Generalized Solution for Three-Dimensional Transient Heat Conduction Problems with Partial Heating

2019 ◽  
Author(s):  
Robert L. McMasters ◽  
Filippo de Monte ◽  
James Beck
Keyword(s):  
Heat Conduction ◽  
Three Dimensional ◽  
Transient Heat Conduction ◽  
Generalized Solution ◽  
Partial Heating

Generalized Solution for Two-Dimensional Transient Heat Conduction Problems With Partial Heating Near a Corner

2019 ◽  
Vol 141 (7) ◽  
Author(s):  
Robert L. McMasters ◽  
Filippo de Monte ◽  
James V. Beck
Keyword(s):  
Boundary Condition ◽  
Heat Flux ◽  
Heat Conduction ◽  
Steady State ◽  
Separation Of Variables ◽  
Transient Heat Conduction ◽  
Generalized Solution ◽  
Convective Heating ◽  
Two Dimensional ◽  
Partial Heating

A generalized solution for a two-dimensional (2D) transient heat conduction problem with a partial-heating boundary condition in rectangular coordinates is developed. The solution accommodates three kinds of boundary conditions: prescribed temperature, prescribed heat flux and convective. Also, the possibility of combining prescribed heat flux and convective heating/cooling on the same boundary is addressed. The means of dealing with these conditions involves adjusting the convection coefficient. Large convective coefficients such as 1010 effectively produce a prescribed-temperature boundary condition and small ones such as 10−10 produce an insulated boundary condition. This paper also presents three different methods to develop the computationally difficult steady-state component of the solution, as separation of variables (SOV) can be less efficient at the heated surface and another method (non-SOV) is more efficient there. Then, the use of the complementary transient part of the solution at early times is presented as a unique way to compute the steady-state solution. The solution method builds upon previous work done in generating analytical solutions in 2D problems with partial heating. But the generalized solution proposed here contains the possibility of hundreds or even thousands of individual solutions. An indexed numbering system is used in order to highlight these individual solutions. Heating along a variable length on the nonhomogeneous boundary is featured as part of the geometry and examples of the solution output are included in the results.

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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