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Multiangular Absorption Measurements in a Methane Diffusion Jet ROBERT J. SANTORO and H . G. SEMERJIAN Thermal Processes Division, National Bureau of Standards, Washington, D.C. 20234 P. J. E M M E R M A N , R. GOULARD, and R. SHABAHANG School of Engineering and Applied Science, The George Washington University, Washington, D.C. 20052 The mathematical r e c o n s t r u c t i o n of a property field, F ( x , y ) , from its p r o j e c t i o n in the θ direction is the b a s i s o f "Computer i z e d Tomography" ( 1 , 2 ) . An identical technique can be used t o r e c o n s t r u c t a field of linear a b s o r p t i o n c o e f f i c i e n t functions i n a combusting flow field from m u l t i a n g u l a r path i n t e g r a t e d absorp tion measurements. The linear a b s o r p t i o n c o e f f i c i e n t is the f a m i l i a r N Q product, where N is the c o n c e n t r a t i o n of species i and Q is the a b s o r p t i o n cross s e c t i o n of species i at the frequency ν. The Bouguer-Lambert-Beer law s t a t e s that i
i
i
i
where we assume a s i n g l e monochromatic light source of frequency ν and that Ν is the only c o n s t i t u e n t in the measured domain w i t h absorption at this frequency. Taking the n a t u r a l l o g a r i t h m of both s i d e s y i e l d s i
If two-dimensional functions in the coordinate system defined i n Figure 1 are considered and letting
where the coordinate systems are r e l a t e d by
it can be shown that the F o u r i e r transforms of the f u n c t i o n taken in the two coordinate systems are r e l a t e d by
0-8412-0570-l/80/47-134-427$05.00/0 © 1980 American Chemical Society
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where r e p r e s e n t s a Fourier transform operation. Thus t h e two F o u r i e r t r a n s f o r m s a r e t h e same i f t h e t r a n s f o r m a x e s ( r , s ) a r e a r o t a t i o n o f (χ,5θ i n t h e f r e q u e n c y domain b y t h e a n g l e θ Λ
Laser Probes for Combustion Chemistry Downloaded from pubs.acs.org by NATL UNIV OF SINGAPORE on 05/05/18. For personal use only.
?(ω,θ) « FCa>,e)
(6)
where ω i s 2-n t i m e s t h e s p a t i a l f r e q u e n c y . This i s the central s l i c e theorem o f F o u r i e r t r a n s f o r m s , which s t a t e s t h a t t h e t r a n s form o f a p r o j e c t i o n taken a t angle θ i s equal t o a l i n e through t h e c e n t e r o f t h e t w o - d i m e n s i o n a l t r a n s f o r m domain o f t h e f u n c t i o n F w h i c h makes a n a n g l e θ w i t h t h e χ a x i s . T h e r e f o r e knowledge o f a l l p r o j e c t i o n s w o u l d d e f i n e t h e t r a n s f o r m o f t h e f u n c t i o n and b y t a k i n g t h e c o r r e s p o n d i n g i n v e r s e t r a n s f o r m , t h e f u n c t i o n can be e v a l u a t e d a t any p o i n t i n i t s domain. A p p l y i n g a c o n v o l u t i o n t h e o r e m (_3,^4) y i e l d s t h e f o l l o w i n g e q u a t i o n w \ Ιο Γ F ( x , y ) = ^-2 /
r°° C»/
π
de
a\
P(jw,6)e
ίω(χ c o s θ + y s i n θ) ι ι , |a>|du> J
(7)
The l i n e a r a b s o r p t i o n c o e f f i c i e n t f u n c t i o n s c a n t h u s be r e c o n structed using only p r o j e c t i o n data. Experiment A methane-argon asymmetrical d i f f u s i o n j e t has been analyzed f o r t h e methane c o n c e n t r a t i o n mapping i n a s t e a d y f l o w c o n d i t i o n . The j e t a p p a r a t u s c o n s i s t s o f a 12.7mm i . d . b r a s s t u b e l o c a t e d 19mm f r o m t h e c e n t e r l i n e o f a 15.25cm c i r c u l a r b r a s s p l a t e . The j e t i s s u p p l i e d f r o m a 10cm d i a m e t e r c y l i n d r i c a l chamber c o n t a i n i n g a g l a s s bead m i x i n g s e c t i o n . A combination o f f l o w s t r a i g h t e n e r s and s c r e e n s a r e c o n t a i n e d i n t h e j e t s e c t i o n t o provide a uniform e x i t flow p r o f i l e . The j e t / p l a t e c o m b i n a t i o n i s mounted o n a m i l l i n g b e d w h i c h a l l o w s f o r a c c u r a t e t h r e e dimensional p o s i t i o n i n g of the unit. A b s o r p t i o n b y methane o f t h e 3.39μ l i n e o f a HeNe l a s e r i s u s e d t o d e t e r m i n e t h e methane c o n c e n t r a t i o n a c r o s s t h e j e t . The o p t i c a l a r r a n g e m e n t u s e d i s shown i n F i g u r e 2. The HeNe l a s e r o u t p u t (^lrnw) i s a t t e n u a t e d w i t h a 1.0 n e u t r a l d e n s i t y f i l t e r s o as n o t t o e x c e e d t h e maximum i r r a d i a n c e s p e c i f i c a t i o n o f t h e d e t e c t o r . The beam i s i n t e r r u p t e d u s i n g a m e c h a n i c a l l i g h t c h o p p e r o p e r a t i n g a t 1015 H z . T r a n s m i t t e d r a d i a t i o n i s d e t e c t e d by a n u n c o o l e d PbSe d e t e c t o r . A p r e a m p l i f i e r w i t h a g a i n o f t e n i s u s e d t o impedance m a t c h t h e r e s u l t i n g s i g n a l t o an o s c i l l o s c o p e and l o c k - i n a n a l y z e r . The o u t p u t f r o m t h e l o c k - i n a n a l y z e r i s d i s p l a y e d on a d i g i t a l v o l t m e t e r . I n t h e p r e s e n t e x p e r i m e n t s a 1 0 % CH^ - 90% A r m i x t u r e was u s e d . A l l e x p e r i m e n t s w e r e done u n d e r c o n d i t i o n s o f atmos p h e r i c p r e s s u r e and room t e m p e r a t u r e . The f l o w r a t e was m e a s u r e d t o b e 0.57 l i t e r s / s . A s e r i e s o f a b s o r p t i o n measurements w e r e t a k e n a c r o s s t h e j e t a t i n t e r v a l s o f .64 mm (.025 i n c h e s ) a t a h e i g h t o f 12.7 mm above t h e j e t . The j e t was t h e n r o t a t e d 15° and t h e e x p e r i m e n t was r e p e a t e d w i t h t h e r e s u l t t h a t d a t a f o r
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1.0 N . D . FILTER
HE-NE LASER
λ = 3.39μ
PB-SE OPTICAL CHOPPER
DETECTOR JET & MILLING BED
PRE-AMP
LOCK-IN AMP DVM
OSCILLOSCOPE
Figure 2.
Schematic of apparatus for methane jet experiment
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t w e l v e s e p a r a t e a n g l e s was o b t a i n e d t o map o u t t h e j e t r e g i o n . S p a t i a l r e s o l u t i o n was l i m i t e d b y t h e 2 mm l a s e r beam diameter. A v e r a g e a b s o r p t i o n measurements w e r e o b t a i n e d b y u s i n g a 4.0 s t i m e c o n s t a n t o n t h e l o c k - i n d e t e c t o r . The a b s o r p t i o n d a t a o b t a i n e d was i n p u t t o a m i n i c o m p u t e r f o r a n a l y s i s , u s i n g a d i s c r e t e f o r m o f e q u a t i o n 7.
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Results An o b j e c t i v e o f t h i s s e r i e s o f e x p e r i m e n t s i s t o p r o v i d e a p r e l i m i n a r y assessment o f the tomographic r e c o n s t r u c t i o n approach for combustion d i a g n o s t i c s . I n order t o minimize experimental d i f f i c u l t i e s introduced by combustion, a simple flow c o n f i g u r a t i o n has been chosen f o r t h i s i n i t i a l study. I t i s mathem a t i c a l t r u i s m t h a t any b a n d l i m i t e d f u n c t i o n can be a c c u r a t e l y r e c o n s t r u c t e d f r o m i t s p r o j e c t i o n s i f b o t h t h e number a n d t h e s i g n a l t o n o i s e r a t i o o f these p r o j e c t i o n s approach i n f i n i t y . In any r e a l c o m b u s t i o n s i t u a t i o n , b o t h o f t h e s e c o n d i t i o n s w i l l b e severely limited. The p r e s e n t r e s u l t s p r o v i d e i n s i g h t s i n t o t h e measurement c a p a b i l i t i e s o f t h e t o m o g r a p h i c r e c o n s t r u c t i o n a p proach under such l i m i t a t i o n s . The r e c o n s t r u c t e d l i n e a r a b s o r p t i o n v a l u e s f o r a c r o s s s e c t i o n o f t h e j e t (12.7mm above t h e e x i t p l a n e ) f o r t h e c a s e o f t w e l v e a n g l e s i s shown i n F i g u r e 3. Q u a l i t a t i v e l y t h e r e c o n s t r u c t e d f i e l d shows t h e e x p e c t e d ^ j e t D r o f i l e . U s i n g an a b s o r p t i o n c o e f f i c i e n t o f 10 atm~ cm" e x t r a p o l a t e d f r o m t h e r e s u l t s o f McMahon e t . a l . ( 5 ) , t h e c e n t e r l i n e c o n c e n t r a t i o n o f methane i s f o u n d t o b e 9.6%. S i n c e t h e measurements were t a k e n c l o s e t o the j e t e x i t the p o t e n t i a l core o f the flow i s observed t o s u r v i v e t o t h e measurement p o i n t . Thus t h e methane c o n c e n t r a t i o n would be expected t o be the 10% i n i t i a l l y i n t r o d u c e d . T h e r e f o r e good agreement i s o b s e r v e d w i t h r e s p e c t t o c o n c e n t r a t i o n measurement. The p o s i t i o n o f t h e j e t c e n t e r l i n e was o b t a i n e d from t h e l o c a t i o n o f t h e peak v a l u e s o f t h e l i n e a r absorption coefficients. This approach y i e l d e d a p o s i t i o n o f 19±1.3 mm w h i c h compares q u i t e w e l l w i t h t h e known p o s i t i o n o f 19 mm. T h i s i s f u r t h e r c o n f i r m a t i o n o f t h e a c c u r a c y o f t h e reconstruction algorithm. The a n a l y s i s was r e p e a t e d u s i n g p r o j e c t i o n s f o r o n l y s i x a n g l e s a n d t h e r e s u l t s a r e shown i n F i g u r e 4. A s c a n b e s e e n t h e r e i s a marked i n c r e a s e i n t h e a b s o l u t e v a l u e s o f t h e l i n e a r a b s o r p t i o n c o e f f i c i e n t i n r e g i o n s away f r o m t h e j e t a s compared to t h e twelve angle case. S i n c e t h e r e i s no methane p r e s e n t i n these r e g i o n s , these values are a r e s u l t o f " r i n g i n g " generated by the a l g o r i t h m . These r e s u l t s a r e i n q u a l i t a t i v e agreement w i t h p r e v i o u s a n a l y t i c a l s t u d i e s u s i n g s i m u l a t e d absorption functions (6). Work i s i n p r o g r e s s t o o b t a i n r e s u l t s w i t h i m p r o v e d s p a t i a l r e s o l u t i o n a n d t o e x t e n d t h e r a n g e o f measurements t o much l o w e r concentrations. F u r t h e r w o r k w i l l a l s o c o n s i d e r more c o m p l i c a t e d f l o w geometries w i t h a l o n g e r range g o a l o f s t u d y i n g combustion
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Figure 4.
Reconstruction of methane jet cross section for six-angle case
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experiments. I n these experiments the goal i s t o achieve "instantaneous t i m e c o n c e n t r a t i o n p r o f i l e s i n two a n d t h r e e d i m e n s i o n s w h i l e u s i n g measurement a p e r a t u r e t i m e s a s s h o r t a s 50 y s e c .
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Literature Cited 1. Gordon, R., Herman, G.T., Johnson, S.A., Sci. Am. (1976), 233, 56. 2. Brooks, R.A., DiChiro, G., Phys. Med. Bio., (1976), 21, 689. 3. Ramachandran, R.N., Lakshmenarayanan, Proc. Natl. Acad. Sci. U.S., (1971), 68, 2236. 4. Shepp, L.A., Logan, B.F.. IEEE Trans. Nucl. Sci., (1974), 21, 21. 5. McMahon, J., Troup, G.J., Hubbert, G., Kyle, T.G., J. Quant. Spectrosc. Radiat. Transfer, (1972), 12, 797. 6. Goulard, R., Emmerman, P.J., 17th Aerospace Sciences Meeting, (1979), 79-0085. R E C E I V E D February
1, 1 9 8 0 .
