Hyperbolic Heat Conduction Equation for Materials With a Nonhomogeneous Inner Structure

1990 ◽  
Vol 112 (3) ◽  
pp. 555-560 ◽  
Author(s):  
W. Kaminski
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Heat Conduction ◽  
Physical Meaning ◽  
Heat Conduction Equation ◽  
Hyperbolic Heat Conduction ◽  
Temperature Profiles ◽  
Using Data ◽  
Penetration Time

The physical meaning of the constant τ in Cattaneo and Vernotte’s equation for materials with a nonhomogeneous inner structure has been considered. An experimental determination of the constant τ has been proposed and some values for selected products have been given. The range of differences in the description of heat transfer by parabolic and hyperbolic heat conduction equations has been discussed. Penetration time, heat flux, and temperature profiles have been taken into account using data from the literature and our experimental and calculated results.

Experimental Demonstration of Inverse Heat Transfer Methodologies for Turbine Applications

2019 ◽  
Author(s):  
David G. Cuadrado ◽  
Francisco Lozano ◽  
Guillermo Paniagua
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Heat Conduction ◽  
Digital Filter ◽  
Heat Conduction Equation ◽  
Metal Temperature ◽  
Temperature Dependent ◽  
Sensitivity Coefficients ◽  
Inverse Heat Transfer ◽  
Transfer Method

Abstract Gas turbines operate at extreme temperatures and pressures, constraining the use of both optical measurement techniques as well as probes. A strategy to overcome this challenge consists of instrumenting the external part of the engine, with sensors located in a gentler environment, and use numerical inverse methodologies to retrieve the relevant quantities in the flowpath. An inverse heat transfer approach is a procedure used to retrieve the temperature, pressure or mass flow through the engine based on the external casing temperature data. This manuscript proposes an improved Digital Filter Inverse Heat Transfer Method, that consists of a linearization of the heat conduction equation using sensitivity coefficients. The sensitivity coefficient characterizes the change of temperature due to a change in the heat flux. The heat conduction equation contains a non-linearity due to the temperature-dependent thermal properties of the materials. In previous literature, this problem is solved via iterative procedures that however increase the computational effort. The novelty of the proposed strategy consists of the inclusion of a non-iterative procedure to solve the non-linearity features. This procedure consists of the computation of the sensitivity coefficients in function of temperature, together with an interpolation where the measured temperature is used to retrieve the sensitivity coefficients in each timestep. These temperature-dependent sensitivity coefficients, are then used to compute the heat flux by solving the linear system of equations of the Digital Filter Method. This methodology was validated in the Purdue Experimental Turbine Aerothermal Lab (PETAL) annular wind tunnel, a two minutes transient experiment with flow temperatures up to 450K. Infrared thermography is used to measure the temperature in the outer surface of the inlet casing of a high pressure turbine. Surface thermocouples measure the endwall metal temperature. The metal temperature maps from the IR thermography were used to retrieve the heat flux with the inverse method. The inverse heat transfer method results were validated against a direct computation of the heat flux obtained from temperature readings of surface thermocouples. The experimental validation was complemented with an uncertainty analysis of the inverse methodology: the Karhunen-Loeve Expansion. This technique allows the propagation of uncertainty through stochastic systems of differential equations. In this case, the uncertainty of the inner casing heat flux has been evaluated through the simulation of different samples of the uncertain temperature field of the outer casing.

Numerical simulation of non-Fourier effects in combined heat transfer

2010 ◽  
Vol 225 (2) ◽  
pp. 429-436
Author(s):  
E Izadpanah ◽  
S Talebi ◽  
M H Hekmat
Keyword(s):  
Heat Transfer ◽  
Heat Conduction ◽  
Heat Conduction Equation ◽  
Splitting Method ◽  
Thermal Relaxation ◽  
Conduction Equation ◽  
Hyperbolic Heat Conduction ◽  
Wave Nature ◽  
Fourier Heat Conduction ◽  
Combined Heat Transfer

The non-Fourier effects on transient and steady temperature distribution in combined heat transfer are studied. The processes of coupled conduction and radiation heat transfer in grey, absorbing, emitting, scattering, one-dimensional medium with black boundary surfaces are analysed numerically. The hyperbolic heat conduction equation is solved by flux splitting method, and the radiative transfer equation is solved by P1 approximate method. The transient thermal responses obtained from non-Fourier heat conduction equation are compared with those obtained from the Fourier heat conduction equation. The results show that the non-Fourier effect can be important when the conduction to radiation parameter and the thermal relaxation time are larger. Further, the radiation effect is more pronounced at small values of single scattering albedo and conduction to radiation parameters. Analysis results indicate that the internal radiation in the medium significantly influences the wave nature.

scholarly journals Analysis of temperature distribution in a pipe with inner mineral deposit

2014 ◽  
Vol 35 (2) ◽  
pp. 37-49
Author(s):  
Magda Joachimiak ◽  
Michał Ciałkowski ◽  
Jarosław Bartoszewicz
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Heat Transfer Coefficient ◽  
Heat Conduction ◽  
Temperature Distribution ◽  
Mineral Deposit ◽  
Mineral Deposits ◽  
Heat Conduction Coefficient ◽  
Temperature Distributions

Abstract The paper presents the results of calculations related to determination of temperature distributions in a steel pipe of a heat exchanger taking into account inner mineral deposits. Calculations have been carried out for silicate-based scale being characterized by a low heat transfer coefficient. Deposits of the lowest values of heat conduction coefficient are particularly impactful on the strength of thermally loaded elements. In the analysis the location of the thermocouple and the imperfection of its installation were taken into account. The paper presents the influence of determination accuracy of the heat flux on the pipe external wall on temperature distribution. The influence of the heat flux disturbance value on the thickness of deposit has also been analyzed.

Growth and Decay of a Thermal Pulse Predicted by the Hyperbolic Heat Conduction Equation

1983 ◽  
Vol 105 (4) ◽  
pp. 902-907 ◽  
Author(s):  
B. Vick ◽  
M. N. O¨zisik
Keyword(s):  
Heat Flux ◽  
Heat Conduction ◽  
Wave Front ◽  
Heat Conduction Equation ◽  
Heat Propagation ◽  
Heat Diffusion ◽  
Conduction Equation ◽  
Hyperbolic Heat Conduction ◽  
Thermal Front ◽  
Wave Nature

The wave nature of heat propagation in a semi-infinite medium containing volumetric energy sources is investigated by solving the hyperbolic heat conduction equation. Analytic expressions are developed for the temperature and heat flux distributions. The solutions reveal that the spontaneous release of a finite pulse of energy gives rise to a thermal wave front which travels through the medium at a finite velocity, decaying exponentially while dissipating its energy along its path. When a concentrated pulse of energy is released, the temperature and heat flux in the wave front become severe. For situations involving very short times or very low temperatures, the classical heat diffusion theory significantly underestimates the magnitude of the temperature and heat flux in this thermal front, since the classical theory leads to instantaneous heat diffusion at an infinite propagation velocity.

Inverse Analysis of In-Cylinder Gas-Wall Boundary Conditions: Investigation of a Yittria Stabilized Zirconia Thermal Barrier Coating for Homogeneous Charge Compression Ignition

2016 ◽  
Author(s):  
Ryan O’Donnell ◽  
Tommy Powell ◽  
Mark Hoffman ◽  
Zoran Filipi
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Heat Conduction ◽  
Surface Temperature ◽  
Inverse Analysis ◽  
Fast Response ◽  
Thermal Barrier ◽  
Temperature Profiles ◽  
Inverse Heat Conduction ◽  
Cycle Efficiency

Thermal Barrier Coatings (TBC) applied to in-cylinder surfaces of a Low Temperature Combustion (LTC) engine provide opportunities for enhanced cycle efficiency via two mechanisms: (i) positive impact on thermodynamic cycle efficiency due to combustion/expansion heat loss reduction, and (ii) enhanced combustion efficiency. Heat released during combustion elevates TBC surface temperatures, directly impacting gas-wall heat transfer. Determining the magnitude and phasing of the associated TBC surface temperature swing is critical for correlating coating properties with the measured impact on combustion and efficiency. Although fast-response thermocouples provide a direct measurement of combustion chamber surface temperature in a metal engine, the temperature and heat flux profiles at the TBC-treated gas-wall boundary are difficult to measure directly. Thus, a technique is needed to process the signal measured at the sub-TBC sensor location and infer the corresponding TBC surface temperature profile. This task can be described as an Inverse Heat Conduction Problem (IHCP), and it cannot be solved using the conventional analytic/numeric techniques developed for ‘direct’ heat flux measurements. This paper proposes using an Inverse Heat Conduction solver based on the Sequential Function Specification Method (SFSM) to estimate heat flux and temperature profiles at the wall-gas boundary from measured sub-TBC temperature. The inverse solver is validated ex situ under HCCI like thermal conditions in a custom fabricated radiation chamber where fast-response thermocouples are exposed to a known heat pulse in a controlled environment. The analysis is extended in situ, to evaluate surface conditions in a single-cylinder, gasoline-fueled, HCCI engine. The resulting SFSM-based inverse analysis provides crank angle resolved TBC surface temperature profiles over a host of operational conditions. Such metrics may be correlated with TBC thermophysical properties to determine the impact(s) of material selection on engine performance, emissions, heat transfer, and efficiencies. These efforts will also guide next-generation TBC design.

Experimental Demonstration of Inverse Heat Transfer Methodologies for Turbine Applications

2020 ◽  
Vol 142 (6) ◽  
Author(s):  
David Gonzalez Cuadrado ◽  
Francisco Lozano ◽  
Guillermo Paniagua
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Heat Conduction ◽  
Digital Filter ◽  
Heat Conduction Equation ◽  
Metal Temperature ◽  
Temperature Dependent ◽  
Sensitivity Coefficients ◽  
Inverse Heat Transfer ◽  
Transfer Method

Abstract Gas turbines operate at extreme temperatures and pressures, constraining the use of both optical measurement techniques as well as probes. A strategy to overcome this challenge consists of instrumenting the external part of the engine, with sensors located in a gentler environment, and use numerical inverse methodologies to retrieve the relevant quantities in the flowpath. An inverse heat transfer approach is a procedure that is used to retrieve the temperature, pressure, or mass flow through the engine based on the external casing temperature data. This manuscript proposes an improved digital filter inverse heat transfer method, which consists of a linearization of the heat conduction equation using sensitivity coefficients. The sensitivity coefficient characterizes the change of temperature due to a change in the heat flux. The heat conduction equation contains a non-linearity due to the temperature-dependent thermal properties of the materials. In previous literature, this problem is solved via iterative procedures that however increase the computational effort. The novelty of the proposed strategy consists of the inclusion of a non-iterative procedure to solve the non-linearity features. This procedure consists of the computation of the sensitivity coefficients in the function of temperature, together with an interpolation where the measured temperature is used to retrieve the sensitivity coefficients in each timestep. These temperature-dependent sensitivity coefficients are then used to compute the heat flux by solving the linear system of equations of the digital filter method. This methodology was validated in the Purdue Experimental Turbine Aerothermal Laboratory (PETAL) annular wind tunnel, a two-minute transient experiment with flow temperatures up to 450 K. Infrared thermography is used to measure the temperature in the outer surface of the inlet casing of a high-pressure turbine. Surface thermocouples measure the endwall metal temperature. The metal temperature maps from the IR thermography were used to retrieve the heat flux with the inverse method. The inverse heat transfer method results were validated against a direct computation of the heat flux obtained from temperature readings of surface thermocouples. The experimental validation was complemented with an uncertainty analysis of the inverse methodology: the Karhunen–Loeve expansion. This technique allows the propagation of uncertainty through stochastic systems of differential equations. In this case, the uncertainty of the inner casing heat flux has been evaluated through the simulation of different samples of the uncertain temperature field of the outer casing.

Thermal Modeling and Analysis of a Thermal Barrier Coating Structure Using Non-Fourier Heat Conduction

2012 ◽  
Vol 134 (11) ◽  
Author(s):  
Stephen Akwaboa ◽  
Patrick Mensah ◽  
Ebubekir Beyazouglu ◽  
Ravinder Diwan
Keyword(s):  
Heat Flux ◽  
Heat Conduction ◽  
Temperature Distribution ◽  
Thermal Barrier Coating ◽  
Thermal Wave ◽  
Heat Conduction Equation ◽  
Thermal Barrier ◽  
Hyperbolic Heat Conduction ◽  
Barrier Coating ◽  
Fourier Heat Conduction

This paper presents a numerical solution of the hyperbolic heat conduction equation in a thermal barrier coating (TBC) structure under an imposed heat flux on the exterior of the TBC. The non-Fourier heat conduction equation is used to model the heat conduction in the TBC system that predicts the heat flux and the temperature distribution. This study presents a more realistic approach to evaluate in-service performance of thin layers of TBCs typically found in hot sections of land based and aircraft gas turbine engines. In such ultrafast heat conduction systems, the orders of magnitude of the time and space dimensions are extremely short which renders the traditional Fourier conduction law, with its implicit assumption of infinite speed of thermal propagation, inaccurate. There is, therefore, the need for an advanced modeling approach for the thermal transport phenomenon taking place in microscale systems. A hyperbolic heat conduction model can be used to predict accurately the transient temperature distribution of thermal barrier structures of turbine blades. The hyperbolic heat conduction equations are solved numerically using a new numerical scheme codenamed the mean value finite volume method (MVFVM). The numerical method yields minimal numerical dissipation and dispersion errors and captures the discontinuities such as the thermal wave front in the solution with reliable accuracy. Compared with some traditional numerical methods, the MVFVM method provides the ability to model the behavior of the single phase lag thermal wave following its reflection from domain boundary surfaces. In addition, parametric studies of properties of the substrate on the temperature and the heat flux distributions in the TBC revealed that relaxation time of the substrate material, unlike the thermal diffusivity and thermal conductivity has very little effect on the transient thermal response in the TBC. The study further showed that for thin film structures subject to short time durations of heat flux, the hyperbolic model yields more realistic results than the parabolic model.

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