Mathematical Modelling of Granulation: Static and Dynamic Liquid ...
Mathematical Modelling of Granulation: Static and Dynamic Liquid ...
Mathematical Modelling of Granulation: Static and Dynamic Liquid ...
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P. Rynhart et al., <strong>Mathematical</strong> <strong>Modelling</strong> <strong>of</strong> <strong>Granulation</strong> 201<br />
<strong>and</strong> that <strong>of</strong> the slope at the point <strong>of</strong> contact is<br />
R0 ′ = cot(α + θ).<br />
Using the information above, an analytic solution to equation (3) is possible upon making the substitution<br />
Differentiating U with respect to X gives<br />
U =<br />
<br />
1+R ′2 − 1<br />
2<br />
. (4)<br />
dU<br />
dX = − R′ R ′′<br />
3 . (5)<br />
1+R ′2 2<br />
Rearranging the right h<strong>and</strong> sides <strong>of</strong> (4) <strong>and</strong> (5), substituting these equations into (5) <strong>and</strong> applying the chain<br />
), (3) can be written as the following first order differential equation,<br />
rule (where dU dX<br />
dX dR<br />
Integrating (6) gives<br />
= dU<br />
dR<br />
dU U<br />
+ =∆P. (6)<br />
dR R<br />
U =<br />
R ∆P<br />
2<br />
where the constant <strong>of</strong> integration E is the energy <strong>of</strong> the liquid bridge surface. Equation (3) defines a Hamiltonian<br />
dynamical system <strong>and</strong> hence the energy E is conserved. By combining (4) <strong>and</strong> (7),<br />
<br />
<br />
E = R<br />
1 R∆P<br />
−<br />
1+R ′2 2<br />
. (8)<br />
Substituting (4) into (7) <strong>and</strong> rearranging gives R ′ as<br />
R ′ = dR<br />
dX<br />
= ±<br />
+ E<br />
R<br />
<br />
R2 − ∆PR2 + E2<br />
∆P R 2<br />
2<br />
2<br />
+ E<br />
Rearranging the above, the shape <strong>of</strong> the bridge (where R0 ≤ R ≤ R1)isgivenby the integral<br />
X =<br />
If E =0then (9) can be solved to give<br />
R<br />
R0<br />
<br />
∆PR 2<br />
2<br />
+ E<br />
R 2 − ∆PR 2<br />
2<br />
X 2 + R 2 2 2<br />
=<br />
∆P<br />
showing that the liquid bridge then has a spherical shape.<br />
(7)<br />
+ E2 dR. (9)<br />
If E = 0, (9) can be completed using integral tables [9]. The following parametric solution in terms <strong>of</strong> X<br />
is produced,<br />
<br />
R<br />
X = F<br />
ξ ,χ<br />
<br />
R0<br />
− F<br />
ξ ,χ<br />
<br />
η + 2E<br />
<br />
∆P η<br />
<br />
R0<br />
+ η E<br />
ξ ,χ<br />
<br />
R<br />
− E<br />
ξ ,χ<br />
(11)<br />
(10)