Simple deconvolution of time varying signal of gas phase variation in a 12 MW CFB biomass boiler

Transcription

1 Simple econvolution of time varying signal of gas phase variation in a MW CFB biomass boiler Byungho Song*, Anreas Johansson, Lars-Erik Åman, Filip Johnsson, an Bo Leckner *Department of Chemical Engineering, Kunsan National University Gunsan, Korea Department of Energy Conversion, Chalmers Univeristy of Technology, Götenburg, Sween January 9, 5

2 Introuction A gas concentration measure by gas analyzer may not be the same as the concentration at the probe tip in the boiler. There is a time elay for gas transport an gas species can be isperse in some extent uring the transport. The measure concentration can be restore (econvolute) to its original concentration by help of a flow moel that woul give the mean resience time an RTD of flui elements which escribes the extent of gas ispersion. A simple econvolution scheme for time series signals by tanksin-series moel has been evelope in this stuy.

3 The convolution of signals A blurring or smoothing by a weighte function, RTD of flui Tracer response represents convolution of the unit impulse of the system Cout ( t) = t Cin( t t') E( t') t' To econvolute output, to fin E(t) or input uner integral is ifficult [Levenspiel, 999].

4 Deconvolution methos,. Direct metho yk y x = = zk + ε k xe = e y N = xk je j j = k =,,..., N Matlab has its Fourier transform version But very limite success Mills an Duukovic, Computers Chem. Engng., 3(8), (989). Linear flow moel an parameter estimation (AR or ARX moel) x(t) G(t) y(t) Nameche an Vassel, Water Sci. Tech., 33(8), 5-4 (996)

5 3. Flow moels Levenspiel, O., Chemical Reaction Engineering, 3r, John Wiley & Sons, New York (999) Moels are useful for representing flow in vessels an other purposes. There are two moels for small eviations from plug flow, they are roughly equivalent. 3-. Dispersion moel All correlations for flow in real reactors use this moel. 3-. Tanks-in-series moel Simple, it can be easily extene to other arrangements.

6 Tanks-in-series moel C C i- C i C N 3 N- N A mass balance at i th tank, in a series of N tanks, gives: Vtot Ci = υci υci N t C i = M C i = = δ ( t), at t = υ Let τ = total resience time for all N tanks, τ C N t i = Ci C i (B-3) with Laplace transform, the final output is C N C = τ + s N N

7 In imensionless forms t θ = τ E( θ ) = N N( Nθ ) ( N )! e Nθ (B-7) σ θ σ = τ = N (B-8) Fig. B. RTD curves for the tanks-in-series moel.

8 Step response test The gas transport system for measurement of gas concentration in a MW CFB boiler with a tracer input part. Air P 3-way valve Gas probe Transport tube Pump Mass spectro meter Tracer bag bypass Tracer = methane of ppm Dimension of tube =.4 m ia x 6 m length (probe=m) Flow rate of L/min gives resience time of 3. sec

9 E(t) curve from experiment Step response.9 8 C(t) curve.8 F(t) curve pm p n, tratio n e C onc 4 F (t) t, sec t, s ec.4.5 Data calculate by tanks moel The experimental E curve gives N=33 by tank moel,.35.3 N= σ θ σ = τ = N E (t) variance=.8344 area = R T D n l e ss o n si e m i ) ta, E (the. E curve gives resience time of 5 sec time.5.5 Theta, imensionless time

10 Deconvolution by tanks-in-series moel C C C C 3 3 A mass balance at i th tank, in a series of N tanks, gives: τ Ci C (B-3) i = + Ci N t If N = 3 the following equations can be use to obtain input concentration C from C 3. C C = τ + N t 3 C3 C C = τ + N t C C C = τ + N t C

11 In somecase the signal get noise at a certain point an the noise grow quickly to make the signal vibrate too much N = r = N =4 r = N =6 r = N =8 r = N = r = N = r = N =4 r = N =6 r = N =8 r = N = r = N = r = N =4 r = N =6 r = N =8 r = N =3 r = N =3 r = N=33 r= Figure D3. Deconvolution of a simulate curve (FT4) by 33 tanks.

12 N = r = N = r = N =3 r = N =4 r = N =5 r = N =6 r = N =7 r = N =8 r = N =9 r = N = r = Figure D4. Deconvolution of a simulate curve (FT4) by tanks (tou = 5)

13 Measure of error e j y stanar eviation of = t stanar eviation of y in j th tank (C-4), f j y stanar eviation of N = t stanar eviation of y τ in j th tank (C-5) E = N e j j = F = N j = f j (C-6)

14 r o g er E, av 4 3 Ranom Multiple peaks Sine curves One peak Step response 3 4 N r o g er F, av 5 5 Ranom Multiple peaks Sine curves One peak Step response 3 4 N Figure D9. The changes of average ratio E an F with variation of N from the econvolution of the various output signals with total resience time of 5 s.

15 The signal can be smoothene before the signal is ifferentiate. Sample frequency= 4Hz y 4 analytic DIFF y y ( t ) - y / t t,sec 4 analytic graient y t,sec 4 analytic fftiff y y / t - y / t - y / t t,sec 4 - analytic fiting y y / t t,sec 4 - analytic graient of mov avg t,sec t,sec Figure E4. The effect of signal smoothening on the ifferentiation of a sine curve with a noise (sample frequency = 4 Hz).

16 Moving average was foun to be very effective to reuce the noise in the calculate graient. y ( t ) Sample frequency= Hz y y / t 4 - analytic DIFF y t,s ec 4 analytic graient y t,s ec 4 analytic fftiff y y / t - y / t - y / t t,s ec 4 - analytic fiting y y / t t,s ec 4 - analytic graient of mov avg t,s ec t,s ec Figure E5. The effect of signal smoothening on the ifferentiation of a sine curve having a noise (sample frequency = Hz).

17 Application of moving average to the signal The best size of winow for the moving average shoul be etermine by trial an error. One peak signal coul be econvolute without any noise by aapting 3 points (winow size) average to the signal before the ifferentiation.

18 Larger winow size gives more stable signal, whereas the peak height ecrease so much that the obtaine peaks may not be able to count gas ispersion properly.

19 The time series gas concentration (methane) The length is usually 899 sec an sampling frequency 4 Hz. It was clear that the effect of sampling frequency is very strong. Smaller frequency is better for the stable econvolution. First the frequency of the real signal has been reuce half an fortunately the change of 4 to Hz almost oes not affect the signal shape 5 4 Hz ata pm p n e, a M eth Time, s 5 Hz ata pm p n e, a M eth Time, s

20 The gas concentration signal of s (methane) was econvolute by tanks of τ = 5 an N=33 with various winow size of moving average.

21 Full length of signal, 899 s can be treate, but just not so goo to look at.

22 The gas concentration signal of 7 s (benzene) was econvolute by tanks of τ = 5 an N=33 with winow size of 9.

23 De convolution by ta nks 5 5 y=file freq= tou=5 N=33 moving=9 N=

24 Conclusions Fast varying signal is easy to have inherent signal perturbation error that is coming from the repeate numerical ifferentiation of signal through each tank. The signal growth error greatly epens on the shape of signal to be recovere. The parameter stuy tells that the numerical error can be reuce if tanks system has very small variance (very short resience time an larger number of tanks), however it woul be a plug flow tube. A compartment moel consiste of a plug tube an tanks-in-series has been evise to reuce the number of ifferentiation in the system. This moel was effective in reucing the chance of signal perturbation, but the moel coul not remove signal error completely. Fast time varying signal of gas concentration from the fluiize be boiler coul be successfully recovere to its original shape by the econvolution proceure with a moving average filter propose in this stuy.