By Angiolo Farina, Axel Klar, Robert M. M. Mattheij, Andro Mikeli´c, Norbert Siedow, Antonio Fasano
This quantity offers a evaluation of complex technological difficulties within the glass and of the maths concerned. it's remarkable that this type of probably small examine zone is intensely wealthy and demands an impressively huge number of mathematical equipment, together with numerical simulations of substantial complexity. the issues handled listed below are very ordinary of the sphere of glass production and canopy a wide spectrum of complementary matters: injection molding through quite a few ideas, radiative warmth move in glass, nonisothermal flows and fibre spinning. The ebook can definitely be worthwhile not just to utilized mathematicians, but in addition to physicists and engineers, who can locate in it an summary of the main complicated types and techniques.
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Extra info for Mathematical Models in the Manufacturing of Glass: C.I.M.E. Summer School, Montecatini Terme, Italy 2008
By the chain rule of differentiation, dVp dVp (t) = (z)Vp (z). dt dz (79) As for the initial condition, let z = 0 and Vp = V0 at t = 0. Then the motion of the plunger involves the following initial value problem: ⎧ 2 2π μ L2 ⎨ 1 dVp (z) = Vp (z) Fg,1 (z) + Fg,0 (z) + Fg,0(z) + Fe 2 dz m pV ⎩ Vp (0) = V0 (80) Since the plunger velocity does not need to be known to determine the glass domain at given plunger position z, an implicit time integration scheme can be used to solve (80), thus overcoming the stiffness problem .
For more advanced heat modelling during the parison press the reader is referred to [23, 38, 64]. M. Groot et al. 2) can be specified for the parison press: u∗ p ez ) u∗ T∗ n (u∗ − V · n = 0, · n = 0, · n = 0, · n = 0, · · · · T∗ n t t t t T∗ n + β p∗ (u∗ − Vpez ) T∗ n + βm∗ u∗ T∗ n = 0, = 0, = 0, = 0, on on on on Γs , Γg,p , Γg,m , Γo . (40) In the remainder of this section it is assumed that the glass gob initially has an uniform temperature distribution, so that with (38) it follows that μ = μ¯ is constant.
Figure 30 visualises the temperature distribution during pressing. The initial temperature of the glass is 1, 000 ◦ C and the initial temperature of the air and the equipment is 500 ◦ C. The glass gradually heats up the equipment at the contact surface. 5 s. By contrast, the glass rapidly heats up the air. In order to verify the accuracy of numerical solutions, the volume conservation of the glass is examined. Figure 31 depicts the ratio of the glass volume to the initial glass volume as a function of time.