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Volume Flow Rate in Liquid and Gas Measurement

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On this blog, we have discussed in the past various flow meter technologies where a volumetric flow rate is required. It is also important that we discuss the unit of flow measurement in some of these flow meter technologies. This post intends to increase your understanding of volumetric flow rates in liquid and especially gas flow measurements.

As we may have seen, the majority of flow meter technologies operate on the principle of interpreting fluid flow based on the velocity of the fluid. Some of the flow meter technologies using this principle include:
(a) Ultrasonic flow meters
(b) Turbine Flow meters
(c) Orifice Flow meters etc.

In these velocity-based flow meters, fluid velocity can easily be translated into volumetric flow by using the continuity equation below:

                               Q = AV
Q = Volumetric Flow rate
A = Cross-sectional area of flow meter throat
V = Average fluid velocity at throat section

Volumetric Flow rate in Liquid Flow Measurement
Many industrial fluid flow applications involving liquids are in volumetric units because liquid measurement is relatively simple. Volumetric flow rate measurements in liquids are mostly in cubic feet per unit time (e.g. ft3/min), cubic meter per unit time (e.g. m3/min) or gallons per unit time (e.g. gallons/min). Liquids are essentially incompressible: that is, they do not easily yield in volume to applied pressure. This makes volumetric flow measurement relatively simple for liquids: one cubic meter of a liquid at high pressure and temperature inside a process vessel will occupy approximately the same volume (≈ 1m3) when stored in another process vessel at ambient pressure and temperature. That is to say that volumetric flow rates in most liquid systems are practically independent of pressure and temperature changes.

Volumetric Flow Rate in Gas Flow Measurement
Gases and vapours easily change their volume under the influences of pressure and temperature. In other words, a gas will yield to an increasing pressure by decreasing in volume as the gas molecules are forced closer together, and it will yield to a decreasing temperature by decreasing in volume as the kinetic energy of the individual molecules is reduced.

This makes volumetric flow measurement more tricky and complex for gases and vapours than for liquids. One cubic meter of gas at high a pressure and temperature inside a process vessel will not occupy one cubic meter under different pressure and temperature conditions in the same vessel. This implies that volumetric flow measurement for gas is virtually meaningless without the accompanying data on pressure and temperature.
Standardized Volumetric Flow Measurement
Since a gas occupies different volumes at different conditions of temperature and pressure, gas volumes are specified at some agreed-upon set of pressure and temperature known as standard conditions and the gas volumes referred to as standardized volumetric flow measurement.

To distinguish actual volumetric flow rate from standardized volumetric flow rate, we commonly preface each unit with a letter “A” or letter “S” as the case may be e.g. ACFM and SCFM. Here, ACFM means actual cubic foot per minute which is the volume of the gas at the flowing conditions. SCFM means standard cubic foot per minute which is the volume of the same gas now at standard conditions of temperature and pressure.

Standard Conditions Used to Determine Standardized Volumetric Flow Rates
Various standard conditions exist for determining standardized flow rates in most custody transfer applications for gases around the world especially natural gas:
(a) API (American Petroleum Institute) uses 14.7PSIA and 60 degree Fahrenheit equivalent to 519.67 degree Rankine as their standard conditions for calculating gas volumetric flow rates
(b) ASME (American Society of Mechanical Engineers) uses 14.7 PSIA and 68 degrees Fahrenheit (527.67 degrees Rankine) as their standard conditions for calculating
gas volumetric flow rates.
(c) The American CAGI (Compressed Air and Gas Institute) uses 14.5 PSIA and 68 degrees Fahrenheit (527.67 degrees Rankine) as their standard conditions for calculating gas volume flow rates.

Actual Versus Standard Volumetric Flow Rates

As shown above, consider a gas with an actual flow rate of VA, at pressure PA  and temperature TA  at flowing condition. Suppose the gas is allowed to expand to standard conditions and we now want to determine the volume of the gas, VS, at this condition of pressure PS and temperature TS . We can use the ideal gas equation to determine this:

P =  Pressure
V = Volume
Z = Compressibility factor of the gas
R = Universal gas constant
T = Temperature

At conditions far from their critical phase-change points, most real gases behave like ideal gases i.e Z =1 hence we have a fair approximation of the ideal gas law thus:
                       $PV = nRT$

Now at actual flowing conditions, we have: 
 $P_AV_A = nRT_A $  -----------(1)
At standard conditions, we have: 
$P_SV_S = nRT_S$ -------------(2)
Dividing equation (2) by (1) we have:
 $\frac{P_SV_S}{P_AV_A} = \frac{nRT_S}{nRT_A}$            
Which now reduces to:
 $\frac{V_S}{V_A} = \frac{P_AT_S}{P_ST_A}$              
Since we know that the definition of volumetric flow (Q) is volume over time (V/t), we may divide each Volume ,V, variable by t to convert this into a volumetric flow rate thus:

 $\frac{Q_S}{Q_A} = \frac{P_AT_S}{P_ST_A}$                  
The above equation gives us the ratio of standardized volumetric flow rate (Qs) to actual volumetric flow rate (Q), for any known pressures and temperatures.
From this equation, we can see that standard volumetric flow rate is given by:

 $Q_S = Q_A\frac{P_AT_S}{P_ST_A}$    
To use the above formula, pressure and temperature must be in absolute units as envisaged by the ideal gas law.
Let us consider an example to illustrate the application of the above formula in a natural gas custody transfer application:
A natural gas metering station is producing 200,000ACFH(actual cubic feet per hour) of gas at an average temperature of 350C and pressure of 18barg.
(a) What is the volumetric flow rate in standard cubic feet per hour(SCFH)
(b) How much volume of gas can be  delivered in two(2) days in
(i) SCF (standard cubic feet) (ii) SCM(standard cubic meters)
(Use P = 14.7PSIA and $T = 68^0F$ as standard conditions
(a) To get the volumetric flow rate in SCFH we convert the volume to standard conditions (P = 14.7PSIA and $T = 68^0F$) using the formula below:
Where :
$Q_S$ = Volumetric flow rate in SCFH
$Q_A$ = 200,000ACFH
PA  = 18barg = (18*14.7 + 14.7) = 279.3psia 
Note Absolute Pressure = Gauge Pressure + 14.7
$T_A = 35^0C = (1.8*35 + 32) = 95^0F = 95 + 459.67  = 554.67^0R$
Note ( degree F = 1.8*C + 32) and (degree R = F + 459.67)
$P_S$ = 14.7psia
$T_S$  = 68 + 459.67 = 527.67degree Rankine
$Q_S$  = 200,000 x (279.3*527.67)/(14.7*554.67) = 3,615,025.15008SCFH =     
             = 3.615025MMSCFH
(b)(i) Total volume of gas in SCF delivered in two(2) days :
         = 3,615,025.15008 x 2 x 24 = 173,521,207.20384SCF [1 day = 24 hours]
         = 173.521MMSCF
(b)(ii) Total volume of gas in SCM delivered in two (2) days:
          = 173,521,207.20384SCF x 0.0283168466SCM/SCF
          = 4,913,573.40624SCM = 4.9136MMSCM

Conversion Factors:
As far as volumetric flow rate calculations are concerned, you will find the following conversion factors and formula useful:
1SCF = 0.0283168466SCM
1SCM = 35.314666711SCF
MSCF = 1000SCF
MMSCF = 1,000,000SCF
MSCM = 1,000SCM
MMSCM = 1,000,000SCM
Degree F = 1.8*C + 32  , F denotes Fahrenheit, C denotes degree celsius
Degree R = F + 459.67
Absolute Pressure(psia) = Gauge Pressure(psia) + 14.7psia

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