013 Review Problem - Volume of water inside the Venturi meter

Problem 13
The accompanying figure represents the longitudinal view of a Venturi meter, a device designed to measure the flow of water in pipes. If the throat of the of the meter is 6 in. long and has an inside diameter of 4 in., find the volume of water in the meter which is used in 12-in. pipe line if the altitudes of the tapering parts are in the ratio 1:3 and the smaller altitude measures 12 in.
 

013-venturi-meter.gif

 

Frustums

Frustum of a pyramid (or cone) is a portion of pyramid (or cone) included between the base and the section parallel to the base not passing through the vertex.
 

$V = \frac{1}{3}\left( A_1 + A_2 + \sqrt{A_1A_2} \right)h$

 

Frustum of a cone and frustum of a pyramid

 

Derivation of formula for volume of a frustum of pyramid/cone

Frustum of a pyramid and frustum of a cone
 

Frustum of a pyramid and frustum of a cone

 

The formula for frustum of a pyramid or frustum of a cone is given by
 

$V = \dfrac{h}{3} \left[ \, A_1 + A_2 + \sqrt{A_1A_2} \, \right]$

 

Where:
h = perpendicular distance between A1 and A2 (h is called the altitude of the frustum)
A1 = area of the lower base
A2 = area of the upper base
Note that A1 and A2 are parallel to each other.