Background of the Invention One of the most significant changes in l. ight weight coated (LWC) paper production is the use of the pressurized pond coater. The pressurized pond coat : +Esuch as short-dwell coaters has enabled the paper maker to improve while maintaining coated paper quality. The term "short-dwell" ref*r-s c. o the relative : Ly short period of time that the coating is in contact. wit'. h.". v'eb of paper material before the excess is metered off by a trailing doct. ! blade. Prior art short-dwell coaters consi. st of a captive pond just : prior to a doctor blade. The pond is approximately 5 cm in length and is pressurized to promote adhesion of the coating to the paper web. The e--xcest. =Oati. rlg supplied to the sheet creates a backflow of coating. This coating backflow provides a wetting line and thus, to some extent, excludes the boundary layer of air entering with the sheet and eliminates skip coating.

The excess coating is typically channeled over an overflow baffle and collected in a return pan before returning to tanks to be screened.

While. pond coaters are extensively used in coating paper webs, such coaters suffer from a major problem. The flow in the coating chamber of the pond upstream of the doctor blade contains recirculating eddies or vortices which can result in coat-weight nonuniformities and wet streaks or striations i. n severa] ways. For example, these eddies can become unstable due to centrifugal forces and result in the generation of unsteady flow and rapidly fluctuating vortices, which deteriorate the coating uniformity and its quality. Also, the vortices tend to entrap small air bubbles which result in the buildup of relatively large air inclusions in the coating liquid which tend to accumulate in the core region of the eddies. Vortex fluctuations tend to force these ai. r inclusions into the blade gap. This adversely affects the coating quality. Usually, the presence of air inclusions results in regions of lower coat weight which are 2-4 cm wide and about 10-100 cm long, known in the industry as"wet streaks". These problems are discussed in an article "Principles of Hydrodynamic Instability : Application'in Coating Systems", C. K.

Aidun, Tappi Journal, Vol. 74, No. 3, March, 1991.

Previously, geometries utilizing streamlined boundaries in coating devices have been employed to eliminate the formation of recirculating eddies or vortices. See, e. g., Aidun, U. S. Patent No. 5,366,551 entitled"Coating Device for Traveli. ng Webs,"wherein curvilinear geometries are employed for the elimination of vortices and flow instability due to centrifugal forces, and for the avoidance of harrnful pressure fluctuations which could result in coat-weight nonuniformities. The elimination of recirculating eddies or vortices also reduces the possibility of entrapping air pockets or air bubbles in the core of the vortices which could reach the blade gap and could result in coat-weight nonuniformities and wet streaks.

Additionally, the walls of the coating composition application chamber in conventional coating devices are considered rigid and do not deform under tin ? effect of hydrodynamic pressure, and thus exert shear stress by the flow on the boundaries in contact with the coating liquid. Such wall shear stress on the coating liquid creates flow separation from the applicator walls in the application chamber which also results in coat-weight nonuniforrnities and wet streaks, as well as, recirculating eddies and vortices. Pranckh, F. R., and Scriven, L. E.,"The Physics of Blade Coating of Deformable Substrate,"1988 Coating Conference Proc., TAPPI Press, Atlanta, GA, (1988) have provided a detailed analysis of blade coating using a finite element approximation method including the complex interactions of the boundary in addition to the solution of the flow fi. eld and free surface location. The blade was modeled as a thin, inextensible, elastic solid and the substrate deformed due to normal stresses.

In Aidun, U. S. Patent No. 5, 354,376 entitled"Floatation Coating Device for Traveling Webs,"one of the applicator walls is designed to be a floating or moving wall or belt. The effect of the floating applicator wall is to reduce vortices through the use of a moving substrate, e. g. a suspended belt, as the applicator wall which moves with a given speed with the liquid to prevent flow separation and recirculation inside the application chamber. The floatation coating device for traveling webs seeks to alleviate recirculations in a fixed domain pressurized pond coating system. The combination of a moving applicator wall and a sufficient flowrate allow for the design of a vortex- free coater configuration.

Development of high speed blade coating is of particular interest in the industry to enhance production, and to reduce cost the analysis of the coating process which is complex because the governing equations of fluid motion are non-linear and the free-surface position is part of the unknown. Moreover, the non-linear constitutive behavior of typical coating fluids increases the complexity.

It would be desirable to provide a coating device which has the coating advantages of a short-dwell coater, but which did not have the problems associated with recirculating eddies or vortices and the entrapment of air pockets or air bubbles in the core of the vortices.

It would be further desirable to provide a coating device with reduced shear stress on the flowing stream of the liquid coating composition in the apllivat : ion chamber a, the coating composi. tion downstreams.

It is another object of the present invention to provide a coating device which receives a liquid flow of a carrier fluid introduced in the direction of the travel of the web positioning the liquid flow of the liquid co- :, - : ing composition between the carrier fluid and the web with reduced shear stress on the flowing stream of the liquid coating composition in the application chamber as the coating composition downstreams.

It is a further object of the present invention to provide a coating device which receives the flow of carrier fluid through a channel for directing air flow into the coating composition application chamber below the flow of the liquid coating composition reducing shear stress on the flowing stream of the liquid coating composition.

Accordingly, it is a principal object of the present invention to provide a vortex-free short-dwell coating device.

These and other objects will become more apparent from the following description and the appended claims.

Summary of the Invention The invention relates to coating devices for application of coating material to the surface of a web or a flexible substrate. Such coating devices employ a pressurized channel where a flowing stream of the coating liquid comes into contact with the substrate. The coating liquid first enters at the upstream side of the channel wetting the substrate as it flows in the same direction with the substrate. A doctor element is positioned at the downstream side of the channel where the excess coating in the channel follows the contour of the boundary formed by the doctor element and leaves the channel.

The present invention is further directed toward the study of flow patterns in blade coating to develop high-speed coaters, wherein the coater may be modified to provide an air layer between the coating liquid and any lower boundary. The air layer thus serves as a carrier fluid.

'TI-i (- coater devices of the described embodiments provide two inlet channels and an outlet channel. The first inlet channel carries the coating liquid, and the second channel can be used to pump the carrier fluid, e. g. air, into the coating head to pressurize the chamber and to keep the contact wetting line at the upstream section attached to the substrate. The air pressure can vary from zero to any level appropriate for the coating operation. The air layer serves as a carrier fluid removing the wall shear stress on the coating liquid in the channe], and thus the coating flow for the operation of the device may proceed without flow separation from the wall (i. e., in a vortex- free mode) at relatively low flow rates appropriate for commercial applications. The exc'esF coating liquid and all of the air leave the coaler head at the outlet channel. The blade is used to meter the excess coating from the substrate.

Accordingly the pressure inside the channel may be increased above ambient pressure, if necessary, in order to prevent air entrainment into the coating liquid. However, the system may also operate at ambient pressure if air entrainment is not an issue. The revised vortex-free coater and computation simulation of the flow in the system are presented below. The computation -ii ; ions assume ambient pressure in the air layer and, therefore, the coating layer just upstream of the blade.

Briefly summarized, the present invention relates to high speed coating methode and apparation for applying a liquid coating composition on a web of mineral as the web travels along a path throngh the device from an apstream dtt'-ction to a downstream direction with a doctor element being spaced from the web and extending across the path of the web transversely of the direction of travel of the web. A coating composition application chamber receives the liquid flow of the liquid coating composition from the upstream direction to the downstream direction, and comprises an upstream interior side wall and an upstream boundary wall for directing the liquid-coating composition flow into t'he application chamber, and the doctor element for spreading and defining the @@@@@@@@ the liquid coqting composition on the web at the downstream side of the application chamber. The coating composition application chamber is further adapted for receiving a liquid flow of a carrier fluid introduced at. the upstream side of the application chamber in the direction of the travel of the web positioning the liquid flow of the liquid coating composition between the carrier fluid and the web, the liquid coating composition flowing from the upstream side of the application chamber in the direction of the travel of wen to the doctor element defining a path which the flowing strearn cL t ln.. liquid coating composition dowstreams in the diretion of travel of the web with reduced shear stress on the flowing stream of the liquid coating @@@@@@@ the application chamber as the coating composition downstream<BR> @@@@@ Description of the Drawings Figure lais a schematic cross-sectional view of an embodiment of a short- well coating device according to the invention ; Figure 1H is a schematic cross-sectional view of another embodiment of the short dwell coating device according to the invention ; Figure 1C represents a domain description in cross-section for the described studies of the short-dwell coating devices according to the r, U Fi. qurc 2 represents a gap region description of the domain for short-dwell c)) tinQ devices : Figure 3 illustrates the effect of flowrate variation shown as a mesh @@@@@@@@ representali@@ of the domains: Figure 4 @@@@@@ the effect of flowrate variation shown as streamlines in the domains : Figure 5 illustrates the effect of flowrate variation shown as mesh of applicator channel exit ; Figure 6 illustrates the effect of flowrate variation shown as streamlines in applicator channel exit ; Figure 7 illustrates the effect of flowrate variation shown as pressure contours in applicator channel exit; Figure 8 illustrates the effect of flowrate variation shown as mesh of gap region ; Figure 9 illustrates the effect of flowrate variation shown as streamlines in gap region ; Figure 10 illustrates the effect of flowrate variation shown as velocity field in gap region; Figure 11 illustrates the effect of flowrate variation shown as pressure contours in gap region ; Figure 12 illustrates the effect of flowrate variation shown as mesh of blade t ip region ; Figure 13 illustrates the effect of flowrate variation shown as streamlanes at blade @@@ @@@@@ : Figure 14 illustrates the effect of flowrate variation shown as pressure contourn in blade tip region : Figure Figure 15 illustrates the effect of flowrates variation shown as horizontal velocity profiles at midôint of blade tip ; ; Figure 16 illustrates the effect of flowrate variation shown as horizontal velocity profile at endpoint of blade tip; Figure T ? illustrates the effect of flowrate variation shown as horizontal velocity profile at @@ ; ; Figure 18 illustrates the effect of flowrate variation shown as pressure distribution along the blade ; Figure 19 illustrates the effect of flowrate variation shown as pressure distribution along the substrate ; Figure 20 illustrates the effect of flowrate variation schown as pressure distribution a] ong the blade tip ; Figure 21 illustrates the effect of flowrate variation shown as coating <BR> @@@@@@ @@ inlet flowrates<BR> @@@@@@ness vs inlet flowrates :<BR> Figure 22 illustrates the effect of flowrate variation schwn as f@@@@@ flowrate vs inlet flowrate; Figure 23 illustrates the effect of flowrate variation shown as coating thickness vs thickness under web; Figure 24 illustrates the web speed variation shown as coating thickness vs web speed; Figure 25 illustrates the web speed variation shown as coating thickness vs reynolds number; and Figure 26 illustrates the web speed variation shown as coating thickness vs capillary number.

Detailed Description of the Embodiments As shown in Figure 1A, the short-dwell coating device 10 of the present invention includes of a first continuous channel 12 for receiving a liquid coating composition material 14 which passes through a coating application chamber 16 which is in contact with a roll or web 18 of material which is to be coat. ed. The coating device 10 further includes of a second continuous channel 20 for receiving a liquid flow of a carrier fluid such as air 22 which ale--'passes through a coating application chamber 16 positioning the liquid flow of the liquid coating composition 14 between the carrier fluid 22 and the vaele 1 of material which is to be coated. For purposes of orientation and discussion, the coating chamber has an upstream side and a downstream side ;, : il'lespect to movement-of the web with the upstream side being to the left : of Figure 1A. The use of the terms"horizontal"and"vertical"are with respect to a horizontal orientation of the web 18. The web 18, however, is usually supported on a counter roll and has a slight curvature in the region of the'coating application chamber 16.

The coating devices described herein include a blade or doctor element 24 which is spaced from the web 18 for defining the thickness of the coating on the web 18. The doctor element 24 extends across the 18 web transversely to the direction of the web motion. The doctor element also forms a downstream boundary wall of the coating chamber 16 and extends downwardly for a further distance to define the downstream wall of an exit plenum or outlet channel 26 formed between the doctor element 24 and a downstream interior wall 28 in the embodiment of Figure 1A, for the circulation of the liquid flow of the carrier fluid, e. g., air 22 which circulates with the I. iquid flow of the liquid coating composition 14 through the coating application chamber 16 as the web 18 of material which is coated.

In Figure 1A, an upstream boundary wall 30 defines the upstream side of the coatinq device 10. The upstream boundary wall 30 extends downwardly for a further distance to define the upstream side of an entrance plenum of the first channel 12. The upstream boundary wall 30 terminates at its uppermost end in contact with the web 18 via a contact line or wetting line 32 of the liquid coating composition 14, thus preventing air entrainment at the upstream section 34. As shown, the terminal end 36 of the upstream boundary wall 30 preferably has a curvilinear shape so that this terminus of the upstream boundary wall is substantially tangential. to the web 18. The upstream boundary wall 30 and its terminal end 36 also extend across the web transversely to the direction of the web motion.

The coating device 10 and particularly the coating application chamber 16 are represented in cross--section in Figure 1A. The embodiment of Figure 1A provides interior walls including an upstream interior side wall 38, an interior. top wall 40 and an downstream interior side wall 42. The. inferior wal1s 38, 40 and 42 in comb : irlat-ion with the upstream boundary wall 30 and the doctor element 24 define the coating composition application chamber 16 of the embodiment. The coating composition application chamber 16 is further adapted for receiving the liquid flow of the carrier fluid 22 as a fluid layer introduced from the upstream side of the application chamber substantially parallel to and in the direction of the travel of the web supporting the liquid flow of the liquid coating composition 14 between the fluid layer 22 and the web 18.

The fluid layer opposite the web defines a top interior fluid layer wall above the interior top wall 40 and the fluid layer opposite the doctor blade defining a downstream interior fluid layer wall adjacent the downstream interior side wall 42. The top interior fluid layer wall of the carrier fluid 22 provide a layer which substantially conveys the liquid coating composition 14 from the terminating curvilinear section of the upstream interior wall in the direction of the travel of the web to the doctor element 24. The coating device 10 also provides the upstream boundary wall 30 and the upstream interior side wall 38 as upwardly inclined in a direction toward the downstream side ; the downstream interior wall 42 and the doctor element 24 being downwardly inclined in a direction toward or away from the upstream side. Accordingly, the upstream walls 30,38, the top interior fluid layer wall and web 18, the downstream interior fluid layer wall and doctor element 24 thus define a path in which the flowing stream of the liquid coating composition 14 downstreams in the direction of travel of the web 18 to at least reduce wall shear stress on the flowing stream of the liquid coating composition from the interior fluid layer wall as the coating composition downstreams thereon, reducing the formation of recirculating eddies and vortices in the coating composition.

Figure 1B shows an another embodiment of a short-dwell coating device 50 of the present invention which includes of a first continuous channel 52 for receiving the liquid coating composition material 14 which passes through a coating application chamber 56 in contact with the web 18 to be coated. The coating device 50 also includes of a second continuous channel 54 for receiving a liquid flow of the carrier fluid, e. g, air 22 which also passes through the coating application chamber 56 positioning the liquid flow of the liquid coating composition 14 between the carrier fluid 22 and the web 18 of material which is to be coated, as in the embodiment of Figure 1A discussed abovr'. The Figure 1B embodiment however does not utilize the interior top wall 40 and downstream interior side wall 42 of Figure 1A, and thus allows the carrier fluid 22 to exit into the open area of the coating application chamber 56, which may be provided under pressure. At an upstream opening 58 of the second continuous channel 54, the liquid coating composition material 14 is pressed as a layer against the web 18. The flow rate of the liquid coating composition material 14 is reduced in the Figure 1B embodiment, with respect to the Figure 1A embodiment, and an approximately 1 mm. thick layer the liquid coating composition material 14 adhering to the web 18 travels the 5 to 10 centimeters in the coating application chamber 56 to a doctor element 60 biased with a load 62 to spread and define the thickness of the liquid coating composition 14 on the web 18. As in the Figure 1A embodiment, the doctor element 60 also extends across the path of the web 18 transversely of the direction of travel of the web 18.

Pressure provided at the upstream opening 58 of the second continuous cohal-llle] 54 is desirable where the liquid coating composition material 14 is layered against the web 18 to prevent air entrainment by maintaining the contact or wetting line of the liquid coating composition 14 with the web 18, as discussed above. Advantageously however, any pressure provided in the coating application chamber 56 of the Figure 1B embodiment is reduced downstream of the opening 58, and thus the likelihood of downstream entrainment by the carrier fluid itself is reduced.

The coating device 50 and particularly the coating application chamber 56 are represented in cross-section in Figure 1B. The embodiment of Figure 1B provides an upstream interior side wall 64 and an upstream boundary wall 66 for directing the liquid coating composition flow into the application chamber 56. The coating composition application chamber 56 also is adapted for receiving the liquid flow of the carrier fluid 22 introduced at the upstream side of the application chamber 56 in the direction of the travel of the web 18 positioning the liquid flow of the liquid coating composition 14 between the carrier fluid 22 and the web 18. The liquid coating composition 14 thus flow from the upstream side of the application chamber in the direction of the travel of the web 18 to the doctor element 60 defining a path which the flowing stream of the liquid coating composition downstreams in the direction oi travel of the web with reduced shear stress on the flowing stream of the liquid coating composition in the application chamber as the coating composition dowstreams.

The embodiments described concern the study of modified vortex-free coater configurations in an effort to investigate the hydrodynamic behavior of the current system at very low flow rates. Avoidance of flow separation and recirculation is shown in studies by way of computer modelling. The flow field and the free surface boundary location are solved using a Galerkin finite element approach for web speeds ranging from 15m/s to 30rn/s and flow rates from 4 to 7 li. ter/sec./mete (1/s/m). Several mechanisms of instability are present due to the complexity of the domain in coating devices. The non- linear constitutive behavior of typical coating fluids increases the complexity. Boundaries within such high speed coating devices are typically flexible, permeable, and unknown in different regions. Accordingly, the flow is modeled as being nearly parallel throughout the majority of the domain, with the important ; exception of the region in which the web and the blade converge forcing some of the liquid under the blade tip and the rest to curve and flow down the blade.

In the gap region, between the substrate and the blade tip, the flow is nearly parallel and experiences high shear rates. Squires theorem requires that the first instability in parallel shear flows occur due to a two- dimensional instability. In the returning flow, the possibility of centrifugal instabilities to three-dimensional disturbances exist. The flow field of a blade coater with a lower free surface is'examined. The flow is assumed to be incompressible, two-dimensional and steady. The effects of flowrate and web speed variation on the design will provide insight into the optimal operating conditions. A further analysis of the stability of the resulting solutions to 2-D and 3-D disturbances will provide additional information. The velocity field, pressure field, and location of the two free surfaces of the blade coater is depicted in Figure 1C with parameters detailed in Tables 1 and 2. The region of particular interest is shown in Figure 2, here the blade (G4) and the web (G,), converge to form a gap with a vertical cross-section length (blade gap) of 50 microns. A portion of the fluid pumped in at the inlet (Gì) proceeds through the gap and coats the substrate, while the-excess is scraped off and flows nearly parallel to the blade.

Table 1: Fluid Parameters P density 1200 kg/m p. zero shear rate 1.0 kg/ (m-s) viscosity infi. infinite shear rate 0.05 kg/ (m-s) viscosity surface tension 0.05 kg/s' c Carreau exponent 0.65 time constant 0.01 s web web velocity varies from 15-30 m/s U [,,, centerline velocity on varies from 2-5 m/s inlet. q inl-t flowrate varies frorn 4-7 1/s/m Table 2: Geometry Parameters inlet length 0.0025 m Lgap gap length 50 E-6 m applicator channel 0.5 mm exit Lthick blade thickness 1.25 mm blade length (modeled) 60.104 mm web length (modeled) 59.551 run angle of blade 45° C, coating thickness O (10 Um) M, vertical distance frorn O (100 Um) web to free surface at C-C The problem can be defined in a dimensionless manner. The inlet cross-section length and web velocity are used as the length and velocity scales. Table 3 relates the dimensiorlless quantities to the parameters given in Tables 1 and Table 3: Dimensionless Quantities <BR> <BR> Fv Reynolds Number Re = #UwebLin/#<BR> P<BR> Ca Capill. ary Number y We Weber Number We<BR> ReCa #U²webLm#U<BR> The equations governing the flow in the coaler are continuity and momentum ### = u### = 0 (1) Here Ajj denotes the stress tensor, is assumed to be of the form ## = - p#n + ## Where T,. denotes the deviatoric stress tensor with the constitutive relation Tjj = ### = 2### Where Ei, is the rate of strain tensor, given by <BR> <BR> l<BR> ## = 1/2(u## + u##) The f] ujd for the current application is assumed to be shear thinning, the dynamic viscosity is approximated by the Carreau constitutive model where ## and ## denote the and inf#note shear rate viscosities. The <BR> <BR> parameters in the ####a# model are determined based in the behavior of<BR> t. ypicai coating colors. the above equations are non-dimensionalised uding the velocity of the web and the width of the inlet channel as the velocity and length scales respectively Uwen, L##=Lunder The velocity and pressure are scaled using the velocity and dynamic pressure scales <BR> <BR> '--'-JP<BR> p.= P/#u² The superscript * denotes di. mensi. onless variable. The independent'varialles, <BR> <BR> posit. ion and time, are scaled using the velocity and length scales<BR> X#+Xi/L##, t'=t/U4/Ls Ly The body force f, is non-dimensionalized The continuity, momentum, and constitutive relations can respectively be expressed in dimensionless form as u## = 0(4) where ## = 1/2(u##+ u##)<BR> K'=, K-U"<BR> L# The Dirichlet boundary conditions for this coating system are specified as <BR> <BR> <BR> U##/## = Uunlet/Us #l => inlet<BR> r. Us I 2 = > web<BR> U5<BR> => appl. icator channel, => blade<BR> <BR> ###### conditions are applied at the outflow boundaries<BR> g => exit, 1 6 => gap exit<BR> On the free-surfaces and ra, the kinematic condition is given by dt* at'ax*<BR> + ##/## u²<BR> dt' #t #xj When the flow is independent of time this condition reduces to =0 (7) where ni is the unit vector normal to the surface.

The dynamic boundary condition requires the stress to be continuous across the interface, therefore the normal and tangential stresses are respectively given <BR> <BR> by<BR> ## = ti##/##i<BR> #t=ti##/##i<BR> dx ; The fluid surface tension, y, is constant, therefore the tangential component of the traction vector is zero. The above dynamic boundary condition is non-- dimensionalized by <BR> <BR> 2H* 2H*-##/p#<BR> ## = ReCa We<BR> <BR> =o The above non-dimensional equations (4) and (5) with the constitutive relation (6) and appropriate boundary conditions completely describe the flow field. The finite element method is employed via FIDAP to solve for the velocity and pressure at discrete points within the domain. The unknown boundary location is determined in a fully coupled manner by simultaneously requiring the condition (7) be satisfied on the free surfaces.

The governing equations, constitutive relation, and boundary conditions completely define the given blade coating problem. The domain is discretized using 9-nodeS, isoparametric, quadrilateral elements. The velocity is approximated over the element using biquadtratic basis functions and the pressure with bilinear basis functions. The free surface boundary is determined by satisfying the steady state kinematic and dynamic conditions in ai fully coupled The nonlinearity of the governing equations requires an iterative solution approach. The stokes flow in the fixed domain provides an initial guess for the Newton-Raphson iteration procedure. Parameter continuation methods are used to assist in the variation of the parameters to reach the desired solution for given boundary conditions. Convergence is achieved when the norm of the solution change in between iterations is less than 10-3.

The resulting coater configurations and streamlines are shown in Figures 3 and 4 for the cases listed in Table 4. A noticeable change in the free surface location is apparent as the flowrate is varied. An increase in flowrate results in a larger vertical cross-section under the web, a decrease in exit cross-section width on GsE and an increase in the exit velocity magnitude on the same boundary.

The desire to avoid recirculating flow and minimize surface defects leads us to examine closely three regions where flow separation and recirculation is possible; the meniscus just aft of the applicator channel, the corner where the blade and web converge to construct the gap, and the blade tip where a meniscus forms and the substrate is coated. The mesh, streamlines, and pressure contours are plotted for these three regions in Figures 5-14. As demonstrated in these figures, the results show no flow separation or flow recirculation. A true vortex-free coating flow system exists at low flo rates (4 1/s/m) and high coating speeds (20 m/s).

The'velocity profiles in the gap region provide insight into the coating quality. Figure 15 shows the horizontal, non-dimensional velocity profile at ri l locatiorl A A on the blade tip while Figure 16 depicts the profile at location B-B, the endpoint of the blade tip. Figure 17 illustrates the effect of f flowrate variatiot-i shown a. horizontal velocity profile at !"., the gap exit Rt the static contact line it is clear that the formation of the meniscus slightly affects the velocity profile. The apparently linear pressure distribution along the blade tip, Figure 20, indicates an almost constant pressure gradient in the gap that increases with the flowrate. These velocity profi : les and pressure distribution demonstrate a nearly Poiseuille-- Co, uette velocity distribution, the linear combination of flow between two walls at a relative velocity to one another and flow between stationary walls with a constant pressure gradient. Thus, the coating flowrate and thickness increase slightly wit ! ; the increase in the inlet flowrate due to the larger pressure gradient, see Figures 21., 22 and 23. The portion of the coater where the blade and web form a converging channel is much more affected by the flowrate variation.

Examination of the corner region formed by web and blade, presented in Figure 8, shows significant free surface shape variation with flowrate variation. As the flowrate is decreased the free surface migrates toward the gap threatening to entirely disappear into the gap with further reduction of the inlet flowrate. The corresponding streamlines are shown in Figure 9.

The pressure along the blade and substrate are shown in Figures 18 and 19, all graphed quantities are non-dimensionalized. Table 6 can be used to convert all variables to dimensional quantities. Away from the gap the pressure remains fairly constant. Within the gap region the pressure peaks at the leading edge of the blade, just upstream of the gap. The maximum pressure increases as flowrate increases. At higher flowrates, the pressure increases in a more gradual manner, exhibiting a more distinct plateau. Following the peak the flow field experiences sub-ambient pressures and then adjusts to the ambient exit. pressure. The pressure contours in the gap region, shown in <BR> <BR> Figure 11, Jndicate thaf a decrease jn flowrate causes a larger pressure<BR> gradient ; but decreases the value of the maximum pressure. table 5: Case Study-Effect of Web Speed Variation <BR> <BR> Case Umet U# qin/et Qm/t C1 Re Ca We<BR> m/s m/s l/s/m #m l/ReCa C6VI5 15 3.6 6 0. 409921 27.42438 45 300 1/13500 C6V20 20 3.6 6 0. 552128 27.66575 6 400)/24000 C6V25 25 3.6 6 0.695813 27.873 75 5(X) 1/37500 C6V30 30 3.6 6 0. 0841083 28.0655 9 90 600 1/54000 C7V15 15 4.2 7 0.410793 27.48275 45 300 1/13500 C7V20 20 4.2 7 0. 553462 27.7325 60 400 1/24000 C7V25 25 4.2 7 0.698024 27.9615 75 500 1/37500 C7V30 30 4. 2 7 0. 844202 28. 1695 90 600 1/54000 T ; lhlc 4: Case Study-Effect of Flowrate Variation (ase U### U### Qr/m C1 W1 Rc Ca We m/s m/s m/s Us/m l/s/m l/s/m #m #m l/ReCa C#V20 20 24 4 5481175 3.61508 27.465 208.4447 60 400 1/24000 CSV20 20 3 5. 550354 4.611883 27.575 27.575 259. 0522 60 400 «) C6V20 20 3.6 6. 552128 5.60895 27.66575 309.472 60 400 1/24000 C7V20 20 4.2 7 553462 6.52 27.7325 354.6727 60 400 1/24000 Table 6 : Conversion to Dimensional Units dimensionle scale web speed multiply by dimensional ssquantity units #### = ##²wel 15 m/s 0.270 E+6 Pa ### = ### 20 m/s 0.480 E+6Pa pU, ##² = ##²wel. 25 m/s 0. 750 E+6 Pa p' ##² = ##²wel 30 m/s 1.080 E+6 Pa q' U#L## = UwelL#### 15 m/s 37.5 l/s/m 15 m/s 3'7. 5 1/s/m U3L3 = UwehLin.et 20 m/s 50.0 l/s/m UgLs = U###L### 25 rn/s 62.5 1/s/m q' U#L# = U###L#### 30 m/s 75.0 1/s/m u U UW 5 m/s 15 rn/s u, Us = Unwell 20 m/s 20 m/s uii Us = Uweh 25 m/s 25 m/s ## Us = time,, 30 m/s 30 m/s X L# = L#### all 0. 0025 m Table 5 gives resu]. t. s for t-. he variation of the web speed for two flowratps ; 6 and # ####. The increase in act speed is effectively an increase in the two non-dirnenF. ional parameteTs characterizinq the flow, the Reynolds Number and the Capillary Number. Here we find that as the inertial. effects are magnified, t)) e pressure qradient increases while the maximum pressure decreases. Along the web, a gradual pressure adjustment followed by a sharp pressure peak is observed at lower Reynolds Numbers. The effects of. increase in web speed appear to have a qualitative relation to the effects of decreasing the flowrate. vel. ocily profil. e is again present in the qap ####. Increaseing web #### forces a greater amount of fluid ## exit the gap. t.. through viscous shear and the nearly monstant pressure gradient. Coating thickness increase is observed with art increase of web speed, as shown in Figures 24, 25 and 26.

The results of the present analysis exhibit qualitative agreement with those of Pranckh & Scriven (1988), as discussed above in connection with background of the invention. The graphical flow solution in the present. study, Figure 8#14, schould the compared to those of Pranckh & Scriven for the and pressure contours of their base case.

Pranckh & Scriven looked at the pressure distribution along the substrate for their base case and another case where both the Reynolds Number and flowrate were increased. In their base case Pranckh & Scriven found the pressure distribution had an inflection point, or plateau, followed by a peak just prior to the leading edge of the blade. Pranckh & Scriven found increasing the Reynolds Number and flowrate decreased the maximum pressure and eliminated the pressure plateau.

In the described embodiments it is determined that the pressure profile along the substrate has a peak just prior to the gap. The slope of the pressure plateau and the dimensionless pressure peak were also found to decrease with increasing Reynolds Number. The described embodiments also investigate the effects of the variation of the web speed (or Re|q=cOrst and ) and flowrate on the coating thickness, see Figures 24, 25 and 26. Similar to Pranckh & Scriven, it is found that the coating thickness varies nearly linearly with the increase in Reynolds Number, Capillary Number, and flowrate.

While preferred embodiments of the invention has been shown and described for the apparatus and method for coating devices for traveling webs in which a flowing stream of liquid coating composition flows in the same direction as the web movement in a vortex-free coater reducing wall shear stress on the coating material, other embodiments of the present invention will be-readily apparent to those skilled in the art from consideration of the specification and practice of the invention disclosed herein. It is intended specification and examples be considered as exemplary only, with a true scope and spirit of the invention being indicated by the claims.

Appendix : Nomenclature Kronecker delta rate of strain tensor surface tension boundary ll height of free surface t dynamic viscosity |t zero shear rate viscosity § infinite shear rate viscosity density 0stress tensor (7,. normal component of the traction vector 0. tangent : ial component of the traction vector Tdeviatoric stress tensor Ca Capillary Number C. coat. ing thickness c Carreau exponent f, component of gravitational acceleration H Gaussian mean curvature of the free surface Ktime constant applicator channel-exit L#### blade length (modeled)<BR> L### gap length<BR> ###### inlet length ## le#### ##### Lblack blade thickness Lweb length (modeled) lis/m (liter/sec)/meter m/s meter/sec n1 unit normal vector p pressure Pa ambient pressure q#### flowrate exiting along blade flowrate exiting gap i. nlet. flowrate Re Reynolds Number <BR> <BR> Psingularity<BR> 'time'<BR> T tare<BR> ## ##### ######<BR> ### #### velocity on intet Poiss#### profile<BR> ## length scale<BR> ! web velocity velocity We Weber Number t-;, vel-tical distance from web to free surface at C C x, Cartesian coordinate angle of blade denotes dimensionless variable