Optimal sensors placement for monitoring a steam condenser of the distillation column using bond graph approach
Automatic Laboratory of Setif, Electrical Engineering Department, University of Setif 1, 19000, Setif, Algeria
Emails: ksamia2002@yahoo.fr, mostefai@univsetif.dz, mabroukkhemliche@yahoo.fr
^{* }Corresponding author, Phone: +213668769714; Fax: +213 36 92 51 02
Abstract
This paper deals with monitoring of a process engineering system. The steam condenser was monitored by bond graph tool. The model was constituted by nine capacitive and resistive elements which needed minimum of sensors. This method was based on Analytical Redundancy Relations which were generated from a condenser model and represented residuals. After substitution, we obtained the placement of six sensors which guaranteed the monitoring of nine components. A fault is created by the abrupt annulment of the fluid flow value provided by the source. The block diagram is elaborated on SYMBOLS software and we supervised the residuals evolution.
Keywords
Optimization; Sensors placement; Monitoring; Condenser; Distillation Column; Bond graph
Introduction
The monitoring system with rigid structure knows progress during the last years. The monitoring of the thermal system is difficult because of the phenomenon complexity [1]. These industrial processes have a strongly nonlinear behaviour due mainly to the mutual interaction of several phenomena with various nature and the association of the technological components. The system modelled and monitored, is a condenser of the distillation column which is produced thermal, chemical and hydraulic phenomena. In this case, the bond graph tool reduces all difficulties because of its causal and structural properties. After the validation of the model, the placement of the sensors, is made and followed by the research of the optimal case [2], and [3]. The advantages of applying bond graph model to the condenser of distillation column are the ability to build a graphical circuit, the use of detailed equations, the encapsulation of pieces of the model into submodels and the representation of the system power flows in details. The main drawback of using bond graphs is that they cannot describe distillation equations. In this work, both bond graph modelling and monitoring by sensors placement are proposed for a condenser of a distillation column.
Description of the system
The condenser schematic is shown in Figure 1. The steam enters at the left side, condensed on the pipes filled with cooling water.
It falls to the floors and leaves from there to the receiver through control valves [1]. The water outflow (condensate) is controlled by three valves in order to keep its level constant. The condenser is a module of four inputoutputs. It is composed by a coolant circuit made up of a unit of pipes which lead the liquid. The liquid phase of cold fluid accumulates heat while passing in steam (vapor) phase. When the steam flow attempts the condenser, a contact is obtained between the steam and the pipes of the fluid circuit in which circulate both the cold fluid and the steam condensed in fine droplets which are formed at the condenser bottom [4].
Figure 1. The condenser operating in the distillation column
Steam circulating in the column is from a portion of the feed when it is vaporized. In the boiler, the liquid mixture of the beginning is heated for the vaporizer. The steam which arrives in the line of column heading is condensed completely or partially by a condenser [5], [6]. When the steam arrives in contact with the fluid pipes in which circulates a cold fluid, it was condensed in liquid which runs out downwards by simple gravity and collected at the bottom of the condenser. The liquid obtained is generally returned in the column under the name of backward flow by tubular of feed. The other part is collected like a distillate and the nonvaporized part comes to contribute to the liquid flow. The operation is repeated, by ensuring the condensation of the steam of column heading [2].
Sensors placement algorithm
This procedure consists of an optimal sensors placement for the components monitoring, i.e. to detect and to isolate the components failures. From a physical process we elaborate a bond graph model. In bond graph model, sensors are placed only at the junction 0 and 1, we consider a virtual placement of the sensor at the position j, which is represented by the variables x and y. N_{0}, and N_{1} are respectively the junction numbers 0_{Ci}, 1_{Rj}. i, j, n and m are respectively the junction orders, the number of the bonds’ fixed on the junction 0_{Ci}, 1_{Rj}. The structural equations of the junction s0_{Ci}, which describe the effort equality and the pressure conservation produced in C element, are given by (1).
_{} 
(1) 
where f_{k} and e_{k}
are respectively the flow and the effort of the junction number k; a_{k}=+1
if the bond graph semi flow is toward junction; a_{k}=1
otherwise. For the nonlinear functions Ф_{Ci} and s the
_{} 
(2) 
where i=1, N_{0 }; De_{i} effort detectors, x_{i} binary variables x_{i}=1 if we place detector; x_{i}=0 otherwise. The structural equations of the junctions 1_{Rj} are given by equation (3).
_{} 
(3) 
For the nonlinear functions Ф_{Rj}, the flow and effort variables are determined by (4):
_{} 
(4) 
where j=1, N_{1}; Df_{j} flow detectors, y_{j} binary variables y_{j} =1 if we place detector; y_{j} =0 otherwise. The combinations of the variables x_{i} and y_{j} make it possible to generate analytical redundancy relations which give the structures of the residuals. The bond graph model of the condenser in the distillation column representing hydraulic and thermal phenomena with steam and liquid phases and using sensors placement (effort and flow detectors) is represented by Figure 2. This model is composed by 5 junctions type 0 attached to 5 components C_{V}, C_{L}, C_{T1}, C_{T3} and C_{TS }noted 0_{1}, 0_{2, }0_{3,} 0_{4, }0_{5 }and 4 junctions type 1 attached to 4 components R_{T1}, R_{T3}, R_{TS} and R noted 1_{1}, 1_{2}, 1_{3}, 1_{4} [2]. These components are: C_{V} the pipe capacity traveled by fluid in vapor phase, C_{L }the pipe capacity of fluid in liquid phase; C_{T1} the tank_{1} capacity; C_{T3} the tank_{3} capacity; C_{TS} the tank capacity of a distillate; R_{T1}, R_{T3}, R_{TS} and R are respectively, the valves of tank_{1} and tank_{3}, the distillate and the reflux drum.
Variables 
Bond graph descriptions 
Physical descriptions 
E 
Effort 
Pressure P, Temperature T 
F 
Flow 
Masse flow dm/dt, Enthalpy flow dH/dt 
De 
Effort detector 
Measured effort 
Df 
Flow detector 
Measured flow 
Se, MSe 
Effort source 
Thermal or Hydraulic effort 
Sf, MSf 
Flow source 
Thermal or Hydraulic flow 
R 
Resistive component 
Valve 
C 
Capacitive component 
Tank 
ARR 
Analytical Redundancy Relation 
Residual (Failure indicators) 
S 
Derivative operator 
d/dt 
1/s 
Integral operator 
∫dt 
Φ 
Nonlinear components evolution 
Φ_{R}; Φ_{CV}; Φ_{CL}; Φ_{CT1}; Φ_{CT3}; Φ_{CTS}; Φ_{RT1}; Φ_{RT3}; Φ_{RTS} 
C_{V} 
Vapour (steam) capacity 
Pipe capacity travelled by fluid in vapour phase 
C_{L} 
Liquid capacity 
Pipe capacity travelled by fluid in liquid phase 
C_{T1} 
Tank1 capacity 
Level of fluid into tank1 
C_{T3} 
Tank3 capacity 
Level of fluid into tank3 
C_{TS} 
Capacity of distillate tank 
Distillate into reflux drum 
R_{T1} 
Tank1 resistor 
Valve of fluid into tank1 
R_{T3} 
Tank3 resistor 
Valve of fluid into tank3 
R_{TS} 
Resistor of distillate tank 
Distillate valve 
The bond graph model is described as follows:
Figure 2. Bond graph model of the condenser with sensors placement
Constitutive and structural equations
The calculation of the residuals needs application of the previous algorithm by generating flow f and effort e of all junctions. For example, the junctions 0_{De1} and 1_{Df1} are described by the equations (5) and (6).
_{} 
(5) 

(6) 
where Msf_{1} and Msf_{2} represent the controlled (modulated by signals) flow source of bond 1 and bond 2. From the previous equations we can deduce the system (7) which permits us to generate residuals:
_{} _{} _{} _{} _{} _{} _{} _{} _{} _{} 
(7) 
Optimization technique
The equations above allow according to binary variables x_{i} and y_{j}_{ }determining the final system structure to supervise. The optimal case is with how many sensors to supervise this model. There exist 2^{n} = 2^{9} = 512 combinations to obtain the optimal case.
From the combination with 9 sensors placed in the 9 positions, the ARRs are calculated until to obtain the optimal combination. The detection is defined by columns of the signature table different to zero. The isolation is guaranteed by the columns different each other.
In the second step, the substitution is used to reduce the number of ARRs and to verify the monitoring from the signature table. In our case, 6 ARRs satisfied that all the components are monitored. The generated ARRs are given by the following relations (the sensors are placed only if x_{i}_{ }and y_{j} = 1).
Substitution with 9 sensors
All the sensors are placed, so each component is monitored by one sensor and the system (7) becomes as follows for [x_{1} y_{1} x_{2} y_{2} x_{3} y_{3} x_{4} y_{4 }x_{5 }x_{6} x_{7}]^{T}^{ }= [11011111110]^{T}
_{} _{} _{} _{} _{} _{} _{} _{} 
(8) 

C_{V} 
C_{L} 
C_{T1} 
R_{T1} 
C_{T3} 
R_{T3} 
C_{TS} 
R_{TS} 
R 
ARR_{1} 
1 



_{ } 
_{ } 
_{ } 
_{ } 
_{ } 
ARR_{2} 
1 


1 
_{ } 
_{ } 
_{ } 
_{ } 
_{ } 
ARR_{3} 





1 



ARR_{4} 


1 






ARR_{5} 
1 






1 

ARR_{6} 




1 




ARR_{7} 








1 
ARR_{8} 






1 


ARR_{9} 
1 








In this case faults on all the components are detectable and localizable.
Substitution with 8 sensors
One of the possible combinations is chosen, so 8 ARRs are generated for monitoring 9 components. The combination used in the system (7) is [01101111110]^{T}.
_{} _{} _{} _{} _{} _{} _{} _{} 
(9) 

C_{V} 
C_{L} 
C_{T1} 
R_{T1} 
C_{T3} 
R_{T3} 
C_{TS} 
R_{TS} 
R 
ARR_{1} 



1 
_{ } 
_{ } 
_{ } 
_{ } 
_{ } 
ARR_{2} 
1 



_{ } 
_{ } 
_{ } 
_{ } 
_{ } 
ARR_{3} 


1 






ARR_{4} 







1 

ARR_{5} 




1 
1 



ARR_{6} 








1 
ARR_{7} 






1 


ARR_{8} 

1 



1 



In this case faults on all the components are detectable and isolable.
Substitution with 7 sensors
Not localizable case [01000111111]^{T}
_{} _{} _{} _{} _{} _{} _{} 
(10) 

C_{V} 
C_{L} 
C_{T1} 
R_{T1} 
C_{T3} 
R_{T3} 
C_{TS} 
R_{TS} 
R 
ARR_{1} 
1 

1 
1 
_{ } 
_{ } 
_{ } 
_{ } 
_{ } 
ARR_{2} 




1 
1 
_{ } 
_{ } 
_{ } 
ARR_{3} 








1 
ARR_{4} 






1 


ARR_{5} 






1 


ARR_{6} 

1 







ARR_{7} 
1 






1 

The fault on (C_{T1} and R_{T1}) and the fault on (C_{T3} and R_{T3}) are not localizable (the same column vectors).
Detectable and isolable case [00101111110]^{T}
_{} _{} _{} _{} _{} _{} _{} 
(11) 

C_{V} 
C_{L} 
C_{T1} 
R_{T1} 
C_{T3} 
R_{T3} 
C_{TS} 
R_{TS} 
R 
ARR_{1} 
1 


1 
_{ } 
_{ } 
_{ } 
_{ } 
_{ } 
ARR_{2} 


1 



_{ } 
_{ } 
_{ } 
ARR_{3} 







1 

ARR_{4} 




1 
1 



ARR_{5} 








1 
ARR_{6} 






1 


ARR_{7} 

1 



1 



Substitution with 6 sensors [01001111010]^{T}
_{} _{} _{} _{} _{} _{} 
(12) 

C_{V} 
C_{L} 
C_{T1} 
R_{T1} 
C_{T3} 
R_{T3} 
C_{TS} 
R_{TS} 
R 
ARR_{1} 
1 


1 
_{ } 
_{ } 
_{ } 
_{ } 
_{ } 
ARR_{2} 


1 



_{ } 
_{ } 
_{ } 
ARR_{3} 
1 





1 
1 

ARR_{4} 




1 
1 



ARR_{5} 






1 

1 
ARR_{6} 

1 



1 



All the combinations with 5 sensors placed cannot ensure the detection and the isolation of the failures that affect this system. The optimal case is to monitor 9 components by only 6 sensors.
The table (6) of the residuals structure permit to notice that the residuals structures are different and the faults signatures are different and not equal to zero, thus the components C_{V}, C_{L}, C_{T1}, R_{T1}, C_{T3}, R_{T3}, C_{TS}, R_{TS} and R are monitored.
Sensitivity of the detectors
Under normal operating, the residuals must be constantly equal to zero but it is not always the case because of the modelling errors and the unavailability of the technical specifications of the real system.
From SYMBOLS software (System Modelling Bond graph Language Simulation), we have implanted the bond graph model [5] associated to the block diagram of the ARR that contain affected components.
We create fault on R_{T1} component monitored by the detector Df_{1}_{ }(disturbance from the association of Dirac pulse and Unit echelon). Their values are not considered only their appearances in the relation are taking into account with evaluation term (1 for the presence and 0 for the absence).
In the first time, the fault is created between the instant t_{1}=2s and t_{2}=2.01s by the annulment of fluid flow provided by the source [9], [10].
In the second time, the block diagram of the ARR_{1}, ARR_{2 }and ARR_{3} expressions that contain Df_{1} detector (see Figure. 3) is elaborated. The three outputs are represented by three scopes.
The failure is injected on 1_{RT1} junction monitored by the detector Df_{1}_{ }(Figure. 4) and not sensitive to ARR_{4}, ARR_{5} and ARR_{6} (Figure. 5). It represents sealing in the valve and leakage in the tank.
Figure 3. Block diagram representing the ARRs function of Df_{1}
_{ }
Figure 4. Sensitivity of Df_{1} detector with failed operating of ARR_{1}, ARR_{2} and ARR_{3}
_{ }
Figure 5. Sensitivity of Df_{1 }detector with normal operating of ARR_{4}, ARR_{5} and ARR_{6}
The residuals ARR_{1}, ARR_{2}_{ }and ARR_{3} are sensitive to the failures, it is due to the fact that the Df_{1} detector appears strongly in these residuals, on the other hand residuals ARR_{4}, ARR_{5} and ARR_{6} are null because the detector does not appear in the relations of these residuals it is the normal operating.
The bond graph tool used for modelling and monitoring is very efficient because of the complexity of the phenomena that are produced in the steam condenser as hydraulic and thermal phenomena and the separation between steam and liquid phases.
The Analytical Redundancy Relations are deduced from graphical model of the steam condenser of the distillation column.
The structural junction equations and constitutive element laws for generating these relations are used as the failures indicators.
The algorithm used for researching the optimal case is very efficient.
The simulator of Symbols Software allows the validation and simulation of the model. The technique applied to steam condenser for components monitoring can be extended to detect and locate failures actuators and sensors.
Acknowledgements
Corresponding author would like to thank Prof. Mohammed Mostefai my PhD supervisor and Prof. Mabrouk Khemliche the Head of Automatic Laboratory of Setif –Algeria for performing the present work.
References
1. Raudensky M., Hnizdil M., Hwang J. Y., Lee S. H., Kim S. Y., Influence of the water temperature on the cooling intensity of mist nozzles in continuous casting, Journal of materiali in tehnologije / Materials and Technology, 2012, 3, p. 311315.
2. Latreche S., Khemliche M., Mostefai M., Modeling and simulation of the distillation column boiler, Asian Journal on Information Technology, 2006, 5(8), p. 823828.
3. Nan T., Jinjia W., Jiabin F., Experimental and numerical study on the thermal performance of a water/steam cavity receiver, Energies Journal, 2013, 6(3), p. 11981216.
4. Ould Bouamama B., Thoma J. U., Cassar J.P., Bond graph modeling of steam condensers, In: IEEE International conference on systems, man, and cybernetic Orlando (USA), 1997, p. 490494.
5. Cicile J.C., Distillation, Absorption, Généralités sur les colonnes de fractionnement, Technique de l’Ingénieur, Traité de Génie des Procédés, J 2 621, 1999.
6. Thoma J.U., Ould Bouamama B, Modelling and simulation in thermal and chemical engineering. Bond graph approach, Springer  Verlag, Berlin, Germany, 2000.
7. Medjaher K., Samantary A. K., Ould Bouamama B, Bond graph model of a vertical Utube steam condenser coupled with a heat exchanger, International Journal of Simulation Modelling Practice and Theory, 2009, 17(1), p. 228239.
8. Khemliche M., Ould Bouamama B., Haffaf H, Sensors placement for diagnosability on bond graph model, Sensors and Actuators Journal, Elsevier Aphysical, 2006, 4, p. 9298.
9. Kozak D., Ivandic Z., Kontajic, P, Determination of the critical pressure for a hotwater pipe with a corrosion defect, Materiali in tehnologije / Materials and Technology, 2010, 6, p. 385390.
10. Badoud A., Khemliche M., Latreche S, Modeling, simulation and monitoring of nuclear reactor using directed graph and bond graph, Proceedings of World Academy of Science, Engineering and Technology, 2009, 37, p. 199208.