Measurement of instantaneous flow rates in periodically operating injection systems

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. List of symbols cj C.C. D fo i Io It L m rh ~P/~z Po, P~ r R Re t T Ta U V complex Fourier coefficients complex conjugate of different variables inner diameter of pipe repetition frequency of valve motion complex unit Bessel function of zeroth order Bessel function of first order distance between valve and measuring volume integrated fluid mass within one period mass flow rate pressure gradient along the pipe axis normalized amplitude of the complex pressure gradient radial position pipe radius Reynolds number time time of the period of the repetition valve opening Taylor number (= Rx/~) axial velocity integrated fluid volume within one period

Study on an injection quantity sensor. II: Evaluation of the sensing element

“…Their deceasing amplitude can be explained by energy losses due to frictional effects. These pressure oscillations are induced by the sharp deceleration of the fuel column at the end of the pressure pulse and by additional reflection effects, as reported by Durst et al using laser Doppler anemometry (LDA) measurements at a periodically operating injection system [10]. When these variations in fluid density are amplified by a positive interference within the hydraulic system, the needle may be lifted unintentionally, leading to the after-injection of a small fuel quantity into the combustion chamber in an uncontrolled way, resulting in an increased emission of unburnt hydrocarbons [11].…”

Development and assessment of a new operating principle for the measurement of unsteady flow rates in high-pressure pipelines

“…As a consequence, the harmonic components that have ω j < Ω in Eq. (19) and are not present in the spectrum of p(t)…”

“…(13) over a complete working cycle and taking an ideally periodic flow, one obtains the following result by a few analytical steps: (20) where p up − p down av is the time-averaged pressure drop between should be added to the Fourier series that is given by Eq. (19). In this way the disturbances due to the piezoelectric transducer zero offsets are also removed from the flow rate calculation.…”

“…Such a flow can be computed from a reduced linear form of the Navier-Stokes' momentum balance equation. The linearity of this equation permits the resultant velocity field to be computed for any arbitrary time-varying periodic laminar flow as a superposition of Lambossy's solutions [19]. Actually, for the reconstruction of the velocity field, the time history of the centerline velocity is measured by an LDA technique, and then the volumetric flow rate is obtained by integrating the velocity over the cross-section [22].…”

“…An LDA-based technique has recently been developed for the measurement of unsteady flow rates [19] and it has been applied to the analysis of fluid dynamic transients in a gasoline injection system [20]. In [21], Lambossy derived an analytical solution for the velocity field of a harmonically time-oscillating fully developed laminar pipe flow.…”