Ebook Casting: An analytical approach - Part 2

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;...

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, U s =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 U f =2 m/s will be used in the shot sleeve, the gate velocity of U g =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

Table 4.1 Estimates of the dimensionless numbers for typical gravity and high-pressure die

casting

Fast shot Gravity pour Slow shot (shot sleeve)

Shot sleeve Gate

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

5 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

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

6 By dividing the Bond number by the Weber number, we get another dimensionless relationship,Gl U2 , between gravity and inertia The ratio is small, less than 0.01, for flow in high-pressure die casting.

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

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 U 0= 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 U 0 = 5 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

x

U y

u

y

3 0 0

332

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

b R

0

3 0664

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 U 0= 5 m/s, the total force due to friction at the walls of the runner is

about 1 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

7 The viscosity coefficient P is set equal to zero in Equation 2.8 for the inviscid flow model.

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:

C gh U

P  U 2  U2

1

, (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:

1 2 1 1

2

2

12

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

2 1 1

2

2

12

1

U P

U

As fluid accelerates along a streamline, pressure drops so that the sum on the

left-hand 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 1/2UU 2 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

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

1 2 1 2

2

12

1

gh U gh

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 U 1§0, then for the velocity

elsewhere along the streamline, we get

h g h

h g

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 2 m/s, irrespecive of the alloy in the ladle

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 h 1 , U 1 and P 1 are known and C 1 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,

1 2

2

1

C gh U

At the same time, due to the incompressibiity of metal, the flow rate, Q, at any

cross section must be the same:

Q

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

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, P a

Then according to Equation 4.9, the initial velocity of the metal at a gate, U g, is

a 1

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,

a g

P C

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 C 1 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

A

Q gh C P

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

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

A

Q C P

2

1

U

The flow rate Q in high-pressure die casting is primarily defined by the

pre-programmed 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

A R 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, P a 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

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:

A

Q dt

dh

(4.16)

Pressure P R in Equation 4.15 will vary in time as the level of metal in the cavity

increases and can be approximated as

gh P

where P R0 is the value of the pressure at point R at the moment when metal enters

the mould cavity Finally, velocity U R can be expressed using the flow rate and the

area A R :

R R

A

Q

Substituting Equations 4.15, 4.17 and 4.18 in Equation 4.16 and solving the

resulting differential equation for h yields the following expression for h

h

E

where E 2(P a PR0) U and t 0 is the time of metal entry into the mould cavity

The value of PR0 depends on variations in the cross-sectional area in the runner and

gates between point R and the cavity For example, if the area is constant and equal

to A R , then P R0 can be estimated as Ugh 1 , where h 1 is the height of the bottom of the

mould cavity above point R (see Figure 4.4)

From Equations 4.16 and 4.19, one can find the flow rate as a linear function of

From Equation 4.19, one can also calculate the time required to fill a mould If

h max is the elevation of the top of the mould cavity, then the fill time, t fill is given by

1

max 0 0

A

A g

Given the values of the average horizonal cross-sectional area A of the casting and

E, Equation 4.21 gives the fastest possible fill time Viscous and other flow losses

in the runner and gate system increase the fill time Nevertheless, these equations

are useful in providing rough estimates of the flow parameters, short of performing

detailed numerical or experimental studies

Figure 4.4. Derivation of the filling rate as a function of time

4.4 Flow in a Shot Cylinder

The speed of the plunger in a horizontal shot cylinder must be carefully controlled

to avoid unnecessary entrainment of air in the metal If the plunger moves too fast,

it creates large overturning waves on the surface of the metal that faciliate

turbulent mixing of the air into the bulk metal A plunger moving too slowly

results in waves travelling along the length of the shot sleeve in both directions,

preventing proper expulsion of air into the die cavity In either case, the outcome is

excessive porosity in the final casting In an ideal situation, the slope of the metal

free surface should be directed away from the plunger everywhere along the length

of the cylinder and at the same time, should not be too steep

The dynamics of waves in a horizontal shot sleeve can be analyzed by drawing

an analogy with the flow in an open channel From the start, we will approximate a

cylindrical shot sleeve by a channel of rectangular cross section filled with liquid

metal to depth h0 This simplification of the shape of the cylinder is justified for

initial fill fractions of 40-60% [Lopez et al, 2003] and allows for some useful

solutions For a shallow wave travelling along the free surface due to gravity g, the

speed of the wave, c0, is given by

0

Equation 4.22 is valid for waves that are long and shallow compared to the mean

depth of the fluid Note that the wave speed is independent of the properies of the

metal If the speed of the plunger is too slow, these waves will travel a distance

equal to several times the length of the sleeve, reflecting off the moving plunger

and the opposite end of the sleeve, before the transition to the fast shot

Figure 4.5. Schematic illustration of a propagating surface wave when the plunger is

moving slowly (upper image) and the hydrauic jump forming ahead of the fast moving

plunger

As the plunger accelerates, it first catches up, then overtakes the waves, that is,

the flow becomes supercritical As a result, metal piles up to the top of the sleeve

in front of the plunger, creating a flow condition called hydraulic jump The

hydraulic jump moves ahead of the plunger and is similar to a shock wave in a gas

where flow undergoes a sharp transition through its relatively thin front Since

waves cannot overtake it, metal in front of the jump “does not know” about its

violent approach and continues to move slowly As soon as the hydraulic jump

engulfs a volume of metal ahead of it, the metal is quickly accelerated to a much

higher speed

Figure 4.5 schematically illustrates the two flow zones in a short sleeve

separated by a hydraulic jump If the plunger is fast enough, then the metal will

pile up to the top of the sleeve If we neglect the relatively slow speed of the metal

in front of the jump, then the speed of this front, D, can be estimated from the

Equation 4.23 shows that the hydraulic jump always moves faster than the

plunger and that, just like the wave speed in Equation 4.22, its speed is

independent of metal properties

Equations 4.22 – 4.23 provide some guidance to what the plunger speed can be

during the slow shot stage A more detailed analysis is possible by modelling the

flow of metal in a rectangular shot sleeve of length L and height H using the

shallow water approximation [Lopez et al, 2000] In this approximation, the flow

in the vertical direction is neglected in comparison with the horizontal velocity

component The flow is modeled in two dimensions, with the x axis directed in the

direction of motion of the plunger, and the z axis pointing upwards If viscous

forces are omitted, then the flow has only one velocity component, u, along the

length of the channel Pressure at every point in the flow is then hydrostatic:

P

where h(x,t) is the height of the fluid at point x and time t, as shown in Figure 4.6

Figure 4.6. Schematic representation of the flow in a shot sleeve and the coordinate system

With these assumptions about the flow Equations 2.8 and 2.9 reduce to

,0)2()()2(

,0)2()()2(

w

w



w

w

w

w



w

w

x

c u c u t

c u

x

c u c u t

c u

(4.25)

where

gh t

x

The plunger speed in the positive x direction is given by dX/t = X’(t), where

X(t) defines the position of the plunger at time t>0 At the surface of the plunger,

)(')),(

At all other walls of the channel, including the end at x=L, the normal velocity

component is equal to zero The initial conditions at t=0 are

.)0,(

,0)0,(

,0)0('

,0)0(

0

h x h

x u X

X

(4.28)

Equation 4.25 defines two sets of waves travelling at the respective speeds of

u+c and u-c along the metal surface The quantity u+2c is conserved in the first set

of waves, and u – 2c is conserved in the second set Combined with the boundary

and initial conditions, Equations 4.27 and 4.28, this yields the following solution

for a wave that separates from the surface of the plunger at time t = t p

.)('2

11

),(

),('),(

),()('2

3)

()(

2 0

p p

p

t X gh

g t x h

t X t x u

t t t X c

t X t x

(4.29)

Figure 4.7. Geometric representation of the slow shot soluion, Equation 4.29 The thick

solid line represents the position of the plunger X(t) as any given time, with its slope

defining the plunger velocity X’(t) The waves on the metal surface, or charactersitics, are

represented by thin solid lines The tangent of the slope - of a characteristic is equal to the

speed of the wave As the plunger accelerates, the speed of the waves originating at its

surface, and hence the slope -, increases After time t c , the speed of the plunger becomes

constant, so all charactersitics that are created after this time have a constant slope When

two characteristics intersect each other at a point P, the metal surface slope becomes vertical

causing overturning of the waves The horizontal dashed line repesents the end of the shot

cylinder at x=L.

As the plunger moves along the length of the channel, it sends waves forward

Each wave represents a small segment of the metal free surface and the column of

metal directly below it (Figure 4.6) Flow parameters in each such wave are

constant and depend only on the time of separation from the plunger, t p According

to Equation 4.29, once a wave detaches from the plunger, it travels ahead of it at

the constant speed of

)('2

3)

(

X c

The metal depth, h, and velocity, u, are constant in the wave They both increase

with the speed of the plunger Therefore, to maintain a monotonic profile of the

metal surface away from the plunger, the latter must not decelerate, that is, its

acceleration must not be negative at any time:

0)(' t t

If the plunger accelerates, then each successive wave will move faster than waves

generated earlier This will lead to the steepening of the surface slope as the waves

travel further down the channel, and can result in overturning of the waves

The solution given by Equation 4.29 can be illustrated geometrically Figure

4.7 depicts the motion of the plunger and of the waves on a graph with time plotted

along the horizontal axis and distance along the vertical one The thick line

represents the function X(t) The thin lines represent travelling waves Since their

respective speeds are constant, these lines are straight and are called

characteristics The slope of each line is related to the wave speed given by

where Tis the angle between the characteristic and the horizontal axis The

characteristic that originates at the (0,0) point corresponds to the wave in a

quiescent pool of metal with its speed given by Equation 4.22

As the plunger accelerates, the slope of the characteristics increases, therefore,

they intersect each other at some point down their paths The intersection point of a

pair of characteristics is where the faster wave catches up with the slower one

Having two solutions at the same location can be interpreted as the slope of the

free surface becoming vertical, at which point the wave is likely to overturn and

trap air However, if the intersection occurs beyond the end of the shot sleeve at x

= L, then we can say that the two waves never meet and the surface slope does not

reach the vertical

Figure 4.8. Illustration for calculating the slope of the metal’s free surface

Let us analyze the evolution of the surface slope between two waves generated

at the plunger at close instances, t 2 > t 1 (see Figure 4.8) The slope is given by

2 1

2 1

)tan(

x x

h h dx

3)

()('2

3)

()

(

)('2

1)

('2

11

2 2

0 1 1

0 2 1

2 2 0

2 1 0

t t t X c

t t t X c

t X

t

X

t X c

t X c

'

,

)('2

1)

(')()(

1 1

2

2 1 1

1 2

'

˜



'

˜

'

˜



t t X t X t X

t t X t t X t X t

X

(4.35)

Substitution of Equation 4.35 in Equation 4.34 and omitting higher-order terms

with respect to 't yields

))(

('2

3)('21

)(')('2

11

)tan(

1 1 1

0

1 1

0

t t t X t

X c

t X t X c

Equation 4.36 gives the expression for the free surface slope as a function of

time t Note that if the plunger moves at a constant speed, i.e X’’(t 1 )=0, then the

right-hand side of Equation 4.36 becomes zero and the slope of the free surface is

horizontal

If the plunger accelerates, then the denominator on the right-hand side of

Equation 4.36 decreases and the slope grows with time When the denominator

turns zero, the slope becomes vertical The maximum slope, Dmax, is achieved when

the wave reaches the end of the shot sleeve at t=t L This time can be computed

from

)('23)(1 0

1 1

t X c

t X L t

tan(

)('2

3)

('2

1

1

)tan(

)('2

3)

('21

)

(

1 max

1 0

1 0

max 1

0 1 0

1

''

max

t X L t

X c

t X

c

g

t X c

t X c

(4.38)

Equation 4.38 represents the final and most useful form of the solution for the

flow of metal in a shot sleeve It can now be used to calculate the velocity of the

plunger to maintain a certain slope of the metal surface during the slow shot stage

For example, if Dmax is set equal to 10°, then the plunger acceleration given by

Equation 4.38 ensures that the slope of 10° is not exceeded anywhere and anytime

during the motion of the plunger

Equations 4.31 and 4.38 can be combined to give a range of values for the

plunger acceleration at any give time

)()

('

X t

Two things are achieved when the plunger acceleration stays within this range The

slope of the metal surface is directed away from the plunger and toward the

opposite end of the shot cylinder Second, the slope will not exceed the angle

defined by Dmax at anytime during the slow shot process

To obtain the solutions for X(t) and X’(t), Equation 4.38 needs to be integrated

numerically with respect to t 1 using the initial conditions for X and X’ given by

Equation 4.28 Figure 4.9 shows numerical solutions for the plunger position, X(t),

acceleration, X’’(t), and speed, X’(t) (the latter is shown as a function of both time

and distance along the channel length) for several values of Dmax The integration

was done for a shot cylinder of length L = 0.7 m and height of H = 0.1 m and an

initial fill fraction of 40%, i.e., h0= 0.04 m

Note that the plunger motion is slower for smaller values of Dmax It takes the

plunger 1.66 seconds to get to the end of the shot sleeve for the most conservative

case considered with Dmax = 5o; for Dmax = 90o, the time is 0.83 seconds The

difference between these two extreme cases is just 0.83 seconds However, these

times will be longer if we add an additional constraint of the plunger velocity not

to exceed the critical velocity at which the metal surface reaches the ceiling of the

channel at h = H [Garber, 1982] The critical velocity of the plunger can be derived

from the solution for h(t,x) given by Equation 4.29 [Tszeng and Chu, 1994]:

 0

and is shown in Figure 4.9 by the horizontal dashed line For the selected

parameters of the shot sleeve, X’ cr= 0.73 m/s Even for Dmax= 5o, the plunger

velocity reaches the critical value after it moved just over 60% of the channel

length at t c = 1.35 s For steeper surface slopes, the critical velocity is reached at

earlier times, for example, for Dmax= 90ot c = 0.58 s and the plunger position is 22%

of L.

Figure 4.9 Solutions of Equation 4.38 for the plunger position a., acceleration b., velocity

c and velocity as a function of distance along the length of the shot channel (d), at different

maximum surface slopes Dmax : 1 – 90o, 2 – 60o, 3 – 45o, 4 – 30o, 5 – 15o and 6 – 5o The

horizontal dashed lines in plots c and d represent the critical plunger velocity given by

Equation 4.40

When the plunger reaches the critical velocity, the metal surface comes in

contact with the ceiling of the shot cylinder Beyond this point, the solution given

by Equation 4.29 becomes invalid It can also be argued that if the plunger

continues to accelerate, then the potential for creating an overturning wave

increases since all the energy of the flow is now redirected forward by the walls

and ceiling of the channel It is usually recommended to keep the plunger velocity

below the critical value The CFD simulation of the slow shot process, presented in

Section 4.10.1, provides more details of the flow before and after the critical

velocity is reached

It is often assumed that the overturning of the metal surface that causes air

entrainment occurs when the wave profile becomes vertical, that is, Dmax= 90o In

reality, the breaking of the wave surface may happen at more moderate angles, as

can be seen while observing ocean waves Equation 4.38 allows the engineer to

define any maximum permitted wave slope, obtaining a sufficient safety margin to

avoid any air entrainment

Being able to define a safety margin for the surface slope is also important

because we made some simplifications in arriving at Equation 4.38, such as

replacing the cylindrical channel with a rectangular one Obviously, the curved

walls of the shot cylinder will exacerbate the potential for wave overturning as the

metal level rises Besides, the critical velocity is attained faster in a cylindrical

channel than in a rectangular one of the same width and, therefore, an extra safety

margin must be used in an estimating of the critical velocity by Equation 4.40

Equation 4.36 can be used to obtain the slope Dmin of the metal surface right at

the plunger This can be done by setting t = t 1 Then it reduces to a very simple

form

g

t

X '( ))

Equation 4.41 gives the initial surface slope for a wave detaching from the plunger

at time t = t 1; it is a function of only the plunger’s acceleration and not its position

or even velocity As the wave propagates along the length of the channel, it

steepens reaching the maximum slope, Dmax , at the end of the channel at x = L,

given by Equations 4.36 and 4.37

One can easily get carried away in preventing air entrainment by defining a

very small value of Dmax in Equation 4.38 This would lead to a very slow plunger

motion Of course, in real life situations, the requirement of minimum air

entrainment must be combined with other criteria that control the quality of the

casting, such as a specific filling rate and minimal early solidification in the shot

sleeve and runner system

4.5 Gas Ventilation System

The size of the cross-sectional area of the ventilation system has to be based on the

volume of the gas to be evacuated and the parameter of the die casting process

They will define how much time is available to evacuate the air from the cavity of

the die-cast die The gas ventilation system has to be calculated as a converging–

diverging nozzle

For subsonic flow in the air stream, the static pressure just inside the ventilation

channel exit will be equal to the pressure in the downstream region There are two

reasons that explain that equality

There is no internal mechanism that can produce the difference in static

pressure between the inlet and outlet of the ventilation channel For subsonic flow,

the pressures in the inlet and outlet are equal

Any disturbances in the outlet of the ventilation channel can be propagated

back to the inlet through the entire flow field A continuous increase in the

upstream pressure will cause an increase in the mass flow as well as in velocity of

the air until the local Mach number at the exit reaches unity One-dimensional compressible flow theory can be applied to a ventilation channel The effect of viscous friction in this model is neglected The flow can be assumed to be isentropic when no shocks are present The critical pressure ratio at which the flow becomes sonic can be determined [Munson, Young and Okiishi 2006]:

1 CR

p

p

Using isentropic theory, the relation between mass flow rate m and

cross-sectional area A can be described as

JJ

1

1 2 /

1

1

2 1 1

11

2

p

p p

p RT A p

d

) 1 ( 2 1 1

RT A p

equal to or less than (p 2 /p 1 )CR, then the velocity at the throat becomes sonic There

is no convenient rule to apply define mass flow when the ratio of downstream to upstream pressure is more than critical The general rule for obtaining mass flow through the ventilation channel is

1 If the ratio of downstream to upstream pressure is less than the critical mass flow, it can be determined using Equation 4.43

2 If the ratio of downstream to upstream pressure is less than the critical mass flow, it can be calculated using Equations 4.43 and 4.44 The lesser of the two values is used

Let us determine how the flow in the throat section of the nozzle develops As

is shown in Figure 4.10:

1 The shock wave curves upstream

2 The shock first originates upstream of the M =1 line of the supersonic zone

Now let us look at the actual amount of mass flow compared with that one determined from one-dimensional theory The actual flow, even disregarding friction, is as much as 5% less than that predicted by one-dimensional theory The choked flow is about 0.7% less than that predicted by one-dimensional theory These numbers will vary with the shape of the nozzle [Shapiro, 1953, V1]

Steady-state flow can be used for most engineering calculations During the high pressure die-cast process, a sudden flow (within 10–15 m/s) changes from slow (20–30 m/s) to fast (150–200 m/s) In this case, unsteady flow has to be considered to describe the behaviour of the gas as it moves through the ventilation system

Figure 4.10 Shock wave formation in a divergent nozzle

To avoid mathematical difficulties, it will be necessary to make a number of assumptions:

1 The flow will be considered geometrically one-dimensional, implying that all fluid properties are uniform over each cross section of the passage, and that changes in cross-sectional area take place very slowly

2 The viscosity and thermal conductivity of the gas will be neglected That means that all parts of the gas are related through the isentropic relations unless the shock of changing strength appears

3 It will be assumed that the equation of state is that of a perfect gas

4 The gravity effects are negligible

5 The fluid can be treated as a continuum

Flow characteristics have to be established based on the normal shock relations for a perfect gas

The air ventilation system of the die-cast die is a diverging–converging nozzle with an abrupt decrease in cross-sectional area Due to limitation of the measurement equipment in being unable to measure temperature, pressure, and

humidity inside of the cavity of the die, measurements of the gas exiting die have

to be used for further analysis To evaluate the pressure and temperature inside the cavity of the die-cast die, equations that describe changes in the thermodynamic state due to abrupt changes in the cross-sectional area of the ventilation channel have to be derived

Basic thermodynamic equations for the conservation of energy, mass and conservation of momentum are used to define correlations between area contraction and variations in pressure, density and temperature The Mach number

is used as an initial parameter

The equation of conservation of energy can be written as

2 2 2

2 1 1 2

1

12

1

JU

Subscripts 1 and 2 indicate quantities before and after an abrupt contraction

Constant acceleration of gravity:

2

1

U (4.45) For a perfect gas,

RT

p U For an ideal gas in isentropic flow,

The constant C can be evaluated at any point on a streamline:

1

/ 1 1 / 1

1 1 2 /

1 /

J J

p p

C dp

2 1

2 1 1

1

21

2

V p gz

V p

J

For one-dimensional flow z 1 =z 2 and Equation 4.45 can be recast as

21

21

2 2 2 2 2

1 1

J

The continuity equation is

2 2 2 1 1

1 2 1

1 2 1

2

12

1

U

JJ

U

J

V V

V p

Subscript s is used to designate that the partial differentiation occurs at constant

entropy Equation 4.50 suggests that the speed of sound can be calculated by

determining the partial derivatives of pressure with respect to density at constant

entropy For an isentropic ideal gas,

UJJUU

JUU

J J

C p c

1

2 2

1 2 1

2

12

1

U

JJ

V V

V c

2 2 1

2 1 2

1

2 2

1

2 1

2

11

2

1

U

UJ

J

p

p c

V V

V c

2 1 1

2 2 1 2

1

2 2

11

2

1

U

UJ

J

p

p M V

2 2 1 2 1 1

A

A M

A

A V

V M A

A p

p

U

UJ

And from the equation of continuity,

2 1 2 1 1

V

(4.54)

By using the perfect gas equation of state, the temperature ratio can also be

computed:

1 2

1 2 1

2

/

/

UU

p p T

T

(4.55) Now, using Equation 4.54,

1 2 1 2

2 2 1 1 2 2 1 2

1

2 2

1 1

2 2 2 2

1

//

U

UU

U

p p A A p

p V

V c

V

c V M

1 2 1 2

U

U

p p

M A

2

1 2

1 2

1 2 1 2

2 1 2

JU

UU

U

J

J

M A

A M

M A A

M

(4.57)

which can be recast into a quadratic equation for the density ratio:

02

2)1(1

2

1 2

1 2 1 2

1 2

1 2

UJ

U

A

A M M

A

A

(4.58) Equation 4.58 yields the following solution:

 1

1

2 1 2 2 1

2 1 2 1 2

U

M A A

A A M

(4.58)

)1()/(

)/()

/(21)/1(/

2

2 1 2 2 1

4 1 2 2 1 2 1 2 2 1 2

1 2 1 2 1

M A A M A A A

A M A A

To obtain the equation for the pressure ratio, the result of the Equation 4.58 must

be applied to the inverted form of Equation 4.53:

1 2

Differentiation of Equation 4.59 with respect to A 1 /A 2 yields the expression for

the area ratio that gives the maximum pressure difference:

2 1

2 1 CR

2

1

)1(2

12

1

M

M A

(4.60)

Equations 4.56 – 4.60 establish relationships between pressure, density, area

and Mach number

Figure 4.11 Relationship between air pressure ratio and area ratio of a ventilation channel

The curves shown in Figure 4.11 were derived by using Equation 4.60

After the curves shown on Figure 4.11 were computed, the maximum pressure

differential was found by using Equation 4.60 and the MATLAB® code In prior

discussions, it was established that transient flow can occur when the Mach

number reaches 0.9 Using the curves in Figure 4.7 the maximum pressure

differential found for Mach number 0.9 was p 1 /p 2 = 0.72 This value corresponds to

the area differential = 0.6 As shown before, during die-cast operation, 50% of the

gas ventilation system can be plugged up To account for the reduction in the area

of the ventilation system and still maintain maximum possible gas mass flow, the

area differential variation was taken as 0.3 Using Equation 4.60 with a Mach

number equal to 0.9 and the area differential equal to 0.3, the calculated pressure

differential was 0.78

A general criterion for calculating the gas evacuation from a die cast die is

established Now two separate procedures can be outlined

1 Calculation of the conventional gas evacuation system

2 Calculation of the vacuum assist gas ventilation system

Calculation of the conventional gas evacuation system

1 The volume of the die-cast part has to be calculated

2 The fill time has to be calculated The fill time is the time necessary to

fill the cavity of the die cast die with aluminium

3 Using the density of steam at standard atmospheric conditions 1.23

kG/m3, the cavity fill time and the volume of the part, the mass flow of

the gas can be calculated As shown above, mass flow that takes into

account two-dimensional gas flow is about 5% less than that calculated

using one-dimensional theory To account for this difference, mass flow

has to be increased by 5% Ideally, all the gas has to be evacuated from

cavity of the die cast-die

4 Now Equation 4.43 can be rewritten to calculate the cross-sectional area

of the gas evacuation system:

J

J

1

1 2 /

1 1 2

1

11

2

p

p p

p

RT p

m A

4 The maximum length of the ventilation channel for a calculated cross

section can be determined from Equation 4.44 [Munson, Young and Okiishi,

2006]:

2 max

)1(2

)1(ln2

11

M M

M f

D L

J

JJ

There are two different types of the ventilation blocks:

- valve

- valveless

A valve system uses a valve shutoff to prevent metal from flowing outside of the die-cast die Valveless systems rely on the metal to solidify before it reaches the end of the block

A vacuum assist gas ventilation system consists of a die-cast die, vacuum tank, vacuum pump, gate, and runner system that connect the cavity of the die-cast die with the vacuum tank During the cycle, gas is forced from the cavity of the die- cast die into the vacuum tank To maintain pressure in the vacuum tank within specified values, the vacuum pump is used to evacuate the gas into the atmosphere The number of die-cast machines that can be connected to the same vacuum system is limited by the volumetric pumping speed of the vacuum pump Several die-cast machines can be connected to the same vacuum tank

There are two types of problems that must be solved:

1 Calculation of the maximum number of machines that can be connected to the same vacuum tank based on gas pump productivity, the volume of gas flowing into the tank and total process time

2 Calculation of the size of the vacuum tank and volumetric pumping speed of the vacuum pump based on the number of die-cast machines connected to the same vacuum assist gas ventilation system

Queueing theory can be used to solve both problems

Before proceeding any further, the fundamentals of queueing theory must be defined

4.7 Little’s Formula

John D Little developed one of the fundamental relations of queueing theory He

related the steady-state mean system size to steady-state average customer waiting

times Letting T q represent the time the customer (transaction) spends waiting in

the queue prior to entering service and T represent the total time a customer spends

in the system,

S T

where

S – the service time

T, T q , S are random variables

Two often used measures of system performance with respect to customers are

]

[ q

q E T

W and W E[T], the mean waiting time in the queue and the mean

waiting time in the system

Little’s formulas are

O = average rate of customers entering the queueing system

W = the mean waiting time in the system

W q= the mean waiting time in the queue

4.8 Poisson Process and the Exponential Distribution

The most common stochastic queueing models assume that interarrival times and

service times obey an exponential distribution or, equivalently, that the arrival rate

and service follow a Poisson distribution The general formula for a Poisson

probability distribution with mean O is t

t n

n

t t

!

)()

Thus, if we consider the random variable defined as the number of arrivals to a

queueing system by time t, this random variable has the Poisson distribution given

by Equation 4.66 with mean of O arrivals, or a mean arrival rate of O [Gross, t

1998]

We can calculate the vacuum system where several die-cast machines are

connected to the same vacuum tank Every die-cast machine can start the cycle

according to a Poisson distribution with mean N/min (where N = the number of

machines connected to a vacuum system) After every cycle, gas from the die-cast

die is forced into the vacuum tank When the pressure in the tank reaches the

specified limit, the vacuum pump evacuates gas into the atmosphere The amount

of time it takes the pump to evacuate gas from an upper to a lower limit of pressure

in the vacuum tank is considered a server’s service time The criterion used in

choosing the maximum number of die-cast machines that can be connected to the

same vacuum system will be a portion of the pump’s busy time (it should be less

than 1)

A simple Markovian birth-death queueing model can be employed to estimate

optimal size and to ensure steady-state operating conditions for the vacuum assist

*

*4

*

2

f ss

where

D = diameter of the shot sleeve

L = length of the shot sleeve

P = percent of fill of the shot sleeve f

2 The amount of gas in cavity, gates, and overflows (V c)

The total volume of gas is

c shs

As molten metal fills the cavity of the die-cast die, it forces gas into the vacuum

tank Pressure rises in the vacuum tank with every shot When the pressure in the

vacuum tank exceeds the specified limit, the vacuum pump pumps air out of the

tank Gas flows into the vacuum tank at a very high velocity The average fill time

is 0.055-0.075 s The vacuum pump operates much more slowly For the vacuum

system to operate within the specified parameters, the amount of gas that flows into

the vacuum tank and the volume of gas that is extracted from the tank must be in

balance

It can be assumed that die-cast machines cycle according to a Poisson

distribution with a mean rate of N/min (N = the number of machines under

consideration) We will consider this system a single server, since there is only one

vacuum pump The mean time to complete the service is the amount of time it

takes the pump to evacuate gas from the vacuum tank The assumption will be that

the service time of the vacuum pump is exponential

Example 4.1

Pump volumetric speed is 22 SCFM

Average amount of gas flow into the tank (each shot) = 0.274 ft3

Average cycle time = 0.41min

Number of machines under consideration = 10

Nominal pressure in the vacuum tank = 2.5 PSI

Maximum acceptable pressure rise in the vacuum tank = 0.25 PSI

The MATLAB® code to calculate the maximum number of die-cast machines that can be connected to the same vacuum system

The number of machines that can be connected to the vacuum system will be based on the pump’s busy time For the vacuum pump to keep up with the amount of gas entering the vacuum tank, the pump’s busy time must be less than 1

Ps=input('Volumetric pumping speed (ft^3/min)= ');Nm=input('Number of die cast machines under

Table 4.2 Tabulated results for Example 4.1 Number of die cast

When traffic congestion exceeds 1, the queue continues growing Since the goal

is to achieve steady-state conditions, traffic congestion should never exceed 1

4.9 Cooling

Two analytical solutions for the heat transfer process are presented in this section

The first describes a simple cooling process when temperature is assumed uniform

throughout the cooling body, that is, the lumped-temperature approach is used In

the second solution, we look at one-dimensional, transient solution for heat transfer

from a fixed-temperature boundary into a semi-infinite medium

4.9.1 Lumped-temperature Model

When two objects at different temperatures are put in contact over a period of time,

their average temperatures will gradually draw closer to each other due to heat

exchange During this process the colder object acquires internal energy from the

hotter one The energy exchange occurs by collisions between the molecules at the

contact surface By definition temperature is the measure of the average kinetic

energy of chaotic molecular motion As a body gains or loses kinetic energy, its

temperature increases or decreases, respectively

In our case, when energy is transferred from the hot casting to cold water in the

cooling system, assuming that there is no heat lost into the environment, the

conservation of energy principle implies that the exact amount of energy lost by

one object must be gained by the other Let us denote the amount of heat gain or

loss by the objects as Q Then the conservation principle can be described as

LOSS GAIN Q

Although the internal energy of an object is directly proportional to its mass, it

does not necessarily mean that the two objects of the same mass and temperature

have the same amount of internal energy If they are made of different materials,

then their specific heats are generally different The specific heat is defined as the

quantity of heat required to raise the temperature of 1 gram of a substance by 1°C

If T1and T2 are the initial temperatures of the two objects, and T0is the terminal

temperature after a prolonged contact, then the respective amounts of heat lost or

gained by the objects are

),(

),(

2 0 2 2 2

1 0 1 1 1

T T C M Q

T T C M Q



 (4.71)

where M denotes mass and C specific heat, with the indices referring to the two

objects According to Equation 4.70, Q 1 is equal to -Q 2, a condition that allows us

to calculate the final temperature:

2 2 1 1

2 2 2 1 1 1 0

C M C M

T C M T C M T





(4.72)

Equation 2.72 gives the steady-state solution for the heat transfer problem of

two bodies exchanging heat, which is achieved after a very long (theoretically-

infinite) period of time To find the transient solution for temperature, we can

simplify the problem to have only a single body cooling to the

constant-temperature environment Let us consider a casting of mass M, specific heat C and

initial temperature, T IN , cooling to the air at a constant temperature T ENV We will

employ here Newton’s law of cooling, which states that the rate of heat loss of a

body is proportional to the difference in temperature between the body and its

surroundings It can be expressed as

T TENV

hA dt

ENV IN

Equation 4.74 shows the evolution of the casting average temperature with

time It starts at the initial temperature and approaches the temperature of the

environment at a rate defined by the constant D Moreover, the rate is higher for

larger surface area and heat transfer coefficient or for smaller mass and specific

heat

The transient solution for cooling can also be used to evaluate the time it takes

to cool the casting to a certain temperature T FIN Rearranging terms in Equation

4.74 gives the desired formula

ENV FIN

T T

T T t

According to this expression, it will take an infinite amount of time for the

temperature to become equal to the surrounding temperature T ENV because the

argument of the logarithm becomes zero at that temperature However, it takes a

finite period of time to cool to a temperature above T ENV, no matter how close to

T ENVthat temperature may be For smaller values of D the wait is longer

After the casting is extracted from die-cast die, the last stage of the process is to

cool it to room temperature There are passive and active ways to cool a casting A

casting can be left in the open area to be cooled by the air, or it can be quenched in

a tank of water When two objects at different temperature are kept together over

the certain period of time, eventually they will reach the same temperature This

process is called heat exchange The energy that is transferred from one object to

another is called internal energy In our case when energy is being transferred from

the casting into the water and we can assume that there is no heat lost into the

environment, then the conservation of energy principle implies that the energy lost

by one object must be gained by the other

By definition, temperature is the measure of the average kinetic energy of

molecular motion As a body gains or loses kinetic energy, its temperature will

increase or decrease Although the internal energy lost or gained by an object is

directly proportional to its mass, it does not means that two objects of the same

mass and temperature have the same amount of internal energy Temperature

reflects only the kinetic energy portion of the internal energy, so an object with a

greater fraction of its internal energy in the form of potential energy will have a

greater internal energy at a given temperature

This property is reflected in the quantity called the specific heat The specific

heat is defined as the quantity of heat required to raise the temperature of 1 gram of

a substance by 1°C

Now we can write an equation that allows us to define the quantity of heat lost

or gained by a body:

)(T T0MC

To calculate the time required to cool casting to a room temperature we will

use Newton’s law of cooling, which states that rate of the heat loss of the body is

proportional to the difference in temperature between the body and its

surroundings It can be expressed as

)(T C T ENV k

T C = temperature of the body

T ENV = temperature of the environment

k = constant

Separating variables in Equation 4.78 yields

kdt T

T

dT ENV C



2 2

2 1 2< /small>

1

12

1

JU

Subscripts and indicate quantities before and after an abrupt... 2< /small> and Equation 4.45 can be recast as

21

21

2 2 2< /small>

1 1

J

The continuity equation is

2 2 1... can be determined from Equation 4.44 [Munson, Young and Okiishi,

20 06]:

2