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# Расчет изменения температуры радиального ребра при лучистом теплоотводе с его поверхности.

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```…
.,
.
536.24
.,
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, .
,
.
.
.
,
.
,
-
.
,
.
.
.
:
,
,
,
,
-
.
, .
-
(1)–(3)
[1].
.
-
.
,
0
r
0;
(1)
R2
(3)
r,
2
, ; R1, R2 –
www.vestnik.magtu.ru
4
,
0
;
; T0 –
, ; ,
–
–
-
-
2 T03 R22
1; Sk
–
.
(2)
T=T(r) –
,
-
,
R1
R2
2
dT
dr
-
r
–
R2
,
:
d T 1 dT 2
T4
2
dr
r dr
T T0
r R1 ;
T –
T0
:
(1) – (3)
d2
d
2
d
d
1 d
d
2
1
0;
(4)
;
(5)
1.
(6)
0
0
4
Sk
(4)
.
(4)–(6)
—————————————————————————————————————————————
101
.
,
U
,
-
,
,
.
,
2
Sk
2
U
-
F )=0
2
2
ln
2
(12)
,
,
( )
.
0
-
(12)
,
-
(8)
.
(4),
,
U=U ),
[2, 3].
1
( )
3
2
3Sk
1
ln
2
2
0
(13)
.
2
0
( )
( )
U( )
( )
1
(1
3
d
4
1
3
),
(7)
F ),
(9),
-
,
,
-
,
( )
3
1
.
1 3U ( )
(8)
(7)
F( )
dU
4
d
2
,
(4)–(6)
.
2
1 dU
d
dU
d 2
4
U=0
dU
0
d
dU
d
3
3
,
2
Sk
0,
(9)
0,
(10)
=1.
(11)
( ) 1.
(9)
d 2U
d 2
1 dU
d
2
dU
4
d
Sk
0.
(14)
-
(14), (10)–(11)
F( )
4
3
dU
d
,
(9),
,
4
(4),
-
2
-
dU
d
W.
(15)
(15)
(11).
F ),
,
,
,
(11)
-
(9),
-
.
(14), (10)–(11)
dW 1
W 4W 2
d
W=0
=1.
Sk
0,
(16)
(17)
(9)–
-
U ),
(16),
[4],
-
( ),
-
(17)
.
102 ————————————————————————————————————
. . .
. 2015.
1
…
W
a K1 (a) I1 (a ) I1 (a ) K1 ( a )
,
4 K1 (a ) I 0 (a ) I1 ( a) K 0 (a )
a
.,
,
(18)
(21).
,
,
(21)
(23),
,
.
2 Sk .
,
(10),
.
-
(15), (18)
«
U )
K (a) I 0 (a ) I1 (a) K 0 (a )
U ( ) ln 4 1
.
K1 (a) I 0 (a 0 ) I1 (a) K 0 (a 0 )
»,
«
»(
In(x)
(19)
.
).
-
Kn(x)
(n=0; 1)
,
(19)
[5–7].
(8),
[5–7]:
max
3
(20)
1
.
K1 (a ) I 0 (a 0 ) I1 (a ) K 0 (a 0 )
1 3ln 4
K1 (a ) I 0 (a ) I1 (a ) K 0 (a )
1
x2
2
(3!) 4
(20)
max
(
( =1)
1
1)
3
1 3ln 4
K1 (a ) I 0 (a 0 ) I1 (a ) K 0 (a 0 )
K1 (a ) I 0 (a ) I1 (a ) K 0 (a )
,
(21)
.
x
2
K1 ( a) I 0 (a)
1 x
3!4! 2
n
n 0
(25)
1
x2
2
4
0 ( n !)
3
..
1 x
2!3! 2
4
n
K
n
;
5
(26)
x
1
n
n
!(
1)!
2
0
x2
1
2
4
1 ( n !)
1
1K
2n 1
;
ln
x
I 0 ( x );
2
(27)
1
x
ln ) I1 ( x)
2
x
K1 ( x) (
(21)
n
7
2
1
x2
(2!) 2 4
...
1 x
2! 2
n
-
1
,
a
I1 (a) K 0 (a)
I1 ( x)
K0 ( x)
,
[8]
1 x2
(1!) 2 4
I 0 ( x) 1
2n 1
1
x
n !( n 1)! 2
n 1
K 1
(28)
1
K
1
,
2n 2
,
max
(
1)
3
1
1 3ln a[ K1 (a ) I 0 (a 0 ) I1 ( a ) K 0 ( a
4
0
,
)]
.
(22)
K1(x)
3
3 K1 ( a ) I 0 ( a 0 ) I1 ( a ) K 0 ( a 0 )
ln
b
K1 ( a ) I 0 ( a ) I1 ( a ) K 0 ( a )
1
a
a
bSk ; b
4
3
1
I1 ( x ) K 0 ( x )
x
.
I 0 ( x)
K1 ( x)
1
(
(23)
,
,
,
(25)–(28)
1),
,
1)
4
I0 ( x)
n
3
1
3
ln
2
1
1
2
0
1
(24)
.
4
I1 ( x)
n
1
x2
2
4
0 ( n !)
.
n
(29)
,
1
x
0 n !(n 1)! 2
2n 1
.
(30)
2
0
x<2
,
www.vestnik.magtu.ru
-
,
(13)
1)
x
.
(25), (26)
b
(
(
:
-
(13),
min
=0,57721566.
(23)
(29)
(30)
—————————————————————————————————————————————
.
103
.
=0,75
Sk=1,0; 1,5; 2,0
-
Sk=1
(13)
(23)
(22)
0,900
0,912
0,917
0,878
0,890
0,895
Sk=1,5
0,862
0,882
0,893
0,835
0,854
0,864
.
=1
0=0,5.
1.
Sk=2,0
0,830
0,856
0,874
2.
0,800
0,822
0,840
3.
:
.,
.
, 1977. 461 .
.
.
:
.
.
4.
5.
1.
.
2.
170 .
6.
7.
«
»
.
-
.
. .:
. .:
.
:
.,
. .:
8.
-
.,
.
-
, 1974. 145 .
.
.
.
, 1992. 96 .
, 1976. 576 .
-
.
, 2007.
.
, 1979. 830 .
.,
.:
-
.
, 1962. 450 .
.
,
. .:
, 1963. 1100 .
,
-
INFORMATION ABOUT THE PAPER IN ENGLISH
CALCULATION OF TEMPERATURE CHANGES OF A RADIAL RIB
IN CASE OF RADIANT HEAT REMOVING FROM ITS SURFACE
Vidin Yury Vladimirovich – Ph.D. (Eng.), Professor, Siberian Federal University, Russia.
Kazakov Roman Vladimirovich – Ph.D. (Eng.), Associate Professor, Siberian Federal University, Russia. E-mail:
roman.kazakov@list.ru
Abstract: An approximate analytical calculation of
temperature changes in the radial rib in case of radiant heat
transfer on its surface is prepared. The proposed method is
based on higher and lower estimate of the required temperature field. The authors use an integral linearizing transformation which contributes to a significant decrease in the
effect of a linear component in the original differential
equation. It is rather easy to evaluate maximum and minimum temperature along the rib radius using the resulted
analytical expressions. The calculations use modified Bessel functions, whose numeric values are tabulated in detail.
According to the calculations, a difference in limit temperature values is minor even at increased values of the Stark
radiation number. A temperature arithmetic mean of limit
temperatures is rather close to a real one.
Keywords: temperature field, radiant heat transfer,
radial ribbed surface, analytical method, modified Bessel
functions.
1.
References
Kern D., Kraus A. Razvitye poverkhnosti teploobmena [Developed
heat exchange surfaces]. Moscow: Energiya, 1977, 461 p.
2.
3.
4.
5.
6.
7.
8.
Vidin Yu.V. Inzhenernye metody rascheta protsessov teploobmena [Engineering methods of calculating heat transfer processes]. Krasnoyarsk: KrPI, 1974, 145 p.
Vidin Yu V. Inzhenernye metody teploprovodnosti [Engineering
methods of thermal conductivity]. Krasnoyarsk: Krasnoyarsk State
University, 1992, 96 p.
Kamke E. Spravochnik po obyknovennym differentsialnym
uravneniyam [Handbook of ordinary differential equations]. Moscow: Nauka, 1976, 576 p.
Vidin Yu.V. Krankiy spravochnik po teplomassoobmenu [Quick
reference guide for heat and mass transfer]. Krasnoyarsk: Siberian Federal University, 2007, 170 p.
Abramovits M., Stigan I. Spravochnik po spetsialnym funktsiyam
[Handbook of special functions]. Moscow: Nauka, 1979, 830 p.
Segal B.I., Semendyaev K.A. Pyatiznachnye matematicheskie tablitsy [Five-figure mathematical tables]. Moscow: Gosudarstvennoe izdatelstvo fiziko-matematicheskoy literatury,
1962, 450 p.
Gradshtein I.S., Ryzhik I.M. Tablitsy integralov, summ, ryadov i
proizvedeniy [Tables of integrals, sums, series and products].
Moscow: Gosudarstvennoe izdatelstvo fiziko-matematicheskoy literatury, 1963, 1100 p.
104 ————————————————————————————————————
. . .
. 2015.
1
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температура, теплоотводе, радиального, расчет, ребра, лучистое, поверхности, изменения
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