Liquid, Steam and Gas  Flow Coefficients C_{v}
Calculate flow coefficients for the design of control valves  Imperial units.
With the flow coefficients capacities of valves at different sizes, types and manufacturers can be compared. The flow coefficients are in general determined experimentally and express the
flow capacity in imperial units  GPM (US gallons per minute) that a valve will pass for a pressure drop of 1 lb/in^{2} (psi)
The flow factor  Kv  is also commonly used with capacity in SIunits.
The flow coefficient  C_{v}  required for a specific application can be estimated by using specific formulas for the different fluids or gases. With an estimated C_{v} value  the correct size of control valve can be selected from the manufacturers catalogs.
Note that an oversized control valve may hurt process variability by putting too much gain in the valve leaving less flexibility for the controller. An oversized valve operates more frequently at lower openings with increased dead band as result.
Flow Coefficient  C_{v}  for Liquids
For liquids the flow coefficient  C_{v} is expressed with water flow capacity in gallons per minute (GPM) of 60^{o}F with pressure drop 1 psi (lb/in^{2}).
Flow expressed by volume
C_{v} = q (SG / dp)^{1/2} (1)
where
q = water flow (US gallons per minute)
SG = specific gravity (1 for water)
dp = pressure drop (psi)
or alternatively in metric units:
C_{v} = 11.6 q (SG / dp)^{1/2} (1b)
where
q = water flow (m^{3}/hr)
SG = specific gravity (1 for water)
dp = pressure drop (kPa)
Water Control Valves  flow coefficient C_{v} diagram
Flow expressed by weight
C_{v} = w / (500 (dp SG)^{1/2}) (1c)
where
w = water flow (lb/hr)
SG = specific gravity (1 for water)
dp = pressure drop (psia)
or alternatively in SI units:
C_{v} = 5.8 w / (500 (dp SG)^{1/2}) (1d)
where
w = water flow (kg/hr)
SG = specific gravity (1 for water)
dp = pressure drop (kPa)
Example  Flow Coefficient Liquid
The flow coefficient for a control valve which in full open position passes 25 gallons per minute of water with a one pound per square inch pressure drop can be calculated as:
C_{v} = (25 gpm) (1 / (1 psi))^{1/2}
= 25
Flow Coefficient  C_{v}  for Saturated Steam
Since steam and gases are compressible fluids, the formula must be altered to accommodate changes in density.
Critical (Choked) Pressure Drop
With choked flow and critical pressure drop, the outlet pressure  p_{o}  after the control valve is aprox. 58% of the inlet pressure  p_{i } before the control valve. The flow coefficient at choked  or critical  flow can be expressed as:
C_{v} = m / 1.61 p_{i} (2)
where
m = steam flow (lb/hr)
p_{i} = inlet steam absolute pressure (psia)
Non Critical Pressure Drop
For non critical pressure drop the outlet pressure  p_{o}  after the control valve is more than 58% of the inlet pressure  p_{i } before the control valve. The flow coefficient for non critical flow can be expressed as:
C_{v} = m / (2.1 ( (p_{i} + p_{o}) dp) ^{1/2}) (2b)
where
p_{o} = outlet steam absolute pressure (psia)
Flow Coefficient  C_{v}  Superheated Steam
The flow coefficient for superheated steam should be multiplied with a correction factor:
C_{v} = C_{v_saturated} (1 + 0.00065 dt) (3)
where
dt = steam temperature above saturation temperature at the actual pressure (^{o}F)
Example  flow coefficient superheated steam
The flow coefficient for steam superheated with 50^{o}F can be calculated as:
C_{v} = C_{v_saturated} (1 + 0.00065 (50^{o}F) =1.0325 C_{v_saturated}
Flow Coefficient  C_{v}  Saturated Wet Steam
Saturated wet steam includes non evaporated water particles reducing the "steam quality" and a flow coefficient for very wet saturated steam should be multiplied with a correction factor:
C_{v} = C_{v_saturated} ζ^{1/2} (4)
where
ζ = dryness fraction
Example  Flow Coefficient Wet Saturated Steam
For steam with moisture content 5% the dryness fraction can be calculated as:
ζ = w_{s} / (w_{w} + w_{s})
= 0.95 / (0.95 + 0.05)
= 0.95
where
w_{w} = mass of water
w_{s} = mass of steam
The flow coefficient can be calculated as:
C_{v} = C_{v_}_{saturated} (0.95)^{1/2}
= 0.97 C_{v_saturated}
Flow Coefficient  C_{v}  Air and other Gases
Note!  there is a difference between critical and non critical pressure drops.
Critical pressure drop is approximately 47% of inlet p_{i} absolute pressure. With critical pressure drop the outlet absolute pressure p_{o} is approximately 53% of inlet p_{i} pressure.
Pressure Drop Larger than Critical Pressure Drop
For pressure drop larger than the critical pressure drop the flow is choked. The flow coefficient can be calculated as:
C_{v} = q [SG (T + 460)]^{1/2}/ (F_{L} 834) p_{i} (5)
where
q = free gas per hour, standard cubic feet per hour (Cu.ft/h)
SG = upstream specific gravity of flowing gas gas relative to air (SG = 1.0) at 14.7 psia and 60^{o}F
T = flowing air or gas temperature (^{o}F)
F_{L} = pressure recovery factor
p_{i} = inlet gas absolute pressure (psia)
Pressure Drop Less than Critical Pressure Drop
For pressure drop less than the critical pressure drop the flow is not choked. The flow coefficient can be calculated as:
C_{v} = q [SG (T + 460)]^{1/2}/ [1360 (dp p_{o})^{1/2}] (5b)
where
dp = (p_{i}  p_{o})
p_{o} = outlet gas absolute pressure (psia)
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