Revisited the Mathematical Derivation Wall Thickness Measurement of Pipe for Radiography

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Elisabeth Bryant
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1 DI Inenaional Sympoium on Digial induial adiology and Compued Tomogaphy, June 5-7, 007, Lyon, Fance eviied he Mahemaical Deivaion all Thickne Meauemen o Pipe o adiogaphy Ab.. Hamzah and Sii M. M. Ami Non Deucive Teing(NDT) Goup, Induial Technology Diviion, Malayian Nuclea Agency, Bangi, Kajang, Selango, Malayia; Phone: , Fa: ; azak_hamzah@nucleamalayia.gov.my, madiha_ami@nucleamalayia.gov.my Abac all hickne meauemen o pipe i vey impoan o he ucual inegiy o he induial plan. Hoeve, he adiogaphy mehod ha an advanage becaue he abiliy o peneaing he inulaed pipe. Thi ill have economic benei o induy. Moeove, he ea o digial adiogaphy ha moe advanage becaue he peed o adiogaphic ok, le epoue ime and no chemical ued o ilm developmen. Eihe he convenional adiogaphy o digial adiology, he all hickne meauemen i uing he angenial adiogaphy echnique (TT). In cae, o a lage diamee, pipe (moe han inche) he deeminaion maimum peneaion all hickne mu be aken ino he conideaion. Thi pape i eviied he mahemaical deivaion o he deeminaion o all hickne meauemen baed on angenial adiogaphy echnique (TT). The mahemaical appoach ued in hi deivaion i he Pyhagoa heoem and geomeical pinciple. In ode o deive he maimum peneaion all hickne a imila appoach i ued. Keyod : all hickne meauemen, angenial adiogaphy, maimum peneaion all hickne. Inoducion The mo pominen advanage in adiogaphy a Non-Deucive Teing (NDT) echnique egadle he convenional o digial adiogaphy i he abiliy o peneae he inulaed pipe. A e kno induial adiogaphy i moe popula o deec deec in he elding and caing. Hoeve, a e yea back, all hickne meauemen ha becoming moe popula and i i vey cucial in he aemen o ucual inegiy o induial plan. Thee ae o mehod in aeing all hickne; he double all mehod and Tangenial adiogaphy Technique (TT). The double all mehod i an eablihed mehod ued by he adiogaphe. Fo angenial adiogaphy, a ceain geomeical eup o pipe inpecion i needed. In he pape, e ae eviied he mahemaical deivaion he all hickne meauemen o pipe uing adiogaphy mehod.. Tangenial adiogaphy Tangenial adiogaphy i a mehod employed o pipe inpecion o monio cooion and evaluae all hickne ihou emoving he inulao. In hi mehod, a ceain geomeical e-up o pipe inpecion i needed. In hi mehod, only pa o he adiogaph, hich lie belo he angenial locaion, ae inepeed. The middle pa o he image i ignoed []. In obaining he good image a he angenial locaion ha i he chod, he elecion o opimum enegy i vey impoan. Thi i due o he chod egmen, hich ha dieence in hickne ha vaie om he hinne o he maimum peneaed all hickne. The elecion o opimum enegy i cucial o peven bun-o eec a he

2 hinne poion o he ample and caeing o oo bigh image a he hicke egmen. Theeoe, he elecion o he iadiaion ouce ued mu be accoding o he maimum o he peneaed hickne.. all Thickne Deivaion eviied [] The main pupoe o he Tangenial adiogaphy echnique i o evaluae he all hickne. In deiving he all hickne equaion, he mahemaical appoach on Pyhagoa heoem and geomeical pinciple ae ued. The deeminaion o he all hickne uing angenial mehod i epeened in Figue. Y(0,y) Z(0,z) B (,h) (-) D(,) h I-9 (-) O(0,0) C H A(,0) T (-) Figue. The chemaic diagam o deeminaion o all hickne Fom Fig., AY i he angen o he cicle o adiu a poin B (,h). h i he lengh o he pependicula dopped om B o H on OA. Uing Pyhagoa heoem, alo h AB AC CB ( ) ( ) ( ) AB AH..() Aea o ABC h AC ( ) ( )h bu alo aea o ABC BC AB and olving o h,. Hence, h o h ( ) ( )

3 h [ ] ( ) ( ). () Theeoe () and () imply ( ) ( ) ( ) (3) Ne, ince he gadien o AB i he ame a ha o BY, y h 0 h h y. Thi implie 0 ( ) ( ) ( ) ( )..... (4) ( ) Fom Fig., AZ i he angen o he cicle o adiu - a poin D(,). i he lengh o he pependicula dopped om D o T on OA. By Pyhagoa heoem, AD AC CD ( ) ( ) and by Pyhagoa heoem, AD AT (5) bu alo aea o ACD CD AD The aea o ACD AC ( ) ( ) ( ) ( ) ( ). Thi implie ( ) ( ) ( ) and olving o, e obain o ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( )... (6)

4 Theeoe (5) and (6) implie [ ] o.... (7) The gadien o AD i he ame a ha o DZ, 0 z 0. Thi implie, z ) (. (8) No z y o ha and quaing boh ide o he equaion,. Le θ. No θ implie θ θ + and ubiuing o θ, +.(9)

5 . Maimum Peneaion all Thickne [3] Tangenial adiogaphy i a mehod ued o aeing he eidual all hickne o pipeline hee cooion o eoion i likely o occu. Hoeve, hee ei limi in adoping hi mehod epecially in deemining he ouce ued. In obaining he coec value o he eidual all hickne he calculaion on maimum peneaion all hickne, L ma mu ake ino conideaion, a hi i o enue he appopiae adiaion enegy being ued. A (D a /)- B D a / Figue. The chemaic diagam o deemine L ma. Le AB be he maimum peneaed all hickne L ma., i he ue all hickne and D a i he oue diamee. Uing Pyhagoa heoem, Da + D a Da 4 Da 4 + D a 0.(3) Solve (3) and yield D a..(4) Squae oo (4) on boh ide o he equaion. Thi yield, Since ABL ma and AB, hence, Da...(5) L ma.(6)

6 Subiue (6) in (5) o obain 3. Concluion L ma Da (7) I i impoan o induial adiogaphe o undeand he deivaion o he all hickne meauemen a lea hey kno he baic paamee and he limiaion o he echnique hen peoming he epeimen. Acknoledgemen The auho ould like o hank he IAEA o coodinaing and he inancial uppoing on hi eeach pojec. eeence. Validaion o Poocol o Cooion and Depoi Evaluaion in Lage Diamee Pipe by adiogaphy, epo o he nd CM o he CP, Ianbul Tukey, pp 3, Mach 004, IAEA.. S.M.M.Ami, A..Hamzah, H.A.Kaim and M.B. Zubi, Analyical Sudie O Tangenial adiogaphy Mehod eviied, peened a he Coneence on Advance in Theoeical Science (CATS 003) /Advanced Technology Conge (ATC 003) S.M.M.Ami, A..Hamzah, S.Sayui, A.Ama and S.ejab, Tangenial adiogaphy Technique(TT) Fo Pipe all Thickne Meauemen Uing adiaion Mahemaical Appoach, poceeding MINT &D Semina eeach and Developmen, Semina B, pp70-76, MINT, July 004.

8.5 Circles and Lengths of Segments

8.5 Circles and Lengths of Segments LenghofSegmen20052006.nb 1 8.5 Cicle and Lengh of Segmen In hi ecion we will how (and in ome cae pove) ha lengh of chod, ecan, and angen ae elaed in ome nal way. We will look a hee heoem ha ae hee elaionhip