52
Appendix A
Meniscus volume
Figure 1 Volumes
Developing the Matlab program with the purpose of simulate the behaviour of a picking and releasing system we had to calculate the volume of the meniscus between the plane and the sphere. As we supposed the liquid-vapour interface to have a circular shape (Figure 1), we estimated the meniscus volume as difference between the volume of the frustum of cone (bounded by the fuchsia line) and the sum of the spherical cap (hatched area in blue) and the rotation solid obtained from the circular segment around the axis (hatched area in red).
53 The calculus of volume of frustum of cone and of the spherical cap is executed using the simple formulae written afterwards related to Figure 2:
(
)
π[
(
θ)
]
(
θ θ)
π 1 cos sin sin
3 1 ' ' ' ' ' 3 1 2 2 2 2 2 R x R x R h r R r R h
Vfrustumofcone = + + = + − a + + a (A.1)
(
)
π[
(
θ)
]
[
(
θ)
]
π 1 cos 3 1 cos 3 1 ' ' ' ' 3 ' ' 3 1 2 − = − 2 − − = h R h R R RVsphericalcap (A.2)
In order to calculate the volume of the rotation solid, we used the Gouldino theorem that relates volume V to the plane section area A and to the distance xg of the barycentre from the axis:
A x
Vrotational solid =2π g (A.3)
First of all we had to determine the length RG1 that represents the distance of the barycentre G1 of the rotation solid’s section from the centre of the meniscus arc (Figure 3).
Figure 3 Barycentres
The barycentres of the sector of circle (in blue) and of the triangle (red figure) and of the circular segment (green) that compose it are linked by the relation:
T G G GT A A Y A R R = 1 1+ 2 2 (A.4)
The centres of gravity and the areas have been calculated as follows:
2 2 a T r A =ε a r r T GT r dr rd dr rd r dA dA r dA r A R a a 3 2 1 0 2 2 0 2 2 = = = =
∫ ∫
∫ ∫
∫
∫
∫
− − ε ε ε ε ξ ξ54
( )
( )
( )
2 sin 2 cos sin 2 2 2 2 2 2 ε ε ε a a a r r r H b A = = = = = 2 cos 3 2 3 2 2 ε a G H r Y 2 1 A A A = T − (A.5)Note that ε is the angle of the circular sector that corresponds to the arc of the meniscus and it’s value is ε =π −
(
α+θ +β)
.Substituting the previous formulae into equation (A.4) and deducing RG1:
( )
( )
( )
− − = − − = ⇒ ε ε ε ε ε sin cos sin 3 2 2 2 2 2 1 a T G T GT G r A A A Y A R RThe distance xG1 of the barycentre from the axis is (Figure 4):
γ β sin sin 1 1 a a G G x r R x = − − (A.6)
where γ is the angle in figure and it’s value is γ =ε +β
2
Figure 4 Geometry of the meniscus
Using (A.5) and (A.6) in the equation (A.3) we determined the volume of the rotation solid. Finally we achieved the objective to estimate the meniscus volume:
solid rotational cap spherical cone of frustum meniscus V V V V = − −
