Variable Area Flow Meter Explanation

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EMEM 416: Thermal-Fluids Laboratory I John D. Wellin

04/17/09 Page 1 of 4

An Explanation of Variable Area Flow Meters

The figure on the right shows the basic configuration of the variable

area flow meters used in the Vortex Tube Characterization, made by

Omega Engineering, Inc. A metal float is suspended in the glass tube

by the fluid forces acting on it as the flow passes over the float. The

calibrated flow rate is then read from a scale etched or printed on the

outside of the tube. The tube itself is tapered, so that as the flow rate

increases, the float will stabilize at a higher position in the tube; and it is

tapered by the appropriate amount so that the flow scale is linear. The

tapered design is what gives the variable area flow meter its name. To

understand the operational principle, consider the free-body diagram of

the float shown below. Assume the float has reached a stable

equilibrium point in the tube because of some flow over it. By a

balance of forces, we have:

buoydragfloatFFW +=

This result is quite general, but can be

simplified when the fluid is a gas. Consider

the relationship between the weight of the

float and the buoyancy force:

floatfloatbuoyfloatVg)(FW =

Where Vfloat is the volume of the float, float is its density, is the

fluid density, and g is the acceleration of gravity. Because float istypically much greater than for gases, it is generally safe to neglect

the buoyancy force in comparison to the floats weight. In our

specific case, the float is made from stainless steel, and the fluid is

air. Because the density of stainless steel is anywhere from 7000 to

8000 times the density of air, we will indeed neglect Fbuoy in our

specific case, so we have to an excellent approximation:

dragfloatFW =

Now, the fluid drag force acting on an object is generally given by an expression like:

front

2

21

DdragAvCF

Where is the fluid density (assumed uniform within the flow meter), v is the velocity, CD is the

drag coefficient, andAfront is thefrontal area of the object; that is, the projected, cross-sectional

area that lies perpendicular to the oncoming flow. Let Q to be the volumetric flow rate through

the flow meter, and let A be the flow area, which is actually the annular region contained

between the tube wall and the float itself. By the relationship that Q=vA, we have:

Fluid Drag

Force: Fdrag

Fluid Buoyancy

Force:Fbuoy

Float Weight:

Wloat


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front221

Ddragfloat A

AQCFW =

Because the float has a finite extent in the flow direction, there is some ambiguity as to where the

flow area A (and therefore v) is determined, but we could certainly assume without loss of

generality that it is taken at the measurement edge of the float (the lower lip of the conical capdepicted in the free-body diagram above). Over narrow flow ranges, the drag coefficient for a

bluff body like the float is nearly constant, and Afront and Wfloatare constant by definition. As a

matter of fact, the float is often designed in such a way as to achieve a constant drag coefficient,

which has the added benefit of making the flow meter relatively insensitive to fluid viscosity.

Thus we have, after re-arrangement:

AKQ =

Where Kis some constant value for the flow meter. This is the key operational principle sought:

it says that the indicated flow rate Q will be linearly distributed along the height of the meter, ifthe tube cross-sectional area is linearly distributed along the height. We also have that the

indicated flow rate depends upon the square root of the density, which is important when

considering a change in thermodynamic conditions or a change in fluid. Note that this derivation

parallels those of Figliola & Beasley (2006) and Holman (2001). More complete derivations

following Bernoullis principle are given by Brain & Scott (1982) and Doebelin (2004), but the

simplifying conditions relevant to our investigation yield the same basic results above.

In practice, a variable area flow meter is designed for use with a particular fluid, and then

calibrated with that fluid. Especially in the case of a gas, the thermodynamic conditions of the

calibration must be carefully controlled and documented since, as noted, the density is importantfor proper indication. Any situation that involves a different density than that used for

calibration of the flow metereither because of compressibility, or a change in fluid type

requires that a correction be applied to the indicated flow rate. For our purposes, consider the

following scenario: we wish to measure the volumetric flow rate of air at some density , set by

the conditions of temperature T and pressure P, with a variable area flow meter that was

calibrated at conditions T0 , P0, and corresponding 0. If we define Q0 to be the flow rate directly

indicated by the meter, we know that the true flow rate Q at Tand P will not be Q0. The next

figure illustrates this scenario.

Floats at same

height, so flow area

A is the sameFlow conditions at

T0 and P0 :

Q0 indicated by

scale is correct

Flow conditions at

Tand P: scale

indicates Q0, but

true value is Q


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Because the height of the float determines the value ofKA, which is therefore the same value in

the two cases shown, we have:

QQ00

=

Treating air as an ideal gas with P =RTgenerally (R = gas constant), and solving for Q yields:

0

0

0 T

T

P

PQQ =

Recall that Q is the true flow rate at the actual conditions ofTand P. This bears distinction

from the concept of an actual flow rate vs. a standard flow rate. The volumetric flow rates

Q0 and Q refer to actual flow rates, given the context of the derivation. Further conversion

would be required to change the true, actual volumetric flow Q into its standardized counterpart.

For the sake of further discussion, lets resort to the nomenclature used for the Vortex Tube

Characterization, where we will be working with standardized volumetric flow rates in SCFM,

and actual flow rates in ACFM. The flow meters from Omega are conveniently calibrated at

standard conditions Tstandard and Pstandard, which means that the values for Q0 will be the same in

ACFMor SCFM, so we will take the latter. Thus, lets refer to Q0, the indicated flow rate of the

meter, as SCFMmeter, and the generic true value for Q asACFM. Instead ofTand P, lets refer to

the measured conditions as Tactual and Pactual. By these re-assignments, the previous equation

translates to:

dardtans

actual

actual

dardtans

meter T

T

P

PSCFMACFM =

The Omega flow meters were designed to indicate standardized values, so it is appropriate to

label the output as SCFMmeter. In practice, we will want the true flow rate in standardized terms

as well, so we need to convert the trueACFMto SCFMaccording to:

actual

dardtans

dardtans

actual

T

T

P

PACFMSCFM =

Combining, there results:

actual

dardtans

dardtans

actual

meter T

T

P

P

SCFMSCFM =

The end result is nothing more than a flip of the square-root terms in the original relationship.

This is the expression we need most for the Vortex Tube Characterization; it lets us convert the

indicated SCFMmeter to the true SCFM through the vortex tube, for the actual operating

conditions Tactual and Pactual.


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References

Brain, T. J. S., & Scott, R. W. W. (1982). Survey of Pipeline Flowmeters.Journal of Physics E:

Scientific Instruments, 15, 967-980.

Doebelin, E. O. (2004). Constant-Pressure-Drop, Variable-Area Meters (Rotameters). In

Measurement Systems: Application and Design (5th ed., pp. 633-635). New York, NY:McGraw-Hill Higher Education.

Figliola, R. S., & Beasley, D. E. (2006). Rotameters. In Theory and Design for Mechanical

Measurements (4th ed., pp. 410-411). Hoboken, NJ: John Wiley & Sons, Inc.

Holman, J. P. (2001). Flow Measurement by Drag Effects - Rotameter. InExperimental Methods

for Engineers (7th ed., pp. 306-309). New York, NY: McGraw-Hill Higher Education.

Variable Area Flow Meter Explanation

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