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Problem 1 In operations such as drilling, a long cylindrical shaft has to be rotated against the viscous resistance offered by a viscous liquid in a narrow gap surrounding the cylinder. The following data is to be used for designing a drilling application: - Drilling mud specific gravity: 2 (water reference) - Viscosity: $0.1Pas$ - Drilling shaft diameter: $168mm$ - Average gap width (between the shaft and earth material): $10mm$ - Maximum operating length: $30m$ - Maximum rotational speed: $2000rpm$ (a) Plot the power-vs-rotational speed characteristics of the motor required to drive the drilling shaft. Provide a detailed justification of your calculation. (b) What is the maximum operating flow-induced pressure difference between the drilling shaft surface and the earthen wall of the bore hole? How does this pressure difference compare with the (underground) overall operating pressure of $200atm$ ?

(a) To calculate the power required to rotate the drilling shaft, we can use the following formula:

where P is the power required in watts, N is the rotational speed in revolutions per minute (rpm), and T is the torque in newton-meters (Nm).

The torque required to rotate the shaft against the viscous resistance of the drilling mud can be calculated using the following formula:

where F is the force required to overcome the viscous resistance of the drilling mud and r is the radius of the drilling shaft.