Prediction of Coating Thickness

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Hilary Foster
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1 Prediction of Coating Tickness Jon D. Wind Surface Penomena CE 385M 4 May 1

2 Introduction Tis project involves te modeling of te coating of metal plates wit a viscous liquid by pulling te plate vertically troug a pool of te liquid. Te goal of te modeling is to predict te coating tickness as a function of te operational parameters in te system (plate velocity, temperature gradients, viscosity, density, and surface tension). Te liquid coating will cure or arden by te application of eat. To speed up te cure time, a proposed idea is to eat te plate by conducting eat troug it. Te concern is tat because of te temperature gradients, te coating tickness may be affected because of a flow field derived from surface tension gradients. Tus, te momentum and energy balances are coupled. A scematic of te system is sown in Figure 1. Gravity γ 2 U Apply constant eat flux at z=l T 1 Fluid T 1 > T 2 L z Viscous Stress Viscous Liquid T 2 Z= x Figure 1: Scematic of Coating Apparatus Assumptions Coating ardens at z > L (out of region for modeling) No temperature gradients for z > L No eat loss from edges of liquid coating to surrounding air Linear temperature profile from z = to z = L Tin coating (dt/dx = ) Newtonian fluid Laminar flow (no rippling) Fully developed flow at z = 2

3 Velocity Profile and Film Tickness For tis system, gravity and surface tension forces are balanced by viscous forces. Te flow is unidirectional in te z-direction, so te Navier-Stokes equation reduces to µ 2 v z x 2 = ρg z We wis to solve for te velocity profile in order to solve for te coating tickness. Te appropriate boundary conditions are: at x =, v z = U at x =, τ xz = γ z = γ z = µ v z x (no slip) (Marangoni stress) We also know te following empirical relationsip for te temperature dependence of surface tension: γ = γ 1 T T C For many systems n = 11/9 but we will assume tat n = 1 for simplicity. n Tis leads to te velocity profile Te constants of integration are: v z = ρg 2µ x2 + C 1 x + C 2 C 1 = 1 γ µ z + ρg and C 2 = U v z = 1 µ ρgx 2 + γ 2 T c z + ρg x + U (1) Te dimensionless velocity profile for typical operating parameters (sown in Table 1 on page 5) is sown in Figure 1: 3

4 Figure 1: Velocity Profile for Typical Coating Operation Te tin film equation, sown in Equation 2, can be used to estimate te film tickness. t + v zdx = (2) For te steady-state value of, we can set d/dt equal to zero and solve for. d v z dx d = (3) By inserting te velocity profile from Equation 1, te integral in Equation 3 is evaluated below: 1 µ ρgx γ T c z + ρg x + U dx = ρg3 3µ 1 γ 2µ T c z 2 +U and is found by: or = b ± b 2 4ac 2a were a = ρg µ and b = γ µt c z and c = U (4) Te positive root for is te only one tat is pysically realizable. 4

5 Stress Balance We wis to find a dimensionless number tat describes te effect of Marangoni forces on te film tickness. In te absence of surface tension, te film tickness can be calculated by Equation 5. = µu 1 / 2 ρg (5) Tis equation was derived in a similar fasion to tat of Equation 4, except tat te sear stress is zero at te liquid/air interface. Tis represents a balance between viscous and gravitational forces, as sown below: ρg = µ v z x = µ U Wen Marangoni stresses are considered, viscous forces are balanced by gravitational and surface tension forces: ρg + γ z = µ U As a result te dimensionless Wind number 1, is described in Equation 6. Wi = γ / z ρg (6) Figure 2 sows ow te film tickness is affected by te Marangoni stress were te tickness is normalized relative to te case were tere are no Marangoni stresses. 1 Not well known in te literature 5

6 γ ρ Figure 2: Effect of Marangoni Stress on Film Tickness Practical Implications Typical operating parameters for a coating operation are sown in Table 1: Table 1: Operating Parameters for a Typical Coating Operation Plate Velocity (U ) Viscosity (µ) Density (ρ) γ T c dt/dz Wi (dγ/dz)/ ρg Coating Tickness m/s cp g/cm 3 dyne/cm K C/m dimensionless mm E Te value of te Wind number is so small tat te Marangoni effect as a negligible effect on te coating tickness (see Figure 2). Tere would ave to be a uge surface tension gradient to significantly affect te coating tickness. 6

Homework #4 Solution. μ 1. μ 2

Homework #4 Solution. μ 1. μ 2 Homework #4 Solution 4.20 in Middleman We have two viscous liquids that are immiscible (e.g. water and oil), layered between two solid surfaces, where the top boundary is translating: y = B y = kb y =