**Microchannel Simulation**

Mohammad Hassan Saidi, Omid Asgari and Hadis Hemati *Sharif University of Technology Iran* 

#### **1. Introduction**

In the current state of fast developing electronic equipment, having high speeds and at the same time high heat generation rates, researchers are trying to find an effective microcooling method. The main problem in the cooling of microdevices is their high heat generation rate in a limited space. In this way, much research has been conducted on the analysis of different novel microcooling methods such as microheat pipes, microjet impingements, microcapillary pumped loops, microelectrohydrodynamic coolers, and microchannel heat sinks. Among the different microcooling devices the microchannel heat sink has been of special consideration due to its capabilities such as high capacity of heat removal. In principle, microchannels can be machined at the back of a substrate of electronic chips and, therefore, can reduce the internal (contact) thermal resistance of the heat sink practically to zero. The cooling rates in such microchannel heat exchangers should increase significantly due to a decrease in the convective resistance to heat transport caused by a drastic reduction in the thickness of thermal boundary layers.

The overall excellent potential capacity of such a heat sink for heat dissipation is based on the large heat transfer surface-to-volume ratio of the microchannel heat exchangers. The use of microchannel heat sink was first introduced by Tuckerman and Pease in 1981 [1]. Their research was based on an experiment where they showed that high heat rates of 790 *W/cm2* could be removed by microchannel heat sinks.

Currently, research in the field of microchannels is going on in three aspects of experimental methods [2,3,4], numerical methods [5,6,7,8], and analytical methods [9,10,11,12]. It is well known that the cross-sectional shape of a channel can have significant affect on the fluid flow and heat transfer characteristics in microchannels. Peng and Peterson [13] performed experimental investigations of the pressure drop and convective heat transfer for water flowing in rectangular microchannels, and found that the cross-sectional aspect ratio had significant influence on the flow friction and convective heat transfer in both laminar and turbulent flows. Kawano et al. [14] provided experimental data on the friction and heat transfer in rectangular, silicon based microchannel heat sinks, and more recently Wu and Cheng [15,16] conducted a series of experiments to measure the friction factor and convective heat transfer in smooth silicon microchannels of trapezoidal cross-section. The experimental methods has its own values and the new developments in the micromachining techniques help the researchers to perform precise experiments, but because of its high cost it fails to be commonly applied.

Microchannel Simulation 191

The micro-heat sink modeled in this investigation consists of a 10 mm long silicon substrate with a silicon cover. The rectangular microchannels have a width of 57 µm and a depth of 180 µm. The hydraulic diameter of microchannel under development is about 86.58 µm and is expected to decrease to 10 µm. This yields a typical Knudsen number for water as a coolant to be between 3.5×10-5 and 3.5×10-4 which lies in the continuous flow regime (*Kn*<10-3) [20]. Hence, the conservation equations based on the continuum model (Navier-

A schematic of the rectangular microchannel heat sink is shown in Figure 1 where a unit of cell consisting of one channel was selected because of the symmetry of the structure. The unit cell of the heat sink can be represented by an asymmetric rectangular channel with the cross-sectional dimensions as shown in Table 1. The channel geometry is similar to that employed in the experimental work of Kawano et al. [14] and in the numerical investigations of Qu and Mudawar [5] and Fedorov and Viskanta [6]. It is assumed that the

Consider a steady 3D flow in a silicon microchannel heat sink with heating from below and with adiabatic conditions at the other boundaries, as showed in Figure 1. There are some assumptions in this numerical simulation, e.g. the transport processes are considered to be steady-state and three-dimensional, the flow is incompressible and laminar, thermal radiation is neglected, the thermophysical properties are temperature dependent. Under the stated assumptions, the governing equations and related boundary conditions for a fully

( ) ( ) ( )0 *uvw*

Fig. 1. Part I

(1)

*xyz* 

Stokes equations of motion) can still be used to describe the transport processes.

heat flux input at the bottom of the heat sink is uniform.

**2.2 Model equations and boundary conditions** 

developed 3D flow heat transfer are given as [21],

Continuity equation

**2. Analysis** 

**2.1 Problem description** 

There has been much effort to analytically model the microchannel heat sink. Knight et al. [9] used empirical correlations to evaluate the performance of a microchannel heat sink. Koh and Colony [17] first modeled microstructures as a porous medium using Darcy's law. Later on Tien and Kuo [18] developed a model for heat transfer in microchannels using the modified Darcy equation and the two-equation model. Another method to model the microchannel heat sink analytically is to use the fin and plate theory. In this way, the solid walls are assumed to be as fins that are connected to the base plate. Applying this assumption, the thermal resistance and other thermal characteristics of the system could be derived using the fin and plate heat transfer relations. Because of limiting and less accuracy of the analytical approach in some phenomenons, e.g. conjugate heat transfer the numerical method is being used vastly to simulate the performance of microchannels like other fields of heat transfer.

Webb and Zhang [19] claim that the accepted single-phase flow correlations adequately predict their experimental data for round and rectangular tubes with hydraulic diameter of the order of 1 mm. Wesberg et al. [8] solved a two-dimensional conjugate heat transfer problem for microchannel heat sinks to obtain detailed spatial distributions of the temperature of the heat sink cross-section along the length of the channels. Fedorov and Viskanta [6] developed a three dimensional model to investigate the conjugate heat transfer in a microchannel heat sink with the same channel geometry used in the experimental work done by Kawano et al. [14]. This simulation showed that the average channel wall temperature along the flow direction was nearly uniform except in the region close to the channel inlet, where very large temperature gradients were observed. Qu and Mudawar [5] conducted a three-dimensional fluid flow and heat transfer analysis for a rectangular microchannel heat sink with a geometry similar to that of Kawano et al. [14] using a numerical method similar to that proposed by both Kawano et al. [14] and Fedorov and Viskanta [6]. This model considered the hydrodynamic and thermal developing flow along the channel and found that the Reynolds number will influence the length of the developing flow region. It was also found that the highest temperature is typically encountered at the heated base surface of the heat sink immediately adjacent to the channel outlet and that the temperature rise along the flow direction in the solid and fluid regions can both be approximated as linear.

In this study the SIMPLE pressure correction approach has been chosen. For simplicity the equations are solved in a segregated manner. An effective multigrid solver is developed for pressure correction equation. Methods for calculating cell-face velocities are studied. Convection and diffusion terms are modeled using upwind and the central difference stencil, respectively. The variables are located in the cell-centred colocated grid. The general coordinates are described using the pure control volume approach with vector notations. In the method developed herein, the velocity field is first solved and then the mean velocity derived. The thermophysical properties are chosen at a reference temperature (an estimated averaged liquid bulk temperature) determined from the energy balance and then a new final velocity field and temperature distribution for the microchannel is defined. A numerical model with fully developed flow is presented and used to analyze the heat transfer in a microchannel heat sink for low Re numbers. The numerical model is based upon a three dimensional conjugate heat transfer approach (3D fluid flow and 3D heat transfer). Computations were performed for different total pressure drops in the channel. The system of three-dimensional Navier-Stokes equations for conservation of mass, momentum, and energy based on the continuum flow assumption is used as a mathematical model of the process.
