#
Liquid, Steam and Gas - Flow Coefficients *C*_{v}

_{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 SI-units.

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)

*Water flow (US gal per minute)*

*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 flow (m ^{3}/hr)*

*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 =(1 for water)specific gravity

dp = pressure drop (psia)

*Water flow (lb/hr)*

*Pressure drop (psi)*

or alternatively in SI units:

C_{v}= 5.8 w / (500 (dp SG)^{1/2})(1d)

where

w = water flow (kg/hr)

SG =(1 for water)specific gravity

dp = pressure drop (kPa)

*Water flow (kg/hr)*

*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*before the control valve. The flow coefficient at choked - or critical - flow can be expressed as:

_{i }-

C_{v}= m / 1.61 p_{i}(2)

where

m = steam flow (lb/hr)

p_{i}= inlet steam absolute pressure (psia)

*Steam flow (lb/hr)*

*Inlet Steam 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*before the control valve. The flow coefficient for non critical flow can be expressed as:

_{i }-

C_{v}= m / (2.1 ( (p_{i}+ p_{o}) dp)^{1/2}) (2b)

where

p_{o}= outlet steam absolute pressure (psia)

*Steam flow (lb/hr)*

*Inlet Steam Pressure (psia)*

*Outlet Steam Pressure (psia)*

### Flow Coefficient - *C*_{v} - Super-heated 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 super-heated steam

The flow coefficient for steam super-heated 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*is approximately 53% of inlet

_{o}*p*pressure.

_{i}#### 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)

*Gas flow (Cu.ft./h) (f.a.d)*

*Inlet Gas Absolute Pressure (psia)*

*Pressure recovery factor*

*upstream Specific Gravity Gas*

*Gas Temperature ( ^{o}F)*

#### 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)

*Gas flow (Cu. ft./h) (f.a.d)*

*Inlet Gas Pressure (psia)*

*Outlet Gas Pressure (psia)*

*upstream Specific Gravity Gas*

*Gas Temperature ( ^{o}F)*