Đo nhiệt độ P15 ppsx

For this reason, any influence of the dynamic properties of indicating instruments may be neglected in most cases.Knowledge of the dynamic properties of a temperature sensor is necessary

a medium at a constant temperature using a temperature sensor immersed in the medium Determination of the dynamic errors of a thennometer, requires knowledge of its dynamic properties In many non-electric thermometers where the sensor and indicator form one inseparable unit, the dynamic properties to be described must refer to the whole device Electric thermometers are mostly used when it is essential to know the dynamic error so that it can be taken into consideration Consequently, the dynamic parameters of electric sensors will be the main topic for discussion in this chapter It must be stressed that dynamic errors in temperature measurement are principally caused by the sensor For this reason, any influence of the dynamic properties of indicating instruments may be neglected in most cases.

Knowledge of the dynamic properties of a temperature sensor is necessary for the following main cases:

" to determine the necessary immersion time, while measuring a constant medium temperature,

" to determine the dynamic errors while measuring temperatures changing with time,

" to compare the dynamic properties of different temperature sensors, so that the one best suited for a specific application, may be chosen,

" to determine the true temperature variations of temperatures changing in time by correcting known indicated values,

" to describe the dynamics of a sensor when it is part of a closed loop temperature control system as described by Michalski and Eckersdorf (1987),

" to choose the type and optimum settings of a corrector of dynamic errors.

Temperature Measurement Second Edition

L Michalski, K Eckersdorf, J Kucharski, J McGhee

Copyright © 2001 John Wiley & Sons Ltd ISBNs: 0-471-86779-9 (Hardback); 0-470-84613-5 (Electronic)

As the problem of the dynamics of temperature sensors is approached in different ways,the number of existing references concerned with these dynamics is very large.Consequently only concise principles are presented in this chapter.

15.1.1 Dynamic errors

Dynamic error can be defined as the difference between the sensor temperature, t9T(t), andthe temperature, 0(t) , measured by another inertia free sensor, exhibiting the same staticerrors Consequently the dynamic error is that part of the systematic error which varies withtime To simplify the problem, assume that the static error equals zero Using Figure 15.1,the dynamic error is then defined as:

AOdy (t) = OT(t) -t9(t) (15.1)Measuring the stepwise changing temperature, the relative dynamic error is defined as:

Sz9dyn(t)= A6dy (t) - t9T(t)-6(t) (15.2)

where A9 is the value of the temperature step

The response time, tr, after which the relative dynamic error does not exceed a certainvalue, is closely connected with the relative dynamic error For instance at t >_ tr,5 % theabsolute value of the relative error is 60d,(t) <<-5% The dynamic error can also bedefined for other, non-periodic temperature variations Its value is then mostly related to themaximum change of the measured temperature

In dynamic temperature measurement Hofmann (1976) asserts that it is necessary todetermine the dynamic errors in two cases The first occurs when the measured, indicatedtemperature and the sensor's dynamic properties are known Another occurs when themedium temperature, the input, is known as a function of time as well as the dynamicproperties of the sensor In both cases it is convenient to represent the dynamic error usingthe Laplace transform to obtain:

GENERAL INFORMATION 281When measuring a sinusoidally changing temperature 6(CV) = A6sin 0ot +tip thedynamic errors consist of amplitude and phase errors Amplitude error, AA(o)), is given bythe difference of the amplitudes of the sensor temperature 019T (c)) and its true value A6,

15.1.2 Dynamic properties of temperature sensors

Modelling temperature sensors consists of four main steps decribed by Jackowska-Strumillo

et al (1997) Applying this process eventually allows the dynamic properties of atemperature sensor to be described by the following equation:

F[Yn(t),Yn-1(t) Y(t);6m (t),6m-1 (t) t9(t)] = 0 (15 7)where y(t) y" (t) are the sensor output signal and its time derivatives, and13(t) 0m (t) are the measured temperature and its time derivatives

In the case when the dynamic behaviour is linear, equation (15.7) becomes:

d lY(t)

t-o dt` j-o dtiwhere ai and bj are constant coefficients and m < n

The transfer function, G7(s), can be used to describe and present the dynamic properties

of a temperature sensor It is the ratio of the Laplace transform of the sensor output signal,

y(s), to the Laplace transform of the measured temperature signal, 9(s), when the initial conditions are zero so that:

as a function of time (Doetsch, 1961).

Each electric temperature sensor may be regarded as composed of a thermal conversion stage and an electrical conversion stage as shown in Figure 15.2 In the thermal conversion stage, the temperature, 9(t), of the medium whose temperature is being measured is converted into the sensor's temperature, t9T(t) The sensor's temperature, OT (t), is converted into the electrical output signaly(t) (e.g thermal emf) in the electrical conversion stage This second conversion stage has a purely static character Thus, the sensor transfer function GT(s) of equation (15.9) can be expressed as a product of the transfer function of the thermal conversion stage, FT(s), and ofthe coefficient KT, representing the properties of the electrical conversion stage, and called the sensor gain so that equation (15.11) is obtained:

(a)

4(t) THERMAL (t) ELECTRICAL y (t) CONVERSION CONVERSION

t-DOMAIN PRESENTATION (b)

In the case of steady-state periodic variations of measured temperature, the frequency

response, GT (jco) , of the sensor may be considered instead of the sensor transfer function,

GT(s) The sensor frequency response is the ratio of the phasor values of the output signal y(jro) to the phasor value of the variable component of the sinusoidally changing measured temperature 6(jo))

Nja)) where co is the angular frequency and j =

The frequency response ofa sensor can be obtained by substitutingjco in equation (15.9)

in place of the operator, s Consequently from equation (15 10) the frequency response becomes:

m

Y,bi(jo))j

~) nYai (jco)i M(j~)

where Ay((o) is the amplitude of the output signal y(jw), AO is the amplitude of the first harmonic of measured temperature O(jco) and (p(o)) is the phase shift between y(jO)) and 6(j))

Equations (15.15) and (15.16) are related as :

Ay A~ ) = JGT(jp)j =KT P2 (p)+Q2 (0 ) (15.17) (p(co) = argGT Go)) _ -arctan Q

Ay((o) l AO of equation (15.17) and (p(o-)) of equation (15.18) are respectively called the

amplitude and phase characteristics of a temperature sensor Sometimes it is more convenient to use the amplitude of the sensor temperature AOT((O) instead of the amplitude ofthe output signal Ay((o)

IDEALISED SENSOR 285

further assumed, that the thermal capacity, mc, of the sensor is negligibly small compared with the total thermal capacity of the medium and that the heat transfer coefficient a , between the sensor and the medium is constant.

15.2.1 Transfer function

To set up the differential equation, which describes the sensor's dynamics and thus its transfer function, the method of heat balances will be used Assume, at the time t = 0-, an infinitesimally small time before zero, that the sensor is in a steady state, with its temperature equal to the ambient temperature OT = 0,At t = 0+ immerse the sensor in the medium at temperature 0 , higher than the ambient temperature so that 0 > Oa For the temperature excess over the given reference value the notation O is introduced In this book O is also simply referred to as temperature The initial conditions at t = 0- are given by:

OT =ZT -Oa =0 and O=6-6a >0 According to Newton's law, when the sensor is immersed in the medium, the heat transferred to the sensor in the time interval dt will be:

where m is the mass of the sensor, and c is the specific heat of the sensor material.

From equations (15.20) and (15.21) it follows that

a4(0- OT)dt = mcdOT (15.22) or

a4 dt Introducing the notation:

aA

Equation (15.23) can be expressed as :

The frequency response of the thermal stage of an idealised sensor may be written as :

and of the sensor as a whole:

IDEALISED SENSOR 28715.2.2 Measurement of time varying temperature

If the changes of measured time varying temperature, 0(t), can be described by elementary functions, the temperature indicated by the sensor 6r (t) , and the dynamic measurement error can be determined in a simple way by using the Laplace transform As an example, consider a step temperature input from an initial temperature, 6b, to a final temperature

oe In practice this case corresponds to the immersion of a temperature sensor with a temperature, Ob, into a medium with a temperature, 6e , which is written mathematically as:

~9(t) Ob

fort <_ 0

- {6e ort>0 or

z9(t) = (Oe - Ob )1 (t) + Ob where 1(t) is a unit step at t = 0 (15 33)

As it is necessary to obtain zero initial conditions for the Laplace transform, the excess temperature O = Oe - 6b , will be used, to obtain:

for which the Laplace transform is:

s From (15.27) it follows that:

OT(t) = OX-e-t/NT) (15 39) The final result is then:

OT (t) =(Ve - Ob)(I -e_tIAIT)+Ob (15 40) From equations (15.38) and (15.40) it follows that the step input response of an idealised temperature sensor is an exponential curve, having the time constantNT as shown

in Figure 15 3 From this curve, the time constantNTcan be found in a graphical way from the tangent to the curve OT(t)=f(t) at any point, or as the time after which

OT(t= NT) = 0.6320,

Also the half-value time, or 50 % rise timeto 5,which is the time when OT = 0.50, can

be used to determine the time constant From equation (15.39) att = to.5,it is clear that:

IDEALISED SENSOR 289 From equation (15.1) (15 39) and (15 40) the dynamic error will be:

AOdyn (t) =OT(t) - 0e =-(tie-f%)e-t/NT (15 43) or

AOdyn(t) = OT (t) - Oe = -pee-t/NT (15.43a) Thus it is clear that:

From equation (15 2) (15.39) and (15.40) the relative dynamic error will be:

e or

6Odyn (t) =OT(O- pe = -e-tlNT (I 5.44a)

" Ramp input response,

" Exponential input response,

" Sinusoidal input response,

" Periodic non-sinusoidal input response.

Table 15.2 Measurement of time-varying temperature by an idealised temperature sensor with thetransfer function given in equation (15.30)

Input Mathematical expressions Eq'n no Input and output comparison

cpi =-27c-=-arctancoNTt

(15.62)

TO

REAL SENSORS 291

Numerical example

The medium temperature, 9e = 220 °C, is to be measured by a temperature sensor with a timeconstant,NT = 30 s and an initial ambient sensor temperature of9b = 20 °C How long should theimmersion time be, to ensure that the indication error is less than2 °C?

Solution:

A9d, <_2 °C corresponds to

Is1~dy°I< 2202-20 = 0.01From Table 15.1 for c59d,=-0.01, the necessary immersion time ist= 4.6NT,so that:

t > 4 6NT= 4.6x30 = 138s

Numerical example

A sinusoidally varying medium temperature was measured by a temperature sensor withNT= 30s.The period of temperature oscillations was TO =100 s, while the amplitude of the sensortemperature, AOT= 5 °C Determine the true amplitude of the temperature oscillations of themedium

15.3.1 Sensor design

The temperature indications of an idealised sensor depend upon the average temperature ofits whole mass As the thermal conductivity, A, , of the material was assumed to be infinitelyhigh, the sensor temperature is the same all over its volume In real industrial sensors,

which are mostly cylindrical, the sensitive part of the sensor is not always the whole of thesensor Figure 15 4 gives a way of relating the dynamic step response behaviour of thesensor to its sensitive physical structure

1-SENSOR SENSITIVE

PA RT 2-SHEATH EATH 3-INSULATING

Figure 15.4 Step response of real temperature sensors

Each of the three general kinds of response is due to one of the following sensitive partsofthe sensor, relevant for the measurement :

" in volumetric response sensors the whole mass, as in mercury-in-glass thermometersneglecting the extremely thin glass layer, as shown in response (a) ofFigure 15.4,

" in surfacial response sensors the surface of the sensor, as in bare resistance temperaturedetectors with the resistance wire wound on the surface, as in (b) of Figure 15.4,

" in central response sensors the centre of the cross-section, as in sheathed thermocouplesensors represented in (c) of Figure 15.4

A reasoned understanding of the step responses of all three structures, shown inFigure 15.4, is possible using a qualitative explanation of the heat transfer processesinvolved

Volumetric units exhibit a response which is closest to the response of the idealisedsensor This arises from their whole volume acting as their sensitive part A good example

of a sensor which exhibits this kind ofbehaviour is a mercury-in-glass thermometer

In surfacial types, such as bare RTDs, the surface of the unit is the sensor sensitive part.During the initial part of the response, it heats up quickly Some time later, the centralregions of the assembly start to absorb more and more of the heat input As it obviouslyslows down the rate of temperature increase at the surface, the sensitive part of the sensor

"slows" to its final steady value

In the central response type, it takes a finite time for heat to diffuse into the centralregion of a sensor structure such as that of a sheathed thermocouple Hence, with centralresponse sensors, there is a noticeable delay before the centrally located sensor starts to heat

up After the main volume of the assembly has been heated the sensitive part starts to absorbheat fairly quickly because of its relatively low thermal mass

As proposed by Lieneweg (1975) all of the sensors can be classified by the ratio oftheirresponse times, tO.9 / tO.5 , so that for:

1 sensors with volumetric response, tO.9 / tO.5 = 3 32 ,

2 sensors with surfacial response, tO.9 / tO.5 > 3.32 ,

3 sensors with central response, tO.9 / tO.5 < 3 32

REAL SENSORS 293

Defining a sensor to be within one of the above three groups, depends on its step response From this reasoning, it can be seen that the step response of any real sensor depends upon its design, its working temperature and the heat transfer conditions at this temperature.

15 3 2 Changing heat transfer coefficient

In the derivation of the differential equation (15.25), which describes the dynamic properties of an idealised sensor, it was assumed that both the sensor time constant and the heat transfer coefficient between the sensor surface and the surrounding medium were constant and temperature independent In reality, the overall heat transfer coefficient, which must include convection, conduction and radiation, is a function of the medium temperature

as well as of the instantaneous sensor temperature Hence, time constant of the sensor will vary with the varying heat transfer conditions.

These problems are especially apparent while measuring temperatures due to a predominantly radiative heat exchange, as occurs inside chamber furnaces working above

600 °C (Hackforth, 1960; Michalski, 1966) The radiant heat flux between the chamber walls at temperature, T2, and the temperature sensor at Ti , is given by equation (8.24a)) as:

[( 100)4-Co1\4]

where A1 is the sensor heat exchange surface andE1 is the sensor emissivity.

This formula which is valid for walls with surfaceA2 > 3A 1, can be rewritten as :

If the sensor temperature, T1, does not differ from T2 by more than ±10 % of T2, corresponding to :

0.9<T1 < 1.1

T2

then equation (15.64) can be replaced by the following approximate dependence ofaccuracy to within 1%.

3

ar = el Co x 0.5 x 10-z(TI + T (15 65)

100 100)According to Eijkman (1955) and Lieneweg (1975) the coefficient, ar , increases as thetemperatures of the furnace walls and the sensor increase, as shown in Figure 15 5,achieving far higher values than the convective heat exchange coefficient, ak Thetemperature dependence of ar causes a considerable difference between the step responses

of bare thermocouples in an electric furnace and exponentially curved responses On theother hand high values of A means that they closely approach ideal sensors

15.3.3 Equivalent transfer function

The application of the concept of the transfer function to real temperature sensors leads tothe idea of an equivalent transfer function, which is only valid under the followingnecessary simplifying assumptions:

" The dynamic properties of the sensor are linearised within the given temperature range

" A mean time constant is used as a value to describe the bi-directional heat flow betweenthe sensor and the surrounding medium as described by Skoczowski (1982)

" The sensor is represented by a lumped parameter model

The most commonly used sensor transfer functions are given in Table 15.3 Thesemodels, which take the design of the sensor and the heat transfer conditions into account,have been considered by Bliek and Fay (1979), Eckersdorf (1980), Hofmann (1965, 1966,1967b), Rubin and Feldman (1968), Schwarze (1964) and Souksounov (1970) The

600

T2=1500K z

Figure 15.5 Radiant heat transfer coefficient, %, versus sensor temperature, Ti, for a sensor of

ai = 0.8, at given temperature, TZ, ofthe furnace walls

majority of industrial thermocouples and resistance thermometer sensors may be simulated

by the second order system given by equation (15.67) in Table 15.3 Low inertia sensors areusually represented by a first order system model in accordance with equation (15.66) inTable 15 3

15.3.4 Calculation of dynamic properties of sensors

It is assumed that the temperature sensor can be represented by a homogeneous cylinder ofthermal conductivity, A, specific density, p, and specific heat, c The sensor is totallyimmersed in the medium whose temperature is to be measured Its thermal capacity isinfinitesimally small compared with that of the medium, while the heat transfer coefficientbetween the sensor and the medium, as well as the sensor parameters, are constant Internalheat sources are also assumed to be non-existent in the sensor Under the aboveassumptions, Hofman (1976) and Jakob (1957, 1958) show that the Fourier differentialequation of an infinitesimally long cylinder is:

dOa(r,t) =a[d2OYr,t) + I dO(r,t) (15.71)

2

where a, which is the thermal diffusivity of the cylinder material, is equal to A/pc

For real temperature sensors ofradius R, the solution ofthis equation follows that for theboundary condition ofthe third kind which have the general form:

d O(R,t) a

a0(O,t) - 0

(15 71b)ar

and also with zero initial conditions given by:

This solution, considered by Hofmann (1976), gives the temperature any chosen point ofthe cylinder at a distance, r, from the cylinder axis (0 < r < R) for a step change of theambient temperature from 0 to Oe

For some sensors it is also possible to use other simple models such as that of a sphere

or plate The corresponding solutions of the Fourier equation for boundary conditions of thethird kind are given in Grober et al (1963), Hofmann (1976) and Jakob (1957, 1958) Theabove theory, which is valid for a simple sensor, can also be extended to multi-layer sensors

as shown by Hofmann (1976), Lieneweg (1938a, 1938b, 1941, 1962) and Yarishev (1967)

To determine the dynamic properties of real sensors, it is necessary to set up theirequivalent models first As temperature sensors with tubular sheaths are widely used inpractice, most publications concern this type of sensors The majority of authors, such as

REAL SENSORS 297 Caldwell et al (1959), Eijkman (1955), Eijkman and Verhagen (1958), Meyer-Witting (1959) and Yarishev (1967), use equivalent circuits based on a second order system, as shown in Figure 15.6 The thermal capacity, m1cl, of the sheath is represented by a capacitor, Cl, and that of the sensitive part of the sensor, m2c2, by C2 The thermal resistances between the sheath and its environment, Ilk,, and between the sheath and the temperature sensitive part, 1/k2, are similarly represented by the resistors R1 and R2 respectively.

All these capacitances and resistances are given as relative values per unit of the sensor length provided that the sensor is sufficiently long to neglect any heat exchange along its length The thermal resistance across the sheath wall has also been neglected Element pairs

of the analogous quantities of the thermal and electric models are given in Table 15 4 (Gr6ber et al., 1963) The input voltage, V, corresponds to the measured mean temperature,

O, the output voltage, VT, to the temperature, OT, of the sensor's sensitive part, so that the transfer function ofthe analogue circuit, given in Figure 15.6, is :

Figure 15.6 Real temperature sensor and its electric analogue circuit

Table 15 4The analogy between electric and thermal systems

Thermal system Electrical system Scale factor

Temperature above ambient O,(°C) Voltage V, (V) of voltage K,, = VIO (V/°C)

Thermal resistance 1/k, (°C/W) ResistanceR, (S2) of resistance KR= Rk, (f2W/°C)

Thermal capacitance mc, (J/°C) Capacitance C (F) of capacitance KC= C/mc, (F°C/J) Timet, (s) Model timetej ,(s) of time Kt = te 1lt,(s/s)

EXPERIMENTAL DETERMINATION OF THE DYNAMIC PROPERTIES OF SENSORS 299

15.4 Experimental Determination ofthe Dynamic

Properties of Sensors

15.4.1 Classification and application of the methods

Dynamic properties of real sensors are usually determined experimentally by adapting stepresponse and frequency response methods (Rake, 1980) When testing for operationalintegrity and performance checking, the dynamic parameters of temperature sensors should

be found in the same environment and at the same temperature where the sensor will beused Nevertheless, it may be just as important to measure the parameters for design ormodel validation purposes Before the development of digital techniques it was essential toensure that analogue recorders used for recording the measurements, had a response time of

at least five times shorter than the sensor's response time This condition is easily met usingany of the various digital data acquisition systems described in Chapter 13

McGhee et al (1992a, 1992b) provide a review ofthe dynamic properties and testing ofcontact temperature sensors A classification of experimental identification methods, which

is given in Figure 15.7, shows that the two main methods rely upon external or internal

electnc two-section lowinertiafurnace furnace furnace

INTERNAL " - ~~ STEP TEMPERATURE

INPUT heating \~~,SINUSOIDAL,

METHODS `, current ;SQUARE WAVE,

TEMPERATURES waterpipe-lineor waterpipe-lineor

(convective heat two pipe-lines double pipe-line

liquid bathFigure 15.7 Classification ofmethods for the experimental determination of sensor dynamics

stimulation by an interrogating signal As shown in Section 15.4.4, the results obtained by both methods are not identical Moreover, the aperiodic or periodic nature of the signal respectively determines whether the method used belongs to the time domain or frequency domain group of testing procedures

External temperature input methods are widely used to measure the dynamic properties

of temperature sensors, which are subjected to known changes of the temperature of the surrounding medium For example, for step input response the sensor should be immersed

in a medium at a different temperature from its initial temperature As this means that the sensor is not in its real operating conditions, this method is used for the purposes of design

or model validation At low temperatures, below about 100 °C, where heat exchange by radiation is negligibly small, a convective heat exchange coefficient , a k , can explicitly describe the working conditions of the sensor Most methods are based on measurements in either flowing water or air In practice, at temperatures over 600 °C, with free convection conditions, there is only radiant heat exchange characterised by the source temperature as a final value In all other cases it is necessary to specify the heat exchange conditions precisely, by giving the values of the heat exchange coefficient a and of the final temperature 9,.

In most of the older publications such as those of Higgins and Keim (1954), Hofmann (1976), Huhnke (1973), Jakob (1958) and Kondratiev (1947), the experimental determin- ation of the sensor dynamic properties is performed at low temperatures using well known methods, although the associated instrumentation is now largely obsolete More contemporary, microprocessor based instrumentation was used by McGhee et al (1989) in the first application of MBS signals to temperature sensor testing Further development of this method appear in McGhee et al (1993) and Jackowska-Strumillo et al (1992, 1996, 1997) These low temperature applications for external input testing are described in Section 15.4.2 for the time domain approach, while external low temperature frequency domain testing is presented in Section 15.4 3 Only a few publications deal with external input in the temperature range above 600 °C (Rubin and Feldman, 1968; Bernhard, 1979 ; Eckersdorf, 1980; Eckersdorf and Michalski, 1984) These methods are presented in Section 15 4.4.

Dynamic testing of electric sensors in real operating conditions uses in situ testing by the internal input method An electric current, flowing in the electrical part of the sensor, causes direct ?R self-heating in the resistance winding of an RTD or in the lead wires and measuring junction of a thermocouple These methods are described in Section 15.6.3

15.4.2 External input, time domain testing with convective heat

transfer

Step input method The simplest and most popular method of measuring the step response of a temperature sensor is to immerse the sensor by plunging it into a well-stirred water bath at a temperature in the range of 20 to 60 °C above the ambient temperature This method belongs to the time domain group of testing methods Berger and Balko (1972) advise the mechanisation of the immersion procedure to ensure that the transition in the external temperature of the sensor is as fast as possible The heat transfer coefficient, a, between the sensor and the medium is some thousands of W/m °C This method gives explicit and reproducible measuring conditions, because any occurring variations of the

EXPERIMENTAL DETERMINATION OF THE DYNAMIC PROPERTIES OF SENSORS 301

high values of a are usually so small that they do not exert a significant influence on thesensor dynamics (Kondratiev, 1947; Yarishev, 1967)

To measure sensor dynamics with different values of a, some special channels andpipe-lines are constructed, in which a gas, mostly air, or liquid, mostly water, flows at anadjustable velocity This velocity is measured in order to enable a comparison of the resultsobtained Hofmann (1976) describes an air channel in which the flow velocity may beadjusted between 0 and 20 m/s It is intended for sensors, having sheath diameter, D, up to

10 mm and an immersion length up to 100 mm The highest achievable values of a were

150 W/m2 °C, for D = 10 mm, and 400 W/m2 °C, for D = 1 nun Chohan and Natour

(1988) describe an air channel, which was designed for sensor testing at air velocities up to

is pushed in the direction of the flowing medium and simultaneously the flow of hot-waterstops A deciding factor is the step input duration, which has to be shorter than 0.1 of thehalf-value time, to 5 ,of the tested sensor In this set-up a mechanical device ensures that the

95 % step response time, tO.95, is shorter than 20 to 100 ms

A step change in temperature can also be realised using a low-inertia mesh heatingelement Such a mesh is mostly used for gas heating, because in liquid heating far greaterheating power would be needed The air-channel shown in Figure 15.9 (Huhnke, 1973) has

a main construction consisting of a thin wire mesh heating element placed parallel to thetested temperature sensor and perpendicular to the direction ofthe air stream A step change

in the temperature of the flowing medium is achieved by switching the heating power onand off In most cases temperature changes of about 10 °C, which are applied, should beregarded as exponential approximations to step temperature changes characterised by atime-constant, N (Huhnke, 1973) For example, at an air velocity of v = 2 to 15 m/s, the

TESTED SENSOR FLOWING

BLOWER

-D-Figure 15.8 Pipe line with a movable shield Figure 15.9 Air channel with mesh heating element

time constant of the heating wires of d = 0.02 mm, is N = 16 to 7 ms, corresponding to aresponse time, to.95 = 50 to 20 ms Application of a low inertia heating element, also makes

it possible to generate other temperature test signals, especially sinusoids The method ofinterpreting the recorded step-input response of a temperature sensor depends upon itscharacter and upon the necessary precision and may be as follows :

" determination of tr, considered in Section 15 1 1, or to.5 and to.9, from Section 15.2.2,directly from the step input response,

" determination of the equivalent transfer function choosing the right sensor model fromTable 15.3 based on the to.9/t0.5 ratio

All ofthe following variations of sensor response are summarised in Table 15 5

Volumetric response sensors, with to.91to.5 z 3 32, which also have dynamic propertiessimilar to those of the ideal sensor of Section 15 2.2, are regarded as first order inertiaelements Their equivalent transfer function is given by equation (15 66) ofTable 15.3 Central response sensors, with to.9lto.5 < 3,32, which are characterised by s-shaped stepresponse curves, are regarded as a series connection of a first order inertia element and apure lag Their equivalent transfer function is given by equation (15 68) in Table 15.3 Thevalues of the pure lag parameter, LT, the time constant, NT, and thus LTINT, can be found inthe way shown in Figure 15.10 In some cases, an appropriate alternative is to use a modelwith a second order inertia element with an equivalent transfer function given byequation (15.67) of Table 15.3 In this case, the values of NTi and NT2 can be found fromFigure 15.11

A more precise method of determining NTI and NT2 , when NTI >> NT2, is theYarishev (1967) logarithmic technique shown in Figure 15 12 When the step-inputresponse of equation (15.77) ofTable 15.5, is given as the relative unit-step response:

h(t) = OT(t) (15.79a)

e

it allows a definition of the function, h * (t) , given by:

h * (t) =1- h(t) (15 79b)The relations in equations (I 5.79a) and (15.79b) are shown in Figure 15 12(a)

When t >> NT2, h * (t) will be transformed into hi (t)

h1(t) = NTI e-ONTI (15 80)

NT1 - NT2Taking logarithms on each side of equation (15.80) gives the straight line:

lnhj (t) =1n NTI _ t (15 81)

NT I -NT2 NT I

Plots In h*(t) are shown in Figure 15 12(b) Solving equation (15.80) forthe instants tj and t2, shown in Figure 15.12(b) gives the value of the time constant, NTI, as:

. temperature sensor with central response constantsNTIandNT2when LTINTis known and

has n found using the method of Figure 15 10

~(t ,h(t)'' RESPONSEEXPERIMENTAL STEP-INPUT e'*"' OF STEP-INPUT RESPONSE

tr,(t) LOGARITHMIC DISPLAY - - - - - -

1,0 :

.

TIME t TIME t

Figure 15.12 Logarithmic method ofdetermining the time constants, NT t andNT2

EXPERIMENTAL DETERMINATION OF THE DYNAMIC PROPERTIES OF SENSORS 305

The time constant, NT2 , can be determined by reading the value of h*(O) fromFigure 15 12(b) and inserting it into equation (15.80) at t = 0, to obtain:

NTI - NT2Since NTI has already been found it can be used in equation (15.83) to obtain NT2 from:

hI (0) -1

The time constant NT2 can also be found in a similar way as NTI by drawing the functionh2* (t) = hl (t) - h * (t) with a semi-logarithmic scale This method is then specially usefulfor checking if the second-order inertia system gives a sufficiently precise model of thesensor If the function h2 (t) is not a straight line, the sensor should be approximated by ahigher order system

Sensors with surfacial response, where t0.9/t0.5 > 3,32, are mostly approximated by order inertia systems, as described by equation (15.66) in Table 15 3 If a more preciseapproximation is necessary, a transfer function as given in equation (15.70) from Table 15 5

first-is used:

GT(s)=KT (I+sNT3)

(1+SNTI )(1 + sNT2 )

The transfer function zero, at s= - 1/NT3, takes account of the thermal layer from the inside

of the sensitive sensor surface to the core ofthe sensor

A method to find the time constants, NTI, NT2 and NT3, which is based on the recordedstep input response, is described in Hofmann (1976) and Yarishev (196/) However, forsensors with surfacial response it is usually precise enough to represent their dynamicproperties by an equivalent transfer function with a single inertia:

G s) -=KT( T I+sNTIISubsequently, the value of NTI is determined in the same way as for sensors with volumetricresponse

To determine the parameters from a step input response of sensors with volumetricresponse under changing heat-transfer coefficient the Lieneweg method is used to find acharacteristic parameter, Y, from Figure 15.13 This method is based on experimentallydetermined step input responses of the tested sensor in two different media The values ofthe surfacial heat transfer coefficients, a, must be known in each case In most cases themeasurements are conducted in water and air to find either the half-value times, tO.5,w, in