Chapter 8, Problem 8.14P

(a)

Interpretation Introduction

Interpretation:

The gas velocity at the centre of the duct for a pitot tube is to be calculated for a particular manometer reading.

Concept Introduction:

Pitot tube is a type of flow meter used to measure the local velocity of the fluid at a particular point. The formula for the velocity of a pitot tube is given as,

  $\text{u}\text{=}\sqrt{\frac{{\text{p}}_{\text{a}}\text{-}{\text{p}}_{\text{b}}}{{\text{ρ}}_{\text{gas}}}}$ ..... (1)

p_a − p_b = Pressure drop due to manometer

  ${\text{ρ}}_{\text{gas}}$ = Gas density

The pressure drop due to manometer is,

  ${\text{p}}_{\text{a}}\text{-}{\text{p}}_{\text{b}}\text{=}{\text{ρ}}_{\text{w}}\text{gh}$ ..... (2)

The notations used here are,

g = Acceleration due to gravity

h = Manometer reading

  ${\text{ρ}}_{\text{w}}$ = Density of water

The gas density is given as,

  ${\text{ρ}}_{\text{gas}}\text{=}\frac{\text{PM}}{\text{RT}}$ ..... (3)

P = Pressure of gas

T = Temperature of gas

R = Universal gas constant

M = Molecular weight of the gas

(b)

Interpretation Introduction

Interpretation:

The percent error in the maximum velocity calculated in the previous part is to be determined for changes in dependent parameter.

Concept Introduction:

The error is same for negative and positive sign so we can calculate only for the one sign.

% Error = $\frac{\text{Final}\text{Value}\text{-}\text{Initial}\text{Value}}{\text{Initial}\text{Value}}\text{×100}$ ..... (1)

The velocity varies with different factors as,

  $\begin{array}{l}\text{u}\text{α}{\text{(ρ)}}^{\text{-}\frac{\text{1}}{\text{2}}}\mathrm{.......}\text{(2)}\\ \text{u}\text{α}{\text{(M)}}^{\text{-}\frac{\text{1}}{\text{2}}}\mathrm{.......}\text{(3)}\\ \text{u}\text{α}{\text{(T)}}^{\frac{\text{1}}{\text{2}}}\mathrm{.......}\text{(4)}\\ \text{u}\text{α}{\text{(P)}}^{\frac{\text{1}}{\text{2}}}\mathrm{.......}\text{(5)}\end{array}$

P is the pressure