Lynne Wong's PhD thesis

mW K2aw
+ K1
=
1800 +
m
2 2
2
K2
( aw
) + K1K2aw
− K1K2
( aw
)
( 1 − K a ) ( 1 K K a )
2
a w
⎡ 1 + ( K K − K ) a − K K ( a )
=
=
⎢
⎣
2
2
w
1
2
1
2
w
⎤
⎥
⎦ 1800
2
1 2 2 w 1 2 w W
K
( + K K )
+ K K
( K1K2
+ K2
) aw
( K + K K )
2 1 2
1800
2 1 2
1800
2
2 2
1K2
( aw
)
( K + K )
W W
W K
+
−
1800 K
K
The parameters K 1 , K 2 and W can then be calculated from their algebraic relationship to b,
c and d.
2
1
2
hence
b
c
W ( K1 K2
− K2
) 1800( K2
+ K1K2
)
= ×
= K1K2
− K2
1800( K2
+ K1K2
) W
(6)
and
d
b
=
− W K
1800( K +
2
2
1 K2
1800( K2
+ K1K2
)
×
K K )
1
2
W
=
K K
1
2
2
(7)
from equation (6), K 2 =
c / b
( K ,
1)
1 −
substituting K 2 into equation (7),
K1
( c / b)
( K − 1)
1
1
K
1
( K − 1)
2
2
= d / b
d / b
= = z
2
( c / b)
2
zK1 − ( 2z
+ 1) K1
+ z =
0
2
The quadratic equation K 1 Ax + Bx + C = 0
was found to be A = z, B = − 2z − 1 and C = z
and the root of the equation is
−
B ±
2 −
B
2A
4AC
K 1 =
±
2z
+ 1 ±
( − 2z
− 1)
2z
2 −
4z
2
The positive value is taken for K 1 , and this value was used to calculate K 2 from the
equation given above.
W is calculated from the equation b =
W
1800 K K
( + K )
2
1
2
W = b × 1800 ( K + K K )
2
1
2
265