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ASME/NIST Long-Radius Flow Nozzle

Picture of an ASME type gas flow nozzle

The flow nozzle, as specified by ASME/NIST, exhibits better consistency and pressure-recovery characteristics than the sharp-edged orifice, and is more compact than the converging-diverging venturi. The long-radius flow nozzle,which has been precisely defined and its flow characteristics studied, is the subject of this theoretical analysis.

The differential-pressure fluid flow meter, using thin-plate orifice, flow nozzle or venturi sensor, is widely used.

The operating principle of all three primary elements - orifice, nozzle and venturi - involves the Bernoulli relationship between the pressure drop through a flow restriction and the mass rate of flow. The relationship includes, in addition to the sizes of the passages and properties of the fluid, the upstream and downstream pressures and upstream temperature. Here "upstream" and "downstream" refer to the restriction.

It has been more than 2 1/2 centuries since Daniel Bernoulli developed the theory for ideal behavior of differential-pressure devices. Although the orifice plate, nozzle and venturi meter are all based on Bernoulli's equations, they differ in their ability to recover pressure drop, and in the magnitude of their departure from simple theoretical characteristics.

IDEAL FLOW THEORY

The flow of fluids, both liquids and gases (incompressible and compressible, respectively) is governed by the equations of continuity (mass flow rate, mdot) and energy. Let us write these mass and energy equations for two stations - (1) entrance to the nozzle and (2) the throat of restriction.

The mass continuity equation for mass flow rate (mdot) is:AV<sub>1</sub>ρ<sub>1</sub> = aV<sub>2</sub>ρ<sub>2</sub>where:
A = the duct area a = throat areaV<sub>1</sub> and V<sub>2</sub> = fluid velocities at points (1) and (2)ρ<sub>1</sub> and ρ<sub>2</sub> = fluid densities at points (1) and (2)         (1)

The Energy Equation for the same two cross-section points is:

P1/ρ1 + V1^2/2gc + ui1 = Ps/ρs + V2^2/2gc +ui2where:P<sub>1</sub> and P<sub>2</sub> = fluid static pressure at points (1) and (2)u<sub>i1</sub> and u<sub>i2</sub> = internal energy at points (1) and (2)g = local gravity acceleration<sub>c</sub> = gravitational constant            (2)

For incompressible fluids, density and temperature may be taken as unchanging (ρ1 = ρ2 = ρ ) and ui1 = ui2(assuming no appreciable change in liquid temperature). Equations (1) and (2) are then readily combined to yield the classical result for nozzle exit velocity:

V2^2 = 2gc ( δ-P/ρ )(1/(1- Β^4))where: δP = P<sub>1</sub>-P<sub>2</sub>Β = d/DΒ<super>2</super> = a/A                               (3)
The mass flow rate, mdot = ρaVs, is then expressed as: mdot^2 = a^2(2gc ρ δP /(1 - Β^4) )

COMPRESSIBLE FLOW

The case of gaseous (compressible) flow, including viscous friction, is more involved. We are obliged to introduce, in addition to the mass continuity and energy equations, an equation of state relating the intensive properties of the fluid - temperature, pressure, density. For moderate temperatures and pressures (the majority of cases), we can use the ideal gas law:

P = ρRT 

where:R = gas constant, ft * lb<sub>f</sub>/l<sub>m</sub>°R   = <i>R</i>/Molecular Weight   = 1545.33/29 (for air)   = 53.3

Another reasonable assumption is that the flow through the nozzle is unaccompanied by heat transfer between the fluid and the nozzle material, and that the internal friction inthe fluid is negligible - i.e., it is an isentropic process. In this isentropic case:

(6)P1/ρ1^γ = P2/ρ2^γ = CONSTANT  
where: γ = ratio of specific heat (c<sub>p</sub>/c<sub>v</sub>)

If equations (1), (5) and (6) are incorporated into (2), and taking as zero the work done on or by the system (δw = du + pdv = 0), we can arrive at:

(V2^2/2gc - V1^2/2gc = P1/ρ1(γ/(γ-1))(1 - r^((γ-1)/γ))where: pressure ratio r = p<sub>2</sub>/p<sub>1</sub>

From the continuity equation, (1), and using

V<sub>1</sub>/V<sub>2</sub>= (Beta^2)r^1/gamma

Solving for V1 and substituting into (7):

Equation is too complex to render adequately in text

Knowing the exit velocity of the nozzle flow, we can compute the mass flow rate:

Equation is too complex to render adequately
 in text

To bring (10) into agreement with the form for the incompressible case, we can substitute:

P<sub>1</sub> = δP/(1-r)

Note that this equation for compressible mass flow rate is similar to that for incompressible flow (eq. 4) with the addition of the last term in equation (11), which is called the expansion factor.

DISCHARGE COEFFICIENT

The actual mass flow rate through the nozzle is diminished from the theoretical value (11)by the existence of a fluid boundary layer. Therefore, the application of a "discharge coefficient" (CD) to the theoretical value is required to obtain the actual flow rate.

For a particular differential-pressure flow device, the preferred technique of obtaining the proper discharge coefficient is to run tests of the device over a large flow range using independent methods for determining the actual mass flow. In the absence of such extensive procedures, various empirical formulations have been suggested for arriving at suitable values of CD. From elementary boundary-layer theory it is known that the discharge coefficient is a function of Reynolds Number (Rd).

One formulation recommended by "Fluid Meters and Their Applications" (ASME, Equation II-III-42, page 220) is based on flat-plate boundary-layer theory:

where: a = .5 for R<sub>d</sub> < 10<super>6</super> (laminar flow)             = .2 for R<sub>d</sub> > 10<super>6</super> (turbulent flow)

The transition from laminar to turbulent boundary layer is actually a zone around Rd = 106 ( more like .3 X 106 to 3 X 106)rather than a point at Rd = 106. Therefore, we prefer to vary the exponent "a" linearly over the zone, as shown in Figure 1.

Figure 1, Graph -  Value of exponent

It is more convenient to measure static temperature and pressure at the nozzle inlet than the corresponding stagnation values. Hence, the stagnation values called for in some of the flow equations must be computed by iterative methods. Equation I-5-107 on page 67 of "Fluid Meters"can be used iteratively to compute the ratio p1/p01, using 1.0 as the initial guess for that ration.

Although most nozzle applications are designed specifically for either subsonic flow or for critical flow, this computational procedure facilitates the detection of the pertinent flow regime, as determined from input data. First, the critical pressure ratio (p2/p1)crit is computed from equation I-5-110 (page 68) and is solved iteratively with an initial "guesstimate" of 0.5 for the unknown. Comparison is made with the actual pressure ratiop2/p1, thereby sending the computational procedure to one or the other path corresponding to subsonic or critical flow.

Fluid viscosity of a gas is a strong function of its temperature. "ASME Fluid Meters"contains a set of curves (Fig. II-I-8, p. 162) which gives the viscosity-temperature relationship for eight common gases: air, CO2, nitrogen, oxygen, hydrogen, argon, carbon monoxide and methane. Fitting of quadratic polynomials to this data results in good representations for machine computation; they are included in a BASIC program (available from Delta-T). The viscosity values used are based on the computed value of nozzle exit temperature, since it is the exit Reynolds Number which determines the discharge coefficient (CD).

The ideal-gas assumption is valid for most ordinary ranges of gas temperature and pressure. A gas compressibility factor (z) which accounts for the deviation of the real gas from the ideal (as discussed on p. 26 of ASME Fluid Meters) may be incorporated into the code for very low temperatures and high pressures. For example, at 70°F and 270 psia, the compressibility factor z is approxiately 0.995. This is a 0.5% error in mass flow, small but not negligible.

Appendix 1 shows use of the basic flow equations and available ASME Type Gas Flow Nozzles.

Manufacturers and performance evaluators of aircraft engines have certain unique requirements which are met by the use of ASME-type gas-flow nozzles. These nozzles are used as standards or masters, against which to calibrate engine components through which air flows are critical. These components are run in test facilities, with the flow nozzles as references. By this procedure, an Effective Flow Area (EFA) is experimentally established, as compared against a reference nozzle. The computational procedure outlined has been shown to yield consistent accuracies of better than 0.1%, according to August Fleming of Fleming & Associates, a producer of engine test rigs installed at various locations.*

CONCLUSION

The procedure described is a treatment of the theory and its experimental implementation of the ASME/NIST short-radius gas-flow nozzle. The computer code is available at no charge from Delta-T Company.

*(Delta-T Company is a manufacturer of ASME gas flow nozzles used in the facilities mentioned above).

THE AUTHOR

DAVID WALD is President of Delta-T Company in Santa Clara, California. As an Aeronautical Research Scientist at NASA Ames Research Center, he helped develop instrumentation for re-entry simulation systems and plasma-arc technology. He received his B.Mech.Eng.from Brooklyn Polytechnic Institute and his M.Aero.Eng. from Rensselaer Polytechnic Institute. He can be reached at Delta-T Company.