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Pressure-Based Flowmeters

All masses require force to accelerate (we can also think of this in terms of the mass generating a reaction force as a result of being accelerated). This is quantitatively expressed by Newton’s Second Law of Motion:

 

Newton's Second Law of Motion Illustrated

 All fluids possess mass, and therefore require force to accelerate just like solid masses. If we consider a quantity of fluid confined inside a pipe1, with that fluid quantity having a mass equal to its volume multiplied by its mass density (m = ρV , where ρ is the fluid’s mass per unit volume), the force required to accelerate that fluid “plug” would be calculated just the same as for a solid mass:


Newton's Second Law of Motion with a solid mass

 

Since this accelerating force is applied on the cross-sectional area of the fluid plug, we may express it as a pressure, the definition of pressure being force per unit area:

Force Equation

Since the rules of algebra required we divide both sides of the force equation by area, it left us with a fraction of volume over area (V/A) on the right-hand side of the equation. This fraction has a physical meaning, since we know the volume of a cylinder divided by the area of its circular face is simply the length of that cylinder:


Pressure Equation

When we apply this to the illustration of the fluid mass, it makes sense: the pressure described by the equation is actually a differential2 pressure drop from one side of the fluid mass to the other, with the length variable (l) describing the spacing between the differential pressure ports:


Pressure Drop Illustrated

This tells us we can accelerate a “plug” of fluid by applying a difference of pressure across its length. The amount of pressure we apply will be in direct proportion to the density of the fluid and its rate of acceleration. Conversely, we may measure a fluid’s rate of acceleration by measuring the pressure developed across a distance over which it accelerates.

We may easily force a fluid to accelerate by altering its natural flow path. The difference of pressure generated by this acceleration will indirectly indicate the rate of acceleration. Since the acceleration we see from a change in flow path is a direct function of how fast the fluid was originally moving, the acceleration (and therefore the pressure drop) indirectly indicates fluid flow rate.

3What really matters in Newton’s Second Law equation is the resultant force causing the acceleration. This is the vector sum of all forces acting on the mass. Likewise, what really matters in this scenario is the resultant pressure acting on the fluid plug, and this resultant pressure is the difference of pressure between one face of the plug and the other, since those two pressures impart two forces on the fluid mass in direct opposition to each other.

A very common way to cause linear acceleration in a moving fluid is to pass the fluid through a constriction in the pipe, thereby increasing its velocity (remember that the definition of acceleration is a change in velocity). The following illustrations show several devices used to linearly accelerate moving fluids when placed in pipes, with differential pressure transmitters connected to measure the pressure drop resulting from this acceleration:


Restriction to measure differential pressure

Another way we may accelerate a fluid is to force it to turn a corner through a pipe fitting called an elbow. This will generate radial acceleration, causing a pressure difference between the outside and inside of the elbow which may be measured by a differential pressure transmitter:

 

Pipe Elbow

The pressure tap located on the outside of the elbow’s turn registers a greater pressure than the tap located on the inside of the elbow’s turn, due to the inertial force of the fluid’s mass being “flung” to the outside of the turn as it rounds the corner.

Yet another way to cause a change in fluid velocity is to force it to decelerate by bringing a portion of it to a full stop. The pressure generated by this deceleration (called the stagnation pressure) tells us how fast it was originally flowing. A few devices working on this principle are shown here:

 

Deceleration to measure flow

The following subsections in this flow measurement chapter explore different primary sensing elements (PSE’s) used to generate differential pressure in a moving fluid stream. Despite their very different designs, they all operate on the same fundamental principle: causing a fluid to accelerate or decelerate by forcing a change in its flow path, and thus generating a measurable pressure difference. The following subsection will introduce a device called a venturi tube used to measure fluid flow rates, and derive mathematical relationships between fluid pressure and flow rate starting from basic physical conservation laws.


Venturi Tubes and Basic principles - The standard “textbook example” flow element used to create a pressure change by accelerating a fluid stream is the venturi tube: a pipe purposefully narrowed to create a region of low pressure. As shown previously, venturi tubes are not the only structure capable of producing a flow-dependent pressure drop. You should keep this in mind as we proceed to derive equations relating flow rate with pressure change: although the venturi tube is the canonical form, the exact same mathematical relationship applies to all flow elements generating a pressure drop by accelerating fluid, including orifice plates, flow nozzles, V-cones, segmental wedges, pipe elbows, pitot tubes, etc... Click here to read more...

Volumetric Flow Calculations - As we saw in the previous subsection, we may derive a relatively simple equation for predicting flow through a fluid-accelerating element given the pressure drop generated by that element and the density of the fluid flowing through it... Click here to read more...

Mass Flow Calculations - Measurements of mass flow are preferred over measurements of volumetric flow in process applications where mass balance (monitoring the rates of mass entry and exit for a process) is important. Whereas volumetric flow measurements express the fluid flow rate in such terms as gallons per minute or cubic meters per second, mass flow measurements always express fluid flow rate in terms of actual mass units over time, such pounds (mass) per second or kilograms per minute... Click here to read more...

Square-Root Characterization - It should be apparent by now that the relationship between flow rate (whether it be volumetric or mass) and differential pressure for any fluid-accelerating flow element is non-linear: a doubling of flow rate will not result in a doubling of differential pressure. Rather, a doubling of flow rate will result in a quadrupling of differential pressure... Click here to read more...

Orifice Plates - Of all the pressure-based flow elements in existence, the most common is the orifice plate. This is simply a metal plate with a hole in the middle for fluid to flow through. Orifice plates are typically sandwiched between two flanges of a pipe joint, allowing for easy installation and removal... Click here to read more...

Other Differential Producers - Other pressure-based flow elements exist as alternatives to the orifice plate. The Pitot tube, for example, senses pressure as the fluid stagnates (comes to a complete stop) against the open end of a forward-facing tube. A shortcoming of the classic single-tube Pitot assembly is sensitivity to fluid velocity at just one point in the pipe, so a more common form of Pitot tube seen in industry is the averaging Pitot tube consisting of several stagnation holes sensing velocity at multiple points across the width of the flow... Click here to read more...

Proper Installation - Perhaps the most common way in which the flow measurement accuracy of any flowmeter becomes compromised is incorrect installation, and pressure-based flowmeters are no exception to this rule. The following list shows some of the details one must consider in installing a pressure-based flowmeter element... Click here to read more...

High-Accuracy Flow Measurement - Many assumptions were made in formulating flow equations from physical conservation laws. Suffice it to say, the flow formulae you have seen so far in this chapter are only approximations of reality. Orifice plates are some of the worst offenders in this regard, since the fluid encounters such abrupt changes in geometry passing through the orifice. Veni tubes are nearly idtureal, since the machined contours of the tube ensure gradual changes in fluid pressure and minimize turbulence... Click here to read more...

Equation Summary - from Volumetric flow rate, Mass flow rate, equations related are discussed... Click here to read more...

 

1Sometimes referred to as a plug of fluid.

Go Back to Lessons in Instrumentation Table of Contents


 

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