MECH.5810 Homework 1 Additive manufacturing has exploded over the last decade, with myriad applications and techniques for a wide variety of ma- terials. A very interesting application is the printing of high- viscosity fluids such as food or wet clay in a so-called cold paste printing process. Many printers of this type are directly driven with a piston/plunger connected to a stepper motor to control the printing process. However, there is another, indi- rect, option to drive the printing, and that is to supply high- pressure air to the top of the printer head containing the work- ing material. The high pressure then forces out the printer ma- terial from the nozzle. A UK artist Jonathan Keep has built his own 3D printer using this process to use in his art. You can find several examples of his printer in action on his web- site: http://www.keep-art.co.uk/Self-build.html. In this problem you will analyze this type of indirect, pneumatically driven cold-paste printing process. A simplified printer-head schematic is provided in 1. The veloc- ity of the clay at the exit is described as the following function pressure: of D2 D3 D1 High pressure air enters here Air g(t) Clay h(t) Clay is ejected here Figure 1: Schematic of cold paste printing with clay as the printing medium D Pelay Patm L Uclay 32 clay where Pclay is the pressure of the clay, which may be assumed to equal the pressure of the compressed air, and Patm is the standard atmospheric pressure (Patm = 101325Pa). You may also assume that all chambers have circular cross-sections and straight sides, i.e. they are perfect cylinders, with the diameters provided. You may neglect the effects of gravity and assume the temperature of the air and clay remain constant at 22C. Clay may be assumed to be an incompressible fluid, while the air in the print-head chamber can be assumed to follow the ideal gas law (P = pRT). You may take the density of clay to be pc 1750 K, and viscosity c = 25Pa s. Use the following dimensions for your analysis: D1 = 0.0025 m D2 = 0.05 m D3 = D1 hinit = 0.1 m Ginit = 0.05 m L = 0.01 m = a Plot the velocities of both the clay exiting the printer head and air entering as functions of the air pressure. Assume steady operation for each pressure. b During printer start-up, and other times the printer-head velocity is changing, the material feed rate must be modified in order to keep the amount of material deposited uniform as a function of position. Determine expressions that relate the pressure and velocity of air to the velocity of clay leaving the printer head when the printer is accelerating. Plot the air pressure and velocity versus time for the following conditions: Initial clay velocity uc = Final clay velocity uc = 0 m/s 0.1 m/s Accelerations of the print head corresponding to clay velocity accelerations of: 0.1, 0.2, 0.4 m/s