Final answer:
a. To compute the frictional pressure gradient, use the power law model. b. The shear rate at the wall of the drill pipe can be calculated using a specific equation. c. Plot the flow velocity vs. pipe radius and the shear stress vs. pipe radius.
Step-by-step explanation:
a. To compute the frictional pressure gradient resulting from a 50-gal/min flow rate in a 2.9 in. ID drill pipe, we can use the power law model. The power law model relates the pressure gradient to the flow rate, pipe diameter, fluid viscosity, and fluid behavior index. Since the flow pattern is laminar, the power law model simplifies to:
dP/dz = (128 * η * Q * n * r^((n-1)/n)) / (π * D^4)
Where:
dP/dz is the pressure gradient
η is the fluid viscosity
Q is the flow rate
n is the fluid behavior index
r is the pipe radius
D is the pipe diameter
To compute the pressure gradient, we substitute the given values into the equation:
dP/dz = (128 * 1.00 Pa.s * 50 gal/min * (0.70-1) * (1/2.9)^(0.30/0.70)) / (π * (2.9 in.)^4)
b. To compute the shear rate at the wall of the drill pipe, we can use the expression:
γ = (2 * Q) / (π * D)
To find the rotational readings that would give the best n and k factors for the pressure gradient computation, it is necessary to experimentally determine the values of n and k that best fit the data to the power law model.
c. To make a plot of flow velocity v vs. pipe radius, we can use the Poiseuille's law for laminar flow:
v = (ΔP * r^2) / (4 * η * L)
Where:
v is the flow velocity
ΔP is the pressure difference
r is the pipe radius
η is the fluid viscosity
L is the pipe length
To make a plot of shear stress vs. pipe radius, we can use the expression:
τ = (ΔP * r) / (2 * L)