Electrical Capacitance Tomography with a Square Sensor

Transcription

1 Electrcal Capactance Tomography wth a Square Sensor W Q Yang * Department of Electrcal Engneerng and Electroncs, Process Tomography Group, UMIST, P O Box 88, Manchester M60 QD, UK, emal w.yang@umst.ac.uk S Lu Insttute of Engneerng Thermophyscs, Chnese Academy of Scences, P O Box 2706, Bejng 00080, P R Chna, emal shlu@hotmal.com Abstract In the past, electrcal capactance tomography (ECT) wth crcular sensors has been extensvely nvestgated. For some ndustral applcatons such as crculatng fludsed beds, square sensors are requred. Research nto ths specfc area has been carred out at UMIST for the frst tme. To generate senstvty maps, the Laplace equaton s solved usng a fnte dfference method (FDM). Both the lnear back-projecton (LBP) and an LBP-based teratve algorthm have been mplemented for mage reconstructon. Expermental results are very promsng. Keywords: Square sensor, Fnte dfference method, Iteratve algorthm, Capactance tomography. INTRODUCTION Electrcal capactance tomography (ECT) has an ablty to present concentraton dstrbutons n two-phase delectrc processes and has been deployed n vsualsaton of flow patterns, e.g. n crculatng fludsed beds [, 2] and pneumatc conveyors [3, ]. Over the past few years, mage qualty and measurement accuracy of ECT have been mproved sgnfcantly. For example, n a recent expermental nvestgaton the dfference between the results obtaned from an ECT system and pressure measurements s below 3% under certan condtons [5]. In the past, all ECT systems have been desgned for use wth crcular sensors, normally havng 6, 8, 2 up to 6 electrodes. A large number of ndustral applcatons, however, nvolve square or rectangular geometry, such as most ndustral bolers and crculatng fludsed beds n thermal engneerng. Ths paper reports, for the frst tme, the feasblty of ECT wth a square sensor. 2. SENSOR STRUCTURE The sensor s depcted n Fg.. It conssts of 2 measurement electrodes, a plastc frame and a copper sheldng. The electrodes of 0 cm n length are mounted onto the outer surface of the plastc frame wth 2 mm spacng between neghbourng electrodes. The cross secton of the copper sheldng s 80 by 80 mm. The nsde dmensons of the plastc frame are 60 by 60 mm. The thckness of the plastc frame s 2 mm Fg. Square sensor copper sheldng measurement electrode plastc frame 3. GENERATION OF SENSITIVITY MAPS Senstvty dstrbutons of a sensor, whch are called senstvty maps, are essental for mage reconstructon. In the lnear back-projecton (LBP) algorthm, for example, an mage s obtaned by supermposng all senstvty maps together usng capactance measurements as weghtng factors. Although senstvty maps can be expermentally measured [6], n most cases they are generated from numercal solutons of the Laplace equaton. For electrcal felds, t s * W Q Yang s a member of the Vrtual Centre for Industral Process Tomography 33

2 2 2 φ φ + 0 () 2 2 x y where, φ s the potental dstrbuton. Because t s dffcult to fnd analytcal solutons of the equaton, fnte element methods (FEM) are often utlsed to solve ths problem [7]. Due to the feature of the square sensor, fnte dfference methods (FDM) seem more sutable than FEM for solvng the equaton [8, 9]. For the sensor shown n Fg., the sensng doman s dvded nto 60 by 60 square mesh grds and central dfferencng scheme s used. The procedures are summarsed as follows. One electrode s set a voltage as a source electrode and the remanng electrodes are kept at the earth potental. For a typcal nternal grd pont as shown n Fg.2, ts value s determned by (a) Electrode energsed φ o φ (2) φ2 φ3 φο φ (b) Electrode 2 energsed Fg.3 Typcal potental dstrbutons φ Fg.2 A typcal nternal grd pont An teratve approach s used to solve the potentals over the whole doman. For the 2- electrode sensor, 2 potental dstrbutons are obtaned. Fg.3 shows two typcal potental dstrbutons when electrode and electrode 2 are energsed, respectvely. The senstvty of electrode par -j at a spatal locaton (x, s calculated by dot-multplyng the two electrc felds. S j p( x, r r E x y E x y (, ) j (, ) dxdy V V j (3) where, E (x, s the electrc feld dstrbuton when electrode s the source electrode wth an extaton voltage V appled, whle other electrodes reman at the earth potental, and P(x, s the area of the pxel at (x,. Fg. llustrates some of the senstvty maps, showng the hgher senstvty between neghbourng electrods (e.g. and 2) and the symetrc feature between two opposte electrodes (e.g. 2 and 8).. IMAGE RECONSTRUCTION. Lnear back-projecton LBP s a smple mage reconstructon algorthm and has commonly been used n ECT wth crcular sensors. For the 2-electrode square sensor, t takes the same format as the crcular sensors. G 2 Cj Sj Sj j + 2 j + () where, G (x, s the grey level n a pxel (x,,.e. the reconstructed mage, and C j s the normalsed capactance measurement. 3

3 for mage reconstructon wth the square sensor. An ntal mage G k s obtaned usng the LBP algorthm as shown n equaton (). The estmated capactance vector D k s obtaned Gk C LBP + Qk + α S + T f - G k+ Dk S LFP Gk Fg.5 Iteratve loop (a) Between electrodes and 2 from the current mage by lnear forward projecton (LFP). Dk, j Gk Sj Sj x y x y (5) The error vector Q k s used to reconstruct an error mage usng the LBP algorthm. The error mage s added to the current mage and then constraned to get a modfed mage G k+. (b) Between electrodes and 0 ( x) x ( x < 0) (0 x ) f (6) ( x > ) The mathematcal expresson of the algorthm s as follows. G k + T [ G + S ( C SG )] f α (7) k 5. EXPERIMENTAL EVALUATION k (c) Between electrodes 2 and 8 Fg. Senstvty maps calculated usng the fnte dfference method.2 Iteratve algorthm To obtan hgh-qualty mages, teratve methods have to be used to overcome non-lnear problems caused by electrc feld dstorton. An LBP-based teratve algorthm has recently been developed at UMIST and has been used for ECT wth crcular sensors successfully. Its prncple s llustrated n Fg.5 [0]. Ths algorthm s chosen The two algorthms dscussed above, LBP and teratve, have been mplemented usng the MATLAB. Experments were carred out usng an ECT system to evaluate the square sensor performance and to nvestgate the effcency of teratve mage reconstructon. The sensor was calbrated by ar (ε r.0) and plastc pellets (ε r 2.0) as the lower and hgher permttvty materals. It was then tested wth plastc pellets and plastc rods (ε r 2.5). Fg.6 shows some mages wthout teratons: () horzontally half-flled wth plastc pellets, (2) dagonally half-flled wth plastc [pellets, and (3) a round plastc rod n the centre. Fg.7 shows another set of mages for three crcular rods placed n the magng area, one wthout teratons and another wth 0 teratons. Obvously the mage produced by LBP only shows a blurred area, but not three objects. In contrast, the mage produced by the teratve 35

4 algorthm shows three objects clearly. Ths ndcates sgnfcant mprovement by teratons. (a) Before teratons (a) Horzontally half-flled wth plastc pellets (b) After teratons (b) Dagonally half-flled wth plastc pellets (c) A round plastc rod n the centre Fg.6 Reconstructed mages wthout teraton Fg.7 Reconstructed mages wth three plastc rods 6. CONCLUSION AND DISCUSSION Some ndustral applcatons requre square sensors, such as crculatng fludsed beds. The senstvty dstrbutons of a square sensor can be calculated usng FDM whch s easer to be mplemented than FEM. Once senstvty maps have been obtaned, mages can be reconstructed usng MATLAB wth easy. The teratve algorthm based on LFP and LBP has demonstrated very good performance. Many expermental results show that the qualty of mage from the square sensor s better than that from a crcular sensor. As can be seen n Fg., the senstvty dstrbutons of the square sensor are more unform between two sensng electrodes, than those of a crcular sensor. It s beleved that the senstvty n the centre (the weakest area) of a square sensor s hgher than that of a crcular sensor, and hence a square sensor can provde better mage resoluton than a crcular sensor. For a square sensor to be used n a real-sze ndustral fludsed bed, t s necessary to cope wth a large scale, say 2 by 2 meters. A dffculty wth ths s that the current ECT system would not 36

5 be approprate for so long sheldng cables. Ths problem wll be addressed n the near future. ACKNOWLEDGEMENTS The authors would lke to thank Dr. M.Wang and Prof. H.McCann of UMIST, Mr. M.Byars of Process Tomography Ltd, Dr. D.M.Spnk of Unversty of Leeds and Prof. H.X.Wang of Tanjn Unversty (on vstng to UMIST) for ther help n ths work. REFERENCES [] S.Wang, T.Dyakowsk. and M.S.Beck, "Applcaton of electrcal capactance tomography to measure gas-sold moton n fludzed bed", n Heat-transfer, AICHE symposum seres, Houston, 996, pp [2] J.S.Halow, "Capactance magng of fludzed beds", n Process Tomography -- Prncples, Technques and Applcatons, edted by R.A.Wllams and M.S.Beck, Butterworth Henemann, 995, pp 7-86 [3] S.L.McKee, T.Dyakowsk, R.A.Wllams, T.A.Bell and T.Allen, "Solds flow magng and attrton studes n a pneumatc conveyor", Powder Technology, 995, 82, pp 05-3 [] S.P.Luke and R.A.Wllams, "Industral applcatons of electrcal tomography to solds conveyng", Measurement + Control, 997, 30, pp [5] S.Lu, S.Wang, D.Mason, T.Dyakowsk and D.Geldart, "Measurement of solds concentraton n gas-sold flows usng capactance tomography and pressure sensors", n Sensors and ther Applcatons VIII, Proc. of the 8th Conf. on Sensors and ther Applcatons, Glasgow, September 997, edted by A.T.Augoust and N.M.Whte, Insttute of Physcs Publshng, Brstol and Phladelpha, pp [6] W.Q.Yang, and W.F.Conway, "Measurement of senstvty dstrbutons of capactance tomography sensors", Rev. Sc. Instrum., 998, 69, pp [7] D.M.Spnk, "Drect fnte element soluton for the capactance, conductance or nductance, and force n lnear electrostatc and magnetostatc problems", COMPEL, 996, 5, pp 70-8 [8] J.D.Anderson, Computatonal flud dynamcs -- The bascs wth applcatons, McGraw-Hll Inc., Sngapore, 995 [9] C.H.Qu, Electrcal resstance tomography for magng sub-seabed sedment porosty, MSc Thess, UMIST, 995 [0] W.Q.Yang, D.M.Spnk, T.A.York and H.McCann, "An mage reconstructon algorthm based on Lanweber teraton method for capactance tomography", (submtted to IEEE Trans. Image Processng) 37