Mathematic modeling of a new inlay buoyancy artefact


ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 42 

ACTA IMEKO 
ISSN: 2221-870X 
December 2020, Volume 9, Number 5, 42 - 46 

 
 

MATHEMATIC MODELING OF A NEW INLAY BUOYANCY ARTEFACT 
 

K. Lehrmann1, F. Härtig1, R. Tutsch2 

 
1 Physikalisch-Technische Bundesanstalt, Braunschweig, Germany, katharina.lehrmann@ptb.de  

2 Technical University Braunschweig, Institute of Production Metrology, Braunschweig, Germany, r.tutsch@tu-bs.de  

 

Abstract: 

The novel buoyancy artefact concept for the use 

in mass metrology enables the determination and 

correction of systematic air buoyancy effects caused 

by measurements outside a vacuum. In contrast to 

other buoyancy artefacts with a nominal mass of 

1 kg, the new artefact, called inlay artefact, has a 

smaller volume resulting from an enclosed tungsten 

core. The theoretical design of the new artefact 

consists of dismountable discs with spherical 

distance pieces and a cylinder. This work focuses on 

the design and mathematical model considering 

limiting factors caused by the handling and the 

geometry of the comparators used. 

Keywords: buoyancy artefact; kilogram; silicon; 

inlay artefact; revised SI 

1. INTRODUCTION 

Highly accurate mass calibrations require high 

sophisticated mass comparators which are operated 

in vacuum.  Outside a vacuum the measurements are 

being strongly influenced by environmental 

conditions, such as buoyancy effects. This is 

particularly evident, if mass standards of the same 

nominal value made of different materials such as 

steel, platinum-iridium, or silicon are calibrated on 

the basis of a substitution method or compared 

against each other. In this case buoyancy effects 

significantly influence the findings because of the 

resulting different volumes [1]. One way to correct 

the buoyancy effects is to determine the air densities 

by using what is known as buoyancy artefacts 

calibrated by volume [2]. Most accurate results for 

the determination of air densities are achievable 

with a relative uncertainty in the order of magnitude 

of 5 × 10-5 [3]. Typically, aside the same nominal 

weight such artefacts have nearly the same surface 

areas and surface properties but different volumes. 

The greater the volume difference of buoyancy 

artefacts used, the more reliably the air density can 

be determined. A design is presented in order to 

achieve a large volume difference between 

buoyancy artefacts used. 

2. DESIGN OF A NEW KIND OF 
BUOYANCY ARTEFACT 

Figure 1 shows a sketch of the proposed new 

buoyancy artefact. It consists of several discs made 

of monocrystalline silicon and one cylinder made of 

silicon with a tungsten core enclosed. The entire 

standard is referred to as an “inlay artefact”. 

According to the principal approach of buoyancy 

artefacts, the nominal mass of the artefact 𝑀ref  is 
1 kg and its nominal surface area must be identical 

to the surface area of the other buoyancy artefacts 

𝐴ref  used. Due to the tungsten core the effective 

density of the artefact is much higher than the 

density of the buoyancy artefacts made of pure 

silicon. This leads to a smaller volume and thus, to 

less buoyancy. Distance pieces, which are part of a 

statically determined coupling, allow the stackable 

and demountable individual parts of the artefact to 

be dismantled for cleaning purposes. 

 
Figure 1: Exemplary figure of an inlay buoyancy artefact 

with two discs (top and center disc) and a cylinder with 

enclosed tungsten core 

For modeling the new buoyancy artefact, the 

following variable input parameters apply: 

− the surface area ratio between discs and the 
cylinder 

− the number of discs 

− the diameter of the discs 

− the diameter of the cylinder 

http://www.imeko.org/
mailto:katharina.lehrmann@ptb.de
mailto:r.tutsch@tu-bs.de


ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 43 

For practical handling the radius of the discs and 

the cylinder should be equivalent. To achieve the 

specific surface areas, the surface area ratio of discs 

to cylinder should be at least two. First 

investigations have shown that the required surface 

area can be manufactured by three identical discs 

made of pure silicon. In order to provide enough 

volume to include a tungsten core the size of the 

ground disc have to be enlarged. From now on the 

ground disc was named cylinder (silicon disc with 

tungsten core). An additional estimation shows that 

the resulting surface area could be realized by a 

minimum of two further discs. Further requirements 

for the new buoyancy artefact are shown in Table 1 

and depend on the geometric specifications of the 

comparators used, practical handling and the 

considerations of tilting stability. 

Table 1: Settings for numerical iteration 

Parameter Range 

max. total height ℎmax ≤ 105 mm 

tilt stability 20 mm ≤ 𝑟dsc, 𝑟cyl ≤ 45 mm 

number of discs 2 – 5 

 

The silicon cylinder and the plug enclosing the 

inlay core are detachably bonded to each other 

according to the specifications of conical tapered 

ground joints [4] as used for gas-tight silicon oxide 

compounds. 

For all calculations, the density of silicon is 

assumed to be 2 328.8 kg/m³ [5] and for tungsten is 

assumed to be 19 250.0 kg/m³ [6]. 

3. MATHEMATIC MODELING OF A NEW 
KIND OF BUOYANCY ARTEFACT  

For the sophisticated calculation of the inlay 

artefact, a number of limiting conditions have to be 

considered. For example, the surface area must 

correspond to a specified reference surface area and 

at the same time the mass must correspond to a 

specified nominal mass. In addition to the basic 

cylindrical geometry, the geometries of the 

spherical distance pieces and chamfers have to be 

taken into account for constructive reasons. Based 

on all these complex and correlated specifications, 

the radii and heights of the discs and the cylinder 

must be precisely dimensioned. In this section, first 

the calculations for the surface areas and then the 

masses are outlined. 

The corrections caused by chamfers are 

addressed comprehensively in section 4, while 

corrections relating to distance pieces are explained 

in section 5. In the following, this refers to all 

subsequent calculations that correlate with chamfers 

and distance pieces. 

3.1. Calculation of the Surface Areas for 
Cylinder and Discs 

The pure surface area 𝐴dsc of all discs, without 
any corrections for chamfers and distance pieces, 

depends on the given surface area of the reference 

𝐴ref, and the surface area ratio of discs to cylinder, 
denoted 𝑟𝑎𝑡𝑖𝑜. This results in: 

𝐴dsc = 𝐴ref
 𝑟𝑎𝑡𝑖𝑜

𝑟𝑎𝑡𝑖𝑜 + 1
− 𝐴corr,dsc,cal + 𝐴corr,dsc,cha  (1) 

and includes the correction for chamfers 

𝐴corr,dsc,cha, and the correction for distance pieces 
𝐴corr,dsc,cal. The surface of the cylindrical part is 
obtained as: 

𝐴cyl = 𝐴ref
1

𝑟𝑎𝑡𝑖𝑜 + 1
− 𝐴corr,cyl,cal + 𝐴corr,cyl,cha (2) 

including the reference surface 𝐴ref , surface area 
ratio of discs to cylinder 𝑟𝑎𝑡𝑖𝑜 and the correction 
for distance pieces 𝐴corr,cyl,cal  as well as the 

correction for chamfers 𝐴corr,cyl,cha. The height of 

a disc ℎdsc is calculated from: 

ℎdsc  =

𝐴dsc
𝑛dsc

− 2 π ∙ 𝑟dsc
2

2 π ∙ 𝑟dsc
 

(3) 

with number of discs 𝑛dsc and radius of the disc(s) 
 𝑟dsc. The height ℎcyl of a cylinder with radius 𝑟cyl 

can be calculated as: 

ℎcyl  =
𝐴cyl − 2 π ∙ 𝑟cyl

2

2 π ∙ 𝑟cyl
 (4) 

with the surface area of the cylinder 𝐴cyl calculated 

in equation (2). 

3.2. Calculation of Masses of Cylinder and Discs 
The calculation of the total mass shall be 

composed of the calculation of the mass of the 

individual discs 𝑀dsc  and the mass of the cylinder 
𝑀cyl . The discs are designed on the basis of the 

previously determined heights and radii.  

The total mass of the cylinder 𝑀cyl consisting of 

the silicon shell and the inlay core of tungsten is 

calculated, knowing the reference mass 𝑀ref  and 
the mass of all discs 𝑀dsc, by means of: 

𝑀cyl = 𝑀ref − 𝑀dsc   (5) 

The further steps are simplified if only 

cylindrical geometries are used for calculations. In 

order to obtain these principles, the masses of the 

distance pieces and chamfers are extracted from the 

surrounding silicon cylinder. The pure cylinder 

mass 𝑀cyl,pure is given by: 

𝑀cyl,pure = 𝑀cyl − 𝑀corr,cal + 𝑀corr,cha   (6) 

http://www.imeko.org/


ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 44 

As previously stated, the corrections for the 

distance pieces 𝑀corr,cal  and chamfers 𝑀corr,cha  
will be shown in sections 4 and 5.  

Having the volume of the cylinder 𝑉cyl,Si,pure , 

the mass of the tungsten core 𝑀cyl,W , can be 

described by: 

𝑀cyl,W =
𝑀cyl,pure− 𝜌Si∙𝑉cyl,Si,pure

𝜌W−𝜌Si
∙ 𝜌W  (7) 

including the corresponding densities of silicon 𝜌Si 
and tungsten 𝜌W.  

According to: 

𝑉cyl,Si,pure = 𝑉cyl,Si − 𝑉corr,cyl,cal + 𝑉corr,cyl,cha (8) 

the value of the reduced volume 𝑉cyl,Si,pure of the 

silicon cylinder considers the volumetric 

corrections influenced by both, the spherical 

distance pieces 𝑉corr,cyl,cal  and the chamfer 

𝑉corr,cyl,cha. 

For reasons of symmetry and to simplify the 

calculation elegantly, the geometric proportions of 

the enclosing silicon cylinder 𝑟cyl , ℎcyl  and the 

internal tungsten core are assumed to be identical, 

so that: 

𝑟cyl

ℎcyl
=

𝑟cyl,W

ℎcyl,W
 (9) 

Both the radius 𝑟cyl,W  of the inlay cylinder, 

given by: 

𝑟cyl,W = √
𝑉cyl,W

π

𝑟cyl

ℎcyl

3

  (10) 

and the height of the inlay cylinder ℎcyl,W, given by: 

ℎcyl,W = 𝑟cyl,W ∙
𝑟cyl

ℎcyl
 (11) 

are calculated from the fixed ratio of the geometry 

of the silicon cylinder for reasons of symmetry. 

The effective density of the inlay artefact 𝜌ily is 

calculated as: 

𝜌ily =
𝑀dsc +  𝑀cyl,Si + 𝑀cyl,W

𝑉dsc +  𝑉cyl
 (12) 

derived from the mass of the discs and the mass of 

the cylinder with the inlay tungsten artefact. 

4.  CORRECTION OF CHAMFER 

To correct the chamfers, volume, mass and 

surface area differences must be determined in 

comparison with the uncorrected discs and cylinder. 

The adjustment is classified by cylinder and 

chamfer. 

Figure 2 shows a section through a cylinder or 

disc with a circumferential chamfer. According to 

the mathematical description in section 2, the radii 

𝑟dsc  or 𝑟cyl  must be substituted for the variable 𝑟. 

The ring surface 𝐴rng, the lateral surface 𝐴shl of the 

uncorrected cylinder and the resulting chamfer 

surface 𝐴cha are shown under the chamfer angle α. 

 
Figure 2: Schematic cross section through a cylindrical 

object with straight chamfers 

4.1. Calculation of the Surface Areas for 
Chamfer 

The correction 𝐴corr,cha depends on the surface 
of the cylinder shell surface area 𝐴shl , the ring 
surface area of the disc 𝐴rng and the surface area of 

the chamfer 𝐴cha. As a result: 

𝐴corr,cha = (𝐴rng + 𝐴shl − 𝐴cha). (13) 

The surface area of an annular ring of the top 

face of a disc can be expressed as: 

𝐴rng = π (𝑟
2 − (𝑟 −

𝑎

tan 𝛼
)

2

) (14) 

The surface area of a disc of solid material 

without chamfer can be calculated from: 

𝐴shl =  2 π ∙ 𝑟 ∙ 𝑎 (15) 

For the surface correction of a chamfer the 

general approach for calculating rotational 

symmetric bodies [7]: 

𝐴cha = 2 π ∫ 𝑓(𝑧) √1 + (𝑓
′(𝑧))

2
𝑑𝑧

𝑎

0
  (16) 

is chosen. Herein the function 𝑓(𝑧) describes the 
mathematical cross section of the rotatory body. 

𝑓(𝑧) = 𝑟 − 𝑧 ∙ tan 𝛼  (17) 

Hence, the correction of the chamfer surface area 

results in: 

𝐴cha = 2 π ∫ (𝑟 − 𝑧 ∙ tan 𝛼) ∙ √1 + (tan 𝛼)
2 𝑑𝑧

𝑎

0

 (18) 

After resolving the integral, the final correction 

for all chamfers is: 

𝐴corr,cha = 𝑛dsc ∙ 𝑛cha ∙  

(
−2 π √1 + (tan 𝛼)2 ∙ (𝑎 ∙ 𝑟 −

𝑎2

2
) +

2 π ∙ 𝑟 ∙ 𝑎 + π (𝑟2 − (𝑟 −
𝑎

tan 𝛼
)

2
)

)  
(19) 

http://www.imeko.org/


ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 45 

4.2. Calculation of Volume and Mass of 
Chamfers 

The mathematical theory results from the 

approach to calculate a quarter torus. The general 

formula for the volume of a rotational symmetric 

object is [7]: 

𝑉 = π ∫ (𝑓(𝑧))
2

𝑑𝑧
𝑎

0

 (20) 

The resulting correction for volume of chamfers 

is described by the uncorrected cylinder volume 

𝑉cyl  subtracted by the chamfer volume correction 

𝑉corr,cha which results in: 

𝑉cha = 𝑉cyl − 𝑉corr,cha (21) 

The detailed relation of this correction reads: 

𝑉corr,cha 

= π ∫ 𝑟2 𝑑𝑧
𝑎

0
 − π ∫ (𝑟 − 𝑧 ∙ tan 𝛼)2𝑑𝑧

𝑎

0
  

(22) 

After solving the integrals, the final volume 

correction 𝑉corr,cha becomes: 

𝑉corr,cha = 𝑛cha ∙  

(π 𝑟 2𝑎 − π (𝑟 2 ∙ 𝑎 +
𝑎3

3
∙ (tan 𝛼)2 − 𝑟 ∙ 𝑎2 ∙ tan 𝛼)) 

(23) 

Assuming the known density 𝜌Si  and the 
determined correction 𝑉corr,cha, the adjustment for 
the mass of the chamfers is: 

𝑀corr,cha = 𝜌Si ∙ 𝑉corr,cha (24) 

5. CORRECTION OF DISTANCE PIECES  

The distance pieces enlarge the mass and surface 

area of the entire inlay artefact. Hence, corrective 

calculations are necessary.  

Distance pieces are placed on the top and bottom 

of the discs and the cylinder in relation to each other 

as shown in Figure 3. Distance pieces can be 

attached firmly to a disc in various methods, for 

example with a bond connection or high-vacuum 

compatible adhesive. 

 

Figure 3: Illustration of gap with spherical distance 

pieces (silicon spheres, 𝑟 = 2 mm, 𝛽 = 45°) 

In this context, only the upper half of the distance 

pieces must be considered for the correction of mass 

and surface area.  

Table 2 lists the relation between the number of 

distance pieces and the disc type or cylinder. Good 

mechanic design points out that a six-point 

mounting is the most stable stacking method for 

statically determined bearings. This avoids 

wiggling or instability of couplings. The selected 

stacking method requires exactly nine distance 

pieces per gap. This results in six distance pieces 

which are fixed on the top side of the cylinder. In 

addition, for any center disc, there are six distance 

pieces on the bottom side, and three further distance 

pieces on the top side. Finally, there are three 

distance pieces left for the top disc. 

Table 2: The total number of distance pieces per top disc, 

centre disc or cylinder and numbers for variable 𝑛, where 
𝑛 depends on the number of total discs 

Type of 

discs/cylinder 

Number of  

distance pieces (𝒌) 
Numbers for 

variable (𝒏) 
top disc 3 2 

center disc(s) 9 𝑛dsc 

cylinder 6 2 

5.1. Calculation of the Surface Areas for 
Spherical Distance Pieces 

The surface area correction 𝐴corr,cpl  derives 

from the spherical surface area 𝐴cpl. However, the 

surface area of the cylinder 𝐴cir which is covered by 
the base of the hemisphere must be subtracted as: 

𝐴corr,cpl = 𝐴cpl − 𝐴cir (25) 

Using the relevant parameters regarding the 

number of distance pieces 𝑘 the correction becomes: 

𝐴corr,cpl =
𝑘

2
∙ 4 π ∙ 𝑟cpl

2 − k ∙ π ∙ 𝑟cpl
2 (26) 

5.2. Calculation of Volume and Mass of 
Spherical Distance Pieces 

The volume correction through all distance 

pieces is made considering the number of discs 𝑛. 
Polished silicon spheres with a radius of 2 mm serve 

as distance pieces.  

Depending on the type of disc or cylinder, 

different numbers of the spherical distance pieces 𝑘 
must be inserted into the calculations. This results 

in the equation for the volume correction 𝑉corr,cpl 
according to: 

𝑉corr,cpl = (𝑛 − 1) ∙
𝑘

2
∙

4

3
 π ∙ 𝑟cpl

3 (27) 

Knowing the density 𝜌Si  and the calculated 
correction 𝑉corr,cpl leads to the adjustment for the 

mass of the distance pieces 𝑀corr,cpl as: 

𝑀corr,cpl = 𝜌Si ∙ 𝑉corr,cpl (28) 

http://www.imeko.org/


ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 46 

6. RESULTS  

The parameters for height and radius of the discs 

and the cylinder are determined numerically by a 

minimizing fit algorithm (Newtonian method) 

provided by Mathematica [8]. The solution 

fulfilling all constraints can be achieved, if discs and 

the cylinder have an identical diameter and a 

minimum number of discs is used. This results in a 

volume difference of approximately 315 cm³ 

between the new inlay artefact and a silicon 

buoyancy hollow cylinder with a volume of 

approximately 646 cm³ [1]. To sum up, 1690 

iteration runs were evaluated.   

Table 3 shows one solution for the fitted values 

for the designed inlay artefact, which fulfills all 

constraints. 

Table 3: Model parameters for an inlay artefact 

Input parameters Value Unit 

mass  1 kg 

ratio 2 - 

radius of disc(s) 0.034 m 

radius of cylinder 0.034 m 

chamfer angle 45 ° 

chamfer length 0.000 5 m 

Fitted parameters Value Unit 

mass of tungsten inlay 0.261 42 kg 

mass of silicon 0.738 58 kg 

number of discs 2 - 

total height 0.096 689 8 m 

height of one disc 0.030 187 1 m 

height of cylinder 0.030 658 8 m 

height of tungsten core 0.015 204 4 m 

radius of tungsten core 0.016 861 4 m 

radial wall thickness  

Si cylinder 

0.017 138 6 m 

axial wall thickness  

Si cylinder 

0.007 727 2 m 

Results Value Unit 

volume 0.000 330 60 m³ 

effective density  

of new inlay artefact 

3024.78 kg/m³ 

7. SUMMARY 

The aim of this theoretical design study of a new 

inlay buoyancy artefact was to create a buoyancy 

artefact of nominal mass 1 kg with a small volume 

compared to other buoyancy artefacts. The small 

volume of the artefact, resulting in a particularly 

low air buoyancy was achieved by a combination of 

discs and a cylinder made of silicon, with a tungsten 

core inside. In contrast to conventional artefacts 

made of stainless steel, the newly designed artefact 

can be efficiently and gently cleaned the same way 

as the reference, like a silicon sphere. 

Special focus is given on the correction of 

chamfer and spherical distance pieces. A version of 

the inlay artefact is exemplarily calculated and 

illustrated. 

The theoretical design study for parameters, i.e. 

radius and height of the silicon parts and the 

tungsten core, were determined by numerous 

constraints using a variation method and could be a 

basis for future manufacturing. 

8. REFERENCES 

[1] K. Lehrmann, R. Tutsch, F. Härtig, “Design of 
sorption and buoyancy artefacts made of silicon”, 

IMEKO, Cavtat-Dubrovnik, 2020, not yet 

published. 

[2] D. Knopf, Th. Wiedenhöfer, K. Lehrmann, 
F. Härtig, “A quantum of action on a scale? 

Dissemination of the quantum based kilogram”, 

Metrologia 56 024003, 2019. 

[3] M. Kochsiek, M. Gläser (eds.), “Comprehensive 
Mass Metrology”, ISBN 3-527-29614-X, p. 255, 

VCH, Berlin, 2000. 

[4] DIN 12242-1, Laboratory glassware; 
interchangeable conical ground joints, dimensions, 

tolerances, 1980. 

[5] PTB internal density determination by hydrostatic 
weighing at 20 °C, May 2020. 

[6] N. Greenwood, A. Earnshaw, Chemie der 
Elemente, Edition 1, VCH, Weinheim, ISBN 3-

527-26169-9, S. 1291, 1988. 

[7] T. Ahrens et al., Mathematik, Ch. 11 Heidelberg, 
ISBN 978-3-8274-1758-9, 2008. 

[8] www.wolfram.com/mathematica/ (2020-05-08). 
 

http://www.imeko.org/
http://www.wolfram.com/mathematica/