STATISTICAL CHARACTERIZATION OF MULTI-PHASE FLOW BY
DYNAMIC TOMOGRAPHY
Gerd-Rudiger Tillack1, Ulazimir Samadurau1, Valentin Artemiev2, and Alexander Naumov2
federal Institute for Materials Research and Testing (BAM), Unter den Eichen 87,
12205 Berlin, Germany
Institute of Applied Physics of National Academy of Science of Belarus, Academiceskaya
str. 16, Minsk, 220072, Belarus
ABSTRACT. The paper presents a special reconstruction algorithm that is capable to monitor density
differences in multi-phase flows. The flow cross section is represented as discrete dynamic random
field. A fixed gray value is assigned to each flow phase characterizing the material property of the
phase. The image model is given by a set of non-linear stochastic difference equations. The corresponding inversion task is not accessible by common tomographic techniques applying reconstruction
algorithms like filtered backprojection or algebraic reconstruction technique (ART). The developed
algorithm is based on the Kalman filter technique adopted to non-linear phenomena. The average velocity distribution together with the corresponding covariance matrix of the liquid flow through a pipe
serves as prior information in statistical sense. To overcome the non-linearity in the process model as
well as in the measurement model the statistical linearization technique is applied. Moreover the Riccati equation, giving the error covariance matrix, and the equation for the optimal gain coefficients
can be solved hi advance and later used in the filter equation. It turns out that the resulting reconstruction or filter algorithm is recursive, i.e. yielding the quasi-optimal solution to the formulated inverse problem at every reconstruction step by successively counting for the new information collected
hi the projections. The applicability of the developed algorithm is discussed in terms of characterizing
or monitoring a multi-phase flow in a pipe.
INTRODUCTION
Nondestructive evaluation (NDE) techniques are mainly applied to quality control
during the production process as well as to ensure the safe operation of industrial constructions and engineered components (in-service inspection). As a result the evaluation provides a snapshot of the condition of the investigated object or structure at a specific time.
However any dynamic behavior is neglected in standard NDE. On the other hand, the dynamics of the investigated system becomes important for applications like process monitoring or process control. Special statistical reconstruction techniques are capable to trace
the system properties as a function of time. For this purpose a modified Kalman filter approach is presented here.
In recent years the computerized tomography (CT) has been applied to monitor dynamic processes, called process tomography [1]. Process tomography is analogous to medi-
CP657, Review of Quantitative Nondestructive Evaluation Vol. 22, ed. by D. O. Thompson and D. E. Chimenti
© 2003 American Institute of Physics 0-7354-0117-9/03/S20.00
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cal tomography (body scanners). Using an array of sensors placed around the periphery of a
process vessel, it images the concentration and movement of components inside, both costeffectively and in real-time. Measurements are reconstructed to form 2D or 3D images providing information to monitor processes. Process tomography can be applied to many types
of processes and unit operations, including pipelines, stirred reactors, fluidized beds, mixers, and separators. Depending on the sensing mechanism used, it is non-invasive, inert and
non-ionizing. It is therefore applicable in the processing of raw materials and materials degradation, hi large-scale and intermediate chemical production, and in the food and biotechnology areas. Today, the most applied technique is electrical tomography due to short data
acquisition time. Other sensing mechanisms can be used if the detector read-out and data
processing is fast enough.
In fact, state-of-the-art process tomography techniques are based on classical principles of tomography [2]. These approaches assume that the image properties are constant
during the acquisition of a complete data set in relation to the assessed spatial resolution.
This requirement restricts its applicability to slow varying processes, high speed acquisition
systems, or the use of multiplex detection systems in order to gain the required data set.
This paper discusses a different approach to real-time quasi-optimal reconstruction of nonlinear dynamic images in a new framework for process tomography. The principle for the
reconstruction of the dynamic image properties is based on the Bayesian approach to the
Markov stochastic estimation theory [3,4]. This approach leads to a modified Kalman filter
reconstruction algorithm. The case of linear dynamic reconstruction has been studied
in [5,6].
RECONSTRUCTION ALGORITHM
The following restrictions are made to focus the investigation on specific processes
like monitoring liquid and/or gas flows and on-line production control of modern materials.
In general, those processes can be treated as stochastic processes, i.e. its properties can be
described in terms of probability theory. Hence the considered system or process properties
are supposed to be dynamic random fields or images. In the estimation theory [4] the following form is introduced to describe dynamic random fields:
±X=F[t,X(t),u(t),w(t)]
(1)
where x(t)=[xi(t), X2(t),..., xs(t)]T is the S-dimensional image or process vector, F[...] is a
known non-linear function, u(f) is the regular component, and w(t) gives the stochastic influence, i.e. the image noise. If w(t) is given by Gausian distributed white noise with zero
mean, the process described by eq. (1) is a Markov process, i.e. given the process state x at
time t no additional data concerning states of the system at previous times can alter the
probability of the states jc at a future time. For the further discussion the process is assumed
to be a Markov process. Supposing the time variable t takes discrete values fc=l,2,... the
stochastic process turns into a Markov sequence given by the following equation
x
k+i=ak(xk) + uk + wk.
(2)
In order to describe a multi-phase flow the non-linear dynamic image model in the
state space is subdivided into two steps: (i) a linear Gaussian model for the image forming
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vector Xk and (ii) a non-linear transformation to obtain the multi-level image vector yk. The
corresponding mathematical formulation of the image model transfers to
(3)
where A* represents the image model matrix and/is a non-linear vector function. Assuming a 3-phase flow the components /=1,2,...,S of the non-linear vector function in eq. (3)
takes the following form
X
x
,(2)
V
X
<
~
,(3)
v
X
v
X
(2)
(4)
(2)
with v(/) (/= 1,2,3) giving the three levels of the image defined by the choice of the thresholds jt,£(/). Correspondingly, the image probability distribution function (PDF) p(yik) is defined by
-/ 3) )
(5)
with the phase probabilities
xw
(6)
In eq. (6) <3> represents the Gaussian distribution function with the image mean value m&
and the variance pik.
To transform the non-linear image model (3-6) into an appropriate linear model the
statistical linearization technique is used as described in [7]. The statistical linearization
vector tk is given by
3
(7)
/=!
and the statistical linearization matrix follows as
(8)
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Introducing eqs. (7) and (8) into the image model (3-6) the linearized model equations are
given by
xk+l
k+l =Akxkk+ukk+wkk
,,
Finally the observation model can also be non-linear but here it is assumed to be
linear for simplification purpose given by
ZM =Hk+iyk+i +v*+1 = Hk+ltl+l +Hk+}Fk+lxk+l +vk+l =h*k+l +H*k+lxk+l +vt+1
(10)
with the mxS observation matrix Hk and the mxl observation noise v*.
For the image reconstruction task m<S the corresponding inverse problem, i.e. the
reconstruction or estimation of the image vector jt*, is ill-posed and can be solved in the
framework of probabilistic reconstruction techniques introducing prior knowledge. To
overcome the restrictions of state-of-the-art process tomography techniques as discussed
above, the proposed algorithm is based on the Kalman filter [8]. For the case discussed
above, the standard linear Kalman filter has to be modified. Assuming a quadratic cost
function the modified Kalman filter is given by the following equations:
(0 the equation for the optimal estimation
x
k+i ~ xk+\,k + Kk+i \_zk+i ~hk+\ ~ Hk+\xk+i,k J '
(11)
(ii) the Riccaty equation for the error covariance matrix Pk
*k+\ ~ *k+l,k ~~ **k+l**k+l*k+l,k '
\^^J
and (Hi) the equation for the gain matrix Kk
'MPk^(HlJ +RM]~l.
(13)
Here xk+ltk represents the one-step prediction of the estimation
and the Pk+l k is the one-step prediction for the error covariance matrix
Qk.
(15)
The initial condition for the solution of the estimation equation (1 1) is the prior mean value
m0 and of the Riccaty equation (12) is the prior covariance matrix P0 . This equation does
not depend on the projection data and can be calculated independent on eq. (1 1) in advance.
The error covariance matrix characterizes the estimation accuracy at the current estimation
step and controls the choice of the gain matrix. The gain matrix Kk depends on the error
covariance matrix Pk but does not depend on the projection data. Therefore the gain can be
calculated in advance together with the error covariance matrix.
Eqs. (11-15) are used to reconstruct the stochastic image Xk- The final goal of reconstruction is to access the three phases in the flow.
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FIGURE 1. Structure of modified Kalman filter for non-linear process and observation model.
The phases are separated by minimizing the average costs for the three phase system applying a quadratic cost function
yk=argtnmj(f(xk)-yk)T(f(xk)-yk)p(xk)dxk.
(16)
The structure of the above described modified Kalman filter algorithm can be summarized
as shown in Fig. 1.
RECONSTRUCTION RESULTS
This article addresses mainly the development and applicability of a new reconstruction algorithm for process tomography. Therefore and because of the requirement on
the speed of the data acquisition system, the input data used for the reconstruction task are
simulated data. This also opens up the opportunity to study the stability of the algorithm as
function of system variables that can not be separated experimentally. The data acquisition
setup is shown in Fig. 2. A linear responding line detector is used for the projection measurement. The source-detector system is lateral rotated around the pipe with angle incre-
iateral rotation of
flow motion
line detector
FIGURE 2. Data acquisition setup.
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ments of 7.2 degree. At every position a radiographic fan beam projection is measured. Additionally it is assumed that the speed of the data acquisition system is fast enough to neglect motion unsharpeness. An X-ray simulator has been used that counts for attenuation
but neglects the influence of scattered radiation. The projection data have been stored and
evaluated separately. As stated above, the Riccaty equation for the error covariance matrix
Pk and the equation for the gain matrix Kk do not depend on the projection data and can
be calculated in advance and stored. Therefore the reconstruction procedure makes only use
of these results and is reduced to the solution of the filter equation (11) that is recursive.
The result of the reconstruction is summarized in Fig. 3. The first row shows the
forming image Xk at different time. From this image the three-phase flow j* is determined
from eq. (4). The rows 3 and 4 represent the reconstruction results for the three-phase flow
if one or 50 projections have been used per reconstruction step. Comparing the reconstruction results with the original given in row 2 qualitatively it turns out that the accuracy of reconstruction is increased if more projections per reconstruction step are employed. To
quantify the accuracy of reconstruction Fig. 4 gives for phase 2 (oil) a direct comparison of
the concentration with the original for the evaluated time k. Fig. 5 gives the absolute error
of phase separation for all three phases. These results show that the proposed reconstruction
algorithm is capable to monitor the concentration of different phases in a flow with acceptable accuracy.
forming
ttiree-ptiase flow
(original)
low
(cane projection per
reeosasttuctioa
three-phase flow
(50 projection per
time
fe*lQ5
k»110
to*! 15
FIGURE 3. Result of reconstruction for stationary stratified flow of the three components (1) air, (2) oil,
and (3) water at different time steps k.
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1.0
Phase 2
0.8
A
original
—— reconstruction
I 0.6
; 0.4
0.2
0.0
0
50
100
time A:
150
FIGURE 4. Comparison of reconstruction for stationary stratified flow for phase 2 (oil) with original data.
0.050 0.025 0.000
! -0.025 -0.050 0
100
50
time A:
FIGURE 5. Error of phase separation for stationary stratified flow.
CONCLUSIONS
The presented investigation has shown that the proposed dynamic reconstruction algorithm is capable to handle tune dependent processes in the framework of process tomography. The introduced dynamic image model is non-linear. The projection model can also
be non-linear but is here assumed to be linear. Accordingly, the resulting inverse problem is
ill-posed. The stochastic estimation theory employing the Bayesian approach is applied to
regularize the above problem. As a result a modified time-discrete Kalman filter algorithm
is derived for quasi-optimal dynamic reconstruction. The presented example shows the
principle capability of the algorithm to reconstruct the dynamic properties of the investigated process from restricted number of projections. Two tasks have been distinguished.
On one hand the algorithm allows the reconstruction of the object or process properties in
the three dimensional Cartesian space like standard CT applications having access only to a
limited number of projections and views. On the other hand the algorithm can be used for
system monitoring as for monitoring the phase concentration as function of time. These results can be applied to automatic process control to ensure the specified product properties
in a production line.
Computing expenses turn out not to be problematic because for known prior statistics the equations for the error covariance function and the gain can be calculated in ad-
649
vance and are stored. This computation takes most of the expenses. The reconstruction itself makes use of the pre-calculated functions and is not a limiting factor for the application
of the algorithms.
The presented example has shown that non-linear processes and/or observations can
be used. The corresponding reconstruction task to be solved turns to a non-linear optimization problem. Compared to earlier investigations assuming linear process and observation
models yielding a linear reconstruction algorithm, here a non-linear generalization of the
Kalman filter has been proposed. This allows in principle to apply other physical phenomena than the absorption of X-rays for the data acquisition like sound propagation in solids
or liquids and electrical impedance measurements.
ACKNOWLEDGEMENT
This work was funded by the Federal Ministry of Economics of Germany and the
National Academy of Sciences of Belarus.
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