Analog solution for heat flux using a time dependent source temperature of the form T=at N

TECHN\SC

.E

HOGE~;Ct

.

OOl

DElFT

Vt.lEGTUIGt",'JVH~U, DE

. .. ~v

-

8 DEC.

19

66

tslL:>LlV 11 'L.L."

von

KARMAN INSTITUTE

FOR FLUID DYNAMICS

TECHNICAL NOTE 23

ANALOG SOLUTION FOR HEAT FLUX

USING A TI

M

E DEPENDE

N

T SOURCE TE

M

PERATURE

OF THE FOR

M

T

=

atn

by

A. BENOIT and J.-M. CHANOINE

RHODE-SAINT-GENESE, BELGIUM

NOVEMBER

1964

ven KARMAN INSTITUTE FOR FLUID DYNAMICS

TECHNICAL NOTE 23

ANALOG SOLUTION FOR HEAT FLUX

USING A TIME DEPENDENT SOURCE TEMPERATURE

OF THE FORM T

=

at

n

by

A. Beneit and

Jo-MD Chaneine

i

ACKNOWLEDGEMENTS

We wish to express our thanks to Mr. W. F. Campbell

for the opportunity to conduct this work at the von Karman

Inst±tute~

One of the authors is indebted to the "Centre National

d'Etudes et de Recherches

A~ronautiques",

Belgium

a

ror a research

fellowship.

Thanks are due to Dr

.

J

.

D. Haygood for his critical

ii

SUMMARY

The errors resulting in the determination of

temperatures and heat fluxes by a finite one-dimensional

analog network are computed for the following conditions.

The boundary temperature. at the origin.

~s

of the

n

form t , where t is the time and n is a non-negative integer,

the end of the slab is held at constant temperature or is

perfectly insulated.

The results for both cases are given in

tablee and represented graphically for values of n ranging

from zero to ten.

i i i ~ABLE OF CONTENTS Page Acknowledgements l. Summary l.l.

..

Table of Contents l.l.l. Notation l.V List of Fie;ures v

List of Tables Vlo

1. Introduction 1

2. Theoretical Considerations 2

3 • Presentation of the Results

₉

4.

Computer Program 10

Refcrences

Figures

iv

NOTATION I

Symbols

a

conductivity

A

temperature

coefficients

f9r z

=

0

AI

heat

flux coefficiente

for z

=

0

B

temperature

coefficients

for z

=

~

B'

heat

flux coefficients

for

z

=

~

D

error

coefficients

K

cell

o

rder

1

non-negative

integer

N

number of

cells

0

integer

at least equal to three

n

non-negative

integer

r

non-negative integer

t

non-dimens

i

onal

time

,

Y

temperature

or heat

flux

z

terminal

i

mpedance

ö

d

ifference

between exact and

analog

solutions

~ non~dimensional

abs

ciss

a,

0

'

~ ~ 1

Spbscripts

1

temperature

2

hea~

flux

N

ana

log system

of N

cells

~

continuous system

superscripts

(N)

analog

system of N cells

LIST OF FIGURES

1. Configuration of the analog network

2. Coefficients At(~) versus ~ for several values of 2.

3. Coefficients A~ (~) versus~ for several values of 2.

4.

Coefficients Bt(~) versus ~ for several values of 2.

5.

6.

7.

10.

Coefficients B' · (~) versus ~ for several values of 2. t

Comparison between Atand its analog

Comparison between AU and its analog (N)Aw

R. 2.

. . (N)B

Compar~sqn between B2. and 1ts analog 2.

Comparison between BU _{and it s analog (N)B}9

t X.

Error coeffici ents for t emperat ure versus reduced distance (z

=

0)

110 Error coefficients for heat fl ux versus reduced distance (z

=

0)

12. Error coefficients for temperature versus reduced

distance (z =co)

13. Error coefficients for heat flux versus reduced distance (z

= ...

)

vi

LIST OF TABLES

10

Ideal thermal short circuit

lol Temperature

coe~~icients

Ai

Analog temperature coefficients (N)A i

Error coefficients for temperature (N)D

l

• i

1 02 Heat

~lux coe~~icients A~

Analog heat

~lux

coefficients (N)At

. . (U)

Error coeff1c1ents

~or

heat

~lux

D2,i

20

Perfect thermal

insulation

201

Temperature

coe~~icients

Bi

Analog temperature

coe~~icien

ts (N) B i

Error coefficients for temperature(N)D

l

2

0

2

Heat

~lux coe~~icients

B

U

i

Analog heat

~lux

coef~icient

(N)B'i

• i

. • (N)

Error

c·oe~~1c1ents ~or

heat

~lux

_{D2 i}

10

10 Introducti2n

The temperature distribution and heat flux in a

one-dimensional finite slab of homogeneous material is determined by the analog method for a boundary temperature of the form

tnt n being apon-negative integero The interest of considering

such a temperature form arises from the fact that under relatively

general assumpti onsa a gi ven function of t can be approximated

by a suitable polynomialo The analog solution is compared

with the exact one and the resulting errors can be deduced

from tables given at the end of the reporto

are performed for the two l imit ing cases

The calculations

a )o The boundary of the slab consists of a perfect

thermal short-circuit j or

b)o The boundary of the slab ~s a perfect thermal

insulatoro

In both casesa t he computat ions are done for values of the integer n ranging from zero t o t eno This i nt erval wi ll gene rally be sufficient for practical purposes, only the so-called "st eady state " response wi l l be consi deredo

20

Theoretical

Considerations

The

continuous slab

and

the

analog network are shown

in Figure lo

This figure also

indicates

the correspondence

between the two systems o

The temperature measured at the

th

I

output of the K

cell corresponds

to the actual temperature

at the point

~such

that

~ = l _

f

_N

where N

is

the total number of cells

o

On the other hand.

the heat flux through the

resistor

of the Kth cell is to be

compared with

the

heat flux at

~'

such that

~'

=

t

+

2N

1

The expressions for the

temperature

an~

heat flux for the

t~o

systems havp been

de~ived

in

a

previous

paper

(see

Reference l)

and will only be mentioned

br

i

eflyo

201

Perfect

therm~l

short

circuit

In

this case we have

n

n-R.

Yl

pO> ( ~ i)

t)

=

n

X

I

AR. (~ )

t

(

n-R.

)

! R. =0

r

(N)A U;)

n-R.

Y

liN

(

~

,t)

n

X

t

=

(n-R.)I

R. =0 R. (~)

Y

1

is

written

for the temperature

and the

indexes

N and

~

refer respectively

to

a network of N cells and to the

contin~Qus

The functi ons of E; t A R,(

d

and. their analog eN) A

R,(~)

are

obtained as fQ.;Llows ~

a) 0 For A R,( ~) the recurrence relati ons are used

A ₀

=

~ 2 R,+l AR,

=

~. (2R,+1) 1 R,-l

L

m=O A m ( 2 ( R.-m)

+

1) 1

blo and thei r analogs are computed thraugh the

relations where CR. (~) ~r

=

2N sin

2N

r 'Tr 2 (,R,+ 1) c N ~r s~n

c)o The numerical val ues of the AR.( ~) can also be found by using t he following convergent s~ries

obtained as the l imi t of

(4)

when

N

approache~

infinity~ = sin (rn~) (rn)2R.+l (:ja) (3b)

( 4 )

(6 )

If we write

=

_•

(8 )

the

~rror

(difference between

Y1~

and Y

1N

) can be written

in the form

n

=

n!

!

1=0

n-1

t

(n-1) !

The determination of the

(N)n

(~)

is thus sufficient to

1.R.

obtain

61JN(~.t)

for any va1ue of to

The errorfor the heat flux can be written in a simi1ar

form o

Indeed. we have

where

aL

-1

(

n

_!

_AD

₍ ~

,

)

t

n-R.

a

YZ

~\lt)

= n!

(n-R.)I

I~ R.=O R.

(10)

-1

(

n

_I

(N)A!

_{( ~ ! )}

t

n-R.

NA

_{Y2 • N}

~lIt )

=

nl

(n-R.)r

1=0

(11)

=

These functions are computed using the recUrrence

relations derived

from

(3)i

i

o

e

o

A'

=

1

0

(lla)

~2r

r-1

A'

( ~ )

A

9 (~ )

=

I

m

(2r)

!

(

2 ( r- m ) +

1) I

r

_m=O

(llb)

b).

whi1e their ana10gs are obtained from

N-1

=

r

r=l

with

C' (~') tir

=

2N (_l)r sin

C' (~') &ar

2t+ 1

c

N,r

(~) 0

cos

(rn~') N 5 •

(13)

c).

Here

aga~n,

the numerical va1ues obtained for the

A',(~)

can be checked using the series 1imi ts of (12)

1

when N approaches

infinityo

This gives

A' (~) 1

Writing

=

we have

where

=

n nl

L

1=0 n-.t t '

(n-R.)

I

['hese

fun"C·t~ons

have

b

een

computed

and

can

be used to

~etermine

°

₂

N for any va1ue of

t

o

. I

(14)

6.

2.2

Per~ect therma~

isolation

The temperaturesin the continuous medium and in

the analog network are given by the expressions

n

n-i

Yl ...

(~·t)

=

nl

l

Bi(~)

t

_(n-i)!

i=O

n

n-i

(N)B

(~)

Y

_{1 N(~·t)}

=

nl

l

t

_(n-i)1

•

i=O

i

The functions

Bi(~)

and their

analOg~(N)Bi(~)

are

obtained as

~ollows

a) •

b) •

The B

(~)

1S

compu~ed

from

i

B

(d

=

1 0 ~2r

r-l

~m

(

~)

B

(q

₌

(2r)1

l

(2

( r-m

f)

I

r

_m=O

Their

analogs

(N)B

(~)

are obtained from

i

where

=

N-l

=

_l

r=O

=

2N

(

-1

) r

S1n

.

(2r+l)

~1T

·cos

( 2 r + l )

~1T~

2N sin(..!

•

2r+l

)

2

'2N

(18)

(208.)

(20b)

(21 )

(22)

(23)

c)o

For the values of

Bi(~)

we have the verification

=

( 2

r+l

)

cos

~1T~

[(2~+l)1T]

2i+l

For the errors, we write again (N)n

(d

_{= B}

_(d

(N)B _{i (}

_E)

l , i i then _{n- i} n (N)n (E;) 6 1 N(E;,t) nl

L

t = _{(n- i)} !

,

_i=O 1 i i

As before, the determination of the (N)D (E;) will allow

l . i the computation of 6

1

,

N(E;,t) for any value of t .

For the heat flux we have

-1 ( V ) _nl n

_L

_{B' ( E;' )} t n- i a Y2 E; at :;: (n- iN

,00

_i=O i -1 ( V ) _nl n

_l

(N)B' _{( E;' )} t n- i NA _{Y2 N}

_,

E; , t = _(n-i)l i=O i (25) (26) (28)

a) 0 The B~ (E;) are the derivatives of the B i(

0

with

respect to E;o They can be computed from

Be(E;) = 0 0 (29a) Bi ( E;) = E; (29b) E;2 Rr-l . i -1 B ' -( E; ) Bi ( E;) =

L

m (29c) (2i-l) 1

-

( 2 ( i -m)

)t

m=O b) 0 Their anal~g_s (N)B8

i (E; ) are calculated through

the relation

(N)B'(E;V) ₌ (_l)i+l N-l

_L

CUisr(E;) (30)

2i+l

r=O _{,cN r} t

where

(31)

c) 0 The limit of (30) for N approaching infinity gives

another expression for B'

1 thus B' (E;')

=

2(-1) 1+1 1 CD

,r

r=O

A~ for the previous cases we write

( ) -1 ( W ) 02 • N E; u ~ t

=

liL Y 2 a CD E; \) t

=

n 02 , N ( E; 0 • t) = n I

r

R.=O (2r+l cos -1TE;U) 2 \ 21

(~

1T) 2

This farm is similar to the previous ones and from the

values of D₂ R; the error can be deducedo

\)

(32)

(33)

(34)

9.

30 Presentation of the Results

The funct ion s

the ir analogs

( N ) A R; ( f; ), ('N) A' R, ( f; ) 11 ( N ) B i ( f;), (N) B I i ( f;) and the differences

(i

=

1, 2)

have been computed~ using an IBM 1620 computer, for values of

n ranging from 1 to 10 or 110 The calculations have been

performed for th ree values of

H,

ioeo

M

=

5,

10 and 20. The

calculating program is described briefly in Section

40

Figures 2 to

5

respectively show the functions AR.' A'R.'

Bi' B'R. versus the reduced distance f;o These Figu~es show

that the functions AR. and AQR. decrease roughly by an order

of magnitude when the index increases from R. to .R.+l. They

also become more symmetrical ~ith respect to f;

=

0.5) when

R. increaseso The functions BR. and B~ R. do not decrease so rapidly as the fun~tions AR. and AgR. when R. increases (compare

Figures

4

and

5

with Figures 2 and 3)0 In Figures

6

to

9

the previous functions AR.~ AUt~ BR.~ BgR. are çompared with

t~~i~

analogs

(N)AR.~ (N)AijR.~ N

)

BR.~

(N)Bft R.fOr the three cases

N

=

5,

10 and

2po

In Figures 10 to 13 the differences D . .

. l.,J

are represente4 versus ~ for j

=

2t

3

and

40

These values

have been chosen as examples~ bec~use for sufficiently large

values of t . the corresponding coefficients mainly determine

~he behayior of ai N0 It is also to be temembered that in all

'cases (N}D . .

(~)

='0 for j =

O~l

whatever the number of cells N.

40

Computer Program

A program has been written for an IBM 1620 computer

in order to calculate the temperature and heat flux

coefficien~

and their analog valueso

For practical reasons the range of

variation of the

integer

n has been limited to the interval

1·~ n,

20

0

The general flOw

diagram .

is gi ven in Figure 14

followed by the

~isting

of the program o

The automatic printing operatiQ.n

~s

obtained through

the following instructions

N2

=

72 - Ne

DO IOJ

=

l.,N2

10 PRINTING

6 FORMAT

72 is the number of lines of a page of which NC have

a1ready been used o

Of course

~his

sequence will only be

performed if NC has reached a minimum value which, for

presentation reasons has been taken

eq~a1

to 630

A sequence of

instructions

makes

it

possible to print

(or compute) any set of one

i

two or three of the three

q uantities A

i"

(N)

A

(N)D

l '

1,1

DO

5J.~1

=

1,3

IF

(11-1)

56.54,56

54

ACCEPT TAPE M4,Il

56

~F

(63-NC)

9,9,11

wher, the initial

va~ue

for 11

~s

one and M4 takes the .alues

11.

The correspondence between the sets of va1ues (Ii.M4) and

the printed quantities is given in the tab1e

M4 =

0

Ii =

1

..

A

_II

(N)A

_•

(N)D

1

0

2

..

A

_•

(N)A

_•

0

3

..

A

_•

0 0

1

2

..

(N)A

_•

(N)D

1

1

3 0

(N)A

•

0

2

3

..

(N)D

₁

M4=o

_•

Ii=3 and M4=2,I1=3

..

A

_•

(N)D

1

A1though the practical va1ues of

~

wi11 normai1y be of the

form

(fini~e

difference equation)

~

=

1 -

i

K

1

~

K(integer)< N

i t

1S

possibie. using the manua

1 s

w

i

t

c

h

1 (

on position "on")

to introduce any vaiue for

~ 0

For the three other sets of

( B. (

N )

B.

(N) Dl) an d (

Ba. (

N )

B v •

re sul t

6

«

A

v 11 (

N )

Á.' •

(N)

D

2

)). the program is

similar éxcept

p

of course

_a

for the calcu1ation

o~

the

A. BENOIT

REFERENCES

Erreur résultant de la détermination des

flux thermiques par la méthode analogique

dans Ie cas de températures superficielles

de la forme t

n

(n entier non négatif).

Institut von Karman de Dynamique des Fluides,

Table 1.1 - Temperatu~e coefficients for an ideally conducting terminal.

Ai : Temperature coefficients

(N)A

i Analog temperature coefficients (li)n Error coefficients

l,i for temperature Yl,~(Ç,t)

N= +5 +.800 +.600 +.400 +.200 +.800 +.600 +.400 +.200 +.800 +.600 +.400 +.200 +.800 +.600 +.400 +.200 +.800 +.600 +.400 +.200 +.800 +.600 +.400 +.200 Al ( t) -. 48000E -0 1 -.64000E-Ol -. 56000E -01 -.32000E-Ol A6 ( t) +.40493E-06 +.65511E-06 +.65502E-06 +.404 77E-06 (N)Al(t) -.48000E-0 1 -. 64000E -01 -.56000E-Ol -.32000E-Ol (N)A 6(t) +.47733E-06 +.77219E-06 +.77200E-06 +.47703E-06 {N)n {tl 1.2 -. 1 6000 E -0 3 -.21333E-03 -.18666E-03 -.10666E-03 {N)n (c) 1.7 +.89531 E":'08 +. 11~484E-07 +. 14480E -07 +.89483E-08 A 2 ( c) +.40640E-02 +.631L_.6E-02 +.60853E-02. +.3669 3E -02 A7 (c ) -.41022E-07 -.66373E-07 -.66371E-07 -.41018E-07 {N)A (C) 2 +.42240E-02 +.65280E-02 +.62720E-02 +.377 60E -02 {N)A (C) 7 -.49975E-07 -.80857E-07 -.808 52E -07 -.49966E-07 (N)n (C) 1.3 +.27946E-04 +.43235E-04 +.41564E-04 +.25031 E-04 {N)n (C) 1.8 -.10768E-08 -.17423E-08 -.17422E-08 -.10767E-08 A3 ( C) -.3944 SE -0 3 -.63260E-03 -. 62659E -03 -~38456E-03 A8{ C ) +.41562E-08 +.67249E-08 +.67249E-08 .+.41561 E-08 (N)A

Ct>

3 -.42240E-03 -. 67584E -03 -. 668 1 6E -03 -.40960E-03 {N)A (C) 8 +.52331 E-08 +.84673E-08 +.84671E-08 +.52329E-08 {N)n (C) 1.4 -.41396E-05 -. 66280E -0 5 -.65563E-05 -.40205E-05 {N)n (C) 1.9 +. 12689E -09 +.20532E-09 +.20532E-09 +. 12689E-09 n n-i n! }: Ai(Ç) _t _ _ i=O (n-i)! n ( ) n-i h!}: NA ( o t i=O i (n-i)! n n! L i=O A4 ( C) +.39 564E -04 +.63884E-04 +.63731E-04 +.3931 SE -04 A9 (C) -.42111E-09 -.68137E-09 -.68137E-09 -.42111E-09 (N)A (C) 4 +.43 704E-04 +.70512E-04 +.70287E-04 +.43335E-04 {N)A (C) 9 -.54801 E -09 -.88670E-09 -.88669E-09 -.54800E-09 {N)n (C) 1.5 +.56316E-06 +.90886E-06 +.90620E-06 +. 55883E-06 -.14720E-l0 -.23818E-l0 -.23818E-l0 -. 14720E -10 A5 (t ) -.39989E-05 -. 64672E -05 -. 64633E -05 -.399 26E -05 AlO (t ) +.42667E-l0 +.69038E-l0 +. 69038E -1 0 +.42667E-l0 (N) A { " 5 -.45621E-05 -.73760E-05 -.73695E-05 -.45514E-05 (N)A (r) 10 .. +.57388E-l0 +.928 56E -1 0 +.928 56E -1 0 +.57388E-l0 (N)n

(c>

1.6 -.72401E-07 -.11707E-06 -.11697E-06 -.72253E-07 +.16866E-l1 +.27290E-ll +.27290E-l1 +.16866E-ll

N=+10

C Al ( C) A

2 ( C) A3 ( C) A4 ( C) A5 (C )

+.900 -. 28 50 0 E -0 1 +.21707E-02 -.20

4

96E-03 +. 20813E -04 -. H026E-05

+.800 -.L~8000E-Ol +.40640E.:.o2 -.' 39 4 5E -0 3 +.39 564E -04 """ .• 39989E-05 +.700 -. 59500 E -0 1 +.54839E-02. -.54070E-03 +.54405E-04 .:.:. 5 5028 E -05

+.600 -. 64000E -01 +.631lt6E-{)2 .-. 63260E -{) 3 +. 63884E -04 -.64672E-05

+.500 -.62500E-Ol +. 65104E -02 -.66189E-03 +.67090E-04 -. 679 79 E -0 5 +.400 -. 56000E -01 +.60853E-02 -.62659E-{)3 +.63731E-04 -.64633E-05 +.300 -. 45500E -01 +.5103 5E -02 -.53091 E -03 +.54156E-04 -.5L_f965E-05

+.200 -. 32000E -01 +.3669 3E-{)2 -.38456E-03 +.39315E-04 -.39926E-05 +.100 -.16500E-Ol +.19167E-{)2 -.20179E-03 +.20658E-{)4 -.20987E-05

C A6 ( C) A7 (C ) Aa (t ) A

9(C) A10(

t>

+.900 +.21289E-06 -.21566E-07 + • 218 50 E -08 -.22139E-09 +.22431 E-l0 +.800 +.40493E-06 -.41022E-07 +.41562E -08 -.42111E-{)9 +.42667E-l0 +.700 +.55731E-{)6 -.56461 E -07 +. 57206E -08 -.57961E-{)9 +.58727E-l0 +.600 +.65511E-06 -.66373E-07 +. 67249E -08 - • 68 1 37 E -{)9 +.69038E-l0 +.400 +.68878E-06 -.69788E-07 +.70710E-08 -. 71644E -09 +.72591E-l0 +. 00 +. 65502E -06 - • 663 71 E -07 +.67249E-08 -. 68137E -09 +. 69038E -10 +.300 +. 55715E -06 -.564 57E-{)7 +.57205E-08 -.57961E-09 +.58 727E -10 +.200 +.404 77E-{)6 -.410 18E -07 + • 4 1 ~ 61 E -08 -.42111E-{)9 +.42667E -10 +.100 +.21279E-06 -.21564E-07 +.21 50E -08 -.22139E-{)9 +.22431E-l0

t (N) A ₁ (t) (N)A ₂ (t) (N)A ₃ (F;) (N)A ₄ (t) (N)A ₅ (t)

+.900 -. 28500E -0 1 +.21945E-02 -.21161E-03 +.21339E-04 -.21732E-05 +.800 -.48000E-Ol +.41040E-02 - • 40 1 28 E -03 +.40562E-04 -.41330E-05 +.700 -.59 500E -01 +.55335E-02 -.54990E-03 +.55773E-04 -.56871E-05 +.600 -.64000E-ol +.63 680E -02 -. 64320E -<l 3 +.65484E-04 -.66836E-05 +.400 -.62500E-Ol +.65625E-<l2 -. 6728 1 E -<l 3 +. 68764E-<l4 -.70252E-05 +. 00 -.56000E-Ol +. 61320E-<l2 -.63680E-03 +.6531 5E -04 -.66791E-05 +.300 -.45500E-<l1 +.51415E-<l2 -.53946E-03 +.55498E-04 -.56799E-05 +.200 -. 32000E -0 1 +. 36960E -02 -.39072E-03 +.40287E-04 -.41258E-<l5 +.100 -.16500E-Ol +.19305E-02 -.20501E-03 +.21168E-04 -. 21687E -05

t (N)A₆(t) (N)A ₇ (t) (N)A _a (t) (N)A (t)

9 (N)A 10 (t)

+.900 +. 22 18 3 E -06 -. 22658E -07 +.23146E-<l8 -.2364 5E -09 +.24155E-1O +.800 +.42194E-<l6 -. 43098E -07 +.44026E-<l8 -.449 76E-09 +.45947E-l0 +.700 +. 58072E -06 -.59318E-07 + • 60 59 7 E -08 -. 61904E -09 +.63240E-l0 +.600 +.68262E-06 -• 69 7 3 1 E -0 7 +.71235E-<l8 -.72773E-<l9 +.74344E-l0 +.500 +.71769 E -06 -.73318E-07 +.74901E-<l8 -.76518E-09 +.78169E-l0 +.400 +. 68250E -06 -. 69728E-<l7 +.71234E-<l8 -. 72772E-<l9 +.74343E-l0 +.300 +.58053E-06 -.59313E-<l7 +.6059 5E-<l8 -. 61904E-<l9 +.63240E-l0 +.200 +.42175E-06 -.43093E-07 +.4402 5E -08 -.44976E-<l9 +.4594 7E -10 +.100 +.22172E-<l6 -.22655E-07 +.23145E-<l8 -.2364 5E-<l9 +.24155E-l0

t (N)D _1.2 (I;) (N)D _1.3 (t) (N)D _1.4 (t) (N)D _1.5 (t) (N)D 1.6 (t)

+.900 -.23751E-04 +. 36497E -05 -.52631E-<l6 +.70518E-07 -.89431E-08 +.800 -.40002E-04 + • 68 2 69 E -0 5 -.99811E-06 +.13404E-<l6 -.17008E-07 +.700 -.49586E-<l4 +.920641t-05 -. 1 3 679 E -<l 5 +.18431 E-06 -.23404E-07 +.600 -. 53337E -04 +.10596E-04 -. 1 6001 E -0 5 +.21641E-06 -.27505E-07 +.500 -. 52087E~04 +. 10920 E -04 -.lU38E-05 +. 22726E -06 -.28

4

11E-07

+.400 -.46670E-04 +.10204E-04 -.1 43E-05 +. 21586E -06 -.27 86E-<l7 +.300 -.37919E-04 +.85569E-05 -.13422E-<l5 +.18343E-06 -.23375E-07

+.200 -.26668E-04 +.61513E-05 -.97221E-<l6 +.1331áE-<l6 -.1697

4

E- O

é

(N) D (E;) (N) D (() (N) D (() (N)D (() (N)

E; D

111(()

1.7 1.8 1.9 1.10

_•

+.900 +. 10912E -08 -.129 52E-09 +. 1 5060 E -1 0 -. 17240E -11 +.19490E-12 +.800 +.20755E-08 -.24635E-09 +.28647E-10 -.32792E-l1 +.37073E-12 +.700 +.28565E-08 -.33907E-09 +. 39429 E -10 -.45134E-11 +.51027E-12 +.600 +.33578E-08 -.39860E-09 +.46352E-10 -. 53058E -11 +.59986E-12 +.500 +.35303E-08 -.4191OE-09 +.48736E-1O -.55789E-11 +.63072E-12

+.LfOO +.33572E-08 -.39857E-09 +.46350E-10 -. 53057E -11 +.59985E-12 +.300 +.28557E-08 -.33905E-09 +.39428E-10 -.45134E-ll +.51027E-12 +.200 +.20746E-08 -.24633E-09 +. 286 Lf6E -1 0 -. 32792E -11 +.37073E-12 +.100 +.10906E-08 -.12950E-09 +. 15060E-1O -.17239E-l1 +.19490E-12

N=+20

( Al ( () A

2 ( () A3 ( () A4 ( () A5 «( )

+.950 -. 1 5437 E -0 1 +.11044E-02 -.10535E-03 +.10537E-04 -.1 0644E -05 +.900 -. 28500E-0 1 +.21707E-02 -.20796E-03 +. 20813 E -04 -.21026E-05 +.850 -.39312E-Ol +.31662E-02 -.30515E-03 + • 30 5 69 E -04 -. 30889E -05 +.800 -.48000E-Ol +. 40640E -02 -.3944 5E -0 3 +.39564E-04 -.39989E-05 +.750 -.54687E-01 +.48421E-02 -.4l361E-03 +.4l~75E-04 -. 48102E -05 +.700 --. 59500 E -0 1 +.54839 E -02 -.5 070E-03 +.5 05E-04 -. 55028E -05 +.650 -.62562E-Ol +.5977 3E-02 -. 59410E -0 3 +. 59886E-04 -.60 597E -05 +.600 -.64000E-Ol +.63 JLf6E -02 -.63 260E -0 3 +. 63884E -04 -.64672E-05 +.550 -.63937E-01 +.64923E-02 -.65534E-03 +.66304E-04 -.67153E-05

+.500 -.62500E-Ol +.65104E-02 -.66189E-03 +.67090E-04 -.679 79E -05 +.450 -.59812E-Ol +.63725E-02 -. 65219 E -03 +. 66224E -04 -. 67132E -05 +.400 -. 56000E -01 +.60853E-02 -. 62659E -0 3 +.63731E-04 -.6463 3E -05 +.350 -.51187E-Ol +.56583E-02 -.5858 1 E -0 3 +. 59674E -04 -.60543E-05 +.300 -.45 500E -01 +.51035E-02 -.53091 E -03 +.54156E-04 -.54965E-05 +.250 --. 39062E-0 1 +.443 52E -02 -.46327E-03 +.47313E-04 -. 48036E -05 +.200 -.32000E-Ol +.36693E-02 -.38456E-03 +.39315E-04 -.39926E-05 +.150 -. 24437E-0 1 +.2823 5E -02 -.29670E-03 +.30356E-04 -.30835E-05 +.100 -.16500E-Ol +.19167E-02 -.20179E-03 +. 20658E-04 -.20987E-05 +.050 -. 83 1 2 5E -0 2 +.96875E-03 -.10210E-03 +. 104 56E -04 -.10624E-05

( A6 ( () A7 «( ) A8 (E;) A9 «() AlO «( )

+.950 +.107é 7E - 06 --. 1 09 17 E -0 7 +. 11 06 1 E -08 -. 11207E-09 +. 11 35 5E -1 0

+.900 +.2129E-06 -.21566E-07 +.21850E-08 -.22139E-09 +.22431 E-10

+.850 +.31276E-06 -.31684E-()7 +.32102E-08 -.32525E-09 +.32955E-1O

+.800 +.Lf01_{f93E-06 -.41022E-07 +. Lf 1562E -08 -.42111E-09 +.42667E-1O}

+.750 +. Lf8 712E -06 -.49349E-07 +. 50000E -08 -. 50660E-09 +.51329E-l0 +.700 + • 5 573 1 E -06 -. 56461 E -07 +.57206E-08 -.57961 E -09 +.587 27E-1O +.650 +. 61377E -06 --.62183E-07 +.63003E-08 -.63835E-09 +. 64679E-1O +.600 +.65511E-06 -.66373E-07 +.67249E-08 - • 68 1 37 E -09 +.69038E-1O +.550 +.680 32E -06 -.68929E-07 +.69839E-08 -.70762E-09 +. 71 697 E -1 0

+·500 +.68878E-06 ",".69788E-07 +.7071OE-08 -. 71644E-09 +. 72 59 1 E -1 0 +.450 ~-. 68027E-06 -.689 28E-07 +.69839E-08 -.70762E-09 +. 71 697 E -1 0 +.400 +.65502E-06 -.66371E-07 +.67249E-08 -. 68137E -09 +.69038E-l0 +.350 +.6136LfE-()6 -.62180E-07 +.63002E-08 -.63835E-09 +.64679 E-1O +.300 +.55715E-06 -. 564 57 E -0 7 +. 57205E-n8 -.57961E-0:) -I- • 58 7 27 E -1 0

-:-.250 +. ',U _ :)E-06 -.lf9345E-07 +.49999E-08 -.~0660E-09 +. ~ 1 3 29 E -1 0

+.200 +.40Lf77E-06 -. Lf 10 18E-07 +.Lf1561E-08 -. 2111E-09 +. 2667E-l0 +.150 + • 31 263 E --() 6 --. 3 1 68 1 E -07 +.32101E-08 -.32525E-09 +.32955E-l0

+.100 + • 2 1 27':) E --() G -. 2 1 5 6Lf E -Cl 7 +. 21850 E -() 8 -.22139E-()9 +.22/-I31E-l0

t (N) A ₁ (t) (N)A ₂ (t) (N)A (!;l ₃ (N)A 4 (t) (N)A 5 (t) +.950 -.15L_{f37E-ol +. IlO 7 6E -0 2 -. 10 58 1 E -0 3 +. 10604E -04 -.10733E-05} +.900 -.2B500E--(J1 +.21766E-02 -.20886E-03 +. 209 Lf3E -04 -.21200E-05

+.850 -.39312E-Ol +.317LfLfE-02 -.30647E-03 +.3076OE-04 -.31145E-05

+.800 -.48000E-Ol +. 407 LfOE -02 -.39615E-03 +.39811E-04 -. 40320E -05 +.750 -. 54687E-0 1 +.48535E-02 -.L.7563E-03 +.47872E-04 -.48 500E -05 +.700 -.59 500E -0 1 +.54963E-02 -. 54299E -0 3 +.54744E-04 -.55483E-05 +.650 -.62562E-Ol +. 59903E -,-02 -.59660E-03 +. 60258E -04 -.61098E-05

+.600 -.64000E-Ol +. 63280E -02 -.63524E-03 +.64281 E -04 -.65206E-05

+.550 -.63937E-Ol +. 65056E -02 -. 65805E -0 3 +.66715E-04 -. 67707E -05 +.500 -. 62500E -01 +.65234E-02 -.66460E-03 +.67505E-04 -. 68 54 1 E -05

+.450 -. 59812E-0 1 +.63849E-02 -.6548 SE -0 3 +.66633E-04 -.67686E-05

+.400 -. 56000E-0 1 +.609 70E -0 2 -.62913E-03 +.64123E-04 -.65166E-05

+.350 -.51187E-01 +.56690E-02 -. 58817E -03 +.60041 E-04 -.61043E-05

+.300 -. 45500E -01 +.51130E-02 -.53304E-03 +.54489 E -04 -.55418E-05

+.250 -. 39062E -0 1 +.44433E-02 -.46512E-03 +.47604E-04 -.48432E-05

+.200 -.32000E-ol +.36760E-02 -.3861OE-03 +.39 556E -04 -.40255E-05

+.150 -.24437E-ol +.2828 6E -02 -.29788E-03 +.30 542E-04 -.31089E-05

+.100 -.16500E-ol +.19201 E-02 -.20259E-03 +.20785E-04 -.21160E-05

+.050 -.83124E-02 +. 97048E -0 3 -.10251E-03 +.10520E-04 -.10711E-05

t (N)A₆(t) (N)A ₇ (t) (N)A ₈ U;) (N)A (C) ₉ (N)A ₁₀ (C)

+.950 +.10889E-06 -.11053E-07 +.11222E-08 -.11393E-09 +.11567E-l0

+.900 +.21509E-06 -.21834E-07 +.22167E-08 -.22506E-09 +.22850E-l0

+.850 +. 31600E -06 -.32078E-07 +.32567E-08 -. 33065E -09 +.33571E-1O

+.800 +.409 12E -06 -.41531E-07 +.42165E-08 -.4281OE-09 +.43465E-l0

+.750 +.49216E-06 -.49962E-07 +.50725E-08 -.51500E-09 +.52288E-l0

+.700 +. 56307E -06 -.57162E-07 +.58035E-08 -.58923E-09 +.59824E-l0

+.650 +.62012E-06 -.62955E-07 +.63917E-08 -. 64894E -09 +. 65887E -1 0

+.600 +.66189E-06 -.67197E-07 +.68224E-08 -.69268E-09 +.70327E-l0

+.550 +.68735E-06 -. 69785E -07 +.70852E-08 -.719 36E -09 +.73036E-l0

+.500 +.69589E-06 -.70654E-07 +.71735E-08 -. 72832E-09 +.73947E-l0

+.450 +.68730E-06 -.6978 3E -07 +.70852E-08 -.71936E-09 +. 73036E-1O

+.400 +.66178E-06 -.67195E-07 +.68224E-08 -. 69.268E-09 +.70327E-l0

+.350 +.61998E-06 -.62951E-07 +.63916E-08 -. 64894E-09 +. 65887E -1 0

+.300 +.56291 E -06 -.571 58E -0 7 +.580 34E-08 -.58922E-09 +. 59824E -1 0

+.250 +.49198E-06 -. 49958E -07 +.50724E-08 -.51500E-09 +.52288E-l0

+.200 +.40895E-06 -.41527E-07 +.42164E-08 -.42809 E -09 +.43464E-l0

+.150 +. 31586E -06 -. 32074E-07 +.32566E-08 -. 33065E -09 +.33571E-l0

+.100 +.21/f99E-06 -.21832E-07 +.22167E-08 -. 22506E -09 +.22850E-1O

+.050 +. 108B3E-06 -.11052E-07 +.11221 E-08 -.11393E-09 +.11567E-l0

t (N)n _1,2 (C) (N) n _1,3 (C) (N)n _1.4 (C) (N)n 1,5 (C) (N)n 1.6 (!;l

+.950 -.321 á7E-05 +.46123E-06 -.66051E-07 +. 88179E -08 -. 11146E-08

+.900 -.593 7E-05 +. 90662E -06 -. 1 3038 E -0 6 +.17416E-07 -.22017E-08

+.850 -.8191 LfE-05 +.13222E-05 -.19132E-06 +.25580E-07 -.32344E-08

+.800 -.10001 E-OLf +.16968E-05 -.24730E-06 +.33107E--{17 - • 4 18 Z ~ E -08

+.750 -.11394E-04 +.20215E-05 -. 29692E-06 +.3981OE-07 -.5036 E-08

+.700 -.12398E-04 +.2289 3E -05 -.33897E-06 +.45525E-07 -.57620E-08

+.650 -.13035E-04 +. 2Lf9 51 E -0 5 -.37243E-06 +.50 11OE-07 -.63449E-08

+.600 -.13336E-04 +.26360E-05 -. 39658E -06 +.53458E-07 -. 67719E -08

-r·550 -.1332 2E -{14 +.2709 7E -0 5 - • Lf 1 079 E -06 +.55479E-07 -. 70312E-08 +.500 -.13023E-04 -r. 27174E -05 -.41491 E-06 +. 56137E -07 -.71180E-08 +.Lf50 -.12463E-0/. +.26597E-05 -.4088 2E -06 +. 554 1 3 E -0 7 -.70293E-08 +.400 -. 11 668 E -{I I. +.25396E-05 -. 39 27 5 E -0 6 +.53323E-07 - • 67 671 E -08

+.350 -.10666E-oI. +.23614E-05 -.36720E-06 +.49932E-07 -.63393E-08

+.300 -.94808E-()5 +. 2 1 299 E -0 5 -.33278E-06 +.4531LfE-07 -.57553E-08

+.250 -.81392E-{15 +.18509E-05 -. 29038E -0 6 +.39589E-07 -.50297E-08

+.200 -.66676E"'-05 +.15312E-05 -.24104E-06 +.32895E-07 -.41805E-08 +.1

sn

-.50921E-05 +.11783E-05 -. 18 59 7 E -06 +. 2540 1 E -0 7 -.3228 7E -08

+.100 -.3Lf381E-05 +.79989E-06 -.12648E-{16 +.17285E-07 -.21975E-08

(ri)D (0 (N)D (~) (N)D (~) (N)D (~) (N)

C _{1,7 1,8 1,9 1,10} Dl 11 (~)

•

+.950 +.13556E-09 -. 16038E-1O +.18590E-l1 -.21212E-12 +.23906E-13

+.900 +.26779E-09 -.31683E-l0 +.36725E-ll -.41905E-12 +.47225E-13

+.850 +.39342E-09 -.46547E-l0 +.539 54E -11 -.61564E-12 +.69380E-13

+.800 +.509 35E-09 -. 60264E -1 0 +.69854E-l1 -. 79707E -12 +.89827E-13

+.750 +.61273E-09 -.72497E-l0 +.84034E-ll -.95887E-12 +.1 0806E -12

+.700 +.701 02E -09 -.8294 SE -1 0 +.96145E-l1 -.10970E-ll +.12363E-12

+.650 +.77201 E-09 -.91346E~10 _{+. 10588E -10 -.12081E-l1 +.13615E-12}

+.600 +.82405E-09 -.97507E-1O +.11302E-l0 -.12896E-l1 +.14534E-12

+.550 +.85572E-09 -.10125E-09 +.11737E-l0 -.13393E-l1 +. 1 5093 E -1 2

+.500 +.86637E-09 -.10252E-09 +.11883E-l0 -.13560E-l1 +. 1 528 1 E -1 2

+.450 +.85567E-09 -.10125E-09 +.11737E-l0 -.13393E-ll +. 1 ~093E -12

+.400 +.82385E-09 -.97495E-l0 +. 11 30 1 E -1 0 -.12895E-l1 +.1 533E-12

+.350 +.77186E-09 -.91343E-l0 +.10588E-l0 -.12082E-ll +.13616E-12

+.300 +. 7008 1 E -09 -.82939E-l0 +.96tLf3E-ll -.10970E-ll +.12363E-12

+.250 +.61252E-09 -.72490E-l0 +.84032E-ll -.95886E-12 +.10806E-12

+.200 +. 50911_{fE -09 -.60257E-l0 +.69851 E -11 -.79705E-12 +.89826E-13}

+.150 +.39324E-09 - • 4654 1 E -1 0 +.53952E-ll -.61563E-12 +.69380E-13

+.100 +.26766E-09 -. 3 1679 E -1 0 +.36723E-l1 -.41904E-12 +.47224E-13

Table 1.2 - Heat flux coefficients for an ideally conducting terminal. A~ : Heat flux coefficients

(N)A~ : Analog heat flux coefficient (U)D _{2 R. :} E _{'rror coeff~c~ents} . .

• for heat flux

N= +5 +.900 +.700 +.500 +.300 t, +.900 +.700 +.500 +.300 t' +.900 +.700 +.500 +.300 C' +.900 +.700 +.500 +.300 t' +.900 +.700 +.500 +.300 t' ' +.900 +.700 +.500 +.300 +.23833E-oO +. 78333E -01 -.41666E-Ol -.12166E-00

At;

(C' ) -. 20583 E -0 5 -.12717E-05 +.52799E-09 +.12720E-05 (N)A'(C') 1 +.23999E-oO +.79999E-Ol -.39999 E -01 -. 12000 E -00 (N)A'(C') 6 -.23866E-05 -.14742E-05 +.9 5027E -09 +.14748E-05 {N)D ( t ' ) ·2,2 +.40 194E -03 +.13527E:"03 -.64723E-04 -.19805E-03 (N)D (C') 2,7 -.41352E-07 -.25545E-07 +.14161E-l0 +.25554E-07 A' (t ' ) 2 -.20718E-Ol '-.11384E-01 +.12152E-02 +.12281E-Ol

Af

(t ' ) +.20852E-06 +.12886E-06 -.13363E-10 -.12887E-06 (N)A'(t') 2 -.21119E-01 -.11520E-01 +. 12800E -02 +.12480E-01 (Jf)A'(C') 7 +.24987E-06 +.15441 E-06 -.27525E-10 -. 15442E -06 (N)D ( t ' ) 2,3 -.10540E-03 -.5753 SE -04 +.63647E-05 +.62298E-04 (N)D (C') 2,8 +.5038 5E-08 +.31136E-08 -.45972E-12 -.31139E-08 n n-R. a- lY

2.",,(I,;'.t) = n!

L

_R.=o A'(I,;,)t_{R. (n-R.)I} ·

n ( ) n-R. NA- 1Y2 N(I,;'.t) = nl L N A' (1,;' )_t _ _ • R.=O (n-R.)I ó 2 H ( f; , • t) = n I

î

(N ) D ( f; , ) tn - R. • R.=o 2.R. (n-R.)I +.20065E-02 +.12096E-02 -.32035E-04 -.12305E-02 -.21127E-07 -.130 57E -07 +.33719E-12 +.13057E-07 (N)A'(C') 3 +.21119E-02 +.12672E-02 -.38400E-04 -. 12928E -02 (N)A'(C') 8 -.26165E-07 -.16170E-07 +.79691E-12 +.16171E-07 (N)D ( t ' ) 2.4 +.17371E-04 +.10435E-04 -. 30615E -06 -.10638E-04 (N)D ( t ' ) 2.9 -.59945E-09 -.37047E-09 +.14661E-13 +. 37'048E-09

A4

(t ' ) -.20115E-03 -. 12360E -03 +.82024E-06 +.12412E-03 A

9

(t') +. 21406E -08 +. 13229E -08 -.84070E-14 -. 13229E -08 (11)

A4

(C ' ) -. 2 18 52 E -03 -.13404E-03 +.11264E-05 +.13475E-03 (II)A'(t') 9 +.27400E-08 +.16934E-08 -.23068E-13 -.169 34E-08 (N)D (C') 2,5 -.24822E-05 -.15235E-05 +.11933E-07 +.15310E-05 +.70053E-1O +.43295E-l0 -.46837E-15 -.43295E-l0 +.20328E-04 +.12546E-04 -.20834E-07 -. 12559E -04 A' _{10 ..} (r') -. 21688E-09 -. 13404E -09 +.19935E-15 +.13404E-09 (II)A'(t') 5 +.2281OE-04 +.140 69E -04 -.32768E-07 -.14090E-04 (I)A' (r') 10 .. -.28694E-09 -.17734E-09 +.66773E-15 +. 1 77 34 E -09 (II)D (t') 2,6 +.328 29E -06 +.20254E-06 -.42228E-09 -.20280E-06 -.80735E-ll -.49897E-ll +. 15655E -16 +.49897E-l1

N=+10

C' A' (C ' ) ₁ A' (C' ) ₂ A' (C ' ) _{A4 ( C ' ) A' (C ' )}

3 ₅

+.950 +.28458E-00 -. 21826E-o 1 +.20887E-02 -.20900E-03 +. 21113E -04 +.850 +.19458E-oO -.19013E-Ol +.18728E-02 -.18829E-03 +.19041 E -04 +.750 +.11458E-oO -.14246E-Ol +.14684E-02 -.14901E-03 +. 1 5101 E -04 +.650 +.4458 3E -01 -.83261E-02 +.92249E-03 -.9 5176E -04 +.9ó830E-05 +.550 -.15416E-Ol -.19511E-02 +. 29369E -03 - • 3 21 78 E -04 +.33209E-05 +.450 -.65416E-Ol +.42780E-02 -.354 74E -03 +. 33739E -04 -.33605E-05 +.350 -.10541E-00 +.98613E-02 -.96094E-03 +.96145E-04 -.97075E-05 +.250 -.13541E-00 +.14398E-ol -.14694E-02 +. 14902E -03 -.15101E-04 +.150 -.15541E-00 +.17590E-Ol -.18350E-02 +.18732E-03 -.19016E-04

E;' A6 (E;') A'(C') ₇ AS (C') A' (C' ) A' (E;')

9 10

+.950 -. 21377E -05 +.21655E-06 -.21941E-07 +. 22230E -08 -. 22524E -09 +.850 -.19 283E -0 5 +.19535E-06 -.19793E-07 +.20054E-08 -. 20319E -09 +.750 -.15300E-05 +.1 5503E -06 -.15707E-07 +.15915E-08 -. 1 61 2 5E -09 +.650 -.98206E-06 +.99527E-07 -.1 0084E -07 +.10218E-08 -. 10353E -09 +.550 -.33800E-06 +.34284E-07 -.34747E-08 +.35209E-09 -.35674E-l0 +.450 +.33900E-06 -.3431OE-07 +.34753E-08 -.3521OE-09 +.35675E-l0 +.350 +.98268E-06 -.99543E-07 +.1008 5E -07 -.10218E-08 +.10353E-09 +.250 +.15300E-05 -.15503E-06 +.15707E-07 -. 1 59 1 5E -08 +. 161 2 5E -09 +.150 +.19277E-05 -.19534E-06 +.19792E-07 -.200 54E -08 +. 20319E -09

E;' (N)A'(E;') (N)A'(C') (N) A' (E; , ) (N)A

4(E;') (N)A'(E;')

1 2 3 5

+.950 +.28500E-oO -.21945E-Ol +.21161E-02 -.21339E-03 +.21732E-04 +.850 +.19500E-00 -.19095E-Ol +.18966E-02 -.19223E-03 +.19 598E -04 +.750 +.11500E-oO -.14295E-Ol +.14862E-02 -.15210E-03 +.15541 E -04 +.650 +.45000E-ol -.83450E-02 +. 93292E -03 -.97114E-04 +.99644E-05 +.550 -.14999E-Ol -. 19450E -02 +.29612E-03 -.32794E-04 +.34159E-05 +.450 -.65000E-Ol +.43050E-02 -.36012E-03 +.34487E-04 -.34604E-05 +.350 -.10500E-00 +.99050E-02 -.97332E -03 +.98167E-04 -.99919E-05 +.250 -.13500E-00 +.14455E-Ol -.14874E-02 +.15211E-03 -.15541E-04 +.150 -.15500E-oO +.17655E-Ol -.18570E-02 +.19118E-03 -.19570E-04

E;' (N)A'(E;') (N)A'(E;') _{(N)A'(E;') (N) A' (E;' )}

_(ln

A, (E;')

6 _{7 8}

9 10

+.950 -.22183E-05 +.22658E-06 -.23146E-07 +.2364 5E -08 -.24155E-09 +.850 -.200 10E -0 5 +.20439E-06 -.20880E-07 +.21330E-08 -.21791E-09 +.750 -.15877E-05 +.16220E-06 -.16570E-07 +.16928E-08 -.17293E-09 +.650 -.10190E-05 +. 1 04 1 3 E -06 -. 10638E -07 +.10868E-08 -.II103E-09 +.550 -.35067E-06 +.35869E-07 -.3665 5E -08 +.37449E-09 -.38258E-l0 +.450 +.35184E-06 -.35899E-07 +.36663E-08 -.37451E-09 +. 38259E -10 +.350 +.10197E-05 -.10415E-06 +.10639E-07 -.1 0868E -08 +.11103E-09

+~250 +.15877E-05 -.16220E-06 +.16570E-07 -.16928E-08 +.17293E-09 +.150 +.20003E-05 -.20437E-06 +. 20879E-07 -. 21330E -08 +.2179 1 E -09

E;' (N)D _2.2 (C') (N)D (c') (N) D (c' ) (N) D (E; , ) (N)D (E;')

2.3 2.4 . 2.5 2,6

+·S~O +.~ 188~E-Ol -.2~417E-04 +.43939E-05 -.61820E-06 +.80644E-07 +. 0 +. 137 E-O -.2 859E-04 +.39386E-05 -.55691 E -06 +.72726E-07 +.750 +.48044E-04 -.17863E-04 +.30868E-05 -.44069 E -06 +.57675E-07 +.650 +.18875E-04 -.10429E-04 +.19380E-05 -. 28139E -06 +.36979E-07 +.550 -.61294E-05 -.24320E-05 +.61561E-06 -.950 59E -07 +.12678E-07 +.450 -.26964E-04 +.53782E-05 -.747 39E -06 +.99874E-07 -.12839E-07 +.350 -.43634E-04 +.12376E-04 -.20211E-05 +.28439E-06 -.37078E-07 +.250 -.56137E-04 +.18062E-04 -.30892E-05 +.44072E-06 -.57676E-07 +.150 -.644 72E -04 +.22062E-04 -.38571E-05 +.55394E-06 -.72628E-07

t' (N)D (E;')

2.7 (lI)D 2.8 (t') (N)D 2.9 (1.;') (N)D 2.10 (t') (N)D 2.11 ( ' ) t

+.950 -.10022E-07 +.12050E-08 -.14147E-09 +.16314E-l0 -.18553E-ll +.850 -.90408E-08 +.108 70E -08 -.12762E-09 +.14717E-l0 -.16737E-ll +.7SO -. 71 73 5E -08 +.86268E-09 -.10128E-09 +.11680E-l0 -.13282E-l1 +.650 -.46041 E -08 +.5538 2E -09 -. 65027E -1 0 +.74990E-ll -.85280E-12 +.550 -.15844E-08 +. 19077E -09 -.22404E-l0 +.25839E-ll -.29385E-12 +.450 +.15895E-08 -.19093E-09 +.22409E-l0 -. 25840E -11 +.2938 SE -12 +.350 +.46072E-08 -. 55392E-09 +.65030E-l0 -.74991E-ll +. 85280E -12 +.250 +.71736E-08 -.86268E-09 +. 1 0 1 28 E -09 -.11680E-l0 +.13282E-ll +.150 +.90377E-08 -.10870E-08 +.12762E-09 -.14717E-l0 +.16737E-l1

N=+20

t' A' ₁ (t ' ) A' ₂ (t ' ) A' ₃ (t ' )

_{A4 (t' )}

A' (t' )

5

+.975 +.30864E-00 -.22120E-Ol +.210~4E-02 -.21 097E -03 +.21311 E -04 +.925 +.2611L~E-00 -.21353E-Ol +.205 3E-02 -.20571 E -03 +.20785E-04 +.875 +.21614E-00 -.19933E-Ol +. 19459E -02 -.19533E-03 +.19745E-04 +.825 +.17364E-oO -.17972E-Ol +. 17878E -02 -.18008E-03 +.18218E-04 +.775 +.13364E-00 -.15576E-Ol +.15848E-02 -.16038E-03 +.16242E-04 +.725 +.961L~5E -01 -.12845E-Ol +.13430E-02 -.13673E-03 +. 13866E-04 +.675 +. 611 L~ 5E -0 1 -.98745E-02 +.10691E-02 -.10972E-03 +.11148E-04 +.625 +.2864 5E-0 1 -.67498E-02 +.77062E-03 -.80051 E -04 +.81577E-05 +.575 -.13541 E-02 -.35529E-02 +.45522E-03 -.48453E-04 +.49667E-05 +.525 -. 28854E -01 -.35892E-03 +.13094E-03 -.157 20E -04 +.16549E-05 +.475 -.538 54E -01 +.27634E-02 -. 19425E -03" +.17340E-04 -. 1 69 61 E -0 5 +.425 -.76354E-Ol +.57517E-02 -.51256E-03 +. 49917E-04 -.50039E-05 +.375 -.963 54E -0 1 +.85496E-02 -.81654E-03 +.81 21 SE -04 -.81872E-05 +.325 -.11385E-oO +.11107E-Ol ":.10991 E -02 +.11047E-03 -.11167E-{l4 +.275 -.12885E-00 +.13380E-Ol -.13541 E-02 +.13699E-03 -.13872E-04 +.225 -.l Lf135E-00 +.15332E-Ol -.15756E-02 +.16013E-03 -. 16236E-{l4 +.175 -.15135E-00 +. 1 69 3 1 E -0 1 -.17589E-02 +. 17934E -03 -.18199E-{l4 +.125 -.15885E-00 +.18152E-Ol -.19000E-02 +.19416E-03 -.19716E-04 +.075 -.16385E-00 +.18977E-Ol -.19957E-02 +.20424E-03 -.20748E-04

t, At (f; , )

6 A' 7 (f; , ) A' 8 (f;' )

A'

9 (f;') A' 10 ( t ' )

+.975 -.21577E-05 +.21858E-06 -.22146E-07 +.22438E-08 -.22734E-09 +.925 -.21045E-05 +.21320E-06 -.21600E-07 +. 21885E -08 -.22175E-09 +.875 -.19995E-05 +. 20256E-06 -.20523E-07 +.20794E-08 -.21069E-09 +.825 -.18452E-05 +.18694E-06 -.18940E-07 +.19191E-08 -.19444E-09 +.775 -. 16455E -05 +.16671E-06 -.16891 E-07 +.1711 SE -08 -.17341E-09 +.725 -.14052E-05 +. 14238E -06 -.14426E-07 +.14617E-08 -. 14810E -09 +.675 -.11303E-05 +.11454E-06 -.11606E-07 +.1176OE-08 -.11915E-09 +.625 -.82

4

70 E- 06 +.83892E-07 -.85007E-08 +.86132E-09 -.87271E-l0 +.575 -.50 67E-06 +.51170E-07 -.518 55E-08 +.52542E-09 -. 53237E -10 +.525 -.169 25E -06 +.17188E-07 -.17425E-08 +. 17658E -09 -.17892E-l0 +.475 +.17029E-06 -.17215E-07 +.17432E-08 -.17660E-09 +.17892E-l0 +.425 +.50561E-06 -.51193E-07 +. 518 6 1 E -08 -.52544E-09 +.53237E-l0 +.375 +.82844E-06 -.83911E-07 +.85012E-08 -.86134E-09 +.87271E-1O +.325 +.11308E-05 -.11456E-06 +.11607E-07 -. 11760E -08 +.11915E-09 +.275 +.14054E-05 -.14239E-06 +.14427E-07 -.14617E-08 +.14810E-09 +.225 +.16453E-05 -.16671E-06 +. 1 689 1 E -0] -.17115E-08 +.17341E-09 +.175 +.18447E-05 -.1869 3E -06 +.18940E-07 -. 19190E -08 +.19444E-09 +.125 +.19987E-05 -.20254E-06 +.20523E-07 -. 2°é94E-{l8 +.21 069E-09 +.075 +.21036E-05 -.21317E-06 +.21600E-07 -.21 85E-08 +.22174E-09

t' (N)A'(t') (N) A' ( t ' ) (ti) A' ( t ' ) (N)A

4

( t ' ) (N) A' ( t ' )

1 2 3 ₅

+.975 +.308 7L_{fE -00 -. 22152E -0 1 +.21163E-02 -. 2 1 208 E -03 +.21466E-n4}

+.925 +.26125E-00 -.21380E-Ol +.2061OE-02 -. 20678E -03 +.20935E-04

+.875 +.21625E-00 -.19955E-Ol +.19521E-02 -.19634E-03 +.19888E-n4

+.825 +.17375E-00 -.17990E-ol +.179 34E -02 -. 18 1 02 E -0 3 +.18350E-04

+.775 +.13375E-00 -. 15590E -0 1 +. 1 589 7 E -02 -. 1 6 1 2 1 E -03 +. 16360E -04

+·725 +.96249E-Ol -.12855E~01 +.13470E-02 -.13743E -03 +. 139 66E -04

+.675 +. 61 249 E -0 1 -.98809E-02 +.10722E-02 -.11 028E-03 +. 11 229 E -04

+.625 +.28749E-Ol -. 67528E -02 +.77273E-03 -.80453E-04 +.82162E-05

+.575 -. 1 2500 E -0 2 -.35528E-02 +.45633E-03 -.48691 E-04 +.50022E-05

+.525 -.28750E-ol -. 35593E -03 +.13105E-03 -.15788E-04 +. 16664E -0 5

+.475 -.53750E-Ol +. 27690E -02 -.19511E-03 +.17442E-04 -.17088E-05

+.425 -.76250E-Ol +. 57596E -02 -. 51436E -03 +.50 184E -04 -.50404E-05

I +.375 -.96250E-Ol +.85596E-02 -.81921E-03 + • 81 64 1 E -04 -.82466E-05

+.325 -.11375E-00 +.11119E-Ol -.11026E-02 +. 111 0 5E -0 3 -.11248E-04

+.275 -.12875E-00 +.13394E-Ol -.13583E-02 +.13770E-03 -.13973E-04

+.225 -.14125E-00 +.15347E-Ol -.15804E-02 +.1609 5E -03 -.16353E-04

+.175 -.15125E-00 +.16947E-Ol -.17642E-02 +.18026E-03. -.18331E-04

+.125 -.15875E-00 +. 18 1 69 E -0 1 -. 19057E -02 +.19515E-03 -.19858E-04

+.075 -.16375E-00 +.18994E-ol -.20017E-02 +. 20528E -03 -.2089 7E -04

t' (N)A'(t') ₆ (N)A'(t') ₇ (N)A'(t') ₈ (N)A'(t') ₉ (N)A' ( t ' ) 10

+.975 -. 21777E -05 +. 22107E -06 -.22444E-07 +.22787E-08 -.23135E-09

+.925 -.21241E-05 +.21562E-06 -.21891E-07 +. 22226E -08 -.22566E-09

+.875 -.20181E-05 +.20487E-06 -.20799E-07 +.21117E-08 -. 21440E-09

+.825 -.18624E-05 +.18907E-06 -.1919·5E-07 +.19489E-08 -.1)787E-09

+.775 -.16607E-05 +. 1 686 1 E -0 6 -.17119E-07 +. 1 738 1 E -08 -.17646E-09

+.725 -.14182E-05 +.14400E-06 -.14621E-07 +. 14844E -08 -.15071E-09

+.675 -.11408E-05 +.11585E-06 -.11762E-07 +.11943E-08 -.12125E-09

+.625 -.83538E-06 +.84846E-07 -.86151E-08 +.87471E-09 -.88810E-l0

+.575 -.5093 5E -06 +.51751E-07 -.52552E-08 +.53359E-09 -.54176E-l0

+.525 -.17081E-06 +.17 383E -07 .:... 17660E -08 +.17932E-09 -. 18207E-1O

+.475 +.17189E-06 -.17411E-07 +.17667E-08 -.17934E-09 +.18208E-l0

+.425 +.51032E-06 -. 51 77 6E -07 +.52559E-08 -. 53360E -09 +. 54 1 7 6E -1 0

+.375 +.83615E-06 -.84865E-07 +.86156E-08 -.8747 2E -09 +.88810E-l0

+.325 +. 11 4 13 E -05 -.11586E-06 +.11763E-07 -.11943E-08 +.12125E-09

+.275 +.14184E-05 -.14401E-06 +.14621E-07 -.14844E-08 +.15071 E-09

+.225 +.16606E-05 -.16861E-06 +.17119E-07 -.17381E-08 +. 17646E -09

+.175 +.18619E-05 -.18905E-06 +.19195E-07 -.19489E-08 +.19787E-09

+.125 +.20173E-05 -.20485E-06 +.20799E-07 -.21117E-08 +.21440E-09

+.075 +.21231E-05 -.21560E-06 +.21890E-07 -.22225E-08 +.22566E-09

t' (N)D _2.2 (f;') (N) D (f;' ) (N)D (E;') (N) D ( t ' ) (N) D ( t ' )

2.3 2.4 2.5 2.6

+.975 +.32159E-04 -.69211E-05 +.11013E-05 -.15438E-06 +.20072E-07

+.925 +.27233E-04 -.66816E-05 +.10725E-05 -.15053E-06 +.19576E-07

+.875 +022534E-04 -.62361E-05 +.10160E-05 -.1429 3E -06 +.18597E-07

+.825 +.181IOE-04 -.~6221E-05 +.93342E-06 -. 13178 E -0 6 +.17159E-07

+.775 +.13942E-04 -. 8721E-05 +. 82738E -06 -. 1 1 73 6E -0 6 +.15298E-07

+.725 +.10035E-04 -.40176E-05 +.70113E-06 -.10005E-06 +.13060E-07

+.675 +.63874E-05 -. 30877E -05 +.5 5808E -06 -.80287E-07 +.10500E-07

+.625 +.30024E-05 -.21102E-05 +.40220E-06 -.58570E-07 +.76829 E-08

+.575 -.12530E-06 -.11101E-05 +.23751 E -06 -.35447E-07 +.46774E-08

+.525 -.29896E-05 -.11114E-06 +.68226E-07 -.11494E-07 +.15582E-08

+.475 -.55934E-05 +.86538E-06 -.10152E-06 +.12696E-07 -.15979E-08

+.425 -.79373E-05 +.17998E-05 -.26768E-06 +.36~33E-07 -.47133E-08

+.375 -.10022E-04 +.26751E-05 -.42638E-06 +.59 35E-07 -.77116E-08

+.325 -.11845E-04 +.34748E-05 -.57389E-06 +.80843E-07 -.10518E-07

+.275 -.13409E-04 +.41860E-05 -. 70699E-06 +.10025E-06 -.13066E-07

+.225 -.14711E-04 +.47964E-05 -.82264E-06 +.11718E-06 -.15292E-07

+.175 -.15752E-04 +.52962E-05 -.91831E-06 +.13123E-06 -.17141E-07

+.125 -.16532E-04 +.56781 E-05 -.99192E-06 +.14208E-06 -.18570E-07

ct (N)D (ct)

2.7 (ll)D 2.8 (ct) (N)D 2.9 (ct) (N)D 2.10 (ct) (N)D 2.11 C ( t )

+.975 -.24865E-08 +.29801E-09 -. 34875E-1O +.40089E-ll _-.4~44~E-12

+.925 -.24252E-08 +.29067E-09 -.34016E-l0 +.39102E-ll -.4 32 E-12

+.875 -.23042E-08 +.27617E-09 -. 32320E -10 +.37152E-ll -.42115E-12

+.825 -.21265E-08 +. 25488E-09 -. 29828E -10 +. 34288E-11 -.38868E-12

+.775 -.18962E-08 +. 22730E -09 I -.26601E-1O +.30578E-ll -.34663E-12

+.725 -. 16194E-08 +. 19413E -09 -. 22720E -1 0 +.26116E-11 -.29606E-12

+.675 -.13026E-08 +.15617E'-09 -. 18278E -1 0 +.21011E-11 -.23818E-12

+.625 -.95381E-09 +. 11437E-09 -. 1 338 6E -1 0 +.15388E-11 -.17444E-12

+.575 -. 58154E-09 +.69761 E-l0 -.81658E-11 +.93871 E -1 2 -.10641E-12

+.525 -.19501E-09 +.23432E-l0 -.27439E-11 +.31547E-12 -.35763E-13

+.475 +.19627E-09 -.234 72E -1 0 +.27454E-11 -.31554E-12 +.35768E-13

+.425 +.58269E-09 -.69798E-l0 +.81672E-11 -.93878E-12 +.10641E-12

+.375 +.95472E-09 -.11440E-09 +.13387E-10 -.15389E-11 +.17444E-12

+.325 +.13031E-08 -.15618E-09 +.18278E-l0 -.21011E-11 +.23818E-12

+.275 +.16196E-08 -. 19414E -09 +.22720E-10 -.26117E-ll +. 29606E -12

+.225 +. 18961 E -08 -.22730E-09 +.26601E-l0 -.30579E-ll +.34664E-12

+.175 +. 21 259 E -08 -.25486E-09 +. 29828E -10 -.34288E-ll +.38868E-12

+.125 +.23034E-08 -.2761 5E -09 +.32320E-l0 -.37153E-11 +.42116[-12

Table 2.1 - Temperature coefficients for a perfect isolation terminal. Bt : Temperature coefficients

(J)B

t : Analog temperature coefficient

(N) . • • Dl t : Error coeff~c~ent , for temperature N= +5 t +.800 +.600 +.400 +.200

c

+.800 +.600 +.400 +.200 C +.800 +.600 +.400 +.200

c

+.800 +.600 +.400 +.200 t +.800 +.600 +.400 +.200 +.800 +.600 +.400 +.200 Bl (t ) -.l8000E-00 -.32000E-oO -.42000E-oO -.48000E-00 B6 (C) +.17436E-02 +.33165E-02 +.45648E~02 +.53663E-;02 (N)B (C) 1 -.18000E-00 -.32000E-00 -. 42000E-oO -.48000E-00 (N)B 6(C) +.18168E-02 +. 34557E -02 +.47564E-02 +.55915E-02 (N)D (C) 1.2 -.60004E-03 -.10667E-02 -.14000E-02 -.16001E-02 (N)D (C) 1.7 , +.35748E-04 +.67997E-04 +.93590E-04 +.11 002E -03 B 2 (t ) +. 65400E -01 +.l2373E-00 +.16940E-oO +.19840E-oO _ B7 (t ) -.70666E-03 -.13441E-02 -.18500E-02 -.21748E-02 (N)B (C) 2 +.66000E-Ol +.12480E-00 +.17080E-oO +. 20000E -00 (N)B (t) 7 -.74241E-03 -.14121E-02 -.19436E-02 -.22849 E-02 (N)D (t) 1.3 +.43922E-03 +.83061E-03 +.lI368E-02 +.13312E-02 (N)D (C) 1.8 -.16975E-04 -. 32288E -04 -.44441E-04 -.52244E-04 n n-t Y l

,

.. (c,t)

=

nl

L

Bt(C)t . t=Q . (n-t) I B3 (t ) -.26224E-ol -.49857E-ol -. 68583E -01 -. 80588E -01 Ba(C) +. 28640E -03 +.54476E-03 +.74980E-03 +..88145E-03 (N)B (C) 3 -.26664E-Ol -. 50688E -01 -.69720E-ol -.81920E-Ol (N)Ba(C) +. 30337E -03 +.57705E-03 +.79424E-03 +.93369E -03 (N)D (C) 1.4 -.26533E-03 -.50440E-03 -.69381 E -03 -.81523E-03 (N)D (t) 1.9 +.78961E-05 +.15019E-04 +.20672E-04 +.24301E-04 n ( ) n-t n!

L

N B (C)..:;t _ _ t=Q t (n-t)1 n!

I

(N)D (C)tn-t t=o l , t (n-t)! B4 (C ) +.10616E-Ol +.20193E-Ol +.27791E-01 +.32669E-ol B

₉

(t) -.11607E-03 -.22078E-03 -.30388E-03 -.35723E-03 (N)B 4 (C) +.10882E-ol +.20697E-Ol +.2848 5E -01 +.33484E-Ol (N) B (C) 9 -.12397E-03 -.23580E-03 -.324 55E-03 -.38154E-03 (N)D (C) 1.5 +.14385E-03 +.27361 E -03 +.37657E-03 +.44266E -03 (N)D (C) 1.10 -.361 S4E -05 -.68770E-05 -.94653E-05 -.11127E-04 B5 (t ) -.43022E-02 -.81834E-02 -.11263E-o 1 -. 13240E -0 1 B10(t) +.47043E-04 +.89481 E-04 +.12316E-03 +.14478E-03 (N)B (C) 5 -.44461 E -02 -.84S70E-02 -.11ó39E-Ol -.13683E-Ol (N)B (c) 10 .. +.50658E-04 +.96358E-04 +.13262E-03 +.15591E-03 (N)D (t) 1.6 -.73189E-04 -. 1 39 2 1 E -03 -. 19160E-03 -.22524E-03 (N) Dl.11 (C ) +.16349E-05 +. 31099E -05 +.42804E-05 +.50319E-05

N=+lO

t B

1 (t ) B2 (t ) B3 (t ) B4 (t) B5 (t )

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(t) B

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(N)B (t) 10

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(N)D (t) (N)D (t) (N) D (t ) (N) D (t) (N)D (t)

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1,7 1,8 1,9

,

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I; B6 (I; ) B7 (I; ) B8 (I; ) B

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(E;) BlO (I;)

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(n)B (C)

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