Abstract
Research and development work carried out to provide a method to measure accurately instantaneous flow rates in periodically operating injection systems is summarized. The instantaneous flow rate is reconstructed from axial velocity time series measured by a laser Doppler anemometer on the center-line of a capilary pipe flow. The theoretical background, on which the evaluation of the instantaneous flow rate is based is provided. It is shown that the axial mean velocity is sufficient to reconstruct the periodically varying flow rate and the pressure gradient. The application of the proposed method is described and an instrument is suggested that can be employed in many fields where fast, periodically varying flow rates occur and instantaneous information is needed.
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Abbreviations
- C j :
-
complex Fourier coefficients
- C.C. :
-
complex conjugate of different variables
- D :
-
inner diameter of pipe
- f 0 :
-
repetition frequency of valve motion
- i :
-
complex unit
- J 0 :
-
Bessel function of zeroth order
- J l :
-
Bessel function of first order
- L :
-
distance between valve and measuring volume
- m :
-
integrated fluid mass within one period
- ∂P/∂z:
-
mass flow rate
- δP/δz :
-
pressure gradient along the pipe axis
- p 0,p n :
-
normalized amplitude of the complex pressure gradient
- r :
-
radial position
- R :
-
pipe radius
- Re :
-
Reynolds number
- t :
-
time
- T :
-
time of the period of the repetition valve opening
- Ta :
-
Taylor number \(( = R\sqrt {\omega /v} )\)
- U :
-
axial velocity
- V :
-
integrated fluid volume within one period
- \(\dot V\) :
-
instantaneous volumetric flow rate
- μ :
-
viscosity
- ν :
-
kinematic viscosity
- ε :
-
fluid density
- τ :
-
normalized open valve time
- ω :
-
angular velocity of Fourier modes
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Durst, F., Ismailov, M. & Trimis, D. Measurement of instantaneous flow rates in periodically operating injection systems. Experiments in Fluids 20, 178–188 (1996). https://doi.org/10.1007/BF00190273
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DOI: https://doi.org/10.1007/BF00190273