Flexible blade coating.
Abstract Flexible blades are often employed to spread liquid
coating thinly onto a flat smooth substrate. In this study, we derive a
fifthorder nonlinear ordinary differential equation for the thickness
of a Newtonian coating and for the corresponding blade deflection. After
solving this equation numerically, a graph is produced to help engineers
predict the coating thickness. We find that blade deflection and coating
thickness are governed by the blade angle and by a new dimensionless
group called the blade flexibility. For values of blade flexibility less
than one, the coating thickness matches that of a rigid blade. The
results of this analysis provide an engineer with the ability to design
a flexible blade configuration to deliver the desired coating thickness
onto a substrate.
Keywords Flexible blade coating, Squeegee Introduction The use of flexible blades to apply coatings of specific thickness onto a substrate finds wide use in industry. The simple design and reliability of a flexible blade operation make it an ideal choice for coating application. Table 1 summarizes the literature on blade coating. The heading "Constrained blade" in Table 1 refers to physical restrictions placed on the flexible blade. While prior studies have focused on the constrained blade, here we focus entirely on the unconstrained flexible blade. Table 1: The literature on flexible blade coating analysis Blade Unsupported Constrained Coupling crosssection blade blade Follette and r Fowells (5) Windle and r Beazley(6) Saita and r X Scriven(7) Saita(8) r X Scriven and r x X Pranckh(9) Alheid(10) Saita and X Corvalan(11) Alam and X X Toivakka(12) This article r X X NonNewtonian Fluid 1 Film height Adimensionalization pressure prediction profile Follette and X Fowells (5) Windle and X X Beazley(6) Saita and X X X Scriven(7) Saita(8) X X X Scriven and X X Pranckh(9) Alheid(10) Saita and X x X Corvalan(11) Alam and Toivakka(12) This article X X X FEM Analytical Numerical solution solution of ODE Follette and Fowells (5) Windle and Beazley(6) Saita and X Scriven(7) Saita(8) X Scriven and X Pranckh(9) Alheid(10) Saita and x Corvalan(11) Alam and X Toivakka(12) This article X Legend: r = rectangular Our numerical solution of the fifthorder nonlinear differential equation that governs flexible blade coating provides insight into the factors controlling the coating thickness. This method removes the need for elements that previous FEM solutions utilized to model flexible blade coating. The derivation of a dimensionless group called blade flexibility* is outlined and its effect on coating thickness and blade behavior is studied. Blade flexibility represents the ratio of fluid viscosity to the blade stiffness. We model a flexible, unsupported rectangular blade coating with a Newtonian fluid to explore the effect of blade angle on the coating behavior and blade deflection. A graph of the relation between blade angle, blade flexibility, and coating height is presented. A worked example illustrates how to apply the results by solving a coating problem. All dimensionless variables are defined in Table 2, while the dimensional ones are shown in Table 3. Analysis Figure 1 illustrates the peculiar nature of flexible blade coating, sometimes called squeegee coating. We have a flow through a converging channel where the shape of the convergence is governed by the blade deflection. The distributed load from the coating pressure, exerted under the blade, causes this blade deflection. Since the blade deflection governs the shape of the convergence, the blade deflection thus governs the pressure profile. Problems of this general type are sometimes called elastohydrodynamic (Fig. 2). Table 2 constructs dimensionless groups from the dimensional quantities defined in Table 3. Table 2: Dimensionless groups Coating H=H/ [B.sub.L] thickness Position along L [equivalent to] L/L length of blade Pressure P [equivalent to] W [B.sub.L.sup.3] [rho]/EI Reynolds number Re [equivalent to] [rho] HV/[mu] Blade V [equivalent to] [mu] flexibility VW[L.sup.5]/EI[B.sub.L.sup.3] Blade Y [equivalent to] Y/[B.sub.L] deflection Blade length [beta] [equivalent to] L/[B.sub.L] Poisson ratio v Table 3: Dimensional variables Gap between deflected blade and substrate B Gap between free end of blade (e= 0) and substrate B(0) Gap between undeflected blade position and substrate [B.sub.0] Gap between cantilevered end of blade and substrate [B.sub.L] Young's modulus of flexible blade E Blade stiffness El Final coating thickness H Blade moment of inertia about e axis I Position along length of blade e Undeflected blade length L Pressure P Volumetric flow rate 0 Substrate velocity V Blade width w Position along substrate surface X Blade deflection y Blade base inclination 0 Newtonian viscosity of coating [mu] Coating density p Results For the fluid pressure exerting the distributed load on the flexible blade, we follow a flow analysis for laminar flow between parallel plane surfaces based on Bird et al., example 1.31 pp. 1314(1): Q=1/2 WB(x)VW[[B(x)].sup.3]/12[micro] dp/dx (1) [FIGURE 1 OMITTED] [FIGURE 2 OMITTED] Hence, dp/dx=12[micro]/W[[B(x)].sup.3] (1/2 WB(x)VQ) (2) As fig 1 shows that this pressure causes the normal blade deflection, y(e), and thus governs the gap. The undeflected blade position is given by [B.sub.0](e) [equivalent to] [B.sub.L]cos([theta]) (Le); 0 [less than or equal to] [theta] [less than or equal to] [PI]/w (3) where [theta] is the constant base inclination, and B'(L) [equivalent to] dB/de=tan([theta]) (4) Where B(e) [equivalent to] [B.sub.0] (e)+y(e) cos([theta]) (5) Substituting (3) into (5) yields B(e) [equivalent to] [B.sub.L]cos([theta]) (Le) +y(e) cos ([theta]) (6) And the deflection of the cantilevered blade under its distributed load is given by [d.sup.4]y/d[e.sup.4]=W/EI p(e) (7) Adimensionalizing the blade deflection with Y [equivalent to] y/ [B.sub.L] (8) yields [d.sup.4]Y/d[e.sup.4]=W/EI[B.sub.L] p(e) (9) and position along the blade with L [equivalent to] Le/L (10) so that de=LdL [(de).sup.4]=d[e.sup.4]=[(LdL).sup.4]=[L.sup.4]d[L.sup.4] (11) yields [d.sup.4]Y/d[e.sup.4]=1/[L.sup.4] [d.sup.4]Y/d[L.sup.4] (12) Substituting (12) in to (9) thus yields the following dimensionless blade deflection equation: [d.sup.4]Y/d[L.sup.4]=P (13) where the dimensionless pressure is P [equivalent to] W[L.sup.4]p/EI[B.sub.L] (14) [mu]QL[=]EI[=]Pa.s([m.sup.3]/s)m[=]Pa.[m.sup.4] However, (2) gives p(x), where x and e are related geometrically, through the Pythagorean theorem by [[[B.sub.L][B.sub.0](e)].sup.2]+[x.sup.2]=[(Le).sup.2] (15) To convert p(x) in (2) to p(e), we need x(e). Substituting (3) into (15) and solving for yields: X=(Le) sin [theta] (16) or by differentiating with respect to e dx=sin [theta] de (17) substituting (17) into (2), we get dp/de=qw[mu]/W[[B(x)].sup.3] (Q1/2 WB(x)V) sin [theta] (18) and substituting (16) into (6) yields B(e) which in turn can be substituted into (18) to give dp/de=[6[micro]sin [theta]/W] (2QW[[B.sub.L](Le)cos[theta]+y(e)cos[theta]]V)/[[[B.sub.L](L.e)cos[theta]+y(e)cos[theta]].sup.3] (19) Adimensionalizing with the variables in Table 2 yields the dimensionless pressure gradient in the fluid under the flexing blade as follows: dP/dL=(6[micro][L.sup.5]sin[theta]/EI[B.sub.L.sup.4]) (2QVW[B.sub.L][1[beta]L cos[theta]+Y cos [theta]])/[[1[beta]L cos[theta]+Y cos [theta]].sup.3] (20) where [beta] is defined in Table 2 and where the volumetric flow rate is Q=HWV [d.sup.5]Y/d[L.sup.5]=6V sin [theta] (2H[1[beta]L cos [theta]+Y cos [theta]])/[[1[beta]L cos [theta]+Y cos [theta]].sup.3] (21) Where V [equivalent to] [mu]VW[L.sup.5]/EI[B.sub.L.sup.3] (22) Equation (21) is our main result. We find that dimensionless blade deflection Y and the dimensionless coating thickness, H, are governed by [theta] and by the dimensionless group, V, which we call the blade flexibility. Equation (21) is a fifthorder ordinary differential equation in two unknowns subject to the following six boundary conditions: two for the blade's cantilevered base, Y(1)=0 (23) Y'(1)=0 (24) two for its free tip, y" (0)=0 (25) y"' (0)=0 (26) and two pressure boundary conditions, respectively, for its leading and trailing edges: Y"" (1,0)=0 (27) Since equation (21) is not directly in the dimensionless neutral axial coordinate, U its closed analytic solution subject to the six boundary conditions (23)(27) is especially challenging, and is the subject of ongoing research. We thus evaluate (21) numerically. We specify a trial value for H, and then solve (21) subject to any five of the six boundary conditions in (23)(27), and then adjust the trial value for H, until the sixth boundary condition is met. We arrive at a plot of H for 0 = (0  [PI]/2) over the range [10.sup.1] [less than or equal to]V [less than or equal to] [10.sup.1]. Restrictions Equation (1) applies to gently tapered flow channels only. An order of magnitude analysis of the equations of motion and continuity following the method of Bird et al example 1.33, pp. 16181 restricts equation (1) to [B.sup.L]B(0)/L sin [theta] [much less than] 1 (28) Re(B(0)/L sin [theta]) [much less than] 1 (29) where Table 2 defines Re. These restrictions (equations 28 and 29) apply to equations (1), (13), and (21) too. Equation (28) does not restrict our analysis to small deflections, rather restricts our results to gap differences [B.sub.L]  B(0) subceding L sin([theta]). Special case: rigid blade We can enhance our understanding of what governs the coating thickness by solving for H in (21): H=[1[beta]L cos [theta]+Y cos [theta]]/2[[1[beta]L cos [theta]+Y cos [theta]].sup.3] /12V sin [theta] [d.sup.5]Y/d[L.sup.5] (30) and compare it to the solution for the rigid blade (2), (3): H=1/1+([B.sub.L]/[B.sub.0]) (31) From the blade geometry, we get [B.sub.0]=[B.sub.L](1[beta] cos [theta]) (32) Substituting (32) into (31), for the rigid blade, we get H=1/1+1/1=[beta]cos [theta] (33) From (33), it is clear that H only depends on 0 and [beta] for the rigid blade. In particular, the coating thickness exhibits no dependence on fluid viscosity, unlike the case of the flexible blade. Discussion The numerical solution employed the dsolve subroutine in Maple.4 In addition to the behavior of H, the solution of (21) explains the behavior of the other dimensionless groups, Y and P. The rigid blade solution, (33), was used to develop the initial guess for the solution of H. The criterion for convergence was Y(1)<1 x [10.sup.4]. With [beta] and V set, H was varied until this convergence criterion was met. The results of this process arc found in Figs. 35 where V = 0.1, 1.0, and 10, respectively. This family of curves shows that the behavior of H is generally the same across this range of V. Tables 48 provide a closer look at H as V varies. From these tables, it is seen that for very small values of H, the departure from the rigid blade solution is small. The exception to this is found in Table 8, the parallel blade solution. As expected, H is very large, which is consistent with the rigid blade solution Table 4: Numerical solutions to equation (21) for [theta] = 0.0444([PI]/8); [beta]= 1.0 V H V(0) H/HRb P (1/2) 0.1 0.00215 0.00072 1.001 0.01 1 0.00215 0.007 1.001 0.1 10 0.00215 0.065 1.001 0.93 Table 5: Numerical solutions to equation (21) for [theta] = ([PI]/8) [beta] = 1.08 V H V(0) H/HRb P (1/2) 0.1 0.0022 0.015 0.998 0.22 1 0.0022 0.13 0.998 1.85 10 0.0022 0.67 0.998 9.3 Table 6: Numerical solutions to equation (21) for [theta] = ([PI]/4) [beta] = 1.41 V H V(0) H/HRb P (1/2) 0.1 0.00297 0.028 1.0 0.4 1 0.00297 0.225 1.0 3.21 10 0.00296 1.08 0.996 14.7 Table 7: Numerical solutions to equation (21) for [theta] = (3[PI]/8) [beta] = 2.61 V H V(0) H/HRb P (1/2) 0.1 0.00119 0.037 0.996 0.54 1 0.00119 0.315 0.996 4.49 10 0.00119 1.64 0.996 22.5 Table 8: Numerical solutions to equation (21) for [theta] = 3.566([PI]/8) V H V(0) H/HRb P (1/2) 0.1 0.47664 0.00064 1.0 0.0077 1 0.476659 0.0059 1.0 0.077 10 0.476825 0.059 1.0 0.78 [FIGURE 3 OMITTED] [FIGURE 4 OMITTED] [FIGURE 5 OMITTED] In an alternate view of the results, Fig. 6 compares H vs V at [theta] = 3[PI]/8. As expected, H decreases as $ increases. The main purpose of Fig, 6 is to examine the effect of V on H. In general, H increases as V increases, or more exactly, H for the flexible blade reduces to that of the rigid blade Hrb as V vanishes. However, Table 7 shows that H < Hrb as H approaches zero. Figures 79 were constructed to explore this phenomenon. In this case, the ratio of dimensionless coating thickness for flexible and rigid blades H/Hrb is plotted vs (1 over the range of 0. The graphs in these three figures show that as V increases from 0.1 to 10, the behavior of a flexible blade as measured by H closely resembles that of a rigid blade for V [greater than or equal to] 1. For V = 10, H of the flexible blade departs from that of a rigid blade for all values of 0, but as discussed above, still approaches the rigid blade solution for very small values of H. The exceptions to this are found at 6 = 0.0444(7c/8) and 0 = 3.9556([pi]/8). At these two values of 0, the thickness of the coating H deposited by the flexible blade nearly matches that of a rigid blade H over the range of V and [beta] studied. The behaviors of P and Y at 0 = 0.0444(tt/8) for values of [i= 0.5 to 1.0 are depicted in Figs. 10 and 11, respectively. Figure 10 shows that decreasing H (by increasing [beta]) skews the dimensionless pressure distribution P with its maximum at L ~ 0.9. The shift in maximum P, from L ~ 0.5 to L ~ 0.9, as H decreases, is a result of the fluid pressure being relieved by the blade deflection. The corresponding graph of Y in Fig. 11 shows that as H decreases, the dimensionless deflection increases. Figures 12 and 13 examine P and Y, respectively, for increasing values of 6 at fl = 1.41 and V = 10. The behavior of P and Y under these conditions is consistent with that at 6 = 0.0444([pi]/8). [FIGURE 6 OMITTED] Worked example An engineer is assigned the task of modifying an existing flexible blade coating operation for use with a new thin coating. This coating exhibits nearly Newtonian behavior with a viscosity (/*) of 5.0 Pas. The substrate is 180 cm wide, and the line will run at 50 m/min. The required coating thickness ([mu]) is 0.02 mm. The gap between the substrate and the fixed end of the blade, B^, is 1.0 cm. It is desirable to maintain this gap as it is used for several products run on this same line, and varying it would add to set up time and operational control. The engineer is asked to calculate the coating thickness. For the given dimensions of H and [[beta].sub.L], H = 0.002. She uses equation (22) to calculate the dimensionless flexibility, V, for the process. With [B.sub.L], [mu], V, and W specified, the remaining variables required to define V are blade material modulus, blade thickness, and blade length. The rubber blade has a Young's modulus of 3.0 x 104 psi. Blade thickness is 3.0 mm, and the blade length, L, is 2.75 cm. Under these conditions, the dimensionless flexibility of V = 9.3. With B^ and L specified above, ft = 2.75. Now with fi and H known, the engineer uses Fig. 5 to obtain 6 by interpolation. For H = 0.002 and 0 = 2.75, 6 =3.03(tt/8). Under these conditions, the flexible blade will deposit a coating thickness of 0.02 mm onto the substrate. [FIGURE 7 OMITTED] [FIGURE 8 OMITTED] [FIGURE 9 OMITTED] [FIGURE 10 OMITTED] The engineer checks the restrictions of these results with equations (28) and (29). She finds that the condition for equation (29) is met since it is of order 106 and the restriction for equation (28) is close with its value being 0.39. [FIGURE 11 OMITTED] [FIGURE 12 OMITTED] [FIGURE 13 OMITTED] Conclusion This article uncovers a fifthorder nonlinear ordinary differential equation for blade deflection and for the corresponding coating thickness in Newtonian flexible blade coaling. The solution of equation (21) provides insight into the behavior of a flexible blade coating system. Among the key insights is the dimensionless blade flexibility, V, defined in equation (22). This group represents the ratio of the fluid viscous forces to the resistive forces from the blade flexure. For values of V > 1, the fluid forces acting on the blade are sufficient to overcome the blade stiffness and cause deflection. The role of the blade angle was also explored. The coating thickness produced by a flexible blade was found to approach that of a rigid blade as blade angle approached zero and as it approached [PI]/2 for all values of V [less than or equal to] 10. For blade angles between 0 and [PI]/2, the coaling thickness produced by the flexible blade departed from that of a rigid blade when V>1. By considering both V and [theta], one can obtain the blade deflection and coating thickness in flexible blade operations. Dimensionless plots are provided to help practitioners predict coating thickness, and a worked example is included showing how to use the results. We note that the literature contains no experimental measurements of coating thickness for the case studied here, the unconstrained blade. Our study highlights the need for these experiments. Acknowledgments The authors are grateful to the Piacon Corporation of Madison, Wisconsin for its sustaining sponsorship of the Rheology Research Center. The authors are also indebted to Professor R. Byron Bird of the Rheology Research Center at the University of Wisconsin for helpful discussions. The authors gratefully acknowledge the support received from Jonathan Zivku at Mapiesoft's Technical Support Center in Waterloo, Ontario, for his assistance in refining the numerical solution. References (1.) Bird, RB, Armstrong, RC, Hassager, O, Dynamics of Polymeric Liquids, 2nd ed., Vol. 1. Wiley, New York, 1987 (2.) Middleman, S, Fundamentals of Polymer Processing. McGrawHill, New York, 1977 (3.) Tadmor, Z, Gogos, CG, Principles of Polymer Processing, 2nd ed. Wiley, Hoboken, NJ, 2006 (4.) Maple 13.0 by MapleSoft. Waterloo Maple Inc, Waterloo, 2009 (5.) Follette, WJ, Fowells, RW, "Operating Variable of a Blade Coater." TAPPI, 43 953957 (1960) (6.) Windle, W, Beazley. KM, "The Role of Viscoelasticitv in Blade Coating." TAPPI, 51 340348 (1968) (7.) Saita, FA, Scriven, LE, '"Coating Flow Analysis and the Physics of Flexible Blade Coating." TAPPI Coating Conference Proceedings, pp. 1321, 1985 (8.) Saita, FA, "Simplified Models of Flexible Blade Coating." Chem. Eng. Set, 44 817825 (1988) (9.) Scriven, LE, Pranckh, FR, "Elastohydrodynamics of Blade Coating." AIChE J., 36 587597 (1990) (10.) Alheid, RJ, "Flexible Blade Coating Arrangement and Method with Compound Blade Loading." U.S. Patent Number 5,077,095, 1991 (11.) Saita, FA, Corvalan, CM, "Blade Coating on a Compressible Substrate." Chem. Eng. Sci., 50 17691783 (1995) (12.) Alam, P, Toivakka, M, "Deflection and Plasticity of SoftTip Beveled Blades in Paper Coating Operations." Mater. Des., 30 871877 (2009) A. J. Giacomin (*), J. D. Cook, L. M. Johnson, A. W. Mix Rheology Research Center, University of WisconsinMadison, Madison, WI 537061572, USA email: giacomin@wisc.edu A. J. Giacomin, J. D. Cook, A. W. Mix Mechanical Engineering Department, University of WisconsinMadison, Madison, WI 537061572, USA L. M. Johnson Material Science Program, University of WisconsinMadison, Madison, WI 537061572, USA * We note that blade flexibility is related to the Elasticity Number [N.sub.Es] in Scriven and Pranckn (9) by the following relation: V=[N.sub.Es] [L.sup.3]/(1[v.sup.2])[B.sub.L.sup.3] DOI 10.1007/sl199801193666 

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