Liquid cooling electronics using microchannels integrated in the chips is an attractive alternative to bulky aluminum heat sinks. Cooling can be further enhanced using nanofluids. The goals of this study are to evaluate heat transfer in a nanofluid heat sink with developing laminar flow forced convection, taking into account the pumping power penalty. The proposed model uses semi-empirical correlations to calculate effective nanofluid thermophysical properties, which are then incorporated into heat transfer and friction factor correlations in literature for single-phase flows. The model predicts the thermal resistance and pumping power as a function of four design variables that include the channel diameter, velocity, number of channels, and nanoparticle fraction. The parameters are optimized with minimum thermal resistance as the objective function and fixed specified value of pumping power as the constraint. For a given value of pumping power, the benefit of nanoparticle addition is evaluated by independently optimizing the heat sink, first with nanofluid and then with water. Comparing the minimized thermal resistances revealed only a small benefit since nanoparticle addition increases the pumping power that can alternately be diverted towards an increased velocity in a pure water heat sink. The benefit further diminishes with increase in available pumping power.

Power and semiconductor electronic systems find widespread application in residential, commercial, military, and space environments. In everyday life, these systems are commonly used in televisions, automobiles, telephones, computers, and so forth. Due to their widespread use, electronics chips need to operate reliably under a wide variety of environmental conditions. One of the key factors that affects reliability is thermal management. The difference between the input and the output energy in an electronic system is converted to heat, which must be removed efficiently to prevent overheating and chip failures. Efficient thermal management will be a key enabling technology for the future growth of electronics. This work is motivated by the need to address this issue at the component (chip) level. The methodology presented here can be used for optimal design of on-chip microchannel heat sinks with nanofluid flow. The current study takes into consideration that the flow is not fully developed due to the short length of these microchannels (the literature review reveals that often the optimization is done using fully developed heat transfer and pumping power correlations). Furthermore, the current study takes into consideration the increase in pumping power due to nanoparticle addition (whereas several studies in the literature perform this optimization by only considering the benefit of particle addition at a given flow rate and not the pumping power penalty). Finally, the current study evaluates the benefit of nanoparticle addition by independently optimizing the nanofluid and base fluid heat sinks at a specified pumping power.

Microchannels are compact cooling elements that can provide increased heat dissipation rates and reduced temperature gradients across electronic components. Tuckerman and Pease [^{5} W/m^{2}^{2} using water as the coolant. Unlike traditional heat sinks that need a large surface area to increase heat dissipation rates, microchannels use small diameter channels to increase the heat transfer coefficient by forcing the coolant in close contact with the channel walls. The heat transfer coefficient (and therefore the heat dissipation) increases as the flow channel diameter is decreased. Based on channel diameter, Mehendale et al. [

The removal of heat using microchannel heat sinks can be enhanced using nanofluids (liquid solutions with dispersed nanometer-sized particles) as reported by numerous studies in the literature, a summary of which is presented in recent key review articles [

It is extremely difficult to theoretically estimate the thermophysical properties of nanofluids (since the details of the microstructure and the small scale effects are usually not known accurately). This incomplete understanding also makes the modeling of the nanoparticle flow challenging. Two modeling approaches are summarized next.

A first potential approach to modeling nanofluids uses discrete phase modeling (DPM) and is referred to as the Euler-Lagrangian method. The fluid is treated as a continuous media, and the flow field is solved based on Navier-Stokes equations. The nanoparticles are individually tracked in a Lagrangian reference frame. The motion of each nanoparticle is determined by trying to take into account all local forces on the particle (gravity, thermophoretic, Saffman lift, drag, Brownian, Soret and Dufour, etc.). The nanoparticles can exchange momentum, mass, and energy with the Euler frame fluid phase, and vice versa (if two-way coupling is specified). The DPM approach can be computationally very time consuming, especially if there are a large number of particles. Therefore, the current work uses another modeling approach (called the single-phase approach) that implements experimental data to find empirical thermophysical property correlations (usually polynomials) that best fit the data. The particles and the base fluid mixture are treated as a single fluid with enhanced thermophysical properties, where the enhanced thermophysical properties are evaluated using experimental correlations rather than simple binary mixture rules. However, experimental data in the current literature is scarce, and reliable empirical correlations are only available for thermal conductivity and viscosity of nanofluids. Several researchers [

An integrated circuit (IC) heat sink is shown in Figure

The IC heat sink.

In cases where the heat generation in the chip is nonuniform (such as when the heat generation is at the bottom surface of the chip), the channel walls will be subjected to a nonuniform circumferential and axial heat flux. For such cases, evaluating the heat transfer capabilities and optimization of the heat sink will require the use of a 3D finite element or finite volume software package. Since 3D optimization studies are computationally expensive and time consuming, the current model can be used to obtain an approximate solution (initial guess) for further detailed evaluations.

The thermophysical properties used in this study are based on the correlations from the literature described later. In the absence of available data in the literature, the effective density (^{−23} J/K), and

Maiga et al. [

Thermophysical properties used in the model [

Water | Alumina | |
---|---|---|

Density (kg/m^{3}) |
996.54 | 3989.22 |

Specific heat (J/kg |
4177.78 | 778.92 |

Thermal conductivity (W/m |
0.61 | 34.63 |

Viscosity(kg/m |
0.000866 | — |

Churchill and Ozoe [

After obtaining the thermal resistance, we also need to evaluate the pumping power needed to move the coolant through the channels. By considering the force balance between the pressure that drives the flow and the opposing friction force due to wall shear stress, we can write an expression for pressure drop

In the present study, the thermal resistance given by (

In order to validate our model, the heat transfer coefficient predictions of the proposed model are compared with the experimental data from the literature. Anoop et al. [^{2} on the tube wall. A 3 percent alumina-water nanofluid at 22°C was used as the coolant. The Reynolds number was 1460. The average size of the alumina nanoparticle was 35 nm. The local heat transfer coefficient was measured at various positions along the length of the tube.

The results of model validation are presented in Figures

Average heat transfer coefficient over the channel length versus Reynolds numbers.

Local heat transfer coefficient along the channel versus dimensionless axial distance.

Different estimation methods were considered; however, due to the flatness of the fitness function, a GA was selected because of its robust nongradient-based optimization approach. More information regarding the principles of GAs can be found in Goldberg [

The GA solves for the optimal values of the parameters by maximizing the

The GA is programmed in MATLAB scripting language and based on the algorithm by Goldberg [

The inverse of thermal resistance (see (

For each level of maximum pumping power considered in this study, the minimized thermal resistance of the nanofluid heat sink and the corresponding values of the parameters are presented in Table

Thermal resistance for microchannel heat sink with nanofluid coolant.

Power |
Diameter |
Velocity |
Channel count | Volume fraction |
Reynolds number | Resistance |
---|---|---|---|---|---|---|

0.1 | 175 | 1.73 | 88 | 2.38 | 303 | 0.136 |

0.5 | 175 | 3.35 | 88 | 3.49 | 538 | 0.088 |

0.9 | 175 | 4.22 | 88 | 4.22 | 634 | 0.076 |

1.5 | 100 | 4.58 | 198 | 2.18 | 466 | 0.066 |

2.5 | 100 | 5.75 | 198 | 2.19 | 585 | 0.057 |

3.5 | 100 | 6.64 | 198 | 2.34 | 668 | 0.052 |

Thermal resistance for microchannel heat sink with base fluid coolant.

Pumping power (W) | Diameter |
Velocity |
Channel count | Reynolds number | Resistance |
---|---|---|---|---|---|

0.1 | 175 | 1.87 | 88 | 381 | 0.139 |

0.5 | 175 | 3.76 | 88 | 766 | 0.091 |

0.9 | 175 | 4.83 | 88 | 983 | 0.079 |

1.5 | 100 | 4.93 | 198 | 574 | 0.068 |

2.5 | 100 | 6.18 | 198 | 719 | 0.059 |

3.5 | 100 | 7.15 | 198 | 833 | 0.054 |

Minimized thermal resistance versus pumping power for heat sink using base fluid and alumina-water nanofluid coolant.

The solutions at relatively low values of pumping power (0.1, 0.5, and 0.9 W) converge towards an optimal diameter of 175

In general, the thermal resistance can be lowered with more particles (higher volume fraction) added to the flow and also with higher flow rate or velocity. Both of these will impose some additional pumping power penalty. Under the constraint of a given pumping power, the benefit of nanoparticle addition must be evaluated by independently minimizing the thermal resistance of the base fluid and nanofluid heat sink. The minimization is achieved through some optimal combination of velocity and volume fraction for the case of nanofluid heat sink (Table

A semianalytical model is presented to evaluate the heat transfer and flow characteristics of a nanofluid heat sink with developing flow laminar forced convection. The model is used to predict the thermal resistance and pumping power as a function of four design variables that include the channel diameter, the flow velocity, number of channels, and the particle volume fraction. The robust nongradient-based approach of a genetic algorithm was successfully applied to find optimal design with minimum thermal resistance as the objective and fixed specified value of pumping power as the constraint. The study revealed that for a given pumping power, only a small benefit is achieved through nanofluid coolants when the comparison is made by independently optimizing the heat sink, first with nanofluid and then with base fluid. This is because the nanoparticles increase the pumping power which can alternately be diverted towards an increased velocity in a pure fluid heat sink. The benefit of adding nanoparticles is further decreased as the available pumping power is increased. The methodology and results presented here would be useful for the understanding and optimal design of microchannel heat sinks with nanofluid flow. The current model and estimation methodology are flexible enough to be extended to the case of nonuniform heat flux on the channel walls in future studies. For more accurate results that take into account the effect of axial conduction along the channels, three-dimensional finite element model can be created with a single-phase coolant in channels that have effective nanofluid properties.

Lower limit value for the search interval of a design parameter

Upper limit value for the search interval of a design parameter

Specific heat

Differential length along the channel wall (m)

Channel diameter (m)

Graetz number

Heat transfer coefficient (W/

Height (meter)

Thermal conductivity (W/m

Length (m)

The equivalent decimal value of a design parameter in binary form

The number of bits used to represent a design parameter in binary form

The number of channel layers accommodated by heat sink cross-section

The total number of channels in heat sink

Nusselt number

Pressure (N/

Pumping power (W)

Poiseuille number

Prandtl number

Heat transfer (W)

Thermal resistance

Reynolds number

Temperature

Velocity (m/s)

Width (m)

Axial distance (m).

Thermal diffusivity (

Density (kg/

Gradient operator

Viscosity (kg/m.s)

Model output

Percent volume fraction

Kinematic viscosity (

Reference or inlet value

Average

Base fluid (water)

Property evaluated based on diameter or hydraulic diameter

Nanofluid (

Mean/average

Nanoparticle.

Dimensionless quantity.