Problem

Consider a 5-cm-diameter shaft rotating at 5600 rpm in a 25-cm-long bearing with a clearance of 0.5 mm. Determine the power required to rotate the shaft if the fluid in the gap is (a) air, (b) water, and (c) oil at 40°C and 1 atm.

Step-by-step solution

1. Step 1 of 7

   Calculate the linear velocity of the shaft by using the relation,

   

   Here, D is the diameter of the shaft and N is the speed of the shaft.

   Substitute  for D and 5600 rpm for N.

   

   Comment

2. Step 2 of 7

   Calculate the surface area of the shaft by using the relation,

   

   Here, L is the length of the shaft.

   Substitute  for D and 0.25 m for L.

   

   Comment

3. Step 3 of 7

   Assume the shaft rotating in a bearing is equal to the flow between parallel plates with one plate moving and another plate is stationary. Then write the temperature distribution in the gap as follows:

   

   Differentiate the equation with respect to y.

   

   Find the temperature gradient at the surface, 

   

   The power required to rotate the shaft is equal to the heat energy dissipated to the surrounding fluid.

   Comments (3)

4. Step 4 of 7

   Write the equation for power required to rotate the shaft by using the relation,

   

   Here, k is the thermal conductivity of the shaft and is the temperature gradient.

   Substitute for 

   

   Here,  is the dynamic viscosity of the surrounding fluid and V is the velocity of the shaft.

   Comment

5. Step 5 of 7

   a)

   Obtain the dynamic viscosity of air at  and 1 atm pressure from appendix A (Table A-15).

   

   Calculate the power required to rotate the shaft when surrounded by air.

   

   Substitute  for ,  for ,  for V, and 0.25 m for L.

   

   Therefore, the power required to rotate the shaft when surrounded by air is 

   Comment

6. Step 6 of 7

   b)

   Obtain the dynamic viscosity of water at  and 1 atm pressure from appendix A (Table A-9).

   

   Calculate the power required to rotate the shaft when surrounded by water.

   

   Substitute  for ,  for ,  for V, and 0.25 m for L.

   

   Therefore, the power required to rotate the shaft when surrounded by water is 

   Comment

7. Step 7 of 7

   c)

   Obtain the dynamic viscosity of engine oil at  and 1 atm pressure from appendix A (Table A-13).

   

   Calculate the power required to rotate the shaft when surrounded by engine oil.

   

   Substitute  for ,  for ,  for V, and 0.25 m for L.

   

   Therefore, the power required to rotate the shaft when surrounded by engine oil is 

   Comment