Читать книгу Fundamentals of designing hydraulic gear machines - Jarosław Stryczek - Страница 10

As Figure 2.1 has already presented, in the pump, mechanical energy E_mech delivered by the shaft from the motor is transformed into hydraulic energy of pressure E_hydr accumulated in the working fluid. Hence, following the energy conservation law:

E_mech = E_hydr

In the ideal pump, there is no energy loss, therefore:

M_gt · φ_g = V_gt · Δp_g

(2.1)

where:

M_gt – theoretical torque on the shaft of the pump (the torque of the generator),

φ_g – angle of the revolution of the shaft of the pump,

V_gt – theoretical volume of the working fluid displaced from the pump,

Δp_g = p_o - p_I – difference of pressure at the outlet p_o and the inlet p_I (suction and charging pressure difference).

After differentiating relative to time, Equation (2.1) takes the following form:

(2.2)

As a result, the following dependency is obtained:

M_gt · ω_g = Q_gt · Δp_g

(2.3)

where:

ω_g – angular velocity of the shaft of the pump,

Q_gt – flow generated by the pump, i.e. the theoretical displacement of the pump.

Theoretical displacement of the pump Q_gt regardless of the volumetric loss, is defined by the dependency:

Q_gt = q_g · n_g

(2.4)

where:

q_g – specific delivery understood as the maximum obtainable delivery of the working fluid, expressed in cm³, which is generated by the real pump after one revolution at the outlet pressure equal to the inlet pressure, namely Δp = p_o - p_I ≈ 0, i.e. with no volumetric loss,

n_g – rotational velocity of the shaft of the pump.

By implementing dependency (2.4) in the Formula (2.3), taking ω_g = 2πn_g into account, and modifying the formula, the following theoretical torque M_gt on the shaft of the pump formula is created:

(2.5)

The process of energy transformation in the pump is described by two characteristic quantities:

 – theoretical delivery Qgt of the pump (see Formula 2.4),

 – theoretical torque Mgt delivered onto the drive shaft of the pump (see Formula 2.5). In that case it is assumed that the shaft of the pump revolves at a constant speed ng, and in the working fluid delivered from the inlet to the outlet of the pump, an increase in the Δpg = po - pI is generated.

The process of energy transformation should be analysed for two kinds of the pump:

 – for the ideal pump featuring no energy loss,

 – for the real pump featuring energy loss.

Figure 2.4a presents the energy balance for the ideal pump. The figure shows that the streams representing theoretical delivery Q_gt and theoretical torque M_gt flow through the pump with no loss.

Fig. 2.4. Balance of the characteristic quantities of the pump.a) ideal pump, b) real pump.

What stems from Formula (2.4) is that theoretical delivery Q_gt of the ideal pump, assuming constant rotational velocity of the shaft n_g = const, is constant and does not depend on loading the pump with pressure Δp_g. Therefore, Figure 2.5a depicts characteristics of theoretical delivery Q_gt of the ideal pump depending on its load Δp_g, namely Q_gt = f (Δp_g). The characteristics is illustrated with the straight horizontal line. Formula (2.5) shows that theoretical torque M_gt, assuming constant specific delivery q_g changes in direct proportion to loading of the pump Δp_g. Hence, Figure 2.5b shows the characteristics of theoretical torque M_gt of the ideal pump depending on load Δp_g, namely M_gt = f (∆p_g). The characteristic is a straight line coming from the beginning of the coordinate system.

Fig. 2.5. Characteristics of the pump.a) theoretical delivery Q_gt and real delivery Q_g, b) theoretical torque M_gt and real torque M_g,c) volumetric efficiency η_vg, hydraulic – mechanical efficiency η_h–mg, total efficiency η_g

Figure 2.4b presents the energy balance for the real pump. The figure shows that the stream indicating theoretical delivery Q_gt is reduced by the value of volumetric loss ∆Q and, consequently, on the outlet of the pump, real delivery Q_g is generated. The volumetric loss depends on the leakage of the working fluid through the clearances G in the pump, formed between the rotating displacement chamber T and the fixed elements of the housing. Hence, Figure 2.5a depicts the characteristics of real delivery Q_g depending on the load of the pump ∆p_g, namely Q_g = f (∆p_g). The characteristics resembles the shape of a parabola. Between theoretical characteristics Q_gt and real characteristics Q_g, volumetric loss ∆Q_g is marked. Figure 2.4b shows that the stream indicating theoretical torque M_gt is increased by hydraulic-mechanical loss ∆M_g and, as a result, the shaft of the pump has to be loaded with real torque M_g. Hydraulic-mechanical loss ∆M_g torque results both from the resistance of the working fluid flow through the channels and the clearances in the pump, and the friction between those parts of the pump which remain in the relative motion during the operation of the pump. Taking that into consideration, Figure 2.5b presents the characteristic of real torque M_g depending on pressure ∆p_g working on the pump, namely M_g = f (∆p_g).

Characteristic M_g is still a straight line but it is shifted relative to the characteristic of theoretical torque M_gt by the value of torque loss ∆M_g.

The analysis of Figure 2.5a allows to observe the following dependency:

Q_g = Q_gt - ∆Q_g

(2.6)

The figure shows that volumetric loss ∆Q_g grows proportionally to the growth of pressure ∆p_g working on the pump. It results from the theory of flow through the clearances, which states that the flow through the clearances of the displacement chamber and volumetric loss ∆Q_g grow proportionally to the growth of the pressure difference ∆p inside and outside the displacement chamber.

By juxtaposing real delivery Q_g with theoretical delivery Q_gt, the volumetric efficiency of the pump can be defined as:

(2.7)

The analysis of Figure 2.5b allows to the determine the following dependency:

M_g = M_gt + ∆M_g

(2.8)

The figure shows that loss torque ∆M_g grows along with the growth of pressure ∆p_g in the pump. This occurs mainly due to an increase in the mechanical friction and an increase in the resistance of the working fluid flow in the channels of the pump.

The comparison of theoretical torque M_gt with real torque M_g enables the determination of the hydraulic-mechanical efficiency of the pump:

(2.9)

Considering the real pump as a machine which collects power from the motor and transfers it to the hydraulic system, it is necessary to determine the power balance.

Driving power N_g, which should be supplied from the motor, can be defined knowing real torque M_g, which ought to be applied on the shaft of the pump, as well as the angular velocity ω_g of the shaft:

N_g = M_g · ω_g

(2.10)

Effective power N_e, which can be utilized in a hydraulic system, is determined based on the knowledge of real delivery Q_g, and load ∆p_g of the pump, namely:

N_e = Q_g · ∆p_g

(2.11)

By comparing effective power N_e to driving power N_g, the total efficiency η_g of the pump can be defined as:

(2.12)

By implementing dependencies (2.10), (2.11) and then (2.7), (2.9) and (2.5) in the formula, the following equation is obtained:

η_g = η_vg · η_hmg

(2.13)

It means that the total efficiency of the pump is a product of the volumetric efficiency η_vg and the hydraulic-mechanical efficiency η_hmg.

The curves η_hmg of all the three, namely of efficiency η_vg, η_hmg, η_g relating to load ∆p_g of the pump, are presented in Figure 2.5c.