逆向辊涂，剪切速率

In this study an incompressible flow with non-Newtonian fluids in the reverse roll coating process was investigated. Non-Newtonian behavior of the coating fluid was accounted by using power law model with power index, n, ranging from 0.8 to 1.2. Effect of roll speed ratio (V2/V1) of the panel roll to the applicator roll and gap distance on the coating film thickness were also investigated. Numerical results were in good agreement with those of experimental data within 15%-20%. Results indicated that the film thickness ratios are function of power-law index, roll speed ration and ratio of roll radius to gap distance. The equations for film thickness ratio were obtained from numerical results as shown below: t2 t1 =0.89n ( V2 V1 −0.83 H0 Rm 0.0025 , t3 t1 =65.6n ( V2 V1 −0.043 H0 Rm 0.624 , where t1, t2 and t3 are inlet film, transferred film and leakage film thickness, respectively, n is power-law index, V1 and V2 are roll speed of applicator roll and panel roll, Rm is average radius of two rolls and H0 is gap distance. The correlations are accurate within 10% for 0.8≤n≤1.2, 0.5≤V2/V1 ≤2, and 7.7×10 ≤H0/Rm ≤1.54×10 −4. AMS subject classifications: 76A05

… The region of interest in the present analysis of the reverse roll-coating line is identified in Fig. … span the full range of shear rates encountered within the actual coating process, as the nip …

… gap of a reverse roll coater, Thisflow is crucial in determining the thickncss and the quality … infinite shear rate viscosity to the zero shear rate viscosity, r0 is the characteristic shear rate at …

… The Fluid Dynamics of Reverse Roll Coating The flow in the metering gap of a reverse roll coater is examined by experiments and finite element solutions of the Navier-Stokes equations…

… in shear stress that diminished when increasing the imposed shear rate; at a shear rate of … They concluded that in the roll coating process, where the shear rates are much higher, one …

The Galerkin finite element method of solution formulated by Coyle et al. is applied to analyse the coating flow field between two reverse rotating rolls. A wide range of operating conditions that are of practical importance were studied. Many quantitative results that can benefit a practising engineer are presented and physical insight is gained by the analysis of the data. It is shown that the reverse roll coating process is affected by various factors, among which the roll speed ratio is the most significant. Regardless of other parameters chosen, a critical speed ratio always exists at which the metered film thickness experiences a minimum. At this critical speed ratio, the dynamic contact line is exactly located at the centre of the gap between the two rolls. The significance of the Reynolds and the Capillary numbers increases with increasing speed ratio. For speed ratios beyond the critical value, a thinner metered film can be achieved by means of either increasing the film thickness of the entering layer or by decreasing the principal roll radius. Furthermore, the computational predictions indicate that changing the roll radius ratio has no obvious effect on reverse roll coating and that gravity effects, quantified by the Stokes number, can be ignored under normal operating conditions. Copyright © 1999 John Wiley & Sons, Ltd.

… liquids under the deformation rates of interest. (Typical range of shear rates in the experimental … purpose, to illustrate the general features of reverse roll coating, has been achieved. It is …

… , all liquids exhibit a power-law behaviour in the high shear rate region – a region of relevance to roll coating. Hence, only the power-law parameters determined in the high shear rate …

Abstract Reverse roll coating is probably the most widely used coating operation, yet its full potential has not been exploited as it is shown in this paper which considers operation with a negative gap. We demonstrate through a wide range of experimental data that such operation can yield very thin and stable films with no ribbing or cascade instabilities when low viscosity fluids are used. Typically, stable film thickness less than 5 μm can be obtained at speeds up to 150 m/min when a rubber roller is used at −100 μm gap with fluids of viscosity in the range 10–200 mPa s. These film thicknesses can be made to decrease further down to 1 or 2 µm with a judicious choice of speed ratios (applicator to metering roller) and rubber hardness. Such new findings make this simple coating method an attractive roll to roll technique for application in the newer coating technologies, such as in the production of solar cells and plastic electronics. The data obtained in this study have been underpinned by a model based on the classical lubrication theory, well developed for such flow situations. Essentially it is shown that the film thickness non dimensionalised with respect to the set negative gap is controlled through a single parameter, the elasticity number Ne which combines all the operating parameters. Of course, this flow problem has complexities, particularly at high speed ratios and at zero gap so the data obtained here can serve as a basis for more comprehensive modelling of this classical fluid mechanic problem.

… The key feature of a reverse roll coater is the fluid flow in the gap between twoparallel rigid … two rolls side-by-side and half-submerged in the coating liquid. Over a certain range of roll …

This article presents a mathematical model and theoretical analysis of coating of a thin film of non-Newtonian polymers as they travel through a small space between two reverse-rotating rolls. The dimensionless forms of the governing equations are simplified with the help of the lubrication approximation theory (LAT). By using the perturbation technique, the analytical solutions for velocity, flow rate and pressure gradient were obtained. From an engineering point of view, some significant results such as thickness of the coated web, pressure distribution, separation points, separation force and power input were computed numerically. The effect of velocities ratio k and Weissenberg number We on these physical quantities is presented graphically; others are shown in tabular form. It is noted that the involved material parameters provide a mechanism to control the flow rate, pressure distribution, the thickness of coating, separation force and power input. Moreover, the separation point is shifted toward the nip region by increasing velocities ratio k.

… viscosity of the coating colors at low shear rates (shear-thinning behavior) decreases significantly the air entrapment in the nip, although the air content in the coating color may still be a …

… profiles in both forward and reverse roll coating (Young 1997; … the coating colors exhibit shear thinning up to a shear rate of … The onset of shear thickening appears at a larger shear rate …

This article demonstrates a mathematical model and theoretical analysis of the Micropolar fluid in the reverse roll coating process. It is important because micropolar fluids account for the microstructure and microrotation of particles within the fluid. These characteristics are significant for accurately describing the behavior of complex fluids such as polymer solutions, biological fluids, and colloidal suspensions. First, we modeled the flow equations using basic laws of fluid dynamics. The flow equations are made modified using low Reynolds number theory. The simplified equations are solved analytically. The exact expression for velocity and pressure gradient are obtained, while pressure is calculated numerically using Simpson Rule. Graphical depictions are carried out to comprehend the impact of the newly emerged physical constraints. The influence of micropolar and microrotation parameters on the velocity, pressure and pressure gradient are elaborated with the help of different graphs.

Reverse roll coating is a high-speed coating process used in many applications. The single largest application for reverse roll coating is applying architectural paint to coils of sheet metal; however, in the converting industry reverse roll coating is used for some thermal coatings, adhesives, color laser paper receivers, wax coatings, magnetic coatings, vinyl dispersions for wallcovering, some specialty light sensitive applications, and many specialty products that require a high precision application process. This work presents a mathematical model for the steady flow of non-isothermal, incompressible, micropolar fluid in the metering nip of a reverse roll coater with velocity slip applied at both roll’s surfaces. Lubrication approximation theory (LAT) is utilized to reduce the complexity of the non-dimensional equations. Exact solutions for pressure gradient, velocity, and temperature are achieved. At the same time, numerical techniques (Simpson’s 3/8 rule with Newton Raphson method) are used to compute the pressure distribution, coating thickness, flow rate, and separation points. Effects of coupling number, slip parameter, microrotation, and velocity ratio on velocity, pressure, temperature, and pressure gradient are presented in graphs. It is observed from the current analysis that the intrinsic rotations of the fluid particles prove to be the controlling parameter of the pressure gradient, which significantly varies the coating thickness. Also, as a result of velocity slip, the flow rate declines and the coating thickness reduces as the fluid moves faster along the boundary walls.

… This study investigates the reverse roll coating technique for an incompressible non-isothermal upper-convected Jeffery's fluid model, and the flow equations are simplified using (LAT) …

… roll coating (co-rotating rolls in the film splitting region) or the reverse roll coating (counter-rotating rolls in … , whereas the flow in 3-roll full roll coating is based on the reverse roll coating …

… The rheology of the magneto-hydrodynamic (MHD) hybrid nanomaterial on the final coating thickness is studied by using the reverse roll coating technique. In the coating industry, the …

This article describes the development of a mathematical model of the reverse roll coating of a thin film for an incompressible non-isothermal magnetohydrodynamics (MHD) viscoplastic fluid as it passes through a small gap between two rolls rotating reversely. The equations of motion required for the fluid added to the web are constructed and simplified using the lubrication approximation theory (LAT). Analytical results are obtained for the velocity profile, pressure gradient, and temperature distribution. The pressure distributions and flow rate are calculated numerically using the trapezoidal rule and regular false position method, respectively. Some of these results are presented graphically, while others are shown in a tabular form. From the present analysis, it has been observed that the magnitude of pressure distributions increases by increasing the value of the involved parameters. It is worth mentioning that the velocities ratio and Brickman’s number are controlling parameters for the temperature distributions. The results indicate the strong effectiveness of the viscoplastic parameter and velocities ratio for the velocity and pressure distributions. It is also concluded that the coating of Casson material has been remarkably affected by the magnetohydrodynamics effects.

The non-isothermal couple stress fluid inside a reverse roll coating geometry is considered. The slip condition is considered at the surfaces of the rolls. To develop the flow equations, the mathematical modelling is performed using conservation of momentum, mass, and energy. The LAT (lubrication approximation theory) is employed to simplify the equations. The closed form solution for velocity, temperature, and pressure gradient is obtained. While the pressure and flow rate are obtained numerically. The impact of involved parameters on important physical quantities such as temperature, pressure, and pressure gradient are elaborated through graphs and in tabular form. The pressure and pressure gradient decreases for variation of the couple stress parameter and velocity ratio parameter K. While the variation of the slip parameter increases the pressure and pressure gradient inside the flow geometry. Additionally, flow rate decreases for the variation of the slip parameter as fluid starts moving rapidly along the roller surface. The most important physical quantity which is responsible for maintaining the quality of the coating and thickness is flow rate. For variation of both the couple stress parameter and the slip parameter, the flow rate decreases compared to the Newtonian case, consequently the coating thickness decreases for the variation of the discussed parameter.

A semi‐implicit Taylor‐Galerkin/pressure‐correction algorithm of a transient finite element form is applied to analyse the flow instabilities that commonly arise during reverse‐roller coating. A mathematical model is derived to describe the solvent coating applied to the underside of the sheet, assuming that the lacquer is a Newtonian fluid and considering the flow between application roller and foil. Here, we have investigated the effects of temporal instabilities, caused by adjustment of nip‐gap width and foil‐position, extending our previous steady‐state analysis. Foil shifting is found to have a significant influence upon pressure and lift on the foil, drag on the roller, and free coating profiles. This would result in process instabilities, such as chatter and flow‐lines. In contrast, nip‐gap adjustment has no influence on the coating finish.

… pressure distribution of fluids flowing through a deformable nip of rolls counter-rotating at … predicting the pressure distribution in forward or reverse roll coating, the results obtained when …

Reverse roll coating, one of the most widely adopted coating techniques, is considered as a self‐metered process. Its volume flow rate in the coating bead is analytically unpredictable due to the presence of the dynamic contact point. In an attempt to approximate the volume flow rate in the coating bead, the position of the dynamic contact point was numerically approximated by computational fluid dynamics (CFD) and then introduced to Navier‐Stokes equations. As a result, this semi‐analytical approach represented a non‐linear correlation of the volume flow rate with speed ratio and capillary number.

… here is the viscous flow past an air bubble in a narrow nip region between rolls, as shown in … case, as is the case in reverse roll coating. The contact line moves toward the inlet diverging …

… , the nip gap, and the transfer roll. The color fed to the chamber is strongly accelerated when reaching the transfer roll upstream from the metering nip and can be subjected to flow …

… SUMMARY The flow in a fixed gap reverse roll coater is … As flow rate, A, is increased the metered downstream film … Fluid is dragged through the nip by the lower roller and metered by the …

… localised nip-pressures. The majority of this study focuses on the secondary nip-vortex and its influences on the contact point and dynamic wetting line. This aspect of the flow provides …

A two-roller system, consisting of an image bearing roll and a biased metering roll, is a common design for liquid ink electrophotography. To stabilize toned images, the development system is required to minimize the liquid content of the toned images. The reverse metering roll, moving in a direction opposite to the process direction, provides a mechanical shearing that reduces the liquid film thickness of the toned image. In this report, a discussion of the hydrodynamic metering mechanism is presented based on the study of modeling results as well as experiments. Formulation of computational models, based on the Navier-Stokes equations for free surface flows, is given followed by the description of experimental apparatus and methods. The critical parameters identified through this study include process parameters as well as geometrical and material properties (roller and gap sizes, process and metering speeds, viscosity, surface tension, etc.). The impact ofthe fluid flow characteristics on the final toned image attributes, such as the background cleanliness and the potential image degradation due to viscous shearing, is then discussed. Such issues can be made more apparent by examining the calculated fluid flow details.

… is motivated by the quadratic dependence of the velocity field on x2 in a Newtonian flow. When the viscosity is shear thinning, the velocity field is no longer quadratic; however, a more …

… the shear thinning, the farther out of the gap the splitting meniscus moves, the higher the flow rate through the gap, and … An important point is that the entire flow field and free surface are …

On the way toward a carbon-neutral economy, a rise in the demand for separation technologies can be expected. This holds especially for membranes, which are energy efficient and thus very promising technologies. However, this challenges membrane researchers to consider the sustainability and scalability of their membrane fabrication processes. At our institute, we employ a roll-to-roll coating process for the production of thin film composite membranes. This procedure is relatively easy to scale up and can be adapted for different polymers/solvents and thus applications. To increase the production efficiency and optimize the process for any polymer of interest, it is necessary to develop a solid understanding of the physics of this production system. Therefore, we would like to present a numerical model based on computational fluid dynamics that can predict the film thickness of a polydimethylsiloxane-based polymer coated in a roll-to-roll setup. In the future, this model should improve the production efficiency and fine-tuning of process parameters. We verify the numerical procedure with a mesh refinement study and validate the predicted film thicknesses with experimental results for different roll speeds and polymer concentrations. The predicted variability of the thickness is assessed by a design of experiments study and compares relatively well to the measured variations of the coated membrane thickness.

… Ó American Coatings Association 2020 Abstract This paper focuses on the flow stability of reverse roll coating flows employed in practical applications to coat thin liquid films on …

This article is concerned with the numerical simulation of a reverse roller‐coating process, which involves the computation of Newtonian viscous incompressible flows with free‐surfaces. A numerical scheme is applied of a transient finite element form, a semi‐implicit Taylor‐Galerkin/pressurecorrection algorithm. For free‐surface prediction, we use kinematic boundary adjustment with a mesh‐stretching algorithm. In the present work, an alloy sheet (foil) passes over a large roller and then a smaller applicator roller, which provides the in‐feed. In combination, the applicator roller, the foil and the fluid form part of the underside coating mechanism. The aim of this study is to investigate fundamental aspects of the process, to ultimately address typical coating instabilities. These may take the form of chatter and starvation. A uniform coating thickness is the desired objective. A mathematical model is derived to describe the solvent coating applied to the underside of the sheet, assuming that the lacquer is a Newtonian fluid. In particular, the work has concentrated on the flow patterns that result and a parameter sensitivity analysis covering the appropriate operating windows of applied conditions. Effects of independent variation in roll‐speed and foil‐speed are investigated, to find that maxima in pressure, lift and drag arise at the nip and are influenced in a linear fashion.

… coating and reverse roll coating. … CFD software package CFD-RC [14] which uses finite volume method to solve the governing equations. The two rolls of forward roll coater have a roll …

… ) has advanced as a class of Computational Fluid Dynamics (CFD) methods used to solve … that reverse roll coating has better advantages compared to forward roll coating in terms of …

… data for forward roll coating given by Gaskell et al. (1998a) for a system with σ = 0.0285 N m … accurate at significantly higher roll speeds/Capillary numbers in the related case of reverse …

… features of the nip flow and is numerically more manageable than the entire representation of the free surface elastohydrodynamic reverse roll coating problem in three dimensions. …

Abstract In the current investigation, we propose the analytical and numerical solutions of the Navier–Stokes equation for the reverse roll coating process (RRCP) using a bath of the Sisko fluid model (SFM). A suitable transformation is applied to the partial differential equation-based mathematical model of SFM-RRCP, resulting in a set of nonlinear ordinary differential equations. The perturbation method has been employed to find the analytical solutions for velocity, flow rate, pressure gradient, pressure, and temperature distribution. The Newton–Raphson method has been used to find coating thickness. Furthermore, numerical integration has been applied to compute some mechanical quantities of interest, including power input and roll separation force. Sensitivity analysis is another approach implemented using response surface methods to examine the impacts of velocity ratio and non-Newtonian parameters on coating thickness, power input, and roll separating forces. The residual graphs and contour diagrams are also shown. It has been observed that as the value of non-Newtonian fluid parameters increases, the velocity profile decreases. However, the coating thickness on the web is a decreasing function. Further, it has been observed that separation points moved toward the nip region when the value of the velocity ratio was raised. Moreover, the sensitivity of roll separating forces and power input to input parameters is negative.

… revisiting the fluid mechanics of reverse roll coating, a simple coating operation, and assessing … CFD method. As the data of Coyle et al.3 were obtained from a model reverse roll coater (…

Roll coating plays a major role in industrial coating, including wallpapers, plastic and photographic films, sticky tapes, magnetic recordings, wrapping magazines and books, and so on. The current study proposes a mathematical model for a non‐isothermal, incompressible Sutterby fluid flowing through a narrow gap between two heated, counterrotating rollers. Lubrication approximation theory is used to simplify nondimensional expressions. The perturbation technique provides exact results for velocity profile, temperature, flow rate, and pressure gradient, whereas the numerical technique (Simpson rule) is used to compute the pressure profile and flow rate, respectively. The effects of the involved parameters on various physical characteristics like pressure, flow rate, temperature, pressure gradient, force, and power input are depicted in graphs and tabular form. A mechanism for controlling the coating thickness, power input, flow rate, separation force, and pressure distribution is provided by the material properties involved. With variations to a Sutterby fluid parameter β $\beta $ and the velocity ratio K, both the pressure gradient and pressure decrease. It is significant to note that the temperature distribution is controlled by the velocities ratio and Brinkman number. Moreover, the separation point is shifted towards the nip area and the coating thickness on the web reduces with increasing velocities ratio K.

… They employed lubrication approximation and obtained both the numerical and analytical solutions for Newtonian and non-Newtonian fluids. The reverse roll coating was also …

The flow of non-Newtonian fluids is extremely important in a variety of industrial applications and processes, including painting, printing, and coating. The goal of this paper is to develop a mathematical model for an incompressible, non-isothermal third-grade fluid as it travels through a minor gap between two heated rolls that are revolving in the opposite direction. Using appropriate dimensionless parameters, the dimensionless version of the governing equations is constructed. To simplify the nondimensional form of the governing equations, the LAT (lubrication approximation theory) is applied. It is mentioned that the perturbation approach should be used. It is a very important tool in modern mathematical physics and practical fluid mechanics, and it is used a lot. Through perturbation techniques, exact solutions for velocity distribution, flow rate, temperature distribution, and pressure gradient are offered. To observe the flow pattern, streamlines are also drawn. Numerical outcomes of some substantial engineering quantities like flow rate, roll separation force, power input, and pressure distribution are found. Some outcomes are displayed graphically although others are presented in tabular form. We have observed that the third-grade parameters give a way to control velocity, pressure gradients, flow rate, and coating thickness. Further, increasing the velocities ratio values lowers the coating thickness on the web. Furthermore, the Brickman number has a significant influence on the temperature profile and has increased with the increase in the Brickman number. Both the model and the numerical calculations are in agreement with the viscous fluid model that is already present in the literature.

PurposeThe coating process is broadly employed in the manufacturing of wallpapers, adhesive tapes, wrapping, protection of fabrics and metals, X-ray and photographic films, beautification, books and magazines, film foils, magnetic records, coated paper, etc.Design/methodology/approachIn this study, an incompressible flow of non-Newtonian fluid is modeled to inspect the rheological behavior of finite coating thickness in the reverse roll coating process. With the assistance of lubrication approximation theory (LAT), the dimensionless form of governing expressions is simplified. Exact solutions for distributions for velocity, flow rate, temperature and pressure gradient attained utilizing perturbation technique and their variation is presented as well as discussed in graphs. Meanwhile, some important factors from an engineering perspective including coating thickness and transition point were calculated mathematically and are displayed in a tabular manner. Also, streamlines are drawn to observe the flow pattern.FindingsPrandtl fluid parameters provide a controlling factor to regulate the flow rate, velocity, coating thickness, and pressure gradient leading to an efficient coating process. Moreover, the Brinkman number and Prandtl fluid parameters significantly improve the temperature distribution.Originality/valueIn the literature, this study fills a gap in the theoretical prediction of coating thickness rheologically influenced by Prandtl fluid in reverse roll coating process.

Roll-coating process plays an important role in many industries for its practical applications such as paint, PVC coated fabrics and plastic industries. There are several roll-to-roll coating methods including forward and reverse roll-coating. However, the roll-over-web coating study of the Rabinowitsch model is presented in this paper. The flow equations for the problem are developed and converted into dimensionless form with the help of dimensionless variables and then finally simplified by a well-known lubrication approximation theory. We employ the regular perturbation technique to get analytical expressions for velocity, pressure, and pressure gradient. Engineering quantities such as power input function and roll-separating force are calculated by Runge-Kutta method. The dimensionless Rabinowitsch parameter effect on velocity, pressure, pressure gradient, load-carrying force, and power input are shown graphically. It is interesting to note that for the shear thickening case, the Rabinowitsch model predicts 35% higher pressure, while in the shear thinning case it predicts 29% less pressure in the nip region when compared to the Newtonian model. The force and power show a decreasing trend on increasing the dimensionless Rabinowitsch parameter a. Moreover, the separation point shifts right of its Newtonian value when fluid behaves like shear thickening and volumetric flow rate increases which causes the coating thickness to increase.

… models to analyze meniscus roll-coating. The zero flux … reverse roll-coating using an optical sectioning technique. They performed four experiments for forward and reverse roll coating …

An industrially-important variant of reverse roll coating is studied in which the metering gap … via complimentary experimental, analytical (lubrication) and computational (finite element) …

… the lubrication approximation theory is used to provide numerical results in roll coating over … differential equation during reverse roll coating process under lubrication approximation …