3.2.2.2.1. Aims of current study¶

1. Link between deposit-generated streak-line defects and nature of associated underlying flow structure

2. Provide data to assist in predicting whether defects will occur

Physics Investigated:

• Single and multi-layer flows

• Effect of slot exit

• Slide lip geometry

• Phenomena of back-wetting caused by static contact line migration past the upper slot corner

3.2.2.3.1. Two dimensional flow apparatus¶

Geometry variation:

• Square edge

• Chamfered edge

Method for visualization:

• Hydrogen bubble technique

• Local velocities measured by including a chopper in the illumination

• Deposit growth mimics by injecting a hardening agent into the flow

3.2.2.3.2. Cylindrical flow apparatus¶

Variables:

• Surface finish

• Top corner profile

• Step height

3.2.2.3.3. Method for visualising specific regions¶

• Bouncing probe method used to calculate surface profile at slot exit

• Upper meniscus: Same method used in bead forming region between lip and moving web

• Lower meniscus: Illumination with a sheet of light from a laser diode source

• Profile of interface was determined by casting a shadow of a surface bubble track (the lower layer scattered light, which helped)

• Streakline formation was assessed by: coating many rolls under known conditions and recording the average number of sharp streak-lines per unit width.

3.2.2.4.1. Overall equations and assumptions¶

Assumptions:

• Steady

• Isothermal

• 2D

• Incompressible

• Inelastic

• Coating liquids are often shear thinning however these effects are small - so assuming a Newtonian fluid is valid

Equations:

• 2D Navier-Stokes

• Continuity Equation

Dimensionless Numbers:

• Reynolds Number

• Stokes Number

Complications:

• Presence of free surfaces

• Internal interfaces

For multi-layer systems, only a double layer system is considered as it shows the key features, namely:

• Upper and lower free surfaces

• One internal interface

• Presence of both static and dynamic contact lines

Approach is a Galerkin weighted residual formulation

3.2.2.4.2. Single-layer slot exit flow¶

Geometry:

• Chamfered downstream corner

• Raised back slot

• Modified slot exit channel

Non-dimensional version of Navier-Stokes, length scale and velocity scale are:

• Slot width

• Flow rate per unit width / Slot width

The velocity profile at the inlet is given by:

• 1D form of Navier-Stokes Equations

Outlet boundary obtained using:

• Stress evaluation method

Two cases:

1. Static contact line is pinned at the corner, while static contact angle at slot exit is predicted

2. Static contact angle is specified, while static contact line is predicted

3.2.2.4.3. Double-layer slot exit flow¶

More complex than single-layer slot exit flow, it has:

• Internal interface - the nature of the boundary conditions which apply at an internal static contact line still remains a matter of debate

• Inter-layer diffusion occurs

Assume:

• Diffusion process is negligible

• No interfacial tension - velocity is continuous, but pressure discontinuous

Difficulty:

• Lack of interfacial tension results in an over-specified problem if the contact angle is imposed

Solution 1:

• If interfacial contact line is known to locate close to the corner, the preferred course of action is to pin it there

• If interfacial contact line locates away from the corner, solutions include:

  • geometric extrapolation

  • imposition of pressure continuity

  • imposition of small interfacial tension which decays to zero within a short distance downstream, this enabling the specification of a contact angle

Macroscopic flow field is independent of the choice of method, however adding small interfacial tension was chosen

Boundary conditions:

• Velocity profile at slot inlet is obtained as for single layer case

• Outflow determined by using stress evaluation method for 2 layers

• Inflow at upper slot determined from solution of 1D Navier Stokes solution

3.2.2.4.4. Bead forming flow¶

Boundary conditions:

• Inlet velocity profile obtained from 1D solution to Navier-Stokes equations

• Zero traction condition applied at outlet

Physics:

• Simplification is possible via lack of internal contact line

• Lower free surface meets slide feed device at static contact line

• Lower free surface meets moving web at dynamic contact line

• Static contact line is rarely pinned to the tip of the slide feed device, so is computed

• Static contact angle is prescribed

Difficulty:

• No slip hypothesis breaks down near dynamic contact angle

Solution:

• Dynamic contact angle is prescribed

• This allows slip between the web and the liquid close to the dynamic contact line

3.2.2.5.1. Single layer slot exit flow¶

Eddy physics:

• The top, layer forming slot can be responsible for disturbances

• Flow near the static contact line would be recirculating in nature

• Agreement between theory and experiment is good

• A slow recirculation near the contact line is predicted when a high backstep is used

Experimental proof of eddy presence causing streak lines:

• Gelatin containing hardening agent introduced into flow

• Hardened deposit found in eddy generating region

• Mechanism for generating streaklines was the gradual formation of brittle deposit due to long residence time associated with eddy presence.

• Deposits were dislodged by periodic cleaning to form minute fragments, which then produced streaklines

Back wetting: migration of contact line upstream:

• Even a small increase in flowrate causes back wetting

Implications of back wetting:

• Can cause the contact line to be other than straight

• Can cause recirculation near the static contact line

• Both of the above are potential for defect formation

Axi-symmetric rig:

• Question What are the maximum sustainable flow rates before back wetting occurs for three different slot exit designs?

Method 1 in axi-symmetric rig:

• Cut back the top face to form a sharper edge

• Use a hydrophobic coating on the top face

Advantages of this:

• Approximately double the maximum sustainable flow rate before back wetting occurs

Disadvantages:

• Sharp top edge is impractical, due to hazard and difficult machining

Method 2 in axi-symmetric rig:

• Height of slot is reduced

• Upstream of slide coated to prevent back wetting

Advantages of this:

• Causes static contact angle to increase to about 90 degrees

• Substantial reduction in deposit growth

• Question How is the flow structure affected by flow properties?

• Reducing viscosity increases eddy size near downstream slot exit

• Increasing flux also increases eddy size

Operability diagram:

• Parameters: Reynolds Number and Capillary Number

• Indicates operability window for a particular slot geometry

• Desirable range: Low Re and Medium to High Ca

Operability diagram \(\Rightarrow\) to coat a low viscosity liquid, the slot geometry must be changed to enable eddy free regime

Geometric changes:

1. Chamfered slot exit \(\Rightarrow\) recirculations are going to be small

2. Stepped slot:

   • Decrease in liquid velocity and its associated momentum due to widening the slot in turn leads to a reduction in size of eddies

   • Widening slot on upstream side is better than widening slot on downstream side

   • Sharp step may generate eddies \(Rightarrow\) widening slot gradually or using chamfer is better

3.2.2.5.2. Slot exit flow for two merging layers¶

Comparison with numerical simulations and:

1. Two layer flow experiments for the case of a chamfered slot with identical layers of aqueous glycerine

2. Free surface and internal interface profiles for flow out of a chamfered slot

Good comparison.

A parametric study showed that fluxes and viscosities have greatest influence on flow structure:

• When upper flux \(\lt\) lower flux, then increases in upper flux cause minimal change

• When upper flux \(\lt\) lower flux, then eddies occur upstream

Geometric study:

• Small chamfer reduces size of downstream eddy

• Larger chamfers reduce size of upstream and downstream eddies to undetectable sizes

Curved diffuser:

• Flow is free from eddies

• Layer thicknesses change smoothly without humps observed previously

Why don’t they use curved diffusers? Too expensive? Hard to clean/manufacture?

Maybe it’s the sharp edge - could be a hazard?

3.2.2.5.3. Bead forming flow¶

Validation of numerical model against:

• Experimental upper and lower free surface profiles

Eddies can occur in bead forming region and are increased by:

• Coating gap increase

• Suction applied to lower meniscus is increased

Eddies cause:

• Prolonged residence time of coating liquid within the recirculation region, resulting in deposits forming at the lip face

• The deposits become grazed and break off by the passage of the web splices

• This leads to sharp streak lines

The model requires:

• Sufficiently refined computational mesh in the lower free surface region

Physics:

• Correlation between primary eddy strength and number of sharp streaklines per metre width

Affected by:

• Static contact angle

• Lip radius

Practical advice:

• Maintain a lip free of mechanical imperfections

• Ensure the surface properties of the lip are such that a high static contact angle is possible

• Lip radius is also an important factor

3.2.2.6.1. In slot exit region¶

• To minimise/prevent eddies modify slot exit geometry using curvature, chamfers and/or a diffuser

3.2.2.6.2. In bead forming region¶

• An area close to static contact line at the junction of the lower free surface and underside of lip surface is where eddies form

• Eddies will exist near the static contact line if the static contact angle is less than 40 degrees \(\Rightarrow\) regular cleaning of the surface of the underside lip region is needed to avoid solid deposition

• Or increase static contact angle with a pre-surface treatment, thus weakening eddy structure.