In this chapter the modeHing of microchannels will be discussed. Section 2.4.1 is an introduetion into modeHing of microchannels in which two models are explained. The remainder of section 2.4 discusses one of these models in more detail.
2.4.1 Introduetion
The thermal and hydraulic behaviour of microchannels is modelled. The goal is to
determine whether a model can be used to predict the thermal and hydraulic properties of microchannel cooling. A model that accurately prediets the cooling performance can be useful in optimising the cooling for a certain product geometry.
Two models have been analysed by Aubry [7].
One of thesemodelsis adapted from Bau and Weisberg [8]. In this model only one channel is considered. The geometry is depicted in tigure 2-6. Only half the channel is modelled because of symmetry of the channel. The bottorn is assumed to be insulated because of the low thermal conductivity of glass. At the top a uniform heat flux is applied. The flow inside the channel is assumed to be hydraulically and thermally fully developed. Furthermore channel walls are assumed to be smooth. Por both the velocity and the temperature field analytical formulas are solved by using a Matlab code[7].
,.. .. ;··, , L __ _...
A - A
~
...• tt!tn
... ~./
"
A - A
Figure 2-6 Microchannel geometry in analytica/ model of Bau
/
The second model is made with the commercial software package Flotherm. Flotherm can solve complex heat transfer probieros that occur in microelectronics. This is done by solving the conservation equations for momentum, thermal energy and mass. These equations are then solved by means of numerical integration. The complete domain for which the equations have to be solved is subdivided into small volumes. For each of these volumes the conservation equations are expressed in algebraic form. After setting the boundary conditions and other specitications, such as fluid and material properties, the problem can be solved.
The geometry, as depicted in tigure 2-7, is simulated in Flotherm. A uniform flow, insteadof a developed flow, is applied at the entrance since the fluid enters the channels from a large reservoir. Furthermore all material properties, such as conductivity and viscosity, are linearly temperature dependent, instead of constant, as is the case with the model presented by Bau. Since fluid properties depend on temperature, and often linearly on a certain temperature domain, the Flotherm approximation will be better than the analytica! model.
Moreover in the analytica! model the assumption is made that the flow is both hydraulically and thermally fully developed. The hydraulic entrance length can be calculated by [9]:
v·D2 ze
=
0.05. ReD . DH=
0.05*--H-H V (2.12)
In the experiments using water as a cooling fluid the Reynolds number ranges from 80-1200, where the Reynolds number is detined as:
Re= v·Dh =p·v·Dh (2.13)
V J1
where visthefluid mean velocity in the microchannels (m/s), Dh the hydraulica!
diameter of a microchannel, v the kinematic viscosity ofthe fluid (m2/s), p the fluid density (kg/m3) and J..l the total viscosity (Nm/s2).
The defmition of the hydraulica! diameter is:
Dh = -4·A (2.14)
p
where A is the cross sectional area of one channel (m2) and P the perimeter of a channel (m).
For a mean value of Re of 700 this results in a hydraulic entrance length of 5.25 mm, which is about one third ofthe total channellength. For higher Reynolds numbers the entrance length is even longer.
Aubry[7] performed simulations with a geometry, comparable to tigure 2-7. At a flow rate of 1 V min the flow was not even fully thermally developed at the end of the channel.
Neglecting the influence ofthe thermalentrance length would lead to an overestimation of the thermal resistance. Aubry [7] showed a 16 % higher thermal resistance for the analytica! model, compared to Flotherm simulations. He contributed the difference to the assumption of fully developed flow in the analytica! model. When camparing to
performed experiments the Flotherm model was in better agreement than the analytica!
model.
Purthermore in Plotherm it is possible to simulate more complex situations, as will be the case when exploring the possibilities of microchannel cooling of an actual device.
Therefore simulations will be done with Plotherm instead of the analytica! model.
2.4.2 Flotherm modelling
Simulations done with Plotherm will be compared to experiments. This is done to determine whether model and experiments are in agreement. Since simulating the experimental geometry with multiple channels is very time consuming
(> 1 day CPU time), it is worthwhile to explore the possibilities of single channel
simulations. lf single channel simulations prove to be accurate the simulation time can be reduced to approximately 1 hour CPU time.
2.4.2.1 Single channel geometry
In tigure 2-7 the geometry of a single channel is given. The dimensions are the same as in the experiment, except for the fin width. This is only half ofthe experimental value since symmetry boundary conditions are applied at both sides. The inside walls of the channel are covered with a 100 nm layer of Si
02
to create a hydrophilic surface.Because the Reynolds number in the performed calculations is below 2300 the flow type is assumed to be laminar.
The topside of the channel is uniformly heated over a distance of 1 cm, as in the case of the experimental device.
The fluid flow at the entrance is assumed to be uniform over the cross section of the channel since the fluid enters the channel from a large reservoir.
In the experiment, the thermal resistance is determined by measuring the supplied power
·and a temperature difference, as explained insection 2.3.4.2.
~:
... .'.:· . .'·.: .. ·.;_:::.~.:-... ···- ...
···-
~-: '·'Figure 2-7 Cross section (left) and top view (right) ofthe single channel model geometry
The heater temperature is determined in Plotherm calculating the mean temperature of the heater. The value of Tj-out is determined by calculating the mean temperature of a small area on the silicon at the place where the thermocouple is approximately positioned, as indicated in tigure 2-7.
Physical properties of water, air and solids have to be defined befere launching the simulation. Values ofviscosity and thermal conductivity are modelled as linearly temperature dependent, the value of specitic heat can only be detined as a constant. The data for waterviscosity on the temperature range used are within 9% accuracy. All other data is within 3 % accuracy of the souree values. More information can be found
appendix E.
2.4.2.2 Multichannel geometry
The multichannel geometry is modelled as depicted in tigure 2-8. The main differences with the single channel model are the addition of channels and the non-uniform heat source. Purthermere the inlet and outlet hole of the microchannel structure have been added. These holes are square instead of round as in the experiment since Plotherm can only model rectangular surfaces. Only half of the experimental geometry is modelled.
This is possible because of the symmetry of the setup and has the advantage of saving computation time.
/
n n - '
~~ r j
/ : !=
\-' \ ..
~~Î ~~
.. )
~-- .1 I
! i
; i
Figure 2-8 Cross section (left) and top view (right) of rnultichannel model geornetry