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1、1,微機電糸統分析期末報告(二)Inkjet 授課教師:李旺龍 學 生:李聰瑞,2,Inkjet Model,Inkjet printers are attractive tools for printing text and images because of their low cost,high resolution,and acceptable speed.The working principle behind inkjet technology is to eject small droplets of liquid from a nozzle onto a sheet of pa
2、per.Important properties of a printer are its speed and the resolution of the final images.Designers can vary several parameters to modify a printers performance.For instance,they can vary the inkjet geometry and the type of ink to create droplets of different sizes.The size and speed of the ejected
3、 droplets are also strongly dependent on the speed at which ink is injected into the nozzle.Simulations can be very useful to improve the understanding of the fluid flow and to predict the optimal design of an inkjet for a specific application.Although initially invented to produce images on paper,t
4、he inkjet technique has since been adopted for other application areas.Instruments for the precise deposition of microdroplets often employ inkjets.These instruments are used within the life sciences for diagnosis,analysis,and drug discovery.Inkjets have also been used as 3D printers to synthesize t
5、issue from cells and to manufacture microelectronics.For all of these applications it is important to be able to accurately control the inkjets performance.This example demonstrates how to use COMSOL Multiphysics to model the fluid flow within an inkjet.The model uses the Navier Stokes equations to
6、describe the momentum transport and conservation of mass.Surface tension is included in the momentum equations.A reinitialized,conservative level set method represents and moves the interface between the air and ink.,3,Figure4-61 shows the geometry of the inkjet studied in this example.Because of it
7、s symmetry you can use an axisymmetric 2D model.Initially,the space between the inlet and the nozzle is filled with ink.Additional ink is injected through the inlet during a period of 10 s,and it consequently forces ink to flow out of the nozzle.When the injection stops,a droplet of ink snaps off an
8、d continues to travel until it hits the target.,Model Definition,Figure 4-61:Geometry of the inkjet,4,Transport of Mass and Momentum,The Navier-Stokes equations describe the transport of mass and momentum for fluids of constant density.It is possible to model both ink and air as being incompressible
9、 as long as the fluid velocity is small compared to the speed of sound.In this model,you must add a term to account for surface tension.The Navier-Stokes equations with surface tension are,Here,denotes density(kg/m3),equals the dynamic viscosity(Ns/m2),u represents the velocity(m/s),and p denotes pr
10、essure(Pa).Fst is the surface tension force,which is,where I is the identity matrix,n is the interface normal,equals the surface tension coefficient(N/m),and equals a Dirac delta function being nonzero only at the fluid interface.In COMSOL Multiphysics,the Navier-Stokes equations are already formula
11、ted in the Incompressible Navier-Stokes application mode,which is available in the MEMS Module.You must add only the surface-tension force expressed in cylindrical coordinates.The following table gives the physical parameters of ink and air:,5,REPRESENTATION AND CONVECTION OF THE FLUID INTERFACE,In
12、this model you use a reinitialized,conservative level set method to describe and convect the interface.The 0.5 contour of the level set function defines the interface,where equals 0 in air and 1 in ink.In a transition layer close to the interface,goes smoothly from 0 to 1.The normal to the interface
13、 is,The interface moves with the fluid velocity,u,at the interface.The following equation describes the reinitialized convection of the level set function:,The thickness of the transition layer is proportional to.For this model you can use=2h,where h is the mesh size.You then obtain a sharper interf
14、ace in the regions where the mesh is finer.In this example you use a structured mesh.For unstructured meshes,avoid letting depend on h.,The parameter determines the amount of reinitialization.If the velocity gradients are small at the fluid interface you can choose=1.In this example the velocity gra
15、dients at the interface are significant,and you must therefore choose a larger value for to keep the thickness of the transition layer constant.,You can use the level set function to smooth the density and viscosity jump across the interface by letting,To simplify the calculation of the surface tens
16、ion force,set,6,Initial Conditions,Figure4-62 shows at t=0,that is,the initial distribution of ink and air.The velocity is initially 0.,Figure 4-62:Initial distribution of ink.Black corresponds to ink and white corresponds to air.,Boundary Conditions,Inlet,The inlet velocity in the z-direction incre
17、ases from 0 to the parabolic profile,during the first 2 s.The velocity is then v(r)for 10 s and finally decreases to 0 for another 2 s.You can obtain this effect by using the smooth step function H(t 1,2),which is 0 for t 1+2 as shown in Figure4-63.,7,Figure 4-63:Smooth step function f(t)=H(t 1,2).,
18、The time-dependent velocity profile in the z-direction can then be defined as,For the level set equation,use=1 as the boundary condition.,Outlet,Set a constant pressure at the outlet.The value of the pressure given here is not important because the velocity depends only on the pressure gradient.You
19、thus obtain the same velocity field regardless of whether the pressure is set to 1 atm or to 0.Use convective flux as the boundary condition for the level set equation.,8,Walls,On all other boundaries except the target,set no slip conditions.If you use them at the target,the interface cannot move al
20、ong this boundary.You can resolve this problem by replacing the no slip condition with a Navier slip condition.In this case,the Navier slip condition for the velocity component in the r-direction,u,is given by,where is a small parameter called the slip length.Note that should be of the same order as
21、 the size of the mesh.Use insulation/symmetry as the boundary condition for the level set equation on all walls including the target.This prevents any outflow of water or ink through the walls.,Results and Discussion,Figure4-64 and Figure4-65 show the ink surface and the velocity field at different
22、times.The droplet hits the target after approximately 160 s.,9,Figure 4-64:Position of air/ink interface and velocity field at various times.,Figure 4-65:Position of air/ink interface and velocity field at various times.,10,Figure4-66 illustrates the mass of ink that is further than m from the inlet
23、.The figure shows that the mass of the ejected droplet is approximately kg.,Figure 4-66:Amount of ink just above the nozzle.,This example studies only one inkjet model,but it is easy to modify the model in several ways.You can,for example,change properties such as the geometry or the inlet velocity
24、and study the influence on the size and the speed of the ejected droplets.You can also investigate how the inkjet would perform if the ink were replaced by a different fluid.It is also easy to add forces such as gravitation to the model.,11,Modeling in COMSOL Multiphysics,To set up the model in COMS
25、OL Multiphysics you use two 2D axisymmetric application modes:Incompressible Navier-Stokes,and Convection and Diffusion.In the Navier-Stokes equations you must add the surface-tension force.You must also modify the Convection/Diffusion equation to obtain the level set equation.You use a structured m
26、esh and refine it in the regions where the fluid interface passes.Before starting the calculations,you must reinitialize the level set function.Do so by solving only for the level set function with zero velocity.Then use the resulting solution as initial data.To calculate the droplets mass use an in
27、tegration coupling variable.To visualize the droplet in 3D,revolve the 2D axisymmetric solution to a 3D geometry.,References,Jyi-Tyan Yeh,“A VOF-FEM Coupled Inkjet Simulation,”Proc.ASME FEDSM01,New Orleans,Louisiana,2001.2.E.Olsson and G.Kreiss,“A Conservative Level Set Method for Two Phase Flow,”J.
28、Comput.Phys.,vol.210,pp.225-246,2005.,12,Modeling Using the Graphical User Interface,Model Navigator,13,Geometry Modeling,1 From the Draw menu select Specify ObjectsLine2 Select Closed polyline(solid)from the Style list.3 In the r edit field,enter 0 1e-4 1e-4 2.5e-5 2.5e-5 1e-4 1e-4 2e-4 2e-4 0,and
29、in the z edit field 0 0 2e-4 5.75e-4 6e-4 6e-4 1.5e-3 1.5e-3 1.6e-3 1.6e-3.4 Click OK.5 Click the Zoom Extents button on the Main toolbar.6 Shift-click the Rectangle/Square button on the Draw toolbar.Enter 1e-4 in the Width edit field,2e-4 in the Height edit field,and 0 in both the r field and the z
30、 fields.Make sure Corner is selected in the Base list.Click OK.7 Repeat the previous step three times to specify three more rectangles according to the following table:,14,OPTIONS AND SETTINGS,1 From the Options menu select Constants.2 Enter the following constant settings:,3 Click OK.4 From the Opt
31、ions menu select ExpressionsScalar Expressions.5 Insert the following expressions:6 Click OK.,15,16,Boundary ConditionsIncompressible Navier-Stokes,1 In the Model Tree,right-click Incompressible Navier-Stokes(mmglf)and select Boundary Settings.2 Press Ctrl and select all the boundaries at the symmet
32、ry line,that is,Boundaries 1,3,5,7,and 9.Select Axial Symmetry from the Boundary condition list.3 Select the inlet(Boundary 2)and select the boundary condition Inflow/Outflow velocity.Type inletvel in the z-velocity edit field.4 Select Boundary 24(the outlet)and specify the boundary condition Outflo
33、w/Pressure.5 Select Boundaries 11,18,and 23(the target).Select the boundary condition Inflow/Outflow velocity and type lambdaslip*uz in the r-velocity edit field.6 Verify that the boundary condition for the remaining boundaries(12,13,15,19,20,and 22)is No slip.7 Click OK.,17,18,19,Subdomain Settings
34、Convection and Diffusion,1 In the Model Tree window,right-click Convection and Diffusion(cd)and select Subdomain Settings.2 Make sure that all the subdomains are selected.3 Enter 0 in the Diffusion coefficient edit field,u in the r-velocity edit field,and v in the z-velocity edit field.4 Click the I
35、nit tab and enter phi0 in the edit field.5 Click the Element tab and select Lagrange Linear in the Predefined elements list.6 Click OK.,20,Adjust the convection/diffusion equation to obtain the conservative level set equation:1 Select PhysicsEquation SystemSubdomain Settings.Click the f tab.2 Replac
36、e the expression in the fourth row with 0.3 Click the tab.4 Replace the expression in the first column,fourth row with r*u_phi_cd*phi+gamma*(r*phi*(1-phi)*normr-2*h*r*phir).Replace the expression in the second column,fourth row with r*v_phi_cd*phi+gamma*(r*phi*(1-phi)*normz-2*h*r*phiz).5 Click OK.,2
37、1,Boundary ConditionsConvection and Diffusion,1 In the Model Browser,right-click Convection and Diffusion and select Boundary Settings.2 Select Boundaries 1,3,5,7,and 9,then select Axial symmetry in the Boundary condition list.3 Select the inlet(Boundary 2)and set Concentration as the boundary condi
38、tion.Enter 1 in the Concentration edit field.4 Select the outlet(Boundary 24)and set Convective flux as the boundary condition.5 The default boundary condition Insulation/Symmetry holds for all other boundaries.6 Click OK.,22,Mesh Generation,1 From the Mesh menu select Mapped Mesh Parameters.2 Click
39、 the Boundary tab.3 Select Boundaries 1,2,3,5,7,9,15,and 22,then select the Constrained edge element distribution check box.Next select each boundary separately and specify the Number of edge elements according to this table:4 Click Remesh and then click OK.,23,Computing the Solution,First reinitial
40、ize to obtain the correct shape of in the transition layer.1 From the Solve menu select Solver Parameters.2 Click the General tab and enter 0 2e-6 in the Times edit field.Click OK.3 Select SolveSolver Manager.4 Click the Solve For tab.Select Convection and Diffusion(cd).Click OK.5 Select SolveSolve
41、Problem.This creates a good initial solution to the level set function.6 To visualize the initial level set function,click the Plot Parameters button on the Main toolbar.Click the Surface tab,then select Convection and DiffusionConcentration,phi from the Predefined quantities list.Click OK.,Use the
42、obtained solution as an initial condition to the simulation of the droplet motion.1 Click the Solver Manager button on the Main toolbar.Click the Initial Value tab.2 Click the Store Solution button.Select the time 2e-6.Click OK.3 In the Initial value area select Stored solution.4 Select 2e-6 from th
43、e Solution at time list.5 Click the Solve For tab.Click Geom1(2D)to select all the variables.Click OK.6 From the Solve menu select Solver Parameters.7 Click the General tab,then enter 0:1e-5:2e-4 in the Times edit field.Click OK.8 Click the Solve button on the Main toolbar.,24,Postprocessing and Vis
44、ualization,To visualize the inkjet surface in 3D,perform these steps:1 From the Draw menu select Revolve.2 In the Objects to revolve list select all the elements(select any one and then press Ctrl+A).Click OK.3 In the Model Tree click Geom1 to return to the 2D model.4 Select OptionsExtrusion Couplin
45、g VariablesSubdomain Variables.5 Select any subdomain and then press Ctrl+A to select all the subdomains.6 Enter phi3d in the Name field and phi in the Expression field.7 Click the General transformation option button.8 Locate the second row,then enter vel3d in the Name field and sqrt(u2+v2)in the E
46、xpression field.9 Select General transformation for this variable.10 Click the Destination tab.In the Geometry list select Geom2.In the Level list select Subdomain.11 Select any subdomain and press Ctrl+A to select all the subdomains.12 Select the Use selected subdomains as destination check box.13
47、Enter sqrt(x2+z2)in the x edit field,then click OK.14 Select phi3d from the Variable list.15 Select the Use selected subdomains as destination check box and enter sqrt(x2+z2)in the x edit field.Click OK.16 From the Solve menu select Update Model.17 Click the Plot Parameters button on the Main menu.1
48、8 Select the General tab.Clear the Geometry edges check box and select the Isosurface check box.19 Clear the Element refinement Auto check box and type 4 in the edit field.20 Click the Slice tab.Enter vel3d in the Expression edit field.Enter 0 in the x levels edit field and 1 in the z levels edit fi
49、eld.21 Click the Isosurface tab.Enter phi3d in the Expression field.Select Vector with isolevels and type 0.5 in the corresponding edit field.22 Select Uniform Color and click the Color button.Choose a gray color to plot the ink/air interface.Click OK.23 Click the General tab.In the Solution at time
50、 list select 0.Click OK to plot the initial position of the interface.24 Click the Scene Light button on the Draw toolbar.25 To plot the solution at other times,click the Plot Parameters button and select other values in the Solution at time list.Click Apply to plot the solution at the corresponding
