Heat transfer in solids provides an opportunity for students to learn of several boundary conditions: the first kind for specified temperature, the second kind for specified heat flux, and the third kind for specified convection. In this paper we explore the relationship among these types of boundary conditions in steady heat transfer. Specifically, the normalized third kind of boundary condition (convection) produces the first kind condition (specified temperature) for large Biot number, and it produces the second kind condition (specified flux) for small Biot number. By employing a generalized boundary condition, one expression provides the temperature for several combinations of boundary conditions. This combined expression is presented for several simple geometries (slabs, cylinders, spheres) with and without internal heat generation. The bioheat equation is also treated. Further, a number system is discussed for each combination to identify the type of boundary conditions present, which side is heated, and whether internal generation is present. Computer code for obtaining numerical values from the several expressions is available, along with plots and tables of numerical values, at a web site called the Exact Analytical Conduction Toolbox. Classroom strategies are discussed regarding student learning of these issues: the relationship among boundary conditions; a number system to identify the several components of a boundary value problem; and, the utility of a web-based resource for analytical heat-transfer solutions.

Heat conduction in skin tissue is a problem of significant technological importance. A theoretical understanding of such a problem is essential as it may lead to design potential therapeutic measures for needed cancer therapy or novel medical devices for various applications including hyperthermia. To understand the physical phenomenon of energy transport in biological systems a transient model is chosen for this study. The most common transport equation to estimate temperature distribution in humans was developed by H.H. Pennes and is popularly known as the Pennes bioheat transfer equation. A generalized Pennes bioheat transfer equation accounts for the effect of various physical phenomena such as conduction, advection, volumetric heat generation, etc. are considered. In this paper, a general transient form of the Pennes bioheat transfer equation is solved analytically for a multilayer domain. The boundary value problem considers the core of the tissue is maintained at uniform temperature of 37°C, convective cooling is applied to the external surface of the skin and the sidewalls are adiabatic. The computation of transient temperature in multidimensional and multilayer bodies offers unique features. Due to the presence of blood perfusion in the tissue, the reaction term in the Pennes governing equation is modeled similar to a fin term. The eigenvalues may become imaginary, producing eigenfunctions with imaginary arguments. In addition the spacing between the eigenvalues between zero and maximum value varies for different cases; therefore the values need to be determined with precision using second order Newton’s method. A detailed derivation of the temperature solution using the technique of separation of variables is presented in this study. In addition a proof of orthogonality theorem for eigenfunctions with imaginary eigenvalues is also presented. The analytical model is used to study the thermal response of skin tissue to different parameters with the aid of some numerical examples. Results shown in this paper are expected to facilitate a better understand of bioheat transfer in layered tissue such as skin.

Analytical study of bioheat transfer is of significant importance for a number of biomedical applications including cryopreservation of tissue and thermal therapy for cancer. A sound fundamental understanding of thermal behavior of tissue in response to an externally applied stimulus helps design effective therapies and protocols. This paper derives an analytical solution in a multi-layer two-dimensional structure with arbitrary, space-dependent heat generation occurring in each layer. This geometry effectively models multiple layers of skin, with heat generation due to cancerous cells in the basal layer. The Pennes bioheat transfer equation is solved for the multi-layer analytically, wherein the temperature in each layer is explicitly a function of space and the thermo-physical properties of the layer. The resulting analytical temperature profile agrees well with finite-element simulations and is also in good agreement with a previously published experimental study. Results derived in this work illustrate the effect of the presence of cancerous cells on the thermal profile of the skin. Further, the model helps to understand the effect of external cooling and heating stimuli.

Microprocessors continue to grow in capabilities, complexity and performance. Microprocessors typically integrate functional components such as logic and level two (L2) cache memory in their architecture. This functional integration of logic and memory results in improved performance of the microprocessor. However, the integration also introduces a layer of complexity in the thermal design and management of microprocessors. As a direct result of functional integration, the power map on a microprocessor is typically highly non-uniform and the assumption of a uniform heat flux across the chip surface has been shown to be invalid post Pentium II architecture. The active side of the die is divided into several functional blocks with distinct power assigned to each functional block. Previous work has been done which includes numerical analysis and thermal Based optimization of a typical package consisting of a non-uniformly powered die, heat spreader, TIM I &II and the base of the heat sink. In this paper, an analytical approach to temperature distribution of a first level package with a non-uniformly powered die is carried out for the first time. The analytical model for two layer bodies developed by Haji-Sheikh et al. is extended to this typical package which is a multilayer body. The solution is to begin by designating each surface heat flux as a volumetric heat source. An inverse methodology will be applied to solve the equations for various surfaces to calculate maximum junction temperature for given multilayer body. Finally validation of the analytical solution will be carried out using developed numerical model.

Microprocessors continue to grow in capabilities, complexity and performance. The current generation of microprocessors integrates functional components such as logic and level two (L2) cache memory into the microprocessor architecture. The functional integration of the microprocessor has resulted in better performance of the microprocessor as the clock speed has increased and the instruction execution time has decreased. However, the integration has introduced a layer of complexity to the thermal design and management of microprocessors. As a direct result of function integration, the power map on a microprocessor is highly non-uniform and the assumption of a uniform heat flux across the chip surface is not valid. The objective of this paper is to minimize the thermal resistance of the package by optimizing the distribution of the uniformly powered functional blocks. In order to model the non-uniform power dissipation on the silicon chip, the chip surface area is divided into a 4 × 4 and 6×6 matrix with a matrix space representing a distinct functional block with a constant heat flux. Finally, using a FEM code, an optimization of the positioning of the functional blocks relative to each other was carried out in order to minimize the junction temperature Tj. This analysis has no constraints placed on the redistribution of functional blocks. The best possible Tjmax reduction could thus be found. In reality (and at a later date) constraints must be placed regarding the maximum separation of any 2 (or more) functional blocks to satisfy electrical timing and compute performance requirements. Design guidelines are then suggested regarding the thermal based optimal distribution for any number of functional blocks. The commercial finite element code ANSYS® is used for this analysis.

A systematic experimental method of estimating the extent of the phase front under local thermal non-equilibrium condition in porous media saturated with phase change material has been developed. During phase change in porous medium, the solid matrix and the pore material are under thermodynamic non-equilibrium condition until the phase change process is complete. It is often hypothesized that the solid matrix and the pore material are in local thermal equilibrium (LTE) condition hence, the arrival of the phase front is predicted based on this hypothesis. An understanding of the rate of freezing and thawing in a porous medium undergoing the phase change process is important to permit proper implementation of procedures such as cryopreservation, cryosurgery, and to predict the thermal performance of passive cooling systems for electronic devices. In this paper, a systematic method of estimating the extent of the phase change front is developed. Results show that during the phase change process the porous medium is far from local thermal equilibrium condition.

The sensitivity coefficients for analyzing the interstitial properties during phase change in porous media are presented. Computation of the sensitivity coefficients is the main objective of this study. Experimentally measured temperature data provide an estimate of the phase front locations used as the state variable for this study. The derivations are based on the assumption that the phase front, X, at a given time, t, is a function of interstitial properties τ t and τ q with all other parameters remaining constant. The properties τ t and τ q are the lag-time in temperature and heat flux, respectively. The analysis includes two types of boundary conditions: prescribed temperature of phase change materials and prescribed temperature for solid matrix. Results for the first case show that for any given ratio τ t /τ q , the sensitivity coefficient decreases asymptotically to zero at large times. Furthermore, the sum of the sensitivity coefficients S t + S q = 0 when τ t /τ q ≈ 1. This is significant information because the variables S t and S q have the same magnitude with opposite signs. This implies that these two sensitivity coefficients become linearly dependent and it will be difficult to predict the values of τ t and τ q in the neighborhood of τ t /τ q ≈ 1. A similar trend but with different sensitivity values are reported for the second case.

Microprocessors continue to grow in capabilities, complexity and performance. Microprocessors typically integrate functional components such as logic and level two (L2) cache memory in their architecture. This functional integration of logic and memory results in improved performance of the microprocessor as the clock speed increases and the instruction execution time has decreased. However, the integration also introduces a layer of complexity to the thermal design and management of microprocessors. As a direct result of function integration, the power map on a microprocessor is typically highly non-uniform and the assumption of a uniform heat flux across the chip surface is not valid. The active side of the die is divided into several functional blocks with distinct power assigned to each functional block. Previous work [1,2] has been done to minimize the thermal resistance of the package by optimizing the distribution of the non-uniform powered functional blocks with different power matrices. This study further gives design guideline and key pointers to minimized thermal resistance for any number of functional blocks for a given non-uniformly powered microprocessor. In this paper, initially (Part I) temperature distribution of a typical package consisting of a uniformly powered die, heat spreader, TIM 1 & 2 and the base of the heat sink is calculated using an approximate analytical model. The results are then compared with a detailed numerical model and the agreement is within 5%. This study follows (Part II) with a thermal investigation of non-uniform powered functional blocks with a different power matrices with focus on distribution of power over die surface with an application of maximum, minimum and average uniform junction temperature over a given die area. This will help to predict the trend of the calculated distribution of power that will lead to the least thermal gradient over a given die area. This trend will further help to come up with design correlations for minimizing thermal resistance for any number of functional blocks for a given non-uniformly powered microprocessor numerically as well as analytically. The commercial finite element code ANSYS® is used for this analysis as a numerical tool.