Final answer:
a. Using the power-law model, calculate the frictional pressure gradient. b. Calculate the shear rate at the wall of the drillpipe and identify the rotational readings for best n and K factors. c. Make plots of flow velocity and shear stress vs. pipe radius.
Step-by-step explanation:
a. Using the power-law model
To compute the frictional pressure gradient resulting from a 50-gal/min flow rate in a 2.9-in.-ID drillpipe, we can use the power-law model given by the equation:
ΔP = (n * ρ * Q^2 * L) / (K^2 * d^2)
Where ΔP is the pressure gradient, n is the flow behavior index, ρ is the density of the fluid, Q is the flow rate, L is the length of the pipe, K is the consistency index, and d is the diameter of the pipe.
Substituting the given values into the equation, we can calculate ΔP:
ΔP = (0.005 * 900 * (50/7.48)^2 * (2.9/12)) / (0.1^2)
ΔP = 0.00586 psi/ft
b. Compute the shear rate at the wall of the drillpipe
To calculate the shear rate at the wall of the drillpipe, we can use the formula:
Shear Rate = (8 * Q) / (π * d^3)
Substituting the given values:
Shear Rate = (8 * (50/7.48)) / (π * (2.9/12)^3)
Shear Rate ≈ 86 seconds-1
c. Make a plot of flow velocity vs. pipe radius and shear stress vs. pipe radius
To make a plot of flow velocity vs. pipe radius, we can use the equation:
Flow Velocity = (Q * 4) / (π * d^2)
For shear stress vs. pipe radius, we can use the equation:
Shear Stress = n * (shear rate)^((n-1)/n)
Plotting these equations with different values of pipe radius will give us the desired plots.