202

CALIBRATION OF DIGITAL PRESSURE GAUGES, COMBINED PRESSURE GAUGES AND VACUUM GAUGES

Tatyana Petkova, Ivan Ivanov, Lyuboslav Hristov

Summary: The report describes the methods for the calibration of digital pressure gauges, combined pressure gauges and vacuum gauges developed by the Laboratory for Pressure, Flowrate and Level Measurement in Metrology Assurance Department. The methods were developed in order to provide the traceability of the measurement results for the above pressure gauges at Kozloduy NPP. The calibration is performed with a piston pressure gauge (further in the text referred to as reference pressure gauge).

The purpose of the methods is to find out the difference (deviation) between the pressure measured by the calibrated and reference pressure. In the process of calibration, the real values of pressure are identified in the measuring unit of the calibrated pressure gauge both in pascal , [Ра], or its multiple from the SI international system.

The main sections defining the conditions, methods and nature of calibration, as well as processing of calibration results are reviewed.

Кey words: digital pressure gauges, difference (deviation) between the measured pressure and reference pressure, traceability

1. IntroductionThe calibration with digital pressure gauges is

made according to the method of direct comparison of the pressure measured by the reference pressure gauge and calibrated pressure gauge.

Technically, the method is used as a source of pressure and creates pressure in the measuring system, which pressure is measured simultaneously by the reference and calibrated pressure gauge.

The presented technique defines the conditions, methods and equipment to calibrated the digital pressure gauges according to [1].

In the process of calibration, the real values of pressure are identified in the measuring unit of the calibrated pressure gauge both in pascal , [Ра], or its multiple from the SI international system.

The purpose of the technique is to find out the difference (deviation) between the measured pressure by the pressure measured by the calibrated pressure gauge and reference pressure according to [1].

Pressure gauges with class of accuracy higher or equal to 0,025 in the measuring range of minus 95kP to 60 MPa are subject to calibration under the current technique.

2. Applied reference pressure gauges- Piston pressure gauges, combined pressure

gauges and vacuum gauges with uncertainty (from the latest calibration certificate) less than or equal to 1/2 of the allowable error of the calibrated pressure gauge

The measuring range of the reference pressure gauge should comply with the range of the cali-brated pressure gauge.

The reference pressure gauge should be tracea-ble to national or international reference.

3. Calibration operation sequenceThe indication of the digital pressure gauge

is reset when pressure is not set prior to the first series according to the technical description and operational procedure. The set up of zero is not allowed between the separated measurement series.

The pressure is changed gradually from the lower to the upper measuring range limit.

The indications for the different values (points) of the set pressure, if possible, evenly distributed along the measuring range are recorded. The upper and lower limit values should be included.

The operations listed above will be repeated also when the pressure is decreased taking into consideration the indications of the calibrated device at the same pressure values.

The measuring series are performed according to Table 1 depending on the accuracy class of the calibrated pressure gauge [1].

An additional measurement for repeatability for reinstallation is made for digital pressure gauges with accuracy class of less than 0.1.(type A)

Calibration sequence

203

Additional measurement of the repeatability for the installation:

p Максимално натоварване

M5 M6

t

Figure No. 2

p Максимално натоварване

M1 M2 M3

t

Figure No. 3

p

Максимално натоварване

t

M1 M2

Figure No. 4

4. Processing of calibration resultsThe processing of the calibration results is

performed according to the requirements of doc-ument [2].

• Mathematical modelThe evaluation of the value of the deviation P∆

of the pressure measure by the calibrated pressure gauge, the pressure measured by the reference pressure gauge and the input quantities is:

възпр.нула... PP dddd ++++−=∆ повтхистетизм PPPPP (1)

whereasP∆ - deviation of the measured pressure by the

calibrated pressure gauge from the actual pressure measured by the reference pressure gauge

измP – arithmetic mean value measured for every j-th point of the calibrated pressure gauge

етP – reference pressure from the calibration certificate for the j-th point

хистPd - evaluation of the correction of the hys-teresis of the indications of the calibrated pressure gauge for the j-th point

повтPd - evaluation of the correction for repeat-ability of the indications of the calibrated pressure gauge

нулаPd - evaluation of the correction of the de-viation from the zero point

възпрPd - evaluation of the correction for repeat-ability of the indications of the calibrated pressure gauge.

For measurement equipment with accuracy class higher than 0,1 in the mathematical model

възпрPd does not take part. •Evaluation of the input quantities, Pi > Evaluation of the input quantities, Pi, as a

result of the pressure measured by the calibrated device:

- evaluation of the pressure recorded by the calibrated pressure gauge is defined as mean value измP

Calibration sequence Type A Figure No. 2.3

Type B Figure No. 4

Type C Figure No. 5

Accuracy class of the calibrated pressure gauge ? 0.1 0.1 0.6 > 0.6

Number of measuring points with zero 9 9 5

Preliminary pressure increase to the maximum value (pieces)

3 2 1

Pressure change + time to establish the condition (seconds)

> 30 > 30 > 30

Waiting to reach the final value of measuring series (minutes)

2 2 2

Number of measuring

series, М1 to М6

ascending order 3 2 1

descending order 3 1 1

Table 1

t

p

Настройка на нулева точка

Предварително натоварванеМ1...М6 измервателна последователност

Максимално натоварване

2 минути

M1 M2 M3 M4

Figure No. 1

204

∑=

=6

1,,

1k

jkjизм PМ

P (2)

whereasjизмP , - mean value of the pressure for the j-th

point from all seriesjkP , - indications of the calibrated pressure

gauge for the j-th point for every series of meas-urements

М – a number of complete measuring series (ascending and descending sequences). They are selected from Table 1

- evaluation of the correction of the hysteresis of the indications хистPd - rectangular distributed quantity with zero value and dissipation area hj calculated according to the formula:

( ) ( ) ( ) ( ){ }0,5,50,5,60,3,30,3,40,1,10,1,2 )()(1 PPPPPPPPPPPPn

h jjjjjjj −−−+−−−+−−−= (3)

whereashj – hysteresis of the j-th point;n – a number of the measuring cycles, one cycle

consists of the ascending and descending series;Indications of the calibrated pressure gauge:P1,0 – at pressure equal to zero for the first series

of measurementP3,0 – at pressure equal to zero for the third

series of measurementP5,0 – at pressure equal to zero for the fifth series

of measurementP1,j – for the j-th point for the first series of

measurementP2,j – for the j-th point for the second series of

measurementP3,j – for the j-th point for the third series of

measurementP4,j – for the j-th point for the fourth series of

measurementP5,j – for the j-th point for the fifth series of

measurementP6,j – or the j-th point for the sixth series of

measurement- evaluation of the correction for repeatability

of the indications of the calibrated pressure gauge δPrepeatability. - rectangular distributed quantity with value zero and dissipation area .повтb , calculated according to formulae:

|)()(| 0,1,10,3,3,' PPPPb jjjвъзх −−−= (4)

|)()(| 0,2,20,4,4,' PPPPb jjjнизх −−−= (5)

{ }jнизхjвъзхповт bbb ,'

,'

. ,max= (6)

whereasjвъзхb ,

' - difference of the measured value with reset zero signal for ascending third and first series

jнизхb ,' - difference of the measured value with

reset zero signal for ascending third and first series.повтb - maximum value from the differences

between the values measured in the relevant meas-uring series corrected with a zero point according to formulae 4 and 5

- evaluation of the correction of the deviation from the zero point нулаPd is a rectangular distrib-uted quantity with zero value and dissipation area f0, formula (7).

The zero point is set up prior to the first measur-ing cycle and should be recorded before and after the measuring cycle. The deviation from the zero point is calculated according to the formula:

{ }|)(),(),(|max 0,50,60,30,40,10,2 PPPPPPfo −−−= (7)

- evaluation of the correction of the repeatability of the indications of the calibrated pressure gauge δPrepeatability for reinstallation and the same conditions is rectangular distributed quantity with value zero and dissipation area calculated accord-ing to the formula:

|)()(| 0,1,10,5,5, PPPPb jjjвъзх −−−= (8)

|)()(| 0,2,20,6,6, PPPPb jjjнизх −−−= (9)

{ }jнизхjвъзхвъзп bbb ,,. ,max= (10)

whereasjвъзхb , - difference of the measured value with

reset zero signal for ascending fifth and first seriesjнизхb , - difference of the measured value with

reset zero signal for descending sixth and second series

.възпb - maximum value from the differences between the values measured in the relevant meas-uring series corrected with a zero point according to formulae (8) and (9).

> Evaluation of the input quantities, Pi, as a result of the pressure measured by the reference pressure gauge

The real pressure value measured by the ref-erence pressure gauge Pет,,j, [Ра], in every cali-brated point is calculated according to the formula (11), in compliance with [3]:

( )( )[ ] HgtA

mmgP

флp

q

ss

m

в

jет ∆+−++

+−=

∑= ..

.20.1.

)()1.(

.

00

, ρβα

ρρ

(11)

205

whereasg – gravity acceleration at the place of oper-

ationρв – conventional air density, kg/m3

ρm– conventional density of the material of the weights, kg/m3

m0 – mass of the measuring piston and и hoist-ing disk, kg

ms – mass of the weights on the hoisting disk, kg

q – number of weightsАp – piston effective cross section, m2

α – temperature factor of linear expansion of measuring piston, ºС-1

β – temperature factor of linear expansion of measuring cylinder, ºС-1

t – ambient temperature during the measure-ment, °С

ρфл – density of operating fluid, kg/m3

ΔH – difference between the levels where the nozzle of the calibrated device is located and the basis of the measuring piston, m

- evaluation of the gravity acceleration g, - value of gravity acceleration at the location of calibration,rectangular distributed quantity with dissipitation area 2ag, equal to 2.10-4 m/s2 ac-cording to [3]

- evaluation of the air density,rectangular dis-tributed quantity with dissipitation area 2aρв, equal to 0,012 kg/m3

- evaluation of the material of weights,rectan-gular distributed quantity with dissipitation area 2aρm, equal to 50 kg/m3

- evaluation of the mass of weights, mi, -from the certificate for calibration of the weights, normal distributed quantity

- evaluation of effective cross section of the piston, AP, -from the certificate for calibration of the reference pressure gauge, normal distributed quantity

- evaluation of the temperature factors of linear expansion of the system cylinder - piston α and β - from the documentation of the piston pressure gauge, rectangular distributed quantities with dis-sipitation area 2аα,β , defined as 5% of the value of α and β. In the event of missing information about these factors, the values of the temperature factor for linear expansion of steel α+β = 1,134.10-4 °С-1

are used according to [3]- evaluation of the ambient temperature t meas-

ure during calibration, rectangular distributed value with area of dissipitation of the possible values 2at,

equal to 0,1 °С.The second member of the sum of the formula

(11) reflects the impact of the calibration method on the calibration result:

- evaluation of the density of the operating fluid ρфл - rectangular distributed quantity- reference pressure gauges used for calibration according to the current methods use two operating fluids with characteristics given in Table 2.

Table 2

Operating fluid

Density , ρфл , kg/m³

Dissipation area, 2α ρфл , kg/m³

Oil 874 5 Air 1.12 0.012

- evaluation of the difference in the levels of the

nozzle of the calibrated device and the reference level of the piston ΔH - the measuring diagram for the calibration ensures that the during the calibra-tion ΔH is equal to zero, rectangular distributed quantity with zero value and dissipitation area 2aΔH, equal to 5.10-3 m.

• Root square uncertainty of the input quanti-ties, u(xi)

> Root square uncertainty of the input evalua-tions of the quantities characterized by rectangular distribution are calculated according the formulae:

- for the evaluation of the measured pressure измP , the end resolution of the calibrated pressure

gauge r is defined and it is rectangular distributed quantity with zero value and dissipitation area 2r, equal to the least meaning digit of the relevant pressure gauge:

( )3

rru = (12)

- of the correction of the hysteresis of the in-dications:

3)( hPu хист =d (13)

- of the correction of the recurrence of the indications:

3)( .

.повт

повтb

Pu =d (14)

- of the correction of the deviation from the zero point of the indications:

3)( 0f

Pu нула =d (15)

206

- of the evaluation the correction of the repeat-ability of the indications of the calibrated pressure gauge for reinstallation under the same conditions :

3)( .

.възп

възпb

Pu =d (16)

- of the evaluation of the gravity acceleration at the location of calibration:

3)( gа

gu = (17)

- of the evaluation of the air density:

3)( в

аu в

ρρ = (18)

- of the evaluation of the density of the material of weights:

3)( m

аu m

ρρ = (19)

- of the evaluation of the temperature linear expansion of the system cylinder - piston:

3)( βαβα +=+

аu (20)

- of the evaluation of the ambient temperature:

3)( tаtu = (21)

- of the evaluation of the density of the oper-ating fluid

3)( фл

аu фл

ρρ = (22)

- of the evaluation of the difference in the level of the nozzle of the calibrate device and the basis of the measuring piston

3H)( ∆=∆ H

аu (23)

> Root square uncertainty of the affecting quantities Xi with normal distribution is defined according to the formulae:

- of the evaluation of the weighing of

kU

тu sms

,)(Σ

= (24)

whereasUm,s -expanded uncertainty of the masses of

the weights from the calibration certificate;k - coverage factor, from the calibration certif-

icate; - evaluation of effective cross section of the

piston;

k

UAu pA

p =)( (25)

whereas

pAU - expanded uncertainty of the effective cross section of the piston from the calibration certificate.

• Sensitivity factors> Functional dependence between the measured

pressure by the calibrated measuring equipment and impact of the input quantities ( )ixfP = is linear and the sensitivity factors 1=ic .

The input quantities Xi, which take part into the pressure measurement by the calibrated device

.измP are:- resolution;- hysteresis; - recurrence; - deviation from zero point;- repeatability.> For input quantities Xi, which take part into

the measurement of pressure, Pет, by the reference pressure gauge , the formula (11), the sensitivity factors ci are defined as private derivatives:

ii x

fc∂∂

= (26)

whereasdf - private derivative of the dependence

Pет = f (Xi);dxi - independent variable (input quantity Xi),

used for differentiation , i.e. :- mass of the weights, kg:

[ ])20)((1

)1(

−++

−−=

tA

gc

p

m

в

m βαρρ

(27)

- gravity acceleration at the location of calibra-tion, m/s2:

207

[ ])20)((1

)()1(1

0

−++

+−−=

∑=

tA

mmc

p

q

ss

m

в

g βαρρ

(28)

- conventional air density, kg/m3

[ ])20)((1

)(1

0

−++

+−=

∑=

tA

mmgc

pm

q

ss

в βαρρ (29)

- ρm– conventional density of the material of the weights, kg/m3

[ ])20)((1

)(.

21

0

−++

+−=

∑=

tA

mmgc

pm

q

ssв

m βαρ

ρρ (30)

- Аp – piston effective cross section, m2;

[ ])20)((1

)().1(

21

0

−++

+−−=

∑=

tA

mmgc

p

q

ss

m

в

Ap βαρρ

(31)

- α+β – temperature factor of linear expansion of measuring system cylinder -piston, ºС-1:

[ ]21

0

)20)((1

)().20)(1(

−++

+−−−=

∑=

+ tA

mmgtc

p

q

ss

m

в

βαρρ

βα (32)

- t – ambient temperature during the calibration, °С;

[ ]21

0

)20)((1

)().)(1(

−++

++−−=

∑=

tA

mmgc

p

q

ss

m

в

t βα

βαρρ

(33)

- ρфл– density of the operating fluid, kg/m3 (as a result of calibration method):

Hgcфл

∆= .ρ (34)

- g – gravity acceleration at the location of operation, m/s2 (as a result of calibration method):

Hc флg ∆= .ρ (35)

- ΔH – difference between the levels where the nozzle of the calibrated device is located and the

basis of the measuring piston, m, of the reference pressure gauge (as a result of the calibration meth-od):

gc флH .ρ=∆ (36)

• Contributions to the measurement uncertain-ties, ui(P)

Contributions of the input quantities u(Pi) for the uncertainty of the measured pressure are cal-culated as:

( ) ( )iii PucPu .= (37)

• Combined root square uncertainty, uc(p) When all input quantities are non-correlated

or correlation between them is negligibly low, the combined root square uncertainty of the measured pressure is calculated according to the formula:

222)( мететизмc uuupu ++= (38)

whereas- uизм. uncertainty with which the calibrated

pressure gauge measures the pressure value; - ueт is uncertainty with which the reference

pressure gauge measures the actual value of the pressure Pет under specific working conditions ;

- uмет. is a contribution to the combined root square uncertainty from the calibration method described by the second addend to the right place of the formula (11).

• Expanded uncertainty, U It is defined by the formula:

( )pukU c.= (39)

whereask - coverage factor, defined according to the se-

lected confidential interval, for confidential interval of 95 % - k is accepted to be equal to 2.

5. Reference[1] German Calibration Service DKD. Calibra-

tion of pressure measuring devices DKD-R 6-1[2] BDS ENV 13005 Guide to the expression

of uncertainty in measurement[3] Radev, R., Metrology of Pressure, issued

Standardization, Sofia, 1986

208

Information about the authorTatyana P. Petkova, Engineer, Master's Degree

in Technology of Machine Building and Metal Cutting Machines, 1989, Metrology Assurance Department, Kozloduy NPP plc

e-mail: tpetkova@npp.bg

Ivan P. Ivanov, Engineer, Master's Degree in Automation of Production, 1993, Metrology As-surance Department, Kozloduy NPP plc

e-mail: ipivanov1@npp.bg

Lyoboslav Hr. Hristov, Engineer, Master's Degree in Computer Technologies in Machine Engineering, 2015, Metrology Assurance Depart-ment, Kozloduy NPP plc

e-mail: LHHristov@npp.bg