Chapter 2, Problem 2.1P

Interpretation Introduction

Interpretation: The pressure difference across a simple U-tube manometer is to be calculated in the appropriate units.

Concept introduction: A simple U-tube Manometer is a device that can measure the pressure difference across a U-tube filled with fluids of different or the same density.

The pressure difference across the Manometer is given as,

  ${\text{p}}_{\text{a}}{\text{-p}}_{\text{b}}\text{}\text{= }\text{}{\text{gR}}_{\text{m}}{\text{(ρ}}_{\text{a}}{\text{-ρ}}_{\text{b}}\text{)}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}$ ……. (1)

In the above formula, the notations used are,

  $\begin{array}{l}{\text{p}}_{\text{a}}{\text{-p}}_{\text{b}}\text{}\text{= Pressure difference}\\ \text{g = Acceleration due to gravity = 9}{\text{.8 m/s}}^{\text{2}}\\ {\text{R}}_{\text{m}}\text{}\text{=}\text{}\text{Manometer }\text{}\text{Reading}\\ {\text{ρ}}_{\text{a}}\text{}\text{=}\text{}\text{Density }\text{}\text{of }\text{}\text{substance }\text{}\text{A}\\ {\text{ρ}}_{\text{b}}\text{}\text{=}\text{}\text{Density }\text{}\text{of }\text{}\text{substance }\text{}\text{}\text{}\text{B}\end{array}$

The density of the substance is given as,

  $\text{Density }\text{}\text{}\text{}\text{}\text{}\text{}\text{= }\text{}\left(\text{Specific }\text{}\text{Gravity }\text{}\text{×1000}\right)\text{}{\text{kg/m}}^{\text{3}}$ ……. (2)

Answer to Problem 2.1P

The pressure difference across a U-tube Manometer is 23544 N/m².

Explanation of Solution

The U-tube Manometer mentioned in the question can be given as:

The fluid A is Mercury and the fluid B is Carbon tetrachloride here.

  

The given data is,

  $\begin{array}{l}{\text{R}}_{\text{m}}\text{}\text{= }\text{}\text{}\text{0}\text{.2 m}\\ \text{Specific }\text{}\text{gravity }\text{}\text{of }\text{}\text{mercury }\text{}\text{(Substance }\text{}\text{A) }\text{}\text{= }\text{}\text{13}\text{.6}\\ \text{Density }\text{}\text{of }\text{}\text{Mercury }\text{}\text{= }\text{}\text{13}\text{.6×1000 }\text{}\text{= }\text{}\text{13600 }\text{}{\text{kg/m}}^{\text{3}}\\ \text{Specific }\text{}\text{gravity }\text{}\text{of Carbon tetrachloride }\text{}\text{(Substance }\text{}\text{B) }\text{}\text{= }\text{}\text{1}\text{.6}\\ \text{Density }\text{}\text{of Carbontetrachloride }\text{}\text{= }\text{}\text{1}\text{.6×1000 }\text{}\text{= }\text{}\text{1600}\text{}{\text{ kg/m}}^{\text{3}}\end{array}$

Substitute these values in equation (1) above,

  $\begin{array}{l}{\text{p}}_{\text{a}}{\text{-p}}_{\text{b}}\text{}\text{= }\text{}\left(\text{9}{\text{.81m/s}}^{\text{2}}\right)\text{×0}{\text{.2 m×(13600 kg/m}}^{\text{3}}{\text{-1600 kg/m}}^{\text{3}}\text{)}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\\ {\text{p}}_{\text{a}}{\text{-p}}_{\text{b}}\text{}\text{= }\text{}\text{23544 }\text{}\text{}{\text{kg/ms}}^{\text{2}}\\ {\text{p}}_{\text{a}}{\text{-p}}_{\text{b}}\text{}\text{= }\text{}\text{23544 }\text{}{\text{N/m}}^{\text{2}}\end{array}$

Conclusion

The pressure difference across a U-tube Manometer for the given readings of density and manometer is 23544 N/m².