A semi analytical approach in thermal analysis of Hydrodynamic lubrication of journal bearing

Materials Today: Proceedings xxx (xxxx) xxx

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A semi analytical approach in thermal analysis of Hydrodynamic lubrication of journal bearing P. Sudam Sekhar a,⇑, Venkata Subrahmanyam Sajja b, V.R.K. Murthy a, S. Parthiban a a b

Division of Mathematics, Vignan’s Foundation for Science, Technology and Research, Guntur, India Department of Mathematics, Koneru Lakshmaiah Education Foundation, Guntur 522502, India

a r t i c l e

i n f o

Article history: Received 16 July 2019 Received in revised form 15 October 2019 Accepted 18 October 2019 Available online xxxx Keywords: Hydrodynamic lubrication Power law fluid Temperature effects Non-Newtonian Journal bearing

a b s t r a c t Hydrodynamic lubrication of journal bearing considering rotation of the journal including the effects of temperature is studied in this work. It describes the theoretical analysis of power law fluid film lubrication of journal. A semi analytical solution is obtained by solving the continuity equation and momentum equation along with thermal energy equation under isothermal boundaries. The obtained results are compared and found that they are in good agreement with the results available in the literature. Further, the delta profile which is the location of the points of zero velocity gradients is also presented in order to get the simplified form the Reynolds and energy equations. The obtained results are compared with the experimental results and seen to be a good agreement with the effect of temperature. Ó 2019 2019 Elsevier Ltd. All rights reserved. Selection and peer-review under responsibility of the scientific committee of the National conference on Functionality of Advanced Materials.

1. Introduction Hydrodynamic lubrication with Newtonian/non-Newtonian proposes that the load carrying surfaces of the bearing are isolated by a moderately thick film of lubricant, to avert metal to metal contact, and the stability subsequently acquired may be clarified by the laws of mechanics of fluid. In this situation the pressure state of the lubricant is difficult to guarantee better performance of the machine elements such as journal bearings. Many work has been done to study the different aspect of journal bearing at heavily loaded condition. Sun, Yongping et al. [1] studied the effect of temperature by solving Reynolds equation numerically for the high speed hydrodynamic lubricating bearing and oil film pressure distribution. Borse, N. V et al. [2] studied the journal bearing under thermal boundary conditions in steady-state. Finite difference method is used to solve Reynolds equation, viscosity-temperature relationship and two dimensional energy equations simultaneously. Chatterton, Steven, et al. [3] investigated experimentally to a 160-mm diameter cylindrical journal bearing with two axial grooves and observe the influence of the applied static load and rotational speed. ⇑ Corresponding author.

Mahdi, M. A et al. [4] considered a cavitated finite journal bearing with the effect of both journal speed and couple stress fluids and investigated theoretically. Also by using Elrod cavitations algorithm the performance characteristics are studied by finding the solution to the modified Reynolds equation. It has been concluded that the non- Newtonian lubricants exhibits improved in the fill-film pressure and the load-capacity. In the light of the above discussion, thermal effects in hydrodynamics lubrication considering the rotation of the journal has been discussed with the assumption that pressure and mean temperature of the fluid film depends on the consistency of the Powerlaw fluid. This concept is so far never discussed in any literature hence an attempt is made to analyze this problem. The modified Reynolds equation along with energy equations are numerically solved simultaneously [9,10] and discussed the temperature and pressure relationships. 2. Mathematical model The governing equations of fluid flow with respect to the hydrodynamic lubrication under some common assumptions [5,9,11] are

@ @y




s

 @u dp ¼ @y dx

ð1Þ

E-mail addresses: [email protected], [email protected] (P. Sudam Sekhar). https://doi.org/10.1016/j.matpr.2019.10.117 2214-7853/Ó 2019 2019 Elsevier Ltd. All rights reserved. Selection and peer-review under responsibility of the scientific committee of the National conference on Functionality of Advanced Materials.

Please cite this article as: P. Sudam Sekhar, V. Subrahmanyam Sajja, V. R. K. Murthy et al., A semi analytical approach in thermal analysis of Hydrodynamic lubrication of journal bearing, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.10.117

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P. Sudam Sekhar et al. / Materials Today: Proceedings xxx (xxxx) xxx

@u @v þ ¼0 @x @y

ð2Þ 

where

n1

@u s ¼ m 

and m ¼ m0 ea pbðT m T 0 Þ

@y

Z

ð3Þ

 1=n h i nþ1 nþ1 n 1 dp1 ðd  yÞ n  ðdÞ n ; 0 6 y 6 d n þ 1 m1 dx

for the region, h1 < h 6

 u3 ¼

h

T dy ¼ h T m

With

 u2 ¼



n nþ1

ð11Þ 

e cos h Þ

ð5Þ

where the radial clearance is c and e is the distance between the centre of the journal and the centre of the bearing (Fig. 1.). The assumed boundary conditions related to the problem under consideration are:

u ¼ U at y ¼ h and u ¼ 0 at y ¼ 0

ð6Þ

where the journal velocity is denoted as U. @u@u = 0 at y ¼ d in the region ðp 6 h 6 pÞ which is divided @y @y into two sub regions

p 6 h < h1 and  h1 6 h <

p

The boundary conditions for the velocity for the given geometry are:

@u1 @u2 P 0; d 6 y 6 h; 6 0 ; 0 6 y 6 d; I :  p @y dy 6 h < h1

ð7Þ

@u3 @u4 P 0 ; 0 6 y 6 d; 6 0 ; d 6 y dy @y 6 h; II :  h1 < h < From Eq. (1) u1 p 6 h < h1 as-



u1 ¼ U þ

n nþ1



p, u3 and u4 may be obtained as

1=n h i nþ1 nþ1 1 dp2  ðdÞ n  ðd  y Þ n ; 0 6 y 6 d m2 dx

ð4Þ

0

And h=c ¼ ð1 

and

1 dp1 m1 dx

ð8Þ

p u2

is

obtained

in

the

ð10Þ

n nþ1

u4 ¼ U þ

 

1 dp2 m2 dx

1=n h

ðh  dÞ

nþ1 n

 ðy  dÞ

nþ1 n

i

;d6y6h ð12Þ

Since the fluid passes through every cross section is constant, it follows from Eq. (2) as

@ @x

Z

h

u dy ¼ 0 ) 0

and with

dp dx

dQ ¼0 dx

¼ 0 ; at h ¼ h1 ; h ¼ h1 we get Q = U h1/2 and

#n  n " dp1 2n þ 1 U ðh  dÞ  U ðh1 =2Þ ; p < h 6 h1 ¼ m1 R 2nþ1 2nþ1 n dh ðdÞ n þ ðh  dÞ n

ð13Þ

#n  n " dp2 2n þ 1 U ðh1 =2Þ  U ðh  dÞ ; h1 < h 6 p ¼ m2 R 2nþ1 2nþ1 n dh ðdÞ n þ ðh  dÞ n

ð14Þ

Using the velocity matching conditions: u1 = u2 and u3 = u4 at y = d, the following equation is obtained:

h i 2 nþ1 nþ1 3   n n 2n þ 1 4½ðh  dÞ  ðh1 =2Þ ðh  dÞ  ðdÞ 5 ¼1 2nþ1 2nþ1 nþ1 ðdÞ n þ ðh  dÞ n

ð15Þ

region: 3. Energy equation

1=n h

i nþ1 nþ1 ðy  dÞ n  ðh  dÞ n ; d 6 y 6 h

The equation of thermal conductivity with some assumption [6,7] considered in this problem are -

ð9Þ k

@2T @u ¼ 0 þs @y2 @y 

where

n1

@u s ¼ m   @y

ð16Þ @u @y

This may be solved by using the following boundary conditions-

T ¼ T 11 at y ¼ h ; T ¼ T 12 at y ¼ 0

ð17Þ

From Eq. (16) in the regionp 6 h < h1 , T 11 and T 12 can be obtained as:

T 11 ¼ 

m  1 dp nþ1 n 3nþ1 n2 1 1 ðy  dÞ n m1 dx k ð2n þ 1Þð3n þ 1Þ

þ c1 y þ d1 ; d 6 y 6 h T 12 ¼ 

ð18Þ

m  1 dp nþ1 n 3nþ1 n2 1 1 ðd  yÞ n m1 dx k ð2n þ 1Þð3n þ 1Þ

þ c2 y þ d2 ; 0 6 y 6 d

ð19Þ

In the same way in the region

T 21 ¼  þ

m  1 dp nþ1 n 3nþ1 n2 2 2  ðd  yÞ n m2 dx k ð2n þ 1Þð3n þ 1Þ a1 y þ b1

ð20Þ

Fig. 1. Journal bearing Geometry.

Please cite this article as: P. Sudam Sekhar, V. Subrahmanyam Sajja, V. R. K. Murthy et al., A semi analytical approach in thermal analysis of Hydrodynamic lubrication of journal bearing, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.10.117

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P. Sudam Sekhar et al. / Materials Today: Proceedings xxx (xxxx) xxx

T 22 ¼ 

m  1 dp nþ1 n 3nþ1 n2 2 1  ðy  dÞ n m2 dx k ð2n þ 1Þð3n þ 1Þ

þ a2 y þ b2

ð21Þ

Further T m1 and T m2 can be defined as:

T m1

1 ¼ h

T m1

Z

h

0

1 T dy ¼ h

"Z

d

0

1 T 12 dy þ h

Z

#

h

T 11 dy

or

d

" # 4nþ1 4nþ1 1 nA ðh  dÞ n þ ðdÞ n ¼ ðT 11 þ T 12 Þ þ 2 ð4n þ 1Þ h " 3nþ1 3nþ1 # ðdÞ n þ ðh  dÞ n ; p 6 h < h1 A 2

where A ¼ 

m1  1

dp1 m1 dx

k

nþ1  n

n2 ð2nþ1Þð3nþ1Þ

ð22Þ



Similarly,

T m2

" # 4nþ1 4nþ1 1 nB ðh  dÞ n þ ðdÞ n ¼ ðT 21 þ T 22 Þ þ 2 ð4n þ 1Þ h " 3nþ1 3nþ1 # n n ðdÞ þ ðh  dÞ ; h1 < h < p B 2

 where B ¼  mk2  m12

dp2 dx

nþ1  n

n2 ð2nþ1Þð3nþ1Þ

ð23Þ Fig. 2. Pressure profile for different values of n.



The non-dimensional ization scheme applied to the above equations are



p ¼ ap; m ¼ mcn a; cn ¼ 

 n  n 2n þ 1 U R n c c





T m ¼ b T m ; d ¼ d=c ; h ¼ h=c

ð24Þ

Equations (13), (14), (15), (22), and (23) can be rewritten as 

 n  dp1 ¼ m1 ðf Þ dh

ð25Þ



 n  dp2 ¼ m2 ð f Þ dh

ð26Þ 



 nþ1   1  n m1 ðT 11 þ T 12 Þ  g pr ðf Þ 2 ð3n þ 1Þ



 nþ1   1  n m2 ðT 21 þ T 22 Þ  g p ð f Þ 2 ð3n þ 1Þ r

T m1 ¼



T m2 ¼

pr ¼ 

f ¼





ð29Þ 

ðh  dÞ  ðh1 =2Þ 



2nþ1 n



þ ðdÞ 2

g ¼

ð28Þ

  Ubc c ka R

ðh  dÞ 

ð27Þ



ð30Þ

2nþ1 n



4nþ1 n



4nþ1 n

n þ ðdÞ 6ðh  dÞ 4  ð4n þ 1Þ h

3

2



6ðdÞ 7 5 4

3nþ1 n





3nþ1 n

þ ðh  dÞ 2

parameters. For different values of n as well as for different values

3 7 5

Fig. 3. Pressure profile for Newtonian fluid with different values of epsilon.



ð31Þ



of e the dimensionless pressure P and temperature T m is calculated. The set of numerical values used in these calculations are 

R ¼ 4 : 5774 ; a ¼ 1 : 6 X 106 dyne 1 m 2 ; 

4. Result and discussion

c ¼ 0:4 ; e ¼ 0:66 ; 0:86 ; Pr ¼ 300:875 

R.K. Fehlberg Method is used to solve the Eqs. (25)-(31) numerically. The pressure and temperature are obtained and the variations are shown in Figs. 2–5 with respect to different

Fig. 2 shows the pressure curve P along h for various values of 

power-law index ‘n’. Pressure P increases continuously from -p to –h1, then decreases from –h1 to h1 and then increases

Please cite this article as: P. Sudam Sekhar, V. Subrahmanyam Sajja, V. R. K. Murthy et al., A semi analytical approach in thermal analysis of Hydrodynamic lubrication of journal bearing, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.10.117

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P. Sudam Sekhar et al. / Materials Today: Proceedings xxx (xxxx) xxx

5. Conclusion Hydrodynamic non-Newtonian lubrication of journal bearing including thermal effect is studied and only journal is considered to move with certain velocity. Pressure and temperature profiles are analyzed for both Newtonian and non-Newtonian cases. Further, d- profile has been added in order to reduce the mathematical complexity of the problem that yields semi analytical solution. It has been observed that a significant increase in Pressure with power-law index ‘n’ for fixed value of epsilon. Also it is clear that temperature increases with increasing values of n and with increase of epsilon temperature decreases for Newtonian fluid. Moreover it has been clearly seen that journal bearing exhibit the same trends of pressure and temperature lubricated with Newtonian as well as non- Newtonian fluid. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Appendix. A: Mathematical symbols (Nomenclature) Fig. 4. Temperature profile with different value of n with fixed epsilon.

c m p R T V 

P h n Q t u,v h 

Tm

radial clearance consistency index hydrodynamic pressure radios of the journal temperature squeeze velocity Non dimensional pressure oil film thickness flow behaviour index flow flux time of approach velocity components angular co-ordinate Non dimensional mean temperature

References

Fig. 5. Temperature profile for newtonian fluid with different values of epsilon.

continuously up to p. Similar trends of Peng and Khonsari [7] has 

been noticed for each n for the pressure P against h. If the pressure profile in Fig. 3 is considered for 0 6 h 6 p instead ofp 6 h 6 p then it becomes very similar to [8,10]. 

Figs. 4 and 5 represents the temperature distribution T m against h for various values of ‘n’ and various values of epsilon. It is 

observed that T m decreases as h increases except in the vicinity 

of zero where the T m increases with h from h1 to zero and then again decreases as h increases from zero to þh1 and finally 

increases with h but marginally. The change in T m near the inlet and outlet shows the effect of the convection is left. Chatterton, Steven, et al. [3,9,11] also presented the similar trends.

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Please cite this article as: P. Sudam Sekhar, V. Subrahmanyam Sajja, V. R. K. Murthy et al., A semi analytical approach in thermal analysis of Hydrodynamic lubrication of journal bearing, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.10.117