07 hydrometers calibration cuckow's method_claude jacques

1

MWG7 Workshop on HydrometersQuerétaro (México) 2006-11-21 / 23SISTEMA

INTERAMERICANODE METROLOGIA

SISTEMAINTERAMERICANODE METROLOGIA

Calibration of hydrometers with the Cuckow method

Claude JACQUESNRC, CanadaChair of SIM MWG72006-11-21

2

IntroductionSISTEMAINTERAMERICANODE METROLOGIA

SISTEMAINTERAMERICANODE METROLOGIA

The Cuckow method is a calibration method forhydrometers.

3

IntroductionSISTEMAINTERAMERICANODE METROLOGIA

SISTEMAINTERAMERICANODE METROLOGIA

Advantages over comparison methods with master hydrometers and many liquids:

• It uses a single liquid that covers all the ranges

• It doesn’t need a set of fragile, breakable, master hydrometers

4

Experimental setupSISTEMAINTERAMERICANODE METROLOGIA

SISTEMAINTERAMERICANODE METROLOGIA

The hydrometer is suspended from a balance.

The balance prevents the hydrometerfrom sinking.

The reading of the balance is a functionof the density of the liquid in which thehydrometer floats freely at the samescale mark on its stem.

5

Experimental setupSISTEMAINTERAMERICANODE METROLOGIA

SISTEMAINTERAMERICANODE METROLOGIA

The ideal is a liquid with a lower density than the scale ofthe hydrometer, to make sure the hydrometer sinks.

Liquids used:

• Pure water (known density, but high surface tension)

• Hydrocarbons: n-nonane, n-dodecane, … n-tetradecane, etc. (density to be calibrated, but low surface tension)

6

Experimental setupSISTEMAINTERAMERICANODE METROLOGIA

SISTEMAINTERAMERICANODE METROLOGIA

If the density of the liquid is higher than the hydrometer, a ballast weight load is added to the hydrometer to make itsink.

7

Experimental setupSISTEMAINTERAMERICANODE METROLOGIA

SISTEMAINTERAMERICANODE METROLOGIA

The level of immersion can be varied, so many points can be calibrated on the stem.

8

Steps to be performedSISTEMAINTERAMERICANODE METROLOGIA

SISTEMAINTERAMERICANODE METROLOGIA

1. Clean the hydrometerThe cleaning of the hydrometer is done by a simple wiping

with ethyl alcohol or acetone. If the surface is contaminated with hydrocarbons, as could be the case for LPG — Liquefied Petroleum Gas — hydrometers, a more complete sequence of solvents is required : varsol, naphta, and acetone. Prior to cleaning, the compatibility of the hydrometer material and the solvents must be verified.

9

Steps to be performedSISTEMAINTERAMERICANODE METROLOGIA

SISTEMAINTERAMERICANODE METROLOGIA

2. Measure the stem diametersThe stem diameters are measured at the points of calibration

to evaluate the stem circumferences. The diameters are measured twice at right angles with a caliper, and the mean is calculated.

The force applied downward to the hydrometer by the surface tension is proportional to the circumference.

10

Steps to be performedSISTEMAINTERAMERICANODE METROLOGIA

SISTEMAINTERAMERICANODE METROLOGIA

3. Measure the hydrometer mass in airBefore doing any weighing, the balance must be exercised.

It is a good habit also to calibrate it, noting the density of air at calibration.

Then, the hydrometer is weighed on the pan of the balance, producing a mass reading.

11

Steps to be performedSISTEMAINTERAMERICANODE METROLOGIA

SISTEMAINTERAMERICANODE METROLOGIA

4. Supplementary ballast weight loadIf the density of the hydrometer is less than that of the

standard liquid, it must be loaded with a supplementary ballast weight, so that it will sink.

The value of the effective mass in liquid of the weight must be deducted from the balance readings.

12

Steps to be performedSISTEMAINTERAMERICANODE METROLOGIA

SISTEMAINTERAMERICANODE METROLOGIA

5. Calibration – installation of the hydrometer

The body of the hydrometer is partly immersed in the standard liquid with the stem still in air.

The calibration is made from the bottom of the scale to the top to avoid the build up of liquid drops on the stem. The stem is lowered into the liquid to the point at which it is to be calibrated.

13

Steps to be performedSISTEMAINTERAMERICANODE METROLOGIA

SISTEMAINTERAMERICANODE METROLOGIA

5. Calibration – measurementCare must be taken to go down sufficiently to have the

graduation below the surface of the liquid, and to lift so as to build the meniscus systematically on a wet stem surface.

The reading is made with the graduation levelled to the main surface of the liquid.

The surface tension of the liquid must be well known for the correction to the calibration to be made for a corresponding zero value.

14

Steps to be performedSISTEMAINTERAMERICANODE METROLOGIA

SISTEMAINTERAMERICANODE METROLOGIA

5. Calibration – volumeWith the same set-up, it is possible to immerse the

hydrometer completely and measure its density and volume.

15

Theory – balance calibrationSISTEMAINTERAMERICANODE METROLOGIA

SISTEMAINTERAMERICANODE METROLOGIA

Before starting a calibration, the balance must be exercised by loading it a few times with a weight.

Then it is calibrated, either by an internal weight or an external weight.

16

Theorybalance calibrationSISTEMA

INTERAMERICANODE METROLOGIA

SISTEMAINTERAMERICANODE METROLOGIA

The calibration factor of the balance is

'

'

3cal

1

where air density during balance calibration8000 kg/m density of calibration weight

a

cal

a

ρβ

ρ

ρ

ρ

= −

=

= =

17

Theoryfirst of three equationsSISTEMA

INTERAMERICANODE METROLOGIA

SISTEMAINTERAMERICANODE METROLOGIA

Weighing of the hydrometer in air

( )

( )where

balance reading of weighing in air absolute mass of the hydrometer total volume of the hydrometer

immersed volume of the hydrometernon-immersed volume of the hydromete

a a

a

R m V v

Rm

V v

Vv

β ρ= − +

==

+ =

== r

density of airaρ

=

18

Theorysecond of three equationsSISTEMA

INTERAMERICANODE METROLOGIA

SISTEMAINTERAMERICANODE METROLOGIA

Hydrometer floating freely in liquid at a scale reading

0

where diameter of the stem at scale reading surface tension of liquid to be measured local acceleration of gravity density of liquid to be measured

LL a

L

L

Dm V vg

D

g

π γ ρ ρ

γ

ρ

= + − −

====

19

Theorythird of three equationsSISTEMA

INTERAMERICANODE METROLOGIA

SISTEMAINTERAMERICANODE METROLOGIA

Hydrometer suspended from balance in standard liquid

wherebalance reading of suspended hydrometer

surface tension of calibration liquid density of calibration liquid

ss s a

s

s

s

DR m V vg

R

π γβ ρ ρ

γρ

= + − −

=

=

=

20

Theorythe three equationsSISTEMA

INTERAMERICANODE METROLOGIA

SISTEMAINTERAMERICANODE METROLOGIA

( )

0

a a

LL a

ss s a

R m V v

Dm V vg

DR m V vg

β ρ

π γ ρ ρ

π γβ ρ ρ

= − +

= + − −

= + − −

21

Theorysolution of three equations

(simplified)SISTEMAINTERAMERICANODE METROLOGIA

SISTEMAINTERAMERICANODE METROLOGIA

1) We neglect surface tensions γ

2) We set m’ = m – vρa (to simplify the equations)

22

Theorysolution of three equations

(simplified)SISTEMAINTERAMERICANODE METROLOGIA

SISTEMAINTERAMERICANODE METROLOGIA

'

'

'

0

a a

L

s s

R m V

m V

R m V

β ρ

ρ

β ρ

= −

= −

= −

23

Theorysolution of three equations

(simplified)SISTEMAINTERAMERICANODE METROLOGIA

SISTEMAINTERAMERICANODE METROLOGIA

a s s aL

a s

R RR Rρ ρ

ρ−

=−

(Neglecting surface tensions of liquids)

24

Theorysolution of three equationsSISTEMA

INTERAMERICANODE METROLOGIA

SISTEMAINTERAMERICANODE METROLOGIA

But life is never that simple;

so, we take into account the surface tensions:

25

Theorysolution of three equationsSISTEMA

INTERAMERICANODE METROLOGIA

SISTEMAINTERAMERICANODE METROLOGIA

( )

0

a a

LL a

ss s a

R m V v

Dm V vg

DR m V vg

β ρ

π γ ρ ρ

π γβ ρ ρ

= − +

= + − −

= + − −

26

Theorysolution of three equationsSISTEMA

INTERAMERICANODE METROLOGIA

SISTEMAINTERAMERICANODE METROLOGIA

( ) ( )( )

( )

a s s a s a L s a

L

a s s

DR Rg

DR Rg

πβ ρ ρ γ ρ γ ρ ρρ

πβ γ

− + + −=

− +

27

Theorysolution of three equationsSISTEMA

INTERAMERICANODE METROLOGIA

SISTEMAINTERAMERICANODE METROLOGIA

( ) ( )( )

( )

s a L s a

L

a s s a

a s s

R R

R R

Dg

Dg

πβ γ ρ γ ρ ρρ

β γ

ρ

π

ρ + +

+−

−=

−

Remark: neglecting the surface tensions, we go back to previous simpler solution (in red)

28

Theorydensity of hydrometerSISTEMA

INTERAMERICANODE METROLOGIA

SISTEMAINTERAMERICANODE METROLOGIA

By immersing completely the hydrometer, it is possible to measure its density:

h

where density of the hydrometer balance reading at full immersion of the hydrometer

ca s s a

h ca s

cs

R RR R

R

ρ ρρ

ρ

−=

−

=

=

29

Theorydensity of hydrometerSISTEMA

INTERAMERICANODE METROLOGIA

SISTEMAINTERAMERICANODE METROLOGIA

By immersing completely the hydrometer, it is possible to measure its absolute mass:

1 1

ca s

a s

h h

R Rm β βρ ρρ ρ

= = − −

30

Theoryvolume of hydrometerSISTEMA

INTERAMERICANODE METROLOGIA

SISTEMAINTERAMERICANODE METROLOGIA

By immersing completely the hydrometer, it is possible to measure its volume:

ca s

h h a h s

R RmV v β βρ ρ ρ ρ ρ

+ = = =− −

31

ConclusionSISTEMAINTERAMERICANODE METROLOGIA

SISTEMAINTERAMERICANODE METROLOGIA

Advantages of the Cuckow method:

1. One liquid only. No dangerous and toxic liquidsrequired for high density calibrations, like acids.

2. More accurate and faster than comparisons withreference liquids and reference hydrometers.

3. Calibrations can be made exactly at any scale marks on the stem.