Approximate Analytical Equations Research Articles

Numerical evaluation of a set of analytical infiltration equations

In this paper, a set of analytical infiltration equations that are commonly used to evaluate one‐ and three‐dimensional water infiltration from a surface disc source is studied. Both the quasi‐exact analytical formulation and the related approximations for short and long times are assessed. The analytical properties of the quasi‐exact formulation are evaluated using a proposed scaling procedure in order to define the validity domains of related approximations. Both quasi‐exact and approximate analytical equations are then studied with respect to their ability to reproduce numerically generated cumulative infiltrations from a 10 cm radius disc source for four soils (sand, loam, silt, and silty clay) at several initial saturations. The quasi‐exact formulation is suitable for sand, loam, and silt when their soil‐dependent and saturation‐independent shape parameters, γ and β, are properly chosen (between 0.75 and 1 and 0.3 and 1.7, respectively). Approximations derived for the same shape parameters can also be used, provided that their use is restricted to proposed validity intervals. However, none of these equations applies for silty clay, since its hydraulic properties do not fulfill the conditions required for the use of the quasi‐exact formulation.

Localized light absorption by hemoglobins of an erythrocyte suspension

A correction factor accounting for the differences between the absorption spectra of an erythrocyte suspension and hemolyzed blood (hemoglobin solution) has been calculated. Erythrocytes and their aggregates are simulated by "soft" cylindrical (disk-shaped) particles. The anomalous diffraction approximation is used. At equal absorbing mass, the absorption coefficient of the suspension in the blue spectral region is shown to be capable of being several times lower than that of the solution due to localization of the absorber in erythrocytes. The hemoglobin localization effect is less important for λem > 600 nm. Approximate analytical equations that are valid for cylindrical particles with a length-to-diameter ratio l/d > 3–4 are proposed for calculating the correction factor. More cumbersome formulas of anomalous diffraction must be used at smaller l/d values. It is shown that these formulas transform at the limit into the analytical equations. The spectra of the correction factor for cylindrical and spherical particles are compared. The simple approximation for spherical particles is demonstrated to work well for rather long aggregates when the cylinder volume-to-surface area ratio is the same as for a spherical particle. The results can be used to solve various problems of erythrocyte optics, e.g., to reproduce sizes of aggregates by measuring their spectral absorption.

A parametric study of stability and resonances around L 4 in the elliptic restricted three-body problem

The size of the stable region around the Lagrangian point L 4 in the elliptic restricted three-body problem is determined by numerical integration as a function of the mass parameter and eccentricity of the primaries. The size distribution of the stable regions in the mass parameter-eccentricity plane shows minima at certain places that are identified with resonances between the librational frequencies of motions around L 4. These are computed from an approximate analytical equation of Rabe relating the frequency, mass parameter and eccentricity. Solutions of this equation are determined numerically and the global behaviour of the frequencies depending on the mass parameter and eccentricity is shown and discussed. The minimum sizes of the stable regions around L 4 change along the resonances and the relative strength of the resonances is analysed. Applications to possible Trojan exoplanets are indicated. Escape from L 4 is also investigated.

Distribution of non-sinusoidal currents in parallel conductors used in three-phase four-wire networks

This paper presents an analysis of the distribution of non-sinusoidal currents among parallel-connected cables in low-voltage, three-phase TN–S systems. The cables are assumed to be laid on metallic cable trays in free space, which means that the influence of earth is neglected. An iterative algorithm is developed which can accurately estimate the distribution of line and neutral currents among parallel-connected cables at various harmonic frequencies provided that the impedance matrix of the whole cable system is given. The impedance matrices of the cable systems are calculated by a Finite Element Method (FEM) model, because – as shown – the approximate analytical equations presented in the literature introduce significant errors at frequencies above the 7th harmonic. On the contrary, a FEM model can take into account the variation with frequency of the ohmic resistance, self and mutual inductance of a cable system. The FEM model is further used to estimate the temperature at the surface of the cables. It is shown that the non-uniform distribution of the current among parallel-connected cables can be very high and that it increases with frequency and with the number of parallel cables. The proposed model is used to estimate the cable arrangement that yields almost uniform current distribution without having to transpose the cables. Examples of the distribution of the current of two common non-sinusoidal loads are presented for various cable configurations. The proposed calculation method is validated through comparison with the measured distribution of the current among eighteen 300 mm 2 single-core cables in an industrial distribution substation. Measurements and calculations of the surface temperature of the individual cables showed also very good agreement which further validates the method.

Power series like relation of power law and coupled creep constrained grain boundary cavitation under strain gradient plasticity analysis

The continuum damage theory of Kachanov and Rabotnov has limitations since the mechanical properties of a material (especially plastic deformation and fracture) are determined by its microstructure. When a solid deforms at high temperature its microstructure may in some sense be altered- holes and cracks may nucleate and grow inside the solid by various mechanism controlled by diffusion and by power law creep or by a combination of these mechanisms. Considering a coupled diffusion power law creep mechanism using a mechanistic model approximate analytical equations for the growth rate under multi-axial stress states are developed. These results are related to the power law mechanistic results in a power series like form, which are used to analyze the crack, tip fields for the coupled mechanism using a strain gradient plasticity analysis. The Kachanov-Rabotnov results and the HRR results are shown to be special cases of these results. Journal of the Nigerian Association of Mathematical Physics Vol. 8 2004: pp. 25-34

Gas-surface accommodation coefficients from viscous slip and temperature jump coefficients

Analytical equations for the viscous slip and temperature jump coefficients for a gas flowing over a surface, obtained from a variational approach, are evaluated and compared with highly accurate numerical results for rigid sphere gas interactions. The gas-surface interaction is assumed to be governed by either the elastic-diffuse (Maxwell) one-parameter model or the Cercignani-Lampis two-parameter model. Approximate analytical equations obtained for solving the inverse problem to estimate the accommodation coefficient for the one-parameter model or the tangential momentum accommodation coefficient for the two-parameter model from an error-free measurement of the viscous slip coefficient are almost always accurate to within 3%. The corresponding estimate of the accommodation coefficient for the one-parameter model from the temperature jump coefficient is almost always accurate to within 5%.

Impact of microstructure on grain-boundary diffusion in polycrystals

The impact of a complex microstructure on polycrystalline diffusion is investigated using both numerical and analytical methods. In particular, the diffusion equation is numerically integrated using the finite-difference method to obtain the concentration profile for a diffusant in a simplified microstructural representation. The methodology is first validated for an idealized model of diffusion in a prototypical, single grain-boundary system and then applied to a Voronoi model of a microstructure resulting from homogeneous nucleation and growth. The diffusive behavior is quantified by obtaining uptake curves as a function of time for different ratios of grain boundary to lattice diffusivities. Such curves can be used to estimate an unknown grain-boundary diffusivity, given certain microstructural assumptions. Finally, approximate analytical equations describing a diffusant uptake in polycrystalline microstructural models are developed and found to agree well with the numerical results.

Different models for a hard-aperture function and corresponding approximate analytical propagation equations of a Gaussian beam through an apertured optical system

Beam profiles that consist of a sum of complex-Gaussian functions, a sum of polynomial-Gaussian functions and a sum of multi-Gaussian functions offset by some fixed amount are proposed as three types of model for a hard-aperture function. By expanding an aperture function into these models, approximate analytical propagation equations for a Gaussian beam through an apertured ABCD optical system are obtained. Comparison among these models themselves and among propagation characteristics of a Gaussian beam through these models are made. It is shown that the first and third types of model for a hard-aperture function are more suitable than the second type, in terms of calculation efficiency and simulation results, for application to such diffraction problems. Moreover, there are some differences in the applicability of the first and the third models.

Formulation based on artificial neural network of thermodynamic properties of ozone friendly refrigerant/absorbent couples

This paper presents a new approach based on artificial neural networks (ANNs) to determine the properties of liquid and two phase boiling and condensing of two alternative refrigerant/absorbent couples (methanol/LiBr and methanol/LiCl). These couples do not cause ozone depletion and use in the absorption thermal systems (ATSs). ANNs are able to learn the key information patterns within multidimensional information domain. ANNs operate such as a ‘black box’ model, requiring no detailed information about the system. On the other hand, they learn the relationship between the input and the output. In order to train the neural network, limited experimental measurements were used as training data and test data. In this study, in input layer, there are temperatures in the range of 298–498 K, pressures (0.1–40 MPa) and concentrations of 2%, 7%, 12% of the couples; specific volume is in output layer. The back-propagation learning algorithm with three different variants, namely scaled conjugate gradient (SCG), Pola–Ribiere conjugate gradient (CGP), and Levenberg–Marquardt (LM), and logistic sigmoid transfer function were used in the network so that the best approach can find. The most suitable algorithm and neuron number in the hidden layer are found as SCG with 8 neurons. For this number level, after the training, it is found that maximum error is less than 3%, average error is about 1% and R 2 value are 99.999%. As seen from the results obtained the thermodynamic equations for each pair by using the weights of network have been obviously predicted within acceptable errors. This paper shows that values predicted with ANN can be used to define the thermodynamic properties instead of approximate and complex analytic equations.

Investigation of thermodynamic properties of refrigerant/absorbent couples using artificial neural networks

This paper presents a new approach to determine the properties of liquid and two phase boiling and condensing of two alternative refrigerant/absorbent couples (methanol–LiBr and methanol–LiCl), which do not cause ozone depletion for absorption thermal systems (ATSs) using artificial neural networks (ANNs). The back-propagation learning algorithm with three different variants and logistic sigmoid transfer function were used in the network. In order to train the neural network, limited experimental measurements were used as training and test data. In input layer, there are temperatures in the range of 298–498 K (with 25 K increase), pressures (0.1–40 MPa) and concentrations of 2, 7, and 12% of the couples; specific volume is in output layer. After training, it is found that maximum error is less than 3%, average error is about 1% and R 2 values are 99.999%. As seen from the results obtained the thermodynamic properties have been obviously predicted within acceptable errors. This paper shows that values predicted with ANN can be used to define the thermodynamic properties instead of approximate and complex analytic equations.

Approximate analytical propagation equations of Gaussian beams through hard-aperture optics

By using the method of expanding the aperture function into a finite sum of complex Gaussian functions, approximate closed-form propagation equations of Gaussian beams through a paraxial multi-apertured ABCD optical system are derived and illustrated with numerical examples. The numerical results find an agreement with those by straightforward integral of the Collins formula, but the computing time is strongly reduced.

Effect of alloying elements on the γ to α transformation in steel. I

A newly constructed computer program was used to simulate partitionless growth of α in γ of Fe-Ni-C alloys, taking into account so-called solute drag by evaluating the dissipation of Gibbs energy due to diffusion inside the interface and in the nickel spike being pushed in front of it. It could be shown how the conditions at the α/γ interface vary with the velocity. A continuous change from paraequilibrium to quasi-paraconditions could be illustrated in the phase diagram. By combination with an approximate analytical growth equation, it was possible to derive the thickness of α as function of time. The growth velocity was assumed to start at very high values but decreased due to the pile-up of carbon. For alloy compositions outside the limit for quasi-paraconditions and just inside it the growth was predicted to stop suddenly when critical conditions are approached during the reaction. For alloy compositions further inside the limit, there was no such stop, except by the action of impingement of diffusion fields for carbon. Somewhere between the lines for paraequilibrium and quasi-paraconditions there is a rather rapid drop of the final thickness of α. In order to explain experimental information on this drop one must accept an appreciable tendency of segregation of nickel to the α/γ interface.

Approximate analytical equations for the deformation of an inclusion in a viscoplastic matrix

The strain rate in an inclusion embedded in a matrix undergoing large strains is investigated theoretically. The inclusion is either a spheroid (axisymmetric loading), or an elliptical cylinder (plane strain loading), and both are assumed to be made of incompressible power law viscous materials, including the limit case of perfect plasticity. Localization factors (i.e., normalized strain rates) derived from various approaches (viz, linearization models, variational principle, finite element method) are first compared for a sphere and a circular cylinder. The influence of the inclusion aspect ratio is then investigated by means of the variational approach. Finally, an original and practical purpose of the paper is to propose approximate analytical equations to fit the dependence of the localization factor on the strain rate sensitivity parameter, as well as on the inclusion hardness and aspect ratios.

Bifurcations of Periodic Motions of Two-Link Wheeled Transport Robots

Approximate analytic equations of limit cycles on invariant manifolds are constructed. Evolution of the system is studied.

The convective model of cross-flow microfiltration is not inconsistent

The approximate analytical equations of the convective model and of the model of Romero and Davis are compared. Introducing the approximations used by Romero and Davis in the convective model gives the same functional dependence of the average filtration flux, 〈ν〉, on particle radius and solute volume fraction and a different dependence on the shear rate, γ s. The convective model gives an explicit dependence of 〈ν〉 on transmembrane pressure and non-Newtonian ‘apparent’ viscosity of the cake. It is shown that the pseudoplastic rheological behaviour of concentrated colloidal solutions (cakes) rationalizes the experimentally established dependence 〈nu;〉 ∼ γ m s with 0.4 < m < 1.5.

A new correlation for bench-scale piloted ignition data of wood

This paper presents the results of a combined experimental and theoretical study of piloted ignition of cellulosic materials. The main objective is to present an engineering solution to the piloted ignition problem for wood exposed to radiant heat in a bench-scale piloted ignition test. This has been motivated by the need to have simple models of ignition for use in a computational fluid dynamics (CFD) model of fire spread and extinguishment in building fires. The experiments were conducted on oven-dry and moisture conditioned samples of three wood species using a cone calorimeter. As expected, the experimental data revealed that the effect of moisture content on the piloted ignition process is significant. It was also found that the ignition temperature depends on the external heat flux, which supports other recent studies. Based on the experimental observations, an approximate analytical equation was derived and then used for correlating the ignition data, as well as extracting the piloted ignition properties. The chief distinguishing feature of the present equation over other similar equations is that it takes into account the variation of the ignition temperature with external heat flux.

Regular and Chaotic Solutions of the Sitnikov Problem near the 3/2 Commensurability

Regular solutions at the 3/2 commensurability are investigated forSitnikov’s problem. Utilizing a rotating coordinate system and theaveraging method, approximate analytical equations are obtained for thePoincare sections by means of Jacobian elliptic functions and 3πperiodicsolutions are generated explicitly. It is revealed that the system exhibitsheteroclinic orbits to saddle points. It is also shown that chaotic regionemerging from the destroyed invariant tori, can easily be seen for certaineccentricities. The procedure of the current study provides reliable answersfor the long-time behavior of the system near resonances.

Electro-optic characterization of nonlinear-optical guest–host films and polymers

The Pockels coefficients r33 of four dyes for electro-optic applications are reported, including ones with stilbene, butadiene, azo, and hexatriene linkages between the charge donors and acceptors, with the charge donor being the amino group and the acceptor groups being the nitro and dicyanovinyl groups. r33 are reported for dyes as guest–hosts in poly(methyl methacrylate) and 50/50 copolymers with methyl methacrylate. The largest r33 is observed for the copolymer containing the amino donor, hexatriene linkage, and the dicyanovinyl acceptor group with a value of r33 = 18 pm/V at 140 V/µm at a wavelength λ = 1.3 µm. We also find evidence that the dicyanovinyl hexatriene dyes in the copolymer are interacting with one another, causing a relative decrease in the copolymer’s electro-optic efficiency compared with its value as a guest–host in poly(methyl methacrylate). The Pockels coefficient is measured by the ellipsometric reflection technique. We derive approximate analytic equations to analyze the reflected intensity and verify the validity of these approximations when the analyzing wavelength is far from the absorption of the dye. We also derive an approximate analytic expression for the optical retardation as a function of the ratio of the off-diagonal to the diagonal components of the Pockels coefficients r13/r33 and also the finite birefringence of the poled film. We show that accurate knowledge of the ratio r13/r33 is critical to obtaining accurate values of r33.

A robust algorithm for fixed-bed reactors with steep moving temperature and reaction fronts

A spatially and temporally adaptive completely implicit finite difference algorithm, which can accurately capture the moving temperature and reaction fronts in fixed-bed reactors with highly exothermic reactions, is developed and presented. For spatial adaptation linear interpolation proved to be more robust than higher-order interpolations. Node placement based on the magnitude of the second spatial derivative was implemented and shown to lead to more accurate results and a more robust algorithm than the traditional node placement based on the magnitude of the first derivative. For improved accuracy the interdependence of the spatial mesh size and the time step size is taken into account by coupling spatial and temporal adaptation. Choice of the time step is also related to the physics of the problem. Advantages of the developed algorithm over traditional finite difference and polynomial approximation methods are discussed. The validity of the developed algorithm is proven by comparing the computed results with analytical solution of a linear heat regenerator model. A nonlinear model of a fixed-bed reactor with highly exothermic reaction occurring in the gas phase is also solved and the results are presented. The reasonableness of the solution for this nonlinear model is shown by comparison of model calculated and approximate analytical equation for the front velocity. The robustness of the developed algorithm in solving multiple coupled equations is demonstrated by solving a nonlinear model of a fixed-bed reactor in which both exothermic and endothermic reactions occur in the gas phase.

Kinetics of Irreversible Adsorption of Interacting Spheroidal Particles

The kinetics of the localized, irreversible adsorption of interacting spheroidal particles on homogeneous interfaces was analyzed theoretically. By applying the random sequential adsorption (RSA) approach, numerical MC simulations were performed in the case of the short-ranged Yukawa-type potential of interaction. Both the surface-blocking parameter (available surface function) and adsorption kinetics were determined for various particle elongations and for a broad range of the χα parameter characterizing the range of the interaction potential. It was demonstrated that the exact numerical results can well be described for not too high surface concentrations by the approximate analytical equations derived using the equivalent hard-particle concept. On the other hand, for higher surface concentrations close tojamming, adsorption kinetics of soft particles can well be approximated by the power-law dependence analogous to hard particles. The theoretical analysis revealed that adsorption kinetics of colloid particles of spheroidal shape is considerably diminished by the lateral electrostatic interactions.