Proc Eng I Modelling
151
Transcript of Proc Eng I Modelling
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# + ##% #
) #! #*
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% 5 ( %% % % & 67 %#%# ) 607 %!% , !% 4 8 ! %% !
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* 9 : ( * 9 +
#
; # < ( % # " = %
9 : ! *
$ # > 10? %
# < ! # " * # > * # @ # ## #!
6 7 " : 8 8 A 8 8 8 0 6 A( 8 8 7 : " B @ 6 A 7 8 8 0 " 3 . .8 8 8 . : >
8 8 (A $ : 6(7 6C78 6178 0 67 A B
Aζ−→ 2B
ζ & A < ( 8 6 7 " 67 1 A B8 67 8 67 : 6C7 (
#
67 67 6 7
6#7
67 1
B : A
67 678 8 6#7 B8 A8
8 A 0 A BB Xi 6i ∈1, 2, 3, 4, 57, X1,A = 100 .D8 X1,B = 0 .D8 pR =
4 8 TR = 320?8 . TS = 345?8 A ηA = 0.988 0 8 A Xi,A8 Xi,B 8 ζ 8 pS . E " TR
pR 8 xA xB F B
ζ = 0.93 exp
(−0.76
pRTR
− 0.22xBxA
)
/0 . :
PyA = γAxApA
PyB = γBxBpB
x 8 y 8 p 0 T
lg pA = 2.033− 77.4246
T − 230
lg pB = 1.0044− 123.14
T − 230
' γ
lg γA =0.176(
1− xAxB
)2
lg γB =0.176(
1− xBxA
)2
) :
2 . ( 8 x = g(x)B
X2,A = X1,A +X4,A
X2,B = X1,B +X4,B
X3,A = (1− ζ)X2,A
# X3,B = X2,B + 2ζX2,A
ζ = 0.93 exp
(−0.76
pRTR
− 0.22X2,B
X2,A
) X4,A = X3,A −X5,A
% X4,B = X3,B −X5,B
! X5,A = (1− ηA)X3,A
* X5,B = X5,ApAγApBγB
X4,B
X4,A
pA = 102.033− 77.4246
TS − 230
pB = 101.0044− 123.14
TS − 230
γA = 10
0.176(1− X5,A
X5,B
)2
γB = 10
0.176(1− X5,B
X5,A
)2
# pS =γAp
AX5,A + γBp
BX5,B
X5,A +X5,B
- 8 (8 #
" B 1 n X2,A X2,B X3,A X3,B X4,A X4,B X5,A X5,B
# # # #%! # ** % #*# * #! *! # ! % # # !! * # !% ! #! # ## ! # #%!* %% # ##% !% ## ! # ! ! *!% % # % ## #! # #% % *%%* # # # ##! * # ## % *%%
8 8 8 . :
8 .@ A 0 - ' 8 8 !
C %G 8 G ) . %
0 4 0. " (
< ( - ' .8 0
" 8 8 08 4 8 '
67 A8 8 8
2 ( → B ! Xki,A = Xbe1,A +Xbe2,A
Xki,B = Xbe1,B +Xbe2,B
"
%
xBxA
=Xbe,B
Xbe,A
ζ = 0.93 exp
(−0.76
pRTR
− 0.22xBxA
) Xki,A = (1− ζ)Xbe,A
# Xki,B = Xbe,B + 2ζXbe,A
# $"
pA = 102.033− 77.4246
TS − 230
pB = 101.0044− 123.14
TS − 230
B xA8 xB = 1− xA# VA = ηAXbe,A
LA = (1− ηA)Xbe,A
LB =xBxA
LA
% VB = Xbe,B − LB
! γA = 10
0.176(1− xA
xB
)2
* γB = 10
0.176(1− xB
xA
)2
yA =VA
Vki,A + VB yB = 1− yA ε = |yAγBpBxB − yBγAp
AxA|
$ ε < %
# x()A =
Lki,A
Lki,A + Lki,BH x
()B = 1− x
()A
xA = 5 (xA, x()A ) H xB = 1− xA
1 #% pS = γAp
AxA + γBp
BxB
% & 9 6#7 X4,A = 141.64 kmol/h8
X4,B = 95.655 kmol/h 67 : 6 7
" . 6#∗78 678 . " . A 6#7 6#∗7 (
!
" B 1 n X2,A X2,B X3,A X3,B X4,A X4,B X5,A X5,B
## * # #%! * # ! % %% # ## !% ** % *! # % % #* *%#* !# # * # %! ! ##% *%
$ 8 6#78
1 " * ) 0 . %
%
- A 8 A 6 7 (8 A 6-78 6<78 6)78 8 678 6 78 6#78
A B C67 67 6 7 6#7
B - A
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( 8 C 67 A8 . A 8 67 8 8 (8 ( A
3 ( 0 67 8
*
67 67 6 7
6#7
67C 1 .
B ( A
#
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#
%! *
B (
0 (8 ( ' " 67
- ! " 6 ,. 7 $ . 8 . 8 ) 67 0 ) 6 (7 B I8%J8 I*8J8 I8 8*8!J8I8 86*878!J8 I8 8*8!J8 I8 86*878!J8 I88 8*8!J8 I88 86*878!J
%( 3 .8 $ . 8 6 7 8 "
) ( A ' 0 3 ' - .
- 8 0 0 & 0 8 0 0 '
" ( ! B& Xki,A = Xbe1,A +Xbe2,A
& Xki,B = Xbe1,B +Xbe2,B
B 5
" B& Xki,A = (1− ζ)Xbe,A
&# Xki,B = Xbe,B + 2ζXbe,A
B
ζ = 0.93 exp
(−0.76
pRTR
− 0.22Xbe,B
Xbe,A
)
# $ B& VA = ηAXbe,A
& LA = (1− ηA)Xbe,A
&% VB = ηBXbe,B
&! LB = (1− ηB)Xbe,B
B
pA = 102.033− 77.4246
TS − 230
pB = 101.0044− 123.14
TS − 230
B xA8 xB = 1− xA
# γA = 10
0.176(1− xA
xB
)2
γB = 10
0.176(1− xB
xA
)2
α =γAp
A
γBpB% ηB =
1
1 + α
(1
ηA− 1
)! VA = ηAXbe,A
* LA = (1− ηA)Xbe,A
VB = ηBXbe,B
LB = (1− ηB)Xbe,B
yA =VA
VA + VB yB = 1− yA# ε = |yAγBpBxB − yBγApAxA| $ ε < *
x()A =
Lki,A
Lki,A + Lki,BH x
()B = 1− x
()A
% xA = 5 (xA, x()A ) H xB = 1− xA
! 1 * pS = γApAxA + γBpBxB
1 0 pS ηB
)* & - = A
" B 1 n X2,A X2,B X3,A X3,B X4,A X4,B X5,A X5,B
## *! ##* * ## *! ! *! !* #! # !%% !* #! *%* # # # ! *% # ! *%# # ! # !* # *%#!# # % # !! # *%#
6& &!7B⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
1 0 0 0 −1 0 0 00 1 0 0 0 −1 0 0
−(1− ζ) 0 1 0 0 0 0 0−2ζ −1 0 1 0 0 0 00 0 −ηA 0 1 0 0 00 0 0 −ηB 0 1 0 00 0 −(1− ηA) 0 0 0 1 00 0 0 −(1− ηB) 0 0 0 1
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦∗
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
X2,A
X2,B
X3,A
X3,B
X4,A
X4,B
X5,A
X5,B
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦=
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
X1,A
X1,B
000000
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
. = - =K K LKK . =M N= .= 0
=K 0 '( (8 ζ %8 ηB H ' ( : 0 A " 0 0 '0 (B ζ ηB " 0
1 ) . %
#&
)' ( = A 8 # #$ " 0 8
" . 8 8 8 8
#
U1
U2
R
S1
S2
B 0A ( A
0A ( + " 67 = (0 = 0 : E1
E2 2× 2 " RF B
R =
(1− ζ 02ζ 1
)
" .F S1 S2 B
S1 =
(ηA 00 ηB
)
S1 =
(1− ηA 0
0 1− ηB
)
0A 8 8 0A ( ,
" B
- . " 5 8 y = ax z = by z = abx
- . " / 8 y = ax+ bx z = (a+ b)x
- . " 0 $ y = ax + by
y =a
1− bx $ ( 8
B y = (I−B)−1
A x
" + B X2 = (I − E2 S1 R)−1E1X1 8X3 = "X28 X4 = S1X38 X5 = S2X3
#
#
#
1− ζ
2ζ
ηA1− ηA
ηB
1− ηB
%B 0A (
xx y zza b a× b
!B 1 0A
y yxx
a
b
a+ b
*B 1 0A
xx yya
b a
1− b
B 1 0A
% & ' (
4 8 )0 : 8 ( p = 5 4 V 8 67 E8 0 & & F :
F = V − E
F = 0 $ F > 0 F 8 8 F " ' (
W18 t1 W28 t2
W38 t3
W48 t4
B 2 (
(8 : ( 6 7
) (* β ( Q A # U ' W1 A W2 A % W3 A ! W4 A * t1 t2 t3 t4
% & ' (
∆tln # ch : cc :
+ , * Q = UA∆tln
∆tln =(t1 − t4)− (t2 − t3)
lnt1 − t4t2 − t3
W1 =W2
# W3 =W4
Q =W1ch(t1 − t2) Q =W3cc(t4 − t3)% U = U(W1,W2,W3,W4, t1, t2, t3, t4, β)
+ *! W1 = . . .* t1 = . . . t2 = . . . t3 = . . . ch = . . . cc = . . .
% : 8 E =13 5 V = 158 F = 15 − 13 = 2" 0: : $ 8 0: β8 Q8 A8 U 8 W28 W38W48 t48 ∆tln
" = 6 L7
50 ( 0 ( ( β (8 00 (8 (8 8 00 0 8 0 0 (8 =8 8 6 (78
3 β 8 . $ A W3 . . . $8 8 A W2 Q 8 8
2 ( O
% & ' ( %
Q t4 ∆tln
W4 U
A
W2
f1
f2
f3
f4
f5 f6
f7
B - β W3
Q
t4
∆tln
W4
U
W3
W2
f1
f2
f3
f4f5
f6
f7
B β A
% & ' ( !
$ . )8 (8 67 6V = 88 E = 48F = 47B
f1( x18 x28 x3 ) = 0f2( x38 x48 x5 ) = 0f3( x48 x58 x6 ) = 0f4( x28 x78 x8 ) = 0
67
x1 x2 x3 x4 x5 x6 x7 x8
f1 f2 f3 f4
#B < 67
/ . " ( 678 - 67
x1 x2 x3 x4 x5 x6 x7 x8f1 f2 f3 f4
67
( " : ( 8 ( x1 8 f1( x1 - . x18 f1 6 '7
" ( : ( FF 6FF78 66 77
% & ' ( *
x1 x2 x3 x4 x5 x6 x7 x8
f1 f2 f3 f4
B :
x1 x2 x3 x4 x5 x6 x7 x8
f1 1
f2 f3 f4
6 7
" 66#7 +7
x2 x3 x4 x5 x6 x7 x8
f2 f3 f4
B , :
DDDDx1DD x2 x3 x4 x5 x6 x7 x8
DDDf1DD DDDDDD DDDDDD DDDDDD DDDD DDDD DDDD DDDD DDDDf2 DDDDD f3 DDDDD f4 DDDDD
6#7
% & ' (
" ( 667 ,7B
x2 x3 x4 x5 x6 x7 x8
f2 f3 f4
%B "
DDDDx1DD x2 x3 x4 x5 x6 x7 x8
DDDf1DD DDDDDD DDDDDD DDDDDD DDDD DDDD DDDD DDDD DDDDf2 DDDDD f3 DDDDD
f4 DDDDD 1
67
" B
x3 x4 x5 x6 x7 x8
f2 f3
!B "
% & ' (
DDDDx1DD DDDDx2 x3 x4 x5 x6 x7 x8
DDDf1DD DDDDDD DDDDDD DDDDDD DDDD DDDD DDDD DDDDD DDDDD
f2 DDDDD DDDDD 1
f3 DDDDD DDDDD DDDf4DD DDDDD DDDDDD DDDDD DDDD DDDDD DDDD DDDDDD DDDDDD
x4 x5 x6 x7 x8
f3
*B
#
DDDDx1DD DDDDx2 DDDDx3 x4 x5 x6 x7 x8
DDDf1DD DDDDDD DDDDDD DDDDDD DDDDD DDDDD DDDD DDDDD DDDDD DDDf2DD DDDDD DDDDD DDDDDD DDDDDD DDDDDD DDDD DDDDD DDDDD
# f3 DDDDD DDDDD DDDDD 1
DDDf4DD DDDDD DDDDDD DDDDD DDDDD DDDDD DDDD DDDDDD DDDDDD
/ ( 8 B x58 x68 x78 x8 < 8 6 01
$ 8 ( " 8 8 >
$ " : : .
- $ 1 + k 6k > 07 8 k : E
% & ' (
x1
x2
x3
x4
x5
x6
x7
x8
f1
f2
f3
f4
B -
. k 8 k
& V 8 67 E8 F = V − E " F k -= ( . : k 8 : F + k 8 k 8 0 k 8 F
2 ( & B
x1 x2 x3 x4 x5 x6f1 f2 f3 f4
2 V = 68 E = 48 F = 2 - 8 k = 1 " H 8 L 8 : B
x1 x2 x3 x4 x5 x6f2 f3 f4
- x38 x58 x6
% & ' (
x1
x2
x3x4
x5
x6
f2
f3
f4
B - f1
28 f1 . $
x1
x2
x3x4
x5
x6
f1
f2
f3
f4
B - f1
5 f1 28 . f1 & 8 (8 x1 - 8 x1 8 f1 f28 " . f28 2 8 0: B $ 8 8 $ x3 x5 x6
. & #
x1
x2
x3x4
x5
x6
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f2
f3
f4
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A 0 . 6: 78 8 0 ( (
2 8 . 0H
" 6 7 ' . , ( $ 8 $ 8 . 0 B A 8 ( 8 " A
) 6
7 9 j pj 8 Tj 8 A A Lj + sj 8 xj 8 A 6 7 Vj + Sj 8 yj 4 A hj 8 Hj 9 A sj Lj ( j + 1 $ 8 Sj Vj ( j− 1 j− 1 A A Lj−1
. &
jQjFj 8 Hj
zj
Lj
VjLj−1
Vj+1
sj
Sj
xj8 hj
yj 8 Hj
Tj8 pj
#B "0
j8 j + 1 Vj+18 = " 0 6 7
A Fj 8 zj 8 Hj j 2 j Qj 65 Qj 7
$ C 3C +9
" # 8 0 428 ( " 8 N
2-#3 " B 672 8 - 8 # 6 7 8 3 B
6P2P7B )
Lj−1xi,j−1 + Vj+1yi,j+1 + Fjzi,j
− (Lj + sj)xi,j − (Vj + Sj)yi,j = 0 (i = 1 . . . C; j = 1 . . .N)67
" : 0 M + 16P-P7B 9 " 8 Ki,jB
yi,j = Ki,jxi,j (i = 1 . . . C; j = 1 . . .N) 6%7
. &
6P#P7B 6 Sx Sy7
−1 +C∑i=1
xi,j = 0 (j = 1 . . .N) 6!7
−1 +
C∑i=1
yi,j = 0 (j = 1 . . .N) 6*7
6P3P7B
Lj−1hj−1 + Vj+1Hj+1 + FjHj +Qj
− (Lj + sj)hj − (Vj + Sj)Hj = 0 (j = 1 . . .N)67
" B
Lj−1 + Vj+1 + Fj − (Lj + sj)− (Vj + Sj) = 0 (j = 1 . . .N) 67
8
45 " 0 Ki,j8 %8
Ki,j = K(Tj, pj ,xj ,yj , i) (i = 1 . . . C; j = 1 . . .N) 67
B
hj = h(Tj, pj ,xj) (j = 1 . . .N) 6 7
Hj = H(Tj, pj ,yj) (j = 1 . . .N) 6#7
5 ( 8 N(4C + 9) N(3C + 5) 8 N(C + 4)
) :( L 8 C 0 A 8 8 8 pj : 6N: 7 C Fj 8 zj 8 Hj .8 N(C + 2) - : N − 2 60 78 A 8 : . " A 6 D7 A( R
. :
. & %
- 8 . "
- ( B 0 $ 8 8 : "
- 0 8 3 8
% ( . )0 - : / αi = αi/∗B
yi,j =αixi,j∑C
k=1 αkxk,j
6 & C A D A( RH A B
L0 = RD
V1 = (R+ 1)D
" A ' (B
Lj = Lj−1 + qjFj − sj (j = 2, . . .N)
Vj+1 = Lj + sj + Vj + Sj − Lj−1 − Fj (j = 1, . . . N − 1)
qj j
!
2 67 8 3 67 A :( $ Ki,j 6 7 :( ( yi,j = Ki,jxi,j 8 xi,j . " 8 :( 8 ( . B 8 ( 8 ( B j−18
. & !
j8 j + 1 " ' ( " & '8 8 B⎡
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
b1 c1a2 b2 c2
aj bj cj
aN−1 bN−1 cN−1
aN bN
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦×
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
x1x2xj
xN−1
xN
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦=
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
d1d2dj
dN−1
dN
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
8 # B⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
1 g11 g2
1 gj
1 gN−1
1
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦×
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
x1x2xj
xN−1
xN
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦=
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
u1u2uj
uN−1
uN
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
. ( " . B gj ujB
g1 =c1b1
u1 =d1b1
gj =cj
bj − ajgj−1(j = 2, . . . , N − 1)
uj =dj − ajuj−1
bj − ajgj−1(j = 2, . . . , N)
xN = uN
xj = uj − gjxj+1 (j = N − 1, . . . , 1)
"
( 8 , - . 08 . =" >
. & *
4 . 6 7BTj 8 Lj 8 Vj 8 hj 8 Hj 8 xi,j 8 yi,j 8 Ki,j 6Q8 8 8 C7 6LQ8 8 8 N78 N(3C + 5) " (8 C = 5 N = 70 6 8 7 # " ( 42 $ . 6xi,j 8 yi,j 8 Tj8 Lj 8 Vj77 N(2C + 3)
8 910 × 910 ( 5 6 $( (7 0 4 = 6 (78 8 8 (8 0 . ( 8 8 %8 ' D '
) . ( - 0 $( " (8
Ki,j =γi,j(Tj,xj)xi,jp
i (Tj)
pj
γi,j B
∂Ki,j
∂Tj≈ γi,j
pj
(dpidTj
)Tj
3 .
" : . *% $ H 8
" A
li,j = Ljxi,j
vi,j = Vjyi,j
$ 0 Sx Sy8 0
. &
(
Lj =
C∑i=1
li,j
Vj =
C∑i=1
vi,j
xi,j =li,jLj
yi,j =vi,jVj
" 28 -8 3
li,j−1 + vi,j+1 +Fjzi,j −(1 +
sj∑Ck=1 lk,j
)li,j −(1 +
Sj∑Ck=1 vk,j
)vi,j = 0 62i,j7
vi,j
C∑k=1
lk, j −Ki,j li,j
C∑k=1
vk,j = 0 6-i,j7
hj−1
C∑i=1
li,j−1+Hj+1
C∑i=1
vi,j+1+HjFj+Qj−hj(sj +
C∑i=1
li,j
)−Hj
(Sj +
C∑i=1
vi,j
)= 0
63j7$ ( Ki,j 8 hj8 Hj
N(2C+1) li,j 8 vi,j 8 Tj ( $(
" : A 60 B Lj Vj7 vj $ $( (8 ) A (
" 8 : / N *% 8 xi,j 8 Tj8 Vj / ( 6%78 A ( B
Lj = Vj+1 − V1 +
j∑k=1
(Fk − sk − Sk) 6R7
. &
( 6%7 6R7 678 6!78 67 28 #8 38 $ 0 N(C + 2)
$ Ki,j Tj xi,j H hj Hj Tj 8 " $(
1 Mi,j 8 Sj 8 Hj . = 8 ∆xi,j 8 ∆Tj 8 ∆Vj H Sj B
C∑i=1
∆xi,j = −Sj
08 B⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
∗ ∗∗ ∗ ∗
∗ ∗ ∗
∗ ∗ ∗∗ ∗
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦×
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
∆xi,1∆xi,2∆xi,j∆xi,N−1
∆xi,N
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦+
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
∗ ∗∗ ∗
∗ ∗
∗ ∗∗
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦×
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
∆T1∆T2∆Tj∆TN−1
∆TN
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
+
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
∗ ∗∗ ∗ ∗∗
∗ ∗ ∗∗
∗ ∗ ∗∗ ∗
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦×
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
∆V1∆V2∆Vj∆VN−1
∆VN
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦= −
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
Mi,1
Mi,2
Mi,j
Mi,N−1
Mi,N
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(i = 1, 2, . . .C)
B⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
∗ ∗∗ ∗ ∗
∗ ∗ ∗
∗ ∗ ∗∗ ∗
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
∆T1∆T2∆Tj∆TN−1
∆TN
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦+
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
∗ ∗∗ ∗ ∗∗
∗ ∗ ∗∗
∗ ∗ ∗∗ ∗
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
∆V1∆V2∆Vj∆VN−1
∆VN
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦= −
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
Hi,1
Hi,2
Hi,j
Hi,N−1
Hi,N
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
. &
" .8 : ∆xi,j 2N
#
6 7 . %8 3 :
" 8 8 ' . = ( 8
" <9 ( 6078 1 ( 68 78 $1 ( " 0: 8
7
$ ' " $ A Lj Vj : 8 Tj " . 67 8 <9 Ki,j 8 8
Ki,j . @ 0 i8 B xi,18 xi,28 xi,j 8 xi,N 8 j C 0 H 28 = H = 6Sx 7B
xi,j =
xi,j∑Ck=1 xk,j
(j = 1, 2, . . .N)
< 8 " yi,j =H Sy
" V1 A( L0 A D R Vj+1 N Lj 6jQ8 8 8 N − 17 67 67B
Vj+1 = Lj + sj + Vj + Sj − Fj − Lj−1
. &
)O
S
5
) Lj Vj
)
) hj Hj
< 8 Tj Ki,j
j = 1, 2, . . .N B
5= x
" 6i7B
xi,j 6 (7
Ki,j
Tj 8 Lj8 Vj
B <9
. & #
Lj =(Hj+1 − hj−1)Lj−1 + (Hj+1 −Hj)Fj − (Hj+1 − hj)sj − (Hj+1 −Hj)(Vj + Sj)
Hj+1 − hj
6 N 7" 8
8 8
- A :H : A : 8 A( :8
" H <9 > 68 7
#"
" 1 8 (8 8 8 A
' " (& 0 A Lj B
L j = Lj
C∑i=1
xi,j
" Ljxi,j A 8 1B F# "F 4 Vj ( B
Vj+1 = Lj + sj + Vj + Sj − Fj − Lj−1
Ki,j 0 x 8 = Ki,j
" 8 08 65 75 . = 0 ' ( " '
- (8 1 (
6#"
5 ( $08 2
. &
)O
S
5
1 Ki,j
C Tj 6 7
) hj Hj
) = y
5= x
) Lj Sx
) Vj
" 6i7B
xi,j 6 (7
Ki,j
Tj 8 Lj 8 Vj
B 1
1
H 0 "8 4 42
- <9 "8 : 18 $1 6$ 17
- 8 . BV y A 8
8 68 8 A 78 8 0 : $ A Vj Lj 8 Ki,j V/L 8
5 $18 88 0
(23. 6*%7 $1 ," xi,j (& 0 8 ' = 3 8 A Vj 18 Lj
/ ! ( 6*%7 5 A B
Aj =V j
Lj
=Vjσ
(V )j
Ljσ(L)j
6 Vj7 $ '(
" $
) > > " 0 64 2 2 05 5 61%5785 9::7B
8(. 3 B 1 " H B ±40% . 8 (
# 3B " . A L H B ±25%
1 %
Lj 8 Vj O
S
S
5
5
xi,j 8 yi,j O
) Lj
V j = Vjσj 8 6P 1P7
5= x
) yi,j = Ki,jxi,j
= yB 6σj =∑
i yi,j7
) γi,j
) Ki,j = γ(L)i,j /γ
(V )i,j
6i7B
xi,j 6 (7
Ki,j
Vj Lj
%B $1 0 (23.
1 !
Ki,j O
S
S
5
5
) γi,j Ki,j
5= x y
Lj 8 Vj O
Vj = AjLj
) Lj 6 (7
Aj =Vjσ
(V )j
Ljσ(L)j
yi,j = Ki,jxi,j 8 σ(L)j =
∑i xi,j 8 σ
(V )j =
∑i yi,j
" 6i7B
xi,j 6 (7
Ki,j
Vj 8 Lj8 Ki,j
!B $1 0
1 *
3B < = 8 B 0 " ' H B ±12%
# 9!( 3B 9L " 8 :H B ±6%
9 3B )F $ 8 :8 H ±3%
H 8 ( *%%B
" #B ) C8 *%%
T 44U 440 44U 44 V 44U 8 103 C V V V#9 8 103 C V V V*:8 103 C V V V
" $
% 5
8 8 > 6 7 ?8 ) 8 8
K1
K2=
(C1
C2
)a ( a : (8 ( '8 +$ .8 . 8 & 2
% 5
2 .8 L 0 -
1 #
" B ) (
( 00 ( ## 0(8 #
" B ) (
( . ! = %( #% % *
" ( 6 8 8 7 8 A . ( $ . K1 8 ( 8 8 I1 I28 . (
K2
K1=I2I1
@ ( ( 8 8
" (8 B
2 : #. - 6 5 ;2:#1 $ 67 . -8 #% > ( 88 . 8 67 . -8 ( ! 68 8 8 8 8 8 8 7 $ *
- " % 6 5 ;-"1 - ( 8 *%
1 #
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L 4W 51 51) )) . X X
501 . X3 , Y X
- .
$ 8 : ( " 8 8 8 (8 ,8 KB 8 LB 8 8 DB 8 pB 8 AB 6U8 8 8 m27
" %B )
K = 460L0.91D0.88p0.18
. K = 1260L0.68D0.96
K = 8L0.24D0.5p0.18
00 ( K = 240A0.59
K = 100H0.57D1.72p0.76
" . . 08 8 8 8 " . & , ,
2 B
K = f∑i
Ki
" 6 :7 ; 6 7
1 #
" !B ;
6f7 # #* %
(B
f = ftfpfm
ft >8 fp >8 fm
2
2 B
K =∑i
fiKi
04 . # F (
8 8 8 8 . . .
67 ( B 8 8 8 08 .8 8
67 B 8 8 .
6 7 B 8 8 (8 8 .8 8
< : .8 (B
9 B K = KBfmfp- B K = KB(fS + ft + fm) B K = KB(fS + ft + fm)2 (B K = KB(ft + fp)fm
KB 8 fm 8 fp 8ft 8 fS
< #
" ( H . 8 @ ( . =H ( . 5 $=* & 4 5 =# > 5 966
" . 8 8 ) %) 8 B
Toluene + H2 → Benzene + CH4
2Benzene Diphenyl + H2
2 . $ . 8 ( 8 )
- . 8 ! 8 G
" B
< 9 1 #
, &
3 8 % -8 8 8
9 4 8
(8 ( . - ( : 8 L 8
- : / 67 =6<7 2 ( 8 9: 8 8 8
" : 0 - 8 0
8 ? ##
HHin
T Tin
Hrecycle
Trecycle
Hblowdown
H 8 y
B
D1 0
)
*B
H28 CH4
H28 CH4
"
<=
1
<
)
+Q
+Q
+Q
+Q
−Q
"
<=
=
B
@ A O , E
: #
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H28 CH4
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<=
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% & 0E @ . O @ O
! @ . .O E
B 4 ) C 98 *%*
: #
H28 CH4
H28 CH4
"
<=
1
<
)
"
<=
=
B
: #%
B = . ( 4 / 8 **%
B 4 & 4,028 **
!B 4 4,028 *%
" ! B & 4 4,028*!!
!
6 7 "8 B
T I = T II 67
pI = pII 67
< :( T p8 = ( 8 , " ( G(T, p) >8 ) B
GI = GII 6 7
$ (8 B
µIi = µII
i (i = 1, 2, . . . , c) 6#7
3 6 7) 670 6#7 ' $
. . ' 28 8 : 0 A
#!
%% . & #*
) PVV VT 0 0 " . / V p T H ( B
p = p (V, T ) 67
) 6 7 6U 8 S8 A8 G78 S8 : > 8 1 . . 67 607
8 67
p =NkT
v≡ nNAkT
v≡ RT
V
$ : 0 U 8 8 H 8 62= 7 A8 6, 7 G B
X = X +∆X
X " ∆X 88 :
3 & ' p∗ 6p∗ = p7 T ∗ 6T ∗ = T 7B
X (T, p) = X (T, p) + ∆TpX
∆TpX ≡ X (T, p)−X (T, p)
" 8 B
(dA)T = −p (T, V ) dV
0 B
∆A = − ∫
p (T, v) dv
%% . &
2 v ( V
" @ = : " B 67 : :
/ & ( . ' & ' *
∆Tp∗A = −V∫
∞p (T, v) dv −
∞∫V ∗
p (T, v) dv
V ∗ =RT
p∗
28 " .
V∫∞
RT
vdv
∆Tp∗A = −V∫
∞
[p (T, v)− RT
v
]dv −
V∫∞
RT
vdv −
∞∫V ∗
RT
vdv
∆Tp∗A = −V∫
∞
[p (T, v)− RT
v
]dv −RT ln
V
V ∗
∆Tp∗S = −(∂∆Tp∗A
∂T
)V
=
V∫∞
[∂p (T, v)
∂T− R
v
]dv +R ln
(V
V ∗
)−R
V
V ∗ =V p∗
RT= Z
p∗
p
Z 0 B
Z ≡ pv
NkT≡ pv
nNAT≡ pV
RT
%% . &
/ & & ' 8 B
∆TpA = −V∫
∞
[p (T, v)− RT
v
]dv −RT lnZ 67
∆TpS = −(∂∆TpA
∂T
)V
=
V∫∞
[∂p (T, v)
∂T− R
v
]dv +R lnZ −R
3 . 8 (B
∆U = ∆A+ T∆S
∆H = ∆U +RT (Z − 1)
∆G = ∆A+RT (Z − 1)
%
@ (B
π =8ϑ
3 η − 1− 3
η2
$ ϑ = 0, 75 " π = 0.282463 - = 6ηL = 0.489631 ηV = 5.643057 6LQF&F8 VQF/F7 : " = " . B 3 8
- 8 0 ) 0 = π = 0, 282463
- 6:(7 '8 0= 6 7 A "
∆TV A
RT= − 9
8ϑη− ln (η − 1/3)
6∆TV A/RT 7 0 ϑ = 0.75 . " ηL = 0.489631 ηV = 5.64305 8
%% . &
π
π0
η0
0
# % ! *
B $ 3
π
∆A
RT
η0
0
0
0
0
0
# % ! *
B 1 62=7
%% . &
π
∆G
RT
η0
0
0
0
0
# % ! *
B 1 6,7
" A " 0
" ( 6η = 07 −1.13839 S .
p = −(∂A
∂V
)T
L 6,7 6∆TVG/RT 7 ηL = 0.489631 ηV = 5.64305B
∆TVG
RT= − 9
4ϑη+
3η
3η − 1− ln (η − 1/3)
" ϑ = 0.75 = 0 ! * L ηL = 0.489631 ηV = 5.64305 " L: . 8 8 ( 8 8 , . & & . 8 & & & .
%% . & #
&
" . 6 78 . ( , &
$ . 8 :B
∆GI
RT=
∆GII
RT6%7
) B
(dG)T =RT
pdp = RTd (ln p)
" B
G (T, p) = G (T, p)−G (T, p∗) = RT (ln p− ln p∗) = RT lnp
p∗6!7
& . ( f p " f . .
G (T, p) ≡ RT lnf
p∗6*7
+ ) 6!7 6*7 6 7
∆TPGB
∆TPG ≡ G (T, p)−G (T, p) = RT lnf
p≡ RT lnϕ 67
$ 8 67 : . < ϕB
ϕ ≡ f
p
5 6%7
f I = f II 67
lnϕI = lnϕII 67
" 67 8 67 8 ( 6 %< %<%78 (
& ' ( B
lnϕ ≡ lnf
p≡ ∆TPG
RT≡ ∆TPA
RT+ (Z − 1) 6 7
% ! & 2 , &
< 678 'B
lnϕ (V, T ) = −V∫
∞
[p (T, v)
RT− 1
v
]dv − lnZ + Z − 1 6#7
C ' V T
" # !
%
" : B
( = p =RT
V − b− a
V 267
" 8 b/V σ b " ( > " B
a =27
64
R2T 2c
pc
b =RTc8pc
Zc =3
8
8 B
( = π =8ϑ
8 η − 1− 3
η2
" F 28 a (B a = a(T )
" 0 . 4 B
!"# 6 917B
% ! & 2 , &
" p =RT
V − b+
a(T )
V 2 + 2bV − b2
$ % !&!# : !'# 6 1? 1?7B
#"> p =RT
V − b+
a(T )
V 2 + bV
a = a(T )
a(T ) = acα(T )
ac8 b ( (8 1?B
ac = 0.42748R2T 2
c
pc
b = 0.08664RTcpc
-( 0 B
α(T ) =[1 +(1−√Tr
)f(ω)]2
ω 0 . & 6 78 : %G B
ω ≡ − log10
(pT=0.7Tc
pc
)− 1
f(ω) B
f(ω) = 0.48 + 1.57ω − 0.17ω2
- ) 8 8
< : &
?
5 ( 0 ( $ 8
Z ≡ pV
RT
% ! & 2 , & %
( ρ ρ = 0 6V = ∞7B
Z ≡ pV
RT= 1 +
∞∑n=2
Bn
(1
V
)n−1
" ( ( ( B
?p
RT=
∞∑n=1
Bn
(1
V
)n≡ 1
V+
∞∑n=2
Bn
(1
V
)n
" ρ V ( ρ = 0 6V = ∞78 ": ' 8 8 B1 = 1H ' " '
Bn ≡ 1
n− 1
(∂n−1
∂ρ2
(p
ρRT
))ρ=0
" ' pVV VT 6 pVρVT 7 " ' . " ' BBn(T ) ' B2(T ) : : ' (
/ ( 8 (8 <@1 6($!&) 7 ( B7="
p =RT
V+B0RT −A0 − C0
T 2
V 2+bRT − a
V 3+aα
V 6+
C
T 2V 3
(1 +
γ
V 2
)exp(− γ
V 2
)" % :
3
& A ( $ *$+ B
@;1p
RT=
6
π
[ζ0
1− ζ3+
3ζ1ζ2(1− ζ3)2
+3ζ32 (1− ζ3)
(1 − ζ3)3
]
- $*$+ B
@;1p
RT=
6
π
[ζ0
1− ζ3+
3ζ1ζ2(1− ζ3)2
+3ζ32
(1− ζ3)3
]
%1 ) , ( & !
" , B
%#p
RT=
6
π
[ζ0
1− ζ3+
3ζ1ζ2(1 − ζ3)2
+ζ32 (3− ζ3)
(1 − ζ3)3
]
$
ζn ≡ πσn
6V≡ πσnρ
6
" σ 50 .
$%!
@ Ip8T J ( 678 67 $ V T P >
) ( = V T P 8 ( 3 H L ( " 8 " H 8
$ V 0 T p $ 6V1 < V2 < V37 6V1 < V37 > . A 67 Ip8T 8V1J 67 Ip8T 8V3JH @ Ip8T 8V1J Ip8T 8V3J . 6 7 67 678 ' 6 7 " . B
4 ?)0 ,B T p&. B V
$ T > Tc 6 7 3B
1 V p T $ 8 V V1 < V2 < V3 V1 < V3
%1 ) , ( & *
T
V
p
p
)
V1 V2 V3
pCTC
VC
#B 1 V 3 p
$ V $ V Vc 8 $ B
# 1 V1 8 V3
C 6 78 ' B
lnϕ1 ⇐ lnϕ (V1, T ) 67
lnϕ3 ⇐ lnϕ (V3, T ) 67
$ ' ( 6 > : 7 V1 V3 4 p T -0 8 ( 8 .
% $8 8 ' > p T 8
%1 ) , ( &
6" 7$ ϕ1 < ϕ3 8
0 0 ?)0 65 7 $
" 0 TB p 0 B
4 0 ,B p&. B 67 TB
$ p > pC 6 7 , T < TC ) ?)0# $ : ?)0 TB ⇐ T
3 T 8
4: T ?)0 > > ' 0 67 67 T H
< 6 7 pB . 0 T $ pB p < pC 8 . pB
4 0 ! ,B T&. B 67 pB
$ T > TC , p < pC ) ?)0# $ : ?)0 pB ⇐ p
3 p8
" T . ( (8 - 6 ,,./7B
%< . & -
4 ln p = A− B
T + C
A8 B8 C : 28 :
(
&
# !
6 5 60 7 + F 0 $ 8 N Ni . i . N − Ni " 67 8 "8 08 : 67 B
U id (V, T,y) =
c∑i
yiUi (V, T )
pid (V, T,y) =
c∑i
yipi (V, T )
& & - . & - ( .5 ( ( @ 5 ( 5 6 B %<1 %<<7
- B
V id (p, T,y) =
c∑i
yiVi (p, T )
( ( "
%< . & -
B
Sid =
c∑i
yiSi −R
c∑i
yi ln yi
Aid =c∑i
yiAi +RT
c∑i
yi ln yi
Gid =
c∑i
yiGi +RT
c∑i
yi ln yi
6, 7 8 8 8 ( B
µidi = µ
i +RT ln yi
8 6*7 B
µi ≡ G
i ≡ RT lnfi
p∗
6" 7
"8 . f idi " 0
: B
µidi ≡ RT ln
f idi
p∗≡ RT ln
fi
p∗+RT ln yi ≡ RT ln
yifi
p∗6%7
"8 " B
f idi = yif
i ≡ yiϕ
i p
2 . < ϕidi " '0
B
ϕidi ≡ f id
i
yip=fi
p≡ ϕ
i
$ ( F 8 ' B
f,idi = yip
ϕ,idi = ϕ,
i ≡ 1
# ' ! (
) " 6%7B
µi ≡ RT lnfip∗
6!7
%< . & -
id . fi (0 " : fi 9 p
∗ 6" 7 @ . pVV VTVy 9 : 6!7 > B
/ & ' ϕi '
ϕidi (8 . < ϕi
( Bfi ≡ yiϕip
< ' ϕi8 ( $8 H
9 ' L 0 " "8 '8 . B
p = p (V, T,y)
" 6#7 B
lnϕi (V, T,y) = −V∫
∞
[1
RT
(∂ [np (T, v,y)]
∂ni
)T,(nv),ni
− 1
v
]dv− lnZ 6*7
(
yi ≡ nic∑
j=1
nj
≡ ni
n
8 :8 5 v nv . >H v (
- ' - (8 . 8 " " 8 : a8 b8 c8 8
p = p (V, T ; a, b, c, . . . ) 67
a8 b8 c8 B
a = a(y), b = b(y), c = c(y), 67
%< . & - #
( 8 0 a8 b8 c8 " 67 67 . a8 b8 c8 y : 0 67 " & $ " (
- 8 ( < γi 0B
fi ≡ γifidi ≡ γiyif
i 67
' ( (8 > pVV VT 8 > ( " : " 65 - 7 (
> 8 ( ' 8 ' ( > B
γi ≈ γi (x, T ) 6 7
x y " (
- ' ( ( 6, 7
∆EG ≡ G−Gid = G−c∑
i=1
xiGi −RT
c∑i=1
xi lnxi
E -
9 ' ' H B
yiγifi ≡ fi ≡ yiϕip
28 ' 8 '
%< . & -
- : B
ai ≡ fifi
" ( ' B
ai ≡ γixi
"8 B
µi = µi +RT ln ai
# $ !
@ ' ( (8 . B
< 6 7 ' 8 ( ' $8 paref 8 " ' "
fLi
(T, paref ,x
) ≡ γi (T,x)xif,Li
(T, paref
)6#7
fLi (T, p,x) ≡ fL
i
(T, paref ,x
)ξ 67
ξ
ln ξ =
p∫paref
V Li (T, π,x)
RTdπ 67
- ξ 0( 8
- ' 0 , & " : " F T pi 8 " paref 6#7 0 67B
f,Li (T, pi (T )) = f,V
i (T, pi (T )) = ϕ,Vi (T, pi (T )) p
i (T )
fLi
(T, paref
)= ηf,L
i (T, pi (T ))
%< . & -
η 8 8 B
ln η =
paref∫pi
V Li (T, π)
RTdπ 6%7
- η 0( 8
4 67 ln ξ 6%7 ln η 0 ( H ( - ( V L
i 67 : > V L
i 6%7H V L
i ln ξ $ 8 B
ln ηξ ≈p∫
pi
V Li (T, π)
RTdπ ≈(p− pi (T ))V
Li (T, p)
RT
lnP ≡(p− pi )VLi (T, p)
RT
9 ( B
fLi (T, p,x) ≈ γi (T,x)xiϕ
,Vi (T, pi (T )) p
i (T )P 6!7
B
fLi ≈ γixiϕ
i p
iP 6!7
) / 0 8 : . 1 paref H 6!706!7" 8 F / . F H ( 6 7 $ 0
#" ) *
( '8 0 pi . " ( $ 8
%< . & - %
8 ( 0 6 7 8
$ 0 ( F $8 ( B . $ 8 : "8 0 . ' 67B
limxi→1
γi = 1 6*7
. ' B
limxi→0
γi = 1 6 7
" 8 : 8 ∞
L 8 '
0 8 f,Li
f∞,Li Hi $ / FF8
Hi : B
f∞,Li ≡ Hi ≡ lim
xi→0
fLi
γixi≡ lim
xi→0
fLi
xi
) / FF 8 8 $ 8 /FF : paref
fLi
(paref)= γixiHi
(paref)
6 7
< / FF 8 j "8 / FF . B
Hi (T ) = Hi
(p∀j)
p∀j (T ) ≡
∑∀j:
xjpj (T )
%< . & - !
6j ( 7 $ 8 0 8 p∀j : > F " / FF T p∀j paref 6 7B
ln fLi
(paref)= ln γi + lnxi + lnHi (T ) +
paref∫p∀j
V∞i
RTdπ
" p B
ln fLi (p) = ln γi + lnxi + lnHi (T ) +
p∫p∀j
V∞i
RTdπ
paref 8 : B
ln fLi (p) ≈ lnxi + lnHi (T ) +
(p− p∀j
)V∞i
RT
/
## +
) 8 : 8 ( miB
mi ≡ ni
nWMW
W ( 8 @ 6@78 MW
6 7 ( .D " 8 6 7D6. 7 -
µi = µrefi +RT ln ai
( ai = γimi 6 7
0 1 : B
limmi→0
γi = 1 6 7
%< . & - *
2 ( ( B
µidi = µ∞
i +RT lnmi
∞ : 4 1/MW
: " ( 8 8 6*7B
µidW = µ
W +RT lnxW
& 1− h |h| 1 ( −hB
lnxW = ln
⎛⎝1−∑
i=W
xi
⎞⎠ ≈ −
∑i=W
xi [= − (1− xW )]
µidW = µ
W −RT (1− xW ) = µW −RT
∑i=W
xi
" ( ( : : B
xi ≈ ni
nW
µidW = µ
W −RT1
nW
∑i=W
ni
, 6 7
gid = nW
⎡⎣µ
W −RT1
nW
∑i=W
ni
⎤⎦+∑
i=W
[ni
(µ∞i +RT ln
ni
nWMW
)]
$ : ': 6 7 6 78 8 8 0 < φ B
µi = µ∞i +RT ln (γimi)
µW = µW −RT
φ
nW
∑i=W
ni
%8 = - , & %
g = nw
⎡⎣µ
W −RTφ
nW
∑i=W
ni
⎤⎦+∑
i=W
[ni
(µ∞i +RT ln
γini
nWMW
)]
" ' 0 ' ( ( ,
'& !
3 5
" ( > 3 ( ζ B
ζn =π
6
∑i
σni
Vi≡ π
6
∑i
(σni ρi)
%. 5A 5
) ( & ai8 bi8 ci8 i8 0 ( a8 b8 c8 ( 5
. .
b ( , H H (8 B
b =∑i
xibi 6 #7
$ bij 8 . B
b =∑i
∑j
xixjbij
a B
a =∑i
∑j
xixjaij 6 7
aij
%8 = - , & %
9 ( - H . (8 b B
bij =
(√bi +√bj
2
)2
bij =
(3√bi + 3√bj
2
)3
- 6 #7" a
aij =√aiaj
6 7 $ ( >0 0> . 0 kij B
aij = (1− kij)√aiaj 6 7
" : 8 >
" . 8 (B
kij = kij +m(xi − xj)
kij = xihij + xjhji
kij = kij + nij(xi + xj)hijxi − hjixjhijxi + hjixj
" 0
- 0 6 7B
a =∑i
∑j
xiwjiaji
wji ≡ τijxj∑k
τikxk
. . 5 (8 6 #7 6 7 @ 67 '
lnϕi = − ln
(ZV − b
V
)+
biV − b
− 2√ai
V RT
∑j
[xj (1− kij)
√aj]
%: = - - %
/ ( ( " ' . 6 7B
B2(T ) =∑i
∑j
xixjB2,ij(T )
'& &
( ' . 0 B
RT ln γi =
(∂n∆EG
∂ni
)T,p, N
- 8 ( , ( + - 0 / - # (( . - ( - (. - >0 . - < ( 4( . . > . > .8 " 0 6 7 6 . &. 7 9 6 7
, - !
- 8 ( 0 @ ' ' qi " 0H 0 < > @ ' ' & B
Φi ≡ xiqi∑j
xjqj
- !&"#8 ( ( B
∆EG
RT∑i
qiΦi=∑ij
aijΦiΦj +∑ijk
aijkΦiΦjΦk + · · ·
> 8 a 8 8 ( : (8
%: = - - %
( B
∆EG
RT= (qAxA + qBxB)(2ΦAΦBaAB + 3Φ2
AΦBaAAB + 3ΦAΦ2BaABB)
B
∆EG
RT= AABΦ
2BxA +ABAΦ
2AxB
1 > < ( qB/qA ABA/AAB8 6 !7 < ( qB/qA VB/VA8 $$ 6!0 7B
ln γA =Φ2B
[AAB + 2ΦA
(ABA
ρBρA
−AAB
)]
ln γB =Φ2A
[ABA + 2ΦB
(AAB
ρAρB
−ABA
)]
< ( qB/qA 8 1 6!0 7B
2ln γA =x2B [AAB + 2xA (ABA −AAB)]
ln γB =x2A [ABA + 2xB (AAB −ABA)]
$ $ % !& !0'#8 28 ( (B∆EG
RT= xAxB
[AAB +BAB (xA − xB) + CAB (xA − xB)
2+DAB (xA − xB)
3+ · · ·]
6 %7 ( 0 :
,
3 ( 67B
( )
ln γA =AABx
2B(
AAB
ABAxA + xB
)2
ln γB =ABAx
2A(
xA +ABA
AABxB
)26 !7
" ) AAB N
%: = - - %#
ABA H : $
, . /
& 9 & 8 8 8 "8 & 6 7 - (8 . 5 H > > 8 > >- (8 & & 2 & ( 8 8
25 . *
" )0 ( " ' 0 z = 4 6 7
--
-
--
<
<<
<
B - <
)8 8 ( NA . A8 NB . B " NAA8 NBB8 NABH 8 uAA8 uBB8 uAB
%: = - - %
" ;* 5 1 " 0( 62 !0' 7 ( H 6 (7 - 8 .8 : . A . A " ( ( B
∆EG ≈WxAxB 6 *7
2 ' B
γA =exp
(Wx2BRT
)
γB =exp
(Wx2ART
)
" 1 6 *7 : $% (0
A ;! 5 1 50 κB
N∗AB =κ z
NANB
NA +NB
κ ≡N∗AB(NA +NB)
zNANB
- , 62 !0' 78 . 8 6 FF 7B
N2AB
(zNA −NAB)(zNB −NAB)= exp
(− 2w
zRT
)
$ (8 NAB B
2
κ= 1 +
(1 + 4xAxB
[−1 + exp
(2w
zRT
)])1/2 (
∆EG ≈ RTz
2
(xA ln
1− κxBxA
+ xB ln1− κxAxB
)
%: = - - %
ln γA =z
2ln
1− κxBxA
ln γB =z
2ln
1− κxAxB
$ ( (H
25 . B *
" . . = (8 0( > )23 )22 - 0 6 0. 7 B )23
)3 " 6 )228 )38 5228 )337. ( . > (8 . .
-
-
-
<<
<
B "0 --- <<<
" (8 0 ! % . 8 + " = 8 0 z
0 0 ' > = = " 8 > > > 0 " 0 B
∆EG = ∆EcombG+∆E
resG 6#7
%: = - - %%
ln γi = ln γcombi + ln γresi 6#7
$ . . ;18 ( 5κ = 1 B
∆EcombG = RT
(∑i
xi lnΦi
xi+z
2
∑i
qixi lnΦi
θi
)6#7
ln γcombi = 1− Φi
xi+ ln
Φi
xi− z
2qi
(1− Φi
θi+ ln
Φi
θi
)2 Φi 0 8 θi 0 8 ri 67 . iB
Φi ≡ xiri∑j
xjrj6#7
θi ≡ xiqi∑j
xjqj6# 7
8 0
N2AB
(zNAqA −NAB)(zNBqB −NAB)= exp
(−2wAB
zRT
)6##7
2 (B
∆EresG ≈ RT
z
2
(xAqA ln
1− κθBθA
+ xBqB ln1− κθAθB
)
ln γresA =z
2qA ln
1− κθBθA
ln γresB =z
2qB ln
1− κθAθB
κ ( 6##74 :H
:8 8 8 8
$ . B ; 18 > . 0 " ' ( 6 %7
%: = - - %!
-
-
-
--
-
--
<
<
<<
<
<
<<
%B ) B - <
," $ .(/
$ A B 0 678 . " 3 0 > A8 = " B "8 A B ;!,1 - A A A B 8 A : " A B 28 ( ( A B H ,
. > > ( " ( / & & ( & & A & , & 9 . $ AA AB ( A A 678
25 . *
$ N∗AB NAB
NBA H
NAB = N∗AB = NBA = z
NANB
NA +NB
%: = - - %*
-
∆EG ≈ ∆EA = RTz
2[xA ln (xA + xBABA) + xB ln (xB + xAAAB)]
ln γA =− z
2xA ln (xA + xBABA) +
z
2xB
[ABA
xA + xBABA− AAB
xB + xAAAB
]
ln γB =− z
2xB ln (xB + xAAAB) +
z
2xA
[AAB
xB + xAAAB− ABA
xA + xBABA
]
25 . B *
6 !"& 7 ( @ B
∆GE
RT= −xA ln (xA + ΛABxB)− xB ln (xB + ΛBAxA)
=A
ln γA = − ln (xA + ΛABxB) + xB
(ΛAB
xA + ΛABxB− ΛBA
xB + ΛBAxA
)
ln γB = − ln (xB + ΛBAxA) + xA
(ΛBA
xB + ΛBAxA− ΛAB
xA + ΛABxB
)
$
=∆GE
RT= −∑i
xi ln∑j
Λijxj
= ln γi = − ln
⎛⎝∑
j
Λijxj
⎞⎠+ 1−
∑k
Λkixk∑j
Λkjxj
Λij ≡ exp
(−uii − ujj
RT
)$ 8 Aij = uij − uii : "
8 . 8 8 0 .
!"# ( 8 0 α 0 " . " , "5 ; $
τij ≡ gij − gjjRT
%: = - - !
Gij ≡ exp (−αijτij)
.
") ∆GE
RT=∑i
xi
∑j
τjiGjixj∑k
Gkixk
") ln γi =
∑j
τjiGjixj∑k
Gkixk+∑j
Gijxj∑k
Gkjxk
⎡⎣τij −
∑mτmjGmjxm∑k
Gkjxk
⎤⎦
$ : Aij = gij − gjj α8 α :( 0.20 0.3
$+ !0 !# 66#7V6#77 " 6#7 0 8 0 ( 2 " C5$+C-) 6 0 7 . 8 $ B 6C 4%
∆EG = ∆EcombG+∆E
resG
∆EcombG = RT
(∑i
xi lnΦi
xi+z
2
∑i
qixi lnΦi
θi
)
∆GEres
RT= −∑i
⎡⎣qixi ln
⎛⎝∑
j
τjiθj
⎞⎠⎤⎦
ln γi = ln γcombi + ln γresi
ln γcombi = 1− Φi
xi+ ln
Φi
xi− z
2qi
(1− Φi
θi+ ln
Φi
θi
)
ln γresi = qi
⎡⎢⎣− ln
⎛⎝∑
j
τjiθj
⎞⎠+ 1−
∑k
τkiθk∑j
τjkθj
⎤⎥⎦
τji ≡ exp
(−uji − uii
RT
)
%: = - - !
- B
ln γcombi = ln
Φi
xi− z
2qi ln
Φi
θi+ li − Φi
xi
∑j
xj lj
li =
z
2(ri − qi)− (ri − 1)
6 0 74 r N q C5$+C-) H 0
. ( )0 z . 8 Aji = uji − uii :
4 ( ' ,
,# %
> = > 0 6R7 6Q7 - ( k 8 k i νkiH i :
ri ≡∑k
νkiRk 6#7
qi ≡∑k
νkiQk 6#7
" k i
Qik ≡ νkiQk
, = 8 08 k 4k 8 0 8 (8 )228 522 8 )38 328 )338 8 3 8 H 8 : 28 ( 0 R Q - 08 . = " Rk8 Qk8 Akm : > (
5 !"'# : 0 " ( "
%: = - - !
- ' ( B
ln γi = ln γcombi + ln γresi
k γi i i " Γk . T wB
wk ≡∑i
νkixi∑i
∑mνmixi
Γk = fk (w, T )
8 > > (
ln γrezi =∑k
νki
(ln Γk − ln Γ
(i)k
)
(i) iB
Γ(i)k = fk
(w(i);T
)
w(i)k ≡ νki∑
mνmi
- -3, 6 7 8 8 28 6 0. 6 *%7 C5$+C-) . C5$-)B 64%
ln γcombi = ln
Φi
xi− z
2qi ln
Φi
θi+ li − Φi
xi
∑j
xj lj
li =z
2(ri − qi)− (ri − 1)
0 z . 8 Φ θ : 6#7 6# 78 r N q 6#7 6#7 " B
ln Γresk = Qk
⎡⎣− ln
(∑m
ΨmkΘm
)+ 1−
∑m
ΨkmΘm∑nΨnkΘn
⎤⎦
%: = - - !
( m N n 8 B
Θm =wmQm∑nwnQn
(
Ψnm = exp
(−Anm
RT
)
C5$-) $ Rk Qk Amn = Anm - : : H : : (8 ( H
,0 1 2
" : ' ; 8 = 8 B 8 ;8 C/?C!8 C/!
( ) : 0 0 ; C/?C! 0 0 8 8 : 0 0 H " 0 0 ( ∆EG 0 8 (8 =8 : ( 0 0 = " ( C/! ( 8
,,
- ( 0 4 0 ' "
%6 4 , ( !#
( ) !
3 4
( ( 0 >
9 ( = 4 6 7 / $ : 6 7 = ( " 8 x = 1 y = 1 67 8 x = 0 y = 0 6 7
#
@ . 6x y7 6T p7 8 6 7
/
/ &
&
x, yx, y
pQ TQ
D
D
D
D
xx
yy
pT
xx
xx
y∗y∗
y∗y∗
pBTB
!B <
%6 4 , ( !
) p , x8 L 67 TB " = L 67 y∗ . x
/
/ &
&
x, yx, y
pQ TQ
D
D
D
D
xx
yy
pT
x∗x∗
x∗x∗
yy
yy
pDTD
*B
) T , x8 L 607 pB " = L 67 y∗ . x
$ B, y 6 /7 p8
L 67 TD 67 " = L 67 x∗ . y
, y 6 / 7 T 8 L 67 pD 67 " = L 67 x∗ .
%6 4 , ( !
y$ (
H . ' 8 L .B 8 08 8 8
;&1
@ . ; D&D1 0 z ( 8 0 ) 0 & p0T 0z . - T p p0T 0z 8 F 8 8 8 8 , ) q 0 z " 60=7 "8 q = 0 8 q = 1
" 0
, z8 p T 8 8 8 8 8 8 8 8 6 :7 " 8 8 . $ 8 8
3 A H = =
. 3 8 8 "8 8 8 A
" A 2 J 8h 8 H 8 fi vi A 8
%6 4 , ( !%
/
&
x, y
pQ
x∗ y∗
T
T
z
TB
TD
B
V
L
V + L x
y
p
p
T
TTz
B A
%6 4 , ( !!
F, z, J
L, x, h
V, y,H
p
T
TQ
B A
" B " A , 6:7 &. 5z8 p8 T J 8 V/L8 x8 y8 h8 H 0 6FF7
Az8 p8 J T 8 V/L8 x8 y8 h8 H 0 6FF7
Az8 J 8 T p8 V/L8 x8 y8 h8 H 0 Az8 p8 V/L8 J T 8 x8 y8 h8 H z8 T 8 V/L8 J p8 x8 y8 h8 H z8 p8 xi8 J T 8 V/L8 x8 y8 h8 H z8 T 8 xi8 J p8 V/L8 x8 y8 h8 H z8 p8 yi8 J T 8 V/L8 x8 y8 h8 H z8 T 8 yi8 J p8 V/L8 x8 y8 h8 H z8 p8 vi/fi8 J T 8 V/L8 x8 y8 h8 H z8 T 8 vi/fi8 J p8 V/L8 x8 y8 h8 H z8 yi8 vi/fi8 J p8 T 8 V/L8 x8 y8 h8 H
%6 4 , ( !*
3 ϕ/ϕ2 ϕ/γ γ/γ
@ 8 B
ϕ/ϕ B $ 67 B
yIi ϕIi = yIIi ϕ
IIi 6#7
' 67 0 . " B
KII/Ii ≡ yIIi
yIi=ϕIIi
ϕIi
≡ exp(lnϕII
i − lnϕIi
)6#7
$ 0 F B
yiϕVi = xiϕ
Li 6#7
yi 8 xi 8
Ki ≡ KV/Li ≡ yi
xi=ϕLi
ϕVi
≡ exp(lnϕL
i − lnϕVi
)6#7
ϕ/γ B $ 8 ( ( > B
yiϕVi p = γixiϕ
i p
iP 6#%7
" 8 8 pi " 9 ' . F y - ' . F xH i F8 ' F F 8 / B
Ki ≡ yixi
=γiϕ
i p
iP
ϕVi p
6#%7
γ/γ B $ ( ( B
xIi γIi = xIIi γ
IIi 6#!7
%6 4 , ( *
- ' . F F x B
KII/Ii ≡ xIIi
xIi=γIIiγIi
≡ exp(ln γIIi − ln γIi
)6#!7
4 ϕ/ϕ ϕ/γ 0 0 8 γ/γ 0 - ϕ/ϕ γ/γ $ 8 ϕ/γ 0 > 8 8 : - 8 0 8 ' H ( / ' " 0 + 5B
yip = γixipi 6#*7
Ki ≡ yixi
=γip
i
p6#*7
- 6#706#*7 (
f Ii = f II
i (i = 1, 2, . . . , c)
c ) 6T p7
8 . " 8 . c+1 < . H . c $ 0 (8 (8 : =67 0 $ 8 : 8 . H .
$ 8 8 c 8 8 " 8 8 8
%6 4 , ( *
3 4( ϕ/γ
6 5
$ B
Ki = Ki(T, p) =pip
" " : B
4 7 ,B T x&. B pB y,B 0
) pi . ) pBB
pB ⇐∑i
xipi
) B
yi ⇐ xipi
pB
#
" H $ 8
4 9 #$ ,B T y&. B pD x,B pD p
) pi . ) B
xi ⇐ yip
pi
) B
σ ⇐∑i
xi
%6 4 , ( *
# 5= B
xi ⇐ xiσ
) B
pB ⇐∑i
xipi
$ p pB : p B pD ⇐ p8
% 3B 4 p8
" 0 B
4 9 #$ ,B T y&. B pD x,B pD p
) pi . ) KiB
Ki ⇐ pip
) B
xi ⇐ yiKi
# ) B
σ ⇐∑i
xi
5= B
xi ⇐ xiσ
$ σ : p B pD ⇐ p8
% 3B 4 p8
$ 8 8 8 "
%6 4 , ( *
4 7 ,B p x&. B TB y,B TB T
) pi T ) B
pi ⇐ xipi
B
π ⇐∑i
pi
# $ π p : T B TB ⇐ T 8
3B 4 T 8 ) B
yi ⇐ piπ
$ π < p T 8 " 0 B
4 7 ,B p x&. B TB y,B TB T
) pi T ) B
Ki ⇐ pip
) B
yi ⇐ Kixi
# B
σ ⇐∑i
xi
$ σ : T B TB ⇐ T 8
% 3B 4 T 8
%6 4 , ( *#
5= B
xi ⇐ xiσ
$ σ < 1 T 8
%
" ' 8 < γi
4 7γ ,B p x&. B TB y,B TB T
) pi T ' γi T x ) B
Ki ⇐pi
p
) B
yi ⇐ Kixi
# B
σ ⇐∑i
xi
$ σ : T B TB ⇐ T 8
% 3B 4 T 8 5= B
xi ⇐xi
σ
%6 4 , ( *
$ σ < 1 T 8
4 9γ #$ ,B p y&. B TD x,B TD T H γi
) pi T ) B
xi ⇐ yip
γipi
B
σ ⇐∑i
xi
# 5= B
xi ⇐ xiσ
) γi T = ) B
π ⇐∑i
γixipi
% $ p π : T B TH ⇐ T 8
! 3B 4 T 8
$ π < p T 8
E ϕ/γ
$ 8 . '8 8 8 8 (
3" 4( ϕ/ϕ
' F 8 8 $ 6 7 8 '
%6 4 , ( *
8 8: y : ' " L
8 F
4 79ϕ % $ ϕ/ϕ,B 6T p7 6 7 yI 6 N. K7&. B 6p∗ T ∗7 6 7 yII
,B 6p∗ T ∗7 6p T 7H ϕI ϕII
) ' ϕIi
n n⇐ 0 ) B
Ki ⇐ ϕIi
ϕIIi
# ) B
yIIi ⇐ KiyIi
B
σ ⇐∑i
yIIi
5= B
yIIi ⇐ yIIiσ
% ) ' ϕIIi
! ) B
Ki ⇐ ϕIi
ϕIIi
* $ n = 0 3B ) 8 0
8 B
∆ ⇐∑i
∣∣yIIi − yeIIi∣∣
$ ∆ : δ #6 7
%6 4 , ( *%
3B B
yeIIi ⇐ yIIi
$ nBn⇐ n+ 1
6 7# $ σ : ε
6p T 7B p∗ ⇐ p T ∗ ⇐ T 8 3B 4 6p T 78 6 7
3# 4(
) .8 < .
& F 8 V 8 L 8 8 6 A 78 0 8 λ λ = L/F H
Fzi = V yi + Lxi (i = 1, 2, . . . , c)
F = V + L
( λB
zi = (1− λ)yi + λxi (i = 1, 2, . . . , c)
zi = (1 − λ)Kixi + λxi (i = 1, 2, . . . , c) 67
< ( 8 KiB
xi =zi
λ+ (1− λ)Ki(i = 1, 2, . . . , c) 67
yi =Kizi
λ+ (1− λ)Ki(i = 1, 2, . . . , c) 67
4 B∑i
ziλ+ (1− λ)Ki
= 1 6 7
∑i
Kiziλ+ (1− λ)Ki
= 1 6#7
$ Ki λ 3 :
%6 4 , ( *!
" B $ ψ(λ) : 67ψ(0) ψ(1) K=K ψψ ≡ ψ(0)ψ(1)< 0 < 0 > 0< 0 = 0 = 0< 0 > 0 < 0= 0 > 0 = 0> 0 > 0 > 0
6 7 6#78 . 67 67 3 6 λ7 H = 6 7 6#7 0 ≤ λ ≤ 1H ' " λB
ψ(λ) ≡∑i
yi −∑i
xi =∑i
(Ki − 1)ziλ+ (1− λ)Ki
= 0 67
5 H =H ψ(λ) : 67 0 ≤ λ ≤ 1
dψ(λ)
dλ=∑i
(Ki − 1)2zi
[λ+ (1− λ)Ki]2 > 0 67
" . : - 8 67
ψ(λ) λ = 0 λ = 18 "
3 . H $ 8 Ki . 8 >
% . &
$ H 6$ . : 7
4 ?) & $ ϕ/ϕ ϕ/γ,B T 8 p8 z&. B λ8 x D y8 K ,B K
%6 4 , ( **
) 6 8 7
) ψ(0) ψ(1)8 ψψ ⇐ ψ(0)ψ(1) $ ψψ ≥ 0 8
3B " 67 6 7
# 67 KB
Kei ⇐ Ki
λ 67 ) 6 7 6#7% 5= ! ) = 6$
B ( 8 8 8 : ' $ ( B 0:8 7
* !8 ϕ/ϕ ϕ/γ 8 >B
σ ⇐∣∣∣∣yIi ϕI
i − yIIi ϕIIi
yIIi ϕIIi
∣∣∣∣
σ ⇐∣∣∣∣∣−p+
∑i
γixiϕ,Li piP
yiϕVi
∣∣∣∣∣ :
$ σ : ε x8 y8 λ8
3B !8 ϕ/ϕ ϕ/γ 8 0 Ki
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)= 0
n∑i=1
(fi − ϕ(xi, a, b, c, . . . )
σ2i
∂ϕ(xi, a, b, c, . . . )
∂b
)= 0
$ : 8
$ . 8
- .8 $ . σi σ∗ x∗8 σ∗
i σ∗ σi σ∗
.8
% 4 +
.
$ - .8 (
$ . : ( 38 =
-5 A
C "
8 . :8. 9 > 8 "
$ 8 8 $
, L 3 8 (8 " - 8 L - . :
$ . $8 N 0 8 FF FF 8 . : " .
mina,b,c,...
n∑i=1
N∑j=1
(xi,j − ξi,j(a, b, c, . . . ))2
σ2i,j
f(ξ) = 0
x 8 ξi,j ( j i8 fk(ξ) = 0 k = 1, 2, ...,M7 : 0 " = .
/ ' (
.
- 8 8 > ( 8 F ( . " .
" (8 < 8 8
> 6307 "
F (t, x,dx
dt,d2x
dt2,d3x
dt3, . . . ) = 0 6 %7
t 8 x 8 x(t) . 08 F (t, x, x′, x′′, . . . ) 8 .8 ( $ 6x18x28 x38 7 t8 38 ( Fi xi8 x
′i8 x
′′i 8
. xj 8 x′j 8 x
′′j 8 8 8
> > 2 3 : 3 0
(8 y ≡ dx
dt8 z ≡ d2x
dt28 8
3 6 %7
F (t, x, y, z, . . . ) = 0
dx
dt= y
dy
dt= z 6 !7
- 3 : ( . N 0 3 6 N : 378 N
" N t8 $ 8 ( > t $88 t0 $ Gj(t0, x, x
′, . . . ) =bj bj '8 . ' (
/ ' ( #
3 ( : $ B
dx
dt= f(t, x)
$ B
dxidt
= fi(t, x1, x2, . . . ) (i = 1, 2, . . . )
$ 3
= t 6=7 h 6 =78 t0 = 08 t1 = h8 t2 = 2h8 " 6 .7 x0 = x(t0)8 x1 = x(t1)8 x2 = x(t2) ( . x(t) ti 6& @ 78 : B
xi+1 = xi + h
(dx
dt
)i
+ o(h2)
o(h2) (
h $( i ti ( 3 0
8 F B
xi+1 ≈ xi + hf(ti, xi)
8 Bxi+1 ≈ xi + hfi
" h = F (0 > : > x1 : ( 8 ( " ( ( 8 x28 8 8 8 5 0 " B 67 " 8 L 67 " 8
F )8 (8 : 3
dx
dt= −B2x
/ ' (
$ ( x(t) = −x(0)B2 exp(−B2t)
x(0) 6 x(0)7 :
xi+1 = xi −B2xih =(1−B2h
)xi
B h 8
xi+n =(1−B2h
)nxi = Anxi
A =(1−B2h
) & A
An → 1
1−A |A| < 18 An → ∞ |A| > 0 $ h >
2
B2 1−B2h < −18∣∣1−B2h
∣∣ > 18 : $ .
( . 8 = 2
B2
- = - " 8
(2 @ 8 (0 . B
xi = xi+1 − h
(dx
dt
)i+1
+ o(h2)
< B
xi+1 ≈ xi + hfi+1
8 3B
xi+1 = xi − hB2xi+1
: B
xi+1 =xi
1 + hB2
" h 1 + hB2 > 1
8
" = = "
/ ' (
-5 B
d2x
dt2= 100x
x(0) = 1
dx
dt(0) = −10
t " (
x(t) = A exp(−10t) +B exp(10t)
A B < A = 18 B = 08 ( B
x(t) = exp(−10t)
/ 8 ( t @ 3 8 8 8 8 " ε BB
x(t) = exp(−10t) + ε exp(10t)
- ε 8 t 8 0 -5 B : 3
dx
dt= 998x+ 1998y
dy
dt= −999x− 1999y
x(0) = 1 y(0) = 0 " (
x(t) = 2e−t − e−1000t
y(t) = −e−t + e−1000t
" ' ( . 6 =78 ( > " : ( " 8 = 3 >
$ > 0
< x =1
y ( 8 3 y >
/ ' ( %
3 : 8 . x
C ( 0 68 7 0 8 A 08 : 0 = > = 6 A8 7 67 "8 '
' (9
- 8 (
3 (8 0 - 8 . x h f(ti−1)8 > f(ti−1+ h)8 f(ti) 5 8
f ( B
x(t+ h) = x(t) + hf
" 3 > f ( (
" 3 ( x h 6 k = f(t, x(t)h)78 . 0 t, x(t) t + h, x(t) + k) (t + h/2, x(t) +k/2) " ( 0 "
k1 = hf(ti, xi)
k2 = hf(ti +1
2h, xi +
1
2k1)
xi+1 = xi + k2
2. 3 - 00 1? . :
1 F . ' ( !
B
k1 = hf(ti, xi)
k2 = hf(ti +h
2, xi +
k12)
k3 = hf(ti +h
2, xi +
k22)
k4 = hf(ti + h, xi + k3)
xi+1 = xi +k16
+k23
+k33
+k46
/
$ > 0 L . ( . ' (
5
. 8 8 : 8 - . 8 8 $ .8 > 8 '
$
" . 00 (
# 4 A8 8 ( 8 A
4 . >
0: 6.8 84 8 48 2 7 4 > 8
" > 2 x8 x -
1 F . ' ( *
8 8 U 2 U 6.7( 8 u (
" (8 . . 8 8 L 8 0 8 " Ω / & ' ' (5 & Ω & ( - * 5 & 5 F . ' ' ( . > ( . ( @ , . ,
"
" 0
d2U
dx2= f(x, U,
dU
dx)
a ≤ x ≤ b8 U(a) = Ua U(b) = Ub ( f(x, U, U ′) .8 . U(x) .
$ $ U ′(a) Ub : u(x) U ′ .8 B U ′ ≈ V 8 - 8 u(x)8 : u(b) ub " ub U ′ ≈ V H ub(V )
$ V U ′(a) : B ub(V ) ≈ Ub $8 8 V = U ′(a)8 ub(V ) = Ub " 8 ub(V ) = Ub8
" & 8
5 U(x) . $ & + & x = U(x) xi " ui8 0 " > ui
1 F . ' ( #
# ( ! B
B Ixi8 xi+1J h (0 . B
Ui+1 = Ui + h
(dU
dx
)i
+ o(h2)
2 ( : (dU
dx
)i
=Ui+1 − Ui
h+ o(h)
(dU
dx
)i
≈ ∆(1)i
h=ui+1 − ui
h
2 ( B
(d2U
dx2
)i
≈ ∆(2)i
h2=
∆(1)i+1 −∆
(1)i
h2=ui+2 − 2ui+1 + ui
h2(d3U
dx3
)i
≈ ∆(3)i
h3=
∆(2)i+1 −∆
(2)i
h3=ui+3 − 3ui+2 + 3ui+1 − ui
h3
% B Ixi8 xi+1J h ( . B
Ui+ 12= Ui +
h
2
(dU
dx
)i
+(+h)2
8
(d2U
dx2
)i
+ o(h3)
Ui− 12= Ui − h
2
(dU
dx
)i
+(−h)2
8
(d2U
dx2
)i
+ o(h3)
> B
Ui+ 12− Ui− 1
2= h
(dU
dx
)i
+ o(h3)
o(h3) ( 2(dU
dx
)i
=Ui+ 1
2− Ui− 1
2
h+ o(h2)
> : 8 > - > = 6 7
1 F . ' ( #
) > . = 2h 8 ( B(
du
dx
)i
=ui+1 − ui−1
2h(d2u
dx2
)i
=ui+2 − 2ui + ui−2
4h2(d3u
dx3
)i
=ui+3 − 3ui+1 + 3ui−1 − ui−3
8h2
2( ! B
9 ( ( " ( 0 8 = h . $ 8 >
8 8 (8 : ( (
∂2u
∂x21
)i,j
=ui+2,j − 2ui,j + ui−2,j
4h2
4( : > 8 (B(
∂2u
∂x1∂x2
)i,j
=ui+1,j+1 − ui−1,j − ui,j−1 + ui−1,j−1
4h2
4 ( ( (8 ; (
∇2ui,j ≡(∂2u
∂x21
)i,j
+
(∂2u
∂x22
)i,j
=ui,j+1 + ui−1,j + ui,j−1 + ui+1,j − 4ui,j
4h2
# B
∇2ui,j =ui+1,j+1 + ui−1,j+1 + ui−1,j−1 + ui+1,j−1 − 4ui,j
8h2
> = > 8 6 07 8 0 1 ( . 0
1 F . ' ( #
. ! B
< : > > 67 . ui ui,j,... &>
- 58 . >
dU
dx+ 2U = 1
Ω = x|0 ≤ x ≤ 1 U(0) = 1 " : > 6@ 7
$ [0, 1] #8 = h = 0.258 8 . 8 x1 = 0.258 x2 = 0.58 x3 = 0.758 x4 = 1# .5 & + @ - + ( & . " .
x0 = 0 - > > B
u1 − u00.25
+ 2u0 = 1
u2 − u10.25
+ 2u1 = 1
u3 − u20.25
+ 2u2 = 1
u4 − u30.25
+ 2u3 = 1
" ". u0 = 18 ( B ⎛
⎜⎜⎝4 0 0 0−2 4 0 00 −2 4 00 0 −2 4
⎞⎟⎟⎠⎡⎢⎢⎣u1u2u3u4
⎤⎥⎥⎦ =⎡⎢⎢⎣3111
⎤⎥⎥⎦
$ (u1, u2, u3, u4
)=
(3
4,5
8,9
16,17
32
) B 8 > '
U D 8 0 u $ '
" Ω
< 8 $ 8
1 F . ' ( #
( $ Ω ' 8 8 8 Ω " ' : >
" :
$ : >8 : 0( ' 8 L 8 . U(x) ( ( 6 7 0 ( & ϕi(x) . ' αiB
U(x) =
∞∑i=1
αiϕi(x)
" ( ' αi ( : > 08 . : '
$ : (8 ( u . B
u(x) =
N∑i=1
αiϕi(x)
> 6 7 D [U(x)] = 0 D [.] 0 > " ( U(x) : > Ω8 u(x) B
D
[ ∞∑i=1
αiϕi(x)
]= 0
D
[N∑i=1
αiϕi(x)
]= 0
4 . ' αi . Ω = N > & wj(x) (j = 1, 2, . . . , N) 8 B
∫Ω
D
[N∑i=1
αiϕi(x)
]wj(x)dx = 0 (j = 1, 2, . . . , N)
" N N . ' αi)8 (B
U(x)d2U
dx2+
(dU
dx
)2+ sin(x)
d2U
dx2= 0
1 F . ' ( ##
( U(x) 0 >
u(x) =N∑i
αiϕi(x) ϕi(x) . .
wj(x) 6(j = 1, 2, . . . , N)78 ( j8 =B
N∑i=1
N∑k=1
αiαk
∫Ω
ϕi(x)
(d2ϕ
dx2+
dϕ
dx
)wj(x)dx+
+
N∑i=1
αi
∫Ω
sin(x)ϕi(x)wj(x)dx = 0
" . " 0 .αi
$ B . αi8 > L [.]8 > L[U(x)] = p(x)8 p(x) . " F F B
∫Ω
L
[N∑i=1
αiϕi(x)
]wj(x)dx =
∫Ω
p(x)wj(x)dx
8 ' 8 αi
N∑i=1
aj,iαi = bj (j = 1, 2, . . . , N)
aj,i =
∫Ω
L [ϕi(x)]wj(x)dx
bj =
∫Ω
p(x)wj(x)dx
" .8
& 4. B
1 F . ' ( #
2 . " 8 3 N > xj 8 B
wj(x) = δ(x− xj)
6" : 8 ( xj 7
f(x)B∫ ∞
−∞f(x)δ(x− xj)dx = f(xj)
8 8 xj B∫Ω
L [u(x)] δ(x− xj)dx = L [u(xj)]
2 . $
[wj(x)] = xj−1
18 x8 x28 x38
2 . E$ # 2 L ϕi(x) B
wj(x) = ϕj(x)
" '
-5
8
d2U
dx2+ U = −x
Ω = 0 ≤ x ≤ 1 U(0) = 0 U(1) = 028 8
L[U ] =d2U
dx2+ U
p(x) = −x" (
U(x) =sin(x)
sin(1)− x
1 F . ' ( #
8
ϕi(x) = xi − xi+1 ≡ xi(1− x) (i = 1, 2, . . . ) 6 *7
8 B
ϕ1(x) = x− x2
ϕ2(x) = x2 − x3
u(x) = α1ϕ1(x) + α2ϕ2(x)
> > B
L [ϕ1(x)] = −2 + x− x2
L [ϕ2(x)] = 2− 6x+ x2 − x3
# @ & x1 = 0.25 x2 = 0.5 @ B
a1,1 = −2 + 0.25− 0.252 a2,1 = 2− 6 ∗ 0.25 + 0.252 − 0.253
a1,2 = −2 + 0.5− 0.52 a2,2 = 2− 6 ∗ 0.5 + 0.52 − 0.53
b1 = −0.25 b2 = −0.5
" (− 2916
3564− 7
4 − 78
)(α1
α2
)=
(− 14− 12
)" B
u(x) =x(1− x)(42 + 40x)
217
# " B
w1(x) = 1, w2(x) = x
B
a1,1 =1∫0
(−2 + x− x2) ∗ 1 dx a2,1 =
1∫0
(2− 6x+ x2 − x3
) ∗ 1 dx
a1,2 =1∫0
(−2 + x− x2) ∗ x dx a2,2 =
1∫0
(2− 6x+ x2 − x3
) ∗ x dx
b1 =1∫0
(−x) ∗ 1 dx b2 =1∫0
(−x) ∗ x dx
" 1B(− 116 − 11
12− 1112 − 19
20
)(α1
α2
)=
(− 12− 13
) B
u(x) =x(1 − x)(122 + 110x)
649
1 F . ' ( #%
# . E$ @ B
w1(x) = x− x2, w2(x) = x2 − x3
B
a1,1 =1∫0
(−2 + x− x2) (x− x2
)dx a2,1 =
1∫0
(2− 6x+ x2 − x3
) (x− x2
)dx
a1,2 =1∫0
(−2 + x− x2) (x2 − x3
)dx a2,2 =
1∫0
(2− 6x+ x2 − x3
) (x2 − x3
)dx
b1 =1∫0
(−x) (x− x2)dx b2 =
1∫0
(−x) (x2 − x3)dx
" 1B(− 310 − 3
20− 320 − 13
105
)(α1
α2
)=
(− 112− 120
)
B
u(x) =x(1− x)(71 + 63x)
369
% . " x = 0.258 x = 0.58 x = 0.75 B
%( C ### *%
) ##* %# 4 # * !! *!,. ##! *## *
9
" > 4 ( & - 8
" ( C 8 : .8
/'' ( . , . " . ' αi > D[U(x)] = 0 . = Ω P1[U(x)] = 08 P2[U(x)] = 0 δΩB
∫Ω
D
[N∑i=1
αiϕi(x)
]wj(x)dx+
∑k
∫δΩ
Pk
[N∑i=1
αiϕi(x)
]wj(x)dx = 0
1 F . ' ( #!
(j = 1, 2, . . . , N)
2 ( :
"" &
4 : 8 ,.8 : (8 " " 8 :( A " " " 8 8 8 @ . O 2 . 8> O
8 Ω 8 8 B 8 8 " :
( & ' - . ' ' -5 0 ' & + ( ' -5 0 ' . +
" 0 : 0= 6 08 # 8 ! 78 Ω
48 8 " : & 3 : H > (
2 0 2 0 8 6 7 8 /
8 & >0 > 8
" =8 ' ( 8 .
1 F . ' ( #*
x
x
x
x
x
x
#
#
#
#
#
#
!
!
!
!
!
!
ϕ1
ϕ2
ϕ3
ϕ4
ϕ5
ϕ6
ω1 ω2 ω3 ω4 ω5
*B
<
, ! B = >(2 & 4,028 5 S.8 *
"B G 4 G ! & + ) C 98 )8 *!!
45B >(2 & ! ( = ) ///* = * 5 Y @ W 8 5S.8 *!
)% % " !&B ! = Y @ W 8 5 S.8 **
