**3.5. Computation procedure**

14 Performance Evaluation of Bearings

and

At , *<sup>U</sup> y hT T*

0

*h h* 

> 

> >

 

4 4

*Kxx x x Kz*

2 2

2 12

0

120 120

*T T T T Ch u u Ch u u T T zz z K x K xx*

The temperature in bush is determined by using the Laplace equation within the bearing

In this equation, *r* stands for bush radius, and *<sup>b</sup> T* stands for bush temperature. The equation (13) is then set into finite differences by using central difference technique. The final form is

( , , ) 1 ( 1, , ) 1 ( 1, , ) 2 ( , 1, ) 2 ( , 1, )

*E F EE F EE F E* 1 11 / ; 2 22 / ; 3 33 /

 ; 2 <sup>1</sup> *<sup>F</sup>*<sup>33</sup> *dy* ;

*T ijk ET i jk ET i jk ET ij k ET ij k*

*bb b b b*

; 2 2 <sup>1</sup> *<sup>F</sup>*<sup>11</sup> *r d* ; 2 <sup>1</sup> *<sup>F</sup>*<sup>22</sup> *dz*

222 <sup>222</sup> <sup>0</sup> *bbb TTT xyz* 

*L U m P LU P U L m U L*

*P L U m P*

Where,*TL* , *TU* and *Tm* represent temperatures of the lower bounding surface (journal), upper bounding surface (bearing), and mean temperature across the film respectively.

*Ch T P P T T C h TT T*

*U L*

426 336 *L LUm LUm <sup>y</sup> <sup>y</sup> TT T T T T T T*

*mT Tdy <sup>h</sup>*

2

(13)

(11)

(12)

(14)

1 *h*

After algebraic manipulations, the equation (10) takes the following form:

Final form of the energy equation is represented as:

12

 

3 ( , , 1) 3 ( , , 1)

*b b*

 

Where, 22 2 2 2 22 *<sup>E</sup> r d dy dz* 

*ET ijk ET ijk*

*LU m*

12

reproduced here.

**3.4. Heat conduction equation** 

material as given below Hori [2006]:

2 2 <sup>2</sup> <sup>4</sup>

*hP P u u Kx z K*

6 6 12 12

Coupled numerical solutions of Reynolds, energy and heat conduction equations are obtained for offset-halves and elliptical journal bearings. The temperature of upper and lower bounding surfaces have been assumed constant throughout and set equal to oil inlet temperature for first iteration. For subsequent iterations the temperatures at oil bush interface are computed using heat conduction equations and appropriate boundary conditions. The numerical procedure adopted for obtaining the thermohydrodynamic solution is discussed below.

#### a. Reynolds Equation

A suitable initial value of attitude angle is assumed and film thickness equations (1-4) are solved. Then equation (6) has been used to find isothermal pressures. The initial viscosity values are assumed to be equal to the inlet oil viscosity.

### b. Energy equation

The solution for the determination of temperature begins with the known pressure distributions obtained by solution of Reynolds equation. Viscosity variation in the fluid film domain corresponding to computed temperatures and pressures is calculated using equation (8). With new value of viscosity, equation (7) has been solved for thermal pressure. These values of pressure and viscosity, are used to further solve energy equation (12). Mean temperatures obtained by solving equation (12) are substituted in equation (11) to find the temperature in the oil film. Now, this temperature is used to solve the equation (13) to obtain the temperature variation in the bush. The computation is continued till converged solutions for thermal pressure loop and temperature loop have been arrived. The load carrying capacity is obtained by applying the Simpson rule to the pressure distribution. In computation, wherever reverse flow arises in domain, upwind differencing has been resorted to. Power losses have been evaluated by the determination of shear forces, and then employing the Simpson rule.

The boundary conditions used in the solution of governing equations are:


$$T(0, y) = T\_0; \ T(\mathbf{x}, 0) = T\_0; \quad k\_{\text{oil}} \left( \frac{\partial T}{\partial \mathbf{y}} \right)\_{\text{bunched}}^{\text{upper}} = k\_s \left( \frac{\partial T\_s}{\partial y\_s} \right)\_{y\_s = 0};$$

$$-k\_s \left( \frac{\partial T\_s}{\partial y\_s} \right)\_{y\_s = t} = h\_c \left( T\_s \left( \mathbf{x}\_s, t \right) - T\_a \right); \qquad -k\_s \left( \frac{\partial T\_s}{\partial \mathbf{x}\_s} \right)\_{x\_s = 0} = h\_c \left( T\_s \left( 0, y\_s \right) - T\_a \right) \tag{5}$$

$$-k\_s \left( \frac{\partial T\_s}{\partial \mathbf{x}\_s} \right)\_{x\_s = l} = h\_c \left( T\_s \left( l\_s, y\_s \right) - T\_a \right); \quad k\_s \left( \frac{\partial T\_s}{\partial \mathbf{z}\_s} \right)\_{z\_s = 0} = h\_c \left( T\_s \left( \mathbf{x}\_s, y\_s, 0 \right) - T\_a \right);$$

$$-k\_s \left( \frac{\partial T\_s}{\partial \mathbf{z}\_s} \right)\_{z\_s = b} = h\_c \left( T\_s \left( \mathbf{x}\_s, y\_s, b \right) - T\_a \right)$$

Where, *Ks* denotes thermal conductivity of bearing, *<sup>c</sup> h* denotes convection heat transfer coefficient of bush, *l* denotes length of the bearing, *s* denotes bearing surface, *t* denotes thickness of bearing, *b* denotes width of bearing, and *<sup>a</sup> T* ambient temperature.

The solution of governing equations has been achieved by satisfying the convergence criterion given below:

For pressure:

$$\frac{|\left(\sum P\_{i,j}\right)\_{n-1} - \left(\sum P\_{i,j}\right)\_n|}{|\left(\sum P\_{i,j}\right)\_n|} \le 0.0001\tag{15}$$

Thermal Studies of Non-Circular Journal Bearing Profiles: Offset-Halves and Elliptical 17

*m* ±50μm

*m* ±50μm

**Parameter Dimension Tolerance Roughness** 

**Parameter Dimension Tolerance Roughness** 

*m* ±50μm

*m* ±50μm

**Oil 2** (Mak 2T)

*C* 0.126 *W m*/ <sup>0</sup>

**Oil 3**  (Mak Multigrade)

*C* 0.126 *W m*/ <sup>0</sup>

*C*

Outer diameter of the bearing, OD 85mm ±0.2mm 10μm Inner diameter of the bearing, ID 65mm ±0.2mm 10μm Length, L 65mm ±0.2mm 10μm

**Table 1.** Input parameters used to study the performance characteristics of offset-halves journal bearing

Outer diameter of the bearing, OD 85mm ±0.2mm 10μm Maximum inner diameter of the bearing, DImax 65.4mm ±0.2mm 10μm Minimum inner diameter of the bearing, DImin 65.2mm ±0.2mm 10μm Length, L 65mm ±0.2mm 10μm

**Table 2.** Input parameters used to study the performance characteristics of elliptical journal bearing

**Oil 1** (Hydrol 68)

880 <sup>3</sup> *Kg m*/ 868 <sup>3</sup> *Kg m*/ 885 <sup>3</sup> *Kg m*/

*C* 230 94 200

*C* -9 -24 -21

**Table 3.** Properties of the bush material (Methyl Methacrylite) and grade oils under study [Chauhan et

2.3e-8 <sup>1</sup> *Pa*

0.034 <sup>1</sup> *K*

Viscosity index\* 98 135 110

*C* ) 0.075 *Pas* 0.065 *Pas* 0.200 *Pas*

*C* ) 0.00771 *Pas* 0.004861 *Pas* 0.01239 *Pas*

Oil hole 6.35mm ±0.15mm Relative sensor position 45º ±1º

Oil hole 6.35mm ±0.15mm Relative sensor position 45º ±1º

Radial Clearance, C 500

Minimum Clearance, Cm 200

Radial Clearance, C 300

Minimum Clearance, Cm 200

Thermal conductivity\*\*, *Koil* 0.126 *W m*/ <sup>0</sup>

Thermal conductivity of bush\*\*, *Kbush 0.22 W m*/ *Deg C*. Coefficient of thermal expansion of bush, *bush h 75e-6* <sup>1</sup> *K*

[Chauhan: 2011]

[Chauhan: 2011]

(at *<sup>o</sup> <sup>T</sup>* =33 <sup>0</sup>

(at *<sup>o</sup> <sup>T</sup>* =100 <sup>0</sup>

Barus viscosity-pressure index\*\*,

Temperature viscosity- coefficient\*\*,

Viscosity,

Viscosity,

Density,

Flash point\*, <sup>0</sup>

Pour point\*, <sup>0</sup>

al. 2011]

For temperature:

$$\frac{|\left(\sum T\_{i,j}\right)\_{n-1} - \left(\sum T\_{i,j}\right)\_n|}{|\left(\sum T\_{i,j}\right)\_n|} \le 0.0001\tag{16}$$

Where, n represents number of iterations.

Here, the authors have made an attempt to present some performance parameters of offsethalves and elliptical journal bearing which have been evaluated using computer program developed by them based on method discussed in the previous articles of the chapter. Input parameters of offset-halves and elliptical journal bearing, and the properties of the grade oils and material used to manufacture the bearings are given in Tables 1, 2, and 3. The study has been carried at oil inlet temperature of 33 <sup>0</sup> *C* (which has been used as reference inlet temperature of the oil by most of the researchers) for the eccentricity ratios equal to 0.6 and speeds ranging from 3000 *rpm* -4000 *rpm .*


criterion given below:

For pressure:

For temperature:

Where, n represents number of iterations.

has been carried at oil inlet temperature of 33 <sup>0</sup>

speeds ranging from 3000 *rpm* -4000 *rpm .*

<sup>0</sup> *TyT* (0, ) ; 0 *Tx T* ( ,0) ;

*s*

*s y t*

*y*

*s css a*

*s*

*s x l*

*x*

*<sup>T</sup> k hTxt T*

*s*

*s*

,

*s cs s a*

*<sup>T</sup> <sup>k</sup> h T ly T*

,

*s c s s s a*

Where, *Ks* denotes thermal conductivity of bearing, *<sup>c</sup> h* denotes convection heat transfer coefficient of bush, *l* denotes length of the bearing, *s* denotes bearing surface, *t* denotes

The solution of governing equations has been achieved by satisfying the convergence

*<sup>T</sup> k hTxyb T*

*s*

thickness of bearing, *b* denotes width of bearing, and *<sup>a</sup> T* ambient temperature.

 , , <sup>1</sup> ,


*P P P* 


 , , <sup>1</sup> ,


*T T T* 


Here, the authors have made an attempt to present some performance parameters of offsethalves and elliptical journal bearing which have been evaluated using computer program developed by them based on method discussed in the previous articles of the chapter. Input parameters of offset-halves and elliptical journal bearing, and the properties of the grade oils and material used to manufacture the bearings are given in Tables 1, 2, and 3. The study

temperature of the oil by most of the researchers) for the eccentricity ratios equal to 0.6 and

*s z b*

*z*

*s*

0 *<sup>s</sup>*

0,

, ,0

*s*

*bounding <sup>s</sup> <sup>y</sup> surface*

*y y* ;

; 0

*s*

*s cs s a*

;

 ; 0

*s c s s s a*

;

*<sup>T</sup> k hTxy T*

*<sup>T</sup> k hT y T*

*s*

*s*

0.0001

0.0001

(15)

(16)

*C* (which has been used as reference inlet

*s z*

*z*

*s x*

*x*

*oil s upper*

*s*

, ,

*<sup>T</sup> <sup>T</sup> k k*

**Table 1.** Input parameters used to study the performance characteristics of offset-halves journal bearing [Chauhan: 2011]


**Table 2.** Input parameters used to study the performance characteristics of elliptical journal bearing [Chauhan: 2011]


**Table 3.** Properties of the bush material (Methyl Methacrylite) and grade oils under study [Chauhan et al. 2011]

Figures 5, and 6, show the variation of oil film temperature in the central plane of the bearing for eccentricity ratio, 0.6 at oil inlet temperature of 33 <sup>0</sup> *C* for all the three grade oils under study at speeds 4000 *rpm* for offset-halves and elliptical journal bearing respectively. It has been observed that oil film temperature rise is very high in lower lobe in comparison to oil film temperature rise in upper lobe for offset-halves, and the oil film temperature rise is though high in lower lobe but it is comparable with the rise in upper lobe of the elliptical journal bearing. A high temperature rise in Oil 3 compared to the other grade oils has been observed which may be because of its high viscosity value. Similarly, figures 7 and 8, show the variation of thermal pressure in the central plane of the bearing for eccentricity ratio, 0.6 at oil inlet temperature of 33 <sup>0</sup> *C* for all the three grade oils under study at speeds 4000 *rpm* for offset-halves and elliptical journal bearing respectively. It has been observed that thermal pressure rise is very high in lower lobe in comparison to thermal pressure rise in upper lobe for both offset-halves and elliptical journal bearing.

Thermal Studies of Non-Circular Journal Bearing Profiles: Offset-Halves and Elliptical 19

**Figure 6.** Variation of oil film temperatures in the central plane of the elliptical bearing for different

**Figure 7.** Variation of thermal pressure in the central plane of the offset-halves bearing for different

*C* and eccentricity ratio=0.6

*C* and eccentricity ratio=0.6

grade oils at 4000 *rpm* , oil inlet temperature=33 <sup>0</sup>

grade oils at 4000 *rpm* , oil inlet temperature=33 <sup>0</sup>

**Figure 5.** Variation of oil film temperatures in the central plane of the offset-halves bearing for different grade oils at 4000 *rpm* , oil inlet temperature=33 <sup>0</sup> *C* and eccentricity ratio=0.6

eccentricity ratio, 0.6

bearing for eccentricity ratio, 0.6

grade oils at 4000 *rpm* , oil inlet temperature=33 <sup>0</sup>

at oil inlet temperature of 33 <sup>0</sup>

pressure rise in upper lobe for both offset-halves and elliptical journal bearing.

Figures 5, and 6, show the variation of oil film temperature in the central plane of the

oils under study at speeds 4000 *rpm* for offset-halves and elliptical journal bearing respectively. It has been observed that oil film temperature rise is very high in lower lobe in comparison to oil film temperature rise in upper lobe for offset-halves, and the oil film temperature rise is though high in lower lobe but it is comparable with the rise in upper lobe of the elliptical journal bearing. A high temperature rise in Oil 3 compared to the other grade oils has been observed which may be because of its high viscosity value. Similarly, figures 7 and 8, show the variation of thermal pressure in the central plane of the bearing for

study at speeds 4000 *rpm* for offset-halves and elliptical journal bearing respectively. It has been observed that thermal pressure rise is very high in lower lobe in comparison to thermal

**Figure 5.** Variation of oil film temperatures in the central plane of the offset-halves bearing for different

*C* and eccentricity ratio=0.6

at oil inlet temperature of 33 <sup>0</sup>

*C* for all the three grade

*C* for all the three grade oils under

**Figure 6.** Variation of oil film temperatures in the central plane of the elliptical bearing for different grade oils at 4000 *rpm* , oil inlet temperature=33 <sup>0</sup> *C* and eccentricity ratio=0.6

**Figure 7.** Variation of thermal pressure in the central plane of the offset-halves bearing for different grade oils at 4000 *rpm* , oil inlet temperature=33 <sup>0</sup> *C* and eccentricity ratio=0.6

Thermal Studies of Non-Circular Journal Bearing Profiles: Offset-Halves and Elliptical 21

**Figure 9.** Variation of oil film temperature in the central plane with circumferential angle for Oil 1 at eccentricity ratio=0.6 and speed=3000 *rpm* for offset-halves and elliptical profile bearings (d= 0.1 *m* ,

**Figure 10.** Variation of thermal pressure in the central plane with circumferential angle for Oil 1 at eccentricity ratio=0.6 and speed=3000 *rpm* for offset-halves and elliptical profile bearings (d= 0.1 *m* ,

l/d=1, *C =* 200

l/d=1, *C =* 200

*m* , *<sup>m</sup> C* = 120

*<sup>m</sup>* , *<sup>o</sup> T =*<sup>33</sup> <sup>0</sup>

*C* )

*m* , *<sup>m</sup> C* = 120

*<sup>m</sup>* , *<sup>o</sup> T =*<sup>33</sup> <sup>0</sup>

*C* )

**Figure 8.** Variation of thermal pressure in the central plane of the elliptical bearing for different grade oils at 4000 *rpm* , oil inlet temperature=33 <sup>0</sup> *C* and eccentricity ratio=0.6

The comparative study of performance characteristics like oil film temperature, thermal pressure, load capacity, and power loss for the bearing configurations under study has also been presented in figures 9, 10, 11, and 12. Fig. 9 shows, that the oil film temperature rise is on much higher side for upper lobe of elliptical journal bearing when compared to the oil film temperature rise of same lobe of the offset-halves journal bearing, whereas the oil film temperature in lower lobe of offset-halves journal bearing is little higher the oil film temperature rise in lower lobe of elliptical journal bearing. Hence, the overall oil film temperature rise and thermal pressure has been observed high for elliptical journal bearing. The load capacity and power losses have been found on higher side for Oil 3 and on lower side for Oil 1. The trend remains same for both the journal bearings. The load capacity and power loss has also been observed high for elliptical journal bearing when compared with offset-halves journal bearing. It can be concluded from the above discussion that the lubricating oil with higher viscosity value results in high oil film temperature rise, high thermal pressure, high load capacity and also high power loss value, whereas the lubricating oil with low viscous value results in low oil film temperature rise, low thermal pressure, little low load capacity and power loss value.

oils at 4000 *rpm* , oil inlet temperature=33 <sup>0</sup>

pressure, little low load capacity and power loss value.

**Figure 8.** Variation of thermal pressure in the central plane of the elliptical bearing for different grade

The comparative study of performance characteristics like oil film temperature, thermal pressure, load capacity, and power loss for the bearing configurations under study has also been presented in figures 9, 10, 11, and 12. Fig. 9 shows, that the oil film temperature rise is on much higher side for upper lobe of elliptical journal bearing when compared to the oil film temperature rise of same lobe of the offset-halves journal bearing, whereas the oil film temperature in lower lobe of offset-halves journal bearing is little higher the oil film temperature rise in lower lobe of elliptical journal bearing. Hence, the overall oil film temperature rise and thermal pressure has been observed high for elliptical journal bearing. The load capacity and power losses have been found on higher side for Oil 3 and on lower side for Oil 1. The trend remains same for both the journal bearings. The load capacity and power loss has also been observed high for elliptical journal bearing when compared with offset-halves journal bearing. It can be concluded from the above discussion that the lubricating oil with higher viscosity value results in high oil film temperature rise, high thermal pressure, high load capacity and also high power loss value, whereas the lubricating oil with low viscous value results in low oil film temperature rise, low thermal

*C* and eccentricity ratio=0.6

**Figure 9.** Variation of oil film temperature in the central plane with circumferential angle for Oil 1 at eccentricity ratio=0.6 and speed=3000 *rpm* for offset-halves and elliptical profile bearings (d= 0.1 *m* , l/d=1, *C =* 200*m* , *<sup>m</sup> C* = 120*<sup>m</sup>* , *<sup>o</sup> T =*<sup>33</sup> <sup>0</sup> *C* )

**Figure 10.** Variation of thermal pressure in the central plane with circumferential angle for Oil 1 at eccentricity ratio=0.6 and speed=3000 *rpm* for offset-halves and elliptical profile bearings (d= 0.1 *m* , l/d=1, *C =* 200*m* , *<sup>m</sup> C* = 120*<sup>m</sup>* , *<sup>o</sup> T =*<sup>33</sup> <sup>0</sup> *C* )

Thermal Studies of Non-Circular Journal Bearing Profiles: Offset-Halves and Elliptical 23

performance. The presently available design data/handbooks do not provide any direct analytical methods for the design and analysis of these non-circular journal bearings. However, the present methodology to a large extent has succeeded in standardizing the

[1] Banwait SS., and Chandrawat HN., "Effect of misalignment on thermohydrodynamic

[2] Basri S., and Gethin DT., "A comparative study of the thermal behaviour of profile bore

[3] Booker JF., And Govindachar S., "Stability of offset journal bearing systems", Proc. of

[4] Chandra M., Malik M., and Sinhasan R., "Comparative study of four gas-lubricated noncircular journal bearing configurations", Tribology International, 1983; 16: 103-108. [5] Chauhan A., "Experimental and theoretical investigations of the thermal behaviour of some non-circular journal bearing profiles", Ph. D Thesis, NIT Hamirpur, 2011. [6] Chauhan A., and Sehgal R., "An experimentation investigation of the variation of oil temperatures in offset-halves journal bearing profile using different oils", Indian

[7] Chauhan A., Sehgal R., and Sharma RK., "Thermohydrodynamic analysis of elliptical journal bearing with different grade oils", Tribology International, 2010; 43: 1970-1977. [8] Chauhan A., Sehgal R. and Sharma RK., "Investigations on the Thermal Effects in Non-

[9] Crosby WA., "Thermal considerations in the solution of finite journal bearings", Wear

[10] Fuller DD., "Theory and practice of lubrication for engineers", John Wiley and Sons,

[12] Hussain A., Mistry K., Biswas S., and Athre K., "Thermal analysis of Non-circular

[13] Ma MT. and Taylor M., "An experimental investigation of thermal effects in circular and elliptical plain journal bearings. Tribology International, 1996; 29(1): 19-26.

Circular Journal Bearings", Tribology International, 2011; 44; 1765-1773.

[11] Hori Y., "Hydrodynamic lubrication", Springer-Verlag Tokyo, 2006.

equations for design and analysis of these bearings.

*University Institute of Engineering and Technology, Sector-25,* 

*Department of Mechanical Engineering, NIT Hamirpur (HP), India* 

bearings", Tribology International, 1990; 23: 265-276.

analysis of elliptical journal bearings", IE (I) J 2000; 81: 93-101.

**Author details** 

*Panjab University, Chandigarh, India* 

IMechE 1984; C283/84: 269-275.

Journal of Tribology, 2008; 2: 27-41.

bearings", Trans ASME 1996; 118: 246-254.

1980; 64: 15-32.

New York, 1956.

Amit Chauhan

Rakesh Sehgal

**5. References** 

**Figure 11.** Variation of load capacity with speed for Oil 3 at eccentricity ratio=0.7 and speed=3000 *rpm* for offset-halves and elliptical profile bearings (d= 0.1 *m* , l/d=1, *C =* 200*m* , *Cm* = 120*<sup>m</sup>* , *<sup>o</sup> <sup>T</sup> <sup>=</sup>*<sup>33</sup> <sup>0</sup> *C* )

**Figure 12.** Variation of power loss with speed for Oil 3 at eccentricity ratio=0.7 and speed=3000 *rpm* for offset-halves and elliptical profile bearings (d= 0.1 *m* , l/d=1, *C =* 200*m* , *Cm* = 120*<sup>m</sup>* , *<sup>o</sup> <sup>T</sup> <sup>=</sup>*<sup>33</sup> <sup>0</sup> *C* )

## **4. Conclusion**

This chapter deals with the thermal studies related to offset-halves and elliptical journal bearings. For each bearing configuration, the use of grade Oil 2 gives, the minimum temperature rise, and power losses which implies that the oil with low viscosity should be preferred as compared to high viscosity oil from thermal point of view and temperature rise is low for offset-halves bearing. An attempt has been made to compare the performance of two bearing configurations namely: offset-halves and elliptical journal bearings of same geometrical size by using three common commercial grade oils under similar operating conditions. Further, load carrying capacity of elliptical journal bearing has been found high in comparison to offset-halves journal bearing for same operating conditions. Among the two bearing configurations power losses are found to be minimum in case of offset-halves journal bearing. The chapter presents different performance parameters like oil film temperature, thermal pressures, load carrying capacity, and power losses which will help the designer to design this type of non-circular journal bearings as well as analyze their performance. The presently available design data/handbooks do not provide any direct analytical methods for the design and analysis of these non-circular journal bearings. However, the present methodology to a large extent has succeeded in standardizing the equations for design and analysis of these bearings.
