Formation of bubbles at single nonwetted nozzles in mercury
- PDF / 208,167 Bytes
- 2 Pages / 604 x 784 pts Page_size
- 66 Downloads / 202 Views
=(2n+ 1)"
j=rt
J"
k+7+
)
1 "
[9]
T h i s m e t h o d for the d e t e r m i n a t i o n of the nth L e g e n d r e coefficient is dependent only on the coefficients of the nth L e g e n d r e p o l y n o m i a l and the coefficients of the p o w e r s e r i e s g r e a t e r than or equal to n. F o r e x a m p l e , to c o n v e r t a s e c o n d o r d e r power s e r i e s of the f o r m 4+2x+
3x 2
[10]
to a s e c o n d o r d e r modified L e g e n d r e p o l y n o m i a l s e r i e s , the c o e f f i c i e n t s Co, C1, and C2 m u s t be e v a l u ated. F r o m Eq. [6a], it is s e e n that Bo = 1 and hence 2
1, =1.
i
:6.
[11]
F o r the coefficient C~, we find that Bo = - 1 and B~ = 2 f r o m Eq. [6hi, which gives 2
c,=3.
Aj.
--2.5.
[12]
e s t a b l i s h e d by S h a r k e y e t al, but as an e m p i r i c a l r e p r e s e n t a t i o n the L e g e n d r e s e r i e s has m a n y m o r e advantages. Eq. [9] also allows for a slightly d i f f e r e n t a p p r o a c h in d e t e r m i n i n g the L e g e n d r e c o e f f i c i e n t s for a set of d a t a . S t a r t i n g with a set of o r i g i n a l data or a set of d u m m y " d a t a p o i n t s " as suggested by Bale and P e l ton, one can d e t e r m i n e a l e a s t s q u a r e s p o l y n o m i a l r e g r e s s i o n f r o m a s t a n d a r d r o u t i n e found in m o s t of the s t a t i s t i c a l p a c k a g e s a v a i l a b l e in c o m p u t e r c e n t e r s . The r e s u l t f r o m t h i s p o l y n o m i a l r e g r e s s i o n is a s i m ple power s e r i e s . It can then be e a s i l y c o n v e r t e d into a L e g e n d r e s e r i e s by Eq. [9]. F o r low o r d e r r e p r e s e n t a t i o n s , this s e e m s to be a s i m p l e r method than that of a l e a s t s q u a r e s r e g r e s s i o n u s i n g the L e g e n d r e polynomials. T h i s study was p e r f o r m e d by P . M. N. in an U n d e r g r a d u a t e R e s e a r c h P a r t i c i p a t i o n P r o g r a m at Lehigh U n i v e r s i t y . A p p r e c i a t i o n is e x p r e s s e d to the National S c i e n c e F o u n d a t i o n (under G r a n t E P P 7 5 - 0 4 6 9 5 ) for s u p p o r t of this w o r k . 1. C. W.Baleand A. D. Pelton:Met. Tran~, 1974,vol. 5, pp. 2323-37. 2. J. D. Esdaile:Met. Trans., 1971,vol.2, pp. 2277-82. 3. R. L. Sharkey,M. J. Pool and M. Hoch:Met. Tran~, 1971,vol.2, pp. 3039-49.
And f i n a l l y for C2, we know that Bo = 1, B~ = - 6 , and Be = 6 f r o m Eq. [6c]. T h e r e f o r e , 2
Cz = 5 "j
(1
Aj 9 1 + j
0
2 +j +
= 0.5.
[13]
The e x a m p l e power s e r i e s of Eq. [10] is the s a m e as the L e g e n d r e s e r i e s 6 + 2 . 5 P l ( x ) + 0.5P2(x).
[14]
A s i m p l e c o m p u t e r p r o g r a m can be w r i t t e n which u s e s Eq. [9] to c o n v e r t the c o e f f i c i e n t s of a power s e r i e s into the L e g e n d r e c o e f f i c i e n t s . As a r e s u l t , s o l u t i o n t h e r m o d y n a m i c data which have a l r e a d y been e x a m i n e d and put into a power s e r i e s r e
Data Loading...
