Problem Statement
A pipe contains an oil of sp. gr. 0.8. A differential manometer connected at the two points A and B of the pipe shows a difference in mercury level as 20 cm. Find the difference of pressure at the two points.
Given Data & Constants
- Specific gravity of oil, \(S_{\text{oil}} = 0.8\)
- Difference in mercury level, \(h_{\text{m}} = 20 \, \text{cm} = 0.2 \, \text{m}\)
- Density of water, \(\rho_{\text{water}} = 1000 \, \text{kg/m}^3\) (standard value)
- Specific gravity of mercury, \(S_{\text{Hg}} = 13.6\) (standard value)
- Density of mercury, \(\rho_{\text{Hg}} = 13600 \, \text{kg/m}^3\) (standard value)
- Acceleration due to gravity, \(g = 9.81 \, \text{m/s}^2\) (standard value)
Solution
1. Calculate the Density of Oil (\(\rho_{\text{oil}}\))
The density of the oil is its specific gravity multiplied by the density of water.
2. Calculate the Pressure Difference (\( \Delta P \))
The pressure difference between points A and B is determined by the height difference in the manometer and the difference in densities of the two fluids (mercury and oil).
The difference of pressure at the two points is \( \Delta P = 25113.6 \, \text{N/m}^2 \) or \(25.114 \, \text{kPa}\).
Principle of the Differential Manometer
A differential manometer is a device used to measure the pressure difference between two points in a pipe or in two different pipes. It works by balancing the pressure from the two points against a column of a heavier, immiscible fluid (the "manometric fluid"), which is mercury in this case.
The pressure at a common reference level within the manometer must be equal. By creating a pressure balance equation, we can isolate the pressure difference (\(P_A - P_B\)) and relate it to the height difference (\(h_m\)) of the manometric fluid. The formula used here assumes the pipe is horizontal.
Physical Meaning: Pressure Drop
The calculated pressure difference of 25,113.6 N/m² represents the pressure drop between points A and B. In a flowing fluid system, this pressure drop is primarily caused by two factors:
- Frictional Losses: As the oil flows through the pipe, friction between the fluid and the pipe wall causes a loss of energy, which manifests as a drop in pressure.
- Change in Elevation: If point B is at a higher elevation than point A, some pressure energy is converted into potential energy, also contributing to a pressure drop. (In this problem, we assume the pipe is horizontal).
Measuring this pressure drop is crucial in engineering for determining flow rates, calculating pumping power requirements, and analyzing the efficiency of fluid transport systems.
