Thermal boundary layers in critical flow venturis

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Highlights

  • The discharge coefficients of small critical flow venturis are sensitive to the environmental temperature conditions due to the thermal boundary layer of the gas flowing near the venturi wall.

  • Experiments with critical flow venturis made of thermally conductive materials were compared to Geropp's theoretical calculations of thermal boundary layer effects.

  • A correction for thermal boundary layer effects that is a function of the Reynolds number and the temperature difference between the venturi wall and the flowing gas reduced the temperature sensitivity of a mass flow measurement from 0.12 % to <0.01 %.

Abstract

We improve the usefulness of small (diameter < 10 mm) critical flow venturis (CFVs) as transfer standards for gas flow by measuring and explaining how their discharge coefficients depend on the temperature T of their environment. At Reynolds numbers Re < 2.5 × 105 (e.g., a 2 mm diameter throat; inlet air at 1 MPa), CFVs exhibit sensitivity to the environmental temperature of approximately 0.02 % K−1 due to biased measurements of the stagnation temperature T0 (temperature “sampling” error) and from ignoring the low-density, annular, thermal boundary layer generated by heat transfer from the CFV's body to the gas flowing through the CFV. To reduce temperature sampling errors, we used a non-metallic approach pipe and a temperature sensor with a low stem-conduction error. To correct for thermal boundary layer effects on the flow, we used Geropp's functional form: CT=1+KTRe1/2ΔT/T0 where ΔT is the difference between the CFV's inner wall temperature and the stagnation temperature. For CFVs made of stainless steel and copper with diameters of d = 0.56 mm, 1.1 mm, and 3.2 mm we measured KT ≈ −7 while theoretical predictions of KT by Geropp and Ding et al. are −1.7 and −3.845 respectively. Introducing the correction for room temperature changes (CT) measured in this work, reduces the room temperature sensitivity of the flow measured with the 0.56 mm diameter CFVs from 0.02 % K−1 to less than 0.003 % K−1. Smaller, but significant, improvements are achieved with larger CFVs.

Section snippets

Introduction to critical flow venturis

Toroidal critical flow venturis (CFVs) consistent with documentary standards have a contracting inlet with a radius of curvature approximately twice the throat diameter followed by a conical outlet. If a sufficiently large pressure ratio is maintained across the CFV (conservatively Pup/Pdown > 2 for air), the gas entering the CFV expands and reaches sonic velocity at the throat. The commonly used “0th order” physical model assumes isentropic flow and adiabatic wall conditions to calculate mass

Introduction to thermal effects on CFVs

CFVs are often used as working standards to calibrate other meters or as transfer standards during inter-laboratory comparisons because of 1) their excellent calibration stability over time [13], and 2) the well-developed physical model that accounts for their sensitivity to gas properties and to the gas temperature. However, smaller CFVs (d < 10 mm) show significant, unaccounted sensitivity to the temperature of the CFV's environment. During the CCM.FF-K6 2002 key comparison, the temperature

Geropp's similarity solution for the non-adiabatic boundary layer

The adiabatic wall assumed in Tang's 1969 [7] and Geropp's 1971 [8] solutions for the CFV boundary layer is a simplifying approximation. In most applications, the gas flowing along the wall of a thermally conductive CFV is warmer than gas near the wall of a thermally insulating (adiabatic) CFV. For Re < 106, there is a significant heat flux from the CFV body into a thermal boundary layer (Fig. 3). The thermal boundary layer is warmer than the free stream (or core flow) and its lower density

Correcting thermal boundary layer effects in CFVs CT

Geropp's theory accounts for thermal boundary layer effects in CFVs. We now consider a practical approach for applying Geropp's theory, particularly in situations where CFV users have neither accurate values for the CFV's geometry (d and Ω) nor values for the CFV's wall temperature Twall. The need for accurate values of d and Ω can be circumvented by calibrating CFVs using a reference flow standard. Also Twall of a typical metal CFV can be well approximated by the CFV's body temperature (See

Temperature distributions in the bodies of CFVs

We conducted thermal boundary layer measurements using heated CFVs made of stainless steel, copper, and a machinable ceramic (Macor3). For the copper CFVs, the thermal resistance between the CFV wall and the gas flow is much greater than the thermal resistance of the copper body; therefore, the temperature anywhere within the CFV body is a good approximation to the temperature of the inner wall of the CFV (See Fig. 4.). Section 5.1 describes a finite-volume numerical model used to estimate the

Experimental measurement of CT

The goal of the CFV holder and approach-pipe design was to accurately measure the gas temperature entering a CFV with an elevated body temperature. CFVs with d = 3.2 mm, 1.1 mm, and 0.56 mm were machined from copper, stainless steel, and a machinable ceramic material. The 9 CFVs were calibrated against the NIST PVTt flow standards [19] using an experimental arrangement designed to minimize temperature sampling errors. The temperature of the CFV body was proportional-integral-derivative (PID)

Analysis of experimental results

The calibration data consisted of the reference mass flow from the PVTt standard, the composition of the dried air, the static pressure (P0) and temperature upstream (T0 based on Tz) from the CFV, and the temperature of the CFV body. The corrections from static to stagnation pressure were all small (<0.012 %) because the ratio of the approach-pipe and throat diameters was >6 for these experiments.

The mass flow calculated via the CFV, accounting for thermal expansion of the CFV material is m˙R*,α

Discussion and conclusions

Temperature sampling errors are the largest concern when trying to achieve CFV flow measurements that are reproducible under variable room temperature or gas temperature conditions. The gas expanding through the CFV cools the CFV body and connecting piping which causes temperature gradients in the gas and can lead to stem conduction errors in the gas temperature sensor. In these experiments, a special design improved the measurement of T0. Improved designs for measuring the temperature of gas

Author statement

John Wright: conceptualization, methodology, software, validation, formal analysis, investigation, writing -original draft, writing – Review and editing, visualization, supervision, Woong Kang: software, validation, formal analysis, investigation, writing – Review and editing, visualization, Aaron Johnson: formal analysis, investigation, software, writing – Review and editing, Vladimir Khromchenko: software, validation, formal analysis, investigation, Michael Moldover: methodology, formal

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was partially supported by a Cooperative Research and Development Agreement with WEST, Inc. and their subsidiary Flow Systems, Inc. We also wish to acknowledge Gina Kline of the NIST Fluid Metrology Group for her assistance in the laboratory. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

References (25)

  • N. Bignell et al.

    Thermal effects in small sonic nozzles

    Flow Meas. Instrum.

    (2002)
  • J.G. Pope et al.

    Performance of coriolis meters in transient gas flows

    Flow Meas. Instrum.

    (2014)
  • E.W. Lemmon et al.

    NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties-REFPROP

    (2007)
  • J.D. Anderson

    Modern Compressible Flow

    (2004)
  • Measurement of gas flow by means of critical flow venturi nozzles

    (2005)
  • Measurement of gas flow by means of critical flow venturis and critical flow nozzles

    (2016)
  • G.W. Hall

    Transonic flow in two-dimensional and axially-symmetric nozzles, Quart

    J. Mech. Appl. Math.

    (1962)
  • J.R. Kliegel et al.

    Transonic flow in small throat radius of curvature nozzles

    AIAA J.

    (1969)
  • S.P. Tang

    Discharge coefficients for critical flow nozzles and their dependence on Reynolds number

    (1969)
  • D. Geropp

    Laminare Grenzschichten in Ebenen und Rotationssymmetrischen Lavaldusen, Deutsche Luft- und Raumfahrt Forschungsbericht 71 – 90

    (1971)
  • A.N. Johnson et al.

    Comparison between theoretical CFV models and NIST's primary flow data in the laminar, turbulent, and transition flow regimes

    ASME J. Fluid. Eng.

    (July, 2008)
  • M. Ishibashi et al.

    Effect of inlet curvature on the discharge coefficients of toroidal-throat critical-flow venturi nozzles, fedsm2005–77470

  • Cited by (3)

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