Solutions of Heat-Conduction Problems With Nonseparable Domains

1963 ◽  
Vol 30 (4) ◽  
pp. 493-499 ◽  
Author(s):  
Daniel Dicker ◽  
M. B. Friedman
Keyword(s):  
Heat Conduction ◽  
Separation Of Variables ◽  
Three Dimensional ◽  
Transient Heat Conduction ◽  
Steady State Solution ◽  
Zero Temperature ◽  
Rapid Convergence ◽  
Curved Boundaries ◽  
Convex Quadrilateral ◽  
The One

A method is presented for obtaining eigenfunctions of and solutions to the transient heat-conduction equation for a wide class of three-dimensional convex hexahedral domains and two-dimensional convex quadrilateral domains having straight or curved boundaries for which separation of variables cannot be applied. The method is employed to solve for the temperature distribution in a trapezoidal domain, initially at zero temperature, the boundaries of which are subjected to suddenly applied values at the initial instant. The solution is obtained in the form of a series and an examination of successive terms indicates fairly rapid convergence; it is found that the one-term solution yields almost as good values as a four-term solution, which is significant since the former is obtained with little effort. An independent method is utilized for obtaining the steady-state solution, i.e., t → ∞, and it is found that all approximations by the former method are substantially equal to the correct value for this case.

One-Dimensional Transient Heat Conduction in Composite Living Perfuse Tissue

2013 ◽  
Vol 135 (7) ◽  
Author(s):  
S. M. Becker
Keyword(s):  
Heat Conduction ◽  
Initial Conditions ◽  
Separation Of Variables ◽  
Composite Layer ◽  
Circulatory System ◽  
Transient Heat Conduction ◽  
Bioheat Equation ◽  
Perfusion Rate ◽  
One Dimensional ◽  
Perfuse Tissue

Modeling the conduction of heat in living tissue requires the consideration of sudden spatial discontinuities in property values as well as the presence of the body's circulatory system. This paper presents a description of the separation of variables method that results in a remarkably simple solution of transient heat conduction in a perfuse composite slab for which at least one of the layers experiences a zero perfusion rate. The method uses the natural analytic approach and formats the description so that the constants of integration of each composite layer are expressed in terms of those of the previous layer's eigenfunctions. This allows the solution to be “built” in a very systematic and sequential manner. The method is presented in the context of the Pennes bioheat equation for which the solution is developed for a system composed of any number of N layers with arbitrary initial conditions.

Simple Explicit Equations for Transient Heat Conduction in Finite Solids

2008 ◽  
Vol 131 (1) ◽  
Author(s):  
A. G. Ostrogorsky
Keyword(s):  
Fourier Series ◽  
Heat Conduction ◽  
Biot Number ◽  
Series Solution ◽  
Transient Heat Conduction ◽  
Simple Equation ◽  
Lower Error ◽  
Transient Conduction ◽  
The One ◽  
Cylinders And Spheres

Abstract Based on the one-term Fourier series solution, a simple equation is derived for low Biot number transient conduction in plates, cylinders, and spheres. In the 0<Bi<0.3 range, the solution gives approximately three times less error than the lumped capacity solution. For asymptotically low values of Bi, it approaches the lumped capacity solution. A set of equations valid for 0<Bi<1 is developed next. These equations are more involved but give approximately ten times lower error than the lumped capacity solution. Finally, a set of broad-range correlations is presented, covering the 0<Bi<∞ range with less than 1% error.

A New Method for Thermal Analysis of Die Casting

1993 ◽  
Vol 115 (2) ◽  
pp. 284-293 ◽  
Author(s):  
M. R. Barone ◽  
D. A. Caulk
Keyword(s):  
Heat Conduction ◽  
Heat Conduction Problem ◽  
Die Casting ◽  
Three Dimensional ◽  
Transient Heat Conduction ◽  
Steady Solution ◽  
Cavity Surface ◽  
Varying Coefficients ◽  
Transient Heat Conduction Problem ◽  
Time Varying Coefficients

A new approach is developed for solving the initial value, steady periodic heat conduction problem in steady-state die casting. Three characteristics found in nearly all die casting processes are exploited directly: The casting is thin compared with its overall size, its thermal conductivity is high compared with that of the mold, and the cycle time is short compared with the start-up transient of the process. Under these conditions, it is reasonable to neglect the transverse temperature gradients in the casting and assume that all die temperatures below a certain depth from the cavity surface are independent of time. The transient die temperatures near the cavity surface are represented by a polynomial expansion in the depth coordinate, with time-varying coefficients determined by a Galerkin method. This leads to a set of ordinary differential equations on the cavity surface, which govern the transient interaction between the casting and the die. From the time-averaged solution of these equations, special conditions are derived that relate the transient solution near the cavity surface to the three-dimensional steady solution in the die interior. With these conditions, the steady temperatures in the bulk of the die can be determined independently of the explicit surface transients. This reduces the effort of solving a complex transient heat conduction problem to little more than finding a steady solution alone. The overall approach provides a general analytical tool, which is capable of predicting complex thermal interactions in large multicomponent dies.

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