CALIBRATION OF FLOWMETERS B. V. Biryukov, M. A. Danilov, and S. S. Kivilis

UDC 681.121.089.6

Calibration is a most important flowmeter test and it is carried out for all types of testing* and at all the flowmeter development and production stages as well as in the course of their utilization [i]. It can be carried out as a separate test or as an intermediate checking of the flowmeter-characteristic variations included in reliability testing (AllUnion State Standard (GOST) 16504-74). In the first case the calibration results are used for evaluating the most important characteristics of flowmeters: the static conversion characteristic Q(X) (where Q is the flow and X is the information parameter of ~he flowmeter output signal); the error ~(A); the flow measurement ranges Qmin...Qmax; and the output-signal information parameters XQmin.-. XQma x. Moreover, during calibration as a separate test it is necessary to form an aggregate (or range) of initial influencing-quantity values ZII, ZI2 , ...., Zli , ..., Zlm (i.e., to establish the initial utilization conditions). This aggregate is a most important characteristic for evaluating the influence functions (see GOST 81009-72 "State System for Ensuring Uniform Measurements (GSI). Normalized metrological characteristics of measuring equipment"). Calibration results obtained at the rough project stages are used for forming an idea about the degree to which the flowmeter characteristics correspond to the technical tasks and for determining -- at the technical project stages during the development of working documentation --~the specification requirements, and -- finally -- in the course of production and utilization-- for inspecting the flowmeters and rejecting them if necessary. In the second case (in testing for reliability) the calibration results (several of them) represent the initial data for evaluating the reliability indexes (including metrological reliability). Calibration is expected to provide trustworthy and comparable results, and to be efficient, since it is a basic, often-repeated test. It is possible to meet these incompatible requirements by standardizing calibration as an experimental process and automating the collection and processing of the obtained measuring information. An instrument suitable for this purpose consists of the universal calibration model shown in Fig. i, which comprises sequential operations 01, 0~, 03, 0~, 05, and 06 entailing the conversion of the initial information 11, ..., Is into the final output information FI, ..., Fs by means of analytical or experimental tests T,, ..., T6. The output information serves as the input one for subsequent operations or as material for arriving at conclusions about the flowmeter properties (a posteriori information P). The operation 0,, which entails the forming of an a priori model for the calibrated flowmeter, comprises evaluating, during calibration, the flowmeter characteristics and establishing their expected measurement boundaries. This sets the nomenclature of the output information FI; the range of (Qmin"'Qmax)init cl; the type of Q(X)appx; the expected variations boundaries (XQmin)ex and (XQmax)ex; the expected total error ~(A)ex and its systematic ~(As)ex m and random ~(~lex components; (Zli)i=1 ; the type and properties of the simulating medium; and the hydraulic resistance P(Q) of the primary flowmeter transducer. Here the subscripts "initcl" denote the initial values set during calibration; "appx" --approximating; and "ex" --expected. The initial information 11 comprises requirements for technical tasks or specifications for flowmeter characteristics design data, and information on suitable analogs. This infor*According to GOST 16504-74 "Control and testing. Basic terms and definitions," the term "type of testing" comprises a set of tests one of which is calibration. The type of testing should strictly correspond to the development stage according to GOST 2.103-68 "Combined Design-Document System (ESKD). Development stages." Translated from Izmeritel'naya Tekhnika, No. 12, pp. 41-45, December, 1980.

1114

0543-1972/80/2312-1114507.50

9 1981 Plenum Publishing Corporation

~

List of influencing quantities | Meansfor ' I reproducing flows

l

~repr

l

IFlow-mea- ~ suring ? e a n s ~

Fig. 1

Testing flowmeter equipment ~ ITested flowt meter ~ ~X

vleans for repro.,I meas. influenc- I ing quantifies ]

Fig. 2

mation is included in the mass data A, but, as it will be shown later, it is selected and systematized in the course of the test TI, which is the central item of the operation. The test TI comprises the automated readout of the mass data A for collecting initial information and, forming alternatives (systematizing the information Iz and adopting corresponding decisions). Effective readout as an element of the test Tx is facilitated by preliminary systematization of the mass data A, whose every element should include the designation of the characteristic, its quantitative determination, address of its supplement containing detailed information on its characteristics and the key term for finding the supplement. As alternatives it is possible to take, for instance, various types of approximating equations Q = bx, Q = a + bx, or Q = a + bx + cx 2, where a, b, and c are constants. Arriving at a decision consists in selecting an alternative on the basis of the programmed criterion, by computing the expected characteristic by means of the programmed algorithm (e.g., determining the relationship between the random and systematic error components). The test Tx program contains means for qualimetric forecasting, which is reduced to comparing requirements with the actual characteristics of analogs. The initial information I= of the operation 02 consisted of the data F~ (simulating media, initial measurement range, expected systematic error) which was used for selecting the basic testing equipment element, the flowmeter installation [2]; as well as the type and variation boundaries of the output-signal information parameter (for selecting the recorder); list and measurement range of the influencing-quantities initial values (for selecting their measuring and reproducing equipment). Moreover, the initial information I2 can also include data on the availability of measuring-system elements potentially suitable for utilization and on their characteristics. The test T2 contains the selection of optimal measuring equipment by means of criteria representing the requirements for the instruments' characteristics. The test is completed by the formation of a structural scheme for the testing equipment on the basis of the obtained output information I2 (see Fig. 2). The operation 03 is aimed at determining the optimal number n c of fixed-flow values reproduced in the calibration range and the number n r of repeated measurements for each calibration. These operations provide the output information F3. The initial information I3 is made up of the tolerated flowmeter random and systematic error-components estimates (qo and qA, respectively) which are found in the course of T3 testing on the basis of the expected values of 8(As) and 8(~); of the expected total error 8(A); of the approximating equation; and of the measurement range. In other words the information 13 is formed from the data Ft. The test Ts comprises the computation algorithms qo and qA or the interrogation program of the mass data A if the latter contains recommendations on the calibration volume and the algorithm for taking decisions in the form of design formulas or criteria on the selection of alternatives from the recon=nended solutions (selected beforehand from the mass data A)~ The operation 03 requires certain explanations. As the result of implementing 03 the optimal volume of calibration should be selected, i.e., the minimum n~ = ncn r of measurements which would ensure trustworthy results at a given precision level, since a striving to raise

1115

TABLE i Flowmeter expectedt~176 imit

below

V o l u m e of n c - nr calibra-

J

Measure- tions for an a p p r o x i m a t i n g ment-range e q u a t i o n of t y p e type

0,5

f r o m 0.5 to 1.5

q=bx

Q=a+bx Q=a+bx+ 4c~[ ..~cx a "

~

.~'~5

5X5

8X5

3X5

3X5

--

W N

5x5 3x5

5x5 5><5

8• -

L

L

3X5

5X5

3•

f r o m 1.5

W

sxs

5x5

t o 3"

N

3x5 3x5

5•

W

3X5

5X5

above

3

N h

3x3 --

3x5 --

5X5

-

-

---

trustworthiness by increasing n E leads to a deterioration of metrological reliability, more expensive calibration, and a reduction in the useful life. A precise solution of the calibration-volume optimization problem provided in [3] requires the knowledge of the flowmeter-error correlation function. Such a priori information, especially at the flowmeter development stages, is normally not available. Therefore it is necessary to give preference to the simplified computation method [4], in which the calibration-results trustworthinessocriteria are represented by qA and qo expressed in fractions of the mean-square deviation (oA). The design formulas incorporated in the test T3 program have the form Z~ . ,r> qa h e > 2q---~- ;

where Z~ is the quantile of the measurement-results distribution which depends on the adopted fiducial probability u. With the assumption of normal distribution (which is admissible in a priori computations) the value of Zu can be determined from the relationship 2e0(Za) = =, where ~o(Ze) is the Laplace function. The values of qA and q~ are set in the mean-square value range of 0.1-0.3. As an alternative to the above design method it is possible to propose the method for selecting the calibration volume from the reco~endations contained in the mass data A, for instance, given in the form of the Table i compiled on the basis of the generalized experience gained in calibrating flowmeters of different types. Notation W in Table i stands for wide flowmeter measurement range (Qmin:Qmax less than i:i0), N for narrow range (Qmin:Qmax from i:I0 to 1:5), and L for local range (Qmin:Qmax >

1:5). The initial information I4 of the 04 operation includes the produced testing system (see F2) and calibration volume (see F,), as well as the specially developed program which specifies the procedure for changing from one flow to another and the sequence in forming groups with the n r volume. Test T~ comprises sequential reproduction of the fixed flows Qmin, Q2, ..., Qi, ..., Qmax whose numerical values contain the information F3. Flows are determined by means of indirect methods entailing the results obtained in measuring the amount of liquid flowing through the flowmeters and the duration of the flow (mean flow duration) T. Each flow Qij obtained in this manner (j is the ordinal measurement number in the i-th group; j = i, ...,

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TABLE 2 Normalized characteristic

Normalized characteristic parameter

Estimate of parameter and manner of presenting in documentation

Conversion characteristic

Approximating equation of the type Q = bx, Q = a + bx, or Q = a + bx + cx 2

Equation in terms of symbols; numerical values of parameters a, b, and c with an indication of units

Measurement range* Qmin..'Qmax

Upper Qmax and lower Qmin measurement-range boundaries

Numerical arithmetic-mean values of flows in groups Qmin and Qmax

Output-signal information parameter X

Physical quantity; lower Xmi n and upper Xma x measurement boundaries

Name of physical quantity; numerical values of Xmi n and Xmax, determined in solving the approximating equation with respect to x in the case of Q = Qmin and Q = Qmax, respectively

Total error characteristic ~(A)

Relative value limit of

Symmetrical% interval represented by the equation 6(A) -- ~A(Q) or by a table of discrete values of 6(A)I, ~(A)2, 9 .., ~ (A)i , .... ~ (A)n c and evaluated from (6)

Systematic error characteristic ~(As)

Relative value limit of ~(A s)

Equation of the interval boundary # ~(As) = ~As(Q) or table of ~(As)1,...,

~(As)nc Random-e~ror characteristic d(A)

Relative mean-square errors of deviations ~(~) and mathematical-expectation of fluctuations

Equation of t h e interval boundary # (Ao) = ~oA(Q) or table of ~(~)i, ...,

~(A)nc

~(~)

*In the All-Union Standard (GOST) 8.009-72 is not normalized. %~(A), ~(As), and~(~) are interpreted as • • s), and •

o

n r) corresponds to an information-parameter value of the flowmeter output signal measured for the same duration Tij: !

xiJ

ri#

0

The obtained set of paired values of Qij and xij constitutes the output information F~. The initial information Is of operation 05 comprises the already-mentioned combination of Qij and xij. The test T5 is carried out in three phases. The first phase consists in determining the approximating equations and it begins with the primary processing of blunders, which is carried out roughly and experimentally, with the number of rejected points being very limited. At the same time it is possible to determine in a preliminary manner whether the selected type of approximation equations corresponds to the experimental data. This operation is also carried out experimentally (visually, according to the location of points) and it serves to eliminate the very gross errors, i.e., it registers only complete discrepancy between the selected type of approximation equation and experimental data. The computation proper of the approximation equation parameters allows for an alternative. In practice three competing computation methods are used, namely, the graphicoanalytical, the mean-values, and the least-squares methods9 It is advisable to use the first two [5, 6] for "rough" flowmeters (~(A)ex > 3%) whose static characteristic is known to be suitable for approximation with a linear equation of the type Q = bx or Q = a + bx. The third method is the most precise and universal. It consists essentially in determining the approximate-equation parameters which would provide for a minimum sum of the squares of the deviations of the experimental points Qij from the approximating curve Q(X)appx, i.e., for

1117

nC

nr

~.d~ [Q,i--Q(X)xl:,]~ =

min.

i='

i=l

where Q(X)xijis the ordinate of the approximation-curve point with the abscissa xij. This method has been well studied and, therefore, in developing the T5 program it is possible to use materials referring to the method as a whole (e.g., [7, 8]) or materials containing methods specific to certain flowmeters (e.g., [9, i0] which provide algorithms for computing parameters of equations of all the three recommended types). The second Ts phase consists in static testing of the hypotheses about the existence of rough deviations of experimental points from the approximating curve, about the correct selection of the approximating equation, and about the normal distribution of the experimental points. The hypotheses can be checked by means of various existing agreement criteria, whose application has been described, e.g., in [6, 7] and their numerical values provided in [ii13]. Without describing in detail the technique for testing the hypotheses, let us provide general recommendations on utilizing the agreement criteria. For checking the hypothesis on the presence of blunders it is possible to use the criteria of maximum deviations (criterion 3~), or of Romanovskii. The hypothesis on correct selection of the approximating equations can be evaluated by establishing the values of the group mathematical-expectation deviations from the mathematical expection of the entireaggregate of experimental points. Agreement criteria can consist of the Fisher or Student criteria, or else of the nonparametric Abbe criterion which is based on estimating the critical length of a series and establishes the presence of a systematic shift in observations. For the purpose of estimating normal distributions it is possible to use the Pearson asymmetry and excess criterion or (in the case of large samplings) the Kolmogorov criterion. The third T5 phase consists of estimating the flowmeter errors and it is of the greatest interest for the subsequent qualimetry. The total error includes its systematic and random components which are added algebraically:

8 (~) = 8 (As) + 8 (~). The component ~(~s) consists in the instrumental error of the reproduction and measuring equipment, i.e., of the flowmeter installation error ~fmi and the measured-information recorder error drec" The moduli of both errors are known [i0], since the source of information on their numerical values consists in the measuring equipment rated values. The random component also consists of the sum of two independent quantities: the deviation of the experimental points from their approximating curve and their mathematical expectation fluctuations, i.e., those of the approximation curve proper. The first one has a normal distribution with the mean-square deviation of

l

o (~) = -~-

//

nc

nr

~=I j=l

"z-/

nc

where

nz

----~ nri

is the total number of the experimental points (nri is the number of the

i-th group points after the sorting of blunders; f is the number of parameters of the approximating equation. The mathematical expectation of fluctuations is estimated by means of the empirically obtained mean-square deviation which can be calculated, depending on the type of the approximating equation, from one of the following formulas [14]: l

1118

(x~ + nzx" - - 2x

[xl}

(i)

or O.~

where

[x"z] =

ne

nr

~

~ ;

~l

.f=l

_

_

(~)

= c

1

(/~) -7-

(2)

[All -~- A22x2 J- A$$@' .-~- 9 (--AlsX + AIzx 2 - - A23x3)],

A

for m = i, 2, 3, 4; x is the current value of the flowmeter output-

signal information parameter; A is the equation determinant; A::, A2=, As3, A1a, A:s, and A23 are the second-order minors. The determinant A has the form

A_-

1.I ,..I1

b ~-] Ix~i. [x2] [x3] [x41

The error is estimated for approximations of the type of Q = bx and Q = a + bx and from (2) for Q = a + bx + cx 2. The above estimate is distributed according to the Student law. The values of ~(~) are estimated from the formula

5

--

to I / < 9~.~-Io: d) +

o (A),

assuming that their distribution in normal. Here

Z~.,z_1

is the dispersion distribution coefficient; t~ is the quantile of the nor-

mal distribution for the fiducial probability a. The last calibration operation O6 consists in normalizing the obtained experimental characteristics and presenting them in utilization (specifications, certified records) or in a normative-technical documentation (standards for general requirements and general specifications). Normalization is carried out in accordance with the All-Union State Standard (GOST) 8.009-72, but with simplifications due to the specific features of the tested objects. The recommended manner for presenting the characteristics* is shown in Table 2whose data constitute the output information F6. Information F6 is used as a base for forming the mass data of a posteriori information P. Moreover, P incorporates a list of influencing quantities ZI, ..., Zm (from FI), their initial values Zi, , ..., Zim , which are finally established in the experimental cycle 06, the noninformative output-signal characteristics A x optionally established in the course of the 06 process, and the hydraulic resistance $ of the flowmeter primary transducer. These quantities can also be normalized. LITERATURE CITED i. 2. 3. 4. 5. 6. 7.

B . V . Biryukov et al., Izmer. Tekh., No. i0 (1979). B . V . Biryukov et al., Flowmeter Testing Installations [in Russian], ~nergiya, Moscow (1976). A . S . Nemirovskii, Probability Methods in Measurement Techniques [in Russian], Standartoy, Moscow (1964). A . N . Kartashova, Trustworthiness of Measurements and Criteria for the Quality of Instrument Testing [in Russian], Standartov, Moscow (1967). D . N . Khorafas, Systems and Simulation [in Russian], Mir, Moscow (1967). I . V . Dunin-Barkovskii and N. V. Smirnov, Probability Theory and Mathematical Statistics in Technology [in Russian], GITTL, Moscow (1955). A. Hald, Mathematical Statistics with Technical Applications [Russian translation], IL, Moscow (1956).

*Only for characteristics evaluated from calibration results obtained in separate testing and not in the course of reliability testing. 1119

8. 9. i0. ii. 12. 13. 14.

Yu. V. Linnik, Least-Squares Method and Foundations of Observations Processing Theory [in Russian], Fizmatgiz, Moscow (1962)o L. L. Boshnyak and L. N. Byzov, Tachometric Flowmeters [in Russian], Mashinostroenie, Leningrad (1968). B. V. Biryukov et al., Precise Liquid Flow Measurements [in Russian], Mashinostroenie (1977). L. N. Bol'shov and N. V. Smirhov, Mathematical-Statistic Tables [in Russian], Nauka, Moscow (1965). D. B. Owen, Collection of Statistical Tables [Russian translation], VTs AN SSSR, Moscow (1966). G. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill (1967). E. F. Dolinskii, Measurement Results Processing by Means of the Least-Squares Method [in Russian], Standartov, Moscow (1971).

TURBINE TRANSDUCERS FOR WELL-TYPE FLOW-RATE METERS V. I. Bar-Sliva

UDC 681.121:622:53.087.92

Well-type flow-rate meters of the "Terek" type with turbine transducers are designed for hydrodynamic studies on running and injection wells. In the upper part of the housing for flow meters having a diameter of 42 mm and a length of about i000 mm there is a cable adapter with an armored cable that is used to lower the instruments into a well, and in the middle part a measuring channel with the transducer. Within the housing between the input and output openings of the measuring channel there is a bunching device that directs the flow from a w e l l h a v i n g a diameter of 125-150 mm into the measuring channel. This device i s n o t driven [i], thereby ensuring operation at high temperatures for a wide range of flow rates. The readings are obtained by means of a permanent magnet secured to the upper part of the turbine's shaft and type KEM-2A ("Terek-l") and type MUKIA-I ("Terek-3") hermetically sealed relays that are connected with the armored cable. A characteristic of well measurements is the wide range of flow rates measured and the variable viscosity of the working medium, A change in the fluid's viscosity alters the thickness of the boundary layers on the turbine's blades, thus causing a change in the fluidts flow pattern around the blades and in the area of the impeller's free cross section. A survey and classification of the existing methods for reducing the influence of viscosity on the readings from turbine flow-rate meters is presented in [2]. The impeller blades used in the "Terek-l" meter have a slotted profile, each blade being composed of a main and an auxiliary with the height of the latter equal to that of the main part but with a lesser width and positioned so that they form a slot. This design has a number of advantages: First of all, the flow through the slot repels the boundary layer on the main blade and thus reduces its thickness; secondly, the boundary layer formed on the auxiliary blade only needs to overcome part of the pressure increment on the upper surface of the profile; thirdly, owing to the high fluid velocity the boundary layer that breaks loose from the slot is very forcefully carried away by the external flow in comparison with the design having no slot. A turbine transducer having blades with a slotted profile was tested with an experimental model of a type "Ob'-l" deep flow meter designed according to a compensating circuit and using mechanical recording for the readings. In this case the transducer functions in a retarding mode so that the impelier's torque change can be followed. Experiments were performed on the hydraulic flow-rate metering installation at the All-Union Oil and Gas Scientific-Research Institute with two working fluids: water (~ = i mm=/sec) and a mixture of transformer oil with water (~ = 40 mm2/sec). The effect of the slot dimension e on the transducer's characteristic was investigated. It was shown experimentally that the value of e, within certain limits, had only a very small influence on the variation of the calibration curve. In [4] the results of comparison tests are shown for various types of impellers with and without an auxiliary blade. Translated from Izmeritel'naya Tekhnika, No. 12, pp. 45-46, December, 1980.

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0543-1972/80/2312-1120507.50

9 1981 Plenum Publishing Corporation