An inverted U-tube manometer is connected to two horizontal pipes A and B through which water is flowing. The vertical distance between the axes of these pipes is 30 cm. When an oil of specific gravity 0.8 is used as a gauge fluid, the vertical heights of water columns in the two limbs of the inverted manometer (when measured from the respective centre lines of the pipes) are found to be the same and equal to 35 cm. Determine the difference of pressure between the pipes.

Inverted Manometer Pressure Difference Calculation

Problem Statement

Given Data

Specific gravity of oil, $S_{oil} = 0.8$
Vertical distance between pipes, $h_{oil} = 30 \, \text{cm} = 0.3 \, \text{m}$
Height of water column in each limb, $h_{water} = 35 \, \text{cm} = 0.35 \, \text{m}$

Solution

1. Define Densities

Density of water ($\rho_w$):

$$ \rho_w = 1000 \, \text{kg/m}^3 $$

Density of oil ($\rho_{oil}$):

$$ \rho_{oil} = S_{oil} \times \rho_w $$ $$ \rho_{oil} = 0.8 \times 1000 \, \text{kg/m}^3 $$ $$ \rho_{oil} = 800 \, \text{kg/m}^3 $$

2. Set up the Manometer Equation

Let's set the datum line at the level of the oil-water interface in the left limb (point C). We can equate the pressure at this level with the pressure at the same level in the right limb (point D).

$$ p_C = p_D $$ $$ p_A - \rho_w g h_{water} = p_B - \rho_w g h_{water} - \rho_{oil} g h_{oil} $$

The term $ \rho_w g h_{water} $ appears on both sides and can be cancelled out.

$$ p_A = p_B - \rho_{oil} g h_{oil} $$

Rearranging to solve for the pressure difference $p_B - p_A$:

$$ p_B - p_A = \rho_{oil} g h_{oil} $$

3. Calculate the Pressure Difference

Now we substitute the known values into the equation.

$$ p_B - p_A = 800 \, \text{kg/m}^3 \times 9.81 \, \text{m/s}^2 \times 0.3 \, \text{m} $$ $$ p_B - p_A = 2354.4 \, \text{N/m}^2 $$

Converting to N/cm²:

$$ p_B - p_A = 2354.4 \, \text{N/m}^2 \times \frac{1 \, \text{cm}^2}{100^2 \, \text{m}^2} $$ $$ p_B - p_A \approx 0.2354 \, \text{N/cm}^2 $$

Final Result:

The difference of pressure ($p_B - p_A$) is $ 2354.4 \, \text{N/m}^2 $ or $ 0.2354 \, \text{N/cm}^2 $.

Explanation of the Manometer Setup

An inverted manometer uses a light fluid (oil) to measure the pressure difference between two points containing a heavier fluid (water). The key to solving this problem is to establish a pressure balance equation.

We choose a horizontal reference line (datum) at the highest point of the heavier fluid, which is the oil-water interface in the left limb. The pressure at this level must be the same in both limbs. By writing expressions for the pressure at this level starting from pipe A (for the left limb) and pipe B (for the right limb), we can set them equal and solve for the unknown pressure difference.

Physical Meaning

The positive result for $p_B - p_A$ means that the pressure in pipe B is higher than the pressure in pipe A. The calculated value of 2354.4 N/m² is the magnitude of this difference.

This pressure difference is what supports the 30 cm column of the lighter oil. Because pipe B is at a lower elevation than pipe A, we would naturally expect its pressure to be higher due to the hydrostatic head of the water between them, even if there were no flow. The manometer allows for a precise measurement of this pressure difference, which is crucial for analyzing the hydraulic gradient and energy losses in the piping system.