Approximate Integral Method Research Articles

Melting and resolidification of a subcooled metal powder particle subjected to nanosecond laser heating

Melting and resolidification of a subcooled spherical metal powder particle subjected to nanosecond laser heating are investigated analytically. The problem is divided into three stages: preheating, melting and resolidification, and thermalization. The temperature distributions in the liquid and solid regions and the location of the solid–liquid interface are obtained using an integral approximate method. The effects of the laser intensity and pulse width, initial subcooling, and particle radius on the melting and resolidification of powder particles were investigated. The Selective Laser Sintering (SLS) process for a pulsed laser can be simulated by repeating the three stages but varying the initial conditions.

Analysis for Slip Flow Over a Single Free Disk With Heat Transfer

Analysis for Slip Flow Over a Single Free Disk With Heat Transfer

Experimental Investigation of Separated Nozzle Flows

A detailed experimental study of separated nozzle flows has been conducted at the NASA Langley Research Center 16-Foot Transonic Tunnel Complex. As part of a comprehensive static performance investigation, force, moment, and pressure measurements were made and schlieren flow visualization was obtained for a subscale, nonaxisymmetric, two-dimensional, convergent-divergent nozzle. For reference, experimental results were compared with theoretical predictions based on one-dimensional gasdynamics and an approximate integral momentum boundary-layer method. Results from this study indicate that overexpanded nozzle flow was dominated by shock-induced boundary-layer separation, which was divided into two distinct flow regimes: three-dimensional separation with partial reattachment and fully detached two-dimensional separation. The test nozzle was observed to go through a marked transition in passing from one regime to the other. In all cases, separation provided a significant increase in static thrust efficiency compared to the ideal prediction. Results suggest that, with controlled separation, the entire overexpanded range of nozzle performance would be within 16% of the peak thrust efficiency. By offering savings in weight and complexity over a conventional mechanical variable geometry exhaust system, a fixed geometry nozzle may be able to cover an entire flight envelope in some applications.

Heating process characteristics and kinetics of rice straw in different atmospheres

Rice growing engenders large quantities of straw, which is usually burnt in the open field, a practice implying a public health risk. There are, however, several ways of making use of this material, including its conversion into a valuable fuel. The aims of this research are to describe the thermal characteristics of the heating processes of rice straw in different atmospheres of air and nitrogen and to fit the heating profiles to two mathematical models proposed. This research was followed by thermogravimetric analysis, visual observation of the heating profiles revealing three stages in the rice straw heating process: drying, devolatilization and burning. The models described were found to adequately describe the weight loss of rice straw. The approximate integral method (AIM) is less complex than the direct method (DM), but the latter shows the order of reaction. A better fit is obtained for the heating atmospheres with a lesser proportion of oxygen. The presence of oxygen increases the activation energy and decreases the final temperature of each stage considered.

Studies on indigo dyeing and diffusion by the film roll method

The influence of reducing agent, indigo concentration and dyebath pH on indigo dyeing and diffusion was studied for the film roll method. An almost linear relationship between the indigo concentrations in the dyed film and in the dyebath was maintained throughout the experiments. Two different calculation methods were used to obtain the diffusion coefficient and the surface indigo concentration in the film. Similar surface indigo concentrations were obtained from the two methods, but a lower diffusion coefficient was obtained by the approximate integral method. It was found that the dyebath pH is the dominant factor influencing the rate of diffusion of indigo and the colour yield of the dyed film. The dyebath pH also had a considerable influence on the reduction of indigo to leuco‐indigo, a step that is essential to the dyeing process.

Vaporization, melting and heat conduction in the laser drilling process

Melting and vaporization phenomena during the laser drilling process are investigated in this paper. The locations of the solid–liquid and liquid–vapor interfaces were obtained by solving energy conservation equations at the interfaces. The dependence of saturation temperature on the back pressure is taken into account by using the Clausius/Clapeyron equation. The conduction heat loss to the solid is also included in the model and is solved by using an integral approximate method. The results show that the fraction of the heat lost through conduction to the solid is very small and its effect on the vaporization process is not significant. On the other hand, the conduction heat loss significantly reduces the thickness of the liquid layer, which becomes the recast layer after drilling.

An approximate analytic model for flow through capillary tubes

An analytn model for working fluids flowing through capillary tubes has been established with the approximate integral method. All the possible flow regimes in the capillary tubes, including subcooled, two-phase and superheated regimes, are covered in the model, and different analytic solutions are given respectively for each flow regime. As examples, the mass flow rates of refrigerants R12, R134a and R600a through capillary tubes are predicted by the model, and compared with those evaluated by a general distributed-parameter model. The mean bias falls into 1% and the maximum bias is 2.2%, while the computation speed of the new model is more than one order of magnitude higher than that of the distributed-parameter one.

A Thermal Model for Mushy Zone Formation in Binary Solutions

In this paper, the solidification of a binary solution on a cold isothermal surface will be studied by using an inetgral approximate method. The advantage of solving the solidification of a binary solution using an integral approximate method is that the mushy zone thickness will be able to be expressed as a simple algebraic equation which can be easily solved

The impact of the resolution of the identity approximate integral method on modern ab initio algorithm development

The computation of the two-electron four-center integrals over gaussian basis functions is a significant component of the overall work of many ab initio methods used today. Improvements in the computational efficiency of the base algorithms have provided significant impact. Somewhat overlooked are methods that provide approximations to these integrals and their implementation in application software. A partial review of approximate integral techniques focused on the resolution of the identity (RI) four-center, two-electron integral approximation is given. The past and current uses of the RI algorithms are presented along with possibilities for further exploitation of the technology.

Analysis of Freezing in an Eccentric Annulus

Freezing in an eccentric annulus is investigated numerically by using a temperature transforming model. Since the effect of the heat conduction along the circular direction on the growth of the freezing layer is very small, an analytical solution by employing integral approximate method is proposed. The freezing rate obtained by the analytical solution agreed very well with that of the numerical solution, although the analytical solution is much simpler than the numerical solution. The effects of the eccentric annulus geometric structure on the freezing process is also investigated.

Heat Transfer During Solidification Around a Horizontal Tube With Internal Convective Cooling

Theoretical solution for the solidification process around a horizontal tube with internal convection cooling is described in this paper. Boundary layer analysis and the approximate integral method were applied to obtain the solution for the natural convection in the liquid region and the transient temperature profile in the solid region. Effects of Biot number on the wall temperature and the volume fraction solidified are also discussed in this paper.

A Multi-Angle Averaging Theorem Applied to the J 2 Problem

A particular multi-angle averaging theorem for systems admitting a finite Fourier expansion of the field is presented, together with its application to the problem of motion around an oblate planet (the J2 problem), in harmonic oscillator formulation. This method of approximate integration has the advantage of working with (close on) directly measurable elements.

Semi-analytical solution of thermal energy storage system with conjugate laminar forced convection

A thermal energy storage system with a hollow cylinder of Phase Change Material (PCM) was studied semi-analytically. The melting of the PCM was solved by using an integral approximate method, and heat transfer in the container wall was treated as a radial one dimensional conduction problem. The forced convective heat transfer inside the tube was solved by an analytical method, which was coupled with the heat conduction in the PCM and the container wall. The results show that the laminar forced convective heat transfer inside the tube never reached the fully developed state, even for a very long tube. For a transfer fluid of moderate Prandtl number, the laminar forced convective transfer inside the tube must be solved simultaneously with the phase change of the PCM.

Application of integral methods to ablation charring erosion - A review

To predict ablation, charring, and erosion of heat-shield materials, approximate heat balance integral (HBI) methods offer speed and versatility; however, traditional HBI articles treat only simple, idealized models of material response. This paper reviews application of HBI methods to more realistic models of material response, specifically, for carbon-carbon and carbon-phenolic heat shields on reentry vehicles. The review shows that HBI successfully extends to most such simulations of ablation, charring, and erosion in hypersonic flow, but unexpected problems can crop up and trade-offs exist. Pertinent material models are also summarized, including efficient expressions that fit material thermal properties and carbon-air thermochemical ablation functions.

An approximate integral method for calculating the diffracted component from multiple two-dimensional objects

A method for determining the diffracted or scattered components due to the interaction of a wave with multiple two-dimensional objects is given. The method presented uses approximate boundary conditions to simplify the numerically cumbersome set of integral equations associated with nonapproximate analytic solutions of the Helmholtz integral. The approximate boundary conditions, defined here, are shown to decouple the N by N set of simultaneous equations. The approximate boundary conditions are used to calculate the insertion loss for various sizes of two-dimensional objects. The predictions are compared to measured results and limitations are discussed.

An approximate solution for the dynamic response of wall transfer probes

An approximate solution for the transient Lévêque problem has been developed based on the approximate integral method and the method of weighted residuals. This solution can be used to explore the dynamic response of thermal or mass boundary layers in large amplitude, but non-reversing, unsteady flows. The accuracy of this approximate solution has been proven over a wide range of dimensionless frequencies ( ω = Ω(L 2/S 2 0D) 1,3) . The approximate solution forms the basis of a simple method for calculating the instantaneous shear rate from the measurement of instantaneous heat or mass flux at the surface of a wall transfer probe when the quasi-steady state assumption is not valid.

Darcy-Forchheimer natural, forced and mixed convection heat transfer in non-Newtonian power-law fluid-saturated porous media

The governing equation for Darcy-Forchheimer flow of non-Newtonian inelastic power-law fluid through porous media has been derived from first principles. Using this equation, the problem of Darcy-Forchheimer natural, forced, and mixed convection within the porous media saturated with a power-law fluid has been solved using the approximate integral method. It is observed that a similarity solution exists specifically for only the case of an isothermal vertical flat plate embedded in the porous media. The results based on the approximate method, when compared with existing exact solutions show an agreement of within a maximum error bound of 2.5%.

Analytical solution of melting in a subcooled semi-infinite solid with boundary conditions of the second kind

The melting problem in a semi-infinite region with constant heat flux boundary condition is solved by a semi-exact method and an integral approximate method. Effect of subcooling on the transient solid-liquid interface location and surface temperature are also discussed in this paper.

Forced convection heat transfer to an elastic fluid of constant viscosity flowing through a channel filled with a Brinkman-Darcy porous medium

An analysis is presented of fully developed flow and heat transfer in a channel confined by two parallel walls subjected to uniform heat flux in a highly porous medium saturated with an elastic fluid of constant viscosity. The Brinkman-extended Darcy model is used for studying the effect of the boundary viscous frictional drag on the heat transfer characteristics. The approximate integral method is employed to obtain a solution.

Acoustic diffraction by multiple two-dimensional objects; its calculation using an approximate integral method

A method for determining the diffracted or scattered components of an acoustical wave by multiple two-dimensional objects is given. The method presented uses approximate boundary conditions to simplify the cumbersome set of integral equations associated with nonapproximate analytic solutions of the Helmholtz integral. The approximate boundary conditions are used to calculate the insertion loss for multiple two-dimensional objects of various sizes. The predictions are compared to measured results and limitations are discussed.