CFD Simulation and Experimental Verification for non-newtonian Falling Film on Inclined Plate

Transcription

1 CFD Simulation and Experimental Verification for non-newtonian Falling Film on Inclined Plate S. HAERI, S.H. HASHEMABADI Computational Fluid Dynamics Research Laboratory, Department of Chemical Engineering Iran University of Science and Technology Narmak, 16846, Tehran, Tel: +98(1) Iran Abstract: - In this study extensive unsteady 3-D simulations are carried out to predict velocity and thickness of the non-newtonian falling film flow on inclined plates which have been measured experimentally. Different concentration carboxymethyl cellulose (CMC) solution (1.1, 1.5 and %), has been selected as the operating fluid in experimental works. The rheological properties were characterized by PolyVisc Visco Star Series L, surface tension was measured in ambient temperature and a single static contact angle was measured by the goniometry method. The power-law fluid model is iplemented to account for non-newtonian behavior of the operating fluid. Inclined plates with different inclination angles α (0<α<π/), and assorted surfaces (ceramics, aluminum and glass) were used to study the effect of inclination and contact angle on the vital parameters. Simulation is performed by a version of VOF family of algorithms, CICSAM, which is used to track the air-liquid interface. Surface tension and contact angle effects are also considered in formulations. All numerical implementations are carried out under OpenFOAM 1.3 CFD toolbox frame work. Good agreement between experimental data and CFD simulation results is observed. Keywords: - CFD, VOF, CICSAM, Film flow, Non-Newtonian, Inclined Plate 1 Introduction Flow processes often involve the presence of free surfaces, the tracking of which has significant impact on the manufacturing and the final quality of the product. Examples abound, e.g., casting processes, mold filling, thin film processes, extrusion, spray deposition, fluid jetting devices in which material interfaces are present. Often, properties that depend on the shape of the interface itself (e.g., surface tension) play an important role in the dynamics of the problem, and the physics (e.g., capillarity) can drive the flow making it essential to accurately determine the position, curvature, and topology of the interface. While the hydrodynamics of thin film flow of Newtonian liquids has been extensively studied for several decades, see e.g. [1], only modest attention has been devoted to gravity-driven films of non- Newtonian liquids. The flow of non-newtonian liquid is not often fully understood in many processes and equipment units and so the designs are not properly optimized. Only a few studies have experimentally investigated the flow of non-newtonian fluids down inclined plates or planes. Astarita et al. [] measured the film thickness of non-newtonian fluids in fully developed laminar flow down inclined plates for a range of flow rates and plate angles. Therien et al. [3] measured the film thickness for power law fluids flowing down inclined plates and compared the results with an analytical expression with good agreement. Sylvester et al. [4] compared experimental and predicted film thicknesses for power law fluids flowing down a vertical wall in the laminar and wavy fully developed regimes. CFD modeling of flows over inclined planes is a difficult task owing to the presence of free surface. Although many researchers studied this phenomenon ISBN: page 48 ISSN:

2 numerically a full CFD approach is utilized less often. Flows down inclined planes were numerically investigated in the work by Ruyer-Quil and Man- Neville [5, 6]. They used a simplified model of the flows based on the so-called boundary layer approximation. Several CFD methods have been developed in recent years with the aim of simulating such complex flows. One such development is the volume of fluid (VOF) and level set methods to cope with the presence of material interfaces and gaining increasing acceptance by researchers although older algorithms like MAC [7] are still receiving attention. The VOF was initiated since the early 1980 s due to Hirt and Nichols [8]. The level set method is a more recent development dating back to Osher and Sethian [9]; see also Sethian and Smereka [10] for a summary. Comparing these two methods individually, each has its strength and limitations. Recently, a combined VOF and level set has been developed [11]. Using these reliable numerical models and algorithms, one can accurately calculate and study the flow behavior and phenomena, and to gain considerable insight into the physics behind these. These methods are recently employed to simulate film flow of non-newtonian fluids. Sutalo et al. [1] measured the film thickness of a power-law fluid and predicted the results by CFX-4 code with a version of VOF method including surface tension by CSF (Continuum Surface Force) method of Brackbill [13]. Miyara [14] used MAC method to simulate the flow of wavy liquid film down vertical and inclined planes and the results showed good agreement with experimental measurements. Dou et al. [15] used projection/level set method to predict the shape of the front of film including surface tension and contact angle effects. All mentioned CFD studies are restricted to D simulation. In this study a high resolution differencing scheme for capturing sharp fluid-fluid interface based on normalized variable diagram (NVD) of Leonard [16] devised by Ubbink [17] named CICSAM (Compressive Interface Capturing Scheme for Arbitrary Meshes) is employed to simulate film flow of non-newtonian fluids. Results are compared with experimental measurements. imaging 1% iodine solution was added to the prepared CMC solution. Although CMC s power law parameters can be found in the literatures [18, 19] colored CMC apparent viscosity was measured by PolyVisc Visco Star Series L and power-law model has been fitted to the experimental data. Surface tension was measured in 4 C and a single static contact angle was measured by the goniometry method and its variation was neglected. Being far from critical point and very minor changes in temperature corroborates the assumption of constant surface tension which is used throughout this paper. Table 1 summarizes measured properties of CMC solution used in numerical simulations. The experimental setup consisted mainly of a plate with adjustable angle of inclination which is shown in Fig. 1. Image acquisition and processing techniques were used to track the dynamic of fluid film and effects of different parameters. Whereas the fluid is highly viscous and the velocity of the falling film is very slow an ordinary camera, Canon EOS 400 (3 frame per second), has been used in this work. After every run the plates were thoroughly washed and dried to minimize the possibility of contact angle changes. Also all experiments were carried out in ambient temperature( 4 ± o C). This setup was used to investigate the effect of different parameters on dynamics of falling film of a non-newtonian fluid. Fig. 1. Simple schematic of experimental setup Experimental Setup Food grade CMC solution with 1.1, 1.5 and weight percent were prepared by solving required amount of CMC in double distilled Tehran city water. For better ISBN: page 49 ISSN:

3 C Table 1 Rheological properties of three different CMC solutions m n σ θ ( ) (Pas n ) (N/m) * ** NA 44.5 NA * Glass Surface, ** Ceramic surface, *** Aluminum surface, NA: Not available 3 Problem Formulation *** NA 58.5 NA 3.1 Governing Equations In VOF frame work for two incompressible fluids flow in the computational domain, the conservation of mass and momentum are respectively: u =0 (1) ρu + ( ρuu) = P + τ + ρg + f σ t () For generalized non-newtonian fluids: τ = μ & γ γ&, ( ( ) ) Where ( ) ( ) γ& = u + u is the rate-of-strain tensor. In this study power-law model is n 1 implemented and μ ( & γ) = m & γ. This model is applicable in a specific range of shear rates and generates unphysical values where shear rate is very small so it is necessary to bound it between an upper and lower bound. In this study these values are set to minimum and maximum apparent viscosities measured by the viscometer. The momentum equation for non-newtonian fluid flow can be simplified: ρu + ( ρuu) ( μ u) = t P + ρg+ f + u σ ( ) ( μ ) (3) A transport equation for indicator functions is also added to the system of equations which takes the following form: α + u α = 0 (4) t Where 1 1 = for the po int ( x,t )inside fluid α 0 for the po int ( x,t )inside fluid In order to model the two fluid as a continuum by using Eq. (1) and Eq. (3) the indicator function should be continuous and differentiable over the hole domain this restriction is even more stringent for calculation of curvature to calculate f σ, needing to be twice differentiable [13, 0]. A possible way is to give the transitional area between two fluids a small finite thickness so 0< α δ < 1 is defined in transitional area. Density and dynamic viscosity are defined as mean values in cells with both fluids present by utilizing the indicator function as follow: ( 1 ) ( 1 ) ρ = αρ + α ρ (5) 1 μ = αμ + α μ (6) 1 Surface tension creates a pressure jump across the interface and in equilibrium its gradient must be equal to extra body force inserted in momentum equation. The momentum equation is written for the whole domain but this force manifests itself only at interface and the discontinuous pressure jump must be calculated. These difficulties overcame by Brackbill [13] who defined this body force Eq. (7): σ α = σ α ( α ) f (7) 3. Boundary Conditions Schematic representation of physical domain is depicted in Fig.. A fixed velocity with parabolic distribution is implemented on the inlet: ISBN: page 50 ISSN:

4 &Q ( ) r wr = 1 (8) π R 0 R0 All flow quantities are extrapolated from inside of the domain for outlet boundary condition. All other boundaries are implemented as wall boundaries meaning no slip condition for velocity and it is enough to set a zero gradient boundary condition for indicator function [13]. Although this is not the case in experimental set up, low velocity of liquid and insignificant impact of air around the falling film corroborates this assumption. Wall adhesion is implemented on the wall on which the liquid flows. This is done by addition of contact angle effects and consequently defining the unit normal to the interface by Eq. 9 [13]. nˆ = nwcosθs + nt cosθs (9) A value of θ s = 90 reduces this formulation to zero gradient and θ s <90 means that the fluid wets the wall. 3.3 Discretisation Physical domain is discretised by about 500,000 hexahedron and wedge shape nodes. Initial stages of calculation is very sensitive to mesh and time step size due to splash of liquid against the plate. So an optimization between process time (time step size) and memory usage (mesh size) need to be performed. This is overcome by dividing the physical domain into different zones and meshing each partition with resolution to guaranty Co < 1 with a minimum tolerable time step size of sec. Fig.. Schematic representation of physical domain used for numerical simulation Temporal discretisation is implemented implicitly. Although this method cause numerical diffusion in the direction of flow [1, ] it is tolerated over the more computationally expensive second order Crank- Nicholson method. Convection term in momentum equation is discretised with a High Resolution (HR) differencing scheme proposed by Jasak [3] based on NVD diagram of Leonard [16]. In this method face value of velocity magnitude is calculated with Eq. (10) u % < 0 % D ud orud> 1 fu + ( 1 ) β < % x D fxua m ud< 1 % u = ud f 1 ( 1 fx) ud βm u% D + ( 1 f ) 0< % x ua ud< βm βm Where (10) ISBN: page 51 ISSN:

5 u u = 1 d A D u% D ( u) D In Eq. (10), also A and D subscripts are determined by direction of flow (i.e. Ff 0 D= C&A= DW or if Ff 0 D= DW&A= C ), β m is a prescribed constant, 0< β m <0.5, with recommended value of 0.1, f x is central differencing weighting factor and is set to a constant value of 0.5 in this study. Diffusive terms in momentum equation are divided to orthogonal and non-orthogonal terms. The non-orthogonal terms can be calculated by previous time step values and corrected in an internal loop however due to computational expense of this method and because the non-orthogonality of meshes is small and time step size is also small with respect to other time scales of the system no correction is executed in solution algorithm. Indicator function needs to be smoothed in order to calculate the curvature and consequently f σ, Brackbill used a B-spline as an interpolation function to smooth the indicator function and his method is used in this study. Face values of indicator function transport equation are calculated by a high resolution differencing scheme of Ubbink [17], specially devised for arbitrary meshes. This scheme actually introduces a blending factor base on the angle between the interface and direction of motion for prediction of normalized face values of CICSAM, using Hyper-C [16] and ULTIMATE QUICKEST [4] differencing schemes: % α = γ % α + 1 γ % α, ( ) ( θ f ) f f fcbc f fuq cos + 1 (11) γ f = min k γ, 1 Where k γ >0 is constant to control the dominance of different schemes with recommended value of 1. θ f is the angle between unit vector in direction of gradient of α and unit vector connecting two cell centers. All numerical implementations are carried out under OpenFOAM 1.3 CFD toolbox frame work. 4 Results and Discussion 4.1 Thickness of Film Dimensionless group parameters characterizing the flow behavior are the Reynolds, Webber and Froude numbers. These dimensionless groups are defined for the purpose of the current study n n by: Rex ρu = x m, Wex = ρu x σ sl and Fr = u x g cosϕ. Experimentally, these groups x are effectively summarized in a correlation [5, 6] for thickness of falling film: Re We x x Frx 0.1 Re We Fr < 100 x x x δ x = (1) Re We x x Fr x 100 Re We Fr 5 10 x x x Film thickness shows a descending trend with respect to time which is a result of dispersion of fluid over the surface. It can also be related to shear thinning properties of the operating fluid meaning as the fluid begins to flow over the surface shear rate increases resulting in a thinner fluid which consequently causes decreasing thickness. This phenomenon last till the flow reaches a steady velocity. In Fig. 4 contours of falling film are depicted which clearly shows the decreasing film thickness. 3 Fig. 4. Variation of Film thickness with time φ=30, 1.5% CMC, 11ccpm, θ=58.5 Fig. 5 shows film thickness calculated from the correlation and thickness predicted from the ISBN: page 5 ISSN:

6 simulation. Good agreement between experimental and simulation data is evident from this figure. Average error in calculating film thickness is 10%. Fig. 5. Comparison of film thickness from simulation and experimental measurements Fig. 6. Velocity distribution on different surfaces. φ=30, 1.5% CMC, 11ccpm. 4. Falling Film Velocity Velocity can be nondimensionalized by the velocity of the corresponding fully developed inviscid fluid at the same length. Flow time can also be nondimensionalized in the same manner [7]. This is done for the purpose of this study and velocity versus time is plotted in Fig. 6. which shows the dimensionless velocity versus dimensionless time for different surface tension. Velocity changes on aluminum surface shows the biggest error of 15% which can be related to variable contact angle on this surface that varies between 55 and 65 for receding a advancing contact angles respectively and is not taken into account in this study. Fig. 7. shows the same plot for different flow rates and show very good agreement between experimental data and predicted CFD simulation values. Fig. 7. Velocity profile with different flow rates. φ=30, 1.1% CMC, θ= Conclusion In this study flow of a falling film is studied numerically and experimentally. Dynamics of falling film is studied using image capturing and processing techniques and a correlation devised relating the film thickness to non-dimensional numbers. A version of VOF algorithm is also utilized to study this phenomenon numerically and good agreement among devised correlation, other experimental measurements and numerical simulation is observed. ISBN: page 53 ISSN:

7 Nomenclature C Concentration (%Weight) d Position vector connecting two cell centers F Face Flux f x weighting factor f Body force vector g Gravitational acceleration vector m Consistency index n Power-Law index n Normal Vector P Pressure Q & Flow Rate R 0 Inlet pipe radius t Time u Velocity vector V Velocity magnitude x Film length at any time Greek Symbols α Indicator function β Weighting factor δ Film thickness θ Contact angle φ Inclination angle γ& Shear rate γ& Rate-of-Strain tensor μ Viscosity, Apparent viscosity ρ Density σ Surface tension τ Stress tensor Superscripts * Dimensionless variable - Mean variable ~ Normalized variable Subscripts A Acceptor cell C Central cell D Donor cell DW Downwind cell F Face value S Static sl Solid-Liquid t Tangential w Wall x Variable calculated at x References: [1] H.I. Anderson H.I., The momentum integral approach to laminar thin film flow. In: Proc. ASME Symp. on Thin Fluid Films, Cincinatti, OH, FED [] G. Astarita, Non-Newtonian gravity flow along inclined plane surfaces, Ind. Eng. Chem. Fundam. (I&EC Fundamentals) 3 (4) (1964) [3] N. Therien, B. Coupal, J.L. Corneille, Verification experimental de l epaisseur du film pour des liquides non-newtonien s ecoulant pargravite sur un plan incline, Can. J. Chem. Eng. 48 (1970) [4] N.D. Sylvester, J.S. Tyler and A.H.P. Skelland, Non-Newtonian thin films: theory and experiment, Can. J. Chem. Eng. 51 (1973) [5] C. Ruyer-Quil, P. Manneville, Improved modeling of flows down inclined planes, Eur. Phys. J. B. 15 (000) [6] C. Ruyer-Quil, P. Manneville, Further accuracy and convergence results on the modeling of flows down inclined planes by weighted-residual approximations, Phys. Fluids 14 (00) [7] F.H. Harlow, J.E. Welch, Numerical calculation of time dependent viscous incompressible fluid flow with free surface, Phys. Fluids 8(1) [8] C.W. Hirt, B.D. Nichols, Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys., 39 (1981) [9] S. Osher, J.A. Sethian, Fronts propagating with curvature-dependant speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys., 79 (1988) [10] J.A. Sethian, P. Smereka, Level set methods for fluid interfaces, Annu. Rev. Fluid Mech. 35 (003) [11] M. Sussman, E.G. Puckett, A coupled level set and volume-of-fluid method for computing 3D and axisymmetric incompressible two-phase flows, J. Comput. Phys. 16 (000) [1] I.D. Sutalo, The flow of non-newtonian fluids down inclines, J. non-newtonian Fluid Mech. 136 (006) [13] J.U. Brackbill, D.B. Kothe, C. Zemach, A continuum method for modeling surface tension, J. Comput. Phys. 100 (199) [14] A. Miyara, Numerical simulation of wavy liquid film flowing down on a vertical wall and an inclined wall, Int. J. Therm. Sci. 39 (000), [15] H.S. Dou, N. Phan-Thien, B.C. Khoo, K.S. Yeo, R. Zheng, Simulation of front evolving liquid film flowing down an inclined plate using level set ISBN: page 54 ISSN:

8 method, Computational Mechanics 34 (004) [16] B.P. Leonard, The ULTIMATE conservative difference scheme applied to unsteady onedimensional advection, Comp. Meth. in Appl. Mech. and Eng., 88 (1991) [17] O. Ubbink, R.I. Issa, A method for capturing sharp fluid interfaces on arbitrary Meshes, J. of Comp. Physics, 153( 1999) [18] T. Yamamoto, R. Kimoto, N. Mori, Tip velocity of viscous fingers in shear thinning flows in a Hele- Shaw cell, JSME 48 (005) [19] F. Yasar, H. Togrul, N. Arslan, Flow properties from cellulose and carboxymethyl cellulose from orange peel, J. of Food Engineering 81 (007) [0] B. Lafaurie, C. Nardone, R. Scardovelli, S. Zaleski and G. Zanetti, Modelling merging and fragmentation in multiphase flows with SURFER, J. Comput. Phys., 113 (1994) [1] C.A.J. Fletcher, Computational techniques for fluid dynamics, Vol., Specific techniques for different flow categories, nd ed (1991) New York: Springer-Verlag. [] H. Jasak, Error analysis and estimation for the finite volume method with applications to fluid flows, PhD Thesis 1996, University of London. [3] H. Jasak, H.C. Weller, R.I. Issa and A.D. Gosman, High resolution NVD differencing scheme for arbitrarily unstructured meshes, Int. J. Numer. Meth. Fluids 31(1999) [4] B.P. Leonard, A stable and accurate convective modeling procedure based on quadratic upstream interpolation. Comp. Meth. in Appl. Mech. and Eng. 19 (1979) [5] S. Haeri, S.H. Hashemabadi, Numerical analysis of two phase flow for interface shape prediction, MSc. Thesis 007, Iran Uni. Sci. and Tech. Tehran, Iran. [6] S. Haeri, S.H. Hashemabadi, Experimental Study of non-newtonian Falling Film Dynamics, Accepted paper in Iranian Chemical Engineering Conference, IChEC 008. [7] H.I. Anderson, D.Y. Shang, An extended study of hydrodynamics of gravity driven film flow of power law fluids, Fluid Dynamic Research (1998) ISBN: page 55 ISSN: