Modeling of conjugate heat transfer using Galerkin approach (original) (raw)
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Calculation of conjugate heat transfer in a heat sink using Volume Averaging Technique (VAT)
A fast running computational algorithm based on Volume Averaging Technique (VAT) is developed and solutions obtained using the Finite Volume Method (FVM) and the Galerkin Method (GM). The goal is to improve computational capability in the area of heat exchangers and help eliminate some of the empiricism involved in their design that leads to overly constrained designs with resulting economic penalties. VAT is tested and applied to a simulation of air-flow through an aluminum (Al) chip heat sink. Using VAT, the computational algorithm is fast running, but still able to present a detailed picture of temperature fields in the air-flow as well as in the solid structure of the heat sink. The calculated whole-section drag coefficient, Nusselt number and thermal effectiveness were compared with the experimental data to verify the computational model and validate numerical code. The comparison also shows a good agreement between GM and FVM results although different thermal boundary conditions at the bottom were used. The constructed computational algorithm enables prediction of cooling capabilities for the selected geometry. It also offers possibility for geometry improvements and optimization, to achieve higher thermal effectiveness.
Numerical technique for modeling conjugate heat transfer in an electronic device heat sink
International Journal of Heat and Mass Transfer, 2003
A fast running computational algorithm based on the volume averaging technique (VAT) is developed to simulate conjugate heat transfer process in an electronic device heat sink. The goal is to improve computational capability in the area of heat exchangers and to help eliminate some of empiricism that leads to overly constrained designs with resulting economic penalties.
Modeling of forced convection in an electronic device heat sink as porous media flow
An algorithm for simulation of conjugate heat transfer used to find the most suitable geometry for an electronic chip heat sink is described. Applying Volume Averaging Theory (VAT) to a system of transport equations, a heat exchanger structure was modeled as a homogeneous porous media. The interaction between the fluid and the structure, the VAT equation closure requirement, was accomplished with drag and heat transfer coefficients, which were taken from the available literature and inserted into a computer code. The example calculations were performed for an aluminum heat sink exposed to force convection airflow. The geometry of the simulation domain and boundary conditions followed the geometry of the experimental test section. The comparison of the whole-section drag coefficient and Nusselt number as functions of Reynolds number shows a good agreement with the experimental data. The calculated temperature fields reveal the local heat flow distribution and enable further improvements of the heat sink geometry.
APPLICATION OF GALERKIN METHOD TO CONJUGATE HEAT TRANSFER CALCULATION
Numerical Heat Transfer, Part B: Fundamentals, 2003
A fast-running computational algorithm based on the volume averaging technique (VAT) is developed and solutions are obtained using the Galerkin method (GM). The goal is to extend applicability of the GM to the area of heat exchangers in order to provide a reliable benchmark for numerical calculations of conjugate heat transfer problems. Using the VAT, the computational algorithm is fast-running, but still able to present a detailed picture of temperature fields in air flow as well as in the solid structure of the heat sink. The calculated whole-section drag coefficient C d and Nusselt number Nu were compared with finite-volume method (FVM) results and with experimental data to verify the computational model. The comparison shows good agreement. The present results demonstrate that the selected Galerkin approach is capable to perform heat exchanger calculations where the thermal conductivity of the solid structure has to be taken into account.
Numerical investigation of chip cooling using volume averaging technique (VAT)
Advanced Computational Methods in Heat Transfer VII
The present paper describes construction of an algorithm for conjugate heat transfer calculations in order to find the most suitable form for a heat sink. Applying Volume Averaging theory (VAT) to a system of transport equations, a heat exchanger structure was modeled as a homogeneous porous media. The example numerical simulations were performed for test sections with isothermal structure as well as with heat conducting Al pin-fins. The geometry of the simulation domain and boundary conditions followed the geometry of the experimental test section used in the Morrin-Martinelli-Gier Memorial Heat Transfer Laboratory at University of California, Los Angeles. The comparison of the drag coefficient as a function of Reynolds number reveals good agreement with already published data, whereas the comparison of the Nusselt number distributions shows much larger discrepancies. Finite conductivity of a solid phase decreases the heat transfer coefficient and the Nusselt number. The influence of conductivity becomes larger with increasing Reynolds number.
The Galerkin method solution of the conjugate heat transfer problems for the cross-flow conditions
A conjugate heat transfer model of fluid flow across a solid heat conducting structure has been built. Two examples are presented: a.) air-stream cooling of the solid structure and b.) flow across rods with volumetric heat generation. To construct the model, a Volume Average Technique (VAT) has been applied to the momentum and the energy transport equations for a fluid and a solid phase to develop a specific form of porous media flow equations. The model equations have been solved with the semi-analytical Galerkin method. The detailed velocity and temperature fields in the fluid flow and the solid structure have been obtained. Using the solution fields, the whole-section drag coefficient Cd and the whole-section Nusselt number Nu have been also calculated. To validate the developed solution procedure, the results have been compared to the results of the finite volume method and to the experimental data. The comparison demonstrates an excellent agreement.
Multiphase Science and Technology, 2013
Air-and water-cooled heat sinks are still the most common heat rejection devices in electronics, making their geometric optimization a key issue in thermal management. Because of the complex geometry, the use of finite-difference, finite-volume, or finite-element methods for the solution of the governing equations becomes computationally expensive. In this work, volume averaging theory is applied to a general heat sink with periodic geometry to obtain a physically accurate, but geometrically simplified, system model. The governing energy and momentum equations are averaged over a representative elementary volume, and the result is a set of integro-partial differential equations. Closure coefficients are introduced, and their values are obtained from data available in the literature. The result of this process is a system of closed partial differential equations, defined on a simple geometry, which can be solved to obtain average velocities and temperatures in the system. The intrinsic smoothness of the solution and the simplified geometry allow the use of a modified Fourier-Galerkin Method for efficient solutions to the set of differential equations. Modified Fourier series are chosen as the basis functions because they satisfy the boundary conditions a priori and lead to a sparse system of linear equations for the coefficients. The validity of the method is tested by applying it to model the hydraulic and thermal behavior of an air-cooled pin-fin and a water-cooled micro-channel heat sink. The convergence was found to be O(N −3.443 ), while the runtime was ∼0.25 s for N = 56. The numerical results were validated against the experimental results, and the agreement was excellent with an average error of ∼4% and a maximum error of ∼5%.
Application of Fourier-Galerkin Method to Volume Averaging Theory Based Model of Heat Sinks
Volume 8C: Heat Transfer and Thermal Engineering, 2013
Efficient analysis of heat sink performance is a crucial step in the optimization process of such devices. Accurate analysis of these complex geometric systems with CFD and FEM methods requires fine meshes which imply significant computational time. In this study, Volume Averaging Theory (VAT) is rigorously applied to obtain a geometrically simplified but physically accurate model for any periodic heat sink geometry. The governing equations are averaged over a Representative Averaging Volume (REV) to obtain a set of integro-differential equations. Some information about lower level phenomena is lost in every averaging process and a closure scheme is required to model these behaviors. Experimental data for friction factor and Nusselt number in an REV is used to close the set of PDEs. This mathematical process replaces the complex geometry of the heat sink with a fictitious continuous medium and smoothens the quantities of interest throughout the system. These system features allow the use of a global Fourier-Galerkin method to efficiently solve the resulting equations and accurately predict the performance of the system. The effectiveness of the method is proven by applying it to model thermal behavior for laminar flow over an aircooled pin-fin heat sink and a water-cooled micro-channel heat sink. The convergence in the Nusselt number in the case of constant heat flux is found to be quadratic with respect to the number of basis functions. The accuracy of the method is validated by comparing the numerical results obtained to existing experimental data. The maximum difference between the predicted Nusselt number and the experimental measurements is found to be only 4% for both cases.
Applied Mathematical Modelling, 2005
A mathematical model of fluid flow across a rod bundle with volumetric heat generation has been built. The rods are heated with volumetric internal heat generation. To construct the model, a volume average technique (VAT) has been applied to momentum and energy transport equations for a fluid and a solid phase to develop a specific form of porous media flow equations. The model equations have been solved with a semi-analytical Galerkin method. The detailed velocity and temperature fields in the fluid flow and the solid structure have been obtained. Using the solution fields, a whole-section drag coefficient C d and a whole-section Nusselt number Nu have also been calculated. To validate the developed solution procedure, the results have been compared to the results of a finite volume method. The comparison shows an excellent agreement. The present results demonstrate that the selected Galerkin approach is capable of performing calculations of heat transfer in a cross-flow where thermal conductivity and internal heat generation in a solid structure has to be taken into account. Although the Galerkin method has limited applicability in complex geometries, its highly accurate solutions are an important benchmark on which other numerical results can be tested.
Numerical Study of Conjugate Heat Transfer for Cooling the Circuit Board
Journal of Electronics Cooling and Thermal Control, 2016
In this paper, a 3D model of a flat circuit board with a heat generating electronic chip mounted on it has been studied numerically. The conjugate heat transfer including the conduction in the chip and convection with the surrounding fluid has been investigated numerically. Computational fluid dynamics using the finite volume method has been used for modeling the conjugate heat transfer through the chip and the circuit board. Conjugate heat transfer has broad applications in engineering and industrial applications in design of cooling off electronic components. Effects of various inlet velocities have been studied on the heat transfer variation and temperature of the circuit board. Numerical results show that the temperature of the chip reduces as the velocity of the inlet fluid flow increases.