Математическое моделирование технологического оборудования 5-94275-107-2
В монографии рассматриваются вопросы разработки аппаратурного оформления многоассортиментных химических производств. При
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ȿ.ɇ. Ɍɭɝɨɥɭɤɨɜ
ɆȺɌȿɆȺɌɂɑȿɋɄɈȿ ɆɈȾȿɅɂɊɈȼȺɇɂȿ ɌȿɏɇɈɅɈȽɂɑȿɋɄɈȽɈ ɈȻɈɊɍȾɈȼȺɇɂə ɆɇɈȽɈȺɋɋɈɊɌɂɆȿɇɌɇɕɏ ɏɂɆɂɑȿɋɄɂɏ ɉɊɈɂɁȼɈȾɋɌȼ
ɆɈɋɄȼȺ «ɂɁȾȺɌȿɅɖɋɌȼɈ ɆȺɒɂɇɈɋɌɊɈȿɇɂȿ-1» 2004
ȿ.ɇ. Ɍɭɝɨɥɭɤɨɜ ɆȺɌȿɆȺɌɂɑȿɋɄɈȿ ɆɈȾȿɅɂɊɈȼȺɇɂȿ ɌȿɏɇɈɅɈȽɂɑȿɋɄɈȽɈ ɈȻɈɊɍȾɈȼȺɇɂə ɆɇɈȽɈȺɋɋɈɊɌɂɆȿɇɌɇɕɏ ɏɂɆɂɑȿɋɄɂɏ ɉɊɈɂɁȼɈȾɋɌȼ
«
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66.01.011 11-1 116 81
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1 ɁȺȾȺɑȺ ɊȺɁɊȺȻɈɌɄɂ ȺɉɉȺɊȺɌɍɊɇɈȽɈ ɈɎɈɊɆɅȿɇɂə ɋɈȼɊȿɆȿɇɇɈȽɈ ɉɊɈɆɕɒɅȿɇɇɈȽɈ ɏɂɆɂɑȿɋɄɈȽɈ ɉɊɈɂɁȼɈȾɋɌȼȺ ,
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2 ɁȺȾȺɑȺ ɈɉɊȿȾȿɅȿɇɂə ɄɈɇɋɌɊɍɄɌɂȼɇɕɏ ɏȺɊȺɄɌȿɊɂɋɌɂɄ ɉɊɈɂɁȼɈȾɋɌȼȿɇɇɈȽɈ ɈȻɈɊɍȾɈȼȺɇɂə ɂ ɊȿɀɂɆɈȼ ȿȽɈ ɊȺȻɈɌɕ , ,
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. . . 2.1. ɗɥɟɦɟɧɬɚɪɧɚɹ
Ʉɨɠɭɯɨɬɪɭɛɱɚɬɵɟ
ȿɦɤɨɫɬɧɨɟ
Ʉɨɥɨɧɧɵɟ
Ⱥɩɩɚɪɚɬɵ ɫ
Т
ɋɨɪɛɰɢɨɧɧɨɟ
А
К
Ʌɟɧɬɨɱɧɨɟ
ɋɭɲɢɥɶɧɨɟ
Т
Р
,
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, .
, -
Ɍɪɚɧɫɩɨɪɬ
Ɋɢɫ. 2.1 Ʉɥɚɫɫɢɮɢɤɚɰɢɹ ɷɥɟɦɟɧɬɚɪɧɵɯ ɨɛɥɚɫɬɟɣ . –
[5].
-
, ,
: ρ
τ;
t = t (α, β, γ, τ)
∂t – ∂τ
–
dp dt = div (λ ∇ t ) + QV + + η ΦV + SV , dτ dτ
(2.1)
α, β, γ
,
; div (λ ∇ t ) –
x, y , z :
div (λ ∇ t ) =
;ρ–
– ;
V
–
;τ–
∂ 2t ∂ 2t ∂ 2t ∂λ ∂t ∂λ ∂t ∂λ ∂t +λ 2 + + + + ∂x ∂x ∂x ∂y ∂y ∂z ∂z ∂ y2 ∂ z2
;λ– ; dp / dτ –
; QV – ;η–
; SV – ; vx, vy, vz –
. :
;
(2.2) -
∂t d t ∂t ∂t ∂t + vz +vy + vx = ∂z d τ ∂τ ∂y ∂x
.
(2.3)
,
-
. :
• • • • • •
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, . 3 ɂɋɉɈɅɖɁɈȼȺɇɂȿ ɆȿɌɈȾȺ ɄɈɇȿɑɇɕɏ ɂɇɌȿȽɊȺɅɖɇɕɏ ɉɊȿɈȻɊȺɁɈȼȺɇɂɃ ȾɅə Ɋȿɒȿɇɂə ɆɇɈȽɈɆȿɊɇɕɏ ɂɅɂ (ɂ) ɆɇɈȽɈɋɅɈɃɇɕɏ ɁȺȾȺɑ ɌȿɉɅɈɉɊɈȼɈȾɇɈɋɌɂ , ( )
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, ,
(
. 4.1). , .
-
∂ t (r , τ) ∂ ti (ri , τ) 2 ∂ 2ti (ri , τ) + Qi (ri , τ); + Ak ,i i i = ai 2 ∂r ∂ ri ∂τ i
i = 1, 2, ..., N ;
λ1
ti (ri , 0 ) = f i (ri ) ;
(
(
λ i , ai2
i-
;τ– Qi(ri, τ) –
;
(4.2)
)
)
(
)
λj
j = 1, 2, ..., N − 1.
(
∂t j Rj,τ ∂ rj
)=λ
j +1
α1 < 0 ;
(4.3)
)
(4.4)
(
∂ t j +1 R j , τ ∂ r j +1
ii-
,
(4.5)
; ti(ri, τ) – ; ; Ak, i –
: k = 0, A0,i = 0; k = 1, A1, i = 1/ri; k = 2, A2, i = 2/ri;
α1, αN – tc1(τ), tcN(τ) – .
; ; Ri – 1, Ri –
i-
r, :
, U (µ, τ) =
–
,
(
),
;N–
–
ρ(r )
τ > 0;
∂ t N (R N , τ ) + α N t N (R N , τ) − t c N (τ) = 0 . ∂ rN
t j R j , τ = t j +1 R j , τ ;
r–
k = 0, 1, 2;
∂ t1 (R0 , τ) + α1 t1 (R0 , τ) − t c1 (τ ) = 0; ∂ r1
λN
(
Ri −1 ≤ ri ≤ Ri ;
(4.1)
µ–
∑ N
λm
∫ ρ (rm )tm (rm , τ)Wm (rm , µ)drm ,
Rm
2 m =1 am 0
d ρ(rm ) − Ak , m ρ(rm ) = 0. dr
Wm (rm , µ )
(4.6)
(4.7) –
-
, ):
d 2Wm (rm , µ ) d rm2
+ Ak ,m
m = 1, 2, ..., N ,
d Wm (rm , µ ) d rm
+
Rm −1 ≤ rm ≤ Rm ;
µ2 am2
Wm (rm , µ ) = 0 ;
(4.8)
λ1 λN
(
)
d W1 (R0 , µ ) + α1 W1 (R0 , µ ) = 0 ; d r1
d W N (R N , µ ) + α N W N (R N , µ ) = 0 ; d rN
(
)
W j R j , µ = W j +1 R j , µ ; λ j
(4.8)
(
d Wj R j ,µ d rj
)=λ
j = 1, 2, ..., N − 1.
j +1
(
d W j +1 R j , µ d r j +1
r + C 2 m exp − m 2
–
2 A + A2 − 4 µ k m k m , , 2 am
2 A − A2 − 4 µ k m k m , , a m2
+ .
(4.11)
(4.12)
µ µ Wm (rm , µ )= C1m sin rm + C 2m cos rm . a m am
(4.13)
µ µ Wm (rm , µ ) = C1m J 0 rm + C 2 m Y0 rm , am am
(4.14)
,
. Wm (rm , µ )=
1m
),
(4.10)
[3]: r Wm (rm , µ ) = C1m exp − m 2
J 0 ( z ), Y0 ( z )
(4.9)
µ µ 1 C1m sin rm + C 2m cos rm . zm am am
2m
(4.9) – (4.11),
(4.15)
11 = 1.
(4.6)
(4.1)
(4.2). (4.6)
(4.3)
– (4.5). :
d U (µ n , τ) + µ n2 U (µ n , τ) = G (µ n , τ) + dτ
α α + N W (R N , µ n )t c N (τ) − 1 W (R1 , µ n )t c1 (τ); λ λ
U (µ n , 0) =
τ):
(4.17) –
; G(µn, τ) –
∑ N
λm
∫ ρ(zm ) f m (rm )Wm (rm , µn ) drm ,
(4.16)
Rm
2 m =1 am Rm −1
(4.17)
Qm(zm,
G(µ n , τ) =
(4.16) – (4.17)
∑ N
∫ ρ(zm )Qm (rm , τ)Wm (rm , µn ) drm .
λm
Rm
2 m =1 am Rm −1
[3]:
(
)
U (µ n , τ ) = exp − µ n2 τ ×
( )
τ × U (µ n , 0) + ∫ (G (µ n , τ) + FW ( µ n , τ) ) exp µ n2 τ dτ , 0
FW (µ n , τ) =
(4.18)
(4.19)
αN α W (RN , µ n )tcN (τ) − 1 W (R0 , µ n )tc1 (τ) . (4.20) λ λ
: t m (rm , τ ) = Sn =
∑ N
m =1
∫ ρ (rm ) Wm (rm , µ n ) dz m .
λm
Rm
a m2
Rm −1
2
U (µ n , τ ) Wm (rm , µ n ) , Sn n =1
∑ ∞
(4.21)
(4.22)
5 Ɋȿɒȿɇɂȿ ɁȺȾȺɑɂ ɌȿɉɅɈɉɊɈȼɈȾɇɈɋɌɂ ȾɅə ɉɈɅɈȽɈ ɈȽɊȺɇɂɑȿɇɇɈȽɈ ɐɂɅɂɇȾɊȺ ɋ ɎɍɇɄɐɂɈɇȺɅɖɇɈ ɆȿɇəɘɓȿɃɋə ɌȿɆɉȿɊȺɌɍɊɈɃ ɈɄɊɍɀȺɘɓȿɃ ɋɊȿȾɕ ɋɈ ɋɌɈɊɈɇɕ ȻɈɄɈȼɕɏ ɉɈȼȿɊɏɇɈɋɌȿɃ , . ,
,
-
, . (
. 5.1). -
: ∂ t (x, r , τ) 2 ∂ 2t (x, r , τ) ∂ 2t (x, r , τ) 1 ∂ t (x, r , τ) =a + + , 2 ∂ x2 ∂τ r ∂r r ∂ 0 ≤ x ≤ l , R0 ≤ r ≤ R1 , τ > 0 ; x
t0
α2
1 tv(x, τ) α0
λ c
t (x, τ) αc
(5.1)
Ɋɢɫ. 5.1 ɉɨɥɵɣ ɨɝɪɚɧɢɱɟɧɧɵɣ ɰɢɥɢɧɞɪ ɫ ɩɟɪɟɦɟɧɧɵɦɢ ɬɟɦɩɟɪɚɬɭɪɚɦɢ ɫɬɨɪɨɧɵ ɛɨɤɨɜɵɯ ɩɨɜɟɪɯɧɨɫɬɟɣ
ɨɤɪɭɠɚɸɳɟɣ ɫɪɟɞɵ ɫɨ
t ( x, r , 0) = f ( x, r ) − t c 0 ;
λ
∂ t (0, r , τ) + α1 t (0, r , τ) = 0; α1 < 0 ; ∂x
(5.3)
∂ t (l , r , τ) + α 2 t (l , r , τ) = 0 ; ∂x
(5.4)
λ λ
(5.2)
∂ t (x, R0 , τ) + α 0 (t (x, R0 , τ ) − t v (x, τ) + t c 0 ) = 0; α 0 < 0 ; ∂r λ
∂ t (x, R1 , τ ) + α c (t (x, R1 , τ ) − t c (x, τ ) + t c 0 ) = 0 . ∂r
(5.5)
(5.6) -
. (5.1) – (5.6)
-
,
. T (r , τ ) = ∫ t (x, r , τ) S (µ, x ) dx ; l
(5.7)
0
S (µ, x) –
,
-
: d 2 S ( x) = − µ 2 S ( x) ; dx
(5.8) – (5.10)
(5.8)
λ
d S ( 0) + α1 S (0) = 0 ; dx
(5.9)
λ
d S (l ) + α 2 S (l ) = 0 . dx
(5.10) :
S ( x) = sin(µ x + ϕ) ,
λµ ; ϕ = − arctg α1
µn – n-
(5.11)
(5.12) α 3 sin (µ l + ϕ) + λ µ cos (µ l + ϕ) = 0 .
t ( x , r , τ) =
∫
∑ ∞
n =1
N n = S 2 ( µ n , x ) dx =
(5.13)
T ( r , τ ) S (µ n , x ) , Nn
(5.14)
l
( (
) (
)
( ) ( ))
1 sin µ nl + ϕ n cos µ nl + ϕ n − sin ϕ n cos ϕ n = 0,5 l − µn 0
(5.1) – (5.6)
.
(5.15)
: ∂ 2 T ( r , τ) 1 ∂ T ( r , τ) ∂ T ( r , τ) + µ 2 T ( r , τ) ; + = a2 ∂τ r ∂r ∂r T ( r , 0) =
∫ ( f ( x, r ) − t c0 ) S (µ, x ) dx ;
(5.16)
l
(5.17)
0
λ
λ U (τ ) =
∫ (t v (x, τ) − tc0 ) S (µ, x ) dx ;
∂ T (R0 , τ ) + α 0 (T (R0 , τ ) − U (τ )) = 0 ; ∂r
∂ T (R1 , τ) + α c (T (R1 , τ) − W (τ)) = 0 , ∂r
(5.18)
(5.19)
l
(5.20)
0
W (τ ) =
∫ (tc (x, τ) − tc0 ) S (µ, x ) dx . l
(5.21)
0
r
-
∫
V (τ ) = T (r , τ ) r P (η, r ) dr ; R1
R0
T (r , τ ) =
P (η, r )
∑ ∞
k =1
P (η k , τ ) V (τ ) , Zk
(5.22)
(5.23)
d 2 P( r ) 1 d P (r ) 2 + + η P(r ) = 0 ; r dr d r2 λ
λ
(5.24)
∂ P(R0 ) + α 0 P(R0 ) = 0 ; ∂r
(5.25)
∂ P(R1 ) + α c P(R1 ) = 0 . ∂r
(5.26)
[3]: P (r ) = J 0 (η r ) + A Y0 (η r ) ,
A=
ηk
η λ J1 (η R0 ) − α 0 J 0 (η R0 ) , − η λ Y1 (η R0 ) + α 0 Y0 (η R0 )
(5.27)
(5.28)
– kη λ J1 (η R0 ) − α 0 J 0 (η R0 ) η λ J1 (η R1 ) − α c J 0 (η R1 ) . (5.29) = − η λ Y1 (η R1 ) + α c Y0 (η R1 ) − η λ Y1 (η R0 ) + α 0 Y0 (η R0 )
Z k = ∫ r P 2 (r ) dr = R1
R0
(
)
R12 (J 0 (ηk R1 ) + Ak Y0 (ηk R1 ))2 + (J1 (ηk R1 ) + Ak Y1 (ηk R1 ))2 − 2 R2 − 0 (J 0 (η k R0 ) + Ak Y0 (η k R0 ))2 + (J 1 (ηk R0 ) + Ak Y1 (η k R0 ))2 . 2
=
(
(5.22)
)
(5.30)
(5.16) – (5.19),
:
α R2 α R2 d V (τ ) + η k2 a 2 V (τ) = 0 0 P(R0 )U (τ) + c 1 P(R1 )W (τ); dτ λ λ V (0) =
∫ ∫ ( f (x, r ) − tc0 ) S (x ) r P(r ) dx dr .
(5.31)
R1 l
(5.32)
R0 0
(5.31) – (5.32)
(
)
R1 l V ( τ ) = exp − η 2k a 2 τ ∫ ∫ ( f (x, r ) − t c0 ) S (x ) r P (r ) dx dr + R 0 0
(
)
α a 2 R0 + 0 P(R0 ) ∫ ∫ (t v (x, τ) − t c0 ) S (x ) exp η 2k a 2 τ dx dτ + λ 00 τ l
+
(
)
τ l α c a 2 R1 P(R1 ) ∫ ∫ (t c (x, τ) − t c 0 ) S (x ) exp η 2k a 2 τ dx dτ . λ 00
(5.33)
t ( x, r , τ) =
∑∑
V (τ) P(r ) S ( x) . Nn Zk n =1 k =1 ∞
∞
(5.34)
,
, -
, . ∂ t ( x, r, τ) 2 ∂ 2t ( x, r, τ) ∂ 2t ( x, r, τ) 1 ∂ t ( x, r, τ) ; =a + + 2 ∂ x2 r ∂τ ∂r r ∂ 0 ≤ x ≤ l , R0 ≤ r ≤ R1 , τ > 0 ; t ( x, r , 0) = f ( x, r ) − t c 0 ;
λ
∂ t (0, r , τ) = 0; ∂x
(5.37)
∂ t (l , r , τ) = 0; ∂x
(5.38)
∂ t (x, R1 , τ ) + α c (t (x, R1 , τ ) − t c (x, τ ) + t c0 ) = 0 . ∂r
,
(5.10):
(5.40)
(5.41)
∂ S (l ) = 0. ∂x
(5.42)
(5.11): S ( x) = cos (µx) .
(5.43)
(5.12): ϕ = 0 . (5.13): µ n =
(5.39)
.
∂ S (0) = 0. ∂x
(5.9):
(5.36)
∂ t ( x, R0 , τ) + α 0 (t ( x, R0 , τ) − t v ( x, τ) + t c 0 ) = 0 ; α 0 < 0 ; ∂r λ
(5.35)
(5.44)
πn . l
(5.45)
(5.15):
∫
1 N n = S 2 ( µ n , x ) dx = 0,5 l + sin (µ n l ) cos (µ n l ) . µn 0 l
(5.17): T (r , 0) = ∫ f ( x, r ) S (µ, x ) dx . l
0
(5.47)
(5.46)
(5.20): U (τ) = ∫ t v ( x, τ) S (µ, x) dx . l
(5.48)
0
(5.21): W (τ) = ∫ tc ( x, τ) S (µ, x) dx . l
(5.49)
0
(5.32): V (0) =
∫ ∫ f ( x, r ) S ( x) r P(r ) dx dr .
R1 l
(5.50)
R0 0
(5.33):
(
)
R1 l V (τ) = exp − η 2k a 2 τ ∫ ∫ f ( x, r ) S ( x) r P (r ) dx dr + R 0 0 +
+
(
)
α 0 a 2 R0 P(R0 ) ∫ ∫ t v ( x, τ) S ( x) exp η 2k a 2 τ dx dτ + λ 00 τ l
(
)
(5.51)
τ l α c a 2 R1 P(R1 ) ∫ ∫ t c ( x, τ) S ( x) exp η2k a 2 τ dx dτ . λ 00
6 Ɋȿɒȿɇɂȿ ɁȺȾȺɑɂ ɋɌȺɐɂɈɇȺɊɇɈɃ ɌȿɉɅɈɉɊɈȼɈȾɇɈɋɌɂ ȾɅə ɋɈɋɌȺȼɇɈȽɈ ɄɈɇȿɑɇɈȽɈ ɐɂɅɂɇȾɊȺ
; ,
;
, . (
.
,
, ,
. .
x
α3 tc3
1
α1 tc1 0
R
r α2 tc2
h y
α4 tc4
. 6.1)
.
4.
,
Ɋɢɫ. 6.1 ɋɨɫɬɚɜɧɨɣ ɤɨɧɟɱɧɵɣ ɰɢɥɢɧɞɪ
∂ 2 t1 (x, r ) ∂ 2 t1 (x, r ) 1 ∂ t1 (x, r ) + = 0, 0 ≤ x ≤ l , 0 ≤ r ≤ R ; + r ∂r ∂ r2 ∂ x2 ∂ 2 t 2 ( y, r ) ∂ 2t 2 ( y, r ) 1 ∂ t 2 ( y, r ) + + = 0, 0 ≤ y ≤ h ; ∂r r ∂ y2 ∂ r2 λ1
∂ t1 (l , r ) + α 3 (t1 (l , r ) − t c3 ) = 0 ; ∂x
λ1
∂ t1 (x, R ) + α1 (t1 (x, R ) − t c1 ) = 0 ; ∂r ∂ t1 (x, 0 ) = 0; ∂r ∂ t 2 (h, r ) + α 4 (t 2 (h, r ) − t c 4 ) = 0 ; ∂y
λ2
∂ t 2 ( y, R ) + α 2 (t 2 ( y, R ) − t c 2 ) = 0 ; ∂r ∂ t 2 ( y, 0) = 0; ∂r
(6.3)
(6.4)
(6.6)
(6.7)
(6.8)
t1 (0, r ) = t2 (0, r ).
(6.9)
∂ t1 (0, r ) ∂ t (0, r ) = − λ2 2 . ∂x ∂y
. .
(6.2)
(6.5)
λ2
λ1
(6.1)
(6.10)
: ∂ 2 t1 (x, r ) ∂ 2 t1 (x, r ) 1 ∂ t1 (x, r ) = 0, 0 ≤ x ≤ l , 0 ≤ r ≤ R ; + + r ∂r ∂ r2 ∂ x2
λ1
∂ t1 (l , r ) + α 3 (t1 (l , r ) − t c3 ) = 0 ; ∂x
λ1
∂ t1 (x, R ) + α1 (t1 (x, R ) − t c1 ) = 0 ; ∂r
(6.12)
(6.13)
∂ t1 (x, 0 ) = 0; ∂r
λ1
(6.14)
∂ t1 (0, r ) = − m (r ). ∂x
(6.15)
(6.11) – (6.15) .
(6.11)
r
:
∫
T (x, µ ) = ρ(r )t1 (x, r ) P(r , µ ) dr , R
(6.16)
0
ρ( r ) = r
–
,
[4] d ρ(r ) ρ(r ) − = 0. dr r
(6.17)
P ( r , µ)
(
µ–
): d 2 P(r , µ ) 1 d P(r , µ ) 2 + + µ P(r , µ ) = 0 ; r dr d r2 λ1
d P (R , µ ) + α1 P ( R , µ ) = 0 ; dr
(6.19)
d P(0, µ ) = 0. dr
(6.18) – (6.20)
(6.20)
: P(r , µ ) = J 0 (µ r ),
µ–
(6.18)
(6.21)
α1 J 0 (µ R ) − µ λ1 J 1 (µ R ) = 0 .
(6.22)
:
∑
t1 (x, r ) =
∞
n =1
T (x, µ n ) P(r , µ n ) , Zn
(
(6.23)
)
Z n = ∫ r P 2 (r , µ n ) dr = ∫ r J 02 (r , µ n ) dr = 0,5 R 2 J 02 (µ n R ) + J 12 (µ n R ) . R
R
0
0
(6.24) J 0 ( x), J1 ( x)
–
, (6.16)
.
(6.11) – (6.15),
: d 2 U ( x, µ ) d x2
− µ n2 U (x, µ ) = 0 ;
d U (l , µ ) + α 3 U (l , µ ) + S = 0 ; dx
λ1
λ1 U ( x, µ ) = T ( x, µ ) − K ;
d U (0, µ ) = M. dx
(6.25)
(6.26)
(6.27)
(6.28) K =
α1 R t c1 λ1 µ 2
J 0 (µ R );
R t c3 S = α 3 K − J 1 (µ R ) ; µ M = − ∫ r m(r ) J 0 (µ r ) dr .
(6.29)
(6.30)
R
(6.31)
0
(6.25) – (6.27)
[3]: U (x, µ ) = A sh (µ x ) + B h (µ x ),
A
B
(6.32)
(6.26) – (6.27): A=
M ; λ1 µ
(6.33)
B=− C = h (µ l ) +
α3 sh (µ l ) ; λ1 µ
S+M C , D
(6.34)
(6.35)
D = α 3 ch (µ l ) + λ1 µ sh (µ l ).
(6.36)
(6.11) – (6.15):
t1 (x, r ) =
∑ ∞
n =1
J 0 (r µ n ) M n S + M n Cn sh (µ n x ) − n h (µ n x ) + K n Zn Dn λ1 µ n
.
(6.37) , (6.15):
(6.11) –
∂ 2 t 2 ( y, r ) ∂ 2t 2 ( y, r ) 1 ∂ t 2 ( y, r ) + + = 0, 0≤ y ≤ h, 0 ≤ r ≤ R; ∂r r ∂ y2 ∂ r2
λ2 λ2
∂ t 2 (h, r ) + α 4 (t 2 (h, r ) − t c 4 ) = 0 ; ∂y
∂ t 2 ( y, R ) + α 2 (t 2 ( y, R ) − t c 2 ) = 0 ; ∂r ∂ t 2 ( y, 0) = 0; ∂r
λ2
(6.38)
(6.39)
(6.40)
(6.41)
∂ t 2 (0, r ) = − m (r ). ∂y
(6.42) ,
(6.11) – (6.15): y → x, λ 2 → λ1 , α 2 → α1 , α 4 → α 3 , t c 2 → tc1 , t c 4 → t c 3 , γ n → µ n , M 1 n → M n , K1 n → K n ,
S1 n → S n , C1 n → Cn , D1 n → Dn , Z1 n → Z n .
(6.43)
(6.9)
1
(6.10)
:
∞ J 0 (r µ n ) Sn + M n Cn ∞ J 0 (r γ n ) S1n + M1n C1n − − + K n = + K1n ; Dn D1n n =1 Z n n =1 Z1n ∞ ∞ J 0 (r µ n ) J 0 (r γ n ) M =− M1n . Zn Z1n n =1 n =1
∑ ∑
∑
∑
r J 0 (r µ k )
0
R.
:
(6.44)
1 S +M C R 2 k k − k + K k r J 0 (r µ k ) d r = Dk Zk 0 R ∞ 1 S1n + M 1n C1n − r J 0 (r µ k ) J 0 (r γ n ) d r ; + K = n 1 D1n 0 n =1 Z1n R R ∞ 1 Mk 2 µ = r J r d r r J 0 (r µ k ) J 0 (r γ n ) d r . ( ) k 0 Z n =1 Z1n 0 k 0
∫
∑
∑
∫
∫
∫
(6.45)
:
∫ r J 0 (µ n r ) J 0 (µ k r ) d r = 0,
n≠k .
R
(6.46)
0
N ,
,
(6.45) M 1k , M k ; 1 ≤ k ≤ N . , (6.45)
∑
:
N − G M H nk G1n M 1n = Lk , 1 ≤ k ≤ N ; k k n =1 N M − H M = 0 , k nk n n =1
∑
Gk =
Ck ; Dk
G1k =
C1k ; D1k
(6.47)
(6.48) H nk =
R µ n J1 (µ n R ) J 0 (γ k R ) − γ k J 0 (µ n R ) J1 (γ k R ) ; Zn µ 2n − γ 2k
Lk = −
(6.47) 1n:
∑
N S Sk + K k + H nk 1k − K1k . Dk D1k n =1
,
k N
n =1
(6.50)
N
∑ H nk (Gk + G1n ) M 1n = − Lk ,
(6.49)
-
1≤ k ≤ N .
(6.51)
k:
Mk =
∑ H nk M n , N
n =1
1≤ k ≤ N .
(6.52)
7 ȾɂɎɎȿɊȿɇɐɂȺɅɖɇɈȿ ɍɊȺȼɇȿɇɂȿ ɌȿɉɅɈɉɊɈȼɈȾɇɈɋɌɂ ȼ ɗɅȿɆȿɇɌȺɏ ɈȻɈɊɍȾɈȼȺɇɂə, ɂɆȿɘɓɂɏ ɎɈɊɆɍ ɋɌȿɊɀɇȿɃ ɂ ɉɅȺɋɌɂɇ , ,
.
• •
: ; ;
• • •
; ; . .
∆
. .
Q1 – Q2 –
, ,
; .
,
,
Q1 − Q2 = α f (t ( ) − tt ),
α– ;
; f = ∆x – ; t( ) –
– ; –
; tt –
(7.1)
,
.
Q1 − Q2 = F (q1 − q 2 ),
F–
; q1, q2 –
, .
∆
,
: q1 − q 2 = −
λ
d 2t (x ) dx 2
F dx = α
(t ( ) − tt ) dx .
T (x ) = t (x ) − tt d 2T (x )
:
dx
2
k2 =
d 2t (x ) d dt (x ) dq (x ) dx . ∆x = − − λ dx = λ dx dx dx dx 2
α , λF
(7.5)
− k 2T ( x ) = 0 .
(7.6)
T (x ) = C1 h (k x ) + C 2 sh (k x ).
: −λ −λ
q0 –
;L– C1 =
(7.3)
(7.4)
:
:
(7.2)
q0 (α t sh (kL ) + λ k h (kL )) λk ; α t h (kL ) + λ k sh (kL )
(7.7)
dT (0) = q0 ; dx
(7.8)
dt (L ) = α t (t (L ) − tt ) , dx
; αt –
(7.9) ,
α t tt +
(7.10)
C2 = −
q0 λk
.
(7.11)
. h. ∆ × ∆у × h.
.
. Q1–
, ;
Q2–
, ;
Qу1 – Qу2 –
,
у;
,
у.
,
α1
f = ∆x ∆y
,
Qx1 − Qx 2 + Q y1 − Q y 2 = (α1 + α 2 ) f (t ( , y ) − tt ),
(7.12)
α2 – –
; ;
t( , у) –
; ,у–
; tt –
.
, Qx1 − Qx 2 = Fx (q x1 − q x 2 ), Q y1 − Q y 2 = Fy (q y1 − q y 2 ) , Fx = h ∆y , Fy = h ∆x
(7.13) у
–
; q 1, qу1, q 2, qу2 –
, ∆
.
∆у
,
: q x1 − q x 2 = −
∂t (x, y ) ∂ ∂q (x, y ) ∂ 2 t ( x, y ) dx ; ∆x = − − λ dx = λ ∂x ∂x ∂x ∂x 2
∂ 2 t ( x, y ) ∂q (x, y ) ∂ ∂t (x, y ) dy = λ dy . q y1 − q y 2 = − ∆y = − − λ ∂y ∂y ∂y ∂y 2
T ( x, y ) = t ( x, y ) − t t
:
-
∂ 2T (x, y ) ∂x 2
+
:
∂ 2T (x, y ) ∂y 2
∂ 2 t ( x, y ) ∂ 2 t ( x, y ) h dx dy = (α1 + α 2 )(t ( , y ) − tt ) dx dy . λ + ∂x 2 ∂y 2 (α + α 2 ) , (7.16) k2 = 1 λh − k 2T (x, y ) = 0 .
(7.17)
(7.14)
(7.15)
∂T (0 , y ) = q x0 ; ∂x
−λ
∂T (x , 0 ) = q y0 ; ∂y
−λ −λ −λ
(7.18)
(7.19)
∂T (Lx , y ) = α x t T (L x , y ) ; ∂x
(
∂T x , L y ∂y
)=α
yt
(
T x , Ly
),
(7.20)
(7.21)
у; L , Ly –
q 0, qy0 – у; α t, αуt –
,
,
-
.
у,
, d 2 P( y ) dy 2
:
+ µ 2 P( y ) = 0
(7.22)
dP (0) = 0; dy
−λ
(7.23)
( ) = α P (L ) . yt y dy
dP L y
(7.24)
P ( y ) = cos (µ y ),
µ–
(
)
(7.25)
(
)
µ λ sin µ L y = α yt os µ L y .
U (x ) =
ρ ( y ) = 1.
∫ T (x, y )ρ ( y ) P ( y ) dy ,
(7.26)
Ly
(7.27)
0
T ( x, y ) =
µn N =
2 2 ∫ ρ ( y ) P ( y ) dy = ∫ cos (µ y ) dy =
Ly
Ly
0
0
(
(
U (x ) P ( y ) , N n =1
∑ ∞
) (
))
1 µ L y + sin µ L y cos µ L y . 2µ
(7.28)
(7.29) (7.17) – (7.21):
d 2U (x ) dx
2
− ν 2U (x ) +
−λ
Qx 0 =
∫ q x0 P ( y ) dy =
Ly
0
(
−λ
)
q x0 sin µ L y . µ
q y0 λ
= 0, ν 2 = µ 2 + k 2 ;
(7.30)
dU (0 ) = Qx 0 ; dx
(7.31)
dU (Lx ) = α xt U (Lx ), dx
(7.32)
(7.33)
: U (x ) = A sh (νx ) + B h (νx ) +
А
В
q y0
λν 2
(7.34)
.
: A=−
B=−
,
Qx 0 ; νλ
A (λν ch (ν Lx ) + α xt sh (ν Lx )) +
(7.35)
α xt q y 0
α xt ch (ν Lx ) + λν sh (ν Lx )
λν 2
(7.36)
.
(7.28) . 8 ȾɂɎɎȿɊȿɇɐɂȺɅɖɇɈȿ ɍɊȺȼɇȿɇɂȿ ɉȿɊȿɇɈɋȺ ɌȿɉɅȺ ɀɂȾɄɈɋɌɖɘ, ȾȼɂɀɍɓȿɃɋə ȼ ɊȿɀɂɆȿ ɂȾȿȺɅɖɇɈȽɈ ȼɕɌȿɋɇȿɇɂə ɉɈ ɄȺɇȺɅɍ ,
-
. . ,
,
[7].
:
• • • •
; ; ; . -
, . . , . : – τ– ; t(x, τ) – G– –
; ; ; ;
-
qi – ∆Fi = i∆ – i– αi – tFi(x, τ) – i = 1, 2 –
; i-
; i-
; i-
i-
;
; .
∆
. . ∆τ
,
Q0 = G c t (x, τ ) ∆τ ;
∆τ
,
(8.1)
Q3 = − G c t (x + ∆ , τ ) ∆τ ;
,
(8.2)
(
)
Q1 = q1∆F1 = α1 t F1 (x, τ ) − t (x, τ ) ∆F1 ∆τ ;
,
(
(8.3)
)
Q2 = q 2 ∆F2 = α 2 t F2 (x, τ) − t (x, τ ) ∆F2 ∆τ .
:
(
)
Q1 = α1 t F1 (x, τ) − t (x, τ)
(
)
Q2 = α 2 t F2 (x, τ) − t (x, τ )
1∆
2
(8.4)
∆τ ;
(8.5)
∆ ∆τ .
(8.6)
∆τ:
S–
Q = S ∆x ρ c (t (x, τ + ∆τ)− t (x, τ)) ,
;ρ–
(8.7)
. ,
-
: Q = Q0 + Q1 + Q2 + Q3
S ∆x ρ c (t (x, τ + ∆τ ) − t (x, τ )) =
( + α 2 (t F (x, τ ) − t (x, τ ))
)
= G c t (x, τ ) ∆τ + α1 t F1 (x, τ ) − t (x, τ ) 2
2∆
∆τ +
∆τ − G c t ( x + ∆ , τ ) ∆τ .
∆x∆τ Sρc
1∆
(8.8)
(8.9)
,
∂t ( x, τ ) ∂t (x, τ ) = − Gc + ∂τ ∂x + α1 t F1 (x, τ ) − t (x, τ )
(
)
(
:
)
1 + α 2 t F2 ( x, τ ) − t ( x, τ )
(8.10) 2.
,
Sρc
∂t (x, τ ) ∂t (x, τ ) α1 1 + α 2 2 t ( x, τ ) = +W + S ρc ∂τ ∂x α1 1t F1 (x, τ ) + α 2 2 t F2 (x, τ ) = , S ρc
:
W=
G Sρ
–
,
. :
∂ t ( x, τ ) ∂ t ( x, τ ) + K t ( x , τ ) = F ( x, τ ) ; +W ∂x ∂τ
t (0, τ ) = t 0 (τ ); t (x, 0) = f (x ) ,
F ( x, τ ) =
(8.12)
K = α1
α1
1 + α2
Sρc 1t F1 ( x, τ ) + α 2
2
Sρc
; 2 t F2
( x, τ )
(8.14) .
[1]. ∂t (x, τ ) : ∂τ
-
-
∂t (x, τ ) t (x, τ) − t (x, τ − dτ) ≈ . ∂τ dτ
(8.15)
dτ
P=
K dτ + 1 ; W dτ
V (x ) =
1 W
(8.12) (8.13)
(8.12) – (8.13),
f( ) –
(8.11)
:
dt (x ) + Pt (x ) = V (x ) , dx
f (x ) ; F ( x, dτ ) + d τ
(8.16)
(8.17)
t (0) = t 0
. (8.16):
x t (x ) = xp (− Px ) t 0 + ∫ V (x ) xp (P x ) dx . 0
(8.18)
∆ t =
1 ∆x
∫
t (x ) dx =
∆x 0
1 ∆x
∫
∆x 0
x xp (− P x ) t 0 + V (ξ ) xp (P ξ ) dξ dx . 0
∫
t F ( x, τ ) ,
(8.19)
K =
;α–
–
dt (x ) + K t (x ) = S (x ) , dx
S (x ) =
t (0 ) = t 0
V =
(8.24)
1t F1
α1
(x ) + α 2
1 + α2
2 t F2
K = α1
α1
1 + α2
Gc ( t x 1 F1 ) + α 2
(8.21)
(8.20)
2
(8.21)
; 2 t F2
(x )
Gc
(8.22) .
x t (x ) = exp (− Kx ) t 0 + S (x ) exp (Kx ) dx . 0
, α1
α t F ( x, τ ) , Sρc
. :
t (x ) = V + (t 0 − V ) xp (− K x ),
F ( x, τ ) =
α ; Sρc
(x )
∫
t F1 (x ) = const = t F1
. .
(8.23)
t F2 (x ) = onst = t F2 ,
(8.25)
.
2
∆ t =
(t − V ) (1 − exp (− Kx )). 1 t (x ) dx = V + 0 ∆x 0 K∆x
∫
∆x
t F (x ) ,
x t (x ) = exp (− K1 x ) t 0 + K1 ∫ t F (x ) exp (K1 x ) dx , 0
K1 =
α . Gc
(8.27)
(8.28)
, . . t F (x ) = onst = t F ,
t (x ) = t F + (t 0 − t F ) exp (− K1 x ) ,
∆ t =
,
(8.26)
1 ∆x
(4.1) – (4.5), (5.1) – (5.6) ,
∫ t (x ) dx = t F +
∆x 0
(t 0 − t F ) (1 − exp (− K x )). 1 K1∆x
(8.29)
(8.30)
(8.12) – (8.13) -
. 9 Ɋȿɒȿɇɂȿ ɈȻɊȺɌɇɈɃ ɁȺȾȺɑɂ ɌȿɉɅɈɉɊɈȼɈȾɇɈɋɌɂ ,
,
-
, . (
)
(
-
).
, (10
). . . ,
-
, ,
, ,
. ,
-
, ,
, 30 – 50 %. 900 %.
. .
.
,
-
,
. . ,
:
1) , ; 2)
, , . . ,
,
-
; ; 3)
. ,
, . «
. ,
», . . .
. ,
, .
.
-
,
, -
. [7]
, . (
)
,
. -
. :
∂ 2 t ( x, τ ) ∂t (x, τ ) , 0 ≤ x ≤ R, τ ≥ 0 ; =a ∂τ ∂x 2
(9.1)
t (x, 0) = t 0 = onst
λ
(9.2)
∂t (R, τ ) + α (t (R, τ ) − t c ) = 0 ; ∂x
(9.3)
∂t (0, τ ) =0 , ∂x
–
, ; , ;t – .
λ a= cρ
(9.4)
;τ–
; t(x, τ) –
; λ, , ρ –
–
T ( x , τ ) = t ( x, τ ) − t 0
; t0 – ;α–
;R–
-
∂T (x, τ ) ∂ 2T (x, τ ) ; =a ∂τ ∂x 2
:
T ( x, 0 ) = 0 ;
λ
∂T (R, τ ) + α (T (R, τ ) − t c + t 0 ) = 0 ; ∂x ∂T (0, τ ) = 0. ∂x
∫
S (x, p ) = T (x, τ) exp (− p τ) dτ .
(9.5) (9.6) (9.7)
(9.8)
∞ 0
:
(9.9)
a λ
d 2 S ( x, p ) dx 2
− p S ( x, p ) = 0 ;
t +t ∂S (R, p ) + α S (R, p ) − c 0 p ∂x
∂ S (0, p ) = 0. ∂x
p S (x, p ) = A sh a
sh(z) ch(z) – A B
(9.10) = 0 ;
(9.11)
(9.12)
p x+ B h a
x ,
(9.13)
. (9.3), (9.4): λ
p a
A ch p R + B sh p R + a a
p p t −t + α A sh R+ B h R − α c 0 = 0; a a p
λ
p a
А = 0,
p 0 + B sh a
0 = 0 ,
(9.15)
(9.16) B=
p α (t c − t 0 ) h x a . S ( x, p ) = p p p pλ R+α h R sh a a a
A ch p a
(9.14)
α (t c − t 0 )
p p p sh pλ R+ α h R a a a
.
(9.17)
(9.18)
S(x, p),
-
: α=
S(x, p)
(tc − t0 )
p p sh R a a . p p x − p S ( x, p ) h R a a
S ( x , p ) pλ h
(9.19)
S ( x, p ) =
∫ (ti (x, τi ) − t0 )exp(− p τi ) dτi =
∞
t ( x, τ k ) − t 0 exp(− p τ k ), = (t i (x, τ i ) − t 0 ) exp(− p τ i ) dτ i + k p 0
∫
0 τk
ti(x, τi), i = 1, 2, …, k –
(9.20)
, .
,
,
,
. -
. .
, .
,
. ,
-
(9.1) – (9.4). T ( x, τ ) = t ( x, τ ) − t ,
(9.21)
:
∂T (x, τ ) ∂ 2T ( x , τ ) ; =a ∂τ ∂x 2
T ( x, 0 ) =
λ
0
= t0 − tc ;
∂T (R, τ ) + α T (R , τ ) = 0 ; ∂x ∂T (0, τ ) = 0. ∂x
∫
W (τ) = T (x, τ)ρV (x ) dx ,
(9.22) (9.23) (9.24)
(9.25)
R
(9.26)
0
V( ),
, –
:
d 2 S (x ) dx 2
λ
+ µ 2 S (x ) = 0 ;
dS (R ) + α S (R ) = 0 ; dx
dS (0 ) = 0. dx
-
(9.27)
(9.28)
(9.29)
S (x ) = sin (µ x + ϕ).
dS (x ) = µ cos (µ x + ϕ) . dx µ os ( ϕ) = 0 , (9.29)
(
(9.31)
) ϕ=
S (x ) = os (µ x ) .
(9.32)
π , 2
(9.33) (9.34)
(9.28)
ρ
(9.30)
− λ µ sin (µ R ) + α cos (µ R ) = 0 ,
(9.35)
α = λ µ tg (µ R ) .
(9.36)
, a ρ′ = 0 ,
ρ = onst,
, ρ = 1.
(9.37)
(9.38) :
dW (τ) + a µ 2 W (τ) = 0 ; dτ
(9.39)
(9.23):
∫
W (0 ) = T0 cos (µ x ) dx = R
0
(9.22) – (9.23)
:
(
)
W (τ) = W (0) exp − µ 2 a τ =
T0 sin (µ R ) . µ
(9.40)
(
)
T0 sin (µ R ) exp − µ 2 a τ . µ
(9.41)
:
∫
W (τ) = Ti cos (µ x ) dx , R
Ti = T (xi , τ)
(9.42)
0
–
;
i = 1, 2, …, N –
-
. ,
,
. .
[2]
∫
I = Ti cos (µ x )dx ≈ h (G2 − 7 (G1 + GN )+ GN −1 ) / 12 + h R
0
; Gi = Ti os (µ xi ) –
h– µ
; xi –
. I =
τ
α.
∑ Gi , N
(9.43)
i =1
i-
(
-
)
T0 sin (µ R ) exp − µ 2 a τ . µ
(9.44) .
(9.36) .
10 ɆȿɌɈȾɂɄȺ ɊȺɋɑȿɌȺ ɌȿɉɅɈɈȻɆȿɇɇɈȽɈ ɈȻɈɊɍȾɈȼȺɇɂə , . ,
,
. . ,
.
∆
, (
,
,
-
. 10.1). , ,
,
,
,
-
.
∆x Ɋɂɋ. 10.1 ɗɅȿɆȿɇɌȺɊɇȺə ɈȻɅȺɋɌɖ ɄɈɀɍɏɈɌɊɍȻɑȺɌɈȽɈ ɌȿɉɅɈɈȻɆȿɇɇɂɄȺ
. , .
,
,
-
. . , 1. G1, G2 –
; α1, α2, α , αoc –
; t1, t2 –
; 1, 2, ρ1, ρ2, λ1, λ2 – , , ,ρ,ρ,ρ,λ,λ,λ – , ,
;
,
-
, , ;δ,δ,δ – ;t –
,
;d,d –
. ,
-
,
. , ,
. 10.2. ,
-
. G1
t1
t1
δ1 λ G2
α1
t1 U(r)
α2
t2
t2
t2
δ
λ
δ
λ
t
r
α
r
S1(r) t S2(r)
t
α
t ∆x
Ɋɢɫ. 10.2 ɋɬɪɭɤɬɭɪɚ ɷɥɟɦɟɧɬɚɪɧɨɣ ɨɛɥɚɫɬɢ ɤɨɠɭɯɨɬɪɭɛɱɚɬɨɝɨ ɬɟɩɥɨɨɛɦɟɧɧɢɤɚ, ɪɚɛɨɬɚɸɳɟɝɨ ɜ ɫɬɚɰɢɨɧɚɪɧɨɦ ɬɟɦɩɟɪɚɬɭɪɧɨɦ ɪɟɠɢɦɟ
,
. , , .
•
: ;
, 2–3 ;
,
•
,
-
. • • • • •
,
:
t1( ) – U(r) – t2( ) – S1(r1) – S2(r2) –
; ; ; ; . ,
.
dt1 (x ) + K1t1 (x ) = V1 (x ), dx
0≤ x≤ ∆ x ;
(10.1)
t1 (0 ) = t10 ;
(10.2)
d 2U (r ) 1 dU (r ) + = 0, r dr dr 2
;
(10.3)
dU (r ) + α1 (U (r ) − t1 ) = 0 ; dr
λ
λ
r ≤r ≤r +δ
dU (r + δ dr
) + α (U (r 2
(10.4)
+ δ )− t2 ) = 0 ;
(10.5)
0≤ x≤∆ x ;
(10.6)
dt 2 (x ) + K 2 t 2 ( x ) = V2 ( x ) , dx t 2 (0 ) = t 20 ;
d 2 S i (ri ) 1 dSi (ri ) + = 0, ri dri dri2 λ
V1 (x ) = K1t F1 (x ) ;
Ri −1 ≤ ri ≤ Ri ;
dS1 (R0 ) + α (S1 (R0 ) − t 2 ) = 0 ; dr1
S1 (R1 ) = S 2 (R2 ); α1 1 , G1c1
i = 1, 2,
λ
(10.8)
(10.9)
dS 2 (R2 ) + α oc (S 2 (R2 ) − t oc ) = 0 ; dr2
λ
K1 =
(10.7)
(10.10)
dS1 (R1 ) dS 2 (R1 ) =λ , dr1 dr2
(10.11)
(10.12) K2 =
α
+ α2 G2 c 2
1
= 2πr n,
R0 = r ,
2
,
V2 (x ) = 2
α
t F (x ) + α 2
= 2π (r + δ ) n,
R1 = r + δ ,
G2 c 2
2 t F2
(x )
= 2π r ;
R2 = r + δ + δ .
; (10.13) (10.14) (10.15)
, . . t F1 (x ) = onst = t F1 , t F2 (x ) = onst = t F2
t F (x ) = onst = t F ,
-
:
(
)
t1 (x ) = t F1 + t10 − t F1 exp (− K1 x ) ;
(10.16)
U (r ) = AU + BU ln (r ) ;
(10.17)
t 2 (x ) = V2 + (t 20 − V2 ) exp (− K 2 x ) ; S i (ri ) = Ai + Bi ln (ri ),
V2 =
α
α
tF + α2 + α2
2 t F2
;
(10.18)
i = 1, 2 ,
(10.19)
(10.20)
2
BU =
δ ln 1 + r
t 2 − t1 1 ( α r 2 +δ
+ λ
λ AU = t1 − BU ln (r ) + α1 r B1 =
λ λ ln(R0 )+ − R0α λ
;
(10.21)
(10.22)
t2 − toc
1 1 λ ln(R2 )+ + λ ln (R1 ) − R2αoc λ λ
B2 =
;
(
(10.25) .
(10.26)
)
∆x t10 − t F1 1 t1 = t1 (x ) dx = t F1 + (1 − exp (− K1∆x )) ; ∆x 0 K1∆x
t2 =
; (10.23)
(10.24)
λ B1 ; λ
λ A2 = A1 + B1 ln(R1 )1 − λ
∫
;
λ A1 = t 2 − B1 ln (R0 ) + α R0
:
1 − ) α1r
(t − V ) 1 t 2 (x ) dx = V2 + 20 2 (1 − exp (− K 2 x )) . K 2 ∆x ∆x 0
∫
(10.27)
∆x
(10.28)
:
t F1 = U (r ); t F2 = U (r + δ ); t F = S1 (r ).
(10.29)
α1
α
-
, . :
α1 = α1 (t1 ,U (r
α = α (t 2 , V1 (r
)) ,
)) ,
α 2 = α 2 (t 2 , U (r + δ
)) ;
α oc = α oc (t oc , V2 (r + δ + δ
)) . . ,
-
. , . 1 , (10.27), (10.28).
-
2 (10.17)
(10.19).
3
.
4 5
2
3. (10.16), (10.18). -
(
). ∆Q1 = ∆Q2 + ∆Q ,
∆Q1 –
,
; ∆Q2 –
∆Q –
; .
∆ Q1 = G1c1 (t1k − t10 ) ;
(10.30) ,
(10.31)
∆Q21 = G2 c2 (t 2 k − t 20 ) ;
∆ Q = α oc 2π R2 ∆x (V2 (R2 ) − t oc ).
:
(10.32) (10.33) -
, ∆Q , rf
∆G =
rf –
-
(10.34)
. ,
-
. ,
.
,
,
-
. • •
,
: ; ;
• • • •
; ,
; ; . :
• t1k, t2k – • G 1, G 2 – • ∆Q –
; ,
;
,
.
,
-
, .
1 2 3 4
. . . . . . . 1.
,
-
. . , ∆τ.
,
-
,
,
,
. :
∆Q1 = ∆Q2 + ∆Q + ∆Q + ∆Q + ∆Q , ∆Q1 –
, ;
∆Q –
;
∆Q – ,
,
; ∆Q2 – ,
(10.35)
,
-
; ∆Q –
; ∆Q –
-
. -
, 3(
), ,
-
( ).
,
-
, . ,
–
. ,
,
,
-
. ,
, ,
,
-
:
∂t1 (x, τ ) ∂t (x, τ ) + W1 1 + K1t1 ( x, τ ) = F1 (x, τ ), ∂τ ∂x
t1 (0, τ) = t10 (τ);
0 ≤ x≤ ∆ x;
t1 (x, 0 ) = f1 (x );
∂ 2U (r , τ) 1 ∂U (r , τ) ∂U (r , τ) , = at2 + ∂r 2 r ∂r ∂τ
(10.37)
r ≤ r ≤ r + δ , τ > 0;
U (r ,0 ) = f t (r );
∂U (r , τ ) + α1 (U (r , τ ) − t1 ) = 0 ; ∂r
λ
α1 < 0 ;
(10.40)
∂U (r + δ , τ) + α 2 (U (r + δ , τ ) − t 2 ) = 0 ; ∂r ∂t 2 (x, τ ) ∂t ( x, τ ) + W2 2 + K 2 t 2 ( x, τ ) = F2 ( x, τ ), 0 ≤ x ≤ ∆ x ; ∂τ ∂x
t 2 ( x, 0 ) = f 2 ( x ) ;
∂ 2 S i (ri , τ) 1 ∂S i (ri , τ) ∂S i (ri , τ) , + = ai2 ∂r 2 r r ∂ ∂τ i i i
λ
K1 =
α1 1 G1c1
,
F1 (x, τ ) = K1t F1 (x, τ ) ; K2 =
(10.45) (10.46)
dS 2 (R2 , τ ) + α oc (S 2 (R2 , τ ) − t oc ) = 0 ; dr2
(10.47)
dS1 (R1 , τ ) dS 2 (R1 , τ ) =λ , dr1 dr2
λ
(10.42)
i = 1, 2, Ri −1 ≤ ri ≤ Ri ; (10.44)
dS1 (R0 , τ ) + α (S1 (R0 , τ ) − t 2 ) = 0 ; dr1
S1 (R1 , τ ) = S 2 (R2 , τ );
(10.41)
(10.43)
S i (ri , 0) = ϑi (ri ) ;
λ
(10.38)
(10.39)
λ
t 2 (0, τ ) = t 20 (τ);
(10.36)
(10.48)
(10.49) α
+ α2 G2 c 2
2
,
V2 ( x , τ ) =
α
t F ( x, τ ) + α 2 G2 c 2
2 t F2
( x, τ )
; (10.50)
1
= 2π r n ,
2
R0 = r ,
t1 , t 2
= 2π (r + δ ) n ,
= 2π r
R2 = r + δ + δ
R1 = r + δ ,
;
.
(10.51)
(10.52)
, t F1 (x, τ), t F2 (x, τ), t F (x, τ ) –
– ,
.
(10.36) – (10.48)
. x t1 (x, dτ) = exp (− P1 x ) t10 (dτ) + ∫ θ1 (x, dτ) exp (P1 x ) dx , (10.53) 0
P1 =
B=
t1 − t 2 1 1 ln(r ) − ln(r + δ ) + λ − ( ) r r α + δ α 1 2
;
θ1 (x, dτ) =
K 1 dτ + 1 ; W1dτ
f (x ) 1 F1 (x, dτ ) + 1 . W1 dτ
ξ(ν n , τ)ζ(r , ν n ) , Zn n =1
U (r , τ) = A + B ln(r ) + ∑ ∞
(10.54) (10.55)
(10.56)
λ A = t1 − B ln(r ) + α1 r
(
;
ξ(ν n , τ ) = ξ(ν n ,0 ) exp − ν 2n τ
(10.57)
);
(10.58)
ν r ν r ζ (r , ν n ) = J 0 n + Dn Y0 n ; a a ξ(ν n , 0 ) =
(10.59)
∫ r ( f t (r ) − A − B ln (r )) ζ(r , ν n ) dr ;
r +δ
(10.60)
r
ν λ νn νn J 1 r − α1 J 0 n r at at at Dn = λ νn νn ν α1 Y0 n r − Y1 r a at at t νn
;
(10.61)
– ν (r + δ J 0 at
ν (r + δ + D Y0 at
) −
ν (r + δ νλ a α J1 at t 2
) −
ν (r + δ νλ a α Y1 at t 2
) +
) = 0 ;
(10.62)
Zn =
∫ r ζ (r , ν n ) dr = ∫
r +δ r
ν (r + δ × J 0 n at
) Y
ν r − 0,5 r 2 J 02 n at
ν r − r 2 Dn J1 n at
0
ν (r + δ + δ ) + J 12 n at at r
ν n (r + δ at
νnr at
ν r + J1 n Y1 at
ν r + J 12 n a t
Y1
2
ν (r + δ + δ ) + Y12 n at at
)2 Dn2 Y02 ν n (r
)2 J 02 ν n (r
= 0,5 (r + δ
+ 0,5 (r + δ
ν r ν r r J 0 n + D n Y0 n dr = at at
r +δ
2
−
νnr at
) + J
) −
) + (r
ν n (r + δ at
1
) Y
1
+δ
)2 Dn ×
ν n (r + δ at
) +
−
ν r − 0,5 r 2 D 2n Y02 n at
.
ν r + Y12 n at
(10.63)
x t 2 (x, dτ ) = exp (− P2 x ) t 20 (dτ ) + θ 2 (x, dτ ) exp (P2 x ) dx , 0
∫
P2 =
K 2 dτ + 1 ; W2 dτ
θ 2 ( x , dτ ) =
1 W2
f (x ) V2 (x, dτ) + 2 . dτ
(10.65) S i (ri , τ ) = Ai + Bi ln(ri ) +
B1 =
ln (R0 ) +
σ(µ n , τ ) ωi (ri , µ n ) , Yn n =1
∑ ∞
t 2 − t oc
λ λ λ ln(R2 ) + − R0 α R2 α oc λ
1 1 + λ ln (R1 ) − λ λ
λ A1 = t 2 − B1 ln (R0 ) + α R0 B2 =
∑ a m2 0,5 Rm2 Cm2 , n J 02 2
λ
m =1 m
µ R µ n Rm + J12 n m a m am
µ R + Rm2 Cm , n Dm , n J 0 n m am
µ n Rm Y0 a m
+
µ R + J1 n m a m
Y1
µ n Rm a m
+
;
λ B1 ; λ
λ A2 = A1 + B1 ln(R1 )1 − λ Yn =
(10.64)
(10.66)
; (10.67)
(10.68)
(10.69) .
(10.70)
µ R + 0,5 Rm2 Dm2 , n Y02 n m am
µ R + Y12 n m a m
µ R + J12 n m −1 a m
−
µ R µ n R m −1 + J 1 n m −1 a a m m
Y1
µ n Rm −1 − 0,5 Rm2 −1Cm2 , n J 02 am − Rm2 −1C m , n Dm , n ×
µ n Rm −1 Y1 × J1 a m
−
µ n Rm −1 µ R + Y12 n m −1 − 0,5 Rm2 −1 Dm2 , n Y02 am am
µ n R m −1 − a m
.
(10.71) ωm (rm , µ n ) =
m, n
C1, n
D1,n
C 2, n
D2, n
µ r J 0 n m am
µ r + Dm , n Y0 n m a m
;
= 1;
(10.73)
µ λ1 µ n µ n J 1 R0 − α1 J 0 n R0 a1 ; a1 a1 = µ λ µ µ α1 Y0 n R0 − 1 n Y1 n R0 a a a 1 1 1
µ R2 C 2 J 0 a2
µ R2 µ λ2 − J1 a2 a2 α
(10.74)
µ µ µ C1, n J 0 n R1 + D1, n Y0 n R1 − D2 , n Y0 n R1 a 2 ; (10.75) a1 a1 = µ Y0 n R1 a 2
µ µ λ1 a21 µ n J 0 R1 C1, n J1 n R1 + D1, n Y1 n R1 − λ 2 a1 a2 a1 a1 = → µn µn J 0 R1 Y1 R1 − a2 a2 µ µ µ − J1 n R1 C1, n J 0 n R1 + D1, n Y0 n R1 a2 a1 a1 → ; µn µn R1 Y0 R1 − J1 a2 a2
µ n – n-
(10.72)
µ R2 + D2 Y0 a2
µ R2 µ λ2 − Y1 a2 a2 α
(10.76)
= 0 .
(10.77) σ(µ n , 0) =
∑ 2
λm
∫ (ϑm (rm ) − Am − Bm ln(rm )) ×
Rm
rm 2 m =1 a m Rm −1
×
m, n
µ r µ r J 0 n m + Dm , n Y0 n m drm ; am am
(
σ(µ n , τ ) = σ(µ n , 0 ) exp − µ n2 τ
).
(10.78)
(10.79)
(10.71) – (10.78)
1
,
2–
-
; R0 = r ; R1 = r + σ ; R2 = r + σ + σ .
t1 (dτ ) =
U (τ ) = A − +
∑ν ∞
n =1
x − τ + exp ( P1 x ) t10 (d ) θ1 (ψ, dτ) exp (P1ψ ) dψ dx ; 0 0
∫
(
B B + (r + δ 2 (r + δ )2 − r 2
(r + δ Z n n a
ν r − r J1 n at
) J1 ν n (r
ν r + Dn Y1 n a t ×
+δ
at
∫
∆x
1 ∆x
)2 ln (r
) + D
(
)) +
+ δ ) − r 2 ln (r ν n (r + δ at
nY1
)
exp − ν n2 τ ×
) −
∫ r ( f (r ) − A − B ln(r ))ζ (r , ν n ) dr .
r +δ
(10.80)
(10.81)
r
S i (τ ) = Ai − +
∑µ ∞
n =1
R i Y n n ai
− Ri −1
× exp
(
i,n
− µ 2n
τ
(
(
))
Bi B + 2 i 2 Ri2 ln (Ri ) − Ri2−1 ln Ri −1 + 2 Ri − Ri −1
µ n Ri ai
i , n J1
µ R + Di , nY1 n i a i
−
∆x x 1 t2 (dτ) = exp (− P2 x ) t20 (dτ) + θ2 (ψ, dτ)exp (P2 ψ ) dψ dx ; (10.82) ∆x 0 0
∫
∫
µ n Ri −1 µ R + Di , n Y1 n i −1 × J1 a i ai
)∑ a ∫ r λm
2
Rm
m 2 m =1 m Rm −1
(ϑm (rm ) − Am − Bm ln(rm ) ) ωm (rm , µ n ) drm . (10.83) , ,
-
. , -
, . ,
(
∆Q = c π (r + δ
)2 − r 2 )∆x n ρ
:
(U (∆τ) −U b (∆τb )) ,
(10.84)
U b (∆τb ) –
.
(
, S1b (∆τ b )
∆Q = c π (r + δ
)2 − r 2 )∆x ρ
:
(S1 (∆τ)− S1b (∆τb )) ,
(10.85)
–
.
(
, S 2b (∆τ b )
∆Q = c π (r + δ
)2 − r 2 )∆x ρ
:
(S 2 (∆τ)− S 2b (∆τb )) ,
(10.86)
– . ,
-
, . , . 1 , (10.80), (10.82). 2 (10.55)
(10.66)
.
3
.
4
2
3.
5
(10.53), (10.64).
6
(10.35). . , :
t1 , t 2 , t1 , t 2
–
( νn , µn
G1 , G2
–
-
);
ξ(ν n , ∆τ ) σ(µ n , ∆τ ) Dn , Ci , n , Di , n , , Zn Yn
–
; –
,
;
A , B , Ai , Bi ∆τ
–
,
;
–
. : ;
;
-
U-
;
;
;
-
. ,
, .
-
. 1
:
– d – δ – δ – δ – δz – n – m – lp – se – λ , λ , λ , λz –
;
D
; ; ; ; ; ; ; ; ,
; ,
,
-
. 2
,
– ρ1(t), ρ2(t) – 1(t), 2(t) – λ1(t), λ2(t) – µ1(t), µ2(t) – β1(t), β2(t) – σ1(t), σ2(t) – r1(t), r2(t) – t 1( ), t 2( ) – t – 3 , k1 – , F – G1, G2 – t1 , t2 – t1 , t2 – 1,
: ;
2
; ; ; ; ; ; ; ; . : ; ; ; ; . ,
,
–
.
,
. (F, G1, G2, t1 , t2 , t1 , t2 )
, , t1 ; t2
1
2
. : G1
G2; t1
t2 . ,
, ,
. , , , ( ). ,
,
, ,
,
-
.
, ,
-
, , .
(
),
(
,
)
-
( ). ,
-
,
,
(
)
t1 (x ) = t F1 + t10 − t F1 exp (− k t K1 x ) ;
. (10.87)
t 2 (x ) = V2 + (t 20 − V2 ) exp (− k m K 2 x ) ; t 1 = t F1 + k t t 2 = V2 + k m
(t10 − t F ) (1− exp (− k K1∆x
1
t
(t 20 − V2 ) (1 − exp (− k K 2 ∆x
(10.88)
K1 ∆x )) ;
m
(10.89)
K 2 x )) ;
(10.90)
∆ Q1 = k t G1 c1 (t1 − t10 ) ;
(10.91)
∆Q21 = k m G2 c2 (t 2 − t 20 ) ,
kt (
) km ( k1 , k2 , k3
(10.92)
)
-
:
k t = k 3 max (k1 , k 2 );
k1 , k2 , k3
k m = k1 k t .
(10.93)
:
k1 = 1 k2 = 1 k3 = 1 = –1.
,
k1 = –1 – ,
; k2 = –1; ,
k3
(
)
,
. . U-
,
,
.
(
)
, .
.
-
, . : – 0,462 ; – 0,009 ; – 0,07 ; – 0,0025 ; – 0,0002 ; –2 ; – 110.
– 150 ° ;
, – 25 ° .
. – 20 ° ;
0,034
/ ;
–5° ;
. – 15 ° .
0,278
-
/ .
. 10.3
.
10.4 – 10.8
.
. : –
;r–
,
-
. ,
. , ,
-
, ( ): 140
150
t1 t1
t2 5
125
0
x
2
Ɋɢɫ. 10.3 ɂɡɦɟɧɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪ ɩɨ ɞɥɢɧɟ ɚɩɩɚɪɚɬɚ: t1 –
; t2 –
0
0,05
Ɋɢɫ. 10.4 ɂɡɦɟɧɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɩɚɪɨɜ ɛɟɧɡɨɥɚ ɜ ɷɥɟɦɟɧɬɚɪɧɨɣ ɨɛɥɚɫɬɢ
5,5
t2
x
U
-
8,5
5,2 0
0,0125
0,05
x
Ɋɢɫ. 10.6 ɂɡɦɟɧɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɩɨ ɬɨɥɳɢɧɟ ɫɬɟɧɤɢ ɬɪɭɛɤɢ ɬɪɭɛɧɨɝɨ ɩɭɱɤɚ ɜ ɷɥɟɦɟɧɬɚɪɧɨɣ ɨɛɥɚɫɬɢ
Ɋɢɫ. 10.5 ɂɡɦɟɧɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜɨɞɵ ɜ ɷɥɟɦɟɧɬɚɪɧɨɣ ɨɛɥɚɫɬɢ
5,19
15,0
S1
S2
5,18
5,0
0,213
0,222
r
0,0145
y
Ɋɢɫ. 10.7 ɂɡɦɟɧɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɩɨ ɬɨɥɳɢɧɟ ɫɬɟɧɤɢ ɤɨɪɩɭɫɚ ɚɩɩɚɪɚɬɚ ɜ ɷɥɟɦɟɧɬɚɪɧɨɣ ɨɛɥɚɫɬɢ t 0,t 0,t 0 –
0,222
0,272
r
Ɋɢɫ. 10.8 ɂɡɦɟɧɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɩɨ ɬɨɥɳɢɧɟ ɫɥɨɹ ɬɟɩɥɨɢɡɨɥɹɰɢɢ ɚɩɩɚɪɚɬɚ ɜ ɷɥɟɦɟɧɬɚɪɧɨɣ ɨɛɥɚɫɬɢ ,
t10(x), t20(x) – ρ,ρ ,ρ – , , –
; ;
,
; ,
.
. . ,
-
, . . ,
-
. , . , ,
-
, . ,
,
,
-
, .
aτ Fo = 2 < 0,5 R
.
;τ–
-
λ a= – cρ
(
;R–
-
)
-
. ,
. . .
,
, . -
,
. , ,
-
. ,
, ,
.
,
,
, -
. . ,
,
;
; .
(
). , ,
,
-
, . .
•
,
-
,
•
. ,
-
.
•
(
,2–4
).
•
,
. -
, , ;
,
-
. . . ,
, ,
-
t 0 = onst = t (0 ) =
,
(r
+δ
r t (r ) dr . )2 − r 2 r∫ r +δ
2
(10.94)
, t (r , 0 ) = t b (τ bk ) =
tb(r, τ) –
(r
+δ
r t (r , τ ) dr = t 0 = )2 − r 2 r∫ b bk r +δ
2
onst ,
(10.95)
; τbk –
.
,
-
: 1 ∆Q1 = α1 ∆F1 τb
t (r , τ b ) dτ − t1 ; 0
∫
τb k
(10.96)
,
: 1 ∆Q = α 2 ∆F2 τb
(
,
(10.96)
0
+ δ , τb ) dτ − t 2 ;
(10.97)
)2 − r 2 )∆x n ρ (t (τ k ) − tb (τbk )) = ∆Q1 − ∆Q . (10.98) :
∆Qc = cc π (r + δ
,
∫ t (r
τb k
(10.97)
:
∫
1 t (r , τ) dτ = A + B ln (r ) + τ0 τ
+ ×
∑Z ∞
n =1
(
(
))
µn r µn r 2 J0 a + Dn Y0 a 1 − exp − µ n τ ×
2 n µnτ
1
(10.99)
µn r µn r ∫ r ( f (r ) − A − B ln(r ) ) J 0 a + Dn Y0 a dr .
r +δ
r
. ,
-
,
. ,
, . ,
,
,
. ,
,
,
-
. ,
,
.
.
-
(
,
tbk r , τb k
)
tb (r + δ , τ bk ) ,
,
-
. -
,
, , . .
,
.
, , ,
,
,
-
,
. (10.44) – (10.48) .
(10.35) , :
1 ∆ Q3 = α ∆F τb
t (r , τb ) dτ − t 2 , 0
∫
τb k
(10.100)
∆F = π d ∆ x .
, : 1 ∆Q = α ∆F τb
∆F = π (d + δ ) ∆ x .
t (r + δ , τ b ) dτ − t , 0
∫
τbk
,
:
(
∆Q = c π (r + δ t kb (τ bk )
-
(
∆Q = c π (r + δ
,
b
)2 − r 2 )∆x n ρ (t (τ ) − t kb (τbk )) , (10.102)
– .
t
(10.101)
(τbk ) –
)
2
−r
2
)∆x n ρ
(t (τ ) − t b (τbk )) , (10.103) :
. ,
: ∆Q2 = ∆Q − ∆Q − ∆Q − ∆Q = ∆Q − ∆Q3 .
,
(10.100) – (10.101)
(10.104)
(10.99).
11 ɆȿɌɈȾɂɄȺ ɊȺɋɑȿɌȺ ɇȿɋɌȺɐɂɈɇȺɊɇɈȽɈ ɌȿɆɉȿɊȺɌɍɊɇɈȽɈ ɉɈɅə ȿɆɄɈɋɌɇɈȽɈ ȺɉɉȺɊȺɌȺ . ;
: ;
,
,
; ;
-
;
. .
-
,
. . ,
, . .
-
. , ,
. –
–
. , ( (
,
),
),
-
,
, –∆ .
.
, ,
∆ 1.
-
, -
-
. :
•
(
,
.);
•
(
,
,
,
,
-
.).
,
. .
1 D – – Dd – Dr – Dz – dz – hz – δ – δ – δt – δ – δz – n – m – Ne –
: ; ; ; ; ; ; ; ; (
,
,
.);
; ; ; ; ; ;
ω – λ,λ,λ,λ – 2
-
,
; ; , ,
. :
. 3 «2» –
( ,
«1»
,
«3» –
,
«4» –
-
): 1, 2, 3 – ρ1(t), ρ2(t), ρ3(t), ρ4(t) – 1(t), 2(t), 3(t), 4(t) – λ1(t), λ2(t), λ3(t), λ4(t) – µ1(t), µ2(t), µ3(t), µ4(t) – β1(t), β2(t), β3(t), β4(t) – σ1(t), σ2(t), σ3(t) – r1(t), r2(t), r3(t) – t 1( ), t 2( ), t 3( ) – qr1 – dqc1 – qv1 – t – ,
;
; ; ; ; ; ; ; ; ; ; ; , . -
. , ,
. [9]: ρ
=
∑
1
i
–
xi
i-
xi ρi
,
(11.1)
,
(11.2)
.
: µ
=
∑ i
xvi
–
i-
1 xvi µi
. .
-
,
-
. ,
.
,
: 1 = α i ∆Fi τj i =1
∑ N
Q1 j
Q1 j
– ;N– ; ti, j (r , τ) –
,
ti , j (r , τ) dτ − t pj , 0
∫
τj
(11.3)
j, ,
-
i-
; α i , ∆Fi –
j-
-
; τj –
i; t pj –
j-
j-
. :
Q
ti , j (r + δ , τ) Q
j
=
∑
–
j
ti , j (r + δ , τ) dτ − t , 0
1 α i oc ∆Fi oc τj i =1
∫
τj
N
(11.4)
j-
;
–
,
i-
–
; α , ∆F
j-
i;t
–
.
,
. ,
. 1 Qvj = α vi ∆ Fvi τj i =1
∑ M
–
Qvj
t vi, j (r , τ) dτ − t pj , 0
∫
τj
(11.5)
,
j; tvi, j (r , τ) –
; M –
,
,
-
i-
; α vi , ∆Fvi –
j-
-
i. ,
j-
.
: Q j = Q1 j + Qvj + Qhj + Qdj + Qcj + Qmj + Qsj + Qej + Q pj + Q
Q1j –
,
,
,
j
, (11.6)
; Qvj – ; Qhj –
(
)
; Qdj – ; Qmj –
; Qsj –
; Qej – ,
; Qpj –
,
; Qcj – , -
,
, ; Q
j
–
. : ∆I j =
Qj τ j G
.
(11.7)
,
,
. ,
, . 12 ɆȿɌɈȾɂɄȺ ɊȺɋɑȿɌȺ ɋɈɊȻɐɂɈɇɇɈȽɈ ɈȻɈɊɍȾɈȼȺɇɂə .
∆ , ,
, .
-
∆τ.
-
. ,
-
, , . ,
-
, ,
.
,
,
-
. , , . .
,
. . . ,
.
, . : G – – t – Dg, Dc –
; ; ; -
-
; α1, α , αoc – , ) 1, ρ1, λ1 – , ,ρ,ρ,λ,λ – ; r – d – δ – t –
,
(
; ,
; ,
; ; ; . -
,
-
. , ,
-
, . ,
:
,
,
. . :
• t1 – • t(r, τ) – • t (r1) – • 1– • (r, τ) –
(r, τ) –
t(r, τ)
; ; ; ; . t (r1)
, . ∂ 2t (r , τ) 2 ∂t (r , τ) ∂t (r , τ) + q , = ac2 + 2 ∂τ r ∂r cc ρ c ∂r
0 ≤ r ≤ R,
τ > 0 ; (12.1)
t (r , 0 ) = f (r ) ;
(12.2)
∂t (0, τ) = 0; ∂r
λc
(12.3)
∂t (R, τ) + α c (t (R, τ) − t c ) = 0 , ∂r
αc < 0 .
(12.4)
∂ 2t (r1 , τ) 1 ∂t (r1 , τ) ∂t (r1 , τ) , r ≤ r1 ≤ r + δ , τ > 0 ; (12.5) = a2 + ∂r 2 ∂ r r ∂τ 1 1 1 t (r1 ,0 ) = f (r1 ) ;
λ λ
∂t (r , τ) + α (t (r , τ ) − t1 ) = 0 ; ∂r1
(12.6) α < 0;
∂t (r + δ , τ) + α oc (t (r + δ , τ) − t oc ) = 0 ; ∂r1
∂ 2c(r , τ) 2 ∂c (r , τ) ∂c(r , τ) , 0 ≤ r ≤ R , τ > 0 ; = Dc + ∂r 2 r ∂r ∂τ c (r , 0) = f c (r ) ; ∂c (0, τ) 30,
Nu D =
)
),
PrD =
ν Dg
.
Nu D = 0,395 Re 0,64 PrD0,333 ;
(12.38)
Nu D = 0,725 Re 0, 47 PrD0,333 ;
(12.39)
Nu D = 0,515 Re 0,85 PrD0,333 .
(12.40)
(12.41) α1
α ,
-
. 1,5 Re Re 0,33 , < 200 ; 0,0035 Pr ε ε Nu = 0, 67 Re Re 0 , 4 Pr 0,33 , ≥ 200 . ε ε
(12.42)
-
0, 5 0,5 (1 − ε ) , 0,31 Re ε Nu = 0, 2 0,8 (1 − ε ) 0 , 1 Re , ε
(12.43) Nu =
(12.42) ω–
ωd αd , Re = , d = ν λ
;ε–
Re∈(1,5 , 57 );
Re∈(57 , 150).
(12.43)
F , π
;F– . .
1
, , ,
-
,
. 2 ,
-
,
. 3 4
,
. .
-
. 1–4
5 .
(
). ∆Q1 = ∆Q2 + ∆Q3 + ∆Q + ∆Q ,
∆Q1 –
; ∆Q2 –
; ∆Q3 –
,
,
,
; ∆Q –
(12.44) -
; ∆Q –
. , -
, . ɁȺɄɅɘɑȿɇɂȿ
,
,
-
, ,
,
-
. , ,
-
, . , ,
,
« "
"»
. . .
,
«
»,
-
«
». -
,
, ,
,
. :
• •
; ,
• •
; ;
• •
; . :
• • • • • • • •
-
;
; ; ; ; ,
,
-
; ,
; ; . ,
,
. .
n;
;
,
-
,
-
. ,
,
. .
.
ɋɉɂɋɈɄ ɅɂɌȿɊȺɌɍɊɕ 1
. ., . 2
. . . . .
3
-
.:
, 1996. 496 . .
.:
, 1978. 512 .
.
. 2-
4
. ., .
.:
5 6
. . . ., // 7
.
.:
.
-
, 1961. 703 . . ., . . , 1970. 712 . : . .: , 1972. 560 . . . . 1985. № 2. . 118 – 123. . ., . ., . . ,
-
// 8 / 9 ,
. 1995. . 1, № 1 – 2. . 39 – 52. . ., . ., . . . . . . .: , 1981. 264 . . . 5 ./ . . . .
, .
.:
, , 2000. . 1. 480 .
,
-