Design peculiarities and mathematical model of an enhanced low-frequency vibratory cavitation device

2.1. Design peculiarities of an enhanced low-frequency vibratory cavitation device

Most of the existent vibratory cavitation devices are restricted to operate at the temperatures higher than 70 °C and pressure of over 0.1 MPa [5, 10, 11, 16, 17]. In Fig. 1, there is presented the design diagram of the enhanced low-frequency vibratory cavitation device equipped with the electromagnetic drive for purifying liquids at the pressure of 0.35…0.4 MPa and temperature over 75…80 °C. The main unit of the proposed cavitator is the processing (purification) chamber 8 with two symmetrical brackets (flanges) 6, to which two cylindrical cases 3 equipped with the inlet pipes 1 for supplying the cooling liquid are connected. Inside each case 3, there are installed the stators with coils and the armatures of the electromagnetic drives 4. The drives are sealed from the working chamber with the help of the rubber rings (gaskets) 5 and sealing (pressurizing) casings (housings) 7 designed with central holes and glands for moving the armature rods of the electromagnets 4. The cylindrical cases 3 are sealed using the gaskets placed between the cases 3 and the corresponding brackets (flanges) 6. The armature rod is fixed to the disk-type spring element (rubber membrane) at its central hole. The membrane allows the armature rod to oscillate at the amplitude of 3.5…4 mm. The initial distance between the armature and the core of the electromagnet can be regulated by the number of washers (spacers) installed between the armature body and the disk-type spring element (rubber membrane). On the ends of the armature rods, there are fixed the concaves (grates) 9 for generating cavitation. The flat or conical surface of a concave is made with a system of holes ensuring the liquid flows through. The working chamber is equipped with the pipes 10 and 11 for supplying and disposal of the processing gases (nitrogen, air, etc.) and liquid to be treated. Considering the safety regulations, the cavitator is covered by the protective housings 2, which are fixed in the supports 12 with the help of clamps. The supports are mounted on the base through the rubber dampers. In addition, the working chamber 8 is equipped with the peephole (observing window) 13 bolted to the chamber through the gasket 14.

The best performance of the vibratory cavitation-purification process is observed at the near-resonance regimes when the forced frequency of the electromagnetic exciter is close to the natural frequency of the cavitator’s oscillatory systems under the predefined operational conditions (liquid type, pressure, and temperature). These regimes can be experimentally tested by adjusting the forced frequency and the oscillation amplitude of the armature rods in a wide range with the help of a special control system. The system allows for adjusting the excitation conditions in accordance with technological requirements and with the aim of providing the best performance of the cavitation-purification process or minimal power consumption of the electromagnetic drive.

Fig. 1Design diagram of an enhanced low-frequency vibratory cavitation device: 1 – inlet and outlet pipes for cooling liquid of the electromagnets; 2 – protective housings; 3 – cylindrical cases; 4 – electromagnetic drives; 5 – rubber rings (gaskets); 6 – brackets (supports, flanges) for the electromagnets fixation; 7 – sealing (pressurizing) casings (housings) with glands; 8 – processing (purification) chamber; 9 – concaves (grates); 10, 11 – pipes for supplying and disposal of processing gases and liquid to be treated; 12 – clamps with supports; 13 – peephole; 14 – gasket

2.2. Dynamic diagram and mathematical model of the cavitator’s oscillatory system

The dynamic diagram of the cavitator’s oscillatory system is presented in Fig. 2. The external moving body (mass $m_1$) simulates the cavitator suspended from the stationary base by the system of spring-damper elements characterized by the stiffness and damping coefficients $c_1$, $k_1$, respectively. The internal oscillating bodies (masses $m_2$, $m_3$) are set into motion due to the action of the forces $F_1$ and $F_2$ applied between the corresponding masses and the cavitator’s body (see Fig. 2). The internal bodies can slide along the horizontal longitudinal axis of the cavitator and their motion is restricted by the spring-damper elements characterized by the stiffness and damping coefficients $c_2$, $k_2$, respectively. In order to uniquely describe the motion conditions of the cavitator’s oscillatory system, the following generalized coordinated are adopted: $x_1$, $x_2$, $x_3$– absolute displacements of the cavitator’s body (mass $m_1$) and internal masses ($m_2$, $m_3$) in the inertial coordinate system (relative to the stationary base, see Fig. 2).

While performing further investigations, the pull-push forces of the electromagnets are considered as periodic ones: $F_1\left(t\right)=F_{1\mathrm{m}\mathrm{a}\mathrm{x}}\sin \left({\omega }_{1}t+{\phi }_1\right)$, $F_2\left(t\right)=F_{2\mathrm{m}\mathrm{a}\mathrm{x}}\sin \left({\omega }_{2}t+{\phi }_2\right)$, where $F_{1\mathrm{m}\mathrm{a}\mathrm{x}}$, $F_{2\mathrm{m}\mathrm{a}\mathrm{x}}$ are the maximal (amplitude) values of the excitation forces; ${\omega }_1$, ${\omega }_2$ are the forced frequencies; ${\phi }_1$, ${\phi }_2$ are the initial phases. Neglecting the complicated hydrodynamic processes occurring during the cavitation processes, let us consider the simplified case when the damping (viscous friction) forces are approximately modeled as functions of the cavitator’s body absolute speed and concaves (grates) relative speeds: $F_{d1}\left(t\right)=c_1{\dot{x}}_1$, $F_{d2}\left(t\right)=c_2{\left({\dot{x}}_1-{\dot{x}}_2\right)}^{{\alpha }_2}$, $F_{d3}\left(t\right)=c_2{\left({\dot{x}}_1-{\dot{x}}_3\right)}^{{\alpha }_2}$, where ${\alpha }_2$, ${\alpha }_2$ are the correction coefficients depending on the liquid to be purified. According to Hooke’s law, the spring forces are considered as functions of the cavitator’s body absolute displacement and concaves relative displacements: $F_{s1}\left(t\right)=k_1{x}_1$, $F_{s2}\left(t\right)=k_2\left(x_1-x_2\right)$, $F_{d3}\left(t\right)=k_2\left(x_1-x_3\right)$. Therefore, the simplified mathematical model describing the motion conditions of the cavitator’s oscillatory system can be presented as follows:

Fig. 2Dynamic diagram of the cavitator’s oscillatory system

3.1. Studying the influence of the excitation frequency on the system’s dynamic behavior

Numerical modeling of the cavitator’s oscillatory system motion conditions is performed in the Mathematica software using the Runge-Kutta methods and considering the following inertial, stiffness, damping, and excitation parameters [10, 11, 16, 17]: $m_1=$ 29.5 kg, $m_2=m_3=$ 0.85 kg, $c_1=$ 50 (N∙s)/m, $c_2=$600 (N∙s)/m, $k_1=$1.9∙10⁴ N/m (considering the natural frequency of the corresponding single-mass system of about 4 Hz), $k_2=$8.8∙10⁴ N/m (considering the natural frequency of the corresponding single-mass system of about 52 Hz), $F_{1\mathrm{m}\mathrm{a}\mathrm{x}}=F_{2\mathrm{m}\mathrm{a}\mathrm{x}}=$ 250 N, ${\alpha }_1=1$, ${\alpha }_2=1$. The forced frequencies ${\omega }_1$, ${\omega }_2$ and initial phases ${\phi }_1$, ${\phi }_2$ of the electromagnetic excitation are considered as controllable parameters. The following cases are studied in the present paper: ${\omega }_1={\omega }_2=$314, 628 rad/s (50, 100 Hz); ${\phi }_1=$0; ${\phi }_2=$ 0, $\pi$ rad. The corresponding results of the numerical modeling are presented in Fig. 3 in the form of time dependencies of the cavitator’s body and armature rods displacements. The amplitudes of vibrations ($x_2\left(t\right)$, $x_3\left(t\right)$) decrease from about 2.8 mm to 1.2 mm with an increase in the forced frequency from 50 Hz to 100 Hz. Therefore, the proposed low-frequency operational conditions of an enhanced device allow for generating larger displacements of the concaves (grates) and for intensifying the cavitation and purification process. In order to provide the same amplitudes of concaves at larger forced frequencies, it is necessary to equip the device with a more powerful electromagnetic drive reducing the machine’s efficiency due to larger power consumption.

In order to reduce vibrations that can be transmitted from the oscillating parts of the cavitator to the supports and nearby equipment, the armature rods of the actuating electromagnets should be subjected to synchronous antiphase oscillations (${\omega }_1={\omega }_2$, ${\phi }_2=$$\pi$ rad). This can be achieved by the corresponding connection of the stators’ coils of the electromagnets to the power supply and control system. Herewith, the reactive forces generated due to oscillatory movements of both armatures, being synchronous and oppositely directed, mutually cancel each other and minimize the transmission of vibrations to the machine’s base.

Fig. 3Time response curves of the cavitator’s body and concaves (grates) oscillations at different operational conditions

a)${\omega }_1={\omega }_2=$ 314 rad/s (50 Hz), ${\phi }_2=$ 0 rad

c)${\omega }_1={\omega }_2=$ 314 rad/s (50 Hz), ${\phi }_2=\pi$ rad

b)${\omega }_1={\omega }_2=$ 628 rad/s (100 Hz), ${\phi }_2=$ 0 rad

d)${\omega }_1={\omega }_2=$ 628 rad/s (100 Hz), ${\phi }_2=\pi$ rad

3.2. Analyzing the influence of the viscous friction on the system’s vibrations

In order to analyze the cavitator’s operation while purifying different liquids, the influence of the viscous friction (damping) coefficient on the concaves (grates) motion conditions is analyzed. The input parameters are the same as considered above. The forced frequency is set to ${\omega }_1={\omega }_2=$ 314 rad/s (50 Hz), and the phase shift angle ${\phi }_2=\pi$ rad. The numerical modeling is performed at the following values of $c_2=$100, 300, 600, 900, 1200 (N∙s)/m. The decrease in the $c_2$ from 1200 to 100 (N∙s)/m causes a rapid increase in the concaves (grates) amplitude. Therefore, in order to ensure the safe and reliable operation of the cavitator, its control system should provide the possibility of regulating the forced frequency and the voltage supplied to the electromagnets.

Fig. 4Time response curves of the concaves (grates) oscillations at different viscous friction (damping) conditions