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					        Mathematical modelling of curtain coating for the paper industry
                                                        Rosemary Dyson (OCIAM)
                 with Peter Howell, Chris Breward (OCIAM), Peter Herdman (ArjoWiggins), and Tim Boxer (Smith Institute)

                                                                                                     1      What is Curtain Coating?
                               Slot
                                                                                                     Curtain coating is a process used mainly in the photographic industry to coat photographic
                                                                                                     film. ArjoWiggins would like to transfer this technology to coat the paper which they make.
                                                                                                     A reservoir of coating mix, which is typically an aqueous solution containing 20-50% solids
                                                                                                     and surfactants, is formed into a curtain of fluid using either a slot or a slide, as shown in
                                Fluid in the curtain falling onto the substrate                      Figures 1 and 2 respectively. A substrate or “web” is passed very quickly underneath and the
                                                                                                     curtain falls under gravity until it hits the web. The coated substrate is passed through driers
                                and being pulled along.                                              to evaporate off the water and thus the finished paper is left with a uniform solid coating. The
                                                                                                     coater machines have edge guides with water running in them to fix the width of the curtain
                                                                                                     as, without them, the curtain is observed to “neck in” laterally.
                                                                                                         Current methods used for coating include blade coating and air-knife coating.

                                                                                                     Curtain coating advantages
                                                                               Moving substrate
                                                                                                         • Substrate can be moved very fast - 10ms−1
                                                                                                         • Low shear to substrate
                                                                                                         • Low coat weight can easily be applied
                     Figure 1: A curtain formed by a slot.
                                                                                                         • Coating is surface following
                                                                                                         • Possibility of multi-layer coating - where a curtain is formed using a slide as in Figure
                                                                                                           2 and applied to the substrate as for the single layer case. This has many advantages
                                                                                                           such as being able to apply several layers simultaneously, reducing the time to produce a
                                                                                                           finished product, or applying a very thin layer of expensive mix by using a cheaper one as
Slide                                                                                                      a base to form a curtain. The photographic industry can coat with curtains consisting of
                                                                                                           upwards of 15 layers, but these layers tend to comprise of the same mix, just with different
                                                                                                           colour pigment in them, whereas ArjoWiggins would like to be able to coat several layers
                             A curtain with several different layers of mix.                               of widely varying rheologies.

                                                                                                     Disadvantages/Problems
                                                                                                         • Stability of curtain
                                                                                                         • Air entrainment - when the substrate is moving too fast whereupon the fluid in the curtain
                                                                                                           does not wet the substrate properly, leads to streaks/bubbles/pin holes
                                                                                  Moving substrate       • Skip coating - where the mix bounces off the web
                                                                                                         • Formation of a heel - leads to recirculation, and so ageing and degradation, of the coating
            Figure 2: A multilayer curtain formed on a slide.                                              mix
                                                                                                         • The “teapot” effect - also leads to recirculation
2      Steady state model                                                                                • curtain moves at substrate speed: u = U at s = l
                                                                                                                                                     q
                                                                                                         • velocity of fluid at coater set: u =      h0
                                                                                                                                                         at s = 0
                             h0
                                                    Coater
                                                                                                         • distance of coater from substrate gives a condition for the (unknown) curtain length:
                                                                                                             l
                             s=0
                                                                                                            0
                                                                                                               sin θ ds = −H
                                   h

                                                                                                      Nondimensionalisation We now dot the equation (2) with the normal and tangent vectors,
                                                                   s (arc length)
                                                                                                      rearrange our equations to give the derivatives explicitly, nondimensionalise using
                      H                        −θ


                                                                                                                                     ˆ                                          ρgql ˆ
                                                                                                                              h ∼ h0 h,             ˆ
                                                                                                                                              u ∼ U u,              s
                                                                                                                                                               s ∼ lˆ,   T ∼        T
                                                                                                                                                                                 U
                                                             s=l

                                   ρg
                                                                                                      and drop hats to give
                                                                                                                                                       q
                                                                                                                                               uh =         = v,                                                  (4)
                                                                                                                                                      U h0
                                                                                                                                                      Re
                                                                                                                                               u    =      T u,                                                   (5)
                                                                                                                                                      F r2
                                                             U                                                                                                   sin θ
                                                                                                                                               T    = Re T u +         ,                                          (6)
                                                                                                                                                                   u
                           Figure 3: The coordinates in the curtain.                                                                                       cos θ
                                                                                                                                                θ   =                  ,                                          (7)
                                                                                                                                                      u (T − F r2 u)
    • Uniform flow across the thickness of the curtain since we have a small aspect ratio              where
       = h0 /l ≈ 0.01
                                                                                                                                        ρU l
    • Parametrise with respect to arc length s along the centre-line of the curtain                                                Re =      is the Reynolds number,
                                                                                                                                         4µ
    • θ = angle, measured upwards, that the centre-line makes with the horizontal                                                        U
                                                                                                                                   Fr = √    is the Froude number,
                                                                                                                                          gl
    • T = the tension from the viscosity
                                                                                                      and v is the draw ratio. Note here, we are nondimensionalising with the unknown value of l
    • l = the a priori unknown length of the curtain
                                                                                                      and so finding Re and F r is part of the problem, we will have to “back them out”.
    • q = the constant volumetric flux supplied through the nozzle of the coater                          The boundary conditions give

    Conservation of mass and momentum immediately give us                                                                                         θ = 0, u = 1 at         s = 1,                                  (8)
                                                                                                                                                        q
                                                                                                                                                  u=        = v at        s = 0,                                  (9)
                                                   hu = q,                                      (1)                                                    h0 U
                                                                                                                                              1
                 d        cos θ                  0       d                          cos θ                                                                    H
                      T                + ρgh          =                  ρhu2               ,   (2)                                             sin θ ds = − .                                                   (10)
                 ds       sin θ                 −1      ds                          sin θ                                                   0                l

Here the first term on the left hand side of (2) is the tension force, the second term is the          Immediate observations
gravity force and the right hand side is the momentum flux.
                                                                                                                                  dx                  dy
   We assume the fluid is Newtonian for a constitutive relationship, i.e.                                 • Centre-line given by   ds
                                                                                                                                       = cos θ and    ds
                                                                                                                                                           = sin θ.

                                          T = 4µhu ,                                            (3)                                     1                  1
                                                                                                         • Free edges given by (x ±     2
                                                                                                                                            h sin θ, y ±   2
                                                                                                                                                               h cos θ) where     = h0 /l is the aspect ratio.
                                     d
as in the Trouton model, where = ds . Other properties of the fluid such as surface tension or            • By adding the T and u equations and integrating we get
viscoelasticity could be incorporated by modifying this relationship.
    For boundary conditions we assume                                                                                                          (T − F r2 u) cos θ = constant.

    • curtain arrives tangentially at substrate: θ = 0 at s = l                                            This is a equivalent to a horizontal force balance and so does not exist when θ = ±π/2.
3     Inertia-dominated limit
                                                                                                                                          1

The parameter regime in which ArjoWiggins is interested is F r ≈ 10, Re ≈ 1000 and so we
consider the limit Re  1, F r = 0(1).
                                                                                                                                      0.8


Boundary conditions For a jet of water emerging from a nozzle, we prescribe θ(0), but we
cannot “pull” the jet to prescribe θ at the contact point. So here, in this similar case, we expect                                   0.6

to prescribe θ(0) not θ(1). We have confirmed this rigorously by formulation of the unsteady                                       u
model. This showed that where we apply the θ boundary condition depends on the sign of the                                            0.4
constant T − F r 2 u, where u is the velocity of the fluid minus the velocity of the arc-length
                   ¯         ¯
based coordinate system, i.e. whether we are in the inviscid- or viscous-dominated limit.
                                                                                                                                      0.2


Solution
                                                                                                                                              0    0.2   0.4       0.6   0.8    1
    • We expect the velocity of the fluid to be closer to the free-fall velocity, so we rescale                                                                 s
          ¯
      u = u/F r
                                                                                                                               Figure 4: The composite expansion for u.
                               ¯
    • Also need to rescale T = T /Re
                                                                                                                                      2
    We only give our work on the simplest case in which θ(0) = −π/2. Then the θ equation is
satisfied identically by θ(s) = −π/2, and taking the limit Re   1, F r = 0(1) gives us
                                       2            √
                                 ¯ Fr
                                                                                                                                  1.5
                                 T = 2 ,       u=
                                               ¯     2s + v 2
                                                          ¯                                   (11)
                                     ¯
                                     u
       ¯
where v is the rescaled, nondimensional initial velocity. This is precisely a free fall velocity.                                     1
In this limit the substrate does not have any effect on the flow of the curtain. However, the                                   T
                                                        ¯
solution (11) does not satisfy the boundary condition u(1) = F r, so we look for a boundary
layer near s = 1 across which the velocity adjusts. We find we must return to our original T
                                                                                                                                  0.5
                                     ˆ
and u scalings, and rescale s = 1 − s/Re. Then taking the limit Re → ∞ we see that θ = 0
                                    ˆ
and, since we require θ → −π/2 as s → ∞ to match, we must have
                                                π                                                                                     0
                                             θ≡− ,                                                                                        0       0.2    0.4       0.6    0.8       1
                                                2                                                                                                              s
i.e.   the curtain can never turn around onto the substrate in this one-dimensional model.
Solution of the equations with appropriate boundary conditions yields                                                          Figure 5: The composite expansion for T .

                                                ˆ
                                           V eV s                                                     Conclusions
                               u =        ˆ
                                                    ,
                                       eV s + V − 1                                                     • The curtain can never turn from the vertical in this over-simplified one-dimensional model.
                                                            ˆ
                                         2             V eV s                                             Since the curtain clearly does turn, we must therefore consider a two-dimensional region
                              T = Fr          −V + V s        ,
                                                  e ˆ+V −1                                                close to the impingement point in which θ goes from − π to 0.
                                                                                                                                                                    2
                                    π
                               θ = − ,                                                                  • In this inertia dominated limit we can see that the substrate does not exert any influence
                                    2                                                                     over the fluid in the curtain until we reach a boundary layer in which the viscosity of the
            √
                2
              ¯ +2                                                                                        fluid takes effect. In this boundary layer the tension comes into play and the velocity
where V = v r is the impingement speed into the boundary layer. If we plot a composite
              F                                                                                           increases sharply, but away from this region the tension is negligible and we have a free
                                                                                ¯
expansion of the outer and inner solutions, taking Re = 1000, F r = 10 and v = 0.6 as shown               fall velocity.
in Figures 4 and 5, we see that the velocity increases slowly until it reaches the boundary near
s = 1 where it increases sharply to the substrate speed. We also see that the tension is virtually      • In practice the substrate speed is important for the stability of the process, but this may
zero and decreasing until the boundary layer is reached where it too increases sharply as the             have more to do with air entrainment. Also, it is not clear whether our assumption of
substrate exerts influence.                                                                                no-slip on the boundary is correct.
4     The air flow behind the curtain                                                                                               1.4
                                                                                                                                                              θ=α=     π
                                                                                                                                                                       4



We investigate the air flow in a wedge with moving walls as a possible mechanism for air                                            1.2

entrainment. We consider both the viscous- and inertia-dominated air flows, but do not match                                         1

them together yet.
                                                                                                                                   0.8


                                                                                                                                   0.6



                                                              ur = −V , uθ = 0                                                     0.4


                                                                                                                                   0.2


                                                                                                                                    0
                                                                                                                                         0         0.5    1      1.5           2
                                                                                                                                                                                    θ=0



                                             α
                                                              ur = −U , uθ = 0                                  Figure 7: The streamlines for the viscous-dominated air section.
                                                                                                                                                                  θ = α = π/4
                                                                                                                                         1.4


                              Figure 6: The Stokes’ flow setup.                                                                           1.2


                                                                                                                                          1

Viscous-dominated air flow Upon solution of Stokes’ equation in a domain as shown in
                                                                                                                                         0.8
Figure 6, assuming the curtain is a straight line moving with constant velocity, and that the
air is incompressible we find the streamlines as plotted in Figure 7, U = V = 1. We see the                                               0.6



air is pulled in towards the impingement point by the substrate and the curtain, whereupon it                                            0.4


turns and flows out along the line of symmetry.                                                                                           0.2

    The pressure and stress caused by the air are then of the form
                                                                                                                                          0                                        θ=0
                                                                                                                                               0    0.5   1     1.5        2


                                      2Cµ            2Aµ
                                p =          cos θ +      sin θ,                         (12)
                                        r             r
                                      2µ                                                                           Figure 8: The streamlines of the inviscid flow taking α = π/4.
                              τrθ   =      (B sin θ + A cos θ) ,                         (13)
                                       r
                                         2µ                                                     5     Future Work
                              τθθ   = − (A sin θ + B cos θ) ,                            (14)
                                          r
                                                                                                Impingement zone
where A, B, and C are known constants.                                                              • Air entrainment - Including physical properties of the substrate such as porosity, and
                                                                                                      including air forces into the curtain model and coupling these with the air flow equations
    • 1/r type singularity as r → 0. Close to the impingement point the pressure and stress
      caused by the air flow need to be taken into account in our model for the flow in the           • How the fluid turns onto the substrate - We have shown we need a two-dimensional
      curtain.                                                                                        region for the fluid to turn onto the substrate. Some preliminary work on this has been
                                                                                                      completed.
    • Stokes’ flow approximation is only valid when Re = U l/ν is small - i.e. up to 10−6 m
      from the impingement point.                                                               Curtain stability
                                                                                                    • In what operating conditions will the curtain remain stable?
 Inviscid air flow Repeating a similar calculation for the inviscid air flow, with U = −V = 1
and taking a line of symmetry θ = β = α/2 we find the streamlines as shown in Figure 8.              • Surface tension - Effect of surfactants and whether slot or slide coater (which gives unequal
We see here that the air is pushed out along the line of symmetry before being pulled back in         surface age and so unequal surface tension)
towards the impingement point by both the substrate and the fluid in the curtain.
   Upon calculating the pressure we find                                                         Multilayer coating
                                         1   π 2 D2                                                 • Photographic industry can apply up to 15 layers simultaneously, but their layers have
                                    p = − ρ 2 2(π/β+1) ,                                 (15)         very similar rheologies
                                         2 β r
                                                                                                    • Layers of widely different rheology could be analytically and practically interesting
where D is a known constant, and so p → −∞ as r → 0; the air pressure is “sucking” the
curtain down onto the substrate. Note this is the opposite of the viscous-dominated air flow.        • Stability?

				
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