Injection molding is a manufacturing process in which a mold is filled with a liquid that solidifies after cooling. The mold is then opened and the product removed. Such molds have openings for injecting the liquid and venting the displaced air. Consider the simple prototype in Fig. P8.15, in which the space to be filled is a gap of thickness H between disks of radius R₂. Applying a constant pressure Pin at an injection port of radius R₁ in one of the disks results in a volume flow rate Q(t). The radius of the filled region at a given instant is R(t). Assume that the flow is pseudosteady, that inertia, surface tension, and gravity are all negligible, and that R₂ ≫ R₁ or H. It is desired to relate the filling time tf to the pressure drop |ΔP| = Pin − P₀, the viscosity, and the linear dimensions.

(a) Relate the height-averaged radial velocity 

to Q(t) and the dimensions.

(b) Solve for v_r(r, z, t) in terms of ∂P/∂r.

(c) Evaluate ∂P/∂r and relate Q(t) to |ΔP| and R(t).

(d) Obtain the differential equation that governs R(t) and find the relationship between t_f and |ΔP|.

(e) Approximately how large must t_f be to justify the pseudosteady approximation?

Transcribed Image Text:

Z H Q(t), Pin Liquid R(1) R₂ Air Po 2R₁ R(t) (a) (b) Figure P8.15 Injection of a liquid at volume flow rate Q between disks of radius R₂ that are separated by a distance H. The radius of the injection port is R₁ and the pressure there is Pin