Shock tube based dynamic calibration of pressure sensors
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1 Shock tube based dynamic calibration of pressure sensors C. E. Matthews, S. Downes, T.J. Esward, A. Wilson (NPL) S. Eichstädt, C. Elster (PTB) 23/06/2011 1
2 Outline Shock tube as a basis for calibration Modelling the calibration process System identification by optimisation Investigation of generated pressure inputs Conclusions and future work 2
3 Shock tube Suitable periodic (e.g. sinusoidal) pressure signal difficult to generate Shock tube therefore used to generate dynamic pressure signal Diaphragm housing Adaptor Driver section Driven section High pressure Low pressure Diaphragm Expansion wave Contact surface Shock front Sensors for shock speed estimation 3 Test sensor
4 Shock tube: System output 4
5 Shock tube: System output 5
6 Calibration method Input System Output h( t) x( t) x(t) y(t) y t h( t) x( t) t KNOWN INFERRED MEASURED System frequency response inferred by deconvolution Aim to also derive dynamic model for system (test sensor + charge amplifier) 6
7 Calibration: System input Simulations predict near-ideal step input followed by reflections and disturbances Therefore, simulate pressure input ( ) as near-ideal step Estimate amplitude of pressure step (P) from initial pressure (P 1 ) and temperature (T 1 ) in driven section of shock tube, and shock-wave velocity (V s ): 2 7 Vs 298 P P T1 x P Pressure /units Finite-volume simulation Time /ms WiSTL University of Wisconsin (ISA ) 7
8 Frequency response Step input x P Measured output y DFT Frequency response H Meas y, x P Simulated Inferred Measured Aim: derive dynamic model for system 8
9 Calibration: System model Model system (sensor and charge-amplifier) response Models developed using data sheets and experimental data Input System: Sensor + Amplifier Output Simple 2 nd order system with delay Low-pass filter Parameters (): gain (S 0 ) damping () resonance ( 0 ) time delay () Parameters (): cut-off frequency (f c ) Transfer function: H 2 S 2 2 s 0 0 s exps s Transfer function: H s n b1s b s n s a n1 2 n1 2s b a n1 n1 9
10 Fit system model parameters Step input x P Measured output y DFT H Frequency response Meas y, x P Optimise System model H Model For given, simulated input ( x P ) and measured output (y), optimise system response parameters () min H Meas y, x H P Model 10
11 Fit system model parameters: multiple data sets Run optimisation on six shock tube data sets Input pressure step amplitude estimated from each data set Same sensor and charge amplifier used for each measurement Error bars show standard uncertainties associated with optimisation process only Values quoted in data sheets for sensor and charge-amplifier: resonance 0.4 MHz cut-off frequency 0.2 MHz Gain Damping Resonance (MHz) Cut-off (MHz) 11
12 Model validation Step input x P Measured output DFT Frequency response H y, x Meas P Optimise System model H Model Model response Modelled output y, x Model P y Meas Capture some features: rise, over-shoot Later disturbances not explained 12
13 Optimisation and validation: conclusions and next steps Conclusions Degree of consistency across data sets suggests system model may be appropriate But, many remaining features in measured data not explained by current model y t h( t) x( t) Next steps Current model assumes input x(t) is near-ideal step Deviations from ideal step input exist due to non-instantaneous opening of diaphragm; non-uniform shock tube; interactions between shock front and boundary layer; side-wall reflections Investigate effects of allowing non-ideal (more realistic) input x(t) t 13
14 Develop model for pressure input Develop model x(,t) for pressure input generated by shock tube Studies of non-ideal diaphragm burst suggest oscillations follow nearideal step (e.g. Guardone 2007) Model as 2 nd order system response to nearideal-step gain rise-time damping resonance Model parameters (): step time, step rise-time, resonance frequency, gain, damping step-time 14
15 Sensitivity analysis on input model Input model: vary DFT Frequency response Optimise System model: optimise Measured output Perform sensitivity analysis to identify effects of non-ideal input assumption Mean SD Vary input model parameters Monte Carlo Simulation with 3500 trials Input model parameters drawn from uniform distributions Limits determined from initial sensitivity analysis on individual parameters Varying rise-time, resonance, damping 15
16 Sensitivity: Monte Carlo simulation Input model: vary Measured output DFT Frequency response Optimise System model: optimise For reasonable parameter ranges in chosen input model: little variation in system frequency parameters larger variation in system damping parameter 16
17 Sensitivity: Monte Carlo simulation Input model: vary Measured output DFT Frequency response Optimise System model: optimise Input model 17
18 Sensitivity: Monte Carlo simulation Input model: vary Measured output DFT Frequency response Optimise System model: optimise Input model 18
19 Sensitivity: Monte Carlo simulation Input model: vary Measured output DFT Frequency response Optimise System model: optimise Input model 19
20 Sensitivity: Monte Carlo simulation Input model: vary Measured output DFT Frequency response Optimise System model: optimise Input model Measured output Modelled output 20
21 Sensitivity: Monte Carlo simulation Input model: vary Measured output DFT Frequency response Optimise System model: optimise Input model Measured output Modelled output 21
22 Sensitivity: Monte Carlo simulation Input model: vary Measured output DFT Frequency response Optimise System model: optimise Input model Measured output Modelled output 22
23 Conclusions Using shock tube as primary calibration method requires accurate knowledge of generated input signal. Current models of input and system response reproduce some features observed in experiments, but cannot fully explain data. Models produce degree of consistency across multiple data sets, with frequency parameters easier to identify than damping. Input and/or system response models need to be extended to explain additional disturbances. Outlook Use experiments and simulation to investigate remaining discrepancies between the model and data. Derive more complex model to account for the deviations. Identify techniques developed that may be applied to other dynamic measurement problems. 23
24 References ISA A guide for the Dynamic Calibration of Pressure Transducers Guardone A. (2007) Three-dimensional shock tube flows for dense gases, J. Fluid Mech. (583) Wisconsin Shock Tube Laboratory: 24
25 Shock tube based dynamic calibration of pressure sensors C. E. Matthews, S. Downes, T.J. Esward, A. Wilson S. Eichstädt, C. Elster (PTB) 23/06/
26 Sensitivity: individual parameters Input model: vary i Measured output DFT System response Optimise System model: optimise Rise-time : 0.1 μs Gain : 1.21 Damping : 0.2 Step-time : ms Rise-time : 0.1 μs Gain : 1.21 Damping : 0.2 Resonance : 0.5 MHz Limits used in subsequent MCS 26
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