RELACIONES DE LARGO PLAZO DE LA BALANZA COMERCIAL,

Figure 6.19 is an exaggerated schem atic representation o f the dynamic and static spring profile. It is im portant to note that for a given spring tension, the low er and upper bounds o f the size o f the particles discharged from the spring are respectively equal to the static, ds and dynamic, da intercoil distances. W hereas the former is easily obtained from the spring tension, the calculation o f the latter is not straight forward. The follow ing describes the developm ent o f a sem i-theoretical technique for the prediction o f the dynamic intercoil distance.

R eferring to the figure 6.19, the relation betw een ds and da can be w ritten as:

0/2

Figure 6.19 An exaggerated schematic representation of the dynamic and static spring profile. The static intercoil distance (ds) and its corresponding dynamic intercoil distance (da) as well as curvature (p) and maximum deflection (c) are also shown in the figure.

t a n ( e / 2 ) = — — -— (6 .5 4 ) 2 (p - c)

where, p is the radius o f curvature o f the spring m ode m easured from the spring main axis and c, is the spring maximum deflection. Taking tangents from both sides o f the above equation results in

0 = 2 ta n ’^[d s/2 (p - c)] (6.55)

On the other hand, the dynamic gap, d j can be expressed as:

dd = ( p + R)0 (6.56)

where, R is the spring mean coil diam eter.

Substituting 0 given by equation 6.55 into equation 6.56 gives:

dd = 2 ( p + R )ta n '^ [d s /2 (p - c)] (6.57)

Referring to the above equation, it is clear that for a given spring intercoil distance, dg calculation o f the upper discharge particle size, dd requires a know ledge o f the radius o f curvature, p as well as the maximum deflection, c.

The radius o f curvature, p can m athem atically be w ritten in the following form (see for example G oodm an 1980):

p = l / f ' ( x ) ( 6 . 5 8 )

where; f'(x ) is the second derivative o f any im plicit function, f(x) at any stationary point.

R eferring to equation 6.40, w e have:

f(x) = cie"* + C2e '“^ + C3C0 s(Px) + C4sin(Px) -B

from which:

f(x ) = C iae“^^^ - C2a e ‘“^^^ - C3psin(pl/2) + C4Pcos(pI/2) (6.59)

f (x) = + C2a^ e '“^^^ - C3p^cos(pi/2) - C4P"^sin(pI/2) (6.60)

Substituting equation 6.60 into equation 6.58 produces:

p = [cia^e“^^ + C2a^e‘“^^^ - C3p^cos(pl/2) - C4p^sin(pi/2)]‘^ (6.61)

The above equation can be used for the calculation o f the radius o f curvature, p as all the term s on the R.H.S are know n. Now, according to equation 6.57 the only term rem aining for the calculation o f the dynamic intercoil distance is the spring am plitude o f vibration, c.

As discussed earlier, (in accordance to equations 6.41-6.44) c approaches infinity at resonance. This is an intrinsic characteristic

o f resonance. In practice how ever, the spring resonant am plitude is o f the order o f a few m illim eters. On the other hand, it is rem arkable that when operating away from resonance, it is possible to obtain a reasonably accurate prediction o f the spring profile and the am plitude o f vibration. Figure 6.20 provides a graphical presentation dem onstrating the above phenom enon. The data show theoretical predictions (equation 6.40) o f various profiles obtained at, in close vicinity, and aw ay from resonance for a loaded spring (5 gr). The experim ental data presented in the same graph were obtained in conjunction w ith 5 gr o f 1000-1200 pm glass Ballotini. As it may be observed, a reasonable degree o f agreem ent betw een theory and experim ent is obtained when the system is operated away from resonance (36 Hz).

The im plication o f the above data is that when operating the particle sizer at resonance, for any given tension, the prediction o f the dynamic intercoil distance requires a direct m easurem ent o f the spring resonance am plitude o f vibration.

Figure 6.21 shows a com parison betw een theory (curve A) and experim ent (curve B) for the variation o f the upper discharge particle size plotted against the static intercoil distance, dg. Each data point has been obtained in conjunction w ith different sieve cuts corresponding to 5 gr batches o f glass Ballotini. In the case o f theory, the upper discharge particle size w as directly equated to the spring dynamic intercoil distance as calculated from equation 6.57. The upper particle size obtained from experim ent refers to the sieve

£

.£

20

40

60

80

X coortinate (mm)

Figure 6.20 Various mode shapes of the loaded spring as calculated from theory as well as obtained from experiment. The data relate to a fixed spring length (107 mm) which is loaded with 5 gr glass Ballotini in the size range 1000-1200 pm and vibrated at frequencies of 45, 39 and 36.3 Hz ( under the experimental conditions, resonant frequency of the spring is 36 Hz).

750

% 600

8 450

O 300

c 150

0

150

300

450

600

750

Spring static intercoil distance (pm)

Figure 6.21 Variation of the spring dynamic intercoil distance in terms of the corresponding static intercoil distance as obtained from theory (curve A) and experiment (curve B) . The data relate to 5 gr sieved batches of glass Ballotini vibrated at 37 Hz frequency.

upper size. The static intercoil distance in both cases is simply the ratio o f the spring extension and num ber o f active coils (equation 4.1). A 45° line is drawn to allow a com parison betw een ds and dd. As it is evident from the data, a reasonable agreem ent betw een theory and experim ent is obtained.

The fact that in each and every case the spring dynamic intercoil distance is larger than the corresponding static intercoil distance underlines the im portance o f a know ledge o f the spring vibration characteristics w hen perform ing size analysis.

The finite disagreem ent betw een theory and experim ent may be attributed to a number o f factors some o f w hich are described below:

I) A pplication o f slender beam s theory to springs. This requires the fulfilment o f a number o f assum ptions relating to the deform ation o f the spring during transverse vibration (see 6.2.2).

II) The validity o f the assum ption associated with the spring boundary conditions. Theory is only applicable to boundary conditions relating to either clam ped, hinged or free ends. In practice, each spring end may experience a com bination o f these effects.

III) The application o f an average value for the m ass correction factor. As it was shown, this is found to be a w eak function o f the

spring tension.

IV) Assuming uniform intercoil extension o f the spring in response to the applied tension. This introduces inaccuracies in the determ ination o f the spring static intercoil distance.

V) Errors associated with the experim ental determ inations o f spring characteristics including its stiffness, moment o f inertia and density (see the appendix).

VI) Errors associated with sieving. This is potentially im portant as the perform ance o f the system is evaluated by com parison with sieve data.

C H A PTER 7

Evaluation o f The System ’s Response in Conjunction With Irregular Particles

In chapter 4 it was shown that in the case o f irregular particles, a relatively poor degree o f agreem ent was obtained w hen com paring size analysis data produced using the present technique as com pared to those from sieves.

In the same chapter, it w as postulated that the m ost likely explanation was the difference betw een the orientation o f the particles as they exit the discharge aperture in each system.

The purpose o f this chapter is to first determ ine an appropriate conversion factor for correlating particle size data obtained from the current system to those from sieves and to determ ine, using electron m icroscopy, the m ost statistically appropriate particle dimensions m easured in conjunction w ith either technique.