**A rotameter calibrated for metering has a scale ranging from 0.014 m ^{3}/min to 0.14 m^{3}/min. It is intended to use this meter for metering a gas of density 1.3 kg/m^{3} with in a flow range of 0.028 m^{3}/min to 0.28 m^{3}/min. What should be the density of the new float if the original one has a density of 1900 kg/m^{3}? Both the floats can be assumed to have the same volume and shape**.

Data:

Old:

Density of float (r_{f})= 1900 kg/m^{3}

Density of gas (r) = 1.3 kg/m^{3}

Q_{min} = 0.014 m^{3}/min

New:

Density of gas (r) = 1.3 kg/m^{3}

Q_{min} = 0.028 m^{3}/min

**Formula:**

Where Q = volumetric flow rate

A_{2} = area of annulus (area between the pipe and the float)

A_{1} = area of pipe

A_{f} = area of float

V_{f} = volume of float

C_{D} = rotameter coefficient

Calculations:

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

Q = k (r_{f} - r)^{0.5}

where k is a proportionality constant

Q_{new} / Q_{old} = ( (r_{fnew} - r) / (r_{fold} - r) )^{0.5}

i.e. 0.028 / 0 .014 = ( (r_{fnew }- 1.3) / (1900 - 1.3) )^{0.5}

2 = (r_{fnew }- 1.3)^{0.5} / 43.574

(r_{fnew }- 1.3)^{0.5} = 87.15

(r_{fnew }- 1.3) = 7595.1

r_{fnew} = **7596 kg/m ^{3}**