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Файл:Методология расчётов гидродинамических параметров шахтных автоматизированных стационарных установок с центробежными нагнетателями. Монография
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Ɉɝɪɚɧɢɱɢɜɚɹɫɶ ɧɚ ɞɚɧɧɨɦ ɷɬɚɩɟ ɢɫɫɥɟɞɨɜɚɧɢɣ ɩɟɪɜɵɦ ɫɩɨɫɨɛɨɦ
t
ɨɩɪɟɞɟɥɟɧɢɹ ɩɟɪɟɞɚɬɨɱɧɵɯ ɮɭɧɤɰɢɣ ɡɜɟɧɶɟɜ ɫɯɟɦɵ ɪɢɫ. 3.11, ɛ, ɩɪɨɢɡɜɟɞɺɦ ɬɟɨɪɟɬɢɱɟɫɤɢɟ ɢɫɫɥɟɞɨɜɚɧɢɹ ɧɚɢɛɨɥɟɟ ɭɩɪɨɳɟɧɧɨɣ ɫɬɪɭɤɬɭɪɵ ɨɛɴɟɤɬɚ, ɩɪɟɞɫɬɚɜɥɟɧɧɨɣ ɧɚ ɪɢɫ. 3.11, ɜ.
ɉɪɟɞɫɬɚɜɥɟɧɢɹ ɫɥɨɠɧɨɣ ɫɢɫɬɟɦɵ, ɩɪɢɜɟɞɟɧɧɨɣ ɧɚ ɪɢɫ. 3.11, ɚ, ɜ ɜɢɞɟ
ɞɜɭɯ ɭɩɪɨɳɟɧɧɵɯ ɦɨɞɟɥɟɣ (ɪɢɫ. 3.11, ɛ ɢ ɪɢɫ. 3.11, ɜ) ɩɨɡɜɨɥɹɟɬ ɨɩɢɫɚɬɶ
ɤɚɠɞɵɣ ɤɚɧɚɥ ɨɬɞɟɥɶɧɨ (ɛɟɡ ɭɱɺɬɚ
ɜɥɢɹɧɢɹ ɫɨɫɟɞɧɢɯ ɤɚɧɚɥɨɜ), ɚ ɡɚɬɟɦ,
ɢɫɩɨɥɶɡɭɹ ɩɪɢɧɰɢɩ ɫɭɩɟɪɩɨɡɢɰɢɢ, ɲɢɪɨɤɨ ɢɫɩɨɥɶɡɭɟɦɵɣ ɜ ɬɟɨɪɢɢ ɚɜɬɨɦɚɬɢɱɟɫɤɨɝɨ ɪɟɝɭɥɢɪɨɜɚɧɢɹ ɞɥɹ ɥɢɧɟɣɧɵɯ ɢ ɤɜɚɡɢɥɢɧɟɣɧɵɯ ɫɢɫɬɟɦ, ɨɤɨɧɱɚɬɟɥɶɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɢɡɭɱɚɟɦɵɣ ɨɛɴɟɤɬ ɜ ɜɢɞɟ ɩɪɟɨɛɪɚɡɨɜɚɧɧɨɣ ɨɩɪɟɞɟɥɟɧɧɵɦ ɨɛɪɚɡɨɦ ɫɬɪɭɤɬɭɪɧɨɣ ɫɯɟɦɵ ɢɡ ɪɹɞɚ ɩɟɪɟɞɚɬɨɱɧɵɯ ɮɭɧɤɰɢɣ. Ɍɚɤɨɣ ɩɨɞɯɨɞ ɤ ɪɟɲɟɧɢɸ ɫɥɨɠɧɵɯ ɞɢɧɚɦɢɱɟɫɤɢɯ ɨɛɴɟɤɬɨɜ, ɤɚɤɢɦɢ ɹɜɥɹɸɬɫɹ ɝɢɞɪɨɬɪɚɧɫɩɨɪɬɧɵɟ ɭɫɬɚɧɨɜɤɢ,
ɧɟ ɩɪɨɬɢɜɨɪɟɱɚɬ ɬɟɨɪɟɬɢɱɟɫɤɢɦ ɨɫɧɨɜɚɦ
ɫɢɧɬɟɡɚ ɫɢɫɬɟɦ ɪɟɝɭɥɢɪɨɜɚɧɢɹ ɢ ɩɨɡɜɨɥɹɟɬ ɩɪɟɞɫɬɚɜɢɬɶ ɪɟɡɭɥɶɬɚɬɵ ɢɫɫɥɟɞɨɜɚɧɢɣ ɜ ɡɚɤɨɧɱɟɧɧɨɦ ɢɧɠɟɧɟɪɧɨɦ ɜɢɞɟ ɫ ɭɱɺɬɨɦ ɤɨɧɤɪɟɬɧɵɯ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɬɪɟɛɨɜɚɧɢɣ, ɩɪɟɞɴɹɜɥɹɟɦɵɯ ɤ ɮɭɧɤɰɢɨɧɢɪɨɜɚɧɢɸ ɩɪɨɢɡɜɨɞɫɬɜɟɧɧɵɯ ɦɚɲɢɧ ɩɨɞɨɛɧɨɝɨ ɤɥɚɫɫɚ.
ɉɨɫɤɨɥɶɤɭ ɜ ɦɚɬɟɦɚɬɢɱɟɫɤɨɦ ɨɩɢɫɚɧɢɢ ɡɜɟɧɶɟɜ ɝɢɞɪɨɬɪɚɧɫɩɨɪɬɧɵɯ
ɭɫɬɚɧɨɜɨɤ ɦɧɨɝɨ ɨɛɳɟɝɨ ɫ ɜɨɞɨɨɬɥɢɜɧɵɦɢ ɭɫɬɚɧɨɜɤɚɦɢ, ɚɜɬɨɪɵ ɫɨɱɥɢ
ɜɨɡɦɨɠɧɵɦ ɧɟ ɩɨɜɬɨɪɹɬɶ ɦɧɨɝɢɟ ɨɛɳɢɟ ɜɵɤɥɚɞɤɢ
, ɫ ɤɨɬɨɪɵɦɢ ɦɨɠɧɨ
ɨɡɧɚɤɨɦɢɬɶɫɹ ɜ ɩɨɞɪɚɡɞɟɥɟ 2.2, ɚ ɨɫɬɚɧɨɜɢɬɶɫɹ ɬɨɥɶɤɨ ɧɚ ɫɩɟɰɢɮɢɱɟɫɤɢɯ
ɦɨɦɟɧɬɚɯ, ɯɚɪɚɤɬɟɪɧɵɯ ɞɥɹ ɝɢɞɪɨɬɪɚɧɫɩɨɪɬɧɵɯ ɭɫɬɚɧɨɜɨɤ.
3.11.3. Ɇɚɬɟɦɚɬɢɱɟɫɤɨɟ ɨɩɢɫɚɧɢɟ ɨɬɞɟɥɶɧɵɯ ɡɜɟɧɶɟɜ ɨɛɴɟɤɬɚ
Ʉɚɧɚɥ I – I. Ɋɚɫɱɺɬɧɚɹ ɝɢɞɪɚɜɥɢɱɟɫɤɚɹ ɫɯɟɦɚ ɩɪɢɜɟɞɟɧɚ ɧɚ ɪɢɫ. 3.12, ɚ.
ȼɨɡɦɭɳɚɸɳɢɦ ɮɚɤɬɨɪɨɦ, ɞɟɣɫɬɜɭɸɳɢɦ ɧɚ ɨɛɴɟɤɬ ɩɨ ɷɬɨɦɭ ɤɚɧɚɥɭ, ɹɜɥɹɟɬɫɹ ɢɡɦɟɧɟɧɢɟ ɩɪɢɬɨɤɚ ɝɢɞɪɨɫɦɟɫɢ ɜ ɩɪɢɺɦɧɭɸ ɺɦɤɨɫɬɶ ɝɢɞɪɨɬɪɚɧɫɩɨɪɬɧɨɣ ɭɫɬɚɧɨɜɤɢ. ɉɨɞ ɞɟɣɫɬɜɢɟɦ ɷɬɨɝɨ ɮɚɤɬɨɪɚ ɭɪɨɜɟɧɶ ɝɢɞɪɨɫɦɟɫɢ ɜ ɩɪɢɺɦɧɨɣ ɺɦɤɨɫɬɢ ɢɡɦɟɧɹɟɬɫɹ ɫɨɝɥɚɫɧɨ ɭɪɚɜɧɟɧɢɸ:
tdh
F
)(
d
tQtQ
ɍɉ
, (3.104)
ɝɞɟ F – ɩɥɨɳɚɞɶ ɡɟɪɤɚɥɚ ɜɨɞɵ ɜ ɩɪɢɺɦɧɨɣ ɺɦɤɨɫɬɢ;
h(t) – ɭɪɨɜɟɧɶ ɝɢɞɪɨɫɦɟɫɢ;
Qɉ(t) ɢ Qɍ(t) – ɩɪɢɬɨɤ ɢ ɪɚɫɯɨɞ ɝɢɞɪɨɫɦɟɫɢ.
ɂɡɦɟɧɟɧɢɹ ɭɪɨɜɧɹ ɝɢɞɪɨɫɦɟɫɢ ɜ ɩɪɢɺɦɧɨɣ ɺɦɤɨɫɬɢ ɜɵɡɵɜɚɟɬ ɢɡɦɟɧɟɧɢɹ ɜɵɫɨɬɵ ɜɫɚɫɵɜɚɧɢɹ ɭɝɥɟɫɨɫɚ, ɱɬɨ ɜɥɢɹɟɬ ɧɚ ɫɬɚɬɢɱɟɫɤɭɸ ɯɚɪɚɤɬɟɪɢɫɬɢɤɭ ɧɚɩɨɪɧɨɣ ɫɟɬɢ, ɢɡɦɟɧɟɧɢɹ ɤɨɬɨɪɨɣ ɢɥɥɸɫɬɪɢɪɭɸɬɫɹ ɝɪɚɮɢɤɨɦ,
ɩɪɟɞɫɬɚɜɥɟɧɧɵɦ ɧɚ ɪɢɫ. 3.12, ɛ. ȼ ɨɛɳɟɦ ɫɥɭɱɚɟ ɭɪɚɜɧɟɧɢɟ ɯɚɪɚɤɬɟɪɢɫɬɢ-
ɤɢ ɫɟɬɢ ɢɦɟɟɬ ɜɢɞ:
111
U
O
U
ɝɞɟ Ɋ
– ɞɚɜɥɟɧɢɟ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɟ ɜɵɫɨɬɟ ɧɚɝɧɟɬɚɧɢɹ, ɉɚ;
Ƚ
Ƚɋ
ɚ – ɨɛɨɛɳɺɧɧɵɣ ɤɨɷɮɮɢɰɢɟɧɬ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɬɪɭɛɨɩɪɨɜɨɞɚ, ɉɚ/(ɦ
Q – ɩɨɞɚɱɚ ɭɝɥɟɫɨɫɚ ɜ ɞɚɧɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ, ɦ
Qɉ(t)
Qɍ(t)
2
, (3.105)
QɚɊɊ
6/ɫ2
3
/ɫ.
H
Ɇ
H
Ƚ
H
ɜɫ
H
h(t)
ɚ)
y
1'
2
Q
1U
P'
1
2'
2Q1
Ɋ
2
0
1
bQ
Ɋ
0
aQ
ȱȱ
Ɋ
Ƚ
0
Ɋ
Ƚ
ȱ
Ɋ
Ƚ
Q
Q
2Q0Q1
ɛ)
Ɋ
P'
C
2
2
P'
2U
2
2
Ɋ
Ƚ
x
ɜ)
Ɋɢɫɭɧɨɤ 3.12. Ɋɚɫɱɟɬɧɚɹ ɫɯɟɦɚ ɢ ɝɪɚɮɢɤɢ ɫɬɚɬɢɱɟɫɤɢɯ ɯɚɪɚɤɬɟɪɢɫɬɢɤ ɨɛɴɟɤɬɚ
bQ
aQ
);
Q
2
1
2
1
ɚ
8
L
,
2
Z
R
Ƚ
ɝɞɟ Ȝ – ɤɨɷɮɮɢɰɢɟɧɬ Ⱦɚɪɫɢ;
ȡ – ɩɥɨɬɧɨɫɬɶ ɬɪɚɧɫɩɨɪɬɢɪɭɟɦɨɣ ɠɢɞɤɨɫɬɢ (ɜ ɞɚɧɧɨɦ ɫɥɭɱɚɟ ɜɨɞɵ), ɤɝ/ɦ
3
;
L – ɩɪɨɬɹɠɟɧɧɨɫɬɶ ɬɪɚɫɫɵ ɬɪɭɛɨɩɪɨɜɨɞɚ, ɦ;
Ȧ – ɩɥɨɳɚɞɶ ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ ɬɪɭɛɵ, ɦ2;
– ɝɢɞɪɚɜɥɢɱɟɫɤɢɣ ɪɚɞɢɭɫ ɡɚɢɥɟɧɧɨɣ ɬɪɭɛɵ, ɦ.
R
Ƚ
112
PHgɊɊ
, (3.106)
)(thɆȽ
ɡɞɟɫɶ ɊɆ = const;
U
U
H = const;
= var.
P
h(t)
ɋ ɭɱɺɬɨɦ ɮɨɪɦɭɥɵ (3.106), ɭɪɚɜɧɟɧɢɟ (3.105) ɩɪɢɦɟɬ ɜɢɞ:
U
thɆɋ
)(
2
QɚPHgɊɊ
. (3.107)
ɍɪɚɜɧɟɧɢɟ (3.107) ɧɟɥɢɧɟɣɧɨ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɚɪɚɦɟɬɪɚ Q.
ɉɪɨɢɡɜɟɞɟɦ ɥɢɧɟɚɪɢɡɚɰɢɸ ɱɥɟɧɚ
ɨɤɪɟɫɬɧɨɫɬɢ ɬɨɱɤɢ 0 (ɪɢɫ. 3.12, ɛ) ɜ ɪɹɞ Ɍɟɣɥɨɪɚ:
Ɉɬɛɪɨɫɢɜ ɱɥɟɧ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɦɚɥɨɫɬɢ, ɧɚɯɨɞɢɦ:
2
2
2)( QQɚQQQaQaQaQf |
0
2
2
2 QQQaQaQa |
0
2
Qɚ , ɪɚɡɥɨɠɢɜ ɮɭɧɤɰɢɸ f(Q) ɜ
.
00
2
.
000
Ɍɨɝɞɚ ɥɢɧɟɚɪɢɡɨɜɚɧɧɨɟ ɭɪɚɜɧɟɧɢɟ ɫɬɚɬɢɱɟɫɤɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɫɟɬɢ
ɩɪɢɦɟɬ ɜɢɞ:
U
thɆɋ
2
2 QQQɚQɚPHgɊɊ
0)(
, (3.108)
0
ɝɞɟ Q0 – ɡɧɚɱɟɧɢɟ ɩɨɞɚɱɢ ɭɝɥɟɫɨɫɚ ɜ ɬɨɱɤɟ ɪɚɡɥɨɠɟɧɢɹ, Q0 = const;
Q – ɡɧɚɱɟɧɢɹ ɩɨɞɚɱɢ ɜ ɬɟɤɭɳɢɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ.
ɇɚɩɨɪɧɚɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ ɭɝɥɟɫɨɫɚ ɦɨɠɟɬ ɛɵɬɶ ɚɩɩɪɨɤɫɢɦɢɪɨɜɚɧɚ
ɡɚɜɢɫɢɦɨɫɬɶɸ:
QkɊɊ
ɇ
0
, (3.109)
ɝɞɟ Ɋ0 – ɞɚɜɥɟɧɢɟ, ɪɚɡɜɢɜɚɟɦɨɟ ɭɝɥɟɫɨɫɨɦ ɩɪɢ ɧɭɥɟɜɨɣ ɩɨɞɚɱɟ, ɉɚ:
HgɊ
– ɫɜɹɡɶ Ɋ0 ɧɚɩɨɪɨɦ ɇ0;
00
k – ɭɝɥɨɜɨɣ ɤɨɷɮɮɢɰɢɟɧɬ ɧɚɩɨɪɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ, ɉɚ ā ɫ/ɦ3.
ɉɪɢɪɚɜɧɢɜɚɹ ɜɵɪɚɠɟɧɢɹ (3.108) ɢ (3.109) ɢ ɞɢɮɮɟɪɟɧɰɢɪɭɹ ɨɛɟ
ɱɚɫɬɢ ɩɨɥɭɱɟɧɧɨɝɨ ɜɵɪɚɠɟɧɢɹ ɩɨ ɩɟɪɟɦɟɧɧɵɦ P
dQkdQQɚdɊ
2 . (3.110)
th
0)(
ɢ Q, ɢɦɟɟɦ:
h(t)
ɉɪɟɞɫɬɚɜɢɜ ɥɟɜɭɸ ɱɚɫɬɶ ɭɪɚɜɧɟɧɢɹ (3.104) ɜ ɟɞɢɧɢɰɚɯ ɞɚɜɥɟɧɢɹ,
ɩɨɫɥɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ, ɢɦɟɟɦ:
g
>@
dɊ
)(
F
ɉth
dttQtQ
. (3.111)
ɋɞɟɥɚɜ ɩɨɞɫɬɚɧɨɜɤɭ ɜ ɭɪɚɜɧɟɧɢɟ (3.110) ɢɡ (3.111) ɢ ɩɪɨɢɡɜɟɞɹ
ɝɪɭɩɩɢɪɨɜɤɭ ɱɥɟɧɨɜ, ɧɚɯɨɞɢɦ:
113
t
ɉ
ȱ
Ɋɚɡɞɟɥɢɜ ɨɛɟ ɱɚɫɬɢ ɭɪɚɜɧɟɧɢɹ (3.112) ɧɚ ȡÂg, ɩɨɥɭɱɢɦ:
ɡɞɟɫɶ
Ɍ
2 . (3.112)
0
Ɍ
2
ȱ
1
U
0
g
tdQ
FkQɚ
UU
dt
tdQ
1
d
FkQɚ
– ɩɨɫɬɨɹɧɧɚɹ ɜɪɟɦɟɧɢ ɤɚɧɚɥɚ 1 – I, c.
(t)QQ(t)
, (3.113)
ɉȱ
(t)QgQ(t)g
ɉ
ȼ ɨɩɟɪɚɬɨɪɧɨɣ ɮɨɪɦɟ ɭɪɚɜɧɟɧɢɹ (3.113) ɦɨɠɟɬ ɛɵɬɶ ɩɪɟɞɫɬɚɜɥɟɧɨ
ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
(s)QQ(s)Q(s)sɌ
1
.
ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɩɟɪɟɞɚɬɨɱɧɚɹ ɮɭɧɤɰɢɹ ɤɚɧɚɥɚ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ
ɜɨɡɦɭɳɚɸɳɟɝɨ ɜɨɡɞɟɣɫɬɜɢɹ ɢɦɟɟɬ ɜɢɞ:
1
ȱ
sW
Q(s)
1
1
ȱɉ
. (3.114)
1
sɌ(s)Q
Ʉɚɧɚɥ 2 – I. Ɋɟɝɭɥɢɪɭɸɳɢɦ ɮɚɤɬɨɪɨɦ, ɞɟɣɫɬɜɭɸɳɢɦ ɧɚ ɨɛɴɟɤɬ ɩɨ
ɞɚɧɧɨɦɭ ɤɚɧɚɥɭ, ɩɪɢɧɹɬɨ ɢɡɦɟɧɟɧɧɵɟ ɤɨɷɮɮɢɰɢɟɧɬɚ ɝɢɞɪɚɜɥɢɱɟɫɤɨɝɨ
ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɞɪɨɫɫɟɥɹ (ɡɚɞɜɢɠɤɢ). ɉɨɞ ɜɥɢɹɧɢɟɦ ɷɬɨɝɨ ɮɚɤɬɨɪɚ
ɢɡɦɟɧɹɟɬɫɹ ɨɛɳɟɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɧɚɩɨɪɧɨɣ ɫɟɬɢ, ɜ ɪɟɡɭɥɶɬɚɬɟ ɱɟɝɨ
ɩɪɨɢɫɯɨɞɢɬ ɫɦɟɳɟɧɢɟ ɪɚɛɨɱɟɣ ɬɨɱɤɢ, ɨɩɪɟɞɟɥɹɸɳɟɣ ɭɫɬɚɧɨɜɢɜɲɟɣɫɹ
ɪɟɠɢɦ ɪɚɛɨɬɵ ɝɢɞɪɨɬɪɚɧɫɩɨɪɬɧɨɣ ɭɫɬɚɧɨɜɤɢ (ɪɢɫ. 3.12, ɜ).
ɉɭɫɬɶ ɞɨ ɧɚɱɚɥɶɧɨɝɨ ɦɨɦɟɧɬɚ ɜɪɟɦɟɧɢ t0, ɡɚɞɜɢɠɤɚ ɧɚ ɧɚɩɨɪɧɨɦ ɬɪɭɛɨɩɪɨɜɨɞɟ ɛɵɥɚ ɩɨɥɧɨɫɬɶɸ ɨɬɤɪɵɬɚ, ɚ ɜ ɫɢɫɬɟɦɟ ɢɦɟɥ ɦɟɫɬɨ ɭɫɬɚɧɨɜɢɜɲɢɣɫɹ ɪɟɠɢɦ, ɩɚɪɚɦɟɬɪɵ ɤɨɬɨɪɨɝɨ ɨɩɪɟɞɟɥɹɥɢɫɶ ɪɚɛɨɱɟɣ ɬɨɱɤɨɣ, ɧɚɯɨɞɹɳɟɣɫɹ ɧɚ ɧɚɩɨɪɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɟ ɜ ɩɨɥɨɠɟɧɢɢ 1. ɉɪɢ ɷɬɨɦ ɩɨɬɟɪɹ
ɞɚɜɥɟɧɢɹ ɜ ɫɟɬɢ Ɋ1ɋ ɛɵɥɚ ɩɨɥɧɨɫɬɶɸ ɤɨɦɩɟɧɫɢɪɨɜɚɧɚ ɞɚɜɥɟɧɢɟɦ Ɋ1ɍ,
ɪɚɡɜɢɜɚɟɦɵɦ ɧɚɫɨɫɨɦ, ɢ ɢɦɟɥɨ ɦɟɫɬɨ ɫɥɟɞɭɸɳɟɟ ɪɚɜɟɧɫɬɜɨ Ɋ
Ɋ
= Ɋ0 – kÂQ1; Ɋ1ɋ = ɊȽ + ɚÂQ
1ɍ
2
. ȼ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t = t0 ɩɪɨɢɫɯɨɞɢɬ
1
= Ɋ1ɋ, ɝɞɟ
1ɍ
ɦɝɧɨɜɟɧɧɨɟ ɩɪɢɤɪɵɬɢɟ ɡɚɞɜɢɠɤɢ, ɜ ɪɟɡɭɥɶɬɚɬɟ ɱɟɝɨ ɩɨɬɟɪɹ ɞɚɜɥɟɧɢɹ ɜ
2
ɫɟɬɢ ɜɨɡɪɚɫɬɚɟɬ ɧɚ ɜɟɥɢɱɢɧɭ bÂQ
, ɝɞɟ b = ȟÂȡ/2ÂȦ2, ɤɝ/ɦ7. Ɂɞɟɫɶ ȟ –
1
ɤɨɷɮɮɢɰɢɟɧɬ ɝɢɞɪɚɜɥɢɱɟɫɤɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɡɚɞɜɢɠɤɢ ɨɬ ɬɢɩɚ ɡɚɞɜɢɠɤɢ
ɢ ɫɬɟɩɟɧɢ ɟɺ ɡɚɤɪɵɬɢɹ.
ɉɨɫɤɨɥɶɤɭ ɫɱɢɬɚɟɬɫɹ, ɱɬɨ ɧɚɩɨɪɧɚɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ ɦɚɲɢɧɵ ɨɫɬɚɺɬɫɹ
ɩɪɢ ɷɬɨɦ ɧɟɢɡɦɟɧɧɨɣ, ɬɨ ɜɟɥɢɱɢɧɚ bÂQ
2
ɜ ɦɨɦɟɧɬ t = t0 ɨɤɚɡɵɜɚɟɬɫɹ ɧɟ
1
ɫɤɨɦɩɟɧɫɢɪɨɜɚɧɧɨɣ ɩɪɢɪɚɳɟɧɢɟɦ ɞɚɜɥɟɧɢɹ, ɪɚɡɜɢɜɚɟɦɵɦ ɭɝɥɟɫɨɫɨɦ. ɗɬɨ
ɨɛɭɫɥɨɜɥɢɜɚɟɬ ɩɨɹɜɥɟɧɢɟ ɮɢɤɬɢɜɧɨɣ ɬɨɱɤɢ 1', ɡɚɧɢɦɚɸɳɟɣ ɧɟɭɫɬɨɣɱɢɜɨɟ
ɩɨɥɨɠɟɧɢɟ ɧɚ ɧɨɜɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɟ ɫɟɬɢ, ɭɪɚɜɧɟɧɢɹ ɤɨɬɨɪɨɣ ɜ ɨɛɳɟɦ
ɫɥɭɱɚɟ ɢɦɟɟɬ ɜɢɞ:
114
t
Ƚɋ
2
.
QbɚɊɊ
ɇɟɭɫɬɨɣɱɢɜɨɫɬɶ ɩɨɥɨɠɟɧɢɹ ɪɚɛɨɱɟɣ ɬɨɱɤɢ 1' ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t > t0
ɜɵɡɵɜɚɟɬ ɩɨɹɜɥɟɧɢɟ ɩɟɪɟɯɨɞɧɨɝɨ ɪɟɠɢɦɚ ɪɚɛɨɬɵ ɭɫɬɚɧɨɜɤɢ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɟɝɨɫɹ ɡɚɦɟɞɥɟɧɢɟɦ ɩɨɬɨɤɚ ɜ ɧɚɱɚɥɶɧɨɦ ɫɟɱɟɧɢɢ ɬɪɭɛɨɩɪɨɜɨɞɚ ɞɨ ɬɟɯ
ɩɨɪ (t = t
ɤɨɬɨɪɨɦ ɩɨɬɟɪɹ ɞɚɜɥɟɧɢɹ ɜ ɫɟɬɢ Ɋ
ɞɚɜɥɟɧɢɟɦ Ɋ
ɜɪɟɦɟɧɢ t = t
ɦɟɬɪɵ ɤɨɬɨɪɨɝɨ ɨɩɪɟɞɟɥɹɸɬɫɹ ɪɚɛɨɱɟɣ ɬɨɱɤɨɣ 2, ɝɞɟ Q
Ɋ
= Ɋ2ɋ, Ɋ2ɍ = P0 – kÂQ2:
2ɍ
), ɩɨɤɚ ɬɨɱɤɚ 1' ɧɟ ɡɚɣɦɟɬ ɧɨɜɨɟ ɭɫɬɨɣɱɢɜɨɟ ɩɨɥɨɠɟɧɢɟ 2, ɜ
1
ɛɭɞɟɬ ɩɨɥɧɨɫɬɶɸ ɤɨɦɩɟɧɫɢɪɨɜɚɧɚ
2ɋ
, ɪɚɡɜɢɜɚɟɦɵɦ ɭɝɥɟɫɨɫɨɦ. ɉɪɢ ɷɬɨɦ, ɧɚɱɢɧɚɹ ɫ ɦɨɦɟɧɬɚ
2ɍ
, ɜ ɫɢɫɬɟɦɟ ɜɧɨɜɶ ɜɨɡɧɢɤɚɟɬ ɭɫɬɚɧɨɜɢɜɲɢɣɫɹ ɪɟɠɢɦ, ɩɚɪɚ-
1
< Q1, P2 > P1,
2
Ƚɋ
2
.
QbɚɊɊ
22
Ⱦɥɹ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɨɩɢɫɚɧɢɹ ɩɨɜɟɞɟɧɢɹ ɫɢɫɬɟɦɵ ɜ ɩɟɪɟɯɨɞɧɨɦ
ɪɟɠɢɦɟ ɜ ɩɪɨɦɟɠɭɬɤɟ ɨɬ t ɞɨ t
, ɜɨɫɩɨɥɶɡɭɟɦɫɹ, ɤɚɤ ɢ ɜ ɩɪɟɞɵɞɭɳɢɯ
1
ɫɥɭɱɚɹɯ ɫ ɧɚɫɨɫɚɦɢ ɢ ɜɟɧɬɢɥɹɬɨɪɧɵɦɢ ɭɫɬɚɧɨɜɤɚɦɢ – ɨɫɧɨɜɧɵɦɢ
ɭɪɚɜɧɟɧɢɹɦɢ ɦɟɯɚɧɢɤɢ. Ⱦɥɹ ɷɬɨɝɨ ɩɨɦɟɫɬɢɦ ɧɚɱɚɥɨ ɤɨɨɪɞɢɧɚɬ ɜ ɬɨɱɤɭ 1 ɢ
ɩɪɢɦɟɦ ɧɚɩɪɚɜɥɟɧɢɟ ɨɫɟɣ ɬɚɤɢɦ, ɤɚɤ ɭɤɚɡɚɧɨ ɧɚ ɪɢɫ. 3.12, ɜ, ɬɨɝɞɚ
ɭɪɚɜɧɟɧɢɟ ɡɚɦɟɞɥɟɧɢɹ ɩɨɬɨɤɚ ɦɨɠɟɬ ɛɵɬɶ ɡɚɩɢɫɚɧɨ ɜ ɜɢɞɟ:
m
dV
d
, (3.115)
Z
'
Ɋ
1U
ɝɞɟ m – ɦɚɫɫɚ ɫɬɨɥɛɚ ɠɢɞɤɨɫɬɢ ɜ ɬɪɭɛɨɩɪɨɜɨɞɟ, ɤɝ;
V – ɫɪɟɞɧɹɹ ɫɤɨɪɨɫɬɶ ɠɢɞɤɨɫɬɢ ɜ ɬɪɭɛɨɩɪɨɜɨɞɟ, ɦ/c;
Ȧ – ɩɥɨɳɚɞɶ ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ ɬɪɭɛɨɩɪɨɜɨɞɚ, ɦ2;
– ɩɪɢɪɚɳɟɧɢɟ ɞɚɜɥɟɧɢɹ, ɪɚɫɯɨɞɭɟɦɨɝɨ ɧɚ ɡɚɦɟɞɥɟɧɢɟ ɩɨɬɨɤɚ
Ɋ'
1U
ɠɢɞɤɨɫɬɢ, ɉɚ (ɩɨ ɚɧɚɥɨɝɢɢ ɫ ɧɚɫɨɫɚɦɢ, ɪɚɫɫɦɨɬɪɟɧɧɵɦɢ ɧɚɦɢ ɪɚɧɟɟ, ɷɬɚ
ɜɟɥɢɱɢɧɚ ɤɜɚɥɢɮɢɰɢɪɭɟɬɫɹ ɤɚɤ ɢɧɟɪɰɢɨɧɧɨɟ ɞɚɜɥɟɧɢɟ). Ⱦɥɹ ɧɟɡɚɢɥɟɧɧɨɝɨ
ɬɪɭɛɨɩɪɨɜɨɞɚ ɢɦɟɟɦ: m = ȡÂLÂȦ, V = Q/Ȧ.
ɂɡ ɝɪɚɮɢɱɟɫɤɨɝɨ ɩɨɫɬɪɨɟɧɢɹ ɫɥɟɞɭɟɬ (ɫɦ. ɪɢɫ. 3.12, ɜ):
2
11
,
ɊQbɊ ' '
ɋU
ɝɞɟ ǻPC – ɪɚɡɧɨɫɬɶ ɩɨɬɟɪɶ ɞɚɜɥɟɧɢɹ ɜ ɫɟɬɢ ɜ ɞɜɭɯ ɭɫɬɚɧɨɜɢɜɲɢɯɫɹ
ɪɟɠɢɦɚɯ, ɩɪɢɱɺɦ:
.
ɊɊɊ
'
ɋɋɋ
12
ɂɫɩɨɥɶɡɭɹ ɭɪɚɜɧɟɧɢɹ ɧɚɩɨɪɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɦɚɲɢɧɵ (3.109) ɢ
ɩɪɢɧɢɦɚɹ ɜɨ ɜɧɢɦɚɧɢɟ, ɱɬɨ Ɋ
= Ɋ1ɍ, Ɋ2C = Ɋ2ɍ, ɧɚɯɨɞɢɦ:
1C
QQkɊ
' ,
ɋ
21
115
ɬɨɝɞɚ
Z
Z
t
ȱ
Z
U
k
ȱ
Z
k
[
Ɋ
'
U
1
2
U
Q
1
2
QQk
2
21
,
ɝɞɟ k – ɭɝɥɨɜɨɣ ɤɨɷɮɮɢɰɢɟɧɬ ɧɚɩɨɪɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɭɝɥɟɫɨɫɚ
Z
3
.
= Q0 = const,
1
2
ZU
ZU
tdQL
)(
[
dt
Q
t
2
2
0
)(
Z
QQk
.
t
0
(ɫɦ. (3.109)), ɉɚ ā ɫ/ɦ
ɉɨɥɚɝɚɹ ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ ɩɪɢ ɡɚɦɟɞɥɟɧɢɢ ɩɨɬɨɤɚ, ɱɬɨ Q
ɚ Q
= Q(t), ɭɪɚɜɧɟɧɢɟ (3.115) ɡɚɩɢɲɟɦ ɜ ɜɢɞɟ:
2
Ɋɚɡɞɟɥɢɜ ɨɛɟ ɱɚɫɬɢ ɩɨɥɭɱɟɧɧɨɝɨ ɜɵɪɚɠɟɧɢɹ ɧɚ kÂȦ ɢ ɩɪɨɢɡɜɟɞɹ
ɝɪɭɩɩɢɪɨɜɤɭ ɱɥɟɧɨɜ, ɚ ɡɚɬɟɦ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ, ɧɚɯɨɞɢɦ:
ɝɞɟ
Ɍ
– ɩɨɫɬɨɹɧɧɚɹ ɜɪɟɦɟɧɢ ɤɚɧɚɥɚ 2 – I:
2
k
– ɤɨɷɮɮɢɰɢɟɧɬ ɩɨɞɚɱɢ ɬɨɝɨ ɠɟ ɤɚɧɚɥɚ:
2
Ɍ
tdQ
d
Ɍ
k
[
2
ȱ2
ȱ
ktQQ(t)
, (3.116)
ȱȱ
202
L
, c;
2
U
Q
0
, ɦ2/ɫ.
2
2
Ⱥɧɚɥɨɝɢɱɧɨ ɩɨɥɭɱɚɟɦ ɭɪɚɜɧɟɧɢɟ, ɨɩɢɫɵɜɚɸɳɟɟ ɞɢɧɚɦɢɤɭ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɪɟɝɭɥɢɪɭɸɳɟɝɨ ɜɨɡɞɟɣɫɬɜɢɹ ɩɨ ɤɚɧɚɥɭ 2 – I ɩɪɢ ɭɫɤɨɪɟɧɢɢ
ɩɨɬɨɤɚ ɜ ɬɪɭɛɨɩɪɨɜɨɞɟ. ɉɪɢ ɷɬɨɦ ɫɱɢɬɚɟɦ, ɱɬɨ ɞɨ ɧɚɱɚɥɶɧɨɝɨ ɦɨɦɟɧɬɚ
ɜɪɟɦɟɧɢ t
ɡɚɞɜɢɠɤɚ ɧɚ ɧɚɩɨɪɧɨɦ ɬɪɭɛɨɩɪɨɜɨɞɟ ɛɵɥɚ ɩɪɢɤɪɵɬɚ ɧɚ ɨɩɪɟɞɟ-
0
ɥɟɧɧɭɸ ɜɟɥɢɱɢɧɭ, ɚ ɜ ɫɢɫɬɟɦɟ ɛɵɥ ɭɫɬɚɧɨɜɢɜɲɢɣɫɹ ɪɟɠɢɦ, ɩɚɪɚɦɟɬɪɵ ɤɨɬɨɪɨɝɨ ɨɩɪɟɞɟɥɹɥɢɫɶ ɪɚɛɨɱɟɣ ɬɨɱɤɨɣ, ɧɚɯɨɞɹɳɟɣɫɹ ɜ ɩɨɥɨɠɟɧɢɢ 2
(ɫɦ. ɪɢɫ. 3.12, ɜ), ɝɞɟ Ɋ
ȼ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t = t
= Ɋ2ɋ. Ɂɞɟɫɶ Ɋ2ɍ = P0 – kÂQ2, Ɋ2ɋ = PȽ + (a+b)ÂQ
2ɍ
ɩɪɨɢɫɯɨɞɢɬ ɦɝɧɨɜɟɧɧɨɟ ɨɬɤɪɵɬɢɟ ɡɚɞɜɢɠɤɢ,
0
ɜ ɪɟɡɭɥɶɬɚɬɟ ɱɟɝɨ ɩɨɬɟɪɹ ɞɚɜɥɟɧɢɹ ɜ ɫɟɬɢ ɭɦɟɧɶɲɚɟɬɫɹ ɧɚ ɜɟɥɢɱɢɧɭ bÂQ
2
.
2
2
, ɚ
2
ɬɨɱɤɚ 2 ɡɚɣɦɺɬ ɧɟɭɫɬɨɣɱɢɜɨɟ ɩɨɥɨɠɟɧɢɟ 2' ɧɚ ɧɨɜɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɟ ɫɟɬɢ,
ɭɪɚɜɧɟɧɢɟ ɤɨɬɨɪɨɣ ɜ ɞɚɧɧɨɦ ɫɥɭɱɚɟ ɢɦɟɟɬ ɜɢɞ:
Ƚɋ
2
.
QɚɊɊ
ȼ ɫɥɟɞɭɸɳɢɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t > t0 ɜɨɡɧɢɤɚɟɬ ɩɟɪɟɯɨɞɧɵɣ ɪɟɠɢɦ
ɪɚɛɨɬɵ ɭɫɬɚɧɨɜɤɢ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɢɣɫɹ ɭɫɤɨɪɟɧɢɟɦ ɩɨɬɨɤɚ ɜ ɧɚɱɚɥɶɧɨɦ
ɫɟɱɟɧɢɢ ɬɪɭɛɨɩɪɨɜɨɞɚ ɞɨ ɬɟɯ ɩɨɪ (t = t1), ɩɨɤɚ ɬɨɱɤɚ 2' ɧɟ ɡɚɣɦɟɬ ɧɨɜɨɟ
ɭɫɬɨɣɱɢɜɨɟ ɩɨɥɨɠɟɧɢɟ 1, ɩɨɫɥɟ ɱɟɝɨ ɜ ɫɢɫɬɟɦɟ ɜɧɨɜɶ ɜɨɡɧɢɤɚɟɬ ɭɫɬɚɧɨɜɢɜɲɢɣɫɹ ɪɟɠɢɦ. ɉɨɦɟɫɬɢɜ ɧɚɱɚɥɨ ɤɨɨɪɞɢɧɚɬ ɜ ɬɨɱɤɭ 2 ɢ ɫɨɯɪɚɧɢɜ ɩɪɢɧɹɬɨɟ ɪɚɧɟɟ ɧɚɩɪɚɜɥɟɧɢɟ ɨɫɟɣ, ɡɚɩɢɲɟɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ
ɭɫɤɨɪɟɧɢɹ ɩɨɬɨɤɚ ɜ ɜɢɞɟ:
116
Z
Z
ȱ
ȱ
[
[
ɯ
t
U
dV
m , (3.117)
Z
' 2UɊ
dt
ɡɞɟɫɶ
Ɋ' – ɩɪɢɪɚɳɟɧɢɟ ɞɚɜɥɟɧɢɹ, ɪɚɫɯɨɞɭɟɦɨɝɨ ɧɚ ɪɚɡɝɨɧ ɩɨɬɨɤɚ
2U
ɠɢɞɤɨɫɬɢ ɜ ɬɪɭɛɨɩɪɨɜɨɞɟ.
ɂɡ ɝɪɚɮɢɱɟɫɤɨɝɨ ɩɨɫɬɪɨɟɧɢɹ ɫ ɭɱɺɬɨɦ ɧɚɩɪɚɜɥɟɧɢɹ ɤɨɨɪɞɢɧɚɬɧɵɯ
ɨɫɟɣ, ɫɥɟɞɭɟɬ:
ɝɞɟ
Ɍɨɝɞɚ
ɊɊɊ
' , ɢɥɢ
ɋɋɋ
21
Ɋ
ɋ
U
[
'
2
U
2
22
2
Q
2
2
2
,
ɊQbɊ ' '
ɋU
. (3.118)
QQk
12
QQkQkɊQkɊɊ
' .
122010
ɉɨɥɚɝɚɹ, ɱɬɨ ɩɪɢ ɪɚɡɝɨɧɟ ɩɨɬɨɤɚ Q2 = Q0 = const, ɭɪɚɜɧɟɧɢɟ (3.117) ɫ
ɭɱɺɬɨɦ ɮɨɪɦɭɥɵ (3.118) ɡɚɩɢɲɟɦ ɜ ɜɢɞɟ:
ZU
Z
dt
)(
tdQL
)(
[
t
2
Q
ZU
0
2
2
Z
QQk
,
t
0
ɱɬɨ, ɜ ɤɨɧɟɱɧɨɦ ɫɱɺɬɟ, ɩɪɢɜɨɞɢɬ ɤ ɭɪɚɜɧɟɧɢɸ (3.116), ɤɨɬɨɪɨɟ ɜ ɨɩɟɪɚɬɨɪɧɨɣ ɮɨɪɦɟ ɡɚɩɢɫɢ ɢɦɟɟɬ ɜɢɞ:
ȼ ɭɪɚɜɧɟɧɢɢ (3.119) ɡɧɚɤ ɦɢɧɭɫ ɭɤɚɡɵɜɚɟɬ ɧɚ ɬɨ, ɱɬɨ ɫ ɪɨɫɬɨɦ
skQ(s)sQ(s)Ɍ
22
. (3.119)
s
ɩɨɞɚɱɚ ɭɝɥɟɫɨɫɚ ɭɦɟɧɶɲɚɟɬɫɹ ɢ ɧɚɨɛɨɪɨɬ. ɉɨɷɬɨɦɭ ɩɪɢ ɜɵɜɨɞɟ ɩɟɪɟɞɚɬɨɱɧɨɣ ɮɭɧɤɰɢɢ ɤɚɧɚɥɚ 2 – I ɡɧɚɤ ɦɢɧɭɫ ɦɨɠɧɨ ɨɩɭɫɬɢɬɶ:
2
sW
ȱ
Q(s)
(s)
[
k
2
ȱ
2
ȱ
. (3.120)
1
sɌ
Ʉɚɧɚɥ I – II ɮɢɡɢɱɟɫɤɢ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɧɚɩɨɪɧɵɣ ɬɪɭɛɨɩɪɨɜɨɞ
ɝɢɞɪɨɬɪɚɧɫɩɨɪɬɧɨɣ ɭɫɬɚɧɨɜɤɢ, ɜ ɤɨɬɨɪɨɦ ɞɢɧɚɦɢɤɚ ɩɪɨɢɫɯɨɞɹɳɢɯ ɩɪɨɰɟɫɫɨɜ ɜ ɭɫɬɚɧɨɜɢɜɲɟɦɫɹ ɪɟɠɢɦɟ ɨɩɢɫɵɜɚɟɬɫɹ ɫɢɫɬɟɦɨɣ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɜ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ [12]. ɗɬɢ ɭɪɚɜɧɟɧɢɹ ɢɦɟɸɬ ɜɢɞ:
P
ɯ
P
ɫ
w
w
V
w
2
2
t
w
V
U
w
½
U
Vɚ
°
°
¾
°
°
¿
w
w
w
117
ɉɪɢ ɨɩɪɟɞɟɥɟɧɢɢ ɩɟɪɟɞɚɬɨɱɧɨɣ ɮɭɧɤɰɢɢ ɬɪɭɛɨɩɪɨɜɨɞɚ ɜɵɞɟɥɹɸɬ
ɯ
t
ɯ
ɞɜɚ ɯɚɪɚɤɬɟɪɧɵɯ ɫɥɭɱɚɹ: ɬɪɭɛɨɩɪɨɜɨɞɵ ɫ ɛɨɥɶɲɢɦɢ ɢ ɦɚɥɵɦɢ ɡɚɬɭɯɚɧɢɹɦɢ. Ƚɪɚɧɢɰɚ ɦɟɠɞɭ ɧɢɦɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɜɟɥɢɱɢɧɟ ɛɟɡɪɚɡɦɟɪɧɨɝɨ ɩɚɪɚɦɟɬɪɚ Į, ɩɪɟɞɫɬɚɜɥɹɸɳɟɝɨ ɫɨɛɨɣ ɨɬɧɨɲɟɧɢɟ ɩɨɬɟɪɢ ɞɚɜɥɟɧɢɹ ɨɬ ɬɪɟɧɢɹ ɧɚ
ɭɱɚɫɬɤɟ ɬɪɭɛɨɩɪɨɜɨɞɚ ɞɥɢɧɨɣ L ɩɪɢ ɭɫɬɚɧɨɜɢɜɲɟɦɫɹ ɬɟɱɟɧɢɢ ɫɨ
ɫɤɨɪɨɫɬɶɸ V
ɇ.ȿ. ɀɭɤɨɜɫɤɨɝɨ:
ɤ ɭɞɚɪɧɨɦɭ ɞɚɜɥɟɧɢɸ, ɩɨɞɫɱɢɬɚɧɧɨɦɭ ɩɨ ɮɨɪɦɭɥɟ ɩɪɨɮ.
0
Ɋ
'
D
'
2
O
VL
ɬɪ
Ɋ
16
0
Ɍɚ
ɫR
Ƚɭɞ
,
ɝɞɟ Ɍ – ɩɟɪɢɨɞ ɤɨɥɟɛɚɧɢɣ ɞɚɜɥɟɧɢɹ ɜ ɬɪɭɛɨɩɪɨɜɨɞɟ: ɫLɌ 2 , ɫ;
L – ɞɥɢɧɚ ɬɪɚɧɫɩɨɪɬɧɨɝɨ ɬɪɭɛɨɩɪɨɜɨɞɚ, ɦ;
c – ɫɤɨɪɨɫɬɶ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɭɞɚɪɧɨɣ ɜɨɥɧɵ ɜ ɬɪɭɛɨɩɪɨɜɨɞɟ ɫ ɭɩɪɭɝɢɦɢ
ɫɬɟɧɤɚɦɢ, ɦ/ɫ;
R
– ɝɢɞɪɚɜɥɢɱɟɫɤɢɣ ɪɚɞɢɭɫ ɬɪɭɛɵ, ɦ;
Ƚ
Ȝ – ɤɨɷɮɮɢɰɢɟɧɬ Ⱦɚɪɫɢ ɞɥɹ ɞɚɧɧɨɝɨ ɬɪɭɛɨɩɪɨɜɨɞɚ.
ȼ ɪɚɛɨɬɟ [12] ɭɫɬɚɧɨɜɥɟɧɨ, ɱɬɨ ɜɨɥɧɨɜɨɣ ɯɚɪɚɤɬɟɪ ɩɟɪɟɯɨɞɧɨɝɨ ɩɪɨɰɟɫɫɚ ɜ ɬɪɭɛɨɩɪɨɜɨɞɟ ɫ ɪɨɫɬɨɦ Į ɩɟɪɟɫɬɚɺɬ ɛɵɬɶ ɹɜɧɨ ɜɵɪɚɠɟɧɧɵɦ ɢ ɭɠɟ
ɩɪɢ Į > ʌ ɩɪɚɤɬɢɱɟɫɤɢ ɢɫɱɟɡɚɟɬ. ȼ ɫɜɹɬɢ ɫ ɷɬɢɦ, ɩɪɢ Į > ʌ, ɩɨɬɟɪɹ ɞɚɜɥɟɧɢɹ ɨɬ ɬɪɟɧɢɹ ɡɧɚɱɢɬɟɥɶɧɨ ɩɪɟɜɨɫɯɨɞɢɬ ɢɧɟɪɰɢɨɧɧɨɟ
ɫɥɭɱɚɟ ɫɢɫɬɟɦɚ ɭɪɚɜɧɟɧɢɣ ɩɪɢɜɨɞɢɬɫɹ ɤ ɜɢɞɭ:
w
P
2
w
ɯ
w
P
2
ɫ
w
½
U
Vɚ
°
°
¾
V
U
w
°
°
¿
w
ɞɚɜɥɟɧɢɟ. ȼ ɷɬɨɦ
ɗɬɢ ɭɪɚɜɧɟɧɢɹ ɩɪɟɞɫɬɚɜɥɹɸɬ ɫɨɛɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɩɚɪɚɛɨɥɢɱɟɫɤɨɝɨ
ɬɢɩɚ, ɫɜɨɞɹɳɢɟɫɹ ɤ ɢɡɜɟɫɬɧɨɦɭ ɭɪɚɜɧɟɧɢɸ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ:
P
w
t
w
2
Ɋ
w
,
2
w
2
ɫ
EE
. (3.121)
2
ɚ
ɇɟɭɫɬɚɧɨɜɢɜɲɟɟɫɹ ɞɚɜɥɟɧɢɟ ɠɢɞɤɨɫɬɢ ɜ ɤɨɪɨɬɤɢɯ ɬɪɭɛɨɩɪɨɜɨɞɚɯ,
ɩɪɢ Į > 0,5, ɧɨɫɢɬ ɹɪɤɨ ɜɵɪɚɠɟɧɧɵɣ ɤɨɥɟɛɚɬɟɥɶɧɵɣ ɯɚɪɚɤɬɟɪ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ, ɤɚɤ ɭɤɚɡɵɜɚɸɬɫɹ ɜ ɪɚɛɨɬɟ [12], ɜ ɢɫɯɨɞɧɨɣ ɫɢɫɬɟɦɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ ɱɥɟɧɨɦ
Vɚ U2 , ɨɩɪɟɞɟɥɹɸɳɢɦ
ɩɨɬɟɪɢ ɞɚɜɥɟɧɢɹ ɨɬ ɬɪɟɧɢɹ. Ɍɨɝɞɚ ɞɥɹ ɤɨɪɨɬɤɢɯ ɬɪɭɛɨɩɪɨɜɨɞɨɜ ɦɨɠɧɨ
ɡɚɩɢɫɚɬɶ ɫɥɟɞɭɸɳɭɸ ɫɢɫɬɟɦɭ:
118
ɯ
t
U
ɯ
t
P
w
ɯ
w
P
w
w
V
w
t
w
U
2
ɫ
w
½
°
°
¾
V
w
°
°
¿
Ⱦɚɧɧɚɹ ɫɢɫɬɟɦɚ ɫɜɨɞɢɬɫɹ ɤ ɭɪɚɜɧɟɧɢɸ ɦɚɥɵɯ ɩɨɩɟɪɟɱɧɵɯ ɤɨɥɟɛɚɧɢɣ
ɫɬɪɭɧɵ:
2
V
w
2
w
2
V
w
UU
2
ɫ
. (3.122)
2
w
ɏɚɪɚɤɬɟɪ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɜɨɡɦɭɳɟɧɢɣ ɜ ɬɪɭɛɨɩɪɨɜɨɞɟ ɜ ɞɚɧɧɨɦ
ɫɥɭɱɚɟ ɨɩɪɟɞɟɥɹɟɬɫɹ ɢɡ ɭɪɚɜɧɟɧɢɣ (3.121) ɢ (3.122) ɩɪɢ ɫɥɟɞɭɸɳɢɯ
ɧɚɱɚɥɶɧɵɯ ɢ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɹɯ.
ɇɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ: V = 0, P = 0 ɩɪɢ t 0, 0 x L, ɬɨ ɟɫɬɶ ɞɨ
ɦɨɦɟɧɬɚ ɜɪɟɦɟɧɢ t = 0 ɞɜɢɠɟɧɢɹ ɛɵɥɨ ɭɫɬɚɧɨɜɢɜɲɢɦɫɹ. ɉɪɢ ɷɬɨɦ ɩɨɞ V
ɢ P ɩɨɞɪɚɡɭɦɟɜɚɸɬ ɢɡɛɵɬɨɱɧɵɟ ɡɧɚɱɟɧɢɹ ɫɤɨɪɨɫɬɢ ɢ ɞɚɜɥɟɧɢɹ ɜ ɭɫɬɚɧɨ
-
ɜɢɜɲɟɦɫɹ ɪɟɠɢɦɟ.
Ƚɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ: x = 0, V = f(t), x = L, P = P0 = const, ɬɨ ɟɫɬɶ, ɤ
ɨɞɧɨɦɭ ɤɨɧɰɭ ɬɪɭɛɨɩɪɨɜɨɞɚ ɩɪɢɫɨɟɞɢɧɟɧ ɚɝɪɟɝɚɬ, ɢɡɦɟɧɹɸɳɢɣ ɪɚɫɯɨɞ
ɠɢɞɤɨɫɬɢ ɩɨ ɢɡɜɟɫɬɧɨɦɭ ɡɚɤɨɧɭ ɢ ɩɪɢɫɨɟɞɢɧɟɧɧɵɣ ɤ ɬɪɭɛɨɩɪɨɜɨɞɭ ɱɟɪɟɡ
ɪɟɝɭɥɢɪɭɸɳɭɸ ɡɚɞɜɢɠɤɭ. ɇɚ ɞɪɭɝɨɦ ɤɨɧɰɟ ɬɪɭɛɨɩɪɨɜɨɞɚ ɩɪɨɢɫɯɨɞɢɬ
ɫɜɨɛɨɞɧɨɟ ɢɫɬɟɱɟɧɢɟ ɠɢɞɤɨɫɬɟɣ ɩɪɢ ɩɨɫɬɨɹɧɧɨɦ ɞɚɜɥɟɧɢɢ, ɪɚɜɧɨɦ ɚɬɦɨɫɮɟɪɧɨɦɭ. ɉɪɢ ɬɚɤɢɯ ɧɚɱɚɥɶɧɵɯ ɢ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɹɯ ɪɟɲɟɧɢɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ (3.121) ɢ (3.122) ɞɚɺɬ ɜɵɪɚɠɟɧɢɹ, ɩɪɢɜɟɞɟɧɧɵɟ ɜ
[13], ɤɨɬɨɪɵɟ ɫɨɨɬɜɟɬɫɬɜɭɸɬ ɩɟɪɟɞɚɬɨɱɧɵɦ ɮɭɧɤɰɢɹɦ ɞɥɹ ɞɥɢɧɧɵɯ ɢ ɤɨɪɨɬɤɢɯ ɬɪɭɛɨɩɪɨɜɨɞɨɜ:
k
ɝɞɟ kɩɬ – ɩɟɪɟɞɚɬɨɱɧɵɣ ɤɨɷɮɮɢɰɢɟɧɬ ɬɪɭɛɨɩɪɨɜɨɞɚ:
, ɊɊ – ɞɚɜɥɟɧɢɹ ɜ ɧɚɱɚɥɟ ɬɪɭɛɨɩɪɨɜɨɞɚ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɜ ɧɚɱɚɥɶɧɨɦ
0010
k
ɩɬ
ȱȱȱ
ȱȱȱ
sW
sW
ɩɬ
2
sɌ
1
k
ɩɬ
22
V
'
W
s
, (3.123)
ɟ
1
s
ɟ
sɌsɌ
1
1
ɿɿLɊɊ
100010
W
, (3.124)
;
ɢ ɤɨɧɟɱɧɨɦ ɭɫɬɚɧɨɜɢɜɲɢɯɫɹ ɫɨɫɬɨɹɧɢɹɯ;
119
i0, i1 – ɭɞɟɥɶɧɵɟ ɩɨɬɟɪɢ ɞɚɜɥɟɧɢɹ, ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ, ɜ ɧɚɱɚɥɶɧɨɦ ɢ
W
ɤɨɧɟɱɧɨɦ ɭɫɬɚɧɨɜɢɜɲɢɯɫɹ ɫɨɫɬɨɹɧɢɹɯ;
ǻV – ɢɡɦɟɧɟɧɢɟ ɫɤɨɪɨɫɬɢ ɩɨɬɨɤɚ ɜ ɬɪɭɛɟ;
ɫL
– ɜɪɟɦɹ ɱɢɫɬɨɝɨ ɡɚɩɚɡɞɵɜɚɧɢɹ;
2
1
ɜ ɬɪɭɛɟ;
2
ɥɟɛɚɧɢɣ.
Ʉɨɷɮɮɢɰɢɟɧɬɵ ɜ ɭɪɚɜɧɟɧɢɹɯ (3.123) ɢ (3.124) ɦɨɝɭɬ ɛɵɬɶ ɪɚɫɫɱɢɬɚɧɵ ɬɟɨɪɟɬɢɱɟɫɤɢɦ ɩɭɬɺɦ ɬɨɥɶɤɨ ɥɢɲɶ ɜ ɫɥɭɱɚɟ ɬɟɱɟɧɢɹ ɨɞɧɨɪɨɞɧɨɣ ɠɢɞɤɨɫɬɢ, ɢɦɟɸɳɟɣ ɦɟɫɬɨ ɜ ɡɚɤɥɸɱɢɬɟɥɶɧɨɣ ɮɚɡɟ ɪɟɚɥɢɡɚɰɢɢ ɫɩɨɫɨɛɚ ɪɟɝɭɥɢɪɨɜɚɧɢɹ ɝɢɞɪɨɬɪɚɧɫɩɨɪɬɧɨɣ ɭɫɬɚɧɨɜɤɢ ɫ ɧɟɩɨɥɧɨɣ ɩɪɨɦɵɜɤɨɣ ɬɪɚɧɫɩɨɪɬɧɨɝɨ ɬɪɭɛɨɩɪɨɜɨɞɚ. ȼ ɞɪɭɝɢɯ ɫɥɭɱɚɹɯ ɡɧɚɱɟɧɢɟ ɷɬɢɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ
ɨɩɪɟɞɟɥɹɸɬ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɦ ɩɭɬɺɦ
ɟɤɬɟ, ɥɢɛɨ ɧɚ ɟɝɨ ɦɨɞɟɥɢ ɫ ɫɨɛɥɸɞɟɧɢɟɦ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɤɪɢɬɟɪɢɟɜ ɩɨɞɨɛɢɹ [14].
– ɩɨɫɬɨɹɧɧɚɹ ɜɪɟɦɟɧɢ ɞɟɦɩɮɢɪɨɜɚɧɢɹ ɤɨɥɟɛɚɧɢɣ ɞɚɜɥɟɧɢɹ
ɚɫLɌ
2
ɚ
2
ɫLɌ
– ɩɨɫɬɨɹɧɧɚɹ ɜɪɟɦɟɧɢ «ɪɚɫɤɚɱɢɜɚɧɢɹ» ɫɨɛɫɬɜɟɧɧɵɯ ɤɨ-
!3
ɧɚ ɪɟɚɥɶɧɨɦ ɩɪɨɦɵɲɥɟɧɧɨɦ ɨɛɴ-
120
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