f_ MEMORRNDUM

To: ChE 486 Class -- Spring 1997Date: January 21, 1997 Re:

From: Prof. R. BaratGas Rotameter Corrections

Theory shows us that, for variable area rotameters, the mass flowrate is approximately proportional to the square root of the meteredfluid mass density according to

(1)

where w = mass flow rate, CR = constant coefficient, 52 = minimumcross sectional area for flow between the float and the tube wall, g =gravitational constant, Vf = volume of the float, pf = mass density ofthe float, P = mass density of the fluid, and 51 = appfoximately thecross sectional area of the float at its maximum diameter. For allpractical pu{poses, (pr- p) = pf. Equation (1) becomes

w=C52pO.s (2)

and thewhere C = lumped constant. Now, as the flow rate increasesfloat rises, 52 increases to allow for the fluid to pass.

For a fluid metered at a fixed density p, the rotameter calibrationbasically depends on w and 52; hence, almost all variable arearotameters have a linear calibration curve. However, if the operatingfluid density of the rotameter is significantly different from thecalibration density, then the flow rate for a given reading must becorrected. In our lab, this would be irrelevant for liquids since theyate essentially incompressible and metered at room temperature.

The corrections must be made for g_u. flows. Now, in yourexperiment, take note where the rotameter flow control valve islocated. If located on the inlet, and if the rotameter outlet flow isessentially at the calibration pressure (usually atmospheric), then nocorrection is needed. However, if the control valve is on the outlet.then the rotameter is likely operating at an elevated pressurerelative to the calibration pressure.

w=cRsr@;o's

For a given rotameter reading (i.e. Sz value), we begin with Equation2 in which we can sav

wz =*, (f) i

(3)

(5)

Let state I = state of the gas (i.e. temperature, pressure, and identityof the gas) at which the rotameter is calibrated. Let state 2 = state ofthe gas at which the rotameter is operated. We will be able tocorrect the calibration curve of any rotameter for temperature,pressure, and type of gas.

It is desirable to convert from mass flow rates to volumetric flowrates at STP conditions. For a given mass rate w, we can say, usingthe ideal gas law, that

w=pv=p*v*=(-?)**u* (4)

where V = volumetric flow rate, * = STP conditions, W = molecularweight, P - pressure, and R - gas constant. Substituting Equation 4into Equation 3, we get

This reduces to our working relationship for calibration correction

Yz* - vr. (H) (f)" (6)

We will consider three cases.

Case I: Isothermal. Same Gas. Different Pressure

We must evaluate the density ratio.

(. ,)* *rrr* = (#)* *, u,* (ff)"

r=(*+)(++) = fiEquation 6 becomes

(7)

D 0.5yz* = vl. (ri) (8)

Case 1/: Isothermal. Different Gas. Samdj Pressure

We must evaluate the density ratio.

Pz-11-w1\ / R T \ - w2pl

= \ R T/ \ p w,/

= \h

(9)

Equation 6 becomes

,Wr. o '5Vz* = ut- (*;) (10)

Case 11l.' Different Temperature. Same Gas. Same Pres.sure

We must evaluate the densitv ratio.

pz ;rwrrRrr \ 11;= \ R 12/ \T-w/ = rz ( l l )

Equation 6 becomes

T 0.5

Yz* - vr* (il) Gz)

General Case ; Different Temperature. Gas. and Pressure

We must evaluate the densitv ratio.

ff=(*#)(+-fu) =#fi# (11)Equation 6 becomes

yz* = vr* 5f:lur;os Gz)