 12 are quite complicated and in many practical applications it may be desirable to replace them by simpler approximate equations. The construction of such approximations, of adequate accuracy, is a field still largely unexplored. Nevertheless, it is important to understand the structure of the complete equations if approximations are to be made deliberately, rather than by inadvertent omission. 

A similar approximation should be applied to the components of the equation of motion and the significant terms (with respect to ) consistent with the expanded constitutive equation identified. This analy.sis shows that only FI and A appear in the zero-order terms and hence should be evaluated up to the second order. Furthermore, all of the remaining terms in Equation (5.29), except for S, appear only in second-order terms of the approximate equations of motion and only their leading zero-order terms need to be evaluated to preserve the consistency of the governing equations. The term E, which only appears in the higlier-order terms of the expanded equations of motion, can be evaluated approximately using only the viscous terms. Therefore the final set of the extra stress components used in conjunction with the components of the equation of motion are... 

The conductor-like screening model (COSMO) is a continuum method designed to be fast and robust. This method uses a simpler, more approximate equation for the electrostatic interaction between the solvent and solute. Line the SMx methods, it is based on a solvent accessible surface. Because of this, COSMO calculations require less CPU time than PCM calculations and are less likely to fail to converge. COSMO can be used with a variety of semiempirical, ah initio, and DFT methods. There is also some loss of accuracy as a result of this approximation. 

In this section we proceed to study the plate model with the crack described in Sections 2.4, 2.5. The corresponding variational inequality is analysed provided that the nonpenetration condition holds. By the principles of Section 1.3, we propose approximate equations in the two-dimensional case and analytical solutions in the one-dimensional case (see Kovtunenko, 1996b, 1997b). 

In this section we deal with the simplified nonpenetration condition of the crack faces considered in the previous section. We formulate the model of a plate with a crack accounting for only horizontal displacements and construct approximate equations using penalty and iterative methods. The convergence of these solutions is proved and its application to the onedimensional problem is discussed. Analytical solutions for the model of a bar with a cut are obtained. The results of this section can be found in (Kovtunenko, 1996c, 1996d). 

Natural-draft cooling towers are extremely sensitive to air-inlet conditions owing to the effects on draft. It can rapidly be estabUshed from these approximate equations that as the air-inlet temperature approaches the water-inlet temperature, the allowable heat load decreases rapidly. For this reason, natural-draft towers are unsuitable in many regions of the United States. Figure 10 shows the effect of air-inlet temperature on the allowable heat load of a natural-draft tower for some arbitrary numerical values and inlet rh of 50%. The trend is typical. 

The basic approximate equation for the separation process gives the water flux, m" (kg/m /s) across an RO membrane, in the absence of fouling,... 

Equation (5-62) predicts the point of maximiim velocity for laminar flow in annuli and is only an approximate equation for turbulent flow. Brighton and Jones [Am. Soc. Mech. Eng. Basic Eng., 86, 835 (1964)] and Macagno and McDoiigall [Am. Inst. Chem. Eng. J., 12, 437 (1966)] give more accurate equations for predicting the point of maximum velocity for turbulent flow. 

An approximate equation for the faUing-rate period may be obtained by integration of Eq. (12-33). This gives an equation for materials in which moisture movement is controlled by dimision ... 

Values of enthalpy constants for approximate equations are not tabulated here but are also computed for each stage based on the initial temperature distribution. 

The empirical parameters method uses simple mathematical approximation equations, whose coefficients (empirical parameters) are predetermined from the experimental intensities and known compositions and thicknesses of thin-film standards. A large number of standards are needed for the predetermination of the empirical parameters before actual analysis of an unknown is possible. Because of the difficulty in obtaining properly calibrated thin-film standards with either the same composition or thickness as the unknown, the use of the empirical parameters method for the routine XRF analysis of thin films is very limited. 

Equation (4.137) is almost exactly the same as the approximation equation (4.123) derived for wet bulb temperature. When the partial pressure of water vapor is low compared with the total pressure—in other words when the humidity x is low—the specific heat of humid air per kilogram of humid air, Cp, and the specific heat of humid air per kilogram of dry air, Cp, are al most the same Cp = Cp. Therefore, in a situation where the humidity is low and Le s 1, the thermodynamic wet bulb temperature is very nearly the same as the technical wet bulb temperature dy... 

The mechanics of materials approach to the micromechanics of material stiffnesses is discussed in Section 3.2. There, simple approximations to the engineering constants E., E2, arid orthotropic material are introduced. In Section 3.3, the elasticity approach to the micromechanics of material stiffnesses is addressed. Bounding techniques, exact solutions, the concept of contiguity, and the Halpin-Tsai approximate equations are all examined. Next, the various approaches to prediction of stiffness are compared in Section 3.4 with experimental data for both particulate composite materials and fiber-reinforced composite materials. Parallel to the study of the micromechanics of material stiffnesses is the micromechanics of material strengths which is introduced in Section 3.5. There, mechanics of materials predictions of tensile and compressive strengths are described. 

We approximate differentials with increments, and all subsequent expressions are. therefore, approximate. Equation (3-155) becomes... 

Since for T just less than Tc, Mq is small but nonzero, we can approximate tanh (Mq) by Mo + Mq/3. Solving this approximate equation for Mo, we find that... 

By the usual approximations, i.e. by assuming that the activity coefficients of the un-ionised molecules and, less justifiably, of the ions are unity, the following approximate equation is obtained ... 

The Clausius-Clapeyron equation The Clapeyron equation can be used to derive an approximate equation that relates the vapor pressure of a liquid or solid to temperature. For the vaporization process... 

In this section, some of the important aspects of non-Newtonian behaviour will be quantified, and some of the simpler approximate equations of state will be discussed. An attempt has been made to standardise nomenclature in the British Standard, BS 511 8 1 i. 

Methods have been given for the calculation of the pressure drop for the flow of an incompressible fluid and for a compressible fluid which behaves as an ideal gas. If the fluid is compressible and deviations from the ideal gas law are appreciable, one of the approximate equations of state, such as van der Waals equation, may be used in place of the law PV = nRT to give the relation between temperature, pressure, and volume. Alternatively, if the enthalpy of the gas is known over a range of temperature and pressure, the energy balance, equation 2.56, which involves a term representing the change in the enthalpy, may be employed ... 

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