Three-dimensional Pipe Fouling Layer Estimation by
Using Conjugate Gradient Inverse Method
by
Wen-Lih Chen*, Yu-Ching Yang, and Haw-Long Lee
Clean Energy Center
Department of Mechanical Engineering
Kun Shan University
Yung-Kang City, Tainan 710-03, Taiwan
Republic of China
*Corresponding author. E-mail: [email protected]
-1-
Abstract
In this study, a conjugate gradient method based inverse algorithm is applied to
estimate the unknown three-dimensional fouling-layer profiles on the inner wall of a
pipe system using simulated temperature measurements taken on the pipe wall. The
temperature data obtained from the direct problem are used to simulate the exact
temperature measurements. Results show that an excellent estimation on the
fouling-layer profiles can be obtained for the two cases investigated in this study. The
technique presented here can be used as an alternative to ultrasonic waves fouling
detection techniques to provide crucial information for the optimization of cleaning
schedule for piping systems.
Keywords: inverse problem; fouling layer; pipe system; conjugate gradient method
Nomenclature
h
convection heat transfer coefficient (Wm-2K-1)
J
functional
J
gradient of functional
k
thermal conductivity (Wm-1K-1)
L
length of the pipe (m)
M
total number of measuring positions in the  -direction
N
total number of measuring positions in the z-direction
p
pressure (Nm-2)
q
direction of descent
R1
inner radius of the pipe (m)
R2
outer radius of the pipe (m)
r
spatial coordinate (m)
T
temperature (K)
Tin
fluid inlet temperature (K)
T
ambient temperature (K)
vr
fluid velocity in the r-direction (ms-1)
-2-
vz
fluid velocity in the z-direction (ms-1)
v
fluid velocity in the  -direction (ms-1)
Y
measurement temperature (K)
z
spatial coordinate (m)

small variation quality
α
thermal diffusivity (m2s-1)

step size
γ
conjugate coefficient

very small value

spatial coordinate (radian)

variable used in adjoint problem
μ
fluid dynamic viscosity (kgm-1s-1)
ρ
density (kgm-3)
Superscripts
K
iterative number
Subscripts
f
fluid
s
solid
1. Introduction
As soon as any system involving piping is put into operation, fouling layers start
to build up on the inner walls of its pipes. The existence of fouling layers poses
several negative impacts to the overall performance of the system. First, it reduces the
flow cross-section area of the pipes, leading to an increase in the pressure drop.
Second, the fouling layer itself acts as an additional thermal resistance to the pipes’
inherent design thermal resistance and decreases the overall heat transfer rate if the
pipes are used for heat exchange purpose. Consequently, the pumping power needs to
increase to compensate the excessive pressure drop, and the pipes need to be designed
in a larger diameter at the first place to increase the heat-transfer surface to anticipate
-3-
the downgrading of heat-transfer rate due to the build-up of fouling layers. Both result
in more capital required for construction and operation of the system. Third, after a
certain period of operation time, the system inevitably needs to be shut down and its
pipes cleaned to remove the fouling layers. This could further interrupt the production
schedule of a factory and results in more loss in revenue. Nowadays, fouling has
become a costly problem that plagues many industries and causes tremendous
economy loss worldwide in each year [1].
Although the fouling-layer problems can, to a certain extent, be mitigated by
regular cleaning according to some maintenance schedule, this practice, however,
produces some others problems on its own. First, unnecessary cleaning can be
conducted if the fouling layers are still thin but a maintenance schedule is due. Second,
the cleaning is actually performed blindly without the knowledge of the locations of
fouling layers, so regions with very thick fouling cannot be thoroughly cleaned. In
recent years, some researchers reported numerical models to plan for an optimal
cleaning schedule instead of adopting a regular maintenance schedule for a heat
exchanger network (HEN) [2-3]. In these optimization models, the information
regarding the growth as well as the behaviors of the fouling inside a pipe is very
important for the optimization scheme to perform well. However, this crucial
information is predicted yet by another numerical or mathematical fouling model.
That is, a sensible outcome of the optimization scheme is largely pending on the
predictive accuracy of the fouling model. Unfortunately, fouling is such a complicated
phenomenon that when it comes to the prediction of fouling, most models are more
art than science and can only achieve limited degree of predictive accuracy [4]. To
this end, precise detection of fouling layers appears to be one of the most essential
elements to tackle the problems of fouling.
To address the need for fouling detection, ultrasonic wave techniques have been
-4-
implemented by several researchers to detect the thickness of fouling layers in
food-processing pipes [5-7]. In these techniques both the longitudinal and shear waves
were employed to make use of their unique characteristics of propagation inside pipes
and fouling layers. Using shear waves even makes on-line detection possible because
it is less sensitive to the liquid flowing inside a pipe. However, due to the fact that a
fouling layer is often not uniform in thickness, and its properties are various,
ultrasonic techniques can only provide qualitative diagnosis on the condition of
fouling build-up such as whether the fouling is severe, moderate, or minimal.
Additionally, most ultrasonic detection techniques are only capable of detecting
fouling-layer thickness one point at a time. To acquire a big picture of the fouling
condition in a piping system by using ultrasonic techniques will prove to be a
daunting and lengthy undertaking. These techniques still need years of research to
perfect.
Inverse methods have recently been applied to various engineering problems. A
great number of applications of inverse methods are continuously being proposed for
different technical fields [8–10]. Despite the ill-posed nature of inverse problems, the
solutions of these problems are important for theoretical studies and measurement
techniques, especially in cases where measurement is difficult, instruments for
measurement are expensive, or the measurement process to directly measure certain
physical quantities is complicated. Under these circumstances, a satisfactory
estimated result can be easily obtained by using a numerical method and some simple
instruments, for examples, using thermal couples to measure the inner wall
temperatures of burners, the temperatures of cutting tool tips, and so on. One of the
very interesting topics of inverse problems, attracting a lot of attentions in recent
years, is the technique of inverse geometry problem (or shape identification problem).
The applications of shape identification problem have been widely used in various
-5-
industrial fields, for examples, the prediction of frost thickness in refrigeration
systems, the prediction of the geometry of blast furnace inner wall, the prediction of
crevice and pitting in furnace wall, and the optimization of geometry [11]. In the past,
there have been many researchers devoted to the study of inverse geometry problems
using a variety of numerical methods. Huang and Chen [12] developed a modified
model to estimate the outer boundary configurations of a multiple region domain
without confining the search directions. Park and Shin applied the coordinate
transformation technique with the adjoint variable method to a shape identification
problem in determining unknown boundary configuration for heat conduction systems
[13] and natural convection systems [14]. Divo et al. [15] used the genetic algorithm
and a singular superposition technique to detect the unknown sphere cavity in a 3D
inverse geometry problem. Kwag et al. [16] followed a new algorithm to estimate the
phase front motion of ice in a thermal storage system. Recently, Su and Chen [17]
utilized the reversed matrix method with both the linear least-squares error method
and the concept of virtual area for a shape identification problem to identify the
geometry of inner wall in a furnace. To the authors’ best knowledge; there is no report
in the open literature documenting three-dimensional fouling layer estimation using
an inverse method.
The objective of the present inverse geometry problem is to estimate the unknown
irregular three-dimensional fouling profile on the inner wall of a pipe, which involves
conjugate heat transfer, based on the simulated temperature measurements taken on
the pipe wall. This technique can provide an alternative to ultrasonic waves and other
techniques to accurately detect the three-dimensional fouling layer thickness along the
inner wall of a pipe. It is non-destructive and allows on-line measurement to perform
so that the production schedule wouldn’t be interrupted. In reality, the variation in the
slop of the fouling-layer profile might be large, and the fouling layer might be
-6-
irregular in shape; thus the pipe flow will never be fully developed, and there is a
possibility of forming some localized separation bubbles in the pipe flow. Therefore,
the flow field needs to be solved by the Navier-Stokes equations, which have to be
incorporated into the inverse procedure. In this, the Navier-Stokes equations are
coupled with the inverse algorithm through the perturbation of fouling-layer profile.
That is, as the fouling-layer profile is modified after an inverse iteration, a new flow
field needs to be solved by the Navier-Stokes equations because the boundary of the
flow domain has been changed. Then the updated flow field affects the temperature
distributions both in fluid and solid materials via the mechanism of conjugate heat
transfer, hence altering the course of the inverse algorithm and, in turns, resulting in
yet another new fouling-layer profile. The iteration cycle then goes on and on until a
convergence criterion is met. Theoretically, the current inverse method is able to cope
with arbitrary fouling profiles, making it a valuable tool for acquiring the critical
information needed by the cleaning optimization schemes mentioned earlier. Here, we
employ the conjugate gradient method (CGM) [18-22] and the discrepancy principle
[23] to the inverse geometry problem to determine the 3D fouling layer configuration
in the pipe system. The conjugate gradient method with an adjoint equation, also
called Alifanov’s iterative regularization method, belongs to a class of iterative
regularization techniques, which means the regularization procedure is performed
during the iterative processes, thus the determination of optimal regularization
conditions is not needed. On the other hand, the discrepancy principle is used to
terminate the iteration process in the conjugate gradient method.
2. Analysis
2.1 Direct problem
To illustrate the methodology of developing expressions for use in determining the
unknown three-dimensional irregular fouling profile f(θ,z), on the inner wall of a pipe,
-7-
the following 3D steady-state heat transfer problem is considered. Fig. 1 shows a
schematic representation of the considered pipe flow. The length of the pipe is L, with
inner and outer radii of R1 and R2, respectively. The fluid temperature at the inlet is
Tin (r ) . After a period of operation, a layer of fouling is assumed built up on the pipe’s
inner wall, and its profile f(θ,z) is unknown. Then, the mathematical formulation of
this steady-state forced-convection heat transfer problem, covering the fluid domain,
fouling layer, and solid pipe material, respectively, can be expressed as [24]:
Navier-Stokes equations and boundary conditions:
1 
1 v vz
(rvr ) 

0,
r r
r  z
 (vr
(1a)
vr v vr v2
v
v
p
1  vr
1  2 vr 2 v  2 vr

  v z r )    [
(r
)  r2  2

 2 ],
r
r 
r
z
r
r r
r
r
r  2 r 2 
z
 ((vr )
(1b)
v v v vr v
v
v
1 p
1  v
1  2v
2 vr  2v


 vz  )  
 [
(r
)  2  2

 2 ] , (1c)
r
r 
r
z
r 
r r
r
r
r  2 r 2 
z
vz v vz
vz
p
1  vz
1  2 vz  2 vz
 (vr

 vz
)    [
(r
) 2

],
r
r 
z
z
r r r
r  2 z 2
r
vz (r ,  , 0)  2.0  vz ,av [1  ( ) 2 ] , at 0  r  f ( , 0) , 0    2 , and z = 0,
R1
v  vz  0 , at 0  r  f ( , 0) , 0    2 , and z = 0,
vr v vz


 0 , at 0  r  f ( , L) , 0    2 , and z = L,
z
z
z
vr  v  vz  0 , at r = f ( , z ) , 0    2 and 0  z  L.
(1d)
(1e)
(1f)
(1g)
(1h)
Energy equations and boundary conditions:
In the fluid region:
T f
2
2
T f
v T f
1  T f
1  Tf  Tf
vr

 vz
f[
(r
) 2
 2 ],
r
r 
z
r r
r
r  2
z
(2a)
T f  Tin (r ) , at 0  r  f ( , 0) , 0    2 and z = 0,
(2b)
T f
z
 0 , at 0  r  f ( , L) , 0    2 and z = L.
-8-
(2c)
In the fouling region:
1  Ts1
1  2Ts1  2Ts1
(r
) 2
 2 0,
r r
r
r  2
z
Ts1
 0 , at f ( , 0)  r  R1, 0    2 , and z = 0,
z
Ts1
 0 , at f ( , L)  r  R1, 0    2 , and z = L.
z
(3a)
(3b)
(3c)
In the pipe wall region:
1  Ts 2
1  2Ts 2  2Ts 2
(r
) 2

 0,
r r
r
r  2
z 2
Ts 2
 0 , at R1  r  R2, 0    2 , and z = 0,
z
Ts 2
 0 , at R1  r  R2, 0    2 , and z = L,
z
ks 2
Ts 2
 h(Ts 2  T ) , at r = R2, 0    2 , and 0  z  L.
r
(4a)
(4b)
(4c)
(4d)
At the interface between the regions of fluid and fouling:
T f  Ts1 , at r = f ( , z ) , 0    2 , and 0  z  L,
kf
T f
n
 ks1
Ts1
, at r = f ( , z ) , 0    2 , and 0  z  L.
n
(5a)
(5b)
At the interface between the regions of fouling and pipe wall:
Ts1  Ts 2 , at r = R1, 0    2 , and 0  z  L,
k s1
Ts1
T
 k s 2 s 2 , at r = R1, 0    2 , and 0  z  L.
r
r
(6a)
(6b)
where k is the thermal conductivity, and vz ,av is the average velocity. The direct
problem considered here is concerned with the determination of the medium
temperature when the irregular fouling-layer configuration
-9-
f ( , z ) , thermal
properties, and boundary conditions are known.
2.2 Inverse problem
For the inverse problem, the irregular fouling-layer configuration f ( , z ) is
regarded as being unknown, while everything else in Eqs. (1)–(6) is known. In
addition, temperature readings taken at some appropriate locations of the pipe wall are
considered available. The objective of the inverse analysis is to predict the unknown
irregular fouling-layer profile f ( , z ) from the knowledge of these temperature
readings.
Let
the
measured
temperature
at
the
measurement
position
(r ,  , z )  (rm , i , z j ) be denoted by Y (rm ,i , z j ) . Here i = 1~M and j = 1~N, while M
and N are the total measurement points in the  and z directions, respectively. In
addition, rm represents the radius of which the thermocouples are located. Then this
inverse problem can be stated as follows: by utilizing the above mentioned measured
temperature data Y (rm ,i , z j ) , the unknown fouling-layer configuration f ( , z ) is to
be estimated over the specified domain.
The solution of the present inverse problem is to be obtained in such a way that
the following functional is minimized:
N
M
J [ f ( , z )]  [Ts 2 (rm ,i , z j )  Y (rm ,i , z j )]2 ,
(7)
j 1 i 1
where Ts 2 (rm ,i , z j ) is the estimated (or computed) temperature at the measurement
location (r ,  , z )  (rm ,  j , zi ) . These quantities are determined from the solution of
the direct problem given previously by using an estimated f K ( , z ) for the exact
f ( , z ) . Here f K ( , z ) denotes the estimated quantities of f ( , z ) at the Kth
iteration. In addition, in order to develop expressions for the determination of the
unknown f ( , z ) , a “sensitivity problem” and an “adjoint problem” are constructed
- 10 -
as described below.
2.3 Sensitivity problem
The sensitivity problem is obtained from the original direct problem defined by
Eqs. (2)–(6) in the following manner: It is assumed that when f ( , z ) undergoes a
variation f ( , z ) , T (r , , z ) is perturbed by T+ T. Then replacing in the direct
problem f by f + f and T by T + T, subtracting from the resulting expressions the
direct problem, and neglecting the second-order terms, the following sensitivity
problem for the sensitivity function T can be obtained.
In the fluid region
vr
T f
2
2
T f
v T f
1  T f
1  T f   T f

 vz
f[
(r
) 2

],
r
r 
z
r r
r
r  2
z 2
T f  0 , at 0  r  f ( , 0) , 0    2 and z = 0,
T f
 0 , at 0  r  f ( , L) , 0    2 and z = L.
z
In the fouling region:
(8a)
(8b)
(8c)
1  Ts1
1  2 Ts1  2 Ts1
(r
) 2

 0,
r r
r
r  2
z 2
(9a)
Ts1
 0 , at f ( , 0)  r  R1, 0    2 , and z = 0,
z
(9b)
Ts1
 0 , at f ( , L)  r  R1, 0    2 , and z = L.
z
(9c)
In the pipe wall region:
1  Ts 2
1  2 Ts 2  2Ts 2
(r
) 2

 0,
r r
r
r  2
z 2
(10a)
Ts 2
 0 , at R1  r  R2, 0    2 , and z = 0,
z
(10b)
Ts 2
 0 , at R1  r  R2, 0    2 , and z = L,
z
(10c)
- 11 -
k s 2
Ts 2
 hTs 2 , at r = R2, 0    2 , and 0  z  L.
r
(10d)
At the interface between the regions of fluid and fouling:
T f  Ts1 , at r = f ( , z ) , 0    2 , and 0  z  L,
T f
kf
 ks1
n
(11a)
Ts1
, at r = f ( , z ) , 0    2 , and 0  z  L.
n
(11b)
At the interface between the regions of fouling and pipe wall:
Ts1  Ts 2 , at r = R1, 0    2 , and 0  z  L,
k s1
(12a)
Ts1
Ts 2
 ks 2
, at r = R1, 0    2 , and 0  z  L.
r
r
(12b)
The sensitivity problem of Eqs. (8)-(12) can be solved by the same method as the
direct problem of Eqs. (2)-(6).
2.4 Adjoint problem and gradient equation
To obtain the adjoint problem, Eqs. (2a), (3a), and (4a) are multiplied by the
Lagrange multipliers (or adjoint functions)  f (r ,  , z ), s1 (r , , z), and s 2 (r , , z ),
respectively, and the resulting expressions are integrated over the correspondent space
domains. Then the results are added to the right hand side of Eq. (7) to yield the
following expression for the functional J [ f ( , z )] :
N
M
J [ f ( , z )]  [Ts 2 (rm ,i , z j )  Y (rm ,i , z j )]2
j 1 i 1

L
2
f


T f


r
 f  vr
z 0   0  r 0


L
2
R1
z 0
 
r f
L
2
R2
z 0
0
 
0
r  R1
s1  [

2
2
T f
v T f
1  T f
1  Tf  Tf
 vz
 f [
(r
) 2

r 
z
r r
r
r  2
z 2


]drd dz


1  Ts1
1  2Ts1  2Ts1
(r
) 2
 2 ]drd dz
r r
r
r  2
z
s 2  [
1  Ts 2
1  2Ts 2  2Ts 2
(r
) 2

]drd dz .
r r
r
r  2
z 2
- 12 -
(13)
The variation J is obtained by perturbing f
by Δf and T by T in Eq. (13).
Subtracting from the resulting expression the original Eq. (13) and neglecting the
second-order terms, we thus find
J [ f ( , z)] 
2
N
M
     2[T
L
z 0

R2
0
r  R1
2
L
j 1 i 1
f

2
L
R1
z 0
 
r f
L
2
R2
z 0
0
 
0
r  R1
(r , , z )  Y (r ,  , z )]Ts 2   (r  rm )   (  i )   ( z  z j )drd dz


Tf


r
 f  vr
z 0   0  r 0

s2
s1  [

2
2
T f
v T f
1  T f
1  T f  T f
 vz
 f [
(r
) 2

r 
z
r r
r
r  2
z 2


]drd dz


1  Ts1
1  2 Ts1  2 Ts1
(r
) 2

]drd dz
r r
r
r  2
z 2
s 2  [
1  Ts 2
1  2 Ts 2  2 Ts 2
(r
) 2

]drd dz .
r r
r
r  2
z 2
(14)
where  is the Dirac function. We can integrate the second to fourth double integral
terms in Eq. (14) by parts. Utilizing the boundary conditions of the sensitivity
problem, then J is allowed to go to zero. The vanishing of the integrands containing
T leads to the following adjoint problem for the determination of  f (r ,  , z ),
s1 (r , , z ), and s 2 (r , , z ) :
In the fluid region:
2
2

1 

1   f
1  f  f
(vr  f ) 
(v  f )  (vz  f )   f [
(r
) 2
 2 ]0,
r
r 
z
r r
r
r  2
z
(15a)
 f  0 , at 0  r  f ( , 0) , 0    2 and z = 0,
(15b)
f
 f
z
 vz  f  0 , at 0  r  f ( , L) , 0    2 and z = L.
(15c)
In the fouling region:
1  s1
1  2s1  2s1
(r
) 2

0,
r r
r
r  2
z 2
(16a)
- 13 -
s1
 0 , at f ( ,0)  r  R1, 0    2 , and z = 0,
z
(16b)
s1
 0 , at f ( , L)  r  R1, 0    2 , and z = L.
z
(16c)
In the pipe wall region:
1  Ts 2
1  2Ts 2  2Ts 2 N M 2[Ts 2 (r , , z )  Y (r ,  , z )]
(r
) 2

 
  (r  rm )   (  i )   ( z  z j )  0 ,
r r
r
r
r  2
z 2
j 1 i 1
(17a)
s 2
 0 , at R1  r  R2, 0    2 , and z = 0,
z
(17b)
s 2
 0 , at R1  r  R2, 0    2 , and z = L,
z
(17c)
ks 2
s 2
 hs 2 , at r = R2, 0    2 , and 0  z  L.
r
(17d)
At the interface between the regions of fluid and fouling:
k s1 f  k f s1 , at r = f ( , z ) , 0    2 , and 0  z  L,
ks1
 f
n
 kf
s1
, at r = f ( , z ) , 0    2 , and 0  z  L.
n
(18a)
(18b)
At the interface between the regions of fouling and pipe wall:
ks 2s1  ks1s 2 , at r = R1, 0    2 , and 0  z  L,
(19a)
s1 s 2

, at r = R1, 0    2 , and 0  z  L.
r
r
(19b)
Then the adjoint problem can be solved by the same method as the direct problem.
Finally the following integral term is left:
J  
L
z=0
 T  
  s1 s1 
 f ( , z )d dz .
0
 n n  r  f ( , z )
2

(20)
From the definition used in the reference [18], we have
J  
L
z=0
2

0
J ( , z)f ( , z)d dz ,
(21)
- 14 -
where J ( , z ) is the gradient of the functional J [ f ( , z )] . A comparison of Eqs.
(20) and (21) leads to the following form:
 T  
.
J ( , z )    s1 s1 
 n n  r  f ( , z )
(22)
2.5 Conjugate gradient method for minimization
Assuming the functions of T (r , , z ) , T (r ,  , z ) ,  (r ,  , z ) , and J ( , z ) are
available at the Kth iteration, the iteration process based on the conjugate gradient
method is now used for the estimation of f ( , z ) . By minimizing the above
functional J [ f ( , z )] , the function f ( , z ) can be evaluated at the (K+1)th step by
f K 1 ( , z )  f K ( , z )   K q K ( , z) , K  0, 1, 2,... ,
(23)
where  K is the search step size in going from iteration K to iteration K+1, and q K
is the direction of descent (i.e., search direction) given by
q K ( , z )  J K ( , z )   K q K 1 ( , z) ,
(24)
which is conjugation of the gradient direction J K ( , z ) at iteration K and the
direction of descent q K 1 ( , z ) at iteration K-1. The conjugate coefficient 
K
is
determined from
N
K 
M
 [ J 
j 1 i 1
N M
 [ J 
j 1 i 1
K
( , z ) (   i ) ( z  z j )]2
with
K 1
 0  0.
(25)
( , z ) (   i ) ( z  z j )]
2
The convergence of the above iterative procedure in minimizing the functional J is
proved in the reference [18]. To perform the iterations according to Eq. (23), we need
- 15 -
to compute the step size  K and the gradient of the functional J K ( , z ) .
The functional J [ f K 1 ( , z )] for iteration K+1 is obtained by rewriting Eq. (7) as
N
M
J [ f K 1 ( , z )]  [Ts 2 ( f K   K q K )  Y (rm ,i , z j )]2 ,
(26)
j 1 i 1
where we replace f K 1 by the expression given by Eq. (23). If temperature
Ts 2 ( f K   K q K ) is linearized by a Taylor expansion, Eq. (26) takes the form
N
M
J [ f K 1 ( , z )]  [Ts 2 ( f K )   K Ts 2 (q K )  Y (rm , i , z j )]2 ,
(27)
j 1 i 1
where Ts 2 ( f K ) is the solution of the direct problem at (r ,  , z )  (rm , i , z j ) by using
estimated f K ( , z ) for exact f ( , z ) . The sensitivity function Ts 2 ( q K ) is taken
as the solution of Eqs. (8)–(12) at the measured position (r ,  , z )  (rm , i , z j ) by
letting f ( , z )  q K ( , z ) [25]. The search step size  K is determined by
minimizing the functional given by Eq. (27) with respect to  K . After rearrangement,
the following expression is obtained:
N
K 
M
 T
j 1 i 1
s2
(q K )[Ts 2 ( f K )  Y (rm , i , z j )]
N
.
M
 [T
j 1 i 1
s2
K
(28)
2
( q )]
2.6 Stopping criterion
If the problem contains no measurement errors, the convergence condition for the
minimization of the criterion is:
J ( f K 1 )   ,
(29)
- 16 -
where  is a small specified number, can be used as the stopping criterion. However,
the observed temperature data contains measurement errors; as a result, the inverse
solution will tend to approach the perturbed input data, and the solution will exhibit
oscillatory behavior as the number of iteration is increased [26]. Computational
experience has shown that it is advisable to use the discrepancy principle [23] for
terminating
the
iteration
process
in
the
conjugate
gradient
method.
Assuming Ts 2 (rm ,i , z j )  Y (rm ,i , z j )   , the stopping criteria  by the discrepancy
principle can be obtained from Eq. (7) as
  MN 2 ,
(30)
where  is the standard deviation of the measurement error. Then the stopping
criterion is given by Eq. (29) with  determined from Eq. (30).
2.7 Computational procedures
The computational procedure for the solution of this inverse problem may be
summarized as follows:
Suppose f K ( , z ) is available at iteration K.
Step 1 Solve the flow field given by Eqs. (1a)–(1h).
Step 2 Solve the direct problem given by Eqs. (2)–(6) for T f (r ,  , z ), Ts1 (r , , z ), and
Ts 2 (r , , z ), respectively.
Step 3 Examine the stopping criterion given by Eq. (29) with  given by Eq. (30).
Continue if not satisfied.
Step 4 Solve the adjoint problem given by Eqs. (15)–(19) for  f (r ,  , z ), s1 (r , , z),
and s 2 (r , , z ), respectively.
Step 5 Compute the gradient of the functional J ( , z ) from Eq. (22).
- 17 -
Step 6 Compute the conjugate coefficient  K and direction of decent q K ( , z ) from
Eqs. (25) and (24), respectively.
Step 7 Set f ( , z ) = q K ( , z ) and solve the sensitivity problem given by Eqs.
(8)-(12) for T f (r , , z ), Ts1 (r , , z ), and Ts 2 (r , , z ), respectively.
Step 8 Compute the search step size  K from Eq. (28).
Step 9 Compute the new estimation for f K 1 (r , , z) from Eq. (23) and return to Step
1.
3. Results and discussion
The objective of this article is to validate the present approach when used in
estimating the 3D irregular profile of the fouling layer built on a pipe’s internal wall
accurately without prior information on the functional form of the unknown profile, a
procedure called function estimation. In the present study, we assume that the material
of the pipe wall is steel, and that of the fluid is water. Then the material properties,
geometric parameters, and thermal parameters of the system are listed as follows:
 f = 1.4×10-7 m2s-1, ν = 8.9×10-7 m2s-1,
kf = 0.5 Wm-1K-1, ks1 = 2.0 Wm-1K-1, ks2 = 42.0 Wm-1K-1,
R1 = 0.0125 m, R2 = 0.013 m, L = 0.15 m, Tin (r )  T0  constant, T  T0  20 K,
h = 100 Wm-2K-1, vz,av = 0.05 ms-1.
The numerical procedure in this paper is based on the unstructured-mesh, fully
collocated, finite-volume code, ‘USTREAM’ developed by the corresponding author.
This is the descendent of the structured-mesh, multi-block code of ‘STREAM’ [27].
One of the numerical challenges of the current problem is that the fouling layer profile,
i.e. the solid/fluid interface, changes iteration after iteration. This affects the boundary
profiles of both the fouling layer and the pipe flow regions. As a result, the meshes for
both regions need to be re-built after each inverse iteration. Therefore, a mesh
- 18 -
reconstruction algorithm has been incorporated into the code to re-build the part of
mesh covering the fouling layer and the pipe flow (the mesh for solid pipe wall is
untouched). In order to test the accuracy of the present inverse analysis, we first
consider a simple simulated exact profile of the fouling layer, f(θ, z), as:
f ( , z )  0.12  0.2sin( ) sin(
z
0.15
) m.
(31)
As seen, equation (31) defines a three-dimensional surface that has a single valley at
the middle of the pipe. That is, the thickest fouling layer is located at the θ = 180° (or
 ) in the middle of the pipe, then the thickness gradually reduces towards θ = 0° and
θ = 360° (or 2 ), along the circumferential direction, as well as towards the two
openings of the pipe, along the axial direction. Some mesh cross sections of the direct
problem are given in Fig. 2(a). This mesh consists of 252,000 cells where there are 35,
80, and 90 cells allocated in the r-, θ-, and z-directions, respectively. It is also
noticeable that the mesh has been compressed towards the fouling-layer/fluid
interface in r-direction to increase the numerical resolution at the adjacent regions of
the interface. The mesh has been selected for its cell density being dense enough to
return a grid-independent solution after a mesh-density test. Fig. 2(b) shows the
temperature contours of the direct problem with the exact fouling-layer profile at the
same cross sections as those in Fig. 2(a). The plot clearly shows that thick fouling
layer can locally reduce the temperature gradients near the pipe wall and hence
decreases the rate of heat transfer. In this study, the thermocouples are assumed
located on the inner surface of the solid pipe, namely rm = R1 = 0.0125 m. Here, we do
not have a real experimental set up to measure the temperature Y (rm ,i , z j ) in Eq.
(7).
Instead, we assume a real profile of the fouling layer, f(θ, z) of Eq. (31), and
substitute the exact f(θ, z) into the direct problem of Eqs. (2)-(6) to calculate the
temperatures at the locations where the thermocouples are placed. The results are
- 19 -
taken as the measured temperature Y (rm ,i , z j ) , so-called the simulated temperature
measurement.
The three-dimensional exact and inverse fouling-layer profiles obtained from the
numerical experiments with an initial guess fouling-layer profile, a uniform
fouling-layer thickness,
f 0 ( , z )  0.012 m are shown in Fig. 3(a) and 3(b),
respectively. The plot shows that the estimated values of f(θ, z) seem to be in very
good agreement with the exact values. To further validate the agreement, quantitative
comparison of the axial fouling-layer profile at three different circumferential angles,
namely, θ = 9° (or


), 90° (or
), and 180° (or  ) is given in Fig. 4. The plot
20
2
indicates that the agreement between the estimated and the exact profiles are generally
excellent, especially at θ = 90°. The inverse method only slightly underestimated the
fouling-layer thickness at θ = 9° and 180°. It also can be noted that the inverse profile
somehow develops very slight zigzag oscillation, a phenomenon commonly observed
in other inverse solutions for different problems, towards the outlet of the pipe at θ =
9°. The comparison of exact and inverse measurement temperatures at the same three
circumferential angles is depicted in Fig. 5. Here, one can see that the inverse values
are almost identical to the exact values at all locations. Once the fouling-layer
boundary profile has been correctly estimated, accurate temperature distributions
within the entire domain can also be obtained. This is demonstrated in Fig. 6, the field
temperature contours at the cross section right at the middle of the pipe (z = 0.075 m).
The plot shows that the predicted temperature distributions are almost identical to the
exact values.
To prove the current method’s ability to estimate any functional form of the
fouling-layer profile, an additional and more complicated profile is tested:
f ( , z )  (0.012  0.01z ) sin( ) m, 0≦z＜0.025 m;
- 20 -
f ( , z )  [0.0095  0.01( z  0.025)]sin( ) m, 0.025≦z＜0.05 m;
(32)
f ( , z )  [0.12  0.005( z  0.1)]sin( ) m, 0.05≦z＜0.1 m;
f ( , z )  [0.1  0.005( z  0.2)]sin( ) m, 0.1≦z≦0.15 m.
Along the axial direction, equation (32) is a piecewise linear function which defines a
W-shape profile; along the circumferential direction, on the other hand, the
fouling-layer thickness gradually increases then decreases as shown in Fig. 7(a). The
piecewise linear function has three corners at z = 0.025, 0.05, and 0.1 m, respectively,
which might be difficult for the inverse method to approximate. Furthermore, the
profile creates two contraction and two diffusion sections within the pipe flow and a
cavity between the two valleys of the fouling layer. Under certain flow conditions,
adverse pressure gradient acting on the first diffusion section can result in a separation
bubble trapped within the cavity, making the fluid flow much more complicated.
The
estimated
profile,
again
obtained
with
the
flat
initial
profile
of
f 0 ( , z )  0.012 m is given in Fig. 7(b). Even with this highly complicated
fouling-layer profile, the inverse method appears to return quite satisfactory
agreement with the exact profile although the agreement seems to deteriorate slightly.
The quantitative comparison between the exact and inverse profiles along the axial
direction at the same three circumferential angles is given in Fig. 8. Comparing Fig. 4
and Fig. 8, it is clear that the discrepancy between the exact and inverse profiles has
indeed slightly more pronounced at some localized regions for this complicated
profile. In this case, the inverse method underestimates the depth of first valley,
especially at θ = 180°, and the zigzag oscillation has spread to wider areas.
Nevertheless, all the important features of the exact fouling-layer geometry are
captured by the inverse method, and the inverse profile is still in good agreement with
the exact profile for most regions. The comparison of measurement temperatures of
the second case is given in Fig. 9. Despite the slight inconsistency between the exact
- 21 -
and inverse fouling profiles, the inverse method again returns almost identical
temperature distributions with the exact values at the measurement locations. To this
end, the capability of the current inverse method to accurately estimate arbitrary
fouling-layer profiles inside a pipe is well demonstrated.
4. Conclusion
The conjugate gradient method was successfully applied for the solution of the
inverse geometry problem to determine the unknown three-dimensional irregular
boundary profiles of fouling layer on the internal wall of a pipe system with the
knowledge of temperature at some measurement locations. Subsequently, the
temperature distributions in the system can be calculated. The current inverse
procedure involves solving the Navier-Stokes equations to acquire for the pipe flow
field, thus enabling it to cope with arbitrary fouling-layer geometries. The results
show that the accuracy of the current method is still satisfactory even when it is used
to estimate a highly complicated fouling-layer profile. The technique presented in this
study is promising and can be used as an alternative to ultrasonic wave techniques to
detect the detailed three-dimensional pipe fouling layer throughout the pipe (not
confined to a localized region at a time). It is non-destructive and allows on-line
detection to be performed, hence not causing any interference to the production
schedule.
Acknowledgements
This work was supported by the National Science Council, Taiwan, Republic of
China, under the grant number NSC 97-2221-E-168-041.
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- 22 -
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Parameters by Solving the Two-dimensional Inverse Heat Conduction Problem,
J. Eng. Phys., vol. 41, pp. 581–586, 1981.
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[27] F. S. Lien, W. L. Chen, and M. A. Leschziner, A Multiblock Implementation of
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- 25 -
Figure Captions
Fig. 1: Schematic of the configuration of the pipe system.
Fig. 2: The computational mesh and temperature contours of the direct problem at
several cross sections for the first case: (a) the computational mesh and (b) the
temperature contours.
Fig. 3: The exact and the inverse fouling-layer profiles for the first case: (a) the exact
profile and (b) the inverse profile.
Fig. 4: The fouling-layer profile along the axial direction at three different
circumferential angles for the first case.
Fig. 5: The measurement temperature distributions along the axial direction at three
different circumferential angles for the first case.
Fig. 6: Comparison of the exact and inverse field temperature contours at the middle
cross section of the pipe for the first case.
Fig. 7: The exact and the inverse fouling-layer profiles for the second case: (a) the
exact profile and (b) the inverse profile.
Fig. 8: The fouling-layer profile along the axial direction at three different
circumferential angles for the second case.
Fig. 9: The measurement temperature distributions along the axial direction at three
different circumferential angles for the second case.
- 26 -
solid pipe, Ts2
fouling layer, Ts1
r=f(θ,z)
vin
Tin
fluid, Tf
r
z
R1
CL
R2
Fig. 1
- 27 -
(a)
T-T0
18
16
14
12
10
8
6
4
2
(b)
Fig. 2:
- 28 -
1.25
1.25
1.2
1.2
1.15
1.15
r(cm)
r(cm)
1.1
1.1
1.05
1.05
1
360
12
270
1
15
9
180
 (o)
6
90
z (cm)
3
0
0
(a)
1.25
1.25
1.2
1.2
1.15
1.15
r(cm)
r(cm)
1.1
1.1
1.05
1.05
1
360
12
270
1
15
9
180
 (o)
6
90
3
0
(b)
Fig. 3:
- 29 -
0
z (cm)
1.2

1.15
f(cm)
1.1

1.05
1


exact
inverse
0.95
0.9
0
2
4
6
8
z(cm)
Fig. 4:
- 30 -
10
12
14
20



T-T0(K)
19.98

19.96

19.94
exact
inverse
19.92
19.9
0
2
4
6
8
z(cm)
Fig. 5:
- 31 -
10
12
14
T-T0
T-T0
20
18
16
14
12
10
8
6
4
2
0
20
18
16
14
12
10
8
6
4
2
0
inverse
exact
Fig. 6:
- 32 -
1.25
1.25
1.2
1.2
1.15
1.15
1.1 r(cm)
r(cm) 1.1
1.05
1.05
1
1
0.95
360
12
270
0.95
15
9
180
( )
o
6
90
z (cm)
3
0
0
(a)
1.25
1.25
1.2
1.2
1.15
1.15
1.1 r(cm)
r(cm) 1.1
1.05
1.05
1
1
0.95
360
12
270
9
180
( )
o
0.95
15
6
90
3
0
(b)
Fig. 7:
- 33 -
0
z (cm)
1.2

1.15
f(cm)
1.1

1.05
1


exact
inverse
0.95
0.9
0
2
4
6
8
z(cm)
Fig. 8:
- 34 -
10
12
14
20

19.98

T-T0(K)


19.96

19.94
exact
inverse
19.92
19.9
0
2
4
6
8
z(cm)
Fig. 9:
- 35 -
10
12
14
- 36 -