Various industrial processes require that a flat sheet be coated with a uniform liquid film. Dip coating is a procedure in which an immersed substrate is withdrawn through a liquid–gas interface. In the process in Fig. P1.6, in which a substrate is pulled upward at a constant velocity V, the coating achieves a uniform thickness H at a certain height above the liquid. It is desired to predict H from V and the liquid properties (μ, ρ, and γ).

(a) Representative values are H = 0.02 to 0.2 cm, V = 0.4 to 3 cm/s, μ = 0.3 to2 Pa · s, ρ = 900 kg/m³, and γ = 30 mN/m. By examining the ranges of dimensionless groups, show that viscosity, gravity, and surface tension are all important, but inertia is not.

(b) A theoretical result that is applicable here is (White and Tallmadge, 1965)

Show that this may be rewritten as a relationship between Bo and Ca.

(c) If one group is chosen now as N = (Bo/Ca)^1/2 = [H²g/(νV)]^1/2 and Ca is the second group, show that

Use this to derive explicit expressions for N for Ca → 0 and Ca → ∞. How does H vary with V in these limits?

Transcribed Image Text:

VT Solid H Air Liquid Figure P1.6 One side of a solid sheet being withdrawn upward from a liquid.