## 1

12m dx 2

Note that qx is the flow per unit width across the film. The flow in the axial direction is

Now let us consider a flow balance through an elemental volume across the film (see Figure 6). The net outflow through the volume equals the net reduction in volume per unit time:

dqx \ f dqz \ 0h qx + —— dx b dz — qx dz +1 qz + —— dz b dx — qz dx = —— dx dz (11)

Thus, this gives us

0qz dz

0h dt

Substituting Equations 9 and 10 into Equation 12, we obtain

_d_ (_L_ h3 -d^b — — ( — h3 -O^b = — — — U Oh

dx \ 12m dx y dz \ 12m dz y dt 2 dx and the final equation becomes

12 dh

Equation 14 is the general form of Reynolds' equation used for laminar, two-dimensional lubrication problems.

Reynolds' equation is a flow balance equation. The left-hand side represents pressure-induced flows in the x and z directions through the differential element. The first term on the right-hand side represents the shear flow of the fluid induced by the surface velocity of the journal u. Note that this term contains the derivative of clearance with respect to distance. If this term is zero, then there is zero pressure produced by hydrodynamic action, h h q x

and the term 3h/3x is the mathematical representation of the tapered wedge. The second term on the right-hand side refers to a time rate of change of the film thickness, which can be translated to a normal velocity of the center of the journal. It produces pressure by a fluid velocity normal to the bearing surfaces that attempts to squeeze fluid out of a restricted clearance space. This phenomenon is called the squeeze film effect in bearing terminology. Since it is proportional to the velocity of the center of the journal, it is the phenomenon that produces viscous damping in a bearing.

The solution to Reynolds' equation (refer to Equation 14) provides the pressure at all points in the bearing. The application of the digital computer has enabled a rapid solution of Reynolds' equation over a grid network representing the bearing area.2,3

Once the pressures have been obtained, a numerical integration is applied to determine the performance parameters (in other words, the load capacity):

w = J| prdU dr The flow across any circumferential line is

The flow across any axial line is qz

The viscous frictional moment is obtained by integrating the shear stress over the area and can be shown to be

where « = journal surface speed, rad/s.

Typical computer program output includes the following:

• Pressure distribution throughout the grid network

• Load capacity

• Side leakage and carryover flows

• Viscous power losses

• Righting moments due to misalignments

• Attitude angles

• Cross-coupled spring and damping coefficients due to displacements and velocity perturbations of the journal center

• Clearance distribution

Turbulence Equation 14 is for laminar conditions. For very high speed bearings, operations beyond the turbulent regime may occur and Reynolds' equation must be modified. The turbulent theory has been developed, and the literature on this topic can enable performance predictions for turbulent bearings.5,6

The onset of turbulence is determined by examining the bearing's Reynolds number, which is the ratio of inertia to viscous forces and is defined as

## Renewable Energy 101

Renewable energy is energy that is generated from sunlight, rain, tides, geothermal heat and wind. These sources are naturally and constantly replenished, which is why they are deemed as renewable. The usage of renewable energy sources is very important when considering the sustainability of the existing energy usage of the world. While there is currently an abundance of non-renewable energy sources, such as nuclear fuels, these energy sources are depleting. In addition to being a non-renewable supply, the non-renewable energy sources release emissions into the air, which has an adverse effect on the environment.

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