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 fifth-order nonlinear ordinary differential equation ordinary differential equation
Equation containing derivatives of a function of a single variable. Its order is the order of the highest derivative it contains (e.g., a first-order differential equation involves only the first derivative of the function). for the thickness of a Newtonian coating and for the corresponding blade deflection deflection /de·flec·tion/ (de-flek´shun) deviation or movement from a straight line or given course, such as from the baseline in electrocardiography.
1. . 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 (jargon) for values of - A common rhetorical maneuver at MIT is to use any of the canonical random numbers as placeholders for variables. "The max function takes 42 arguments, for arbitrary values of 42". "There are 69 ways to leave your lover, for 69 = 50". 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
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 Non-Newtonian 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 fifth-order nonlinear differential equation differential equation
Mathematical statement that contains one or more derivatives. It states a relationship involving the rates of change of continuously changing quantities modeled by functions. that governs flexible blade coating provides insight into the factors controlling the coating thickness. This method removes the need for elements that previous FEM FEM Female
FEM Finite Element Method
FEM Finite Element Model
FEM Fédération Européenne des Métallurgistes (European Metalworkers' Federation)
FEM Faculdade de Engenharia Mecânica (Brasil) solutions utilized to model flexible blade coating. The derivation derivation, in grammar: see inflection. 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.
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 Dimensionless groups
A dimensionless group is any combination of dimensional or dimensionless quantities possessing zero overall dimensions. Dimensionless groups are frequently encountered in engineering studies of complicated processes or as similarity 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
For the fluid pressure exerting the distributed load on the flexible blade, we follow a flow analysis for laminar flow laminar flow
Fluid flow in which the fluid travels smoothly or in regular paths. The velocity, pressure, and other flow properties at each point in the fluid remain constant. between parallel plane surfaces based on Bird et al., example 1.3-1 pp. 13-14(1):
Q=1/2 WB(x)V-W[[B(x)].sup.3]/12[micro] dp/dx (1)
[FIGURE 1 OMITTED]
[FIGURE 2 OMITTED]
dp/dx=12[micro]/W[[B(x)].sup.3] (1/2 WB(x)V-Q) (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 Theta
A measure of the rate of decline in the value of an option due to the passage of time. Theta can also be referred to as the time decay on the value of an option. If everything is held constant, then the option will lose value as time moves closer to the maturity of the option. ]) (L-e); 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)
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]) (L-e) +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)
[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] L-e/L (10)
[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:
where the dimensionless pressure is P [equivalent to] W[L.sup.4]p/EI[B.sub.L] (14)
However, (2) gives p(x), where x and e are related geometrically, through the Pythagorean theorem Pythagorean theorem
Rule relating the lengths of the sides of a right triangle. It says that the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse (the side opposite the right angle). by
To convert p(x) in (2) to p(e), we need x(e). Substituting (3) into (15) and solving for yields:
X=(L-e) 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] (Q-1/2 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] (2Q-W[[B.sub.L]-(L-e)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 atmospheric sciences (meteorology, climatology and related fields), the pressure gradient (typically of air, more generally of any fluid) is a physical quantity that describes in which direction and at what rate the pressure changes the most rapidly around a particular location. in the fluid under the flexing blade as follows:
dP/dL=(-6[micro][L.sup.5]sin[theta]/EI[B.sub.L.sup.4]) (2Q-VW[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 In fluid dynamics and hydrometry, the volumetric flow rate, also volume flow rate and rate of fluid flow, is the volume of fluid which passes through a given surface per unit time (for example cubic meters per second [m3 s-1 is Q=HWV HWV Höhere Wirtschafts- und Verwaltungsschule (University of Zurich business school)
HWV Verzeichnis der Werke Georg Friedrich Händel (German: Händel's music catalog)
HWV Hardware Version
[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)
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 fifth-order ordinary differential equation in two unknowns subject to the following six boundary conditions: two for the blade's cantilevered base,
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 axial /ax·i·al/ (ak´se-al) of or pertaining to the axis of a structure or part.
1. Relating to or characterized by an axis; axile.
2. 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 boundary condition
The set of conditions specified for behavior of the solution to a set of differential equations at the boundary of its domain. 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].
Equation (1) applies to gently tapered flow channels only. An order of magnitude A change in quantity or volume as measured by the decimal point. For example, from tens to hundreds is one order of magnitude. Tens to thousands is two orders of magnitude; tens to millions is three orders of magnitude, etc. analysis of the equations of motion and continuity following the method of Bird et al example 1.3-3, pp. 16-181 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):
From the blade geometry The term Blade geometry refers to the physical properties of a sword blade: cross-section and taper. Cross-Section
The cross section of a blade is the primary way of determining its function and place in history. , 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.
The numerical solution employed the dsolve subroutine A group of instructions that perform a specific task. A large subroutine might be called a "module" or "procedure." Subroutine is somewhat of a dated term, but it is still quite valid. 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. 3-5 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 4-8 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 7-9 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]
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 interpolation
In mathematics, estimation of a value between two known data points. A simple example is calculating the mean (see mean, median, and mode) of two population counts made 10 years apart to estimate the population in the fifth year. . 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 10-6 and the restriction for equation (28) is close with its value being 0.39.
[FIGURE 11 OMITTED]
[FIGURE 12 OMITTED]
[FIGURE 13 OMITTED]
This article uncovers a fifth-order 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 viscous /vis·cous/ (vis´kus) sticky or gummy; having a high degree of viscosity.
1. Having relatively high resistance to flow.
2. Viscid. forces to the resistive resistive /re·sis·tive/ (re-zis´tiv) pertaining to or characterized by resistance. forces from the blade flexure flexure /flex·ure/ (flek´sher) a bend or fold; a curvation.
caudal flexure the bend at the aboral end of the embryo.
cephalic flexure the curve in the midbrain of the embryo. . 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 Madison is the capital of the U.S. state of Wisconsin and the county seat of Dane County. It is also home to the University of Wisconsin–Madison.
The 2006 population estimate of Madison was 223,389, making it the second largest city in Wisconsin, after Milwaukee, and 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 Coordinates:
Waterloo is a city in Ontario, Canada. It is the smallest of the three cities in the Regional Municipality of Waterloo, and is adjacent to the larger city of Kitchener. , for his assistance in refining the numerical solution.
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1. Having the properties of a polymer.
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New York, Middle Atlantic state of the United States. It is bordered by Vermont, Massachusetts, Connecticut, and the Atlantic Ocean (E), New Jersey and Pennsylvania (S), Lakes Erie and Ontario and the Canadian province of , 1987
(2.) Middleman mid·dle·man
1. A trader who buys from producers and sells to retailers or consumers.
2. An intermediary; a go-between. , S, Fundamentals of Polymer Processing. McGraw-Hill, New York, 1977
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(4.) Maple 13.0 by MapleSoft. Waterloo Maple Waterloo Maple Inc. is a privately held Canadian software company, headquartered in Waterloo, Ontario. It is best known as the manufacturer of the Maple computer algebra system. Corporate History
Waterloo Maple Inc. Inc, Waterloo, 2009
(5.) Follette, WJ, Fowells, RW, "Operating Variable of a Blade Coater." TAPPI TAPPI Technical Association of the Pulp and Paper Industry , 43 953-957 (1960)
(6.) Windle, W, Beazley. KM, "The Role of Viscoelasticitv in Blade Coating." TAPPI, 51 340-348 (1968)
(7.) Saita, FA, Scriven, LE, '"Coating Flow Analysis and the Physics of Flexible Blade Coating." TAPPI Coating Conference Proceedings, pp. 13-21, 1985
(8.) Saita, FA, "Simplified Models of Flexible Blade Coating." Chem. Eng. Set, 44 817-825 (1988)
(9.) Scriven, LE, Pranckh, FR, "Elastohydrodynamics of Blade Coating." AIChE J., 36 587-597 (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 com·press·i·ble
That can be compressed: compressible packing materials; a compressible box.
com·press Substrate." Chem. Eng. Sci., 50 1769-1783 (1995)
(12.) Alam, P, Toivakka, M, "Deflection and Plasticity of Soft-Tip Beveled bev·el
1. The angle or inclination of a line or surface that meets another at any angle but 90°.
2. Two rules joined together as adjustable arms used to measure or draw angles of any size or to fix a surface at an angle. Blades in Paper Coating Operations." Mater. Des., 30 871-877 (2009)
A. J. Giacomin (*), J. D. Cook, L. M. Johnson, A. W. Mix
Rheology Research Center, University of Wisconsin-Madison “University of Wisconsin” redirects here. For other uses, see University of Wisconsin (disambiguation).
A public, land-grant institution, UW-Madison offers a wide spectrum of liberal arts studies, professional programs, and student activities. , Madison, WI 53706-1572, USA e-mail: email@example.com
A. J. Giacomin, J. D. Cook, A. W. Mix
Mechanical Engineering Department, University of Wisconsin-Madison, Madison, WI 53706-1572, USA
L. M. Johnson
Material Science Program, University of Wisconsin-Madison, Madison, WI 53706-1572, 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 (Digital Object Identifier) A method of applying a persistent name to documents, publications and other resources on the Internet rather than using a URL, which can change over time. 10.1007/sl1998-011-9366-6