A journal bearing is idealized as a stationary flat plate and a moving flat plate that moves parallel to it. The space between the two plates is filled by an incompressible fluid. Consider such a bearing with the stationary and moving plates at 10°C and 20°C respectively, the distance between them is 3 mm, the speed of the moving plate is 5 m/s, and there is engine oil between the plates.
(a) Calculate the heat flux to the upper and lower plates
(b) Determine the maximum temperature of the oil.
GIVEN
Journal bearing: Two flat plates, one stationary, one moving with oil between them
Stationary plate temperature (Ts) = 10°C
Moving plate temperature (Tm) = 20°C
Distance between plate (H) = 3 mm = 0.003 m
Speed of the moving plate (Up) = 5 m/s
FIND
(a) Heat flux (q/A) for the plates
(b) The maximum temperature of the oil
ASSUMPTIONS
Steady state
Constant physical properties
Negligible edge effects
Oil is incompressible
SKETCH
PROPERTIES AND CONSTANTS
From Appendix 2, Table 17, for engine oil at 15°C
Thermal conductivity (k) = 0.145 W/(m K)
Absolute viscosity (?) = 1.561 (Ns)/m2
Prandtl number (Pr) = 196
(a) The temperature distribution for this geometry was derived in Problem 5.20
For this case
This is plotted below
The first derivative of the temperature is
The heat flux at the top plate (y = H) is
The heat flux at the bottom plate (y = 0) is
(b) The maximum temperature occurs where the first derivative is zero
Checking the second derivative
This is negative throughout the region, therefore, T(y) is concave down throughout the region and the temperature at y = 1.6 mm is indeed the maximum temperature. These calculations are verified by the graph of T(y). Inserting ymax into the expression for T(y) yields: Tmax = T(0.00161 mm) = 48.8°C.
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