The proposed hydrostatic weighing system for volume and density of solids above 1 kg at NMISA

Bongani Ndlovu

Scientist: Mass, Volume and Density

T&M2019

17 September 2019

Mass metrology

• Volume and density measurements are critical for high accurate mass determination (e.g., OIML class E1, E2)

• OIML weights have irregular shapes

• Hydrostatic weighing to determine the volume and density

Hydrostatic weighing

Weighing in air

𝑚𝑇 − 𝜌𝑎1𝑉𝑇 = 𝑚𝑅1 − 𝜌𝑎1𝑉𝑅1 + 𝐼𝑇𝑎𝑖𝑟 − 𝐼𝑅1

𝑎𝑖𝑟(1)

where 𝑚𝑇 mass of the test weight,

𝜌𝑎1 air density while weighing in air

𝑉𝑇 volume of the test weight

𝑚𝑅1 mass of the reference weight while weighing in air

𝑉𝑅1 volume of a reference weight while weighing in air 𝑚𝑅1

𝜌𝑅1

𝐼𝑇𝑎𝑖𝑟 balance indication of the test weight in air

𝐼𝑅1𝑎𝑖𝑟 balance indication of the reference weight while weighing in

air.

Weighing in liquid

𝑚𝑇 − 𝜌𝑤𝑉𝑇 = 𝑚𝑅2 − 𝜌𝑎2𝑉𝑅2 + 𝐼𝑇𝑙𝑖𝑞

− 𝐼𝑅2𝑎𝑖𝑟

(2)

where 𝜌𝑤 liquid (water) density

𝑚𝑅2 mass of the reference weight while weighing the test in liquid

𝜌𝑎2 air density while weighing in liquid

𝑉𝑅2 volume of a reference weight while weighing in liquid 𝑚𝑅2

𝜌𝑅2

𝐼𝑇𝑙𝑖𝑞

balance indication of the test weight in liquid

𝐼𝑅2𝑎𝑖𝑟 balance indication of the reference weight while weighing

the test in liquid.

Volume of a solid• Volume 𝑉𝑇 of a test weight at a measurement temperature

(1) – (2):

𝑉𝑇 =𝑚𝑅1 1−

𝜌𝑎1𝜌𝑅1

+ 𝐼𝑇𝑎𝑖𝑟−𝐼𝑅1

𝑎𝑖𝑟 − 𝑚𝑅2 1−𝜌𝑎2𝜌𝑅2

+ 𝐼𝑇𝑙𝑖𝑞−𝐼𝑅2

𝑎𝑖𝑟

𝜌𝑤−𝜌𝑎1(3)

• At a reference temperature of 20 °C, 𝑉𝑇 20 is calculated by

𝑉𝑇 20 =𝑚𝑅1 1−

𝜌𝑎1𝜌𝑅1

+ 𝐼𝑇𝑎𝑖𝑟−𝐼𝑅1

𝑎𝑖𝑟 − 𝑚𝑅2 1−𝜌𝑎2𝜌𝑅2

+ 𝐼𝑇𝑙𝑖𝑞

−𝐼𝑅2𝑎𝑖𝑟

𝜌𝑤−𝜌𝑎11 − ϒ 𝑡 − 20 (4)

where:

ϒ is the thermal expansion coefficient of the test weight and 𝑡 is the temperature during measurement

Current system

• For 1 g – 1 kg solid weights: VC1005

• Extend the range to cover 2 kg - 10 kg with a new Hydrostatic weighing system

Hydrometer system• Same measurement

principle (hydrostatic weighing)

• To use the same bath that is used for the hydrometer calibrations

Proposed hydrostatic weighing system

Proposed equipment

• Balance• Capacity 10 kg, res. 0.2 mg, under-balance weighing port

• Environmental logger

• Stainless steel suspension wire• Roughness at air-water interface

• Temperature-controlled thermostatic bath• Maintained at T ≈ 20 °C

• Inner liquid bath• Weight holder

• Loading mechanism to operate submerged in water

• Thermometer

Loading/unloading in liquid• Weight to remain inside the liquid for the duration of the

experiment• Temperature stability• Air bubbles (removed just once)

• Loading mechanism• Operate inside the liquid for the duration of experiment• Motor operated with remote control• Ensure the liquid level does not change due to loading/unloading

Suspension wire

• Non repeatable meniscus on a smooth suspension wire • Affects the accuracy of the results

• Apply coating (roughness) on the wire to get a constant meniscus

• Electrolytic coating solution:• Distilled water (H2O) 200 parts• Platinum (IV) chloride (PtCl4) 5 parts• Lead acetate (Pb(C2H3O2)2) 1 part• Passing a current of 30 mA

Bonhoure A., 1958. Suspension wire for hydrostatic weighing

Accessed on: https://www.bipm.org/utils/common/pdf/CIPM-PV-OCR/CIPM1958.pdf

https://www.bipm.org/utils/common/pdf/CIPM-PV-OCR/CIPM1958.pdf

High accuracy mass calibration in air

𝑚𝑇 = 𝑚𝑅 + 𝜌𝑎(𝑉𝑇−𝑉𝑅) + 𝐼𝑇𝑎𝑖𝑟 − 𝐼𝑅

𝑎𝑖𝑟

where:

𝐵 = 𝜌𝑎(𝑉𝑇−𝑉𝑅) is the air buoyancy correction

• Therefore accurate 𝜌𝑎(𝑇, 𝑃, 𝑅𝐻), 𝑉𝑅 =𝑚𝑅

𝜌𝑅and 𝑉𝑇 =

𝑚𝑇

𝜌𝑇lead to more accurate 𝐵

• 𝑉𝑅 or 𝜌𝑅 and 𝑉𝑇 or 𝜌𝑇 are determined by hydrostatic weighing

Conclusions

• The Hydrostatic weighing project for solids ranging 2 – 10 kg is on-going at NMISA

• Accurate volume/density leads to even more accurate air buoyancy corrections

• Available hydrostatic weighing system calibration services which at NMISA:

• Volume and density of solids ranging 1 g – 1 kg

• Hydrometer calibration