Читать книгу Fundamentals of designing hydraulic gear machines - Jarosław Stryczek - Страница 11

As Figure 2.2 presents, in the motor hydraulic energy E_hydr accumulated in the working fluid is transformed into mechanical energy E_mech transferred onto the shaft of the working unit. It is a reverse transformation than the one observed in the pump. In an ideal motor, there is no energy loss, and based on the energy conservation law the following equation is created:

M_st · ω_s = Q_st · ∆p_s

(2.14)

where:

M_st – theoretical torque on the shaft of the motor (torque of the motor),

ω_s – angular velocity of the shaft of the motor,

Q_st – flow rate of the working fluid through the motor, namely the theoretical capacity of the motor,

Δp_s = p_I - p_o – pressure difference on the inlet and the outlet (supply and discharge pressure difference).

Theoretical capacity Q_st of the motor regardless of the volumetric loss is determined by the dependency:

Q_st = q_s n_s

(2.15)

where:

q_s – specific capacity of the motor, understood as the minimum volume of the working fluid expressed in cm³ which should be delivered to the motor in order the shaft to perform one revolution, at the supply pressure equal to the discharge pressure Δp_s = p_I - p_o = 0,

n_s – rotational velocity of the shaft of the motor.

After implementing dependency (2.15) in Formula (2.14), and taking into account that ω_s = 2πn_s, and after all the necessary transformations, the formula for calculating torque M_st on the shaft of the motor is as follows:

(2.16)

When analysing the process of energy transformation in the motor, it is necessary to consider two characteristic quantities:

– theoretical capacity Q_st of the motor (see Formula 2.15),

– theoretical torque M_st on the shaft of the motor (see Formula 2.16).

At the same time, it is assumed that the shaft revolves at the constant speed n_s, and in the working fluid flowing through the motor, a decrease in pressure Δp_s = p_I - p_o is observed.

The process of energy transformation should be analysed regarding two kinds of motors:

– an ideal motor without the energy loss,

– a real motor with energy loss.

Figure 2.6a presents the energy balance for the ideal motor. The figure depicts streams, standing for theoretical capacity Q_st and theoretical torque M_st, which flow through the motor without any loss.

Fig. 2.6. Balance of the characteristic quantities of the motor.a) ideal motor, b) real motor.

What results from Formula (2.15) is that theoretical capacity Q_st of the ideal motor, assuming the constant rotational velocity of the shaft (n_s = const), is constant and does not depend on the decrease in pressure δp_s. Figure 2.7a presents the characteristic of the theoretical capacity of the motor relating to the pressure decrease in the motor, in a form of a straight horizontal line. What results from Formula (2.16), however, is that theoretical torque M_st on the shaft of the motor, assuming constant capacity q_s, changes in direct proportion to the decrease in pressure Δp_s. The characteristic of theoretical torque relating to the decrease in the pressure is shown in Figure 2.7b in a form of a straight line coming from the beginning of coordinate system.

Figure 2.6b presents the energy balance for the real. The figure shows that the stream which indicates the real capacity Q_s of the motor is larger than the stream of theoretical capacity Q_st by the value of volumetric loss ΔQ_s. The volumetric loss are, similarly to the loss of in pumps, caused by the leakage through the clearances. Hence, Figure 2.7a shows the characteristic of real capacity Q_s depending on the load of the motor, namely Q_s = f (Δp_s). The characteristic is of a parabolic shape. Between real and theoretical characteristics the volumetric loss ΔQ_s is marked. Figure 2.6b shows also that the stream representing real torque M_s of the motor is smaller than theoretical torque M_st by the value of hydraulic–mechanical torque ΔM_s.

The hydraulic-mechanical loss torque, as in the case of pumps, is a consequence of resistance of the motor. Respectively, Figure 2.7b shows the characteristics of real torque M_s depending on the pressure which loads the motor (Δp_s), namely M_s = f (Δp_s). The characteristics is still similar to a straight line, however shifted in relation to the characteristics of the theoretical torque M_st by the value of loss ΔM_s.

Based on Figure 2.7a, the following dependency is derived:

Q_s = Q_st + ΔQ_s

(2.17)

The figure shows that in the motor, just as in the pump, the volumetric loss ΔQ_s increases along with an increase in the loading pressure Δp_s.

By comparing theoretical capacity Q_st to real capacity Q_s, volumetric efficiency η_vs of the motor can be defined as:

(2.18)

From the analysis of Figure 2.7b, the following dependency is derived:

M_s = M_st - ΔM_s

(2.19)

The figure shows that the characteristics is shifted in relation to the coordinate system origin by value Δp_{s min}. Hence, the motor will be able to start only if the pressure Δp_{s min} necessary to deal with the resistance of the working fluid flow through the channels, and the resistance of the motion of the moveable elements of the motor

Fig. 2.7. Characteristics of the motor.a) theoretical capacity Q_st and real capacity Q_s, b) theoretical torque M_st and real torque M_s,c) volumetric efficiency η_vs, hydraulic – mechanical efficiency η_h–ms, total efficiency η_s

is provided. The figure also shows that loss torque ΔM_s increases along with an increase in the pressure difference Δp_s.

By comparing real torque M_s to theoretical torque M_st of the motor, the hydraulic – mechanical efficiency of the motor can be expressed as:

(2.20)

Power is calculated for both the pump and for the motor.

The inlet power N_s delivered by the working fluid stream flowing into the motor is defined as a product of real capacity Q_s and pressure difference Δp_s, namely:

N_s = Q_s · Δp_s

(2.21)

The effective power N_e transferred into the system is defined as a product of torque M_s on the shaft of the motor and angular velocity ω_s, namely:

N_e = M_s · ω_s

(2.22)

By comparing effective power N_e to inlet power N_s, the total efficiency η_s of the motor is:

(2.23)

After transformations similar to the ones utilized for the pump, the following dependency is obtained:

η_s = η_vs · η_hms

(2.24)

It means that the total efficiency η_s of the motor is a product of volumetric efficiency η_vs and hydraulic – mechanical efficiency η_hms. Figure 2.7c presents the curves of the efficiency of the motor which are similar to the curves of the efficiency of the pump shown in Figure 2.5c.