The comparative performance of some finite difference equations for transient heat conduction

The established finite difference equations to estimate one-dimensional transient heat flow in solids are the ‘classical’ form (explicit), and the Crank–Nicolson and ‘pure-implicit’ forms (both implicit). They are all based on finite difference approximation to the Fourier continuity equation. To these are now added three more explicit forms: exponential linear, exponential inverse cosine and polynomial, which are based on exact solutions to the Fourier equation. The performance of each of the six equations is tested against the exact results of a well known step excitations problem (the Groeber model). The tests consist of examining (i) finite difference errors arising from a single implementation of each form at different stages in the transient cooling process, (ii) the errors that accumulate during part or all of the cooling process (both as regards any bias that is introduced, and also a measure of variance) and (iii) the run times in executing the various forms. The nondimensional time step r was treated as the independent variable, and can be made arbitrarily large, by use of a simple time-division procedure (otherwise r < ½ for use with the classical form). It is shown that having regard to both error and run time, the polynomial form appears to be the most efficient estimator.     

Vapor-Phase Helmholtz Equation for HFC-227ea from Speed-of-Sound Measurements

This work presents measurements of the speed-of-sound in the vapor phase of 1,1,1,2,3,3,3-heptafluoropropane (HFC-227ea). The measurements were obtained in a stainless-steel spherical resonator with a volume of 900 cm³ at temperatures between 260 and 380 K and at pressures up to 500 kPa. Ideal-gas heat capacities and acoustic virial coefficients are directly produced from the data. A Helmholtz equation of state of high accuracy is proposed, whose parameters are directly obtained from speed-of-sound data fitting. The ideal-gas heat capacity data are fit by a functions and used when fitting the Helmholtz equation for the vapor phase. From this equation of state other thermodynamic state function are derived. Due to the high accuracy of the equation, only very precise experimental data are suitable for the model validation and only density measurements have these requirements. A very high accuracy is reached in density prediction, showing the obtained Helmholtz equation to be very reliable. The deduced vapor densities are furthermore compared with those obtained from acoustic virial coefficients with the temperature dependences calculated from hard-core square-well potentials.     

Dielectric-Constant Gas Thermometry and the Relation to the Virial Coefficients

At PTB new dielectric-constant gas thermometry (DCGT) measurements were performed at the temperature of the triple point of water. As discussed recently in an accompanying paper, the main goal was the determination of the Boltzmann constant \(k\) as a contribution to the international efforts directed to a new definition of the base unit kelvin via fixing the value of  \(k\) . Besides the linear term in the series expansion used for fitting the results of measurements of DCGT isotherms that reveals \(k\) , in this paper the higher-order terms are analyzed. For retrieving highly accurate virial coefficients of helium from the data obtained at gas pressures up to 7 MPa, an extended DCGT working equation is developed. Applying this equation, information is deduced on the viral coefficients up to the fourth density virial coefficient. Finally, comparisons with the latest ab initio calculations for the second and third density virial coefficients as well as the second dielectric virial coefficient are performed.     

On continuous lifetime distributions with polynomial failure rate with an application in reliability

It is shown that the Laplace transform of a continuous lifetime random variable with a polynomial failure rate function satisfies a certain differential equation. This generates a set of differential equations which can be used to express the polynomial coefficients in terms of the derivatives of the Laplace transform at the origin. The technique described here establishes a procedure for estimating the polynomial coefficients from the sample moments of the distribution. Some special cases are worked through symbolically using computer algebra. Real data from the literature recording bus motor failures is used to compare the proposed approach with results based on the least squares procedure.     

A Rational Fundamental Equation of State for Pentafluoroethane with Theoretical and Experimental Bases

A fundamental equation of state for pentafluoroethane was established on the basis of not only assessment of the experimental data but also by introducing parameters for virial coefficients having a theoretical background in statistical thermodynamics. The equation of state has a range of validity for temperatures from the triple point up to 500 K and pressures up to 70 MPa. The estimated uncertainties of the equation are 0.1% for the vapor pressure, 0.15% in density for the saturated-liquid phase, 0.5% in density for the saturated-vapor phase, 0.1% in density for the liquid phase, 0.1% in pressure for the gaseous phase, 0.5% in density for the supercritical region, 0.01% in speed of sound for the gaseous phase, 0.9% in speed of sound for the liquid phase, 0.5% in isobaric specific heat for the liquid phase, and 1.2% in isochoric specific heat for the liquid phase. The derived specific heats in the gaseous phase are close to the values from the virial equation of state with the second and third virial coefficients derived from intermolecular potential models and precise speed-of-sound measurements.     

Periodically distributed parallel cracks in a functionally graded piezoelectric (FGP) strip bonded to a FGP substrate under static electromechanical load

In this paper, the Fourier integral transform–singular integral equation method is presented for the problem of a periodic array of cracks in a functionally graded piezoelectric strip bonded to a different functionally graded piezoelectric material. The properties of two materials, such as elastic modulus, piezoelectric constant and dielectric constant, are assumed in exponential forms and vary along the crack direction. The crack surface condition is assumed to be electrically impermeable or permeable. The mixed boundary value problem is reduced to a singular integral equation over crack by applying the Fourier transform and the singular integral equation is solved numerically by using the Lobatto–Chebyshev integration technique. The analytic expressions of the stress intensity factors and the electric displacement intensity factors are derived. The effects of the loading parameter λ, material constants and the geometry parameters on the stress intensity factor, the energy release ratio and the energy density factor are studied.     

An equation of state formulation of the thermodynamic properties of R134a (1,1,1,2-tetrafluoroethane)

New correlations for the thermodynamics properties of R134a are presented. A classical equation for the molar Helmholtz energy is used with temperature and density as the independent variables. The equation is accurate for both the liquid and vapour phases at pressures up to 70 Pa, and for a temperature range from the triple point to 450 K. Temperatures are given on the new International Temperature Scale of 1990 (ITS 90). The equation was developed by using experimental data for pressure-volume-temperature (PVT) properties, isochoric heat capacity, second virial coefficients, speed of sound and coexistence properties. Comparisons with experimental data and with two other equations of state are given. Ancillary equations representing the saturated liquid and vapour densities and the vapour pressure are also presented.     

A Rational Helmholtz Fundamental Equation of State for Difluoromethane with an Intermolecular Potential Background

A new fundamental thermodynamic equation of state for difluoromethane was developed by considering the intermolecular potential behavior for improving the reliability in the gaseous phase. Reliable second and third virial coefficients are introduced in accordance with the principle of a unified relation of the intermolecular potential energy and the fundamental equation of state. The fundamental equation of state is able to provide reliable thermodynamic properties even at low temperatures or in the region near saturation where precise and accurate experimental data are not available. The estimated uncertainties of calculated properties from the equation of state are 0.07% in density for the liquid phase, 0.1% in pressure for the gaseous phase, 0.35% in pressure for the supercritical region, 0.07% in vapor pressure, 0.2% in saturated-liquid density, 0.7% in saturated-vapor density, 0.01% in speed of sound for the gaseous phase, 0.7% in speed of sound for the liquid phase, and 0.6% in isochoric specific heat for the liquid phase. The equation is valid for temperatures from the triple point to 450 K and pressures up to 72 MPa.     

Stability analysis of Trefftz methods for the stick-slip problem

The stick-slip problem is a two-dimensional Stokes flow problem, and is classified into biharmonic equation with crack singularities. The collocation Trefftz method (CTM) is used to provide the very accurate solutions and leading coefficients. In this paper, the error analysis is made, to show the exponential convergence rates, and the new stability analysis is explored more in detail. We derive the bounds of effective and traditional condition numbers, to have the polynomial and the exponential growth rates, respectively. The moderate effective condition number is a suitable criterion of stability for the CTM solution of the stick-slip problem, while the huge condition number is misleading. Besides, numerical experiments are carried out to support the stability analysis. Hence the effective condition number becomes a new trend of stability for numerical partial differential equations.     

Study on Isochoric Specific Heat Capacity of Liquid R-410A and HFE-347pcf2

The isochoric heat capacity (c _v ) of R-410A [a mixture of 49.81 mass% difluoromethane (HFC-32) + 50.19 mass% pentafluoroethane (HFC-125)] and 1,1,2,2-tetrafluoroethyl-2,2,2-trifluoroethylether (HFE-347pcf2) was measured at temperatures from 277 K to 400 K and at pressures up to 30 MPa. The reported density measurements for R-410A and HFE-347pcf2 are in the single-phase region and cover a density range above 0.92 g·cm^?3 and 1.33 g·cm^?3, respectively. The measured data of R-410A are compared with data reported by other researchers. Also, the measured data of R-410A are examined with an available equation of state. As a result, it is found that the present c _v data for R-410A agree well with those by other researchers and the calculated values with the equation of state in the measurement range except near the critical isochore.     

A thermodynamic property formulation for cyclohexane

A formulation for the thermodynamic properties of cyclohexane is presented. The equation is valid for single-phase and saturation states from the melting line to 700 K at pressures up to 80 MPa. It includes a fundamental equation explicit in reduced Helmholtz energy with independent variables of reduced density and temperature. The functional form and coefficients of the ancillary equations were determined by weighted linear regression analyses of evaluated experimental data. An adaptive regression algorithm was used to determine the final equation. To ensure correct thermodynamic behavior of the Helmholtz energy surface the coefficients of the fundamental equation were determined with multiproperty fitting, Pressure-density-temperature (P-p-T) and isobaric heat capacity (C _p -P-T) data were used to develop the fundamental equation, SaturationP-p-T values, calculated from the estimating functions, were used to ensure thermodynamic consistency at the vapor-liquid phase boundary. Separate functions were used for the vapor pressure, saturated liquid density, saturated vapor density. ideal-gas heat capacity. and pressure on the melting curve, Comparisons between experimental data and values calculated using the fundamental equation are given to verify the accuracy of the formulation. The formulation given here may be used to calculate densities within ±0.1 %, heat capacities to within ±2 %. and speed of sound to within ± 1 %, except near the critical point.Paper presented at the Twelfth Symposium on Thermophysical Properties, June 19–24, 1994, Boulder, Colorado, U.S.A.     

Development of a New Alpha Function for the Peng–Robinson Equation of State: Comparative Study of Alpha Function Models for Pure Gases (Natural Gas Components) and Water-Gas Systems

Numerous modifications have been suggested for the temperature dependence of the attractive term of the Peng–Robinson equation of state (PR-EOS), through the alpha function. In this work, a new alpha function combining both exponential and polynomial forms is proposed. Pure-compound vapor pressures for different molecular species were fitted and compared using different alpha functions including the Mathias–Copeman and Trebble–Bishnoi alpha functions. The new alpha function allows significant improvements of pure compound vapor pressure predictions (about 1.2% absolute average percent deviations) for all the systems considered, starting from a reduced temperature of 0.4. In addition, a generalization of the classical Mathias–Copeman alpha function was proposed as a function of the acentric factor. These alpha functions were used for VLE calculations on water+various gases including gaseous hydrocarbons. A general procedure is presented to fit experimental VLE data. The corresponding thermodynamic approach is based on the Peng–Robinson equation of state with the above cited alpha functions. It includes the classical mixing rules for the vapor phase and a Henry's law approach to treat the aqueous phase.     

Toward an analytic equation of state for fluorine

There are more than 1000 good experimental measurements of fluid fluorine, which cover the range from the triple point to 300 K for pressures up to 23 MPa. The development of an equation of state to fit these data is described. The form of equation used is that of a reduced Helmholtz energy function which includes an equation to represent the behavior of the ideal gas. The coefficients in the equation are determined by a least-squares method where the terms used are selected statistically from a given bank, thus reducing intercorrelation. Two banks of terms were chosen and various fits made using PT data alone, and multiproperty fits with both banks. The serious consequence of using PT data alone is demonstrated, and the difference in quality between the two banks is described. The problems of inconsistency between the single-phase and the saturated density data are discussed. New equations are given which represent the ideal-gas properties.     

Global sensitivity analysis by polynomial dimensional decomposition

This paper presents a polynomial dimensional decomposition (PDD) method for global sensitivity analysis of stochastic systems subject to independent random input following arbitrary probability distributions. The method involves Fourier-polynomial expansions of lower-variate component functions of a stochastic response by measure-consistent orthonormal polynomial bases, analytical formulae for calculating the global sensitivity indices in terms of the expansion coefficients, and dimension-reduction integration for estimating the expansion coefficients. Due to identical dimensional structures of PDD and analysis-of-variance decomposition, the proposed method facilitates simple and direct calculation of the global sensitivity indices. Numerical results of the global sensitivity indices computed for smooth systems reveal significantly higher convergence rates of the PDD approximation than those from existing methods, including polynomial chaos expansion, random balance design, state-dependent parameter, improved Sobol's method, and sampling-based methods. However, for non-smooth functions, the convergence properties of the PDD solution deteriorate to a great extent, warranting further improvements. The computational complexity of the PDD method is polynomial, as opposed to exponential, thereby alleviating the curse of dimensionality to some extent.     

A three parameter equation of state for gases

This paper presents an equation of state having the following features. It holds for many gases for temperatures from near critical to about five times critical, and for densities from zero to a little above critical, with the exception of the immediate vicinity of the critical point. It also holds for mixtures. Its accuracy is excellent, in most cases within experimental error. Exceptions are H₂O, CO₂, H₂ at high density. In the best cases the rms deviations in the compressibility factor are a few parts in ten thousand. In relatively poor cases the deviations are a few parts per thousand.The equation has a single reduced form for all gases, and requires only three characteristic parameters to identify each gas: a reducing parameter, T_B; a reducing density, d_o; and a dimensionless parameter, k_B. These three parameters are readily (though often not uniquely) evaluated from experimental data. Besides determining the PVT behaviour of a gas, including its virial coefficients, these three parameters are directly related to those in the common three-parameter intermolecular potentials.     

求解变系数对流扩散反应方程的指数型高精度紧致差分方法

本文给出了一种数值求解变系数对流扩散反应方程的指数型高精度紧致差分方法.我们首先将模型方程变形,借助常系数对流扩散方程的指数型高精度紧致差分格式,采用残量修正法得到变系数对流扩散反应方程的指数型高精度紧致差分格式;并从理论上分析了当Pelect数很大时,本文格式达到四阶计算精度时网格步长的限制条件;离散得到的代数方程组可采用追赶法直接求解.数值实验结果与理论分析完全吻合,表明了本文格式对于边界层问题或大梯度变化的物理量求解问题具有的高精度和鲁棒性的优点.     

On the spatial decay for the dynamical problem of thermo-microstretch elastic solids

This paper derives spatial decay bounds for a dynamical problem of thermo-microstretch elasticity defined on a semi-infinite cylindrical region. Previous results for isothermal elastodynamics and the parabolic heat equation lead us to suspect that the solution of the problem should tend to zero faster than a decaying exponential of the distance from the finite end of the cylinder. We prove that an energy expression is actually bounded above by a decaying exponential of a quadratic polynomial of the distance.     

An analysis of transport phenomena for multi-solid particle systems at higher Reynolds numbers by a 5th order polynomial Karman-Pohlhausen method

Using the concentric spheres free surface model and a 5th order polynomial Karman-Pohlhausen method of the laminar boundary layer theory, the dimensionless tangential stress distributions, the dimensionless pressure distributions around a solid sphere in a swarm and the viscous, form and total drag coefficients for multi-solid sphere systems were numerically computed at higher Reynolds numbers, based on the first assumption that the pressure distribution equals that of potential flow between concentric spheres up to the separation point, and behind the separation point in the wake region the pressure does not recover and keeps constant, and on the second assumption that the pressure distribution varies according to the measurement of Flachsbart.The theoretical drag coefficient of single solid spheres in an infinite medium based on the second assumption agreed with the experimental data in the range of the Reynolds numbers from $\text{3 × 10}{}^2\text{to 10}{}^5$.The friction factor for multi-solid particle systems based on the first assumption is almost the same as that on the 4th order polynomial and agreed with the experimental data of packed and distended beds.The void functions obtained from the drag coefficients for multi-solid particle systems based on both first and second assumptions were almost the same as the one on the 4th order polynomial.Using the velocity profiles based on concentric spheres free surface model and a 5th order polynomial Karman-Pohlhausen method of the laminar boundary layer obtained previously, the diffusion equation was solved numerically at higher Reynolds numbers on the first assumption of the pressure distribution around a solid sphere in a swarm equals that of potential flow between concentric spheres from the frontal stagnation point to the separation one, and the pressure does not recover, but keeps constant behind the separation point in the wake region. The mass transfer rate for multi-solid particle systems so computed was almost the same as that on the 4th order polynominal and agreed with the experimental data of single Solid spheres, and packed and particulate fluidized beds at higher Reynolds numbers.     

Exponential finite elements for diffusion–advection problems

A new finite element method for the solution of the diffusion–advection equation is proposed. The method uses non‐isoparametric exponentially‐varying interpolation functions, based on exact, one‐ and two‐dimensional solutions of the Laplace‐transformed differential equation. Two eight‐noded elements are developed and tested for convergence, stability, Peclet number limit, anisotropy, material heterogeneity, Dirichlet and Neumann boundary conditions and tolerance for mesh distortions. Their performance is compared to that of conventional, eight‐ and 12‐noded polynomial elements. The exponential element based on two‐dimensional analytical solutions fails basic tests of convergence. The one based on one‐dimensional solutions performs particularly well. It reduces by about 75% the number of elements and degrees of freedom required for convergence, yielding an error that is one order of magnitude smaller than that of the eight‐noded polynomial element. The exponential element is stable and robust under relatively high degrees of heterogeneity, anisotropy and mesh distortions. Copyright © 2005 John Wiley & Sons, Ltd.