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Robert L. McMasters

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Journal Articles

Journal: Journal of Heat Transfer

Article Type: Research-Article

*. July 2019, 141(7): 071301.*

*J. Heat Transfer*Paper No: HT-18-1491

Published Online: May 14, 2019

Abstract

A generalized solution for a two-dimensional (2D) transient heat conduction problem with a partial-heating boundary condition in rectangular coordinates is developed. The solution accommodates three kinds of boundary conditions: prescribed temperature, prescribed heat flux and convective. Also, the possibility of combining prescribed heat flux and convective heating/cooling on the same boundary is addressed. The means of dealing with these conditions involves adjusting the convection coefficient. Large convective coefficients such as 10 10 effectively produce a prescribed-temperature boundary condition and small ones such as 10 −10 produce an insulated boundary condition. This paper also presents three different methods to develop the computationally difficult steady-state component of the solution, as separation of variables (SOV) can be less efficient at the heated surface and another method (non-SOV) is more efficient there. Then, the use of the complementary transient part of the solution at early times is presented as a unique way to compute the steady-state solution. The solution method builds upon previous work done in generating analytical solutions in 2D problems with partial heating. But the generalized solution proposed here contains the possibility of hundreds or even thousands of individual solutions. An indexed numbering system is used in order to highlight these individual solutions. Heating along a variable length on the nonhomogeneous boundary is featured as part of the geometry and examples of the solution output are included in the results.

Journal Articles

Journal: Journal of Heat Transfer

Article Type: Research-Article

*. July 2018, 140(7): 071301.*

*J. Heat Transfer*Paper No: HT-17-1507

Published Online: March 30, 2018

Abstract

A desirable feature of any parameter estimation method is to obtain as much information as possible with one experiment. However, achieving multiple objectives with one experiment is often not possible. In the field of thermal parameter estimation, a determination of thermal conductivity, volumetric heat capacity, heat addition rate, surface emissivity, and convection coefficient may be desired from a set of temperature measurements in an experiment where a radiant heat source is used. It would not be possible to determine all of these parameters from such an experiment; more information would be needed. The work presented in the present research shows how thermal parameters can be determined from temperature measurements using complementary experiments where the same material is tested more than once using a different geometry or heating configuration in each experiment. The method of ordinary least squares is used in order to fit a mathematical model to a temperature history in each case. Several examples are provided using one-dimensional conduction experiments, with some having a planar geometry and some having a cylindrical geometry. The parameters of interest in these examples are thermal conductivity and volumetric heat capacity. Sometimes, both of these parameters cannot be determined simultaneously from one experiment but utilizing two complementary experiments may allow each of the parameters to be determined. An examination of confidence regions is an important topic in parameter estimation and this aspect of the procedure is addressed in the present work. A method is presented as part of the current research by which confidence regions can be found for results from a single analysis of multiple experiments.

Proceedings Papers

*Proc. ASME*. HT2017, Volume 2: Heat Transfer Equipment; Heat Transfer in Multiphase Systems; Heat Transfer Under Extreme Conditions; Nanoscale Transport Phenomena; Theory and Fundamental Research in Heat Transfer; Thermophysical Properties; Transport Phenomena in Materials Processing and Manufacturing, V002T15A004, July 9–12, 2017

Paper No: HT2017-4702

Abstract

A desirable feature of any parameter estimation method is to obtain as much information as possible with one experiment. However, achieving multiple objectives with one experiment is often not possible. In the field of thermal parameter estimation, a determination of thermal conductivity, volumetric heat capacity, heat addition rate, surface emissivity and convection coefficient may be desired from a set of temperature measurements in an experiment where a radiant heat source is used. It would not be possible to determine all of these parameters from such an experiment; more information would be needed. The work presented in the present research shows how thermal parameters can be determined from temperature measurements using complementary experiments where the same material is tested more than once using a different geometry or heating configuration in each experiment. The method of ordinary least squares is used in order to fit a mathematical model to a temperature history in each case. Several examples are provided using one-dimensional conduction experiments, with some having a planar geometry and some having a cylindrical geometry. The parameters of interest in these examples are thermal conductivity and volumetric heat capacity. Both of these parameters cannot be determined simultaneously from one experiment but the practice of using two complementary experiments allows each of the parameters to be determined. An examination of confidence regions is an important topic in parameter estimation and this aspect of the procedure is addressed in the present work. A method is presented as part of the current research by which confidence regions can be found for results from a single analysis of multiple experiments.

Journal Articles

Journal: Journal of Heat Transfer

Article Type: Research-Article

*. February 2018, 140(2): 021301.*

*J. Heat Transfer*Paper No: HT-17-1191

Published Online: September 13, 2017

Abstract

This paper provides a solution for two-dimensional (2D) heating over a rectangular region on a homogeneous plate. It has application to verification of numerical conduction codes as well as direct application for heating and cooling of electronic equipment. Additionally, it can be applied as a direct solution for the inverse heat conduction problem, most notably used in thermal protection systems for re-entry vehicles. The solutions used in this work are generated using Green's functions (GFs). Two approaches are used, which provide solutions for either semi-infinite plates or finite plates with isothermal conditions, which are located a long distance from the heating. The methods are both efficient numerically and have extreme accuracy, which can be used to provide additional solution verification. The solutions have components that are shown to have physical significance. The procedure involves the derivation of previously unknown simpler forms for the summations, in some cases by virtue of the use of algebraic components. Also, a mathematical identity given in this paper can be used for a variety of related problems.

Proceedings Papers

Kevin D. Cole, Filippo de Monte, Robert L. McMasters, Keith A. Woodbury, A. Haji-Sheikh, James V. Beck

*Proc. ASME*. IMECE2016, Volume 5: Education and Globalization, V005T06A021, November 11–17, 2016

Paper No: IMECE2016-66605

Abstract

Heat transfer in solids provides an opportunity for students to learn of several boundary conditions: the first kind for specified temperature, the second kind for specified heat flux, and the third kind for specified convection. In this paper we explore the relationship among these types of boundary conditions in steady heat transfer. Specifically, the normalized third kind of boundary condition (convection) produces the first kind condition (specified temperature) for large Biot number, and it produces the second kind condition (specified flux) for small Biot number. By employing a generalized boundary condition, one expression provides the temperature for several combinations of boundary conditions. This combined expression is presented for several simple geometries (slabs, cylinders, spheres) with and without internal heat generation. The bioheat equation is also treated. Further, a number system is discussed for each combination to identify the type of boundary conditions present, which side is heated, and whether internal generation is present. Computer code for obtaining numerical values from the several expressions is available, along with plots and tables of numerical values, at a web site called the Exact Analytical Conduction Toolbox. Classroom strategies are discussed regarding student learning of these issues: the relationship among boundary conditions; a number system to identify the several components of a boundary value problem; and, the utility of a web-based resource for analytical heat-transfer solutions.

Proceedings Papers

*Proc. ASME*. HT2016, Volume 2: Heat Transfer in Multiphase Systems; Gas Turbine Heat Transfer; Manufacturing and Materials Processing; Heat Transfer in Electronic Equipment; Heat and Mass Transfer in Biotechnology; Heat Transfer Under Extreme Conditions; Computational Heat Transfer; Heat Transfer Visualization Gallery; General Papers on Heat Transfer; Multiphase Flow and Heat Transfer; Transport Phenomena in Manufacturing and Materials Processing, V002T15A013, July 10–14, 2016

Paper No: HT2016-7103

Abstract

This paper provides a solution for two-dimensional heating over a rectangular region on a homogeneous plate. It has application to verification of numerical conduction codes as well as direct application for heating and cooling of electronic equipment. Additionally, it can be applied as a direct solution for the inverse heat conduction problem, most notably used in thermal protection systems for re-entry vehicles. The solutions used in this work are generated using Green’s functions. Two approaches are used which provide solutions for either semi-infinite plates or finite plates with isothermal conditions which are located a long distance from the heating. The methods are both efficient numerically and have extreme accuracy, which can be used to provide additional solution verification. The solutions have components that are shown to have physical significance. The extremely precise nature of analytical solutions allows them to be used as prime standards for their respective transient conduction cases. This extreme precision also allows an accurate calculation of heat flux by finite differences between two points of very close proximity which would not be possible with numerical solutions. This is particularly useful near heated surfaces and near corners. Similarly, sensitivity coefficients for parameter estimation problems can be calculated with extreme precision using this same technique. Another contribution of these solutions is the insight that they can bring. Important dimensionless groups are identified and their influence can be more readily seen than with numerical results. For linear problems, basic heating elements on plates, for example, can be solved to aid in understanding more complex cases. Furthermore these basic solutions can be superimposed both in time and space to obtain solutions for numerous other problems. This paper provides an analytical two-dimensional, transient solution for heating over a rectangular region on a homogeneous square plate. Several methods are available for the solution of such problems. One of the most common is the separation of variables (SOV) method. In the standard implementation of the SOV method, convergence can be slow and accuracy lacking. Another method of generating a solution to this problem makes use of time-partitioning which can produce accurate results. However, numerical integration may be required in these cases, which, in some ways, negates the advantages offered by the analytical solutions. The method given herein requires no numerical integration; it also exhibits exponential series convergence and can provide excellent accuracy. The procedure involves the derivation of previously-unknown simpler forms for the summations, in some cases by virtue of the use of algebraic components. Also, a mathematical identity given in this paper can be used for a variety of related problems.

Proceedings Papers

Kevin D. Cole, Filippo de Monte, Robert L. McMasters, Keith A. Woodbury, Junghoon Yeom, James V. Beck

*Proc. ASME*. IMECE2015, Volume 5: Education and Globalization, V005T05A027, November 13–19, 2015

Paper No: IMECE2015-52179

Abstract

Applied computer solutions for conductive heat transfer are a critical component in any modern undergraduate heat transfer course. This need has been addressed in many ways through various textbook exercises and software packages. The present work involves a catalog of analytical solutions organized with a numbering system that describes the boundary conditions and initial conditions for each problem. The solutions are pre-programmed and accessible via a free web site called the Ex act A nalytical C onduction T oolbox, or EXACT. Students can access these solutions for use in homework and project work. In this paper examples of several types of student exercises are given, including a re-creation of the Heisler charts and a two dimensional steady-state example. Additionally, an account is given of classroom use of these tools in a graduate heat transfer course, outlining the education advantages of the EXACT web page. The concept of intrinsic verification is also discussed, focusing on the applicability of this concept to enhancing insight among undergraduate students. General support is also expressed for the need of analytical solutions to heat transfer and diffusion problems in an undergraduate setting.

Journal Articles

Journal: Journal of Heat Transfer

Article Type: Research-Article

*. October 2014, 136(10): 101301.*

*J. Heat Transfer*Paper No: HT-13-1438

Published Online: July 15, 2014

Abstract

There are many applications for problems involving thermal conduction in two-dimensional (2D) cylindrical objects. Experiments involving thermal parameter estimation are a prime example, including cylindrical objects suddenly placed in hot or cold environments. In a parameter estimation application, the direct solution must be run iteratively in order to obtain convergence with the measured temperature history by changing the thermal parameters. For this reason, commercial conduction codes are often inconvenient to use. It is often practical to generate numerical solutions for such a test, but verification of custom-made numerical solutions is important in order to assure accuracy. The present work involves the generation of an exact solution using Green's functions where the principle of superposition is employed in combining a one-dimensional (1D) cylindrical case with a 1D Cartesian case to provide a temperature solution for a 2D cylindrical. Green's functions are employed in this solution in order to simplify the process, taking advantage of the modular nature of these superimposed components. The exact solutions involve infinite series of Bessel functions and trigonometric functions but these series sometimes converge using only a few terms. Eigenvalues must be determined using Bessel functions and trigonometric functions. The accuracy of the solutions generated using these series is extremely high, being verifiable to eight or ten significant digits. Two examples of the solutions are shown as part of this work for a family of thermal parameters. The first case involves a uniform initial condition and homogeneous convective boundary conditions on all of the surfaces of the cylinder. The second case involves a nonhomogeneous convective boundary condition on a part of one of the planar faces of the cylinder and homogeneous convective boundary conditions elsewhere with zero initial conditions.

Proceedings Papers

*Proc. ASME*. HT2013, Volume 4: Heat and Mass Transfer Under Extreme Conditions; Environmental Heat Transfer; Computational Heat Transfer; Visualization of Heat Transfer; Heat Transfer Education and Future Directions in Heat Transfer; Nuclear Energy, V004T14A007, July 14–19, 2013

Paper No: HT2013-17177

Abstract

There are many applications for problems involving thermal conduction in two-dimensional cylindrical objects. Experiments involving thermal parameter estimation are a prime example, including cylindrical objects suddenly placed in hot or cold environments. In a parameter estimation application, the direct solution must be run iteratively in order to obtain convergence with the measured temperature history by changing the thermal parameters. For this reason, commercial conduction codes are often inconvenient to use. It is often practical to generate numerical solutions for such a test, but verification of custom-made numerical solutions is important in order to assure accuracy. The present work involves the generation of an exact solution using Green’s functions where the principle of superposition is employed in combining a one-dimensional cylindrical case with a one-dimensional Cartesian case to provide a temperature solution for a two-dimensional cylindrical. Green’s functions are employed in this solution in order to simplify the process, taking advantage of the modular nature of these superimposed components. The exact solutions involve infinite series of Bessel functions and trigonometric functions but these series sometimes converge using only a few terms. Eigenvalues must be determined using Bessel functions and trigonometric functions. The accuracy of the solutions generated using these series is extremely high, being verifiable to eight or ten significant digits. Two examples of the solutions are shown as part of this work for a family of thermal parameters. The first case involves a uniform initial condition and homogeneous convective boundary conditions on all of the surfaces of the cylinder. The second case involves a nonhomogeneous convective boundary condition on a part of one of the planar faces of the cylinder and homogeneous convective boundary conditions elsewhere with zero initial conditions.

Proceedings Papers

*Proc. ASME*. HT2012, Volume 2: Heat Transfer Enhancement for Practical Applications; Fire and Combustion; Multi-Phase Systems; Heat Transfer in Electronic Equipment; Low Temperature Heat Transfer; Computational Heat Transfer, 981-985, July 8–12, 2012

Paper No: HT2012-58189

Abstract

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 (CBCF). This material exhibits a higher conductivity in the direction in which the fibers are oriented than in the direction perpendicular to the fiber orientation. The two analysis methods being compared in this experiment use different portions of the data in obtaining results. One method utilizes 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 utilizes only the temperature history from a single pixel in the center of the sample, similar to the data which 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 CBCF material. Additionally, both models include a parameter which accounts for the non-uniform heating of the sample surface from the flash. Although the sample surface is ostensibly heated uniformly in flash diffusivity experiments, the heating has been found to be somewhat non-uniform, with more energy deposited more heavily in the center of the sample. This affects the analysis results, particularly in tests on anisotropic materials. The results in this work show very little difference between the thermal parameters arising from the two methods. The robustness of the method using the single-pixel temperature history shows that anisotropic thermal diffusivity can be measured using standard flash diffusivity instruments, avoiding the additional complexity associated with a thermal imaging camera.

Proceedings Papers

*Proc. ASME*. HT2009, Volume 1: Heat Transfer in Energy Systems; Thermophysical Properties; Heat Transfer Equipment; Heat Transfer in Electronic Equipment, 491-496, July 19–23, 2009

Paper No: HT2009-88159

Abstract

The laser flash method, as a means of measuring thermal diffusivity, is well established, and several manufacturers produce equipment for performing these types of experiments. Most analysis methods used for interpreting the data from these experiments assume one-dimensional transient conduction, with insulated surfaces during the time subsequent to the flash. More recently, models of greater sophistication employing nonlinear regression have been applied to flash diffusivity experiments. These models assume an instantaneous flash and are highly accurate for most samples of moderate diffusivity and sample thickness. As samples become thinner and more highly conductive, the duration of the experiments becomes very short. Since the duration of the flash is typically on the order of several milliseconds, the assumption that this period of time is instantaneous becomes less valid for very short experiments. A model accounting for the duration of the flash is applied to three samples of stainless steel of varying thicknesses and analyzed with two different mathematical models. One model accounts for the finite duration of the flash and the other does not. The model accounting for the flash duration generates results that are much more consistent between samples than the model assuming an instantaneous flash. Moreover, the conformance of the mathematical model accounting for flash duration is much closer to the measured data than the model which assumes an instantaneous flash. As part of the finite flash duration model, the length of the flash is estimated by nonlinear regression, optimizing the conformance of the model to the measured data. Additionally, the starting time of the flash is treated as a parameter and is determined simultaneously with flash duration, thermal diffusivity and flash intensity. Statistical methods are also used for showing the validity of the added level of sophistication of the more advanced mathematical model.

Proceedings Papers

*Proc. ASME*. HT2007, ASME/JSME 2007 Thermal Engineering Heat Transfer Summer Conference, Volume 1, 175-185, July 8–12, 2007

Paper No: HT2007-32046

Abstract

The analytical solution for the problem of transient thermal conduction with solid body movement is developed for a parallelepiped with convective boundary conditions. An effective transformation scheme is used to eliminate the flow terms. The solution uses Green’s functions containing convolution-type integrals, which involve integration over a dummy time, referred to as “cotime.” Two types of Green’s functions are used: one for short cotimes comes from the Laplace transform and the other for long cotimes from the method of separation of variables. A primary advantage of this method is that it incorporates internal verification of the numerical results by varying the partition time between the short and long components. In some cases, the long time solution requires a zeroth term in the summation, which does not occur when solid body motion is not present. The existence of this zeroth term depends upon the magnitude of the heat transfer coefficient associated with the convective boundary condition. An example is given for a two-dimensional case involving both prescribed temperature and convective boundary conditions. Comprehensive tables are also provided for the nine possible combinations of boundary conditions in each dimension.

Proceedings Papers

*Proc. ASME*. HT2008, Heat Transfer: Volume 3, 405-419, August 10–14, 2008

Paper No: HT2008-56038

Abstract

The laser flash method for measuring thermal diffusivity is well established and has been in use for many years. Early analysis methods employed a simple model, in which one dimensional transient conduction was assumed, with insulated surfaces during the time subsequent to the flash. More recently, models of grater sophistication have been applied to flash diffusivity experiments. These models have been matched to experimental data using nonlinear regression and assume one-dimensional conduction. The advanced models have achieved highly accurate agreement with experimental data taken from thin samples, on the order of one millimeter in thickness. As samples become thicker, models which neglect edge losses can lose some conformity to the experimental data. The present research involves the application of a two dimensional model which allows for penetration of the laser flash into the sample. The accommodation of the flash penetration is important for porous materials, where the coarseness of the porosity is more than one percent of the sample thickness. Variability of the area of incidence of the flash is also investigated to determine the effect on the model and the results. Statistical methods are used in order to make a determination as to the validity of the two dimensional model, as compared with the one dimensional analysis method.

Journal Articles

Journal: Journal of Heat Transfer

Article Type: Research Papers

*. November 2008, 130(11): 111301.*

*J. Heat Transfer*Published Online: August 28, 2008

Abstract

The analytical solution for the problem of transient thermal conduction with solid body movement is developed for a parallelepiped with convective boundary conditions. An effective transformation scheme is used to eliminate the flow terms. The solution uses Green’s functions containing convolution-type integrals, which involve integration over a dummy time, referred to as “cotime.” Two types of Green’s functions are used: one for short cotimes comes from the Laplace transform and the other for long cotimes from the method of separation of variables. A primary advantage of this method is that it incorporates internal verification of the numerical results by varying the partition time between the short and long components. In some cases, the long-time solution requires a zeroth term in the summation, which does not occur when solid body motion is not present. The existence of this zeroth term depends on the magnitude of the heat transfer coefficient associated with the convective boundary condition. An example is given for a two-dimensional case involving both prescribed temperature and convective boundary conditions. Comprehensive tables are also provided for the nine possible combinations of boundary conditions in each dimension.