HFSR Instrumentation Error Analysis
M. J. Roberts
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This report was prepared as an account of work sponsored by the United States
Government. Neither the United States nor the Energy Research and Development
Administration, nor any of their employees, nor any of their contractors,
subcontractors, or their employees, makes any warranty, express or implied, or
assumes any legal liability or responsibility for the accuracy, completeness or
usefulness of any information, apparatus, product or process disclosed, or represents
that its use would not infringe privately owned rights.
ORNL-TM-5055
UC-37, Instruments
UC-80, General Reactor
Technology
Contract No. W^AOS'-eng-ZS
INSTRUMENTATION AND CONTROLS DIVISION
HFIR INSTRUMENTATION ERROR ANALYSIS
M. J. Roberts
DECEMBER 1975
NOTICE
Tills report was prepared a l an account of work
sponsored b> the United Stales Government. Neither
the Untied Stales nor the United Stttef Enetgy
Research and Desrelopmcnt Administration, nor any ot
their employee
any of their contractors,
subcontractors. or t h t r employees, makes any
vtrr»-:y. express or imvtied, or assumes any legal
liability ot responsibility Tor 'he accuracy, completeness
or uscfulnes of any Informatki, sppaiatus, product or
process disclosed, or represents .<iat its use would not
mfnnic prmtcly owned nfhts.
N O T I C E This document contain! information of a preliminary nature
and was prepared primarily for internal use at the Oak Ridge National
Laboratory. I t is subject to reviiion or correction and therefore does
not represent a final report.
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Oak Ridge, Tennessee
37830
operated by
UNION CARBIDE CORPORATION
for the
ENERGY RESEARCH AND DEVELOPMENT ADMINISTRATION
^DISTRIBUTION f; *
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iii
ABSTRACT
The accuracies of the measurements of operating conditions
made by the electronic safety system at the High Flux Isotope
Reactor were calculated theoretically.
The technique was small-
signal sensitivity analysis of the errors as they accumulated
through cascaded instruments in various systems.
The error
bounds (for 99.7% probability) at typical operation of 100 MW
thermal power (mode 1) were ^4.05 MW in heat power, 1.82°F in
inlet temperature, and 5.95 MW in perceived heat power on fluxto-flow ratio.
The error bound in flow at the nominal flow
trip point of 1500 gpm was *v86 gpm.
The maximum unfavorable
error in perceived heat power in flux-to-flow measurement was
+420 kW for operation in research mode 2 (lower coolant flow
and, hence, lower power; and approximately the same temperature
difference as in mode 1), and +17.1 kW for mode 3 (no coolant
flow, static operation, and, hence, low power).
V
CONTENTS
Page
1.
Introduction
2.
System Description
1
3.
Error Analysis
4
4.
Appendix:
4.1
1
. . . *
HFIR Instrumentation Error Analysis
14
Nominal Relations
14
4.2
Sensitivity
18
4.3
Statistical Methods
24
1
1.
INTRODUCTION
The High Flux Isotope Reactor (HFIR) at Oak Ridge National Laboratory
(ORNL) produces transplutonium elements, principally
science and industry.
252
Cf, for use in
The reactor is also used to determine the amount
of damage to materials due to radiation exposure.
The safety system at
the HFIR, which consists of three redundant electronic analog computer
channels, shuts down, or "scrams," the reactor if any operating parameter
exceeds its safety system setting (SSS).
For a shutdown, or scram, to
occur, two of the three safety system channels must agree that an SSS has
been exceeded.
This type of operation helps to assure reliability and to
minimize unnecessary scrams.
In this work, the author determined the accuracies of all the measurements of operating conditions made by the safety system.
The principal
objective was to combine the accuracies given by the manufacturers of all
instruments in the system to determine the accuracy of the system for the
following scram mechanisms:
Heat power, coolant inlet temperature, cool-
ant flow, coolant pressure, and flux-to-flow ratio.
three modes of operation:
The analysis covered
(1) typical operation of 100 MW(t); (2) lower
coolant flow, lower power, but approximately the same temperature difference as mode 1; and (3) no coolant flow, static operation, and low power.
This analysis forms some of the necessary background information
for writing the technical specifications for the HFIR.
Before SSS's can
be established, one must know the uncertainty of his measurement of the
parameters for which the SSS's are established.
These uncertainties must
be subtracted from the SSS's to ensure that the SSS's are not exceeded.
2.
SYSTEM DESCRIPTION
The scram mechanisms are (1) heat power, (2) coolant inlet temperature, (3) coolant flow, (4) coolant pressure, and (5) flux-to-flow ratio.
Figure 1 is a diagram of one channel of the system that measures coolant
inlet temperature, coolant flow, heat power, and flux-to-flow ratio.
Coolant pressure is measured independently.
Each instrument in the system
of Fig. 1 has an accuracy specified by the Alden Hydraulic Laboratory,
BLANK PAGE
2
F o x b o r o , o r ORNL, as i n d i c a t e d .
fied accuracies.
vidual
Table 1 i s a t a b u l a t i o n of
By u s i n g s t a t i s t i c a l a n a l y s i s
techniques,
i n s t r u m e n t a c c u r a c i e s w e r e combined m a t h e m a t i c a l l y
s y s t e m a c c u r a c y f o r measurement o f c o o l a n t
h e a t p o w e r , and f l u x - t o - f l o w
Table 1.
temperature,
Flow
bulb
Current
Specified
flow,
accuracy
transmitter
repeater
extractor
subtractor
±0,.52 o f s p a n
±0..52 o f s p a n
±0,.52 o f s p a n
±0,.52 o f s p a n
multiplier
±0.. 5 2 o f s p a n
Riiset f l u x
electronics
±3..122 o f
span
±1,.252 o f
span
comparator
flow0
± 0 . ,52 o f s p a n
Heat-power
Trip
to yield a
coolant
:
±0.. 2 5 * o f t r u e
Square-root
indi-
± 0 .. 5 V
transmitter
Current
these
A c c u r a c i e s s p e c i f i e d by t h e m a n u f a c t u r e r s
o f the system instruments
Venturi
Temperature
speci-
ratio.
Instrument
Resistance
these
1
^When u s e d w i t h an e r r o r l e s s t e m p e r a t u r e t r a n s m i t t e r , the output c u r r e n t of the t r a n s m i t t e r w i l l
i n d i c a t e a temperature that i s w i t h i n ±0.5°F of the
true temperature.
The d i f f e r e n t i a l p r e s s u r e o f t h e v e n t u r i w i l l
correspond t o a f l o w w i t h i n 0.252 o f the t r u e f l o w .
According to s t a t i s t i c a l
t h e o r y , an i n s t r u m e n t
c a n n o t be g u a r a n t e e d
a l w a y s f a l l w i t h i n a s p e c i f i e d a c c u r a c y , s i n c e s u c h w o r d s as
and " n e v e r " have no s t a t i s t i c a l s i g n i f i c a n c e .
1.
Methods
G. S. Z a l k i n d and F. G. S h i n s k e y , The Application
in Evaluating
the Accuracy
F o x b o r o C o . , F o x b o r o , MA.
of Analog
"always"
Instrument accuracy
d e s c r i b e d b y s t a t i n g how c l o s e t h e i n s t r u m e n t r e a d i n g i s
Instruments
to the
of
and
to
is
true
Statistical
Systems,
3
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Fig.
1.
One c h a n n e l o f t h e e l e c t r o n i c s a f e t y
system.
4
reading.
The i n s t r u m e n t
accuracies given i n Table 1 are
as t h r e e s t a n d a r d d e v i a t i o n s
(3o)
Gaussian e r r o r d i s t r i b u t i o n .
random e r r o r s ,
f r o m t h e mean on each
That i s ,
the t o t a l error w i l l ,
interpreted
instrument's
considering both systematic
w i t h 99.7% p r o b a b i l i t y ,
the s p e c i f i e d accuracy over the e n t i r e range of
be
within
the i n s t r u m e n t .
When
a number o f i n s t r u m e n t s a r e g r o u p e d t o p e r f o r m a s i n g l e f u n c t i o n ,
their
e r r o r s become combined i n s u c h a way t h a t t h e e r r o r o f t h e w h o l e
appears random, even though t h e i n d i v i d u a l
largely systematic.1
Therefore,
The s y s t e m a c c u r a c i e s
described i n
the Appendix.
applied
to a square-root
that
results.
ERROR ANALYSIS
for
coolant
inlet
temperature, coolant
The r e s u l t s a r e p l o t t e d i n F i g s .
of
2-9.
i n F i g s . 4 , 5 , 8 , and 9 a r e
small-signal sensitivity analysis
extractor.
The s q u a r e - r o o t
when
extractor
converts
t h e measured p r e s s u r e d i f f e r e n c e a c r o s s t h e v e n t u r i t o a s i g n a l
t i o n a l t o f l o w i n t h e f l o w measurement s y s t e m and i s ,
i n h e a t p o w e r and f l u x - t o - f l o w r a t i o
e x t r a c t o r s and m u l t i p l i e r - d i v i d e r s ,
errors for square-root
approach z e r o ,
results
for
functions,
such as
gpm a r e q u i t > c o n s e r v a t i v e .
sensitivity
used
Since the
instrument
calculated
signal
conservative results at a l l
conservative,
square-root
the
e x t r a c t o r s a r e unbounded as t h e i n p u t
f l o w s >1000 gpm a r e o n l y s l i g h t l y
by s m a l l - s i g n a l
therefore,
the calculated error of
the i n p u t s i g n a l l e v e l s .
the analysis y i e l d s
propor-
calculation.
F o r i n s t r u m e n t s t h a t compute n o n l i n e a r
output i s a f u n c t i o n of
flow,
r a t i o were c a l c u l a t e d u s i n g t h e methods
The l a r g e e r r o r s a t v e r y s m a l l f l o w s
caused b y a c h a r a c t e r i s t i c
errors
is a v a l i d approximation
3.
h e a t p o w e r , and f l u x - t o - f l o w
system
e r r o r s may b e
c o n s i d e r i n g each i n s t r u m e n t ' s
as random w i t h a G a u s s i a n d i s t r i b u t i o n
y i e l d s p r a c t i c a l and m e a n i n g f u l
instrument
and
levels
flows.
but f o r
As f l o w a p p r o a c h e s z e r o , t h e e r r o r s
The
<1000
predicted
a n a l y s i s become so i n a c c u r a t e as t o b e mean-
ingless.
F i g u r e s 2 and 3 show t h e b o u n d o n t h e e r r o r o f
The o r d i n a t e s a r e t h e u n c e r t a i n t i e s
in inlet
the i n l e t
temperature.
temperature.
The a b s c i s s a s
5
~
ORNL-DWG 7 5 - 1 0 4 0 9
2.30
2.20
2.10
2.00
1.90
1.80
1.70
h-
1.60
^
1.50
75
100
125
150
175
200
N O M I N A L INLET C O O L A N T T E M P E R A T U R E (°F)
F i g . 2.
E r r o r bound (99.7% p r o b a b i l i t y ) o n i n l e t t e m p e r a t u r e
measurement a s a f u n c t i o n o f n o m i n a l i n l e t t e m p e r a t u r e .
6
ORNL-DWG 75-10410
^
1-80
Z
1.00
75
100
125
150
175
N O M I N A L I N L E T COOLANT T E M P E R A T U R E (°F)
F i g . 3.
E r r o r b o u n d (95.5% p r o b a b i l i t y ) o n i n l e t t e m p e r a t u r e
measurement a s a f u n c t i o n o f n o m i n a l i n l e t t e m p e r a t u r e .
200
7
ORNL-DWG 7 5 - 1 0 4 1 1
12.0
10.0
8.00
6.00
4.00
—
2.00
4.0
8.0
12
(x103)
C O O L A N T FLOW ( g p m )
Fig. 4.
E r r o r b o u n d s (99.7% p r o b a b i l i t y ) o n h e a t power
measurement f o r s e v e r a l t e m p e r a t u r e d i f f e r e n c e s a s a f u n c t i o n
of coolant flow.
8
ORNL-DWG 7 5 - 1 0 4 1 2
12.0
>»
3
o
-O
o
a.
10.0
A7"=
A r =
A7" =
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A 7" =
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A 7"=
AT —
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A 7"=
AT=
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z
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O
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DC
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<r
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a:
UJ
£
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a.
H
72°F
66°F
60°F
54°F
48°F
42°F
36°F
30°F
24°F
18°F
12°F
6°F
2.00
<
LLI
X
4.0
8.0
COOLANT
12
FLOW
16
(gpm)
F i g . 5.
E r r o r bounds (95.5% p r o b a b i l i t y ) o n h e a t power
measurement f o r s e v e r a l t e m p e r a t u r e d i f f e r e n c e s as a f u n c t i o n
of coolant f l o w .
(x103)
9
COOLANT FLOW
(gpm)
Fig. 6. Error bound (99.7% probability) on flow measurement
at low flow as a function of nominal coolant flow.
10
ORNL-DWG 7 5 - 1 0 4 1 4
(x102)
5.00
h-
4.00
h-
3.00
2.00
1.00
h-
1.0
2.0
COOLANT
3.0
FLOW
4.0
U10 3 )
(gpm)
Fig. 7. Error bound (95.5% probability) on flow measurement
at low flow as a function of nominal coolant flow.
11
ORNL-DWG 75-10415
12.0
I
0
I1
X
10.0
LL
>S
Z
Z
o
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h<
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6.00 h-
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o
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2.00 L -
o
cr
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UJ
0
4.0
8.0
12
16
(x103)
COOLANT FLOW (gpm)
Fig. 8. Error bounds (99.7% probability) on perceived heat
power on flux-to-flow ratio trip for maximum and minimum temperature difference as a function of coolant flow.
12
COOLANT FLOW (gpm)
Fig. 9. Error bounds (95.5% probability) on perceived heat
power on flux-to-flow ratio trip for maximum and minimum temperature difference as a function of coolant flow.
13
are the nominal, or expected, temperatures.
At nominal operation of
^120°F inlet temperature, the measurement error could reasonably be ex pected to be <1.82°F with 99.7% probability or <1.21°F with 05.5% probability.
Figures 4 and 5 show the bound on the error in the measurement of
heat power.
The ordinates are the uncertainties in heat power.
For
example, at a usual operation of ^100 MW, one could reasonably expect
that if the heat power safety system were calibrated the error in the
measurement of heat power would be <4.1 MW with 99.7% probability or
<2.73 MW with 95.5% probability.
Figures 6 and 7 are graphs of the bound on the error of flow measurement at low flows.
Figures 8 and 9 are graphs of the maximum error (in MW) of heat
power that might occur on flux-to-flow ratio scram, plotted vs flow.
Most of this error is the error in measuring heat power, including the
reset flux electronics.
The remainder is the uncertainty in measurement
of flow and the uncertainty in the comparison process itself.
Pressure is monitored independently by a safety pressure switch made
by the Barksdale Company.
Its specified accuracy is 0.5% of span.
For modes 2 and 3, the reset gain in the reset flux electronics is
clamped at 1.0 (nominally), and the trip comparator for heat power uses
a fixed voltage as a reference, instead of a signal proportional to flow.
The value of the reference is controlled by a potentiometer; the maximum
voltage is 10 V, which corresponds to 16,000 gpm.
The ionization chamber
current required to cause a trip is inversely proportional to the product
of the feedback resistance in the current-to-voltage conversion amplifier
and the reset gain.
The "worst case" ionization chamber location for
modes 2 and 3 would be where the reset gain would be maximum, or 1.3 in
mode 1.
The feedback resistors are nominally 1, 60, and 1500 Mfl for
modes 1, 2, and 3, respectively.
Thus, the nominal worst-case power at
trip would be 2.82 MW for mode 2 and 113 kW for mode 3.
These calcula-
tions are based on an assumption that the ionization chamber current is
proportional to heat p o w e r — a reasonably good assumption except for th'
period immediately after shutdown.
In any case, including the worst
case just described, there are statistical error bands on the power at
14
trip.
For
2 the 3o uncertainty i s G.1G2 i™,
uncertaint
- i.08 kW.
f o r ^ d z 3 th?
Therefore, one can reasonably say there is a
<99.7% assu\.^nce that the highest heat power at flux-to-flow t r i p that
could occur would be 2.92 MW for mode 2 or 117.1 kW for mode 3, and there
is a >95.5% assurance that the highest heat power at flux-to-flow t r i p
that could occur would be 2.89 MM for mode 2 or 115.7 kW for mode 3.
APPENDIX:
HFIR INSTRUMENTATION ERROR ANALYSIS
4.1
4.1.1
Nominal Relations
Definitions
Tq
temperature of coolant at outlet of reactor pressure
vessel (°F)
T^
temperature of coolant at inlet of reactor pressure
vessel (°F)
Rq
resistance of dynatherm resistance bulb at outlet (£1)
R^
resistance of dynatherm resistance bulb at i n l e t (ft)
I
current from temperature transmitter directly
proportional to outlet coolant temperature
(span = 0-40 mA)
current from temperature transmitter directly
proportional to inlet coolant temperature
(span = 0-40 mA)
V^
voltage directly proportional to i n l e t temperature
(span » 0-4 V)
Ijyp
current proportional to difference between the i n l e t
temperature and 135°F (span « 0-26.67 pA)
1
ox
ground isolated version of I
o
(span - 0-40 mA)
current proportional to (T q — Tj) (span • 0-40 mA)
15
Ah
pressure d i f f e r e n c e across v e n t u r x
Q
flow of coolant
CJJ
coefficient
(£«. K2O)
(cfs)
r e l a t e d t o R e y n o l d s number o f
pipe
(dimens i o n l e s s )
c u r r e n t p r o p o r t i o n a l t o Ah ( s p a n = 0 - 4 0 mA)
Ip
current proportional
I
gr°und i s o l a t e d v e r s i o n of
FX
to V
( s p a n = 0 - 4 0 mA)
( s p a n = 0 - 4 0 mA)
Ijjp
c u r r e n t p r o p o r t i o n a l t o h e a t power
( s p a n = 0 - 4 0 mA)
Vgj,
voltage proportional to reset f l u x
( s p a n = 0 - 1 0 V)
VF
voltage proportional to flow
IDpF
current proportional to
( s p a n = 0 - 1 2 . 4 2 V)
( 1 . 3 x % f l u x ) —% f l o w
( s p a n = - 7 3 t o + 6 7 yA)
current proportional to the pressure
across the v e n t u r i a t low f l o w
V
LLh
* D L F
( s p a n = 0 - 4 0 mA)
voltage p r o p o r t i o n a l t o the pressure
across the v e n t u r i at low f l o w
difference
difference
(span = 0 - 4 V)
c u r r e n t p r o p o r t i o n a l t o t h e d i f f e r e n c e between
pressure d i f f e r e n c e across the v e n t u r i
0.2708 f t H20 (span = - 4 . 3 3 t o +22.33
4.1.2
the
and
pA)
Equations
BT
0
R « ac
0
and
8T
R±
w h e r e a » 2 2 7 . 1 9 fl a n d B -
- ae
1.07169 x
,
10-3/°F.
(1)
16
[A'6]
and
where y = 1194.947 mA, 6 = 0.483267, and
= 100 n
= 263.26 R
-
(3)
T
DT
150 K
150 K
.
'
1 = 1 .
ox
o
X
d - *
Cl
(4)
(5)
ox - V
•
(6)
where x = 1.736.
2
(4)
Ah = f ^ r t
,
(7)
where K = 7.6865 ft 5/2 /sec.
I h = 4>Ah ,
(8)
17
where $ = 1.15942 mA/ft.
where X =
mA .
I
• aI
FX
F
T
= pI
HP
Fxh '
where p = 0.03489/mA.
RF " "°- 2 5 'hp
where V ^ 1s in volts, and
is in mA
V F - -0.3109 I F ,
where V_ is in volts, and I_ is in mA.
r
F
T
_
RF
DFF
150 K
F
170 K
where T g ^ is in mA.
I L L h - (24 mA/ft)Ah
V
LLh -
100
" *LLh
18
L
_ V lLh
1.65 V
DLF
150 K " 150 K
4.2
4.2.1
1
"
;
Sensitivity
Definitions
The sensitivity of the percentage of span change in X to a percentage
of span change in Y is
s
s
*
«
s/
,
s
(is)
where X g is the span of X, and Y g is the span of Y; that is, span change
(Z)
in X = SSy [span change (%) in Y] .
The sensitivity of the percentage of span change in X to a relative
percentage change in Y is
(19)
Rsii-xVr.
s/
that is, span change (%) in X = RS* [relative change
in Y] .
The sensitivity of the percentage of span change in X to an absolute
change in Y is
as
y
=
¥s'/
X
that is, span change (%) in X = ASY
3 Y
>
(20)
(change in Y).
All these sensitivities [Eqs. (18)-(20)] are defined for incrementally small perturbations, but they will hold approximately true for
finite changes if they are small.
19
4.2.1
Calculated Sensitivities
3R
^
BT
= aBe
3R,
°
BT
1
= ctBe
(21)
31
c
= Y
3R
mA/fi
R
< o
!h
3R.,
=
+
V
1
mA/n
Y
( R
i
+
:
v
BT
° = ccBYe °
~3T
31
i
tnA/°F
<Ro
+
V
:
V
1
and
3I±
3T7
l
I
A
s
T
o
o
BT,
=
mA/°F
a6YG
(R, +
s
m .
40
t
ST
o
(22)
R
< o
+
V '
20
As'i
Ab
Ti
=
«gl
40
eT
i
EE
"R
(R +
i
v
:
'F .
(23)
= 100 fi
31.
SS ^ = 1 .
i
(24)
31
DT
3V, = 1/150 K
(25)
i
ai
ox
91
= 1
ss r ox = 1
o
(26)
31.
= T .
31
OX
SSj"
ox
= T .
(27)
21
SS,
(28)
= -T
3(Ah) _
3Q
2Q
(29)
CCDK)<
3I
h
3 (Ah)
9I
=
h _
3Q
•
2<t>Q
" (C D K) 2 •
24>Q2
40(C D K)
RS
_ 4»Ah
2
V i i
Q " 20
20
(30)
a i.
31,
SS,
31
FX
= 1.
3IT
(31)
22
si.
SS
= pi
FX
= P iFX
3I
hp
31. = Pi,
FX
S S t h p = pi. .
d
FX
3 V.
RF
= -0.25
31.
HP
S S , " = -1 .
X
HP
8V.
F
= -0.3109
91.
23
^TlTTTT
a
=
^
1
i
5
b
=
0
-
0 0 6 6 7
U
DFF
SS
u
= 0.47619
.
RF
— =
3V_
Jr
==— = -0.00588 U
170 K
DFF
SS
= -0.521849
F
3ILLh
F(Ah)"
=
24
'
I'
3 V.
LLh
31LLh
= 100 n
.
24
SS,
LLh
31
DLF
3 V,
LLh
DLF
SS,
'LLh
4.3
4.3.1
150 K
1 .
(AL)
Statistical Methods
General Theory
Given a function, D - f(A, B, C), then
if AA, AB, and AC are small.
However, if AA, AB, AC, and AD are
measures of statistical probability of error (i.e., standard deviation),
the following relation holds:
since the variance of a sum of independent random variables is the sum of
the squares of the individual variances.1
25
4.3.2
Applications to HFIR Instrumentation
Figure 10, a diagram of the safety system, illustrates the sources
of error and the way they accumulate.
Symbol k represents the errors
at the various steps in the system (nomenclature adapted from ref. 1).
Next we write
A1
where AI
are ao,,
and 6T
O
o - [(wo
O
AT
o)
+
(k
o>2]
•
<">
a is any real number, o is the standard
A
deviation, and X is a subscript denoting current or temperature, i.e.,
• 3oj
or ATq «
, respectively.
However, a must be the same for
all calculations.
We also write
BT
31
r
°
DT
a
*R
°
<Ro *
V
but
3T
ac
(R
o
+
V
is approximately a constant that equals 0.2498 ± 0.0552 over a temperature
range from 75 to 200°F.
Therefore,
31<
o 0.2498 8Y
3T
- 0.319897 .
26
Al
o
= ["0.102334(AT ) 2 + (k0 ) 2 V / 2
l
o
J
and
(100) - 2 . 5 I 0 . 1 0 2 3 3 4 ( A T q ) 2 + ( k Q ) 2 J
.
(45)
Similarly,
j—
T
(100) = 2 . 5 [ 0 . 1 0 2 3 3 4 ( 4 ^ )
2
I1/2
+ (ki)2J
,
(46)
where k Q and k^ are absolute standard deviation multiples.
By using the methods of Zalkind and Shinskey,1 the following relations
can be derived:
AV.
(100) =
ai
Hy
26
(100)
*
10
+
°)
(kvi) ]
'
- [ ( ^ i x loo)'
(48)
.,1/2
Al
ox ( 1 0 0 ) =
40
(47)
x
1 0 0
)
+
(k
x
) j
,
where k^ is a percentage of span standard deviation multiple.
(49)
27
ORlkL-MC 7S-10376
RATIO TRIP
SIGNAL
Fig. 10. One channel of the electronic safety system diagrammed
to illustrate the sources of error and their accumulation.
28
d
AO
(100) -
|^1.736
40
A I,
736 ^
x 100
x 100
r-wn
(50)
where k^ is a percentage of span standard deviation multiple.
1/2
40
(100)
(51)
where ^ ^ x 100^ is a relative percentage standard deviation multiple.
Additional derived relations are as follows:
1/2
40
(100)
(52)
1/2
th
40
Al.
FX
40
(100) =
th
40
x 100
(53)
2(IF/40)
2
(100) =
10
°)
+
2
k
( y) ]
1 / 2
'
(54)
Al.
~40
(55)
29
(100)
x 100
• I. (nar
( t ? X loo)
2
(100) « | ^0.A7619
+
( ^
+
x
[°- 5 2 1 8 4 9 (i^2
F
)
2
]
(56)
,
lQo)J
x 10
+ k
°).
( n) }
(57)
1/2
2
AI
LLh
(100) =
40
'
1/2
AV
4
(100)
^
^
|/AI
- [ ( ^
IV 40
,
ioo)2
'
+
(59)
and
m
1
-tO-HO ]
dlf
(100) =
26.67
1/2
(60)