Effects of the Wet Steam Non-Equilibrium Condensation on the Dynamics of the Excitation-Relying Bearing-Rotor system

This paper investigated the effects of the wet steam non-equilibrium condensation on the dynamic characteristics of the bearing as well as the bearing-rotor system by constructing and analyzing a non-linear coupled model of the bearing-rotor system. An excitation-relying dynamic model of bearing is established based on the finite difference method, in which the excitation is converted from the pressure pulsation on the surface of rotor blades generated from the non-equilibrium condensation process. The Raccia transfer matrix method is implemented to analyse the dynamic behavior of this coupled system. Results show that the wet steam non-equilibrium condensation process would greatly reduce the bearing stiffness and damping and result in more intense vibration of the system, besides, its induced pulsed displacement would drive the excitation-relying bearing-rotor system more unstable.


Introduction
The instability of the bearing-rotor system would induce serious accidents of turbines.Especially for the wet steam turbine such as nuclear turbine and the last stage of low pressure gas turbine, the nonequilibrium condensation of wet steam (NECS) not only causes the pressure pulsation on the blade surface, but also affects the stiffness and damping of bearing by loading torque to the bearing.Therefore, it is has great engineering significance to analyze the dynamics of the turbines bearingrotor system considering the effects of NECS.
Currently, in the design of bearing, only the gravity of the rotor is regarded as the bearing support reaction force to determine the stiffness and damping of bearing, a lot of studies on this subject used the computational fluid mechanics or the finite difference method have been carried out [1][2][3][4][5].However, the calculated bearing stiffness and damping using above approaches is insufficient to reflect the reality, which is the bearing stiffness and damping will change during the NECS process.The effects of the NECS on the dynamic characteristics of the bearing as well as the bearing-rotor system have not been clarified clearly.
The aim of this paper is to investigate such effects by constructing and analyzing a non-linear coupled model of the bearing-rotor system considering the effects of NECS.An excitation-relying dynamic model of bearing is established based on the finite difference method (FDM), in which the excitation is equivalent from the pressure pulsation on the surface of rotor blades generated from the NECS process.The Raccia transfer matrix method (RTMM) is applied to investigate the dynamic behavior of this coupled system under the NECS excitation.

Calculation models
Taking a low pressure turbine that contains ten cascades as an example, the excitation-relying bearingrotor system working in wet steam as shown in Figure 1.Inertial coordinate system (O-XYZ), bearingrotor system coordinates (o s -xyz) and blade coordinate system (o i -x i y i z i ) are defined to describe the coupled system.The differential equations of motion of the excitation-relying bearing-rotor system can be expressed as where M r , C r , K r is the mass, damping and stiffness of the rotor (include the shaft and the blade where, Φ=θ + φ is the opening angle of the oil film, θ is the angle between the bearing concentric line and the oil film force. The change of the oil film force (F o ) will result in the alter of the thickness h and the pressure p o of oil film, the motion of oil film is governed by the Reynolds equation [6] as follows 3 3 where, c and e are the radius gap and the eccentricity.μ and are the lubricant dynamic viscosity and the journal rotation angular velocity.The oil film force F o can then be calculated by iterately solving Eqs.( 3)-( 4) use the FDM until the balance, which is the oil film force F o is equal to the sum of the NECS excitation (F W (t)) and the rotor gravity (F G ) as given by For a certain turbine, the rotor gravity can be easyly calculated, however, the NECS excitation is hard to be obtained.The approach applied here is to equivalent the pressure (p) generated from the NECS on the blade surface into the torque (T) acting on the axis [6] (as shown in Fig. 1(c)), and then, convert the torque (T) into the NECS excitation (F W (t)) by introducing the torque factors (α x =3.5 and α y =1.42) [7][8], the detailed deduction given as follows.The torque acting on the axis can be expressed as the resultant force moment of the y i direction forces of each blade (Eq.( 6)) because the x i direction forces can be offset over the full cascade.

  cos (
) where, for the blade i, d i is the distance between the blade centroid and hub centroid (o i ).R d is the radius of the shaft.S i is the surface area of blade.β yi is the angle between the y i of the blade coordinates o i -x i y i z i and y of the bearing coordinates o s -xyz.n=10 is the number of cascade in the wet steam turbine.p i is the pressure on the blade surface which can be obtained from the numerical simulation of the NECS performed by ANSYS CFX based on the non-isothermal classical nucleation theory [9][10] and the droplet growth model [11].Then, the NECS excitation F W (t) can be calculated as gave by   Finally, the excitation-relying damping and stiffness of the bearing can be calculated by substituting Eq.( 7) into Eq.( 5) and combining with Eqs.( 2)-(4).Once the NECS excitation F W (t) as well as the bearing stiffness and damping K o (F W , t) and C o (F W , t) are determined, the dynamic characteristics of the coupled bearing-rotor system can be obtained by calculating Eq.(1) based use the RTMM.

Calculation results and discussion
Take the blade at the last stage as an example, the time-varying pressure on the blade surface at different blade spans during the NECS process is obtained as shown in Figure 2. It can be found that the pressure shows great unsteady characteristic in the middle and lower area of the blade (3%~50% span), this pulsation will gradually disappear at the blade tip (90% span).6)-( 7) as shown in Figure 3.It can be found that the NECS excitation in x direction is fluctuating between about 6.3×10 5 ~7.3×10 5 N, and the NECS excitation in y direction is fluctuating between about 4.5×10 5 ~5.2×10 5 N.  4. Besides, in order to estimate the effects of NECS, the steady bearing stiffness and damping that calculated without the NECS excitation (i.e.F W (t)=0 ) are shown in Table 1.The comparison between the bearing stiffness and damping calculated with and without the NECS excitation shows that most components of the excitation-relying damping and stiffness have great decrease (about 70%) except the k yy which is increased by about 30%.Besides, the decreasing amplitude of the bearing stiffness is larger than the bearing damping, which means that the former is more sensitive to the NECS excitation.
Once the excitation-relying damping and stiffness of the bearing are obtained, the dynamic motion Eq.( 1) can then be calculated by the RTMM.The axes contrails are drawn as shown in Figure 5 to show the stability of the bearing-rotor system.The comparison between Case a and Case b shows that the displacement of the excitation-relying bearing-rotor system is almost four times than the non-relying bearing-rotor system in which the effects of the NECS excitation on the bearing are ignored.Furthermore, axes contrails of Case a is pulsating with about 10-4m during the rotation of the system, in which the pulsation displacement in x direction (P x ) is slightly more than in y direction (Py).While axes contrails of Case b is steady during the rotation of the system.It can be concluded that when the effects of NECS excitation on the bearing are considered.The vibration response would greatly increase and the system has more tendency to instability.
In a sum, the NECS excitation would greatly reduce the bearing stiffness and damping and result in more intense vibration of the system.Besides, the pulsation displacement caused by NECS would drive the excitation-relying bearing-rotor system more unstable.This paper constructed a non-linear coupled model of the excitation-relying bearing-rotor system considering the non-equilibrium condensation of wet steam to analyze the effects of wet steam excitation.An excitation-relying dynamic model of bearing was established based on the FDM, in which the excitation is converted from the pressure on the surface of rotor blades generated from the NECS process.The RTMM is applied to investigate the dynamic behavior of this coupled system.The following conclusions are draw: (1) NECS could decrease the stiffiness and damping of the bearing.In the calculated case shown in this paper, mosts components of the bearing stiffiness and damping are decreased by about 70%.
(2) When the effects of NECS excitation on the bearing are considered, the vibration response would greatly increase and the system has more tendency to instability.In the calculated case shown in this paper, the vibration response of the excitation-relying bearing-rotor system is almost four times than the non-relying bearing-rotor system, and the pulsation displacement (about 10 -4 m) was generated in the former.

Figure 1 .
Figure 1.Dynamic model of the excitation-relying bearing-rotor system

Figure 2 .
Figure 2. Pressure on the blade surfaces at different spans The NECS excitation (F W (t)) can be obtained by substituting the pressure on the blade surface into Eqs.(6)-(7) as shown in Figure3.It can be found that the NECS excitation in x direction is fluctuating between about 6.3×10 5 ~7.3×10 5 N, and the NECS excitation in y direction is fluctuating between about 4.5×10 5 ~5.2×10 5 N.

Figure 3 .
Figure 3. Components of the NECS excitation in x and y direction Unsteady excitation-relying damping C o and stiffness K o of the bearing can be calculated use the FDM by substituting the NECS excitation F W (t) into Eq.(5) and combining with Eqs.(2)-(4).The eight components of the non-dimensional C o and K o are shown in Figure4.Besides, in order to estimate the effects of NECS, the steady bearing stiffness and damping that calculated without the NECS excitation (i.e.F W (t)=0 ) are shown in Table1.

Figure 4 .
Figure 4. Components of the non-dimensional excitation-relying damping and stiffness of bearing

Figure. 5
Figure.5 Axes contrails of the bearing-rotor system.Case a: with the NECS excitation, Case b: without the NECS excitation.
). F W (t) is the NECS excitation.C o and K o are the excitation-relying damping and stiffness of the bearing, which can be deduced from the bearing support reaction force (i.e. the oil film force F o ) as given by