# 3.2.1. Thompson et al. (2001) “Investigation of Rigid-Roll Coating”¶

Thompson, H.M., Kapur, N., Gaskell, P.H., Summers, J.L. and Abbott, S.J. “A theoretical and experimental investigation of reservoir-fed rigid-roll coating”, Chemical Engineering Science, vol 56, 2001.

## 3.2.1.1. Abstract¶

A variant of reverse roll coating is studied via:

- Experimental
- Analytical (lubrication)
- Computational (finite element)

Experimental results:

- Flow rate and wetting line position over range of roll speed ratios and capillary numbers
- Provided the wetting line is sufficiently far from the nib, the flow rate depends linearly on reservoir level

What is unique about this study:

- Variation of dynamic contact angle with metering roll speed has been accounted for

Lubrication model boundary conditions:

- Free surface
- Surface tension
- Wetting line effects

Physics:

- The effect of gravity is influential
- Flow in the reservoir is recirculating in nature
- The size and number of recirculations depends on reservoir geometry

Good agreement between models and experimental data

## 3.2.1.2. Introduction¶

Rigid-roll coating systems operating in reverse mode are used for applying thin liquid films, in the production of:

- Magnetic media
- Adhesive tapes
- Films
- Foils

Types of rigid-roll coating systems:

- Pan-fed
- Direct-fed

Physics:

- Flow in the
**positive metering gap**between the co-rotating applicator and metering rolls plays a crucial role in determining the**final wet film thickness transferred to the substrate**

Discrepancy between FE model and experiment:

- Neglect of gravity
- Absence of dynamic wetting line

Discrepancy between lubrication model and experiment:

- At high speed ratio and capillary number
- Importance of the position of the dynamic wetting line relative to the nip - w.r.t onset of cascade instability

Other effects:

- Effect of feed condition on the flow structure

Direct-fed reverse roll coating is investigated. The web thickness is determined by:

- Competition between metering action of the reverse roll configuration
- Influence of gravity in the form of hydrostatic head provided by the reservoir

### 3.2.1.2.1. Aim of this paper¶

Quantify the effect of the reservoir on:

- The coated film thickness
- Dynamic wetting line position

## 3.2.1.3. Experimental Method¶

A rig was designed to:

- Retain the important features of the industrial model
- Be flexible in operation
- Allow optical access

Conditions:

- Large bank of liquid ensured that ambient temperature changes did not affect liquid properties
- 0.5kg weight was applied to the hosing of the tank to seat it on the rolls
- Doctor blade used to remove any residue

Fluid:

- Newtonian mineral oil
- Surface tension measured with a torsion balance
- Viscosity measured with a viscometer
- Density measured by measuring the mass of a known volume of fluid

Visualisation:

- Liquid illuminated
- Illumination of nip region achieved using a mirror
- Flow patterns recorded on a video system via a microscope and CCD camera

### 3.2.1.3.1. Flux and wetting line location measurements¶

Non-contacting methods for measuring film thickness:

- Infra-red absorption
- Microwave absorption
- X-ray fluorescence
- Capacitance probes

Problem with non-contacting methods:

- Not possible to deduce an exact flux due to variations in velocity through the film

Solution:

- Web was scraped clean of liquid using a twin scraper blade
- A mass of liquid was collected over a known time interval, hence flux determined

Wetting line:

- Position of wetting line measured using microscope fitted with a cross hair graticule

### 3.2.1.3.2. Flow visualisation¶

Purpose:

- Highlight flow patterns near dynamic wetting line
- Show how the flow patterns depend on operating parameters

Hydrogen bubble technique impractical due to:

- buoyancy effects
- long residence times
- difficulty of constraining a bubble stream to a narrow planar section

Used dye injection method instead (a laser was not needed)

Flow in reservoir was visualised by:

- discharging dye continuously by traversing flow field at stagnation points
- dye was not confined to a single streamline

Flow in nip region was visualised by:

- releasing a pulse of tracer liquid onto metering roll just upstream of dynamic wetting line

## 3.2.1.4. Mathematical Models¶

Assumptions:

- Isothermal
- 2D
- Newtonian
- Incompressible
- Governed by the Navier Stokes Equations

Two approaches:

- Lubrication model:

- A new approach - where variation of contact angle will roll speed is predicted rather than prescribed

- Numerical model:

- Finite element method for full non-linear problem
- Algebraic mesh generation algorithm

### 3.2.1.4.1. Lubrication Analysis¶

Assumptions:

- Uni-directional
- Away from free surface
- One-dimensional

Physics:

- Used non-dimensional scaling
- In the past the influence of wetting lines has been ignored in previous free surface models
- Lubrication theory is inapplicable in the vicinity of the downstream free surface and the solution domain is terminated by introducing appropriate boundary conditions

#### 3.2.1.4.1.1. Boundary Conditions¶

- At the free surface, the pressure is equal to the capillary pressure due to surface tension
- We assume that the free surface forms an arc of a circle, so it is a function of the dynamic wetting line angle
- Previous studies didn’t model the idea that the dynamic wetting line angle varies with metered roll speed, however
**the nature of the flow near the dynamic wetting line is still a matter of debate** - The variation is accounted for using a hydrodynamic asymptotic model for wetting
- The asymptotic model is based on the assumption that the free surface is planar near the wetting line and requires the velocity of the fluid in the liquid-gas interface to be a function of the dynamic wetting line angle
- A calibration procedure is adopted in order to estimate the variables in the above in terms of a single adjustable parameter - the interfacial thickness.
**Further experimental data is needed to verify the accuracy of the estimates for the parameters** - The calibration results in two equations in terms of Bond Number (which measures the relative importance of gravitational to surface tension force) and Capillary Number
- A third equation is derived to relate the flux to the radius of curvature of the meniscus at the wetting line
- Newtonian iteration is used to solve the three equations

Limitation:

**Lubrication theory is inapplicable near the meniscus where the flow is 2D and often recirculating**

This requires fully non-linear 2D finite element simulation

### 3.2.1.4.2. Finite Element Solutions¶

Flow in the reservoir is solved using FE method (due to it’s topological flexibility)

Models:

- Extends from top of the reservoir into the nip region
- Extends from the nip region to the downstream free surface

Solution method:

- The governing equations are non-dimensionalised w.r.t. velocity, length and pressure
- Solved using Bubnov-Galerkin weighted residual FE formulation

Problems:

- Obtaining FE solutions in the reservoir region is straightforward since the geometry is fixed.
- Flow in the downstream region is more complex because of the existence of the meniscus and the associated wetting line

**Downstream Model**

Solution to boundary conditions for downstream model:

- Boundary conforming mesh based on spine approach
**Conditions at the wetting line are controversial**

Solution:

- Dynamic wetting angle is predicted and fluid is allowed to slip on the roll surface for those nodes adjacent to the wetting line

**Reservoir Model**

- Velocity profile at nip is the same as that specified in the downstream model
- Reservoir surface set to atmospheric pressure
- Parabolic velocity profile set at the surface to ensure mass conservation

**Solution**

- Frontal Method
- Newton iteration
- Leading to second order convergence

Grid refinement was also completed

## 3.2.1.5. Results¶

Variables of interest:

- Flow rate
- Wetting line position is useful w.r.t cascade instability when it migrates upstream of the nip

Two experiments:

- Peripheral speed of the applicator
**fixed**and metering roll speed**varied**. Three different reservoir levels. Capillary number**fixed**, speed ratio**varied** - Peripheral speed of metering roll
**fixed**and applicator roll speed**varied**. Capillary number**varied**and speed ratio**varied**

In both sets of experiments Bond Number is **fixed**

- Experiment 1

- Shows q decreasing as S increases until a critical value is reached, beyond which q increases. This is caused by the wetting line moving upstream of the nip, which increases the effective gap at the meniscus.
- Effect of reservoir level depends on position of wetting line w.r.t. nip - i.e. whether it’s upstream or downstream of the nip

- Experiment 2

- Shows q increases as Ca increases
- Wetting line is located downstream of the nip for Ca > 0.2

**Validation of model:**

- q and wetting line position are validated with good agreement

Results also show the sensitivity of the wetting line position to dynamic wetting line angle

**Problem** we are unable to predict wetting line positions at the higher Ca numbers - possibly due to Bretherton condition
**Suggesting** flow rate insensitive to this condition, whereas wetting line position is more sensitive.
**Reasoning** Circle assumption is not a good representation of the free surface shape for higher Ca numbers - it’s probably more parabolic

Effect of head:

- linear dependence between q and head - where wetting line is safely downstream of the nip

Flow inside the reservoir:

- Recirculations can cause difficulties for the lubrication model because the rectilinear assumption is violated in such regions

## 3.2.1.6. Conclusions¶

- This is a study of the equilibrium flow in a variant of reverse roll coating, where the metering gap sits beneath a large reservoir
- Experimental data shows:

- If wetting line is sufficiently far downstream of the nip: flow rate increases
**linearly**with reservoir level - If wetting line is close to the nip: effect of level on flow rate is
**non-linear**and influences the onset of the**cascade instability**

- Hydrodynamic model:

- Dynamic contact angle with metering roll speed is incorporated using a hydrodynamic model for wetting.
- However, data is lacking for the hydrodynamic model, so a calibration method is proposed, in terms of the interfacial layer thickness.
- This interfacial layer thickness is calibrated by matching the prediction against theory for one data point only.

- FE solutions can predict accurately:

- Flow rate
- Wetting line positions (even at high Capillary Numbers)

- Lubrication model predicts the flow rate over a range of Capillary Numbers, however:

- Wetting line prediction is sensitive to the way the free surface is represented
- Free surface profile should be represented by a parabolic rather than a circular arc
- But the lubrication prediction demonstrates the benefit of incorporating the effects of free surface, surface tension and the wetting line - especially the ability to predict the flow rate minimum caused by wetting line migration through the nip

- Experimental results vs FE solutions show good comparison
- Flow in reservoir should not be overlooked - given the possibility of recirculations there