The uncertainty evaluation for the Presenter: Bongani ... · CUCKOW’S METHOD •Based on hydrostatic weighing •Hydrometer is weighed in air and then in a reference liquid •Scale

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