MATHEMATICAL OPTIMIZATION: APPLICATION TO THE DESIGN OF OPTIMAL MICRO-CHANNEL HEAT SINKS

This paper documents the geometrical optimization of a micro-channel heatsink embedded inside a highly conductive solid, with the intent of developing optimal solutions for thermal management in microelectronic devices. The objective is to minimize the peak wall temperature of the heat sink subject to various constraints such as manufacturing restraints, fixed pressure drop and total fixed volume. A gradient based multi-variable optimization algorithm is used as it adequately handles the numerical objective function obtained from the computational fluid dynamics simulation. Optimal geometric parameters defining the micro-channel were obtained for a pressure drop ranging from 10 kPa to 60 kPa corresponding to a dimensionless pressure drop of 6.5 × 107 to 4 × 108 for fixed volumes ranging from 0.7 mm3 of 0.9 mm3. The effect of pressure drop on the aspect ratio, solid volume fraction, channel hydraulic diameter and the minimized peak temperature are reported. Results also show that as the dimensionless pressure drop increases the maximised dimensionless global thermal conductance also increases. These results are in agreement with previous work found in literature

FLOW ORIENTATION IN CONJUGATE COOLING CHANNELS WITH INTERNAL HEAT GENERATION

ABSTRACT This work presents a three-dimensional geometric optimisation of conjugate cooling channels in forced convection with internal heat generation within the solid for an array of circular cooling channels with different flow orientations based on constructal theory. Three flow orientations were studied: Firstly, an array of channels with parallel flow; secondly, an array of channels in which flow of the every second row is in a counter direction to one another and thirdly, with the every flow in the array of channels in counter direction to one another. The geometric configurations and the flow orientations were optimised in such a way that the peak temperature was minimised subject to the constraint of fixed global volume of solid material. The cooling fluid was driven through the channels by the pressure difference across the channel. The system had hydraulic diameter and channel to channel spacing as degrees of freedom of the design variables. A gradient-based optimisation algorithm was applied to search for the best optimal geometric configurations that improve thermal performance by minimising thermal resistance for a wide range of dimensionless pressure differences. This optimiser adequately handles the numerical objective function obtained from numerical simulations. The effect of porosities, applied pressure difference, flow orientation and heat generation rate on the optimal hydraulic diameter and channel to channel spacing were reported. Results obtained show that the effects of dimensionless pressure drop on minimum thermal resistance were consistent with those obtained in the open literature

Maximum heat transfer density rate enhancement from cylinders rotating in natural convection

In this paper we investigate the thermal behaviour of an assembly of consecutive cylinders in a counterrotating configuration cooled by natural convection with the objective of maximizing the heat transfer density rate (heat transfer rate per unit volume). A numerical model is used to solve the governing equations that describe the temperature and flow fields. The spacing between the consecutive cylinders is optimised for each flow regime (Rayleigh number) and cylinder rotation speed. It was found that the optimized spacing decreases as the Rayleigh number increases and the heat transfer density rate increases, for the optimized structure, as the cylinder rotation speed is increased. Results further show that there is an increase in the heat transfer density rate of the rotating cylinders over stationary cylinders.The National Research Foundation (NRF-DST)http://www.elsevier.com/locate/ichmtai201

Constructal multi scale cylinders with rotation cooled by natural convection

This paper investigated the thermal behaviour of an assembly of multi scale cylinders in a staggered counter-rotating configuration cooled by natural convection with the objective of maximizing the heat transfer density rate (heat transfer rate per unit volume). A numerical model was used to solve the governing equations that describe the temperature and flow fields and a mathematical optimisation algorithm was used to find the optimal structure for flow configurations with two degrees of freedom. The multi scale structure of the cylinder assembly was optimized for each flow regime (Rayleigh number) and cylinder rotation speed for two degrees of freedom. Smaller cylinders were placed at the entrance to the assembly, in the wedge-shaped flow regions occupied by fluid that had not yet been used for heat transfer, to create additional length scales to the flow configuration. It was found that there was almost no effect of cylinder rotation on the maximum heat transfer density rate, when compared to stationary cylinders, at each Rayleigh number; with the exception of high cylinder rotation speeds, which served to suppress the heat transfer density rate. It was, however, found that the optimized spacing decreased as the cylinder rotation speed was increased at each Rayleigh number. Results further show that the maximum heat transfer density rate for a multi scale configuration (without cylinder rotation) was higher than a single scale configuration (with rotating cylinders) with an exception at very low Rayleigh numbers.University of Pretoria and the National Research Foundation (NRF-DST)http://www.elsevier.com/locate/ijhmthb201

Constructal conjugate cooling channels with internal heat generation

This paper presents a geometric optimisation of conjugate cooling channels in forced convection with internal heat generation. Two configurations were studied; circular channels and square channels. The configurations were optimised in such a way that the peak temperatures were minimised subject to the constraint of fixed total global volume. The fluid was forced through the cooling channels by the pressure difference across the channels. The structure has one degree of freedom as design variable: channel hydraulic diameter and once the optimal channel hydraulic diameter is found, optimal elemental volume and channel-to-channel spacing result. A gradient-based optimisation algorithm is applied in order to search for the best and optimal geometric configurations that improve thermal performance by minimising thermal resistance for a wide range of dimensionless pressure difference. This optimiser adequately handles the numerical objective function obtained from CFD simulations. The results obtained show the behaviour of the applied pressure difference on the optimised geometry. There are unique optimal design variables for a given pressure difference. The numerical results obtained are in agreement with the theoretical formulation using scale analysis and method of intersection of asymptotes.NRF, TESP, University of Stellenbosch/ University of Pretoria, SANERI/SANEDI, CSIR, EEDSM Hub and NAC.http://www.elsevier.com/locate/ijhmtai201

Mathematical optimisation of laminar forced convection heat transfer through a vascularised solid with square channels

This paper presents a three-dimensional geometric optimisation of cooling channels in forced convection of a vascularised material with the localised self-cooling property subjected to a heat flux. A square configuration was studied with different porosities. Analytical and numerical solutions were provided. The geometrical configuration was optimised in such a way that the peak temperature was minimised at every point in the solid body. The optimisation was subject to the constraint of a fixed global volume of solid material, but the elemental volume was allowed to morph. The solid material was subject to a heat flux on one side and the cooling fluid was forced through the channels from the opposite direction with a specified pressure difference. The structure had three degrees of freedom as design variables: the elemental volume, channel hydraulic diameter and channel-to-channel spacing. A gradient-based optimisation algorithm was used to determine the optimal geometry that gave the lowest thermal resistance. This optimiser adequately handled the numerical objective function obtained from numerical simulations of the fluid flow and heat transfer. The numerical results obtained were in agreement with a theoretical formulation using scale analysis and the method of intersection of asymptotes. The results obtained show that as the pressure difference increases, the minimised thermal resistance decreases. The results also show the behaviour of the applied pressure difference on the optimised geometry. The use of the optimiser made the numerical results to be more robust with respect to the optimum internal configurations of the flow systems and the dimensionless pressure difference.The NRF, TESP, Stellenbosch University/ University of Pretoria, SANERI/SANEDI, CSIR, EEDSM Hub and NAC.http://www.elsevier.com/locate/ijhmtai201