The coating of Newtonian liquids onto a roll rotating at low speeds

A theoretical treatment is presented for the prediction of the rate at which a liquid is entrained by a roll which is rotating partially submerged in a Newtonian liquid, and also for the prediction of the variation in the thickness of the film around the roll. The method involves the integration of the momentum equation after neglecting inertia forces in the dynamic meniscus region and the matching of the surface curvature with that derived for the static meniscus region near the liquid surface. The resulting equation allows the prediction of the relationship between a dimensionless film thickness parameter, T, and the capillary number, Ca, as a function of the radius of the roll, the withdrawal angle and the position of the roll periphery. The liquid flux on the roll can also be predicted.     

Free coating of viscous and viscoelastic liquids onto a partially submerged rotating roll

Experiments are performed on the dynamics of formation of a liquid coating picked up by a cylindrical roll rotating partially submerged in a free bath. Data on coating thickness for Newtonian fluids are found to correlate according to T = H(ρg/μU)^1/2 = 0.56 for fluids whose viscosities range from 0.11 to 33 poise. Data for strongly non-Newtonian and Viscoelastic fluids (polyacrylamide solutions) can be forced to fit this correlation by defining an “equivalent coating viscosity.” It is clear that this defines a pseudo viscosity, because the “equivalent coating viscosity” is observed to increase with increasing roll speed. This suggests that strongly Viscoelastic fluids respond to the rapid deformation suddenly imposed in the dynamic meniscus near the pickup point in a distinctly elastic manner that alters the flow through the meniscus.     

A semi-empirical model of the forward roll coating flow of newtonian fluids

A theory of the forward roll coating flow of Newtonian fluids has been developed recently by Greener and Middleman [1] utilizing the usual lubrication approximations and some very simple physical notions regarding the region in which the total flux through the nip splits into two films. These basic notions restricted the validity of their model to the simple case of two rolls of equal rotational speeds and equal radii. Their analysis resulted in an expression for the dimensionless total flux through the nip λ as a weak function of a modified capillary number N.The aim of this paper is to extend the model presented by these workers [1] by introducing to it experimental evidence of the flux distribution [2] and thus generalizing its applicability to operations involving rolls of different diameters rotating at different speeds. The results of this extended model give the dimensionless total flux λ as a function of the modified capillary number N, the speed ratio r_s, and the radius ratio r_r. This function is a weak one over the range of N, r_s, and r_r, normally encountered in forward roll coating operations and is in good agreement with experimental data [2].     

Free coating of non-newtonian liquids onto a vertical surface

The coating of non-Newtonian liquids onto a vertical surface continuously withdrawn from the liquid bath is considered. An analytic treatment is presented for purely viscous non-Newtonial liquids using the Ellis and generalised Bingham models both of which may be reduced to a new theory for power-law fluids. The theories give a relationship between the dimensionless film thickness, T₁, and the Capillary number, C₁, as a function of the fluid physical properties and the parameters of the viscous model. The dimensionless groups have been generalised to allow for non-Newtonian behaviour. The power-law and Ellis model predictions are compared with previous theoretical studies and shown to be consistent with known limits. Experimental data are also presented for a wide range of non-Newtonian fluids and compared with the new theories.     

Free coating of pseudoplastic liquids onto a vertical surface

The coating of pseudoplastic liquids onto a vertical surface continuously withdrawn from the liquid bath is considered. The problem is treated in a two-dimensional nonlinear approach incorporating the inertial effects. The model gives a relationship between the dimensionless film thickness, T, and the Capillary number, Ca, as a function of the fluid physical properties and the parameteres of the power-law model. The comparison with the experimental data is satisfactory.     

Viscous flow on the outside of a horizontal rotating cylinder: The roll coating regime with a single fluid

The flow of a viscous fluid on the outside of a horizontal rotating drum partially submerged on the fluid, has many instances of interesting practical applications. The purpose of this paper is to present a well rounded set of theoretical, numerical and experimental results for a set of parameters within the range of interest of coating processes.     

Centrifugal degassing of highly viscous newtonian liquids

The movement of gas bubbles in a Newtonian liquid under centrifugal forces was considered. Taking into account wall effects and the increase of bubble diameter during the movement caused by pressure decrease, equations for the bubble movement time in the case of entirely free and immobile bubble surface were derived. Experiments on centrifuging air bubbles showed that in the centrifuge the bubble retained a free surface even at much smaller diameters than under gravity forces. A dimensionless bubble diameter was defined and its critical value above which bubbles behaved as free ones was evaluated.     

Extrudate swell of newtonian liquids from rectangular dies

Extrudate swell from square and rectangular dies was studied for glycerol solutions. Experiments were performed for a wide range of viscosities 75 < μ < 464 mPa · s, with small changes in surface tension and density. Dies with aspect ratios of 3:1 and 2.25:1 and a wide range of length to hydraulic diameter ratios, i.e., 24 < l/D_h < 62, were used. Results were obtained in the range of Reynolds number 1 < Re < 50. Swell ratios for the short and long faces of a die are below the swell ratio of a plane slit. Swell ratios for the short face are consistently higher than those for the long face for lower Reynolds numbers. Liquid contracted at the corners, while swelling along faces.     

Air entrainment in a roll coating system

Experiments are performed to determine the critical speed at which air is entrained by a roll, rotating partially submerged about an axis parallel to t free surface. Data for Newtonian fluids correlate in terms of a critical Capillary number, Ca* = U* μ/σ, which is found to depend on a parameter γ = σ(?/μ⁴g)^{$\text{1}\text{3}$} over four decades in γ. A model based on simple force balances at the dynamic meniscus reflects these observations.Data obtained with viscoelastic polymer solutions indicate that elasticity is a strong stabilizing factor.     

Blotching in roll coating

After thermoforming, plastic parts are stacked for shipping, and these parts tend to stick together. Called nesting, sheet stock is often first coated with a silicone compound before thermoforming to prevent this. The coating usually consists of a small amount of lubricant dispersed in a majority of carrier fluid, and this fluid must then dry before reaching the sheet winder or else the coating blotches. This coupling of coating and drying to determine when to expect blotching is examined. Roll coating involves a dimensionless group called the elasticity number that governs the thickness of the coating to be dried. The drying section involves the evaporation of the coating carrier fluid, and then diffusion into the dry surrounding atmosphere. When analyzing the drying, a new dimensionless group that governs blotching is discovered, called blotchability. The result of this analysis allows practitioners to determine which operating conditions cause blotching, and how to eliminate it. Roll coating uses a deflecting rubber roll to apply vanishingly thin coatings (≤1 μm), an interesting elastohydrodynamic problem.     

Tri-helical gravure roll coating

A mathematical model is presented and solved for the fluid flow within the coating bead of a tri-helical gravure roll coater, operating in reverse mode. A variety of rolls etched circumferentially with grooves of different cross-sectional shape and aligned at non-zero angles of pitch are investigated. Predictions of fluid pick-out from the grooves are compared with complementary experimental data. Quantitative agreement between the two is found to be very good, showing a linear increase in pickout as a function of web-to-roll speed ratio and groove depths up to the point at which streaking, as observed experimentally, occurs and beyond which the model is no longer valid. In regions of parameter space for which there is no experimental data available for comparison purposes the model predicts that: (i) fluid pick-out decreases with increasing groove depth while the film thickness tends to increase and (ii) an increase in groove aspect ratio leads to a reduction in both pick-out and film thickness for a given land width and groove cross-sectional shape.     

Shear rate dependent viscosity of suspensions in newtonian and non-newtonian liquids

Viscosity measurements in a wide shear rate range, on suspensions of six Newtonian and non-Newtonian liquids, are reported.The shear rate dependency of suspensions in non-Newtonian liquids is compared with that of suspensions in Newtonian ones and an extension to the former of the Krieger—Dougherty theory, which holds for the latter, is attempted. Although some success is obtained, an alternative explanation of the observed behavior of the suspensions of non-Newtonian liquids is discussed.     

Mass transfer to newtonian and non-newtonian fluids from rotating cylinders

Mass transfer from rotating cylinders to Newtonian and non-Newtonian fluids is investigated. Present and previous data are analysed and correlations are presented. Free convection mass transfer correlations for vertical cylinders are found to differ from those for horizontal cylinders available in previous literature. Modified Prandtl analogy is applied, by considering the eddy penetration of the diffusion sublayer, to obtain the following expression for forced convection mass transferSh Sc^{?$\text{1}\text{3}$} = 0.62fRe which is found to correlate the present and previous data in the range of Sc, 2.6–32000, Pr, 0.72–676 and Re, 10²–10⁵ with a standard deviation of 14.6%. Differential viscosity is used to correlate mass transfer data to non-Newtonian fluids. Experimental results are seen to be lower than those predicted by the Newtonian correlation.     

Flow of generalized newtonian liquids through fixed beds of nonspherical particles

The flow of generalized Newtonian liquids through a random fixed bed of particles has been investigated and a universal method of calculation of the creeping flow was suggested. The usefulness of this method has been verified experimentally for the flow of power law, Ellis and Sutterby liquids through fixed beds of different nonspherical particles.     

Heat transfer to newtonian and non-newtonian liquids in a screw agitator and draft coil system

Heat transfer has been investigated for several Newtonian and non-Newtonian liquids mixed in a flat bottomed vessel equipped with a screw agitator and a coil acting simultaneously as draft tube and heat exchanger. Heat transfer rates from coil to liquid were determined for different coil designs, rheological properties and operating conditions in the heating and cooling modes. A circulation Reynolds number is defined, the characteristic length being the effective height of the heat exchanger (the coil) and the characteristic velocity being the circulation velocity of the fluid which was determined experimentally. Using this Reynolds number, it was possible to establish a single correlation for the Nusselt number as a function of Reynolds and Prandtl numbers. Geometric ratios do not appear in this correlation which could adequately represent more than 200 experimental data. A similar single correlation could not be obtained when using the conventional mixing Reynolds number. This novel system is shown to be very efficient for handling rheologically complex fluids.     

Free coating of a Newtonian liquid onto a vertical surface

The entrainment of Newtonian liquid films onto a vertical surface which is continuously withdrawn from a bath of the liquid is considered. A new theoretical treatment is presented which is significantly different from previous theoreis and which predicts accurately the relationship between the dimensionless thickness parameter, T₀, and the Capillary Number, Ca, up to a value of Ca of 2. Experimental results have been obtained using a capacitance technique for film thickness measurement for a variety of Newtonian fluids with viscosities ranging from 0·006 to 2·06 N. sec m^?2.     

Mixing of newtonian and non-newtonian liquids: Screw agitator and draft coil system

Mixing efficiency and power consumption have been investigated for several Newtonian and non-Newtonian liquids in a novel system consisting of a screw agitator rotating in a draft coil. For Reynolds numbers larger than 50, the inertial forces increase the pumping capacity of the screw impeller and hence its mixing efficiency. Shear-thinning effects can be accounted for by taking the effective rate of deformation to be proportional to the impeller rotational speed. Elasticity effects could be satisfactorily correlated with the Weissenberg number. The ratio of the mixing time to the average circulation time is constant. The power number could not be correlated with the Reynolds number; however, the ratio of the power number to the circulation number is inversely proportional to the Reynolds number for Newtonian as well as for shear-thinning inelastic fluids. The power consumption for elastic fluids is much larger.     

Roll coating of purely viscous liquids

Experimental measurements are reported for the flow of Newtonian liquids through the nip of two co- and counter-rotating rolls of various size and speed ratios. Results are presented in a dimensionless form for the total volumetric flux through the nip and for the distribution of this flux between the two rolls. Simple correlations have been developed and these should be useful in the design and analysis of roll coating equipment.Some preliminary data are also presented for a shear thinning inelastic non-Newtonian liquid. The forms of the correlating equations developed for the Newtonian systems are still applicable but the constants in the equations are slightly modified.     

Knurl roll design for stable rotogravure coating

When a thin film of fluid is coated on a web by any type roll or spreader coater such that the fluid splits while in a divergent channel the resulting fluid surface is not smooth but is made up of ridges, running in the direction of coating. At production speeds (50 m/min and higher) there is a natura frequency of this fluid ribbing effect which is independent of the type of coater and fluid rheology, and depends only on the depth of the wet coat of transferred liquid. A study has been made to determine how this natural ribbing frequency is related to coating stability on a rotogravure coater with ruling mill knurled coating roll. It has been observed that many knurl rolls deposit a metastable fluid pattern on the web when the resulting fluid sur matches the knurl pattern of the roll. Under some coating conditions this resulting surface becomes unstable and the fluid is transferred to a conditio fewer ridges per cm, resulting in a non-uniform coated surface. It was found that to enhance stable transfer from the knurl grooves it is necessary to the knurl roll surface to match the natural fluid ribbing frequency. The farther the knurl line frequency deviates from the natural fluid ribbing frequ the more often the coating surface will become unstable.