Calibration of hydrometers

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INSTITUTE OF PHYSICS PUBLISHING MEASUREMENT SCIENCE AND TECHNOLOGY

Meas. Sci. Technol. 17 (2006) 25602566 doi:10.1088/0957-0233/17/10/005

Calibration of hydrometersSalvatore Lorefice and Andrea MalengoINRiMIstituto Nazionale di Ricerca MetrologicaStrada delle Cacce, 91-10135 Torino,Italy

E-mail: S.Lorefice@inrim.it

Received 1 February 2006, in final form 26 May 2006Published 25 August 2006Online at stacks.iop.org/MST/17/2560

AbstractAfter a brief description of the different methods employed in periodiccalibration of hydrometers used in most cases to measure the density ofliquids in the range between 500 kg m3 and 2000 kg m3, particularemphasis is given to the multipoint procedure based on hydrostaticweighing, known as well as Cuckows method. The features of thecalibration apparatus and the procedure used at the INRiM (formerlyIMGC-CNR) density laboratory have been considered to assess all relevantcontributions involved in the calibration of different kinds of hydrometers.The uncertainty is strongly dependent on the kind of hydrometer; inparticular, the results highlight the importance of the density of the referencebuoyant liquid, the temperature of calibration and the skill of operator in thereading of the scale in the whole assessment of the uncertainty. It is alsointeresting to realize that for high-resolution hydrometers (division of0.1 kg m3), the uncertainty contribution of the density of the referenceliquid is the main source of the total uncertainty, but its importance fallsunder about 50% for hydrometers with a division of 0.5 kg m3 andbecomes somewhat negligible for hydrometers with a division of 1 kg m3,for which the reading uncertainty is the predominant part of the totaluncertainty. At present the best INRiM result is obtained with commerciallyavailable hydrometers having a scale division of 0.1 kg m3, for which therelative uncertainty is about 12 106.Keywords: density, hydrometer, calibration(Some figures in this article are in colour only in the electronic version)

1. Introduction

Formerly attributed to Archimedes, but invented by Hypatiaof Alexandria (370415 AD), hydrometers, also known asgravimeters, densimeters or areometers, are the most ancient,simple and highly effective instruments, widely used atdifferent levels of accuracy, to measure the density of liquidsfrom 500 kg m3 to 2000 kg m3.

On the basis of the working principle, hydrometers canbe classified into constant-volume hydrometers and constant-mass hydrometers. A constant-volume hydrometer is madeto float up to a specific mark in liquids of different densitiesby placing weights on it, so that its immersed volume is keptconstant. The constant-mass hydrometers, known as variabledisplacement type, float up to different positions and their massdoes not change.

Leaving apart the constant-volume kind, in this paper theconstant-mass hydrometers have been taken into account sincethey are more employed and several national and internationalorganizations have defined their specifications as well as theprocedures of use [113].

A constant-mass hydrometer consists of a hollow thinglass tube sealed with a stem containing a graduated scaleand with an enlarged lower section, the bulb, weighted at thebottom to keep it floating upright. The density of the liquid isdirectly measured by the depth of immersion of the emergingstem at the point aligned with the horizontal surface of theliquid.

Some of these hydrometers are produced with a built-inthermometer and are known as thermo-hydrometers.

Generally there is no special distinction between primarystandard and secondary standard hydrometers; in common

0957-0233/06/102560+07$30.00 2006 IOP Publishing Ltd Printed in the UK 2560

Calibration of hydrometers

use they are differentiated on the basis of (i) their minimumdistance of graduation, (ii) their range of work and (iii)maximum permissible error.

Their scale can also be graduated in units correlatedwith the liquid density and their name may change fromindustry to industry. In the petroleum and chemical industrieshydrometers determine the density or specific gravity ofliquids, in breweries they are used for assessing the strength ofalcohol (alcoholometers), in the sugar industry for measuringthe percentage of sugar present in sugar cane solutions (Brixhydrometers) and in the dairy industry for measuring thefat content in the milk (lactometers); however a commonclassification is given as

(a) density hydrometers, indicating the density of a specifiedliquid, at a specified temperature, in specified units of theSI system;

(b) specific gravity hydrometers, indicating the specificgravity of a liquid in terms of the density of pure water atthe same temperature, usually at 15.6 C;

(c) percentage hydrometers, indicating at a specifiedtemperature the percentage of salts in a sample of seawater or the percentage of sugar in an aqueous solutionand

(d) arbitrary-scale hydrometers, indicating concentration orstrength of a specified liquid, or its density, referredto an arbitrary scale at a specified temperature (Baumehydrometers and API hydrometers).The hydrometers are exposed to long immersion times

in various liquids which may attack the glass, and in usethey are subjected to mechanical stress and abrasion throughhandling and cleaning. Moreover, the natural ageing processof the glass further results in minute changes to the dimensionsof the instrument. The sum of these factors, although notvisible to the eye, may result in slight changes in the weightand/or displacement of the instrument and cause it to changeits indication.

The correctness of measurements is one of the mostimportant prerequisites for the assurance of the quality ofproducts and services, and the accuracy of the measuringinstruments must be consistent with their intended use.

The relatively low price of hydrometers does not excludethe need for their periodic regular calibration and, if requested,of their periodic verification. These two different actions,although established with separate rules and performed bydifferent metrological infrastructures, are mostly based onthe same measuring procedures, which are essential to assurethe accuracy of the readings, the traceability to internationalstandards in the areas of density measurements and to complywith the requirements of the quality system.

2. Calibration methods

When a hydrometer floats freely in a liquid of density x , incorrespondence with the scale reading r, there are downwardand upward forces acting on it: the force due to gravity andthe force due to the surface tension x of the liquid acting onthe stem of the hydrometer are equilibrated by the buoyantforce due to the liquid displaced by the volume V of thehydrometer below the liquid surface and the buoyant force of

the air displaced by the volume of the hydrometer stem abovethe liquid surface. Under these conditions the equilibriumequation is

mg + Dx = gVx + ga1, (1)where m is the mass of the hydrometer, g is the localacceleration due to gravity, D is the diameter of the stem of thehydrometer at the surface of the liquid and a1 is the densityof the air at the time the hydrometer scale was read.

The measure by hydrometers and the accuracy ofmeasured data depend on the usual conditions under whichthey are used; corrections for the temperature and the surfacetension should be taken into account. In the same way theworking temperature, the kind of liquid and its surface tension,and the correct method of reading are the main conditions thatmust also be taken into account during their calibration toobtain high accuracy.

As a rule, hydrometers graduated in density units orhaving different scale units are both calibrated against densityreadings at the reference temperature t of 20 C and for a rangeof surface tension on the basis of information they bear on it.

Usually hydrometers are calibrated at three or fourgraduation marks of the scale and for each of them thecorrection C is calculated as

C = x r, (2)where x is the density of the liquid in which the hydrometerwould freely float at the scale reading r. Consequently, thebest accuracy achieved in the calibration of hydrometers ismainly due to how both the quantities x and r are estimated.

The hydrometer reading r consists in observing thegraduation mark of the hydrometer under calibration frombelow the surface of the liquid and in aligning the middle ofit with the horizontal plane tangential to the liquid surface.To detect correctly the alignment of the hydrometer scale amagnifier or a monitor is commonly used. Recently the imageprocessing technique has been introduced to help the operatorin calibrating and reducing the scale readings contribution tothe calibration uncertainty.

The estimation of x is particularly linked with the usedcalibration method [14].

2.1. The direct comparison method

This method is the simplest way to test hydrometers using asdifferent standards liquids with known density. It is better ifthe standard physical properties such as density and surfacetension are close to those of the liquids where the hydrometerunder test is intended to float freely. The density of standardscould be measured experimentally at the testing temperature,e.g. (i) by the relation between density and temperatureobtained by hydrostatic weighing of a sinker of known massand volume, (ii) against a more accurate reference hydrometeror by suitable densimeters and (iii) finally, estimated by chartsand tables in handbooks listing detailed information aboutthe densities of solutions as a function of their composition(typically, in terms of per cent solute in the solution). Theenvironmental temperature and the liquid temperature shouldbe kept constant during the observations; otherwise if thetemperature of the liquid changes, it can not only causedifferences in density but also cast doubts about the actual

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temperature. However, the main drawback of the directcomparison method is due to the need to have severalreference liquids with appropriate density in order to coverthe whole liquid density range.

2.2. The ring method

This is based on the principle of the constant-volumehydrometers; the hydrometer under test floats in a single liquidof known density and it is immersed to the tested scale readingr by loading the top of its stem with weights [15, 16].

These weights are metallic rings of suitable material andsize, chosen so that when one is slipped on the hydrometerunder test the latter sinks down to the specified graduation.The mass of the rings required to float the hydrometer tograduation, depends upon the scale range, the volume of thebulb and the value of the density corresponding to the lowestgraduation of the hydrometer under test. The number and theweight of the rings and the feature of the buoyant liquid havelimited the application of this method.

2.3. Hydrostatic weighing

The procedure to calibrate hydrometers based on hydrostaticweighing was introduced by Cuckow [17]. Hydrometers ofany range can be calibrated at different designed graduationmarks, measuring the buoyancy force when the hydrometeris placed in air and immersed in a reference liquid. At first,the hydrometer to be calibrated is weighed in air and then itis sunk into a reference liquid, whose density is known at thereference temperature, with the stem connected to an upperbalance through a metal wire. The depth of immersion ofthe hydrometer is adjusted by a mechanical device so that themiddle of the graduation under calibration is aligned with thehorizontal surface of the reference liquid.

In those situations where the range of the hydrometer tobe calibrated is lower than the density of the reference liquid,additional weights are added to the hydrometer to cause it tosink.

This is the most accurate method and it is in use at mostNMIs (National Metrology Institutes), including the INRiM[1820]. Full details of this method are given in the nextsection.

3. Hydrometer calibration by hydrostatic weighing

The theoretical approach for hydrometer calibration byhydrostatic weighing is based on three equilibrium equationsobtained from different situations or conditions.(a) The hydrometer floats freely in the liquid of density x

at the reference temperature t = T0. Equation (1), forwhich the values x , V , are also referred to T0, and a1is the density of the air at the time the hydrometer scalewas read, describes these conditions.

(b) The hydrometer is weighed in air. There are two forcesacting on it: the downward force due to gravity and theupward buoyant force due to the air displaced by thevolume of the hydrometer. The equilibrium equation is

Mag

(1 a2

s

)= mg g(V + )[1 + (T2 T0)]a2,

(3a)

where Ma is the weight of the hydrometer in air, s isthe density of the mass standard, is the volumetricthermal expansion coefficient of the glass from whichthe hydrometer is made, T2 is the air temperature at thetime the weighing in air was done and a2 is the densityof the air at the time the weighing in air was done.

As the product of (T2 T0) is negligible, fromequation (3a) we obtain

Ma

(1 a2

s

)= m Va2[1 + (T2 T0)] a2.

(3b)(c) The hydrometer is weighed partially immersed to the same

graduation mark as in condition (a), but in a referenceliquid of density L at the temperature T3, which is notnecessarily equal to T0. There are two downward forces:the force due to gravity and the force due to the surfacetension L acting on the stem of the hydrometer. There arealso two upward forces: the buoyant force due to the liquiddisplaced by the volume V and the buoyant force due to theair displaced by the volume . The equilibrium equationis

MLg

(1 a3

s

)= mg + DL

gVL[1 + (T3 T0)] ga3[1 + (T3 T0)],(4a)

where ML is the weight of the hydrometer in the referenceliquid, a3 is the density of the air at the time of thehydrostatic weighing, g is the acceleration of gravity atthe level of the liquid bath. Since the contribution dueto the vertical gradient of gravity is negligible, so thatg = g, equation (4a) becomes

ML

(1 a3

s

)= m + DL

g

VL[1 + (T3 T0)] a3. (4b)Assuming that the changes in the air density a and in the

volume of hydrometer (V + ) are negligible and, in addition,that the terms representing the air buoyancies acting on theemergent stems of the hydrometer at various times a areneglected, the density of the liquid x in which the hydrometeris intended to float freely at the stated graduation mark resultsby combining equations (1), (3b) and (4b):

x = (L a)[Ma(1 a2

s

)+ Dxg

1][Ma(1 a2

s

) ML(1 a3s)

+ DLg1]

[1 + (T3 T0)] + a2. (5)

3.1. Uncertainty equation

The measurement uncertainty theory states that allmeasurements are estimations; the best we can hope wouldrequire that all sources of variation were identified andcontrolled so that we reduce the amount of bias (or distancefrom the true value) as much as possible [21]. Inhydrometer calibration, we determine an algebraic correctionto compensate a systematic effect that significantly influencesthe density readings in the range of the calibrated instrument.The correction, as given in equation (2), may be either positive,

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Table 1. Usual contributions of uncertainty to calibration ofhydrometers.

Contribution area Detailed component

Balance RepeatabilityResolutionLinearityCalibration (or standard weight)

Reference liquid DensitySurface tensionTemperature

Hydrometer Series/kind (rangereferencetemperaturereference surfacetensionwidth scale division)Thermal expansion coefficientDiameter

Environment Atmospheric buoyancyLocal gravity

Process and procedure Operator contributions

negative or zero, and the standard uncertainty associated withthe estimate of correction can be expressed as

uc(C) = n

i=1

(C

xi

)2u2(xi), (6)

where the input estimates xi are the input quantities inequations (2) and (5). There is correlation between somevariables, e.g. due to the use of the same instruments duringthe weighing of hydrometers, both in air and in liquid, or inenvironmental measuring devoted to determine the air density,but the effect of correlation can be assumed to be negligible.The contributors to the uncertainty of the calibration can besubdivided into the components listed in table 1.

3.2. Uncertainty budget

Reliability of calibration and the level of uncertainty isstrictly connected with the equipment and the procedure usedin the measurement; the choice should be based on obtainingthe highest accuracy in the calibration. The evaluation of theinput quantities is described as follows.

(a) Weighing of the hydrometer in air and in the liquid. Theweighings of the hydrometer in air and in the referenceliquid are usually performed by electronic balances;commercial balances with a resolution of 0.1 mg aresuitable to obtain good accuracy.

The weighing method used with the hydrometer in airor in the liquid can be the direct reading of the balance orthe more accurate substitution method. To preclude theinfluence of linear drifts several weighing cycles can becarried out with different successive weighings.

The standard uncertainty associated with the weightof the calibrated hydrometer both in air and in the liquid isobtained from contributions due to the balance calibrationand the reference standards calibration, but also fromthe repeatability, linearity, hysteresis, drifting of themeasurements. Moreover, in agreement with equation (5)the standard uncertainty associated with the density of thestandard weights should be considered.

(b) Atmospheric buoyancy. The measurement of air densityis necessary to allow buoyancy corrections to be madewhen comparing weights of different volumes, of differentmaterials or when making mass measurements to thehighest accuracy. In general, the air density is calculatedby means of environmental measurement of temperature,pressure and humidity, determined during the weighings,using the equation recommended by the CIPM (ComiteInternational des Poids et Mesures) [22]. If the carbondioxide concentration is not measured, the value of 0.04%with an uncertainty of 0.02% (rectangular probabilitydistribution) can be adopted.

The uncertainty contribution of the density of airdepends on the accuracy of the measurements, themeasuring equipment and the CIPM equation.

(c) Temperature influence. Thermal conditions play animportant role during the whole procedure; liquiddensity changes and density gradients due to non-uniformtemperature might have a significant effect on hydrometermeasurement results even in a temperature-controlledset-up. Unfortunately, significant temperature gradientscannot always be avoided for practical reasons, e.g.insufficient uniformity and stability of the referenceliquid and differences between the liquid and ambienttemperature [23]. However, the temperature effect shouldbe expected in changing the apparent weight of thehydrometer and the density of the reference liquid.

(d) The reference liquid. The density of the buoyantliquid depends mainly on temperature. Usually thevessel containing the reference liquid is immersed ina thermostatic bath; very important is the stability andthe uniformity of the temperature inside the bath duringthe measurements. Many laboratories check the densityof the reference liquid before and after the calibrationof the hydrometer. The uncertainty contribution of thereference liquid mainly depends on the method used formeasuring its density; other minor contributions includethe evaporation effect and the compressibility which areusually taken to be negligible.

The buoyant liquid should have the lowest surfacetension in order to have a small and reproducible shapeof the meniscus, and to be relatively immune to self-contamination. The surface tension of the referenceliquids can be known by different ways, usually it ismeasured by applying the ring or the plate method,or, alternatively, by literature reference databases. Themost common liquids used as reference liquids whichbetter meet such requirements are organic hydrocarbons,belonging to the alkane family (n-nonane, n-dodecane,. . . , n-tetradecane). However, the relative standarduncertainty associated with the surface tension is currentlylower than 0.02.

(e) Stem diameter. An accurate caliper or a suitableinstrument with a resolution between 0.01 and 0.1 mmis usually used to measure the diameter of the stem of thehydrometer to be calibrated. The uncertainty contributiontakes into account the uncertainty of the caliper and theexperimental standard deviation of the measurements ofthe diameter at the scale mark.

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S Lorefice and A Malengo

Figure 1. Alcoholometer/hydrometer calibration facility at theINRiM.

(f) Expansion coefficient of hydrometers. Hydrometers areusually made of a glass material with a nominal cubiccoefficient of thermal expansion of 25106 C1 withan uncertainty of 2106 C1 (rectangular distribution).

(g) Gravity influence. The calibrated hydrometers areweighted in air and in the liquid at different heights. It isnot normally necessary to make corrections for variationsin gravitational acceleration.

(h) Operator effects and readings error. Human errors caninclude inaccurate readings or misalignments of the scaleson the stem of hydrometer; they can form a significantcomponent of the uncertainty budget. The uncertaintydepends on the sharpness of the scale, the accuracy andreliability of the alignment and also on the operator skill:fatigue level or concentration. Moreover, it is necessary totake into account the vision method (magnifiers, cameras,etc) and the way used to adjust the vertical position ofthe hydrometer (or of the bath) so that the designed markintersects the liquid surface correctly. In the case thatthe automatic alignment method is adopted, the maincomponents in the readings uncertainty come from thevision system (repeatability in the correct alignment) andfrom the perspective error.

(A) (B)

Figure 2. An ideal vertical segment was drawn through the centre of the hydrometer stem along the image of the immersed part of thehydrometer under test. The evaluation of the image profile along it by grey-intensity values is made. The symmetrical point between theminimum value of the plunged scale mark A and its virtual image (reflected scale mark B) is the scale mark to be calibrated [26].

4. The hydrometer calibration facility at the INRiM

The Istituto Nazionale di Ricerca MetrologicaINRiM,formerly Istituto di Metrologia G Colonnetti (IMGC-CNR),provides measurement services for reference hydrometersgenerally used as laboratory standards.

At present the elements of the calibration station (figure 1)are based around a lifting apparatus driven by a computer-controlled servo motor and a weighing system used tomeasure the buoyancy difference of the calibrated hydrometer.Specifically, the lifting apparatus bears and moves up anddown a glass vessel containing the reference liquid (n-nonane),surrounded by 30 l of temperature-controlled circulating water.An external thermostat is used to provide the temperaturecontrol and circulation of water at 20 C, although differenttemperatures between 10 C and 50 C could be allowed. Anelectronic balance for hydrostatic weighing with a capacity of405 g is used to weigh hydrometers both in air and in n-nonane.The environmental equipment consisting of a portable pressuregauge and a thermo-hygrometer is also included as part of thestation. Moreover, a commercially available tensiometer isused for measuring the surface tension of the reference liquid.

The measurements of temperature during the weighing ofhydrometers, both in air and in liquid, are performed by twocalibrated platinum-resistance thermometers (Pt100) linked toan ac bridge. The mean diameter of the hydrometer stem isdetermined by an accurate caliper measuring it at four differentaxes, approximately at each stated mark. The calibrationapparatus also adopts an automatic alignment system [24]which is composed of a CCD camera and a frame grabber, andit is used to acquire images by home-made image processingsoftware. The operating program has the following functions:(i) to process images from the camera by the analysis of theimage profile (figure 2), (ii) to find and adjust by the movementof the glass vessel the position of the horizontal plane and(iii) to control the stepping motor for the alignment of thehorizontal plane with the particular scale mark. This automaticalignment method is a convenient and precise means ofcalibrating a hydrometer, since it allows us both to automatizethe calibration process and to remove the operators mistakesin detecting correctly the alignment of the hydrometer mark tobe calibrated.

The current procedure of weighing is based upon fiveindependent weighing-in-air sequences and five independent

2564

Calibration of hydrometers

Figure 3. Evaluation of the relative individual uncertainty contributions to the combined standard uncertainty in calibrating hydrometerswith different scale division values: from 0.1 kg m3 to 1 kg m3 and a mark distance of about 1 mm in the density range close to1000 kg m3.

Figure 4. Evaluation of the relative individual uncertainty contributions to the combined standard uncertainty in calibrating hydrometerswith a scale division value of 0.5 kg m3 and a mark distance of about 1 mm in the density ranges of 600650 kg m3, 800850 kg m3 and10001050 kg m3.

hydrostatic weighings at each of the stated levels of immersion.The density of the reference liquid is determined as a functionof temperature over the range of 18 C to 22 C by hydrostaticweighing of a zerodur sphere of known mass and volume. Inorder to check the stability of the density over time and possiblecontamination, the density of the liquid is measured before andafter the calibration of each hydrometer by a vibrating tubedensimeter.

An additional weight (ballast) of stainless steel is mountedon the stem of the hydrometer to cause it to sink if its range islower than 800 kg m3.

Table 2 gives a summary of the main contributions tothe uncertainty due to the equipment used at the calibrationhydrometer station of the INRiM laboratory.

4.1. The combined standard uncertainty obtainedat the INRiM

The standard uncertainty of the correction C is obtained bycombining the standard uncertainties of the input quantitiesu(xi) as shown in table 2 by using equation (6).

In obtaining the combined standard uncertainty, differentresults are expected; they mainly depend on the value of thescale division, the distance between marks and the range of thehydrometer to be calibrated. Generally, for each hydrometer

Table 2. Typical individual contribution of the influence quantitiesin agreement with equation (6).

Standarduncertainty

Influence quantities xi u(xi) Unit

Weighing in air, Ma 0.10 mgWeighing in the liquid, ML 0.15 mgAir density, a 0.003 kg m3Temperature of liquid around the 0.01 Chydrometer, TDensity of the buoyant liquid, L 0.005 kg m3Diameter of the hydrometer stem, D 0.1 mmSurface tension of the buoyant liquid, L 0.2 mN m1Readings error, r 0.02 mmDensity of standard weights, Std.W 30 kg m3Gravitational acceleration, g 1 105 m s2

kind, the distance between marks is approximately constantover the whole range so that the reading uncertainty u(r)increases with the value of the scale division.

Figure 3 shows the individual components of the relativeuncertainty as a function of the combined standard uncertainty(on the right) for hydrometers having different scale divisionvalues from 0.1 kg m3 to 1 kg m3 and a mark distance ofabout 1 mm as well as it has been achieved at the INRiM.The histogram highlights the importance that the density

2565

S Lorefice and A Malengo

of the reference liquid, the temperature of calibration andthe readings assume in the whole assessment of uncertainty.The temperature strongly affects the uncertainty through thethermal expansion coefficient of the reference liquid, so thatonly the density uncertainty and the reading uncertainty havethe greatest importance. It is also interesting to realize thatfor high-resolution hydrometers (division of 0.1 kg m3) thedensity uncertainty contribution of the liquid is the main sourceleading to total uncertainty, but its importance falls belowabout 50% for hydrometers with a division of 0.5 kg m3and becomes somewhat negligible for hydrometers with adivision of 1 kg m3, for which the reading uncertainty isthe predominant part of the whole uncertainty.

Considering hydrometers having the same scale divisionvalue of 0.5 kg m3, the mark distance of about 1 mm andthe density ranges of 600650 kg m3, 800850 kg m3 and10001050 kg m3 respectively, the result is shown in figure 4.The histogram shows that the relative uncertainties due to theliquid density, the weighing in the liquid and the temperatureincrease with the density range. About the relative readinguncertainty a decrease in the range is shown. On the basis ofthis result the combined relative uncertainty is nearly constantover the whole density range. Such considerations are alsoapplicable to hydrometers having different scale divisions.

5. Conclusion

Regular calibration of hydrometers is the most importantprerequisite for exact and traceable measurement of density (orits correlated quantities). Hydrostatic weighing is the commoncalibration method accepted by most primary laboratories forcalibrating hydrometers in the range between 500 kg m3 and2000 kg m3 with uncertainties as low as half the minimumscale division.

The assessment of the calibration uncertainty is mainlyaffected by the individual contributions of the density of thereference liquid, the temperature of calibration and the readingof the meniscus by the operator or an automatic system. Thedensity uncertainty of the reference liquid is the most importantterm for high-resolution hydrometers (division of 0.1 kg m3),but its importance falls below about 50% for hydrometers witha division of 0.5 kg m3 and becomes somewhat negligiblefor hydrometers with a division of 1 kg m3, for which thereading uncertainty is the predominant contribution of the totaluncertainty.

Considering hydrometers with the same scale divisionand different density ranges, the relative uncertainty resultsare nearly constant over the whole range. The best INRiMresult is obtained with commercially available hydrometershaving a scale division of 0.1 kg m3, for which the relativeuncertainty is about 12 106.

References

[1] ISO 649-1 1981 Laboratory glasswaredensity hydrometersfor general purposesspecification

ISO 649-2 1981 Laboratory glasswaredensity hydrometersfor general purposestest methods and use

[2] BS 718 1991 Specification for density hydrometers[3] NF B 35-511 1983 General use density hydrometers[4] DIN 12791-1 1981 Laboratory glassware; density

hydrometers; general requirementsDIN 12791-2 1978 Laboratory glassware; density

hydrometers, standard sizes, designationsDIN 12791-3 1983 Laboratory glassware; density

hydrometers; use and test methods[5] ASTM E-100 2005 Standard specification for ASTM

hydrometers ASTM Standards 14.03[6] ISO 387 1987 Hydrometersprinciples of construction and

standardisation[7] ASTM E-126 2005 Standard test method for inspection and

verification of hydrometers ASTM Standards 14.03[8] ISO 3675 1998 Crude petroleum and liquid petroleum

productslaboratory determination ofdensityhydrometer method

[9] ISO 4801 1979 Glass alcoholometers and alcohol hydrometersnot incorporating a thermometer

[10] OIML R44 1985 Alcoholometers and alcohol hydrometers andthermometers for use in alcoholometry OIML R044-e85

[11] ASTM D-1298-99 2005 Standard method for density, relativedensity (specific gravity) or API gravity of crude petroleumand liquid petroleum products by hydrometer methodASTM Standards 5.01

[12] ASTM D-891-95 2004 Standard test method for specificgravity, apparent, of liquid industrial chemicals ASTMStandards 15.05

[13] BS 734-1 1973 Measurement of the density of milk using ahydrometer. Specification for hydrometers for use in milk

BS 734-2 1959 Measurement of the density of milk using ahydrometer. Methods

BS 734C 1962 Measurement of the density of milk using ahydrometer. Percentage of total solids and non-fatty solidsin milk corresponding to given fat content and observeddensity fat content range 26 per cent (with fat in the liquidstate)

[14] Gupta S V 2002 Practical Density Measurement andHydrometry (Bristol: Institute of Physics Publishing)

[15] Bowman H A and Gallagher W H 1969 An improved highprecision calibration procedure for reference standardhydrometers J. Res. Natl Bur. Stand. 73c 5765

[16] Gupta S V and Nath M 1984 A method for calibrating lowdensity hydrometer using a standard hydrometer calibratedfor higher densities Bull. OIML 54 710

[17] Cuckow F W 1949 A new method of high accuracy for thecalibration of reference standard hydrometers J. Chem. Ind.68 449

[18] Wagenbreth H, Gorski W and Kozdon A 1985 Ein verbessertesprufverfahren fur normalaraometer PTB Mitt. 95 3226

[19] Sommer K D, Machovits A, Poziemski J, Steindl D andToth H 1994 Comparison measurement of two standardhydrometers PTB Mitt. 104 136

[20] Lorefice S, Heinonen M and Madec T 2000 Bilateralcomparisons of hydrometer calibrations between theIMGC-LNE and the IMGC-MIKES Metrologia 37 1417

[21] ISO GUM 1995 Guide to the expression of uncertainty inmeasurement ISO Standards

[22] Davis R S 1992 Equation for the density of moist airMetrologia 29 6770

[23] Heinonen M and Sillanpaa S 2003 The effect of densitygradients on hydrometers Meas. Sci. Technol. 14 6258

[24] Lorefice S and Malengo A 2004 An image processingapproach to calibration of hydrometers Metrologia41 L710

2566

1. Introduction2. Calibration methods2.1. The direct comparison method2.2. The ring method2.3. Hydrostatic weighing

3. Hydrometer calibration by hydrostatic weighing3.1. Uncertainty equation3.2. Uncertainty budget

4. The hydrometer calibration facility at the INRiM4.1. The combined

5. ConclusionReferences