Question

Bernoulli equation

7. Bernoulli?s Equation. Water flows through a pipe reducer as is shown the figure. The static pressures at (1) and (2) are measured by the inverted U-tube manometer containing oil of specific gravity, SG, less than one. Determine the manometer reading, h.

Answer 1

water flows through a pipe reducer as shown in figure above.

To determine the manometer reading, h which is given by following steps :

Step 1 : U tube manometer reading -

the static pressure at (1) & (2) which are measured by the inverted U-tube manometer.

From starting point (1) to point (2) means that :

A $\rightarrow$ B $\rightarrow$ C

p₁ - _$\gamma$ (z₂ - z₁) - ( _$\gamma$ l - _$\gamma$ h) + ( _{$\gamma$ oil}h + _$\gamma$ l) = p₂                               { eq.1 }

where, _{$\gamma$ oil} = oil of specific gravity = SG _$\gamma$

p₁ & p₂ = pressure at point 1 & 2

p₁ - p₂ = _$\gamma$ (z₂ - z₁) + (1 - SG) _$\gamma$ h                                { eq.2 }

Step 2 : from Bernoulli's equation, To determine (p₁ - p₂) -

p₁ + (1/2) _$\rho$ v₁² + _$\gamma$ z₁ = p₂ (1/2) _$\rho$ v₂² + _$\gamma$ z₂

or p₁ - p₂ = _$\gamma$ (z₂ - z₁) + (1/2) [ _$\rho$ v₂² - _$\rho$ v₁²]                     { eq. 3 }

Step 3 : comparing above eq. 2 & 3 -

_$\gamma$ (z₂ - z₁) + (1 - SG) _$\gamma$ h = _$\gamma$ (z₂ - z₁) + (1/2) [ _$\rho$ v₂² - _$\rho$ v₁²]   

(1 - SG) _$\gamma$ h = (1/2) [ _$\rho$ v₂² - _$\rho$ v₁²]   

(1 - SG) _$\gamma$ h = (1/2) _$\rho$ v₂² [ 1 - v₁² / v₂²]

(1 - SG) _$\gamma$ h = (1/2) _$\rho$ v₂² [1 - (v₁ / v₂)²]                        { eq.4 }

Step 4 : from continuity equation, To determine (v₂² - v₁²) -

Q₁ = Q₂

A₁ v₁ = A₂ v₂                               { eq. 5 }

where, A = area of the pipe = $\pi$ (D/2)²

v = volume inside a pipe

inserting the value of 'A' in eq.5,

$\pi$ (D₁/2)² v₁ = $\pi$ (D₂/2)² v₂      

v₁ / v₂ = D₂² / D₁²

v₁ / v₂ = A₂ / A₁ = (D₂ / D₁)²