Jackson M.J. Micro and Nanomanufacturing

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Mechanical Micromachining 231

Veff=M + M, and jU

p— (5.47)

Where

is the eddy viscosity, c is a constant, and is equal to

0.09, k is the turbulent kinetic energy and e is the turbulent eddy dis-

sipation. The turbulence model is given by,

dxj

d)Cj

dxj

dxj dxj

3 dxj dxj

+ P£

And

dx.

dx. a dx. k dx

dx. dx

3 dx. dx,

J J

J J J

J J

(5.49)

Where

= 1.0, a

1.3, C

i = 1.44 and

= 192 are constants

from the k-s model.

Energy Equation,

d(pUjh,o<) _ ™j

+ s

(5.50)

dxj dx j

Where h

tot

is defined as the specific total enthalpy, which for the

general case of variable properties is given in terms of the specific

static enthalpy, h, by:

htot=h+

2;h

h(p,T)

(5.51)

232 Micro- and Nanomanufacturing

And

is the source term, which represents the work done by the

viscous and pressure forces.

Equation of State,

For high-speed flows, the density of the fluid changes with pres-

sure and temperature, i.e., the fluid is compressible, therefore the

equation of state plays an important role. The equation of state for an

ideal gas is given as:

pRT (5.52)

Where p is fluid pressure, p is the density of the fluid, T is the

temperature of the fluid and R is the gas constant and is equal to

286.9 J/kg. K for air as an ideal gas.

5.6.7 Method of Solution

The parameter "pressure coefficient" was defined to determine

the pressure variation across the rotor.

Pressure coefficient =

™*

~ ™" (5.53)

* inlet

Where P

max

maximum pressure exerted by air on the rotor, P

min

= minimum pressure exerted by air on the rotor, and

/>,•„/#

= air pres-

sure at the inlet. Maximum pressure and minimum pressure exerted

by air can be obtained by CFX-Post. Inlet pressure was considered

for determining the pressure coefficient to non-dimensionalize the

value of the pressure coefficient, as the inlet pressure (60 psi) was

standard for all geometries considered for numerical simulation.

Mechanical Micromachining 233

5.7 Mechanical Design of High-Speed Rotors

To estimate and to minimize the pressure distribution across the ro-

tor, various designs of rotor have been proposed for the numerical

simulation of high-speed spindles. The following section describes

the different designs of the rotor:

5.7.1 Basic geometry of the rotor

The outer diameter of the rotor was 0.30 inches (7.6 mm), inner di-

ameter was 0.092 inches (2.34 mm) and height along the z-direction

was 0.1445 inches (3.6 mm). Figure 5.25 shows the three-

dimensional geometry of the rotor. Housing with a diameter of 0.31

inches (7.8 mm) was built around the rotor with a height of 0.1735

(4.4 mm) inches. Three inlets, with a diameter of 0.055 inches (1.4

mm),

that make an angle of 120 degrees with each other, were cre-

ated around the housing. An outlet hole was created at the center of

the housing with a diameter of 0.02 inches. The angle between the

blades was considered as 55 degrees. Figure 5.26 shows the front

view of the geometry with housing, rotor and three inlets.

55 degrees

I'M

Fig. 5.25. Three-dimensional view of the rotor [2]

234 Micro- and Nanomanufacturing

CBS"

Fig. 5.26. Front view of the geometry considered for numerical simulations [2]

5.7.2 Rotor with Fillet Surfaces

The basic geometry of the rotor was modified with fillets on the

blade surfaces and in between the blade surfaces of the rotor. It is

observed (Fig. 5.26) that the air that is entering the housing from the

inlets is impinging on the blades and in between the blades. The

modified geometry allows the flow to be streamlined in order to

avoid the sharp edges of the rotor. The dimensions of this geometry

were considered the same as the basic geometry of the rotor. As in

the basic geometry case, three inlets, which make an angle of 120

degrees to each other were considered around the housing and an

outlet at the center of the housing. The geometry of the rotor with

fillets is shown in Fig. 5.27 and the geometry of rotor along with

housing and three inlets is shown in Fig. 5.28.

Mechanical Micromachining 235

z»-*

Fig. 5.27. Rotor with rounded vanes [2]

i_x

QF35"

Fig. 5.28. Front view of the rounded vane geometry considered for numerical

simulation [2]

236 Micro- and Nanomanufacturing

5.7.3 Rotor with a 70-Degree Blade Tip Angle

The geometry of the rotor with 70° blade angle is shown in Fig.

5.29. The basic geometry was modified by changing the angle be-

tween the rotor blades to 70°. Three inlets and an outlet were mod-

eled similar to the previous geometry. Figure 5.30 shows the geome-

try considered for the numerical simulation.

z*x

Fig. 5.29. Rotor with a 70° blade tip angle [2]

Fig. 5.30. Front view of the geometry shown a rotor with 70° blade tip angle,

housing, and three outlets [2]

Mechanical Micromachining 237

5.7.4 Rotor with 90-Degree Blade Tip Angle

The basic geometry of the rotor was modified by increasing the an-

gle between the blades to 90°. The rotor geometry is shown in Fig.

5.31.

The dimensions of the rotor and housing were similar to the

basic rotor geometry. The geometry of the rotor and housing, along

with three inlets and an outlet is shown in Fig. 5.32. CFD simula-

tions have been carried out for the above geometries of the rotor and

the pressure coefficient was found to be less for rotors with 90°

blade angles. Therefore, the rotor with 90° blade angle geometry

was used in all the subsequently modified cases.

Fig. 5.31. Rotor with a 90° blade tip angle [2]

238 Micro-and Nanomanufacturing

QK§'<

Fig. 5.32. Rotor with a 90° blade tip angle, housing, and three inlets [2]

5.7.5 Rotor with 12 Blades

In this case, the number of blades of the rotor was increased to

twelve and the angle between the rotor blades was 90°. Owing to the

increase of the number of blades, the surface area of the rotor in-

creases and the pressure variation on the rotor decreases. The ge-

ometry of the rotor is shown in Fig. 5.33.

Mechanical Micromachining 239

Z*UC

Fig. 5.33. Rotor with 12 blades [2]

5.7.6 Housing with Inclined Inlets

COS'

The dimensions of the rotor and housing were same as that of the

basic geometry of the rotor. The inlets were modified such that they

make an angle of 45° with the z-axis, so that the air can directly im-

pinge on the rotor blades, thus the torque and the angular speed of

the rotor could be increased. An outlet for the air was created at the

center of the housing with a diameter of 0.02 inches (0.51 mm).

Three inlets and a cylindrical housing are shown in Fig. 5.34. The

rotor with three inlets inclined at an angle of 30 with z-axis was

also considered for the numerical simulations. Figure 5.35 shows the

geometry of the rotor with three inlets inclined at an angle of

with

the z-axis.

240 Micro- and Nanomanufacturing

Inlet

Fig. 5.34. Housing with three inclined inlets at an angle of 45° with Z-axis [2]

•'•••"-

as»

Fig. 5.35. Housing with three inclined inlets at an angle of 30° with Z-axis [2]