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The uncertainty evaluation for the
hydrometer calibration at NMISA
Presenter: Bongani NdlovuDate: 09/10/2018
A HYDROMETER
• Thin graduated hollow tube
• Weighted at bottom end to make it float upright
• USED• To measure specific gravity,
concentration, density of liquids 500 – 2000 kg/m^3
HOW IT IS USED
• Floats in a liquid
• Liquid density is where• Liquid level aligns with
the scale
• e.g., Beer concentration
• http://www.schoolphysics.co.uk/age11-14/Matter/text/Hydrometers_/index.html
• https://static.wixstatic.com/media/b0d342_7e2a75815d5b4b1a809e99475a547ae1~mv2.jpg_256
NEED FOR HYDROMETER CALIBRATION
• Over time, small drifts in scale occur due to• Mechanical stress
• Abrasion
• Ageing process of glass
• Calibrate the hydrometer to quantify these small drifts
• Calibration gives the corrections of the drifts in the scale
• At NMISA, hydrometers are calibrated on the hydrometer system using Cuckow’s method
• Consists of 200 g balance with 0,01 mg readability,
• Tamson TV7000 thermostat, CCD camera,
• Computer with software, visualization screen
HYDROMETER SYSTEM AT NMISA
• Thermostatic bath
• Inner vessel: Distilled water (reference liquid)…. 3 T sensors
• Liquid temperature maintained at ≈20 °C
SCHEMATIC DIAGRAM
CUCKOW’S METHOD
• Based on hydrostatic weighing
• Hydrometer is weighed in air and then in a reference liquid
• Scale to be calibrated is aligned to the liquid level and the reference liquid density is determined at that point
• Wright JD, Bean VE and Aguilera J, 2010. NIST calibration services for hydrometers. NIST Special Publication, 250, 78.
• Measure:• Stem diameter of hydrometer using
a Vernier calliper
• Surface tension of the ref liquid using tensiometer
• Hydrometer weighed in air:• Measure m, air T, RH, P
• In the reference liquid: Alignment: CCD camera magnified image
• Measure m, air T, RH, P
• Liquid T
MEASUREMENTS
CUCKOW’S MODEL
CORRECTION:
• 𝑐 = 𝜌𝑥 − 𝜌𝑠𝑐𝑎𝑙𝑒• where 𝜌𝑠𝑐𝑎𝑙𝑒 is the scale mark to be calibratedS Lorefice et al. Measurement Sc. & tech., 17(10), 2006, 2560
UNCERTAINTY BUDGET
• Depends on several factors including• Equipment used
• Reference liquid
• Mathematical model (Cuckow’s equation)
• Uncertainty calculations are based on the GUM, 2008
HYDROMETER MASS
• Mass in air
• Mass in liquid
• Depend on the reference mass uncertainty and balance properties
Ref. mass std. uncertainty (g)
Repeatability (g)
Resolution (g)
Linearity (g)
Uncertainty contribution
(g)
M_a 0,000017 0,0002 0,0001 -8E-06 2,24E-04
M_L 0,000020 0,00002 0,00001 -3E-05 4,24E-05
ADDITIONAL RING WEIGHTS
• In case of distilled water (ρ ≈ 1 g/ml)• Hydrometer floats when its density range < 1 g/ml
• Requires additional ring weight
• Inner diameter almost equal to the stem diameter
• Weigh the ring weight in the reference liquid
AIR DENSITY• Use the CIPM2007 equation
• Uncertainty depends on the measuring equipment and the equation • 𝑢𝑇, 𝑢𝑅𝐻, 𝑢𝑃 and 𝑢𝑒𝑞𝑛
• A Picard et al., Metrologia 45 (2008) 149–155
Variable Symbol Unit ValueStandard
UncertaintyUncertainty contribution
Air temperature T_a °C 20,70 0,02 -1,13E-03Air pressure p hPa 875,82 0,06 6,21E-10
Relative humidity rh %rh 38,5 1,3 -1,21E-05
Equation 2,28E-08Air density unc. (g/ml) 1,13E-06
WATER DENSITY• Use the Tanaka formula
• Density based on SMOW at 1 atm (see level)
• 𝑎5 = 999,972 kg/m^3 assumed for tap water
• Apply compressibility correction factor
• M Tanaka et al. Metrologia, 2001, 38, 301-309
Variable Symbol Units Value Uncert. S_iUncert.
Contribution
Water temp T_L °C 19,9815 0,06 -0,000206 -1,238E-05
Air pressure p hPa 875,822 0,06 4,58E-10 2,748E-11
Equation g/ml 4,5E-07 1 4,5E-07
Water density unc (g/ml) 1,238E-05
LOCAL GRAVITATIONAL ACCELERATION
• Gravitational acceleration• 𝑢𝑔= 𝑢 of g in the lab
• Gravitational acceleration gradient (Δg) due to height (ℎ) difference• hydrometer in liquid and balance on top
• 𝑢Δ𝑔= Δg ∗ ℎ
OTHER CONTRIBUTORS: IN CUCKOW’S MODEL
• Density of reference mass standard• 𝑢ρ 𝑠𝑡𝑑 = 𝑢 from density certificate
• Hydrometer stem diameter• 𝑢𝐷 = 𝑢 of Vernier calliper
• Surface tension of the reference liquid• 𝑢γ = 𝑢 of the tensiometer
• Liquid temperature during measurement• 𝑢𝑇 = 𝑢 of the thermometer
• Thermal expansion coefficient β (glass) of the hydrometer• 𝑢β = 10% β
ADDITIONAL CONTRIBUTIONS
• Scale alignment to the liquid level• The scale alignment is magnified by CCD camera
• 𝑢𝑎𝑙𝑖𝑔𝑛 = 1
10(ℎ𝑦𝑑𝑟𝑜𝑚𝑒𝑡𝑒𝑟 𝑟𝑒𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛)
• Repeatability of the measurement• 𝑢𝑟𝑒𝑝𝑒𝑎𝑡 = 𝐸𝑆𝐷𝑀
THE UNCERTAINTY BUDGET
GRAPHICAL REPRESENTATION
0,00E+00
2,00E-05
4,00E-05
6,00E-05
8,00E-05
1,00E-04
1,20E-04
M_a M_L m_r ρ_s γ_x γ_L D g ρ_L ρ_a T_L β S R
ρ_x
(g/
ml)
CONCLUSIONS
• The uncertainty evaluation for hydrometer calibrationat NMISA has been presented
• The biggest uncertainty contribution came from thesurface tension of the reference liquid
• The uncertainty for the hydrometer calibration can bereduced drastically by using a more suitable (lowdensity, low surface tension) reference liquid(hydrocarbons e.g., tridecane, nonane) than distilledwater
I THANK YOU