scholarly journals An Exact Solution to a Problem in Heat Transfer

1961 ◽  
Vol 14 (2) ◽  
pp. 317 ◽  
Author(s):  
CHJ Johnson
Keyword(s):  
Heat Transfer ◽  
Exact Solution ◽  
Heat Conduction ◽  
Heat Transport ◽  
Solid Body ◽  
Analytical Solutions ◽  
Heat Conduction Equation ◽  
Transport Equations ◽  
Temperature History ◽  
Moving Fluid

In order to determine the complete temperature history in problems concerning the transfer of heat from a moving fluid to a solid body one is obliged, in general, to solve both the flow and heat transport equations for the fluid and the heat conduction equation for the solid. Analytical solutions to this problem cannot in general be given owing to the (non-linear) way in which the flow and heat transport equations for the fluid are coupled.

scholarly journals Analytic transient solutions of a cylindrical heat equation

Filomat ◽  
2021 ◽  
Vol 35 (8) ◽  
pp. 2617-2628
Author(s):  
K.Y. Kung ◽  
Man-Feng Gong ◽  
H.M. Srivastava ◽  
Shy-Der Lin
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Partial Differential Equation ◽  
Heat Conduction ◽  
Heat Equation ◽  
Heat Conduction Equation ◽  
Separation Of Variables ◽  
Transient Heat Conduction ◽  
Transient Temperature ◽  
Transient Solutions

The principles of superposition and separation of variables are used here in order to investigate the analytical solutions of a certain transient heat conduction equation. The structure of the transient temperature appropriations and the heat-transfer distributions are summed up for a straight mix of the results by means of the Fourier-Bessel arrangement of the exponential type for the investigated partial differential equation.

scholarly journals Heat and mass transfer in the process of heat treatment and drying of natural leather with the convective method of energy supply

2021 ◽  
Vol 66 (1) ◽  
pp. 84-92
Author(s):  
A. O. Ol’shanskii ◽  
A. M. Gusarov ◽  
S. V. Zhernosek
Keyword(s):  
Heat Transfer ◽  
Heat Treatment ◽  
Mass Transfer ◽  
Heat Conduction ◽  
Differential Equations ◽  
Heat And Mass Transfer ◽  
Analytical Solutions ◽  
Transfer Coefficients ◽  
Heat Transfer Coefficients ◽  
Unsteady Heat Conduction

In the work, the authors investigated the possibility of using the results of analytical solutions of the linear differential equations of unsteady heat conduction with constant heat transfer coefficients to calculate the temperature of the material during heat treatment of leathers. Heat treatment of natural leathers as heat-sensitive materials is carried out under mild temperature conditions and high air moisture contents, the temperature does not undergo significant changes, and the heat transfer coefficients change almost linearly. When using analytical solutions, the authors made the assumptions that for small temperature gradients over the cross section of a thin body, the thermal transfer of matter can be neglected and for values of the heat and mass transfer Biot criteria less than unity, the main factor, limiting heat and mass transfer, is the interaction of the evaporation surface of the body with the environment; so, in solving the differential heat equation we can restrict ourselves to one first member of an infinite series. In this case, a piecewise stepwise approximation of all thermophysical characteristics with constant values of these coefficients at the calculated time intervals was applied, which made it possible to take into account the change in the transfer coefficients throughout the entire heat treatment process. Processing of experimental data showed that in low-intensity processes with reliable values of the transfer coefficients, it is possible to use the results of solutions of differential equations of unsteady heat conduction in heat transfer calculations. The results of the study of heat transfer during drying of leather confirm the laws of temperature change established experimentally. Together with experimental studies of drying processes, analytical studies are of great practical importance in the development of new methods for calculating heat and mass transfer in wet bodies.

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.

scholarly journals Determining An Unknown Boundary Condition by An Iteration Method

Symmetry ◽  
2018 ◽  
Vol 10 (9) ◽  
pp. 409
Author(s):  
Dejian Huang ◽  
Yanqing Li ◽  
Donghe Pei
Keyword(s):  
Boundary Condition ◽  
Exact Solution ◽  
Heat Conduction ◽  
Iteration Method ◽  
Heat Conduction Equation ◽  
Heat Conduction Problem ◽  
Variational Iteration Method ◽  
Conduction Equation ◽  
Unknown Boundary ◽  
Initial Condition

This paper investigates the boundary value in the heat conduction problem by a variational iteration method. Applying the iteration method, a sequence of convergent functions is constructed, the limit approximates the exact solution of the heat conduction equation in a few iterations using only the initial condition. This method does not require discretization of the variables. Numerical results show that this method is quite simple and straightforward for models that are currently under research.

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.

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