### Flow Instrumentation: Principles and Formulas

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The measurement of fluid flow is arguably the single most complex type of process variable measurement in all of industrial instrumentation. This is because there are vast array of flow metering technologies that can be used to measure fluid flow each one with its own limitations and individual characteristics. Even after a flow meter has been properly designed and selected for the process application and properly installed in the piping, problems may still arise due to changes in process fluid properties (density, viscosity, conductivity), or the presence of impurities in the process fluid. Flow meters are also subject to far more wear and tear than most other primary sensing elements, given the fact that a flow meter’s sensing element(s) must lie directly in the path of potentially abrasive fluid streams.

Given all these difficulties and complications of fluid flow measurement, it becomes imperative for any end user of any given flow meter technology to understand the complexities of flow measurement. What matters most is that you thoroughly understand the physical principles upon which each flow meter depends. If the “first principles” of each technology are understood, the appropriate applications and potential problems become much easier to identify and solved.
Here some basic principles and common formulas used in flow instrumentation are introduced.

Flow rate
Fluid flow rate can either be volumetric or mass flow rate. The volumetric flow rate is the volume of fluid passing a given point in a given amount of time and is typically measured in gallons per minute (gpm), cubic feet per minute (cfm), liter per minute, and so on. Mass flow rate is the mass of fluid per unit time pass a given point. Mass flow rate measurements are more common in gases (gas density varies more than does liquid density)
If the density ρ of the fluid is known, mass flow rate and volume flow rate are related by :

Mass flow rate = Density of fluid(ρ) x Volume flow rate

Continuity Equation
The flow of fluid at any point in a pipe of constant cross sectional area(shown below) can be determined using the continuity equation. The continuity equation states that if the overall flow rate in a system is not changing with time , the flow rate in any part of the system is constant. From which we get the following equation:

Q = VA

Where
Q = flow rate
V = average velocity of fluid
A = Cross section area of the pipe.

If the fluid is flowing in a pipe with different cross section areas, i.e., A1 and
A2, as is shown below, the continuity equation gives:

$Q = V_1A_1 = V_2A_2.$

The Bernoulli Equation
The Bernoulli equation is an equation for flow based on the law of conservation
of energy, which states that the total energy of a fluid or gas at any one point in a flow stream is equal to the total energy at all other points in the flow. The diagram below illustrates the Bernoulli equation:
The total energy of fluid in a flow system is comprised of three components namely: potential energy, kinetic energy and pressure energy. When described in terms of meters head of the flowing fluid, the Bernoulli equation becomes:

$Z_1ρg + V^2_1ρ/2 + P_1 = Z_1ρg + V^2_2ρ/2 + P_2$       or

$Z_1 + V^2_1/2g + P_1/У = Z_2 + V^2_2/2g + P_2/У$

Where,
z = Height of fluid (from a common reference point, usually ground level)
ρ = Mass density of fluid
У  = Weight density (У = ρg )
g = Acceleration of gravity
v = Velocity of fluid
P = Pressure of fluid

Flow Rate Measurement Across a Restriction in a Pipe:
As shown in the diagram below, when there is a flow restriction in a pipe, the following occurs:

• A pressure drop is created downstream of the restriction. This pressure drop creates an irreversible head loss in the piping system
• The pressure drop associated with this restriction can be measured and used to determine the flow rate through the restriction device.
• The volume flow rate measured can be calibrated as function of the measured pressure drop
Three main types of flow meters use this flow restriction principle to measure fluid flow. They are the orifice meter, flow nozzle and the venture flow meter. They however have different performance characteristics.

Flow and Differential Pressure in a Restriction Flow Meter
Applying Bernoulli principle across the flow restriction shown in the diagram above and neglecting all head loss, we have that:

$Z_1 + V^2_1/2g + P_1/ρg = Z_2 + V^2_2/2g + P_2/ρg$
Now from the diagram above, $Z_1 = Z_2$, the acceleration due to gravity , g, in the equation will cancel out leaving:

$\frac{P_1}{ρ} + \frac{V^2_1}{2} = \frac{P_2}{ρ} + \frac{V^2_2}{2}$

Multiplying through by fluid density, ρ, and solving we have:

$P_1 - P_2 = \frac{ρ}{2}(V^2_2 - V^2_1)$  .....................(A)

From the continuity equation, $Q = V_1A_1 = V_2A_2$. This implies that:
$V_1 = \frac{Q}{A_1}$ , $V_2 = \frac{Q}{A_2}$
Putting these into equation (A) above and solving, we have:

$P_1 - P_2 = \frac{Q^2ρ}{2}(\frac{1}{A^2_2} - \frac{1}{A^2_1})$

For a given flow restriction device, $A_1$ and $A_2$ are constants as they do not change with pressure, density or flow rate. Knowing this, we may re-write  our equation as:

$Q = K\sqrt{\frac{P_1 - P_2}{ρ}}$

which can be reduced to:

$Q = K\sqrt{\frac{\Delta {P}}{ρ}}$

Where:
K = Constant of proportionality
$\Delta {P } = P_1 - P_2$

From the above equation, we can deduce the following:
(a)Flow is proportional to the square root of the differential pressure across a flow restriction device
(b)Differential pressure is proportional to the square of the flow across the restriction device.

This equation governs all flow rate measurement in differential pressure flow meters.