Continued part 1, part 2 of ebook Casting: An analytical approach provide readers with content about: process design; evaluation of dimensionless numbers; flow in viscous boundary layer; quality control; basic concepts of quality control; statistical process control; tabular summarization of data; numerical data summarization;...
4 Process Design 4.1 Evaluation of Dimensionless Numbers Let’s estimate the dimensionless numbers defined in Section 2.3 for an aluminium alloy in a gravity pour and high-pressure die casting For the metal we will assume the density U= 2,500 kg/m3, surface tension coefficient V= 0.9 N/m and dynamic viscosity coefficient P= 0.0015 Pa s [Campbell, 1991] For the gravity casting, we will take that the metal is poured down a sprue of height H=0.25 m, at the bottom of which it gains the velocity of about U 2GH § 2.4 m/s We can expect that velocities of that magnitude will prevail in the horizontal runner and will be somewhat smaller inside the cavity For the characteristic length, we will pick the runner width of 0.02 m A typical high-pressure die-casting process has a range of velocities and length scales because of the slow and fast shot stages, and also due to the small size of the gates compared to the runner and probably the average wall thickness in the die cavity For the slow shot stage, we will use the velocity of the plunger, Us=0.2 m/s, as the approximation of the average metal velocity in the short sleeve of radius 0.3 m We will assume that the shot sleeve is initially filled with liquid metal to 50% For the fast shot, the plunger velocity of Uf=2 m/s will be used in the shot sleeve, the gate velocity of Ug=40.0 m/s and the gate width d = 0.002 m The values of the dimensionless numbers that correspond to these parameters are shown in Table 4.1 98 Casting: An Analytical Approach Table 4.1 Estimates of the dimensionless numbers for typical gravity and high-pressure die casting Re We Bo Fr Gravity pour Slow shot (shot sleeve) 40 000 160 2.7 7.7 100 000 33 450 0.12 Fast shot Shot sleeve Gate 000 000 300 450 1.2 67 000 400 0.03 400 From the values of the Reynolds number it is clear that the flow is turbulent in all cases The highest level of turbulence may be expected in the shot sleeve due to its relatively large size The variations in the values of the remaining parameters warrant a more detailed discussion 4.1.1 Gravity Pour The Weber number of 160 in the gravity pour example sufficiently exceeds the critical value of 60 to expect a breakup of the metal surface and splashing in the sprue and runner There is enough kinetic energy in the flow to overcome surface tension and tear it into individual droplets as the liquid stream shears and turns in the runner and impinges on the walls The Bond number of 2.7 indicates that gravity may be able to stabilize the surface, but only where the flow is horizontal Surface tension is still sufficiently strong to produce a noticeable curvature of the free surface When the metal surface is not horizontal, gravity has either a neutral effect or may actually contribute to its breakup Since the Froude number is equal to 7.7, the flow in the horizontal runner is supercritical If the runner remains only partially filled for a while, then the formation of a hydraulic jump and additional surface turbulence is likely where the flow reaches the dead end If this is inevitable, then measures need to be taken to decrease the intensity and duration of these jumps by filling the runner earlier or to force it where it does the least harm 4.1.2 High-pressure Die Casting: Slow Shot The high values of the Weber and Bond numbers mean that surface tension by far is not an important factor for flow in the shot sleeve Since the surface is mostly horizontal, gravity would flatten out any perturbations of the free surface due to surface tension The Froude number of 0.12 is quite low, so surface waves will travel faster than the plunger If there is no motion in the metal initially, then after the plunger begins to move, one can reasonably expect a single wave travelling ahead of it.5 This wave will typically be shallow, without breaking and overturning of metal As In actual situations there maybe significant motion in the metal prior to the movement of the plunger because of the residual flow left from the shot sleeve filling stage Process Design 99 the plunger moves forward, the level of metal rises and the crest of the wave will at some point start interacting with the ceiling of the shot sleeve By this time, the wave is likely to have travelled the length of the shot sleeve several times, and it is hard to predict its exact location without a more detailed analysis Ideally, the remaining air should be expelled into the runner and not trapped in the sleeve by the wave 4.1.3 High-pressure Die Casting: Fast Shot If there is any free surface still remaining in the shot sleeve at the start of the fast shot, then surface tension is even less relevant here than during the slow shot The Froude number of 1.2 means that the plunger is just fast enough to prevent a wave separation from its tip In reality, the flow and wave formation are complicated by the cylindrical shape of the sleeve, the changing level of metal and by the plunger acceleration from slow to fast stages To avoid air entrainment, the amount of the remaining air in the sleeve at the beginning of the fast shot should be minimized The very high value of the Weber number at the gate, many times over the critical value, points to the very unstable nature of the free surface Since the Bond number is small, gravity does not play any stabilizing role The huge kinetic energy, combined with large perturbations of the flow at the gates due to friction and surface roughness, results in flow atomization Soon after the metal emerges from the gates into the open space of the die cavity, it breaks up into a stream of small droplets The rate and degree of atomization cannot be derived using only the dimensionless numbers, but we can be sure that it does occur For example, the free surface in such flow is perturbed by small-scale subsurface turbulent eddies, and if these eddies have sufficient energy, then further surface rupture would occur The large value of the Froude number confirms that gravity is not going to play a significant role in the flow at the gates Once past the gates, the motion of the metal is controlled mostly by its kinetic energy, momentum and the geometry of the cavity It takes only milliseconds for a small piece of metal to cover the distance between a gate and the far end of the cavity In that time, gravity is only going to alter its velocity by a tiny fraction For the travel time of say 10 ms, gravity adds about 0.1 m/s, which is negligible compared to the gate velocity of 40 m/s.6 The dimensionless numbers discussed in this section give a useful insight into some general aspects of flow But they are not sufficient to obtain details of the interaction between different forces in the fluid, turbulence, waves and instabilities, temporal and spatial variations of the flow parameters For this purpose in the rest of this chapter, we will discuss several simple approximate analytical solutions of Equations 2.8 and 2.9 that help to gain further understanding of metal flow in casting By dividing the Bond number by the Weber number, we get another dimensionless relationship, Gl U , between gravity and inertia The ratio is small, less than 0.01, for flow in high-pressure die casting 100 Casting: An Analytical Approach 4.2 Flow in Viscous Boundary Layer Due to friction and the no-slip boundary condition at the walls of the runnners and mould, a viscous boundary layer develops in the flow Usually the thickness of the boundary layer is thin compared with the overall size of the geometry Unlike the bulk of the flow where gravity and pressure are its main driving forces, flow in the boundary layer is dominated by viscous forces These considerations allowed Ludwig Prandtl to find a set of approximate solutions for laminar flow in the boundary layer [Prandtl, 1928] Consider a one-directional, uniform flow of viscous liquid over a semi-infinite thin plate as shown in Figure 4.1 A viscous boundary layer starts developing downstream from the tip of the plate Fluid velocity transitions within the boundary layer from zero velocity at the surface of the plate to the velocity U0 in the main flow The thickness of the layer, G, increases with distance x along the plate according to G D Px , UU (4.1) where D is a constant dependent on the definition of the transition point between the boundary layer and the rest of the flow For example, if we define this transtion as the point where velocity is equal to 99.5% of U0, then D|5.2 Equation 4.1 shows that the thickness of the viscous boundary layers is not constant, but grows as the boundary layer gradually develops in the flow The growth rate is proportional to the square root of the distance along the plate Although Equation 4.1 was derived for a relatively simple, steady-state flow configuration, it has been successfully applied to much more general cases of transient three-dimensional flows and curved boundaries Moreover, this solution holds up well for Reynolds number values of up to 400 000 We can estimate G in Equation 4.1 using examples in Section 2.3.6 For gravity casting, assuming that the length of the runner is L=0.25 m and flow velocity U0 = 2.4 m/s, the maximum thickness of the boundary layer comes to about 1.3 mm For high-pressure die casting, with the same runner length and velocity of U0 = m/s, it is even smaller, G§ 0.9 mm In the gate, the thickness of the boundary layer is even smaller All these estimates indicate that bulk flow occupies most of the available cross section of the runner system Although the boundary layer in the two examples is thin, this is not necessarily to say that friction at the wall is negligible In fact, the thinner the boundary layer, the larger the gradient of the velocity across it and thus the shear stress Shear stress, W at any point on the surface of the plate is W § wu · áá â wy y P ăă 0.332 UPU 03 x (4.2) Process Design 101 Figure 4.1 Viscous boundary layer in flow over a horizontal plate The magnitude of the shear stress decreases with distance x along the plate The total friction force R acting on one side of a plate of length L and width b is R b ³ L Wdx 0.664b UPLU 03 (4.3) Note that the force is proportional to the 3/2 power of the bulk flow velocty If we stretch the model a bit, then we can probably say that the force in Equation 4.3 would also be acting on the metal in a runner of length L and cross-sectional perimeter b For a high-pressure die casting runner of length L = 0.25 m, width b = 0.15 m and U0 = m/s, the total force due to friction at the walls of the runner is about N, which is far below the pressure force driving the flow This means that flow in the runner can be approximated by the inviscid, or ideal, flow model with a good degree of accuracy.7 In most real world mould filling flow is turbulent Turbulence usually originates within the viscous boundary layer and spreads into the bulk of the flow The boundary layer itself becomes turbulent with a very thin laminar sub layer close to the wall Turbulence results in widening of the viscous boundary layer and The viscosity coefficient P is set equal to zero in Equation 2.8 for the inviscid flow model 102 Casting: An Analytical Approach in steepening of the velocity gradients in it Consequently, the viscous friction forces increase Nevertheless, it is useful to see how the inviscid flow model can be applied to describe certain aspects of liquid metal flow 4.3 The Bernoulli Equation Once of the most commonly used solutions of the general fluid motion equations is the Bernoulli equation It can be derived from Equations 2.8 and 2.9 when the flow is steady and inviscid and can be expressed in the following for: P UU Ugh C, (4.4) where g is the magnitude of the gravity vector and h is the height above a reference point C is an abitrary parameter that is constant along any streamline It can be evaluated by using pressure and velocity at a single point along the streamline: P UU Ugh P1 UU12 Ugh1 (4.5) 4.3.1 Stagnation, Dynamic and Total Pressure If the variation in fluid elevation h is small or gravity forces are negligible compared to pressure and inertia, as in high-pressure die casting or in air, then Equation 4.5 can be reduced to P UU 2 P1 UU12 (4.6) As fluid accelerates along a streamline, pressure drops so that the sum on the lefthand side of Equation 4.6 stays constant The maximum value of pressure occurs at the point where velocity is zero, or at the stagnation point This pressure is called stagnation pressure The term 1/2UU2 is the dynamic pressure, as opposed to the static pressure represented by P The sum of static and dynamic pressures in Equation 4.6 is termed the total pressure The Bernoulli equation in the form of Equation 4.6 led to the development of the theory of the airfoil [Abbott, 1959] The difference between the static pressures on the lower and upper surfaces of an airplane wing creates the lift necessary to keep the plane in the air Process Design 103 4.3.2 Gravity Controlled Flow For gravity casting, gravity is important and cannot be neglected When a streamline is located on the free surface, then, according to Equation 2.16, pressure along it is constant and, therefore, Equation 4.4 can be reduced to U gh 2 U1 gh1 (4.7) Figure 4.2 Examples of flows where the free surface is also part of a streamline Figure 4.2 shows three examples of such flow In the first, metal is poured out of a ladle, in the second, it flows over a weir and in the third case, it flows out of a small hole in a container In the latter case, the streamline is not located completely on the free surface, but its beginning and ending points are In each of these three cases, flow originates in the area where the metal velocity is small compared to the velocity in the jet, where the flow is the fastest If we select that quiescent area to evaluate the right-hand side of Equation 4.7 and assume U1§0, then for the velocity elsewhere along the streamline, we get U g (h1 h) g'h (4.8) This is the classic expression for the velocity in a free flowing fluid, which is named after Torricelli When pouring from a ladle held 'h = 0.2 m above the pouring basin, the velocity at which the jet impinges on the pool of metal in the basin is about m/s, irrespecive of the alloy in the ladle 104 Casting: An Analytical Approach 4.3.3 Flow in the Runner System Once the metal is in the runner system, the Bernoulli principle, Equation 4.5, can be applied to calculate he velocity of metal at various points in the flow Let us assume that flow is inviscid and that all air has been purged, that is, the runner is completely filled by metal If flow originates from the same reservoir, e.g., a pouring basin or a shot sleeve, then the right-hand side of Equation 4.5 is the same for all streamlines within the runner, (Figure 4.3) Figure 4.3 Streamlines during gravity (left) and high pressure die casting filling Let us also assume that the values of h1, U1 and P1 are known and C1 is the value of the expression on the right-hand side of Equation 4.5 Then at any point in the flow inside the runner system, P UU Ugh C1 (4.9) At the same time, due to the incompressibiity of metal, the flow rate, Q, at any cross section must be the same: UA Q, (4.10) where A is the cross-sectional area If the runner splits into multiple channels, then the left-hand side of Equation 4.10 is the sum over all channels Process Design 105 When the metal emerges from the runner system into the casting cavity, the metal pressure at the gates is equal to the initial pressure of the air in the cavity, Pa Then according to Equation 4.9, the initial velocity of the metal at a gate, Ug, is P · ĐC 2ăă gh a áá U U â Ug (4.11) Equation 4.11 says that if there are multiple gates, the initial velocity of metal at these gates will differ only because of the variations in their heights relative to each other and does not depend on the size of each gate For high-pressure die casting, where gravity effects on the flow in the runner system can be ignored, Equation 4.11 becomes Ug §C P · 2ăă a áá U Uạ â (4.12) meaning that metal emerges at all gates at the same speed, irrespective of the size of the gate This does not mean that the size of a gate is not important If one of the gates is made larger, then the flow at each gate will come out more slowly, yet the initial metal velocity at all gates will be equal These conclusions hold for the initial gate velocities As the cavity fills up, gate velocities will vary in less predictable ways because of the back-pressure buildup Another assumption implicitly used in calculating the initial gate velocities was that metal arrives at all the gates at the same time If it does not, then we cannot use the same value of C1 for all gates Note that in none of the examples, where we have applied the Bernoulli equation, is flow really steady-state But temporal variations of the flow are small and, as before, we hope that our approximations are valid to a reasonable degree Let us now look at the variations of pressure in a runner system as a function of flow rate and cross-sectional area Substituting Equation 2.34 in Equation 2.33 and solving for pressure yields P C1 Ugh UQ 2A (4.13) According to Equation 4.13, pressure will be the lowest in the areas with the smallest cross section In sand castings, if pressure drops below one atmosphere, air aspiration through the porous mould may occur, causing porosity in the final casting In a die casting, where the mould is impervious to air, low pressure in the runner system is less likely to cause defects in the casting unless it is near a partition line through which air can seep But if pressure drops below the pressure of the dissolved gasses in the metal, then cavitation may occur, when these gases, mainly air and hydrogen, evolve into bubbles The appearance and violent collapse 106 Casting: An Analytical Approach of these bubbles is usually accompanied by large momentary pulses of pressure that can damage the surface of the die After ignoring the gravity term, Equation 4.13 can be rewritten as P C1 UQ 2A (4.14) The flow rate Q in high-pressure die casting is primarily defined by the preprogrammed motion of the plunger in the shot sleeve and less by the geometry of the runner system If Q is sufficiently large and A small, then the pressure can drop to the level that may initiate cavitation On the other hand, if the cross-sectional area of the runner decreases monotonically from the metal entry point at the shot sleeve end to the gate, then such low pressures are unlikely to occur 4.3.4 Filling Rate Another useful application of the steady-state Bernoulli solution to an unsteady flow is for gravity filling with bottom gating shown in Figure 4.4 Let’s write Equation 2.29 for two points on a streamline beginning in the pouring basin and ending inside the cavity The first point is located at the top of the pouring basin, and the second point R is somewhere inside the runner with the cross-sectional area AR We will assume that the level of metal in the basin is fixed at the height h0 above point R and the metal velocity there is close to zero The ambient air pressures above the basin and inside the cavity are the same, equal to the atmospheric pressure, Pa Finally, we will assume that the free surface inside the cavity is more or less horizontal and h is its time-dependent height above the gate Even though the flow is, of course, time-dependent, once metal enters the mould cavity, the rate of change of pressure and velocity is relatively small compared to the variations of pressure due to gravity This is primarily due to the dissipation of the flow energy in the mould So we can still assume that the Bernoulli equation holds to a good degree of accuracy at every time during filling The Bernoulli equation for a point in the pouring basin and point R on a streamline is Pa Ugh0 PR UU R2 , (4.15) where subscript R refers to the values at point R The rate of change of the level of metal in the mould cavity can be expressed as a function of the flow rate and the average horizontal cross-sectional area in the cavity, A: dh dt Q A (4.16) Pressure PR in Equation 4.15 will vary in time as the level of metal in the cavity increases and can be approximated as Process Design 145 = 1.15 s, the theory predicts 1.12 s, based on the wave speed in undisturbed flow given by Equation 4.26 Given the three-dimensional nature of the simulation and the fact that the flow is viscous, the agreement with the analytical solution is quit remarkable It is important to note that we used the same initial depth, h0, of metal for the round shot sleeve in the simulation and for the rectangular one in the analytical solution In our particular example, with h0 = 0.04 m, the initial fill fraction in the circular cylinder comes to 37.4%, as opposed to 40% for the rectangular channel If we used the same initial fill fraction instead, the results of the two solutions would not be so close This is not to say that there are no differences in the two solutions The velocity contours shown in the plane of symmetry in Figure 4.19 indicate that the main assumption used to obtain the analytical solution that all flow variables vary only in the horizontal direction does not quite hold First, a viscous boundary layer develops at the bottom of the shot sleeve Second, the numerical results show that flow near the free surface moves faster than the bulk of the metal below it, resulting in a sort of a surge wave The slope of the metal surface in this wave is about one and a half to two times larger than 5o It reaches the end of the channel at around 1.3 sec and then reflects back As a result, air may be entrained in the last stages of the process unless, for example, the reflected surge wave is redirected into the runner system Thermal Die Cycling In practice, a die is used for many thousands of shots The heat is constantly added with each shot through the injection of liquid metal and extracted by the cooling channels and, to a lesser extent, by sparaying the die inner surface with lubricant when the die is open After many cycles, this leads to a complex distribution of temperature throughout the die The die cycling simulation serves two purposes First, we want to analyze the thermal balance within the die to ensure that the cooling system is sufficient to prevent the die from overheating and hence to prolong its life Second, the die cycling simulation provides a realistic temperature distribution within the die that can be used for filling simulation The actual filling analysis is usually ignored during a thermal die cycling simulation This is justified by the short duration of the filling several milliseconds compared to the length of each cycle several tens of seconds At the beginning of each cycle, the metal is instantaneously placed into the cavity at the initial uniform temperature of 650°C, and the simulation, therefore, is reduced to a purely thermal problem To describe the changing conditions during a single cycle, the temperature and heat transfer coefficient at the inner surface of the die are vary with time, as shown in Table 4.5 Each cycle is divided into four segments the first corresponds to the closed die with the metal inside The remaining three segments describe the open die 146 Casting: An Analytical Approach Table 4.5 Conditions at the die inner surface during one thermal cycle Cycle segments Die closed Die open to air Spraying Die open to air Temperature °C (initial) 650 30 50 30 Heat transfer coefficient, W/m2 C 20000 10 3000 10 Duratio n, sec 25 10 15 The cooling channels are included in the model since they provide the main means of removing heat from the die They are filled with water at the uniform, constant temperature of 40°C The initial uniform temperature of the die is guessed at 205°C and the simulation is run for twenty five cycles Each cycle lasts 55 s, and the total simulation time is 1375 s Figure 4.20 shows the variation in the total thermal energy of the die as a function of time The thermal cycles are clearly visible, as well as a slow approach to a steady-state solution when the die temperature is the same at the beginning of each cycle Figure 4.20 Total die thermal energy as a function of time during twenty five thermal die cycles Filling Filling is one of the most important stages in die casting that defines the quality of the final part The filling sequence, pressures and velocities that develop in the flow, temperature distribution and solidification, air and oxide film entrainment, efficienty of overflows and vents are some of the important results provided by a numerical simulation In the present simulation, the metal is pushed into the cavity by a plunger with variable speed The maximum plunger velocity is 3.81 m/s, corresponding to a gate Process Design 147 velocity of 40 m/s The total time of the simulation, including the slow shot stage, is 1.775 s The fast shot stage takes 0.175 s The average Reynolds number during the fast shot stage is over 200,000; therefore, the flow is highly turbulent The k-H turbulence model is used in the model to account for this Heat transfer and solidification are also included so that thermal losses and any early solidification can be evaluated The initial temperature of the die is taken from the end of the thermal die cycling simulation The cooling channels are not included in the simulation because the penetration distance of the heat from the metal into the die is significantly smaller than the distance between the cavity surface and the cooling channels This heat penetration distance can be estimated from Figure 4.16; for a fill time of 175 ms, it is about mm, well short of reaching the cooling channels According to Equation 4.92, the die surface temperature rises during filling by almost half of the difference between the initial die temperature and the temperature of the metal, or, in our case, about 75°C Given the thermal boundary layer thickness of only mm, the temperature gradients in the die are quite steep, 25°C per millimetre of the die material Nevertheless, we assume that the die temperature is constant during the simulation, using the temperature distribution obtained at the end of the die cycling simulation This may appear as a stretch in simplifying the model, but it provides a significant benefit for the speed of the simulation The air in the cavity is treated as an adiabatic compressible gas The volume and pressure of each air pocket are computed using the equation of state given by Equation 2.41 For air, J= 1.4 No venting is included in the model, making it the worst-case scenario in terms of the amount of entrained air Figure 4.21 shows selected snapshots of the filling sequence The metal reaches the gates more or less simultaneously, but once inside the cavity, the metal front undergoes severe deformations and breakup As a result, multiple small air pockets, each with its own pressure, are created These bubbles continue to breakup and coalesce throughout filling The images just before the end of filling show the last places to be filled, which can help to place vents and overflows The amount and location of entrained air and oxide film are shown in Figure 4.22 As expected, these are mostly pushed into the last areas to be filled Ideally, the metal contaminated most with entrained oxide film and air should be caught by the overflows The overflow cavity at the centre of the casting appears to be functioning properly in this respect 148 Casting: An Analytical Approach Figure 4.21 Simulated filling sequence for high-pressure die casting Process Design 149 Figure 4.22 Oxide film (left) and entrained air distribution at the end of high-pressure die casting filling The metal with top 45% of oxide concentration is highlighted with a white colour For the entrained air, the white colour shows the metal with top 25% of air entrained during filling Solidification Finally, a solidification simulation is done using the solution from the end of filling as the initial conditions As in the die cycling case, we negelect the residual flow of metal during solidification for faster compuation This can be justified by the assertion that most of the heat is extracted through heat transfer and conduction rather than convection The main purpose of the solidification run is to analyze the solidification sequence and porosity formation and to obtain an estimate of the solidification time, which in turn can tell when it is safe to open the die In this case we cannot ignore the evolution of temperature within the die, so the full energy equation is solved within its domain The predicted solidification time of the part is 5.13 s The biscuit fully solidifies at 17.1 s, therefore, the duration of the closed stage of 25 s used in the die cycling simulation is adequate The maximum porosity of 2.5% is predicted around the thicker central section of the part, near the overflow located at the central opening Air Flow in a Vent In this brief study of air flow through a vent, we consider two types of vents One has a constant rectangular cross section of mm u 21.66 mm, and the other has a variable cross section, where the vent starts at the same cross-sectional area as the first vent and then quickly expands into a large area of 10 mm u 21.66 mm The geometries of the two vents are shown in Figure 4.23 The flow of air at room temperature is generated by a pressure difference between the inlet and outlet of the flow region The pressure drop is varied continuously with time across a wide range of values, simulating different vacuum conditions The variation occurs gradually so that at any given time the flow is close to steady state 150 Casting: An Analytical Approach Figure 4.23 Geometry of the vents: constant area (left) and variable area geometry (right) Figure 4.24 Calculated mass flow rates of air through two types of vent as a function of the ratio of the inlet and outlet pressures The dashed line represents the constant area vent, and the solid line represents the vent with a variable cross sectional area Process Design 151 Figure 4.24 shows the mass flow rates obtained as a function of the ratio of the inlet and outlet pressures In each case, the mass flow rate initially increases with the pressure ratio until it reaches a maximum; beyond that the mass flow rate does not increase as the outlet pressure continues to drop At this critical point, the flow in the vent becomes supersonic The flow patterns and the areas of supersonic flows are shown in Figure 4.25 For the constant area vent, the pressure ratio at the critical point is about 3.0; for the variable geometry vent, it is about 3.5 The maximum flow rate is larger in the vent with variable geometry It is interesting to note that the sudden change in the cross-sectional area results in flow separation in narrowest area, which is reflected in the oscillations of the mass flow rate for that vent in Figure 4.24 If the transition from small to larger area were smooth, these flow instabilities might be avoided all together Figure 4.25 Flow patterns at the maximum flow rate for the constant area (left) and variable geometry vents Dark shading indicates the areas of supersonic flow These results indicate that for simple vent geometry, there is a limit on how much air can be evacuated through it, no matter what vacuum conditions are maintained outside the die A larger mass flow rate can be achieved if the jet of air 152 Casting: An Analytical Approach emerging from the die cavity into the vent is allowed to expand The expansion must be done to prevent flow separation and instabilities in the vent channel 4.10.2 Gravity Sand Casting The complete geometry of sand casting, together with the, risers, sprue and runner system, is shown in Figure 4.26 The casting is a simplified version of an automatic transmission cover Figure 4.26 Geometry of gravity sand casting Two simulations were run for this case a filling and the subsequent solidification The A356 alloy and silica sand mould properties used in the model are listed in Tables 4.3 and 4.4 Filling In the filling simulation, the metal is assumed to be poured from the constant height of 0.1 m above the top of the sprue at temperature of 750ºC The initial mould temperature is 30ºC With these conditions, the model predicted filing time of 5.25 s Figure 4.27 shows selected snapshots of the filling sequence After the initial splashing in the sprue and runner system, the flow stabilizes and progresses smoothly through the remaining part of the casting Process Design 153 Figure 4.27 Simulated filling sequence for gravity casting The distribution of the entrained oxides and air at the end of filling are shown in Figure 4.28 As with high-pressure die-casting filling (Figure 4.22), the areas of the highest concentrations of oxides and air in the metal are somewhat similar and correlate with the filling sequence Given enough time, air maybe able to escape through the porous mould back into the atmosphere, but oxides are likely to stay within the metal and potentially result in leakage or structural defects 154 Casting: An Analytical Approach Figure 4.28 Oxide film (left) and entrained air distribution at the end of gravity casting filling The metal with top 65% of oxide concentration is highlighted with a white colour For the entrained air, the white colour shows the metal with top 15% of air entrained during filling Figure 4.29 Simulated microporosity distribution at the end of solidification of the gravity sand casting Solidification As with die-casting, the solidification simulation was a continuation of the filling run with no flow but using the metal and mould temperatures as the initial conditions The predicted solidification time is minutes The maximum predicted microporosity is 2.3% Figure 4.29 shows the distribution of microporosity through a section of the casting It appears that it is mostly concentrated near the inner Process Design 155 surface of the casting and that there is enough thickness in the metal for the porosity not to cause any leakage problems There is also some porosity in the thicker sections of the casting and runner system, but overall the feeding system appears to its job well Quality Control 5.1 Basic Concepts of Quality Control The meaning of the term quality has developed over time There are seven distinctive interpretations [www.wikipedia.com]: The degree to which a set of inherent characteristics fulfils requirements as ISO-900 Conformance to specifications The difficulty with this is that the specifications may not be what the customer wants.8 Fitness for use Fitness is defined by the customer.9 A two-dimensional model of quality The quality has two dimensions: “must-be quality’’ and “attractive quality”.10 Value to some person (Gerald Marvin Weinber).11 Costs go down and productivity goes up, as improvement of quality is accomplished by better management of design, engineering, testing and Philip B “Phil” Crosby, (June 18, 1926–August 18, 2001) was a businessman and author who contributed to management theory and quality management practices Joseph Moses Juran, (born December 24, 1904), is an industrial engineer and philanthropist Juran is known as a business and industrial quality “guru” 10 Professor Noriaki Kano is the developer of a program whose simple ranking scheme distinguishes between essential and differentiating attributes related to concepts of customer quality This system is referred to as a Kano model 11 Gerald Marvin Weinberg (“Jerry”) is an author and teacher of the psychology and anthropology of computer software development 158 Casting: An Analytical Approach by improvement of processes Better quality at lower price has a chance to capture a market Cutting costs without improvement of quality is futile.12 The loss a product imposes on society after it is shipped Taguchi's definition of quality is based on a more comprehensive view of the production system.13 Energy quality, associated with both the energy engineering of industrial systems and the qualitative differences in the trophic levels of an ecosystem Energy quality is the contrast between different forms of energy, the different trophic levels in ecological systems and the propensity of energy to convert from one form to another One key distinction is that there are two common applications of the term Quality as a form of activity or function within a business One is Quality Assurance which is the “prevention of defects,” such as the deployment of a Quality Management System and preventative activities like FMEA The other is Quality Control which is the “detection of defects,” most commonly associated with testing which takes place within a Quality Management System and is typically referred to as verification and validation 5.2 Definition of Quality The Merriam-Webster definition of quality of an object is “peculiar and essential character, degree of excellence, superiority in kind.” Although the definition of quality is important in general terms, it doesn’t give specific criteria to define the quality of the final product A product produced by different manufacturing processes as well as for various applications will require its own set of criteria Throughout history people struggled to find a universal definition of quality in a single set of criteria to define product quality 5.3 Definition of Control Control is a process that employs a set of procedures to ensure that defined quality standards are met on a consistent basis A process is said to be controlled, if through the use of past experience, we can predict expected variation of the process in the future 12 William Edwards Deming (October 14, 1900 December 20, 1993) was an American statistician, college professor, author, lecturer, and consultant 13 Gen'ichi Taguchi (born January 1, 1924 in Takamachi, Japan) is an engineer and statistician From the 1950s onwards, Taguchi developed a methodology for applying statistics to improve the quality of manufactured goods Taguchi methods have been controversial among many conventional Western statisticians unfamiliar with the Taguchi methodology Quality Control 159 5.4 Statistical Process Control Statistics is a mathematical science pertaining to the collection, analysis, interpretation and presentation of data The mathematical methods of statistics emerged from probability theory, which can be dated to the correspondence of Pierre de Fermat and Blaise Pascal (1654) Probability is the measure that describes the chances that an event will occur Statistical process control is the application of statistical methods to identify and control the special cause of variation in a process Statistical process controlled was first thought of and developed by Walter A Shewhart.14 Statistical Process Control (SPC) is the equivalent of a histogram plotted on its side over time Every new point is statistically compared with previous points as well as with the distribution as a whole to assess likely considerations of process control Forms with zones and rules are created and used to simplify plotting, monitoring and decision making at the operator level SPC separates special cause from common cause variation in a process at the confidence level built into the rules being followed In the foundation of statistical variation and the necessity of statistical control is the basic understanding that, in real life, no single process can have two absolutely identical outcomes Statistics as a science that takes all possible statistical variations of process outcomes into account and helps to predict future behaviour of the process by analyzing the history of the data Statistical data can be summarized and presented in several forms: graphical, tabular, and numerical The main concern of statistics is collecting, analyzing and interpreting data The ability of the organization to predict and control any variation in the manufacturing process has a profound effect on the quality of the final product But it is easier said than done The die-cast process has between 20 – 25 physical characteristics that affect the quality of the casting Control of the process should encompass not only possible variations between these parameters but also understanding of the relations between parameters under specific boundary conditions In the casting process, physical parameters from one cycle have very limited influence on the next Statistical analysis can help to discover simple patterns or patterns of relations between measured parameters and can be used to predict product quality Summary: - Variation is part of reality Controlling manufacturing process means understanding possible variations of the process as well as relations between physical parameters that influence this process - The proper use of statistical tools can enable engineers to predict the quality of the product based on past experience Adjustment of the process can prevent production of defective product before it actually occurs 14 Walter Andrew Shewhart (March 18, 1891 March 11, 1967) was a physicist, engineer and statistician, sometimes known as the father of statistical quality control 160 Casting: An Analytical Approach 5.5 Tabular Summarization of Data Tabular data summarization includes Collecting data Sorting data in specific order Summarizing results Presenting data Table 5.1 shows raw data of impact pressure collected over a period of time Table 5.1 Impact pressure collected overperiod of hours Time 7:00 7:20 7:40 8:00 8:20 8:40 9:00 9:20 9:40 10:00 10:20 10:40 Impact pressure, mPa 13.2 13.7 14.3 14 13.4 14.4 13.6 14.2 13.9 14.1 13.9 13.6 Time 11:00 11:20 11:40 12:00 12:20 12:40 1:00 1:20 1:40 2:00 2:20 2:40 3:00 Impact pressure, mPa 13.4 13.6 14.1 13.9 14.2 14.1 13.9 13.7 14.2 14.3 13.9 14.0 14.1 5.6 Numerical Data Summarization Collected data can be summarized numerically by calculating its central tendency and data dispersion The central tendency is a measure of the middle value of the data set There are many measurements of central tendency The most common is the simple average: N _ X ¦X i N i , (5.1) where _ X = arithmetic mean Xi = individual variable n = number of variables in a data set The most useful measurement of the data dispersion is the standard deviation It can be found by using the formula, ... can also be computed: 122 Casting: An Analytical Approach p2 / p1 U / U1 T2 T1 (4.55) Now, using Equation 4.54, Đ M2 Ã ă ăM â 1ạ 2 V2 / c2 V1 / c1 Đ V2 Ã p1 U ă ăV p U â 1ạ A1 / A2 (4.56) p2... ă1 JM 12 ă A2 áạ â (4.58) Equation 4.58 yields the following solution: U1 U2 JM 12 A1 A2 A1 (4.58) A2 M 12 J 2J A1 / A2 M 12 (1 A1 / A2 ) 2( A1 / A2 ) M 12 ( A1 / A2 ) M 14... Ã U áá M 12 1 JM 12 J ăă áá M 12 ăă áá U2 â A2 â U2 ¹ which can be recast into a quadratic equation for the density ratio: Đ U1 Ã ă ăU â 2? ?? 2 Đ A1 Ã J U1 ă M 12 ăA U2 â 2? ?? Đ A Ã M