Fourier-Bessel Series Model for the Stefan Problem With Internal Heat Generation in Cylindrical Coordinates
Volume 6A: Thermal-Hydraulics and Safety Analyses
A nonlinear, first-order ordinary differential equation that involves Fourier-Bessel series terms... more A nonlinear, first-order ordinary differential equation that involves Fourier-Bessel series terms has been derived to model the time-dependent motion of the solid-liquid interface during melting and solidification of a material with constant internal heat generation in cylindrical coordinates. The model is valid for all Stefan numbers. One of the primary applications of this problem is for a nuclear fuel rod during meltdown. The numerical solutions to this differential equation are compared to the solutions of a previously derived model that was based on the quasi-steady approximation, which is valid only for Stefan numbers less than one. The model presented in this paper contains exponentially decaying terms in the form of Fourier-Bessel series for the temperature gradients in both the solid and liquid phases. The agreement between the two models is excellent in the low Stefan number regime. For higher Stefan numbers, where the quasi-steady model is not accurate, the new model diff...
High reliability is desired in all engineered systems. One way to improve system reliability is t... more High reliability is desired in all engineered systems. One way to improve system reliability is to use redundant components. When redundant components are used, the problem becomes one of allocating them to achieve the best reliability without exceeding other design constraints such as cost, weight, or volume. Systems with few components can be optimized by simply examining every possible combination but the number of combinations for most systems is prohibitive. A computerized iteration of the process is possible but anything short of a super computer requires too much time to be practical. Many researchers have derived mathematical formulations for calculating the optimum configuration directly. However, most of the derivations are based on continuous functions whereas the real system is composed of discrete entities. Therefore, these techniques are approximations of the true optimum solution. This paper describes a computer program that will determine the optimum configuration of...
A detailed experimental study has been carried out to evaluate the heat transfer performance of a... more A detailed experimental study has been carried out to evaluate the heat transfer performance of a solid/liquid phase-change thermal energy storage system. The phase-change material, 99% pure eicosane with a melting temperature of 36.5°C, was contained in a vertically oriented test cylinder that was cooled or heated at its outside boundary, resulting in radially inward freezing or melting, respectively. Detailed quantitative time-dependent temperature distributions and melt-front motion and shape data were obtained. In the freezing case study, a mathematical model was developed based on a one-dimensional analysis, which considered heat conduction as the only mode of heat transfer. In the melting case study, a heat transfer scale analysis was used to help interpret the data and development of heat transfer correlations. In the melting scale analysis, conduction heat transfer in the solid and natural convection heat transfer in liquid were considered. Comparison of experimental data with scale analysis predictions of the solid-liquid interface position and temperature distribution was performed. The analytical results agreed, in the worst case, within 10% of the experimental results in both melting and freezing cases. In the case of melting, scale analysis results agreed within 5% (after initial superheat disappeared in 50 minutes) with experimental results, and experimental results confirm the existence of four melting regions.
h i g h l i g h t s A differential equation modeling the Stefan problem with heat generation is d... more h i g h l i g h t s A differential equation modeling the Stefan problem with heat generation is derived. The analytical solutions compare very well with the computational results. The system reaches steady-state faster for larger Stefan numbers. The interface location is proportional to the inverse square root of the heat generation.
This paper gives the results of a dynamical analysis of the Globesat gravity gradient stabilized ... more This paper gives the results of a dynamical analysis of the Globesat gravity gradient stabilized satellite in a 500 km circular orbit. The linearized equations of motion are developed and the stability of the satellite is investigated. The satellite is equipped with magnetic torquers for the purpose of providing attitude correction torques. These correction torques can be used to effect large changes in orientation and for producing small impulses for damping residual librational motions. The analysis shows that such a satellite can be captured into a gravity gradient stabilized mode, and that residual motions can be damped to small steady state values.
High-accuracy pointing capabilities are desired for many three-axis stabilized small satellites. ... more High-accuracy pointing capabilities are desired for many three-axis stabilized small satellites. Momentum-based attitude control system actuators, initially developed for larger satellites. are being utilized by small satellites to meetthesc pointing requirements. This paper provides an overview of momentum devices available for small satellite applications and three-axis attitude control system (ACS) configurations using these devices. Factors affecting the selection and sizing of ACS components are also addressed. Included are suggestions for potential ACS improvements and cost-saving measures that will make momentum devices more accessible to the small satellite community.
International Journal of Heat and Mass Transfer, 2014
Fundamental understanding of the entropy generation process in characteristic wall shear flows is... more Fundamental understanding of the entropy generation process in characteristic wall shear flows is developed. For entropy generated by fluid friction, the rates are predictable for developed turbulent flows and pure laminar flows. The main concern lies in prediction for flows undergoing so-called ''bypass'' transition from laminar to turbulent state. In this study, the entropy generation process in the bypass transition scenario is investigated for a flat plate boundary layer under effects of adverse and favorable pressure gradients. For the case studied, transition occurs prematurely due to the presence of strong levels of freestream turbulence. Reynolds-Averaged Navier-Stokes (RANS) models and direct numerical simulations (DNS) are implemented to study the local entropy generation in pre-transitional and transitional regions. Three of these RANS models are transitional models such as, traditional SST k-x (2eq), SST k-x (4eq) and k-kl-x. They are used for prediction of the onset of transition and the results are compared with DNS. All the RANS models predicted transition onset prematurely and, consequently, over predict the integral entropy generation rate and the skin friction coefficient in the transition region. Four simulations have been performed for (1) zero, (2) favorable, (3) adverse and (4) strong-adverse pressure gradient cases. The numerical results show that the pressure gradient has a significant effect on the onset of transition and entropy generation. Transition is early and abrupt for the strong adverse pressure gradient but occurs further downstream and is longer with the increasingly favorable flows. Entropy generation rate at the wall increases as the pressure gradient grows.
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, 2013
This study presents the material challenges associated with Advanced Reactor Concepts (ARC) such ... more This study presents the material challenges associated with Advanced Reactor Concepts (ARC) such as the Advanced High Temperature Reactor (AHTR). ARCs are the next generation concepts focusing on power production and providing thermal energy for industrial applications. The efficient transfer of energy for industrial applications depends on the ability to incorporate cost-effective heat exchangers between the nuclear heat transport system and industrial process heat transport system. The heat exchanger required for AHTR is subjected to a unique set of conditions that introduce several design challenges not encountered in standard heat exchangers. The corrosive molten salts, especially at higher temperatures, require materials throughout the system to avoid corrosion, and adverse high-temperature effects, such as creep. Given the very high steam generator pressure of the supercritical steam cycle, it is anticipated that water tube and molten salt shell steam generators heat exchanger will be used. In this paper, the American Society of Mechanical Engineers (ASME) Section III and Section VIII requirements (acceptance criteria) are discussed. Also, the ASME material acceptance criteria (ASME Section II, Part D) for high temperature environments are presented. Finally, lack of ASME acceptance criteria for thermal design and analysis are discussed with potential benefit.
Volume 2: Heat Transfer Enhancement for Practical Applications; Heat and Mass Transfer in Fire and Combustion; Heat Transfer in Multiphase Systems; Heat and Mass Transfer in Biotechnology, 2013
Using a scale/analytical analysis approach, we model phase change (melting) for pure materials wh... more Using a scale/analytical analysis approach, we model phase change (melting) for pure materials which generate constant internal heat generation for small Stefan numbers (approximately one). The analysis considers conduction in the solid phase and natural convection, driven by internal heat generation, in the liquid regime. The model is applied for a constant surface temperature boundary condition where the melting temperature is greater than the surface temperature in a cylindrical geometry. The analysis also consider constant heat flux (in a cylindrical geometry). We show the time scales in which conduction and convection heat transfer dominate.
Integral Solutions of Phase Change With Internal Heat Generation
12th International Conference on Nuclear Engineering, Volume 3, 2004
This paper presents solutions to a one-dimensional solid-liquid phase change problem using the in... more This paper presents solutions to a one-dimensional solid-liquid phase change problem using the integral method for a semi-infinite material that generates internal heat. The analysis assumed a quadratic temperature profile and a constant temperature boundary condition on the exposed surface. We derived a differential equation for the solidification thickness as a function of the internal heat generation (IHG) and the Stefan number, which includes the temperature of the boundary. Plots of the numerical solutions for various values of the IHG and Stefan number show the time-dependant behavior of both the melting and solidification distances and rates. The IHG of the material opposes solidification and enhances melting. The differential equation shows that in steady-state, the thickness of the solidification band is inversely related to the square root of the IHG. The model also shows that the melting rate initially decreases and reaches a local minimum, then increases to an asymptotic...
Heat Transfer Performance of a Porous Metal Foam/Phase Change Material System: Part 1 — Freezing
Heat Transfer: Volume 2, 2005
ABSTRACT A detailed experimental study has been performed to evaluate the heat transfer performan... more ABSTRACT A detailed experimental study has been performed to evaluate the heat transfer performance of a solid/liquid phase-change thermal energy storage system that includes porous metal foam. The phase-change material (PCM) and metal foam were contained in a vertically oriented test cylinder that is cooled at its outside boundary, resulting in radially inward freezing. As the PCM freezes, the solid/liquid interface moves inward from the surface of the test cylinder, and a thermal resistance layer is built up, resulting in a reduced heat transfer rate between the system to be cooled and the PCM. The porous material used in this research was intended to minimize the insulating effect of this thermal resistance layer. In the freezing case study, a one-dimensional mathematical model was developed, which considered heat conduction as the only mode of heat transfer. The effective thermal conductivity of the porous media saturated with solid eicosane was predicted utilizing several models and compared with the measured effective thermal conductivity. The results of this study are discussed in terms of the effectiveness of the metal foam as a heat transfer enhancement device.
Using a scale analysis approach, we model phase change (melting) for pure materials which generat... more Using a scale analysis approach, we model phase change (melting) for pure materials which generate internal heat for small Stefan numbers (approximately one). The analysis considers conduction in the solid phase and natural convection, driven by internal heat generation, in the liquid regime. The model is applied for a constant surface temperature boundary condition where the melting temperature is greater than the surface temperature in a cylindrical geometry. We show the time scales in which conduction and convection heat transfer dominate. NOMENCLATURE c p-specific heat (kJ/kg K) g-gravitational constant (m/s 2) 'h f-Latent heat (kJ/kg) H-height of cylinder (m) k-thermal conductivity (W/m K) q-internal heat generation (W/m 3) r-radial distance (m) Ra +-Rayleigh number (UEg q H 5 /PDk) Ra z-Rayleigh number (UEg q z 5 /PDk) s-distance from CL to phase front (m) St-Stefan number (c p 'T/'h) t-time (s) T-temperature (K) T m-melting temperature (K) u, v, w-velocity components (m/s) x, y, z-spatial components (m) Greek D-thermal diffusivity (m 2 /s) E-coefficient of thermal expansion (1/K) G z-thermal boundary layer thickness (m) P-dynamic viscosity (kg m/s) Q-kinematic viscosity (m 2 /s) U-density (kg/m 3) T-nondimensional time, (q /U'h f)
In this paper, we investigate the effects that the volumetric heat generation has on the movement... more In this paper, we investigate the effects that the volumetric heat generation has on the movement and steady-state location of a solid-liquid phase change front in melting and freezing processes. Volumetric heat generation enhances melting and impedes freezing. This phenomenon occurs in nuclear, geologic, cryogenic and material processing applications. We compare the results from a FLUENT computational model with analytical results of a quasi-static solution of the governing equations. These models are applied for constant surface temperature boundary conditions and various volumetric heat generation values in cylindrical plane wall and spherical geometries. The quasi-static method results in an exact steadystate solution which shows that the location of the phase change front is inversely proportional to the square root of the volumetric heat generation. This method is valid for Stefan numbers less than one and the computational results for this regime give excellent agreement with the analytical model, thereby validating the technique and solutions. For the sake of comparison, we also plot the analytical model and computational results for Stefan numbers of one and greater. The quasi-static analytical solution converges more rapidly to the steady-state value than the computational solution. As expected, at longer time intervals, both the analytical and computational solutions converge to the exact steady-state solution.
Volume 1: Heat Transfer in Energy Systems; Theory and Fundamental Research; Aerospace Heat Transfer; Gas Turbine Heat Transfer; Transport Phenomena in Materials Processing and Manufacturing; Heat and, 2012
This is a preprint of a paper intended for publication in a journal or proceedings. Since changes... more This is a preprint of a paper intended for publication in a journal or proceedings. Since changes may be made before publication, this preprint should not be cited or reproduced without permission of the author. This document was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, or any of their employees, makes any warranty, expressed or implied, or assumes any legal liability or responsibility for any third party's use, or the results of such use, of any information, apparatus, product or process disclosed in this report, or represents that its use by such third party would not infringe privately owned rights. The views expressed in this paper are not necessarily those of the United States Government or the sponsoring agency.
A study was performed to determine the common methods and tools that are available to calculate o... more A study was performed to determine the common methods and tools that are available to calculate or predict a system's reliability. A literature review and seftware survey are included. The desired product of this developmental work is a tool for the system designer to use in the early design phase so that the final design will achieve the desired system reliabilitywithout lengthy testing and rework. Three computer programs were written which provide the first attempt at fulfilling this need. The programs are described and a case study is presented for each one. This is a continuing effort which will be furthered in FY-1992.
Effects of Internal Heat Generation on Solidification
Heat Transfer: Volume 2, 2005
ABSTRACT Solutions using the integral method are presented for solid-liquid phase change in mater... more ABSTRACT Solutions using the integral method are presented for solid-liquid phase change in materials that generate internal heat. This problem is solved for cylindrical, spherical, plane wall and semi-infinite slab geometries. The analysis assumes a temperature profile in the solid phase and constant temperature boundary conditions on the exposed surfaces. We derive differential equations governing the solidification thickness for the geometries as functions of the Stefan number and the internal heat generation (IHG). For cylindrical, spherical, and plane wall geometries, the solidification layer obtains a steady-state value which is related to the inverse of the square root of the IHG. The solutions to the semi-infinite slab geometry problem show that when the surface is cooled to below the freezing point, a solidification layer forms along the edge and begins to grow until it reaches a maximum, then begins remelt. The problem has application to diverse fields such as nuclear energy, materials processing, geophysical fluids, and bioengineering.
Phase Change Heat Transfer Enhancement Using Copper Porous Foam
Journal of Heat Transfer, 2008
A detailed experimental and analytical study has been performed to evaluate how copper porous foa... more A detailed experimental and analytical study has been performed to evaluate how copper porous foam (CPF) enhances the heat transfer performance in a cylindrical solid/liquid phase change thermal energy storage system. The CPF used in this study had a 95% porosity and the phase change material (PCM) was 99% pure eicosane. The PCM and CPF were contained in a vertical cylinder where the temperature at its radial boundary was held constant, allowing both inward freezing and melting of the PCM. Detailed quantitative time-dependent volumetric temperature distributions and melt/freeze front motion and shape data were obtained. As the material changed phase, a thermal resistance layer built up, resulting in a reduced heat transfer rate between the surface of the container and the phase change front. In the freezing analysis, we analytically determined the effective thermal conductivity of the combined PCM/CPF system and the results compared well to the experimental values. The CPF increased...
Using a quasi-static approach valid for Stefan numbers less than one, we derive approximate equat... more Using a quasi-static approach valid for Stefan numbers less than one, we derive approximate equations governing the movement of a phase change front for materials which generate internal heat. These models are applied for both constant surface temperature and constant surface heat flux boundary conditions, in cylindrical, spherical, plane wall and semi-infinite geometries. Exact solutions with the constant surface temperature condition are obtained for the steady-state solidification thickness using the cylinder, sphere, and plane wall geometries which show that the thickness depends on the inverse square root of the internal heat generation. Under constant surface heat flux conditions, closed form equations can be obtained for the three geometries. In the case of the semiinfinite wall, we show that for constant temperature and constant heat flux out of the wall conditions, the solidification layer grows then remelts.
Fourier-Bessel Series Model for the Stefan Problem With Internal Heat Generation in Cylindrical Coordinates
Volume 6A: Thermal-Hydraulics and Safety Analyses
A nonlinear, first-order ordinary differential equation that involves Fourier-Bessel series terms... more A nonlinear, first-order ordinary differential equation that involves Fourier-Bessel series terms has been derived to model the time-dependent motion of the solid-liquid interface during melting and solidification of a material with constant internal heat generation in cylindrical coordinates. The model is valid for all Stefan numbers. One of the primary applications of this problem is for a nuclear fuel rod during meltdown. The numerical solutions to this differential equation are compared to the solutions of a previously derived model that was based on the quasi-steady approximation, which is valid only for Stefan numbers less than one. The model presented in this paper contains exponentially decaying terms in the form of Fourier-Bessel series for the temperature gradients in both the solid and liquid phases. The agreement between the two models is excellent in the low Stefan number regime. For higher Stefan numbers, where the quasi-steady model is not accurate, the new model diff...
High reliability is desired in all engineered systems. One way to improve system reliability is t... more High reliability is desired in all engineered systems. One way to improve system reliability is to use redundant components. When redundant components are used, the problem becomes one of allocating them to achieve the best reliability without exceeding other design constraints such as cost, weight, or volume. Systems with few components can be optimized by simply examining every possible combination but the number of combinations for most systems is prohibitive. A computerized iteration of the process is possible but anything short of a super computer requires too much time to be practical. Many researchers have derived mathematical formulations for calculating the optimum configuration directly. However, most of the derivations are based on continuous functions whereas the real system is composed of discrete entities. Therefore, these techniques are approximations of the true optimum solution. This paper describes a computer program that will determine the optimum configuration of...
A detailed experimental study has been carried out to evaluate the heat transfer performance of a... more A detailed experimental study has been carried out to evaluate the heat transfer performance of a solid/liquid phase-change thermal energy storage system. The phase-change material, 99% pure eicosane with a melting temperature of 36.5°C, was contained in a vertically oriented test cylinder that was cooled or heated at its outside boundary, resulting in radially inward freezing or melting, respectively. Detailed quantitative time-dependent temperature distributions and melt-front motion and shape data were obtained. In the freezing case study, a mathematical model was developed based on a one-dimensional analysis, which considered heat conduction as the only mode of heat transfer. In the melting case study, a heat transfer scale analysis was used to help interpret the data and development of heat transfer correlations. In the melting scale analysis, conduction heat transfer in the solid and natural convection heat transfer in liquid were considered. Comparison of experimental data with scale analysis predictions of the solid-liquid interface position and temperature distribution was performed. The analytical results agreed, in the worst case, within 10% of the experimental results in both melting and freezing cases. In the case of melting, scale analysis results agreed within 5% (after initial superheat disappeared in 50 minutes) with experimental results, and experimental results confirm the existence of four melting regions.
h i g h l i g h t s A differential equation modeling the Stefan problem with heat generation is d... more h i g h l i g h t s A differential equation modeling the Stefan problem with heat generation is derived. The analytical solutions compare very well with the computational results. The system reaches steady-state faster for larger Stefan numbers. The interface location is proportional to the inverse square root of the heat generation.
This paper gives the results of a dynamical analysis of the Globesat gravity gradient stabilized ... more This paper gives the results of a dynamical analysis of the Globesat gravity gradient stabilized satellite in a 500 km circular orbit. The linearized equations of motion are developed and the stability of the satellite is investigated. The satellite is equipped with magnetic torquers for the purpose of providing attitude correction torques. These correction torques can be used to effect large changes in orientation and for producing small impulses for damping residual librational motions. The analysis shows that such a satellite can be captured into a gravity gradient stabilized mode, and that residual motions can be damped to small steady state values.
High-accuracy pointing capabilities are desired for many three-axis stabilized small satellites. ... more High-accuracy pointing capabilities are desired for many three-axis stabilized small satellites. Momentum-based attitude control system actuators, initially developed for larger satellites. are being utilized by small satellites to meetthesc pointing requirements. This paper provides an overview of momentum devices available for small satellite applications and three-axis attitude control system (ACS) configurations using these devices. Factors affecting the selection and sizing of ACS components are also addressed. Included are suggestions for potential ACS improvements and cost-saving measures that will make momentum devices more accessible to the small satellite community.
International Journal of Heat and Mass Transfer, 2014
Fundamental understanding of the entropy generation process in characteristic wall shear flows is... more Fundamental understanding of the entropy generation process in characteristic wall shear flows is developed. For entropy generated by fluid friction, the rates are predictable for developed turbulent flows and pure laminar flows. The main concern lies in prediction for flows undergoing so-called ''bypass'' transition from laminar to turbulent state. In this study, the entropy generation process in the bypass transition scenario is investigated for a flat plate boundary layer under effects of adverse and favorable pressure gradients. For the case studied, transition occurs prematurely due to the presence of strong levels of freestream turbulence. Reynolds-Averaged Navier-Stokes (RANS) models and direct numerical simulations (DNS) are implemented to study the local entropy generation in pre-transitional and transitional regions. Three of these RANS models are transitional models such as, traditional SST k-x (2eq), SST k-x (4eq) and k-kl-x. They are used for prediction of the onset of transition and the results are compared with DNS. All the RANS models predicted transition onset prematurely and, consequently, over predict the integral entropy generation rate and the skin friction coefficient in the transition region. Four simulations have been performed for (1) zero, (2) favorable, (3) adverse and (4) strong-adverse pressure gradient cases. The numerical results show that the pressure gradient has a significant effect on the onset of transition and entropy generation. Transition is early and abrupt for the strong adverse pressure gradient but occurs further downstream and is longer with the increasingly favorable flows. Entropy generation rate at the wall increases as the pressure gradient grows.
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, 2013
This study presents the material challenges associated with Advanced Reactor Concepts (ARC) such ... more This study presents the material challenges associated with Advanced Reactor Concepts (ARC) such as the Advanced High Temperature Reactor (AHTR). ARCs are the next generation concepts focusing on power production and providing thermal energy for industrial applications. The efficient transfer of energy for industrial applications depends on the ability to incorporate cost-effective heat exchangers between the nuclear heat transport system and industrial process heat transport system. The heat exchanger required for AHTR is subjected to a unique set of conditions that introduce several design challenges not encountered in standard heat exchangers. The corrosive molten salts, especially at higher temperatures, require materials throughout the system to avoid corrosion, and adverse high-temperature effects, such as creep. Given the very high steam generator pressure of the supercritical steam cycle, it is anticipated that water tube and molten salt shell steam generators heat exchanger will be used. In this paper, the American Society of Mechanical Engineers (ASME) Section III and Section VIII requirements (acceptance criteria) are discussed. Also, the ASME material acceptance criteria (ASME Section II, Part D) for high temperature environments are presented. Finally, lack of ASME acceptance criteria for thermal design and analysis are discussed with potential benefit.
Volume 2: Heat Transfer Enhancement for Practical Applications; Heat and Mass Transfer in Fire and Combustion; Heat Transfer in Multiphase Systems; Heat and Mass Transfer in Biotechnology, 2013
Using a scale/analytical analysis approach, we model phase change (melting) for pure materials wh... more Using a scale/analytical analysis approach, we model phase change (melting) for pure materials which generate constant internal heat generation for small Stefan numbers (approximately one). The analysis considers conduction in the solid phase and natural convection, driven by internal heat generation, in the liquid regime. The model is applied for a constant surface temperature boundary condition where the melting temperature is greater than the surface temperature in a cylindrical geometry. The analysis also consider constant heat flux (in a cylindrical geometry). We show the time scales in which conduction and convection heat transfer dominate.
Integral Solutions of Phase Change With Internal Heat Generation
12th International Conference on Nuclear Engineering, Volume 3, 2004
This paper presents solutions to a one-dimensional solid-liquid phase change problem using the in... more This paper presents solutions to a one-dimensional solid-liquid phase change problem using the integral method for a semi-infinite material that generates internal heat. The analysis assumed a quadratic temperature profile and a constant temperature boundary condition on the exposed surface. We derived a differential equation for the solidification thickness as a function of the internal heat generation (IHG) and the Stefan number, which includes the temperature of the boundary. Plots of the numerical solutions for various values of the IHG and Stefan number show the time-dependant behavior of both the melting and solidification distances and rates. The IHG of the material opposes solidification and enhances melting. The differential equation shows that in steady-state, the thickness of the solidification band is inversely related to the square root of the IHG. The model also shows that the melting rate initially decreases and reaches a local minimum, then increases to an asymptotic...
Heat Transfer Performance of a Porous Metal Foam/Phase Change Material System: Part 1 — Freezing
Heat Transfer: Volume 2, 2005
ABSTRACT A detailed experimental study has been performed to evaluate the heat transfer performan... more ABSTRACT A detailed experimental study has been performed to evaluate the heat transfer performance of a solid/liquid phase-change thermal energy storage system that includes porous metal foam. The phase-change material (PCM) and metal foam were contained in a vertically oriented test cylinder that is cooled at its outside boundary, resulting in radially inward freezing. As the PCM freezes, the solid/liquid interface moves inward from the surface of the test cylinder, and a thermal resistance layer is built up, resulting in a reduced heat transfer rate between the system to be cooled and the PCM. The porous material used in this research was intended to minimize the insulating effect of this thermal resistance layer. In the freezing case study, a one-dimensional mathematical model was developed, which considered heat conduction as the only mode of heat transfer. The effective thermal conductivity of the porous media saturated with solid eicosane was predicted utilizing several models and compared with the measured effective thermal conductivity. The results of this study are discussed in terms of the effectiveness of the metal foam as a heat transfer enhancement device.
Using a scale analysis approach, we model phase change (melting) for pure materials which generat... more Using a scale analysis approach, we model phase change (melting) for pure materials which generate internal heat for small Stefan numbers (approximately one). The analysis considers conduction in the solid phase and natural convection, driven by internal heat generation, in the liquid regime. The model is applied for a constant surface temperature boundary condition where the melting temperature is greater than the surface temperature in a cylindrical geometry. We show the time scales in which conduction and convection heat transfer dominate. NOMENCLATURE c p-specific heat (kJ/kg K) g-gravitational constant (m/s 2) 'h f-Latent heat (kJ/kg) H-height of cylinder (m) k-thermal conductivity (W/m K) q-internal heat generation (W/m 3) r-radial distance (m) Ra +-Rayleigh number (UEg q H 5 /PDk) Ra z-Rayleigh number (UEg q z 5 /PDk) s-distance from CL to phase front (m) St-Stefan number (c p 'T/'h) t-time (s) T-temperature (K) T m-melting temperature (K) u, v, w-velocity components (m/s) x, y, z-spatial components (m) Greek D-thermal diffusivity (m 2 /s) E-coefficient of thermal expansion (1/K) G z-thermal boundary layer thickness (m) P-dynamic viscosity (kg m/s) Q-kinematic viscosity (m 2 /s) U-density (kg/m 3) T-nondimensional time, (q /U'h f)
In this paper, we investigate the effects that the volumetric heat generation has on the movement... more In this paper, we investigate the effects that the volumetric heat generation has on the movement and steady-state location of a solid-liquid phase change front in melting and freezing processes. Volumetric heat generation enhances melting and impedes freezing. This phenomenon occurs in nuclear, geologic, cryogenic and material processing applications. We compare the results from a FLUENT computational model with analytical results of a quasi-static solution of the governing equations. These models are applied for constant surface temperature boundary conditions and various volumetric heat generation values in cylindrical plane wall and spherical geometries. The quasi-static method results in an exact steadystate solution which shows that the location of the phase change front is inversely proportional to the square root of the volumetric heat generation. This method is valid for Stefan numbers less than one and the computational results for this regime give excellent agreement with the analytical model, thereby validating the technique and solutions. For the sake of comparison, we also plot the analytical model and computational results for Stefan numbers of one and greater. The quasi-static analytical solution converges more rapidly to the steady-state value than the computational solution. As expected, at longer time intervals, both the analytical and computational solutions converge to the exact steady-state solution.
Volume 1: Heat Transfer in Energy Systems; Theory and Fundamental Research; Aerospace Heat Transfer; Gas Turbine Heat Transfer; Transport Phenomena in Materials Processing and Manufacturing; Heat and, 2012
This is a preprint of a paper intended for publication in a journal or proceedings. Since changes... more This is a preprint of a paper intended for publication in a journal or proceedings. Since changes may be made before publication, this preprint should not be cited or reproduced without permission of the author. This document was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, or any of their employees, makes any warranty, expressed or implied, or assumes any legal liability or responsibility for any third party's use, or the results of such use, of any information, apparatus, product or process disclosed in this report, or represents that its use by such third party would not infringe privately owned rights. The views expressed in this paper are not necessarily those of the United States Government or the sponsoring agency.
A study was performed to determine the common methods and tools that are available to calculate o... more A study was performed to determine the common methods and tools that are available to calculate or predict a system's reliability. A literature review and seftware survey are included. The desired product of this developmental work is a tool for the system designer to use in the early design phase so that the final design will achieve the desired system reliabilitywithout lengthy testing and rework. Three computer programs were written which provide the first attempt at fulfilling this need. The programs are described and a case study is presented for each one. This is a continuing effort which will be furthered in FY-1992.
Effects of Internal Heat Generation on Solidification
Heat Transfer: Volume 2, 2005
ABSTRACT Solutions using the integral method are presented for solid-liquid phase change in mater... more ABSTRACT Solutions using the integral method are presented for solid-liquid phase change in materials that generate internal heat. This problem is solved for cylindrical, spherical, plane wall and semi-infinite slab geometries. The analysis assumes a temperature profile in the solid phase and constant temperature boundary conditions on the exposed surfaces. We derive differential equations governing the solidification thickness for the geometries as functions of the Stefan number and the internal heat generation (IHG). For cylindrical, spherical, and plane wall geometries, the solidification layer obtains a steady-state value which is related to the inverse of the square root of the IHG. The solutions to the semi-infinite slab geometry problem show that when the surface is cooled to below the freezing point, a solidification layer forms along the edge and begins to grow until it reaches a maximum, then begins remelt. The problem has application to diverse fields such as nuclear energy, materials processing, geophysical fluids, and bioengineering.
Phase Change Heat Transfer Enhancement Using Copper Porous Foam
Journal of Heat Transfer, 2008
A detailed experimental and analytical study has been performed to evaluate how copper porous foa... more A detailed experimental and analytical study has been performed to evaluate how copper porous foam (CPF) enhances the heat transfer performance in a cylindrical solid/liquid phase change thermal energy storage system. The CPF used in this study had a 95% porosity and the phase change material (PCM) was 99% pure eicosane. The PCM and CPF were contained in a vertical cylinder where the temperature at its radial boundary was held constant, allowing both inward freezing and melting of the PCM. Detailed quantitative time-dependent volumetric temperature distributions and melt/freeze front motion and shape data were obtained. As the material changed phase, a thermal resistance layer built up, resulting in a reduced heat transfer rate between the surface of the container and the phase change front. In the freezing analysis, we analytically determined the effective thermal conductivity of the combined PCM/CPF system and the results compared well to the experimental values. The CPF increased...
Using a quasi-static approach valid for Stefan numbers less than one, we derive approximate equat... more Using a quasi-static approach valid for Stefan numbers less than one, we derive approximate equations governing the movement of a phase change front for materials which generate internal heat. These models are applied for both constant surface temperature and constant surface heat flux boundary conditions, in cylindrical, spherical, plane wall and semi-infinite geometries. Exact solutions with the constant surface temperature condition are obtained for the steady-state solidification thickness using the cylinder, sphere, and plane wall geometries which show that the thickness depends on the inverse square root of the internal heat generation. Under constant surface heat flux conditions, closed form equations can be obtained for the three geometries. In the case of the semiinfinite wall, we show that for constant temperature and constant heat flux out of the wall conditions, the solidification layer grows then remelts.
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