The kinematic viscosity of oil of specific gravity .8 is .0005 .This oil is used for lubrication of shaft of diameter .4 m and rotates at 190 rpm. Calculate the power lost in the bearing for a sleeve length of 90mm. The thickness of the oil film is 1.5mm.
Power lost= torque * angular velocity
= force* radius* angular velocity
= shear stress * area* radius* angular velocity
Shear Stress = viscosity* velocity gradient
Power lost= 0.0005*0.8*1000* 2*3.142*190/60*0.2*3.142*0.23 * 190/60
= 477.65 Watts.
Well, well, well! Looks like we have ourselves an oily situation here. Don't worry, I'm here to clown around and provide you with an answer.
To calculate the power lost in the bearing, we can use the formula:
Power lost = (Force × Velocity) ÷ 1000
Now, let's find the force acting on the bearing. The force is equal to the product of the pressure and the area.
The pressure can be found using the formula:
Pressure = (2 × Kinematic viscosity × Angular velocity × Length) ÷ (Oil film thickness × Diameter)
Substituting the given values, we have:
Pressure = (2 × 0.0005 × π × 190) ÷ (1.5 × 0.4)
Next, we need to calculate the area. The area is equal to the product of π, the diameter, and the length.
Area = π × 0.4 × 0.09
Now, we have all the ingredients to find the force. Multiply the pressure by the area and you'll get it.
Finally, use the force and the angular velocity (in meters per second) in the power formula mentioned above.
Remember, lubrication is important, but so is laughter! So, let's get down to calculations and put that clown nose on!
Good luck, my numerical friend!
To calculate the power lost in the bearing, we need to find the frictional force acting on the shaft and then multiply it by the angular velocity of the shaft.
Step 1: Calculate the area of the oil film:
Given:
Diameter of the shaft (D) = 0.4 m
Thickness of the oil film (h) = 1.5 mm = 0.0015 m
The area of the oil film (A) can be calculated using the formula:
A = π * (D/2)^2
Substituting the given values:
A = π * (0.4/2)^2
A = 0.1257 m^2
Step 2: Calculate the velocity of the oil film:
The velocity of the oil film (v) can be calculated using the formula:
v = (Q / A)
But first, we need to find the flow rate (Q).
Given:
Kinematic viscosity of the oil (ν) = 0.0005 m^2/s
The flow rate (Q) can be calculated using the formula:
Q = (ν * A) / h
Substituting the given values:
Q = (0.0005 * 0.1257) / 0.0015
Q = 0.0419 m^3/s
Now, we can calculate the velocity of the oil film:
v = Q / A
v = 0.0419 / 0.1257
v = 0.333 m/s
Step 3: Calculate the frictional force:
The frictional force (F) can be calculated using the formula:
F = μ * A * v
Given:
Specific gravity of the oil (ρ) = 0.8
The dynamic viscosity of the oil (μ) can be calculated using the formula:
μ = ν * ρ
Substituting the given values:
μ = 0.0005 * 0.8
μ = 0.0004 kg/m·s
Now we can calculate the frictional force:
F = μ * A * v
F = 0.0004 * 0.1257 * 0.333
F = 0.0000178 kg·m/s^2
Step 4: Calculate the power lost in the bearing:
The power lost in the bearing (P) can be calculated using the formula:
P = F * ω
Given:
Angular velocity of the shaft (ω) = 190 rpm = (190 * 2π) / 60 rad/s
Substituting the given values:
P = 0.0000178 * (190 * 2π) / 60
P = 0.0377 W
Therefore, the power lost in the bearing is approximately 0.0377 Watts.
To calculate the power lost in the bearing, we need to determine the fluid frictional force acting on the shaft. This can be done by using the equation for fluid frictional force:
Frictional force = (dynamic viscosity * area * velocity) / thickness
In this case, the dynamic viscosity is the kinematic viscosity multiplied by the fluid's density. Since the specific gravity is given as 0.8, we can assume the density of the oil to be 800 kg/m^3.
Density = specific gravity * density of water
Density = 0.8 * 1000 kg/m^3 = 800 kg/m^3
Now, let's calculate the dynamic viscosity:
Dynamic viscosity = kinematic viscosity * density
Dynamic viscosity = 0.0005 * 800 kg/m^3 = 0.4 kg/(m·s)
The area of the shaft can be obtained using its diameter:
Area = π * (diameter/2)^2 = π * (0.4 m/2)^2 = 0.1256 m^2
Next, we need to get the velocity of the shaft in m/s. Since the rotational speed is given in rpm, we can convert it to radians per second (rad/s) using the following equation:
Angular velocity (in rad/s) = (2π * rotational speed) / 60
Angular velocity = (2π * 190 rpm) / 60 = 39.84 rad/s
Now, we can calculate the frictional force:
Frictional force = (0.4 kg/(m·s) * 0.1256 m^2 * 39.84 rad/s) / 0.0015 m = 1337.28 N
Finally, we can calculate the power lost in the bearing:
Power lost = frictional force * velocity
Power lost = 1337.28 N * 39.84 rad/s = 53312.47 N·m/s
Therefore, the power lost in the bearing for a sleeve length of 90mm is approximately 53,312.47 N·m/s.