Semi-analytical Equation Research Articles

A semi-analytical equation for the Young’s modulus of isotropic ceramic materials

This paper proposes a new relation between the Young’s modulus of ceramics and the volume fraction of porosity. The relation is obtained by upscaling (coarse-graining) the fluctuations of the microstructure. The microstructure is modeled in terms of phase random fields, which upon upscaling lead to elastic coefficients described by continuous spatial random fields. The effective (macroscopic) elastic modulus is then obtained by averaging over the continuum-scale disorder. Using physically motivated arguments, an explicit expression between the Young’s modulus and the porosity is proposed. This expression involves three empirical parameters, i.e. the solid-phase modulus and two perturbation coefficients. The parametric expression is shown to fit well experimental measurements from the literature. Empirical “bounds” for the Young’s modulus are also formulated. These bounds account for variations due to microstructural properties that are not explicitly calculated in the upscaling.

Semianalytical Equations for Transforming Errors into a Local Geodetic Frame

With the development of space-based systems, the results of various applications are generally provided in an earth-centered and earth-fixed global coordinate frame. However, many times it is necessary to transform those results, together with their errors, into local geodetic frames for physical modeling and geometrical interpretation of results as well as to combine them with terrestrial observations. Although it is common practice to transform errors using the full matrix error propagation law, results of many studies in global Cartesian frames are published in compact form without attaching the necessary variance–covariance parameters, therefore preventing a consistent transformation of covariances into a local geodetic frame. This is especially true for coordinates released as time series of sequential epochs. Examination of coordinate/velocity correlations in a global Cartesian frame (GCF) and a local geodetic frame (LGF) reveals that the transformation matrix from GCF into LGF (which depends on the position and origin of LGF) decorrelates the covariance matrix in the GCF up to 90%, depending on design matrix, a priori weights, and constraints. In this study, utilizing the error decorrelation in the LGF, analytical, transformation equations were inverted for correlation parameters in GCF so as to reconstruct the approximate covariance matrix in GCF from a diagonal matrix of variances. Variances in the LGF were then obtained by a basis change through singular value decomposition. Results show that the new semianalytical expressions could be used successfully to transform variances in GCF into LGF especially in the absence of covariance terms.

Well temperature testing—an extension of Slider's method

A new technique has been developed for determination of the formation thermal conductivity, skin factor and contact thermal resistance for boreholes where the temperature recovery process after drilling operations is not completed. Slider suggested a technique for analysing transient pressure tests when conditions are not constant. We extend Slider's method for transient temperature well tests. It assumes that the volumetric heat capacity of formations is known, and the instantaneous heater's wall temperature and time data are available for a cylindrical probe with a constant heat flow rate placed in a borehole. A semi-analytical equation is used to approximate the dimensionless wall temperature of the heater. A simulated example is presented to demonstrate the data processing procedure.

Constant-pressure well test analysis of finite-conductivity hydraulically fractured gas wells influenced by non-Darcy flow effects

Non-Darcy flow effects have long been recognized to have serious adverse impact on the performance of high flow rate gas wells. These effects may mask the presence of fractures around the wellbores of naturally fractured reservoirs and may render the effective fracture conductivity and fracture half-length of hydraulically fractured wells much less than the designed parameters. Even though the effects of non-Darcy flow have been identified in the field and properly acknowledged in the well testing literature, little has been done to improve the well test analysis results. This paper presents a new technique that accurately determines the fracture conductivity of hydraulically fractured gas wells producing at constant-bottomhole pressure and provides direct means to calculate the magnitude of turbulence in the fracture around the wellbore from a single well test. A semi-analytical equation that incorporates the effects of non-Darcy flow in the fracture is presented for the first time. A detailed investigation of the various parameters that influence the flow behavior of real gas in the fracture nearby the wellbore is also illustrated. Furthermore, a systematic method for calculating the fracture conductivity and non-Darcy flow coefficient from a single well test is outlined. The final working equations are presented in such a way that permits a straightforward, simple, yet accurate analysis of the variable flow rate with time. No type-curve matching, multirate tests or correlations are required. The methodology of the proposed technique is illustrated using several synthetic examples.

Computational Modeling of the Stability of Crevice Corrosion of Wetted SS316L

The stability of localized corrosion sites on SS 316L exposed to atmospheric conditions was studied computationally by considering the wetted cathode and the crevice anode separately and linking them via a constant potential boundary condition at the mouth of the crevice. The limitations on the ability of the cathode that are inherent due to the restricted geometry were assessed in terms of the dependence on physical and electrochemical parameters (temperature, electrolyte layer thickness, solution conductivity, and the size of the cathode). An analysis of variance showed that the solution conductivity and the length of the cathode were the most important parameters in determining the total cathodic current capacity of the external surface. A semi-analytical equation was derived for the total current from a restricted geometry held at a constant potential at one end.

A semianalytical sachs model for the flow stress of a magnesium alloy

A semianalytical Sachs-type equation for the flow stress of magnesium-base alloys is developed using the Schmid law, power law hardening, and a sigmoidal increase in the twinning volume fraction with strain. Average Schmid factors were estimated from electron backscattered diffraction (EBSD) data. With these, the equation provides a reasonable description of the flow curves obtained in compression and tension for samples of Mg-3Al-1Zn cut in different orientations from rolled plate. The model illustrates the general importance of basal slip and twinning in magnesium alloys. The significance of prismatic slip in room temperature tension testing is also highlighted. This is supported with EBSD slip line trace analysis and rationalized in terms of a possible sensitivity of the critical resolved shear stress for prismatic (cross) slip to the stress on the basal plane.

An Accommodative Approach Designed for TCP Gold-to-Gold Inner Lead Bonding

This paper proposes an approach that resolves the tradeoff between bond pad crack, the passivation crack, and lead lift failure modes in gold-to-gold inner lead bonding. The bonding head temperature, stage temperature, and bonding force are addressed as three critical recipes. The proposed scenario determines the bonding head temperature according to the maximum compensation flatness, followed by the corresponding stage temperature and required bonding force in sequence. The relevant gold bump hardness, chip warpage, and flatness variation in the bonding tool were evaluated using the finite-element method. The corresponding bonding force was determined using semianalytical equations with empirical factors. A gate driver IC with 280 I/O pins was adopted to conduct the experimental verification in the engineering pilot run. With the optimal recipes, the obtained results demonstrated success in diminishing these three failure modes. The feasibility of proposed approach was verified experimentally.

Laminar flow of soursop juice through concentric annuli: Friction factors and rheology

Laminar axial flow of a pseudoplastic fluid food (soursop juice) in annular ducts has been experimentally investigated. In the first part of the manuscript, the rheological behavior of soursop juice, being essential for the annular flow analysis, was completely determined from 9.3 to 49.4 °Brix and temperatures from 0.4 °C to 68.8 °C, using a rotational rheometer equipped with coaxial cylinders. In order to test the adequacy of the rheology results, pressure loss data in the laminar pipe flow were collected and then experimental and theoretical friction factors were compared, showing excellent agreement, which indicated the reliability of the Power-Law model for describing the soursop juices. In the second part, pressure loss in annular regions was measured and used to estimate friction factors, which were then compared to those resulted from analytical and semi-analytical equations. The principal contributions of this article are to provide a review on the determination of friction factors-Reynolds number of pseudoplastic fluids in annuli, and also supply extensive new experimental data on the rheological properties and pressure loss of an important shear-thinning fluid food, which is of particular interest for the food engineering process design.

Pressure and temperature drawdown well testing: similarities and differences

Temperature and pressure are the most frequently observed physical parameters in boreholes. The same differential diffusivity equation describes the transient flow of incompressible fluid in porous media and heat conduction in solids. The similarities and differences in the techniques of pressure and temperature well testing are discussed. It is shown that the mathematical model of pressure well tests based on the presentation of the borehole as an infinitely long linear source with a constant fluid flow rate in an infinite-acting homogeneous reservoir cannot be used in temperature well testing. A new technique has been developed for the determination of the formation thermal conductivity, initial temperature, skin factor and contact thermal resistance. It is assumed that the volumetric heat capacity of formations is known and the instantaneous heater's wall temperature and time data are available for a cylindrical probe with a constant heat flow rate placed in a borehole. A semi-analytical equation is used to approximate the dimensionless wall temperature of the heater. A simulated example is presented to demonstrate the data processing procedure.

Average Performance Analysis of Circular and Hyperbolic Geolocation

A comparative performance analysis of four geolocation methods in terms of their theoretical root mean square positioning errors is provided. Comparison is established in two different ways: strict and average. In the strict type, methods are examined for a particular geometric configuration of base stations (BSs) with respect to mobile position, which determines a given noise profile affecting the respective time-of-arrival (TOA) or time-difference-of-arrival (TDOA) estimates. In the average type, methods are evaluated in terms of the expected covariance matrix of the position error over an ensemble of random geometries, so that comparison is geometry independent. Exact semianalytical equations and associated lower bounds (depending solely on the noise profile) are obtained for the average covariance matrix of the position error in terms of the so-called information matrix specific to each geolocation method. Statistical channel models inferred from field trials are used to define realistic prior probabilities for the random geometries. A final evaluation provides extensive results relating the expected position error to channel model parameters and the number of base stations.

Determination of formation temperature from bottom-hole temperature logs—a generalized Horner method

A new technique has been developed for the determination of the formation temperature from bottom-hole temperature logs. The adjusted circulation time concept, and a semi-analytical equation for the dimensionless temperature at the wall of an infinite long cylindrical source with a constant heat flow rate, is used to obtain the working formula. It is shown that the transient shut-in temperature is a function of the mud circulation and shut-in time, formation temperature, thermal diffusivity of formations and well radius. The sensitivity of the predicted values of formation temperature to the thermal diffusivity is shown. Two examples of calculations are presented.

Monte Carlo Simulations of Observation Time-Dependent Self-Diffusion in Porous Media Models

Observation time-dependent self-diffusion coefficients can be used to obtain microstructural information of porous media. This paper presents two different kinds of Monte Carlo simulations of the self diffusion process of fluids like water in porous systems, a lattice-free method and a lattice-based method. The results for simple porous media model geometries agree well with each other and with published analytical as well as semi-analytical equations. The use of these equations, which are important for the interpretation of Pulsed Field Gradient-Nuclear Magnetic Resonance (PFG-NMR) time-dependent diffusion data with respect to properties of porous media, is discussed.

Hamiltonian symplectic semi-analytical method and its application in inhomogeneous electromagnetic waveguides

Dual vectors are applied in Hamilton system of applied mechanics. Electric and magnetic field vectors are the dual vectors in electromagnetic field. The Hamilton system method is introduced into the analysis of electromagnetism waveguide with inhomogeneous materials. The transverse electric and magnetic fields are regarded as the dual. The basic equations are solved in Hamilton system and symplectic geometry. With the Hamilton variational principle, the symplectic semi-analytical equations are derived and preserve their symplectic structures. The given numerical example demonstrates the solution of LSE (Longitudinal Section Electric) mode in a dielectric waveguide.

Drainage of Sloping Lands with Variable Recharge Model Validation and Applications

Semianalytical transient equations describing the behavior of water tables in subsurface drained soils when drains rest on a sloping impervious barrier have been derived previously and represent the bases of the SIDRA model. To validate (SIDRA), water table elevations and drainflow rates have been monitored for six years in the French Alps on fields with a slope of 8%. The predicted drain flow rates and water table shapes compare reasonably well with data of a drainage experiment site but the improvement provided by taking the slope into consideration was limited. Running the model with different slopes confirmed that high water table elevations and peak flows were not significantly changed with a slope of 8%. As could be predicted from the analysis in steady state, low water table elevations were the most affected by slope. With the soil parameters of the field experiment and from an analysis in tail recession conditions, it was shown that there is a clear threshold of 12% slope below which the slope has no significant influence and can be neglected in drainage design.

Study of the spread of a cold instantaneous heavy gas release with surface heat transfer and variable entrainment

Air quality models help in developing relationships between the amount of pollutant released into the ambient atmosphere by a source and the corresponding incremental contribution in the atmospheric concentration. Of the various dispersion models, the heavy gas models help in predicting the concentrations due to release of gases heavier than air and the risks associated with the increased concentrations. Several differences exist among the various models developed to study the heavy-gas dispersion phenomena. These differences mainly arise because of the varied treatment given to physical processes involved in the dispersion mechanism. One of these processes, which have not been fully considered in many of the existing models, is the effect of ground heating on the movement of a cloud under windy conditions. In this study, the box model developed by Kunsch and Fannelop [J. Hazard. Mater. 43 (1995) 169] is extended to incorporate variable air entrainment on the dispersion of a heavy gas cloud spreading in a channel under windy conditions. The air entrainment was assumed to be proportional to the cloud frontal velocity. The semi-analytical equations developed were then solved by numerical methods to provide a heavy gas cloud dispersion profile. The popular Runge-Kutta fourth order technique was adopted to solve the differential equations numerically. The model was applied on a cold cloud released instantaneously and the results indicated that the model behavior follows closely the expected dispersion trends and observed cloud characteristics reported in a laboratory study. The trial run carried out in order to model the scenario of no heat transfer, by equating the source temperature to the ambient temperature, followed variations observed in the field. The analysis of cloud behavior indicated that the cloud length is strongly influenced by source density and initial cloud temperature.

Prediction of cooling time in injection molding by means of a simplified semianalytical equation

AbstractA simplified semianalytical equation, used successfully in food freezing/chilling time prediction, is proposed as a potential simple alternative for cooling time prediction in injection molding of polymer parts, amorphous or semicrystalline. This equation is based on a convective boundary condition for the mold‐part interface and requires information on the thermal contact resistance (TCR) or thermal contact conductance (TCC) at this interface, as well as information on the initial and final product temperatures, the mold surface temperature, and the thermal properties of the part. Eighty‐five data points for four polymers, Polystyrene (PS), Polycarbonate (PC), Polypropylene (PP), and Polyethylene (PE) were generated with C‐MOLD™, a commercial injection molding design software, and the performance of the proposed equation was tested. The % mean error and its standard deviation (SD) in cooling time prediction were, respectively, −11.61 and 2.27 for PS, −6.04 and 2.13 for PC, −7.27 and 6.55 for PP, and −8.88 and 2.93 for PE. It was also shown that the accuracy of the proposed equation is not affected significantly by the exact knowledge of the TCC, provided that the latter is not smaller than 1000–2000 W m−2 K−1. Since in this comparison all necessary temperatures were obtained from C‐MOLD™, methods of using the proposed equation independently were tested. The use of the inlet melt temperature as the initial product temperature increased the % mean error by mostly 1.5% while its SD remained practically the same. By incorporating a literature based heat balance method in the proposed equation, it was possible to use it as a stand‐alone predictor of polymer cooling time. The % mean error and its SD calculated this way were, respectively, −9.44 and 0.97 for PS, −9.44 and 0.83 for PC, −14.22 and 5 for PP, and −20.12 and 1.38 for PE. The proposed equation, at least in a preliminary stage, can be used successfully to predict the cooling time of the selected semicrystalline or amorphous polymers with the accuracy being higher for amorphous polymers. © 2003 Wiley Periodicals, Inc. Adv Polym Techn 22: 188–208, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/adv.10048

Drainage of Sloping Lands with Variable Recharge: Analytical Formulas and Model Development

Semianalytical transient equations for shallow subsurface transverse drainage systems installed in sloping lands are developed. They provide a general relationship between drain flow rates, water table elevations, and recharge rates. This relationship demonstrates that, depending on the recharge intensity, several drain flow rates can be observed at a given water table elevation. The recharge contribution is shown to depend on a water table shape factor and to decrease when the water table is low or the slope is steep. For very steep slopes, the recharge intensity no longer influences the drain flow rate. These equations can be used to confirm previous results obtained in steady-state conditions and to determine precisely under which conditions slope needs to be considered in drainage design. They have been incorporated into the field drainage model SIDRA, which simulates hourly values of water table elevations and drain flow rates. The model predictions are compared with the predictions of a steady-state equation and a numerical model, which solves the Boussinesq equation (SLOP model).

A semi-analytical model for truncated-ogive-nose projectiles penetration into semi-infinite concrete targets

The study on the truncated-ogive-nose projectile penetration into semi-infinite concrete targets is described in this paper. With reference to Forrestal's force model of ogive-nose projectile, semi-analytical penetration equations are developed by introducing a resistance constant which takes into account the truncation effect of the ogive-nose projectile on penetration. The semi-analytical model is shown to be in good correlation with limited recently published experimental results.

Sensitivity Analysis for Transonic Unsteady Aeroelastic Constraints

A method for calculating the sensitivities for transonic unsteady aeroelastic constraints with respect to structural design variables (skin thickness, spar thickness, spar cap cross-sectional areas, etc.) is presented. The method requires that the transonic unsteady aerodynamics be represented in the frequency or Laplace domain. This Indicial Response Technique was used to transform time domain aerodynamics found by solving the TSD equations into the Laplace domain. The indicial responses are performed about a static aeroelastic equilibrium found using the TSD equations for the steady aerodynamics. Once in the Laplace domain the unsteady aerodynamics are used to develop semianalytic equations for the constraint sensitivities. These sensitivity equations include a nonlinear term that must be calculated by finite differences at a considerable computational cost. If a static rigid equilibrium is used, this term vanishes producing fully analytic sensitivities. This method of sensitivity calculation will allow formal unsteady aeroelastic optimization in the transonic flight regime. (Author)

Unsteady Aeroelastic Optimization in the Transonic Regime

A methodology for including transonic e utter requirements in the preliminary automated structural design environment is developed and tested. The problem of minimizing structural weight while satisfying behavioral constraints is stated in nonlinear mathematical programming form and is solved using a gradient-based optimizing technique. The structure is modeled by using e nite elements, and the associated design variables consist of the structural properties: thicknesses of skins, spars, and ribs; cross-sectional areas of posts and spar and rib caps; and concentrated masses. The method requires that the transonic unsteady aerodynamic forces be represented in the frequency or Laplace domain. In this work, the indicial response method is used to transform time-domain aerodynamic forces found by solving the transonic small disturbance (TSD) equations into the Laplace domain. The indicial responses are calculated about static aeroelastic equilibriums found using the TSD equations for the steady aerodynamics. Once in the Laplace domain, the unsteady aerodynamic forces are used to determine system dynamic stability by the p-method and in semianalytic equations for the e utter constraint sensitivities. With constraint values and the required gradients, a Taylor series approximation is used to develop an approximate nonlinear mathematical programming problem for weight minimization. This approximate optimization problem is iteratively solved by the method of modie ed feasible directions until convergence of the exact problem is obtained. Examples of the redesign methodology are given for the simultaneous consideration of constraints on transonic e utter, stresses, and displacements. Results found using nonlinear aerodynamics show that designs can differ considerably from those obtained using linear unsteady aerodynamics when in the transonic e ight regime.