A large reciprocating machine has a pressurized lubrication

A large reciprocating machine has a pressurized lubrication system that forces oil to flow between two parallel, planar surfaces as shown in the illustration below. The upper surface moves in the +z direction and the lower surface in the –z direction: v_z (y = 0) = kV₀ and v_z (y = b) = -V₀.

A) Use a shell momentum balance to find an expression for the velocity distribution between the moving surfaces.

B)  Develop an expression for the average velocity,< v_x > in the system.

A large reciprocating machine has a pressurized lubrication system that forces oil to flow between two parallel, planar surfaces as shown in the illustration below. The upper surface moves in the +z direction and the lower surface in the-z direction: y = 0) = kVoand A) Use a shell momentum balance to find an expression for the velocity distribution between the moving surfaces. B) Develop an expression for the average velocity, in the system y=0

Solution

a. This is a problem on turbulent flow as it can be observed that the direction of flow of velocity at the top part and the bottom part are in opposite direction,whereas in a laminar flow the direction of flow of velocity at all the points in the stream are in the same direction.

Nikuradse carried out experiments for different reynolds numbers to determine velocity distribution law. He expressed the velocity distribution in the form as:

(U/Umax)= (y/R)^1/n

where exponent 1/n depends on reynolds number...

So according to the given problem U=(KVo)(b/R)^1/n

b. For smooth pipes:

Avg. velocity(U)=(Q/Area)