Orthotropic Thermal Conductivity Research Articles

Thermal Analysis and Optimization of Substrates With Directionally Enhanced Conductivities

Abstract An approximate analytical solution for the thermal resistance of the axisymmetric chip-on-substrate problem is presented for a substrate with a direction dependent (orthotropic) thermal conductivity. The substrate may be convectively cooled on either, or both, of its planar surfaces. The solution reveals substrate geometries with low maximum substrate temperatures. These optimal substrate sizes are mapped for Biot numbers typical of microelectronic applications. The effects of varying the radial and axial substrate conductivities are investigated. In general, radial conductivity enhancement is beneficial for bottom-side and both-side convective cooling of thin substrates, and for top-side cooling of all substrates. For thin substrates, radial conductivity enhancement provides comparable thermal performance to an equivalent isotropic conductivity enhancement. For electronic packaging applications thin substrates are desirable and radial conductivity enhancement is more beneficial than axial conductivity enhancement.

Estimation of directional-dependent thermal properties in a carbon-carbon composite

This paper presents a laboratory method to measure the thermal properties of a carbon-carbon composite material that is characterized by an orthotropic thermal conductivity and isotropic volumetric heat capacity. Thermal properties are estimated using parameter estimation techniques with measured surface heat flux and temperature histories for an experiment with two-dimensional heat flow. Thermal properties were determined for a temperature range of 65–400°C. The thermal conductivity parallel to the fibers is 12–15 times larger than the conductivity normal to the fiber. The results show excellent agreement with separate one-dimensional results, demonstrating a presumed quadratic relationship with temperature.

Three-dimensional modeling of woven-fabric composites for effective thermo-mechanical and thermal properties

This paper presents effective thermo-mechanical and thermal properties of plain-weave fabric-reinforced composite laminates obtained from micromechanical analyses and a two-scale asymptotic homogenization theory. A unit cell, enclosing the characteristic periodic repeat pattern in the fabric weave, is isolated and modeled. The orthotropic tensors for effective mechanical stiffness, coefficient of thermal expansion and thermal conductivity are obtained by numerically solving appropriate microscale boundary value problems (BVPs) in the unit cell by the use of three-dimensional finite element analyses. Further, analytical models consisting of series-parallel thermal resistance networks are developed in order, to obtain orthotropic thermal conductivity. The numerical and analytical models are explicitly based on the properties of the constituent materials and three-dimensional features of the weave style. Results obtained from the models are compared with experimental values and with models available in the literature. Non-linear mechanical constitutive behavior due to resin stress/strain non-linearity and to transverse yarn damage under in-plane uni-axial loads are also investigated. Parametric studies are conducted to examine the effect of varying fiber volume fractions on the effective thermal properties.

Orthotropic Thermal Conductivity of Plain-Weave Fabric Composites Using a Homogenization Technique

This article presents a new application of two-scale asymptotic homogeni zation schemes to predict the orthotropic thermal conductivity of plain-weave fabric rein forced composite laminates. A unit-cell, enclosing the characteristic periodic repeat pat tern in the fabric weave, is isolated and modeled. A new three-dimensional series-parallel thermal resistance network is developed to solve a steady-state heat transfer boundary value problem (BVP) for this unit-cell. Laminate effective orthotropic thermal conduc tivities are obtained analytically and numerically as functions of (1) thermal conductivity of the constituent materials, (2) fiber volume fraction, and (3) weave style. The analyti cally predicted thermal conductivity values are compared with numerical finite element predictions, with existing models in the literature and with experimentally obtained values.

Transient Heat Conduction from Buried Underground Radioactive Nuclear Waste

An exact analytical solution for transient heat conduction in the ground from buried radioactive nuclear waste has been derived. The heat sources are assumed to be an array of arbitrarily located point sources in an infinite ground. The solution accounts for orthotropic thermal conductivity effects in three dimensions and for the timevarying nature of the heat generation rate of each source due to radioactive decay. Owing to the point source nature of the assumed heat sources, the solution derived is valid only at intermediateand far-field distances from real finite-size sources. Numerical results are presented for several typical waste repository situations in shale, salt, and basalt.

Transient Electrical Heating of Toroidal Coils of Rectangular Cross Section

The exact solution is obtained for the transient thermal response of toroidal coils of rectangular cross section with orthotropic constant thermal conductivity, constant density, and specific heat, and linear boundary conditions. The solution is a rapidly converging doubly infinite eigenfunction expansion whose leading term is generally accurate to within 5 percent. No conclusive results were obtained from our evaluation of the applicability of previously obtained solutions for the infinite rod approximation for the toroidal coils. In some cases, the infinite rod approximation is accurate even for large curvatures, but in other cases it is in substantial error for relatively small curvatures. Hence, the solution presented here is generally recommended to predict the temperature distribution in toroidal coils and a number of design curves are provided for prediction purposes.

Joulean Heating of an Infinite Rectangular Rod With Orthotropic Thermal Properties

The formal solution previously obtained for the transient thermal response of an infinite rectangular rod, with orthotropic thermal conductivity and linear boundary conditions, to Joulean heating under a constant current condition is extended to include the constant voltage case under certain restrictive conditions. The solution is a rapidly converging doubly infinite series whose leading term is an accurate approximation for the thermal response. Graphical aids are presented which allow the designer of such apparatus as solenoids and various coils to quickly obtain estimates of the temperature distribution, the heat flow from the boundaries, the thermal time constant, and the location of the peak temperature. It is found that in certain cases the theory for constant current case predicts no steady-state condition. This condition is unrealistic because it requires an infinite voltage to provide the constant current. The theory predicts that the constant voltage case is always bounded.

Transient and Steady-State Temperature Distribution in Foil-Wound Solenoids and Other Electric Apparatus of Rectangular Cross Section

The formal solution is obtained for the transient thermal response of a rectangular region with orthotropic thermal conductivity, Joulean heating, and general boundary conditions (where the temperature in the surrounding medium is a function of both space and time). In the most practical case, where the surrounding medium is at a constant ambient temperature, the transient solution takes the form of a rapidly converging doubly infinite sum whose leading term is a very simple and accurate approximation for the thermal response. Graphical aids are presented which allow the designer to quickly obtain estimates of the temperature distribution, the maximum temperature, the average temperature, the heat flow from the boundaries, and the thermal time constant. A new and interesting result is that the solution indicates that no steady-state condition exists if the heat transfer coefficients on the boundaries are below certain critical values, or if the current density is above a critical value.