Eindhoven University of Technology MASTER Single and two-phase microscale cooling Eummelen, E.H.E.C.

Eindhoven University of Technology

MASTER

Single and two-phase microscale cooling

Eummelen, E.H.E.C.

Award date:

2004

Link to publication

Disclaimer

This document contains a student thesis (bachelor's or master's), as authored by a student at Eindhoven University of Technology. Student theses are made available in the TU/e repository upon obtaining the required degree. The grade received is not published on the document as presented in the repository. The required complexity or quality of research of student theses may vary by program, and the required minimum study period may vary in duration.

Begeleider: A.J.H. Frijns C.C.S. Nicole . Afstudeerhoogleraar:

A.A. van Steenhoven

Technische Universiteit Eindhoven Faculteit Werktuigbouwkunde Divisie Thermo Fluids Engineering

Single and two-phase microscale cooling

Erik Eummelen Rapportnummer WET 2004.17

Abstract

As microelectranies become faster and smaller, the power density increases and the generated heat becomes a problem. In addition, the area available for heat remaval decreases. As a result, thermal management becomes increasingly critica! to the

electranies industry. Power densities ranging from 80-200 W/cm² are expected for future consumer applications while the maximum temperature should be maintained below 120

oe.

Two cooling techniques have been investigated, namely single phase forced convection through mierocharmels and the pulsating heat pipe, which is a two-phase cooling method.

The goal of this research is to determine whether these two techniques can play a role in cooling of future microelectronics.

Mierocharmels have been etched in silicon. Water and air are employed as cooling fluids.

With water as cooling fluid 105 Watts could be dissipated on a square centimetre. The thermal resistance varied from 0.32 KIW at a flow rate of 0.075 1/min and a pressure drop of 0.1 bar to 0.05 KIW at a flow rate of 11/min (pressure drop 2.6 bar).

Water cooling of the microcharmel structure is also simulated with Flotherm.

Experiments and calculations show good agreement. The temperature distribution calculated with Plotherm showed a maximum temperature of 57

oe

at a flow rate of 0.1 1/min, which is much lower than the maximal allowable temperature. This makes microcharmel cooling with water as cooling fluid an attractive possibility for cooling of microelectronics.

Air has also been used as cooling fluid sirree the combination of water and electranies can cause hazardous situations. The minimal measured thermal resistance is 1.8 KIW at a flow rate of 33 1/min (pressure drop 1.2 bar). The thermal resistance is too high to maintain the maximum temperature below 120

oe

for power of80 or more on a square centimetre.

The second technique that is investigated is the pulsating heat pipe. The main advantage of a PHP is the fact that no extemal pump is required. The fluid motion is caused by thermally driven pressure differences within the system.

Experiments have been conducted on a pulsating heat pipe (PHP) structure made in aluminium.

In vertical operation the thermal resistance is 40 % lower when compared to the thermal resistance of an unfilled heat pipe. The lowest measured thermal resistance is 2.4 KIW. In this case the heat souree temperature is 85

oe

at a rower input of 28

w

on a surface of 5 cm²• Power densities in the order of 80-200 W/cm are not achievable while maintaining the maximum temperature below 120

oe.

It is recommended to focus further research on mierocharmels cooling with water as cooling fluid. More information on design constraints, such as maximal allowable flow rate and pressure drop, should be gained. Further research can then be conducted to optimise the mierocharm el structure. Height and width of the channels can be optimised

to reduce the thermal resistance. Another interesting possibility is the use of non- rectangular channels.

Contents

Abstract 1

---

List of figures 5

Nomendature 7

1 Introduetion 9

---

2 Single Phase cooling: Microchannels 11

2.1 Introduetion 11

2.2 Heat exchange in Microchannels 11

2.2.1 Conductive thermal resistance, Rcand 13

2.2.2 Convective thermal resistance, ^Rcanv 13

2.2.3 Capacitive thermal resistance, Rcap 13

2.3 Microchannels experimental setup 14

2.3.1 Channel test structure 15

2.3.2 Measurement equipment 18

2.3 .2.1 Pressure drop 18

2.3.2.2 Flow rate 18

2.3.2.3 Dissipated power 18

2.3.2.4 Temperature measurements 19

2.4 Modelling of microchannel cooling device 21

2.4.1 Introduetion 21

2.4.2 Flotherm rnadelling 23

2.4.2.1 Single channel geometry 23

2.4.2.2 Multichannel geometry 24

2.5 Microchannel cooling results 26

2.5.1 Comparison single channel vs multichannel 26

2.5.2 Water measurements and calculations 29

2.5.3 Air measurements and calculations 33

2.5.4 Discussion ofthe results 36

2.5.4.1 Comparison of air and water cooling for experimental geometry _ _ 36

2.5.4.2 Consequences of geometrical variation 39

2.6 Conclusions and recommendations 41

--- 3 Two-phase cooling: Pulsating heat pipe _ _ _ _ _ _ 44

3.1 Introduetion 44

3.1.1 Flow within the pulsating heat pipe 46

3.1.2 Heat throughput of pulsating heat pipe _ _ _ _ _ _ _ _ _ _ 46

3.2 Modelling of pulsating heat pipe _ _ _ _ _ _ _ _ _ _ 47

3.2.1 Description ofthe model 47

3.2.1.1 Momenturn equation 48

3 .2.1.2 Continuity equation 52

3 .2.1.3 Energy equation 52

3 .2.1.4 Numerical procedure 53

3.2.2 Results 53

3.3 Pulsating heat pipe experiment 57

3.3.1 Experimental setup 57

3.3.1.1 Platewithpulsatingheatpipe 58

3.3 .1.2 Evaparator 60

3.3.1.3 Condensor 60

3.3.1.4 Filling device 61

3.3 .1.5 Thermal resistance measurement 63

3.3.2 Experimental results on large scale PHP 64

3.3.2.1 Flow visualization: comparison experiment and model 64 3.3.2.2 Effect ofpulsating motion on heat throughput 66

3.3.2.3 Fluid type 68

3.3.2.4 Inclination angle 69

3.3.2.5 Filling ratio 71

3.3.3 Mieraseale silicon pulsating heat pipe 71

3.4 Conciosion and recommendations 73

4 General conclusions and recommendations 76

References 78

Appendix A: Fabrication process of microchannels 81

Appendix B: Pressure losses 84

Appendix C: Heat losses 87

Appendix D: Reater eaUbration 90

Appendix E: Fluid and solid properties in Flotherm 91 Appendix F: Influence of higher flow through centre channels

on temperature 95

Acknowledgements 97

List of figures

Figure 2-1 Schematic overview of microchannels in silicon ... 12

Figure 2-2 Schematic overview of water measurement test setup ... 15

Figure 2-3 Dimensions of the microchannel test structure ... 16

Figure 2-4 Top (left) and bottorn (right) view of the test structure assembly ... 17

Figure 2-5 Power measurement of heater and position of thermocouples ... 19

Figure 2-6 Microchannel geometry in analytica! model of Ba u ... 21

Figure 2-7 Cross section (left) and top view (right) of the single channel model geometry ... 23

Figure 2-8 Cross section (left) and top view (right) of multichannel model geometry ... 24

Figure 2-9 Calculated thermal resistance for water simulations ... 26

Figure 2-10 Calculated thermal resistance for air simulations ... 27

Figure 2-11 Temperature distribution of air cooled (top) and water cooled geometry (bottom) ... 28

Figure 2-12 Pressure drop as a function of water flow rate ... 30

Figure 2-13 Rthheater as a function of water flow rate; P=105 W ... 31

Figure 2-14 RthTj-out as a function of water flow rate; P=105 W ... 32

Figure 2-15 Pressure drop as a function of air flow rate ... 33

Figure 2-16 Rth Tj-out as a function of air flow ra te; P=13.5 W ... 34

Figure 2-17 Absorbed power/input power as a function of air flow rate ... 35

Figure 2-18 Comparison between air and water cooling as a function of pressure drop ... 36

Figure 2-19 Definition of wall, bulk and heater temperature ... 37

Figure 2-20 Contributions of conductive, convective and capacitive resistance to maximum heater temperature with water as cooling fluid; P=105W ... 38

Figure 2-21 Contributions of conductive, convective and capacitive resistance to maximum heater temperature with air as cooling fluid; P=13.5 W ... 39

Figure 2-22 SEM-microphotograph of an array of crosses fabricated in silicon .. .40

Figure 3-1 Schematic overview of a pulsating heat pipe ... .44

Figure 3-2 Heat pipe construction and operation [13] ... .45

Figure 3-3 One-dimensional model of the pulsating heat pipe ... .48

Figure 3-4 Definition of the height difference dh ... .49

Figure 3-5 Advancing (Sa) and receding (Sr) contact angles of liquid slug ... .50

Figure 3-6 Contact angles as a function of liquid slug velocity ... 51

Figure 3-7 Resulting frequency as a function of fluid parameters ... 55

Figure 3-8 Porces acting upon liquid slug ... 56

Figure 3-9 Geometry of the alnminurn pulsating heat pipe ... 58

Figure 3-10 Photograph of pulsating heat pipe assembly ... 59

Figure 3-11 Position of evaparator (heater) and condensor (cooling flow) ... 60

Figure 3-12 Schematic overview of the filling setup ... 61

Figure 3-13 Photograph of pulsating heat pipe filling setup ... 63 Figure 3-14 Squeezing effect in the experiment ... 65 Figure 3-15 Mean evaporator temperature as a function of dissipated power .... 67 Figure 3-16 Thermal resistance as a function of input power for various fluids;

FR=45%, vertical orientation ... 68 Figure 3-17 Definition of incHnation angle

<I> ...

69 Figure 3-18 Thermal resistance as a function of power at various incHnation

angles (FR=20%; ethanol) ... 70 Figure 3-19 Thermal resistance as a function of input power at different filling

ratios (Ethanol; vertical orientation) ... 71

Figure 3-20 Silicon wafer with 4 pulsating heat pipe structures ... 72

Nomenclature

Notations

A Area (mz)

Bo Bondnumber (-)

Cp Specific heat at constant pressure (JikgK)

Cv Specific heat at constant volume (JikgK)

dh Height difference (m)

d Diameter (m)

Dh Hydraulic diameter (m)

f

Friction factor (-)

F View factor (-)

FR Filling ratio (%)

g Gravitational constant (m/sz)

h Enthalpy (Jikg)

he Conveetien coefficient (W/m²K)

hrg Latent heat (JikgK)

k Thermal conductivity (W/mK)

Ke Exit coefficient (-)

Kc Entrance coefficient (-)

L Length (m)

Lc Characteristic length (m)

m Mass (kg)

m Mass flux (kg/s)

Nu Nusselt number (-)

Q Power (W)

r Radius (m)

Y],YJJ Principle radii (rad)

R Gas constant (JikgK)

Rth Thermal resistance (KIW)

Ra Rayleigh number (-)

Re Reynolds number (-)

p _{Pressure (N/mz)}

L1P Pressure drop (N/mz)

Pr Prandtl number (-)

T Temperature (K)

t Time (s)

u Intemal energy (Jikg)

V Velocity (mis)

V Volume (m3)

v

V olumetric flow rate (m3/s)

x

Position (m)

Ze Hydraulic entrance length (m)

Greek symbols

a Thermal diffusivity (m2/s)

f3

Thermal expansion coefficient (liK)

ê Emissivity (-)

rjJ Inclination angle (0)

J.1 Total viscosity (Nm/s2)

V Kinematic viscosity (m2/s)

(} Contact angle (rad)

p Density (kg/m³)

(J Stefan Boltzmann constant (W/m2K⁴⁾

Gt Surface tension (N/m)

<Jf Flow area ratio (-)

't Shear stress (N/m2)

Subscripts

a Advancing

abs Absorbedd

c Cold

cap Capacitive

co Condensor

cond Conduction

conv Convection

ev Evaparator

f Fluid

h Hot

heater Reater

m Inlet

j-out Just behind heater

1

Liquid

0 Object

out Outlet

r Receding

ref Reference

V Vapor

00 Surrounding

Abbreviations

FR Filling ratio

PHP Pulsating heat pipe

1 Introduetion

As integrated circuits become faster and more densely packed with transistors, the power density increases and the generated heat becomes a problem. Heat fluxes ranging from 80-200 W/cm² are predicted, while the maximum temperature of electronic components may not exceed 120

oe.

In addition, the area available for heat removal decreases. As a result, thermal management becomes increasingly critica! to the electranies industry.

The conventional cooling methods will eventually become inadequate. Therefore new cooling techniques will have to be explored. Many promising alternative cooling

techniques involving liquid as a coolant exist [ 1 ], and their methods of heat transfer differ widely. Both single and two-phase cooling techniques are attractive. In this report one cooling technique out ofboth categones will be discussed.

The single phase cooling technique that is explored is forced convection through mierocharmels with the help of an extemal pump. Mierocharmels are channels with a hydraulic diameter in the order of 1-1 OOOJ..Lm [2]. It is shown that large heat flux es are possible without exceeding critica! temperatures [3]. Furthermore mierocharmels can be integrated into the microelectranies itself, which means that it does not require a lot of space.

The two-phase cooling technique that is studied is the pulsating heat pipe. The pulsating heatpipeis a relatively new device, basedon heat pipe technology. Although heat pipe theory is advanced nowadays, the working principles of the pulsating heat pipe are still unknown [ 4]. The pulsating heat pipe consists of a closed tube, which meanders back and forth from one side to the other. In the tube a working fluid is present in both liquid and vapour state. When subjected toa temperature difference self excited oscillation ofthe working fluid occurs. Hence no extemal pump is needed, in contrary to forced convection microchannel cooling. Due to the oscillating motion of the fluid heat is transported from one side to the other. The studies done so far on cooling capabilities give no decisive answer whether this device is can be used in microelectronics cooling.

The goal is to determine whether these two cooling techniques can play a role in cooling of future microelectronics in Philips products. To this end the cooling performance of both techniques will be investigated. However it is noted that the eventual feasibility depends on many factors, such as the available space and the availability of suitable components, such as pumps.

Todetermine the cooling capabilities experiments are performed on both the

microchannels and the pulsating heat pipe. Furthermore numerical calculations will be done on microchannel cooling to predict the cooling performance. An accurate model can be useful for future practical application and optimisation of microchannel cooling.

An attempt is also made to model the fluid behaviour inside a pulsating heat pipe.

After an introduetion in chapter 1, chapter two discusses microchannel cooling. Aftera short introduetion the experimental setup, which is used to measure the cooling

performance of microchannels, is discussed. This is foliowed by a discussion of the model. The results ofthe modeland experiments will be compared. The chapter ends with conclusions and recommendations.

Chapter three is devoted to the pulsating heat pipe. An introduetion is given to explain the general principle ofthe pulsating heat pipe. This is foliowed by a discussion of a model of the pulsating heat pipe and results obtained with the model.

Subsequently the experimental setup of the pulsating heat pipe is explained. A

comparison is made between the fluid behaviour in model and experiment. Experimental results on the cooling performance are also discussed. The chapter ends with conclusions and recommendations.

The fourth chapter contains conclusions and recommendations for further research on both subjects.

2 Single Phase cooling: Microchannels 2.1 Introduetion

A traditional way of cooling microelectronics is by forced single phase convection. Air is then blown over a heat sink, which is placed on top of the heat source. Another way of cooling is forced conveetien through microcharmels. These are very small charmels with a hydraulic diameter in the order of 1-1000 11m.

Higher heat fluxes can be obtained by forced conveetien through microcharmels. For example Gillot et al. [ 5] compared forced air conveetien to liquid microcharmel cooling.

The heat flux was about 10 times higher in case of mierocharm el cooling while reaching the same maximum temperature; 180 W /cm² could be dissipated with mierocharmels cooling.

In this chapter the functionality of forced single phase cooling through mierocharmels is examined. Sectien 2.2 describes the basic principles and geometry of the mierocharm el structure. Subsequently the experimental setup is dealt with in sectien 2.3. The model used to analyse the thermal performance of the mierocharmels is explained in sectien 2.4.

Results ofboth experiments and model calculations are given in sectien 2.5. Conclusions and recommendations are given in sectien 2.6.

2.2 Heat exchange in Microchannels

Mierocharmels can be used to cool microelectronics. By creating several charmels next to each other, a large heat exchanging area, comprised in a small volume, is obtained. The mierocharmels can be machined'into for example an aluminium plate, which is positioned on top of the heat source. However when aluminium is used intermediate layers are present, such as adhesives. These intermediate layers act as thermal harriers, which reduce the cooling performance of such a system.

However the microsize charmels can also be made in silicon. The charmels are then integrated into the chip material and very close to the heat source, without any intermediate layers.

In tigure 2-1 a schematic overview is depicted of a microcharmel structure in silicon. The charmels are separated from each other by fins. The material between the heat souree and the channels is generally called the diffuser, in which heat can spread.

Mierocharmels in silicon will be investigated because of its low thermal resistance.

The overall thermal resistance can be split up in several parts:

1. The thermal resistance due to conduction through the silicon, ^Rcond·

2. The convection resistance, which represents the heat exchange between the microchannel walls and the fluid, ^Rconv·

3. The capacitive resistance, which represents the increase of the fluid temperature between the channel inlet and outlet, ^Rcap·

Heat souree

Glass

Figure 2-1 Schematic overview of microchannels in silicon

2.2.1 Conductive thermal resistance, Rcond

Two cases can be distinguished regarding conductive resistance:

1. The heat flux is unidirectional. The heat flux is only unidirectional when the heat dissipation in the heat souree is uniform, and the heat souree has the same size as the micro-cooler. In the simple case of a monolayer diffuser, the conductive resistance can be expressed as:

L

Rcond =kA' (2.1)

where L the diffuser layer thickness (m), k the thermal conductivity (W /mK) and A the diffuser area through which the heat flux goes (m²). When the diffuser contains several layers, the conductive resistance can be calculated with:

n n L

R = "R = " - i

cond

f;:

^{cond ,i}

i::

^{kiA · (2.2)}

2. The heat flux is 2-Dimensional. This case appears when the heat souree is smaller than the cooler or when the heat dissipation in the heat souree is not uniform.

2.2.2 Convective thermal resistance, Rconv

Heat transfer by means of convection represents heat exchange between the channel walls and the fluid. Convective thermal resistance can be expressed as:

R = -1

- (2 3)

conv hA ' ·

)J-ch

where h the heat transfer coefficient for convection (W/m²K) and Ap-ch (m²) the surface of the microchannels. The heat transfer coefficient for convection depends on numerous parameters such as convection mode (forced or free), flow characteristics (laminar or turbulent), thermal and hydraulic properties etc.

2.2.3 Capacitive thermal resistance, Rcap

Contrary to the former thermal resistances, which deal with temperature differences parallel to the heat flux, this resistance is about the fluid temperature increase between the channel inlet and outlet, perpendicular to the heat flux. This temperature rise is due to the heat quantity absorbed by the fluid while it is flowing through the microchannels. The capacitive thermal resistance can be calculated with:

1

Rcap

= - . -'

(2.4)

pVcP

where p the density ofthe fluid (kg/m³) ,

V

the volumetrie flow rate (m³/s) and cp the specific heat of the fluid (JikgK).

The overall thermal resistance is composed of the above mentioned resistances. Which of these resistances is dominant depends on factors such as geometry, flow rate and coolant fluid.

2.3 Microchannels experimental setup

An experimental setup has been built to test the cooling capabilities of microchannel cooling. Water and air have been chosen as coolant fluids. Water is chosen because of its high specific heat and thermal conductivity. However the combination of water and electronics can cause hazardous situations. Therefore the possibilities of air cooling through microchannels have also been investigated.

The overall thermal resistance of the microchannels is a function of the thermal

resistances as described in the previous section. Due to the capacitive resistance the total thermal resistance will be a function of the cooling flow. Furthermore a certain pressure drop, dependent on flow rate, is present along the microchannels due to friction. The pressure drop as a function of flow rate can give information about the flow type (laminar or turbulent). Since the flow type affects cooling performance the pressure drop will also be measured.

In condusion this means that the following data have to be obtained:

•

•

•

•

two temperatures dissipated power cooling flow rate pressure drop

The experimental setup for water as cooling medium is depicted in tigure 2-2. The cooling flow is obtained by pumping demi waterfroman open reservoir. The water is first directed through a heat exchanger to ensure a constant inlet temperature, set by a thermal bath (NESLAB, RTE-221). A 0.4!Jm filter (PALL, DFA4001 J012) is used to filter out any particles, which can be present in the water to prevent contamination of the channels. After the water has passed through the channels it is fed back to the reservoir to close the loop.

Thermal bath

Filter (0.4 Jlm)

Microchannels {

IIIBI!

14--+- Heat exchanger

Heater

Figure 2-2 Schematic ovefl!iew of water measurement test setup

When air is used as cooling medium the pump is replaced by pressurized air. A valve before the manometer controls the air flow. After the microchannel structure a flow meter is placed to measure the air flow. Since a closed loop is not necessary the reservoir and thermal bath are not present in the air flow measurement.

2.3.1 Channel test structure

The test structure consistsof several parts, which are represented in tigure 2-3. The main part is a layer of silicon, with a thickness of 600 Jlm, in which channels have been etched.

The channels, 75 in total, are 1.5 cm long and have a width of 100 11m and adepthof 300 Jlm. At the beginning and end of the channels a reservoir has been made.

A glass plate is glued on top of the silicon to allow visual inspection. Conneetion tubes are subsequently glued on the glass cover plate. A description of the fabrication process can be found in appendix A.

J.lffi

Cróss-section 1 Glass

Ctoss.,.section 2

.. 4mm ..

^Glass

Figure 2-3 Dimensions of the microchannel test structure

Since measuring thermal resistance is the objective, a heat souree has to he applied to the silicon. Perret[6] made use of an electrical film heater. A layer ofthermal grease was used between the silicon and the heater. Quantitative results could not he obtained because the measured thermal resistance depended largely on the properties of this thermal grease layer.

In order to measure the thermal resistance of the microchannel structure itself such intermediate layers have to he avoided. Therefore a heater is directly deposited onto the silicon. The fabrication process of the heater is also described in appendix A. The eventual test device is shown in tigure 2-4.

Figure 2-4 Top (left) and bottorn (right) view ofthe test structure assembly

2.3.2 Maasurement equipment

Several data have to be obtained to characterize the cooling behaviour of the

microchannels. The way these are obtained are independently described in this paragraph.

2.3.2.1 Pressure drop

The pressure drop of interest is the pressure drop over the microchannels. Therefore a pressure gauge is fitted as close as possible before the channels. The measured pressure can be expressed as:

/1Pmeasured

=

f..F;ubing ,in + ^M.ntrance + f..F;eststructure + ^/1Pexit + f..F;ubing ,out (2.5) In appendix B more information is given on the pressure drop measurement. The pressure drop over the tubing is proven to be negligible. Furthermore it is shown that the influence of entrance and exit effects should be considered when comparing to the pressure drop calculated with standard formulas for laminar channel flow.

2.3 .2.2 Flow rate

When using water as cooling fluid the flow rate is determined by measuring the volume of the fluid, which is collected in a certain amount of time. The water volume is

determined by the sealing on a graduated cylinder. Dividing the volume through the period of time results in the flow rate. The accuracy of the flow measurement is determined from the accuracy of the time measurement and volume reading:

~V ~t ~V - = - + -

V t V ^(2.6)

where ~V,~t and L1Vrepresent the errors in flow rate

V

(Vmin), timet (min) and volume V(l). For all measurements the accuracy is within 3 %.

When determining airflow a different approach is used. The flow rate is then measured by a calibrated air flow meter (Fischer & Porter,10A221632NA2C), which is positioned at the outlet ofthe channels. The extra pressure drop, which is relatively small (0.01 - 0.02 bar; depending on the flow rate) compared to the pressure drop over the channels, will be subtracted from the measured pressure drop. The accuracy of the flow meter is 5 % of full scale.

2.3.2.3 Dissipated power

Heat is dissipated by the copper heater that is deposited directly on the silicon. The power can be obtained by measuring the supplied voltage (Keithley 2000 multimeter) and current (Philips 2535PM multimeter) as indicated in figure 2-5.

Voltmeter

Figure 2-5 Power measurement ofheater and position ofthermocouples

As can beseen from tigure 2-5 the voltmeter measures the voltage over a part ofthe heater; the outer strips on both si des of the heater are not included in the voltage measurement. When comparing a voltage measurement over the whole heater with the measurement as in tigure 2-5 the voltage over the whole heater is 1.20 times the voltage measured as in tigure 2-5. Because of practical considerations the voltage is measured as depicted in the tigure and afterwards multiplied by 1.20.

The power that is supplied by the heater is not necessarily equal to the power that is absorbed by the fluid flow because of heat losses. Heat losses are estimated in

appendix

c.

It is seen that when heating the microchannel structure to 80

oe

^{the total}

amount of heatlossis about 5 W.

2.3.2.4 Temperature measurements

The cooling flow cools the silicon structure, which is heated by the copper heater. To detine a thermal resistance a temperature difference is needed. This temperature difference is the difference between a temperature of the cooling flow and one of the heated structure:

(2.7) where Rth is thermal resistance (KIW), Th temperature ofthe hot side (K), Tc temperature on the cold side (K), and

Q

the power (W).

The inlet temperature is used as the temperature at the cold side. This is done to include the effect of rising fluid temperature along the channel, i.e. the capacitive thermal resistance.

The temperature of the heated structure is not uniform. It will be lower at positions far away from the heater. Even along the heater a temperature gradient exists because the temperature of the cooling flow increases. This temperature gradient is dependent on the applied power and coolant flow rate.

Two different approaches have been chosen to measure a temperature at the 'hot' side.

1. A thermocouple is placed closely to the heater at the outlet side, see tigure 2-5.

The thermal resistance of the structure is then detined as:

R _ (~-out - T;n)

thT]-out - Q (2.8)

where 1J-out is the temperature (⁰C) measured by the thermocouple directly after the heater, Tin the waterinlet temperature (⁰C), and Q the applied power (W). All thermocouples are K-Type ehromel & Alumel, range: -270 to 1372 K. The thermocouples have been calibrated over the temperature range from 20 to 80 oe (temperature range in experiments) by the use of a calibration apparatus. When

comparing the average value of thermocouple Tin to the others values the difference over the whole range is within 0.12 oe. All thermocouples are connected to a computer via a Fluke hydra data logger.

2. The electrical resistance of the heater is measured. From the value of the electrical resistance a temperature can be deduced, since there is a linear relation between electrical resistance and temperature. For more information on heater calibration, see appendix D. The thermal resistance can then be defmed as:

R _ (Theater - T;n)

thheater - Q (2.9)

where Theater is the temperature (⁰C) calculated by measuring the electrical resistance of the heater.

It is noted that the detinitions used in this report forthermal resistance differ from defmitions normally used in textbooks. Normally the thermal resistance as follows:

R

=(T

²

-~)

(2.10)

th

Q

where T2 (K) is the temperature at surface 2 and T1 (K) the temperature at surface 1, and

Q

the heat flowing from one place to the other. In this report the total power input

Q

is used.

The power that is absorbed by the cooling flow can be calculated from the thermocouples positioned in the cooling flow, see tigure 2-5:

Qabs = P

·V·

^CP· (Tout -Tin) (2.11) where Qabs is the power absorbed by the fluid (W), p is fluid density (kglm\

V

^{is the}

flow rate (m³ /s), Cp the specitic heat (J/(kg*K)), and Tout-Tin the temperature difference (K) between inlet and outlet

2.4 Modelling of microchannel coo/ing device

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/s^2).

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

2.5 Microchannel cooling resu/ts

In this section the results of the experiments and Plotherm simulations are presented.

First the calculations done with only a single channel and with the multichannel structure are compared in section 2.5.1. A condusion will be drawn whether the single channel approximation can be used to predict the thermal properties of microchannel cooling.

Section 2.5.2 compares the results for water experiments and calculations. The next section deals with the air measurements and calculations. A discussion of the results is given in 2.5.4.

2.5.1 Comparison single channel vs multichannel

By doing single channel simulations instead of multichannel simulations computation time is reduced significantly. Thus it is worthwhile todetermine whether single channel simulations can be used to accurately predict the thermal resistance of the microchannel structure.

Both water and air calculations have been done. The power per square centimetre is set at the value, which is also used in the experiments; 105 Watt in case of water cooling and 13.5 Win case of air cooling. The heater surface in the single channel model is 1 cm by 0.02 cm. The applied power is then 105

*

0.02

=

2.1 Win case of water cooling. For air cooling the applied power is 13.5

*

^0.02

=

0.27 W. The results can beseen in figure 2-9 and 2-10.

0,3

~ 0,25

• _•

~

•

...

Q) 0,2

• I

+Single channel; Rth l}out I

(,)

t:: 1111

• single channel; Rth Theater!

ra

• ^{I .}

-

^0,15

.!!! ₁₁₁₁

i •

m ultichannel; Rth Tj-out I

Cl)

•

Q)

•

... •

^I

•

iii ^0,1

•

^{i 1111} multichannel; Rth heater i

E

; • _l

... •

Q) J: 0,05 1-

0 i

0 0,2 0,4 0,6 0,8 1,2 1,4

Flow [I/min]

Figure 2-9 Calculated thermal resistancefor water simulations.

16

...

^{14 ...}

~

~ 12 ...

Q)

(,) 10

c CU

-

^tl) tl) ⁸ Q) s..

äi 6

E s..

4 .c Q)

....

2

• i

I

I

_I

•

^{I I}

• • ^I

lil

•

_IJ

•

• ^{• .a}

11111

I

^I

"

^.11!1

I •

single channel; Rth Tj-out ·

'

i • single channel; Rth heater

1

• multichannel; Rth T -jout

! 1111 multichannel; Rth heater

0 ' '

0 5 10 15 20 ²⁵ 30 ³⁵

flow [I/min]

Figure 2-10 Calculated thermal resistancefor air simulations.

In all cases both water and air calculations show a higher thermal resistance when simulating only a single channel.

For air calculations the heater resistance in multichannel simulations is below 40% ofthe resistance of single channel calculations at flows below 10 Vmin. The difference between single channel and multichannel calculations is caused by the effect of heat spreading through the silicon layer.

In tigure 2-11 the temperature ofthe silicon is depicted fora water cooled and air cooled structure. The bottorn picture shows the water cooled structure. Only the silicon in close proximity of the heater has a high temperature. This is caused by the fact that the heat is almost directly absorbed by the water cooled channels situated undemeath the heater.

Only a small amount of heat is absorbed by outer channels, situated next to the heater.

When air is the cooling fluid the cooling capability reduces. The supplied heat will then spread over a larger area (top picture), which enables the outer channels to contribute to the cooling capacity.

Because the heat is spread over a larger area the heat per channel ( or area) is effectively less than what is specified in single channel calculations. Because of the higher power per channel in the single channel model, the air temperature gradient along the channel is higher, which causes an increased capacitive resistance, as defined insection 2.2.4.

-^- -~".~->.;;~:--=-:;..:::::,._ -= -- "'-

" - -

,,:i ,

^4'w'

~- ~ ~ -~

^_

€~~

¹

il "'"' A ".fit, :-: -__

' i ' -=---~~~ --~-~

47631

33.819

•20

gC) 6

31113

Figure 2-11 Temperature distribution of air caoled (top) and water caoled geometry (bottom)

The maximum deviation with water as cooling fluid occurs at a flow of 0.1 1/min. The heater resistance is approximately 12% lower in multichannel calculations. At low flow rates the influence of the outer channels will be higher than at high flow rates. This especially affects the heater temperature sirree the heater strips located at the sides will be cooled by the outer channels too.

In appendix F is shown that the lower

Rth

^Tj-out is also caused by the higher flow through channels located at the centre of the structure.

For air calculations is it concluded that multichannel simulations are necessary sirree the differences in thermal resistance amount up to 60 %.

When water is used as cooling fluid the difference between single and multi channel calculations is within 12% for the performed calculations. At a flow rate of 0.1 1/min the heat spreading will also contribute to the cooling. It is expected that at even lower flow rates the difference between single and multichannel cooling will increase. Therefore only multichannel calculations are compared to experimentally obtained results.

2.5.2 Water measurements and calculations

The cooling capacity of microchannel cooling with water as working fluid has been tested as a function of cooling flow. The power input is set at 105 Watt. Only one power setting is used sirree the thermal resistance is only slightly dependent on the power input, through temperature dependent properties such as conductivity [7].

The meandering shape ofthe heater is deposited on a surface area of 1 cm²• In appendix

e

the maximum heat loss was determined to be 5 Watts at an almost uniform temperature of 80

oe

of the structure. In the conducted experiments the temperature directly after the heater is below 50

oe.

Furthermore a large part of the structure will be even at a lower temperature, which decreases the losses even further. The power removed by forced water mierocharm el cooling will be within 2% of the input power in these experiments.

The pressure drop was measured as a function of the cooling flow as shown in tigure 2-12.

For conventionally sized channels the pressure drop over the channels can be calculated by

!::J>

=

p. v2 . f. L (2.15)

2

DH

where öP is the pressure drop (N/m²), p the density ofthe fluid (kg/m³), v the mean velocity (m/s), fthe friction factor(-), 1 the length ofthe channel (m) and Dh the hydraulic diameter (m). For laminar flow the friction factor is independent on the roughness of the channel, and can be calculated by

f =

^{64 (2.16)}

Re

The results of equation 2.15 with a constant viscosity of 0,001Ns/m² arealso depicted in figure 2-12, together with the pressure drop calculated by Flotherm.

The theoretica! pressure drop is linear while both the Flotherm simulations and

experimental values are showing a non-linear behaviour. At low flowrates the theoretica!

pressure drop values are in excess of the Flotherm simulations and experimental val u es.

This is caused by the fact that at low flow rates the outlet temperature of the water is higher (for flow rate of 0,1 Vmin the outlet temperature is 42

oe).

As can be seen in appendix E the viscosity of water is dependent on temperature and drops with increasing temperature. This will result in a lower pressure drop as calculated with equation 2.15, where a constant viscosity, evaluated at the inlet temperature (20 °C), is used.

At flow rates in excess of 0.4 1/min the experimental value becomes higher than the pressure drop calculated with equation 2.15. The hydraulic entrance length becomes higher with increasing flow rate, and will increase the pressure drop. Furthermore the entrance and exit effects will increase the pressure drop.

The Flotherm calculations are in very good agreement with experiments for flow rates up to 0.8 Vmin. Afterwards the difference increases. At a flow rate of 1 Vmin the measured pressure drop is 15 % higher than the calculated pressure drop. This effect may be caused by a transition from laminar to turbulent flow.

3

11111

;@

2,5 ;@

...

...

.t.

"'

²

.Q

1U ¹¹

•

""' ^-<lc

...

Q. &<·

•

...

0

"

^1,5

'4\i<

q

•

^{i 1®} experiment

I

CD

... · 11111 flotherm; multichannel

...

::::J

IJ)

•

^• theoretica! pressure drop

IJ)

1

...

CD D..

0,5

0

0 0,5 1 1,5 2

Flow [I/min]

Figure 2-12 Pressure drop as ajunetion of water flow rate

The Reynolds number at a flow rate of 0.8 Vmin is 900. For a conventionally sized channel the flow becomes turbulent at Re~ 2300. However it is seen in earlier

experiments by Peng et al. [10] that flow transition between laminar and turbulent flow occurred for Re ~ 200-700.

The cooling capacity is determined by calculating the thermal resistance of the microchannel structure, as explained in sectien 2.3.2. In tigure 2-13 the thermal resistance, determined with the heater resistance, is shown as a function of the cooling flow.

0,4

....

^0,35

~

_0,3

::.:::

...

Gl

(,) 0,25 r: C'CI

-

^U) U) ^0,2

...

Gl

C'CI 0,15

E

...

0,1

Gl .J:.

1-

0,05 0

0 0,2 0,4 0,6 0,8 Flow [I/min]

1 1,2 1,4

I•·

experiment; old heater calib.

111 ^experiment; new heater cal ib. : 1

1oflotherm; multichannel I

Figure 2-13 Rthheater as ajunetion of water flow rate; Q=105 W

The thermal resistance of the microchannel structure shows a decreasing value for increasing flow, which is caused by the decreasing capacitive resistance. In figure 2-13 two series of points are presented as experimental data. The heater calibration, as explained in appendix D, was performed before the measurements were done. It was observed after the experiments that the heater had become partially oxidized. Since this can influence the electrical resistance, the heater calibration was performed again. The calibration showed that the electrical resistance was altered indeed. Using the new calibration data results in a significantly higher thermal resistance. However since it is unknown whether the oxidization has occurred before, during, or after the experiments, the true value can lie anywhere between the two series of points. This makes the thermal resistance data obtained with the heater temperature unreliable.

Therefore the thermal resistance will be determined using the thermocouple temperature Tj-out instead of the heater temperature from this point onwards. The results using

equation 2.8 as definition for the thermal resistance are shown in figure 2-14. In this figure the Flotherm calculations are represented by a mean value with error bars. This is done because the value of Rth Tj-out is determined by taking the mean temperature of a surface of 2x2 mm, since the exact position of the thermocouple is unknown. The maximum value ofthe error barrepresents the value attained with the maximum temperature on the 2x2mm surface. The minimum value represents the value attained with the minimum temperature on the 2x2 mm surface.

The temperature difference is measured with calibrated thermocouples. The calibration showed a maximum deviation of0.12 K between the thermocouple reading ofTmand the other thermocouples for the complete measuring range. The standard deviation on the reading of all thermocouples is about 0.20 K. The overall error ofthe temperature measurement difference is expected to be within 0.5 K. As pointed out in the beginning ofthis section the heat losses are small (within 2% of input power). This results in an overall error in the thermal resistance in the order of 0.01 KIW.

\

0,45

...

^0,4

T

~

^0,35

...

Q) 0,3

u r::

RI 0,25

-

^1/) _1/)

^T

0,2

Q)

'- jij 0,15

E '-

Q) 0,1

.r:.

1- 0,05

0 +---~----~---~----~

0 0,2 0,4 0,6 0,8 1 1,2 1,4 Flow [I/min]

Figure 2-14 RthTj-out as ajunetion of water flow rate; Q=105 W

*'experiment I

A flotherm; multichannel :

The results in figure 2-14 show the sametrend as figure 2-13. The thermal resistance decreases due to the decreasing value of the capacitive resistance. It is seen that the values calculated with Plotherm are in good agreement with the experimentally determined values. From a flow rate of 0.8 1/min the thermal resistance calculated by Plotherm is in excess of the experimental determined resistance. However the difference is smalland within the uncertainty ofthe measurement (0.01 KIW). The thermal

measurements do not give a clear indication of flow transition.

2.5.3 Air measurements and calculations

U sing water cooling in electronics can create hazardous situations. Therefore the possibilities of using air as cooling medium in microchannels are explored. Both

experimental measurements and numerical calculations have been done to determine the cooling capabilities of microchannel aircooling.

The thermal resistance is again measured as a function of flow to characterize the cooling behaviour, now for air cooled microchannels. The pressure drop as a function of flow is depicted in tigure 2-15. The experimental values are measured with the use of a pressure gauge and a flowmeter. The measuring range is limited to the maximum flow rate that can be measured.

The outgoing flow in Flotherm is determined as follows:

. m

vout

=

60 ·1000 · - (2.17)

Pout

where ~ut is the outgoing flow (1/min), mis the massflow (kg/s) and Pout (kg/m³) is the mean density at the end of the channel. Evaluating the Reynolds number at the channel outlet the range ofReynolds numbers varies from 500 <Re< 2050. The experiments and calculations show good agreement on the complete range. A transition from laminar to turbulent flow cannot be seen from these measurements. The experimental pressure drop is not compared to the laminar pressure drop calculated with equation 2.11 because the equation does not account for the compressibility of air.

1,4

... ...

1,2

C'CS 1

.c

...

Q.

0 0,8

"C

...

Q) 0,6

...

:::J 1/) 1/) Q) 0,4

D..

...

t..

0,2 0

0 10

A

•. ,À

4

!Oe>

$A

...

20

Flow [I/min]

+A

"""

+

30

Figure 2-15 Pressure drop as a function of air flow rate

I

I

40

i ^A flotherm; multichannel

I

I 111 experiment •

The thermal resistance calculated with the use ofthe thermocouple temperatures is given in tigure 2-16.

8

7 À

~6

lilt

Cl) 5

u c

-

111

.!!! _1/) 4

Cl)

...

i j

E 3

...

Cl)

J::. 2

1-

1 0

0 10

À

•• _m'

-- ^{~ •• ~rm.}

'

20

Flow [I/min]

!I

'

30

Figure 2-16 Rth TJ-out as ajunetion of air flow rate; Q=l3.5 W

I I

I

I

I I

_I

I

i

!

40

i À flotherm; multichannell 1 m experiment

The thermal resistance values calculated by Flotherm show lower values as the experiments for flow lower than 16 Vmin. At a flow rate of 10 V min the experimental value is approximately 30% lower than the thermal resistance calculated by Flotherm. It can be seen that for air measurements the heat losses become important. These heat losses are discussed in more detail in appendix C.

In tigure 2-17 the quotient of the power absorbed by the air and ingoing power is plotted as a function of flow rate. The ingoing power is determined by the current and voltage measurement. The absorbed power is calculated by equation 2.11. The density in equation 2.11 depends on the temperature and pressure. Assuming that the pressure is equal to surrounding pressure for all measurements the density at the outgoin~ channel is only dependent on the temperature. The density of air at 300 K is 1.177 kg/m [9]. The density at the outlet can be calculated with:

Pref · Tref

Pout

=

^{T (2.18)}

out

where Pour is the density after the channels, Pref is the density at reference temperature Tref (300 K) and T;,ut is the temperature measured at the channel outlet

1,2

1

.. ..

•• ^...

^"...

, _•

0,8

•

...

^I

•

...

c:

ä

_0,6

-

^{.Q lil}

_.. ^•

0 0,4

0,2

0 _'

0 5 10 15 20 25 30 35

Flow [I/min]

Figure 2-17 Absorbed power/input power as a function of air flow rate

For low flow values the power that is absorbed by the air flow is significantly lower than the input power. The thennal resistance of the microchannel structure increases for decreasing flow while the thermal resistance for heat loss is constant. The thermal resistance for heat losses is determined by measuring the temperature of the silicon as a function of heater power without any forced flow through the channels. The results are shown in appendix C. The inverse ofthe slope ofthe curve offigure C-lgives the heat loss thermal resistance, which is about 12KJW. When comparing it to the calculated values in tigure 2-16 it is obvious that heat losses are important at low flow rates.

2.5.4 Discussion of the results

2.5.4.1 Comparison of air and water cooling for experimental geometry

The cooling performance of air is clearly less good compared to water. For pressure drops ranging from 0.3 to 1.2 bars the thermal resistance of the water cooled structure is below 4.2% ofthe thermal resistance ofthe air cooled structure, see tigure 2-18.

10

...

~

~ ... ...

~

..

...

Q) 1

u

s:::::

CU

....

-

⁰

·u; ^~

Q) ...

":. ~

i A Rth Tj-out water I

• Rth Tj-out air I

s..

ftj E 0,1

•• ...

^{I ! I}

s.. Q)

J:

-

^{.... !}

1-

0,01

Pressure drop [bar]

Figure 2-18 Comparison between air and water cooling as aJunetion ofpressure drop The thermal resistance values presented above are calculated using the temperature difference between the water inlet and a thermocouple, placed closely behind the heater.

The most important temperature however is the maximal temperature on the heater, which in practice may not exceed a critica! temperature of 120 °C. Since the heater temperature is not accurately known (tigure 2-13) these values can only be obtained from Plotherm results. These are depicted for water as coolant fluid in tigure 2-20.

It can be seen that the maximum temperature is well below the critica! temperature for microelectronics.

The maximum heater temperature is a consequence of the fluid temperature rise T ^{bulk -} Tin ( capacitive resistance ), T wan - T bulk ( convective resistance) and Theater- T wan (conductive resistance). The detinition of these temperatures is given in tigure 2-19.

Theater is the maximal heater temperature. T wan is the mean temperature of the heat exchanging silicon walllocated beneath the maximum heater temperature.