Integrated silicon flow sensors

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INTEGRATED SILICON FLOW SENSORS

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INTEGRATED SILICON FLOW SENSORS

Geïntegreerde sihcium stromingssensoren

Proefschrift

ter verkrijging van de graad van doctor aan de

Technische Universiteit Delft, op gezag van de Rector Magnificus,

prof.drs. P.A. Schenck, in het openbaar te verdedigen ten overstaan

van een commissie aangewezen door het College van Dekanen op

donderdag 30 november 1989, te 16.00 uur

door

Bastiaan Willem van Oudheusden

Vliegtuigbouwkundig ingenieur

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Dit proefschrift is goedgekeurd door de promotoren

prof.dr.ir. S. Middelhoek

en

prof.dr.ir. J,L. van Ingen

n

dr.ir. J.H. Huijsing

heeft als toegevoegd promotor in hoge mate bijgedragen

aan het totstandkomen van dit proefschrift

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r

'Ach die wind!' riep heer Ollie uit, 'die hebben wij genoeg bestudeerd! Eerst begreep ik het niet zo goed, maar door mijn diepe gedachtenleven weet ik nu precies wat er gebeurt.'

Hij nam een teug en vervolgde: 'Kijk; eenvoudige lieden hebben erg gauw last van wind. Het gaat er maar om uit welke hoek hij waait, als u begrijpt wat ik bedoel. (...) Net wanneer men gewend is aan het ene, verandert de wind en dan komt het andere, en prettig Is het nooit. Maar ik heb de enige manier gevonden om geen last van de wind der verandering te hebben,

en dat is om gewoon maar een heer te blijven, wat jij, jonge vriend?'

Marten Toonder, De Wind der Verandering

Toen ik mij nu wendde tot alle werken die mijn handen hadden gewrocht, en tot het zwoegen waarmee ik mij had afgetobt om dit te volbrengen - zie, alles was ijdelheid en najagen van wind, en er is geen voordeel onder de zon.

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CHAPTER 3 SILICON THERMAL ELOW SENSORS

3.1 THE OPERATION OF SOLID-STATE THERMAL FLOW SENSORS 3.1.1

3.1.2

Total heal loss and average sensor temperature measurement Electronic thermal flow sensors

3.2

3.1.3 Sensor operation modes

AVERAGE THERMAL BEHAVIOUR 3.2.1

3.2.2 3.2.3

Lumped sensor model

Constant—temperature operation

Silicon structures for thermal anemometers 3.3 DIFFERENTIAL THERMAL BEHAVIOUR

3.3.1 Two—branch thermal model

3.3.2 _{Integrated components for on—chip temperature} difference measurement

3.4 DESCRIPTION OF THE INTEGRATED SENSORS 3.4.1

3.4.2 3.4.3 3.4.4 3.4.5

One—dimensional flow sensors Two—dimensional flow sensors Electronic operation

Sensor assembly Offset behaviour

CHAPTER 4 THEORETICAL ANALYSIS 4.1 THERMAL SENSOR MODEL

4.1.1 Convective heat transfer from a heated wall element

4.1.2 The effect of conduction on the sensor temperature distribution 4.1.3 Sensor heat balance

4.1.4 Soludon of the sensor equation

4.2 HEAT TRANSFER IN LAMINAR FLOW

4.2.1 Convective heat transfer in laminar boundary layer flow 4.2.2 Solution for laminar flow

4.2.3 Arbitrary wall temperature distribution \ 4.3 QUASI-ISOTHERMAL ONE-DIMENSIONAL SENSOR ANALYSIS

4.3.1 4.3.2 4.3.3 4.3.4

One—dimensional quasi—isothermal sensor equations Steady—state analysis

Dynamic analysis

Sensor behaviour in laminar flow

45 45 46 46 47 49 49 51 52 56 56 58 62 63 65 67 68 71

4.3.5 Sensor behaviour in turbulent shear flow

75 76 76 77 78 80 82 82 83 84 85 85 87 89 91 91

4.4 HIGHER-ORDER ONE-DIMENSIONAL SENSOR ANALYSIS 4.4.1 Sensor equations

4.4.2 Perturbation solution for almost constant wall temperature 4.4.3 The asymptotic case of zero conduction

4.4.4 Numerical solution method for the general problem

4.5 QUASI-ISOTHERMAL TWO-DIMENSIONAL SENSOR ANALYSIS 4.5.1

4.5.2

Two—dimensional sensor equation Heat transfer

4.5.3 Numerical solution 4.6 SUMMARY

CHAPTER 5 FLOW EXPERIMENTS

5.1 SHEAR FLOW EXPERIMENTS

5.1.1 Description of the test setup

5.1.2 Determination of wall shear stress 5.1.3 Boundary layer measurements

5.1.4 Sensor experiments

DIRECTIONAL FLOW MEASUREMENTS 5.2

5.2.1 Description of the experiments

5.2.2 Analysis of the measurement results

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CHAPTER 6 AN ELECFRONIC WIND METER 6.1 WIND MEASUREMENT

6.2 WIND PROBE DESIGN

6.2.1 Design considerations 6.2.2 Probe configurations 6.3 EXPERIMENTS

6.3.1 Preliminary flow measurements 6.3.2 Sensor experiments

6.3.3 Influence of connection method

96 96 97 98 99 102 103 103 106 109 111 111 112 113 116 121 122 123 126 133 133 135 135 136 139 139 145 145 Vlll _IX -^_

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CHAPTER 7 AN ETCHED 2-D FLOW SENSOR

7.1 SILICON MICROMACHINING FOR SENSOR APPLICATIONS

7.2

7.3

DESCRIPTION OF THE SENSOR 7.2.1 Operating principle

7.2.2 Layout and fabrication EXPERIMENTAL RESULTS 7.3.1 Sensor assembly 7.3.2 Steady—state behaviour 7.3.3 Dynamic behaviour CHAPTER 8 DISCUSSION REFERENCES SUMMARY SAMENVATTING

PRESENTATIONS AND PUBLICATIONS

ACKNOWLEDGEMENTS BIOGRAPHY 152 152 153 153 155 157 157 158 16Ü 165 169 178 180 182 183 184 symbol 1 a A A + B c c Cv

c

C d D on,op F

LIST OF NOTATIONS

quantity 1 L r F [ L

^B

^m

^^^B ^ ^ ^ ^ ^ K 8 G Gr h H H H É I I ^ 1 thermal diffusivity

sensor signal offset components area

sensor signal amplitude

Van Driest sublayer damping constant: 26.0

constant in the turbulent law of the wall, Eq.(2.34) specific heat

velocity of sound

wall shear stress coefficient: Xo/^pf/e^ pressure coefficient: (p—Pred^^?^

specific heat at constant pressure specific heat at constant volume

Nusselt number for isothermal surface and ({) = 0 thermal capacity

tube diameter sensor thickness

diffusion constants (electrons, holes) bandgap energy

electric field

function (in general)

dimensionless stream function

distribution functions for t/f and ^h acceleration of gravity

(themial) feedback conductance Grashof number; gUAT/v^T grid size

enthalpy

thermal conductance

boundary layer shape factor: imaginary unit: fl = —\ electric current 5*/0 unit m2/s V m2 V J/kgK m/s J/kgK J/kgK J/K m m m2/s eV V/m m/s^ W/K J/kg W/K XI

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1

k K l L Ma n N Nu P P^Ps Pi P Pe Pr Q Q R R Re t T AT u " i U+ U t/e X> V12.V34

electric current density themiLil conductivity

Boltzman constant: 1.38-10-23 J/K

dimensionless sensitivity parameter: k[Llk^D reference length

sensor length

Mach number: Vic electron density

number of strips in a thermopile

impurity concentration (acceptor, donor) Nusselt number: ql/kAT

hole density

(static) pressure total pressure electric power

Péclet number: ZQL^/[ia Prandtl number: v/u

electron charge: 1.60 • 10-19 c heat flux

convective surface heat flux local heat production in sensor heat flow

electric resistance thermal resistance

elecrric sheet or square resistance Reynolds number: Ul/v

time

temperature

temperature difference belvi/een sensor and flow flow—induced on—chip temperature differences velocity component parallel to wall

velocity component in Xj direction 1/2 wall shear stress velocity: (Xo/p) dimensionless velocity: 11/11^

flow velocity

free—stream flow velocity

velocity component normal to wall

normalized thermopile output voltages

A/m2 W/Km 1 1 1 1 m

V

^ ^ Vn = ^^34 w

W

voltage bandgap voltage

thermopile output voltages strip width width m ni -3 m -3 m-3 Pa Pa W W/m2 W/m2 W/m2 W Q. K/W

a

s K K K m/s m/s m/s m/s m/s m/s >1 X ; y a

ß

Y 5

e

Os

e

K X \^ V P -Co X X ¥ 0:) 1/3 spatial coordinate parallel to wall

spatial coordinate system

streamwise distance (from leading edge) spatial coordinate parallel to wall

dimensionless coordinate: yujv thermo-electric sensidvity

Seebeck coefficient

temperature coefficient of resistivity

transverse length scale, boundary layer thickness boundary layer displacement thickness

transformed transverse coordinate: y/{Z{J\iax) dimensionless transverse coordinate: _y/5

boundary layer momentum loss thickness

dimensionless temperature variation: {Q^—\)/KC

dimensionless temperature difference: {T — T.^\^)/AT Von Kamian constant: 0,41

mixing length

dynamic viscosity

electron/hole mobility kinematic viscosity

dimensionless coordinate: x/L density, specific mass

wall shear stress time constant

flow direction angle

heat transfer distribution for isothermal surface and (j) integral function of (pf

velocity potential

dimensionless coordinate: xuJvPr pitch angle stream function radial frequency - 0 V V V m m m m m m V/K V/K K-i m m m m Pa s mVV s kg/m^ Pa m^/s m'^/s rad/s XI1 x i u

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t subscripts: amb b c e e

f

g n P

)•

Ps

\- pr

s

t 0 ambient base collector emitter

edge of boundary layer flow, fluid

band gap

normal to surface pressure

static pressure

Preston tube pressure surface, sensor turbulent at surface i f

CHAPTER 1 INTRODUCTION

This thesis describes the development and the testing of thermal sensors for the measurement of the velocity and direction of air flow. The sensors are made of silicon and have been fabricated with standard integrated—circuit techniques.

In the first section of this chapter, the concept of integrated silicon sensors will be discussed briefly. Subsequently, a short review will be given of the basic types of existing flow measuring techniques, focusing on those which are commonly used in wind flow measurements and in wind tunnel research. The history of the integrated flow sensor project of our laboratory will be discussed next, together with the aims which were formulated for

the present research. Finally, a shon outline of the contents of the thesis will be given.

1.1 INTEGRATED SILICON SENSORS

The use of silicon for its electrical properties as a semiconductor has formed the basis for modern micro—electronics, which play a vital part in instrumentation and data systems. This role of silicon can be illustrated with the schematic diagram of Fig.l.la, which relates the functional parts of a general measurement system. Integrated circuit (IC) electronics have resulted in an impressive increase in the performance, in particular, of the system components involving the signal conditioning (amplification, data type conversion), transport and data manipulation (display, storage, processing). Standard fabricadon technologies for integrated circuits are today available, capable of high—volume batch production of high-precision micro-electronic components.

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(b)

sensor

(o)

ampli-fication

con-version

ampli-

consensor — » ,. ,. — *

-iication version

— • - ^

trans-mission

^

manipu-lation

1 i 1 1 1

Figure LI: Functional diagram o f a general measurement system; (a) classical siructure with separate sensor and electronics; (b) "smart" sensor where the sensor and (part of) the signal

conditioning electronics are combined on one chip.

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Recent years have shown an increasing effort in the application of silicon also for the manufacture of sensors. When such sensors are fabricated in the same technology as that used for making integrated circuits, they are usually referred to as integrated silicon sensors. The use of an integrated electronic structure as a sensor for a certain physical quantity relies

on the effects of external influences such as temperature, radiation or mechanical stress, on the electronic operation of the semi-conductor material. Silicon possesses many interesting physical properties which make it suitable for sensor applications [1.1]. The IC processing technology allows mass production of complicated and multi-component devices through the

simultaneous fabrication of many devices on a single silicon wafer (Fig. 1.2).

When the fabrication of the sensor is compatible with standard IC processing, the sensor can be integrated together with electronic circuitry on the same chip [1.2], leading to a

"smart" sensor (Fig.l.lb). Motives for the application of on-chip electronics can be, e.g., the improvement in electronic performance, the reduction of the sensitivity to interference, the

standardization of sensor output-signal formats through on-chip signal conversion, or the possible reduction in size and number of components for the complete measurement system.

1.2 THE MEASUREMENT OF FLOW

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Figure 1.2: Photo of a l-inch wafer containing three different flow sensor designs (each sensor measures 6 by 6 mm)

Man has always been intrigued by flow phenomena, and in many cultures flow has become the very symbol of elusiveness and unpredictability. Also in more recent times interest in flow has not diminished, and much scientific effort has been devoted to the understanding of fluid flow (concerning both liquids and gases). Apart from scientific interest from a fundamental physical point of view, flow phenomena have a direct practical significance in industrial, technical and every-day situations: in meteorology (vvind velocity and direction), civil engineering (wind forces on buildings and constructions), transport and process industry (fluidic transport of media, combusdon, vehicle performance), environmental sciences (dispersion of pollution), biomedics (respiration and blood flow), and indoor climate control (ventilation), to mention but a few.

1.2.1 Measurement techniques

Various methods are available for the measurement of flow, for experimental sciendfic research and for practical observations, and which may be intended for general purposes or, on the other hand, devoted to more specialized applications. In this context it is important to

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realize that "flow" is not a single physical parameter, such as pressure or temperature, but that it is a complex phenomena of a fluid in motion. In general, a flow is characterized by a, normally unsteady, three-dimensional vector field of the flow velocity. In addition, and depending on the character of the fluid and the flow situation, other variables may change throughout the flow field, being influenced by the flow or exerting influence on it: density, pressure, temperature, viscosity, contaminants, etc.

stagnation point (where the velocity is zero). The measurement of flow-related pressure differences has been applied in a large variety of pressure probes [1.4—6],

For the measurement of the pressure difference, standard instruments are available, like various forms of liquid—filled tube manometers, Bourdon—type gauges or diaphragm gauges.

In the following a short review will be given of the various techniques and the related devices which find specific use for the measurement of flow velocity and direction. Such instruments are often referred to as anemometers, derived from the Greek word anemos,

wind. Separate attention will be given to wind measurement and also to methods for the determinarion of surface shear stress (skin friction). The review will be restricted to methods which are common in wind tunnel research or for meteorological applications. A discussion on more special techniques, like those based e.g. on inductive, calorimetric, CorioUs, Coanda or vortex shedding principles, can be found in Ref [1.3].

The physical basis on which most flow sensing techniques rely is either the momentum of the flowing medium or its transporting ability. The former principle is the basis on which mechanical transducers operate, while the latter principle can be applied in a variety of techniques, depending on what specific property is considered to be transported, e.g.: heat,

smoke, dust particles, or a radio—active contaminant.

1.2.2 Mechanical methods

Mechanical flow sensing techniques can consist of the direct detection of the pressure or force which the instrument experiences as a result of the flow. Alternatively, a kinemadc method can be used, where a part of the instrument is set in a rotary motion by the flow, the rate of rotation being a measure of the flow velocity.

Pressure measurement

As will be shown in Chapter 2, the variation of the velocity in a flow can be associated with a change in the local pressure. For incompressible and frictionless flow, the relation is

given by the Bernoulli equation, which states (when neglecting gravitational effects);

1

p + ^ pf/2 = constant = py _(1.1)

where p is the static pressure (in Pa), p the fluid density (in kg/m^) and U the flow velocity (in m/s). The constant p^ is called the total pressure and is equal to the pressure in a

^=^N

^ v^v

T T

(a)

4 (b)

Figure 1.3: The pitot tube; (a) cross—section of a pitot-static tube; {b) three-hole and five-hole pressure probes for ßow direction measurements

The total pressure can be measured with a pitot tube, which is a small tube which has its open end facing the flow. The pitot-static tube consists of two (usually concentric) tubes, see Fig. 1.3. The inner tube detects the total pressure, while the second tube is connected to one or more openings (pressure orifices) in the side of the probe, to record the static pressure. The pressure difference Ap between the two tubes is then equal to the dynamic pressure ^pW^. To be able to calculate the velocity it is sufficient to know the density p of the fluid.

In nomial use a pressure tube should be aligned with the flow (deviations up to 10° are acceptable). For directional flow sensing multi—tube probes have been used [1.5], like the

three—hole and five—hole tubes illustrated in Fig. 1.3b. For its ease in construction and use (if built exactly to geometric specificadons no individual probe calibration is required), pitot and pitot-static tubes have become standard instruments in wand tunnel testing. For low

velocides, however, the pressure differences may be too low to allow accurate measurements.

A second type of pressure instrument can be distinguished, where the flow is led through a channel which varies in cross section. For reasons of continuity, the flow velocity must increase when the cross section is reduced. As a result, a pressure difference which is proportional to plA can be measured between two cross sections with unequal area. An example of this class is the venturi tube. The method of varying cross section is also widely used for flow rate measurements in pipes, in the form of the orifice plate, the nozzle, and the venturi section [1.7].

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)

Force measurement

An object which is exposed to a flow will experience a resistance force F in the direction of the flow. If the force is the resuU of the total effect of the pressure forces, it will

show a quadratic dependence on the flow velocity:

Also ultrasonic methods can be based on such a transit—time measurement or, alternatively, on the Doppler effect [1.3]. The latter principle is also employed in the Laser Doppler anemometer [1.141.

F = cApLßA _(1-2) 1.2.4 Thermal methods

where A is the exposed frontal area. Cj is called the drag or resistance coefficient, which depends on the shape of the body. In the target flow meter the force is measured on a circular disk placed at a right angle to the flow [1.3]. Also a miniature force anemometer has been reported in literature [1.8], which detects the bending of a small silicon candlever beam.

A mechanical instrument based on implicit force measurement is the rotameter [1.9], which is used for flow rate measurements in pipes. It consists of a float placed in a tapered, venical, transparent tube with the diameter increasing upwards. The float is carried upwards by the flow until it reaches a position where the resulting force exactly balances its weight. This position is a measure for the flow rate, which can be indicated on a scale on the tube.

Rotation measurement

There are two major types of rotation flow meters; the vane or propeller anemometer [1.10] which is used for engineering purposes, and the cup anemometer [1.11] for meteorological applications. Both types have a rotor as the working element. When mechanical friction can be neglected, the frequency of the rotation is directly proportional to

the flow velocity.

The vane anemometer consists of a turbine or propeller, with its axis of rotation in the direction of the flow. In the cup anemometer, a number of cups (usually three) have been mounted at the outer ends of light arms, which are connected perpendicular to a vertical rotation axis. The cups are arranged in such a way that when the rotor turns, the cups expose alternatingly their convex and their concave side to the wind flow. As the resistance coefficients for these two situations are unequal, the wind flow will cause a rotation.

1.2.3 Tracer methods

Tracer methods involve the detecdon of the movement of an agent, like a smoke particle, an amount of heat or contaminant, or an ultrasonic pulse, which is transported by the flow. The detection method may be optical, thermal or acoustic, depending on the chiiracter of the tracer agent. In thermal tracer methods the time interval required for a heat pulse to travel the known distance between a sending and receiving station is measured [1.12-13].

Thermal anemometers rely on the cooling of a hot object by the flow. There are two major measurement principles, based either on the total heat transfer of the sensor, or on the measurement of a flow—induced temperature difference caused by the inhomogeneous distribution of the heat transfer over the sensor. For a discussion of the electronic operation of thermal flow sensors the reader is referred to Chapter 3.

Heat transfer measurement

The extent to which a heated object is cooled by a passing flow increases with the flow rate. On this principle the operation is based of the hot—wire anemometer [1.15—17]. Basically, it consists of a thin metal wire, which is heated by passing an electric current

through it. The temperature of the wire can be derived from the electrical resistance of the wire, which is temperature dependent. The relation between heat loss and the flow velocity is usually expressed in the following form, which is known as King's law [1.18]:

P/Ar - ai + aiU 1/2 (1.3)

where P is the power dissipation (in W) in the wire, and A7 the temperature difference between the wire and the flow. The coefficients a^ and (32 depend on the geometry of the probe and on the material properties of the wire and the fluid, and are usually determined from calibration. Hot—wire anemometers find special application in the measurement of very

small flow velocities and fast velocity fluctuations (turbulent flow).

The typical shape of a hot—wire probe is shown in Fig. 1.4. For research probes the wires are normally made from platinum or tungsten, and have a typical length of 1 mm and a thickness of 5 p.m. A single wire which is placed normal to the flow is used when direction variations are unimportant. For directional flow measurements multi—wire probes are used, with wires at different angles (see Fig.1.4b). The heat transfer is mainly determined by the velocity component normal to the wire [1.19—20].

To improve the mechanical strength of the sensor hot-film probes have been developed, which consist of a thin metal film deposited on a quartz tube. Also, thermal anemometers have been made using electronic semiconductor components (see Chapter 3).

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\

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moving components, which is rather undesirable from the viewpoint of reliability and maintenance. In a two-dimensional wind meter a measurement method is employed which allows the simultaneous recording of the wind velocity and direction, usually without moving parts. Instruments have been reported using the pressure, force or heat transfer measurements on a vertical cylinder [1.24—26], or the heal transfer from a horizontal circular disk [1.27]. Also, two—dimensional acoustic wind meters have been developed.

(a)

_(b)

1.2.6 The measurement of wall shear stress Figure 1.4: Hot—wire anemometer, (a) single normal wire; {b) X—wire probe for directional

flow measurements

Temperature difference measurement

A second thermal principle which can be used for flow measurement is the detection of a flow—induced temperature gradient on the surface of a heated sensor. This temperature gradient results from the nonuniformity of the heat transfer and the finite conductivity of the surface. When compared to the total heat loss measurement method described above, this second method has the advantage of a well—defined zero—point (the temperature difference is zero at the absence of flow) and a 360° direction sensitivity.

1.2.5 The measurement of wind velocity and direction

In the case of wind measurement, the flow vector lies predominantly in one (horizontal) plane, in which the direction can vary over the full range of 360°. Especially this second point is essential, as in many other flow measurement problems the variation in flow direcdon is limited. Three classes of instruments can be disdnguished:

— separate velocity and direction indication. The velocity is measured with a direction insensitive method, while the wind direction is indicated separately, e.g. with a wind vane.

Ï

The cup anemometer [1.11] is the common velocity instrument in meteorology. Other principles which can be employed are a circular venturi [1.21] or a thermal sensor in the form of a vertically mounted hot cylinder [1.22] or a hot sphere [1.23].

— self—aligning wind meter. When a velocity probe is pivoted on a vertical axis and provided with a wind vane it automatically aligns itself with the wind direcdon. The velocity probe can, e.g., be a propeller anemometer or a venturi-type instrument [1.21].

— two-dimensional wind meter. All previously discussed instruments contain a number of

An important property in fluid flow is the wall shearing stress, or skin fricdon, which is caused by the fricdon of the flow over the surface. An extensive review of existing experimental techniques has been given by Winter [1.28]. The shear forces are usually very

small, and therefore indirect measurement techniques have been developed. These methods rely on the unique relation between the shear stress and the flow close to the surface, which is known as the law of the wall. Three basic methods, based on the measurement of force, pressure or surface heat transfer, are illustrated in Fig.1.5.

shear profile

thermal boundary layer

I

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(a)

(b)

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Figure 1.5: Shear stress measurement: {a) floating-element; (b) Preston lube; (c) hot-film

Direct force measurement (floating element)

Direct measurement of the shear stress can be realized by detecting the force exerted on a floating surface element or its displacement against a spring, e.g. with capacidve, piezoresisdve or magnetic means [1.29—33].

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>l

N

Pressure measurement (Stanton and Preston tube)

Indirect shear stress measurement can be realized by recording the pitot pressure in the flow close to the surface, as in the case of the Stanton tube [1.34-36] or the Preston tube

[1.37-38]. The Preston tube is simply a circular pitot tube resting on the wall. In the case of the Stanton tube, the inner surface of the pitot tube is formed by the wall itself, which can e.g. be implemented by covering a static pressure hole by the tip of a razor blade [1.36]. Dimensional analysis of the measurement technique leads to the conclusion that, because of the universal structure of the flow near a wall, a unique relation should exist of the form:

pv

_pv-

(1.4)

where ^p is the difference between the pitot pressure and the wall static pressure, TQ the wall shearing stress, d the height or outer diameter of the pitot tube, and v the kinematic viscosity (in m2/s). The exact form of the above relation depends, to some extent, on the geometry of the probe. The use of the Preston tube method will be discussed more extensively in Chapter 5, in relation to the calibration of the integrated sensors in shear flow.

Heat transfer measurement (hot—film)

The relation between the heat transfer from a heated surface element and the wall shear stress was first reported by Page and Falkner [1.39], and has resulted in thermal shear-stress gauges in a large variety of forms [1.40^7]. Special miniature hot-film probes for biomedical applications have been developed, intended for measurements in blood flow [1.48^9]. Today, glue-on hot-film probes are commercially available, which consist of a thin metal film deposited on a plastic foil, and which can be attached to the test surface. The basic calibration relation for surface-mounted hot films has a form similar to that of the hot-wire, Eq.(1.3):

PIKT = hi + b2 To 1/3 _(1.5)

1.2.7 The calibration of flow sensors

For most flow measuring instruments the reiadon between the flow velocity and the instrument signal cannot be predicted with great accuracy, and hence they need to be calibrated. Calibration can be performed in a flow channel against another calibrated or standard instrument. The pitot tube is often used as a standard, because under normal conditions its behaviour is accurately described by Eq.(l.l).

Special mechanical techniques are sometimes used for the calibration at very low flow velocities (below a few m/s), at which pressure measurement does not permit accurate results. The probe is then traversed at a known velocity through a fluid at rest; this can be performed

by a linear or a rotary movement. In the latter case, a compact calibration setup can be obtained in the form of a turntable [1.50].

Also shear-stress probes can be calibrated against a standard instmment, like a Preston tube. The calibrafion of the Preston tube is known from literature [1.38] and will be discussed in section 5.1.2. Alternatively, the probes can be calibrated in a simple flow arrangement for

which the relation is known between the shear stress and another flow parameter which can be measured more easily (e.g., the pressure gradient, or the free stream flow velocity). Commonly used calibration flows are the fully developed flow in a pipe or channel, in which case the shear stress can be derived from the longitudinal pressure gradient [1.33, 4 4 ^ 6 ] , or the boundary layer flow on a flat plate [1.40^2, 4 6 ^ 7 ] .

In addirion to this, special calibration instruments have been developed where a flow is generated for which the shear stress can be predicted theoretically, such as the flow between a rotating flat disk and a stationary flat plate [1.51], or the annular Couette flow between two concentric cylinders [1-52]

1.3 SILICON FLOW SENSORS

Silicon sensors for flow measuring purposes have been developed in various research institutes, employing either a mechanical or a thermal transducdon principle. Two examples of mechanical silicon sensors have been referred to previously, viz. a force anemometer [1.8] and a direct-force shear-stress sensor [1.33]. In addition, silicon diaphragm devices are widely applied in flow measurement for differential pressure measurement [1.53—54].

Thermal principles have proved to be very attractive for the development of silicon flow velocity sensors, because of their structural and electronic simplicity. As no additional

non-electronic components are required, complete integration in silicon is possible. The sensor operation is usually based either on the total heat loss of the sensor, or on the measurement of a flow—induced temperature gradient.

This second principle has been used in the Electronic Instmmentation Laboratory for the development of a class of integrated flow sensors. In the following secdons the operadng principle will be described in more detail, and a short history of the research project will be

given. The major results will be discussed, and the aims which were set for the present research.

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^

1.3.1 Operating principle of the sensor

The heat transfer from a heated sensor to the flow is not distributed uniformly over its surface: at the upstream part the surface is cooled more strongly than at the downstream part, because a thermal boundary layer is built up with an increasing thickness when travelling downstream. Because of the finite thermal conductivity of the sensor, this differential cooling results in a temperature gradient in the sensor in the direction of the flow. The thermal operation of the sensor is illustrated in Fig. 1.6a, which shows the cross section of a sensor chip embedded in a flat wall. Under the influence of the heat transfer to the flow the temperature distribution, which is essentially symmetric at zero flow, will become asymmetric. When the thermal isolation of the sensor is not perfect, also the wall area around the sensor will be heated to some extent, as indicated in the figure. In the theoretical analysis in Chapter 4 this thermal leakage effect will be neglected.

/

As can be seen from the block diagram in Fig. 1.6b, a number of basic transduction steps can be distinguished in the sensor operadon, going from the mechanical via the thermal

to the electrical signal domain. With these transduction steps through the different physical signal domains, three major transduction problems can be associated for the sensor under consideration:

— the flow problem: what is the reladon between the fiow over the sensor and the relevant flow parameter which is to be measured (wind flow, e.g.)?

— the thermal problem: how does the (local) flow over the sensor influence its thermal behaviour?

— the electronic problem: how can the thermal signal (temperature difference) be measured electronically, and eventually be converted into a suitable sensor output format (a data bus

compatible structure, e.g.)?

(a)

flow direction

(b)

Mechonicol

signal

donnain

embedded chip

Thermal

signal

domain

temperature

difference

no flow

with flow

Electrical

signal

domain

Figure 2.6: Transduction principle of the thermal flow sensor; (a) schematic cross section showing the influence of the flow on the temperature distribution; (b) the basic transduction steps in the sensor operation

One—dimensional and two-dimensional flow sensors

By measuring the temperature difference between two fixed points on the chip, a flow sensor results which is sensitive only to flow components along the imaginary line which connects these two points (see Fig.1.7a). In the absence of flow, the temperature difference is zero. The sensor can detect flow reversal, because the polarity of the temperature difference reveals the sense of the flow direction, thus creating a one—dimensional direcdon—sensitive flow sensor principle. When temperature differences are measured in two directions perpendicular to each other (Fig. 1.7b), both flow velocity and direction can be determined, and a two—dimensional sensitivity is obtained. The terms one— and two-dimensional refer in this respect to the plane of the sensor, and should not be confused with the terminology used

in flow theory.

(Q)

(b)

flow vector

Figure 7.7: Basic layout of flow sensors based on the measurement of on-chip temperature differences; (a) one-dimensional sensor; (b) two-dimensional sensor

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^

Let the sensor be heated to a constant temperature difference AT" with respect to the temperature of the flow at a large distance. The flow is assumed to be two-dimensional (see Chapter 4). In approximation, the thermal gradient in the sensor is aligned with the flow direction, while its absolute value ATQ is a function of the flow velocity U, while in addition evidently being proportional to AT. The two temperature differences ATjj and AT24 which are measured on the sensor surface can be regarded as the orthogonal components of this total gradient, so that the sensor operation can be described by the following expressions:

A7^12 = 7-2-r^ = ATo(U) coscj) _(1.6a)

Ar34 = T^-Tj - ATQ(U) sind? _(1.6b)

From these relations it is easy to resolve the flow angle ([), independently of the flow velocity, from the ratio of AT12 and AT^4, while the flow velocity can be derived from the absolute value of the thermal gradient, ATQ:

(}) - tan-i (A734/A7i2) _(l-7a)

ATn = J Arn2 + A7..2 _12' _34' _(1.7b)

For a square sensor geometry the total heat loss is virtually independent of the flow direction, and a function of the flow velocity alone. This may provide an alternative method

to determine the flow velocity.

1.3.2 History of the project

The basic layout of the one-dimensional integrated flow sensor which was realized at the Electronic Instrumentation Laboratory in 1975 is shown in Fig.1.8. The chip contains a large, central heating transistor, while two equally shaped transistors are located near the two opposite sides of the chip to measure the flow-induced temperature differences [1.55-56J.

Figure 1.8: Integrated flow sensor with a central heating transistor and two temperature measuring transistors

{size 2 by 2 mm)

I

In subsequent research, the electronic aspects of various sensor configurations were investigated, e.g. the use of different electronic components for heating the sensor (transistors or resistors) and for measuring the on—chip temperature differences (transistor pairs or integrated thermopiles). It was found that headng transistors could disturb the sensitive temperature difference measurements, because of stray charges resulting from the vertical injection of the electrical current from the emitter. This effect is not present in heating resistors, where the current flows mainly parallel to the surface.

Another problem was that of the electronic offset and drift of the transistor pair measuring the on—chip temperature difference. A temperature-difference sensor which relies on a modulating principle (such as a resistor bridge or a transistor pair) is inherendy suffering from offset: even if the temperature difference is zero, an (offset) signal may be present owing to mismatch in the individual components. This is a serious problem, because as the spatial separation of the temperature sensors on the chip is relatively large, it is hardly possible to obtain components with sufficiendy matched characteristics. This problem can be

avoided by using a temperature—difference sensor relying on a self—generating effect, such as a thermopile.

Parallel to the integrated sensor project, which had been limited to designs with one—dimensional sensitivity, a two—dimensional flow sensor had been realized in a hybrid configuradon, with discrete transistors on a ceramic carrier substrate [1.57], The application which had been in mind was that of a micro—electronic wind meter without moving parts, capable of measuring wind velocity and direction.

1.3-3 Aim of the present work

After the basic electronic sensor principles had been verified, the present research project was initiated with the following aims:

— A theoretical study of the physical transducdon mechanism.

— The design and realization of integrated two—dimensional flow sensors.

— Experimental invesdgation of the sensor behaviour in a well-defined flow.

- T h e design of a suitable probe shape for the application of the .sensor in an electronic wind meter.

/

A number of different sensor designs will be discussed in Chapter 3 of this thesis, with either transistor pairs or integrated thermopiles as on-chip temperature-difference transducers. The experimental part of the research has been focused on the thermopile flow sensors, because of the advantages of these components as far as the pracdcal operanon is concerned. In Chapter 5 only the experimental results of these sensor types will be discussed.

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1.4 ORGANIZATION OF THE THESIS

The audience for which this thesis is intended consists of people involved in the different disciplines of flow research and electronic sensor development. As it cannot be assumed that every reader is equally familiar with both of these fields of science, a review is given in Chapter 2 of the basic theory of fluid motion and of semiconductor electronics. The major concepts and the fundamental equations which are thought to be required for a proper understanding of the thesis will be discussed. This chapter does not contain any new material which cannot be found in standard text books, and the reader which is familiar to either subject may easily omit the corresponding parts of this chapter. Chapter 3 gives a concise review of the general operating principle of electronic thermal flow sensors, and the electronic components with which they can be realized. The last section of this chapter describes the integrated sensors which were used in the present research, and discusses a number of practical aspects of the sensor operation. In Chapter 4 a detailed theoretical analysis is given of the thermal transduction mechanism of the sensor, based upon a simple thermal model of the sensor. The major conclusions of the analysis are summarized in the last section of the chapter. Chapter 5 presents the results of the experimental research. In the First section the calibration of the one—dimensional sensors in a shear flow is described, while the last secdon is devoted to the direction sensitivity of the two—dimensional sensors. Chapter 6 describes the application of the sensor in the form of a micro—electronic wind meter, to which end a suitable aerodynamic probe shape has been developed and tested. In addition to the research which has been performed on wafer—thick sensors, some preliminary experiments have been carried out with a themial flow sensor which contains an etched, on—chip thermal isolation structure (floating membrane). The results are presented in Chapter 7. Finally, Chapter 8 summarizes the major conclusions, and discusses suggestions for further research.

16

CHAPTER 2 BASIC THEORY OF FLOW AND SEMICONDUCTORS

In this chapter a short review is given of the basic theory of fluid motion and of semiconductor electronics. It is intended as an introduction to these fields to make the reader, if necessary, acquainted with the major terminology.

2.1 FLUID MOTION

An important topic in the theory of fluid motion is that which deals with the steady flow around a stationary solid object. At a large distance from the body, an undisturbed flow situation exists, where the fluid moves at a constant speed (parallel flow). Altemadvely, the object may be moving with a constant speed while the fluid is al rest, both problems being fundamentally the same when described with re.spect to a coordinate system which is Fixed to the object. When the structure of the flow in every point of space does not change with time the flow is called stationary. The object which is exposed to the flow will experience a resulting force, while furthermore, the heat transfer between the object and the fluid is influenced to a great extent by the flow around the object. The practical significance of these problems can be illustrated with many examples from everyday life, like the air flow around moving aeroplanes or vehicles, or the wind flow around buildings.

2.1.1 Fundamental equations of motion for a perfect fluid

The dynamics of fluid motion can be described by the action of the contact forces which a surface experiences in a flow. This surface may be a true, physical boundary like the surface of a solid object, or the imaginary interface between two adjoining fluid layers. In

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real fluids there are contact forces which are directed normal to the surface (pressures and normal stresses) and tangential to it (shearing stresses). The stresses arise from the effect of the viscosity of the fluid. In the concept of a perfect fluid the medium is assumed to be incompressible and inviscid (fricdonless), and hence there are no shearing stresses. A change in flow velocity U (expressed in m/s) can be caused only by the local static pressure p (in Pa), when gravitational effects are neglected (their influence may be included in p for cases

without a free surface). Along a streamline, the momentum balance takes the following differential form:

dp - - p f/ dt/

(2.1)

where p is the density of the fluid (in kg/m3). Per definition, p is constant for an incompressible fluid, and in that case Eq.(2.]) can be integrated directly, yielding the well-known Bernoulli equation:

p + iplß ^

_Pi

(2.2)

The quantity p^ is known as the total pressure, which is constant along a streamline in the flow. From Eq.(2.2) it can be seen that p[ is equal to the pressure in a stagnadon point, where

f/= 0. When the undisturbed flow field at infinity is a parallel flow with constant static pressure, p^ is constant throughout the complete flow field. Note that differences in pressure rather than its absolute value are relevant for the fluid motion. Therefore, pressures are usually measured relative to a reference level, like the undisturbed static pressure.

Compressibility

A real fluid (liquid or gas) will always display a certain amount of compressibility. Neglecting these effects may be obvious in the case of liquid flow, but the concept of incompressibility can also be applied for gases at low—subsonic flow velocities. For the adiabatic flow of a perfect gas the speed of sound c is equal to / ^ 7 p , where y - Cp/Cy is the rado between the specific heat at constant pressure and that at constant volume (y - 1.4 for diatomic gases and also for air). Under adiabatic conditions the reladve variation in the density p can be calculated to be:

For small flow velocides the variadons in the thermodynamic properties p and p are small, and hence the value of c remains practically constant. Tne relative change in density with respect to its value at zero velocity, Ap/po, is then approximately equal to \Ma^, where

Ma is called the Mach number, which is defined to the ratio of the local flow velocity to the local sound velocity: Ma ^ Ulc. From this approximation, it can be seen that for Ma up to 0.14, the relative density variations are less than one percent. For air at normal ambient conditions (c = 330 m/s) this corresponds to the velocity range of 0 to 50 m/s, showing that in many practical cases compressibility effects can be neglected.

Equations of motion

The fundamental equadons of motion for fluid fiow are the continuity equation, stadng the conservation of mass, and the momentum equation. Let the flow be described by velocity components U\, «2 'iid W3 in a Cartesian coordinate system with axes x-^, Xi and X3. The equations of motion for steady, incompressible flow without fricdon are given by the

following pardal differendal equations:

3M j /3xj = 0 (2.4)

pMj3Mi/aYj - —dpldX\

(2.5)

Here the tensor notadon has been used, where according to the summadon convendon, summation over all possible values of an index (here: 1, 2 and 3) should be performed if the

index appears twice in a term. Thus, dujdx-^ stands for du\ldx-[ + du-2/dX2 + 3^3/9x3. A free index, like the / in the momentum equadon, indicates a vector equadon. Boundary conditions

for the equations are imposed at solid surfaces, where the reladve velocity component normal to the surface should be zero. Because shearing stresses have been omitted, no condidon can be imposed on the tangendal flow velocity at the surface, and the flow is free to slip past the surface. The undisturbed flow situadon specifies boundary condidons at infinity.

Potential flow

Analytic solutions of the equations of motion can be obtained for irrotational flow fields by introducing a velocity potential O. The velocity vector is defined as the gradient of O, so that u\ = aO/äx;. According to the condnuity equadon, the velocity potential O should satisfy the Laplace equadon: d'^/dx\dx\=0. Together with the flow boundary conditions, which can be translated into boundary condidons for <[> (e.g. at a solid surface the gradient of O normal to the surface should be zero), the velocity field can be determined. The value of the pressure follows from the momentum equadon. As inviscid, irrotadonal flow can be described with a velocity potendal it is sometimes referred to as potential flow.

18

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\

2.1.2 Viscous flow

2.1.3 Thermal effects in fluid flow A fundamental difference between a real fluid and the perfect fluid introduced above is

the existence of tangendal shearing stresses which are caused by the effect of viscosity. When two fluid layers have a different flow velocity, a gradient exists in the average drift velocity of the molecules. As a result, molecules crossing the interface between the layers establish a momentum transport, which can be interpreted as a tangential stress x on the interface. This shearing stress is proportional to the velocity gradient ^u/dx^ normal to the interface (u is in this case the velocity component parallel to the interface):

X = p. du/dXn _(2.6)

the material property p is called the dynamic viscosity (in Pa-s). Because of the viscosity and the molecular interaction between fluid and wall, no slip is assumed to occur for real fluids. Significant shearing forces can occur even in fluids with small viscosity in a region with large velocity gradients. Such a region will exist in the proximity of a solid surface as a result of the no—slip condidon.

The Navier—Stokes equations

The momentum equanons for the stationary, laminar flow of an incompressible viscous fluid are given by [2.1]:

pUjdu-Jdx^ = -dp/dx, + äTjj/arj

where T^ is the stress tensor, which for a Newtonian fluid can be wrillen as:

(2.7)

ij = p (duJ^Xj + dUj/BXi)

Tii = _(2.8)

Substitution of this expression in Eq.(2.7) yields ihe Navier-Stokes equations for an incompressible viscous fluid:

pUjduJdXj = -dp/Bx^ + ^d\jdXjdx^ _(2.9)

An important dimensionless number in the study of viscous flow is the Reynolds number Re, which is defined as Re = Ul/v, where U (in m/s) and / (in m) are velocity and length scales which are relevant to the problem, while v = p./p (in mVs.) is the kinemaUc

viscosity. '

Heal is transported in the flow by convection and by thermal diffusion (which results from thermal conduction). The role of the thermal diffusivity a (in mVs) in the heat transport of the fluid is similar to that of the viscosity v in the momentum u-ansport. In the molecular

theory of gases the diffusion of momentum and of heat are caused by a similar process (the thermal motion of the molecules), hence, a and v will be of the same order. The ratio of v and a is called the Prandtl number Pr (Pr = via), which is a material constant. According to the kineuc gas theory, Pr is independent of temperature, and is predicted to be about 0.74 for diatomic gases. This is found to be in good accordance with experimental data for real gases

[2.2]. For nomial air, which behaves very nearly like a diatomic gas, Pr has a value of about 0.71 at room temperature.

In a real fluid heat production may be present dirough the work of compression and friction. To study the effect of compression, let the flow of a perfect gas be considered without friction and conduction. The energy balance along a streamline states that the sum of the kinetic energy and the enthalpy of the flow remains constant, so that for a perfect gas:

cp r + i ^ = cp r,

(2.10) where Cp is the specific heat at constant pressure (in J/kgK). T^ is known as the total

temperature or the stagnation temperature. The temperature increase in a stagnation point

due to adiabalic compression can be seen to amount to iß/lCp, which for an air flow of 10 m/s is equal to only 0.05 K. The temperature increase due to frictional heat production and that due to compression are comparable when Pr is in the order of one [2.3].

When the surface is heated artificially by means of an internal healing source, the thermal effects of compression and friction may become negligible in comparison to the imposed healing (except for high speed flow). In the present study, this simplifying assumption will be made, and the influences of themial compression and frictional heat generation will therefore be omitted.

The energy equation

In tensor notation, the energy equation for a viscous fluid can be written as [2.4]:

pU[dH/dXi = ü^dplBx\ + Tjj 3i(i/5.tj - dq\ßX\ (2.11)

which Slates that the conveclive change of the enthalpy H (in J/kg) is balanced by the work performed by pressure forces and stresses and by the conducdon of heat. According to Fourier's law, the conductive heat flux per unit area ^ i ( in W/m^) is given by:

20

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\

qi = -kdTldx-,

(2.12) whem k is the thermal conductivity of the fluid (in W/Km). For a perfect gas (for which d// - Cp dV) and neglecting the effects of compression and friction, the energy equation is simplified to the following form:

velocity scale U, shows that similar flow fields exist If the values of the Reynolds number Re = UlN are equal, and that similar temperature fields exist if in addition the values of the

Prandd number Pr are equal [2.51. In general, the solution for any scaled parameter (velocity component, pressure, or temperature, e.g.) can be written as a function ^{x-Jl, Re, Pr).

UidT/Bx, = ad^T/dXidx;

(2.13)

where a = k/pc^ is called the thermal diffusivity of the fluid. It can be seen that when friction and compressibility effects are neglected, the energy equation is reduced to a simple transport equation for the temperature.

In general, the mechanical and thermal problems must be solved simultaneously, which is indeed true for compressible flow and for natural convection. When, however, the influence of temperature effects on the thermodynamic properties of the flow can be neglected, the velocity field can be considered independenUy of the temperature problem. The temperature (or heat) can then be regarded as a passive contaminant which is transported by the flow without influencing it.

Surface heat transfer

The heat transfer from a surface element to the fluid is determined completely by conduction in the fluid, because near the surface all flow velocity components become zero. The local heat transfer QQ per unit area (in W/m ) from the surface to the fluid is proportional to the temperature gradient normal to the surface (where x^ is the distance to the surface):

go = - fc idTldx,)^^ ^ ^

(2.14) The Nusselt number Nu, which is a dimensionless heat transfer parameter, is defined as: Nu = q^l/kM, where AT is the temperature difference between the surface and the fluid, and / a length scale.

Similarity in laminar viscous flow

Under certain conditions it is possible to find identical solutions for different problems if the variables under consideration can be transformed in a suitable way. In this case it is said that similarity exists between the flows under consideration. Similarity may be obtained, e.g., for the viscous flow around bodies of identical shape but with different sizes. A necessary requirement for similarity is obviously the geometrical similarity of the flow under consideration, and the similarity of the boundary conditions. The additional conditions can be derived from the flow equations. Transformation of the vEiriables, using a length scale / and a

2.1-4 The boundary layer concept

The boundary layer concept was introduced by Prandtl in 1904. Prandtl showed that the flow around a body can be divided into two regions (see Fig.2.1): a thin region near and behind the body where viscous effects are important (the boundary layer and the wake), and the remaining region where viscosity can be neglected {inviscid OÏ potential flow region). In the boundary layer, the tangential flow velocity changes very rapidly from zero at the wall to its value at the edge of the boundary layer, which corresponds to the slip velocity if the flow would have been inviscid.

inviscid flow region

boundary layer

z

wake

Figure 2.7: Boundary layer flow around a streamlined object

Because of the small thickness of the boundary layer, gradients along the surface are small with respect to gradients perpendicular to it. As a consequence, the flow equadons can be simplified by neglecdng the diffusion terms for the directions along the surface. Let a coordinate system (xj, X2, y) be defined, where x^ and X2 describe the solid surface which bounds the flow and y is measured normal to the surface, while the corresponding velocity components are denoted by U\, »2 ^rid u. The coordinate system will be assumed to be Cartesian. The boundary layer equations for laminar incompressible flow are given by [2.6]:

22

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^

M

9Uj /Bx^ + dv/dy = O _(2.15)

«j3wi/axj + 1) dujdy =

- ^ ^pf^x i + V

a-w, /a^2

_(2.16) Note that in this case the index summation should be carried out for the values 1 and 2. The momentum equation in the j—direcnon is reduced to the condition that dp/dy - 0, which means that the pressure across the boundary layer is constant, and equal to the value outside the boundary layer. The pressure distribution over the surface is therefore related directly to

the velocity field t/e(-^i) i" the free stream. Using Eq.(2.2), the pressure term in the momentum equation can be expressed as dp/dxi= —pU^We^ldx^. The pressure distribution can be treated as an external condition imposed on the boundary layer, and may be derived from measurements or from inviscid flow calculations.

The complete viscous flow around a body can now be determined by calculating first the inviscid flow, and subsequently the boundary layer development. This method yields an accurate representation of the realistic flow situation as long as the boundary layer is thin and adheres to the surfaces. When the flow separates from the surface this approach fails because the real flow then deviates strongly from the theoretical inviscid flow around the body.

Tliermal boundary layers

For a fluid with a small thermal diffusivity the temperature distribudon in the flow will show a behaviour similar to that of the velocity distribution. The transition from the temperature at the wall to that of the free stream will take place in a narrow region near the body: the thermal boundary layer. It is therefore possible to simplify the temperature equation Eq.(2.13), by neglecting the diffusion terms for the directions parallel to the surface [2.6]:

Uj dTldx-^ + \) dVdy = a d'^T/dy^

components u parallel to the wall and \) normal to it. The value of u at the edge of the boundary layer will be denoted by U^ (see Fig.2.2). For laminar incompressible flow the

two-dimensional boundary layer equations are:

M/dx + 'dv/dy = 0

u du/dx + \) 'óu/dy = t/e dU^ /dx + V d'^u/dy

(2.18)

(2.19)

u dVdx + x> dT/dy = a iP-Tldy'^ (2.20)

Sn^am function

In the case of two-dimensional How a stream function 4^(;t,y) can be introduced, which is defined by w - d'Vldy and OJ - - d'^ldx. By definition, the stream function satisfies the

continuity equation

velocity distribution

temperature distribution

U

X

(2.17)

u(y)

w////////////////////////y///////y/^^^

At the surface a wall temperature T^{x;) may be prescribed, while at the edge of the boundary layer the temperature should reach the free stream value 7 . .

2.1.5 Two—dimensional laminar boundary layers

The following discussion will be restricted to a two—dimensional boundary layer fiow, which means that identical flow patterns occur in parallel planes normal to the surface. Let the flow in such a plane be described in a Cartesian coordinate system (.ï,y), with velocity

Figure 22: Two-dimensional boundary layer flow

Boundary layer parameters

As the velocity distribution u{y) in the boundary layer approaches its free stream value f/e asymptotically, no exact thickness of the boundary layer can be defined. Instead, the thickness is often defined as the distance from the wall where the velocity has reached a certain fraction, e.g. 95% or 99% of the free stream value. For a number of other parameters

which are often used to characterize the boundary layer, the definitions have been given in Tab,2.1. The displacement thickness and the momentum loss thickness are two integral length

24 25

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*>*

scales in the order of the boundary layer thickness. The displacement thickness represent the distance over which the outer flow is displaced by the boundary layer, while the momentum thickness is a measure for the loss of momentum of the flow in the boundary layer. For most parameters generalized expressions can be formulated for three-dimensional boundary layers.

Co^VliV q^llkbJ = -Ul I V 5 " 5

[ay/5"J

4 '^T/^T]

ay/6 J

= Re^^ ^,(xll)

y = Q Jy = 0 (2.21)

= Re^^^ X{xll,Pr)

(2.22)

Table 2.1: The definition of some boundary layer parameters for two-dimensional ßow

displacement thickness:

momentum loss thickness:

momentum loss thickness Reynolds number:

shape factor:

wall shear stress:

wall shear stress coefficient:

5* = / ° " ( i - ^ ) d , 0 0 1 ReQ - Uß/v H - 5*/e

^0

-.

g;

, = 0

Similarity in laminar boundary layer flow

To investigate the similarity for two-dimensional lamin^u boundary layer flow, let the ;c—coordinate be scaled by a length scale /, and the velocity component w by a velocity scale U. To account for the small thickness of the boundary layer, a separate length scale 5 is used for the transformation in the y-direction, and the transversal velocity M is scaled with U- 3/1.

—1/2

When the choice is made that d = lRe , the transformed boundary layer equations are found to be independent of Re - Ul/v. Accordingly, the boundary layer development over geometrically similar bodies (with similar boundary conditions) will be identical if properly scaled, and is completely determined by the dimensionless pressure distribution p/plP, or the free stream velocity UJU-, as a funcdon of xll. Hence, at corresponding posidons on the bodies, equal values of the shape factor H are found, while transversal length scales, such as

* —1/9

the integral boundary layer parameters 5 and 6, are proportional to /-Re . For the wall shear stress XQ and the surface heat transfer <7Q the following general relations can be derived:

0" + iL^PrFQ'

- 0 (2.24)

Eqs.(2.23) and (2.24) can be solved numerically. For the wall shear stress TQ and the surface heat transfer ^Q the following relations can be formulated:

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Table 22: Some characteristic properties of the wedge-type similar boundary layer flows m 1 1/3 0 -^.0904

ß

1.0 0.5 0.0 -0.1988 «0 1.233 1.136 0.664 0.0 H 2.216 2.297 2.591 4.029 stagnation point

constant wall shear stress flat plate separation ( Q )

(b)

m=0

V/////////////////////////////////////A

Figure 2.3: Similar laminar boundary layers, (a) Flow past a wedge; near the leading edge the potential velocity is U^{x) ~ x'". {b) Influence of the similarity parameter m on the shape of the velocity profile

X n / ^ _ pv 2 = ^ 0 t t UJ 3/2 t-1/2 „ ^-"^ = ao(m) 3/2 ^{3m-l)/2 _(2.25)

Hi

= - 0 o ' f ^ l l / 2 ^ - l / 2 ^ „ ^ ^ ^ ^ ^ , , ) Ul 1/2 ^(m-l)/2 _(2.26)

where OCQ = FQ" and o.\ = — 6o'. Note that FQ" is the dimensionless velocity gradient and %' the dimensionless temperature gradient at the surface. The influence of the pressure gradient on the shape of the velocity profile is illustrated in Fig.2.3, whereas Tab.2.2 contains the

values of ÜQ and of the shape factor H for a number of values for m [2.8]. For decelerated flow (m < 0) the velocity profile displays an inflexion point, while for m = -0.0904 a flow profile on the verge of separation is found (XQ = 0).

Fiat plate flow {Blasius solution)

An important example of the class of similar solutions described in the previous section is the boundary layer over a flat plate at zero pressure gradient. The problem was first

formulated by Prandtl in 1904 and solved in more detail by Blasius in 1908, for which reason this solution is often referred to as the Blasius boundary layer soludon. The free stream condition is that of a constant U^ so that JJ^- U and m = 0. A number of properties of the

solution have been collected in Tab.2.3.

Table 2 J : Bounda^ layer properties for the Blasn^ boundary layer profile (Re, = UxN, Re = UlN and ^ = x/l 5* = 1 . 7 2 1 / e . r ^ ^ x ^ l.mRe''fH''^l e - 0.664 ßcx "' X Ree - 0.664/^fix 1/2 H - 2.591 3/2 K-1/2

'Co = 0.332/e^r^^pf;^ = 0332Re'''^-'''9^yr~

Cf - 0.664 Re -'^ = Q.66A Re^^^H'"'^ - 0M\ Re,

-1

Ml

'tiHMi

28

Integrated silicon flow sensors

INTEGRATED SILICON FLOW SENSORS

Bas van Oudheusden

TR diss

1775

s

h

n

C c ^ l l

INTEGRATED SILICON FLOW SENSORS

TR diss

1775

INTEGRATED SILICON FLOW SENSORS

Geïntegreerde sihcium stromingssensoren

Proefschrift

ter verkrijging van de graad van doctor aan de

Technische Universiteit Delft, op gezag van de Rector Magnificus,

prof.drs. P.A. Schenck, in het openbaar te verdedigen ten overstaan

van een commissie aangewezen door het College van Dekanen op

donderdag 30 november 1989, te 16.00 uur

door

Bastiaan Willem van Oudheusden

Vliegtuigbouwkundig ingenieur

Dit proefschrift is goedgekeurd door de promotoren

prof.dr.ir. S. Middelhoek

en

prof.dr.ir. J,L. van Ingen

n

dr.ir. J.H. Huijsing

heeft als toegevoegd promotor in hoge mate bijgedragen

aan het totstandkomen van dit proefschrift

r

CONTENTS

17

17

17

20

21

23

24

30

34

34

37

41

c

LIST OF NOTATIONS

^B

^m

1

V

W

a

ß

e

e

f

)•

Ps

s

CHAPTER 1

INTRODUCTION

(b)

sensor

(o)

ampli-fication

con-version

ampli-

consensor — » ,. ,. — *

-iication version

trans-mission

trans-mission

manipu-lation

manipu-lation

(a)

4

(b)

(a)

(b)

shear profile

_(b)

_pv-