An Exact Solution of the Inverse Problem in Heat Conduction Theory and Applications

1964 ◽  
Vol 86 (3) ◽  
pp. 373-380 ◽  
Author(s):  
O. R. Burggraf
Keyword(s):  
Inverse Problem ◽  
Exact Solution ◽  
Heat Conduction ◽  
Boundary Conditions ◽  
Approximate Solutions ◽  
Convergent Series ◽  
Single Location ◽  
Leading Term ◽  
The One ◽  
Direct Problems

An inverse problem in unsteady heat conduction is one for which boundary conditions are prescribed internally, the surface conditions being unknown. By specifying the boundary conditions at a single location, an exact solution is obtained as a rapidly convergent series with the well-known, lumped capacitance approximation as the leading term. Specific forms of the series are determined for sample inverse problems: solid slab, cylinder, sphere, and transpiration-cooled slab. The solution also is applied to direct problems, involving two-point boundary conditions. By truncating the series, approximate solutions of simple form result. The one-term and two-term approximations compare well with exact solutions.

scholarly journals Simplified Grinding Temperature Model Study

2019 ◽  
Vol 6 (2) ◽  
pp. a1-a7
Author(s):  
N. V. Lishchenko ◽  
V. P. Larshin ◽  
H. Krachunov
Keyword(s):  
Differential Equation ◽  
Heat Conduction ◽  
Heat Source ◽  
Boundary Conditions ◽  
Grinding Temperature ◽  
Temperature Model ◽  
One Dimensional ◽  
Heating And Cooling ◽  
Equation Of Heat Conduction ◽  
The One

A study of a simplified mathematical model for determining the grinding temperature is performed. According to the obtained results, the equations of this model differ slightly from the corresponding more exact solution of the one-dimensional differential equation of heat conduction under the boundary conditions of the second kind. The model under study is represented by a system of two equations that describe the grinding temperature at the heating and cooling stages without the use of forced cooling. The scope of the studied model corresponds to the modern technological operations of grinding on CNC machines for conditions where the numerical value of the Peclet number is more than 4. This, in turn, corresponds to the Jaeger criterion for the so-called fast-moving heat source, for which the operation parameter of the workpiece velocity may be equivalently (in temperature) replaced by the action time of the heat source. This makes it possible to use a simpler solution of the one-dimensional differential equation of heat conduction at the boundary conditions of the second kind (one-dimensional analytical model) instead of a similar solution of the two-dimensional one with a slight deviation of the grinding temperature calculation result. It is established that the proposed simplified mathematical expression for determining the grinding temperature differs from the more accurate one-dimensional analytical solution by no more than 11 % and 15 % at the stages of heating and cooling, respectively. Comparison of the data on the grinding temperature change according to the conventional and developed equations has shown that these equations are close and have two points of coincidence: on the surface and at the depth of approximately threefold decrease in temperature. It is also established that the nature of the ratio between the scales of change of the Peclet number 0.09 and 9 and the grinding temperature depth 1 and 10 is of 100 to 10. Additionally, another unusual mechanism is revealed for both compared equations: a higher temperature at the surface is accompanied by a lower temperature at the depth. Keywords: grinding temperature, heating stage, cooling stage, dimensionless temperature, temperature model.

scholarly journals New Application for the Generalized Incomplete Gamma Function in the Heat Transfer of Nanofluids via Two Transformations

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Abdelhalim Ebaid ◽  
Hibah S. Alhawiti
Keyword(s):  
Heat Transfer ◽  
Exact Solution ◽  
Boundary Conditions ◽  
Gamma Function ◽  
Transfer Equation ◽  
Nonlinear Differential Equations ◽  
Incomplete Gamma Function ◽  
Heat Transfer Equation ◽  
Layer Flow ◽  
The One

The boundary layer flow of nanofluids is usually described by a system of nonlinear differential equations with infinity boundary conditions. These boundary conditions at infinity are transformed into classical boundary conditions via two different transformations. Accordingly, the original heat transfer equation is changed into a new one which is expressed in terms of the new variable. The exact solutions have been obtained in terms of the exponential function for the stream function and in terms of the incomplete Gamma function for the temperature distribution. Furthermore, it is found in this project that a certain transformation reduces the computational work required to obtain the exact solution of the heat transfer equation. Hence, such transformation is recommended for future analysis of similar physical problems. Besides, the other published exact solution was expressed in terms of the WhittakerM function which is more complicated than the generalized incomplete Gamma function of the current analysis. It is important to refer to the fact that the analytical procedure followed in our project is easier and more direct than the one considered in a previous published work.

scholarly journals A perturbed algorithm for strongly nonlinear variational-like inclusions

2000 ◽  
Vol 62 (3) ◽  
pp. 417-426 ◽  
Author(s):  
C.-H. Lee ◽  
Q. H. Ansari ◽  
J.-C. Yao
Keyword(s):  
Exact Solution ◽  
General Setting ◽  
Approximate Solutions ◽  
Strongly Nonlinear ◽  
Special Cases ◽  
The One

In this paper, we define the concept of η- subdifferential in a more general setting than the one used by Yang and Craven in 1991. By using η-subdifferentiability, we suggest a perturbed algorithm for finding the approximate solutions of strongly nonlinear variational-like inclusions and prove that these approximate solutions converge to the exact solution. Several special cases are also discussed.

FICTON THEORY OF DYNAMICAL SYSTEMS WITH NOISY PARAMETERS

1965 ◽  
Vol 43 (4) ◽  
pp. 619-639 ◽  
Author(s):  
R. C. Bourret
Keyword(s):  
Magnetic Field ◽  
Exact Solution ◽  
Dynamical Systems ◽  
Approximate Solutions ◽  
Frequency Parameter ◽  
Time Dependent ◽  
Quantum Mechanical ◽  
Graphical Comparison ◽  
The One ◽  
Random Fluctuations

The theory of randomly perturbed waves described previously (Bourret 1962a, b) is presented in a form applicable to purely time-dependent systems, classical or quantum mechanical. It is then applied to the problem of a spin-[Formula: see text] dipole in a magnetic field with random fluctuations. One- and two-ficton processes are taken into account and a "renormalization" approximation is given also. Graphical comparison of the approximate solutions with the exact solution is presented. As a classical example, the harmonic oscillator with a noisy frequency parameter is analyzed in both the one- and two-ficton approximations.

Approximate solutions in the theory of thermal explosions for semi-infinite explosives

1968 ◽  
Vol 305 (1481) ◽  
pp. 205-217 ◽  
Keyword(s):  
Heat Conduction ◽  
Boundary Conditions ◽  
Heat Conduction Equation ◽  
Numerical Solutions ◽  
Surface Flux ◽  
Approximate Solutions ◽  
Conduction Equation ◽  
Inert Material ◽  
Zero Order Reaction ◽  
Thermal Explosions

If the solution, of the heat conduction equation θ τ ( 0 ) = θ ξ ξ ( 0 ) , ξ > 0 , τ > 0 of a chemically ‘inert’ material is known, then an approximate formula for the explosion time, ד expl. , of an explosive satisfying the heat conduction equation with zero order reaction, θ ד = θ ξξ +exp(-1/θ), ξ > 0, ד 0, and the same initial and boundary conditions as the ‘inert’, is given by the root of the equation, − ∂ θ ( 0 ) ( ξ , τ expt . ) / ∂ ξ | ξ − 0 = ∫ 0 ∞ exp ⁡ [ − 1 / θ ( 0 ) ( ξ , τ expl . ) ] d ξ provided 1/θ (0) (ξ, ד) is suitably expanded about the surface ξ = 0 such that the integrand vanishes as ξ→∞. Similar results hold for one-dimensional cylindrically and spherically symmetric problems. The derivation of the explosion criterion is based on observation of existing numerical solutions where it is seen that (i) almost to the onset of explosion, the solution θ(ξ, ד )does not differ appreciably from θ (0) (ξ, ד ) (ii) the onset of explosion is indicated by the appearance of a temperature maximum at the surface. Simple formulas for ד expl. readily obtainable for a wide variety of boundary conditions, are given for seven sample problems. Among these are included a semi-infinite explosive with constant surface flux, convective surface heat transfer, and constant surface temperature with and without subsurface melting. The derived values of ד expl. are in satisfactory agreement with those obtained from finite-difference solutions for the problems that can be compared.

scholarly journals Influence of Errors in Known Constants and Boundary Conditions on Solutions of Inverse Heat Conduction Problem

Energies ◽  
2021 ◽  
Vol 14 (11) ◽  
pp. 3313
Author(s):  
Sun Kyoung Kim
Keyword(s):  
Exact Solution ◽  
Heat Conduction ◽  
Boundary Conditions ◽  
Material Properties ◽  
Heat Conduction Problem ◽  
Inverse Heat Conduction Problem ◽  
Inverse Heat Conduction ◽  
One Dimensional ◽  
Boundary Location ◽  
Inverse Heat Conduction Problems

This work examines the effects of the known boundary conditions on the accuracy of the solution in one-dimensional inverse heat conduction problems. The failures in many applications of these problems are attributed to inaccuracy of the specified constants and boundary conditions. Since the boundary conditions and material properties in most thermal problems are imposed with uncertainty, the effects of their inaccuracy should be understood prior to the inverse analyses. The deviation from the exact solution has been examined for each case according to the errors in material properties, boundary location, and known boundary conditions. The results show that the effects of such errors are dramatic. Based on these results, the applicability and limitations of the inverse heat conduction analyses have been evaluated and discussed.

Difference Schemes with Nonlocal Boundary Conditions

2001 ◽  
Vol 1 (1) ◽  
pp. 62-71 ◽  
Author(s):  
Alexei V. Goolin ◽  
Nikolai I. Ionkin ◽  
Valentina A. Morozova
Keyword(s):  
Heat Conduction ◽  
Boundary Conditions ◽  
Heat Conduction Equation ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Difference Schemes ◽  
Conduction Equation ◽  
The Stability ◽  
The One ◽  
Necessary And Sufficient

AbstractThe paper deals with the stability, with respect to initial data, of difference schemes that approximate the heat-conduction equation with constant coefficients and nonlocal boundary conditions. Some difference schemes are considered for the one-dimensional heat-conduction equation, the energy norm is constructed, and the necessary and sufficient stability conditions in this norm are established for explicit and weighted difference schemes.

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