Mathematical model of the operation of a weight batcher for dry products/Biriuju produktu svorinio dozatoriaus darbo matematinis modelis.

Bendoraitis, K. ; Paulauskas, L. ; Lebedys, A. 等

1. Introduction

Intensive development of packing technique encourages development of batchers [1-3]. It promotes to necessity increase a speed of dosing process and to necessity increase an accuracy of dosing [4-6]. Dry materials are widely popular materials which have dosed and packed. Sugar, salt, cereals are the dry materials of food products. They dosage and pack fertilizer, fastener, various component and other similar goods they materials have assigned as dry materials. Volumetric batchers are widely using for dosage the dry products [7]. Volumetric batchers usually have high speed of action. However their dosing precision is more dependent of the product's properties. In addition to volumetric batchers there are use weight batchers too [68]. The dosing precision of weight batchers is small dependent of product's properties. Using the weight batcher, you can choose a desirable proportion between speed of dosing and accuracy of dosing. However, the work cycle of weight batcher is more complex. In the paper we will theoretically investigate the work of weight batchers. The mathematical model describes the work of weight batchers for dry products. The sensitive element of weight batchers is regarded as elastic system with one degree of freedom. The body of variable mass is operating to the elastic system [8-11].

2. Structural scheme and physical model for batcher of weight

The weight batcher can achieve the high accuracy of dosing. However, the structure of weight batcher is complex. All components of the batcher affect the dosing accuracy [2, 3, 5, 6, 8]. The weight batcher investigation can begin from their typical structural scheme. It is typical structural scheme the batcher of weight (Fig. 1), where 1-elastic element; 2-shovel; 3-sensor; 4-control unit; 5-unloading mechanism; 6-feeder. The batcher of weight works as follows. Feeder 6 filed the product to the shovel 2, the weight of shovel 2 increases, so elastic element 1 distort, it affects the sensor 3, signal from the sensor 3 is transferred to the control unit 4, which disables the feeder 6, when the quantity of the product in shovel 2 is such as the asked the value. When the feeder 6 is stop, the control unit 4 starts the unloading mechanism 5 and the dose is loss. After a pause the batcher of weight is ready to work again. The batcher of weight combines several processes. These are vibratory transport, weight measurement, preparation, assessment and control technology. All these processes need to be investigated individually. It will also evaluate their interaction. Vibratory transport and vibratory processes are widely used [12-14]. The elastic systems are being payed a lot of attention [15, 16].

[FIGURE 1 OMITTED]

The physical model of dosing process regarded as elastic system with one degree of freedom (Fig. 2), where 1-elastic element with the elastic coefficient k; 2-rigid body with the mass [m.sub.1]; 3-damping element with the coefficient of damping c; 4-filed product with the mass [DELTA]m, 5-discard product with the mass [[DELTA].sub.1]m.

[FIGURE 2 OMITTED]

The working cycle of weight batcher consists of five stages. The first step is the fill with the increased productivity. The second is the fill with the low productivity. The third - the fill is suspended (pause before discharge). The fourth step is discharge of the product. The fifth - the discharge is suspended (pause after discharge). Each of these phases is associated with the characteristics of the product, with the accuracy of dosing and with the efficiency of dosing as well as with other parameters of this system. Such as dosing rate, the systems sensitivity and working range, sensor type, the influence of technological regimes, and others.

When the elastic element is the console with mounted mass M, it reduced mass of shovel [15]

[m.sub.1] = M + 33m'/40 (1)

where [m.sub.1] is reduced mass of shovel, M is mass of shovel, m' is mass of elastic element.

3. Mathematical model for describing the operation of weight batcher for dry products

This article deals with a movement of sensitive element at all five stages. As already mentioned, in this case we deal with the system with one degree of freedom, on this system to operate the body of variable mass. Mass of the dosing product in the shovel increases by linear law m = [[kappa].sub.1]t. Where t is time, [[kappa].sub.1] is constant coefficient (efficiency of feeder). The duration of operation the feeder depends on the dose size to be set up at this stage. Feeder working range in the first stage t [member of] [0 :[t.sub.1]], where [t.sub.1] = [[mu].sub.1]/[[kappa].sub.1] is duration of the feeder work required to form a dose [[mu].sub.1].

The formula of dynamics for body of variable mass is known [9]. This formula lets find the force which acts on the shovel in this case.

N = [[kappa].sub.1][u.sub.1] - [[kappa].sub.1]v - [[kappa].sub.1]gt - [[kappa].sub.1]dv/dt (2)

where [u.sub.1] is the absolute velocity of joining particle, v is the absolute velocity of shovel, dv/dt is the absolute acceleration of the shovel, g is the free fall acceleration. With this assess it is possible to write a differential equation, which describes movement of the shovel [9, 17].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

Eqs. (1) and (2) gives the equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

With the change of variables

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

Equation (4) gives the equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

where solution is expressed in the first kind [J.sub.n] and second kind [Y.sub.n] of Bessel functions [18-20].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where [C.sub.1] and [C.sub.2] are constants. Returning to the previous variables, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

Derivative of Eq. (8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

When the initial conditions of movement t = 0, x = e = [m.sub.1]g/k, [??] = 0 we get that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

It follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

According to [18-20], we obtain

[DELTA] = -1/[pi][square root of [b.sub.1][[tau].sub.1]] (16)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

Since [C.sub.1] = [[DELTA].sub.1]/[DELTA]; [C.sub.2] = [[DELTA].sub.2]/[DELTA], evaluated the equations (16)-(18) gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

The second step is the fill with the low productivity. Mass of the dosing product in the shovel increases by linear law m = [[kappa].sub.2]t. Where t is time, [[kappa].sub.2] is constant coefficient (efficiency of feeder). The duration of work of the feeder depends on the dose size to be set up at this stage. The feeder working range in the second stage t [member of] [0: [t.sub.2]], where [t.sub.2] = [[mu].sub.2]/[[kappa].sub.2]--duration of the feeder work required to form a dose [[mu].sub.2]. It is necessary assess the change mass of the shovel. At this stage reduced mass of the shovel [15]. [m.sub.2] = M + [[kappa].sub.1][t.sub.1] + 33/140 m' where [m.sub.2] is reduced mass of the shovel in the second step. In the second step the differential equation this describes the movement of the shovel.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

where [u.sub.2] is the absolute velocity of joining particle in the second step. The solution [18-20] of Eq. (21) is

From Eqs. (25), (27) and (26), (28) we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

where [J.sub.n] is the first kind and [Y.sub.n] is second kind of Bessel functions, [C.sub.1] and [C.sub.2] are constants. Derivative of Eq. (22) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

When t = 0 , from Eqs. (22), (24) we get that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

At the same time movement of the shovel is described by the equations (7), (10) at the first phase when t = [t.sub.1],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)

With the change of variables

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (30)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (31)

According to [18-20], we obtain

[DELTA] = -1/[pi][square root of [b.sub.2][[tau].sub.2]] (32)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (33)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (34)

Since [C.sub.1]' = [[DELTA].sub.1]/[DELTA]; [C.sup.2]' = [[DELTA].sub.2]/[DELTA], evaluating of the Eqs. (32)-(34) we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (35)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (36)

The third stage starts at the moment when the fill is suspended (pause before discharge). At this moment the shovel mass [mu] = [[kappa].sub.1][t.sub.1] + [[kappa].sub.2][t.sub.2].

Differential duration, which describes movement of the shovel at stage 3 [15] is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (37)

In step 3 t [member of][0: [t.sub.3]], where [t.sub.3] the pause time.

At initial shovel movement conditions t = 0 , x = [x.sub.2], [??] = [[??].sub.2]. According to [15], solution of Eq. (37), at [[omega].sub.0] > [??] shall be

x = g/k ([m.sub.1] + [mu]) + [Me.sup.-[??]t] sin ([omega]t + N) (38)

Derivative of Eq. (38)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (39)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (40)

and accordingly at [[omega].sub.0] < [??] Solution of same Eq. (37) shall be

x = ([m.sub.1] + [mu])g/k + [e.sup.-[??]t] ([M'e.sup.[omega]t] + [N'e.sup.-[omega]t]) (41) Derivative of Eq. (41)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (42)

In this case,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (43)

Dimensions [x.sub.2] and [[??].sub.2] are displacement and speed of the shovel at which ends of the filing of the second stage. These values are derived from Eqs. (27), (28) when t = [t.sub.2].

The fourth step is discharge of the product. Mass of the product being discharged from the shovel decreases at linear law m = -[[kappa].sub.3]t. Here t is time, [[kappa].sub.3] is constant coefficient representing the discharge capacity. The duration of discharge depends on the dose size, i.e. [mu] = [[mu].sub.1] + [[mu].sub.2].

Working range at stage 4 is t [member of] [0: [t.sub.4]], where [t.sub.4] = [mu]/[[kappa].sub.3] is the time needed to fully discharge the dose [mu].

In step 4 a differential equation, which describes movement of the shovel is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (44)

After replacement of variables

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (45)

in the Eq. (44) we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (46)

According to [18] the solution of Eq. (46) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (47)

where [J.sub.1+i] is the first kind and [Y.sub.1+i] is second kind of Bessel functions, [C.sub.1] and [C.sub.2] are constant, i = c/[[kappa].sub.3].

Returning to the previous variables and taking into account that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (48)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (49)

Derivative of Eq. (49)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (50)

[Z.sub.n](x) = [C.sub.1]" [J.sub.n](x) + [C.sub.2]" [Y.sub.n](x) (51)

When t = 0

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (52)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (53)

Shovel movement during third stage, expressed using (38) and (39) at t = [t.sub.3], and [[omega].sub.0] > [??]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (54)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (55)

Accordingly, at t = [t.sub.3] and [[omega].sub.0] < [??] Eqs. (41), (42) give

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (56)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (57)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (58)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (59)

In the fifth step the discharge stops (pause after discharge). Differential equation, which describes movement in this case [15]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (60)

If [OMEGA] = [square root of k/[m.sub.1]] , [THETA] = c/2m and [OMEGA] > [THETA] solution of the Eq. (60) according to [15] shall be

x = [m.sub.1]g/k + [Ae.sup.-[THETA]t] sin(pt + B) (61)

[??] = -A[THETA][e.sup.-[THETA]t] sin (pt + B) + [Ape.sup.-[THETA]t] cos (pt + B) (62)

where [p.sup.2] = [[OMEGA].sup.2] - [[THETA].sup.2], A and B is constants. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (63)

When [OMEGA] < [THETA] solution of Eq. (60) according to [15] shall be

x = [m.sub.1]g/k + [A'e.sup.(p-[THETA])T] + [B'e.sup.-(p+[THETA])T] (64)

where [p.sup.2] = [[THETA].sup.2] - [[OMEGA].sup.2], A' and B' is constant.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (65)

Dimensions [x.sub.4] and [[??].sub.4] are correspondingly the displacement and speed of the shovel at the end of step 4. These values are derived from Eqs. (49), (50) at [t.sub.4] = [mu]/[[kappa].sub.3]. The developed mathematical model allows for analysis and investigation of weight batchers with development stages. Interaction and influence of various system elements on dosing process thus can be evaluated making the batcher design predictable.

4. Conclusions

A typical structural scheme of weight batcher has been analyzed by using physical model comprising sensitive element regarded as elastic system with one degree of freedom with the attached body having variable mass.

Dynamic mathematical model of the batcher has been developed, which describes the batcher performance at its five key working stages. The working stages covered are: filling with the product at increased productivity, accurate low-rate top-up filling, filling suspension (pause before discharge), product discharge and discharge suspension (pause after discharge). Mathematical model allows for dosing process analysis of weight batchers considering influence of its key elements, their interaction and can be used as a working tool in various stages of bathers' development and design.

Received November 10, 2010

Accepted September 09, 2011

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K. Bendoraitis, Kaunas University of Technology, Karaliaus Mindaugo pr. 22, 44295 Kaunas, Lithuania, E-mail: [email protected]

L. Paulauskas, Kaunas University of Technology, Karaliaus Mindaugo pr. 22, 44295 Kaunas, Lithuania, E-mail: [email protected]

A. Lebedys, Kaunas University of Technology, Karaliaus Mindaugo pr. 22, 44295 Kaunas, Lithuania, E-mail: alis.lebedys@ktu

S. Paulauskas, Kaunas University of Technology, Karaliaus Mindaugo pr. 21, 44295 Kaunas, Lithuania, E-mail: [email protected]

E. Milcius, Kaunas University of Technology, Karaliaus Mindaugo pr. 21, 44295 Kaunas, Lithuania, E-mail: [email protected]