### Flow Measurement by Rotameter

A rotameter calibrated for metering has a scale ranging from 0.014 m3/min to 0.14 m3/min. It is intended to use this meter for metering a gas of density 1.3 kg/m3 with in a flow range of 0.028 m3/min to 0.28 m3/min. What should be the density of the new float if the original one has a density of 1900 kg/m3? Both the floats can be assumed to have the same volume and shape.

Data:

Old:

Density of float (ρf)= 1900 kg/m3

Density of gas (ρ) = 1.3 kg/m3

Qmin = 0.014 m3/min

New:

Density of gas (ρ) = 1.3 kg/m3

Qmin = 0.028 m3/min

Formula:

$$\displaystyle Q = C_D A_2\sqrt{\frac{2V_f(\rho_f-\rho)g}{\rho A_f\bigl(1-(A_2/A_1)^2\bigr)}}$$

Where Q = volumetric flow rate

A2 = area of annulus (area between the pipe and the float)

A1 = area of pipe

Af = area of float

Vf = volume of float

CD = rotameter coefficient

Calculations:

From the equation, for the same float area and float volume and the pipe geometry,

Q = k (ρf - ρ)0.5

where k is a proportionality constant

Qnew / Qold = ( (ρfnew - ρ) / (ρfold - ρ) )0.5

i.e. 0.028 / 0 .014 = ( (ρfnew - 1.3) / (1900 - 1.3) )0.5

2 = (ρfnew - 1.3)0.5 / 43.574

(ρfnew - 1.3)0.5 = 87.15

(ρfnew - 1.3) = 7595.1

ρfnew = 7596 kg/m3