Mathematic modeling of hydraulic systems with pneumohydraulic accumulators.

Prodan, Dan ; Gheorghiu, Horia ; Bucuresteanu, Anca 等

Abstract: The present describes the mathematical models necessary for the calculus and the simulation of hydraulic systems that use accumulators. The described mathematical models allow the obtaining of the optimum accumulator for the most used hydraulic plants. One can determine the time interval in which the accumulator can be the only hydraulic energy source from the plant.

Key words: modeling, hydraulics systems, pneumohydraulic accumulators.

1. INTRODUCTION

Accumulators are hydraulic components, which allow receiving, keeping and transmitting the hydraulic energy like volumes of under pressure liquid. Because of the low degree of compressibility of the liquids it is difficult to keep the energy in small volumes, but they allow the transmitting of high efforts. Compared to liquids, gases have another possibilities regarding compressibility, which means keeping a higher energy, in small volumes. Combining liquids and gases, in special constructions, have lead to pneumohydraulic accumulators (Bucuresteanu, 2001).

2. MATHEMATICAL MODELS

We will consider the sketch from Figure 1. Accumulator Ac works between pressures [p.sub.min] and [p.sub.max]. The effective volume is V0 and the initial charging with nitrogen is done al initial pressure p0 (Bucuresteanu, 2001). Usually this pressure verifies the relations:

[p.sub.0] = k x [p.sub.min] (1)

k [member of][0.6/0.9] (2)

The accumulator is charged with oil through sense vent Ss, in the transitory phase, [p.sub.min]. [p.sub.max]. The discharge is made through throttle Dr in the corresponding phases to the pressure lowering, [p.sub.max] [right arrow] [p.sub.min].

[FIGURE 1 OMITTED]

If the oil charging and discharging of the accumulator is done in isothermal conditions (T=ct.), we can write:

[p.sub.0] x [V.sub.0] = [p.sub.max] x [V.sub.min] = [p.sub.min] x [V.sub.max] (3)

In relation (3) we marked: [V.sub.min]--minimal volume occupied by nitrogen, [V.sub.max]--maximal volume occupied by nitrogen.

In the discharging phase the maximal oil volume available is:

[DELTA]V = [V.sub.max] - [V.sub.min] = [p.sub.0] x [V.sub.0]. (1/[p.sub.min] - 1/[p.sub.max])(4)

This volume is being discharged through throttle Dr the throttle characteristic is (Prodan et al., 2005):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

In relation (3) we marked: [Q.sub.D]r--throttle flow, [C.sub.Dr]--throttle constant, [S.sub.Dr]--effective surface of the throttle, [rho]--density of the used oil, [DELTA]p--line drop of the throttle.

If we consider the accumulator in the discharging phase, like in Figure 2, there are two cases:

a. Discharging towards a circuit with negligible pressure, [p.sub.s]=0;

b. Discharging towards a circuit with pressure [p.sub.s], value different from negligible

From the relations above we obtain:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

If we consider the discharge diagram from Figure 2.a., after the integration of the relation (6) we get:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

For the discharge diagram from Figure 2.b., after the integration of the relation (6) we get:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

For determining the time in which the accumulator can be an energy source of the system, is necessary to calculate the integrals for relations (7) and (8). For the first case (Prodan et al., 2000) and for de second (Prodan, 2004) we obtain:

[FIGURE 2 OMITTED]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

where [alpha] is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

The time, for each of the two cases, can also be determined with the help of the simulation programs (Prodan, 2006). After the simulation, for the first case we obtain, for effective flow and nitrogen pressure from the accumulator, the characteristics in Figure 3 and 4. The pressure drops from 80 to 40 bar in approximately 45 s. The flow drops in this time period from 2.3 l/min until approximately 1.7 l/min, after a hyperbola and afterwards, when touching minimum pressure it becomes null. For discharging at a pressure [p.sub.s] =25 bar we will get the flow and pressure characteristics from Figure 5 and 6. This time the dropping is much easier, lasting 65 s. The flow drops from 1.9 l/min to 1.2 l/min hyperbolic, after that it drops suddenly to zero.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

3. EXPERIMENTAL RESEARCHES

For the experimental researches it has been used a mechanical--hydraulic system shown in Figure 7. Tool-holder 1 is clamped on coulisse 2 by disc springs 3 from linear hydraulic motor 4. For a more reliable clamping we insert pressure from B.

For releasing the tool-holder we insert oil under pressure through A and B chamber is tightened to a receiver. The pressure source contains a accumulator (Prodan, 2004) that ensures the losing in both phases, tightening/releasing.

4. CONCLUSION

The accumulators will be dimensioned according to each particular case, in this way assuring the necessary flow and pressure until the plant is stopped without any risks. For the accumulators calculus it is recommend to know all the characteristics of the system components and to create the mathematic models necessary for the simulation. The charging of the circuits by the accumulator can be done depending on the commands received from the relays or pressure transducers.

5. REFERENCES

Bucuresteanu, A. (2001). Acumulatoare pneumohidraulice (Pneumohydraulic accumulators), Printech Publishing House, ISBN973-652-292-X, Bucharest

Prodan, D., Dobrescu, T. & Bucuresteanu, A. (2000). Sisteme hidraulice pentru: blocare, strangere si desfacere (Hidraulic systems for blocking, tightening and unblocking). Hidraulica, No. 2/2000, pag. 13-18, ISSN/1453-7303, 2000, Bucharest

Prodan, D. (2004). Hidraulica masinilor-unelte (Machine-tools hydraulics), Printech Publishing House, ISBN973-718-1093, Bucharest

Prodan, D., Duca, M, & Bucuresteanu, A. (2005). Actionari hidrostatice-organologie (Hydrostatic-organological actuators), AGIR Publishing House, ISBN973-720-011-X 2005, Bucharest

Prodan, D. (2006). Masini-Unelte. Modelarea si simularea elementelor si sistemelor hidrostatice (Machine-tools. Elements and hydrostatical systems modelling and simulation), Printech Publishing House, ISBN 973-718572-2, Bucharest