Mathematical Modelling of Granulation: Static and Dynamic Liquid ...

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200 R.L.I.M.S. Vol. 3, April, 2002 Figure 1: Illustration of the static liquid bridge geometry. Dimensional variables are used (as in equations (1) and (2)). Acontact angle of θ =10 o , α =40 o , β =38 o .The radius of particle A is rA =0.1 mm and particle B rB =0.15 mm. The length of the bridge is 0.12 mm. Static Bridges Consider figure 1 which illustrates a liquid bridge with cylindrical symmetry between two spherical particles ‘A’ and ‘B’ of radii rA and rB. Coordinates r and x define the position of the liquid bridge surface along with the radii of curvature r1 and r2 which lie in the r − x and r − y planes respectively. The curvature in the r-x plane is therefore given by 1 1 ,andthecurvature in the r-y plane by .Allowing ∆p to denote r1 r2 the pressure deficiency caused by the presence of the liquid droplet (∆p >0 when the internal pressure of the bridge is higher than the external (ambient) pressure), the Young-Laplace equation relates the surface tension of the binder γ to the pressure difference ∆p and the mean curvature of the bridge surface, 1 γ + r1 1 =∆p. (1) r2 Gravity does not appear in (1) as the mass of the liquid bridge is very small in comparison with the surface tension force between particles. Upon substitution of the vector calculus results for 1 1 and (see [8] for r1 r2 details), equation (1) can be written 1 γ = −∆p. (2) r ′′ (1 + r ′2 ) 3/2 − r(1 + r ′2 ) 1/2 Equation (2) can be non-dimensionalised by introducing variables X = x r σ and R = σ ,where σ is the scaling variable relating non-dimensional and dimensional variables. In the case of figure 1, σ can be chosen as either rA or rB. Also, a non-dimensional pressure difference ∆P = ∆pσ γ is introduced, enabling the nondimensional version of (2) to be written as R ′′ (1 + R ′2 ) 3/2 − 1 R(1 + R ′2 = −∆P. (3) ) 1/2 In (3) the notation R ′ = dR dX and R′′ = d2R dX2 has been adopted. Initial values for the bridge height R0 and tangent R ′ 0 (occurring on particle A above) are specified. The angle which the bridge makes contact with the tangent plane to the spheres is the contact angle θ and is specified for a given problem. The starting value for the bridge height (occurring at X0)is R0 = rA σ sin α = RA sin α
P. Rynhart et al., Mathematical Modelling of Granulation 201 and that of the slope at the point of contact is R0 ′ = cot(α + θ). Using the information above, an analytic solution to equation (3) is possible upon making the substitution Differentiating U with respect to X gives U = 1+R ′2 − 1 2 . (4) dU dX = − R′ R ′′ 3 . (5) 1+R ′2 2 Rearranging the right hand sides of (4) and (5), substituting these equations into (5) and applying the chain ), (3) can be written as the following first order differential equation, rule (where dU dX dX dR Integrating (6) gives = dU dR dU U + =∆P. (6) dR R U = R ∆P 2 where the constant of integration E is the energy of the liquid bridge surface. Equation (3) defines a Hamiltonian dynamical system and hence the energy E is conserved. By combining (4) and (7), E = R 1 R∆P − 1+R ′2 2 . (8) Substituting (4) into (7) and rearranging gives R ′ as R ′ = dR dX = ± + E R R2 − ∆PR2 + E2 ∆P R 2 2 2 + E Rearranging the above, the shape of the bridge (where R0 ≤ R ≤ R1)isgivenby the integral X = If E =0then (9) can be solved to give R R0 ∆PR 2 2 + E R 2 − ∆PR 2 2 X 2 + R 2 2 2 = ∆P showing that the liquid bridge then has a spherical shape. (7) + E2 dR. (9) If E = 0, (9) can be completed using integral tables [9]. The following parametric solution in terms of X is produced, R X = F ξ ,χ R0 − F ξ ,χ η + 2E ∆P η R0 + η E ξ ,χ R − E ξ ,χ (11) (10)
Page 1: Res. Lett. Inf. Math. Sci., (2002)
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