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Abstract
This research paper describes an experimental method and the corresponding theory which enables the characterization of small viscoelastic pipes. The experimental set-up is a reservoir–pipeline–oscillating-valve configuration, where valve opening and pressure just upstream of the valve were measured. This allows the extraction of pressure amplitude and phase at this point. This configuration can be analytically modelled in the frequency domain with help of the impedance method. The viscoelastic behaviour of the pipe is modelled with a multi-element Kelvin–Voigt model whose moduli are estimated with a least squares method to fit the recorded complex-valued pressure data with the modelled one. Extracting amplitude and phase of the pressure signal is doubling the amount of data which can be used in the estimation algorithm. This and the use of an analytical model provides a robust and fast method to calibrate the viscoelastic properties of a pipe in the frequency domain.
Acknowledgements
The results of this study were achieved through a joint research project between ETH Zurich, Lucerne School of Engineering and Architecture and KNF Flodos AG, a global supplier of liquid diaphragm pumps.
Notation
| A | = | cross sectional inner pipe area (m2) |
| = | valve orifice area (m2) | |
| = | valve orifice area at fully open state (m2) | |
| = | valve orifice discharge coefficient (–) | |
| = | valve orifice discharge coefficient at fully open state (–) | |
| c | = | pipe wall constraint coefficient (–) |
| d | = | inner pipe diameter (m) |
| = | Young's modulus of the isolated spring (Pa) | |
| = | Young's modulus of the kth spring (Pa) | |
| e | = | pipe wall thickness (m) |
| f | = | Darcy–Weisbach steady-state friction factor (–) |
| g | = | gravitational acceleration (m s−2) |
| = | imaginary unit (–) | |
| = | creep compliance of the isolated spring (Pa−1) | |
| = | creep compliance of the kth Kelvin–Voigt element (Pa−1) | |
| = | complex creep compliance (Pa−1) | |
| K | = | bulk modulus of the fluid (Pa) |
| l | = | pipe length (m) |
| = | pipe pressure amplitude at position | |
| = | oscillatory pipe pressure (Pa) | |
| = | measured valve pressure (Pa) | |
| = | oscillatory pressure at upstream end of a pipe (Pa) | |
| = | mean valve pressure (Pa) | |
| = | oscillatory valve pressure (Pa) | |
| = | mean pipe flow (m3 s−1) | |
| = | oscillatory pipe flow (m3 s−1) | |
| = | pipe flow amplitude at position | |
| = | oscillatory flow at upstream end of a pipe (m3 s−1) | |
| = | instantaneous valve flow (m3 s−1) | |
| = | mean valve flow (m3 s−1) | |
| = | oscillatory valve flow (m3 s−1) | |
| t | = | time (s) |
| = | Reynolds number (–) | |
| R | = | total pipe flow resistance per unit length (s m−3) |
| = | steady-state component of pipe flow resistance (s m−3) | |
| = | unsteady-state component of pipe flow resistance (s m−3) | |
| x | = | coordinate along the pipe axis (m) |
| = | characteristic pipe impedance (Pa s m−3) | |
| = | hydraulic pipe impedance at the downstream end (Pa s m−3) | |
| = | hydraulic pipe impedance at the upstream end (Pa s m−3) | |
| = | series impedance per unit length (Pa s m−4) | |
| = | shunt impedance per unit length (Pa s m−2) | |
| γ | = | propagation parameter (m−1) |
| = | oscillatory strain (–) | |
| = | kth dashpot-viscosity (Pa s) | |
| ν | = | kinematic fluid viscosity (m2 s−1) |
| = | Poisson's ratio (–) | |
| ρ | = | fluid density (kg m−3) |
| τ | = | dimensionless valve opening (–) |
| = | mean part of the dimensionless valve opening (–) | |
| = | oscillatory part of the dimensionless valve opening (–) | |
| = | retardation time of the kth Kelvin–Voigt element (s) | |
| = | measured phase between valve opening and valve pressure (rad) | |
| = | phase between valve opening and valve pressure (rad) | |
| ω | = | angular frequency (rad s−1) |