Основы технической теплофизики 5-94275-123-4
В монографии рассмотрены основные положения и законы теплопроводности. Изложен принцип наложения температурных полей, ме
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ȼ.Ɇ. Ɏɨɤɢɧ Ƚ.ɉ. Ȼɨɣɤɨɜ ɘ.ȼ. ȼɢɞɢɧ
ɈɋɇɈȼɕ ɌȿɏɇɂɑȿɋɄɈɃ ɌȿɉɅɈɎɂɁɂɄɂ
ɆɈɋɄȼȺ "ɂɁȾȺɌȿɅɖɋɌȼɈ ɆȺɒɂɇɈɋɌɊɈȿɇɂȿ-1" 2004
ȼ.Ɇ. Ɏɨɤɢɧ Ƚ.ɉ. Ȼɨɣɤɨɜ ɘ.ȼ. ȼɢɞɢɧ ɈɋɇɈȼɕ ɌȿɏɇɂɑȿɋɄɈɃ ɌȿɉɅɈɎɂɁɂɄɂ
"
-1" 2004
536.24 31.312.06 75
, . .
75
Ɏɨɤɢɧ ȼ.Ɇ., Ȼɨɣɤɨɜ Ƚ.ɉ., ȼɢɞɢɧ ɘ.ȼ. : .: " -1", 2004. 172 .
.
, ,
,
-
. ,
-
. , .
.
, . , ,
,
,
-
. 536.24 31.312.06
ISBN 5-94275-123-4
. . . .
, . . , 2004
, "
-1", 2004
ɈɋɇɈȼɕ ɌȿɏɇɂɑȿɋɄɈɃ ɌȿɉɅɈɎɂɁɂɄɂ . . . . 60 × 84/16.
15.04.2004 Times. . : 10,0 . . .; 10,0 .- . . 400 . . 279
" 107076,
-1", ., 4
, -
392000,
,
, 106, . 14
ɉɊȿȾɂɋɅɈȼɂȿ
: "
", "
", "
", "
", " "
. (
(
-
", "
),
)
,
,
,
,
,
. ,
,
.
,
-
, . ,
,
,
. (
,
,
)
, .
-
-, -
-
.
-
,
-
, . :
,
,
.
.
,
. , 7 – 10 − . . ɍɋɅɈȼɇɕȿ ɈȻɈɁɇȺɑȿɇɂə
1–6
T– t– T(0, τ); T – T(R, τ); T – T0 – Tc – T* – ϑ = (T − T0) – θ = T/T0 – , y, z – τ– , 2R – d, D − L, l, δ − f−
,
, , , , , , ,K
, , , ,
2
.
,°
-
F− u− ξ– q− ql − Q− k− kL − – ρ– ( ρ) − p− G− V− m− ω− ν− – λ– α– E− – µn –
2
, , ,
/
2
,
/
, , , ,
/
,
/ ,
,
3
/( · )
3
/( 3· )
, ,
/( 2· ) ,
2
/ ,
3
,
,
/ 3
,
/
, , / , 2/ , 2/ /( · )
,
/( 2· ) ,
/
2
ȼȼȿȾȿɇɂȿ
/( · )
.
, .
-
, , . , ( ,
,
. )−
.
-
,
-
, . , , ,
. ,
. ,
,
,
-
. ,
-
. .
,
,
,
-
. 1. ɈɋɇɈȼɇɕȿ ɉɈɅɈɀȿɇɂə ɢ ɡɚɤɨɧɵ ɌȿɈɊɂɂ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɂ
– ,
),
( . .
.
,
,
-
,
,
. ,
,
. ,
,
,
. ,
.
-
, . −
,
,
-
. , . . (
,
,
,
. .),
,
,
. 1.1.
, , , (
,
).
T = f ( x, y , z , τ ) −
, ( И
-, (
, -
T = f ( x, y , z ) ,
.
,
,
) -
(
. -
)
,
,
. 1.1). –
. , .
.
y
z
z
x
dψ
ds
0
2R2
dz
2R1
Ɋɂɋ. 1.1. ɈȻɔȿɆ ɉȺɊȺɅɅȿɅȿɉɂɉȿȾȺ ɂ ɐɂɅɂɇȾɊȺ
,
T
T1 > T2
T1 T1
T2
r T2
0
T1
T2 x
0
.
1.2.
,
, :
–––––––– – ------ –
,
; ,
,
,
.
.
(
1.2). ( •
. 1.1) :
,
+R
+R
-
+R
3 1 2 1 T = dx dy T ( x, y , z , τ ) dz , 2 R1 2 R2 2 R3 −∫R − R∫ − ∫R 1
2
3
•
T =
1 πR 2 2 L
+ R1
2π
+L
0
0
−L
∫ rdr ∫ dϕ ∫ T (r , ϕ, z, τ ) dz .
1.2. ɁȺɄɈɇ ɌȿɉɅɈɉɊɈȼɈȾɇɈɋɌɂ ɎɍɊɖȿ. ȽɊȺȾɂȿɇɌ ɂ ɉȺȾȿɇɂȿ ɌȿɆɉȿɊȺɌɍɊɕ .
-
,
. gradT =
n–
dT , dn
. : (grad T ) S =
^ dT dT cos (n s ) = . dn ds
,
,
q(
.
-
)
(
. 1.3,
. 1.3).
,
-
q1 = q2 = q3 = q4 = q5 . . 1.3,
, q1 = q5 > q2 = q6 > q3 = q4, .
1
>
q1
2
q2
2
1
q1
q2
q3 1
q3
q6
q4 q5
2
n
0
q4
q5 1
>
2
) ) Ɋɢɫ. 1.3. ɂɡɨɬɟɪɦɵ, ɥɢɧɢɢ ɬɨɤɚ ɬɟɩɥɚ, ɜɟɤɬɨɪɵ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ: –
: q1 = q2 = q3 = q4 = q5;
–
: q1 = q5 > q2 = q6 > q3 = q4 :
-
.
. .
Q(
τ ( ):
F ( 2)
), q = Q / (F τ), . .
1807 .
/(
2
⋅ )
/
2
.
,
,
. . n,
:
-
dT : q = −λ . dn
-
+ q = −λ
q = −λ
dT dn
dT . dn
n,
-
: − q = −λ
q=λ
dT dn
dT . dn
, ,
-
, , . . dQ = − λ
dT dF dτ = q dF dτ . dn
, . . . 1.3. ɄɈɗɎɎɂɐɂȿɇɌ ɌȿɉɅɈɉɊɈȼɈȾɇɈɋɌɂ λ
,
,
λ
Q ( ), τ ( ),
1.
λ=
dQ dT dF dτ dn
λ
– λ ≈ 0,06 – 0,6;
. 1.4. F ( 2),
/
2
n ( ):
⋅( / )⋅
/( ⋅ ).
: – λ≈ – λ ≈ 2…400,
– λ ≈ 0,02..2;
:
λ λ
,
λ
. .
2.
.
,
Ɋɢɫ. 1.4. ɋɯɟɦɚ ɩɪɨɯɨɠɞɟɧɢɹ ɬɟɩɥɨɬɵ Q0, ɱɟɪɟɡ ɟɞɢɧɢɰɭ ɩɨɜɟɪɯɧɨɫɬɢ F, ɩɪɢ ɪɚɡɧɨɫɬɢ ɬɟɦɩɟɪɚɬɭɪ , ɜ ɨɞɢɧ ɝɪɚɞɭɫ, ɧɚ ɟɞɢɧɢɰɭ ɞɥɢɧɵ 1 ɦ
0,05…0,5; /( ⋅ ).
λ
λ .
-
:
λ
, -
.
3.
λ
.
,
-
. 4.
,
λ
5. 6.
λ
, .
λ ≤ 0,23
7.
/( ⋅ )
. , 9.
( (
-
,
,
-
. .) λ
,
,
.
.
λ .
8.
,
λ
.
.) –
-
.
10. , . 1.4. ɆȺɌȿɆȺɌɂɑȿɋɄȺə ɎɂɁɂɄȺ ȼ ɌȿɈɊɂɂ ɌȿɉɅɈɉɊɈȼɈȾɇɈɋɌɂ 1.
qn (
. 1.5),
-
,
: ^ ^ dT dT ; cos( nx ) = − λ q x = qn cos ( nx ) = − λ dx dn
^ ^ dT dT ; q y = qn cos (ny ) = −λ cos(ny ) = − λ dy dn
^ ^ dT dT cos (nz ) = −λ . q z = qn cos(nz ) = − λ dn dz
2. ,
,
M dτ,
-
T – ∆T
n qn
qy T
d2
dy dx
T + ∆T M
y
qx
z x
Ɋɢɫ. 1.5. Ɋɚɡɥɨɠɟɧɢɟ ɜɟɤɬɨɪɚ ɩɨ ɤɨɨɪɞɢɧɚɬɧɵɦ ɨɫɹɦ
q1 = qx1 dy dz dτ + qy1 dx dz dτ + qz1 dx dy dτ, qx1 – dy dz,
; qy1 − dx dy,
y
dx dz,
x ; qz1 –
dτ z
. ,
dτ
,
:
q2 = qx2 dy dz dτ + qy2 dx dz dτ + qz2 dx dy dτ,
qx2 – dy dz,
; qy2 − dx dy,
x y
dτ
dx dz,
; qz2 –
dV = dx d dz
dτ.
z
.
3. dQw
dV dτ
dQw −
,
=W ,
W dQw = W dV dτ .
4.
5.
ξ = f (n, τ), →
ξ = f (x, y, z, τ),
dξ τ =
∂ξ dτ ; ∂τ
dξ n =
∂ξ dn . ∂n
r ∂ξ dξ x = x dx ; ∂x
∂ξ y r dξ y = dy ; ∂y
r ∂ξ dξ z = z dz ; ∂z
r ∂ξ dξ τ = τ dτ . ∂τ
6. : dξ = (ξ 2 − ξ1 )
− dξ = (ξ1 − ξ 2 ) ,
ξ2, ξ1 –
. 1.5. ȾɂɎɎȿɊȿɇɐɂȺɅɖɇɈȿ ɍɊȺȼɇȿɇɂȿ ɌȿɉɅɈɉɊɈȼɈȾɇɈɋɌɂ , , . . T = f ( x, y , z , τ ) ,
, . .
,
. , (
-
. 1.6). ,
−
,
,
.
, .
dV 2
2
V dy
dz dx
+y + z
1 dx
1 0
a)
+x
)
Ɋɢɫ. 1.6. ɗɥɟɦɟɧɬɚɪɧɵɣ ɩɚɪɚɥɥɟɥɟɩɢɩɟɞ ɜ ɨɛɴɟɦɟ V ɢ ɞɜɚ ɪɚɡɥɢɱɧɵɯ ɩɭɬɢ ɞɜɢɠɟɧɢɹ ɷɥɟɦɟɧɬɚɪɧɨɝɨ ɩɚɪɚɥɥɟɥɟɩɢɩɟɞɚ ɨɬ ɬɨɱɤɢ 1 ɤ ɬɨɱɤɟ 2: –
;
–
x, y, z
dV = = dxdydz
( q x1 dydz + q y1 dxdz + q z1 dx y ) dτ + WdVdτ =
= (q x 2 dydz + q y 2 dxdz + q z 2 dxdy ) dτ + (cρ) dVdT .
τ,
W dV dτ
(1.1), 2, −
,
1,
(1.1)
. (W)
(cρ) dV dT −
( W) W = 0. (
,
,
). ,
:
,
dτ,
, dτ.
[
-
(cρ) dVdT = (q x1 − q x 2 ) dydz + (q y1 − q y 2 ) dxdz + + (q z1 − q z 2 ) dxdy ] dτ + WdVdτ .
q1
q2
, .
, q1 –
.
(
: q2 –
. . 1.4) q1 − q2 = − dq.
[
]
(cρ) dVdT = − dq x dydz − dq y dxdz − dq z dxdy dτ + WdVdτ .
∂q y ∂q ∂q (cρ)dVdT = − x dxdydz − dydxdz − z dzdydx dτ + WdVdτ . ∂y ∂z ∂x
dV ∂q y ∂q z ∂q − (cρ) dVdT = − x − dτ + Wdτ . ∂y ∂z ∂x
-
∂T ∂ − λ ∂ 2T ∂x = −λ = . ∂x ∂x ∂x 2
∂q x
,
∂ 2T ∂ 2T ∂ 2T (cρ) dT = λ 2 + 2 + 2 dτ + Wdτ . ∂z ∂y ∂x λ =a (cρ)
,
2
/ .
∂ 2T ∂ 2T ∂ 2T W dT = a 2 + 2 + 2 dτ + dτ . (cρ) ∂y ∂z ∂x
(1.2)
∂T dT = dTτ = dτ . ∂τ
,
,
.
(1.2) ∂T =a ∂τ
∂ 2T ∂ 2T ∂ 2T ∂x 2 + ∂y 2 + ∂z 2
-
W + (cρ) .
(1.3) −
(1.3) .
-
: ∂ 2T ∂ 2T ∂ 2T W + =0. + + ∂x 2 ∂y 2 ∂z 2 λ
,
W
(1.4) ,
-
: ∂ 2T ∂ 2T ∂ 2T = 0. + + ∂x 2 ∂y 2 ∂z 2
(1.3),
(1.4) ,
(1.5) ,
(1.5) , -
. : ∂T =a ∂τ
∂ 2T ∂ 2T ∂x 2 + ∂y 2
;
∂T ∂ 2T =a 2 ; ∂τ ∂x
∂ 2T W + = 0; ∂x 2 λ
∂ 2T = 0. ∂x 2
(1.6)
,
, ,
,
(
(1.4) − (1.6) -
),
. x = r cos , y = r sin
ψ.
(
. 1.1),
z,
r
, ∂ 2T 1 ∂T 1 ∂ 2T ∂ 2T W ∂T + . + + = a 2 + r ∂r r 2 ∂ 2 ∂z 2 (cρ) ∂τ ∂r
(1.7)
(1.7)
.
: ∂T =a ∂τ
(1.7) ( (
∂ 2T 1 ∂T ∂ 2T 1 ∂T W ∂ 2T 1 ∂T ; 0 ; + + = + = 0 . (1.8) + ∂r 2 r ∂r ∂r 2 r ∂r λ ∂r 2 r ∂r
(1.8)
)
, ,
,
,
-
). : ∂T =a ∂τ
, ,
∂ 2T 2 ∂T ∂r 2 + r ∂r .
(1.9) ,
.
-
,
, .
, , ,
.
dT dT = dTτ + dT .
. 1.6 2 = dTx+ dTy + dTz
dT = dTs .
ds = dx + dy + dz. dT
(1.10)
dT = dTτ + dTx + dTy + dTz =
∂T ∂T ∂T ∂T dz = dy + dx + dτ + ∂z ∂y ∂x ∂τ
∂T dx ∂T dy ∂T dz ∂T = dτ + + + . ∂τ dτ ∂x dτ ∂y dτ ∂z
(1.10)
1 dτ, dT
=
=
dV dx = dτ
x,
dn . dτ
dV
x. ∂T + dT = dτ ∂τ
x
∂T + ∂x
y
∂T + ∂y
z
∂T . ∂z
(1.11)
(1.11) (1.2) ∂T + ∂τ
−
x
∂T + ∂x
y
∂T + ∂y
z
∂T =a ∂z
∂ 2T ∂ 2T ∂ 2T W ∂x 2 + ∂y 2 + ∂z 2 + (cρ) . (1.12)
(1.12)
-
. .
1.6. ɌȿɉɅɈɉɊɈȼɈȾɇɈɋɌɖ ɉɅɈɋɄɈɃ ɈȾɇɈɋɅɈɃɇɈɃ ɋɌȿɇɄɂ λ (
2 −
. 1.7).
1
δ,
−
= δ).
F(
,
, F (
= 0);
. -
d 2T =0 dx 2 λ
T
Ɋɢɫ. 1.7. Ɋɚɫɩɨɥɨɠɟɧɢɟ ɤɨɨɪɞɢɧɚɬ ɧɚ ɩɥɨɫɤɨɣ ɫɬɟɧɤɟ: 1− 2−
F(
= 0);
F(
= δ)
T = C1x + C2 ;
dT dx
T1
T2
0
T1 = C1 ⋅ 0 + C2 ; T −T , T = T1 − 1 2 x .
x δ
T2 = C1 + C2 .
=
1.
q = −λ
R=
λ
−
dT λ λ F (T1 − T2 ) = (T1 − T2 ) , Q = qF = F (T1 − T2 ) = . dx R
.
−
.
,
,
-
, . q = Q / F.
1.7. ɌȿɉɅɈɉɊɈȼɈȾɇɈɋɌɖ ɐɂɅɂɇȾɊɂɑȿɋɄɈɃ ɈȾɇɈɋɅɈɃɇɈɃ ɋɌȿɇɄɂ
(
. 1.8):
U=
d 2T 1 dT + = 0; dr 2 r dr
dT ; dr
dU 1 + U =0; dr r
dU dr =− . U r
Ɋɢɫ. 1.8. Ɋɚɫɩɨɥɨɠɟɧɢɟ ɤɨɨɪɞɢɧɚɬ ɰɢɥɢɧɞɪɢɱɟɫɤɨɣ ɫɬɟɧɤɢ: 1−
T λ T1 L
( 2
T2
r = r1);
–
r1
-
r 0
(
r2
r = r2 ) U=
,
T = T1 −
C1 ; T = C1 ln r + C2 . r
T1 − T2 r ln ; r r1 ln 2 r1
q = −λ
dT T −T 1 =λ 1 2 ; r dr ln 2 r r1
Q = q 2 π rL =
πL (T1 − T2 ) ; 1 r2 ln 2λ r1
πL (T1 − T2 ) πL (T1 − T2 ) = , 1 d2 R ln 2λ d1
Q=
R=
1
ln
2λ
d2 − d1
(
,
π
,
).
qL = Q / L. 2. ɊȺɋɑȿɌ ɌȿɆɉȿɊȺɌɍɊɇɕɏ ɉɈɅȿɃ ɂ ɌȿɉɅɈɉȿɊȿȾȺɑɂ
2.1. ɉɊɂɇɐɂɉ ɇȺɅɈɀȿɇɂə ɌȿɆɉȿɊȺɌɍɊɇɕɏ ɉɈɅȿɃ ( ) (
(
)
. 2.1). (−Q); λ −
L,
, ,
L,
" "
"
"
"
, L
-
.
"
.
, "
(+Q), .
, ,
"
"
.
(+ Q ) =
πL (T1′ − T3′ ) ; 1 r3′ ln 2λ r1′
(− Q ) =
r3′′ r1′′ T3′′ = T1′′+ Q . 2 π Lλ
".
-
πL (T1′′− T3′′) , 1 r3′′ ln 2λ r1′′
ln
T′3
T′2
Ɋɢɫ. 2.1. Ɋɚɫɩɨɥɨɠɟɧɢɟ ɢɫɬɨɱɧɢɤɚ ɢ ɫɬɨɤɚ ɬɟɩɥɚ: K−
T′′3
r′3
T′′2
K
T′1
T′′1
r′2 r′1
r′′2 r′′1
+Q
–Q
T3′
K
r′′3
T3′′ .
, "
"
-
, " " K
" "
T0′
T0′′ ,
,
.
-
TK = T3′ + T3′′ − T0 = T1′ + T1′′−
r ′′ r ′ Q ln 3 1 . 2 π Lλ r1′′ r3′
2.2.
,
,
-
,
. ,
-
.
(+ Q ) =
πL (TK′ − T0′ ) , h 1 ln 2λ r1′
(− Q ) =
πL (T0′′ − TK′′ ) , 1 r ′′ ln 2λ h
K TK′ = T0′ +
TK′
Q r ′′ Q h ln . ln ; TK′′ = T0′′ + 2 π Lλ h r′ 2 π Lλ
TK′′ ,
: TK = T0 +
T( x , y ) = T0 +
Q ln 2 π Lλ
x 2 + (h + y )2 x 2 + (h − y )2
Q r ′′ ln , 2 π Lλ r′
.
2R (–Q) T′′0
h r′′
λ
x
0
λ
L >> 2R T′′K
T′0
y
K
h m
r′
x
T′K
(+Q) n 2R y
L−
Ɋɢɫ. 2.2. Ɋɚɫɩɨɥɨɠɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɚ ɜ ɝɪɭɧɬɟ: R; h − ;λ− ; 0−
(
. 2.2)
,
,
, T0
Q.
,
,
,
-
,
.
[
πL T( x , y ) − T0
Q= 1 2λ
ln
]
x 2 + (h + y )2 x 2 + (h − y )2
=
πL [Tm − T0 ] . 1 h ln 2 − 1 2λ R
, Tn = T0 +
n (x = 0; y = h + R),
-
.
-
Q h ln 1 + 2 . 2 π Lλ R
, . 2.3.
, .
,
.
(∆x = ∆y). :
(
. 2.3
-
2.4)
1)
;
2)
∆x
3) :
λ ∆y
2
T
L
1 0
3
4
T = f (x, y)
T1
∆x
T0 T3
–x
1
∆x
0
∆x
3
+x
. 2.3. -
. 2.4.
-
∆y;
,
Qik =
i
λ
F (Ti − Tk ) ,
;k
.
, :
,
-
•
Qik =
λ ∆yL (Ti − Tk ) ; ∆x
Qik =
λ ∆xL (Ti − Tk ) ; ∆y
•
•
Qik = λL (Ti − Tk ) .
(2.1) .
1.
1 2, 3, 4.
T0 =
2.
T1 + T2 + T3 + T4 4
0,
Q10 = Q02 + Q03 + Q04 .
0 (2.1)
λL (T1 − T0 ) = λL (T0 − T2 ) + λL (T0 − T3 ) + λL (T0 − T4 ) ,
.
1 2 : Q10 + Q20 = Q03 + Q04 ,
0,
0
3
4.
λL (T1 − T0 ) + λL (T2 − T0 ) = λL (T0 − T3 ) + λL (T0 − T4 ) ,
T0 =
T1 + T2 + T3 + T4 4
.
3. ).
1, 2, 3, 4
0
(
-
: Q10 + Q20 + Q30 + Q40 = 0 , λL (T1 − T0 ) + λL (T2 − T0 ) + λL (T3 − T0 ) + λL (T4 − T0 ) = 0 , T0 =
T1 + T2 + T3 + T4 4
,
. ,
,
.
:
∆P =
T1 + T2 + T3 + T4 − T0 = 0 . 4
(2.2)
: 1.
,
-
,
,
. 2. ,
.
3.
,
,
-
. 4.
,
-
. 5. ,
-
. ,
. 2.5. (2.1)
(2.2).
: 1)
, –
-
;
2) 3)
−
L >> 7∆x = 7∆y; , b, , ;
4)
, b,
,
, . . 5)
. 2.1. 323
λ
λ, L
3 ∆y
723
3 ∆x
7 ∆y
E
723
e
E
a
b
c
d
L
7 ∆x
323
)
) Ɋɢɫ. 2.5. Ɉɛɳɢɣ ɜɢɞ ( ) ɢ ɪɚɫɱɟɬɧɵɣ ɭɱɚɫɬɨɤ ( ) ɤɥɚɞɤɢ ɩɨ ɦɟɬɨɞɭ ɪɟɥɚɤɫɚɰɢɢ
2.1. Ɋɚɫɱɟɬɧɵɟ ɞɚɧɧɵɟ ɪɚɫɯɨɞɚ ɬɟɩɥɚ a T0, K
∆
b T0, K
;
∆
c T0, K
∆
523
−100
523
423
0
498
0
−25
523
0
523
0
0
523
−6
517
0
423
−13
523
410
0
498
410
0
498
−5
523
0
517
−3
410
−2
493
0
408
0
493
−1
514 514
0
410
−2
−3 −0,5
493
−6
, b
-
:
Q1 = Qb′ + Qc′ = λL (723 − 493) + λL (723 − 514) = 439λL .
,d
: Q2 = Qa′′ + Qb′′ + Qc′′ = λL (408 − 323) +
+ λL (493 − 323) + λL (514 − 323) = 446λL.
Q1
= 442,5λL, ).
Q2
Q
. Q = 0,5 (Q1 + Q2) = = 8Q = 3540λL , .
2.4. ɊȺɋɑȿɌ ɌȿɆɉȿɊȺɌɍɊɇɈȽɈ ɉɈɅə ɆȿɌɈȾɈɆ ɂɌȿɊȺɐɂɂ ,
.
x1, x2, ..., xn, x1 = A1 + B1 x1 + C1 x2 + K + D1 xn , x2 = A2 + B2 x1 + C2 x2 + K + D2 xn , ( N ) LLLLLLLLLLLLLL LLLLLLLLLLLLLL x2 = An + Bn x1 + Cn x2 + K + Dn xn ,
,
-
B1 + C1 + K + D1 < 1, B2 + C2 + K + D2 < 1, (M ) KKKKKKKKK KKKKKKKKK B + C + K + D < 1. n n n
x10, x20, …, xn0 ( (N).
), x11, x21, …, xn1 ( (N).
(
). x12, x22, …, xn2
).
(N)
. .
, 1)
:
,
,
, 2)
; (N)
,
-
.
: 1) ); 2) 3)
(
-
; ,
;
4)
, ,
(N). -
,
. 2.5. a, b, c,
Q a + Qba + Q323a + Q323a = 2Qba + 2Q323a = 0 , Qab + Qcb + Q323b + Q723b = 0 , Qbc + Qdc + Q323c + Q723c = 0 .
(2.1)
:
2λL (Tb − Ta ) + 2λL (323 − Ta ) = 0 ,
λL (Ta − Tb ) + λL (Tc − Tb ) + λL (323 − Tb ) + λL (723 − Tb ) = 0 , λL (Tb − Tc ) + λL (Td − Tc ) + λL (323 − Tc ) + λL (723 − Tc ) = 0 .
Td = Tc,
λL (Td = Tc ) = 0
(N):
Ta = 161,5 + 0,5Tb; Tb = 261,5 + 0,25Ta + 0,25Tc; Tc = 348,5 + 0,333Tb; •
: Ta0 = 523 K; Tb0 = 523 K; Tc0 = 523 K;
•
: Ta1 = 161,5 + 0,5 ⋅ 523 = 423 ; Tb1 = 261,5 + 0,25 ⋅ 523 + 0,25 ⋅ 523 = 522,5 ; Tc1 = 348,5 + 0,333 ⋅ 523 = 522,5 ;
•
: Ta2 = 161,5 + 0,5 ⋅ 522,5 = 424 ; Tb2 = 261,5 + 0,25 ⋅ 423 + 0,25 ⋅ 522 = 498,5 ; Tc2 = 348,5 + 0,333 ⋅ 522,5 = 523 .
•
, :
:
•
Ta6 = 407,5 ;
Tb6 = 491,5 ;
Tc6 = 512,5 ;
Ta7 = 406,5 ;
Tb7 = 491,5 ; Tc7 = 512,5 .
:
, b
-
:
Q1 = Qb′ + Qc′ = λL (723 − 491,5) + λL (723 − 512,5) = 442λL .
,d
:
Q2 = Qa′′ + Qb′′ + Qc′′ = λL (406,5 − 323) +
+ λL (491,5 − 323) + λL (512,5 − 323) = 441,5λL.
Q
Q = 0,5 (Q1 + Q2) = 441,75λL, .
= 8Q = 3534λL,
.
2.5. ȽɊȺɎɂɑȿɋɄɈȿ ɂɁɈȻɊȺɀȿɇɂȿ ɌȿɉɅɈȼɈȽɈ ɉɈɌɈɄȺ , .
-
,
.
-
, : Qi = ξ i λL (T1 − T2 ) ,
ξi –
– ,
/( ⋅ ); L −
;λ−
, 2−
, , ; ,
1
.
, T −T dT dT L λ dS 1 2 , L dS = − T1 − T2 dn dn
Qi = ∫ −
, .
,
dQi = − λ
(2.3)
1 dT λL (T1 − T2 ) . dS dn T1 − T2
(2.3), :
ξi =
,
1 dT − dS . ∫ T1 − T2 dn
,
,
:
T = T1 − T = T1 −
T1 − T2
dT T −T =− 1 2 ; dn
n;
dS = dn;
dT T −T 1 =− 1 2 ; r dn ln 2 n r1
T1 − T2 n ln ; r r1 ln 2 r1
,
dS = n dφ.
h 1 ξ = ∫ dn = ; δ0 h
1 ξ = r ln 2 r1
2π
∫ dϕ = 0
(2.4)
2π . r ln 2 r1
(2.5)
, . .
, .
( (2.4)
. 2.6),
-
(2.5). ,
. 2.6 ,
∆Qi :
∆Qi =
Qi –
,
; Nm – .
λ; L
λ
∆n
r2 / r1 = 3
∆S
T1 h
L
T2
δ h/δ=2
.
T2
2 r2
∆n
Qi , Nm
(2.6) ,
ɚ)
ɛ) . 2.6. ( )
( ) -
,
. , : ∆Ti = −
(T1 – T2) –
T1 −T 2 , Nn
; Nn –
(2.7)
.
(2.7),
-
, . .
-
.
,
.
,
∆T ∆Qi = − λ L (∆S )i , ∆n i
; ∆Qi −
λ, L – (2.6).
,
∆S = f (n) = 1, ∆n
∆Ti = const ∆Qi = − λL (∆T )i .
(2.7)
(2.8)
(2.6),
(2.9)
: :
(2.8)
-
Qi =
(2.3)
-
Nm λL (T1 − T2 ) . Nn
(2.9)
N ξ i = m . N n i
,
. 2.6, ξ =
,
(2.4) ξ =
Nm 8 = =2; Nn 4
. 2.6, ξ =
(2.10)
h
=2.
(2.5)
N m 16 = = 5,33 ; Nn 3
ξ =
2π
ln
r2 r1
= 5,7 .
,
:
1) 2) (2.6); 3)
; ;
4) 5)
(2.10); ∆ ;
6)
1
-
2
(2.9). ,
. 2.7. .
Nm = 8;
Nn = 8; ξ=
400 ; T , Tb
T
(
Nm 8 = = 1, Nn 8
∆ :
∆T = − 1
Ta = T1 − 6,2∆T = 723 − 6,2 ⋅ 50 = 413 ; Tb = T1 − 4,8∆T = 723 − 4,8 ⋅ 50 = 483 ; Tc = T1 − 4,2∆T = 723 − 4,2 ⋅ 50 = 513 .
1
–
2)
-
= 723 – 323 =
T1 −T 2 723 − 323 = = 50 . 8 Nn 2
-
Ɋɢɫ. 2.7. Ɋɚɫɱɟɬλ; L
ɧɵɣ ɭɱɚɫɬɨɤ ɤɥɚɞɤɢ ɩɨ
a
ɦɟɬɨɞɭ ɝɪɚɮɢɱɟ-
723
b
c
ɫɤɨɝɨ ɢɡɨɛɪɚɠɟɧɢɹ
323
ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ: Nm = 8, Nn = 8, (T1 – T2) = 400 Q = ξλL (T1 − T2 ) = 1 ⋅ λL (723 − 323) = 440λL ,
Q
= 8Q = 3520λL,
.
.
2.6. ɗɅȿɄɌɊɈɌȿɉɅɈȼȺə ȺɇȺɅɈȽɂə , ,
. -
.
(
2.8).
. :
∂ 2T ∂ 2T + =0 ∂x 2 ∂y 2
∂ 2 (T − T2 ) ∂ 2 (T − T2 ) l2 + = 0. (T1 − T2 ) ∂x 2 ∂y 2
: X =
x ; l
Y=
y ; l
Θ=
∂ 2Θ ∂ 2Θ + =0. ∂X 2 ∂Y 2
T − T2 . T1 − T2
(2.11)
(2.12)
Ɋɢɫ. 2.8. ɋɟɱɟɧɢɹ ɬɟ-
L
x
2
1
n
ɥɚ ɫɥɨɠɧɨɣ ɮɨɪɦɵ:
l
2
dS
1−1
2−2 −
1
-
x
,
;
L− , dQ = −λ
,
∂T ds L ∂n
dQ = −λ
N=
∂ (T − T2 ) l T −T ds L 1 2 . ∂n l T1 − T2
s n , S= , l l dQ = −
∂Θ dSλL (T1 − T2 ) , ∂N
∂Θ Q = ∫ − dS λL (T1 − T2 ) = ξ λL (T1 − T2 ) , ∂N ∂Θ ξ = ∫ − dS . ∂N
(2.13)
-
∂ 2U ∂ 2U = 0, + ∂x 2 ∂x 2
,
,
dJ = −
∂U dS L . ∂n
, ∂ 2U ∂ 2U + = 0; ∂X 2 ∂Y 2
J =ξ
L (U1 − U 2 ) ,
(2.14)
, ∂U ξ = ∫ − dS ; ∂N
U=
U −U2 . U1 − U 2
(2.15)
(N = N;S =S;X =X;Y =Y) Q = U,
(2.13)
ξ =ξ ,
(2.15)
(2.12)
(2.14),
:
Θ = f ( X ,Y , C, D) ;
(2.11)
U = f ( X ,Y , E, M ) .
(2.12)
,
(
)
Θ1 = f X 1 , Y1 , C , D = 1,
1
2
Θ 2 = f ( X 2 , Y2 , C , D ) = 0.
(2.16): U1 = f ( X 1 , Y1, E , M ) = 1,
U 2 = f ( X 2 , Y2 , E , M ) = 0.
(C = E; D = M),
ξ =ξ =ξ.
Θ = U.
,
-
, ,
-
. 1)
(
,
, .
2.10),
−
. -
, . 2)
, (
ξ
)
3) ξ = h / δ.
, h
.
U1
5)
ξ
)−
(
4)
6)
.
J ( L )= ξ (U1 − U 2 J = (ЭL ) (U1 − U 2
(
). J,
U
).
).
J,
(
U
)
(
: Q = =ξ
7) 1
-
2
)
λ L (T1 − T2 ) ,
-
-
( ) .
2.7. ɌȿɉɅɈɉȿɊȿȾȺɑȺ ɑȿɊȿɁ ɉɅɈɋɄɂȿ ɂ ɐɂɅɂɇȾɊɂɑȿɋɄɂȿ ɋɌȿɇɄɂ . ,
, α1,
2.9. α2. d1, d2
Tf1 Tf2
λ1 λ2 .
α1
W1,
λ2
λ1
Tf1
TW1 TW2
d3.
Tf1 α1
α2
λ1
λ2
TW1
α2
TW2 L
TW3
δ1
δ2
d1 Tf2
)
TW3 d2 d2 d3
Ɋɢɫ. 2.9. Ɋɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɩɥɨɫɤɨɣ ɞɜɭɯɫɥɨɣɧɨɣ ( ) ɢ ɰɢɥɢɧɞɪɢɱɟɫɤɨɣ ( ) ɫɬɟɧɤɟ, ɨɦɵ-
)
Tf2
. δ1
δ2,
W2,
W3.
− -
ɜɚɟɦɨɣ ɝɨɪɹɱɟɣ ɢ ɯɨɥɨɞɧɨɣ ɠɢɞɤɨɫɬɹɦɢ ( (
. 2.9, ),
. 2.9, )
L
.
-
.
,
,
-
. Ɍɟɩɥɨɩɟɪɟɞɚɱɚ ɨɬ ɝɨɪɹɱɟɣ ɠɢɞɤɨɫɬɢ ɤ ɯɨɥɨɞɧɨɣ ɱɟɪɟɡ ɦɧɨɝɨɫɥɨɣɧɭɸ ɩɥɨɫɤɭɸ ɫɬɟɧɤɭ : F (T f 1 − TW 1 )
Q = α1F (T f 1 − TW 1 ) =
Q=
λ1 1
Q=
λ2 2
Rα1
F (TW 1 − TW 2 ) =
F (TW 2 − TW 3 ) =
Q = α 2 F (TW 3 − T f 2 ) =
,
;
F (TW 1 − TW 2 ) ; R1
F (TW 2 − TW 3 ) ; R2
F (TW 3 − T f 2 ) Rα 2
.
, F (T f 1 − T f 2 )
(Q, TW1, TW2, TW3). Q=
n, Q=
Rα1 + R1 + R2 + Rα 2
F (T f 1 − T f 2 )
Rα1 + ∑ Ri + Rα 2 n
,
= kF (T f 1 − T f 2 ) =
. F (T f 1 − T f 2 ) 1
.
(2.17)
k
1
k=
(2.16)
+∑
1
α1 1
n
1
i
λi
+
α2 1
,
/(
2
⋅ ),
(2.18)
( 2)
.
(
),
-
( )
. R =
n 1 1 1 = +∑ i + , k α1 1 λ i α 2
(
2
⋅ )/
,
(2.19)
,
.
Rα =
n 1 , Ri = ∑ i − α 1 λi
. q = Q / F, ,
/ 2. :
TW 1 = T f 1 − k (T f 1 − T f 2 ) Rα1 , TW 2 = T f 1 − k (T f 1 − T f 2 )( Rα1 + R1 ) , TW 3 = T f 1 − k (T f 1 − T f 2 )( Rα1 + R1 + R2 ) .
n,
:
TWi = T f 1 − k (T f 1 − T f 2 )∑ ( Rα1 + Ri ) . i
(2.20)
0
Ɍɟɩɥɨɩɟɪɟɞɚɱɚ ɨɬ ɝɨɪɹɱɟɣ ɠɢɞɤɨɫɬɢ ɤ ɯɨɥɨɞɧɨɣ ɱɟɪɟɡ ɦɧɨɝɨɫɥɨɣɧɭɸ ɰɢɥɢɧɞɪɢɱɟɫɤɭɸ ɫɬɟɧɤɭ :
Q = α1πd1L (T f 1 − TW 1 ) =
πL (T f 1 − TW 1 ) Rα1
Q=
πL (TW 1 − TW 2 ) πL (TW 1 − TW 2 ) = ; d2 1 R 1 ln 2λ1 d1
Q=
πL (TW 2 − TW 3 ) πL (TW 2 − TW 3 ) = ; d3 1 R 2 ln d2 2λ 2
Q = α 2 πd 3 L (TW 3 − T f 2 ) =
;
πL (TW 3 − T f 2 ) Rα 2
,
Q=
πL (T f 1 − T f 2 )
Rα1 + R1 + R2 + Rα 2
,
.
(2.21)
n,
Q=
πL (T f 1 − T f 2 )
Rα1 + ∑ Ri1 + Rα 2 n
= kl πL (T f 1 − T f 2 ) =
πL (T f 1 − T f 2 ) 1
i =1
kl =
d 1 ln i +1 + +∑ di α1d1 1 2λ i α 2 d n +1 1
1
,
.
/( ⋅ ),
( )
(2.23)
) π
(
,
(2.22)
kl
1
n
,
( )
. RL =
1
kL
=
α1d1 1
+∑ n
1
1 2λ i
ln
di +1 1 , ( ⋅ )/ + di α 2 d n +1
(2.24)
,
π. Rα =
αd 1
, Ri = ∑ n
1
1 2λ i
ln
,
d i +1 − di
. qL = Q / L,
/ .
n, TWi = T f 1 − k (T f 1 − T f 2 )∑ ( Rα1 + Ri ) . i
0
3. ɋɌȺɐɂɈɇȺɊɇȺə ɌȿɉɅɈɉɊɈȼɈȾɇɈɋɌɖ ɉɊɂ ɈɋɈȻɕɏ ɍɋɅɈȼɂəɏ
(2.25)
3.1. ɌȿɉɅɈɉɊɈȼɈȾɇɈɋɌɖ ȼ ɇȿɈȽɊȺɇɂɑȿɇɇɈɃ ɉɅȺɋɌɂɇȿ ɉɊɂ ɊȺȼɇɈɆȿɊɇɈɆ ȼɇɍɌɊȿɇɇȿɆ ɌȿɉɅɈȼɕȾȿɅȿɇɂɂ
(2 R l u = 2L + 2δ f = Lδ
Ɋɢɫ. 3.3. ɉɪɹɦɨɣ ɫɬɟɪɠɟɧɶ ɩɨɫɬɨɹɧɧɨɝɨ ɫɟɱɟɧɢɹ (ɪɟɛɪɨ ɨɯɥɚɠɞɟɧɢɹ): Q =0– ;α– ; 0– ; λ– , ( ) dQ = −Wfdx .
W =−
αu (T − T ) . f
(3.1): Q = αF
F = uϑ, υ = T – Tc
l 1l ∫ ϑdx = αu ∫ (T − T ) dx , l 0 0
:
-
(3.1)
dT f. Q = −λ dx x = 0 T = f ( x) .
, : ϑ = ϑ0e− mx ,
∞
Q = αu ∫ ϑ0e − mx dx =
(
0
∂T Q = −λ f = − λ − ϑ0 me − mx ∂x x = 0
(
)
αuϑ0 , m
)x =0 f = λfmϑ0 .
dT dϑ d = = ϑ0e − mx = −ϑ0 me− mx . dx dx dx
: shm (l − x) f, Q = − λ ϑ0 ( − m ) ch (ml) x = 0
Q0 = λmfϑ0 th (ml) ,
dT dϑ d hm (l − x) sh [m(l − x)] = = ϑ0 . = −ϑ0 m dx dx dx ch (ml) ch (ml)
. 3.4. ɌȿɉɅɈɎɂɁɂɄȺ ɉɊɂ ɉȿɊȿɆȿɇɇɈɆ ɄɈɗɎɎɂɐɂȿɇɌȿ ɌȿɉɅɈɉɊɈȼɈȾɇɈɋɌɂ , .
. ,
(
( . 3.4, , ):
), − (dQB − dQ
dQ + dQW = dQB
,
) + dQW = 0.
− (q − q ) dz dy + W dx dy dz = 0 −
dq + W = 0; dx
q = −λ
−
d dT − λ dx dx
dT d =− ; dx dx
+ W = 0.
= ∫ λ(T ) dT ,
:
-
d d + W = 0; dx dx
d2 + W = 0. dx 2
, d2 = 0, dx 2
W=0 =
1
−
∫ λ(T ) dT = ∫ λ(T1 ) dT1 −
W
1
−
2
x;
∫ λ(T1 ) dT1 − ∫ λ(T2 ) dT2
x.
dV dQ
dV
dQB
1
d dy dQW
dz
d 2
)
)
W
dV
d
rA
rB r L 1
dS
dV
r
dS
dSB dQB
dx dQA
dQW
dz
r1 2
r
dr
r2
)
)
. 3.4.
( ) λ = β + kT ,
( ), ( ) ()
. -
(
) (
) (
T + 0,5kT 2 =
T1 + 0,5kT12 −
q=−
(
d = dx
1
−
) (
T1 + 0,5kT12 −
2
T2 + 0,5kT22
) x.
.
. 3.4, , )
, ),
dS dz (
,
dS B dz :
dQ + dQW = dQ
− (dQ − dQ ) + dQW = 0.
− (q r − q r ) dϕ dz + Wr dϕ dr dz = 0, − q = −λ
1 d
r dr
(q r ) + W = 0;
dT d =− , dr dr
−
1 d
r dr
(−λ
dT r ) + W = 0. dr
= ∫ λ (T ) dT
1 d d
r + W = 0. r dr dr
,
.
W=0
d2 1d + = 0, 2 r dr dr =
∫ λ dT = ∫ λ1dT1 −
1−
∫ λ1dT1 − ∫ λ 2 dT2 r ln . r r1 ln 2 r1
1−
2
r ln 2 r1
ln
r r1
λ = be kT ,
e kT = e kT1 −
e kT1 − e kT2 r ln ; r r1 ln 2 r1
Q=−
d πL ( 1 − 2 πrL = 1 d2 dr ln 2 d1
2
)
.
3.5. ɌȿɉɅɈɉɊɈȼɈȾɇɈɋɌɖ ɉɊɂ ɇȺɅɂɑɂɂ ɎɂɅɖɌɊȺɐɂɂ
И G , ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɤɨɥɢɱɟɫɬɜɨ ɯɨɥɨɞɧɨɣ ɠɢɞɤɨɫɬɢ, ɩɪɨɧɢɤɚɸɳɟɣ ɫɤɜɨɡɶ ɤɚɩɢɥɥɹɪɧɨ-ɩɨɪɢɫɬɭɸ ɩɥɨɫɤɭɸ ɢɥɢ ɰɢɥɢɧɞɪɢɱɟɫɤɭɸ ɫɬɟɧɤɭ, ɱɟɪɟɡ ɟɞɢɧɢɰɭ ɩɨɜɟɪɯɧɨɫɬɢ F, ɜ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ. ȿɫɥɢ ɩɪɢ ɷɬɢɯ ɠɟ ɭɫɥɨɜɢɹɯ ɝɨɪɹɱɚɹ ɠɢɞɤɨɫɬɶ ɩɪɨɬɟɤɚɟɬ ɫɤɜɨɡɶ ɫɬɟɧɤɭ ɜ ɨɛɪɚɬɧɨɦ
-
ɧɚɩɪɚɜɥɟɧɢɢ, ɬɨ ɬɚɤɨɣ ɩɪɨɰɟɫɫ ɧɚɡɵɜɚɟɬɫɹ . Ɋɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɩɥɨɫɤɨɣ ɢ ɰɢɥɢɧɞɪɢɱɟɫɤɨɣ ɫɬɟɧɤɟ ɩɪɢ ɢɧɮɢɥɶɬɪɚɰɢɢ ɩɨɤɚɡɚɧɨ ɧɚ ɪɢɫ. 3.5. δ
1
λ
λ
1 1
r1 d
L d
G G
2
0
x x
2
r
dx
r
dr
r2
)
) . 3.5.
( )
( ) Ʉɨɥɢɱɟɫɬɜɨ ɬɟɩɥɚ Q, ɩɨɝɥɨɳɚɟɦɨɝɨ ɜ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ ɩɪɨɬɟɤɚɸɳɟɣ ɫɤɜɨɡɶ ɩɥɨɫɤɭɸ ɫɬɟɧɤɭ F ɠɢɞɤɨɫɬɶɸ (ɨɬɪɢɰɚɬɟɥɶɧɵɣ ɢɫɬɨɱɧɢɤ ɬɟɩɥɚ) ɧɚ ɭɱɚɫɬɤɟ ɩɭɬɢ dx (ɪɢɫ. 3.5, ), ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ dQ = −WdV = −WFdx ,
ɝɞɟ W − ɬɟɩɥɨɬɚ, ɩɨɝɥɨɳɚɟɦɚɹ ɟɞɢɧɢɰɟɣ ɨɛɴɟɦɚ V, ɜ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ. ɋ ɞɪɭɝɨɣ ɫɬɨɪɨɧɵ, ɩɨ ɡɚɤɨɧɭ ɬɟɩɥɨɮɢɡɢɤɢ dQ = − FGdT ,
ɝɞɟ G − ɭɞɟɥɶɧɚɹ ɢɧɮɢɥɶɬɪɚɰɢɹ ɩɥɨɫɤɨɣ ɫɬɟɧɤɢ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, W= G
dT . dx
,
,
d 2T W d 2T G dT + = 2 + = 0. 2 λ dx λ dx dx
P=
G dT , u= λ dx
du + Pu = 0, dx
T =−
D − Px + e P
: T = T1 −
(
; T1 = −
; T2 = −
D + P
D − Pδ + e P
.
)
dT T1 − T2 T −T = λP 1 − P2δ e − Px . 1 − e − Px ; q ( x) = − λ −P dx 1− e 1− e
: Q = q (δ) F = λPF
T1 − T2 − pδ e 1 − e − Pδ
Q=
Q
e
+ Pδ
−1
. ,
P
(
). ,
dQ = −WdV = −WFdr .
(
, dQ = −cFGdT .
ρ − 2 πrL
dr (
dT (cρ) dT = , dr 2 πrL dr
;ρ−
. ,
,
cρ 1 dT d 2T 1 dT W d 2T = 0. + + = 2 + 1 + 2 r dr λ dr 2 πλL r dr dr P =1+
:
. 3.5, ),
, W = cG
G=
)
dT cρ , u= 2 πλL dr
du 1 + P u = 0, dr r
T = −D
(P − 1) r1( P −1)
T1 = − D
(P − 1) r1
T2 = − D
(P − 1) r2
1
1
1
1
+
;
1 ( P −1) +
;
1 ( P −1) +
.
T = T1 − Q (r ) = −λ
Q (r2 ) =
T1 − T2
1 1 ( P −1) − ( P −1)
1 1 r ( P −1) − r ( P −1) , 1
dT 1 πL (T1 − T2 )( P − 1) . 2 πrL = P −1 dr 1 1 1 r − 2λ r1( P −1) r2( P −1) r1
r2
2 πλL (T1 − T2 )( P − 1) = ( P −1) r2 −1
r1
( P −1)
Q
r2 r1
−1
.
. ,
, P = 1−
В
, d T = 0. dr 2
cρ , 2 πλL
ρ = 1, 2 πλL , P=0
-
2
4. ɇȿɋɌȺɐɂɈɇȺɊɇȺə ɌȿɉɅɈɉɊɈȼɈȾɇɈɋɌɖ ȼ ɌȼȿɊȾɕɏ ɌȿɅȺɏ
4.1. ɈȻɓȿȿ Ɋȿɒȿɇɂȿ ȾɂɎɎȿɊȿɇɐɂȺɅɖɇɈȽɈ ɍɊȺȼɇȿɇɂə ɌȿɉɅɈɉɊɈȼɈȾɇɈɋɌɂ -
.
∂ ∂2 = . ∂τ ∂ 2
= f ( , τ) = U (τ) V ( ).
, ∂ = U ′( τ) V ( x) , ∂τ
∂ 2T = U (τ )V ′′( x ) . ∂x 2
(4.1)
U ′(τ ) V ′′( x ) = −k 2 . = U (τ ) V ( x )
V ( x ) U ′(τ ) = V ′′( x ) U (τ )
−k
τ
2
.
,
.
(4.1)
U ′(τ ) + k 2U (τ ) = 0,
,
V ′′( x ) + k 2V ( x ) = 0,
:
, U (τ) = C1 e −
V ( x ) = C2e −ikx + C3e + ikx .
k 2τ
e −ikx = cos kx − i sin kx ,
V ( x ) = C4 cos kx + C5 sin kx .
e + ikx = cos kx + i sin kx ,
= D cos (kx ) e −
k 2τ
+ B sin (kx ) e −
k 2τ
(4.1),
(4.2) D, B, k -
. -
∂ = ∂τ
J1(k ) −
J0(k )
T = DJ 0 (kx ) e −
−
k 2τ
,
+ BJ1 (kx ) e −
(4.3) k 2τ
,
(4.4) ;
.
Ζ = (rT) 4.2.
∂2 1 ∂T ∂r 2 + r ∂r
∂ = ∂τ
∂2 2 ∂T ∂r 2 + r ∂r , ∂ 2Ζ ∂Ζ , = ∂τ ∂r 2
.
,
0
(
τ=0
. 4.1),
α.
. ∂ ∂2 ; = ∂τ ∂ 2
•
∂ ∂
•
=0
(4.5)
= 0;
(4.6) ∂ − λ ∂
•
(4.6) − (4.8)
= α (T −
) , (4.7)
( )τ=0 = .
(4.8)
=R
,
. ,
-
D, B, k . − Tc.
(4.5) − (4.8)
ϑ=T
,
∂ϑ ∂ 2ϑ = , ∂τ ∂x 2
(4.9)
∂ϑ = 0, ∂x x = 0
(4.10)
∂ϑ − λ = αϑ , ∂x x = R
(4.11)
(ϑ)τ=0 = ϑ0 .
(4.9)
(4.12)
(4.1),
ϑ = D cos (kx ) e −
k 2τ
+ B sin (kx ) e −
(4.10) ϑ = D cos (kx ) e −
k 2τ
k 2τ
.
B = 0.
. (4.11)
,
tg µ =
αR µ, µ = kR; Bi = λ Bi 1
: µ1, µ2, µ3, …, µn.
,
τ
x −µ n 2 ϑ = ∑ Dn cos µ n e R . R n =1 ∞
2
D τ=0
D2, D3, ..., Dn.
-
∑ Dn cos µ n R = ϑ0 , ∞
D1 cos µ1
x
x x x + D2 cos µ 2 + D3 cos µ 3 + K = ϑ0 . R R R
cosµ1
x R
(+R) x x x 2 ∫ cos µ1 R dx + D2 ∫ cos µ 2 R cos µ1 R dx + −R −R +R
D1
+ D3
+R
+R
+R
x x x ∫ cos µ3 R cos µ1 R dx + K = ϑ0 ∫ cos µ1 R dx . −R −R
+R
D1
+R
x x 2 ∫ cos µ1 R dx = ϑ0 ∫ cos µ1 R dx , −R −R D1 = ϑ0
2sin µ1 . µ1 + sin µ1 cos µ1
2, 3, …, n, θ= θ =
-
.
n =1
(−R)
(4.12)
T −T τ x ; X= ; F = 2 − R T0 − T R
).
∞ 2 sin µ n ϑ cos (µ n =∑ ϑ0 n =1 µ n + sin µ n cos µ n
, ,
, T −T . θ= T − T0
(4.13)
) e −µ n Fo , 2
(4.13)
( .
-
λϑ ∂T = 0 q = −λ R ∂x x = R
= ∫ θ dX = ∑
θ
1
∞
0
n =1
∑µ ∞
n =1
2µ n sin 2 µ n n
+ sin µ n cos µ n
2sin 2µ n
µ n2 + µ n sin µ n cos µ n
θ=∑ ∞
[
q=
2
2 J1 (µ n )
(µ n ) +
J12
:
(µ n )]
2 r J 0 µ n e −µ n F . R
λϑ0 R
∑µ
2µ n J12 (µ n )
∞
n =1
αR , λ n
[J
2 0
(µ n ) + J12 (µ n )]
e −µ n F , 2
T =
2
R2
∫ (Tr ) dr .
R 0
),
Bi → ∞ , (4.13)
. T →
,
.
.
, cos µ = 0; J 0 (µ ) = 0 , .
, (4.14)
(4.14)
J 0 (µ ) 1 = µ, J1 (µ ) Bi
Bi =
(
2
e −µ n F .
2 n =1 µ n J 0
µn
e −µ n F
,
, θ = ∑ (− 1)n +1 ∞
2
n =1
µn
θ=∑
cos (µ n X ) e − µ n F , 2
2 r J 0 µ n e−µ n F . R n =1 µ n J1 (µ n )
∞
2
. 4.2 – 4.5 Fo
Bi.
4.3. ɆȿɌɈȾ ɉȿɊȿɆɇɈɀȿɇɂə ɌȿɆɉȿɊȺɌɍɊɇɕɏ ɄɊɂɌȿɊɂȿȼ . , (
-
. 4.2). :
θ xy = θ x θ y .
; θy −
θx −
(4.15)
2R1, 2R2,
y.
2 2R2
1
0
2R1
Ɋɢɫ. 4.2. ɉɟɪɟɫɟɱɟɧɢɟ ɞɜɭɯ ɧɟɨɝɪɚɧɢɱɟɧɧɵɯ ɩɥɚɫɬɢɧ, ɨɛɪɚɡɭɸɳɢɯ ɛɪɭɫ ɩɪɹɦɨɭɝɨɥɶɧɨɝɨ ɫɟɱɟɧɢɹ
(4.15)
( , y, τ ) − 0
(4.15) − •
0
•
1
• •
=
−
0
( , y, τ )
−
( , τ) −
=
−
0
θ
=0 y =0
=θ
( y, τ ) −
−
−
0
( , τ) 0
−
−
.
( y, τ )
−
;
0
=0 θ y =0 ;
θ x = R1 = θ x = R1 θ y = 0 ; y =0
θ x = R1 = θ x = R1 θ y = R2 ;
2
y = R2
θ x = 0 = θ x = 0 θ y = R2 .
3
y = R2
. . : θ rz = θ r θ z ,
θz − ) d = 2R1 (
L = 2R2 (
z; θr − )
r.
θxyz = θx θy θz,
θx, θy, θz −
. , . .
,
(
,
,
), -
,
: ,
. .
(
,
) ϑ = f ( ) f (y),
ϑ = (T − Tc) − ,
; ϑ = (Tc − T) − ϑ ( x, y ) =
ϑ( , ϑ ( , 0) −
(4.16)
y)
−
ϑ ( x, 0 ) ϑ (0, y ) ϑ ( x, R2 ) ϑ ( R1 , y ) = , ϑ(0, 0 ) ϑ ( R1 , R2 )
; ϑ (y, 0) − ; ϑ ( , R2) −
; ϑ (0, 0) − R1; ϑ (R1, y) − .
T (x, y) = f ( ) + f (y),
-
(4.17) ;
R2; ϑ (R1, R2) −
T ( , y) −
, (0, y) − T (0, 0) =
T ( , y) = T ( , 0) +
(R1, y) + T ( , R2) − T (R1, R2). (4.18)
(4.17)
,
-
, . .
.
(4.18) (
,
)
, . .
. ϑ (0, 0 ) =
0,24 < Bi < ∞.
"
ϑ (0, R2 ) ϑ ( R1 , 0 ) , ϑ ( R1R1 )
-
(4.19)
"
, , ,
. ,
4.4. ɊȿȽɍɅəɊɇɕɃ ɌȿɉɅɈȼɈɃ ɊȿɀɂɆ
,
.
(4.4) – (4.13)
, ϑ = ∑ An e − µ n F , ∞
2
n =1
1 T −T ϑ= , ctg µ = µ . Bi T0 − T
Bi → ∞, ctg µ = 0
:
Fo > Fo
µ1 =
π
π π ; µ 2 = 3 ; µ 3 = 5 K; µ12 < µ 22 < µ32 < K
2
2
2
*
,
e − µ1 Fo >> e − µ 2 Fo >> e − µ 3 Fo . 2
2
2
,
Fo > Fo* ,
,
, . .
ϑ = A1 e − µ1 Fo ,
-
2
ln ϑ = −µ12 Fo + ln A1 = −µ12
:
a τ + const , R2
ln ϑ = −
τ + const , ψ ln ϑ = − mτ + const .
(4.21) (4.22)
,
-
,
. (4.22),
.
m
= ψ m. ψ,
(4.23) . : • •
•
(4.20)
ψ
ψ
.
=
ψ
.
.
.
R 1 ; = = 2 µ1 π 2R 2
=
1
π 1,31R.
2
; ψ
1
π π + 1,31R L 2
2
;
=
1
π R
2
;
(4.23)
(4.20)
-
ψ
•
, , .
Bi =
(
-
lnϑ1
αR ; λ
lnϑ2
.
=
1
π π π + + 2 R1 2 R2 2 R3 2
2
2
.
lnϑ
Bi → ∞,
τ*
τ1
τ2
R ≠ ∞;
τ
α → ∞.
λ ≠ 0,
).
= const . ϑ1, Ɋɢɫ 4.3. Ƚɪɚɮɢɤ ɢɡɦɟɧɟ-
ɮ
τ1 ϑ2, τ2 (
m
. 4.3).
T1 − T T2 − T ln ϑ1 − ln ϑ2 m= = . τ 2 − τ1 τ 2 − τ1 ln
(4.24)
4.5. ɄȼȺɁɂɋɌȺɐɂɈɇȺɊɇɕɃ ɌȿɉɅɈȼɈɃ ɊȿɀɂɆ ( (
. 4.4)
-
)
: ∂ = ∂τ
∂2 ξ −1 ∂ ; ∂ 2 + ∂
∂ ∂T = 0; λ = qc ; T ∂ ∂
ξ=1
=
ξ=2
0,
.
= b + kτ + ∑ An e − µ n F . ∞
n =1
qc = const
λ; a qc = const
2
λ 2R a
2R
. 4.4. µ1 < µ2 < µ3 < …, (F > F *),
,
.
= b + kτ, 2 qc R ξ 1 x b= 0− − ; λ 2 (ξ + 2 ) 2 R
b =
0
−
qc R ξ ; b = λ 2 (ξ + 2 )
0
−
k =ξ
qc ; λR
1 qc R ξ qc R − = + T0 . λ 2 (ξ + 2 ) 2 (ξ + 2 ) λ
, ,
(qc = const) .
, .
b
-
k, (
. 4.5).
, λ=
qc R ξ qR 1 1 . − = c T0 − b 2 (ξ + 2 ) 2 b − T0 ξ + 2
, λ=
= b + kτ
qc R 2 (b − b
)
, ,
=k
λR . ξ qc
tgϕ = k = b + kτ
b
b
τ
τ*
0
Ɋɢɫ. 4.5. Ɂɚɜɢɫɢɦɨɫɬɶ ɬɟɦɩɟɪɚɬɭɪɵ ɬɟɥɚ ɨɬ ɜɪɟɦɟɧɢ ɩɪɢ ɧɚɝɪɟɜɚɧɢɢ (q = const): − ; − 4.6. ɍɉɈɊəȾɈɑȿɇɇɕɃ ɂɅɂ ɈȻɈȻɓȿɇɇɕɃ ɌȿɉɅɈȼɈɃ ɊȿɀɂɆ : , ,
.
-
. : ,
,
-
. . , :
-
, , .
(
) ,
.
-
,
∂2 ∂ , = ∂τ ∂ 2
(R, τ) = T (τ),
∂ ∂
(0, τ) =
=0
= 0,
(τ),
( , 0) =
0.
(τ) Φ = ln (T − T ) − 1,23∫
(τ),
dT = −2,47 2 τ + const . T −T R
(4.25) .
,
-
.
(4.30)
−
.
-
(4.25) d = 2,47 2 , dτ R = ln (T − T ) − 1,23∫
.
ϕ
= ψ(τ)
. , (
τ
0
dT , T −T
τ
= -
. 4.6).
,
Ɋɢɫ. 4.6. Ɂɚɜɢɫɢɦɨɫɬɶ ɢɡɦɟɧɟɧɢɹ ɬɟɦɩɟɪɚɬɭɪ ɩɨɜɟɪɯɧɨɫɬɢ ɢ ɰɟɧɬɪɚ ɬɟɥɚ ɨɬ ɜɪɟɦɟɧɢ
(4.30)
.
-
. (
).
, . .
.
4.7.
-
, (
. 4.7).
αF (Tc − T ) dτ = cρVdT , αFϑdτ = −cρVdϑ , dϑ αF dτ = − Pdτ . =− cρ V ϑ ϑ = De − Pτ
τ = ϑ = ϑ0 e-Pτ.
0
,
, P=
R−
α αR λ = = Bi 2 , λ cρ R 2 cρ R R αR λ
−
c, ρ, T, V, λ
R
2
α
; θ= ,
− −
T0,
FR − V
, , 1, 2, 3;
=
Ɋɢɫ 4 7 Ʉɨɧɜɟɤɬɢɜ-
−
ξ=
Tc
Tc
θ= τ
Tc
α
F
;
=
0
FR α αF α = =ξ , cρ V V cρ R cρ R
;
α
,
F =
ϑ0
=
Tc
,
Bi =
D
: = −
−
λ cρ
.
ϑ = e − ξBiFo , ϑ0
−
.
0
θ=
− 0−
. ,
:
;
,
.
-
, , ,
αR Bi = < 0,24. λ
. .
-
4.8. ɊȺɋɉɊɈɋɌɊȺɇȿɇɂȿ ɌȿɉɅȺ ȼ ɉɈɅɍɈȽɊȺɇɂɑȿɇɇɈɆ ɉɊɈɋɌɊȺɇɋɌȼȿ −
.
,
,
-
.
,
(
0,
),
-
, . ∂ ∂2 , = ∂τ ∂ 2
); ϑ0 = (
ϑ=( −
0
−
=0
=
τ=0
,
=
0,
∂ϑ ∂ 2ϑ = , ϑ ∂τ ∂ 2
=0
= 0,
ϑ τ = 0 = ϑ0 ,
(4.26)
). ,
∫ π
ϑ = ϑ0
2
−Ζ2
Ζ
dΖ
0
Ζ=
(4.26), ,
4 τ
.
(4.26),
-
. − q = −λ
∂ϑ ; ∂x
∂ϑ dϑ ∂Ζ = , ∂ dΖ ∂x
q=
2 ϑ0
λ
π
e− Ζ . 2
4 τ
(Z = 0) q = b=
2ϑ0
λ
π
4 τ
=
λ cρ ϑ0 = b ϑ0 , πτ
λ ρ . πτ
b ∞ –
.
-
,
0. ϑ.
q (τ) (
.
ϑ=(
− ); ϑ0 = (
−
), 0
0).
:
,
-
Q = ∫ q ( τ ) dτ = ϑ0 ∫ b ( τ ) dτ τ
τ
0
0
Q=
= λ ρ −
ϑ0 τ ,
2
π
(4.27)
. ,
,
-
.
, .
( ρ)
−
.
, ,
ρ,
.
-
,
.
β .
-
,
,
,
.
,
,
-
(4.27).
4.9.
. .
,
-
, .
-
,
.
,
,
-
. .
-
,
,
.
ϑ
ϑ
λ; a; (c; ρ) ϑxmax
max
ϑ0, τ T*
0
ϑ ,τ
T*
x
,
,
-
Ɋɢɫ 4.8. ɂɡɦɟɧɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɩɨɥɭɨɝɪɚɧɢɱɟɧɧɨɦ ɬɟɥɟ:
(
ϑ
=(
,τ
)
,τ
ϑmax = T max − T* −
(
(
)
−
*
)−
;
);
ϑmax = Txmax − T* − x
(
)
4.8
. , . ∂ϑ ∂ 2ϑ ; = ∂τ ∂ 2 2π − ; = Ζ
Ζ−
, ,
2π ϑ0, τ = ϑmax cos τ , Ζ
-
:
.
ϑ x, τ = ϑmax cos ( τ − kx ) e − kx ,
k=
2
. ,
L=
cos ( τ − kx ) = 1,
4,6
k
= 4,6
2
ϑmax x=L = 0,01. max ϑ
ϑmax = ϑmax e − kx . x
.
q
τ
π ∂ϑ = −λ = λk 2ϑmax cos τ + 4 ∂x
q =
λ ρ
. .
τ
π = ϑmax cos τ + , 4
. max q max . τ = ϑ
,
(
= L)
, −
.
-
,
. ,
. ,
.
. -
,
, . , Q=± β=
∫
0,5 Ζ
λ cρ −
q τ dτ = ± ϑmax
2
,
0
.
ρ
,
.
ρ–
.
-
.
, .
-
. ( 4.9),
.
: ϑ
,τ
=
,τ
−
1
−
1
−
2
,
ϑmax = T max − T1 .
,
δ− , ϑ T
− kx ϑmax = ϑmax . 1 e
ϑ
max
T2
0
− k (δ − x ) ϑmax = ϑmax , x 2 e
−
,
T1
-
ϑmax 2
.
δ
-
x
Ɋɢɫ. 4.9. ɂɡɦɟɧɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɫɬɟɧɤɟ ɩɪɢ ɰɢɤɥɢɱɟɫɤɨɦ ɩɨɞɜɨɞɟ ɬɟɩɥɨɬɵ ɤ ɟɝɨ ɩɨɜɟɪɯɧɨɫɬɢ
.
-
,
, 0
( )=
2
−
1
−
1
−
2
.
: 0
( )=
2
−
1
( − )=
2
+
1
−
2
( − ).
,
, -
. , . 4.10.
.
.
-
.
. I –
(
0
( )=
10
−
T 10 − T
20
x.
λ dT ( x ) q0 I = − λ 0 = (T 10 − T dx x =0
II –
, , . 4.10, ).
0−0 ( T
T
III
λ, a
T
II T
10
max
T T
0
δ
λ, a
1
T
10
min 1
T
20
0
.
. 4.10, ).
δ
20
20
). -
)
)
Ɋɢɫ. 4.10. ɂɡɦɟɧɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɩɨ ɬɨɥɳɢɧɟ ɫɬɟɧɤɢ ɩɪɢ ɫɬɚɰɢɨɧɚɪɧɨɦ ( ) ɪɟɠɢɦɟ ɢ ɩɪɢ ɝɚɪɦɨɧɢɱɟɫɤɨɦ ɢɡɦɟɧɟɧɢɢ ɬɟɦɩɟɪɚɬɭɪɵ ɜɧɭɬɪɢ ɩɨɦɟɳɟɧɢɹ ( )
ϑ max ϑ xII = TxII − T0 ( x ); ϑmax 1II = T 1 − T
10 ;
II
− k II x = ϑmax , 1II e
π , ΖII − aΖ II;
k II =
II
.
− k II x = T0 ( x ) + ϑmax . 1II e
dT qII = − λ xII = q0I + λϑmax 1II k II . dx x =0
III –
, ,
0−0. ϑ
max ϑ xIII = TxIII − T0 ( x ); ϑmax 2III = T 2 − T
20 ;
k=
III
III
− k III ( = ϑmax 2III e
− x)
,
π , ΖIII − aΖ III − k III ( = T0 ( x ) + ϑmax II e
.
− x)
dT − k III qIII = −λ xIII q0 I − λϑmax . 2III k III e dx x = 0
IV – . (
), ,
,
,
-
-
(
).
,
(∆q )II = qII − q0I = λϑmax 1x k II . , − k III . (∆q )III = qIII − q0I = −λ ϑmax 2 III k III e
-
II
III
:
-
. .
. -
= q01 + ∆qII + ∆qIII .
q q01 , ∆qII , ∆qIII q
,
=
λ
-
:
q
(
10
−T
20
− k III max . ) + λϑmax 1II k II − λϑ 2III k III e
−
,
,
max 1
10
q
20
,
.
: Q = ∫ q ( τ ) dτ = ϑ0 ∫ b ( τ ) dτ τ
τ
0
0
T max 2 .
Q=
= λ ρ −
ϑ0 τ ,
2
π
(4.27)
. ,
,
-
.
, .
( ρ)
−
.
, ,
ρ,
.
-
,
.
β .
-
,
,
,
.
,
,
-
(4.27).
4.9.
. .
,
-
, .
-
,
.
,
,
-
. .
-
,
,
.
ϑ
ϑ
,
,
λ; a; (c; ρ) ϑxmax
max
ϑ0, τ T*
0
ϑ ,τ
T*
x
Ɋɢɫ 4.8. ɂɡɦɟɧɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɩɨɥɭɨɝɪɚɧɢɱɟɧɧɨɦ ɬɟɥɟ:
-
(
ϑ
=(
,τ
)
,τ
ϑmax = T max − T* −
(
(
)
−
*
)−
;
);
ϑmax = Txmax − T* − x
(
)
4.8
. , . ,
∂ϑ ∂ ϑ = ; ∂τ ∂ 2 2π − ; = Ζ
2π ϑ0, τ = ϑmax cos τ , Ζ
2
Ζ−
,
-
:
.
ϑ x, τ = ϑmax cos ( τ − kx ) e − kx ,
k=
2
. ,
L=
cos ( τ − kx ) = 1,
4,6
k
= 4,6
2
ϑmax x=L max ϑ
.
= 0,01.
ϑmax = ϑmax e − kx . x
.
q
τ
π ∂ϑ = −λ = λk 2ϑmax cos τ + ∂ 4 x
q =
λ ρ
.
τ
π = ϑmax cos τ + , 4
. max q max . τ = ϑ
,
,
(
= L)
−
.
-
,
. ,
. ,
.
. -
,
, . , Q=±
λ cρ −
β=
∫
0,5 Ζ
q τ dτ = ± ϑmax
2
,
0
.
ρ
,
.
ρ–
.
-
.
, .
-
. ( 4.9),
.
: ϑ
,τ
=
,τ
−
1
−
1
−
2
,
ϑmax = T max − T1 .
,
δ− , ϑ T
− kx ϑmax = ϑmax . 1 e
ϑmax 2
,
T1
-
ϑmax x
.
max
T2
0
− k (δ − x ) = ϑmax , 2 e
−
ϑ
δ
-
x
Ɋɢɫ. 4.9. ɂɡɦɟɧɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɫɬɟɧɤɟ ɩɪɢ ɰɢɤɥɢɱɟɫɤɨɦ ɩɨɞɜɨɞɟ ɬɟɩɥɨɬɵ ɤ ɟɝɨ ɩɨɜɟɪɯɧɨɫɬɢ
. ,
,
-
0
( )=
2
−
1
−
1
−
2
.
: 0
( )=
2
−
1
( − )=
2
+
1
−
2
( − ).
,
, -
. , . 4.10.
.
.
-
.
. I –
(
. 4.10, ).
0
( )=
10
−
T 10 − T
20
x.
λ dT ( x ) q0 I = − λ 0 = (T 10 − T dx x = 0
II –
, , . 4.10, ).
0−0 ( T
T
III
λ, a
T
II T
10
max
T T
0
δ
λ, a
1
T
10
min 1
T
20
0
.
δ
20
20
). -
)
)
Ɋɢɫ. 4.10. ɂɡɦɟɧɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɩɨ ɬɨɥɳɢɧɟ ɫɬɟɧɤɢ ɩɪɢ ɫɬɚɰɢɨɧɚɪɧɨɦ ( ) ɪɟɠɢɦɟ ɢ ɩɪɢ ɝɚɪɦɨɧɢɱɟɫɤɨɦ ɢɡɦɟɧɟɧɢɢ ɬɟɦɩɟɪɚɬɭɪɵ ɜɧɭɬɪɢ ɩɨɦɟɳɟɧɢɹ ( )
ϑ max ϑ xII = TxII − T0 ( x ); ϑmax 1II = T 1 − T
10 ;
II
− k II x = ϑmax , 1II e
π , ΖII − aΖ II;
k II =
II
.
− k II x = T0 ( x ) + ϑmax . 1II e
dT = q0I + λϑmax qII = − λ xII 1II k II . dx x =0
III –
, ,
0−0. ϑ
max ϑ xIII = TxIII − T0 ( x ); ϑmax 2III = T 2 − T
20 ;
k=
III
III
− k III ( = ϑmax 2III e
− x)
,
π , ΖIII − aΖ III − k III ( = T0 ( x ) + ϑmax II e
.
− x)
dT − k III qIII = −λ xIII q0 I − λϑmax . 2III k III e dx x = 0
IV – . (
), ,
,
,
-
-
(
).
,
(∆q )II = qII − q0I = λϑmax 1x k II . ,
(∆q )III = qIII − q0I = −λ ϑ2maxIII kIII e − k III . III
II
:
-
. .
. -
q q01 , ∆qII , ∆qIII q
,
=
λ
-
:
= q01 + ∆qII + ∆qIII .
q
(
10
−T
20
− k III max . ) + λϑmax 1II k II − λϑ 2III k III e
−
,
,
max 1
10
q
20
,
T max 2 .
.
5. ɆȺɋɋɈɉɊɈȼɈȾɇɈɋɌɖ ɄȺɉɂɅɅəɊɇɈ-ɉɈɊɂɋɌɕɏ ɌȿɅ
5.1. ɈɋɇɈȼɇɕȿ ɉɈɅɈɀȿɇɂə ɌȿɈɊɂɂ ȼɅȺȽɈɉɊɈȼɈȾɇɈɋɌɂ ɆȺɌȿɊɂȺɅɈȼ , , ,
. .
,
-
.
.
И
. .
,
,
. :
,
,
,
: dG = − i=−
∂U – ∂x
∂U dF dτ ∂x
dG = idF dτ ,
;γ–
,
−
, ∂U = 1, ∂x
,
.
-
,
i = γ, . .
–
. – ρ
.
,
ρ0
γ
. γ,
. (
-
–
).
5.2. ȾɂɎɎȿɊȿɇɐɂȺɅɖɇɈȿ ɍɊȺȼɇȿɇɂȿ ȼɅȺȽɈɉɊɈȼɈȾɇɈɋɌɂ ɉɅɈɋɄɂɏ ɌȿɅ x ,
dx,
dV = Fdx,
.
, F– ,
-
, (
)
(
).
, . x. : dI1x = dI 2 x + dI eρ0 ,
, . . dV dV
F F
dI1x = i1x F dτ –
dτ; dI 2 x = i2 x F dτ – dτ; dI eρ0 = eρ 0 dVdU τ –
dV
, , , dτ ( -
); dU τ –
dτ. eρ 0 dVdU τ = (i1x − i2 x ) Fdτ .
dI eρ 0 = dI1x − dI 2 x
F i2x
i1x
2
, ,
(5.1)
F
dx. .
-
1−
,
-
(i1x − i2 x ) = −dix = − dU τ =
∂i dx . ∂x
∂U dτ . ∂τ
(5.1) eρ 0 Fdx
eρ0
∂U ∂i dτ = − dxFdτ . ∂τ ∂x
∂U ∂− ∂U ∂x eρ 0 =− ∂x ∂τ
=
∂U ∂ 2U . = ∂τ eρ 0 ∂x 2
,
.
∂U ∂ 2U = . ∂τ ∂x 2
(5.2)
(5.2) . (
), . ∂U = 0, ∂τ
(5.2) .
d 2U = 0. dx 2
(5.3)
(5.3)
,
,
,
: ∂ 2U 1 ∂U ∂U , = 2 + ∂τ r r ∂ r ∂
r–
d 2U 1 dU + = 0, dr 2 r dr
(5.4)
. 5.3. ȼɅȺȽɈɉɊɈȼɈȾɇɈɋɌɖ ȼ ɇȿɈȽɊȺɇɂɑȿɇɇɈɃ ɉɅȺɋɌɂɇȿ ɉɊɂ ɋɌȺɐɂɈɇȺɊɇɈɆ ɂ ɇȿɋɌȺɐɂɈɇȺɊɇɈɆ ɊȿɀɂɆȿ (5.2) (
.
(5.3), (5.4) .
. 2)
:
1) 2)
λ
U;
γ;
3) 4) 5)
Q
G; q
i;
χ;
-
6)
ρ
ρ0
,
-
α
7) 8)
β;
;
-
9)
Bi =
,
Bi
=
R
, =
αR , λ
,
τ R2
F
.
=
aτ , R2
,
1) − 4) U = U1 −
U1 − U 2
x (0 ≤ x ≤
) ; G = F (U1 − U 2 ) ; G=
F (U1 f − U 2 f +∑ n
1 1
i
i =1 i
+
)
1
,
( ;F–
;U– ; U1f –
; β1 – ( ); β2 – ; γi – U1f
i-
(5.5)
2
x– i-
(
);
i
); δi – ; U1 U2 – ( ); U2f –
–
i
( .
iU2f
) ,
. 5) – 9) (
-
;
10) Fo
e;
)
,
-
0 ≤ τ ≤ ∞; 0 ≤ x ≤ | ± R|,
-
χτ
∞ 2 sin µ n U −U x −µ n 2 =∑ cos µ n e R , U − U 0 n =1 µ n + sin µ n cos µ n R 2
U– ( τ−
( ) ); x –
x) ; U0 –
; Uc – (
, ;R–
;
; -
χ–
; ctg µ =
1
Bi
µ , Bi
=
R
,β–
.
,
-
U = U1 −
. : U1 − U 2 r ln , r r1 ln 2 r1
G=
G=
πL (U1 − U 2 ) , 1 d ln 2 2 γ d1
πL (U1 f − U 2 f
1 1d1
(5.5)
+∑ n
1
i =1 2 i
d ln i +1 + di
)
.
1
(5.6)
2 d (n +1)
(5.6) . 5.4. ɉȺɊɈɉɊɈɇɂɐȺȿɆɈɋɌɖ ,
∂P i = −µ – ∂x
: dG = i dF dτ,
;µ –
;P –
-
;x–
.
,
,
,
-
: ∂P = ∂τ
M=
-
µ e ρ0
∂2P , ∂x 2
(5.7) ; ρ0 –
,e – .
,
(5.7) d 2P = 0. dx 2
, ∂P = ∂τ
∂ 2 P 1 ∂P ∂r 2 + r ∂r
, ,
d 2 P 1 dP + = 0. dr 2 r dr
: (5.8)
(5.8) ,
. P =P1−
P1−P 2
x
(0 ≤ x ≤ ) ,
G =
G =
F (P 1 f − P 2 f
)
1
1
; ), P
i(
(
; δi – –
);
i
i =1 µ i
; µi P 2f –
1f
+∑ n
– ;P
:
F (P 1 − P 2 ) ,
µ
1
P
-
+
,
(5.9)
2
x
–
P 2–
1
i-
-
;F– ( (
δi – µ i
)
i-
); β 2 –
( .
)
, W =−f
W– .
-
, ∂P , ∂x
-
dD = WdFdτ,
;f–
;P– ,
, τ–
; Φ=
;ρ–
.
(
f e
∂P = 0. ∂τ P = P1 −
ρ
–
(5.10)
,e
x;
–
-
d 2P = 0. dx 2
(5.10) P1 − P2
,
∂P ∂2P =Φ 2 , ∂τ ∂x
D=
) f
F (P1 − P2 ) ;
D =
F (P1 f − P2 f +∑ n
1 1
i
i =1 f i
+
)
,
1
;x– ; P1 –
-
5.5. ȼɈɁȾɍɏɈɉɊɈɇɂɐȺȿɆɈɋɌɖ :
P –
;β1–
; P2 –
2
(5.11)
; δi –
; fi – ( i
fi
ii-
-
; P1f –
); P2f –
–
;F– ;β
i-
( 1
–
); β .
( (
)
2
);
–
-
5.6. ȼɅȺȽɈɉɊɈȼɈȾɇɈɋɌɖ ɂ ɎɂɅɖɌɊȺɐɂə ɉɅɈɋɄɈɃ ɋɌȿɇɄɂ ȼ ɋɌȺɐɂɈɇȺɊɇɈɆ ɊȿɀɂɆȿ ,
-
, -
.
–
. . .
,
j=−
ω−
;ρ−
(5.12)
.
δ
x
dU +ρ , dx
,
(dxF)
,
.
F
, ,
F
,
(Fj1). (Fj2),
j1.
(dxF)
j2 –
F
(dxF) (Fj1) − (Fj2) = 0, . . (j1 – j2) = 0.
(j1 – j2) = − dj = 0. ,
-
j1
j2 .
dU − d − +ρ =0 dx
−
d 2U − dx 2 U=
d 2U ρ − dx 2
. .
G G
dU =0 dx
=
d dU − + ρ = 0. dx dx
dρ = 0. dx
ρ V ρ = ρ V ρ
.
d 2U dU −K = 0. 2 dx dx
dx, -
K=– , U = U1 −
U–
; K=
ρ
(
)
U1 − U 2 + Kx e −1 , e+ K − 1
; U1 − .
; U2 −
ω
-
.
j=−
dU dP −f . dx dx
(5.13)
, .
(5.12),
f d 2 U + P =0 2 dx
ϕ (x),
U = ϕ( x) −
f
(5.13)
f U + P ( x) = ϕ( x) .
P ( x) .
, ,
-
,
,
-
P (x). 5.7. ɋɌȺɐɂɈɇȺɊɇɕɃ ɇȿɂɁɈɌȿɊɆɂɑȿɋɄɂɃ ɊȿɀɂɆ ȼɅȺȽɈɉɊɈȼɈȾɇɈɋɌɂ ɉɅɈɋɄɈɃ ɋɌȿɇɄɂ , ,
,
. .
, -
: j=− H=
∆U – ∆T
dU dT − H , dx dx
(5.14)
. .
d 2U d 2T H + =0 dx 2 dx 2
d2 (U + HT ) = 0, dx 2
(5.14),
x:
(U + HT ) = f ( x) , : U = f ( x) − HT ( x) .
,
. .
,
. -
, "
,
".
, ,
"
,
",
"
,
",
,
"
".
,
.
, .
.
-
6. ɋɉȿɐɂȺɅɖɇɕȿ ȼɈɉɊɈɋɕ ɌȿɉɅɈɉȿɊȿȾȺɑɂ
6.1. ɌȿɉɅɈɉȿɊȿȾȺɑȺ ɉɊɂ ɋɅɈɀɇɈɆ ɌȿɉɅɈɈȻɆȿɇȿ ,
, .
-
. : ∂2 ∂ ξ −1 ∂ = 2 + ∂τ ∂ ∂ ∂ (0, τ ) = 0, ∂
λ
∂
( R, τ )
∂
=
4 4 − , 0 100 100
(x, 0) = ε −
;
−
,
0.
(6.1)
(6.2)
(6.3)
(6.4) ; ξ = 1; 2 − .
(6.3) −
−
,
.
,
(6.1) − (6.4)
,
-
. ,
.
,
-
, T C0 c R 100 . λT 4
i=
. ,
i → ∞,
, . . . −(
=
−
0
i → 0,
− µ 2n Fo
.
n =1
, , . .
.
(cρ ) RTc
τ=− 4ξ T < 0,5, Tc
)∑ ( ) ∞
Tc − T Tc + T0 T T − 2 arctg − arctg 0 . ln Tc Tc T Tc + T Tc − T0 C0 c 100 4
(6.3), T 4 τ∗)
µ12B µ12A
− µ 2n
(7.4)
aτ
x
-
Ɋɢɫ. 7.1. Ƚɪɚɮɢɤ ɢɡɦɟɧɟɧɢɹ ɜɟɥɢɱɢɧɵ β , ɤɨɝɞɚ ɧɚ ɝɪɚɧɢɰɚɯ ɩɪɢɡɦɵ ɞɟɣɫɬɜɭɟɬ ɤɨɧɜɟɤɬɢɜɧɵɣ ɬɟɩɥɨɜɨɣ ɩɨɬɨɤ (ȼiA = 1; ȼiВ = 1,5; ȼiВ = 2): ; – βy – βy , β Fo, iA RA RB . RB → RA, , β . β ( RA = RB)
i , -
.
(
).
, ,
: ∂T ( x, , τ) ∂ 2T ( x, , τ) =2 ; ∂τ ∂ 2
∂T (0, , τ) = 0, ∂
T ( R, , τ ) = T ( , τ) ,
τ > τ∗.
T ( x, , τ * ) = T * ( x, ) ,
,
(7.6), . , (7.7) − (7.10). ,
, ,
(7.6) ,
:
∂θ ∂ 2θ =2 2 , ∂Fo ∂X
∂θ (0, Fo) = 0, ∂X
θ (1, Fo) = θ (Fo), θ ( , , 0) = θ0.
(7.7)
(7.8) (7.9) (7.10) -
(
)
= ln [T ( R, y, τ ) − T (0, y, τ)] − − 1,23 ∫
dT ( R, y, τ) a = −4,94 2 τ + const . T ( R, y, τ) − T (0, y, τ) R
(7.11)
(7.11) .
-
(7.11) . .
-
,
,
.
.
-
, . . .
I–II, III–IV, V–VI
(
. 7.2), : = ln (TII − TI ) − 1,23∫
I − II
III − IV
V − VI
Ɋɢɫ. 7.2. Ɋɚɫɱɟɬɧɵɟ ɬɨɱɤɢ
= ln (TIV − TIII ) − 1,23∫ = ln (TVI − TV ) − 1,23∫
(7.12)
dTIV a = − 4,94 2 τ + const , (7.13) TIV − TIII R
dTVI a = − 4,94 2 τ + const . TVI − TV R
(7.14)
y
2R
ɩɪɢɡɦɵ ɤɜɚɞɪɚɬɧɨɝɨ ɫɟɱɟɧɢɹ: I ≡ (x = y = 0); II ≡ (x = R; y = 0); III ≡ (x = 0, y = 0,5 R); IV ≡ (x = R; y = 0,5 R); V ≡ (x = 0; y = R); VI ≡ (x = R, y = R)
dTII a = − 4,94 2 τ + const , TII − TI R
V
VI
III I
IV
0
II
x
2R
,
[39]:
T −T x − µ 1n R 2 y − µ1m R 2 = D12 cos µ1n e cos µ1m e . T − T0 R R 2
aτ
2
aτ
(7.15)
(7.12) − (7.14)
(7.15)
Bi.
,
-
(7.7) − (7.10)
. .
-
,
:
(
)
θ Fo + ∆Fo,1 = θ1, Fo + 4∆FoΜ 2 θ1+ ∆Y , F0 − θ1, Fo +
(
)
+ 4∆Fo M Ki 1 − θ1,4 Fo ,
(7.16)
θ N , Fo + ∆Fo = θ N , Fo + ∆Fo Μ 2 (θ N +Y ; Fo + θ N −Y , Fo +
(
)
)
+ 2θ N − X , Fo − 4θ N , Fo + 2∆Fo Μ Ki 1 − θ 4N , Fo .
, ( θ
. 7.3
I, II, V, VI
(7.12)
V–VI
(7.14).
∆ /∆Fo,
, Ki
θ
4,94. Bi θVI θV
0,8
0
θII
–1
0,6
θI
–2
0,4
ΦV–VI
–3
ΦI–II
–4
0,2
0
.
Φ
1,0
-
),
. (Fo) . I–II,
(7.17)
0,1
0,2
0,3
0,4
0,5
Fo
Ɋɢɫ. 7.3. ɇɚɝɪɟɜ ɩɪɢɡɦɵ ɤɜɚɞɪɚɬɧɨɝɨ ɫɟɱɟɧɢɹ ɫɭɦɦɚɪɧɵɦ ɩɨɬɨɤɨɦ ɬɟɩɥɚ: (Ki = 0,5; Bi = 0,5; θ0 = 0,2): θI, θII, θV, θVI – ; , – (7.12) (7.14) I–II V–VI (7.12) − (7.14) ,
. ,
,
.
, , -
. V−
,
VI − -
.
. ,
, .
7.3. ɆȿɌɈȾɈɅɈȽɂɑȿɋɄɂȿ ɈɋɇɈȼɕ ɈɉɊȿȾȿɅȿɇɂə ɌȿɆɉȿɊȺɌɍɊɈɉɊɈȼɈȾɇɈɋɌɂ ɉɈ ɂɁɆȿɊȿɇɂəɆ ɌȿɆɉȿɊȺɌɍɊ ɇȺ ɉɈȼȿɊɏɇɈɋɌɂ ɉɊɂɁɆɕ ,
, −4;
;
,
.
:
.
. 7.4
. , )
4
180 .
(
200
-
(
), , .
6 5 4
,
, .
3
,
2
(
)
,
1
. (
– 0,2
),
. .
1 2
I
III (
. 7.2)
, -
II, IV, V, VI
3
. Ɋɢɫ. 7.4. ɋɯɟɦɚ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɨɣ ɭɫɬɚɧɨɜɤɢ: 1– ; 2– ; 3– ; 4– ;5– ; 6–
(
)
80…100 °
. -
τ,
. (I−II, III−IV, V−VI), ,
-
(7.12) − (7.14) I − II
= ln (TII − TI ) − 1,23∫
:
dTII , TII − TI
(7.18)
= ln (TIV − TIII ) − 1,23 ∫
III− IV
V − VI
= ln (TVI − TV ) − 1,23∫
dTIV , TIV − TIII
(7.19)
dTVI . TVI − TV
(7.20)
. 7.5
I, II, III,
IV, V, VI
(2R = 90 (7.18) − (7.20). ∆ /∆τ
I–II,
III–IV, I–II,
=
R∗ −
;
).
V–VI, III–IV,
V–VI
R∗2 ∆ , 4,94 ∆τ
(7.21)
,
−
τ; ∆ /∆τ −
, ,
,
-
. t, °C
Φ
100 VI
90 80
ΦIII–IV
ΦI–II
3
IV
ΦV–VI
II
V III 2
70 I 60
1
50 40 30
0
0
20
40
60
80
100
120
140
160
τ,
Ɋɢɫ. 7.5. ɇɚɝɪɟɜ ɩɪɢɡɦɵ ɤɜɚɞɪɚɬɧɨɝɨ ɫɟɱɟɧɢɹ ɢɡ ɨɪɝɫɬɟɤɥɚ (2R = 90 ɦɦ) ɫɭɦɦɚɪɧɵɦ ɩɨɬɨɤɨɦ ɬɟɩɥɚ: I, II, III, IV, V, VI – ; I–II, III–IV, V–VI − (7.18) − (7.20) . 7.1 : ( 160 ) V−VI, . ( V–VI
)
. (7.18) − (7.20), 4,94 / R2 .
∆ /∆τ (7.21).
I–II,
( III–IV,
), V–VI,
-
∆ /∆τ
7.1. Ɋɚɫɱɟɬ ɤɨɷɮɮɢɰɢɟɧɬɚ ɬɟɦɩɟɪɚɬɭɪɨɩɪɨɜɨɞɧɨɫɬɢ ɩɪɢ ɧɚɝɪɟɜɚɧɢɢ ɨɪɝɫɬɟɤɥɚ (R∗= 0,043 ɦ); ɫɟɱɟɧɢɟ ɩɪɢɡɦɵ (V−VI) τ, TVI, ºC 20 40 60 80 100 120 140 160
61,5 70,0 75,0 80,0 84,0 87,5 91,0 94,0
∆ti = = TV, ºC ln∆ti TVI −TV 47,0 54,5 60,5 66,0 71,0 75,0 79,0 83,0
fi = ∆(t ) = τ, = (t )i+1 − = ∆(t ) (t )i
20 40 60 80 100 120 140 160
8,5 5,0 5,0 4,0 3,5 3,5 3,0
0,568 0,334 0,351 0,297 0,280 0,291 0,261
14,5 15,5 14,5 14,0 13,0 12,5 12,0 11,0
2,674 2,741 2,674 2,639 2,565 2,526 2,484 2,398
1/∆ti 0,069 0,064 0,069 0,071 0,077 0,089 0,083 0,091
Fi = = ln∆ti = Fi−1 + 1,23Fi − fi – 1,23Fi 0,568 0,902 1,253 1,550 1,830 2,121 2,382
0,699 2,042 1,110 1,565 1,541 1,098 1,907 0,659 2,251 0,275 2,609 −0,124 2,930 − 0,532
= = 0,5 [(∆ti )−1 + + (∆ti+1)−1] 0,067 0,067 0,070 0,074 0,080 0,083 0,087
· 106,
2
/
0,14 0,14 0,13 0,11 0,12 0,12
τ
,
, ,
-
∆τ
, -
. , (2R = 90
τ = 60…80 I–II, III–IV, V–VI
),
∆ /∆τ
, ∆ /∆τ (7.21). .
,
∆τ.
90 :
(2R = -
)
I−II
= 0,112 · 10−6,
2
/ ;
III−IV
= 0,118 · 10−6,
2
/ ;
V−VI
= 0,114 · 10−6,
2
/ .
(2R = 90 (2R = 40
),
(2R = 28
),
(2R = 50
),
). -
V−
. VI −
. , . 7.4. ɈɐȿɇɄȺ ɇȺɋɌɍɉɅȿɇɂə ɍɉɈɊəȾɈɑȿɇɇɈɃ ɑȺɋɌɂ ɌȿɉɅɈȼɈȽɈ ɉȿɊɂɈȾȺ ,
,
.
,
–
;
0
−
,
Ψ* =
−
−
0
,
(7.22)
0
( , Ψ∗ 1 %.
: .
. Ψ∗ = f (Bi)
. 7.6) ,
Ψ∗∗
Ψ∗ 1,0 0,8
5 4 3 2
0,6 0,4
1
0,2
0
1,0
0,5
3–
1,5
2,0
2,5
Bi
Ɋɢɫ. 7.6. Ƚɪɚɮɢɤ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬɧɨɲɟɧɢɹ Ψ∗ ɢ Ψ∗ ∗ ɨɬ ɤɪɢɬɟɪɢɹ ɬɟɩɥɨɨɛɦɟɧɚ Bi ɞɥɹ ɬɟɥ ɪɚɡɥɢɱɧɨɣ ɮɨɪɦɵ: 1– ;2– ; ;4– ;5– ɯ– Ψ∗ ; – Ψ∗ ∗
:
ψ∗max > 0,2ξ. ,
Ψ∗
, -
(7.22) , . . −
Ψ** =
Ψ∗ = 0,44; Ψ∗ ∗ = 0,78
Ψ** = f (Bi)
−
. 7.6,
0
.
(7.23)
0
,
1% .
, , .
,
, .
7.5. ɈɉɊȿȾȿɅȿɇɂȿ ɌȿɆɉȿɊȺɌɍɊɈɉɊɈȼɈȾɇɈɋɌɂ ɆȺɌȿɊɂȺɅɈȼ ɄɈɇɌȺɄɌɇɕɆ ɆȿɌɈȾɈɆ ɇȿɊȺɁɊɍɒȺɘɓȿȽɈ ɄɈɇɌɊɈɅə
.
,
-
[59, 68, 76, 78]. , Ɋɢɫ. 7.7. ɋɯɟɦɚ ɭɫɬɚɧɨɜɤɢ ɤɨɧɬɚɤɬɧɵɯ ɬɟɪɦɨɩɚɪ ɧɚ ɩɪɢɡɦɟ ɤɜɚɞɪɚɬɧɨɝɨ ɫɟɱɟɧɢɹ: 1– ;2– ; 3– ; 4– ; 5– ;6–
. 7.7. 1
2
3 4 5 3
4 5
6
6
2
6,
4.
( (
) )
.
5 .
(
)
.
. 7.8 t ,
t (2R = 0,05 ) . 7.9
,
ρ = 1700
,
/ 3. t
t -
-
τ,
,
ρ = 2000
(2R = 0,05 )
,
/ 3.
(
), , Φ = ln (t − t ) − 1,23 ∫
dt
t −t
.
(7.24)
Ɋɢɫ. 7.8. ɇɚɝɪɟɜ ɩɪɢɡɦɵ ɤɜɚɞɪɚɬɧɨɝɨ ɫɟɱɟɧɢɹ ɢɡ ɤɪɚɫɧɨɝɨ ɤɢɪɩɢɱɚ
ρ = 1700 t −
ɜɨɡɞɭɲɧɨ-ɫɭɯɨɣ ɜɥɚɠɧɨɫɬɢ: / ; 2R = 0,05 ; (R∗ = 0,024 ): t − ; − 3
; (7.24)
Ɋɢɫ. 7.9. ɇɚɝɪɟɜ ɩɪɢɡɦɵ ɤɜɚɞɪɚɬɧɨɝɨ ɫɟɱɟɧɢɹ ɢɡ ɫɢɥɢɤɚɬɧɨɝɨ ɤɢɪɩɢɱɚ ɜɨɡɞɭɲɧɨ-ɫɭɯɨɣ ɜɥɚɠɧɨɫɬɢ: ρ = 2000 / 3; 2R = 0,05 ; (R∗ = 0,023 ): t − ; t − ; − (7.24) ∆ /∆τ ψ∗∗
(7.21). . 7.2
,
Excel. 7.2. Ɋɚɫɱɟɬ ɤɨɷɮɮɢɰɢɟɧɬɚ ɬɟɦɩɟɪɚɬɭɪɨɩɪɨɜɨɞɧɨɫɬɢ ɩɪɢ ɧɚɝɪɟɜɚɧɢɢ ɤɪɚɫɧɨɝɨ ɤɢɪɩɢɱɚ ρ = 1700 ɤɝ/ɦ3, (R∗= 0,024 ɦ), t0 = 20 °ɋ τ, TVI, ºC 100 200 300 400 500 600 700 800 900 100 0
∆ti = TV, ºC = ln∆ti TVI −TV
0,172 0,138 0,133 0,133 0,138 0,148 0,157 0,163 0,167
35 41,5 45,5 48 50 51,5 53 54,5 56
30 34,5 38 40,5 42,5 44,5 46,5 48,2 50
5 7 7,5 7,5 7,5 7 6,5 6,3 6
1,609 1,946 2,015 2,015 2,015 1,946 1,872 1,84 1,792
0,2 0,143 0,133 0,133 0,133 0,143 0,154 0,159 0,167
57,5
51,5
6
1,792
0,167
fi = ∆(t ) = τ, = (t )i+1 − = ∆(t ) (t )i
100
(∆ti)–1
= = 0,5 [(∆ti )−1 + + (∆ti+1)−1]
6,5
1,115
Fi = = ln∆ti = Fi−1 + 1,23Fi ψ∗∗ − · 106, fi – 1,23Fi 2 / 0,6 6
-
200
4,0
0,552
1,115
1,37
0,575
300
2,5
0,333
1,667
2,05
− 0,035
400
2,0
0,266
2,000
2,46
500
1,5
0,207
2,266
600
1,5
0,222
2,473
700
1,5
0,255
2,695
800
1,5
0,245
2,950
900
1,5
0,251
3,191
100 0
1,5
3,442
. 7.8 , ρ = 1700
2,78 7 3,04 2 3,31 5 3,62 9 3,93 0 4,23 4 , ,
. 7.2
0,71 1 0,47 8 0,38 1 0,37 7 0,38 4 0,49 5 0,40 2 0,40 7 0,39 3
− 0,445
− 0,772 − 1,095 − 1,443 − 1,788 − 2,138
− 2,442
0,6 7 0,7 0 0,7 3 0,7 5 0,7 8 0,8 0 0,8 2 0,8 3
400 ,
400…900 , 2 = 0,393 · 10−6 / .
(7.21), [64]
/ 3 . 7.9
= 0,390 · 10− 6
2 %, ,
,
2
/ .
300 ,
, ,
(7.21),
300…700 =
(0,023) ( −2,45) − ( −4,62) R∗2 ∆ = = 0,58 ⋅ 10 − 6 4,94 ∆τ 4,94 700 − 300 2
[64]
ИИ
2
/ .
ρ = 2000
= 0,576 · 10− 6
5%
t (2R = 0,04 )
[59, 78].
(7.24).
3
/ .
. 7.10 t ρ = 1200 / 3. τ,
), ,
2
/
(
-
ψ∗∗
-
Excel. .
7.10 1600 ,
, , 1600…2400
(0,018) ( −5,884) − ( −7,463) R2 ∆ = ∗ = = 0,129 ⋅ 10 − 6 4,94 ∆τ 4,94 2400 − 1600 2
2
/ .
(○), -
ψ∗∗ = 0,78
τ = 2200
. . 7.10
,
(∆), ,
1800…2800
1800 -
(0,0195) ( −7,925) − ( −9,572) R2 ∆ = 0,126 ⋅ 10 − 6 = = ∗ 4,94 ∆τ 4,94 2800 − 1800 2
[64] = 0,138 · 10−6
2
ρ = 1220
/
2
/ .
3
-
/ . . ,
,
.
-
,
-
, ,
.
8. ɇȺɍɑɇɈ-ɆȿɌɈȾɈɅɈȽɂɑȿɋɄɂȿ ɈɋɇɈȼɕ ɄɈɆɉɅȿɄɋɇɈȽɈ ɈɉɊȿȾȿɅȿɇɂə ɌȿɉɅɈɎɂɁɂɑȿɋɄɂɏ ɋȼɈɃɋɌȼ ɆȺɌȿɊɂȺɅɈȼ ɆȿɌɈȾɈɆ ɇȿɊȺɁɊɍɒȺɘɓȿȽɈ ɄɈɇɌɊɈɅə
8.1. ɌȿɈɊȿɌɂɑȿɋɄɂȿ ɈɋɇɈȼɕ ɈɉɊȿȾȿɅȿɇɂə И
А
ИА
В . , ρ
:
-
. ∂ϑ ∂ 2ϑ , = ∂τ ∂ 2 ϑ0, τ = ϑmax
− ,
;z−
), ° ;
= 2π / z −
(8.3), ,τ
, = ϑmax cos ( τ − kx) e − kx ,
/2 .
(8.3) ,τ
-
(
, . (8.1) − (8.2)
q
(8.2)
/ ; ϑmax −
ϑ k=
s ( τ),
2
,
–1
(8.1)
π ∂ϑ = − λ = λk 2 ϑmax cos τ + ∂ 4
-
(8.3)
q
−
,τ
π ϑmax cos τ + , 4
=
,
: =
λ−
λ ρ = ρ
(8.4)
,
/( · ); ρ −
,
,
/(
3
⋅ ).
max q max . ,τ = B ϑ
. -
, −
,
(
)
= q max / ϑmax .
(8.5)
, ,
ϑmax
,
q max
-
, ρ=
( ρ) −
.
q max ,τ
ϑmax
(8.6)
.
( ρ) .
-
, . -
,
. 8.2. ɆȺɌȿɆȺɌɂɑȿɋɄɂɃ ɆȿɌɈȾ ɈɉɊȿȾȿɅȿɇɂə ɉɅɈɌɇɈɋɌɂ ɌȿɉɅɈȼɈȽɈ ɉɈɌɈɄȺ ɇȺ ɉɈȼȿɊɏɇɈɋɌɂ ɆȺɌȿɊɂȺɅȺ -
, . , , (
(
),
. 7.4) -
.
∆t ,
.
1.
-
α
= 4,6 + 0,035∆t + 1,7 ∆t 0,333 ,
2
/
⋅ ,
∆t −
.
q = 4,6∆t + 0,035∆t 2 + 1,7 ∆t1,333 ,
−t ,
∆t = t 2.
20 ° ; t λ
0
, 0,05
(8.7) +30 ° .
/ ⋅ , ,
,
,
−15
,
= (5,18 + 0,041t1 ) ,
λ
t1 –
–
2
/
+20 ° ; –
+2
0,25 .
q q = (5,18 + 0,041t1 ) ∆t ,
∆t = t1 – t2, 3.
0
/
2
,
30 ° ; t2 –
. -
: α
= 4,6 + 0,035∆t + 1,5∆t 0,333 ,
2
/
⋅ .
q = 4,6∆t + 0,035∆t 2 + 1,5∆t1,333 ,
t
∆t = t – 4.
–t ,t
− ,
0
+30 ° .
2
/
,
,
+ 40
(8.8) + 400 ° ;
( (
):
α
= 4,6 + 0,035∆t + ∆t 0,333 ,
/
2
)
⋅ . (
. 7.4)
-
(2R = 0,05 ) q = 4,6∆t + 0,035∆t 2 + ∆t1,333 ,
∆t = t –t ;t + 200 ° ; t –
− ,
+ 20
+ 120 ° .
8.3. ɄɈɆɉɅȿɄɋɇɈȿ ɈɉɊȿȾȿɅȿɇɂȿ ɌȿɉɅɈɎɂɁɂɑȿɋɄɂɏ ɋȼɈɃɋɌȼ ɆȺɌȿɊɂȺɅɈȼ ɆȿɌɈȾɈɆ ɇȿɊȺɁɊɍɒȺɘɓȿȽɈ ɄɈɇɌɊɈɅə
,
/
2
,
(8.9) + 20
(
)
t0,
, . t ,
-
t,
, q max
.
t t
q max
. t
t
. (8.8) ϑ = 0,5 (t − t0).
(8.9). z∗ , . . z = 2 z∗.
( ρ)
ρ=
ϑ
q max π
(8.10)
.
z*
, t
t
, a=
R* −
-
∆Φ R*2 , ∆τ 4,94
(8.11)
,
;
−
-
τ.
,
Φ = ln (t − t ) − 1,23 ∫
Φ = ln (t − t ) − 1,23 ∫
∆
/ ∆τ
dt
t −t
dt
t −t
(8.12)
.
(8.13)
.
-
. ,
λ = a ( ρ).
(8.14)
,
,
,
-
. ,
7.5. , t0 = 99 º ,
-
t = 30 º . R = 0,0135 . t
(cp) = 99 − 30 = 69 º .
-
. 8.1 τ
t ,
(8.13)
∆τ = 0,125 · 10−6 2/ λ .
(8.11) 30 .
q max
(8.8) . 8.2.
∆t =
t0 − t =
8.2. Ɋɚɫɱɟɬ ɨɛɴɟɦɧɨɣ ɬɟɩɥɨɟɦɤɨɫɬɢ ɢ ɤɨɷɮɮɢɰɢɟɧɬɚ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɩɨ ɬɟɦɩɟɪɚɬɭɪɧɨɦɭ ɩɨɥɸ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ ɩɪɢ ɨɯɥɚɠɞɟɧɢɢ ɮɬɨɪɨɩɥɚɫɬɚ: t0 = 99 ºɋ, tɠ = 30 ºɋ, qɩmɚɯ = 910 ȼɬ/ɦ2, = 0,125 · 10−6 ɦ2/ɫ
z* ,
t , ºC
ϑ = 0,5( t0 − t )
ρ=
ϑ
q max
/(
π
λ = a ( ρ),
,
z* 3
/( · )
· )
200
76,0
11,5
1786
0,223
400
66,0
16,5
1761
0,220
600
61,0
19,0
1872
0,233
800
57,0
21,0
1952
0,244
1000
54,0
22,5
2040
0,255
,
, (
), ,
.
9. ɈɉɊȿȾȿɅȿɇɂȿ ɌȿɆɉȿɊȺɌɍɊɈɉɊɈȼɈȾɇɈɋɌɂ ɆȺɌȿɊɂȺɅɈȼ ȼ ɌȿɅȺɏ ɄɍȻɂɑȿɋɄɈɃ ɂ ɒȺɊɈȼɈɃ ɎɈɊɆɕ
9.1. ɁȺɄɈɇɈɆȿɊɇɈɋɌɖ ɍɉɈɊəȾɈɑȿɇɇɈȽɈ ɌȿɉɅɈȼɈȽɈ ɊȿɀɂɆȺ ȼ ɄɍȻȿ ,
.
-
∂ 2T ( x, y, z , τ) ∂ 2T ( x, , z , τ) ∂ 2T ( x, y, z , τ) + + = ∂z 2 ∂y 2 ∂x 2 ∂ 2T ( x, y, z, τ) ∂ 2T ( x, , z, τ) 2 2 ∂y 2 ∂ T ( x, , z , τ ) 1 + ∂ z + 2 2 ∂ x2 ∂ T ( x, , z , τ ) ∂ ( , , , τ ) T x z ∂x 2 ∂x 2
∂T ( x, , z, τ) = ∂τ =
(9.1)
:
(9.2),
(9.3) − (9.5)
(9.6)
∂T ( y, z , τ) ∂T ( x, z, τ) ∂T ( x, , τ) = 0; = 0; = 0, ∂x ∂y ∂z
T ( R, y, z , τ) = T ( y , z , τ),
(9.3)
T ( x, R , z , τ ) = T ( x , z , τ ) ,
(9.4)
T ( x, y , R , τ ) = T ( x, y , τ ) ,
(9.5)
T ( x, y , z ,0) = T0 .
(9.6)
∂ 2T ( x, y , z , τ ) ∂T ( x , y , z , τ ) (1 + = 2 ∂τ ∂ x
(9.1)
(9.2)
y
=
grad y q
+
) ,
z
(9.7)
(9.8)
grad x q
y x.
z=
grad z q grad x q
(9.9)
z x. , : θ = 1 − (1 − θ 0 ) ∑ ∞
∞
n =1 m =1 k =1
x cos µ RA
× cos µ
β
∑ ∑D ∞
×
D D
y cos µ RB
z RC
× ex − µ 2 2 τ + µ 2 2 τ + µ 2 RA RB RC2
(9.10),
τ .
(9.8) , RA
R
BiA R ,
. ,
β (
×
(9.10)
(9.9).
Bi RA,
β
.
, RA = RB)
-
. ( , BiA = Bi = Bi , , ,
βz .
). ,
,
RA = RB = RC ,
. β
βz
-
. , : ∂ ( x, , z , τ ) ∂ 2 ( x, , z , τ ) , =3 ∂τ ∂x 2
∂ ( x, 0, z , τ) ∂ (0, , z , τ) = 0; = 0; ∂y ∂x
(9.11)
∂ ( x, , 0, τ) = 0, ∂z
T ( R, y , z , τ ) = T ( y , z , τ ) , T ( x, y, z , τ* ) = T* ( x, y, z ) ,
τ > τ∗.
,
= ln [ T ( R, y, z , τ) − T (0, y, z , τ)] − − 1,23∫
dT ( R, y, z , τ) = −7,41 2 τ + const. T ( R, y, z , τ) − T (0, y, z , τ) R
(9.12)
, . (9.12)
-
. . ,
,
-
.
. , . .
-
. (9.12) Bi,
-
I−II, I−III -
. (
. 9.1),
I − II
= ln (TII − TI ) − 1,23∫
I − III
dTII = − 7,41 2 τ + const, TII − TI R
= ln (TIII − TI ) − 1,23∫
(9.13)
dTIII a = − 7,41 2 τ + const , TIII − TI R*
(9.14)
R*2 = 2 R 2 .
9.2. ɗɄɋɉȿɊɂɆȿɇɌȺɅɖɇɈȿ ɈɉɊȿȾȿɅȿɇɂȿ ɌȿɆɉȿɊȺɌɍɊɈɉɊɈȼɈȾɇɈɋɌɂ ɆȺɌȿɊɂȺɅɈȼ ȼ ɌȿɅȺɏ ɄɍȻɂɑȿɋɄɈɃ ɎɈɊɆɕ (9.12) ,
-
. ,
,
, -
80...100 ° . . ,
5...6 .
,
, ,
,
. -
, . I−
. 9.1 . )
I–II
.
I–II
I–III
II, III − I–III ,
( :
-
z
2RB
x
0 2RC 2RA
t, °C
Φ ΦI–III
100
4
III II
80
2 I
ΦI–II
60
0
40
–2
40
0
80
120
160
200
τ,
Ɋɢɫ. 9.1. ɇɚɝɪɟɜ ɤɭɛɚ ɢɡ ɨɪɝɫɬɟɤɥɚ (2R = 90 ɦɦ) ɫɢɦɦɟɬɪɢɱɧɵɦ ɩɨɬɨɤɨɦ ɬɟɩɥɚ: I– (x = 0; y = 0; z = 0); II – (x= R; y = 0; z = 0); III – (x = R; y = R; z = 0); – (9.15) (9.16) I–II I–III I − II
I − III
= ln (TII − TI ) − 1, 23∫
dTII ; TII − TI
(9.15)
= ln (TIII − TI ) − 1, 23∫
dTIII . TIII − TI
(9.16)
∆ / ∆τ
-
:
2 ∆Φ R ; a= ∆τ 7,41
2 ∆Φ R* . a= ∆τ 7,41
, •
,
:
I–II
= 0,108 · 10–6
(9.17)
2
/ ,
I–III
= 0,119 · 10–6
2
/ ;
-
• I–II
= 0,110 · 10–6
2
/ ,
I–III
= 0,120 · 10–6
2
/ .
,
,
-
, (9.12)
,
-
. 9.3. ɈɉɊȿȾȿɅȿɇɂȿ ɌȿɆɉȿɊȺɌɍɊɈɉɊɈȼɈȾɇɈɋɌɂ ɆȺɌȿɊɂȺɅɈȼ ȼ ɄɍȻȿ ɆȿɌɈȾɈɆ ɌȿɉɅɈȼɈȽɈ ɉɊɈɋɅɍɒɂȼȺɇɂə , (
,
),
. .
,
,
(
)
, .
, ,
. ,
,
, −t
. 9.2 (2R = 90
.
):
t, °C
100
3,0 ɯ
ɯ
ɯ
90
ɯ
3,2
ɯ
t
.
3,4
80 ɯ ɯ
70
3,5
° t
°
.
60
ɯ ɯ
3,6 ɯ
t.
°
50
ɯ
3,7 ° 3,8
40 °
.
−t.. t =
31
º .
30
0
20
t
40
.
60
80
τ,
100
Ɋɢɫ. 9.2. Ɉɯɥɚɠɞɟɧɢɟ ɤɭɛɚ ɢɡ ɨɪɝɫɬɟɤɥɚ (2R = 90 ɦɦ) ɧɚ ɜɨɡɞɭɯɟ: – , (9.18); t . – ;t. – ; − (9.19)
. 9.2
t
t
t
t. −
.
t
.
=t +
.
(t t
. −t
. −t
)2
,
.
(9.18)
,
. ,
,
t
= ln (t
. 9.2
.
∆
− t . ) – 1,23
.
∫t .
dt
−t .
/ ∆τ
.
(9.19)
.
7,41 ∆
.
-
.
/ R2,
/ ∆τ a=
-
R 2 ∆Φ . 7,41 ∆τ
(2R = 90 –6
)
2
= 0,119 · 10 ,
/ . . -
, . 9.4. ɈɉɊȿȾȿɅȿɇɂȿ ɌȿɆɉȿɊȺɌɍɊɈɉɊɈȼɈȾɇɈɋɌɂ ɆȺɌȿɊɂȺɅɈȼ ȼ ɌȿɅȺɏ ɒȺɊɈȼɈɃ ɎɈɊɆɕ
Φ = ln ∆t
.-
− 1,73 ∫
d ∆t (τ) τ = −9,86 2 + const . ∆t .R
(9.20)
, . (9.20)
-
. . ,
, .
.
-
, . . . (9.21)
: = ln ∆θ
.-
− 1,73 ∫
d θ (F ) ∆θ
= −9 ,86F + const .
(9.21)
.-
(9.21) i,
-
. (2R =160
),
~ 100 ° , .
, .
-
.
,
-
,
.
-
.
, . . 9.3 (2R = 160 t, °
)
-
,
. 2,0
100
80 1
60
40
2,5
2
20
3,0
0 0
40
80
120
160
200
240
(
)
Ɋɢɫ. 9.3. Ɉɯɥɚɠɞɟɧɢɟ ɲɚɪɚ ɢɡ ɨɪɝɫɬɟɤɥɚ (2R = 160 ɦɦ) ɧɚ ɜɨɡɞɭɯɟ: 1− ; 2− ; − (9.22)
Φ = ln (t − t ) − 1,73 ∫
∆
/ ∆τ
d ∆t
(t − t )
.
(9.22) -
=
R2 ∆ = 0,108 · 10−6 9,86 ∆τ
2
/ .
,
-
(9.20) .
10. ɆȿɌɊɈɅɈȽɂɑȿɋɄɂȿ ɏȺɊȺɄɌȿɊɂɋɌɂɄɂ ɂ ɉɈȽɊȿɒɇɈɋɌɂ ɈɉɊȿȾȿɅȿɇɂə ɌȿɉɅɈɎɂɁɂɑȿɋɄɂɏ ɋȼɈɃɋɌȼ ɆȺɌȿɊɂȺɅɈȼ ɆȿɌɈȾɈɆ ɇȿɊȺɁɊɍɒȺɘɓȿȽɈ ɄɈɇɌɊɈɅə
10.1. ɄɅȺɋɋɂɎɂɄȺɐɂə ɉɈȽɊȿɒɇɈɋɌȿɃ ɋɊȿȾɋɌȼ ɂɁɆȿɊȿɇɂɃ , , . 1.
,
( "
"
, (
,
)
,
.
)
(
).
−
( (
-
)
),
-
. 2. ,
, ,
,
-
, -
,
. ,
3. ( ),
-
. , :
, ,
-
, .
-
[17, 78].
,
∆t = t − t .
t
t
. ,
, . :
∆t : ∆t = ∆t + ∆t + ∆t .
, ∆t ,
∆t
∆t
∆t -
-
, . (
,
,
),
,
: ,
, .
-
(
∆t
)
-
-
, .
:
.
. -
, . ,
.
∆t
,
.
,
∆t
, ∆t
∆t : ∆t = ∆t
∆t .
∆t
+ ∆t .
∆t
-
,
( ).
,
∆t
∆t ,
. -
,
. ∆t
. ,
-
∆t
-
. , )
,
(
∆t
∆t = ∆t + ∆t . ∆t ,
∆t
,
,
( ).
∆t
∆t
.
∆t : -
= 0. ∆t
:
,
,
(
,
,
,
),
,
, .
∆t
(
),
, ,
,
,
.
∆t
60, 76].
, ,
,
,
, ,
(
. ,
∆t t
) -
, .
,
[59, , . τ
− ,
:
h
4
(
-
,
)
. t
q
u1,
,
,
,
,
b
2
,
. -
1
,
.
u2
, h/2
. tc
3
,0
.
x
,
-
.
∆t (
. ,
,
-
,
),
,
.
( -
, .
),
, ∆t
(
.
). -
, .
, ,
,
−
,
[17, 22, 24, 42, 50, 55, 56, 59, 68, 71, 76, 78] . , . , .
.
,
-
,
,
-
10.2. ɉɈȽɊȿɒɇɈɋɌɂ ɂɁɆȿɊȿɇɂə ɌȿɆɉȿɊȺɌɍɊɕ ɇȺ ɉɈȼȿɊɏɇɈɋɌɂ ɌȿɅȺ
(
−
),
( h
b.
. 10.1 .
)
-
Ɋɢɫ. 10.1. ɋɯɟɦɚ ɪɚɫɩɨɥɨɠɟɧɢɹ ɬɟɪɦɨɩɚɪɵ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ ɦɚɬɟɪɢɚɥɚ: 1− ;2− ( ); 3 − ;4− ; q− ; b, h − ; t − , 0,2...0,8
.
. . ,
,
, .
,
,
,
,
,
,
: 1)
, ,
; 1268–80); 2) 3) ( ),
−
– ,
λ3.
" (
λ1 -
, α,
[78]:
,
(
λ1 λ3 h/2
) [78].
t −
; .
"
t
( ,
6-15;
, (
∆t = t − t =
(t − t ) =
∆t
. 10.1),
(t − t ) / ( −1),
)
(10.1) ;t −
-
. ≈ [1− ϕ ξ (1 + η)] / [1 + µ (1 + η)]. (10.2)
:
(10.2)
ϕ = 1 + h / b; µ = 1 + 2h / b; ξ = α h / λ ; ξ = α h / λ ;
η = ϕ2 ψ (1 + µ) / µ ξ ω; ψ = [1 + πd / 2 (h + b)]2; ω = 1 − πd2 / 4hb;
λ −
d = 0,4 ⋅ 10 t = 200 ° ,
,
;d−
t = 100 ° . −3 , h = b = 0,8 ⋅ 10 . 2 α = 25 /( ⋅ ). , : λ = 1 /( ⋅ ), λ = 0,5 /( ⋅ ). : ϕ = 2; µ = 3; ξ = 0,02; ξ = 0,04; ψ = 1,94; ω = 0,804; η = 322. , (10.2) = − , (10.1), ∆t t = t − ∆t = 101,2 ° . ∆t = ∆t / t = 0,012 = 1,2 %. , ( . 10.2), ∆t ,
−3
,
0,012. = (t − t ) / ( −1) = − 1,18 ° . , [78]:
.
= t (τ) − t (τ) = (ε − ε ) b,
∆t ε −
t (τ)
t (τ) −
(
, ;b− ( )
)
,° / .
ε = (h + b)2 / 8π2 a ,
a −
λ − ; ( ρ)
(10.3)
,
( ρ) −
,
/( ⋅ ); d −
2
[78]:
( ρ) 4 = ( ρ ) + λ 4bh −1 2 πd
πd 2 bh ( ) − 4 2 (b + h) + πd
b ≈ 0,025
= 0,5 ⋅ 10−6
2
/ . (
),
2
,
(10.5)
, ;b h− /( 3 ⋅ ). (
d = 0,4 ⋅ 10−3 ,
[78]:
(10.4)
/ .
, ,
;ε,
. 7.4)
,
/ . , h = b = 0,8 ⋅ 10−3 . 2 : a = 0,12 ⋅ 10−6 / , a = ( ) , : λ . = 0,5 /( ⋅ ), λ . = 0,2 /( ⋅
). ( ρ) . = 1,12 = 1,6 ⋅ 106 /(
⋅ 10 ⋅ ), ( ρ) = 4,35 ⋅ 106
,
6
3
/(
3
,
⋅ ).
/(
3
⋅
),
: ( ρ) . = -
(10.4):
ε = (h + b)2 / 8π2 a = 0,27 ; ε = (h + b)2 / 8π2 a = 0,065
• •
ε . = 0,71 ; ε . = 0,23 .
• • • •
: ∆t
(10.5): ∆t
(10.3)
∆t = (ε − ε . ) b = − 0,011 ; = (ε − ε . ) b = − 0,004 .
∆t 20...80 ° ,
: ∆ζ = ∆t / ∆t = 0,00028 = 0,028 %; ∆ζ = ∆t / ∆t = 0,00007 = 0,007 %.
• •
-
10.3. ɉɈȽɊȿɒɇɈɋɌɂ ɂɁɆȿɊȿɇɂə ɌȿɆɉȿɊȺɌɍɊɕ ɄɈɇɌȺɄɌɇɕɆ ɆȿɌɈȾɈɆ (
)
[59]. ,
-
. ,
t,
t. t
t, . .
−
−
-
. , ,
,
-
,
,
.
, , .
-
. 7.7. ,
,
, , -
.
.
,
[59, 68, 76, 78]. . 10.2 (
)
-
(
-
). ,
(
), .
1, 2, 3
(
. 10.2)
, : r1 = 0,2
∆r
; L2 = 5 = 1,8 ,
.
∆ = L2 −L1 = 4,6
.
; r2 = 2,0 =
r2
, ; L1 = 0,4
−
r1 =
−
, =α /α;
λ2 / λ1;
λ=
100 ∆r =
α,α − (
*λ;
= (cp)2 / (cp)1;
∆r / L1;
); λ2, λ1 − ,
/(
∆=
3
∆ / L 1,
(10.6) ; (cp)2, (cp)1
⋅ ). (
(
)
)
• • •
,
(
-
),
-
,
: : λ1 = 23y /( ⋅ ); (cp)1 = 4350 /( 3 ⋅ ); : λ2 = 0,23 /( ⋅ ); (cp)2 = 1780 /( 3 ⋅ ); : λ3 = 0,5...1,2 /( ⋅ ); (cp)3 = 1600...1900 /( 3 ⋅ ).
L1
2
1 r 3
h L2
−
*λ =
[59]:
r1 r2
Ɋɢɫ. 10.2. Ɇɨɞɟɥɶ (ɫɟɱɟɧɢɟ) ɤɨɧɬɚɤɬɧɨɣ ɬɟɪɦɨɩɚɪɵ, ɢɫɩɨɥɶɡɭɟɦɨɣ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɬɟɩɥɨɮɢɡɢɱɟɫɤɢɯ ɫɜɨɣɫɬɜ ɦɚɬɟɪɢɚɥɨɜ: r1 r2 − ; L1 L2 − ; 1− ( ); 2 − ; 3− ;h− . ,
: ,
; ; ;
. , ,
.
α .
-
α
,
[59, 76]: α = 2λ / (h2 + h3) + 7 · 103 λ / S ,
λ −
(
λ −
−
, .
, (
; -
(10.7)
= 10 2 ; λ2 = 0,23
−6
; /( ⋅ ). .).
: h3 ≈ 100
≈ 60 ° ,
/( ⋅ ).
) λ = 0,029
(
= 4 · 10− 6
2
,
,
≈1 .
/( ⋅ ); h2, h3 − ;Ρ− ;E− 2 / ;S −
), ,
)
(10.7)
−
. (
S
=
Ρ = 0,1
. 10.3)
×
4
10
,
= 0,98 : h2 = 1 = (
(
≈ 10− 4 ; λ3 ≈ 0,8
-
) /( ⋅ ); E ≈ 70 λ
/
2
≈ 70 · 105 / 2. /( ⋅ ).
λ = 2 λ2λ3 / (λ2 + λ3) = 2 · 0,23 · 0,8 / (0,23 + 0,8) = 0,36 ,
-
(10.7),
α = 2 ⋅ 0,029 / (1,01 ⋅ 10−4) + (7 · 103) ⋅ 1 ⋅ 0,36 / (70 · 105) (4 · 10− 6) = = 580 + 90 = 670 /( 2 ⋅ ). ,
, ,
.
, ,
,
.
(
α ≈ 10
/(
2
)
(
⋅ ). (10.6)
:
= α / α = 670 / 10 = 67; λ=
100
*λ =
1;
*λ =
λ2 / λ1 = 0,23 / 23 = 0,01;
= (cp)2 / (cp)1 = 1780 / 4350 = 0,4. :
∆r =
-
∆r / L1 = 1,8 / 0,4 = 4,5;
∆=
∆ / L1 = 4,6 / 0,4 = 11,5.
)
λ
m = 0,3 •
0,41
− 1,3 = − 1.
∆t = (0,046 + 3,74
•
∆t ,
[59]: −0,77
) [(1,38 −
∆t = (0,046 + 3,74 ( , ∆t ,
•
λ
−0,38
)
−0,77
∆r
− 0,011
) [(1,38 − ) :
λ
λ];
−0,38
)
∆
− 0,011
λ ].
(10.8)
∆t = (0,046 + 3,74 / 25,5) [(1,38 − 1) / 4,5 − 0,011] =
= (0,046 + 0,147) [(0,38) / 4,5 − 0,011] = (0,193) [0,073] = 0,014 = 1,4 %; •
∆t = (0,046 + 0,147) [(1,38 − 1) / 11,5 − 0,011] =
= (0,183) [(0,38 − 1) / 11,5 − 0,011] = (0,183) [0,022] = 0,0042 = 0,42 %. ; ∆r = 2,3
=8
; ∆ = 7,6
r2 = 2,5 :
∆r =
= 5 · 10− 6
∆r / L1 = 2,3 / 0,4 = 5,8;
(
∆=
∆ / L1 = 7,6 / 0,4 = 19.
)
S
2
. ,
(10.7),
α = 2 ⋅ 0,029 / (1,01 ⋅ 10−4 ) + (7 · 103) ⋅ 1 ⋅ 0,36 / (70 · 105) ⋅ (5 · 10− 6) = = 580 + 103 = 683
/(
2
⋅ ).
: •
= α / α = 683 / 10 = 68.
∆t
∆t = (0,046 + 0,145) [(1,38 − 1) / 5,8 − 0,011] =
= (0,191) [(0,38 − 1) / 5,8 − 0,011] = (0,191) [0,055] = 0,0104 = 1,04 %; •
; L2
∆t = (0,046 + 0,145) [(1,38 − 1) / 19 − 0,011] = = (0,191) [0,009] = 0,0017 = 0,17 %.
:
=
(10.8)
∆t
: .
[59]:
,
.
δ = L1 {[(0,046 + 3,74
) ∆t
−0,77 −1
, λ]
+ 0,011
.
∆t
,
-
,
(1,38 −
-
−0,38 −1 1/m ) } . λ
= 1 %,
.
δ = L1 / [(0,046 + 0,147)−1 ⋅ 0,01 + 0,011] (1,38 − 1)−1 =
= 0,4 / [(0,193)−1 ⋅ 0,01 + 0,011] (2,63) = = 0,4 / [0,063] (2,63) = 0,4 / 0,165 = 2,4
.
∆t
= 2 %,
.
δ = L1 / [(0,046 + 0,147)−1 ⋅ 0,02 + 0,011] (1,38 − 1)−1 =
= 0,4 / [(0,193)−1 ⋅ 0,02 + 0,011] (2,63) = = 0,4 / [0,115] (2,63) = 0,4 / 0,3 = 1,3
∆t .
∆t
(10.8) ,
.
λ,
, .
) ∆
∆r (
. ,
-
,
.
[78]:
-
ε1 − ( ( ,
), / . ,
(
),
2
= ε1 b,
,
)
,
b ≈ 0,025 / .
[59]: τ1 −
∆t
; a1 −
(10.9) , ; b − -
,
ε1
-
ε1 = τ1 (L1)2 / a1,
(10.10) -
/ . ( a1 = λ1 / (cp)1 = 5,3 ⋅ 10−6
2
/ .
)
[59]:
τ1 = 10
= 0,4;
λ
= 1;
∆r
(F
∆r
= 67;
U
λ
+D
W
τ1 = 10
(F
∆=
11,5;
= 4,5;
−0,5
+ 13,7 ∆
U
− 3,63);
+D
λ
W
+ 13,7
−0,5
− 3,63),
(10.11)
F, U, D, W
: F = 1,57
λ
−0,7
+ 0,34 = 1,57 + 0,34 = 1,91;
U = 1,76 − 0,26 lg
D = 6,28 − 10
W = − (4,1
−1,1
−0,4
λ=
1,76 − 0 = 1,76;
= 6,28 − 10 / 5,37 = 4,42;
+ 0,19) = − (4,1 / 102 + 0,19) = − 0,6. ,
•
τ1 = 10
∆r
(F
U
+D
λ
W
+ 13,7
−0,5
(10.11),
:
− 3,63) =
= 4 (1,91 ⋅ 4,51,76 + 4,42 ⋅ 1−0,6 + 13,7 ⋅ 67−0,5 − 3,63) = = 4 (26,96 + 4,42 + 1,67 − 3,63) = 4 ⋅ 29,4 = 118;
•
τ1 = 10
(F
∆
U
+D
λ
W
+ 13,7
−0,5
− 3,63) =
= 4 (1,91 ⋅ 11,51,76 + 4,7 ⋅ 1−0,6 + 13,7 ⋅ 67−0,5 − 3,63) = 4 ⋅ 143 = 572.
ε1 ,
•
(10.10),
:
ε1 = τ1 (L1)2 / a1 = 118 ⋅ (0,4)2 ⋅ 10−6 / 5,3 ⋅ 10−6 = 3,6 ; •
ε1 = τ1 (L1)2 / a1 = 572 ⋅ (0,4)2 ⋅ 10−6 / 5,3 ⋅ 10−6 = 17,3 . , •
,
(10.9)
∆t •
= ε1 b = 3,6 ⋅ 0,025 = 0,09 ;
:
-
∆t ; ∆r = 2,3
=8
; ∆ = 7,6
r2 = 2,5 ,
τ1, ε1, ∆t
; L2
:
∆r =
= 68;
•
= ε1 b = 17,3 ⋅ 0,025 = 0,43 .
∆r / L1 = 2,3 / 0,4 = 5,8;
∆=
,
∆ / L1 = 7,6 / 0,4 = 19. (10.9) – (10.11)
-
:
τ1 = 10
(F
∆r
U
λ
+D
W
−0,5
+ 13,7
− 3,63) =
= 4 (1,91 ⋅ 5,81,76 + 4,42 ⋅ 1−0,6 + 13,7 ⋅ 68−0,5 − 3,63) = 4 ⋅ 44,6 = 178,4;
ε1 = τ1 (L1)2 / a1 = 178,4 ⋅ (0,4)2 ⋅ 10−6 / 5,3 ⋅ 10−6 = 5,4 ; ∆t
•
τ1 = 10
= ε1 b = 5,4 ⋅ 0,025 = 0,13 ;
(F
U ∆ +
D
λ
W
+ 13,7
−0,5
− 3,63) =
= 4 (1,91 ⋅ 191,76 + 4,42 ⋅ 1−0,6 + 13,7 ⋅ 68−0,5 − 3,63) = 4 ⋅ 342,6 = 1370.
ε1 = τ1 (L1)2 / a1 = 1370 ⋅ (0,4)2 ⋅ 10−6 / 5,3 ⋅ 10−6 = 41,4 . ∆t
= ε1 b = 41,4 ⋅ 0,025 = 1,03 .
∆t = t . max − t . min = 60 − 20 = 40 ° , ∆t* ∆t = 1,03 / 40
, (
. 7.8
∆t* = 0,025 = 2,5 %.
7.9)
∆t
=
.
/
max
(10.9) – (10.11)
∆t
:
∆t*
( )
.
. .
,
[59]:
(%),
,
δ
= L1 [(0,1 τ1
−1
−D
λ
W
− 13,7
−0,5
,
,
ε1 τ1
= ε1
= ∆t
.
b ≈ 0,025 /
∆t
=
.
,
/ b = 1 / 0,025 = 40 .
a1 / (L1)2 = 40 ⋅ 5,3 ⋅ 10−6 / (0,4)2 ⋅ 10−6 = 1325. ,
(10.12),
-
+ 3,63) F−1]1/U. (10.12)
, =1
, =
-
δ
= L1 [(0,1 τ1
−1
−D
λ
W
− 13,7
−0,5
+ 3,63) F−1]1/U =
= 0,4 [(0,1 ⋅ 1325 ⋅ 0,4−1 − 4,42 ⋅ 1−0,215 − 13,7 ⋅ 67− 0,5 + 3,63) ⋅ 1,91−1]1/1,76 = = 0,4 [(331 − 4,42 − 1,67 + 3,63) ⋅ 0,52] 0,57 = 0,4 [(328) ⋅ 0,52] 0,57 = = 0,4 [170,6] 0,57 = 0,4 ⋅ 18,72 = 7,5 . (10.9) – (10.11) ∆t .
,
λ
∆r
,
τ1
.
τ1
[59].
10
. 10 10) 40 %, λ = 1,
,
, (
,
300 K∆ = 2, λ = K∆ = 10
1 %. (
)
. , .
,
.
∆t = [(∆t*
∆t −
)2 + (∆t )2 ± (∆t )2]0,5,
(10.13) К ,
,
∆t = 0,01 К = 0,01 ⋅ 0,5 = 0,005.
(10.13) ∆t
"+"
"−"
-
.
•
, (10.13):
-
∆t = [(∆t*
)2 + (∆t )2 + (∆t )2]0,5 = [(0,025)2 + (0,005)2 + (0,014)2]0,5 =
= [(0,000625) + (0,000025) + (0,000196)]0,5 = = [0,000846]0,5 = 0,029 = 2,9 %. •
∆t = [(∆t*
)2 + (∆t )2 − (∆t )2]0,5 = [(0,025)2 + (0,005)2 − (0,014)2]0,5 =
= [(0,000625) + (0,000025) − (0,000196)]0,5 = = [0,000454]0,5 = 0,021 = 2,1 %. , , ∆t ≤ 0,2 ∆t
-
, .
.
10.4. ɆȿɌɊɈɅɈȽɂɑȿɋɄɂȿ ɏȺɊȺɄɌȿɊɂɋɌɂɄɂ ɂ ɉɈȽɊȿɒɇɈɋɌɂ ɋɊȿȾɋɌȼ ɂɁɆȿɊȿɇɂə ɌȿɉɅɈɎɂɁɂɑȿɋɄɂɏ ɋȼɈɃɋɌȼ ɆȺɌȿɊɂȺɅɈȼ ( , [30]. 138
) :
,
,
,
;
-
3 [30].
; RS–485
, (
−
-
. 10.3).
И
А
-
3 138
Ɋɢɫ. 10.3. ɋɬɪɭɤɬɭɪɧɚɹ ɫɯɟɦɚ ɫɪɟɞɫɬɜ Ɉȼȿɇ ɞɥɹ ɢɡɦɟɪɟɧɢɹ ɌɎɋ ɦɚɬɟɪɢɚɥɨɜ 138
,
[30].
, 138
-
,
-
,
.
− .
-
138
−50…+ 750 ° ,
(L)
0,1 ° ,
-
0,25 %. , ,
,
, ,
-
. (
)
-
.
4…20
,
.
138
.
-
,
. 138
(
):
−
"
4…20
";
200
40 ;
50
300 . 138 ,
,
-
RS−485.
.
. RS−485
3 32
138,
101,
8,
1. ,
( , , ) SCADA (Supervisory, Control and Data Acquisition). SCADA – OWEN PROCESS MANAGER (OPM) – , ( ) , RS−485 OBEH AC 3 [30]. OPM . , , , . OPM : • ; • ; • OWEN REPORT VIEWER (ORV). . OPM COM, OBEH. COM. . COM. COM. COM. RS−485 RS−485 AC 3. 32 , – 256. Ɋɚɛɨɬɚ ɫ ɩɪɨɝɪɚɦɦɨɣ SCADA [30] .
OPM
, .
. ,
:
• • •
(
); ; (
/
/
). ,
,
-
, . ,
−
. "
− ,
,
"
.
. . ,
►
,
-
. .
-
,
,
. . . .
OPM
,
.
. 1–5 .
.
-
,
, . OWEN Report Viewer (ORV).
ORV
,
. ,
. 10.4 ,
, .
Ɋɢɫ. 10.4. ɉɪɨɫɦɨɬɪ ɮɚɣɥɚ ɧɚ ɗȼɆ ɩɪɢ ɨɩɪɟɞɟɥɟɧɢɢ ɬɟɦɩɟɪɚɬɭɪɨɩɪɨɜɨɞɧɨɫɬɢ ɤɪɚɫɧɨɝɨ ɤɢɪɩɢɱɚ ɜ ɝɪɚɮɢɱɟɫɤɨɦ ɜɢɞɟ
,
,
,
,
,
-
.
-
, .
,
,
-
Access, FoxPro, Dbase
Excel.
ɁȺɄɅɘɑȿɇɂȿ
-
. . , . ,
, -
, .
,
,
,
,
.
-
, .
. , (
).
,
,
,
,
, ,
.
− -
, (
,
,
,
.
, -
. -
),
, .
-
, -
ɋɉɂɋɈɄ ɅɂɌȿɊȺɌɍɊɕ
.
1. Ⱥɦɟɬɢɫɬɨɜ ȿ.ȼ., ɋɨɤɨɥɨɜ Ƚ.ə., ɉɥɚɬɭɧɨɜ ȿ.ɋ. . .: , 2000. 242 . 2. Ȼɟɥɨɜ ȿ.Ⱥ. // .1986. № 4. . 756 – 760. 3. Ȼɟɥɹɟɜ ɇ.Ɇ., Ɋɹɞɧɨ Ⱥ.Ⱥ. : 2 . .: , 1982. 671 c. 4. Ȼɨɝɨɫɥɨɜɫɤɢɣ ȼ.ɇ. . .: , 1979. 5. Ȼɨɝɭɫɥɚɜɫɤɢɣ Ʌ.Ⱦ. , : . .: , 1990. 6. Ȼɨɣɤɨɜ Ƚ.ɉ., ȼɢɞɢɧ ɘ.ȼ., Ɏɨɤɢɧ ȼ.Ɇ. . : - , 1992. 172 . 7. Ȼɨɣɤɨɜ Ƚ.ɉ., ȼɢɞɢɧ ɘ.ȼ., ɀɭɪɚɜɥɟɜ ȼ.ɇ. . , 2000. 272 . 8. Ȼɪɨɜɤɢɧ ȼ.Ʌ. // . . . 1980. № 11. . 120. 9. ȼɚɪɝɚɧɨɜ ɂ.ɋ., Ƚɟɪɚɳɟɧɤɨ Ɉ.Ⱥ. // . 1987. № 4. . 77–80. 10. ȼɨɥɶɤɟɧɲɬɟɣɧ B.C. . .: , 1971. 145 . 11. ȼɢɞɢɧ ɘ.ȼ. . , 1974. 144 . 12. ȼɢɞɢɧ ɘ.ȼ., ɂɜɚɧɨɜ ȼ.ȼ. , , 1965. 95 . 13. ȼɢɞɢɧ ɘ.ȼ., ȼɨɪɨɧɤɨɜ Ƚ.ȼ., Ʉɨɧɞɪɚɬɶɟɜ ȿ.Ⱥ. // . I969. № 4. . 795 – 798. 14. ȼɢɤ, ɗɡɢɤɲɢ. // : . 1981. № 1. . 149. 15. ȼɥɚɫɨɜ ȼ.ȼ. . .: , 1977. . 168. 16. Ƚɚɜɪɢɥɨɜ Ɋ.ɇ., ɇɢɤɢɮɨɪɨɜ ɇ.Ⱦ. // . 1983. № 6. . 1023 – 1024. 17. Ƚɨɪɞɨɜ Ⱥ.Ɇ., Ɇɚɥɤɨɜ ə.ȼ., ɗɪɝɚɪɬ ɇ.ɇ. . .: . 1976. 232 . 18. ȽɈɋɌ 23789–79. . .: , I980. 12 . 19. Ⱦɭɥɶɧɟɜ Ƚ.ɇ., ɋɢɝɚɥɨɜ Ⱥ.ȼ. // . 1980. . 39, № 5. . 859. 20. Ⱦɭɥɶɧɟɜ Ƚ.ɇ., Ʌɭɤɶɹɧɨɜ Ƚ.ɇ. , // . 1981. . 40, № 4. . 717. 21. ȿɥɢɫɟɟɜ B.ɇ., ɋɨɥɨɜɨɜ ȼ.Ⱥ. // . 1983. № 5. . 737 – 742. 22. Ɂɚɣɞɟɥɶ Ⱥ.ɇ. . .: , 1974. 108 . 23. ɂɫɚɱɟɧɤɨ ȼ.ɉ., Ɉɫɢɩɨɜɚ ȼ.Ⱥ., ɋɭɤɨɦɟɥ Ⱥ.ɋ. . .: , 1975. 485 . 24. Ʉɚɫɫɚɧɞɪɨɜɚ Ɉ.ɇ., Ʌɟɛɟɞɟɜ ȼ.ȼ. . .: , 1970. 109 . 25. ȿɩɢɮɚɧɨɜ Ƚ.ɂ. . .: , 1977. . 100. 26. Ʉɚɪɫɥɨɭ Ƚ., ȿɝɟɪ Ⱦ. . .: , 1964. 397 . 27. Ʉɨɡɥɨɜ ȼ.ɉ. / . . , . . , . . // . 1987. № 2. . 96 – 102. 28. Ʉɨɡɞɨɛɚ Ʌ.Ⱥ. . .: , 1975. 227 .
29. Ʉɨɧɞɪɚɬɶɟɜ Ƚ.Ɇ. . .: , 1954. 408 . 30. Ʉɨɧɬɪɨɥɶɧɨ-ɢɡɦɟɪɢɬɟɥɶɧɵɟ . , 2003 . 153 . 31. Ʉɭɥɚɤɨɜ Ɇ.ȼ., Ɇɚɤɚɪɨɜ Ȼ.ɂ. . .: 32. Ʉɭɪɟɩɢɧ ȼ.ȼ., ɉɟɬɪɨɜ Ƚ.ɋ., Ʉɚɪɩɨɜ ȼ.Ƚ. . // . 1981. . 3, № 1. . 29 – 31. 33. Ʉɭɪɟɩɢɧ ȼ.ȼ., Ʉɨɡɢɧ ȼ.Ɇ., Ʌɟɜɨɱɤɢɧ ɘ.ȼ. // . 1982. . 4, № 3. . 91. 34. Ʉɭɪɟɩɢɧ ȼ.ȼ., Ⱦɢɤɚɥɨɜ Ⱥ.ɂ.
.
-
//
40, № 6. . 1046. 35. Ʉɭɪɟɩɢɧ B.ȼ., Ʉɚɥɢɧɢɧ ȼ.Ⱥ. // . . 1979. № 2. . 24. 36. Ʉɭɪɟɩɢɧ ȼ.ȼ., ɉɥɚɬɭɧɨɜ ȿ.ɋ., Ȼɟɥɨɜ ȿ.Ⱥ. // 37. Ʉɪɟɣɬ Ɉ., Ȼɥɟɤ ɍ. . .: 38. Ʌɢ, Ɍɟɣɥɨɪ. : 39. Ʌɵɤɨɜ Ⱥ.ȼ. . .: 40. Ʌɵɤɨɜ Ⱥ.ȼ. : . .: 41. Ɇɟɬɨɞɵ , 1973. 336 . 42. Ɇɟɰɢɤ Ɇ.ɋ. . : . . - . 1981. 111 c. 43. Ɇɢɯɟɟɜ Ɇ.Ⱥ., Ɇɢɯɟɟɜɚ ɂ.Ɇ. 44. Ɉɫɢɩɨɜɚ ȼ.Ⱥ.
, 1969. 142 .
. 1981. .
.
-
. 1982. № 4. . 78. , 1983. 256 . . 1978. № 4. . 177 – 182. , 1967. 599 . , 1978. 479 . / . . .
//
-
.
.: -
.
.:
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