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**TECH
NOTE: Solid State Flow Elements** (11/96)

**DISCUSSION:**

Since the turn of the century, "Head" flow devices have been used extensively in virtually all types of flow metering applications. Head meters include; Orifice, Venturi, Nozzle, and Pitot Tube. As a class, they are the still the most popular. This popularity comes from simplicity of design & operation, good accuracy, and cost effectiveness.

With new metering technologies such as Coriolis, Magnetic, and Vortex, it appeared that these meters were to be relegated to more generic and lower precision services. However, all this is changing with today's "SMART" electronics. Now Head meters, when properly outfitted can offer truly remarkable performance complementing their inherent design features. Basic advantages include; no moving parts, NIST traceable accuracy available, no delicate parts, compact & rugged, and familiar operation.

**PRINCIPAL
OF OPERATION:**

Head flow elements
are characterized by a converging-diverging flow path geometry
(excluding Pitot tube), and can be broken into three (3) classes;
*sub-critical* for gases and liquids, *critical* for
gases, and *cavitating* for liquids. The term
"critical" refers to the velocity at the smallest
section. For sub-critical devices; per Bernoulli’s Law,
flow rate is a function of the square root of the differential
pressure developed. Should this velocity reach the speed of
sound, the device is then considered to be in critical flow. Once
critical (or "sonic") flow is reached, a different flow
equation applies. This new equation has flow based on inlet
conditions only, and not differential pressure. In fact, flow is
generally calculated on a mass basis and is a function of the
inlet gas density and individual gas characteristics only. With
the speed of sound a natural barrier, critical flow elements now
make very attractive calibration devices, as well as mass flow
controllers for gases.

A third class is that of the cavitating element (liquids only). Much like the Critical or Sonic gas device, the cavitating Venturi has a natural flow limiting phenomenon. Liquid is accelerated through the device resulting in a lowering of the static pressure at the smallest cross section. If this pressure is sufficiently lowered to or below the liquids' vapor pressure, the liquid changes to the gas phase (boils or cavitates). The resulting dramatic change in volume constricts the flow. Consequently for a given upstream condition, the flow will remain unchanged regardless of downstream pressure during this condition. This then creates an attractive solid state flow limiting or control device.

**Instrumentation
* Controls * Systems * Design * Fabrication**

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Page 2 - Solid State Flow Elements**

**Sub-critical
flow (gases & liquids):**

The basic operational principal for sub-critical (less than sonic velocity) Head meters is based on the Bernoulli energy equation. When fluid (liquid or gas) moves from a larger to a smaller flow area, the velocity (or kinetic energy) increases while the static pressure decreases. The resulting difference in static pressure is then in relation with the flow rate. Pitot Tube meters operate similarly except that the pressure difference is based on the variance in static and impact (total, stagnation) pressures. A basic equation for sub-critical and cavitating flow is;

Q = C (DP / RHO) ^ ^{.5}

Q = Flow / time

C = Coefficient adjusting for a number of factors (1)

DP = Static differential pressure across pressure taps (2)

RHO = Flowing density of fluid

(1) "C" adjusts for parameters such as area, velocity effects, deviation from ideal, thermal and gas effects, etc.

(2) For cavitating flow, P2 (where DP = P1-P2) is such that the static pressure is equal or less than the liquids' vapor pressure at that temperature.

Before modern electronics, the square root function was awkward to handle. In addition, since flow rate is a function of the square root of DP, the DP range had to be quite large to achieve good flow turndown. Today these parameters can be handled with high precision. With "SMART" DP transmitters, turndowns of greater than 10:1 are achievable. On gases with density changes, mass flow rate turndowns can be dramatic.

**Critical
or Sonic flow (gases only):**

For gases a special flow case is realized when the pressure drop through the device is sufficient to cause the velocity at or near the smallest section to reach the speed of sound. This natural barrier is a function of the gas makeup, and inlet density. A basic equation for critical or sonic flow is;

Qm = A C C* Po / [
(R / M) To ] ^ ^{.5}

Qm = Mass Flow / time

A = Section area

C = Coefficient adjusting for a number of factors (1)

C* = Coefficient based on gas characteristics (2)

Po = Inlet total pressure (absolute)

R = Gas constant

M = Gas molecular weight

To = Inlet total temperature (absolute)

(1) "C" adjusts for deviation from ideal in both flow and flow area. Typical "C" values are from 0.85 to near 1.00.

(2) C* is generally known as the critical flow function or coefficient. C* is principally a function of the gas' isentropic (K) coefficient and is well known for most industrial gases.

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Page 3 - Solid State Flow Elements

Note that flow, is calculated without the square root of DP function, and that the flow is in mass units. As above, this equation can be handled with great precision with today's electronics. Coupled with precision instrumentation, an extremely accurate calibration, proving, or control system can be assembled.

**UNCERTAINTY:**

Because of the historical user basis, the performance of most solid state elements is well known. Associations such as ASME / ANSI / AGA and others have well documented "C" factors. Consequently for some applications, calibration may not be necessary. In this case, the accuracy of a properly constructed element may be about +/- 2% (or better) of READING (about +/-1% of READING for Sonic Nozzles). Calibration for sub-critical elements can improve uncertainty to as good as 0.25%. For Sonic Nozzles (critical flow), uncertainty can approach 0.1%. Repeatability regardless of calibration level is near perfect, since we are dealing with no moving or delicate parts. By combining these precision devices with quality instrumentation, good to state of the art flow instrumentation systems can be assembled.

**APPLICATIONS:**

Venturis, nozzles, and specialty orifices are now making a real comeback on tough, high precision, and other demanding applications. Their smooth interior makes for a durable device in erosive services. Ultra clean versions are available for the semi-conductor, aerospace, and food industries. Sonic (critical flow) Nozzles can be packaged to create gas calibration, metering, and control systems with unequaled accuracy. When fitted with a simple pressure regulator, a Sonic Nozzle makes a MASS FLOW controller for gases. Cavitating venturis can be used for flow control of liquids. In addition, specialty orifices such as the quadrant edged or conical inlet provide attractive solutions for very low flow and/or viscous services. Many general industrial applications are improved with the use of these solid state devices.

**DESIGN:**

In general the devices include an inlet section, a reduced area, and a recovery section. The inlet and recovery section may or may not be an integral part of the element. Typically, the system piping will be part of these elements. Ports are included to measure pressure and optionally temperature. Straightening vanes may also be included. The inlet section may be oversized such that static pressure / temperature is a good measure of the total (stagnation) values. This "plenum" effect also minimizes swirl, pulsation, and other flow abnormalities.

Venturis and nozzles typically are constructed to ASME / ANSI standards which will include either a conical or circular (in cross section) inlet, and a conical recovery or diffuser section. Orifices are built to similar standards (including AGA) for the thin sharp edged type. Quadrant and conical styles are designed around historical criteria.

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Page 4 - Solid State Flow Elements

**REFERENCES
and further reading:**

FLOW MEASUREMENT ENGINEERING HANDBOOK, McGraw-Hill, New York, 1983

ASME FLUID METERS, ASME, New York

MEASUREMENT OF FLUID FLOW IN PIPES USING ORIFICE, NOZZLE, AND VENTURI - ASME MFC-3M-1985, ASME, New York

MEASUREMENT OF GAS FLOW BY MEANS OF CRITICAL FLOW VENTURI NOZZLES - ASME/ANSI MFC-7M-1987, ASME, New York

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