ELASTIC-DYNAMIC CHARACTERISTICS OF THE WORKING FLUID OF HYDRAULIC SYSTEMS CONTAINING UNDISSOLVED AIR

I. A. Nemirovskii, O. I. Vasenkov

(Presented by the scientific and technical seminar of the Department of Mining Machines and Mine Transport)

In the working fluid of volumetric hydraulic drives, as a rule, there is some amount of undissolved air [1, 2]. The presence of a gaseous admixture has a substantial effect on the bulk modulus of elasticity of the working fluid, which affects the dynamic characteristics of the hydraulic drive.

Formulas for the analytical calculation of the total bulk modulus of elasticity \(E_\Sigma\) of a gas–liquid mixture have gradually been refined (for example, [5], [6], etc.). The most acceptable formula is given in [6], but it is applicable to the case of an unchanged ratio

\[ \frac{W_{\mathrm{v}0}}{W_{\mathrm{l}0}} \]

(where \(W_{\mathrm{v}0}\) is the volume of the air phase at the initial pressure \(p_0\), and \(W_{\mathrm{l}0}\) is the volume of the liquid phase at \(p_0\)) and an isothermal process in the gas inclusions.

Below are presented the results of an analytical and experimental study of the elastic characteristics of a gas–liquid mixture, carried out by the authors in the hydraulic-automation laboratory of the Tomsk Polytechnic Institute. The dependences obtained make it possible to take into account the polytropic process of deformation of gas inclusions and the variable ratio

\[ \frac{W_{\mathrm{v}0}}{W_{\mathrm{l}0}} . \]

Let us consider two schemes of operation of a gas–liquid mixture in a hydraulic system.

The first scheme (Fig. 1, a) corresponds to the operation of a gas–liquid mixture in a closed rigid cavity, when the initial ratio

\[ \frac{W_{\mathrm{v}0}}{W_{\mathrm{l}0}} \]

remains unchanged (processes are considered during which dissolution of air in the liquid and liberation of air from solution practically do not occur).

According to the second scheme (Fig. 1, b), deformation of the gas–liquid mixture occurs due to a change \(\Delta W_{\mathrm{l}}\) in the initial volume of liquid \(W_{\mathrm{l}0}\), with \(\Delta W_{\mathrm{l}} = Q_1 - Q_2)\Delta t\).

The bulk modulus of elasticity of the gas–liquid mixture \(E_{\Sigma i}\) at pressure \(p_i\) is equal to

\[ E_{\Sigma i} = \frac{\Delta p\, W_{\Sigma i}}{\Delta W_{\Sigma i}} = \frac{\Delta p\left(W_{\mathrm{l}i}+W_{\mathrm{v}i}\right)} {\Delta W_{\mathrm{v}i}+\Delta W_{\mathrm{l}i}}, \tag{1} \]

where \(\Delta W_{B_i}\) and \(\Delta W_{\text{ж}_i}\) are the increments of the volumes of the air and liquid phases corresponding to the pressure increment \(\Delta p\). Let us determine the values of the quantities entering formula (1) for two operating schemes of the hydraulic-liquid mixture.

Fig. 1. Scheme of deformation of a gas-liquid mixture in a hydraulic system:
a) \(\dfrac{W_{B_0}}{W_{\text{ж}_0}}=\mathrm{const}\)—change in the total volume \(W_\Sigma\) due to a change in the cavity volume;
b) \(\dfrac{W_{B_0}}{W_{\text{ж}_0}}=\mathrm{var}\)—change in the ratio \(\dfrac{W_{B_0}}{W_{\text{ж}_0}}\) due to a change in the volume \(W_{\text{ж}_0}\) at constant cavity volume

First scheme (Fig. 1, a)

Let, at pressure \(p_i\), the volume \(W'_{\Sigma i}\) of the hydraulic-system cavity contain a volume \(W'_{\text{ж} i}\) of liquid and a volume \(W'_{B_i}\) of air, that is,
\(W'_{\Sigma i}=W'_{\text{ж} i}+W'_{B_i}\). Then the increment of the air volume
\(\Delta W_B=W'_{B_i}-W'_{B_{i+1}}\), caused by a change in pressure
\(\Delta p=p_{i+1}-p_i\), can be determined from the relation
\(p_i W_i^n=p_{i+1}W_{B_{i+1}}^n=(p_i+\Delta p)(W'_{B_i}-\Delta W_B)\), where \(n\) is the polytropic exponent.

After elementary transformations we obtain

\[ \Delta W_B=\frac{\Delta p\cdot W'_{B_i}}{n(p_i+\Delta p)}. \tag{2} \]

The increment of the liquid volume is equal to

\[ \Delta W_{\text{ж}}=\frac{\Delta p\cdot W'_{\text{ж} i}}{E_{\text{ж}}}, \tag{3} \]

where \(E_{\text{ж}}\) is the bulk modulus of elasticity of the liquid, which in the present work is assumed to be independent of pressure.

For a polytropic process the volume of the gas phase is equal to

\[ W'_{B_i}=W_{B0}\left(\frac{p_0}{p_i}\right)^{1/n}. \tag{4} \]

In turn,

\[ W'_{\text{ж} i}=W_{\text{ж}0}\left[1-\frac{p_i-p_0}{E_{\text{ж}}}\right], \tag{5} \]

From equations (1), (2), (3), (4), and (5) we obtain

\[ E_{\Sigma a_i}= \frac{ 1-\dfrac{p_i-p_0}{E_{\mathrm{liq}}} +\dfrac{W_{\mathrm{air}_0}}{W_{\mathrm{liq}_0}} \left(\dfrac{p_0}{p_i}\right)^{1/n} }{ \dfrac{1}{E_{\mathrm{liq}}} -\dfrac{p_i-p_0}{E_{\mathrm{liq}}^{2}} +\dfrac{W_{\mathrm{air}_0}}{W_{\mathrm{liq}_0}} \cdot \dfrac{p_0^{1/n}}{n p_i^{1/n}(p_i+\Delta p)} }. \tag{6} \]

Formula (6) is suitable for determining both the secant modulus and the tangent modulus of volumetric elasticity of a gas–liquid mixture. For calculations of dynamic processes in hydraulic drives, the tangent modulus is used. Then, according to the definition of the tangent modulus [4], we have

\[ \lim_{\Delta p\to 0} E_{\Sigma a_i}= \frac{ 1-\dfrac{p_i-p_0}{E_{\mathrm{liq}}} +\dfrac{W_{\mathrm{air}_0}}{W_{\mathrm{liq}_0}} \left(\dfrac{p_0}{p_i}\right)^{1/n} }{ \dfrac{1}{E_{\mathrm{liq}}} -\dfrac{p_i-p_0}{E_{\mathrm{liq}}^{2}} +\dfrac{W_{\mathrm{air}_0}}{W_{\mathrm{liq}_0}} \cdot \dfrac{p_0^{1/n}}{n p_i^{1/n+1}} }. \tag{7} \]

It should be noted that if one assumes

\[ \frac{p_i-p_0}{E_{\mathrm{liq}}}\to 0 \]

and \(n \simeq 1\), then expression (7) yields the formula given in [6].

Second scheme (Fig. 1, b)

The volumetric modulus of elasticity of the gas–liquid mixture, as in the first case, may be determined from expression (1), in which \(W_{\Sigma i}=W_{\mathrm{air}_0}+W_{\mathrm{liq}_0}=\mathrm{const}\); \(W_{\mathrm{air}_i}\) is calculated from expression (4). As for the current value of the volume of the liquid phase \(W_{\mathrm{liq}_i}\), with an unchanged cavity volume \(W_{\Sigma}\) (Fig. 1, b), the quantity \(W_{\mathrm{liq}_i}\) may be determined as

\[ W_{\mathrm{liq}_i} = W_{\mathrm{liq}_0}^{\prime} + \left(W_{\mathrm{air}_0}-W_{\mathrm{air}_i}\right) = W_{\mathrm{liq}_0} + W_{\mathrm{air}_0} \left[ 1-\left(\frac{p_0}{p_i}\right)^{1/n} \right]. \tag{5a} \]

From equations (1), (2), (3), (4), and (5a), taking \(\Delta p\to 0\), we obtain

\[ E_{\Sigma b_i}= \frac{ 1+\dfrac{W_{\mathrm{air}_0}}{W_{\mathrm{liq}_0}} }{ \dfrac{1}{E_{\mathrm{liq}}} + \dfrac{W_{\mathrm{air}_0}}{W_{\mathrm{liq}_0}} \cdot \dfrac{1}{E_{\mathrm{liq}}} \left[ 1-\left(\dfrac{p_0}{p_i}\right)^{1/n} \right] + \dfrac{W_{\mathrm{air}_0}}{W_{\mathrm{liq}_0}} \dfrac{p_0^{1/n}}{n p_i^{1/n+1}} }. \tag{8} \]

Figures 2a and 2b show the pressure dependences of the volumetric moduli of elasticity of the gas–liquid mixture \(E_{\Sigma a}\) (first scheme according to Fig. 1, a) and \(E_{\Sigma b}\) (second scheme according to Fig. 1, b). As is seen from the graph in Fig. 2, with an increase in the ratio

\[ \frac{W_{\mathrm{air}_0}}{W_{\mathrm{liq}_0}}, \]

the difference between \(E_{\Sigma a}\) and \(E_{\Sigma b}\) increases. The effect of the polytropic exponent \(n\) on the value of \(E_{\Sigma}\) is also substantial.

The authors carried out an experimental verification of formulas (7) and (8). In contrast to the experiment described in [6], the hydraulic-fluid mixture studied here was one in which the air was not distributed throughout the entire volume of the liquid in the form of macrobubbles, but was concentrated in one place. Such an arrangement of the air inclusion corresponds to real cases of air concentration in dead-end lines, cavities, and recesses of hydraulic systems.

In addition, in this case it is possible to take into account the actual amount of air, since the processes of dissolution and release of air proceed rather slowly. In the case, however, of the distribution of air bubbles throughout the entire volume of the liquid [6], the surface of contact of the air with the liquid is large, and within one cycle of the transient process a substantial change may occur in the ratio \(\dfrac{W_{\mathrm{v}_0}}{W_{\mathrm{l}_0}}\). The presence of mass exchange during one compression cycle of a gas–liquid mixture led the authors of [6] to the conclusion that under rapid compression \(n < 1\).

Fig. 2a. Dependence of the bulk modulus of elasticity of a gas–liquid mixture \(E_\Sigma(p)\) on pressure \(p\) at \(n=1\):

\[ 1-\frac{W_{\mathrm{v}_0}}{W_{\mathrm{l}_0}}=0.01;\qquad 2-\frac{W_{\mathrm{v}_0}}{W_{\mathrm{l}_0}}=0.10;\qquad 3-\frac{W_{\mathrm{v}_0}}{W_{\mathrm{l}_0}}=0.30. \]

Solid lines (deformation scheme \(a\)) in Fig. 1. Dashed lines (deformation scheme \(b\)) in Fig. 1.

In the experimental verification of formulas (7) and (8), both deformation schemes of the working fluid shown in Fig. 1 (a and b) were implemented. The rate of dissolution of air in the working fluid was investigated beforehand. Figure 3 gives the time dependences of the amount of air in the cavity of the experimental setup at a certain pressure and temperature and under various dynamic conditions. The experiments to verify formulas (7) and (8), as well as to determine the exponent \(n\), were carried out after preliminary saturation of the liquid with air, i.e., on that segment of the curves (Fig. 3) where, during one cycle of the periodic deformation process of the mixture, mass exchange between the gas and the liquid practically did not occur.

As a result of the investigations performed, the following conclusions can be drawn:

1. The bulk modulus of elasticity of a gas–liquid mixture depends on the deformation scheme. In particular, for the scheme in Fig. 1,a the calculation of \(E_{\Sigma a_i}\)

Fig. 2b. Dependence of the bulk modulus of elasticity of the gas–liquid mixture \(E_{\Sigma}\) on pressure \(p\) at \(n=1.4\):

\[ 1-\frac{W_{\mathrm{в}_0}}{W_{\mathrm{ж}_0}}=0.01;\qquad 2-\frac{W_{\mathrm{в}_0}}{W_{\mathrm{ж}_0}}=0.10;\qquad 3-\frac{W_{\mathrm{в}_0}}{W_{\mathrm{ж}_0}}=0.30. \]

Solid lines: deformation scheme \(a\) in Fig. 1. Dashed lines: deformation scheme \(b\) in Fig. 1.

Fig. 3. Dependence of the amount of undissolved air in the device on the holding time under pressure. Pressure \(p_{0}=42\) kg/cm\(^2\); \(t=19^\circ\)C; working fluid—oil V 3 in the as-delivered state; 1—static pressure; 2—pressure varying sinusoidally with frequency \(f=8\) Hz and amplitude \(\Delta p=1.4\) kg/cm\(^2\); 3—\(f=25\) Hz; \(\Delta p=1.4\) kg/cm\(^2\); 4—\(f=42\) Hz; \(\Delta p=1.4\) kg/cm\(^2\).

can be carried out by formula (7), and for the diagram in Fig. 1,b the calculation of \(E_{\Sigma a_i}\)—by formula (8).

1. With a dynamic change in pressure in the interval

\[ 0 < \frac{dp}{dt} \leqslant 300\ \text{kg}/\text{cm}^2\ \text{s}, \]

the exponent \(n \approx 1\). Refinement of the value of \(n\) for

\[ \frac{dp}{dt} > 300 \div 500\ \text{kg}/\text{cm}^2\ \text{s} \]

requires additional experiments.

1. For \(n \approx 1\), the amount of undissolved air contained in the working fluid of the hydraulic system can be determined from the formulas: for the diagram in Fig. 1,a

\[ W_{\mathrm{v}_0} = \frac{ W_{\mathrm{zh}_0} \left[ 1-\frac{E_{\Sigma a_i}}{E_{\mathrm{zh}}} \right] \cdot \left[ 1-\frac{p_i-p_0}{E_{\mathrm{zh}}} \right] }{ \frac{p_0}{p_i} \left[ \frac{E_{\Sigma a_i}}{p_i}-1 \right] }, \tag{9} \]

for the diagram in Fig. 1,b

\[ W_{\mathrm{v}_0} = \frac{ W_{\mathrm{zh}_0} \left( 1-\frac{E_{\Sigma b_i}}{E_{\mathrm{zh}}} \right) }{ \frac{E_{\Sigma b_i}}{E_{\mathrm{zh}}} \left( 1-\frac{p_0}{p_i} \right) + E_{\Sigma b_i}\frac{p_0}{p_i} -1 }. \tag{10} \]

LITERATURE

1. T. M. Bashta. Calculation and Design of Aircraft Hydraulic Systems. Mashgiz, Moscow, 1962.

2. D. Blackburn. Hydraulic and Pneumatic Power Control Systems. IL, Moscow, 1962.

3. V. N. Prokof’ev, I. A. Luzanova. Determination of the elasticity criterion of a hydraulic drive. Proceedings of institutes of higher education, “Mechanical Engineering,” No. 7, 1966.

4. Hatton. Fluids for Hydraulic Systems. Khimiya Publishing House, 1966.

5. Rendel D., Allen G. R. Air in Hydraulic Transmission systems. Aircraft Engng., V., XXIII, November, 1951.

6. Hayward A. I. I. How air bubbles affect the compressibility of hydraulic oil. Hydraulic, power transmission, 1962, vol. 8, No. 20.