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Optimal energy transmission analysis through rotating machinery

J. Alan Calderón Ch.γ,1,2, Julio C. Tafur1, Benjamín Barriga1

1 Engineering Department, Mechatronic Master Program,

Pontificia Universidad Católica del Perú

Lima, Perú

2 Applied Nanophysics, Institute for Physics

Technical University of Ilmenau

Ilmenau, Germany

γ. Corresponding author: alan.calderon@pucp.edu.pe

Abstract

Active Magnetic Bearings (AMB) are wide studied and applied nowadays, as for example in mining, petrol companies, power generator stations, etc. It is because this system improves efficiency in energy transmission as the consequence of controlled magnetic force over the shaft, which joins the source of the mechanical energy with the rotating machine (turbine, compressor, pump). Notwithstanding, shafts transmit not only rotating speed, also torque and quite advances of industry forced that shaft rotates at high speed values. Therefore, sensors and actuators must to be faster than this, but whether AMB cannot get an optimal position control, the electrical current, which produces the magnetic field to achieve the controlled magnetic force, will provoke heat through the wires that contain it. For this reason, in this work is analyzed the impact of sensors and actuators that were based in nanostructures in order to get faster response time and robustness, while AMB system can find its desired position control. Furthermore, while the heat is reduced also from their magnets, the total efficiency can be transmitted in better percent than AMB without faster and robust sensors.

Keywords: Rotating machinery, bearings, Active Magnetic Bearings, heat transmission, nanostructures.

Introduction

Rotating machinery are frequently applied to transmit movement and energy, as it was given in electrical energy production, such as in intricate geographical areas, where there are fast flowing rivers (as for example, in Andes mountains rivers). Therefore, it is necessary energy conversion and transmission from mechanical energy to electrical energy by specific systems, such as turbines and electrical transformers. By other side, also there are applications, in which is very important to use rotating machines, because of mechanical movement transmission: mining, agriculture, fishing and every economic activity, which needs mechanical movement transmission. Nevertheless, while there is not good energy transmission in systems as it was described above, it will be necessary to use some mechanisms to reduce the produced heat in mechanical movement conveying that generally pollutes the environment.

To prepare this research, it was used a VARIAC (trademark for Variable Autotransformer) to get speed control in the rotor system that is composed by an alternating control (AC) engine that was coupled with a rotor, for which it is supported by an Active Magnetic Bearing (AMB) control. Also, owing to control algorithm strategies, it was studied some consequences (advantages and disadvantages) from nanostructures over the main control system, while energy transmission process goes through the other mechanical systems that is joined with the rotor [1], [2], [5].

Furthermore, it was necessary to obtain a general mathematical model to describe energy transmission through the proposed system, in which the nonlinear model gave a good solution. The consequence effect to join sensors and actuators that were based in nanostructures support is the energy balance in all the mechanical system. It is because nanostructures robustness and fast response, which are the integration from nanosystem to macro system generate another good consequence: increase the efficiency of mechanical systems, because of reducing energy losses (heat) and in this context there would not be necessary so intense refrigeration complements [2], [3], [4], [5].

Problem and proposed methodological solution

In Figure 1 is described a general system, such as the mechanical movement source, which is the motor (depicted by M1); the machine that receives the movement could be a compressor, turbine, pump, etc. (depicted by M3). Both situations joined by a rotor (depicted by M2), for which represents the axis connection that cross 3 blocks. Furthermore, this figure represents Dynamic Forces Analysis over the rotor. Therefore, its mechanical physical movement can be described by Second Newton Law.

Figure 1

General rotor (shaft) scheme under the equilibrium of forces

Source: Own elaboration.

It means the equation 1, in which M2 is mass of the rotor, Fg is its gravity force, FR represents Reactions Forces and Fc is the inertial effect, because of circumferential movement around axis [5], that generalizes information of forces around the system that is composed complexly way joining M1, M2, M3 and summarizes dynamic analysis over the shaft.

(1)

For which, it is necessary to remember the y, which is the selected coordinate to study the movement of the rotor. Also, every equation is in matrix analysis that means every solution in order to find parameters and coefficients that are in matrix solution too, as it is described in equations 2 and 3. The matrix that is composed for every stiffness coefficient is shown through equation three.

(2)

(3)

By the other hand also, it is depicted the energy transmission in Figure 2, for which by the conservation of energy can be described mechanical movement transmission as in equation 1, as it was described above. Therefore, in Figure 2 are represented the same blocks that do not need specific analysis of force to describe movement transmission. It means to correlate specific coefficients, such as the friction (as the function of heat) is explained better through the energy model, as it is given in equations 4 and 5. Notwithstanding, it is possible to verify the energetic model that can be achieved through a dynamic model, because of Lagrange.

Figure 2

General rotor (shaft) scheme under the energy transmission

Source: Own elaboration.

As a consequence, equation 4 describes energy balance over the shaft, for that V is its speed, and Zf is the imbalance coefficient, due to energy transmission has losses.

(4)

(5)

Otherwise, by Lagrange is obtained equation 5, in which Fi is the coefficient that helps to get losses approximation from the shaft. However, both equations that were described above are given for a component of its matrix form. Therefore, in this work it is proposed a high modulation from sensors to actuators, it means in Figure 3 is represented the effect to improve a mechanic system by nanostructures. It means in Figure 3a is depicted a simple energy transmission among the mechanical systems, for which also must to be given lost energy, such as by the heat. How to avoid this? It could be reduced through many mechanisms of evacuating energy. However, it makes systems to be expensive without total reduction of the heat, because of the natural thermodynamic behavior of systems.

Figure 3

General rotor (shaft) scheme under the energy transmission and control

Source: Own elaboration.

Notwithstanding, while it is controlled by intelligent algorithms through sensors and actuators, the energy transmission enhances, but that transmission gets costing of the produced heat even AMB used over the rotor. It is because the energy transmitted through wires that produce electromagnetic field. Therefore, by faster sensors/actuators as it was proposed from sensors that were based in nanostructures (as in this research), the heat is reduced and movement transmission between rotating machines can get better efficiency. For this reason, it was necessary to quantify the heat that was produced over wires by Joule effect and the evacuated heat as the dependence of geometrical parameters and temperatures, as it was depicted in following equations. For this, Q is the heat evacuation due to friction, T is the temperature, K is the conductivity coefficient, A is cross sectional area to the heat flow, U is the excitation signal and Kp is the proportional gain of the thermal model.

(6)

(7)

For which, the proposal solution is given by equation 7. It means that for steady state:

(8)

(9)

And from which temperatures matrix:

(10)

By other side, it can be improved by control systems. Nevertheless, this is not totally good, due to in sophisticated systems that need fast and robust response, the controller (even so sophisticated) could be that it cannot reduce in total the heat transmission (it is depicted by Figure 3b). Therefore, what to do? It could be enhanced through faster sensors and actuators, as it is depicted in Figure 3c, it because sensors/actuators that were based in nanostructures.

The physical parameters identification and error analysis to get position control are given by a general system identification, as it is shown in equation 11, for a general expression of the rotor dynamic, in which I is the matrix of electrical current values for every component i. Furthermore, the stiffness coefficient Ky and electrical coefficient KIL.

(11)

Such as it was described above, every component of last equation has matrix form. Also the polynomial solution gets the error as the dependence of desired signal in order to identify M2, Ky and KIL that are matrices as the equations 2 and 3, which were described in paragraphs above.

Equation 12 helps to evaluate the error matrix as the dependence of desired signal and the measured signal S in every instance n.

(12)

(13)

For every S is composed by adaptive weight coefficients that was correlated with the input signal (as measured or expected/simulated) X as it was described in equation 13. For this reason, the matrix of error is defined as:

(14)

By other side, the general response y(t) correlated with x(t) and u(t) through a nonlinear function , it is because to look for the optimal trajectory and to get the best position control, as it is represented in equation 15.

(15)

(16)

It means to solve the costing equation 16 by J, the expected trajectory R, in which this expected position is given by equation 17. And the optimal excitation signal in order to find the optimal response is given by equation 18.

(17)

(18)

Anodic Aluminum Oxide (AAO) membranes have quite good mechanical, optical and chemical properties, hence, it helps to study designed sensors that were based in them. By other side, AAO membranes have fast response and robustness that means sensors that were based in this kind of membranes, which can achieve these characteristics. In Figure 4 is shown an AAO membrane and, as indication of yellow raw, it is amplified in nanoscale view (from SEM PUCP) its porous [7].

Figure 4

The position sensor that was based in AAO membranes

Source: Own elaboration.

In Figure 5 is depicted the setup to measure positions of the Active Magnetic Bearing (AMB), due to get its control, which is the machine used to evaluate the algorithms that were proposed in this research.

Figure 5

Experimental setup

Source: Own elaboration.