An analytical model of thermal performance for an eccentric heat source on a rectangular plate with double-sided convective cooling

Three-dimensional vibration solutions are presented for rectangular plates with mixed boundary conditions, based on the small strain linear elasticity theory. The analysis is focused on two kinds of rectangular plates, the boundaries of which are partially fixed while the others are free. One of those studied is a rectangular plate with partially fixed boundaries symmetrically arranged around four corners and the other one is a rectangular plate with partially fixed boundaries around one corner only. A global analysis approach is developed. The Ritz method is applied to derive the governing eigenvalue equation by minimizing the energy functional of the plate. The admissible functions for all displacement components are taken as a product of a characteristic boundary function and the triplicate Chebyshev polynomial series defined in the plate domain. The characteristic boundary functions are composed of a product of four components of which each corresponds to one edge of the plate. The R-function method is applied to construct the characteristic boundary function components for the edges with mixed boundary conditions. The convergence and comparison studies demonstrate the accuracy and correctness of the present method. The influence of the length of the fixed boundaries and the plate thickness on frequency parameters of square plates has been studied in detail. Some valuable results are given in the form of tables and figures, which can serve as the benchmark for the further research.

Acoustic power radiation from a rectangular plate with mixed boundary conditions

The title problem is investigated using the differential quadrature method based on the state‐space formalism. The plates, with mixed boundary conditions, may cross over one‐way internal rigid line supports that impose zero transverse displacement constraints. Differential quadrature procedure is applied in the direction of line supports, while exact solution is sought in the transfer domain perpendicular to the line supports using the state space method. To avoid numerical instability in the transfer matrix method, joint coupling matrices are introduced, mainly according to the continuity conditions at line joints. Natural frequencies of rectangular Kirchhoff plates with general boundary conditions are calculated and compared with the results from other methods. Effects of location of internal line supports and the mixed boundary conditions on the frequency parameters are discussed. Copyright © 2006 John Wiley & Sons, Ltd.

The model problem of a plane angle for a second-order elliptic system subject to Dirichlet, mixed, and Neumann boundary conditions is analyzed. For each boundary condition, the existence of solutions of the form $r^λv$ is reduced to spectral analysis of a particular matrix. Focusing on Dirichlet and mixed boundary conditions, optimal bounds on $|\Re λ|$ are derived, employing tools from numerical range analysis and accretive operator theory. The developed framework is novel and recovers known bounds for Dirichlet boundary conditions. The results for mixed boundary conditions are new and represent the central contribution of this work. Immediate applications of these findings are new regularity results for linear second-order elliptic systems subject to mixed boundary conditions.

Enhancing the thermal performance of heat exchangers while minimizing flow resistance is critical for efficient energy systems. This study focuses on the effect of varying blade counts (2, 4, 6, and 8 blades) and Reynolds numbers (6000, 8000, 10,000, 12,000, and 14,000) on the thermal performance factor (TPF), friction factor, and heat transfer coefficient in thermal boundary layer heat exchangers. Numerical analyses were conducted to generate comprehensive datasets capturing the influence of blade-induced swirl and turbulence intensification across the specified Reynolds number range. An Artificial Neural Network (ANN) was developed and trained using these datasets to predict thermal and hydraulic performance with high accuracy. The model was used to optimize design parameters, identifying configurations that maximize heat transfer enhancement while controlling frictional penalties. Results indicate that increasing blade count significantly improves the heat transfer coefficient and TPF, especially at higher Reynolds numbers, but also leads to a corresponding increase in friction factor. The ANN demonstrated excellent predictive capability, offering a reliable tool for design optimization and performance prediction in thermal boundary layer heat exchangers with high Reynolds numbers and high blade counts. This work provides valuable insights into the interplay between blade geometry and flow conditions and demonstrates the effectiveness of AI-assisted modeling for the next generation of compact, high-performance heat exchangers.

The optical performance and reliability of High power light emitting diode (HP-LED) devices can be affected by the junction temperature, while thermal resistance is one of the main factors which influence the p-n junction temperature, especially for the thermal spreading resistance. So, Thermal analysis was conducted by differential structure functions combining with finite element analysis and calculation of thermal spreading resistance of HP-LED devices. The results show that thermal spreading resistance plays an important role on total thermal resistance of LED devices. The thermal spreading resistance shows a declined and then increased trend with the increase of substrate thickness; the contact area between the heating source and substrate is the main factor which influences thermal spreading resistance; It is beneficial to decrease the thermal spreading resistance when contact area ratio be increased. Thermal-convection resistance becomes the main factor that influences total thermal resistance of HP-LED devices with the changing of substrate bottom area, while the influence of thermal spreading resistance is weak. The eccentric distance between chip and substrate plays a distinct impact on thermal spreading resistance and junction temperature. The results of finite element simulation are in good agreement with calculation of thermal spreading resistance.

The optical performance and reliability of High power light emitting diode (HP-LED) devices can be affected by the junction temperature, while thermal resistance is one of the main factors which influence the p-n junction temperature, especially for the thermal spreading resistance. So, Thermal analysis was conducted by differential structure functions combining with finite element analysis and calculation of thermal spreading resistance of HP-LED devices. The results show that thermal spreading resistance plays an important role on total thermal resistance of LED devices. The thermal spreading resistance shows a declined and then increased trend with the increase of substrate thickness; the contact area between the heating source and substrate is the main factor which influences thermal spreading resistance; It is beneficial to decrease the thermal spreading resistance when contact area ratio be increased. Thermal-convection resistance becomes the main factor that influences total thermal resistance of HP-LED devices with the changing of substrate bottom area, while the influence of thermal spreading resistance is weak. The eccentric distance between chip and substrate plays a distinct impact on thermal spreading resistance and junction temperature. The results of finite element simulation are in good agreement with calculation of thermal spreading resistance.

Thermal spreading resistance is one of the key factors for designing the thermal management systems in microelectronic devices. This type of thermal resistance occurs in most of the microelectronic devices and causes some difficulties for thermal engineers to model the system. One of the common geometries in these devices is the flux channel. Different boundary conditions can be applied on the flux channel based on the designing criteria of the system including the arbitrary distribution of heat sinks over the sink plane. This boundary condition is usually simplified as a constant heat transfer coefficient to facilitate the modeling of the system. In this paper, a flux channel with an arbitrary distributed heat transfer coefficient over the sink plane is studied without simplification of the sink boundary condition. Both adiabatic and convective cooling over the edges of the flux channel are considered. Due to the complexity of the sink boundary condition, the conventional analytical solutions are not applicable and the method of least squares is used. By employing this approach, the effect of a non-uniform heat transfer coefficient on thermal spreading resistance is investigated. The solution is presented in form of a Fourier series expansion which can be used to obtain the temperature all over the channel. Results are validated with Finite Element Models, FEM. This approach is useful for thermal engineers who have some difficulty for modeling complex boundary conditions and presents an effective solution for thermal resistance in the flux channels.

Purpose This paper aims to develop the thermal resistance network model based on the heat dissipation paths from the multi-die stack to the ambient and takes into account the composite effects of the thermal spreading resistance and one-dimensional (1D) thermal resistance. The thermal spreading resistance comprises majority of the thermal resistance when heat flows in the horizontal direction of a large plate. The present study investigates the role of determining the temperature increase compared to the thermal resistances intrinsic to the 3D technology, including the thermal resistances of bonding layers and through silicon vias (TSVs). Design/methodology/approach This paper presents an effective method that can be applied to predict the thermal failure of the heat source of silicon chips. An analytical model of the 3D integrated circuit (IC) package, including the full structure, is developed to estimate the temperature of stacked chips. Two fundamental theories are used in this paper – Laplace’s equation and the thermal resistance network – to calculate 1D thermal resistance and thermal spreading resistance on the 3D IC package. Findings This paper provides a comprehensive model of the 3D IC package, thus improving the existing analytical approach for predicting the temperature of the heat source on the chip for the 3D IC package. Research limitations/implications Based on the aforementioned shortcomings, the present study aims to determine if the use of an analytical resistance model would improve the handling of a temperature increase on the silicon chips in a 3D IC package. To achieve this aim, a simple rectangular plate is utilized to analyze the temperature of the heat source when applying the heat flux on the area of the heat source. Next, the analytical model of a pure plate is applied to the 3D IC package, and the temperature increase is analyzed and discussed. Practical implications The main contribution of this paper is the use of a simple concept and a theoretical resistance network model to improve the current understanding of thermal failure by redesigning the parameters or materials of a printed circuit board. Social implications In this paper, an analytical model of a 3D IC package was proposed based on the calculation of the thermal resistance and the analysis of the network model. Originality/value The aim of this work was to estimate the mean temperature of the silicon chips and understand the heat convection paths in the 3D IC package. The results reveal these phenomena of the complete structure, including TSV and bump, and highlight the different thermal conductivities of the materials used in creating the 3D IC packages.

As a highly efficient heat dissipation unit, a micro heat pipe is widely used in high heat flux microelectronic chips, and its thermal resistance is crucial to heat transfer capacity. Through analyses of the structure and heat transfer performance of a circular heat pipe with sintered wick, the theoretical model of total thermal resistance was established on heat transfer theory, and then simplified, finally a testing platform was set up to test for total thermal resistance performance. The testing results show that when the micro heat pipe is in optimal heat transfer state, its total thermal resistance conform well with that from the theoretical model, and its actual thermal resistance is much lower than that of the rod made of the material with perfect thermal conductivity and of the same geometric size. With the increment of heat transfer capability, the total thermal resistance of a micro heat pipe with sintered wick decreases first, then increases and reaches the minimum when it is in the optimal heat transfer state. The greater total thermal resistance in low heat transfer performance is mainly caused by too much working fluid accumulating in evaporator and the lower velocity in vapor cavity, and the greater total thermal resistance in high heat transfer performance is mainly due to the working fluid drying up in condenser. Total thermal resistance is related to many factors, such as thermal conductivity of tube-shell material, wall thickness, wick thickness, copper powders grain size and porosity, the lengths of condenser and evaporator, and the diameter of vapor cavity etc.. Therefore, the structure parameters of a micro heat pipe with sintered wick should be reasonably designed according to the specific conditions to ensure its heat transfer capacity and total thermal resistance to meet the requirements.

Based on the field test data, vehicle load are simplified as a semi-sinusoidal rectangular distributed load, the expression of vehicle load are obtained by considering the dynamic component of load caused by pavement roughness. By regarding the flexible pavement structure as a three dimensional multi layered system, the dynamic responses of the pavement subjected to the semi-sinusoidal moving load are studied. The governing equations of motion are solved by employing the potential function decomposition and the Fourier transform technique. Considering the mixed boundary condition at the upper surface, the fixed boundary condition at the lowest surface and that the stress and displacement at the interface of the layers are continuous, the theoretical solutions of the components of the responses are deduced with the exact stiffness matrix method. Numerical results are obtained by using FFT algorithm and the calculation results are consistent with ones of laboratory test. The influence of different parameters such as load speed, pavement structural combination and the properties of base layer and subgrade on the responses are carefully discussed. The theory developed herein may be used as a preliminary reference in flexible pavement engineering design.

Maximum frequency design of laminated plates with mixed boundary conditions

Three staggered 90 deg rib geometries corresponding to blockage ratios of 0.133, 0.167, and 0.25 were tested for pitch-to-height ratios of 5, 8.5, and 10, and for two distinct thermal boundary conditions of heated and unheated channel walls. Comparisons were made between the surface-averaged heat transfer coefficients and friction factors for ribs with rounded corners and those with sharp corners, reported previously. Heat transfer coefficients of the furthest upstream rib and that of a typical rib located in the middle of the rib-roughened region of the passage wall were also compared. It was concluded that: (a) For the geometries tested, the rib average heat transfer coefficient was much higher than that for the area between the ribs. For the sharp-corner ribs, the rib average heat transfer coefficient increased with blockage ratio. However, when the corners were rounded, the trend depended on the level of roundness. (b) High-blockage-ratio (e/Dh = 0.25) ribs were insensitive to the pitch-to-height ratio. For the other two blockage ratios, the pitch-to-height ratio of 5 produced the lowest heat transfer coefficient. Results of the other two pitch-to-height ratios were very close, with the results of S/e = 10 slightly higher than those of S/e = 8.5. (c) Under otherwise identical conditions, ribs in the furthest upstream position produced lower heat transfer coefficients for all cases except that of the smallest blockage ratio with S/e of 5. In that position, for the rib geometries tested, while the sharp-corner rib average heat transfer coefficients increased with the blockage ratio, the trend of the round-corner ribs depended on the level of roundness, r/e. (d) Thermal performance decreased with the blockage ratio. While the smallest rib geometry at a pitch-to-height ratio of 10 had the highest thermal performance, thermal performance of high blockage ribs at a pitch-to-height ratio of 5 was the lowest. (e) The general effects of rounding were a decrease in heat transfer coefficient for the midstream ribs and an increase in heat transfer coefficient for ribs in the furthest upstream position.

Experimental investigations have shown that the enhancement in heat transfer coefficients for air flow in a channel roughened with angled ribs is on the average higher than that roughened with 90° ribs of the same geometry. Secondary flows generated by the angled ribs are believed to be responsible for these higher heat transfer coefficients. These secondary flows also create a spanwise variation in heat transfer coefficient on the roughened wall with high levels of heat transfer coefficient at one end of the rib and low levels at the other end. In an effort to basically double the area of high heat transfer coefficients, the angled rib is broken at the center to form a V-shape rib and tests are conducted to investigate the resulting heat transfer coefficients and friction factors. Three different square rib geometries, corresponding to blockage ratios of 0.083, 0.125 and 0.167, with a fixed pitch-to-height ratio of 10, mounted on two opposite walls of a square channel in a staggered configuration are tested in a stationary channel for 5000 < Re < 30000. Heat transfer coefficients, friction factors and thermal performances are compared with those of 90°, 45° and discrete angled ribs. The V-shape ribs are tested for both pointing upstream and downstream of the main flow. Test results show that: a) 90° ribs represent the lowest thermal performance, based on the same pumping power, and is essentially the same for the 2:1 change in blockage ratio, b) low blockage ratio (e/Dh =0.083) V-shape ribs pointing downstream produced the highest heat transfer enhancement and friction factors. Amongst all other geometries with blockage ratios of 0.125 and 0.167, 45° ribs showed the highest heat transfer enhancements with friction factors less than those of V-shape ribs, c) thermal performance of 45° ribs and the lowest blockage discrete ribs are among the highest of the geometries tested in this investigation, and, d) discrete angled ribs, although inferior to 45° and V-shape ribs, produce much higher heat transfer coefficients and lower friction factors compared to 90° ribs.

Stress intensity factors and T-stress inedge-cracked rectangular plates under mixed boundary conditions