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.

Question

GIVEN

A journal bearing

Diameter (D) = 100 mm = 0.1 m

Clearance (H) = 0.5 mm = 0.0005 m

Rotational speed (@) = 3600 rpm

Oil properties

? Density (p) = 800 kg/m3

? Viscosity (?) = 0.01 kg/ms

? Thermal conductivity (k) = 0.14 W/(m K)

Temperature of bearing surface (Tb) = 60°C

FIND

(a) Temperature distribution in the oil film

ASSUMPTIONS

Steady state

Uniform and constant bearing surface temperature

Constant fluid properties

The journal surface is insulated (negligible heat transfer)

SKETCH

Answer 1

Since the clearance is small compared to the bearing diameter, the bearing may be idealized as parallel flat plates with oil between them, one stationary and one moving



(a) As shown in Problem 5.20, the velocity distribution for this geometry is linear





For this geometry, the energy Equation (5.6) with viscous dissipation reduces to



With boundary conditions: at y = 0 T = Tb



Integrating



Applying the second boundary condition



Integrating



Applying the first boundary condition c2 = kTb



This is shown graphically below