WATER JET CONFERENCE - Waterjet Technology Association

2
x 2 ≅ ( e − 2 p + w)/k2 (2-8)
Substituting (2-6), (2-7), (2-8) back into (2-5) and rearranging in terms of Qp gives
p =
n.(1− A) + s(1+ A) + B( e + w)
2(1+ B)
Where A=h/r, B=(h/k)^2
58
(2-9)
e and w are obtained by performing 2nd order Taylor expansions about Q e and Q w
respectively (figure (3)).
w = w ′ − S1
e = e ′ − S1
e ′
r
w ′
r
+ S12
2
+ S12
2
2
w ′
r 2 (2-10)
2
e ′
r 2 (2-11)
The derivatives in equations (2-10), (2-11) are obtained by consideration of 2nd
order Taylor expansions for n w ′ , s w ′ , n e ′ and s e ′ about
For exammple:
w ′ and e ′ respectively.
n w ′ = w ′ + hw
s w ′ = w ′ − hw
w ′
r
w ′
r
+ hw2
2
+ hw 2
2
2
2
w ′
r 2 (2-12)
w ′
r 2 (2-13)
Independently eliminating 1st and 2nd order derivatives from (2-12), (2-13) gives:
similarly
2
w ′
r
w ′
r 2
e ′
r
n w ′ − s ′
= w
2hw
( n w ′ + s w ′ − 2 w ′ )
=
hw
n e ′ − s ′
= e
2he
(2-14)
(2-15)
(2-16)