C O M P U T A T I O N OF T H E T H E R M A L I N E R T I A OF

T H E R M A L D E T E C T O R S

D. F . T a r t a k o v s k i i

Translated from Izmer i t e l ' naya Tekhnika, No. 11, pp. 24-26, November, 1963

C O M M E R C I A L

Constantly-rising requirements for precision in measuring temperature under nonstationary conditions leads to the necessity for further development in the methods for evaluat ing the inert ia of thermal detectors.

These methods, based on expressing parameters character iz ing the iner t ia l properties of thermal detectors by the so-ca l led thermal iner t ia l constants under normal thermal operating conditions of the 1st, 2rid, and 3rd kind, de -

noted respectively by s I, sII, and sIII, have been developed to the fullest extent for thermal detectors made in the form of uniform bodies with a s imple shape of cylinders, spheres, etc. Constants sII and s i i I can be ca lcu la ted for such thermai detectors from the known thermophysicat characteris t ics of a detector and from their boundary condi-

tions [1, 2].

However, commerc ia l thermal detectors have not as ye t been provided with satisfactory methods for comput -

ing their constants and in pract ice, as a rule, only their thermal inert ia constant s I for a normal thermal operating

condit ion of the 1st kind is determined, exper imenta l ly or computed.

Experimental determinat ion of sII and sIII is fraught with considerable difficulties, since for these experiments compl ica ted test equipment is required which serves to reproduce given temperature variat ion characterist ics and

hea t -exchange conditions.

Below we provide formulas for computing thermal inert ia constants for commerc ia l thermal detectors under operating conditions of the 2nd and 3rd kind from coefficients and t ime constants of their transient characterist ics

obtained exper imenta l ly for a s tep-by-s tep variat ion of the ambient temperature.

According to the theory of nonstationary thermal conduction [3], the transient characteris t ic of a thermal de -

tector can be approximated by the sum of an infini te number of exponential components according to the formula

D~ =--- ~ B n e Yn. (1)

n = l

A thermal detector ' s transient function which represents the detector 's react ion to a jump in temperature differs

from its transient character is t ic only by the origin of its coordinates and in the above case can be written as

t (x)--t,_______~ = t - - B~ e r . , (2) tIYI- t~ n : l

where r is t ime; t o is the in i t ia l temperature of the thermal detector and the medium; t m is the temperature of the medium; t ( r ) is the thermal detector 's temperature; B n and T n are the coefficients ( ini t ia l values) and t ime con-

stants of the corresponding exponential components.

Let us assume that the temperature of the medium follows a harmonic law

te (x)=~m+A sin tax.

Let us find an expression for the temperature of a thermal detector whith a transient function of the type of (2). By applying Laplace transforms to (2) and assuming that t m = 1 and t o = O, we obtain the transfer function for a ther-

mal detector .

W(s)=l-- ~ B"Tns n=l Tn$+l'

where s is a parameter of the Laplace transform.

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TABLE 1

Form of transient character is t ic

*11 elll

T~_L - eT~ T~ e r~

Tt-- Tt T,-- Tt

r , - , r : er, r , - r, e r, TI-- T* T1-- T,

T,+ r, 9

r , + r , l - , ~ , r T,

T,+ T~-- Ts T~+ T~-- r, + ~'T,T,T,,

r~ r , + rs e r~ a r,_..+ 1"_.~, # T~+T,+T3 r,+T,+T,+o:T,T,7", Ta--Tj Tx-- T~ 1--tu'(T,T~+ T, Ts+ T, TI)

TABLE 2

~ ' t l , *II, W

7n-~ -d seo sec

r sec

~o,ol ~=o,o6 t o,=o,a

116 120 110

~ 0 54 45

27

20

1160 37

2320 80 14

TABLE 3

101 84 18

43 23 14

26 17 19

11

W

m2~ ~c

w =0,01

B/A a B/A a

116 0.65 0 93 0,21 0 41

_ 0 . 7 0 " 0.51 '

580 0,89 0 97 I o . 4 a o 66

0.92 ' [ 0.66 "

,,oo :!::

~=0,05 ~=0.1

B/A a

:: :; 10.0

The constant component of the receiver ' s t emper -

ature in a s teady-s ta te condition is equal to tav for a constant heat transfer coeff ic ient . , Hence, i t is only nec -

essary to find the var iable temperature component Xout(r).

In our case

1 r, 7+] xin(s)' (3)

where Xin(S) is the Laplace representation of the ambien t - temperature var iable component,

By substituting Aw/(s 2 + w z) for Xin(S) in (3)and re-

vert ing to the original we obtain for the normal opera t - ing condition of the 3rd kind

I

7". c o s ~ n = 1 t (6)

tgl~ =coT.. (7)

By denoting in (4) the ampl i tude and phase of the resulting oscil lat ions respectively by B and Cv, we obtain

t(r ) = B sin (~or - Cv )" (8)

By assuming in expressions (4) and (8) first that w r = 90 ~ and then that w r = 0, we find that: i n the f i r s t instance

( ) A 1-- E Bnsin~fSa = B c o s ~ v , 15=1

Note: The top figures in the Table corresponding to each value of a are ca lcu la ted from (13) and the bot-

tom figures from (18). no

A E Bn sin [3 n cos;[~n=B sin q~v. n = l

(9)

and in the second instance

From the above let us find an expression for the tangent of the phase shift in the detector 's temperature osci l -

lations

Bn sin ~n cos l~n tz~l

tg q~v = (10)

l - - i Bn sln s~n n=-I

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k has been shown in [2] that the thermal inert ia constant 8ii I for normal operating conditions of the 8rd kind

consists of the proportionali ty coeff ic ient in the expression for the tangent of the phase shift in the detector 's t em- perature oscil lat ions

tg {pv=; Bit I r (11)

By substituting their values obtained from (5) and (6) for sinB n and cosB n in (10) we obtain, by taking into con- sideration (11), a formula for 8i i I

n=l 1 + ~ Tn 2 ~111:= (12)

o. 0} 9 ~n2

1-- ~ B n - - n=l I + ~ 7n 2

From equation (9) i t is also possible to obtain a formula for computing the ampl i tude of the detector 's t empera -

ture oscilIations. In fact, by substituting its value for sing n in (9) we find that:

~ ~

1-- B n - - B .=1 1+~' 7~ (18)

A cos q%

The value of 9v which figures in the above formula can be obtained from (10).

It is possible to find the detector 's thermal inert ia constant for a normal temperature condition of the 2nd kind

from (12). It has been shown in [2] that 8ii represents a l imi t to which constant ~III is tending for w-" 0. From the

above assuming in (12) that ~ = 0 we obtain

E ~Ii:= B a Tn" (14) n--1

Thus, the detector 's thermal inert ia constants for a normal operating condition of the 2rid and 3rd kind can be

computed from (12) and (14), provided coefficients B n and t ime constants Tn, which figure in the expression for the

detector 's transient characterist ic , are known. The lat ter can easily be obtained exper imental ly by a s tep-by-s tep

variat ion of the ambient temperature.

Methods for finding coefficients B n and t ime constants T n from an exper imenta l ly -ob ta ined transient charac te r - istic have been developed in [4], and we shall not dwell on them. It should only be noted that the transient charac- teristics of the major i ty of commerc ia l thermal detectors, thermometers and thermocouples, are approximated with adequate precision by a sum of two or three exponents, and the transfer functions are obtained in the form of s imple

expressions whose structure corresponds to the design pecul iar i t ies of the thermal detectors.

Research carried out by N. P. Buvin has shown that commerc ia l thermal detectors can be classified according

to the form of their transfer functions into three types:

1) iner t ia l thermal detectors whose transfer function has the form of

1 (18) W(s)--- ( ; q s + l ) (T~s+ l ) '

for instance, thermocouple TKhK-XIII,

2) iner t ia l thermal detectors with differentiating properties whose transfer function is

T, s + 1 (16) W ( s ) = ( T l s + l ) ( T 2 s + l ) j

for instance, thermocouple TKhK-146 and resistance thermometer ~TP-XXII,

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r, sec

Sa

r

I-

r, �9 ,r -

o mo w~ S. '~C

3) iner t ia l thermal detectors with negat ive di f ferent ia t -

ing properties and a transfer function of

1 - r , s (17) W ( s ) = ( T l s + I ) ( T s s + I ) '

for instance, gas manomet r ic thermometer TG-610.

Table 1 provides formulas for computing constants Sli and ~III for thermal detectors whose transient character is t ic can be represented by a sum of two exponents.

As an example let us compute constants ~II and sIII for thermocouple TKhK-146.

Coefficients B 1 and t~ which figure in the expression for the transient character is t ic of this thermocouple have the form

T,-- r, T~-- T, Bt= "TILT; B~.=--Tt_T 2.

T i m e constants T 1, T z, and T a for various values of the heat transfer coeff ic ient a can be found from the f am- i ly of character is t ic curves (see figure) obtained exper imenta l ly by the method described in [4]. The results of this computat ion are shown in Table 2.

In engineering computations it is normal ly assumed that harmonic variations of the measured temperature pro- duce temperature variations in the detector according to the formula

t (*)=tav+A cos q~v sin (c0,--q%), (18) where

1

V 1+4,, It wil t be seen from (18) that the detector 's ampl i tude of oscil lat ions in this case is reduced by a factor of

cos 9v as compared with the ambient temperature amplitude. However, strictly speaking (18) only holds for the case

when it is possible to neglect the nonuniform temperature distribution over the detector's cross section.

Research carried out for uniform bodies [3] has shown that the oscillations amplitude of the mean volumetric

temperature of uniform thermal detectors ca lcu la ted from (18) deviates increasingly with rising inert ia and frequency from its ac tual values owing to the detector 's nonuniform temperature field, which commonly occurs in its appl icat ion.

This also holds for commerc ia l thermal detectors which a more complex structure.

Table 3 shows the ampli tudes of temperature oscil lat ions in detector TKhK-146 for an ambient temperature os- c i l l a t ion ampl i tude taken as unity. These values were ca lcu la ted from approximate formula (18) and formula (13) which takes into account the pecul iar i t ies of the detector 's temperature field. The Table also shows the values of

coe f f i c i en t a , which should be used for mul t ip ly ing the detector ' s temperature oscil lat ions ampl i tude ca lcu la ted from the s impl i f ied formula, in order to obtain its real value.

It wil l be seen from the data in Table 3 that, for instance, the temperature oscil lat ions ampl i tude for a thermal detector TKhK-146 can be ca lcu la ted from (18) for a frequency w below 0.01 with an error not exceeding 3~ if the measurements are made for a = 580 W / m 2 �9 ~

It should be noted in conclusion that the precision in calcula t ing the corresponding values from (12), (13), and (14) for a given heat transfer coeff ic ient is determined by the precision in ca lcula t ing coefficients B n and t ime con- stants T n which figure in the expression for the transient character is t ic (1). According to the data of [4] the ca l cu l a - tion error amounts to 2-3%.

1.

2.

LITERATURE C I T E D

A. N. Gordov, Transactions of the VNIIIvl, No. 25 (85) (1955).

A, N. Gordov, Transactions of the VNIIM, No. 35 (95) (1958).

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3. G.M. Kondrat'ev, Thermal Measurements [in Russian], Mashgiz (1957). 4. N.P. Buvin, Scientific Papers of Higher Educational Institutions. Series on nElectromechanics and automation,"

No. 2 (1959).

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