**3. Numerical techniques for the thermal analysis of non-circular journal bearing**

In earlier works, the bearing performance parameters have been computed by solving the Reynolds equation only. Over the years, many researchers have proposed number of mathematical models. A more realistic thermohydrodynamic (*THD*) model for bearing analysis has been developed which treats the viscosity as a function of both the temperature and pressure. Moreover, it also considers the variation of temperature across the film thickness and through the bounding solids (housing and Journal). The thermohydrodynamic model also presents coupled solutions of governing equations by incorporating appropriate boundary conditions and considering the heat conduction across the bearing surfaces. Even the importance of *THD* studies in hydrodynamic bearings can be justified by looking at the large volumes of research papers that are being published by researchers using various models. The theoretical investigations have been carried out into the performance of hydrodynamic journal bearing by adopting various methods, which are classified in two categories as: First category is the methods which comprise a full numerical treatment of temperature variation across the lubrication film thickness in energy equation using Finite Difference Method (FDM) or Finite Element Method (FEM). Second one is the methods which incorporate polynomial approximation to evaluate the transverse temperature variation in the lubrication film thickness. Both approaches mentioned can be used for the analysis of hydrodynamic bearings and have certain merits. The first approach is relatively accurate at the expense of computational speed and time, whereas the second is relatively fast at the expense of accuracy.

In present chapter, the thermal studies of non-circular journal bearings: offset-halves and elliptical have been presented using thermohydrodynamic approach.

### **3.1. Film thickness equation**

10 Performance Evaluation of Bearings

at larger coupling numbers.

relatively fast at the expense of accuracy.

**bearing** 

Polynomial yields more accurate results in comparison to Parabolic Profile approximation. However, the former is algebraically more complex to tackle in comparison to the later; the authors also observed that the film temperatures computed by Parabolic Profile approximation are lower in comparison to Legendre Polynomial approximation. Further, it has been concluded by them that the computational time taken in solution of coupled governing equation with both temperature profile approximation have only marginal difference. Mishra et al. [2007] have considered the non-circularity in bearing bore as elliptical and made a comparison with the circular case to analyse the effect of irregularity. It was reported that, with increasing non-circularity, the pressure gets reduced and temperature rise obtained is less. Chauhan et al. [2010] have presented a thermohydrodynamic study for the analysis of elliptical journal bearing with three different grade oils. The authors have made a comparative study of rise in oil-temperatures, thermal pressures, and load carrying capacity for three commercial grade oils. Rahmatabadi et al. [2010] have considered the non-circular bearing configurations: two, three and four-lobe lubricated with micropolar fluids. The authors have modified the Reynolds equation based on the theory of micropolar fluids and solved the same by using finite element methods. It has been observed by the authors that micropolar lubricants can produce significant enhancement in the static performance characteristics and the effects are more pronounced

**3. Numerical techniques for the thermal analysis of non-circular journal** 

In earlier works, the bearing performance parameters have been computed by solving the Reynolds equation only. Over the years, many researchers have proposed number of mathematical models. A more realistic thermohydrodynamic (*THD*) model for bearing analysis has been developed which treats the viscosity as a function of both the temperature and pressure. Moreover, it also considers the variation of temperature across the film thickness and through the bounding solids (housing and Journal). The thermohydrodynamic model also presents coupled solutions of governing equations by incorporating appropriate boundary conditions and considering the heat conduction across the bearing surfaces. Even the importance of *THD* studies in hydrodynamic bearings can be justified by looking at the large volumes of research papers that are being published by researchers using various models. The theoretical investigations have been carried out into the performance of hydrodynamic journal bearing by adopting various methods, which are classified in two categories as: First category is the methods which comprise a full numerical treatment of temperature variation across the lubrication film thickness in energy equation using Finite Difference Method (FDM) or Finite Element Method (FEM). Second one is the methods which incorporate polynomial approximation to evaluate the transverse temperature variation in the lubrication film thickness. Both approaches mentioned can be used for the analysis of hydrodynamic bearings and have certain merits. The first approach is relatively accurate at the expense of computational speed and time, whereas the second is The film thickness equations for offset-halves journal bearing (Fig. 1 (a)) are given as Sehgal et al. [2000]:

$$h = c\_m \left[ \left( \frac{1+\delta}{2\delta} \right) + \left( \frac{1-\delta}{2\delta} \right) \cos\theta - \varepsilon \sin\left(\phi - \theta\right) \right] \text{ (0<\theta<180)}\tag{1}$$

$$h = c\_m \left[ \left( \frac{1+\delta}{2\delta} \right) - \left( \frac{1-\delta}{2\delta} \right) \cos\theta - \varepsilon \sin\left(\phi - \theta\right) \right] \tag{180} \forall \theta \triangleleft \theta \triangleleft \theta \triangleleft \theta$$

In these equations, *h* represents film thickness for circular journal bearing, *C* represents radial clearance, represents eccentricity ratio, and represents angle measured from the horizontal split axis in the direction of rotation. *Cm* denotes minimum clearance when journal centre is coincident with geometric centre of the bearing, denotes offset factor ( / *C C <sup>m</sup>* ), and denotes attitude angle.

The film thickness equations for elliptical journal bearing (Fig. 1 (b)) are given as Hussain et al. [1996]:

$$h = c\_m \left[ 1 + E\_M + \varepsilon\_1 \cos \left( \theta + \phi - \phi\_1 \right) \right], \text{ for } 0 \le \theta \le 180 \tag{3}$$

$$h = \varepsilon\_m \left[ 1 + E\_M + \varepsilon\_2 \cos \left( \theta + \phi - \phi\_2 \right) \right], \text{ for (180\% } \theta \text{ è } \text{360)}\tag{4}$$

Different parameters used in eqns. (3) & (4) are given as:

$$\begin{aligned} \mathcal{E}\_1 &= \left(E\_M\,^2 + \varepsilon^2 - 2E\_M\varepsilon\cos\phi\right)^{\frac{1}{2}}; \; \mathcal{E}\_2 = \left(E\_M\,^2 + \varepsilon^2 + 2E\_M\varepsilon\cos\phi\right)^{\frac{1}{2}} \\\\ \phi\_1 &= \pi - \tan^{-1}\left(\frac{\varepsilon\sin\phi}{E\_M - \varepsilon\cos\phi}\right); \; \phi\_2 = \tan^{-1}\left(\frac{\varepsilon\sin\phi}{E\_M + \varepsilon\cos\phi}\right); \; E\_M = \left(\frac{C\_h - C\_m}{C\_m}\right) \end{aligned}$$

In eqns. (3 and 4), *h* represents film thickness for elliptical journal bearing, *EM* represents elliptical Ratio, 1 2 , represents eccentricity ratio from 0-180 Deg. (upper lobe) and 180-360 Deg. (lower lobe) respectively, 1 2 , represents attitude angles from 0-180 Deg. (upper lobe) and 180-360 Deg. (lower lobe) respectively and *Ch* represents horizontal clearance for elliptical journal bearing. Film thickness represented by eqns. 1, 3 corresponds to upper lobe whereas eqns. 2, 4 represents film thickness for lower lobe of the.

### **3.2. Reynolds equation**

All the simplifying assumptions necessary for the derivation of the Reynolds equation are listed below Stachowiak and Batchelor [1993]:


For steady-state and incompressible flow, the Reynolds equation is Hussain et al. [1996]:

$$\frac{\partial}{\partial \mathbf{x}} \left( \frac{h^3}{\mu} \frac{\partial p}{\partial \mathbf{x}} \right) + \frac{\partial}{\partial z} \left( \frac{h^3}{\mu} \frac{\partial p}{\partial z} \right) = 6Ll \frac{\partial h}{\partial \mathbf{x}} \tag{5}$$

Thermal Studies of Non-Circular Journal Bearing Profiles: Offset-Halves and Elliptical 13

<sup>2</sup> <sup>55</sup> 9

(8)

2 2

12 3 *T a ay ay* (10)

represents Barus viscosity-

(9)

*<sup>H</sup> <sup>E</sup> A* 

<sup>1</sup> <sup>44</sup> 9

*P TT* <sup>0</sup>

represents absolute viscosity of the lubricant at oil inlet temperature,

 

*H H <sup>E</sup> A* ;

Equation (6) assumes viscosity as constant and gives isothermal pressures, whereas eqn. (7) assumes viscosity as a variable quantity and gives thermal pressures. The coefficients appearing in eqn. (6 & 7) are given in Appendix-I. The variation of viscosity with temperature and pressure has been simulated using the following viscosity relation Sharma

> *refe*

pressure index, *T* represents lubricating film temperature, and *<sup>o</sup> T* represents oil inlet

The energy equation for steady-state and incompressible flow is given as Sharma and

*TT T u w Cu w K x zyy y y*

Here, *CP* represents specific heat of the lubricating oil, *K* represents thermal conductivity of the lubricating oil, and *u w*, represents velocity components in X- and Z-directions. The term on the left hand side in eqn. (9) represents the energy transfer due to convection, and the first, second terms on right hand side of the eqn. (9) represents the energy transfer due to conduction and energy transfer due to dissipation respectively. In eqn. (9), x-axis represents the axis along the circumference of bearing, y-axis represents the axis along the oil film thickness and z-axis represents the axis across the width of bearing. The variation of temperature across the film thickness in equation (9) is approximated by parabolic temperature profile. It is pertinent to add here that the temperature computed by this approach have been reported to be on lower side in comparison to those obtained through Legendre polynomial temperature profile approximation by Sharma and Pandey [2006]. The temperature profile across the film thickness is represented by a second order polynomial as:

<sup>2</sup>

In order to evaluate the constants appearing in eqn. (10), the following boundary conditions

 

 

<sup>1</sup> <sup>33</sup> 9

*H H <sup>E</sup> A* ;

represents temperature-viscosity coefficient of lubricant,

*P*

At 0, *<sup>L</sup> y T T*

and Pandey [2007]:

In eqn. (8), *ref*

temperature.

Pandey [2007]:

are used:

**3.3. Energy Equation** 

Here, *P* represents film pressure, represents absolute viscosity of the lubricant, and *U* represents velocity of journal.

This equation is set into finite differences by using central difference technique. The final form is reproduced here.

$$P(i\_{\prime})\_{iso} = A1P(i + 1\_{\prime}j)\_{iso} + A2P(i - 1\_{\prime}j)\_{iso} + A3P(i, j + 1)\_{iso} + A4P(i, j - 1)\_{iso} - A5 \tag{6}$$

$$P(i,j)\_{\text{fit}} = E11P(i+1,j)\_{\text{th}} + E22P(i-1,j)\_{\text{th}} + E33P(i,j+1)\_{\text{th}} + E44P(i,j-1)\_{\text{th}} - E55\tag{7}$$

Where, 3 2 2 <sup>3</sup> <sup>11</sup> 2 *h hh <sup>A</sup> d d* ; 3 2 2 <sup>3</sup> <sup>22</sup> 2 *h hh <sup>A</sup> d d* ; 23 22 2 <sup>3</sup> <sup>33</sup> 2 *rh rh h <sup>A</sup> dz dz z* ; 23 22 2 <sup>3</sup> <sup>44</sup> 2 *rh rh h <sup>A</sup> dz dz z* ; 55 6 *<sup>h</sup> A Ur* ; 3 22 2 2 2 2 *h rh <sup>A</sup> d dz* ; *A A AA A AA A AA A AA A A* 1 11 / ; 2 22 / ; 3 33 / ; 4 44 / ; 5 55 / 3 <sup>6</sup> *<sup>h</sup> <sup>A</sup>* ; <sup>6</sup> <sup>7</sup> *<sup>A</sup> <sup>A</sup>* ; <sup>6</sup> <sup>8</sup> *<sup>A</sup> <sup>A</sup> z* ; 2 2 2 26 2 6 <sup>9</sup> *<sup>A</sup> r A <sup>A</sup> d dz* ; 7 2 *<sup>A</sup> <sup>F</sup> d* ; 2 *<sup>A</sup>*<sup>6</sup> *<sup>G</sup> d* ; *<sup>h</sup> <sup>B</sup>* ; 2

$$H = \frac{r^2 A 8}{2dz};\ H1 = \frac{r^2 A 6}{dz^2};\ H2 = 6 \text{L}\\
r B: \text{E11} = \frac{F+G}{A9};\ E22 = \frac{-F+G}{A9};\ H3 = \frac{F+G}{A9};\ H4 = \frac{F+G}{A9};\ H4 = \frac{F+G}{A9}$$

Thermal Studies of Non-Circular Journal Bearing Profiles: Offset-Halves and Elliptical 13

$$E33 = \frac{H + H1}{A9};\ E44 = \frac{-H + H1}{A9};\ E55 = \frac{-H2}{A9}$$

Equation (6) assumes viscosity as constant and gives isothermal pressures, whereas eqn. (7) assumes viscosity as a variable quantity and gives thermal pressures. The coefficients appearing in eqn. (6 & 7) are given in Appendix-I. The variation of viscosity with temperature and pressure has been simulated using the following viscosity relation Sharma and Pandey [2007]:

$$
\mu = \mu\_{ref} e^{\alpha P - \gamma \left(T - T\_0\right)} \tag{8}
$$

In eqn. (8), *ref* represents absolute viscosity of the lubricant at oil inlet temperature, represents temperature-viscosity coefficient of lubricant, represents Barus viscositypressure index, *T* represents lubricating film temperature, and *<sup>o</sup> T* represents oil inlet temperature.

#### **3.3. Energy Equation**

12 Performance Evaluation of Bearings

**3.2. Reynolds equation** 

fluids.

expansion.

listed below Stachowiak and Batchelor [1993]:

2. Pressure is constant through the film.

5. Lubricant behaves as a Newtonian fluid.

8. There is no vertical flow across the film.

3 2 2 <sup>3</sup> <sup>11</sup>

 

2 *h hh <sup>A</sup> d d* 

23 22

2

<sup>6</sup> <sup>8</sup> *<sup>A</sup> <sup>A</sup> z* ;

2 2 <sup>6</sup> <sup>1</sup> *r A <sup>H</sup>*

2 <sup>3</sup> <sup>44</sup>

<sup>6</sup> <sup>7</sup> *<sup>A</sup> <sup>A</sup>* 

<sup>2</sup> 8 2 *r A <sup>H</sup> dz* ;

;

*rh rh h <sup>A</sup> dz dz z* 

Here, *P* represents film pressure,

represents velocity of journal.

form is reproduced here.

Where,

3 <sup>6</sup> *<sup>h</sup> <sup>A</sup>* ;

same as that of the boundary. 4. Flow is laminar and viscous.

All the simplifying assumptions necessary for the derivation of the Reynolds equation are

1. Body forces are neglected i. e. there are no extra outside fields of forces acting on the

3. No slip at the boundaries as the velocity of the oil layer adjacent to the boundary is the

6. Inertia and body forces are negligible compared with the pressure and viscous terms. 7. Fluid density is constant. Usually valid for fluids, when there is not much thermal

For steady-state and incompressible flow, the Reynolds equation is Hussain et al. [1996]:

<sup>6</sup> *hh h p p <sup>U</sup> x xz z x*

(5)

23 22

2

;

9

*F G <sup>E</sup> A* ;

*<sup>h</sup> <sup>B</sup>* 

;

;

2 <sup>3</sup> <sup>33</sup>

3 22 2 2 2 2 *h rh <sup>A</sup> d dz* 

 

7 2 *<sup>A</sup> <sup>F</sup> d* ; 2 *<sup>A</sup>*<sup>6</sup> *<sup>G</sup> d*;

*rh rh h <sup>A</sup> dz dz z* 

represents absolute viscosity of the lubricant, and *U*

 

This equation is set into finite differences by using central difference technique. The final

( , ) 1 ( 1, ) 2 ( 1, ) 3 ( , 1) 4 ( , 1) 5 *iso iso iso iso iso Pij APi j A Pi j A Pij A Pij A* (6)

( , ) 11 ( 1, ) 22 ( 1, ) 33 ( , 1) 44 ( , 1) 55 *th th th th th Pij E Pi j E Pi j E Pij E Pij E* (7)

*h hh <sup>A</sup> d d* 

; 55 6 *<sup>h</sup> A Ur*

*A A AA A AA A AA A AA A A* 1 11 / ; 2 22 / ; 3 33 / ; 4 44 / ; 5 55 /

*dz* ; 2 6 *H UrB* ; <sup>11</sup>

3 2 2 <sup>3</sup> <sup>22</sup>

2

 

2 2 2 26 2 6 <sup>9</sup> *<sup>A</sup> r A <sup>A</sup> d dz* 

;

;

;

9 *F G <sup>E</sup> A* ; 22

 

3 3

;

The energy equation for steady-state and incompressible flow is given as Sharma and Pandey [2007]:

$$
\rho \mathcal{L}\_P \left( \mu \frac{\partial T}{\partial \mathbf{x}} + w \frac{\partial T}{\partial \mathbf{z}} \right) = \frac{\partial}{\partial y} \left( K \frac{\partial T}{\partial y} \right) + \mu \left| \left( \frac{\partial u}{\partial y} \right)^2 + \left( \frac{\partial w}{\partial y} \right)^2 \right| \tag{9}
$$

Here, *CP* represents specific heat of the lubricating oil, *K* represents thermal conductivity of the lubricating oil, and *u w*, represents velocity components in X- and Z-directions. The term on the left hand side in eqn. (9) represents the energy transfer due to convection, and the first, second terms on right hand side of the eqn. (9) represents the energy transfer due to conduction and energy transfer due to dissipation respectively. In eqn. (9), x-axis represents the axis along the circumference of bearing, y-axis represents the axis along the oil film thickness and z-axis represents the axis across the width of bearing. The variation of temperature across the film thickness in equation (9) is approximated by parabolic temperature profile. It is pertinent to add here that the temperature computed by this approach have been reported to be on lower side in comparison to those obtained through Legendre polynomial temperature profile approximation by Sharma and Pandey [2006]. The temperature profile across the film thickness is represented by a second order polynomial as:

$$T = a\_1 + a\_2y + a\_3y^2\tag{10}$$

In order to evaluate the constants appearing in eqn. (10), the following boundary conditions are used:

At 0, *<sup>L</sup> y T T*

$$\text{At } \ y = h, T = T\_{11}$$

and

$$T\_m = \frac{1}{h} \int\_0^h Tdy$$

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

$$T = T\_L - \left(4T\_L + 2T\_{\rm U} - 6T\_m\right) \left(\frac{y}{h}\right) + \left(3T\_L + 3T\_{\rm U} - 6T\_m\right) \left(\frac{y}{h}\right)^2\tag{11}$$

Thermal Studies of Non-Circular Journal Bearing Profiles: Offset-Halves and Elliptical 15

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

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

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

*P* 0 at 0 *x* and *x l*

*<sup>L</sup> u u* at *y* 0 and 0 *x l*

*u* 0 at *y h* and 0 *x l*

<sup>0</sup> *T T* at 0 *x* and 0 *x h*

*<sup>L</sup> T T* at *y* 0 and 0 *x l*

*<sup>U</sup> T T* at *y h* and 0 *x l*

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

**3.5. Computation procedure** 

solution is discussed below.

values are assumed to be equal to the inlet oil viscosity.

a. Reynolds Equation

b. Energy equation

employing the Simpson rule.

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

Final form of the energy equation is represented as:

$$\begin{aligned} &6\mathcal{T}\_{L} + 6\mathcal{T}\_{\mathrm{U}} - 12\mathcal{T}\_{m} - \frac{\rho\mathcal{C}\_{P}h^{4}}{120K\mu} \frac{\partial\mathcal{P}}{\partial\mathbf{x}} \left(\frac{\mathcal{T}\_{\mathrm{L}}}{\partial\mathbf{x}} + \frac{\mathcal{C}\mathcal{T}\_{\mathrm{U}}}{\partial\mathbf{x}} - 12\frac{\mathcal{C}\mathcal{T}\_{m}}{\partial\mathbf{x}}\right) - \frac{\rho\mathcal{C}\_{P}h^{4}}{120K\mu} \frac{\partial\mathcal{P}}{\partial\mathbf{z}} \\ &+ \left(\frac{\partial\mathcal{T}\_{\mathrm{L}}}{\partial\mathbf{z}} + \frac{\partial\mathcal{T}\_{\mathrm{U}}}{\partial\mathbf{z}} - 12\frac{\mathcal{C}\mathcal{T}\_{m}}{\partial\mathbf{z}}\right) - \frac{\rho\mathcal{C}\_{P}h^{2} \left(u\_{\mathrm{L}} + u\_{\mathrm{U}}\right)}{2K} \frac{\partial\mathcal{T}\_{m}}{\partial\mathbf{x}} - \frac{\rho\mathcal{C}\_{P}h^{2} \left(u\_{\mathrm{U}} - u\_{\mathrm{L}}\right)}{12K} \left(\frac{\mathcal{C}\mathcal{T}\_{\mathrm{U}}}{\partial\mathbf{x}} - \frac{\mathcal{C}\mathcal{T}\_{\mathrm{L}}}{\partial\mathbf{x}}\right) \\ &+ \frac{h^{4}}{12K\mu} \left[\left(\frac{\partial\mathcal{P}}{\partial\mathbf{x}}\right)^{2} + \left(\frac{\partial\mathcal{P}}{\partial\mathbf{z}}\right)^{2}\right] + \frac{\mu\left(u\_{\mathrm{U}} - u\_{\mathrm{L}}\right)^{2}}{K} = 0 \end{aligned} \tag{12}$$

#### **3.4. Heat conduction equation**

The temperature in bush is determined by using the Laplace equation within the bearing material as given below Hori [2006]:

$$\frac{\partial^2 T\_b}{\partial \mathbf{x}^2} + \frac{\partial^2 T\_b}{\partial \mathbf{y}^2} + \frac{\partial^2 T\_b}{\partial \mathbf{z}^2} = \mathbf{0} \tag{13}$$

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 reproduced here.

$$\begin{aligned} T\_b(i,j,k) &= \text{E1T}\_b(i+1,j,k) + \text{E1T}\_b(i-1,j,k) + \text{E2T}\_b(i,j+1,k) + \text{E2T}\_b(i,j-1,k) + \\ \text{E3T}\_b(i,j,k+1) &+ \text{E3T}\_b(i,j,k-1) \\ &= \text{E2T}\_b(i,j,k+1) \end{aligned} \tag{14}$$

Where, 22 2 2 2 22 *<sup>E</sup> r d dy dz* ; 2 2 <sup>1</sup> *<sup>F</sup>*<sup>11</sup> *r d* ; 2 <sup>1</sup> *<sup>F</sup>*<sup>22</sup> *dz* ; 2 <sup>1</sup> *<sup>F</sup>*<sup>33</sup> *dy* ;

$$E1 = F11 / E; E2 = F22 / E; E3 = F33 / E$$
