Optimization of a Porous Microchannel Heat Sink

In the model Performance of a Porous Microchannel Heat Sink, a parameter study is performed to find the optimum value for the thickness of the porous material layer within a microchannel heat sink (MCHS). This example extends the above-mentioned model by another study, in which the parametric sweep is replaced by an Optimization study node. The figure of merit is a measure of performance of the heat sink (in this case relative to an MCHS without porous material) and is therefore used as an objective function that can be maximized. The results agree with those found in the original model version.

Model Definition

The design and operating conditions are described in the original model, Performance of a Porous Microchannel Heat Sink.

Inside the porous domains, the governing equation is the Brinkman equation with a Forchheimer correction term (also known as the Brinkman–Forchheimer or Darcy–Brinkman–Forchheimer equation). The pressure drop depends on the velocity field u as

(1)

where μ (Pa·s) is the fluid viscosity, ρ (kg/m3) the density, and κ (m2) the permeability of the porous substrate.

As the performance of the MCHS should be optimized, the performance parameters are recapitulated here:

•

The pressure drop, that is the pressure difference between inlet and outlet of the porous MCHS

•

The average heat transfer coefficient of the MCHS, given by

(2)

with the average wall temperature at the bottom centerline

•

The Reynolds number is defined as

(3)

with the hydraulic diameter Dh (m) that is defined based on the length and width of the free flow channel, lf and wf respectively, as follows:

•

The Nusselt number describes the ratio of convective to conductive heat transfer according to

(4)

where kf is the fluids thermal conductivity.

•

The Figure of Merit (FOM) compares the performance of two different designs with the following expression:

(5)

The index base refers to the values for the MCHS without the porous structure and Ω = uinlfwfΔp is the pumping power.

Equation 1 is valid for 1 ≤ Re ≤ 1000. An estimation of the Reynolds number (Equation 3) results in Re ∼ 300 such that the choice of the Brinkman-Forchheimer equation is valid.

When the FOM reaches its maximum value, the performance of the porous MCHS is best compared to a standard MCHS without any porous layer. Therefore, the easiest way to use the Optimization study feature is to use the FOM as the objective function and to use as the objective type.

Results and Discussion

The result of the optimization study matches the results of the parametric study in the original model. The best performance compared to a standard MCHS without any porous layer is reached for a porous layer thickness of 0.1 mm. To compare the results of the optimization study with the parameter study performed in the model Performance of a Porous Microchannel Heat Sink, the results for the average pressure drop, the average heat transfer coefficient, and the dimensionless Reynolds and Nusselt numbers are plotted as extra points in the already existing table plots. The extra points are marked with an asterisk. Figure 1 shows that the values for pressure drop and heat transfer coefficient lie well on the curve showing both as a functions of the porous layer thickness.

Figure 1: Pressure drop and average heat transfer coefficient. The results from the parameter study where the porous layer thickness varies from 0.05 to 0.2 mm are taken from the original model “Performance of a Porous Microchannel Heat Sink”. The results of the optimization study are added with an asterisk.

The dimensionless numbers — the Reynolds number and the Nusselt number — also agree well, as can be seen in Figure 2. Figure 3 shows the Figure of Merit and again the optimized value fits perfectly well to the values of the parameter study.

Figure 2: Reynolds number and Nusselt number as a function of porous layer thickness. The results are taken from the original model “Performance of a Porous Microchannel Heat Sink”. The results of the optimization study are added with an asterisk.

Figure 3: The Figure of Merit comparing the performances of the porous and the conventional MCHS. The result of the optimization study was added as an asterisk.

Figure 4 shows the velocity field in a cross section of the channel. The velocity magnitude inside the porous structure is small (dark blue) compared to that of the free flow channel. The pressure along the streamlines is plotted in gray scale. Compare this figure to the corresponding figure in the main model Performance of a Porous Microchannel Heat Sink.

Figure 4: Cross section of the velocity field along the channel (scaled view). The dark blue color indicates the porous structure, because the velocity magnitude is small in this area. The gray scale indicates the pressure along the velocity streamlines.

Notes About the COMSOL Implementation

Note that in COMSOL Multiphysics it is not only possible to use minimization of an objective function, which is the usual way, but also maximization. This makes it possible to just enter the Figure of Merit as the objective function.

Porous_Media_Flow_Module/Heat_Transfer/porous_microchannel_heat_sink_optimization

Modeling Instructions

Root

Start by loading the model file that contains the setup of the microchannel heat sink (MCHS) model with and without porous layer.

From the menu, choose .

Browse to the model’s Application Libraries folder and double-click the file porous_microchannel_heat_sink.mph.

Most of the model parameters are already present after loading the file. Add a few more parameters for the optimization, like the heat transfer coefficient and the pumping power of the reference MCHS. Both values can be found in Table 1.
In the , expand the and then the node and click on to look up the heat transfer coefficient and the pumping power values.

Global Definitions

Now enter these values in the parameters list

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
h_ref	8758[W/m^2/K]	8758 W/(m²·K)	Heat transfer coefficient, reference value
Omega_ref	1.524e-5[W]	1.524E-5 W	Pumping power, reference value

Add a new variable, the figure of merit, to the variable list. In the original model this parameter was only calculated for postprocessing, now it is needed as objective function for the optimization.

Component 1 (comp1)

In the window, expand the node.

Definitions (comp1)

Variables 1

In the window, expand the node, then click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
FOM	h_mchs/h_ref/(Omega/Omega_ref)^(1/3)		Figure of Merit

To keep the results of the original model, add a new study including optimization.

Add Study

In the toolbar, click

to open the window.

Go to the window.

Find the subsection. In the tree, select .

Click

In the toolbar, click

to close the window.

Study 3

Optimization

In the toolbar, click

In the window for , locate the section.

From the list, choose , because the BOBYQA solver is generally the fastest of the derivative-free solvers when the objective function is smooth. Reduce the to reduce the simulation time further.

In the text field, type 0.02.

Locate the section. In the table, enter the following settings:


Expression	Description	Evaluate for
comp1.FOM		Stationary 2

From the list, choose .

Locate the section. Click

In the table, enter the following settings:


Parameter name	Initial value	Scale	Lower bound	Upper bound
th_porous (Porous structure thickness)	0.1[mm]	0.1[mm]	0.05[mm]	0.2[mm]

Step 1: Stationary

In the window, click .

In the window for , click to expand the section.

In the table, enter the following settings:


Geometry	Mesh
Geometry 1	Mesh 1

In the toolbar, click

Results

Velocity (spf) 2

Create a cross-section plot of the velocity (Figure 4) to compare it with Figure 3 of the original model. Therefore create another dataset by following the steps below:

Cut Plane 3

In the window, expand the node.

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Velocity, Cross Section, Optimization

In the window, right-click and choose .

In the window for , type Velocity, Cross Section, Optimization in the text field.

Locate the section. From the list, choose .

Click the

button in the toolbar.

Global Evaluation 2

To analyze the performance of the optimized porous MCHS and to compare it to the solutions of the model ’Performance of a Porous Microchannel Heat Sink’, duplicate the node and apply the new dataset.

Global Evaluation 4

In the window, expand the node.

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. In the table, enter the following settings:


Expression	Unit	Description
FOM	1	Figure of Merit

Click

next to

, then choose .

Heat-Transfer Coefficient and Pressure Drop

Add the values for the heat transfer coefficient, the pressure drop, the dimensionless Reynolds number, the Nusselt number, and the figure of merit as points in the existing table graphs.

Table Graph 3

In the window, right-click and choose .

In the window for , locate the section.

From the list, choose .

In the list, select .

Click to expand the section. Find the subsection. From the list, choose .

In the text field, type 1000. This is necessary because the porous layer thickness as control parameter of the optimization is saved in m whereas in the existing table it is plotted in mm.