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

biomass boiler

January 29, 2005

Byungho Song*, Andreas Johansson, Lars-Erik Åmand, Filip Johnsson, and Bo Leckner

*Department of Chemical Engineering, Kunsan National UniversityGunsan, Korea

Department of Energy Conversion, Chalmers Univeristy of Technology, Götenburg, Sweden

Introduction

A gas concentration measured by gas analyzer may not be the same as the concentration at the probe tip in the boiler. There is a time delay for gas transport and gas species can be dispersed insome extent during the transport.

The measured concentration can be restored (deconvoluted) to its original concentration by help of a flow model that would give the mean residence time and RTD of fluid elements which describes the extent of gas dispersion.

A simple deconvolution scheme for time series signals by tanks-in-series model has been developed in this study.

The convolution of signals

∫ −= tinout dttttCtC 0 ')'()'()( E

To deconvolute output, to find E(t) or input under integral is difficult [Levenspiel, 1999].

A blurring or smoothing by a weighted function, RTD of fluidTracer response represents convolution of the unit impulse of the system

2. Linear flow model and parameter estimation(AR or ARX model)

Deconvolution methods

1. Direct method

x(t) y(t)G(t)

∑−

=−=+=

1

0

N

jjjkkkk exzy ε

,

1....,,1,0 −= Nk

yex

xey1−=

=Matlab has its Fourier transform versionBut very limited success

Mills and Dudukovic, Computers Chem. Engng., 13(8), 881-898 (1989)

Nameche and Vassel, Water Sci. Tech., 33(8), 105-124 (1996)

Models are useful for representing flow in vessels and other purposes. There are two models for small deviations from plug flow, they are roughly equivalent.

3-1. Dispersion modelAll correlations for flow in real reactors use this model.

3-2. Tanks-in-series modelSimple, it can be easily extended to other arrangements.

3. Flow modelsLevenspiel, O., ‘Chemical Reaction Engineering’, 3rd, John Wiley & Sons, New York (1999)

CN

1 2 3 NN-1

CiCi-1C0

Tanks-in-series model

iiitot CC

dtdC

NV υυ −= −1

00 =≠iC )(0 tMCi δυ

==

iii CC

dtdC

N−= −1

τ

A mass balance at i th tank, in a series of N tanks, gives:

, at t = 0

Let = total residence time for all N tanks,

(B-3)

with Laplace transform, the final output is

NN

sN

CC⎟⎠⎞

⎜⎝⎛ +

=τ1

0

τ

In dimensionless forms

τθ t=

θθθ NN

eNNN −

−

−=

)!1()()(

1

E

N1

2

22 ==

τσσθ

(B-7)

(B-8)

Fig. B10. RTD curves for the tanks-in-series model.

Massspectrometer

Tracer bag

P

Pump

3-way valve

Gas probe

Transport tube

Air

The gas transport system for measurement of gas concentration ina 12 MW CFB boiler with a tracer input part.

Step response test

Tracer = methane of 100 ppm

Dimension of tube = 0.004 m dia x 6 m length (probe=1.5m)

Flow rate of 1.5 L/min gives residence time of 3.02 sec

bypass

E(t) curve from experiment

0 2 4 6 8 10-20

0

20

40

60

80

100

t, s ec

Concentration, ppm

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t, s ec

F(t)

45 50 55 60 65 70 75 80-60

-40

-20

0

20

40

60

80

100

120

0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

time

E(t)

tou =5.2682 variance=0.83414 area =1

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

Theta, dimens ionles s time

E(theta), dimensionless RTD

N= 33.2728

Datacalculated by tanks model

The experimental E curvegives N=33 by tank model,

N1

2

22 ==

τσσθ

F(t) curveC(t) curve

Step response

E curve gives residence time of 5 sec.

Deconvolution by tanks-in-series model

If N = 3 the following equations can be used to obtain input concentration C0 from C3.

(B-3)ii

i Cdt

dCN

C +=−

τ1

33

2 Cdt

dCN

C +=τ

22

1 Cdt

dCN

C +=τ

11

0 Cdt

dCN

C +=τ

A mass balance at i th tank, in a series of N tanks, gives:

1 2

C0 C3

3

C2C1

In somecase the signal get noise at a certain point and the noise grow quickly to make the signal vibrated too much.

0 5 10 150

0.5

1

1.5

2

0 5 10 150

0.5

1

1.5

2

N =2 tou =5 r =39.0114

0 5 10 150

0.5

1

1.5

2

N =4 tou =5 r =39.1202

0 5 10 150

0.5

1

1.5

2

N =6 tou =5 r =37.2663

0 5 10 150

0.5

1

1.5

2

N =8 tou =5 r =38.4016

0 5 10 150

0.5

1

1.5

2

N =10 tou =5 r =38.5262

0 5 10 150

0.5

1

1.5

2

N =12 tou =5 r =39.0346

0 5 10 150

0.5

1

1.5

2

N =14 tou =5 r =38.1584

0 5 10 150

0.5

1

1.5

2

N =16 tou =5 r =35.2187

0 5 10 150

0.5

1

1.5

2

N =18 tou =5 r =4.4949

0 5 10 150

0.5

1

1.5

2

N =20 tou =5 r =0.35881

0 5 10 150

0.5

1

1.5

2

N =22 tou =5 r =0.28286

0 5 10 150

0.5

1

1.5

2

N =24 tou =5 r =0.27702

0 5 10 150

0.5

1

1.5

2

N =26 tou =5 r =0.27378

0 5 10 150

0.5

1

1.5

2

N =28 tou =5 r =0.27072

0 5 10 150

0.5

1

1.5

2

N =30 tou =5 r =0.26776

0 5 10 150

0.5

1

1.5

2

N =32 tou =5 r =0.26489

0 5 10 150

0.5

1

1.5

2

N=33 tou =5 r=0.2635

Figure D3. Deconvolution of a simulated curve (FT4) by 33 tanks.

0 5 10 150

0.5

1

1.5

2

0 5 10 150

0.5

1

1.5

2

N =1 tou =5 r =5.6455

0 5 10 150

0.5

1

1.5

2

N =2 tou =5 r =0.87037

0 5 10 150

0.5

1

1.5

2

N =3 tou =5 r =0.55146

0 5 10 150

0.5

1

1.5

2

N =4 tou =5 r =0.44867

0 5 10 150

0.5

1

1.5

2

N =5 tou =5 r =0.3907

0 5 10 150

0.5

1

1.5

2

N =6 tou =5 r =0.35089

0 5 10 150

0.5

1

1.5

2

N =7 tou =5 r =0.32121

0 5 10 150

0.5

1

1.5

2

N =8 tou =5 r =0.29798

0 5 10 150

0.5

1

1.5

2

N =9 tou =5 r =0.27916

0 5 10 150

0.5

1

1.5

2

N =10 tou =5 r =0.2635

Figure D4. Deconvolution of a simulated curve (FT4) by 10 tanks (tou = 5)

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

=y

dtdy

je ofdeviation standard

ofdeviation standardin j th tank (C-4)

Measure of error

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛⎟⎠⎞

⎜⎝⎛

=y

dtdy

Njf

ofdeviation standard

ofdeviation standard τ in j th tank (C-5)

∑=

=N

jjeE

1

,

∑=

=N

jjfF

1(C-6)

100

101

102

103

104

0

10

20

30

40

N

E, avg error

100

101

102

103

104

0

5

10

15

20

N

F, avg error

Random

Multiple peaks

S ine curves

One peak

S tep res pons e

Random

Multiple peaks

S ine curves

One peak

S tep res pons e

Figure D9. The changes of average ratio E and F with variation of N from the deconvolution of the various output signals with total residence time of 5 s.

0 2 4 6 8 10-2

-1

0

1

2

t,s ec

y(t)

S ample frequency= 40Hz

y

0 2 4 6 8 10-4

-2

0

2

4

t,s ec

dy/dt

analyticDIFF y

0 2 4 6 8 10-4

-2

0

2

4

t,s ec

dy/dt

analyticgradient y

0 2 4 6 8 10-4

-2

0

2

4

t,s ec

dy/dt

analyticfftdiff y

0 2 4 6 8 10-4

-2

0

2

4

t,s ec

dy/dt

analyticfiting y

0 2 4 6 8 10-4

-2

0

2

4

t,s ec

dy/dt

analyticgradient of mov avg5

Figure E4. The effect of signal smoothening on the differentiation of a sine curve with a noise (sample frequency = 40 Hz).

The signal can be smoothened before the signal is differentiated.

0 2 4 6 8 10-1

0

1

2

t,s ec

y(t)

S ample frequency= 10Hz

y

0 2 4 6 8 10-4

-2

0

2

4

t,s ec

dy/dt

analyticDIFF y

0 2 4 6 8 10-4

-2

0

2

4

t,s ec

dy/dt

analyticgradient y

0 2 4 6 8 10-4

-2

0

2

4

t,s ec

dy/dt

analyticfftdiff y

0 2 4 6 8 10-4

-2

0

2

4

t,s ec

dy/dt

analyticfiting y

0 2 4 6 8 10-4

-2

0

2

4

t,s ec

dy/dt

analyticgradient of mov avg5

Figure E5. The effect of signal smoothening on the differentiation of a sine curve having a noise(sample frequency = 10 Hz).

Moving average was found to be very effective to reduce the noise in the calculated gradient.

Application of moving average to the signal

The best size of window for the moving average should be determined by trial and error.

One peak signal could be deconvoluted without any noise by adapting 3 points (window size) average to the signal before thedifferentiation.

Larger window size gives more stable signal, whereas the peak height decrease so much that the obtained peaks may not be able to count gas dispersion properly.

The time series gas concentration (methane)The length is usually 899 sec and sampling frequency 40 Hz.It was clear that the effect of sampling frequency is very strong. Smaller frequency is better for the stable deconvolution. First the frequency of the real signal has been reduced half and fortunately the change of 40 to 20 Hz almost does not affect the signal shape

0 5 10 15 20 25-500

0

500

1000

1500

Time, s

Methane, ppm

40 Hz data

0 5 10 15 20 25-500

0

500

1000

1500

Time, s

Methane, ppm

20 Hz data

The gas concentration signal of 120 s (methane) was deconvoluted by tanks of τ = 5 and N=33 with various window size of moving average.

Full length of signal, 899 s can be treated, but just not so good to look at.

The gas concentration signal of 70 s (benzene) was deconvoluted by tanks of τ = 5 and N=33 with window size of 9.

0 10 20 30 40 50 60 700

50

100

150

200

y=file freq=20 tou=5 N=33 moving=9 N= 1

Deconvolution by tanks

Conclusions

Fast varying signal is easy to have inherent signal perturbation error that is coming from the repeated numerical differentiation of signal through each tank. The signal growth error greatly depends on the shape of signal to be recovered.

The parameter study tells that the numerical error can be reduced if tanks system has very small variance (very short residence time and larger number of tanks), however it would be a plug flow tube.

A compartment model consisted of a plug tube and tanks-in-series has been devised to reduce the number of differentiation in the system. This model was effective in reducing the chance of signal perturbation, but the model could not remove signal error completely.

Fast time varying signal of gas concentration from the fluidized bed boiler could be successfully recovered to its original shape by the deconvolutionprocedure with a moving average filter proposed in this study.