Final answer:
To calculate the gauge pressure, the hydrostatic pressure due to the oil column and the dynamic pressure to overcome viscous drag must be considered. The hydrostatic pressure is found by the product of oil density, gravity, and the column height, while the dynamic pressure can be computed from the Hagen-Poiseuille equation.
Step-by-step explanation:
To calculate the gauge pressure at the entrance of a 50.0-m-long vertical pipe when an oil gusher is shooting crude oil 25.0 m into the air through a pipe with a 0.100-m diameter, we must consider the hydrostatic pressure from the oil column and the dynamic pressure required to maintain the flow given the viscosity of the oil.
We will use the following formula which is derived from the Bernoulli's equation taking into account the viscosity:
P = ρgh + ΔP
Where:
- P is the gauge pressure at the entrance of the pipe.
- ρ (rho) is the density of the oil, which is 900 kg/m^3.
- g is the acceleration due to gravity (9.81 m/s^2).
- h is the height of the oil column, which is 50.0 m.
- ΔP is the additional pressure required to overcome the viscous drag in the pipe.
The hydrostatic pressure is calculated using ρgh which is the product of density, gravity, and the height of the oil column. The dynamic pressure can be determined by applying the Hagen-Poiseuille equation for laminar flow in a cylindrical pipe, depending on the oil's viscosity and the pipe's dimensions.
Overall, we will find the gauge pressure by adding the hydrostatic pressure to the extra pressure needed to push the oil to a height of 25.0 m with the given flow conditions.