Теоретическиеосновыметаллургическойтеплотехники:Учебное пособие для вузов. 000-000-000-0

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34.32 24 669.04:533:662.9(07)

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№ 14/18.2-86

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ISBN 000-000-000-0

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– .

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12) 13)

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-

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( . , ,

-

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,

dV ,

. -

,

,

. ,

. .

, (

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) .

10

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) (

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,

-

:

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; 2)

; 3)

–

.

,

-

.

, .

11

-

1. 1.

.

1.1. ,

(

),

[ /

2

=

]. , T [ ].

,

-

P⋅V = M⋅R⋅T, 3

(1.1)

, V; R = Rμ / μ [ /( ⋅ )] – /( ⋅ )– 1 ; Rμ = 8314 ;μ[ / ]– 1 .

V [ ]– ;M[ ]– ,

-

. = 273

P

– P . = 10333

: 101325 = 760 . , : 1 = 98000 = 735,6 =Δ =

-

(

.

. = 10333

/ 2. –

.

/ 2.

. = 10000 ,

. "

, (

=0° .

.

"( . ,

=

)

"

"(

).

< 0, ,

.

. 1.1 ), P ,

)

=

, (

. .

), h

h

, , : 12

h

= ρ⋅g⋅h.

P

h

.

-

. (

. 1.1 ),

-

. / 3

/

2

=

⋅ /

3

= -

. 1.1.

:

;

-

ρ [ / 3]. ,

, 1

3

.

, , /ρ [ ⋅ / ]

. 1

. "z"

z

= ρ⋅g⋅z. ,

-

,

,

-

= ρ⋅g⋅z.

(1.2) ,

,

-

.

0

. z, z 0

=

0

- ρ ⋅g⋅z,

– 13

;ρ –

(1.3) .

,

(1.2)

(1.3) 1

+

=

1

+

2

2

=

0,

(1.4)

. .

-

. ,

(1.4)

1

+ ρ ⋅g⋅z1 =

2

+ ρ ⋅g⋅z2 = const.

, 2

+ ρ ⋅g⋅z2. = =

, , . .

1

z1 = z2,

1

, ,

2 1

(1.5)

>

+ ρ ⋅g⋅z1 > . -

2

,

,

.

"

-

-

-

":

.

1.2.

,

r W

r r , . . W = ∂L / ∂τ ,

r L –

. [1], v

F

=

v

V [ 3] τ.

dV

dτ

, ⎡ 3⎤ ⎢ ⎥, ⎣ c ⎦

(1.6)

, T

-

F

" " T.

V

v -

.

r W

-

v

F

, .

:

14

-

W=

[ ], m=

d2V dv = [ / ]. dF ⋅ dτ dF

(1.7)

dM dV ⋅ ρ = = v ⋅ ρ [ / ]. dτ dτ r qm , F .

m

τ

m F

(1.8)

, -

: qm =

d2M = W ⋅ ρ [ /( 2⋅ )]. dF ⋅ dτ

1.3.

(1.9)

. ,

, :

=

.

ρ ⋅ W2 . 2

(1.10) ,

.

U-

, 1

(

3

V ,

ρ.

. 1.2). W2/2.

1

L ( P = 760 t =0° .

-

)V , , .

. . 1.2.

U–

v0 15

-

t= 0°

,

W0, . . P = 760 . . m [ / ].

-

. v0

v0 = m / ρ0,

ρ0 –

(1.11) .

V = V0 ⋅

T P0 ⋅ T0 P

: v = v0 ⋅

T P0 ⋅ . T0 P

(1.12) (

W W = W0 ⋅

) W0

T P0 ⋅ . T0 P

(1.13)

(1.13) , P = 3-4

-

. v ,

W

(1.13) W = W0 ⋅

α = 1 / 273 K–1 –

T = W0 ⋅ (1 + α ⋅ t ) , T0

(1.14) .

.

-

,

P

=

W02 ⋅ ρ0 P0 T ⋅ ⋅ . 2 P T0

(1.15) -

, P

=

W02 ⋅ ρ0 T W02 ⋅ ρ0 ⋅ = ⋅ (1 + α ⋅ t ) . 2 T0 2

16

(1.16)

.

-

. W=

v [ / ], F

(1.17)

v– t.

.

,

),

(

F1 ⋅ ρ1 ⋅ W1 = F2 ⋅ ρ 2 ⋅ W2 = const = m .

(1.18)

T ≈ const

,

P ≈ const,

F1 ⋅ W1 = F2 ⋅ W2 = const = v . .

(1.19)

, t

1.4.

-

. .

v0

(

-

)

, . . ,

, . , . .

. P

1

+P

1

+

1

>P

2

17

+P

2

+P

2,

1–1 ΔP

2–2 ΔP

.

,

.

. " +

P

1

12

–

ΔP

ρ ⋅ W12 + ρ ⋅ g ⋅ z1 = P 2

2

ρ ⋅ W22 + ρ ⋅ g ⋅ z 2 + ΔP 2

+

,

3

1

12

"

(ρ1 = ρ2)

,

(1.20)

.

(1.20) ΔP

.

12

= 0,

(

2

> P 1,

2


2300

.

, . (

,

) . . -

.

. , , dW /dy=const.

,

, ,

, . 26

,

-

,

, ,

-

, τ =η ⋅ η [

⋅ ]–

,

η

.

dW [ dy

(2.5)

η

η

. η

. [8] .

(

],

)

η

,

τ = (η + η ) ⋅

η = 0.

η,

η

-

η

dWx [ dy

.

].

(2.6)

2.4.

.

, . , . "

-

,

"

.

,

, ( (

),

. 2.7).

,

, , . .

, 1 %.

( =δ)

δ

27

, . . Wδ

-

δ (

,

-

, . 2.7).

-

-

δ ,

, (

. 2.4). :

. 2.7.

, :

13-

; 2; 4-

-

,

;

-

. -

;5-

. . , (

dF . 2.8).

,

,

, .

, , .

,

L ,

=L

– -L ,

L

L –

,–

( ,

– -

. 2.8). – -

, –

.

28

δ max < R δ =R

( . 2.9). L=L , . . .

L= L

. 2.8. L= L .

= -

L L

-L

. (

)

-

, . W0⋅x/ν > 5⋅105

. 2.9.

Re = -

.

, (

)

, .

,

-

, .

,

, . 2.5.

–

-

. . 29

2.5.1.

: . ( [6]). f1, f2 f3 z0, τ); z = f3( 0, 0, z0, τ).

= f1( 0, 0, z0, τ); τ = 0: 0,

.

, . . = f2( 0, z0 0

f5

(

)

⎡ ∂ (ρ ⋅ Wx ) ∂ ρ ⋅ Wy ∂ (ρ ⋅ Wz ) ⎤ ∂ρ = −⎢ + + ⎥ ∂τ ∂y ∂z ⎦ ⎣ ∂x

(2.8)

Dρ ∂ρ ∂Wx ∂Wy ∂Wz = + + + ∂x ∂y ∂z dτ ∂τ

(2.9) .

∂ρ / ∂τ = −div (ρW )

(2.8)

(2.10)

Dρ / dτ = −ρ ⋅ div (W ) ,

div (ρW ) =

-

(2.7)

∂Wy ∂Wz ⎤ ⎡ ∂W Dρ = −ρ ⋅ ⎢ x + + ⎥, dτ ∂y ∂z ⎦ ⎣ ∂x

(2.7)

0,

Wx = f4( , , z, τ); Wy = f5( , , z, τ); Wz = , . -

, f3, f4 f6( , , z, τ).

-

,

(

(2.11)

)

∂ (ρ ⋅ Wx ) ∂ ρ ⋅ Wy ∂ (ρ ⋅ Wz ) + + ∂y ∂x ∂z

(2.12) ,

30

divW =

∂Wx ∂Wy ∂Wz + + ∂y ∂x ∂z

(2.13)

. (2.10)

(2.7)

-

:

. dV ( (

. dV dτ -

, 3

1

),

1

) . "

"

-

. , . . ,

"

z

.

,

"

,

[9].

(2.8)

,

, (

) [10]. :

(

) (

)

(2.7)

.

(2.10), (2.8)

(2.11).

ρ = const

.

∂Wx ∂Wy ∂Wz + =0. + ∂x ∂z ∂y

(2.14)

(2.14) :

.

, "

"

"

. (Wy =Wz = 0) 31

(2.10)

"

∂ (ρ ⋅ Wx ) ∂ρ =− . ∂x ∂τ

(2.15)

∂ (ρ ⋅ Wx ) =0. ∂x

(2.15) (2.16)

Э

2.5.2.

, -

M– M⋅Wx –

d(M⋅Wx) = GΣx⋅dτ [ ⋅ ],

(2.17)

; Wx –

(

; GΣx – M; GΣxxdτ –

); ,

. , . .

, .

-

, :

I = m⋅W = ρ⋅F ⋅W⋅W [ ].

(2.18) -

(2.17) m = dM/dτ m⋅(Wx2 - Wx1) = GΣx,

(2.19)

. .

, ,

-

,

, , , .

, .

GΣx, ,

,

(2.19)

. .

32

-

2.5.3.

( (

-

)

)

(

-

) .

(

,

)

-

,

:

⎡ ∂ (ρWx Wx ) ∂ ρWy Wx ∂ (ρWz Wx ) ⎤ ∂ (ρWx ) ∂P = −⎢ + + η Δ2 Wx , + ⎥ + ρg x − ∂τ ∂ ∂ ∂ x y z ∂ x ⎦ ⎣

(

∂ ρWy ∂τ

) = −⎡ ∂(ρW W ) + ∂(ρW W ) + ∂(ρW W )⎤ + ρg ⎢ ⎣

∂x x

y

∂y y

(

∂z

y

z

)

y

⎥ ⎦

y

−

∂P + η Δ2 Wy , ∂y

⎡ ∂ (ρWx Wz ) ∂ ρWy Wz ∂ (ρWz Wz ) ⎤ ∂ (ρWz ) ∂P + = −⎢ + + η Δ2 Wz , ⎥ + ρg z − ∂ ∂ z ∂τ ∂ x y z ∂ ⎣ ⎦

Δ2 Wx = Δ2 Wy =

Δ2 Wz =

∂ 2 Wx ∂ 2 Wx ∂ 2 Wx , + + ∂z 2 ∂y 2 ∂x 2 ∂ 2 Wy ∂x 2

+

∂ 2 Wy ∂y 2

+

∂ 2 Wy ∂z 2

(2.20)

(2.21)

(2.22)

(2.23)

,

(2.24)

∂ 2 Wz ∂ 2 Wz ∂ 2 Wz + + . ∂x 2 ∂y 2 ∂z 2

(2.25)

(2.20-2.22) 0L,

-

: " ."

"

-

" .

dV = dxxdyxdz 1

1

3

dτ

. 33

(2.20-2.22), ,

:

ρ⋅

∂P DWx = ρ ⋅ gx − + η ⋅ Δ2 Wx , ∂x dτ

ρ⋅

DWy

∂P + η ⋅ Δ2 Wy , ∂y

(2.27)

ρ⋅

DWz ∂P = ρ ⋅ gz − + η ⋅ Δ2 Wz , dτ ∂z

(2.28)

dτ

= ρ ⋅ gy −

(2.26)

DWx ∂Wx ∂Wx ∂Wx ∂Wx , + Wz ⋅ + Wy ⋅ + Wx ⋅ ⋅ = dτ ∂z ∂y ∂x ∂τ

DWy dτ

=

∂Wy ∂τ

+ Wx ⋅ ⋅

∂Wy ∂x

+ Wy ⋅

∂Wy ∂y

+ Wz ⋅

∂Wy ∂z

(2.29)

,

(2.30)

DWz ∂Wz ∂Wz ∂Wz ∂Wz + Wz ⋅ + Wy ⋅ + Wx ⋅ ⋅ = dτ ∂z ∂y ∂x ∂τ

(

)

(2.31)

. (2.26-2.28) ( –

.

-

2.5.1) :

-

0L, 0L. .

-

, -

(2.26-2.28) ,

(

, )

( ,

-

,

. .).

34

. . , -

.

-

. 4

.

-

, ∂Wx ∂Wy ∂Wz + + =0. ∂y ∂x ∂z

(2.32)

(2.26-2.28)

(2.32)

-

. – ( )

. ( ,

123-

20

-

), A + B⋅y + C⋅y' = 0:

( = 0): A + B⋅y = 0; ( = 0): A + C⋅y' = 0; : A + B⋅y + C⋅y' = 0. '–

, ;

. " W = 0) –

, ,

–

,

-

,

, "

( 12-

: . ,

, ,

∂W/∂ = 0,

. ,

. ,

.

,

,

,

.

,

.

, ) dW τ = η⋅ , dL

( . .

35

,

-

2.6.

(

.

. ,

)

,

-

Wz = 0,

: -

. : Wx = f1( , ), Wy = f2( , ). ( ) m = ρ⋅W0⋅δx⋅z, δx.

, ,

: W = f(r). ∂P/∂y

. ,

. . -

: ∂P/∂x=0.

Wx ⋅

Wx ⋅

∂Wy ∂x

+ Wy ⋅

⎛ ∂ 2 Wx ∂ 2 Wx ⋅ ⎜⎜ + 2 ∂y 2 ⎝ ∂x

⎞ ⎟, ⎟ ⎠

(2.33)

⎛ ∂ 2 Wy ∂ 2 Wy ⎞ ⎟, = ν ⋅⎜ + ⎜ ∂x 2 ∂y ∂y 2 ⎟⎠ ⎝

(2.34)

∂Wx ∂Wx =ν + Wy ⋅ ∂y ∂x

∂Wy

∂Wx ∂Wy + =0. ∂x ∂y

(2.33)

(2.35)

(2.34)

. (2.34)

,

. .

, -

(2.33) . . ,

, : ∂Wx/∂y >> ∂Wx/∂x.

36

∂P/∂x = ∂P/∂y = 0

-

-

:

Wx ⋅

∂ 2 Wx ∂Wx ∂Wx , =η ⋅ + Wy ⋅ ∂y ∂x ∂y 2

(2.36)

∂Wx ∂Wy + =0. ∂x ∂y

(2.37)

(2.36-2.37)

.

,

.

-

. (2.36-2.37)

[9, 11]. .

,

-

. :

δ x = 5 / Re 0x,5 ,

Rex = W0⋅x/ν –

(2.38)

; τ = 0,6642 ⋅

:

1 Wo2 ⋅ ρ ⋅ . 2 Re0x,5

(2.39)

δ W.

(

)

W

W=

2 I 1 = ⋅ ∫ 2 ⋅ π ⋅ r ⋅ ρ ⋅ Wr2 dr = W 3 m ρ⋅W⋅F 0 R

. . 2

. .

W = K2⋅W , = 2/3 = 0,667.

⎛ ⎞ E = m ⋅ ⎜⎜ W ⎟⎟ / 2 [ ⎝ ⎠ 2

],

(2.79)

W –

( (

).

)

49

W > W > W.

W

. R ⎛ 2 2 W2 ⎞ ⋅E = ⋅ ∫ ⎜⎜ 2 ⋅ π ⋅ r ⋅ ρ ⋅ Wr ⋅ r ⎟⎟dr = m 2 ⎠ ρ⋅W⋅F 0 ⎝

W=

.

:

3

W = K3⋅W ,

W

W

1 ⋅W 2

(2.80)

=

12 =

0,707. P

1

,

3

E m⋅W ρ⋅W ρ ⋅ (K 3 ⋅ W = = = = v 2⋅v 2 2 2

P

2

W.

=

P

)2

[

/ 3].

=

(2.81)

W,

W

K 32 ⋅ W ⋅ ρ . K12 ⋅ 2 2

(2.82) α.

K 32 / K12

α =

-

0,5 =2, . . 0,52 2

2

, W.

( )

, I = m ⋅ W = m ⋅ K2 ⋅ W 2/

,

= m ⋅ K 2 W / K1 = F ⋅ ρ ⋅ W ⋅ K 2 / K1 [ ]. 2

1

W.

50

α,

(2.83) -

Э

2.12.2.

(1.20)

(2.19)

,

-

. ,

(1.20)

(2.19)

.

-

, ,

–

( . .

)

.

(

)

:

m ⋅ α 2 ⋅ W2 − α 1 ⋅ W1 = G Σ [ ],

P1 + α 1 ⋅

(2.84)

W12 ⋅ ρ W2 ⋅ ρ + ρgz1 = P2 + α 2 ⋅ 2 + ρgz 2 + ΔP 2 2

12

[

].

(2.85)

2.12.3.

2.10, . τ

(2.73),

: ,

-

. , τ =τ

. .

,

,

. –

-

[9]. . , "

"

-

. R = 4⋅103 - 3⋅106

. W ⋅y Wx + 5,5 , = 2,5 ⋅ ln W ν

= R - r, R –

,r–

(2.86) .

51

. 2.13

"

" -

. ,

, .

.

. .

-

, τ =τ ⋅

y . R

(2.87) .

. . ,

.

, -

, W.

. , . -

. ,

. . . -

. 2.13. .

Re

.

, , ,

,

,

.

-

. , .

52

-

–

:"

.

[15] ,

,

".

2.13. 2.13.1.

-

L12 :

ΔP

12

=η ⋅

32 ⋅ L12 ⋅W , D2

(2.88)

. . .

W

,

-

. , ΔP = μ ⋅ μ –

L W2 ⋅ ρ , ⋅ D 2

(2.89)

,

. (2.88)

μ

=

-

(2.89)

64 . Re

(2.90)

Wy = 0,

(2.72)

R

,

–

. (2.89)

D

F [ 2] – "

"

4⋅F

=

;

,

[ ]–

53

-

(2.91) .

2.13.2.

, . Re = 106 Δ–

ε = Δ/R,

,

-

. 2.14 ε ,

.

. μ .

"

-

" . μ

μ

,

-

.

. 2.14. .

Δ/R

,

), ,

( . .

. 2.15 μ = f(Re, Δ/R) 0,0667. μ = f(Re)

"

"

, ).

ε = Δ/R (Δ/R < 0,00197) . μ ( -

μ

Re1,

. .

-

Re2,

Re Δ

. . δ, ,

Δ/R,

μ ,

. ,

(

) .

54

.

Re Δ/R

. 2.15.

2.14.

( W1

W2)

, .

-

-

, . . (

. 2.16).

. 2.16. -

: ;

-

; -

55

900

2-2 Δ

,

12.

α = 180° : ΔP

(

1-1 c 2,

>Δ

ΔP

12

" "

2 = ρ ⋅ W1 / 2 − ρ ⋅ W22 / 2 1

12.

.

(W − W ) ⋅ ρ .

=

)

-

2

1

2

(2.92)

2

. ϕ

ϕ = 6-8°

.

, ϕ

. : ΔP

.

=K

⋅

= (ϕ, D2 /D1)

-

W1 ⋅ ρ . 2 2

⎡ ⎛D ⎞ = ⎢1 − ⎜⎜ 1 ⎟⎟ ⎢ ⎝ D2 ⎠ ⎣ 1.

ϕ = 180°

(2.93)

2

⎤ ⎥ . ⎥ ⎦

.

2

D2 >> D1,

-

( ), ( . 2.16 , 2.16 )

. Wx ΔP Wx (2.93),

=

(W

W2:

)

− W22 ⋅ ρ . 2

2 x

(2.94)

, ,

.

56

. -

. .

-

:

⎡

4 ⎞ ⎤ ⎛ ϕ ⎥ ⋅ ⎜1 − cos ⎞⎟ . 2⎠ ⎝ D 2 ⎠ ⎥⎦ ⎝

(ϕ, D2 / D1 ) = 0,5 ⋅ ⎢1 − ⎜⎜ D1 ⎟⎟ ⎛

⎢ ⎣

K

ϕ = 360°.

K

:Δ

=

.

1 ⎛ R⎞ , ⎜ ϕ , ⎟ = (1 − cos ϕ ) R ⎝ D⎠ 1+ 3 D

R–

(2.95)

. . (2.96)

. ,

(2.95-2.96) . ϕ = 90° (R = 0) W2 ⋅ ρ , . . =α 2

P

=α.

,

. ,

(

.

. 2.17 - ).

-

, – .

,

. 2.17.

,

: .

-

1-

[16]. 57

; 2-

2.15. Э

-

( )

(

,

)

. ,

. -

, , ),

, ( (

,

).

. (

)

,

-

, . ,

,

-

,

,

.

, ,

.

,

,

.

2.15.1.

-

. [5]

X = x / L0; Y = y / L0; Z = z / L0, L0 –

(

(2.97)

,

,

,

), W

/

=

Wy Wx W ; Wy / = ; Wz / = z , W0 W0 W0

W0 –

(2.98)

(

,

,

-

). Wx (2.26, 2.29) ,

gx = 0 (2.26, 2.29)

W0 L0 (∂Wx/∂τ = 0) :

58

W02 L0

⎡ ∂Wx / ⎤ ∂Wx / ∂Wx / + Wz / ⋅ + Wy / ⋅ ⋅ ⎢ Wx / ⋅ ⎥= ∂Z ⎦ ∂Y ∂X ⎣

=−

1 ∂P ν ⋅ W0 + 2 ⋅ ρ ⋅ L 0 ∂X L0

⎛ ∂ 2 Wx / ∂ 2 Wx / ∂ 2 Wx / + + ⋅ ⎜⎜ 2 2 ∂Z 2 ∂Y ⎝ ∂X

⎛W ⎞ ∂⎜⎜ x ⋅ W0 ⎟⎟ 2 / W ∂Wx W ⎠ = W0 ⋅ W / ⋅ ∂Wx = W0 ⋅ x ⋅ ⎝ 0 Wx ⋅ x ∂X ∂x L0 W0 ⎛ x ⎞ ∂⎜⎜ ⋅ L0 ⎟⎟ ⎝ L0 ⎠

⎞ ⎟ ⎟ ⎠

(2.99)

. .

⎡ ⎛ Wx ⎞⎤ W0 ⎟⎟ ⎥ ⎢ ∂ ⎜⎜ 2 / ∂ W ∂ ⎛ ∂Wx ⎞ ∂ ⎠ ⎥ = W0 ⋅ ∂ Wx ⎢ ⎝ W0 = ⎜ ⎟= 2 2 2 ∂ x ⎝ ∂x ⎠ ⎞ ⎥ L0 ⎞⎢ ⎛ x ⎛ x ∂x ∂X L 0 ⎟⎟ ⎢ ∂ ⎜⎜ L 0 ⎟⎟ ⎥ ∂ ⎜⎜ ⎠ ⎥⎦ ⎠ ⎢⎣ ⎝ L 0 ⎝ L0 2

(2.99)

Wx / ⋅ =−

Eu =

(2.101)

W02 / L0,

⎛ ∂ 2 Wx / ∂ 2 Wx / ∂ 2 Wx / + + ⋅ ⎜⎜ 2 2 ∂Z 2 ∂Y ⎝ ∂X

∂2W / ∂2W / ∂Eu 1 ⎛ ∂ 2 W / + + + ⋅ ⎜⎜ ∂X Re ⎝ ∂X 2 ∂Z 2 ∂Y 2

; Re =

P – ρ ⋅ W02

Wx / ⋅

. .

∂Wx / ∂Wx / ∂Wx / = + Wz / ⋅ + Wy / ⋅ ∂Z ∂Y ∂X

1 ν ∂P + ⋅ 2 X L ∂ ρ ⋅ W0 0 ⋅ W0 =−

(2.100)

∂Wy / ∂X

+ Wy / ⋅

W0 ⋅ L 0 – ν

∂Wy / ∂Y

59

+ Wz / ⋅

⎞ ⎟= ⎟ ⎠

⎞ ⎟, ⎟ ⎠

(2.102)

. ∂Wy / ∂Z

Wy =

gy = 0

=−

2 / ∂ 2 Wy / ∂ 2 Wy / ∂Eu 1 ⎛⎜ ∂ Wy + ⋅ + + ∂Y Re ⎜⎝ ∂X 2 ∂Y 2 ∂Z 2

,

⎞ ⎟. ⎟ ⎠

(2.103)

z

, gz = g:

Wz Wx / ⋅

∂ 2 Wz / ∂ 2 Wz / 1 ∂Eu 1 ⎛ ∂ 2 Wz / + + + ⋅ ⎜⎜ − 2 2 Fr ∂Z Re ⎝ ∂X ∂Z 2 ∂Y

= Fr =

∂Wz / ∂Wz / ∂Wz / = + Wz / ⋅ + Wy / ⋅ ∂Z ∂Y ∂X

W02 – g ⋅ L0

⎞ ⎟, ⎟ ⎠

(2.104)

. (2.102) - (2.104) : 1)

–

-

, ,

: X, Y, Z, Re, Fr; 2)

–

, Wx / , Wy / , Wz / , Eu.

. ,

Re, Eu (

-

Fr

). (2.102) - (2.104)

-

:

Wx / = fx(X, Y, Z, Re, Fr);

(2.105)

Wy / = fy(X, Y, Z, Re, Fr);

(2.106)

Wz / = fz(X, Y, Z, Re, Fr);

(2.107)

Eu = fEu(X, Y, Z, Re, Fr).

(2.108)

,

(2.26) - (2.31) Wx, Wy Wz : W = f( , , z, ρ, ν, g, W0, L0). (2.102) - (2.104) , , 60

8

-

∂Wy / ∂Wz / ∂W / = 0, + + ∂Z ∂Y ∂X

(2.109) (

. . 35)

. (

)

.

-

W02 L 0 , . .

1

). Gp

G,

G (

G . G

-

G: Fr =

GP

G: Eu =

G W02 W2 L . = 0 0 ~ G L0 ⋅ g g

(2.110)

(P / ρ) / L0 ~ G . P = 2 G W02 L0 ρ ⋅ W0

(2.111) G

G : Re =

G W0 ⋅ L0 W02 / L0 . ~ = ν ν ⋅ ( W0 / L 0 ) / L 0 G

(2.112)

, ,

( )

,

.

, "

-

". . 1.

, . . . ,

,

, .

61

-

2. . 3. . ,

, -

Wx / , Wy / , Wz /

= 1 / L01 = 2 / L02; Y = y1 / L01 = y2 / L02; Z = Re = W01⋅L01/ν1 = W02⋅L02/ν2 :

z1 / L01 = z2 / L02 Wx / =

Eu

Wy1 Wy 2 Wx1 Wx 2 W W = ; Wz / = z1 = z 2 , ; Wy / = = W01 W02 W01 W02 W01 W02

(2.113) . -

-

. .

:

. – .

–

μ

Re . (

.

-

5)

. 2.15.2.

[1]

: 1.

, ,

. .

,

-

. . 2.

, . .

-

. .

-

. , . ,

. 62

,

,

(

)

. , .

,

.

,

, . -

(Re

= Re )

, ν

(Fr ,

,

. -

10

, (W0

)

10

> Fr ). =ν

-

103

, .

. ,

, .

. Re .

-

. < Re ;

63

3. 3.1.

,

. . . (

, -

< 0,1⋅

), . . 3.1. (

. )

.

Fmin -

. , , . . . ,

,

-

, .

. 3.1.

:

)

; ) ; )

P

1

+

W12 ⋅ ρ1 + z1 ⋅ g ⋅ ρ1 = P 2

; )

1-1

2

+

2-2

W22 ⋅ ρ 2 + z 2 ⋅ g ⋅ ρ 2 + ΔP 2 64

; ) -

12

,

(3.1)

1

= 0, ρ1 = ρ2 = ρ, z1 = z2, Δ W2 = W

2 ⋅ (P

=

−P ρ

1

).

= 0,

12 2

(3.2)

, , . ,

2

(

). , ( " "),

ΔP

12

=

⋅

⋅ ρ2 , 2

W22

W2 = W

ϕ–

( W2 < W

), -

.

(3.1)

2 ⋅ (P

= ϕ⋅

ϕ = 1 / (1 +

−P ρ

1

2

),

(3.3)

).

(3.4) ,

( (

" "). " ")

.

v

= W ⋅F ,

F ε,

.

F

v

= ϕ⋅ε⋅F

2 ⋅ (P

65

−P ρ

1

/F

2

).

-

(3.5)

ϕ⋅ε

μ. . 3.1. 3.1 ϕ 0,98 0,82

,

ε 0,63 1

μ 0,62 0,82 , ,

(3.3)

(3.5) ( )

.

P =P

-

- ρ ⋅g⋅H,

–

(3.6) ,

-

. P =P

- ρ ⋅g⋅H.

(3.7)

P1 - P2 = P - P = g⋅H⋅(ρ - ρ )

v

2 ⋅ g ⋅ H ⋅ (ρ − ρ ρ

= ϕ⋅

W

= μ⋅F

⋅

(3.8)

),

2 ⋅ g ⋅ H ⋅ (ρ − ρ ρ

(3.9)

).

(3.10)

3.2.

,

,

.

, , . . dq 66

= 0.

, (H = q = 0), [Δs = (Δq + Δq ) / T = 0]

.

, -

/ ρ = const, =

p

/

v

–

;

p,

v

(3.11)

/( ⋅ )] – 3

[

.

, ⎛ P ρ = ρ ⋅ ⎜⎜ ⎝P

,ρ –

⎞ ⎟⎟ ⎠

1

K

,

(3.12) .

Δq

.

(1.22), + Δq = 0,

.

L

= ⋅dV

-

,

. (3.12) -

: W=

=

(1.22)

⎡ 2⋅ K P ⎢ ⎛ P ⋅ ⋅ 1 − ⎜⎜ K −1 ρ ⎢ ⎝ P ⎣

⎞ ⎟⎟ ⎠

K −1

K⎤

–

⎥, ⎥ ⎦

(3.13)

.

⎡⎛ ⎞ 2 K ⎛ P P 2⋅K ⋅ P ⋅ ρ ⋅ ⎢⎜⎜ ⎟⎟ − ⎜⎜ m = F ⋅ρ ⋅ W = F ⋅ ⎢⎝ P ⎠ P K −1 ⎝ ⎣

⎞ ⎟⎟ ⎠

K +1

K⎤

⎥. ⎥ ⎦

(3.14)

= const =

. .

67

(3.13), .

-

( = 0) =

W

2

P ⋅ = −1 ρ

2

−1

⋅ R ⋅T .

(3.15) , (3.14) ,

-

, (

. 3.2),

. ,

. 3.2.

, ( ) ( ):

–––––––––––– – –⋅–⋅–⋅–⋅– –

= . ρ = ρ m → 0.

; (3.14)

,

, . . = 0. P =P

, c

c

⎛ 2 ⎞ =P ⎜ ⎟ ⎝ +1⎠

K

−1

.

W,

: , -

(3.16)

.

-

. ,

(

) (

, . ≈ 0,5⋅

3.17

3.20). = const

W ρ

-

,

. 3.3.

(

( (

-

. 3.3). –

), 69

)

,

-

.

. -

. 3.3. : 1-

(3.13).

;2-

m

(3.14), .

70

4.

"

4.1. -

"

(

. 4.1), -

, =

⋅ρ ⋅g,

+

2

(4.1)

= 0). t > t, – ρ = – ρ = ρ0 /(1+α⋅t ), . . ρ0 ≈ ρ0 .

( ρ0 /(1+α⋅t )

>

2

⋅ρ ⋅g.

+

(4.2) , -

-

, 2

>

.

2

, Δ

. 4.1. ; 2-

; 3; 5-

, ; 4-

; 6-

;

7-

1


t. (

ρ –

) G .

, ρ 6)

(

(5.3) .

. 5.2). ,

(5.3) dL = b⋅L.

(5.4)

(

m0 –

)

L m , = 0,316 ⋅ d0 m0

(5.5)

⎞ ⎛ L m = ⎜⎜ 0,316 ⋅ − 1⎟⎟ . d m0 0 ⎠ ⎝

(5.6)

,

[ / ]. ,

–W ,

-

. W

>W ,

, . ,

83

-

. ,

, W =

W

m1 ⋅ W1 + m 2 ⋅ W2 . m1 + m 2

(5.7) , – I0,

(5.7) -

. (5.2)

I

= I0.

(5.4)

. . -

. (5.2), (5.5), . . . W ,

L,

,

mc [18]. , -

. , . ,

,

.

, . 5.2.

. (

,

, ,

,

. .), (

, . 5.3)

. ,

, .

. dV 84

,

-

, -

.

-

, . , . 5.3. . (

.

. 5.4)

-

,

. ,

Δ

.

W2 =

v 2 = F2 ⋅

,

, (

2 ⋅ ΔP ρ2

)

,

2 ⋅ ΔP ρ2

(5.8)

.

(5.9)

-

,

-

. 5.4. ΔP .

. , : 85

⎛ m1 m 2 ⎞ W2 W2 ⎟⎟ ⋅ P2 + m1 1 + m 2 2 = ⎜⎜ + 2 2 ⎝ ρ1 ρ 2 ⎠

⎛m m ⎞ (W − W3 )2 + m (W2 − W3 )2 , W2 = ⎜⎜ 1 + 2 ⎟⎟ ⋅ P3 + (m1 + m 2 ) 3 + m1 1 2 2 2 2 ⎝ ρ1 ρ 2 ⎠

ΔP = P3 − P2 =

(5.10)

m1 ⋅ W1 ⋅ W3 + m 2 ⋅ W2 ⋅ W3 − (m1 + m 2 ) ⋅ W32 . m1 m 2 + ρ1 ρ 2

(5.11)

= m2 / m1 W1, , ΔP

,

, ΔP

. .

W2 .

,

W2 W3 W2, W2

W1

(5.8), ΔP

(5.11),

-

ΔP .

W3

W1.

W2, W1.

,

, W1. ΔP , , W3 .

W3

W2, -

-

. ,

,

–

.

. ,

,

, W2

,

. .

F2 / F1, 3. . F2 / F1, , .

,

(

) (

,

).

86

-

5.3.

, . .

, ,

. .

, -

.

-

, – . ,

-

, ,

. .

,

.

-

. (

,

).

-

. , ,

,

"

", . .

-

, . ,

,

-

. (

. 5.5), . -

( , ,

. 1), 2.14. .

. 5.5. ,

,

,

,

. ( . 1

. 3). 3,

-

2,

. . 87

-

"

" .

,

, .

, , ,

-

, ,

. L v1.

. .

[19, 20]

,

-

. , . . . . 5.6 .

,

,

,

. , , . . 5.6.

-

,

, .

-

,

. (

,

,

)

. , . 88

-

,

,

-

. , .

-

. .

-

, . 5.4.

, , (

. 5.7). .

. , ,

.

,

,

-

. – . . .

. 5.7.

( ) ( ) :

.

,

, -

1-

(

); 2 -

. .

,

, ,

,

. ,

–

.

, . 89

-

, (

) .

.

, ,

-

,

.

90

2. 6. (

)

6.1.

, . 3 ,

:

-

. -

. . Q=

dE [ dτ

/c =

τ [ ].

[ 2] q=

[

]

(6.1)

],

d2E dQ ⎡ , = dτ ⋅ dF dF ⎢⎣ ⋅

2

F

=

2

⎤ ⎥ ⎦

(6.2) -

. . ,

"q",

1 ,

1 ,

q

2

dF

. (

) q = −λ ⋅

∂t , ∂n

91

, (6.3)

λ[

/( ⋅ )] –

, ∂t [ / ]– ; ∂n

-

. , ,

,

-

. . λ

, , . (

q = α ⋅ (m t ± t

)

α [

t –

;t – /( 2⋅ )] –

),

(6.4) . –

,

(

) -

. α

. . "

–

–

, ) ^

-

9. . – " ",

-

4 4 ⎡⎛ ⎞ ⎤ ⎞ ⎛ ⋅ ⎢⎜ ⎟ ⎥, ⎟ −⎜ ⎢⎣⎝ 100 ⎠ ⎝ 100 ⎠ ⎥⎦

q =

– = t + 273;

", (

,

(6.5) 0

= t + 273.

5,67

/( 2⋅ 4);

:

q

-

–" "

–" "(

92

).

6.2.

.

, . , –

-

. -

t >t

4 4 ⎡⎛ ⎞ ⎤ ⎞ ⎛ − ⋅ ⎢⎜ ⎟ ⎥ + α ⋅ (t − t ⎟ ⎜ ⎢⎣⎝ 100 ⎠ ⎝ 100 ⎠ ⎥⎦

q=

.

),

t [0C] –

T [ ]

-

(6.6) . -

4 4 ⎡⎛ ⎞ ⎤ ⎞ ⎛ − ⋅ ⎢⎜ ⎟ ⎥ + α ⋅ (t − t ) . ⎟ ⎜ ⎢⎣⎝ 100 ⎠ ⎝ 100 ⎠ ⎥⎦

q=

(6.6)

(6.7) . , t ,

t

(6.7) q t

.

-

t . q = (t − t

⎧

) ⋅ ⎪⎨α ⎪⎩

t -t

4 4 ⎡⎛ ⎞ ⎤ ⎞ ⎛ ⋅ ⎢⎜ ⎟ ⎥ ⎟ −⎜ ⎣⎢⎝ 100 ⎠ ⎝ 100 ⎠ ⎦⎥

+

q (t − t

(t

−t

⎫

)⎪⎬ ⎪⎭

(6.8) -

α α =

t -t,

)

=

4 4 ⎡⎛ ⎞ ⎤ ⎞ ⎛ − ⋅ ⎢⎜ ⎟ ⎥ ⎟ ⎜ ⎢⎣⎝ 100 ⎠ ⎝ 100 ⎠ ⎥⎦ (t − t )

t -t =

93

,

(6.9)

(

)⋅ (

).

(6.10)

q = (α + α )⋅(t - t ) = α⋅(t - t )

(6.11)

α =

100

4

⋅

2

+

2

+

q = (α + α )⋅(t - t ) = α⋅(t - t ),

(6.12)

α=α +α

(6.13) -

(

). :

.

,

(6.10), t →t , ,

4

α

=

4⋅ ⋅ 1004

.

α

-

3

.

(6.14) ,

α

, 4

,

-

.

6.3. 6.3.1.

. .

-

,

-

. . "

" [15]. . 94

. . , . –

,

. (

)

-

. 6.3.2.

.

,

-

, (

) .

-

(

. 6.1). -

. . 6.1. (t5>t4>t3>t2>t1):

. ,

q-

grad t = n0⋅ n0 –

∂t ⎡ ⎤ , ∂n ⎢⎣ ⎥⎦

(6.15)

, . .

,

-

. -

q.

95

q = −n 0 ⋅ λ ⋅

∂t ⎡ ∂n ⎢⎣

2

⎤ ⎥. ⎦

(6.16)

(6.16)

. ( ,

q

. 6.2). grad t

,

-

. " (6.16).

"

λ . λ

. 6.2.

(

,

) – λ = 0,7-1. λ.

λ[

Q Q = ∫ q ⋅ dF = ∫ λ ⋅ F

F

∂t ⋅ dF, [ ∂n

F,

-

. λ = 30-50 /( ⋅ ), λ < 0,2. /( ⋅ )] (6.16). . q ,

].

(6.17) -

. Q = q⋅F, [

96

].

(6.18)

6.4.

( 6.4.1.

)

. .

(

-

. 6.3).

. t tc . . tc

-

. 6.3.

t . (

-

,

)

. (

)

,

-

Q . . ∂t

∂τ

, . . t x = f (x ) ,

0. , Q

= Q yx = Q x = Q

,

-

(6.19) Q x = q x ⋅ Fx

. . .

, . (6.19)

τ = ∞.

,

97

-

Q , Q x = −λ x ⋅

dt x ⋅ Fx . dx

(6.20)

Fx = const = F . (6.20) : λx = const

λ

,

,

t

, (

-

. 6.3).

t

dt x t −t =− dx S

, ,

(6.21) ,

t −t S

= λ⋅

Q

.

(6.22)

-

: ⋅F .

(6.22)

, =

Q

t − tc . S λ⋅F

(6.23)

(6.23)

-

R

Q

(

. . 6.4).

λ1

=

=

S . λ⋅F

(6.24)

t − tc . R

(6.25)

S1 , ( S2

, 98

,

,

).

2 -

λ2 < λ1.

=∑ n

Si , i =1 λ i ⋅ F

R

(6.25)

(6.26). λi = const λ

.

(6.26)

. 6.4.

qx

F ≠F,

(6.26) F F , F =

,

F −F , ln (F F )

F . -

(6.27)

F = F ⋅F . qr = Q

= const, -

(6.28)

Fr = const Fr

. 6.4. r

(

,

2-

-

. 6.5). q ∂tr/∂r

: - λ 1 > λ 2;

- λ 1 = λ 2; - λ 1 < λ 2

– ,

.

(

: Q = α ⋅ t −t Q =

t

−t R

99

)⋅ F , ,

(6.29)

(6.30)

(

Q =α ⋅ t

−t

α =α +α – ;α =α +α –

)⋅ F ,

(6.31) . t

t (6.29) - (6.31) :

t

Q = t −t α ⋅F Q ⋅R

. 6.5.

, =

Q

=t

Q =t α ⋅F

( ) ( )

t

, (6.32)

−t

, (6.33)

− t . (6.34)

Q =Q =Q =Q ,

t −t . 1 1 +R + α ⋅F α ⋅F

(6.35)

1 α ⋅F

R [ /

],

-

1 – α ⋅F

(

R .

)

Q

(6.33)

=

t −t R +R +R

(6.34),

.

-

(6.36) t ,

100

tc − t R +R

=

Q

.

(6.37)

(6.37)

-

. (6.32)

t t Q :

(6.34) t

= t − Qc ⋅ R ,

(6.38)

t

= t + Qc ⋅ R .

(6.39)

Q : t1 - t2

F1 =

Q

-

F2, R 12

t1 − t 2 . R 12

(6.40)

Q t

t ,

t (6.38)

t

12, tc

23

. .

Q = K ⋅ F ⋅ (t − t

),

:

K

[ K =

t

(6.39)

/( 2⋅ )].

1 . 1 S 1 +∑ i + α α i =1 λ i n

. K

K F.

,

-

λt = λ0 + b⋅t. 101

(6.41)

q ∂t x = x = f (x ) , ∂x λ( x ) ( . 6.6).

λ,

-

. :

=∑ n

R

Si , λ i =1 i ⋅ Fi

Fi – . 6.6.

1 - t↑ λ↑; 2 - t↑ λ=const; 3 - t↑ λ↓

i-

; λi –

λ = f(t): λi –

(6.42)

(6.27)

,

-

(6.28),

-

i, λi = (λi + λi )/2; λi

i-

. -

, . 6.4.2.

.

, R

α.

-

(

), R Q

-

= α ⋅ (t − t ) ⋅ F .

(6.43)

6.4.3.

, . .

, .

( 102

-

)F

(

= ΔI = v ⋅ c 0t ⋅ t − c 0t ⋅ t

Q

Q

= F ⋅q

– ; Δt –

(

(6.45)

Δt

(

t ,

v ⋅ c0 ⋅ t t

(t

=

,t – ); c 0t c 0t – 0– t 0– t .

t ln t

t

.

) − (t

:

−t

−t −t

) = v ⋅ (c

),

(6.47)

,

t 0

-

(6.46)

⋅ t − c 0t ⋅ t

(6.48) t –

).

(6.48) ;t –

t c0

;

t c0

––

0– t .

K =

–

ΔI K ⋅ Δt

−t

− c0 ⋅ t

0– t

α

(6.45) ;t –

F =

(6.47)

(6.44)

= F ⋅ K ⋅ Δt ,

( (6.44)

)

, . .F ≈F: 1 , 1 S 1 + + α λ α

; α

–

-

(6.49)

103

; λ

; S– α

.

α

– -

. , =(

+

) /2,

(6.50)

–

. [23]. , F =

χ =

ΔI χ ⋅ Δt

(6.46): ,

(6.51)

1

α

1 −

⋅ Δτ

+R +

; α

α

−

(

⋅ Δτ

(t − t )− (t ln[(t − t ) (t

–

)– )]

; ΔI = v c 0t ⋅ t − c 0t ⋅ t

; Δt ;v –

1

= −

−t

−t

–

[ 2⋅ / Δτ

]; Δτ

Δτ

-

−

– -

20 % – [ ]; t

; t

–

; α

;R – ,

) ⋅Δτ

–

Δτ

–

104

.

7.

7.1.

(

)

.

-

:

, ,

,

. ∂t/∂n )

. )

, (

(

,

.

[

λ

/( ⋅ )]

∂t = ⋅ ∇2 t , ∂τ

= λ/(ρ⋅ ) [ 2/ ] – ; ∇2 t =

(7.1) ; ρ [ / 3] –

∂2t ∂2t ∂2t – + + ∂x 2 ∂y 2 ∂z 2

. -

. , , "

".

∂t ∂t x , τ = ∂n ∂x :

,

105

t = f( ,τ),

-

q x , τ = −λ ⋅ n ⋅

∂t x ,τ ∂x

(7.2) ,

( q x , τ = −λ ⋅

-

)

∂t x ,τ ∂x

(7.3) , . .

-

,

.

( ↑, t↓,

q

∂t x ,τ

( ↑, t↑,

∂x , . .

Q

.τ.

dV,

-

q

. ,

.τ,

Q

∂τ

-

, Q

∂t x ,τ

< 0, qx,τ > 0),

> 0, qx,τ < 0),

.

(

∂x

.

.τ,

Q

∂t x ,τ

, 0- .

, < 0)

Q

.τ

-Q ∂t x ,τ

(

.τ

∂τ ∂t x ,τ

∂τ

dV,

.τ

-

> 0,

-

= 0). -

, , , . .

Qx+dx,τ

Qx,τ

. ,

dV x

,

dV

Fx+dx, dE

.dV

=Q

.τ

= - Qx+dx,τ ⋅ dτ = - qx+dx,τ ⋅ Fx+dx ⋅ dτ [ 106

d

dτ ],

(7.4)

dE dQτ = Q ⋅ dτ /5/. (7.4) dV : dEyx.dV = Q Q

.τ

(7.4) Qx+dx,τ

Q .

Qx,τ = f( ) Qx+dx

= dE

.dV

(7.5)

(7.5) ,

dV .dV

, -

= - Qx,τ ⋅ dτ = - qx ⋅ Fx ⋅ dτ.

.τ

Q x + dx ,τ = Q x ,τ +

d2E

. Fx,

∂Q x ,τ ∂x

.τ

-

: ,

⋅ dx .

(7.6)

dτ

(7.4) - (7.6):

- dEyx.dV = -Qx+dx,τ ⋅ dτ - (-Qx,τ)⋅ dτ = −

∂Q x ,τ ∂x

dxdτ .

(7.7)

dV, dV,

dtτ:

dτ

d2IdV = dV ⋅ ρ ⋅ c ⋅ dtτ = dV ⋅ ρ ⋅ c ⋅ d2E

.dV

∂t

,τ

∂τ

= d2IdV, =−

∂Q x ,τ ∂x

(7.7)

Qx+dx,τ < Qx,τ.

∂t

,τ

∂τ

dτ .

(7.8)

(7.8)

⋅ dx / (dV ⋅ ρ ⋅

∂Q x 0,

(7.9) ,

. ,

dV = Fx ⋅ dx.

:

Qx,τ = qx,τ ⋅ dτ

dV = Fx ⋅ dx

107

(7.10) (7.9),

-

∂t x ,τ ∂τ

=−

∂Fx

=0;

∂x

Fx = 4⋅π⋅x2 ,

∂Fx

q x ,τ ∂F 1 ∂q x ,τ − ⋅ x . ⋅ ∂x ρ ⋅ ⋅ Fx ∂x ρ⋅ Fx = 2⋅π⋅x ,

= 8 ⋅ π ⋅x.

∂x

q x , τ = −λ

,

∂τ

∂ 2 t x ,τ

=a⋅

∂x 2

∂t

∂x

= 2⋅π ,

(7.11) ,τ

-

, -

(7.12)

⎛ ∂ 2 t x ,τ 1 ∂t x ,τ = a ⋅⎜ + ⋅ ⎜ ∂x 2 ∂τ x ∂x ⎝

⎞ ⎟; ⎟ ⎠

(7.13)

⎛ ∂ 2 t x ,τ 2 ∂t x ,τ + ⋅ = a ⋅⎜ ⎜ ∂x 2 x ∂x ∂τ ⎝

⎞ ⎟, ⎟ ⎠

(7.14)

∂t x ,τ

∂Q x ,τ

∂x

;

∂t x ,τ

= λ/(ρ⋅c) –

∂

∂Fx

λ = const, = const

, ∂t x ,τ

(7.11)

[ 2/ ].

>0

(7.9)

. ∂t x ,τ ∂τ

Qx+dx,τ > Qx,τ, < 0. ,

(

, =0

: Q

Q

.τ

.τ

)

= Qx,τ = qx,τ ⋅Fx > 0;

= Qx+dx,τ = qx+dx,τ ⋅Fx+dx > 0;

Q x + dx ,τ = Q x ,τ +

∂Q x ,τ ∂x

108

⋅ dx ;

-

d2E

.dV

= (Qx,τ - Qx+dx,τ) ⋅ dτ = −

∂Q x ,τ

d2IdV = dV ⋅ ρ ⋅ c ⋅ dtτ = dV ⋅ ρ ⋅ c ⋅ ∂t

∂t x ,τ ∂τ

∂t x ,τ ∂τ

Q "

,τ

∂τ

:

=−

∂Q x ,τ ∂x

⋅ dx / (dV ⋅ ρ ⋅

dxdτ ;

∂x

).

∂t

dτ ,

,τ

∂τ

∂Q x 0,

.

,

Qx+dx,τ > Qx,τ,

,

< 0. ,

,τ

,τ

Q . "

(

Q = α ⋅ (t

.

Q

∂Q x >0 ∂x

= 0,

,τ,

∂t x ,τ

- t ),

∂τ

)

dt x ,τ t x ,τ + dτ − t x ,τ dτ

dτ

,

, =

tx,τ+dτ < tx,τ. Q

,τ

Q ,τ Q x ,τ , Q x + dx ,τ , .

-

,

q x ,τ = ±λ ⋅ " "

-

(7.3):

∂t x ,τ ∂x

(7.15)

109

(7.15) (7.3).

(7.15) ∂t x ,τ < 0, ↑, t↓, ∂x ( ∂t x ,τ > 0, , ↑, t↑, ∂x

. (7.15)

, . . ).

(7.15)

.

q x , τ = f ( x , τ) ,

:

.

,

-

. 7.2.

,

-

, . .

.

"

"

. 7.2.1. , ,

, .

. , -

(

.

τ

r),

,

-

. . . ,

, 110

Δτ.

,

,

-

( ) : t;

tc; t

t (

=

. 7.1). -

. ( t ) .

. 7.1.

Δt = t - t ) ,

( ,

t . Δtτ : Δtτ < Δt ,

, Δt

. .

,

-

-

. (

)

I = M ⋅c⋅t .

(7.16)

, (

τ . 7.2), . .

-

tx = const = t

tx0 = f( ). τ (

).

.

-

, , , . . 111

, -

. , . : (

)

Δt

t

Δt

(

, ,

. 7.2.

, ) -

. I = ∫ t dV ⋅ c ⋅ ρ ⋅ dV .

(7.17)

V

,

(7.5)

(7.17)

t

1 ⋅ t dV ⋅ dV . V V∫

(7.18)

t

.

t =

: t , tc 7.2.2.

= ∂t/∂τ [ / ].

. (7.1)

: C , .

=

∂2t ∂x 2 dt , C dτ

τ. =

dt , C dτ

, ,

112

-

q τ,

=

dt . dτ

-

( τ

q dτ.

τ

,

τ

)

.

(6.5)

,

(6.4)

.

,

, (

,

)

,

= q τ⋅F ⋅dτ.

dQ

(7.19)

–

. dt ,

dτ

-

dI = M ⋅ c ⋅ dt .

dQ

= dI,

(7.19)

τ

=

-

(7.20)

(7.20)

dt q ⋅F = τ . ⋅c dτ

(7.21) F F.

-

F =F . – R. R;

–

R; –

R.

-

/F R. F 1

R ⋅ρ , K1

(7.22)

–

, 1

1

=

= 1,

1

= 2,

= 3. (7.21)

C

τ

=

dt q ⋅ = τ 1. dτ R ⋅ρ ⋅c

113

(7.22)

-

(7.23)

. 7.3

-

.

. 7.3

. 7.3.

q τ = const). ( t ( . 7.3 ). ,

( const) t

τ

)– -

(

= const ,

).

( . 7.3

τ

=

, -

,

.

7.2.3.

, .

-

q

Δt c = q q

=Q

⋅

S , λ

(7.24)

/F .

[21] Δt τ = q Rτ ⋅

R , λ

114

(7.25)

q Rτ –

τ,

-

τ.

, , .

. -

,

. ( q = −λ ⋅

(

∂t/∂x . 7.4).

∂t ), ∂x =0

-

=0 ,

q Rτ ≈

-

q τ + q cτ q τ ≈ , 2 2

Δt τ ≈

,

,

τ=0

[22]. τ′. τ < τ′

τ′ S = R,

R. (

,

) ( q τ = const

(7.27)

.

S

. 7.4. (

q τ ⋅R . 2λ

,

(7.26)

)

tx0 = const

–

. .

τ′ S < R.

τ = τ′ ( ), .

115

. 7.5).

τ′ . .

-

: , , . ,

,

τ=0

τ,

R → ∞. ,

,

τ′

,

.

.

,

,

. 7.5.

.

-

"

"

-

. (7.27)

(

) -

. Δtτ = f(q τ) tx0 = const . 7.6.

. 7.6.

tx0 = const,

(Δt) Δt

-

,

Δt q τ = f(τ).

116

-

7.2.4.

q τ = const

.

(

. -

. 7.7), .

(7.21) (7.27)

, Δtτ

. q τ = const

(

. 7.7.

. 7.7 )

t

τ

=t

0

+

⋅τ = t

λ = const (λ

0

+

q ⋅ 1 ⋅τ , R ⋅ρ ⋅ c

). , .

,

(7.28)

,

-

txτ = f(τ)

. ,

t

τ

Δtτ = f(τ), . .

, ,

. ,

(

τ'

. 7.7 ).

.

117

(

. 7.7 ) .

Δtτ

q

τ

a⋅

∂ 2 t xτ ∂x 2

= q ⋅ 1/(R⋅ρ⋅ ).

= const,

, =

q . R ⋅ρ ⋅c

(7.29)

(7.29),

:

q ⋅x ∂t xτ = + C1 , ∂x λ⋅R t xτ =

∂t xτ ∂x

=0 1,

(7.30)

q ⋅ x2 + C1 ⋅ x + C 2 . 2⋅λ⋅R

=0

(7.31)

t = tc.

(7.30),

,

q ⋅R . 2⋅λ (7.27),

tc.

Δt τ

=

(7.31)

2

=R

t = tc +

,

-

q ⋅R . 2⋅λ

(7.32) q τ = const

t xτ = t

τ

+ Δt τ

-

-

⎛x⎞ ⋅⎜ ⎟ . ⎝R⎠ 2

(7.33) -

. .

118

7.2.5.

q τ,

, . 7.8. (7.30) : q τ = const

-

(

1

. 7.8) q xτ = q τ ⋅ q1R =

x , R

(7.34) . 7.8.

q . 2

(7.35)

τ

. .

[21] τ

q axτ = q a τ − τ ' ⋅

"2" (τ↑, q τ↓) q1 τ − τ ' > q

(qxτ)

τ

x . R

τ

"1". 1

q

,

τ − τ'

q

τ-τ'

-

q τ-τ'. = const, " " ( . 7.8)

-

(7.36) q 2 τ − τ ' > q1 τ ,

"3" (τ↑, q τ↑)

q 2τ − τ '

q 3τ − τ '

-

q 2xτ

-

.

q1x = q ⋅

q τ. "2" (τ↑, q τ↓) x , R

2

qR

119

q . 2

q

3

τ − τ'

>q τ, 1

q

τ

q 3xτ

. .

x 2. ατ = const λ = const q τ/q

2

2

,

τ–τ'.

, Δt τ =

[21]

q τ ⋅R , K 2τ ⋅ λ

(7.38) 2τ.

2τ = 2

.

7.2.6.

(7.33) (7.18) dV = F ⋅d , V = F ⋅R;

– – V = π⋅R2⋅ ;

= 2⋅π⋅ ⋅d ⋅

dV

dV

–

( –

= 4⋅π⋅ 2⋅d , V

t

= tc +

Δt ; 3

t

= tc +

Δt ; 2

120

,

: ), = 4/3⋅π⋅R3, q τ = const

(7.39)

(7.40)

= tc +

t

(7.39)-(7.41)

. . t = tc +

3

(7.41) [21]

3

Δt

-

,

(7.42)

3

= 3,

= 5 / 3 = 1,67 . 3

Δt . 1,67

3

=2,

-

" "

.

(7.42)

, , ,

-

tx = f( ). . 7.9 τ

.

-

. 7.9.

. "2" (τ↑, q τ↓) "3" (τ↑, q τ↑) –

Δt.

"1" (

q τ = = const) .

.

, < K3 ,

K3

"3" –

K3

; "2" -

> K3 . K3

= K3

. 2,

.

3

121

-

-

2

= K2

3

= K3 .

7.2.7.

λ,

, ,

α c

α

. λ,

α

λ,

.

(

) . ,

, :Q

= q ⋅ F ⋅ τ , ΔI = τ=

Δi = i - i – ;

. -

⋅Δi .

M ⋅ Δi R ⋅ ρ ⋅ Δi , = F ⋅q 1⋅q

(7.43)

i = c 0t ⋅ t

;

(7.44)

i = c 0t ⋅ t

.

(7.45) , -

. q

. "i"

-

q

. .

: qi=

q i −q ln (q i / q

122

i i

)

.

(7.46)

τ

q

= q 0 ⋅ e Bτ ,

(7.47)

,q 0–

–

.

τ

= const, λ, α ).

t , ( . . q =

λ,

α

1 1 ⋅ q τ dτ = ⋅ q 0⋅ τ ∫0 B ⋅ τ ∫0 τ

τ

τ

d (Bτ ) =

(7.47),

0

(

τ

B⋅ τ

)= q

−1

τ = ln (q τ / q

:

,

q

0

)

−q B⋅ τ

0

-

.

(7.48) (7.48)

(7.46). 0-τ ,

q = (q

0

-

)/ 2 .

+q

(7.49) -

[23]

"i"

(7.49) .

q = q

0

⋅q

.

(7.50)

(7.50)

-

7.2.8.

τ' τ' =

R ⋅ ρ ⋅ Δi' . 1 ⋅ q'

= const, tx0 = const = t Δi' = c ⋅ t ' −c ⋅ t

0

:

⎛ = c ⋅ ⎜⎜ t ⎝

0

+

(7.51) t 'c = t 0,

0

Δt ' −t K'3

τ'

123

⎞

0⎟ ⎟

⎠

= c⋅

Δt ' K '3

(7.52) (7.38)

Δt' =

q ' ⋅R . K ' 2 ⋅λ

(7.53) (7.52) τ' =

(7.53)

⎛ q' ⎜⎜ ⎝ q'

R2 K 1 ⋅ K ' 2 ⋅K ' 3 ⋅

q' – 0-τ' (

⎞ ⎟⎟ , ⎠

(7.51) (7.54)

τ'; q ' –

. 7.10).

q τ = const

τ' =

R2 . K 1 ⋅ K ' 2 ⋅K ' 3 ⋅

(7.55) τ'

'2

. 7.10.

τ' . .

– τ' τ' = R 2 / (6 ⋅

τ'

t

τ

= R 2 / (10 ⋅

(7.56)-(7.58) = const,

: q ' = 2 ⋅ q' . ( i > 100) τ'

τ', (7.56)

),

(7.57)

).

( i = α⋅R / λ) , τ'

124

'3. ,

),

τ' = R 2 / (8 ⋅

-

τ'

τ' . t

τ

(7.58)

, = const

q τ = const. (7.55).

7.2.9.

S. 2

τ < τ', 3

(7.54)

R

K1S ⋅ K 2S ⋅ K 3S ⋅ a ⋅ τ ,

S= 1S

S 5 , , .

Re . ,

-

. . , , .

, . . -

. . Re ϕ = 82°

. . Re

-

, , 148

-

.

, .

ϕ = 140°. .

-

. ,

. -

. 8.4 .

. 8.4 . 8.4 –

,

. .

-

. 8.4 .

, .

Re -

Re ,

. . 8.4.

. 8.4 .

(α )

-

.

,

:

- Re = 219000;

. Re

2-3

α –

- Re = 70800.

.

-

. .

, .

.

Nu

d

=

⎛ Pr ⋅ Re d ⋅ Pr ⋅ ⎜⎜ ⎝ Prc n

m

149

⎞ ⎟⎟ ⎠

α

0, 25

,

(8.53)

, n, m

Red. -

, α

. , ,

(8.53).

8.12.

[5]

S1

. -

S2

. . , . ,

-

, ,

,

, -

. . .

Re < 1⋅105 ,

–

. – .

. . .

, ,

( "

").

, , . . 8.5

α

,

. α

. ,

α = 50°. ,

-

. , 150

.

. -

. ,

,

,

-

, .

-

. 8.5. (α ) ( 1

60 % .

-

; Re = 14000,

70 %.

; α –

7

): . -

-

. 8.6. -

.

.

. ,

. 8.6. ( αi )

(8.53). -

(i -

. . 151

)

8.13.

[5]

,

, ,

, (

. 8.7). ,

. , .

. , .

, -

,

-

. . 8.7. :

,

δ (

); ;

δ; -

-

. δ

;

,

;

-

, .

, δ

.

δ -

. , (

. 8.7).

.

. 8.7. " , " " "–

t 1 > t 2. ,

. 152

-

. 8.8 .

, ,

.

-

, .

α = A ⋅ 4 t − t , (8.54) = 3,3, = 2,6, (8.54) .

= 1,6.

. 8.8. (

)

Nu = C ⋅ (Gr Pr

)0,25 ⎜⎜ Pr ⎛

⎞ ⎟⎟ ⎝ Pr ⎠

0, 25

.

Pr = Pr = 1

(8.55)

(8.54). -

(

. 8.9). . ,

2

) . , .

(8.55)

. (d = 0,02 153

. 8.9.

.

.

-

. q =

λ ⋅ (t

λ –

δ

1

−t

2

),

(8.56) , .

8.14.

[5]

, . Т

х

.

х.

. ,

-

. .

.

( 8.36). .

Nu = A + C ⋅ (Re⋅ Pr )n ,

α (8.57)

, ,n–

. α

(8.57) . Т

.

ых

х.

,

-

-

-

.

, "

.

. "

"

"

,

,

. . ,

-

. ,

, 154

.

-

. Т

.

ы

ых

ых

. ,

. .

-

: i0 = i +

T0 = T +

W2 , 2

(8.58)

W2 . (8.59) 2⋅c

-

. 8.10.

, -

t0 (

. . 8.10 . 8.10 : . – (

x :

3). -

1-




–

.

-

, . , ⎛ r ⋅ W2 −T q = α ⋅ ⎜T + ⎜ 2 ⋅ cp ⎝

r

, .

155

⎞ ⎟, ⎟ ⎠

-

(8.60)

9. λ2 = 400

λ1 = 0,4

.

-

,

-

. , . . 9.1.

,

, (

. 9.1).

, -

. 9.1.

,

,

, dF, ω = 2π. .

-

Q Q /Q

=Q

+Q

+Q

=Q

+ R + D = 1,

–

(9.1)

;R=Q ; D= Q

/Q

/Q

–

–

-

. D = 0,

R =1-

.

( . A0 = 1, . . D0 = 0 D =1 - .

R0 = 0,

"0" = 0. 156

λ=0

) ( .

100 %) λ = ∞.

: R = 0,

9.2. 9.2.1.

dF , .

:

dF. -

q=

dQ [ dF

/ 2].

(9.2) -

E. Jλ , , Jλ =

dq λ ⎡ dλ ⎢⎣

2

⎤ . ⋅ ⎥⎦

-

(9.3) -

: J 0λ =

1

2

–

dq 0λ C1 ⋅ λ−5 , = C 2 dλ T λ ⋅ e −1

(9.4)

. J0λ = f(T)

,

. 9.2. -

. 9.2.

.

157

9.2.2.

-

q0 =

∫ J 0 λ dλ .

λ=∞

(9.5)

,

σ0 = 5,67⋅10

-8

,

σ0 ,

/( ⋅ ) – 2

(9.5)

λ =0

q0 = σ0⋅T4,

(9.6)

4

. , ,

-

. ⎛ T ⎞ q 0 = C0 ⋅ ⎜ ⎟ , ⎝ 100 ⎠ 4

0

(9.7)

= 5,67

.

(9.7)

1879

-

,

, 1893

.

9.2.3.

ελ =

Jλ(T) –

ελT

J λ (T )

J 0 λ (T )

,

(9.8)

λ

ελT .

,

-

. . , .

(

.

. 9.3).

158

. 9.3.

. 9.4. : )

:

; )

ελ

1-ε = 1 ( ); 2 - ε = 0,67; 3 - ε = 0,33.

2

; )

2

. 9.4

2

.

2

,

Jλ = f(λ) -

–q .

= ∫ J λ dλ = ∫ ε λ ⋅ J 0λ dλ .

q

∞

∞

0

0

(

(9.9)

( q

ε

) =ε

⋅ q 0 = C0 ⋅ ε

) ⎛ T ⎞ ⋅⎜ ⎟ . ⎝ 100 ⎠ 4

(9.10) ε

: 2

. ε

q = = q0

, ελ = f(λ),

∫ ελ ⋅ J 0λ dλ

∞ 0

∫ J 0λ dλ

∞ 0

159

-

.

2

(9.9)

(9.10)

(9.11)

⎛ T ⎞ q c = εc ⋅ q 0 = C0 ⋅ εc ⋅ ⎜ ⎟ . ⎝ 100 ⎠ 4

ελc εc

(9.12) (9.11)

.

" " 9.3.

, . . εc = ε.

.

, . , Jλ , q , Q -

, -

= q ⋅F ε ( . . .

ελ

9.8

Q

9.10),

= R⋅Q .

(

(9.13)

. 9.5)

Q =Q

+Q .

(9.14)

, . 9.5.

. Q

,

-

Q Q =Q

Q

-Q .

(9.15) Q

Q

= ⋅Q , Q

Q

=Q

-Q

,

.

= ⋅Q

-Q ,

160

(9.16)

=

Q

Q

+Q

-Q

-Q

Q

.

(9.17) (9.15),

Q -Q .

=Q

Q =Q (9.17)

+Q

: Q

-Q .

(9.18)

=

(9.18) R = 1- A,

, =

Q

R Q A

+

1

Q

.

(9.19)

(9.19) . Q

0

=Q

0, J0λ

, = 1, R0 = 0,

-

= J0λ.

9.4.

.

F1 (

0

. 9.6).

,

F1,

F2

. F1

,

F2 dλ, J0λ1⋅dλ.

F1

-

F2,

1

λ, 2

. -

. 9.6.

Aλ2, ( F1 ελ2⋅J0λ2⋅dλ, F1. Jλ 2⋅dλ - Jλ

). dλ

2⋅dλ = Aλ2⋅Jλ

: Jλ ⋅dλ ε ⋅J ⋅dλ = A ⋅J ⋅dλ ε ⋅J 2 λ2 0λ2 λ2 0λ1 λ2 0λ2⋅dλ 161

2⋅dλ

=

Jλ

2

λ2⋅J0λ1

=

- ελ2⋅J0λ2. Jλ

T1 = T2, J0λ1 = J0λ2 A λ 2 = ελ 2 ,

2

= 0.

(9.20)

. . . (9.20)

-

[25]. F1 (

. )

-

F2

∫

∞

2

q = q

=

2

λ2

⋅ Jλ

∫ Jλ

∞

0

2

2 dλ

2 dλ

.

(9.21)

0

F1 . Jλ.

Jλ

= ελ⋅ J0λ

= 0, Jλ

λ

= ελ

, (9.19) Q = J 0λ . -

: Jλ

(9.19)

,

(9.21)

(9.11)

λ2

A

= ελ2

Jλ ,

2

2

=ε

, . . 2

≠ ε2

(9.22)

= J0λ1.

: A = ε (Aλ2

(9.21), A 2

,

.

(9.21) ), . .

-

. (

)

,

, Aλ2 = f (λ)),

F2 ( . . F1. F1 162

,

F1,

A2

F2.

ε

≠ ε2

. 9.5.

.

(

).

Iϕ

, dF dω ϕ

L, Iϕ =

⎡ ,⎢ dFM ⋅ dω ⎣ d 2Qϕ

2

-

⎤ ⎥. ⎦

⋅

(9.23) dω -

, , M

dFM dFN (

. 9.7.

. 9.7). : dω =

dFN –

dFN , 2 rMN

(9.24)

, rMN. dFN ,

dω =

ϕN –

-

dFN ⋅ cos ϕ N , 2 rMN

(9.25)

dFN

MN, . . (

rMN ( (9.25) . 9.8)

dFN

N . 9.8). dω.

163

. 9.8.

Iϕ = In ⋅cosϕM,

(9.26)

In – ,

-

. (

. 9.9): dF ,

-

, . . 9.9.

In =

,

qM . π

(9.27)

9.6.

. ( = 5)

,

,

. F = ∑ Fi ,

,

,

K

.

i =1

T,

,

,

-

.

ϕ12

F1,

F2 [26] ϕ12 =

Q F1 − F2 Q F1

.

, dF2,

(9.28) dF1

-

(9.23) ,

"N" –

dF1. F2.

" " (9.23), (9.25) 164

F1, (9.27)

d 2Q dFM dFN = I ϕM ⋅ dFM ⋅ dω = I n ⋅ cos ϕ M ⋅ dFM ⋅ dω = =

dF ⋅ cos ϕ N q q . ⋅ cos ϕ M ⋅ dFM ⋅ dω = ⋅ cos ϕ M ⋅ dFM ⋅ N 2 π π rMN Q F1 − F2 =

∫ ∫ d QdF 2

M dFN

[26]

∫ ∫q

ϕ12 =

⋅

F2 F1

Q F1 = ∫ q M ⋅ dFM

cos ϕ M ⋅ cos ϕ N dFM dFN 2 π ⋅ rMN

∫ q M dFM

.

(9.29)

F1

ϕ 21 =

∫ ∫qN

⋅

F1 F2

-

F1

F1 F2

cos ϕ N ⋅ cos ϕ M 2 π ⋅ rNM

∫ q N dFN

ϕ 21 dFN dFM

.

(9.30)

F2

q M = const

qN :

ϕ12 =

qM ( qN ) ( . 9.10), ( . 9.10 9.10 ) F1. . . (9.29) (9.30)

" " = const.

-

1 cos ϕ M ⋅ cos ϕ N dFM dFN , ⋅∫ ∫ 2 F1 F F π ⋅ rMN

(9.31)

1 cos ϕ M ⋅ cos ϕ N dFM dFN . ⋅∫∫ 2 F2 F F π ⋅ rMN

(9.32)

1 2

ϕ21 =

1 2

ϕ

,

(9.32). 165

ϕ12 , ϕ21

, (9.32).

ϕ

,

(9.31)-

.

ϕ = ϕ

≠ ϕ

, ϕ

,

,

. .

[27] .

: -

[28]

-

. (9.31) (9.32) . . 9.10. ,

-

ϕdF-F = const

(

.

. 9.10 ),

-

[26]: ϕ12 = ϕ22 =

F2 F2 , = F F1 + F2

(9.33)

ϕ21 = ϕ11 =

F1 F1 . = F F1 + F2

(9.34)

. 9.10 ,

ϕ22

9.10

. ,

ϕ12 + ϕ11 = 1 , ϕ21 + ϕ22 = 1 .

166

: (9.35) (9.36)

ϕ22 = 0

, (9.31)

ϕ21 = 1.

(9.32)

-

F1 ⋅ ϕ12 = F2 ⋅ ϕ21 .

(9.37)

. 9.10

(9.37)

9.10

ϕ12 =

F1 . F2

(9.38) -

, . ( (9.33)

, .

. 9.10 , 9.10 , 9.10 ).

(9.34) ,

(9.38)

ϕ12

: ϕ22 = 0

,

-

ϕ21 = 1.

9.7.

, . ϕi = ϕ

, Q

2

Q

1 ⋅ ϕ12

=Q

2

-Q

2

Q

2

=

=

.

i

2⋅Q

2

[

⋅Q

(9.15) 2

-Q

1 ⋅ ϕ12

2.

(9.39)

+Q 1

:Q

2,

Q

1,

Q

2.

167

2

]

⋅ ϕ22 − Q 2

Q

2

(9.40) Q 1, Q

2

⋅ ϕ22 .

-

.

2

(9.40) Q

1,

3 Q

2

-

(9.19), Q Q

1

2

=-Q =

(9.40)

2

⎡⎛ R 1

2 ⎢⎜ ⎜

⎣⎢⎝ A1

Q

1

+

1

Q

1

⎞

1⎟ ⎟ ϕ12

⎠

,

⎛R + ⎜⎜ 2 Q ⎝ A2

2

+

1

Q

2

⎞

2⎟ ⎟ ϕ22

⎠

⎤ ⎥−Q ⎦⎥

2

,(9.41)

R 1 1 − A1 1 R 1 − A2 1 = = −1 ; 2 = = −1 ; A1 A1 A1 A2 A2 A2

Q

1

=

⎛ 1 ⎞ ⎟ ⋅F1; Q 0 ⋅ ε1 ⋅ ⎜ ⎝ 100 ⎠ 4

2

=

⎛ 2 ⎞ ⎟ ⋅F2. 0 ⋅ ε2 ⋅ ⎜ ⎝ 100 ⎠ 4

: ϕ22 = 1 - ϕ21 , ε1 = A1, ε2 = A2 (

Q

12

2

=

12

)

4 4 ⎡⎛ ⎞ ⎛ ⎞ ⎤ ⋅ ⎢⎜ 1 ⎟ − ⎜ 2 ⎟ ⎥ ⋅ F2 ⋅ ϕ21 , ⎣⎢⎝ 100 ⎠ ⎝ 100 ⎠ ⎦⎥

–

-

(9.42) -

C12 =

⎞ ⎛1 ⎞ ⎛1 ⎜⎜ − 1⎟⎟ ⋅ ϕ12 + 1 + ⎜⎜ − 1⎟⎟ ⋅ ϕ21 ε ε ⎠ ⎝ 2 ⎝ 1 ⎠ 0

(9.42) q2 =

C=

Q F2

⎡⎛ ⎞ ⎛ ⎞ = C ⋅ ⎢⎜ 1 ⎟ − ⎜ 2 ⎟ ⎢⎣⎝ 100 ⎠ ⎝ 100 ⎠ 4

2

4

⎤ ⎥, ⎥⎦

0 ⋅ ϕ21 , ⎛1 ⎞ ⎛1 ⎞ ⎜⎜ − 1⎟⎟ ⋅ ϕ12 + 1 + ⎜⎜ − 1⎟⎟ ⋅ ϕ21 ⎝ ε1 ⎠ ⎝ ε2 ⎠

168

.

(9.43)

(9.44)

(9.45)

-

. -

ϕ12 = ϕ21 = 1 ( ) (9.44-9.45) q2 =

(9.44) . -

4 4 ⎡⎛ ⎞ ⎛ ⎞ ⎤ ⋅ ⎢⎜ 1 ⎟ − ⎜ 2 ⎟ ⎥ . 1 1 ⎢⎝ 100 ⎠ ⎝ 100 ⎠ ⎥ ⎦ ⎣ +1+ ε1 ε2

C0

(9.46)

, ,

,

-

. 9.8.

– ,

,

-

, . . 9.11)

(

. . 9.11.

.

–

. -

Q

1

1

"

2

"

4 4 ⎡⎛ ⎞ ⎛ ⎞ ⎤ ⋅ ⎢⎜ 1 ⎟ − ⎜ ⎟ ⎥= ⎣⎢⎝ 100 ⎠ ⎝ 100 ⎠ ⎦⎥

1

=Q

ε1 = ε2 = ε . 2

4 4 ⎡⎛ ⎞ ⎛ 2 ⎞ ⎤ ⋅ − ⎢ ⎥, ⎜ ⎜ ⎟ ⎟ 2 ⎣⎢⎝ 100 ⎠ ⎝ 100 ⎠ ⎦⎥

–

" -

". 1

=

2,

(9.47)

169

-

(9.47) -

4 4 4 1 ⎡⎛ 1 ⎞ ⎛ 2 ⎞ ⎤ ⎛ ⎞ = + ⎢⎜ ⎜ ⎟ ⎟ ⎜ ⎟ ⎥. 2 ⎢⎣⎝ 100 ⎠ ⎝ 100 ⎠ ⎥⎦ ⎝ 100 ⎠

(9.48)

(9.47),

F1 F2 Q Q

2

2

=

1 2

(9.48)

1

-

4 4 ⎡⎛ ⎛ 2 ⎞ ⎤ Q 1 ⎞ ⋅ − ⎟ ⎜ ⎟ ⎥= 2 ⎢⎜ 2 ⎢⎣⎝ 100 ⎠ ⎝ 100 ⎠ ⎥⎦

–

2

,

(9.49)

2-

-

. , 2

ε < ε1 = ε2,

.

2

2

, n Q

2

=

1 n +1

1

-

Δt > 0.

.

4 4 ⎡⎛ ⎛ 2 ⎞ ⎤ Q 2 1 ⎞ . ⋅ − ⎢ ⎥= ⎜ ⎜ ⎟ ⎟ 2 ⎣⎢⎝ 100 ⎠ ⎝ 100 ⎠ ⎦⎥ n + 1

(9.50)

9.9.

.

[29]. (1, . .

. 9.12),

3,

. L

(

,

. .) (

-

. 9.12). .

F1

F3 F2

F1,

ϕ13

, F3.

170

Q

3

-

Q

3

=

4 4 ⎡⎛ ⎛ 3 ⎞ ⎤ 1 ⎞ ⋅ − ⎟ ⎜ ⎟ ⎥ ⋅ F3 ⋅ 0 ⎢⎜ ⎢⎣⎝ 100 ⎠ ⎝ 100 ⎠ ⎥⎦

,(9.51) -

( : L = ∞,

).

L = 0, = 0, . . ,

= 1.

. 9.12.

, .

9.10.

9.10.1.

2

2

( (

,

)

,

)

,

. ,

.

-

, -

. ("

")

. .

.

, . .

.

, (

.

).

.

,

, ( 2

171

2

), .

.

N2

-

2

. ε

, . 2

2

( -

2

.

. 9.4)

Δλ3 = 16,5-12,5 = 4 0,3 , Δλ2 = 3-2,2 = 0,8

2

,

Δλ1 = 3-2,4 = 0,6

, Δλ3 = 8,5-4,8 = 3,7

2

, -

Δλ).

2

1500 (

2

2

, Δλ2 = 4,8-4 = 0,8 , Δλ1 = 2-1,7 = 2 , Δλ4 = 30-12 = 18 . 2 . -

2,

,

2

2.

,

-

, 2

9.10.2.

.

2

.

(

. 9.13)

Iλ dFN

dF d

-

dI λx = − I λx ⋅ λ

λ

⋅ dx ,

(9.52)

– [

-1

]. (9.52)

I λN = I λM ⋅ e −

. 9.13.

172

λ ⋅ rMN

,

rMN –

dFM

dFN.

, A λMN =

I λM − I λN = 1 − e − K λ ⋅ rMN . Iλ ελMN

2

(9.53)

AλMN.

λ

-

rMN (

2

). AλMN

-

,

= ρ⋅R⋅T.

:

, -

, 2 2

, .

2

, PCO 2

2

, 2

dFN,

dFM

PH 2 O

.

2

,

-

ελMN = f(T, , %

2,

%

2

, rMN). ,

ελMN = f(λ), ,

.

-

ε MN = ∫ ελMN dλ . ∞ 0

9.10.3. Э

(

)

dF, – . εVM

dF 173

-

εVF

F . VM), εVF (

εVM (

F

-

VF)

. dF rMN.

dF

F

(9.53),

rVM

ε VM =

VM

= 1− e

ε VF =

VF

= 1− e

− K ⋅ rVM

− K ⋅ rVF

– (9.54)

;

rVF :

(9.54)

,

(9.55) . rVF .

(9.55) rVM S VM = rVM

S VF = rVF .

SVM

λ. . .

S VF ,

-

S VF

εVM

εVF

V

R=S

dF

. , :

εVF = f(T, , %

2,

%

2

, S VF ),

εVM = f(T, , %

2,

%

2

, SVM ). 2

,

2

-

. S .

-

174

S = 1,75⋅ , –

(9.56)

.

S = 0,9⋅D, D–

(9.57)

. :S

= 0,5⋅D.

,

, ,

S

S = 0,6⋅D.

> 0,5⋅D. (9.58)

,

S VF

-

. : =F +F ,

F

= V

V V V –

–

=V

, V

-V ,

–

= 4⋅π ( R

V F

( (

4 ⋅π⋅ R = 3 4⋅π⋅ R

R

(9.61)

)2

=

4 = ⋅π⋅ R 3

V

) )

3

2

(9.60)

(

. F

(9.59)

=

3⋅ V 3 F +F

(9.58),

,

175

V F +F

.

),

,

3

,

(9.61)

S VF = 3,6 ⋅ ε0 ε

2

ε

ε0 = ε

2

+μ⋅ε

2

.

(9.62) εVF

− Δε0 ,

(9.63)

–

2

2

1,55; Δε –

1 2

:

V F +F

. ,

2

;μ–

,

2 2

. (μ ≈ 1, Δε ≈ 0).

(9.63)

9.10.4.

:

,

.

, . (

,

), .

-

. , [30].

-

-

. ,

,

. ( (

"0"

(i = 1) Q

i

=

.

" ")

9.7).

(i = 2)

⎛ ⎞ ⋅ εi ⋅ ⎜ ⎟ ⋅ Fi ⎝ 100 ⎠ 4

0

-

176

Q εi –

ik

=Q

k

(

⋅ ϕki ⋅ 1 − A ki

)

i = 1, 2

k = 1, 2, i-

A ki – i-

; k-

. -

Q

=Q

2

+Q

−Q

(

2

2

=

)

⋅Q

2

1 ⋅ ϕ12 ⋅ 1 − A12 + Q

−Q

(

2

Q Q

2 2

=

1 1

Q

1

Q

2

2.

) ⋅ ϕ22 ⋅ (1 −

Q ⎛ ⎞ ⋅⎜ ⎟ F2 + ⎝ 100 ⎠

1

4

Q

[

1-

⋅ 1 − (1 −

0 ε1

Q Q

1

2

⋅ 1 − (1 −

0ε2

Q =

[

2,

1

1 1 1

4

2 2

⋅

[

2.

(

22

≠Q .

12

1

⎛ 2 ⎞ ⎟ ⋅ F2 . 0 ⋅ ε2 ⋅ ⎜ ⎝ 100 ⎠ 4

2 2

)]− Q

(

ϕ21 ⋅ 1 −

2

21

⋅ (1 −

)+ Q

2

(

⋅ ϕ22 ⋅ 1 −

22

(

⋅ ϕ11 ⋅ 1 −

(9.65) :Q 12

=

21 =

11 = 0

Q

1 =

1,

Q 22

2.

)=

-

(9.65 )

)=

)− 1] . 21

11

3

)− 1] .

12

) ⋅ ϕ21 ⋅ (1 −

[

2

1 1

) ⋅ ϕ12 ⋅ (1 −

[

1

2.

11

(9.64)

(9.64) (9.19), (9.64)

⋅ (1 −

)+ Q 1

⎛ ⎞ ⋅ ε2 ⋅ ⎜ ⎟ ⋅ F2 + ⎝ 100 ⎠ 4

0

2

)]− Q

Q

ϕ12 ⋅ 1 −

) ⋅ ϕ11 ⋅ (1 −

Q ⎛ ⎞ ⋅⎜ ⎟ F1 + ⎝ 100 ⎠

)]

2

2 ⋅ ϕ22 ⋅ 1 − A 22 −

1

:Q

=

2

(9.65 )

: ε1 = ε 2 =

= ε0, ϕ12 = ϕ12 , ϕ21 = ϕ21 , ϕ11 = ϕ11 , ϕ22 = ϕ22 :

-

4 4 4 4 ⎡⎛ ⎡⎛ ⎛ 1 ⎞ ⎤ ⎛ 1 ⎞ ⎤ 0 ⎞ 2 ⎞ ε − + ε − F C ⎟ ⎜ ⎟ ⎥ 1 ⎟ ⎜ ⎟ ⎥ F1ϕ12 , (9.66 ) 0 021 ⎢⎜ 0 201 ⎢⎜ ⎢⎣⎝ 100 ⎠ ⎝ 100 ⎠ ⎥⎦ ⎢⎣⎝ 100 ⎠ ⎝ 100 ⎠ ⎥⎦

177

Q

4 4 4 4 ⎡⎛ ⎡⎛ ⎛ 2 ⎞ ⎤ ⎛ 2 ⎞ ⎤ 0 ⎞ 1 ⎞ ε − + ε − F C ⎟ ⎜ ⎟ ⎥ 2 ⎟ ⎜ ⎟ ⎥ F2ϕ21 , (9.66 ) 0 012 ⎢⎜ 0 102 ⎢⎜ ⎢⎣⎝ 100 ⎠ ⎝ 100 ⎠ ⎥⎦ ⎢⎣⎝ 100 ⎠ ⎝ 100 ⎠ ⎥⎦

=

2

ε021 = ε1 ⋅ ε0 ⋅ [1 − (ϕ 22 − ϕ12 ) ⋅ (1 − ε0 ) ⋅ (1 − ε 2 )] / C ,

(9.67 )

ε012 = ε 2 ⋅ ε0 ⋅ [1 − (ϕ11 − ϕ21 ) ⋅ (1 − ε0 ) ⋅ (1 − ε1 )] / C , ε201 = ε102 = ε1 ⋅ ε2 ⋅ (1 − ε0 ) / C

[

][

(9.67 ) (9.67 )

]

C = 1 − ϕ11 ⋅ (1 − ε0 ) ⋅ (1 − ε1 ) ⋅ 1 − ϕ 21 ⋅ (1 − ε0 ) ⋅ (1 − ε1 ) −

–

"

-

"

− ϕ12 ⋅ ϕ 21 ⋅ (1 − ε 0 ) ⋅ (1 − ε1 ) ⋅ (1 − ε 2 ) .

;

2

.

(9.67 )

: .

,

,

,

, . . .

-

. A 1

(9.66 )

Q

1

=0

4 4⎤ 4 A ⎡ ⎤ ⎡⎛ A ⎞ 4 ⎛ 0 ⎞ ⎛⎜ 1 ⎞⎟ ⎥ ⎢ ⎢⎜ 1 ⎟ − ⎛⎜ 2 ⎞⎟ ⎥ ⋅ F1 ⋅ ϕ12 . ⋅ = ⋅ ε ⋅ F C ⎟ −⎜ 1 0 201 0 ⋅ ε 021 ⋅ ⎜ ⎢⎝ 100 ⎠ ⎝ 100 ⎟⎠ ⎥ ⎢⎜⎝ 100 ⎟⎠ ⎝ 100 ⎠ ⎥ ⎣ ⎦ ⎣ ⎦

. . Q ε 012 –

2

ϕ22 = 0,

(9.66 )

1

=

-

4 4 ⎡⎛ ⎞ ⎛ ⎞ ⎤ ⋅ ε ⋅ − ⎟ ⎜ ⎟ ⎥ ⋅ F2 , 0 012 ⎢⎜ ⎢⎣⎝ 100 ⎠ ⎝ 100 ⎠ ⎥⎦

"

178

-

(9.68) -

"

ε012 =

ε0 ⋅ ε2 ⋅ [1 + ϕ12 ⋅ (1 − ε0 )] . ε0 + ϕ12 ⋅ (1 − ε0 ) ⋅ [1 + (1 − ε 2 ) ⋅ (1 − ε0 )]

F1 / (F2 + F1))

(ϕ12 = ϕ22 = F2 / (F2 + F1), ϕ21 = ϕ11 = :

. .

Q

2

⎡⎛ ⎛ ⎞ 0 ⎞ ⎟ −⎜ 2 ⎟ 0 ⋅ ε 012 ⋅ ⎢⎜ ⎢⎣⎝ 100 ⎠ ⎝ 100 ⎠

=

4

ε012 =

, , . . f0(λ)

2

(9.69)

4

⎤ ⎥ ⋅ ϕ 21 ⋅ F2 , ⎥⎦

(9.70)

ε 2ϕ12 . ε0 + ϕ12 ⋅ ε 2 ⋅ (1 − ε0 )

(9.71)

ελ1 = f1(λ), ελ2 = f2(λ) , . 2 .

ϕ ≠ϕ , ),

ελ0 = -

,

, (

.

179

3.

10.

10.1.

. . .

-

, . , ,

-

.

,

:

1) 2)

; ;

3)

. , ,

,

H,

,

. Fe,

-

l,

S,

Si, ,

. . .

,

,

,

-

,

-

.

. 10.1, . 10.1

,

,

,

,

,

, ,

, ,

,

, ,

180

,

,

, ,

-

10.1.1.

, . : m

, : ) SO2,

n.

(

2,

4

( H2O

)

, : CO , CO c2 , CH c4 ( CO , CO 2 , CH 4

2,

-

N2. . " "( " " " "

. .),

CO , CO 2 , CH 4

. .). ,

-

.

(t = 0

0

f [ / 3], 1 3 . .).

, = 760

ii

=

=

⋅

i

(10.1)

1 , 1 + 0,001242 ⋅ f

–

(10.2)

i

=

f

i

. :

100 − H 2 O , 100

H 2O p =

– (10.4)

,

(10.3)

100 ⋅ f [%] 803,6 + f

, [ / 3]; 803,6 / ( . .).

181

(10.4) . 3

–

-

10.2. 10.2.1.

Qv [

. Q [

/ ]–

. (

,

/ 3] – . – ,

)

-

( ,

). Q

Q .

-

. (

. 10.1.

.

. 10.1).

: 1 13 212 ; 5; 7; 9; 11 -

;4; 6; 8; 10, 14

. v

; ;3 -

. m

. t

-

15 –

t

.

.

,

t t, . .t =t =t.

t Q= Q

-

.

m

⋅c

⋅ (t

−t

)+Q

,

v

–

. 182

-

(10.5)

t < 100 ° , .

,

-

2

.

2

,

2

-

. ,

-

,

,

, -

. ( t = t = t ), Q

,

t =t =t =0° .

Q.

0 °C.

(10.5)

t,t

t

, ,

.

-

2

,

.

20 ° .

Q

-

v 100 ° . Q

, :

Q

VH 2 O [ 3/

2018

/

3

3

(600

Q − Q = 2018⋅ VH 2 O [

]–

/

3

],

(10.6)

,

1

3

,

/ )– 20 ° .

, -

, ,

-

,

Q

,

t < 100 ° .

Q , . .

Q –

,

,

, . ,

, .

183

100 °

-

Q

10.2.2.

. 10.2. 10.2 / 3)

Q ( 12770 H2

10800

CH4

35800

C2H6

63600

C3H8

91300

C4H10

118500

C5H12

146500

10.2.3.

Q = Q

(

)

⋅ Vp + Q

V p = CO p /100 [

CH 4p /100 [

3 4

Q

/

( 2)

3

⋅ V p2 + Q

/

3

]

(

4)

⋅ Vp 4 + Q

( m n)

], V p2 = H 2p /100 [

3

⋅ V pm

3 2

/

n

3

,

(10.7) ], V p 4 =

. .

100,

(i)

Q = 127,7⋅ CO p + 108⋅ H 2p + 358⋅ CH 4p + 636⋅ C2 H 6p + 913⋅ C 3 H 8p +

+ 1185⋅

4

10

p + 1465⋅ C5 H12 [

/ 3],

(10.8)

p p CO p , H 2p , CH 4p , C2 H 6p , C3 H 8p , C 4 H10 , C5 H12 – , 2 6, 5 12 .

. -

, ,

. :

Q = 340⋅

+ 1030⋅

- 109⋅( 184

- SP) - 25⋅WP [

/ ],

(10.9)

,

SP – ,

,

( ( )

,

34100

/ . -

Q ≈ 36000

3

/ . ,

. . : CO ~ 30 %,

.

Q

(

)

= Q

(

)

2

Q

(

)

=Q

(

4)

⋅V

, ,

~ 10 %,

⋅ Vcop = 12770⋅0,30 ≈ 4000

/ 3. H 2p ~ 60 %,

. +Q

4

( 2)

⋅V

2

= 35580⋅0,25 + 10800⋅0,60 ≈ 15000

2 4

/ 3. -

.

. 40000

.

–

. CH 4p ~ 25 %.

; WP – ) .

,

86-87 %

12-13 %

.

-

/ . 10.2.4.

, , Q

(

)

. = 29308

/

(7000

/ ),

-

. ,

-

,

,

= b[

b ( 29308

.

./

/ ]– . ,

1

. 185

),

(10.10)

10.3.

,

( ,

), ,

,

. , 3

1

-

. 3

1

. (

) . ,

-

, .

, . : + 0,5⋅ 2

4

+ 0,5⋅

+ 2⋅

2

=

2

=

2

=

2

2,

(10.11)

,

(10.12)

2

+ 2⋅

.

2

(10.13) :

1

2

1 = 760

. :

.) 22,4

22,4 3. 11,2

3

22,4, : 1

3

1 (t = 0 ° , 22,4 3 2.

2.

1

3 2

3

3

0,5

-

2

2.

. 4

2 m

: m

n

6,

3

8,

4

n⎞ n ⎛ + ⎜ m + ⎟ ⋅O2 = m⋅CO2 + ⋅ H 2O. 4⎠ 2 ⎝

(10.14)

5

12.

(10.14)

. ,

4

,

(

10,

n

). :

186

⎤ ⎡⎛ n⎞ LO 2 = 0,5VCO + 0,5VH 2 + 2VCH 4 + ∑ ⎢⎜ m + ⎟C m H n ⎥ − VO 2 , 4⎠ ⎦ ⎣⎝

LO 2 [ 3

3 2

3

/

3

[ / ]–

1

2

V ,

2,

4

]; VC [ 3/ 3] –

=

1

3

(10.15)

3

; VO 2

. . " ", /100, VH 2 =

, 2/100,

VCH 4 =

.

4/100

. .,

–

,

,

-

(10.15)

⎡ ⎤ ⎡⎛ ⎤ n⎞ L O 2 = ⎢0,5CO + 0,5H 2 + 2CH 4 + ∑ ⎢⎜ m + ⎟C m H n ⎥ − O 2 ⎥ ⋅ 0,01 . 4⎠ ⎣⎝ ⎦ ⎣ ⎦

ƒ 1

(10.16)

: K O 2 = O 2 /100 [ 3]

3

ƒ L [ 3]

;

L O 2 [ 3] L = LO 2 /

O2

[

3

./

3

.

]

(10.17)

⎧ ⎫ n ⎡ ⎤ ⎨0,5CO + 0,5H 2 + 2CH 4 + ∑ ⎢( m + )C m H n ⎥ − O 2 ⎬ ⋅ 0,01 4 ⎣ ⎦ ⎭ , L =⎩ K 2 KO2 –

.

(10.18)

(10.17)

-

, . . . 2

= 21 %

N 2 = 79 % ( K O 2 = 0,21, K N 2

=

0,79). . f

K

(

,

187

). O2

-

KO2 , .

(10.18)

-

" "

. , L .

-

n = L /L .

(10.19)

L

"n"

n nt ,

–t

t .

1.

-

, (

)

(

). η

4. ( K O 2 > 0,21).

t

.

V (

) N2. -

η

-

. .

, . , .

202

-

10.7.

[31]

10.7.1.

, . .

, . : . .

, -

. > Q , , "

Q

" .

, ,

,

-

.

, ,

. 1000-2300

(1-1,5 %).

,

,

-

, .

,

, -

. . .

,

-

. . 5-10 .

-

.

(

)

(

)

-

. , . 203

. . . , 2

2

+

2

=2

2

. .

. . ,

,

-

, 2

+

=2

+

,

–

-

. (

2,

2

) (

-

2

) +

,

2

=

(

)

+ ,

, +

2

=

+

2

;

+

2

=

+

.

2 + 2 +

2

+ 2

2

,

. ,

. 2

2:

204

1) 2) 3)

+

2

+

2

+

=

+ + + ; + 2 .

= 2

=

;

,

-

– .

:

2

700 0

.

,

.

2

. 4

, →

3

(

2)

→

)→

(

→

,

2

,

2,

1

3

.

, . 10.7.2.

.

,

.

-

, ,

. , -

-

. ,

,

. ,

,

(t ). . -

, : .

, ,

,

,

,

.

,

,

, ,

,

.

,

. 205

,

. . -

,

. (

,

-

) ( ,

);

-

. 550

750 °

2,

4

-

. , . 10.7.3.

-

.

-

N. t = 20 ° ,

N , .

-

N ,

. ,

-

, , . ,

-

. . ( K O 2 > 0,21) . .

(t ) (t ) 206

-

, . .

-

. 10.7.4.

V = const (

) ,

,

, ,

,

.

. . ,

, .

–

-

. ,

, .

,

,

(

). , .

(1-3

)

. :

, . .

-

:

,

. ,

. ,

.

t =t .

-

W t

. 10.4. W = 0,2 / .

207

-

10.8. Э 10.8.1. [31]

. . -

,

. 10.4. ,

(W ) (t ): 1-

, .

,

;2;3-

2

;4-

. -

-

–

. . -

4

, . . ,

,

. 10.8.2.

[31]

,

, .

, -

L

,

, . . 3 :

1

.

. .

,

L

-

. "

". 2

.

,

. -

, , , 208

,

. .

-

. .

3

,

-

. . .

, ,

. .

10.8.3.

. -

. , . 10.5 (

).

: .

. ,

-

. . . , .

.

– – -

, . 10.5.

.

.

. 209

-

.

-

. .

,

.

. (

. 10.6). W , . W .

: , . 10.6.

, . 10.6. -

-

.

–

(

, , . 10.7). ,

. 10.7. , 210

-

.

L ,

= f(W )

-

(

. 10.7). Re > 8000-10000.

Re = 3000.

10.8.4.

[17]

,

. -

, [17], W [31].

[17]

.

, -

L ,

L

(

. .

. 10.8). .

, . . 10.8. –

L . L

=L

+L .

(10.55) (5.5)

L m m +m = m0 m

L

=

⎛ L ⋅ρ 1 ⋅ d 0 ⋅ ⎜⎜1 + ρ0 0,316 ⎝

= 1+

0

L ⋅ρ ρ0

0

:

⎞ ⎛ L ⋅ρ ⎟ = 3,16 ⋅ d 0 ⋅ ⎜1 + ⎟ ⎜ ρ0 ⎠ ⎝

211

-

(10.56)

0

⎞ ⎟, ⎟ ⎠

(10.57)

L – (t = t = 0 ° ,

= = 760 ;ρ0– (5.3) ( ) d

1 3 .); ρ 0 –

.

. L

⎛ L ⋅ρ = b ⋅ 3,16 ⋅ d 0 ⋅ ⎜⎜1 + ρ0 ⎝

= b⋅L

L

= ⋅d .

L

= L ⋅(1 + ⋅b) = 3,16 ⋅d0⋅(1 +

0

⎞ ⎟ ⎟ ⎠

-

(10.58)

5.1 L ⋅ρ ρ0

0

)⋅(1 + ⋅b). α = 24°

≈6

b = 2⋅tg(α /2) = 0,425 L

=11⋅(1 +

L ⋅ρ ρ0

0

(10.59)

)⋅d0.

(10.60) -

. , . .

. [31]

10.8.5.

. ,

[17]

: 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11)

d0; Q ;

; ; ; , ;

; ; ; ; . .

1.

. 212

,

, , ,

.

. 10.9

-

, .

,

-

(10.60): d0. .

,

2.

-

. ,

, (10.60),

, .

–

-

, Q

[

3

/ ], Q [

3050 / 3, , 2575 / 3. ρCO = 1,25 3 / Q ( CO ) = 2440 / , / ].

,

ρ H 2 = 0,0893

Q

(H2 )

= 28800 L = 2,3

2

3

/

. 10.9.

/ . 3

– L = 2,38

. / 3

3

; 3

./ :L

(

)

. = 40⋅d0, L

(10.60) ( H 2 ) = 407⋅d0.

,

. –

2 2

L . 3.

.

2

,

2

,

2.

213

,

L ,

d , L

L

. ,

-

,

, . .

.

4-11

-

(10.60) . 4.

. ,

(

)

( ).

. 5.

-

. -

. . , . ,

. ,

-

, . 6.

,

. .

" .

-

" ,

-

, . 7.

. -

d0 . . . ,

, ,

. 214

,

-

8.

. , .

9.

. , . ,

,

-

. , .

,

, . 10.

. , , .

,

,

-

. . ,

. -

.

, , . .

-

450

.

. ,

,

(

), . , .

11.

. .

. , ,

, .

-

. ,

.

215

10.8.6.

. . , .

, . 10.8.5, .

,

216

-

-

........................ 156

.....................................35

............ 24 ...................................... 20 ...................... 74

............................................72 ...................................................173 ................................................172 ..........................................27 ..................................162, 173 ...................12 .......................... 157, 163, 164 ....................... 26, 92, 134, 135

........................................ 92, 95 ........................................... 129 ...................... 112 ........................................... 56 ....................................... 181 ....................................... 205 .................................................... 24 .............................................................. 11 .......................................... 35 .................................... 64 ........................................ 66 ........................ 53 ............................ 95 .................................. 35 1 .................................................. 35 2 .................................................. 35 3 .................................................. 35 ........................................ 35

......................................27 .................102 ..............................................98, 100 ...............................................157 ...................92, 158

................46 ..................................................91 .......173 ................................................156 ..................157, 159 .............................157

........................................... 139 ................................................... 12 ........................................ 12 ................................. 13 .............................. 13 .................................... 15 .................................. 12, 19 ........................................... 12 ....................................... 68 ............................................... 12 ........................................ 13 ............................. 87

..........................157 ...................................................84 ...............................................84 ..............................................85 .............159 ..................................66 ..............................67 ...................26, 42 .........................................42 ..............................32 ...........................................32 ...............................113 ....................134 ................................................134 ....................................139 ..........................................139 ........................................140 .......................................18, 69, 86

................................... 134 .................................. 21, 26 ......................................... 134 ............................... 21, 26 ........................ 188 .......................... 18, 56, 69, 75, 86 ................................ 211 ........... 45, 211 .......................................... 208 .................................................. 11 ........................................... 35

..........................................50 ...........................171

217

.................... 26

...............54 ....................130 ..............................................................6 ...................................................8 ............................8 ......................................8, 64, 84 ..........................................9, 79 ....................................71 ...............................................9 ....................................................8 ........10 ..........................8 ....................................8, 18 ....................................8, 18 ......................................................8 ......................................9 .......................................................8 ....................9 .........................................8, 64 .............................................8 ..............................9, 74 ...........192 ...................91 ........................................72 ..........................67, 69 ..............................................141 ..................................145 ....................................56, 85

.......................................... 27 ........................................... 92 ................................ 158 .......................... 197 ................ 26 .......................................... 50 ..................................... 129 ............................. 172 ......................... 200 ............. 121 ............................................... 66 ............................... 188 ................................................ 65 ............................................. 65 ............... 105 ............ 93, 94 .... 92, 94, 134 ................................. 101 ............................. 92 ................................................. 53 ........................................ 143 ....................... 143 ........................ 49

................143 ...................................138, 144 .............................26

.......................... 120 ...................................... 113 ............................... 86 .................................................. 19

.............................................206 ..............................................206 ......................................48 ....................................................203

.............................................. 37 ........ 105, 110, 133 ............................................ 30 ...................... 105 ................................................ 30

.............................................15 ............................................14 ...................115

............... 110, 116 ................................. 87 ................................... 35

............................................145 ...........................................138 ( ) ...............................26 .....................................................21 ...........................................41

................................................. 115 ................................ 46 ............................... 15

..............................................37 ..............................165, 176 ....................................................14 .....................................45 ....................................................69

................................... 140, 142 .................................. 105 ................. 113

218

....................................... 68 ......................................... 39 .............................................. 112 ........................................ 16 ................................... 39 .............. 207 .................................... 69 ............................. 49 ................................ 17 ....................... 49 ..................................... 76 ................................ 58, 138 ............................................ 69

....................................160 ...........................................8 ..................................97 ......................................97 ....................................94 ......................................................91 .....................182 ..............................................182 ............................................182 ...............................................182 ...........................................182 ..............................................6 .............127 ....................................................180 .....................................180 ..................................180 ..........................................18 ............................26 ........................................22 ..................................24 ..........................22 .............................23 ................................81 ..........................163, 164

............................................ 158 ............ 158 ................................. 156 ....................... 156, 162 ............................. 156 ............................................. 17 .................................... 86 .................. 205 ................................ 191 ......................... 190

..........166 .........164 ...................................171 ..............................82 ..........................212 .....................200

..................... 191 .................... 117 ........................ 92 ............................. 40 ........................... 186 ........................ 203

.............................................18 ............................. 33, 34, 137 .......................32, 85 ..............................................37

.................................. 198 .................................... 198 ................................................ 196 ........................................... 197 ............................................... 196 ............................................. 198 ......................................... 196 ................................ 197 ....................... 100

.......................141 .......................141 ....................... 33, 34, 137 .............. 17, 30, 35, 137 .....................................37, 145 .........................................41

-

.......................................... 135 ........................................ 91

................................................136 ........................147

..................................... 160 ........................................ 160 ................................... 160 ............................. 160 .................................... 160

-

219

.............................136 ................105 ..............................136

............................................. 136

.......................122 ........................................178 .........................................82 ................58, 138 ........................................................169 .........................................169 ...............163 .................................203 .......................174 ....................................................164

............................................. 47 .................................................. 56 – ................................ 53 ...................................... 184 ......................................... 169 .............................................. 161 ......................................... 179 ........................................ 67

220

. . . 1. . .,

1. /

. ., , 1986. – 424 .

2.

. / . ., , 1986. – 320 . . .

3.

: .–

. .

. .:

14 -

: . .,

. . .–

.– .:

-

.:

, 1936. –

21

22

275 c. 4.

.

.–

.:

, 1951. –

22, 24

575 . . ., .:

5. .–

. ., . . , 1981. – 416 .

. .,

6. .–

. . , 1975. – 328 .

.:

8.

.

24, 38, 39, 39, 40, 43, 43, 148, 150, 152, 154 24, 30

.

: :

.–

, 1975. – 228 . : , 1976. – 504 . : . . . – .: : .

. . .:

9. . ., 10.

:

. .

7.

:

. . 1973. – 360 .

.– / .–

. .

11. 12.

. ., , 1987. – 304 .

.– /

. .

.–

24, 43, 43 27, 40

. .,

31, 37, 40, 51

.:

,

31

.:

, 1957. – 784 . , 1965. –

37 40

.:

722 . . . .

13. 14. . IV. 15. . 3. – 16.

., .: . .

.–

.:

, 1974. – 712 .

// . ., . , 1976. – 440 .

.

43 49, 81

.

53, 94

, 1948. – . 3-18.

.–

.:

57

, 1960. –

464 . 17.

. . .

.

III. – .:

. .

18.

// , 1955. – . 83-103. . – .: , 1976. –

81, 83, 211, 211, 211, 212 84

888 . . .1/

19. 20. 21.

, 1964. – 440 . . ., , 1977. – 464 . . .,

//

. .

. .

,– .–

.: .:

. . .

. .

22.

.

.

IV. –

:

// , 1958. – . 3-17.

-

88

-

88

-

114, 119, 120, 121

-

115

, 1937, № 12, . 29–42.

221

23.

.

.

.

. – .:

i ,

104, 123, 131

1969. – 540 . 24.

25. 26.

. . . . ( : .,

130 // . ). , 1999. . 226–235. .

. ., , 1998. – 240 . 27. . ., . , 1961. – 680 . . ., . 28. 294 . 29. . ., .– . ., 30. , . , . 31. . .,

2.–

. .,

. .

.

. . . .2. – .:

, 1975. – 934 . / :

.–

-

.–

.

. .:

.

.:

, 1971. –

166 166

. . .:

162 164, 165, 166

-

170

-

176

-

203, 208, 208, 211, 212

, 1970. – 400 . . .

III, № 4, 1967, . 463–467. . ., . . . – .: , 1965. – 390 .

222

-

Ч

............................................................................................................................................... 3 ...................................................................................................................................................... 6 1. ........................................................................................................................ 12 1. я я я . ........................ 12

1.1. 1.2. 1.3. 1.4.

........................................................................12 ................................................................14 ...........................................................15 ) .........17 ...............................18 , ........................................................................................19

, . ( 1.4.1. 1.4.2.

2. Х

я

2.1. 2.2. 2.3. 2.4.

...........................................................................21 ......................................................................21 ......................................................................................24 . .................................................................................................27 ...........................................................................29 ......................................................................................30 ....................................................................................32 ( )( )...................................................................................................33 ( )................................................................................................................36 . ..............37 . ............................................................................................39 ..............................44 . ......44

.

2.5. 2.5.1. 2.5.2. 2.5.3. 2.6. . 2.7. 2.8. 2.9. 2.10. 2.11.

.

(Wz = 0) 2.12. 2.12.1.

.

2.12.2. 2.12.3. 2.13. 2.13.1. 2.13.2. 2.14. 2.15. 2.15.1. 2.15.2.

3. И

3.1. 3.2. 3.3. 4.

4.1.

я ......... 21

.

...............................................................................................47 ...................................................................48 , ........................................................48 ...............51 .........51 .............................................................................................................53 ......................................................53 ...................................................54 ...........................................................................55 ...............................................................58 . [5] ...............................................................58 [1] ......................................................62

....................................................................................................................... 64

я

(

.........................................................................................64 .............................................................................................66 )..............................................................................69

я

................................................................. 71

..........................................................................................................71

223

4.1.1. 4.1.2.

..............................................................................72 ......................................73 ............................................................................................75 ...............................................................................75 ........................77 ....................................................77

4.2. 4.2.1. 4.2.2. 4.2.3.

Ч

5.

............................................................. 81

5.1. 5.2. 5.3. 5.4.

................................................................................................................81 ............................................................................................................................84 .........................................................................................................87 .........................................................89

2. 6.

.......................................................................................................................... 91 я ( ) .................................... 91

6.1. 6.2.

..............................................................................................91 . ........................................................................................93 ..........................................................................94 ..............................................94 . ..............................................................95 ( )........................................97 . . ....................................................................97 ...............................................................................102 ........................................................................................102

6.3. 6.3.1. 6.3.2. 6.4. 6.4.1. 6.4.2. 6.4.3.

я

7.

7.1. 7.2.

.................................... 105

(

) ..........................................105 ...................................110 ..........................................................................110

7.2.1. 7.2.2.

..................................................................................................................112 7.2.3. ...........................................................................................114 q τ = const .....117

7.2.4. 7.2.5.

............................................................................................119 7.2.6. 7.2.7. 7.2.8. 7.2.9.

............................................................................................................120 ....................................................................122 .......123 S. 2 3 .....................................................................125

7.2.10.

τ → 0......125 ......................................................................................126 " ...........................127 α, λ , ................................................129 .......................................................130 .........................................................131

. 7.2.11. 7.2.12. 7.2.13. 7.2.14. 7.3.

"

8.

................................................................................................. 134

8.1. 8.2.

.................................................................................................................134 ..............................................................................................................135

224

. α .........................................................................................137 .......140 ................................141 ....142 . .............144 ......................................................145

8.3. 8.4. 8.5. 8.6. 8.7. 8.8. 8.9.

.......................................................................................................147 ..................................................148 [5]......148 [5] ........................................150 [5] ..................................................152 [5] ..................................................154

8.10. 8.11. 8.12. 8.13. 8.14. 9.

....................................................................................................... 156

9.1.

, ..................................................................................................156 .................................................................................157 ..........................................................................................................157 ....................................................................................158 .......................................................................................158 . .................................................160

9.2. 9.2.1. 9.2.2. 9.2.3. 9.3. 9.4.

.

9.5. 9.6. 9.7.

.

........................................................................................167 .................................................................................169 . ..............................................................................................................170 ...........................171 2 20 ..........................................................................................171 . ....................................................................................................172 ( ) ...................................................................173

9.8. 9.9. 9.10. 9.10.1. 9.10.2. 9.10.3. 9.10.4. Ч

.............................................161 ...........................................................................................163 ...............................................................................164

.......................................................176 3.

10.1. 10.1.1. 10.2. 10.2.1. 10.2.2. 10.2.3. 10.2.4. 10.3. 10.4. 10.5. 10.5.1. 10.5.2. 10.5.3.

10.

..................................................................................... 180

................................................................................180 ....................................................................................................181 ............................................................................................182 ...................................................182 ............184 .....................................................................184 ................................................................................................185 ............................................................................................186 ...............................................190 ...................................................................................................192 ...............................................192 . .........196 .....198

225

10.5.4. 10.6. 10.7. 10.7.1. 10.7.2. 10.7.3. 10.7.4. 10.8. 10.8.1. 10.8.2. 10.8.3. 10.8.4. 10.8.5. 10.8.6.

,

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......................................................................................................................... 217 .............................................................................................................................................. 221 ............................................................................................................................................. 223

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