Hydrodynamic Lubrication 2009 Part 11 pot

8.5 Temperature Analyses of Tilting Pad Thrust Bearings — Sector Pads 173 [11]), a two-dimensional analysis of a sector pad taking into account the effect of centrifugal force on the lubricating oil (Pinkus and Lund [15]), a three-dimensional analysis of a fixed inclined rectangular pad bearing (Ezzat and Rhode [9]), and a three-dimensional analysis of a tilting sector-pad bearing taking into account only the inclination in pitch mode (Tieu [10]). Fig. 8.2. Sector-type tilting pad bearing [20] 8.5.1 Basic Equations A cylindrical coordinate system (r,θ,z) is considered with the origin at the center of rotation of the disk (see Fig. 8.2). Let the position of the pivot support be (r p ,θ p ). In the direction of pad thickness, a coordinate z 2 with the origin on the lubricating surface of the pad is used. Besides the usual assumptions of lubrication theory, the following assumptions are made: 1. The lubricating surfaces of the pad and the disk are rigid, and their thermal and elastic distortions are disregarded. 2. The velocity gradient and heat conduction in the r and θ directions can be ignored in comparison with those in the z direction. 3. The specific heat at constant pressure c p and the coefficient of thermal expansion α  of the lubricating oil are constant. 174 8 Heat Generation and Temperature Rise 4. The thermal conductivity of the lubricating oil k o and the thermal conductivity of the pad k s are constant. 5. The coefficient of viscosity µ and the density ρ of the lubricating oil are functions of temperature T only. The basic equations in this case are stated below. Let the state when the lubricating surfaces of the pad and the disk are parallel to each other and their separation is h 0R be the standard state. Assume that the pad rotates around the x axis by α p (pitching angle) and around the y axis by α r (rolling angle) from the standard state. Then the gap h between the pad and the disk (film thickness) is given by the following equation, assuming a small inclination of the pad, α p  1 and α r  1: h(r,θ) = (R + ∆R −r cos θ)α r + α p r sin θ + h 0R (8.51) Inclinations of the pad α p and α r are actually automatically determined by the bal- ance of the oil film force. The minimum film thickness h 0 appears at (R+∆R, 0) when α r is positive and at (R, 0) when α r is negative. The flow velocity of the lubricating oil in the radial and the circumferential di- rections is:  r =   z 0 z µ dz − F 1 F 0  z 0 dz µ  ∂p ∂r (8.52)  θ = −rω  1 − 1 F 0  z 0 dz µ  +   z 0 z µ dz − F 1 F 0  z 0 dz µ  1 r ∂p ∂θ (8.53) where F 0 =  h 0 dz µ , F 1 =  h 0 zd z µ The generalized Reynolds’ equation is written as: ∂ ∂r  F 2 ∂p ∂r  + 1 r 2 ∂ ∂θ  F 2 ∂p ∂θ  + F 2 r ∂p ∂r = ω ∂ ∂θ  F 4 − F 3 F 0  (8.54) where F 2 = F 1 F 0 F 3 −  h 0 ρ  z 0 zd z µ dz, F 3 =  h 0 ρ  z 0 dz µ dz, F 4 =  h 0 ρdz The energy equation for the oil film is: c p ρ   r ∂T ∂r +  θ r ∂T ∂θ +  z ∂T ∂z  = k o ∂ 2 T ∂z 2 + α  T   r ∂p ∂r +  θ r ∂p ∂θ  + µ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩  ∂ r ∂z  2 +  ∂ θ ∂z  2 ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ (8.55) 8.5 Temperature Analyses of Tilting Pad Thrust Bearings — Sector Pads 175 The heat conduction equation for the pad is: ∂ 2 T 2 ∂r 2 + 1 r 2 ∂ 2 T 2 ∂θ 2 + ∂ 2 T 2 ∂z 2 2 + 1 r ∂T 2 ∂r = 0 (8.56) The coefficient of viscosity of the lubricating oil is: µ = µ in exp{β (T in − T )} (8.57) The density of the lubricating oil is: ρ = ρ in {1 + α  (T in − T )} (8.58) 8.5.2 Boundary Conditions As boundary conditions for the generalized Reynolds’ equation, it is assumed that the pressure around the pad is atmospheric pressure, i.e., p(r, 0) = p(r,θ 0 ) = p(R,θ) = p(R + ∆R,θ) = 0 (8.59) As boundary conditions for the energy equation of the oil film, it is assumed first that both the inlet oil temperature and the disk surface temperature are equal to T in , i.e., T (r,θ 0 , z) = T(r,θ,0) = T in (8.60) It is also assumed that the oil film temperature changes parabolically in the r direc- tion on the inner and outer circular boundaries of the pad. Furthermore, from the continuity of the heat flux at the interface of the pad and the oil film in the direction normal to the interface, it is assumed that: k o  ∂T ∂z  z=h = k s  ∂T 2 ∂z 2  z 2 =0 (8.61) where T and T 2 are the temperature of the lubricating oil and the pad, respectively, and k o and k s are the thermal conductivities of the lubricating oil and the pad, respec- tively. As boundary conditions for the heat conduction equation of the pad, it is assumed that Eq. 8.61 holds at the lubricating surface and that heat flux is continuous at the other surfaces, the ambient temperature T a and the heat transfer coefficient at the interfaces h c being given. 8.5.3 Numerical Analyses Before numerical computations, each variable is nondimensionalized as follows. Since both the lubricating film and the pad become a cube of side 1, the forms of the computational domain in the finite difference method become very simple: 176 8 Heat Generation and Temperature Rise r = r −R ∆R , θ = θ θ 0 , z = z h , h = h h 0  r =  r Rω ,  θ =  θ rω ,  z =  z Rω , z 2 = z 2 t 2 µ = µ µ in , ρ = ρ ρ in , T = T T in , T 2 = T 2 T in p = ph 0 2 R 2 ωµ in (8.62) Then, the lubricating film and the pad are divided into, for example, 20 divi- sions in the r and θ directions, and into, for example, 15 divisions in the z direction, and Eqs. 8.51 to 8.58 are discretized for the finite difference analysis. The centered difference is used, except in the case of ∂T/∂θ in the energy equation (Eq. 8.55) for which the backward difference is used. This is solved by the successive overre- Fig. 8.3. Flow chart for computation [20] 8.5 Temperature Analyses of Tilting Pad Thrust Bearings — Sector Pads 177 laxation method (SOR method), following the flow chart of Fig. 8.3. Calculation is repeated until the error from the last calculated value becomes, for example, smaller than 10 −5 at all nodal points. Further, to make the center of pressure coincide with the pivot position, the inclination of the pad is adjusted by the Newton–Raphson calculation. The constants in Table 8.1 were used in the following calculations. Table 8.1. Specifications of the sector pad bearing and the oil Inner radius of pad R = 1.6 × 10 −1 m Radial extent of pad ∆R = 7.7 × 10 −2 m Angular extent of pad θ 0 = π/8rad Thickness of pad t = 2.5 × 10 −2 m Thermal conduction of pad k s = 1.08 × 10 2 W/m ◦ C heat transfer at surface of pad h c = 5.81 × 10 2 W/m 2 ◦ C Viscosity of oil (27 ◦ C) µ = 5.0 × 10 −2 Pa · s Viscosity index of oil β = 4.91 ×10 −2 ◦ C −1 Specific heat of oil c p = 2.09 × 10 3 J/kg ◦ C Thermal conductivity of oil k o = 0.214 W/m ◦ C Coefficient of thermal expansion of oil α  = 7.34 × 10 −4 ◦ C −1 Density of oil (27 ◦ C) ρ = 8.55 × 10 2 kg/m 3 8.5.4 Examples of Three-Dimensional Analyses of Temperature Distribution Figure 8.4 shows isothermal lines for the lubricating surface of the pad and for the central cross section of the lubricant film and the pad, parameters being h 0 = 100 µm, N = 3000 rpm, T in = T a = 47 ◦ C, the pivot position in nondimensional radial coor- dinates r p = (r p − R)/∆R = 0.51 and in nondimensional circumferential coordinates θ p = θ p /θ 0 = 0.39. On the lubricating surface of the pad, temperature rises in the circumferential direction gradually near the leading edge and then quicker near the trailing edge. The highest temperature is found on the trailing edge near the outer radius. The temperature does not change very much in the radial direction. The shapes of the isothermal lines are very close to the experimental results under the same conditions (the same pivot position, and so forth) [23]. The highest temperature in Fig. 8.4 is approximately 58 ◦ C, which is 11 ◦ C higher than the entrance oil temperature. Generally speaking, the highest temperature falls if the heat transfer coefficient h c of the pad surface is increased; however, the fall in highest temperature is only 2 ◦ –4 ◦ Corsoevenifh c is increased by 10 – 10 2 times [16]. This probably shows that most of the heat is carried away by the convection of the lubricating oil. In the central cross section of the lubricant film and the pad, the temperature in the lubricant film (the wedge-shaped domain in the lower half; the thickness is 178 8 Heat Generation and Temperature Rise Fig. 8.4. Temperature distribution on the lubricating surface (left) and the central cross section of the lubricant film (lower right) and pad (upper right) [16] exaggerated) rises gradually from the disk surface, and rises quickly near the pad surface. The highest temperature is found at the trailing edge of the lubricant film near the pad surface. The temperature distribution in the pad (the rectangular domain in the upper half) is as shown in the figure. 8.5.5 Comparisons of Three-Dimensional, Two-Dimensional, and Isoviscous Analyses Figure 8.5a-c shows comparisons of the nondimensional load capacity, the tem- perature rise (= highest temperature - entrance temperature), and the nondimen- sional frictional torque calculated by three-dimensional, two-dimensionalal and iso- viscous analyses, under the conditions T in = T d = T a = 27 ◦ C, h 0 = 100 µm, and N = 1500 rpm. The horizontal axis shows the nondimensional circumferential co- ordinates of the pivot position θ p = θ p /θ 0 , and the parameters in the figure are the nondimensional radial coordinates of the pivot position r p = (r p − R)/∆R. The nondimensional load capacity in Fig. 8.5a-ca shows that the three-dimensional analyses give the lowest load capacity, the two-dimensional analyses give an inter- mediate figure, and the isoviscous analyses give the highest load capacities. Fig. 8.5a also shows that there is an optimal pivot position θ p that gives the highest load ca- pacity in each of the three analyses, θ p being slightly dependent on r p . The load capacity, which changes considerably with r p , becomes maximum when r p = 0.51 for any method of analysis here. The load capacity depends not only on the pad incli- nation but also on the oil viscosity, which is a function of temperature and hence of location. This affects the optimal pivot position in a complicated way. One must be aware that the isoviscous and the two-dimensional analyses may give load capacity estimates that are too high. As for the temperature rise shown in Fig. 8.5a-cb, the two-dimensional analysis gives a value considerably lower than the three-dimensional analysis does, although 8.5 Temperature Analyses of Tilting Pad Thrust Bearings — Sector Pads 179 al Torque M ℃ Temperature Rise in Pad Surface ∆T, Fig. 8.5a-c. Comparisons of bearing characteristics using three-dimensional (3-D), two- dimensional (2-D), and isoviscous analyses [20]. a nondimensional load capacity, b temper- ature rise, c nondimensional frictional torgue. r p , nondimensional radial coordinates of the pivot position 180 8 Heat Generation and Temperature Rise it does not consider the heat flow into solid surfaces. The reason for this is that only the average temperature is considered in the film thickness direction in two- dimensional analysis. If the pivot position is moved backward (if θ p is decreased), the temperature rise decreases for both the two-dimensional analysis and the three- dimensional analysis. In the three-dimensional analysis, the large temperature rise falls very rapidly as shown in the figure. The reason why the temperature rise falls when the pivot position is moved backward is that increasing the pad inclination leads to an increase in the rate of oil flow. The two-dimensional analysis can result in temperature estimates that are too small, which requires caution. As for the nondimensional frictional torque shown in Fig. 8.5c, the three- dimensional analysis gives the lowest values, the two-dimensional analysis gives values a little higher, and the isoviscous analysis gives the highest values. This is because the temperature rise is larger in the three-dimensional case and hence the viscosity is lower. It should be noted here that the results obtained by the three-dimensional, the two-dimensional, and the isoviscous analyses differ considerably in terms of load capacity, temperature rise, and frictional torque. Therefore, the analysis of a bear- ing should originally be performed using the three-dimensional analysis; and other approaches cannot substitute for it. When performing the two-dimensional or the isoviscous analyses, it should be kept in mind that they are considerable approxima- tions. 8.5.6 Analysis Considering Inertia Forces In high speed tilting pad thrust bearings, the influence of the inertia force (centrifugal force) on the fluid cannot be disregarded (Kim et al. [21]). Reynolds’ equation for such a case is derived here. In reference to Fig. 8.2, the equations of balance that take inertia into considera- tion are written as follows: ∂ ∂z  µ ∂ r ∂z  = ∂p ∂r − ρ θ 2 r (8.63) ∂ ∂z  µ ∂ θ ∂z  = 1 r ∂p ∂θ (8.64) ∂p ∂z = 0 (8.65) The second term on the right-hand side of Eq. 8.63 is an inertia term. Integrating Eq. 8.64 under the boundary condition  θ = rω at z = 0,  θ = 0atz = h yields:  θ = rω + 1 r ∂p ∂θ  z 0 z µ dz −  rω F 0 + F 1 F 0 1 r ∂p ∂θ   z 0 dz µ (8.66) 8.5 Temperature Analyses of Tilting Pad Thrust Bearings — Sector Pads 181 Subsituting this into Eq. 8.63 and integrating under the boundary conditions  r = 0atz = 0 and z = h gives the following equation:  r = ∂p ∂r  z 0 z µ dz +  1 r G 0 F 0 − F 1 F 0 ∂p ∂r   z 0 dz µ − 1 r  z 0  1 µ  z 0 ρ( θ ) 2 dz  dz (8.67) where F 0 , F 1 , and F 2 are as follows: F 0 =  h 0 dz µ , F 1 =  h 0 z µ dz, G 0 =  h 0  1 µ  z 0 ρ( θ ) 2 dz  dz (8.68) On the other hand, integration of the continuity equation with respect to z from 0 to h, the use of  z = 0atz = 0,  r =  θ =  z = 0atz = h and use of the formula for the change of order of differentiation and integration yields: r ∂ ∂r  h 0 ρ r dz +  h 0 ρ r dz + ∂ ∂θ  h 0 ρ θ dz = 0 (8.69) Substituting Eqs. 8.66 and 8.67 into Eq. 8.69 yields the following Reynolds’ equation, which takes the inertia force and three-dimensional temperature change into consideration, as follows: ∂ ∂r  G 1 ∂p ∂r + G 2  + 1 r  G 1 ∂p ∂r + G 2  + 1 r 2 ∂ ∂θ  G 1 ∂p ∂θ  = 1 r ∂G 3 ∂θ (8.70) where G 1 , G 2 , and G 3 are given as follows: G 1 =  h 0 ρ   z 0 z µ dz − F 1 F 0  z 0 dz µ  dz G 2 =  h 0 ρ  1 r G 0 F 0  z 0 dz µ − 1 r  z 0  1 µ  z 0 ρ( θ ) 2 dz  dz  dz G 3 =  h 0 ρrω  1 F 0  z 0 dz µ − 1  dz (8.71) By means of Eq. 8.70, the three-dimensional thermohydrodynamic lubrication analysis that takes the inertia force into consideration can be carried out. The pro- cedure of numerical computation is the same as before. When µ = ρ = constant, Eq. 8.70 coincides with the Reynolds’ equation (Eq. 4.34) derived in Chapter 4 with reference to cylindrical coordinates. 182 8 Heat Generation and Temperature Rise Examples of the three-dimensional analysis of thermohydrodynamic lubrication taking the inertia forces into consideration are shown below. Specifications of the bearing and the lubricating oil are listed in Table 8.1. Figure 8.6 shows the calculated temperature rise, and illustrates how the differ- ence between the highest and the lowest temperature (entrance temperature) on the pad surface, ∆T, changes with the rotational speed N. The operating conditions of the bearing are: r p = 0.51, θ p = 0.42, h 0 = 100 µm, and T in = 47 ◦ C. If the inertia term is taken into consideration, ∆T is calculated to be larger than that for the case where no inertia effect is considered, and the larger N is, the larger ∆T is. This differ- ence is due to the effect of the velocity component produced in the radial direction. Fig. 8.6. Temperature rise [21] Figure 8.7 shows the relation between the nondimensional load capacity P (see Fig. 8.5a-c) and N under the same operating conditions. If the inertia term is ignored, P decreases monotonously along with the increase in N. This is due to the decrease in viscosity along with the temperature rise in the oil film shown in Fig. 8.6. On the other hand, if an inertia term is taken into consideration, the calculated load capacity tends to be larger due to the flow velocity in the radial direction. As a result, in the area where the curve is upward convex in Fig. 8.7, the load capacity P is about 10% higher than that obtained ignoring the inertia term. If the rotating speed exceeds 2000 rpm, however, the load capacity falls because the influence of the decrease in viscosity due to temperature rise exceeds that of the inertia. Figure 8.8 shows the relation between the inclination (tilt) of a pad and the rotat- ing speed N under the same operating conditions. α p and α r are the circumferential inclination α p and the radial direction inclination α r of the pad multiplied by ∆R/h 0 , respectively. Even though both of the inclinations increase with the increase in N, [...]... disk N and the highest temperature on the pad surface T max determined by analyses that 8.6 Temperature Analyses of Circular Journal Bearings 185 Fig 8 .11 Average bearing pressure and highest temperature on the pad surface [23] THL, thermohydrodynamic lubrication include or neglect the inertia term, together with experimental resutls for comparison For high rotating speeds, it turns out that the analysis... recirculating oil The flow rate of the oil immediately downstream of the oil groove, q st , can be written as a sum of part of the supply oil qin and part of the recirculated oil qr as follows: q st = ηin qin + ηmix qr (8.74) where ηin is the ratio of the supply oil actually involved in lubrication to the supply oil and ηmix is the ratio of the recirculating oil that is mixed with the new oil at the oil... it is rotating and it is difficult to establish boundary conditions because the part of the journal extending outside the bearing may contact seals and other things and so its form is often complicated Therefore, the following simplified method is used to avoid the solution of the heat conduction equation in the journal Let a part of the total amount of heat generated in the oil film Q j = η f j QG be the... [18] However, there has been no paper on perfect circular bearings that takes full account of the mixing of the supply oil and circulation oil, the influence of oil film rupture, and so on In the case of partial bearings, papers by Suganami et al can be mentioned [12] [13] 8.6.1 Basic Equations As basic equations for a lubricant film, the equations for the fluid velocity, the generalized Reynolds’ equation,... embedded at a depth of 0.5 mm from the lubricating surface and were compared with theoretical results (Kim et al [23]) Pivot positions were r p = 0.516 and θ p = 0.436 in nondimensional coordinates Figure 8 .11 shows the relationships between the average pad pressure pm and the highest temperature on the pad surface T max determined by two-dimensional and three-dimensional analyses, together with experimental... and also that by Woolacott and Cooke [6] can be mentioned as those that pursue the temperature characteristics of bearings by precise experiments Theoretical investigations are more common, and those of particular note inculde research by Dowson and March, who solved Reynolds’ equation and the energy equation in the case of an infinitely long bearing on the basis of the above experimental research [5]; . Analyses of Circular Journal Bearings 185 Fig. 8 .11. Average bearing pressure and highest temperature on the pad surface [23]. THL, thermohydrodynamic lubrication include or neglect the inertia term,. sum of part of the supply oil q in and part of the recirculated oil q r as follows: q st = η in q in + η mix q r (8.74) where η in is the ratio of the supply oil actually involved in lubrication. − F 1 F 0  z 0 dz µ  dz G 2 =  h 0 ρ  1 r G 0 F 0  z 0 dz µ − 1 r  z 0  1 µ  z 0 ρ( θ ) 2 dz  dz  dz G 3 =  h 0 ρrω  1 F 0  z 0 dz µ − 1  dz (8.71) By means of Eq. 8.70, the three-dimensional thermohydrodynamic lubrication analysis that takes the inertia force into consideration can be carried out.