Beware 
we're moving in on the world of the mechanical engineer  where
everything is topsyturvy, units are imperial, and nothing is as it
seems.
C_{v} is known as the "flow coefficient", and is frequently
specified for parts or equipment where it is necessary to be able to
predict the rate at which fluid will flow under certain conditions. It
was originally intended for liquid flows, and so it is defined as the
flow rate (in gallons per minute) of water at 60 degrees Fahrenheit,
across a pressure drop of 1 pound per square inch. Let's look at a more
concrete example to get our heads around this.
Fig. 1: Definition
of Cv
In the case of figure 1,
if we were to measure the amount of water that flowed out of the valve
in one minute, and found it to be 2 gallons, then the C_{v} for
that valve would be 2. If we replaced the valve with perhaps a larger
one and repeated this experiment and found that 10 gallons now flowed
in one minute, then the C_{v} of that valve would be 10. From
this simple experiment it is clear that the C_{v} value is an
indication of how much flow a certain valve, constriction, tube,
etc., will allow. The bigger the C_{v}, the higher the flow
rate. Since the C_{v} value is calculated for water, it can't
be used directly to calculate the flow of a gases. C_{v} does,
however, give us enough information to be able to figure out the gas
flow if we know a couple of things about the gas.
The first thing that we need to know about the gas flow is if it is a
critical flow. This has to due with the ratio of inlet to outlet
pressures which is explained on the compressible
flow
page
in the information section. If the inlet pressure is more than about
twice the outlet pressure (1.89 times, to be more accurate) then the
flow is said to be critical. If the flow is critical, then the flow
rate is limited by a shock wave inside the valve and the flow rate is
independent of the outlet pressure. If the flow is critical, the the
flow rate can be calculated using this equation [1]
Eq. 1: Flow Rate
as a
Function of C_{v} for Critical Flows
where Q_{G} is the flow rate in
SCFH (standard cubic feet per hour), C_{v} is the flow
coefficient, P_{IN}
in the inlet pressure in PSIA, G is the specific gravity of the gas,
and T is the absolute temperature (° F + 460.) The specific gravity
is relative to air at 70 ° F and 1 ATM, and is computed by dividing
the density of the gas used by the density of air under the stated
conditions. The densities of various gasses can be found on Wikipedia, among many other places. Wikipedia also
has a good article on specific
gravity.
If the flow is not
critical, then the flow rate does depend on the
outlet pressure, and we need to use this equation [2]
Eq. 2: Flow Rate as a
Function of C_{v} for SubCritical Flows
where Q_{G} is the flow rate in
SCFH (standard cubic feet per hour), C_{v} is the flow
coefficient, P_{IN} in the
inlet pressure in PSIA, P_{OUT}
is the pressure at the outlet in PSIA, G is the specific gravity of the
gas, and T is the absolute temperature (° F + 460.)
There are other forms of
these equations for fluids, and in metric. See
the links at the end of this page, Especially The Engineering Toolbox.
References
[1] Ideal Valve Inc., "Flow Calculations for Needle Valves", August
2007, http://www.idealvalve.com/flowcal.htm
[2] Ideal Valve Inc., "Flow Calculations for
Needle Valves", August 2007, http://www.idealvalve.com/flowcal.htm
Links
The Engineering Toolbox, Control Valves
Ideal Valve Inc., Flow Calculations For
Needle Valves
Prof.
Michael L. Corradini, Department of Engineering Physics, University of
Wisconsin, Fundamentals of Multiphase Flow
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