Isothermal bulk modulus of liquids

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Content ISOTHERMAL BULK MODULUS OP LIQUIDS A T h e s is P r e s e n t e d to The F a c u l t y o f t h e S c h o o l o f E n g in e e r i n g The U n i v e r s i t y o f S o u th e r n C a l i f o r n i a I n P a r t i a l F u l f i l l m e n t o f t h e R e q u ir e m e n ts f o r t h e D e g re e M a s te r o f S c ie n c e i n C h e m ic a l E n g in e e r i n g By C h a r l e s R o b e r t K oppany Ju n e 1965 UMI Number: EP41781 All rights reserved IN FO R M A TIO N TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. Dissertation Publishing UMI EP41781 Published by ProQuest LLC (2014). Copyright in the Dissertation held by the Author. Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 4 8 1 0 6 -1 3 4 6 £ b '<oS K ? 3 This dissertation, written by C h a rle s ...R o b e rt. Kop.psuay. under the guidance of hXs...Faculty Committee and approved by all its members, has been presented to and accepted by the School of Engineering in partial fulfillment of the re quirements for the degree of M a s te r o f S c ie n c e i n C h e m ic a l E n g in e e r i n g Date........ Faculty Committee Chairman ACKNOWLEDGMENT I w is h t o g r a t e f u l l y a ck n o w le d g e t h e f o l l o w i n g , p e o p le f o r t h e i r a s s i s t a n c e i n t h e p r e p a r a t i o n a n d c o m p le t i o n o f t h i s t h e s i s : To Dr# G# J . R e b e r t , my a d v i s e r o n t h i s p r o j e c t f o r h i s e n c o u ra g e m e n t a n d h e l p f u l s u g g e s t i o n s ; To M r. G eorge Ames f o r h i s a s s i s t a n c e o n t h e c o m p u te r p ro g ra m ; To D r . F . J . L o c k h a r t f o r h i s s u g g e s t i o n s and a d v i c e ; and F i n a l l y , t o M rs . R u th Toyama f o r h e r a s s i s t a n c e i n t h e f i n a l p r e p a r a t i o n o f t h i s r e p o r t . ii TABLE OP CONTENTS Pag© INTRODUCTION . . ....... . ................. . . . .............................. 1 , DEVELOPMENT OP PROCEDURE............................ k R educed D e n s i t y C h a r t s . . . . . . . . . . ............... E q u a tio n o f S t a t e 7 W atso n E x p a n s io n F a c t o r M ethod • • . . . . . ...................... 10 S p e e d o f Sound 11 EXPERIMENTAL PVT DATA AND TYPICAL PLOTS USED IN THE CALCULATIONS .................. 13 E x p e r i m e n t a l PVT D a ta f o r n - P e n t a n e ............ 13 T y p ic a l P l o t s U se d i n t h e C a l c u l a t i o n s . . . . . . . . 13 RESULTS ............ 17 DISCUSSION OP RESULTS ....................................................... 28 CONCLUSIONS .............................. 31 BIBLIOGRAPHY ....................... 33 APPENDIX A D a ta and Sam ple C a l c u l a t i o n s ........................... 35 1 . R ed u ced D e n s i t i e s o f L i q u i d s T aken Prom t h e T a b le s o f L y d e r s e n , G re e n h o rn and H ougen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2 . Sam ple C a l c u l a t i o n o f S lo p e s B ased on R ed u ced D e n s i t y C h a r t s f o r Tr = 0 .6 6 .............. 38 3 . E x p e r i m e n t a l P-V D a ta T aken Prom t h e L i t e r a t u r e f o r n - P e n t a n e (7 ) ............ ij.6 ij.. Sam ple C a l c u l a t i o n o f B u lk M odulus f o r n - P e n ta n e U s in g P-V D a ta f o r n - P e n ta n e a t Tp = 0 .6 6 ................................................................................... k l iii I r APPENDIX A ( c o n ’ t . ) Sam ple C a l c u l a t i o n o f B u lk M odulus f o r n - P e n ta n e U s in g B u e h l e r 1s E q u a t io n o f •S t a t e APPENDIX B D a ta and S a m p l e - C a l c u l a t i o n s . ..................... 1* W atso n E x p a n s io n F a c t o r M ethod . . . . . . . . . . . 2 . C o m p a riso n o f B u lk M o d u lu s -V a lu e s . . . . . . . . APPENDIX C M i s c e ll a n e o u s I n f o r m a t i o n ............. 1 . N u m e ric a l D i f f e r e n t i a t i o n U s in g U n e q u a lly - S p a c e d F x v o t a l P o i n t s 2 . R e s u l t s and C o m p a riso n w i t h S lo p e s C a lc u l a t e d b y t h e D o u g la s -A v a k ia n M ethod o f N u m e ric a l D i f f e r e n t i a t i o n • • • • • • • ..................... P age 5 k 55 56 60 61 6 2 6 k LIST OP TABLES T a b le H o. .P age I . S lo p e V a lu e s f o r L i q u i d s H a v in g z G 0 .2 7 0 . 18 I I . C o m p a riso n o f B u lk M odulus V a lu e s f o r n - P e n ta n e (P c = 90 p s i a . , z c = 0 .2 6 9 ) . . . . . . 20 I I I * C o m p a riso n o f B u lk M odulus V a lu e s f o r n - P e n t a n e ............................ 23 IV . B u lk M odulus V a lu e s f o r a Few D i f f e r e n t L i q u i d s 2? V. • E q u a l I n c r e m e n te d D a ta T ak en fro m F i g u r e Jlj.0 V I. E q u a l I n c r e m e n te d D a ta T a k e n .fro m F i g u r e 6 . ij.9 v LIST OF FIGURES F i g u r e Mo* P ag e 1 . R e d u ce d D e n s i t y a s a F u n c t i o n o f R ed uced P r e s s u r e f o r L i q u i d s w i t h Zc = 0.27© (Tr = 0 .9 5 ) ............... 1 * } - 2* P r e s s u r e a s a F u n c t i o n o f S p e c i f i c Volume f o r n - P e n t a n e (Z c = 0 .2 6 9 ) - Tr = 0 .8 8 . . . . 3$ 3 . S lo p e s f o r C la s s o f L i q u i d s H av in g Zc = Q .270 ........................................................... . . . 19 Ij.. C o m p a riso n o f B u lk M odulus V a lu e s f o r n - P e n t a n e a t Tp = 0 .8 8 .............. 26 R ed u ced D e n s i t y a s a F u n c t i o n o f R ed u ced P r e s s u r e f o r L iq u i d s w i t h Zc = 0 .2 7 0 (Tr = 0.66) .......... * . . . .................................... 39 6 . P r e s s u r e a s a F u n c t i o n o f S p e c i f i c Volume f o r n - P e n t a n e (Z q = 0 .2 6 9 ) - Tp = 0 .6 6 . . . . ij.8 7 . W atso n E x p a n s io n F a c t o r a s a F u n c t io n o f R ed u ced P r e s s u r e and R e d u ce d T e m p e ra tu re . . 57 8 . R ed u ced P r e s s u r e a s a F u n c t i o n o f t h e W atso n E x p a n s io n F a c t o r 59 vi : INTRODUCTION ! A l l f l u i d s a r e c o m p r e s s ib le u n d e r th e a p p l i c a t i o n o f p r e s s u r e . The d e g re e o f c o m p r e s s i b i l i t y o f a f l u i d i s m ea s u r e d b y i t s b u l k m o d u lu s . The e f f e c t o f f l u i d c o m p r e s s i b i l i t y m u st be t a k e n i n t o c o n s i d e r a t i o n i n t h o s e p ro b le m s o f f l u i d m o tio n i n w h ic h l a r g e c h a n g e s i n p r e s s u r e o c c u r . B u lk m od ulus c a n be d e f i n e d a s t h e r a t i o o f p r e s s u r e ( s t r e s s ) t o v o l u m e t r i c s t r a i n . When a s p e c i f i c v o lu m e , v , o f f l u i d u n d e r a p r e s s u r e , P , i s s u b j e c t e d t o a n i n c r e a s e i n p r e s s u r e , A P , th e s p e c i f i c volum e d e c r e a s e s b y an a m o u n t, A v . The b u l k m odu lus i s d e f i n e d b y t h e r a t i o B = l im ( A P / A v /v ) = **dP/dv/v Av-*-0 j l f we f u r t h e r r e q u i r e t h e te m p e r a t u r e be k e p t c o n s t a n t , th e r e s u l t i n g q u a n t i t y i s c a l l e d th e i s o t h e r m a l b u l k modu l u s and i s d e n o te d b y : Bt = - v (9 P /2 V ) t Bu l k m odu lus i s e x p r e s s e d i n t h e same u n i t s a s p r e s s u r e j ( p s i a . , a t m . , e t c . ) . Upon I n v e s t i g a t i o n i t h a s b e e n fo u n d t h a t s u c h r e f e r e n c e s a s C h e m ic a l A b s t r a c t s , I n d u s t r i a l an d E n g in e e r i n g C h e m is tr y , and th e I n t e r n a t i o n a l C r i t i c a l T a b le s c o n t a i n v e r y l i t t l e d a t a on i s o t h e r m a l b u l k m o d u lu s o f l i q u i d s . 1 | 2 j B e c a u se o f th e l a c k o f s p e c i f i c d a t a on b u l k m odulus an d i j jth e s m a ll am ount o f PVT d a t a on l i q u i d s , i t h a s b e e n p r o p o s e d to i n v e s t i g a t e th e f e a s i b i l i t y o f b u l k m odulus p r e d i e - I t i o n fro m e m p i r i c a l c o r r e l a t i o n s o f l i q u i d PVT b e h a v i o r . P r e d i c t i n g i s o t h e r m a l b u l k m o d u lu s w ith a c c u r a c y , e v e n fro m e x p e r i m e n t a l PVT d a t a , i s d i f f i c u l t s i n c e f u n d a m e n t a l l y th e c a l c u l a t i o n s r e q u i r e t a k i n g d i f f e r e n c e s o f d a ta p o i n t s . C o n s e q u e n tly , much o f th e a c c u r a c y o f t h e o r i g i n a l d a t a i s l o s t . S e v e r a l m eth o d s f o r p r e d i c t i n g b u l k m odulus a r e c o n s i d e r e d a n d t h e r e s u l t i n g v a l u e s com pared w i t h c o r - ire s p o n d in g v a l u e s c a l c u l a t e d fro m e x p e r i m e n t a l PVT d a t a j |fo u n d i n th e l i t e r a t u r e ( 7 )* I Many f l u i d s su c h a s t h e l i g h t h y d r o c a r b o n s have a j c r i t i c a l c o m p r e s s i b i l i t y o f a b o u t 0 .2 7 0 . The i n v e s t i g a t i o n s w i l l d e a l w i t h t h i s c l a s s o f f l u i d s . One a p p r o a c h to t h e p ro b le m i n v o l v e s t a b l e s o f l i q u i d r e d u c e d d e n s i t i e s p r e s e n t e d by L y d e r s e n , G r e e n k o r n , and Hougen ( I4 -) i n G e n e r a l i z e d T herm odynam ic P r o p e r t i e s o f P u re F l u i d s . T h ese r e d u c e d d e n s i t i e s a r e p r e s e n t e d a s a j j f u n c t i o n o f r e d u c e d t e m p e r a t u r e a n d - p r e s s u r e f o r l i q u i d s w i t h c r i t i c a l c o m p r e s s i b i l i t y f a c t o r s o f 0 . 25> 0 , 0 . 2 ? 0 , and i IO.2 9 O. T h e se t a b l e s c o v e r a ra n g e o f r e d u c e d p r e s s u r e s ! j ifrom 1 t o 30 and re d u c e d t e m p e r a t u r e s fro m 0 .3 0 t o 1 . 0 0 . H i r s c h f e l d e r , B u e h le r , McGee, an d S u t t o n ( 2 ) p r e s e n t a g e n e r a l i z e d e q u a t i o n o f s t a t e f o r th e l i q u i d p h a s e I S (re d u c e d d e n s i t y ^ r£.^» r e d u c e d t e m p e r a t u r e Tr j< l)» A m odi f i c a t i o n o f t h i s e q u a t i o n i s r e a d i l y a d a p t a b l e t o t h e -c a l c u l a t i o n o f i s o t h e r m a l b u l k m o d u lu s . T h is te c h n iq u e o f f e r s a n o t h e r a p p ro a c h to t h e p ro b le m . A n o th e r t e c h n i q u e s u r v e y e d was t h e W atson E x p a n s io n F a c t o r M ethod (5>). T h is t e c h n iq u e i s l i m i t e d to a s m a ll r a n g e o f p r e s s u r e ; t h e r e f o r e , th e r e s u l t s a r e p r e s e n t e d i n A p p e n d ix B , s i n c e t h e t e c h n iq u e was c o n s i d e r e d t o be o f m in o r im p o r ta n c e to t h i s r e p o r t . The W atso n E x p a n s io n F a c t o r h a s b e e n r e v i s e d a n d e x p a n d e d t o become th e L y d e r - - j s e n , G r e e n k o rn , and Hougen r e d u c e d d e n s i t y t a b l e s , i i i i DEVELOPMENT OP PROCEDURE R ed u ced D e n s i t y C h a r ts I n o r d e r to u t i l i z e th e r e d u c e d d e n s i t y c h a r t s , th e e x p r e s s i o n - v ( 3 P / ^ v ) T m u st he o b t a i n e d i n te rm s o f th e r e d u c e d p r e s s u r e , Pr , r e d u c e d t e m p e r a t u r e , T p , and r e d u c e d d e n s i t y , ^ p . r = i / v Bn Bt = 9 P 9 P f 9 ( V f ) p -2 C - L C f 2 J T I P _ 9 P V P e. T 1 ( l n ^ ) ( 1 ) T T H ow ever, Pr Pc , ( = f v ( c > a n d T = Tr T 0 w here Pc = th e c r i t i c a l p r e s s u r e , ^ c = t h e c r i t i c a l d e n s i t y , and Tc = th e c r i t i c a l t e m p e r a t u r e . F i n a l l y B r ~ Pc 9 Pj 9 ( l n ^ r ) T, ( 2 ) A p l o t o f Pr a g a i n s t l n ^ r a t f i x e d Tr a ll o w s one t o e v a l u a t e th e s l o p e 'd P p / ^ ln ^ p a t v a r i o u s v a l u e s o f Pr . The te c h n iq u e u s e d f o r e v a l u a t i n g t h e s e s l o p e s was th e D o u g la s -A v a k ia n m ethod o f n u m e r ic a l d i f f e r e n t i a t i o n ( 9 ) • T h is m ethod em plo ys a f o u r t h - d e g r e e p o ly n o m ia l, k 5 ' w h ic h i s f i t t e d t o s e v e n p o i n t s b y th e m ethod o f l e a s t s q u a r e s . T h is g e n e r a l .p o l y n o m i a l i s y = a + b x + e x 2 + ! dx3 + e x ^ . The s l o p e o f th e p o ly n o m ia l a t th e c e n t r a l I p o i n t (x = 0 ) i s b . The v a l u e s o f t h e in d e p e n d e n t v a r i a b l e a r e s p a c e d a t e q u a l i n t e r v a l s , h , w i t h c o - o r d i n a t e s c h o se n s o t h a t x = 0 f o r t h e c e n t r a l p o i n t o f t h e s e v e n . The s e v e n v a l u e s o f x , t h e n , a r e **3h, - 2 h , - h , 0 , h , 2 h , a n d 3 h . By th e p r i n c i p l e o f l e a s t s q u a r e s : £ R 2 = £ ( a - t b x + cx 2 + dx3 + ex^J- - y )2 ( 3 ) = minimum I l f we d i f f e r e n t i a t e E q u a t io n 3 w i t h r e s p e c t t o e a c h c o n s t a n t , and e q u a t e to z e r o , th e c o n d i t i o n s a r e o b t a i n e d f o r w h ic h ]Tl?2 s h o u ld be a m inim um . T h ese a r e : 7a + b j x + c £ x 2 + d$x3 4 eJFx^- -T y = 0 | a £ x + b £ x 2 + c £ x 3 4 djx^l* 4* e £ x ^ - ^ x y = 0 1 ' i aXx2 + b jx 3 4* cXbJ+- 4- djjc^ + e ^ x ^ - £ x 2y = 0 j a £ x 3 + b£x^4- 4 c j x 5 4 d £ x 6 4- e £ x 7 - £ x 3 y = 0 a ^ x ^ + b$x£ 4- c $ x 6 4- d £ s7 4- e ^ x ^ - £ ? ^ y = 0 L e t k r e p r e s e n t t h e c o e f f i c i e n t o f h i n th e v a l u e s o f x . - 'T hen: £ x = - 3 h - 2 h - h 4- 0 4- h 4- 2 h + 3 h = b£k - 0 = £ x 3 = £ x £ = £ x 7 i S i m i l a r l y : £ x 2 = h 2ILk2 = 28h2 ; = 196h/+ j | = l , £ 88h 6 j £ x 8 = 1 3 >636h8 | i £ 'x y = h l k y j S > 2y = h 2£ k 2y j j £ > 3 y = h 3 £ k 3 y ; Z x ^ 7 s h % k ^ r | ) , I ' I j The f i v e c o n d i t i o n s may now be w r i t t e n : j ! 7a + 2 8 h 2 e + 1 9 6iA e - 51y = 0 j ! 2 8 h 2b + 196h^d - hXky = 0 j 2 8 h2 a + 1 9 6lA c + l , £ 88h8 e - h 2£ k 2y = 0 ! I 9 6 h^b + l , £ 88h 6 d - h3£ k 3y = 0 ! ! 19611^-a + l , £ 88h 6 c + 1 3 ,6 3 6 h 8e - h^X k^r = 0 ! ! . - 1 I A v a k ia n (9 ) g i v e s t h e s o l u t i o n f o r th e c o e f f i c i e n t b a s i j d y /d x = b = ( 3 9 7 £ k y ) / l 5 l 2 h - (7 H k 3 y ) / 2 l 6 h (I4 .) j i w here k = -3» ^ 2 , - 1 , 0 , 1 , 2 , and 3> r e s p e c t i v e l y . ' j | E q u a t io n if c a n be u s e d p r o v id e d t h a t v a l u e s o f th e f u n c - I . 1 > ! t i o n a r e c h o s e n a t t h r e e p o i n t s e q u a l l y s p a c e d on e i t h e r 1 1 j s i d e o f th e p o i n t w h ere th e d e r i v a t i v e i s d e s i r e d . The j m eth o d i s e x a c t p r o v id e d t h e f u n c t i o n c a n b e r e p r e s e n t e d i j b y a f o u r t h - d e g r e e o r s i m p l e r p o ly n o m ia l a n d g iv e s good j j r e s u l t s i n o t h e r c a s e s i f th e f u n c t i o n i s n o t to o c o m p li- ; | e a t e d . j j A c o m p u te r p ro g ram was w r i t t e n u t i l i z i n g t h i s n u - , 1 ' ■ m e r i c a l t e c h n i q u e , y = Pr was t a k e n a s th e d e p e n d e n t 1 j v a r i a b l e and x = l n ^ r a s t h e i n d e p e n d e n t v a r i a b l e . A p p en - J j d i x A c o n t a i n s th e d e t a i l s on th e c a l c u l a t i o n p r o c e d u r e . ; 7 h , th e in c r e m e n t i n th e in d e p e n d e n t v a r i a b l e , w as c h o s e n a s a v e r y s m a ll num ber s u c h a s 0 .0 0 0 5 so t h a t a maximum o f | j a c c u r a c y c o u ld be o b t a i n e d . | | | | E q u a t io n o f S t a t e j I ! j H i r s c h f e l d e r a n d c o -w o r k e rs ( 2 ) p r e s e n t a g e n e r a l - | i z e d e q u a t i o n o f s t a t e f o r t h e l i q u i d r a n g e . T h is e q u a - j ! t i o n i s w r i t t e n a s f o l l o w s : , pI I I = ' W 2 ( £ + 1 + ? ( r * s ( f r - 1 )S/ f ’r + D)Tr | - P „ + P V ' j i w h e re : Pj j j = l i q u i d r e d u c e d p r e s s u r e (T r < l , ^ r - ^ ) ■ PTT = r e d u c e d p r e s s u r e o f a h i g h d e n s i t y g a s a t ! th e d e n s i t y e q u a l t o t h e d e n s i t y o f t h e j s a t u r a t e d l i q u i d ; J = r e d u c e d d e n s i t y , Tr = r e d u c e d t e m p e r a t u r e ^ = a c o n s t a n t f o r a p a r t i c u l a r l i q u i d j s = - 8 . 1 ^- + 1+.50 ^ - 0 . 3 6 3 ^ , k Q = a c o n s t a n t ; p v = r e d u c e d v a p o r p r e s s u r e a t Tr w2 = f ( l - kD -< * + 2 ^ )(1 - T ;2 ) ( 6)a | wx = k o T ;1 + ( B - k0 )T"2 (6 )b ! &C = l i e d e l ' s P a r a m e te r = th e s l o p e o f t h e v a p o r p r e s s u r e c u rv e a t t h e c r i t i c a l p o i n t D = ( ^ r - l P C T r - l J ^ r ^ h o T p 1 + fc^) + h ^ 1* h ^ ) (7 ) h Q = 8 8 .5 - 3 .1 2 f ( 8 ) h x = - i + ( 9 ) 8 h 2 = 8 + lj..06 ^ h 3 = 2 3 .7 - 3 -2 6 ^ (10) ( 11) E q u a t io n s 8 th ro u g h , 11 a r e d e r i v e d m a in ly fro m d a t a t a k e n f o r n i t r o g e n and ammonia i n t h e r e g i o n o f a h i g h d e n s i t y g a s . R i e d e l ' s p a r a m e t e r i s r e l a t e d to t h e v a p o r p r e s s u r e b y t h e e x p r e s s i o n s F o r l i q u i d s w i t h z 0 = 0 .2 7 0 , H i r s e h f e i d e r s u g g e s t s t h a t ^ = 7 and k D = 5 . 5 0 . L y d e r s e n and G re e n k o rn s u g g e s t k Q = 1|_.71. The e q u a t i o n o f s t a t e i s e n t i r e l y g e n e r a l a n d i s d e r i v e d o n t h e b a s i s o f a m o d if i e d p r i n c i p l e o f c o r r e s p o n d i n g s t a t e s . E q u a t io n 5 i s q u i t e u s e f u l f o r h y d r o c a r b o n s and h i g h l y p o l a r s u b s t a n c e s s u c h a s w a t e r and i s b e l i e v e d a d e q u a te f o r p r a c t i c a l l y a l l p u r e l i q u i d s * The e q u a t i o n h a s b e e n fo u n d v a l i d f o r d e n s i t i e s u p t o f o u r tim e s t h e c r i t i c a l d e n s i t y and p r e s s u r e s up t o 190 tim e s t h e c r i t i c a l p r e s s u r e ; h o w e v e r, t h e r e d u c e d t e m p e r a t u r e r a n g e i s r e s t r i c t e d b e tw e e n 0*$0 a n d 0 .9 5 0 . B e in g q u i t e c o m p l i c a te d , t h e g e n e r a l e x p r e s s i o n c a n n o t b e s o l v e d e x p l i c i t l y f o r t h e r e d u c e d d e n s i t y . F o r t h e n o rm a l l i q u i d r e g i o n , t h e a u t h o r s recom m end t h e B u e h le r E q u a t io n : I n pv = o C ln Tr + O.O8 3 8 C pC - 3 . 7 5 ) ( 3 6 t ” 1 - 35 - t | + l| .2 1 n Tr ) (1 2 ) (13) | The te m p e r a t u r e f u n c t i o n s p 0 , ^ o » a nd M w ere ta k e n to be | l i n e a r and f i t t e d t o th e g e n e r a l i z e d t a b l e s o f L y d e r s e n , I G r e e n k o rn , a n d Hougen ( I f ) , As a r e s u l t , th e f o l l o w i n g e x - • ! p r e s s i o n s w ere o b t a i n e d : ) | p 0 = “ 20 + l 5Tr (Ilf) i i ^ o = 8 . 2 8lf - I 8 . 0 7 z c -(lf.lf8 2 - l l f . l z c )Tr (1 5 ) 1 M = - 3 . 1 + 10Tr • • (1 6 ) | ! i [ E q u a tio n 13 h a s b e e n fo u n d s u i t a b l e i n t h e r a n g e Tr = 0 .5 0 j j \ I t o 0 . 9 ^ 0 . I t s m ain a d v a n ta g e i s t h a t i t c a n be s o lv e d e x - [ I ! :p l i c i t l y f o r e i t h e r p r e s s u r e o r d e n s i t y . U s in g e q u a t i o n ! I j1 3 , i s o t h e r m a l b u l k m o d u lu s i s d e r i v e d a s f o l l o w s : ■ p = . . . ____ a. n o r C * -M - r f a - f r p o P r “ Po Brp = Pq “ ^ P r 2 P r I 1 _____ 1 Tr Pc c 1 C , J f ' a P p M 1 <0 * ■ $ 1 Tp 1 0 I I P V )Z C o m b in in g e q u a t i o n s 1 7 , 1 8 , and 1 9 : B «p .= 3 Po M c ( P r - Po) C ° (1 7 ) ( 1 8 ) (1 9 ) ( 2 0 ) i i K now ing th e c r i t i c a l p r o p e r t i e s o f a g iv e n l i q u i d , one c an i , com pute th e i s o t h e r m a l b u l k m od ulu s a t a g iv e n 10 i t e m p e r a t u r e , p r e s s u r e , and d e n s i t y . The r e d u c e d d e n s i t i e s g iv e n by L y d e r s e n , G r e e n k o rn , and Hougen w ere u s e d i n t h e c a l c u l a t i o n s . W atso n E x p a n s io n F a c t o r M ethod W atso n (5>) r e l a t e s t h e m o la l volum e o f a l i q u i d t o an e m p i r i c a l e x p a n s io n f a c t o r w, a s f o l l o w s : V]> w1 = = Vw (2 1 ) w i s r e l a t e d g r a p h i c a l l y to th e r e d u c e d te m p e r a t u r e and r e d u c e d p r e s s u r e . T h is p l o t i s i l l u s t r a t e d a s F i g u r e ? i n A p p e n d ix B . A t th e c r i t i c a l p o i n t , w = 0 .0 i|4 » so t h a t th e m o la l volum e V a t a n y t e m p e r a t u r e and p r e s s u r e may be a p p r o x im a te d a s 0 .0i{i|.Vc/w* We w ould l i k e to o b t a i n a n e x p r e s s i o n f o r i s o t h e r m a l b u l k m o d u lu s i n term s o f t h e e x p a n s io n f a c t o r . The d e r i v a t i o n i s a s f o l l o w s : v = o .o i|4 v 0 „ - o .o l A v 0 dw w w2 bt = - y » p / ^ ) 5 - ( 2 3 , S i m p l i f y i n g e q u a t i o n 23 we g e t : Bip = wC^p/^wJrp = P c w (^P r /^ w ) ^ ( 2I4.) A p p e n d ix B i l l u s t r a t e s t h e u s e o f e q u a t i o n 2ij.. F i g u r e 7 g i v e s w v a l u e s f o r a r a n g e o f r e d u c e d p r e s s u r e s , 0.i|.<Pr < 5j 1 1 H ow ever, we a r e i n t e r e s t e d i n c a l c u l a t i n g b u l k m o dulus v a l u e s i n th e r a n g e , . l<E»g,<30 • F o r t h i s r e a s o n , t h e t e c h n iq u e was c o n s i d e r e d i n a d e q u a t e . S p e e d o f Sound I t i s p o s s i b l e t o d e v e lo p a r e l a t i o n s h i p b e tw e e n t h e s p e e d o f so u n d i n a f l u i d medium and th e b u l k m o d u lu s . The a d i a b a t i c b u l k m odulus i s d e f i n e d a s : Bs = w v 0 2 P /9 v )s s = e n t r o p y (2 £ ) Bs i s r e l a t e d t o B < j> b y th e f a c t o r Cp/Cv , th e h e a t c a p a c i t y r a t i o : Bs = (C p/C V)BT (2 6 ) c , th e s p e e d o f so u n d i n a f l u i d m edium , i s d e f i n e d i n many t e x t s on th e rm o d y n a m ic s a s : c = ^ / ( ^ P / ^ ) 3 ^ = f l u i d d e n s i t y (2 ? ) Bs = + ( 9 P / ^ ) a (2 8 ) T h e r e f o r e , " C Bs - o2 P ( 2 9 ) o r B* (Cp/Cv )Bt (3 0 ) ( ' E q u a t io n 30 th u s g i v e s u s a r e l a t i o n b e tw e e n th e i s o t h e r m a l b u l k m o d u lu s and t h e s p e e d o f s o u n d . S in c e d a t a on th e s p e e d o f so u n d and t h e r a t i o o f h e a t c a p a c i t i e s f o r l i q u i d s w ith a c r i t i c a l c o m p r e s s i b i l i t y f a c t o r o f 0 .2 7 0 i s n o t ! 12 I j r e a d i l y a v a i l a b l e , no c a l c u l a t i o n s w ere p e rfo r m e d u s i n g e q u a t i o n 3 0 . T h is s e c t i o n was a d d e d to t h e r e p o r t j u s t a s a m a t t e r o f i n t e r e s t * ! EXPERIMENTAL PVT DATA AND TYPICAL PLOTS i USED IN THE CALCULATIONS E x p e r im e n ta l PVT D a ta f o r n - P e n ta n e PVT d a t a f o r n - p e n t a n e ( c r i t i c a l c o m p r e s s i b i l i t y f a c t o r z 0 = 0 . 2 6 9 } was fo u n d i n th e l i t e r a t u r e (7 ) f o r a |r a n g e o f r e d u c e d t e m p e r a t u r e s fro m 0 .6 6 t o 0 .9 5 an d r e d u c e d | i p r e s s u r e s fro m 1 t o 2 0 . B u lk m o d u lu s v a l u e s f o r n - p e n t a n e w ere c a l c u l a t e d b y u s e o f th e D o u g la s - A v a k ia n m ethod and com pared w ith t h e c o r r e s p o n d i n g v a l u e s c a l c u l a t e d fro m t h e r e d u c e d d e n s i t y c h a r t s a n d t h e e q u a t i o n o f s t a t e . D e t a i l s |o n th e s e c a l c u l a t i o n s a r e fo u n d i n A p p e n d ix A . A few com- j p a r i s o n s w e re a l s o made a g a i n s t t h e W atso n E x p a n s io n F a c t o r i j M e th o d . T h e s e r e s u l t s a r e t o b e fo u n d i n A p p e n d ix B. T y p ic a l P l o t s U sed i n t h e C a l c u l a t i o n s j F i g u r e 1 i s a t y p i c a l p l o t o f re d u c e d p r e s s u r e a s a f u n c t i o n o f r e d u c e d d e n s i t y ; h o w e v e r, t h e a c t u a l p l o t s u s e d 1 1 i n th e c a l c u l a t i o n s a r e much l a r g e r t h a n shown h e r e . F o r t h i s p a r t i c u l a r c a s e t h e o r d i n a t e (P r ) i s r e p r e s e n t e d b y a o n e - t h i r d s c a l e a n d th e a b s c i s s a ( l n ^ r ) b y a o n e - h a l f j s c a l e . F i g u r e 2 i s a t y p i c a l p l o t o f p r e s s u r e a s a f u n c t i o n o f s p e c i f i c volum e f o r n - p e n t a n e . A g a in t h e s c a l e shown i j h e r e h a s b e e n d e c r e a s e d ; t h e o r d i n a t e (P ) i s r e p r e s e n t e d j 1 • 1 : Iby a o n e - f o u r t h s c a l e a n d th e a b s c i s s a (v ) i s r e p r e s e n t e d ! FIGURE t Reduced D e n s ity a s a R educed P r e ssu r e f o r u lt h Z Q a 0 .2 7 0 26 22 r 20 T2 t o .bo 28 3 6 l o g ^ r _ j FIGURE 2 Pressure as a Function o f S p e c ific Tol’ ume fo r n-Pentane (2 C = 0*269) P (p sia) QGO GOO 000 000 0 0 0 0 .0 2 ? .0 2 7 .029 *031 .033 v (ft3 /tb .) I b y a o n e - e i g h t h s c a l e RESULTS T a b le s I th r o u g h IV a lo n g w i t h F i g u r e s 3 and I4 . show t h e r e s u l t s o f a l l c a l c u l a t i o n s . T a b le I p r e s e n t s th e j s l o p e s c a l c u l a t e d fro m t h e r e d u c e d d e n s i t y c h a r t s a s a I f u n c t i o n o f T r and P r . The b u l k m o d u lu s o f a p a r t i c u l a r \ l i q u i d w i t h z c = 0 .2 7 0 i s com puted by m u l t i p l y i n g t h e a p p r o p r i a t e s l o p e v a lu e b y th e c r i t i c a l p r e s s u r e . F i g u r e 3 i s a p l o t o f o n ly some o f t h e r e s u l t s g iv e n i n T a b le I . T a b l e s I I a n d I I I a lo n g w i t h F i g u r e I 4 . com pare b u l k m o d u lu s v a l u e s f o r n - p e n t a n e c a l c u l a t e d by u s e o f t h e r e d u c e d d e n s i t y c h a r t s , u s e o f th e B u e h le r E q u a tio n o f S t a t e , and fro m e x p e r i m e n t a l PVT d a t a fro m th e l i t e r a t u r e . A l l v a l u e s h e r e 1 ! i j a r e r e p o r t e d i n p s i a . T a b le IV g i v e s b u l k m o d u lu s v a l u e s j f o r a few t y p i c a l l i q u i d s s u c h a s b e n z e n e and t o l u e n e . JT h ese v a l u e s a r e r e c i p r o c a l s o f i s o t h e r m a l c o m p r e s s i b i l i t i e s fo u n d i n t h e l i t e r a t u r e ( 1 , 3 ) , an d a r e r e p o r t e d i n TABLE I* S lo p e V a lu es f o r L iq u id s H aving z G = 0* Pr 1 .0 2.0 1;*0 6 .0 10.0 . 15.0 20.0 25.0 30.0 Tr 0 .3 6 - 651; 872 1032 1299 * 5 3 15*9 1871; 2099 0.1^0 - 1*3 61|6 792 1090 * 3 8 1677 17*3 1918 0 * 6 239 3H; 1;56 598 901 1233 * 1 2 1582 1699 0 .5 2 261 28I4 . 303 32 8 721 lOlf.1 1151 1133 1103 0*56 171*- 202 . 261 351 582 861 999 1087 1128 0.63 67 132 181 221 1;75 633 661 817 86 2 0 .6 6 65 97 151 221 382 580 651 651 6*2 0 .7 3 65 71 111 156 267 1;25 516 519 507 0.80 k -3 5 5 65 83 226 319 381; *25 *13 0 .8 8 21 29 53 77 151 213 279 309 303 0 .9 1 16 22 39 61; 137 183 2*0 302 373 0 .9 5 8 16 30 61 107 151 213 282 329 SLOPE = C r T BULK M ODULUS *3Pj - a in /’ r T H C O PIG’ U R E 3 iSIopes fo r C la ss o f L iq u id s Having Z s. o*270 TTOO toco 5t3o; 20C tGC 28 32 20 TABLE II. Comparison of Bulk Modulus Values for n-Pentane (Pc = I 4.90 psia., z c = 0 . 2 6 9 ) 0.66 BT ( l ) ( F r o m T a b le I ) B t( 2 ) ( P - V D a ta ) % D e v i a t io n * 0 .7 3 31> 700 p s i a 4 7 ,3 0 0 7 4 ,1 0 0 108,200 187,000 286,000 319,000 3 1 ,9 0 0 3 4 ,6 0 0 5 4 ,1 0 0 7 6 ,3 0 0 1 3 1 .0 0 0 2 0 8 ,5 0 0 2 5 2 .0 0 0 6 9 ,8 9 5 p s i a 7 4 ,7 2 0 7 9 ,7 7 9 8 9 ,6 7 5 1 0 3 ,4 9 8 1 4 0 ,5 6 3 1 9 6 ,6 8 3 4 6 , 4 l 8 45,423 5 5 ,5 0 4 6 7 ,3 5 1 86,826 1 1 8 ,5 0 5 2 0 4 ,9 3 0 - 5 4 - 7 - 3 6 .7 - 0 7 .1 +2 0 .6 +7 1 .0 +1 0 3 .5 +5 7 .2 A b s . A vg. 5 0 .0 D e v . - 3 1 .3 - 2 3 .8 - 02.5 + 1 3 .3 +51.0 +76.0 A b s . Avg, D ev . + 2 3 .4 3 1 .6 % D ev . = •* -) x 100 B( 2) A b s o lu te = | D e v . A vg. D ev . I 7 21 Tr = 0 .8 0 P r B^( 1) (Prom T a b le I ) Bip ( 2 ) ( P-V D a ta ) % D e v i a t i o n 1 2 1 ,3 0 0 25,317 - 15.8 2 2 7 ,1 0 0 32,798 -17 4 k 31,900 39>125 - 18.5 6 i|.o,6oo 5 1 4 8 2 - 2 1 .2 10 1 1 0 ,5 0 0 71,666 +51*.. 3 15 1 5 6 ,0 0 0 9 1 ,8 1 9 + 7 0 .0 20 1 8 8 ,0 0 0 108,651 +73.0 I Tr = = 0 .8 8 A b s . A v g . 38.6 D e v . 1 1 0 ,3 1 0 13,575 -2i*..0 2 lij.,200 19>230 - 2 6 ; i k 2 6 ,0 0 0 3 1 ,2 1 5 - 1 6 . 7 6 37,700 38,lij.8 - 0 1 .2 10 7 ^ ,0 0 0 51,814.8 M\2:y 15 io l|.,5 o o 79>103 + 3 2 ;o 20 1 3 7 ,0 0 0 93,085 +lf7 .2 A b s . A v g . 27.2 D e v . 22 i T r = 0 -9 S I Pp Brp( 1 ) ( Prom T a b le I ) B < j(2)(P -V D a ta ) % D e v ia t io n | 1 11,060 5 > 00^ - 18-.8 2 7 >830 1 0 ,7 2 7 - 27-.0 ^ li|.,9 2 0 18*079 - 17-.5 6 2 9 ,9 0 0 2 9 ,9 3 9 -0 0 * 1 3 10 52,lj-00 142,069 +2i}.i6 15 71-4*000 61,8214. +19*7 20 10I4 ., 800 1 0 6 ,5 5 2 +01 *7 A b s; A v g i 15*6 Dev* 23 t a b u ; h i . Comparison of Bulk Modulus Values for n-Pentane Tr ? r = 0 .6 6 B u e h le r BT ( l ) ( E q n . o f S t a t e ) Bt ( 2 )(P -V D a ta ) % D e v ia t io n * 1 4 5 » 4 ° 0 p s i a 6 9 ,8 9 5 p s ia -3 5 ?0 2 5 6 ,3 0 0 7!+,720 - 24?6 4 7 6 ,2 0 0 7 9 ,7 7 9 - 0 4 .5 6 107,000 89,675 +19? 3 10 1 5 3 ,0 0 0 1 0 3 ,4 9 8 +47 ?8 15 2 2 1 ,0 0 0 lI}-0,563 +57?2 2 0 i t 1 i I 3 0 7 ,0 0 0 196,683 A b s . ’ + 5 6 .1 A v g . 35*0 1 Tr = 0 .7 3 I 1 29,100 I4.6, IplS - 3 7 * 4 2 4 7 ,5 o o 4 5 ,4 2 3 +04 • 6 1 h 4 9 ,5 0 0 5 5 ,5 0 4 - 1 0 .8 6 69,000 6 7 ,3 5 1 - 02.45 10 ioij.,000 86,826 +2 0 .9 15 142,700 1 1 8 ,5 0 5 +2 0 .4 20 i 2 0 0 ,0 0 0 2 0 4 ,9 3 0 A b s . - 0 2 .4 A v g . 1 4 .1 % D ev . = — O-J—I , . J H x 100 B (2 ) A b s o lu te - I S D ev . A vg. D e v . 7 2 i ] _ Tr = 0.80 BuehXer* Pr B T ( l) ( E q n . o f S t a t e ) Bt(2)(P-V D a ta ) % D e v i a t i o n 1 18/600 2 5 ,3 1 7 - 2 6 .5 2 23,200 3 2 ,7 9 8 -2 9 -.3 k 3^,100 3 9 ,1 2 5 - 12.8 6 k? >700 51,1*82 - 0 7 . 3 6 10 71}-, 600 71,666 +0l*..l 15 108,800 91,819 +1 8 .5 20 157,000 1 0 8 ,6 5 1 +l4lj..5 A b s. A v g . 2 0 . If Tr = 0 .8 8 1 I 0 , 8lf0 1 3 ,5 7 5 - 2 0 .2 2 ik,k5o 1 9 ,2 3 0 -2lf-.9 y 23,800 3 1 ,2 1 5 - 2 3 .8 6 3 5 ,2 0 0 3 8 , l l }.8 - 0 7 .7 3 10 5 7,800 51,814.8 + 1 1 .5 15 92,000 7 9 ,1 0 3 + 1 6 .3 20 ilj4 ,o o o 9 3 ,0 8 5 +5lj-.8 25 Tr = cr. 95 B u e h le r Pr B iji (1 ) (Ecm • o f S t a t e ) Brp(2) ( P-V D a ta ) % D e v i a t i o n X 6 ,2 0 0 5 , 0 0 k +2I 4 ..O 2 9 ,5 7 0 1 0 ,7 2 7 - 1 0 .8 * 4 - 17,14-50 1 8 ,0 7 9 - 0 3 . 1 4 - 8 6 2 6 , 14.50 2 9 ,9 3 9 - 1 1 .6 10 1 4 . 5 ,0 0 0 1 4 .2 ,0 6 9 +06.97 15 8 0 ,1 0 0 6 1 , 8214. +145.7 20 1 2 6 ,0 0 0 1 0 6 ,5 5 2 +1 8 .2 A b s. A vg. 17*5 26 F IG U R E h Comparison o f Bulk M odtolus Tallies for n-Fentane at fr . = ~ 0.88 B.M.Cpsij* t% 0.000 T20y0G0 TOO.000 B a t a 30,000 20.000 6 18 20 P. V 27 TABLE IV : B u lk M odulus V a lu e s f o r a Few D i f f e r e n t P u re L iq u i d s ( A l l b u l k m od ulus v a l u e s a r e r e p o r t e d i n a tm o s p h e r e s .) I . B enzene Pc = 4 8 .6 a t m . , T c = 5 6 2 . 6 ° k . , z c = 0 .2 7 4 P( atm ) P r T°K T r • K ' Bij B ( l ) B( 2) 9 8 .1 0 2 298 0 .5 3 1 1 ,8 5 0 1 3 ,8 0 0 1 7 ,0 0 0 1 9 6 .2 0 4 298 0 .5 3 1 2 ,6 7 0 1 4 ,7 0 0 2 3 ,3 0 0 2 9 4 .3 0 6 298 0 .5 3 1 3 ,6 0 0 1 5 ,9 0 0 3 4 ,0 0 0 I I . C a rb o n D io x id e Pe = 7 2 .9 a t m . , T c = 3 0 4 .2°K *, z c = 0 .2 7 5 70 1 286 0 .9 4 1 ,0 4 0 583 9 2 4 I I I . C a r b o n t e t r a c h l o r i d e Pc = 45 a t m . , t c = 5 5 6 . 4 ° K ., . z c = 0 .2 7 2 9 8 .1 2 .1 8 298 0 .5 3 6 1 0 ,8 5 0 1 2 ,7 8 0 1 5 ,8 0 0 1 9 6 .2 4 .3 6 298 0 .5 3 6 1 1 ,6 2 0 1 3 ,6 0 0 2 1 ,4 0 0 2 9 4 .3 6 .5 5 298 0 .5 3 6 1 2 ,6 7 0 1 4 ,7 5 0 3 1 ,5 0 0 IV . T o lu e n e Pc = 4 l . 6 a t m . , Tc = 5 9 4 ° k . , z c = 0 0 [— C M • 1 1 4 .5 2 .7 5 298 0 .5 0 0 1 2 ,4 0 0 1 1 ,8 0 0 1 4 ,6 0 0 2 3 0 .5 5 .4 0 298 0 .5 0 0 1 4 ,2 2 0 - - 3 5 5 .0 8 .5 5 298 0 .5 0 0 1 6 ,1 0 0 - - K ey: B”': R e c i p r o c a l o f I s o t h e r m a l C o m p r e s s i b i l i t i e s j F ound i n th e L i t e r a t u r e ( 1 , 3 ) i i j B ( l ) : B u lk M odulus V a lu e s C a l c u l a t e d fro m R ed u ced | D e n s i t y C h a r ts | B ( 2 ) : B u lk M odulus V a lu e s C a l c u l a t e d fro m t h e E q u a t io n o f S t a t e DISCUSSION OF RESULTS T a b le I I shows a maximum d e v i a t i o n o f 1 0 3 .5 $ and a jminimum d e v i a t i o n o f - 0 . 13$ b e tw e e n b u l k m odulus v a lu e s i | c a l c u l a t e d f o r n - p e n t a n e . I t a p p e a r s t h a t th e b e s t a g r e e - j m en t f o r e a c h g iv e n r e d u c e d t e m p e r a t u r e I s a t Pr = I4 . and 6 . A t t h e s e p o i n t s t h e minimum p e r c e n t d e v i a t i o n o c c u r s . The s m a l l e s t a b s o l u t e a v e r a g e p e r c e n t d e v i a t i o n o f 1 5 *6$ o c c u r s f o r b u l k m odulus v a lu e s co m p ared a t a r e d u c e d te m p e ra t u r e o f 0 .9 5 0 . S u c h a r e s u l t w ould be e x p e c te d s i n c e a t h i g h e r t e m p e r a t u r e s , w h ere th e g a s p h a se i s a p p r o a c h e d , i b a s i c volum e m e a su re m e n ts a r e m ore a c c u r a t e t h a n a t lo w e r t e m p e r a t u r e s . A t lo w e r p r e s s u r e s i n th e r a n g e o f Pr = 1 to 16 , b u l k m o d u lu s v a l u e s c a l c u l a t e d fro m t h e r e d u c e d d e n s i t y 1 c h a r t s a r e lo w e r th a n t h e v a l u e s c a l c u l a t e d fro m th e e x - l j p e r i m e n t a l PVT d a t a . A t h i g h e r p r e s s u r e s i n th e r a n g e o f Pr = 1 0 to 3 0 , j u s t th e o p p o s i t e i s t r u e . A l l c o m p a ris o n s I seem t o show t h e same t r e n d . | T a b le I I I show s a maximum d e v i a t i o n o f 5 7 » 2 $ and a jminimum d e v i a t i o n o f - 2 . 4$ . The s m a l l e s t a b s o l u t e a v e r a g e 1 ( d e v i a t i o n i s a t Tr = 0 .7 3 w h ere a v a l u e o f l l ^ . l ^ i s r e p o r t e d . On t h e a v e r a g e i t a p p e a r s t h a t t h e b e s t a g re e m e n t i s |g i v e n a t Pr = 6 f o r e a c h r e d u c e d t e m p e r a t u r e . As a w hole I th e c o m p a ris o n s o f T a b le I I I seem t o show th e same t r e n d i I . 28 i a s t h o s e i n T a b le I I . F i g u r e I 4 . co m p ares b u l k m o d u lu s v a l u e s c a l c u l a t e d by th e t h r e e m e th o d s . A t a ro u n d Pr = 6 th e c u r v e s seem t o a l m o st i n t e r s e c t . F i g u r e ij. i s draw n f o r Tr = 0 .8 8 ; i t i s i j f a i r l y t y p i c a l o f a l l th e c o m p a r is o n s . B etw een Pr = 6 an d } ! an d P r = 18 i t a p p e a r s t h a t th e e q u a t i o n o f s t a t e m ethod a g r e e s m ore c l o s e l y w i t h th e e x p e r i m e n t a l PVT d a t a th a n d o e s t h e r e d u c e d d e n s i t y m e th o d . I n p l o t t i n g d a t a fro m th e r e d u c e d d e n s i t y c h a r t s we w ere u n a b le to draw n an e x a c t jsm o o th c u r v e th r o u g h a l l th e p o i n t s . T h is m ig h t a c c o u n t | f o r th e l a r g e r d e v i a t i o n g iv e n by t h e r e d u c e d d e n s i t y I m e th o d . R e f e r r i n g b a c k to F i g u r e [j. a g a i n , i t i s o b s e r v e d I t h a t a t Pr = 20 th e r e d u c e d d e n s i t y c u rv e an d th e e q u a t i o n j | o f s t a t e c u r v e i n t e r s e c t . Beyond t h i s p o i n t i t a p p e a r s : t h a t t h e r e d u c e d d e n s i t y m ethod now a g r e e s m ore c l o s e l y w i t h th e PVT d a t a . H ow ev er, i n th e c o m p a ris o n s a t o t h e r |r e d u c e d t e m p e r a t u r e s , th e e q u a t i o n o f s t a t e m eth o d a g r e e d I m ore c l o s e l y b e y o n d Pr = 2 0 . No d e f i n i t e c o n c l u s i o n c a n be >made a s t o w h ic h m eth o d i s b e t t e r f o r Pr > 2 0 . B etw een Pr =! 1 to 6 b o t h m eth o d s com pare a b o u t t h e sa m e. i ' ! j The r e s u l t s p r e s e n t e d i n T a b le I s h o w ^ th a t f o r a n y jg iv e n p r e s s u r e , th e b u l k m o d u lu s d e c r e a s e s w ith i n c r e a s i n g I t e m p e r a t u r e ; and f o r any g iv e n t e m p e r a t u r e , th e b u l k j [m odulus i n c r e a s e s w i t h i n c r e a s i n g p r e s s u r e . C o m p a riso n s c o u ld o n ly be made f o r a r a n g e o f I 3 0 I , , r e d u c e d t e m p e r a t u r e s .b etw een 0 ,6 6 and 0 .9 5 b e c a u s e no PVT | d a t a was fo u n d f o r l i q u i d s w i t h z c = 0 .2 7 0 i n th e r a n g e o f I re d u c e d t e m p e r a t u r e s fro m 0 .3 0 t o 0 . 6 6 . The r e d u c e d p r e s s u r e r a n g e was r e s t r i c t e d to v a l u e s fro m 1 to 20 f o r c o m p a ris o n p u r p o s e s . i CONCLUSIONS H ougen, L y d e r s e n , and G re e n k o rn p r e s e n t no d e t a i l s on t h e i r c a l c u l a t i o n o f r e d u c e d d e n s i t i e s . A p p a r e n t l y th e y d e r i v e d v a l u e s fro m a sm o o th ed c u rv e t h r o u g h p o i n t s c a l c u l a t e d f o r v a r i o u s l i q u i d s h a v in g a c r i t i c a l c o m p r e s s i b i l i t y f a c t o r o f 0 .2 7 0 . K now ledge o f t h e i r p r o c e d u r e w ould g iv e some i n d i c a t i o n o f th e e x p e c te d e r r o r i n t h e c a l c u l a t i o n o f b u l k m o d u lu s . The n u m e r ic a l d i f f e r e n t i a t i o n t e c h n iq u e u s e d w ould p r e s e n t a s m a ll e r r o r , t h e r e a s o n b e in g t w o f o l d : ( a ) The |c u r v e s a r e n o t c o m p lic a te d a n d r a t h e r sm o o th , an d (b ) A i I v e r y s m a ll in c r e m e n t i n th e in d e p e n d e n t v a r i a b l e was c h o s e n ! t o a ll o w f o r a n y d i s c o n t i n u i t i e s i n th e c u r v e s . I n th e ra n g e 0 .5Q<Tp<0.95* 1£P i» 5.6, i t c an b e c o n - f |e l u d e d t h a t e i t h e r th e B u e h le r E q u a t io n o f S t a t e o r th e i j s l o p e s d e r i v e d fro m th e r e d u c e d d e n s i t y c h a r t s w i l l g iv e |a b o u t t h e same d e g r e e o f a c c u r a c y i n t h e c a l c u l a t i o n o f jb u lk m o d u lu s . F o r 0 .5 0 5 T r < 0 .9 ^ an d 6£Pr £ 2 0 , th e B u e h le r jE q u a tio n o f S t a t e w ould g iv e th e b e t t e r r e s u l t s . The e q u a t i o n o f s t a t e i s r e s t r i c t e d t o t h i s t e m p e r a t u r e r a n g e b u t c a n b e u s e d f o r r e d u c e d p r e s s u r e s a s h i g h a s 190 and r e d u c e d d e n s i t i e s a s h i g h a s I4 ., a c c o r d i n g to th e l i t e r a t u r e ( 2 ) . I t i s f u r t h e r recom m ended t h a t t h e m o st c o n s i s t e n t 32 j v a l u e s o f b u l k m o dulus a r e c a l c u l a t e d a t Pr = I} , and 6 f o r a n y g iv e n t e m p e r a t u r e i n th e ra n g e Tr = 0 .6 6 t o 0.95>* I t c o u ld a l s o be e x p e c te d t h a t t h e b e s t ^ v alu es o f b u l k m odulus i | a r e g iv e n a t Pr = If. and 6 f o r a r e d u c e d t e m p e r a t u r e ra n g e i 0 .3 0 t o 0 . 6 6 . I t w o u ld b e d e s i r a b l e t o f i n d PVT d a t a i n th e r e d u c e d t e m p e r a t u r e r a n g e 0 .3 0 t o 0 .6 6 * A m ore th o r o u g h e - v a l u a t i o n o f t h e r e s u l t s g iv e n by t h e r e d u c e d d e n s i t y m e th od c o u ld th e n b e m ade. The W atso n E x p a n s io n F a c t o r M ethod ( 5 ) , i l l u s t r a t e d i i n A p p e n d ix B, g i v e s f a i r l y good r e s u l t s b u t d o e s n o t c o v e r ja l a r g e en o u g h r a n g e i n p r e s s u r e * BIBLIO GRA PHY I L 33 _ _ j BIBLIOGRAPHY " I n t e r n a t i o n a l C r i t i c a l T a b l e s , " M c G ra w -H ill B ook Com- ' p a n y , I n c . , Hew Y o rk , 2* 1 9 2 8 . | t H i r s c h f e l d e r , J . C . , B u e h l e r , R . J . ^ M cGee, H. A ., a n d j S u t t o n , J . R . , " G e n e r a l i z e d E q u a tio n o f S t a t e f o r , G ases and L i q u i d s , " I n d . and E n g . Chem. 5 0 , 3 7 5 - 1 3 8 4 ( 1 9 5 8 ) . | i L a n g e , H. A ., H andbook o f C h e m is tr y , H andbook P ub l i s h e r s I n c . , S a n d u s k y , O h io , 1948• L y d e r s e n , A* L . , G re e n k o rn , R . A ., and H ougen, 0 . A *, | G e n e r a l i z e d Therm odynam ic P r o p e r t i e s o f P u re F l u i d s , j U n iv . W is c o n s in , E n g . E x p t . S t a . , Bept,~Lf.9 1 95 5 - 1 I R e id , R . , a n d S h erw o o d , T . , P r o p e r t i e s o f G a se s a n d j L i q u i d s , M c G ra w -H ill Book C o . , I n c . , Hew Y o rk , 1 9 5 8 . I R o u s e , H u n te r , E le m e n ta r y M e c h a n ic s o f F l u i d s , J o h n W ile y and S o n s , I n c . , 1 9 ^ 6 . S a g e , B . H ., and L a c e y , W. N*, "T herm odynam ic P r o p e r t i e s o f H - P e n ta n e , I n d . and E n g . Chem. 3 6 , 7 3 0 -7 ( 1 9 4 2 ) . i S a l v a d o r ! , M. G ., and B a ro n , M. L . , n u m e r i c a l M eth o d s I i n E n g i n e e r i n g , P r e n t i c e - H a l l I n c • , Hew J e r s e y , 1 9 5 2 . I S h e rw o o d , T . K ., and R e e d , G. E . , A p p lie d M a th e m a tic s j i n C h e m ic a l E n g i n e e r i n g , M c G ra w -H ill Book C o ., I n c . , Hew Y o r k a n d L o n d o n , 1 9 3 9 . APPEKDIX A ' D a ta an d Sam ple C a l c u l a t i o n s 1 . R e d u ee d D e n s i t i e s o f L i q u i d s T ak e n fro m t h e T a b le s o f L y d e r s e n j G -reenkorn, a n d H ougen 2 . S am ple C a l c u l a t i o n o f S lo p e s B a se d o n R ed u ced D e n s i t y C h a r t s f o r Tj, — 0 ,6 6 3* E x p e r i m e n t a l P-V D a ta T ak e n fro m t h e L i t e r a t u r e f o r n - P e n ta n e (7 ) S am ple C a l c u l a t i o n o f B u lk M odulus f o r n - B e n ta n e U s in g P-V D a ta f o r n - F e n t a n e a t Tr = 0 .6 6 5>. Sam ple C a l c u l a t i o n o f B u lk M odulus f o r n - F e n t a n e U s in g B u e h le r * s E q u a t i o n o f S t a t e 3£ ! I APPENDIX A | I D a ta and S am ple C a l c u l a t i o n s i • i I !1* R e d u ced D e n s i t i e s o f D i q u i d s T ak e n fro m t h e T a b le s o f L y d e r s e n , G re e n k o rn and H ougen. ! P r Ee 1 . 0 2 . 0 .J+.O 6 . 0 0 .3 6 3 .1 9 3 3 .1 9 8 3 .2 0 7 3 .2 1 2 O.4 0 3 .1 2 3 3 .1 3 2 3 .1 4 3 3 .1 4 8 0*46 3 .0 2 0 3 .0 3 1 3 .0 4 7 3 .0 5 4 0 . 5 2 2 .9 1 1 2 .9 2 2 2 .9 4 1 2 .9 6 0 0 .5 6 2 .8 4 0 2 .8 5 5 2 .8 8 1 2 .8 9 6 0 .6 3 2 .7 0 4 2 .7 3 1 2 .7 6 1 2 .7 9 4 0 .6 6 2 . 6) 4.0 2.672+. 2 .7 1 2 2 .7 4 9 0 .7 3 2 .4 9 8 2 .5 3 3 2 .5 9 0 2 .6 3 9 0 .8 0 2 .3 2 9 2 .3 7 7 2 .4 6 0 2 .5 2 0 0 .8 8 2 .0 9 8 2 .1 7 7 2 .2 9 9 2 .3 8 0 0 .9 1 1 .9 9 0 2 .0 9 2 2 .2 3 5 2 .3 2 5 0 .9 5 1 .8 0 3 1 .9 6 5 2 .1 4 5 2 .2 4 9 i i i r i 37 P r jTr__ 1 0 .0 1 5 .0 2 0 .0 2 5 .0 3 0 .0 1 io .3 6 | 3 .2 2 5 3 .2 3 5 3 .2 if5 3 .2 5 5 3 .2 6 1 1 0 .ij.0 3 .1 6 5 3 .1 7 7 3 .1 8 5 3 .1 9 6 3 .2 0 5 j 0 .if6 3 .0 7 5 3 .0 9 0 3 .1 0 0 3 . H 3 3 .1 2 1 j • 0 .5 2 2 .9 8 5 3 .0 0 0 3 . 0 lif 3 .0 3 0 3 . 0if0 0 .5 6 2 .9 2 5 2.9if9 2 .9 5 9 2 .9 7 6 2 .9 8 9 j 0 .6 3 2 .8 2 8 2.8if9 2 .8 7 0 2 .8 8 9 2 .9 0 6 ■ 0 .6 6 1 2 .7 8 2 2 .8 1 1 2 .8 3 3 2 .8 5 5 2 .8 7 5 ! 0 .7 3 j 2 .6 8 8 2 .7 2 0 2 .7 5 0 2 .7 7 6 2 .8 0 if 1 0 .8 0 1 2 .5 8 7 2 .6 3 if 2 .6 7 3 2 .7 0 2 2.73ij- 1 10*88 2.1f65 2 .5 3 2 2 .5 8 5 2 .6 2 6 2 .6 6 9 i | 0 .9 1 2 . if 18 2.if93 2 .5 5 2 2 .5 9 8 2 .6 3 8 0 .9 5 2 .3 5 £ 2.ifif5 2 .5 1 1 2 .5 6 1 2 .6 0 5 i i i i i t t i r I i 38 2 . S am ple C a l c u l a t i o n o f S lo p e s B ased o n R e d u ce d D e n s i t y C h a r t s f o r T r = 0*66 R ed u ced p r e s s u r e r a n g e s Pp = 1 -3 0 D a ta f o r a p l o t o f Pr v e r s u s l o g ^ r : £ 1 2 4 6 10 15 20 25 30 2.640 2 .6 7 4 2 .712 2 .7 4 9 2.782 2 .8 1 1 2 .8 3 3 2 .8 5 5 2 .8 7 5 l o O.if.2160 0.42716 0 .4 3 3 2 9 0 . 4 3 9 1 ? O.4 4 4 3 6 0 .4 4 8 8 6 O.45225 0 .4 5 5 6 1 0 .4 5 8 6 4 The s c a l e show n i n F i g u r e 5 i s m uch s m a l l e r t h a n t h e s c a l e u s e d i n t h e w o rk in g g r a p h . H ere t h e o r d i n a t e (Pp) i s o n e - t h i r d t h e o r i g i n a l o r d i n a t e s c a l e , an d t h e a b s c i s s a ( lo g ^ p ) i s o n e - e i g h t h t h e o r i g i n a l a b s c i s s a s c a l e . O r i g i n a l s c a l e : One u n i t i n Pp = 2 cm. 0 .0 0 1 u n i t i n lo g f'p = 1 cm. 39 FIGURE 5 Reduced D en sity as a F u nction o f Reduced Pressure fo r liq u id s w ith Z c ~ 0*270 %% 22 r T O TABLE V. E q u a l I n c r e m e n te d D a ta T aken fro m F i g u r e 5 h * 0 * 0 0 1 V a lu e o f Bp W here S lo p e i s D e s i r e d ___ Pr = 1*0 2 .0 4*0 P o i n t s S u r r o u n d in g Pr W here S lo p e i s D e s i r e d I I i f x - l n ^ p 0 .5 7 0 0.1+1860 0* 7 0 0 0.1+1960 0 .8 5 0 0.1+2060 1 .0 0 0 0.1+2160 1 .1 5 0 0.1+2260 1 .3 0 0 0.1+2360 1.1+80 0.1+21+60 1.1+20 0.1+21+16 1 .6 0 0 0.1+2516 1 .8 0 0 0.1+2616 2 .0 0 0 0.1+2716 2 .2 5 0 0.1+2816 2 .5 0 0 0.1+2916 2 .8 0 0 0 . 1+3016 3 .0 5 0 0.1+3029 3 .3 5 0 0.1+3129 3 .6 5 0 0.1+3229 1+.000 0.1+3329 1+.350 0.1+31+29 1+.750 0.1+3529 5 .1 7 0 0.1+3629 41 P o i n t IfSlxere S lo p e i s D e s i r e d y 4*650 0.10617 5 * 0 5 0 0 .4 3 7 1 7 5 .5 0 0 0 .4 3 8 1 7 6 * 0 6 .0 0 0 0 .4 3 9 1 7 6 .5 2 0 0 .4 4 0 1 7 7 .1 0 0 0 .4 4 1 1 7 7 .7 5 0 0 .4 4 2 1 7 7 .6 5 0 0 .4 4 1 3 6 8 * 3 5 0 0 ;4 4 2 3 6 9 ; l 5 0 0 ;^ 4 3 3 6 1 0 .0 1 0 i 0 00 0 ;4 4 4 3 6 1 0 .9 0 0 0 .4 4 5 3 6 1 1 .9 0 0 0 ;4 4 6 3 6 1 3 .0 0 0 0 4 4 7 3 6 1 1 .4 5 0 0 .J 1 2 ;5 0 0 0 4 4 6 8 6 1 3 * 7 5 0 0 ^ 4 4 7 8 6 15.0 15;000 0;44886 1 6 ;4 0 0 0 .4 4 9 8 6 1 7 .8 0 0 0 4 5 0 8 6 1 9 ; 270 0 4 5 1 8 6 P o i n t W here S lo p e i s D e s i r e d 20.0 25.0 30.0 JL JL 1 5 .6 0 0 0 .4 4 9 2 5 1 7 .0 5 0 0 .ij.5025 1 8 .5 0 0 0 *14.5125 2 0 .0 0 0 ■ 0 *14,5 225 2 1 .5 0 0 0 .4 5 3 2 5 2 3 .0 0 0 0 . 1 4 .514.25 2 4 .4 5 0 0 . ij.5525 2 0 .3 5 0 0 .4 5 2 6 1 2 1 .9 5 0 0 .4 5 3 6 1 2 3 .5 0 0 0 .4 5 4 6 1 2 5 .0 0 0 0 .4 5 5 6 1 2 6 .5 0 0 0 .4 5 6 6 1 2 8 .0 0 0 0 .4 5 7 6 1 2 9 .5 0 0 0 .4 5 8 6 1 2 5 .5 0 0 0 .4 5 5 6 4 2 7 .0 0 0 G.4 5 6 6 4 2 8 .5 0 0 0 .4 5 7 6 4 3 0 .0 0 0 0 .4 5 8 6 4 3 l.lj.5 0 0 .4 5 9 6 4 3 2 .9 5 0 0 .4 6 0 6 4 3 k » k$ 0 0 .4 6 1 6 4 Sample Calculation We now h a v e t h r e e p o i n t s e q u a l l y s p a c e d o n e i t h e r i | s i d e o f t h e p o i n t w h e re t h e d e r i v a t i v e i s s o u g h t . T he !c e n t r a l p o i n t i s t h e n t r a n s l a t e d t o t h e o r i g i n (x = 0 ) . F o r e x a m p le , a t Pr = I t J X x ‘= x - 0 . 4 2 1 6 0 k = x * /h k y k 3 y 0 .5 7 0 O .4 1 S6 O - 0 .0 0 3 “ 3 - 1 .7 1 0 - 1 5 .4 0 0 .7 0 0 0 .4 1 9 6 0 - 0 .0 0 2 - 2 - I .4 0 0 5«6o 0 .8 5 0 0 .4 2 0 6 0 - 0 .0 0 1 - 1 -O .8 5 0 - 0 .8 5 1 .0 0 0 0 .4 2 1 6 0 0 .0 0 0 0 0 .0 0 0 0 .0 0 1 .1 5 0 0 .4 2 2 6 0 0 .0 0 1 1 1 .1 5 0 1 .1 5 1 .3 0 0 0 .4 2 3 6 0 0 .0 0 2 2 2 .6 0 0 1 0 .4 0 1 .4 8 0 0 .4 2 4 6 0 0 .0 0 3 3 4 .4 4 0 4 .2 3 0 4 0 .0 0 2 9 .7 0 d y /d x = b = 3 9 7 X k y / l 5 l 2 h - 7 £ k 3 y / 2 l 6 h = (3 9 7 ) ( i t . 2 3 0 ) _ (7 ) ( 2 9 .7 0 0 ) 1 5 1 2 ( 0 . 0 0 1 ) " 2 1 6 ( 0 .0 0 1 ) = lit9 .1 2 7 A l l o f t h e a b o v e o p e r a t i o n i s h a n d le d b y t h e com p u t e r . The n e x t tw o p a g e s i l l u s t r a t e t h e p ro g ra m a n d p r i n t o u t s h e e t . COMPUTER PROGRAM PARAMI9Z PS D I M E N S I O N Y (7 ) PRINT 12. PRINT 16 10 FORMAT ( 7 F 7 .3 * F ij..2 ,F 5 .2 ,F 6 .ii.) 16 FORMAT (1H ,3X ) 1 2 FORMAT ( 1H 1, 7X , 2HTR, 13X , 2HPR, l^ X , ^HSLOPE, 12X , 1 1 H S L 0 P E /2 .3 0 3 ) £ 0 FORMAT (1H ,£ X ,F 1 4 ..2 ,1 0 X ,f 5 .2 , 2 ( 1 0 X , F 1 1 . 5 ) ) 70 FORMAT (1H ,7 ( F 7 .3 » 3 X ) ) LINE=0 1 1 IF(SE N SE SWITCH 0 ) 9 9 ,1 ^ 1If. READ 1 0 ,Y ,T R ,P R ,H ACCUM1=0. S K = -3 .0 DO 3 0 1 = 1 ,7 ACCUM1=SK'*Y ( I ) +AC0UM1 30 SK =SK+1.0 ACCUM2=0 . S K = ~ 3.0 DO lj.0 1=1 ,7 ACCUM2=SK'''~'‘ '3'5 S Y ( I )+ACCUM2 i|.0 SK=SK+1.0 SLOPE=(397 / '5 ‘ A C C U M 1)/(1512.*H )- ( 7 .': c *ACCUM 2)/(2l6 .*H ) d i v s l o = s l o p < e / 2 .3 0 3 PRINT $ 0 , TR,PR,SLOPE,DIVSLO PRINT 7 0 , Y PRINT 16 LINE=LINE+1 IF (L IN E -1 9 ) 1 1 ,8 0 ,8 0 80 PRINT 12 PRINT 16 LINE=0 GO TO 11 99 STOP END SAMPLE PRINT OUT FOR Tr = 0 .6 6 T r 0.66 0 .5 7 0 0 .7 0 0 0.66 1.2*20 1. 0.66 : 3 .0 5 0 3 .3 5 0 0.66 1 2*.650 5.050 0.66 7 .6 5 0 8 .3 5 0 0.66 11.1*50 12.500 0 .6 6 ; l 5 .6 0 0 1 7 . 0.66 2 0 .3 5 0 2 1 . 0.66 2 5 .5 0 0 2 7 .0 0 0 P r 1.00 0 . 2.00 1 .800 4 .0 0 3 .6 5 0 6.00 5 .5 0 0 10.00 9 .1 5 0 15.00 1 3 .7 5 0 20.00 18.500 2 5 .0 0 2 3 .5 0 0 3 0 .0 0 28.500 SLOPE 11*9 .1 2 6 9 8 1 .0 0 0 1 .1 5 0 2 2 2 .3 8 0 9 5 2.000 2.250 31*8.25397 2*.000 2 * .3 5 0 5 0 9 .1 6 6 6 7 6 .0 0 0 6 .5 2 0 879.5632*9 1 0 .0 0 0 1 0 .9 0 0 1336.32*920 1 5 .0 0 0 16.1*00 12*9 9 .8 0 1 6 0 2 0 .0 0 0 2 1 .5 0 0 1500.1982*0 2 5 .0 0 0 2 6 .5 0 0 12*79.56350 3 0 .0 0 0 31.2*50 S L O P E /2 .3 0 3 * 62*.75336 1 .3 0 0 1 . 96.5612*2 2 .500 2. 1 5 1 .2 1 7 5 3 2*.750 5 .1 7 0 2 2 1 . 0882* 1* 7 .1 0 0 7 .7 5 0 3 8 1 .9 2 0 7 5 1 1 .9 0 0 1 3 .0 0 0 580.262*52 1 7 .8 0 0 1 9 .2 7 0 651.23821 2 3 .0 0 0 22*.2*50 651.2*1051 2 8 .0 0 0 2 9 .5 0 0 6 2* 2 .2*5050 3 2 .9 5 0 32*.2*50 '*The d a t a w as p l o t t e d a s Pr v e r s u s l o g ( b a s e 1 0 )&p g i v i n g a s l o p e o f (9 P r / a l o g ^ r ) T . T h en (Q P.p/2 ln ft,)rp = ( 1 / 2 .3 0 3 ) (9 P P/ a i o g ^ p ) T . V 46 3 . E x p e r i m e n t a l P-V D a ta T ak e n fro m t h e L i t e r a t u r e f o r n - P e n t a n e (7 ) T r Pr 1 .0 2 .0 lp .0 6 . 0 0 *66 0 .0 2 6 1 5 0 .0 2 5 9 8 0.02561^ 0 .0 2 5 3 4 0*73 0 .0 2 7 7 8 0 .0 2 7 4 9 0 .0 2 6 9 7 0 .0 2 6 5 3 0 .8 0 0 .0 2 9 7 5 O.0 2 9 2 5 0 .0 2 8 4 6 0 .0 2 7 8 7 0 .8 8 0 .0 3 2 5 3 0 .0 3 1 5 5 0 .0 3 0 2 7 0 .0 2 9 4 4 0 .9 5 0 .0 3 7 2 5 0 .0 3 4 8 0 0 .0 3 2 5 6 0 .0 3 1 2 0 . Pr 1 0 .0 1 5 .* 0 2 0 .0 0 .0 2 4 8 1 f t 3 / l b . 0 .0 2 4 3 3 0 .0 2 3 9 4 0 .0 2 5 8 2 0 .0 2 5 2 3 0 .0 2 4 7 4 0 .0 2 6 9 6 0 .0 2 6 1 7 0 .0 2 5 5 5 0 .0 2 8 1 6 0 .0 2 7 1 1 . 0 .0 2 6 3 6 0 .0 3 9 5 0 0 .0 2 8 0 9 0 .0 2 7 2 1 kl 4» S a m p le C a l c u l a t i o n o f B u lk M o d u lu s f o r n - P e n t a n e U s in g P -V D a ta f o r n * P e n ta n e a t Tt = 0 .6 6 R e d u c e d p r e s s u r e r a n g e s Pp = 1 - 2 0 P c ~ 4 9 0 p s i a D a ta f o r a p l o t o f P v e r s u s v : P ( p s i a ) Pr v ( f t 3 / i b . ) 4 9 0 1 0 .0 2 6 1 5 9 8 0 2 0 .0 2 5 9 8 I 9 6 0 4 0.025614. 6 0 .0 2 5 3 4 10 0 . 0214.81 7 3 5 0 15 0 .0 2 4 3 3 9 8 0 0 20 0 .0 2 3 9 4 F i g u r e 6 o n t h e n e x t p a g e a g a i n i s a r e d u c e d s c a l e p l o t o f t h e a c t u a l w o r k in g p l o t . S lo p e = ( 9 P /? V ) t O r i g i n a l s c a l e : P - 1 00 p s i a = 0 .5 cm V - 0 .0 0 0 0 5 f t 3 / l b = 1 cm 1 * 8 FIGURE 6 P ressu re a s a F u n ctio n of* S p e c if ic Volume f o r n -P en ta n e (Z^ ~ 0 .2 6 9 ) ) 10 0 0 0 0 027 k s TABLE V I. E q u a l I n c r e m e n te d D a ta T a k e n fro m F i g u r e 6 h - 0 .0 0 0 0 5 V a lu e o f Pr W here S lo p e P o i n t s S u r r o u n d in g Pr W here i s D e s i r e d S lo p e i s D e s i r e d _____ Pr v(ftVlb.) G .l8 ij. 0 .0 2 6 0 0 0 .ij4 8 0 .0 2 6 0 5 0 .7 3 5 0 .0 2 6 1 0 Pr = 1 . 0 1 .0 0 0 0 .0 2 6 1 5 1 .2 6 6 0 .0 2 6 2 0 1 .5 7 0 0 .0 2 6 2 5 1 .8 7 8 0 .0 2 6 3 0 1 .1 2 2 0 .0 2 5 8 3 l . i j .10 0 .0 2 5 8 8 1 .6 9 6 0 .0 2 5 9 3 2 .0 2 .0 0 0 0.02598 2 .2 9 0 0 .0 2 6 0 3 2 .5 9 0 0.02608 2 .9 2 0 0 .0 2 6 1 3 50 P o i n t W here S lo p e i s D e s i r e d 4 . 0 6«0 1 0 .0 * P v ( f t 3 / l b 3 .1 0 0 0.025if-9 3 .3 9 0 0 .0 2 5 5 if. 3 .6 7 5 0 .0 2 5 5 9 ij..0 0 0 0 . 0256ij. If. . 320 0 .0 2 5 6 9 If. . 6 3 0 0.0257if. If..9i}.0 0 .0 2 5 7 9 5 .0 0 0 0 .0 2 5 1 9 5 .3 0 0 0 . 0 2 5 2 if. 5 .6 5 0 0 .0 2 5 2 9 6 .0 0 0 0.0253if. 6 .3 6 0 0 .0 2 5 3 9 6 .7 2 0 0 .0 2 5 i|4 7 .0 6 0 0.025if-9 8 .7 6 0 0.02if.66 9 .1 8 0 0.02i|_71 9 .5 5 0 0 . 0214.76 1 0 .0 0 0 0 ,0 2 if.8 l 10.if.10 0.02if,86 1 0 .9 1 0 0.02i}.91 ll.if .2 0 0 . 02if.96 51 ! P o i n t W here S lo p e i s D e s i r e d Pj> v ( f t ^ / l b . ) 1 3 .3 5 0 0 .0 2 4 1 8 1 3 .8 8 0 0 .0 2 4 2 3 34.if.10 0 .0 2 4 2 8 1 5 .0 1 5 .0 0 0 0 .0 2 4 3 3 1 5 .6 0 0 0 .0 2 4 3 8 1 6 .2 0 0 0 .0 2 lf4 3 1 6 .8 0 0 0 .0 2 if4 8 1 7 .8 5 0 0 .0 2 3 7 9 18.500 0.0238k 1 9 .2 3 0 0 .0 2 3 8 9 2 0 ;o 2 0 .0 0 0 0 ;0 2 3 9 4 2 0 .9 0 0 0 ;0 2 3 9 9 2 1 .8 0 0 0 .0 2 4 0 9 2 2 .7 0 0 0 :0 2 4 1 4 52 : i S am p le C a l c u l a t i o n i N e x t, th.e c e n t r a l p o i n t o f each , s e t o f p o i n t s i s 't r a n s l a t e d t o t h e o r i g i n w i t h r e s p e c t t o t h e in d e p e n d e n t I « :v a r i a b l e . F o r Pr = I t :y - Pp x = v ( f t 3 / l b ) x» = x - 0 . 0 2 6 1 5 k k y k ^ y O . I 8 2 4 . 0. 1| 48 0 .7 3 5 1 .0 0 0 ;1 •2 6 6 1 .5 7 0 1 .8 7 8 The c o m p u te r p r i n t o u t i s i l l u s t r a t e d o n t h e n e x t p a g e . 0 .0 2 6 0 0 - 0 .0 0 0 1 5 - 3 - 0 . 5 5 2 - 4 .9 6 8 0 .0 2 6 0 5 - 0 .0 0 0 1 0 - 2 - 0 .9 9 6 - 3.584 0 .0 2 6 1 0 - 0 .0 0 0 0 5 - 1 - 0 .7 3 5 - 0 .7 3 5 0 .0 2 6 1 5 0 .0 0 0 0 0 0 0 .0 0 0 0 .0 0 0 0 .0 2 6 2 0 0 .0 0 0 0 5 1 1 .2 6 6 1 .2 6 6 0 .0 2 6 2 5 0 .0 0 0 1 0 2 3 .3 4 0 1 2 .5 6 0 0 .0 2 6 3 0 0 .0 0 0 1 5 3 5 . 63^ 7.714-7 5 0 .7 0 6 55.214-5 S lo p e - 3 9 7 ( 7 . 7li7) 1 5 1 2 ( .0 0 0 0 5 ) 7 ( 5 5 .2 ii5 ) 2 1 6 ( .0 0 0 0 5 ) S lo p e = 5 i|-5 2 .7 0 % = P e v ( S lo p e ) = (lj.90) ( 0 .0 2 6 1 5 ) ( 5 4 5 2 .7 0 ) BT = 6 9 8 9 5 p s i a 53 SAMPLE PRINT OUT FOR Tr * 0*66 Tr pr SLOPE BULK MODULUS 0*66 1 .0 0 511-52.69814.0 6989l}..9 O . I 8I4 . O.liif.8 0 .7 3 5 1 .0 0 0 1 .2 6 6 1 .5 7 0 I .8 7 8 0*66 2 .0 0 5 8 6 9 .5 2 3 8 0 7lf.720.2 1 .1 2 2 l . l p . 0 1 .6 9 6 2 .0 0 0 2 .2 9 0 2 .5 9 0 2 .9 2 0 0 .6 6 J 4 ..OO 6 3 5 0 .0 0 0 0 0 79 7 7 8 * 9 3 .1 0 0 3 .3 9 0 3 .6 7 5 4 .0 0 0 I4..320 I4..630 ij..9i|.0 0 .6 6 6 .0 0 7 2 2 2 .2 2 2 2 0 89675.ll- 5 .0 0 0 5 .3 0 0 5 .6 5 0 6 .0 0 0 6 .3 6 0 6 .7 2 0 7 .0 6 0 0 .6 6 1 0 .0 0 ' 8513 * 1 4 -9 210 103lj.97.7 8 .7 6 0 9 .1 8 0 9 .5 5 0 1 0 .0 0 0 10.1|.10 1 0 .9 1 0 l l.lj.2 0 0 .6 6 1 5 .0 0 1 1 7 9 0 .lj.7 6 0 0 lij.0 5 6 2 .5 1 3 .3 5 0 1 3 .8 8 0 lij..i4.10 1 5 .0 0 0 1 5 .6 0 0 1 6 .2 0 0 1 6 .8 0 0 0 .6 6 2 0 .0 0 1 6 7 6 6 ,6 6 7 0 0 1 9 6 6 8 3 .1 1 7 .8 5 0 1 8 .5 0 0 1 9 .2 3 0 2 0 .0 0 0 2 0 .9 0 0 2 1 .8 0 0 2 2 .7 0 0 S am p le C a l c u l a t i o n , o f B u lk M o d u lu s f o r n - P e n t a n e U s in g B u s h ie r * s E q u a t io n o f S t a t e : F o r Tr = 0 .8 0 @ Pr = 1 , a n d z e = 0 ,2 6 9 * a n d P c = ij.90 p s i a - M M ( f o ~ { r ) Z ( ° CPr - P o ) Bt = P c p 0 = - 2 0 + l 5 Tr = -2 0 + 1 5 ( 0 . 8 ) = - 8 ,0 M = - 3 . 1 +' 10Tr = - 3 . 1 + 1 0 ( 0 .8 ) = 4 .9 o = 8 . 2 % - I 8 . 0 ? z e - Ci|-.i4-82 - l i j . . l z c )T r c ° - = 2 .8 7 6 5 8 . 28I4 - 1 8 .0 7 ( 0 .2 6 9 ) - (i4 -.i4.82 - 3 4 .1 ( 0 .2 6 9 ) ) (0 . 8 0 ) F ro m t h e r e d u c e d d e n s i t y c h a r t s : 0 2 .3 2 9 Bt = (if.90) i i.9 ( 0 .5 ^ 7 5 ) 2 2 .8 7 6 5 - BT = (ij.90) ( 1 6 .3 ) (2 .3 3 1 5 5 ~ 1 8 * 6 0 0 p s i a APPENDIX B D a ta a n d S a m p le C a l c u l a t i o n s 1 . W a tso n E x p a n s io n F a c t o r M eth o d Z • C o m p a ris o n o f B u lk M o d u lu s V a lu e s 56 1 . W a tso n E x p a n s io n F a c t o r M eth o d F i g u r e 7 o n t h e n e x t p a g e i s d e r i v e d f o r i s o p e n t a n e , c a l c u l a t e d f ro m t h e m e a s u re m e n ts o f Y oung a n d e x te n d e d t o h i g h e r p r e s s u r e s b y t h e d a t a o f S a g e a n d L a c e y o n p r o p a n e a n d n - p e n t a n e . F i g u r e 7 i s a p p l i c a b l e o v e r t h e e n t i r e l i q u i d r e g i o n , e v e n a t p r e s s u r e s g r e a t l y a b o v e t h e s a t u r a t i o n p r e s s u r e . As d e r i v e d e a r l i e r : : I3 t = A p l o t o f Pr a s a f u n c t i o n o f w a t f i x e d t e m p e r a t u r e a l lo w s u s t o e v a l u a t e t h e s l o p e 9 Pj*/9w f o r v a r i o u s v a l u e s o f Pr * C a l c u l a t i o n s w e re p e r f o r m e d f o r Tr = 0 . 6 6 , 0 .7 3 , a n d 0 . 8 8 . F i g u r e 7 r e s t r i c t s u s t o a r e d u c e d p r e s s u r e r a n g e o f fro m O.lf.0 t o 5« OP i n t e r e s t a r e t h e r e d u c e d p r e s s u r e s 1 , 2 , an d ij.. FIGURE 7 Watson Expansion F actor as a Function © f Reduced Pressure and Reduced Temperature 1.0 .8 .7 .■6 . 5 . b i This p lo t has been reproduced from P rop erties o f (rases and L iquids C£) page 5 > 6 * G r i t . p_=Q#^ 08 06 .12 58 S a m p le c a l c u l a t i o n a t Tr = 0 .6 6 f o r n - p e n t a n e (P c = 1+.90 p s i a ) B a t a t a k e n fro m F i g u r e 7 f o r a p l o t o f Pr v e r s u s w: w Q.l|£> 0 .1 1 9 0.80 1.00 0.120 1.20 2 .0 0 0 .1 2 1 3 .0 0 0 .1 2 2 1^.00 0 .1 2 3 $ .0 0 0 .1 2 ij. F i g u r e 8 o n t h e n e x t p a g e i l l u s t r a t e s t h i s p l o t . T h is i s t h e a c t u a l s c a l e u s e d , a n d t h e s l o p e s w e re t a k e n d i r e c t l y fro m i t . Pr w (9 P r /£ w ) T Bt = P0w (2Pr / g w ) T p s i a 1 0 .1 2 0 808 2 0 .1 2 1 1 0 0 0 If. 0 .1 2 3 1 0 0 0 if.7,500 $9,200 60,300 FIGTJKE Seduced P re ssu re a s a F u n ctio n o f th e W atson E xpansion F a cto r 6 0 2 . C o m p a ris o n o f B u lk M o d u lu s V a lu e s P'r B ( l ) B (2 ) b (3 ) X 0 » 6 6 1 3 1 ,7 0 0 % 6 9 ,8 9 5 1 l}-7,500 2 if-7,300 71}., 720 5 9 ,2 0 0 k 71}-,100 7 9 ,7 7 9 6 0 ,3 0 0 0 .7 3 1 3 1 ,9 0 0 if.6 ,l}JL8 1}4,300 2 31}-, 60 0 l}-5,lf.23 5 5 ,7 0 0 k 51}-, 100 55,501}. 5 6 ,7 0 0 0 .8 8 1 1 0 ,3 1 0 13,571}- 1 2 ,0 0 0 2 11}., 200 1 9 ,2 3 0 1 2 ,5 5 0 If- 2 6 ,0 0 0 3 1 ,2 1 5 3 8 ,6 0 0 K eyr B ( l ) C a l c u l a t e d fro m t h e r e d u c e d d e n s i t y c h a r t s ( c o m p u te r) B (2 ) C a l c u l a t e d fro m e x p e r i m e n t a l P-V d a t a (c o m p u te r ) B (3 ) C a l c u l a t e d b y t h e W a tso n E x p a n s io n F a c t o r M e th o d I APPENDIX C M i s c e ll a n e o u s I n f o r m a t i o n 1 . N u m e ric a l D i f f e r e n t i a t i o n U s in g U n e q u a lly S p a c e d P i v o t a l P o i n t s 2 , R e s u l t s a n d C o m p a ris o n w ith . S lo p e s C a l c u l a t e d b y . D o u g la s - A v a k ia n M eth o d o f N u m e ric a l D i f f e r e n t i a t i o n 6 1 » 62! 1 . N u m e ric a l D i f f e r e n t i a t i o n U s in g U n e q u a lly S p a c e d P iv o t a l P o i n t s (U se o f t h e T a y lo r S e r i e s E x p a n s i o n ) (8 ) F o r e q u a l l y s p a c e d p i v o t a l p o i n t s y ( x + h ) — y ( x ) + hy* (x ) + V J.-LS.I + e tc * 2 1 31 T h is e x p a n s io n c a n b e a p p l i e d t o t h e c a s e o f n o n - e q u a l l y s p a c e d p i v o t a l p o i n t s . C o n s id e r t h e f i g u r e b e lo w t a y y = f ( x ) y r y± a h - h E q n . 1 y r = y ( x + a h ) = yj_ + a h y i + a2h -% i + a 3 h 3 y { ' * + ' ’ ' 5 2 k E q n . 2 y x * y ( x - h ) p t i q t t t i. i t i i = y i - h r i * L a . . _ + 2 6 2 + e t c . »i t yr “ yi = (a + D&yi + (a2 ~ D^yi* + (a^ + Dfr'Vi + etc. ’ 2----------- ------ ------g-------- — ’ = yr - y i . Cl - a)hyi' _ C1+ ( a + l ) h Cl + a ) (6 ) 111 i + e t c . E l i m i n a t i n g y £ ’ b e tw e e n e q u a t i o n s 1 a n d 2 we g e t : To g e t t o e q u a t i o n 3 , t h e te r m - { 1 + a ^ h ^ y ^ * 1 (X + a ) ( 6 ) + e t 0 * w as c o n s i d e r e d n e g l i g i b l e a n d d r o p p e d . T h is te r m i s c a l l e d t h e e r r o r . e r r o r — ^ 0 a s h ^ ^-0 f o r a = 1 ( e q u a l i n c r e m e n t s ) a n d f o r a = 1 ( u n e q u a l i n c r e m e n t s ) T h is d i f f e r e n t i a t i o n t e c h n i q u e i s u s e d t o co m p u te t h e s l o p e a t a p o i n t w hen i n f o r m a t i o n i s know n a b o u t a p o i n t t o t h e r i g h t a n d t o t h e l e f t o f t h i s p o i n t . A s a m p le c a l c u l a t i o n w as p e r f o r m e d o n t h e r e d u c e d d e n s i t y d a t a f o r Tr = 0 .6 6 a s a n a l t e r n a t e t o t h e D o u g la s - A v a k ia n m e th o d . F o r e x a m p le , a t Pp = 2:: y-t = 1 x T « 0 . 14.2160 h = 0 .0 0 5 5 6 y ± = 2 x ± = O.lj.2716 a h = 0 .0 0 6 1 a = 1 .1 0 3 y r = 4 x r = O.ij.3329 y i = s l o p e = ( 4 - d - C l . l 0 3 ) 2 )C 2 ) - ( 1 . 1 0 3 ) 2 ( 1 ) ) = 250 T he D o u g la s - A v a k ia n m e th o d g i v e s t h e v a l u e f o r t h i s s l o p e a s 2 2 2 *4 0 . S i m i l a r c a l c u l a t i o n s w e re p e rf o r m e d f o r r e d u c e d p r e s s u r e s o f 4* 1 5 » 2 0 , a n d 2 5 . 64 2 . R e s u l t s a n d C o m p a ris o n w i t h S lo p e s C a l c u l a t e d b y t h e D o u g la s -A v a k ia n M ethod o f N u m e ric a l D i f f e r e n t i a t i o n R e s u l t s an d c o m p a r is o n : (Tr = 0 .6 6 ) ^ r S lo p e d S lo p e ^ 2 2j?0.0 0 2 2 2 .4 0 4 3 3 3 .5 0 3 i|.8 .2 5 6 5 6 7 .0 0 5 0 9 .1 7 10 9 5 2 .0 0 8 7 9 .6 0 15 1 3 1 5 .0 0 1 3 3 6 .3 5 20 1 4 9 3 .0 0 3499 .8 0 25 1 5 6 1 .5 0 1 5 0 0 .2 0 K e y t S lo p e d C o m puted b y m e th o d o f u n e q u a l l y s p a c e d p i v o t a l p o i n t s S lo p e ^ C o m p u ted b y t h e D o u g la s - A v a k ia n m e th o d ( u s e o f e q u a l l y s p a c e d p i v o t a l p o i n t s ) •T he s l o p e s a t Pr - 1 an d 30 c o u ld n o t b e c a l c u l a t e d b e c a u s e p o i n t s w e re n o t know n a t t h e l e f t o r r i g h t o f t h e s e p o i n t s , r e s p e c t i v e l y . I n o r d e r t o u s e t h i s n u m e r i c a l m e th o d , t h e o r i g i n a l d a t a m u s t b e u s e d . One c a n n o t u s e p o i n t s i n t e r p o l a t e d b e tw e e n t h e a c t u a l d a t a p o i n t s .

Asset Metadata

Creator Koppany, Charles Robert (author)

Core Title Isothermal bulk modulus of liquids

Contributor Digitized by ProQuest (provenance)

Degree Master of Science

Degree Program Chemical Engineering

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag engineering, chemical,oai:digitallibrary.usc.edu:usctheses,OAI-PMH Harvest

Language English

Advisor Rebert, Charles J. (committee chair), Lockhart, Frank J. (committee member), Partridge, Edward G. (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c20-311260

Unique identifier UC11259051

Identifier EP41781.pdf (filename),usctheses-c20-311260 (legacy record id)

Legacy Identifier EP41781.pdf

Dmrecord 311260

Document Type Thesis

Rights Koppany, Charles Robert

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus, Los Angeles, California 90089, USA