GATE ME 2017 Set 2 — Question 39

The figure shows a Pitot tube arrangement used to measure the velocity of a gas. The pressure difference measured by the manometer is related to the dynamic pressure of the gas.According to Bernoulli's principle for a Pitot tube, the pressure difference $\Delta P$ between the stagnation point and the static pressure point is given by:
$\Delta P = \frac{1}{2} \rho_{gas} V^2$
where:
$\rho_{gas}$ is the density of the gas
$V$ is the velocity of the gasThis pressure difference is measured by the manometer. The manometer equation relates the pressure difference to the differential head $h$:
$\Delta P = (\rho_{manometer} - \rho_{gas}) g h$
where:
$\rho_{manometer}$ is the density of the manometric fluid (water)
$g$ is the acceleration due to gravity
$h$ is the differential head in the manometerEquating the two expressions for $\Delta P$:
$\frac{1}{2} \rho_{gas} V^2 = (\rho_{manometer} - \rho_{gas}) g h$Given values:
$\rho_{gas} = 1$ kg/m$^3$
$V = 20$ m/s
$g = 9.81$ m/s$^2$
$\rho_{manometer} = 1000$ kg/m$^3$Substitute these values into the equation:
$\frac{1}{2} (1 \text{ kg/m}^3) (20 \text{ m/s})^2 = (1000 \text{ kg/m}^3 - 1 \text{ kg/m}^3) (9.81 \text{ m/s}^2) h$$200 \text{ Pa} = (999 \text{ kg/m}^3) (9.81 \text{ m/s}^2) h$$200 = 9800.19 h$$h = \frac{200}{9800.19} \approx 0.020407$ mTo express the differential head in millimeters (mm), multiply by $1000$:
$h = 0.020407 \times 1000 \text{ mm} = 20.407 \text{ mm}$The differential head $h$ is approximately $20.41$ mm.