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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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Curtain coating, Oxford University, St Catherine's College, Oxford, paper manufacture, boundary conditions, boundary layer, Industrial Mathematics, Smith Institute, the ﬂuid, air ﬂow

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posted: | 9/14/2010 |

language: | English |

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