Robert L. McMasters

METHODOLOGY TO GENERATE ACCURATE SOLUTIONS FOR VERIFICATION IN TRANSIENT THREE-DIMENSIONAL HEAT CONDUCTION

This article describes the development of accurate solutions for transient three-dimensional conductive heat transfer in Cartesian coordinates for a parallelepiped which is homogeneous and has constant thermal properties. The intended use of these solutions is for verification of numerical computer programs which are used for solving transient heat conduction problems. Verification is a process to ensure that a computer code is free of errors and accurately solves the mathematical equations. The exact solutions presented in this article can have any combination of boundary conditions of specified temperature, prescribed heat flux, or imposed convection coefficient and ambient temperature on the surfaces of the parallelepiped. Additionally, spatially uniform nonzero initial condition and internal energy generation are treated. The methodology to obtain the analytical solutions and sample calculations are presented.

Comparing the mathematical models of Lighthill to the performance of a biomimetic fish

The mathematical models for the performance of aquatic animals developed by M Lighthill are compared with the experimental performance of a biomimetic fish. The equations developed by Lighthill are evaluated at steady-state conditions. Equilibrium velocity and mechanical efficiency are calculated using Lighthills mathematical model and compared with experimental results. In both cases, a pattern is found wherein an optimum combination of tail frequency and amplitude maximizes equilibrium velocity. Differences between the theoretical and experimental results are attributed to mechanical limitations in the drive train.

Solution of an initial-boundary value problem for heat conduction in a parallelepiped by time partitioning

Abstract An initial-boundary value problem for transient heat conduction in a rectangular parallelepiped is studied. Solutions for the temperature and heat flux are represented as integrals involving the Greens function (GF), the initial and boundary data, and volumetric energy generation. Use of the usual GF obtained by separation of variables leads to slowly convergent series. To circumvent this difficulty, the dummy time interval of integration is partitioned into a short time and a long time subintervals where the GFs are approximated by their small and large time representations. This paper deals with the analysis and implementation of this time partitioning method.

Estimating the Thermal Conductivity of a Film on a Known Substrate

Estimating the thermal conductivity of a film on a substrate of known thermal properties is examined in this research. The laser flash method, commonly used in the measurement of thermal diffusivity, is applied to a composite sample, which has a film deposited on a substrate. The laser flash is applied to the substrate and subsequent temperature measurements are recorded from the film side of the sample. Both the thermal conductivity and the volumetric heat capacity of the substrate must be known. Additionally, the volumetric heat capacity of the film must be known. The parameter estimation method used includes nonlinear regression of a transient conduction model in the solid material, which includes allowance for convective heat losses. The thermal conductivity is estimated simultaneously with the magnitude of the flash and the convection coefficient. The direct solution model is a two-layer exact solution which brings about very rapid computation, in contrast to numerical solutions. Several experiments are analyzed, with samples having various values of thermal conductivity, demonstrating the range over which the method can be used.

Exact Solution for Nonlinear Thermal Diffusion and Its Use for Verification

An analytical solution is provided to the nonlinear diffusion equation, with the thermal conductivity given as a linear function of temperature. The derivation of the solution, and implications of it, are presented. The boundary and initial conditions associated with the solution provide applicability to specific cases. The solution is useful for verifying numerical (computer) solutions to thermal diffusion with temperature-dependent thermal conductivity. The (nonlinear) analytical solution is compared to a numerical solution from a finite element code to verify the accuracy of the code and to establish the order of convergence for the spatial discretization error

Accounting for Penetration of Laser Heating in Flash Thermal Diffusivity Experiments

Since the early 1960s, the laser flash method of thermal diffusivity measurement has been used on a large variety of materials. Several parameter estimation methods have also been used in analyzing such experiments, employing various levels of sophistication. Estimation of thermal parameters, using the models developed as part of this research, is performed on experimental data from the Oak Ridge National Laboratory in Oak Ridge, TN. The material used is carbon bonded carbon fiber (CBCF) which is designed as an insulating material for atmosphere re-entry applications. Ambient temperatures in the experiments range from 800°C to 1200°C. The approximate thermal diffusivity of the material is 0.3 mm 2 /sec. This research investigates the penetration of the laser flash beyond the surface of the material being heated. Three heat transfer models are presented, each with different assumptions about the initial temperature distribution inside the material. An evaluation is made of the response of the methods to factors which may enter into the experimental process. This is done in quantitative terms so as to assess the adequacy of the models in comparison to one another.

Anisotropic Thermal Diffusivity Measurement Using the Flash Method

A well-established method for determining the thermal diffusivity of materials is the laser flash method. The work presented here compares two analysis methods for flash heating tests on anisotropic carbon bonded carbon fiber. This material exhibits a higher conductivity in the direction in which the fibers are oriented than in the direction perpendicular to the fiber orientation. Of the two analysis methods used, one method uses the temperature data from the entire surface of the sample by examining 201 temperature histories simultaneously, with each temperature history originating from an individual pixel within a line across the middle of the sample. The other analysis method uses only the temperature history from a single pixel in the center of the sample, similar to the data that is traditionally generated using the classical flash diffusivity method. Both analysis methods include accommodations for modeling the penetration of the laser flash into the porous surface of the carbon bonded carbon fiber ...

Diffusion Penetration Time for Transient Heat Conduction

The time duration for processes involving transient thermal diffusion can be a critical piece of information related to thermal processes in engineering applications. Analytical solutions must be used to calculate these types of time durations because the boundary conditions in such cases can be effectively like semi-infinite conditions. This research involves an investigation into analytical solutions for six geometries, including one-dimensional cases for Cartesian, cylindrical, and spherical coordinates. The fifth case involves a heated surface on the inside of a hole bored through an infinite body, which is a one-dimensional problem in cylindrical coordinates. The sixth case involves two-dimensional conduction from a point heat source on the surface of a slab subjected to insulated boundary conditions elsewhere. The mathematical modeling for this case is done in cylindrical coordinates. For each geometric configuration, a relationship is developed to determine the time required for a temperature rise ...

Towards the shortest possible contact time: Droplet impact on cylindrical superhydrophobic surfaces structured with macro-scale features

HYPOTHESIS Recently, it has been shown that the contact time of impacting water droplets on a superhydrophobic cylindrical surface decreases when its radius becomes comparable to that of the droplet, yet the correlation of this reduction with the impact velocity is unclear. Moreover, on a flat surface, experiments involving the addition of a single macrotexture, along with covering the surface with macroscopic cylindrical ridges (ribbed pattern), have been reported to shorten the contact time. Hence, a ribbed-curved surface with an additional macrotexture may logically lead to a very short contact time. Experiments Such a surface was obtained by utilizing an extruder-type 3D printer and a copper wire was used as the additional macrotexture. The bouncing of water droplets of three different volumes on curved and ribbed-curved samples with two different diameters was investigated for varied impact velocities. FINDINGS For the curved surfaces, a scaling model for the contact time reduction with respect to the impact velocity was found. Our data show that adding a wire to the peak of a cylinder containing both micro- and macro-scale roughness (wired ribbed-curved surfaces) yields the contact time even shorter than the inertial-capillary time scale, an unprecedented phenomenon.

Using derivative regularization to solve inverse heat conduction problems

A common approach in dealing with inverse heat conduction problems is to use regularization in an attempt to smooth the estimated heat flux as a function of time. If the unknown heat flux is known to be a series of discrete pulses, however, smoothing is undesirable. In dealing with cases such as these, any reduction in the ill-posed composition of the problem is extremely helpful. The present research, utilizing the method of derivative regularization, employs the principle of matrix pre-multiplication to reduce the ill-conditioned nature of the matrix structure. The use of a pre-conditioning method is fairly straightforward; however, the difficult and most critical aspect of this method is the development of the pre-conditioning matrices for the specific type of problem. As the name implies, derivative regularization employs the time derivatives of the sensitivity coefficients, and of the measured data, in developing a pre-conditioning matrix to reduce the ill-posed nature of the inverse heat conduction problem. While the additive forms of regularization generally introduce bias, the derivative regularization method presented here carries the advantage of being unbiased. However, this unbiased advantage can sometimes come at the cost of larger estimation errors. In this research, the inverse heat conduction problem is used for studying the results of tests over a full range of weighting factors. Additionally, various degrees of measurement errors are added to the data in order to observe the effects of measurement errors on the performance of derivative regularization method.