Chapter I30 Figure 5.6 shows a plot of these relationships for t o l l L = 0.25, 0.5, and 0.75 respectively The principle of superposition can be applied to determine the surface temperature rise for any given function of heat flux It can be determined either as a summation or an integration of the effect of incremental step inputs that convolute the given function An illustration of this is the study of the effect of applying the same total quantity of heat input with a triangular flux with the same maximum value but with different slopes, S1 and S2, during the increase and decrease phases respectively Using the integration approach, therefore: I c = -+ to, A T ( t ) = 11.12S1 Jz: dr JG Figure b.6 Temperature rise as a function of time on the surface of a steel slab subjected to constatn flux for different periods to Thermal Considerations in Tribology 131 Figures 5.7-5.11 show plots of the surface temperature history for S1/S2 9, 4,1, and respectively = Figure 5.12 shows the maximum surface temperature as a function of S1/S2 the same total heat flow qmax can be seen from Fig 5.12 that the for It highest surface temperature rise occurs as the ratio SI/S2 decreases If the heat flux function is determined from experimental data and is difficult to integrate, the temperature rise can be obtained by summing the effects of incremental steps that are constructed to convolute the function as illustrated in Fig 5.13 Better accuracy can be obtained as the number of steps increases The temperature rise at the surface for this case can be calculated as: b, t=t2+ t3, 4, &E1 = T + ( - ) = T AT=- 112 Time (sec) Figure 5.1 Temperature rise on the surface of a steel slab due to a total heat input q applied at different rates ( t O / t l = 0.1) 0 10 Time (sec) Figure 5.8 Temperature rise on the surface due to a total heat input q applied at different rates ( t O / t l = 0.2) Figure 5.9 Temperature rise on the surface due to a total heat input applied at different rates ( t o / t l = 0.5) 132 Figure 5.10 Temperature rise on the surface due to a total heat input q applied at different rates (lo/tl = 0.8) 1.5 3.0 Q) 1.0 9) j - 2.0 c m P C m U s c 1.0 0.5 - g E 6 10 0.0 Time (sec) Figure 5.1 Temperature rise on the surface due to a total heat input q applied at different rates ( t o / t l = 0.9) 133 134 Chapter * * 0.0 } I I I I I S,'% Figure 5.12 Dimensionless surface temperature rise as a function of the rate of the slope ratio for the triangular heat input Figure 5.1 Convolution integration for a general heat input function Thermal Considerations in Tribology 5.5 135 HEAT PARTITION A N D TRANSIENT TEMPERATURE DISTRIBUTION IN LAYERED LUBRICATED CONTACTS This section briefly describes a generalized and efficient computer- based model developed by Rashid and Seireg [23], for the evaluation of heat partition and transient temperatures in dry and lubricated layered concentrated contacts The program utilizes finite differences with the alternating direction implicit method The program is capable of treating the transient heat transfer problem in lubricated layered contacts with any arbitrary distribution of layer properties and thicknesses It takes into consideration the time variation in speeds, load, friction coefficient, fluid film thickness between surfaces, and the effective radius of curvature of contacting solids It calculates the surface temperature distribution in the layered solids in lubricant film Also, the role of the chemical layer on surface and film temperatures in lubricated concentrated contacts can be evaluated Furthermore, the transient operating conditions, which are associated with the performance of such systems, are incorporated in temperature calculations A general model for the contact zone in sliding/rolling conditions can be approximated by two moving semi-infinite solids separated by a lubricant film, as shown in Fig 5.14 The heat generation distribution inside the lubricant film is controlled by the rheological behavior of the lubricant under different pressures, temperatures, and rolling and sliding speed In many concentrated contact problems, the moving solids may have different thermal properties, speeds, bulk temperatures and different chemical layers on their surface All these variables are introduced in the model, as well as any considered heat generation condition in the lubricant film The boundary conditions for this problem are based on the fact that the temperature gradient diminishes away from the heat generation zone Figure 5.1 Model for heat transfer in layered lubricated contacts The temperature field around the contact zone is represented by a rectangular grid containing appropriately distributed nodal points in the two solids and the lubricant tilni normal to the flow and in the flow direction The size of each division c m be changed in such ;i manner that the boundary conditions c m be satisfied for a particular problem without increasing the number o f nodes A larger number o f divisions are used ;icross the lubricant film to xcommodate the rapid chringe in both temperature and velocity across the film The niesh size is progressively expanded in each moving solid with the distances of the node from the heat generation zone The developed program hiis the following special features: The use of finite difference with the alternating direction implicit method provides considerable inodeling flexibility and computing efficiency I t is capable of handling transient variations in geometry loud, spccd, and material properties Thermal Considerations in Tribology 137 It can treat dry or lubricated multilayered contacts with relatively small layer thicknesses Because the program is developed for modeling transient conditions, it can be used for predicting traction characteristics for layered or unlayered solids by incorporation of a proper rheological model for the lubricant Starting from the ambient temperature conditions, the lubricant properties can be iteratively evaluated from the computed temperatures for any particular operating condition The program as developed would be useful in investigating the effect of the different parameters on the temperature distribution in line contacts It can provide a valuable guide for performing experimental studies to generate empirical design equations for layered surfaces It can also be utilized to develop empirical equations for lubricated layered contacts applicable to specific regimens of materials and operating conditions The results for any application can be considerably enhanced by incorporating an appropriate rheological model for the lubricant This would enable the prediction of traction, velocity, profile in the film, and the heat generation distribution in the contact zone 5.5.1 Numerical Results Numerical solutions are carried out to illustrate the capabilities of the program utilizing under the following assumptions: The heat source distributed inside the contact zone follows the dry contact pressure distribution (Hertzian pressure) Then the rate of heat generation distribution per unit volume can be calculated as: where Qmm = 4fWO(UI - U,) e W' f = coefficient of friction WO= load per unit length The solids are homogeneous with no cracks or inclusions The chemical reaction heat sources are negligible compared to frictional heat sources Chapter 138 The heat of compression in the lubricant film and the moving solids has a negligible effect on the temperature rise inside the contact zone Because the lubricant film thickness and the Hertzian contact with (x-direction) are small in comparison to the cylinder width (z-direction), the temperature gradient in the z-direction is expected to be small in comparison with those across and along the film Therefore, the conduction in the z-direction is neglected The thermal properties of the surface layers in lubricated contacts cover a wide material spectrum There are some cases where the surface layer has low thermal conductivity in comparison with the lubricant film (for example, paraffinic and the organic surface layers as compared with oil) At the same time, there are some types of coatings, like silicon carbide (Sic), which are much more conductive than any common lubricant The thermal resistance at the interface between the surface layer and the bulk solid should also be taken into consideration if the thermal boundary layer penetrates the surface layer inside the solid The developed program is utilized to study the variation in maximum film temperature versus oil film thickness for several surface layer thicknesses The attached surface layer to each moving solid is assumed to be identical and the distribution of heat generation is assumed to be uniform across the film ( w = h) For the considered example: where K , KF = thermal conductivities for the lubricant film and the surface layer respectively The maximum film temperature is expected to be strongly dependent on the layer thickness because of its low thermal conductivity in comparison with the lubricant film The results as plotted in Fig 5.15 show a gradual reduction in the influence of the layer thickness on the lubricant film temperature as the film thickness increases for the same friction heat level, as demonstrated by the upper two curves in the figure All the temperature curves for the layered contacts have the tendency to converge to a common level as the lubricant film thickness increases in magnitude This is represented in more detail in Fig 5.16, which shows the dependency of the maximum film temperature upon a wider range of surface layer I800 1600 E I4Oo I / I I I - 1200 Eg I000 - d 800 30 pin, 0.75 p m -6p-in,O.lSp-m pn 0.075 p-m i , : - E f i i 200 1000 600 - - 800 _. ._ -.-.-*_. _ _- - .- .- -_. - ,). /I I I 400 200 I for layered 1800 a 0.00 E ! 1200 : Coating Thloknerw (IO* m) 0.25 o o o I F i b lhicknesm 10 pin, 0.25 p-m 100 w-ln 2.5 u-m - Figure 5.1 Maximum film temperature versus coating thickness (insulative layers, K / K F= 6) Chapter I40 thickness (coating thickness) in thin film lubrication However, thick film lubrication does not show such behavior Any increase in surface layer thickness would initially reduce the temperature diffusion inside the solids until the heat flux leaves the contact zone Beyond this condition, any increase in surface layer thickness does not add any thermal influence to the contact zone, which explains the difference in temperature dependency on surface layer thickness for thin and thick film lubrication The same argument can explain the increase in lubricant film thickness If the lubricant film is less conductive than the surface layer, then the lubricant film thickness has much less influence on the maximum film temperature, as shown in Fig 5.17 Figure 5.18 shows the variation in surface layer temperature versus oil film thickness for different insulative layer thicknesses It should be noted that as the fluid film decreases in thickness, the same friction level will result in a higher surface film temperature Thus, chemical activity may increase to a significant level before bearing asperity surfaces actually achieve contact This has been confirmed experimentally by Klaus [20] such experimental 1800 1600 I c 0.25 I / I Film Thioknema (106 m) 1.25 E e 1200 E f 1000 I 30 p-&I, 0.76 - - - p-in, 0.16 p-m 1400 2.50 I E z 600 - 600 - G e e! E c E -400 - I s - 200 - - 200 = 400 01 - 800 pin, 0.076 y m ;800 L' 1000 / I // I 10 I 50 I 100 ' Film Thickness (10" in) Figure 5.17 Maximum film temperature versus oil film thickness for layered lubricated contacts (conductive layers, K J / K F= 1/6) Thermal Considerations in Tribology 141 observation is difficult to perform using the infrared technique [2 11, because one of the surfaces has to be transparent If the surface layer is less conductive than the lubricating oil, then the maximum surface layer temperature has a stronger dependency on the lubricant film thickness than in the case of conductive layers, as described in Figs 5.18 and 5.19 In the case of boundary lubrication, in which the asperity interaction with the solid surfaces plays a major role, the temperature level becomes even more sensitive to surface layer thickness The small contact width between the asperities generates a shallow temperature penetration across the surface layer, which increases the temperature level even for a very thin layer The following can be concluded from the investigated conditions: In the case of an insulative surface layer, the maximum rise in film temperature is strongly dependent on the surface layer thickness, whereas this is not the case for the conductive surface layer (see Figs 5.15 and 5.17) In both cases, the surface layer temperature decreases with the increase in lubricant film thickness This is attributed to the con- 1800 * // I I I loo0 1400 1600 E 1200 p-in, 0.075 p-m : g 1000 E fi G 800 - - 400 E 600 400 200 - 0 // I 10 I 50 I 100 -0 if 142 , 8oo , ;: / 1400 - ? Film Thkknea8 (10* m) : IS00 E 3e E 1200 1000 f E f E a z 800 , Chapter 30 p-h, 0.75 p-m - - - pin, 0.15 p-m : pin, 0.075 p-m 000 800 G e - 6oo400 - t Figure 5.1 Maximum surface temperature versus oil film thickness for layered lubricated contacts (conductive layers, K / K F= /6) 5.6 vection effects (see Figs 5.18 and 5.19) It should be noted here that this result occurs for the considered smooth surfaces without any asperity interaction This illustrates the importance of the surface layers on convection and consequently, the surface temperatures As can be seen in Fig 5.16, there appears to be a surface layer thickness, for each oil film thickness, beyond which the layer thickness will have no effect on the maximum temperature in the lubricant film DIMENSIONLESS RELATIONSHIPS FOR TRANSIENT TEMPERATURE A N D HEAT PARTITION The use of dimensional analysis in defining interactions in a complex phenomenon is a well-recognized art Any dimensional analysis problem raises two main questions: Thermal Considerations in Tribology 143 The minimum number of the dimensionless groups needed to describe the theoretical analysis; The physical interpretation of these groups and their most appropriate forms Dimensionless relationships for concentrated contact can be of considerable practical importance to the experimentalist and the designer Most of the theoretical analyses are based on computer solutions and the presentation of the results are generally lacking in presenting generalized trends The lack of generality is due to the fact that the presentation of the results is usually in the form of discrete examples, there is no provision of insight into the interaction between variables This section presents dimensionless relationships developed from the computer model described in the previous section which incorporate dimensionless groups representing the system parameters and operating conditions Case 1: Heat Source Moving over a Semi-Infinite Solid (Fig 5.20) This problem is used to check the validity of the modeling approach since an analytical solution by Blok [S] and Jaeger [6] is available for this case The derived equation for the maximum rise in surface temperature is obtained by using a series approximation as: Ts - TB = 1.128 Figure 5.20 Moving semi-infinite solid under a stationary heat source 144 Chapter where Ts= maximum surface temperature TB = bulk temperature A dimensional analysis is carried out by using the n theorem [24, 251 to obtain adequate dimensionless groups for this example Realizing the fact that the heat input to each material element inside the temperature field is balanced by both conductive and convective modes of heat transfer, it can be concluded that: The final form of the dimensionless equation, as a function of Peclet number, can be derived as: where pc UL - Peclet number k The log/log plot of the computed data showed a straight line correlation between the two dimensionless number in Eq (5.6) The equation of this line can be expressed as: (5.7) which is in general agreement with the analytically derived relationship The differences in the constant can be attributed to the numerical approximation in the computer model Case 2: Sliding/Rolling D y Contacts (see Fig 5.21) r The maximum temperatures on the contacting surfaces can be developed from Eq (5.7) as: Thermal Considerations in Tribology Figure 5.21 I45 Two cylinders under dry sliding condition Tsl = Ts2 in this case, therefore, the heat partition coefficient a can be calculated for equal bulk temperatures as: a= (5.10) and accordingly: (5.1 1) Blok [2] derived an identical equation for the flash temperature The contact in Blok's equation is determined analytically as 1.1 instead of 1.03 determined from the developed program Case 3: Heat Source with a Hertzian Distribution Moving over a Layered Semi-Infinite Solid (Fig 5.22) In this case, the relationship for the maximum rise in the solid surface temperature can be obtained using the 7t theorem as: By using the value of the penetration depth D in the solid at the trailing edge [26], Eq (5.12) can be rewritten as: Chapter I46 L U - Figur 5.22 Layered semi-infinite solid moving under a stationary heat source Similarly, the maximum rise in the surface layer temperature is derived as: or by using the penetration depth concept: where = temperature penetration depth at the trailing edge D= UhipocO “=5-= film k0 required entry distance for temperature penetration across the Thermal Considerations in Tribology 147 Tso = maximum rise in the solid surface temperatue for unlayered semi-infinite solids for the same heat input Ts = maximum rise in the solid surface temperature for unlayered semi-infinite solids for the same heat input Case 4: Lubricated Rolling/Sliding Contacts The temperature distribution and heat partition in heavily loaded lubricated contacts is not yet fully understood due to the ill-defined boundary conditions and the modeling complexities in the problem In this part of the work, a number of dimensionless equations are derived for predicting both the maximum film temperature and the heat partition between the contacting solids The model to be analyzed is shown in Fig 5.23 It represents two rollinglsliding cylinders having different radii, thermal properties, and bulk temperatures, which are separated by lubricant film thickness h Because the lubricant is subjected to extremely high pressures and shear stresses, which only act for a very short time, the assumption that the lubricant behaves as Newtonian liquid is not valid Experiments demonstrated that typical lubricants exhibit liquid-solid transitions in elastohydrodynamic contacts [4] and that this transition depends on both pressure and temperature The heat source depth w in the model represents the liquid region where the lubricant undergoes a high shear rate This region ranges between 0.1 and 0.4h At moderate to high sliding speeds, the magnitude of w is approxiamtely lh In order to simplify the derivation of dimensionless equations for this case, w is initially assumed to be equal to zero Now Figure 5.23 Lubricated, heavily loaded sliding/rollingcylinder Chapter 148 the partition of heat in lubricated rollinglsliding contacts can be predicted by using Eqs (5.12) and (5.13) In the practical range of different material combinations and bulk temperature difference, the heat generation zone in Fig 5.23 is assumed to be at the center of the film Accordingly, by referring to Fig 5.14, which represents two layered cylinders rubbing against each other, we can assume that: By assuming that the amount of heat flowing to the upper semi-infinite layered solid is aq,, then the lower one receives (1 - a)q, Since the maximum temperature rise inside the heat generation zone is the same for the two layered semi-infinite solids, then according to Eq (5.9), let: where ko (T) B, = 1.14 Ulplclh -0.013 2.e -1.003 (5;) e) 0.013 exp -900 x 10-6 Similarly: and from Eq (5.3), let: and 71 ' = To2, therefore: (5.14) Thermal Considerations in Tribology 149 Equation (5.14) gives the percentage of heat flowing to the upper layered semi-infinite solid However, the actual amount of heat flowing to each solid surface is expected to be slightly modified by the lubricant film or surface layer existence Because the maximum rise in the solid surface temperature is controlled by the amount of heat flow, then from Eq (5.3) we have: TSOl = T I = Y14,AI B Ts02 = TR2 = Y29rA2 (5.15) (5.16) But from Eq (5.12): and (5.18) where and The percentage of heat flowing to the upper solid can be predicted by substituting Eq (5.17) into Eq (5.15) to get: Y1 = 44 - c1) AI (5.19) and from Eqs (5.16) and (5.18), the percentage of heat flowing to the lower solid is y (A2 = ( I -a) - c2) A2 (5.20) The maximum surface layer temperature in this model, Eq (5.13), is derived without incorporating the influence of the heat source depth 12' and the percentage of heat flow into the semi-infinite layered solid Chapter 150 Therefore, the maximum film temperature in elastohydrodynamic lubrication, can be derived by modifying Eq (5.13) to the following form: Tot - T - B1 = a4,A I + a4,B1exp[-0.5 (31 - (5.21) where Tol = T3 Film Thickness The numerical solution for the minimum film thickness in elastochydrodynamic lubrication for compressible, isothermal, smooth, unlayered, and fully flooded cylinders by a Newtonian lubricant was discussed by Hamrock and Jacobson [27] The equation used for the minimum film thickness in a dimensionless form is written as: The dimensionless groups can be defined as follows: h Dimensionless film thickness Hmin= Re rlo U R Dimensionless speed parameter U = - where U R = rolling velocity R, = effective radius qo = viscosity at atmospheric pressure E, Dimensionless materials parameter Go = a , where E, = effective modulus of elasticity at,= pressure viscosity coefficient of lubricant v = rloea,P P = pressure q = lubricant viscosity W O Dimensionless load parameter PO = E, R, 151 Thermal Considerations in Tribology Since viscosity is strongly influenced by temperature, thermal effects are expected to have a strong influence on the minimum film thickness The modification proposed by Wilson and Sheu [28] can be used for a correction factor for the minimum film thickness by considering the thermal build up at the entrance of the contact zone The viscosity at atmospheric pressure, qo, is based on the average bulk temperatures of the mating solid surfaces A higher average bulk temperature leads to a lower viscosity and consequently, to a thinner lubricant film Under extreme conditions, this may result in severe interaction between the rubbing solids Some work has been devoted to avoid this problem by using different cooling techniques [22, 291 Case 5: Parabolic Heat Source Moving on a Metallic Semi-Infinite Solid with Low-Conductivity Surface Layer The dimensional analysis approach is also used in developing the following dimensionless equations for maximum solid and surface layer temperatures Let: -0.788 (5.23) then: (5.23) and let: (5.24a) then: (5.24b) All the variables in Eqs (5.12) and (5.13), and the above equations have identical definitions However, each set of these equations is valid only for a certain range of thermal properties and surface film thickness 152 Chapter Case 6: Parabolic Heat Source Moving on a Low-Conductivity, SemiInfinite Solid with a Metallic Surface layer By using the same previous procedure, the following equations are derived for this case Let: 0.627 (5.25a) then: (5.25b) and let: (5.26a) then: To - Ts = - CB (5.26b) Equations (5.23b)-(5.26b) are valid for the following range of operating conditions and thermal properties: The conductivity ratio which can be applied for each case is: k Case 5: 5 - 20 k0 k Case 6: 0.05 - 0.2 k0 where ko = conductivity of commonly used metallic solids The speed range is 500 < U < 2000in./sec The limits of the contact zone width are 0.01 l? 0.1 in Thermal Considerations in Tribology 153 The film thickness limits are given by 50 x 10-6 h 200 x Io - ~ in The range of pc covers all the commonly used materials Case 7: Dry Layered Contacts The equations derived in Cases and are utilized to develop dimensionless equations for heat partition and maximum temperatures for four different combinations of thermal properties for contacting solids with surface layers, as identified in Table 5.1 The resulting equations are given as follows: a= A1 +ZI + A + Z ("- + A2 + B ) (5.27) qt TB1 (5.28) (5.29) (5.30a) (5.30b) (5.3 1) The subscripts for each variable refer to the thermal properties and surface layer thickness of the indicated layered solid in Fig 5.24 Table 5.1 Contacts Different Combinations of Thermal Properties for Layered Dry ~~~~~ Combination of thermal properties for the lower layered solid Case Case Combination of thermal properties for the upper layered solid Case Case Chapter 154 Fiaure 5.24 Two rubbing layered cylinders under dry sliding conditions Several cases are investigated from Table 5.1 to show various effects for some operating variables on temperature and heat partition Both denisty and specific heat for all the solid materials in the following examples are assumed to be identical to the corresponding steel properties Numerical ResuIts Sample conditions are considered to illustrate the results obtained from the dimensionless equations Figure 5.25 shows the variation of Tm,,/q, for different slide/roll ratios For dry unlayered contacts, the faster and the slower solid surfaces have equal temperatures because there is no reason for a temperature jump across the interface In the case of dry contacts and constant rolling speed, T m a x / q , almost remains constant for different slide/roll ratios, whereas the lubricated contacts show a considerable dependence on this ratio The slower surface has a higher Tma,/q, as compared to the faster solid if there is a film with low thermal conductivity, such as oil, separating the two solids It can be seen that the film existence would result in a closer heat partition between the two solid surfaces However, the slower solid has a longer residence time [ / U under the heat source as compared to the faster solid, therefore, a higher maximum solid surface temprature On one hand, a thick lubricant film prevents the solids interaction, which eliminates both mechanical and thermal loads between asperities and reduces the friction coefficient On the other hand, it changes the heat partition in an unfavorable manner Figure 5.26 shows a case illustration of the relationships between maximum temperature rise for both lubricant film and solid surfaces and the maximum Hertz pressure A direct proportionality can be seen with a con- ... Lubricated, heavily loaded sliding/rollingcylinder Chapter 14 8 the partition of heat in lubricated rollinglsliding contacts can be predicted by using Eqs (5 .12 ) and (5 .13 ) In the practical range of... = Y14,AI B Ts02 = TR2 = Y29rA2 (5 .15 ) (5 .16 ) But from Eq (5 .12 ): and (5 .18 ) where and The percentage of heat flowing to the upper solid can be predicted by substituting Eq (5 . 17 ) into Eq (5 .15 )... Figs 5 .15 and 5 . 17 ) In both cases, the surface layer temperature decreases with the increase in lubricant film thickness This is attributed to the con- 18 00 * // I I I loo0 14 00 16 00 E 12 00 p -in,