Медведев Г.Г., Максимов В.И., Половников В.Ю. Практикум по гидравлическим расчетам в теплоэнергетике
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11
ɝɞɟ
K
– ɜɟɤɬɨɪ ɦɚɫɫɨɜɵɯ ɫɢɥ, P – ɜɟɤɬɨɪ ɩɨɜɟɪɯɧɨɫɬɧɵɯ ɫɢɥ (ɫɢɥɵ
ɞɚɜɥɟɧɢɹ ɢ ɬɪɟɧɢɹ).
Ɇɚɫɫɨɜɵɟ ɫɢɥɵ ɫɥɟɞɭɟɬ ɫɱɢɬɚɬɶ ɡɚɞɚɧɧɵɦɢ, ɧɚɩɪɢɦɟɪ, ɝɪɚɜɢɬɚɰɢ-
ɨɧɧɵɟ ɫɢɥɵ, ɤɨɬɨɪɵɟ ɱɟɪɟɡ ɩɪɨɟɤɰɢɢ ɧɚ ɨɫɢ ɤɨɨɪɞɢɧɚɬ ɜɵɪɚɠɚɸɬɫɹ ɤɚɤ
ijk
K
XYZ
U
.
ɉɨɜɟɪɯɧɨɫɬɧɵɟ ɠɟ ɫɢɥɵ ɡɚɜɢɫɹɬ ɨɬ ɫɤɨɪɨɫɬɢ, ɫ ɤɨɬɨɪɨɣ ɱɚɫɬɢɰɵ
ɠɢɞɤɨɫɬɢ ɞɟɮɨɪɦɢɪɭɸɬɫɹ ɜ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɦ ɩɨɥɟ ɫɤɨɪɨɫɬɟɣ. ɋɨɜɨ-
ɤɭɩɧɨɫɬɶ ɫɢɥ ɨɩɪɟɞɟɥɹɟɬ ɧɚɩɪɹɠɟɧɧɨɟ ɫɨɫɬɨɹɧɢɟ ɬɟɥɚ. Ɉɩɭɫɤɚɹ ɩɨɞɪɨɛ-
ɧɨɫɬɢ ɭɫɬɚɧɨɜɥɟɧɢɹ ɫɜɹɡɟɣ ɦɟɠɞɭ ɧɚɩɪɹɠɟɧɧɵɦ ɫɨɫɬɨɹɧɢɟɦ ɢ ɫɤɨɪɨ-
ɫɬɶɸ ɞɟɮɨɪɦɚɰɢɢ, ɩɪɢɜɟɞɟɦ ɭɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ ɞɥɹ ɜɹɡɤɨɣ ɧɟɫɠɢɦɚɟ-
ɦɨɣ ɠɢɞɤɨɫɬɢ, ɤɨɬɨɪɵɟ ɧɚɡɵɜɚɸɬɫɹ ɭɪɚɜɧɟɧɢɹɦɢ ɇɚɜɶɟ-ɋɬɨɤɫɚ:
222
222
x
xxx
du P uuu
X
dt x x y z
UU P
§·
w www
¨¸
wwww
©¹
222
222
yyyy
du uuu
P
Y
dt y x y z
UU P
§·
www
w
¨¸
¨¸
wwww
©¹
222
222
z
zzz
ɦɚɫɫɨɜɵɟ
ɫɢɥɵ
ɫɢɥɵ ɫɢɥɵ
ɫɢɥɵ ɜɹɡɤɨɫɬɢ ɢɥɢ ɬɪɟɧɢɹ
ɢɧɟɪɰɢɢ ɞɚɜɥɟɧɢɹ
du P uuu
Z
dt z x y z
UU P
§·
w www
¨¸
wwww
©¹
ȼ ɩɪɢɜɟɞɟɧɧɨɣ ɫɢɫɬɟɦɟ ɭɪɚɜɧɟɧɢɣ, ɤɚɤ ɧɟɬɪɭɞɧɨ ɡɚɦɟɬɢɬɶ, ɢɦɟɟɦ
ɱɟɬɵɪɟ ɧɟɢɡɜɟɫɬɧɵɯ ɜɟɥɢɱɢɧɵ, ɚ, ɢɦɟɧɧɨ, , ,
xyz
uuu ɢ P. Ⱦɥɹ ɬɨɝɨ,
ɱɬɨɛɵ ɡɚɦɤɧɭɬɶ ɫɢɫɬɟɦɭ, ɬ. ɟ. ɩɪɢɜɟɫɬɢ ɜ ɫɨɨɬɜɟɬɫɬɜɢɟ ɱɢɫɥɨ ɧɟɢɡɜɟɫɬ-
ɧɵɯ ɜɟɥɢɱɢɧ ɢ ɭɪɚɜɧɟɧɢɣ, ɧɟɨɛɯɨɞɢɦɨ ɡɚɩɢɫɚɬɶ ɱɟɬɜɟɪɬɨɟ ɭɪɚɜɧɟɧɢɟ,
ɤɚɤɢɦ ɹɜɥɹɟɬɫɹ ɭɪɚɜɧɟɧɢɟ ɧɟɪɚɡɪɵɜɧɨɫɬɢ
0
y
xz
u
uu
xyz
w
ww
www
.
ɉɪɢɜɟɞɟɧɧɚɹ ɫɢɫɬɟɦɚ ɭɪɚɜɧɟɧɢɣ ɨɩɢɫɵɜɚɟɬ ɥɸɛɨɟ ɬɟɱɟɧɢɟ ɜɹɡɤɨɣ
ɧɟɫɠɢɦɚɟɦɨɣ
const
U
ɠɢɞɤɨɫɬɢ, ɬ. ɟ. ɰɟɥɵɣ ɤɥɚɫɫ ɡɚɞɚɱ.
Ⱦɥɹ ɬɨɝɨ, ɱɬɨɛɵ ɢɡ ɷɬɨɝɨ ɤɥɚɫɫɚ ɜɵɞɟɥɢɬɶ ɤɨɧɤɪɟɬɧɭɸ ɡɚɞɚɱɭ, ɧɟ-
ɨɛɯɨɞɢɦɨ ɡɚɞɚɬɶ
ɭɫɥɨɜɢɹ ɨɞɧɨɡɧɚɱɧɨɫɬɢ, ɤɨɬɨɪɵɟ ɜɤɥɸɱɚɸɬ:
x ɝɟɨɦɟɬɪɢɱɟɫɤɢɟ ɭɫɥɨɜɢɹ (ɪɚɡɦɟɪɵ ɢ ɮɨɪɦɚ ɩɨɬɨɤɚ);
x ɮɢɡɢɱɟɫɤɢɟ ɭɫɥɨɜɢɹ;
x ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ (ɡɧɚɱɟɧɢɹ ɫɤɨɪɨɫɬɢ ɢ ɞɚɜɥɟɧɢɹ ɜ ɧɚɱɚɥɶɧɵɣ
ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ
0
tt ). ȼ ɫɥɭɱɚɟ ɭɫɬɚɧɨɜɢɜɲɟɝɨɫɹ ɬɟɱɟɧɢɹ ɧɚɱɚɥɶɧɵɟ
ɭɫɥɨɜɢɹ ɧɟ ɡɚɞɚɸɬɫɹ;
x ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ (ɡɧɚɱɟɧɢɹ ɫɤɨɪɨɫɬɟɣ ɧɚ ɝɪɚɧɢɰɚɯ ɩɨɬɨɤɚ).
12
ɋɥɟɞɭɟɬ ɢɦɟɬɶ ɜ ɜɢɞɭ, ɱɬɨ ɞɨ ɧɚɫɬɨɹɳɟɝɨ ɜɪɟɦɟɧɢ ɜɫɥɟɞɫɬɜɢɟ
ɛɨɥɶɲɢɯ ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɬɪɭɞɧɨɫɬɟɣ ɧɟ ɩɨɥɭɱɟɧɨ ɧɢ ɨɞɧɨɝɨ ɨɛɳɟɝɨ
ɪɟɲɟɧɢɹ ɭɪɚɜɧɟɧɢɣ ɇɚɜɶɟ-ɋɬɨɤɫɚ ɜ ɢɯ ɩɨɥɧɨɦ ɜɢɞɟ, ɬ. ɟ. ɫ ɫɨɯɪɚɧɟɧɢɟɦ
ɜɫɟɯ ɱɥɟɧɨɜ. Ɉɞɧɚɤɨ ɢɡɜɟɫɬɧɵ ɧɟɤɨɬɨɪɵɟ ɱɚɫɬɧɵɟ ɪɟɲɟɧɢɹ, ɤɨɝɞɚ ɭɪɚɜ-
ɧɟɧɢɹ ɇɚɜɶɟ-ɋɬɨɤɫɚ ɭɩɪɨɳɚɸɬɫɹ ɢ ɥɟɝɤɨ ɢɧɬɟɝɪɢɪɭɸɬɫɹ, ɧɚɩɪɢɦɟɪ, ɞɥɹ
ɥɚɦɢɧɚɪɧɨɝɨ (ɫɥɨɢɫɬɨɝɨ) ɬɟɱɟɧɢɹ ɜ
ɬɪɭɛɚɯ.
1.4.2.4. Ɏɫɛɝɨɠɨɣɠ ɟɝɣɡɠɨɣɺ ɣɟɠɛɦɷɨɩɤ (ɨɠɝɺɢɥɩɤ) ɡɣɟɥɩɬɭɣ
Ⱦɥɹ ɭɩɪɨɳɟɧɢɹ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɦɨɞɟɥɢ ɞɜɢɠɟɧɢɹ ɜɜɨɞɢɬɫɹ ɩɨɧɹ-
ɬɢɟ ɢɞɟɚɥɶɧɨɣ ɠɢɞɤɨɫɬɢ, ɧɟ ɫɭɳɟɫɬɜɭɸɳɟɣ ɜ ɩɪɢɪɨɞɟ. ɋɥɟɞɭɟɬ ɨɬɦɟ-
ɬɢɬɶ, ɱɬɨ ɜ ɪɹɞɟ ɫɥɭɱɚɟɜ ɜɥɢɹɧɢɟɦ ɜɹɡɤɨɫɬɢ ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ.
Ⱦɥɹ ɜɵɜɨɞɚ ɷɬɢɯ ɭɪɚɜɧɟɧɢɣ ɢɫɩɨɥɶɡɭɟɦ ɭɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ ɇɚ-
ɜɶɟ-ɋɬɨɤɫɚ. ȼ ɢɞɟɚɥɶɧɨɣ ɠɢɞɤɨɫɬɢ ɫɢɥɵ ɜɹɡɤɨɫɬɢ ɨɬɫɭɬɫɬɜɭɸɬ, ɢ ɩɨ-
ɷɬɨɦɭ, ɩɪɢɪɚɜɧɢɜɚɹ ɢɯ ɤ ɧɭɥɸ, ɛɭɞɟɦ ɢɦɟɬɶ
,
,
.
x
y
z
du P
X
dt x
du
P
Y
dt y
du P
Z
dt z
UU
UU
UU
w
w
w
w
w
w
ɗɬɢ ɭɪɚɜɧɟɧɢɹ ɞɥɹ ɢɞɟɚɥɶɧɨɣ ɠɢɞɤɨɫɬɢ ɧɚɡɵɜɚɸɬɫɹ ɭɪɚɜɧɟɧɢɹɦɢ
ɗɣɥɟɪɚ.
1.4.3. Ɏɫɛɝɨɠɨɣɺ ȼɠɫɨɮɦɦɣ ɟɦɺ ɸɦɠɧɠɨɭɛɫɨɩɤ ɬɭɫɮɤɥɣ
ɣɟɠɛɦɷɨɩɤ ɨɠɬɡɣɧɛɠɧɩɤ ɡɣɟɥɩɬɭɣ
Ɋɚɫɫɦɨɬɪɢɦ ɭɫɬɚɧɨɜɢɜɲɟɟɫɹ ɞɜɢɠɟɧɢɟ ɜɞɨɥɶ ɥɢɧɢɢ ɬɨɤɚ ɩɨɞ ɞɟɣ-
ɫɬɜɢɟɦ ɫɢɥɵ ɬɹɠɟɫɬɢ. ɍɦɧɨɠɚɹ ɤɚɠɞɨɟ ɢɡ ɩɪɢɜɟɞɟɧɧɵɯ ɭɪɚɜɧɟɧɢɣ ɫɨ-
ɨɬɜɟɬɫɬɜɟɧɧɨ ɧɚ ,dx dy ɢ dz ɩɨɫɥɟ ɫɭɦɦɢɪɨɜɚɧɢɹ, ɩɨɥɭɱɢɦ
y
xz
du
du du
dx dy dz
dt dt dz
PPP
Xdx Ydy Zdz dx dy dz
xyz
U
U
§·
¨¸
©¹
§·
www
¨¸
www
©¹
13
Ʌɟɜɚɹ ɱɚɫɬɶ, ɩɨɥɭɱɟɧɧɨɝɨ ɜɵɪɚɠɟɧɢɹ, ɥɟɝɤɨ ɩɪɟɨɛɪɚɡɭɟɬɫɹ, ɟɫɥɢ
,dx dy
ɢ dz ɩɪɟɞɫɬɚɜɢɬɶ ɤɚɤ ,,
xyz
dx u dt dy u dt dz u dt
2
1
2
y
xz
xx yy zz
du
du du
dx dy dz
dt dt dz
udu udu udu du
U
UU
§·
¨¸
©¹
.
ɉɪɢ ɞɟɣɫɬɜɢɢ ɧɚ ɠɢɞɤɨɫɬɶ ɨɞɧɨɣ ɦɚɫɫɨɜɨɣ ɫɢɥɵ – ɫɢɥɵ ɬɹɠɟɫɬɢ,
ɩɪɨɟɤɰɢɢ ɭɫɤɨɪɟɧɢɣ ɧɚ ɨɫɢ ɤɨɨɪɞɢɧɚɬ ɛɭɞɭɬ ɪɚɜɧɵ 0, 0,XYZg
.
ɇɚɤɨɧɟɰ, ɜɵɪɚɠɟɧɢɟ ɜ ɫɤɨɛɤɚɯ ɜ ɩɪɚɜɨɣ ɱɚɫɬɢ ɭɪɚɜɧɟɧɢɹ ɩɪɟɞɫɬɚɜ-
ɥɹɟɬ ɫɨɛɨɣ ɩɨɥɧɵɣ ɞɢɮɮɟɪɟɧɰɢɚɥ ɞɚɜɥɟɧɢɹ dP . Ɍɨɝɞɚ, ɩɨɞɫɬɚɜɥɹɹ ɩɨ-
ɥɭɱɟɧɧɵɟ ɫɨɨɬɧɨɲɟɧɢɹ ɜ ɢɫɯɨɞɧɨɟ ɭɪɚɜɧɟɧɢɟ, ɛɭɞɟɦ ɢɦɟɬɶ
2
11
0
2
gdz dP du
U
.
ɂɧɬɟɝɪɢɪɭɹ ɩɨɫɥɟɞɧɟɟ, ɩɨɥɭɱɢɦ
2
2
Pu
gz const
U
.
Ⱦɚɧɧɨɟ ɭɪɚɜɧɟɧɢɟ – ɟɫɬɶ ɦɚɬɟɦɚɬɢɱɟɫɤɨɟ ɨɬɨɛɪɚɠɟɧɢɟ ɬɟɨɪɟɦɵ
Ȼɟɪɧɭɥɢɢ, ɢ ɨɬɨɛɪɚɠɚɟɬ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɷɧɟɪɝɢɢ ɞɥɹ ɩɨɬɨɤɚ ɢɞɟɚɥɶ-
ɧɨɣ ɠɢɞɤɨɫɬɢ.
Ʉɚɠɞɨɟ ɫɥɚɝɚɟɦɨɟ ɭɪɚɜɧɟɧɢɟ Ȼɟɪɧɭɥɥɢ ɫ ɮɢɡɢɱɟɫɤɨɣ ɬɨɱɤɢ ɡɪɟɧɢɹ
ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɭɞɟɥɶɧɭɸ (ɨɛɧɟɫɟɧɧɭɸ ɤ ɟɞɢɧɢɰɟ ɦɚɫɫɵ) ɦɟɯɚɧɢɱɟ-
ɫɤɭɸ ɷɧɟɪɝɢɸ, ɚ, ɢɦɟɧɧɨ:
gz– ɭɞɟɥɶɧɚɹ ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɩɨɥɨɠɟɧɢɹ, ɬ. ɟ. ɨɩɪɟɞɟɥɹɟɦɚɹ
ɜɵɫɨɬɨɣ
z ɨɬ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɥɨɫɤɨɫɬɢ, Ⱦɠ/ɤɝ;
P
U
– ɭɞɟɥɶɧɚɹ ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɞɚɜɥɟɧɢɹ, Ⱦɠ/ɤɝ;
2
2
u
– ɭɞɟɥɶɧɚɹ ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ, Ⱦɠ/ɤɝ.
ɋɭɦɦɚ ɜɫɟɯ ɫɥɚɝɚɟɦɵɯ ɬɪɟɯɱɥɟɧɚ Ȼɟɪɧɭɥɥɢ ɨɩɪɟɞɟɥɹɟɬ ɩɨɥɧɭɸ
ɭɞɟɥɶɧɭɸ ɷɧɟɪɝɢɸ.
ɇɚ ɪɢɫ. 1.4.3.1 ɩɪɢɜɟɞɟɧɚ ɝɟɨɦɟɬɪɢɱɟɫɤɚɹ ɢɧɬɟɪɩɪɟɬɚɰɢɹ ɫɥɚɝɚɟɦɵɯ
ɭɪɚɜɧɟɧɢɹ Ȼɟɪɧɭɥɥɢ. ɍɪɚɜɧɟɧɢɟ Ȼɟɪɧɭɥɥɢ ɧɚɲɥɨ ɲɢɪɨɤɨɟ ɩɪɢɦɟɧɟɧɢɟ
ɩɪɢ ɪɚɫɱɟɬɟ ɪɚɡɥɢɱɧɵɯ ɬɭɪɛɨ- ɢ ɝɚɡɨɩɪɨɜɨɞɨɜ. ȼ ɩɟɪɜɨɦ ɫɥɭɱɚɟ ɷɬɨ
ɭɪɚɜɧɟɧɢɟ ɩɪɟɞɫɬɚɜɥɹɸɬ ɜ ɟɞɢɧɢɰɚɯ ɧɚɩɨɪɚ:
2
2
Pu
zconst
gg
U
, ɦ.
14
Ɋɢɫ. 1.4.3.1
ȼɨ ɜɬɨɪɨɦ – ɜ ɟɞɢɧɢɰɚɯ ɞɚɜɥɟɧɢɹ
2
2
Pu
gz P const
g
, ɉɚ
Ʉɚɠɞɨɟ ɫɥɚɝɚɟɦɨɟ ɭɪɚɜɧɟɧɢɹ Ȼɟɪɧɭɥɥɢ ɡɚɩɢɫɵɜɚɟɬɫɹ ɜ ɟɞɢɧɢɰɚɯ
ɧɚɩɨɪɚ, ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɜɵɫɨɬɭ ɢ ɦɨɠɟɬ ɛɵɬɶ ɢɧɬɟɪɩɪɟɬɢɪɨɜɚɧɨ
ɝɪɚɮɢɱɟɫɤɢ:
z – ɧɢɜɟɥɢɪɧɚɹ ɜɵɫɨɬɚ, ɦ;
P
g
U
– ɩɶɟɡɨɦɟɬɪɢɱɟɫɤɚɹ ɜɵɫɨɬɚ, ɦ;
P
z
g
U
– ɩɶɟɡɨɦɟɬɪɢɱɟɫɤɢɣ ɧɚɩɨɪ, ɦ;
2
2
u
g
– ɫɤɨɪɨɫɬɧɚɹ ɜɵɫɨɬɚ ɢɥɢ ɫɤɨɪɨɫɬɧɨɣ ɧɚɩɨɪ, ɦ;
2
2
Pu
z
gg
U
– ɩɨɥɧɵɣ ɝɢɞɪɨɞɢɧɚɦɢɱɟɫɤɢɣ ɧɚɩɨɪ, ɦ.
15
1.4.4. ɍɠɲɠɨɣɠ ɣɟɠɛɦɷɨɩɞɩ ɞɛɢɛ
1.4.4.1. Ɍɩɩɭɨɩɳɠɨɣɺ ɭɠɫɧɩɟɣɨɛɧɣɥɣ.
ɉɪɢ ɛɨɥɶɲɢɯ ɫɤɨɪɨɫɬɹɯ ɬɟɱɟɧɢɹ ɝɚɡɚ, ɫɪɚɜɧɢɜɚɟɦɵɯ ɫɨ ɫɤɨɪɨɫɬɶɸ
ɡɜɭɤɚ, ɢɡɦɟɧɟɧɢɟ ɫɤɨɪɨɫɬɢ ɩɪɢɜɨɞɢɬ ɤ ɢɡɦɟɧɟɧɢɸ ɩɥɨɬɧɨɫɬɢ. Ɍɚɤɨɟ
ɞɜɢɠɟɧɢɟ ɢɡɭɱɚɟɬɫɹ ɝɚɡɨɜɨɣ ɞɢɧɚɦɢɤɨɣ.
Ʉɚɤ ɢɡɜɟɫɬɧɨ ɢɡ ɤɭɪɫɚ ɬɟɯɧɢɱɟɫɤɨɣ ɬɟɪɦɨɞɢɧɚɦɢɤɢ, ɨɫɧɨɜɧɵɟ ɩɚ-
ɪɚɦɟɬɪɵ ɫɨɫɬɨɹɧɢɹ ɝɚɡɚ: ɞɚɜɥɟɧɢɟ – P, ɩɥɨɬɧɨɫɬɶ ɢ ɚɛɫɨɥɸɬɧɚɹ ɬɟɦɩɟ-
ɪɚɬɭɪɚ – T . Ⱦɥɹ ɢɞɟɚɥɶɧɵɯ ɝɚɡɨɜ ɨɧɢ ɫɜɹɡɚɧɵ ɭɪɚɜɧɟɧɢɟɦ ɫɨɫɬɨɹɧɢɹ
P
R
T
U
,
ɝɞɟ
R
– ɝɚɡɨɜɚɹ ɩɨɫɬɨɹɧɧɚɹ. Ⱦɥɹ ɜɨɡɞɭɯɚ, ɧɚɩɪɢɦɟɪ, ɜ ɫɢɫɬɟɦɟ ɋɂ
287,1
R
Ⱦɠ/ɤɝɝɪɚɞ.
ȼ ɛɨɥɶɲɢɧɫɬɜɟ ɡɚɞɚɱ, ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɯ ɝɚɡɨɞɢɧɚɦɢɤɨɣ, ɩɪɨɰɟɫɫɵ
ɢɡɦɟɧɟɧɢɹ ɫɨɫɬɨɹɧɢɹ ɢɡ-ɡɚ ɢɯ ɛɵɫɬɪɨɬɟɱɧɨɫɬɢ ɦɨɠɧɨ ɫɱɢɬɚɬɶ ɚɞɢɚɛɚɬɢ-
ɱɟɫɤɢɦɢ, ɬ. ɟ. ɨɧɢ ɩɪɨɬɟɤɚɸɬ ɛɟɡ ɬɟɩɥɨɨɛɦɟɧɚ ɫ ɨɤɪɭɠɚɸɳɟɣ ɫɪɟɞɨɣ.
ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɞɚɜɥɟɧɢɟ ɢ ɩɥɨɬɧɨɫɬɶ ɫɜɹɡɚɧɵ ɫɨɨɬɧɨɲɟɧɢɟɦ
k
Pv const ,
k
P
const
U
ɢɥɢ
0
k
k
P
U
U
U
§·
¨¸
©¹
,
ɝɞɟ
p
c
k
c
Q
– ɩɨɤɚɡɚɬɟɥɶ ɚɞɢɚɛɚɬɵ;
p
c ɢ c
Q
– ɬɟɩɥɨɟɦɤɨɫɬɢ ɩɪɢ ɩɨɫɬɨɹɧ-
ɧɨɦ ɞɚɜɥɟɧɢɢ ɢ ɨɛɴɟɦɟ. Ⱦɥɹ ɜɨɡɞɭɯɚ ɢ ɞɪɭɝɢɯ ɞɜɭɯɚɬɨɦɧɵɯ ɝɚɡɨɜ
1, 4k
, ɚ ɞɥɹ ɩɟɪɟɝɪɟɬɨɝɨ ɩɚɪɚ, ɧɚɩɪɢɦɟɪ
1, 33k
.
ɂɫɩɨɥɶɡɭɹ ɭɪɚɜɧɟɧɢɟ Ɇɟɧɞɟɥɟɟɜɚ-Ʉɥɚɣɩɟɪɨɧɚ ɢ ɚɞɢɚɛɚɬɢɱɟɫɤɨɝɨ
ɩɪɨɰɟɫɫɚ, ɩɨɥɭɱɢɦ
11
1
11
00 0 00 00
;;
kk
k
kk k
TTT
TTT
UUU U
UUU U
§· §· §· §·
¨¸ ¨¸ ¨¸ ¨¸
©¹ ©¹ ©¹ ©¹
.
ȼ ɡɚɞɚɱɚɯ ɨ ɞɜɢɠɟɧɢɢ ɝɚɡɚ ɜ ɞɥɢɧɧɨɣ ɬɪɭɛɟ ɛɟɡ ɢɡɨɥɹɰɢɢ ɫɬɟɧɨɤ
ɩɪɨɰɟɫɫ ɫɨɫɬɨɹɧɢɹ ɦɨɠɧɨ ɫɱɢɬɚɬɶ ɢɡɨɬɟɪɦɢɱɟɫɤɢɦ: ɞɥɢɬɟɥɶɧɵɣ ɤɨɧɬɚɤɬ
ɫɨ ɫɬɟɧɤɚɦɢ ɬɪɭɛɵ ɩɪɢɜɨɞɢɬ ɤ ɬɨɦɭ, ɱɬɨ ɬɟɦɩɟɪɚɬɭɪɚ ɝɚɡɚ ɫɬɚɧɨɜɢɬɫɹ
ɪɚɜɧɨɣ ɬɟɦɩɟɪɚɬɭɪɟ ɫɬɟɧɨɤ. Ⱦɥɹ ɢɡɨɬɟɪɦɢɱɟɫɤɨɝɨ ɩɪɨɰɟɫɫɚ
Pconst
Q
,
ɢɥɢ
P
const
U
.
16
1.4.4.2. Ɍɥɩɫɩɬɭɷ ɢɝɮɥɛ. ɒɣɬɦɩ ɇɛɰɛ
ɋɤɨɪɨɫɬɶ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɦɚɥɵɯ ɜɨɡɦɭɳɟɧɢɣ ɜ ɠɢɞɤɨɫɬɢ ɪɚɜɧɚ
ɫɤɨɪɨɫɬɢ ɡɜɭɤɚ, ɬ. ɟ. ɫɤɨɪɨɫɬɢ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɭɩɪɭɝɢɯ ɤɨɥɟɛɚɧɢɣ, ɤɨ-
ɬɨɪɚɹ ɨɩɪɟɞɟɥɹɟɬɫɹ ɤɚɤ
E
dP
a
U
U
.
ɉɪɨɰɟɫɫ ɢɡɦɟɧɟɧɢɹ ɩɚɪɚɦɟɬɪɨɜ ɝɚɡɚ ɜ ɡɜɭɤɨɜɨɣ ɜɨɥɧɟ, ɤɨɬɨɪɚɹ
ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɪɚɫɩɪɨɫɬɪɚɧɹɸɳɢɟɫɹ ɜ ɝɚɡɟ ɫɥɚɛɵɟ ɜɨɡɦɭɳɟɧɢɹ
ɞɚɜɥɟɧɢɹ ɢ ɩɥɨɬɧɨɫɬɢ, ɫɥɟɞɭɟɬ ɫɱɢɬɚɬɶ ɚɞɢɚɛɚɬɧɵɦ, ɞɥɹ ɤɨɬɨɪɨɝɨ
k
Pc
U
. Ɉɬɫɸɞɚ
1k
dP k c d
U
U
ɢɥɢ
1k
dP P
kc k
d
U
U
U
.
ɉɨɞɫɬɚɜɥɹɹ ɩɨɫɥɟɞɧɟɟ ɜ ɮɨɪɦɭɥɭ ɞɥɹ ɫɤɨɪɨɫɬɢ ɡɜɭɤɚ, ɩɨɥɭɱɢɦ
P
ak kRT
U
.
ɋɤɨɪɨɫɬɶ ɡɜɭɤɚ – ɨɞɧɚ ɢɡ ɜɚɠɧɟɣɲɢɯ ɦɟɯɚɧɢɱɟɫɤɢɯ ɯɚɪɚɤɬɟɪɢɫɬɢɤ
ɝɚɡɚ. Ɂɚɤɨɧɵ ɟɝɨ ɞɜɢɠɟɧɢɹ ɪɟɡɤɨ ɨɬɥɢɱɚɸɬɫɹ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɫɨɨɬɧɨ-
ɲɟɧɢɹ ɫɤɨɪɨɫɬɢ ɝɚɡɚ
u ɢ ɫɤɨɪɨɫɬɢ ɡɜɭɤɚ a.
Ɉɬɧɨɲɟɧɢɟ
u
M
a
ɧɚɡɵɜɚɟɬɫɹ ɱɢɫɥɨɦ Ɇɚɯɚ. Ɍɟɱɟɧɢɹ, ɜ ɤɨɬɨɪɵɯ
ua ɢ 1
M
, ɧɚɡɵɜɚɸɬɫɹ ɞɨɡɜɭɤɨɜɵɦɢ, ɟɫɥɢ ua! ɢ 1
M
! – ɫɜɟɪɯɡɜɭ-
ɤɨɜɵɦɢ.
1.4.5. Ɏɫɛɝɨɠɨɣɺ ɸɨɠɫɞɣɣ
Ɋɚɫɫɦɨɬɪɢɦ ɭɫɬɚɧɨɜɢɜɲɟɟɫɹ ɨɞɧɨɦɟɪɧɨɟ ɞɜɢɠɟɧɢɟ ɝɚɡɚ. ȿɫɥɢ
ɟɞɢɧɫɬɜɟɧɧɨɣ ɦɚɫɫɨɜɨɣ ɫɢɥɨɣ, ɞɟɣɫɬɜɭɸɳɟɣ ɧɚ ɝɚɡ, ɹɜɥɹɟɬɫɹ ɫɢɥɚ ɬɹɠɟ-
ɫɬɢ, ɬɨ ɬɚɤɨɟ ɬɟɱɟɧɢɟ ɨɩɢɫɵɜɚɟɬɫɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɦ ɭɪɚɜɧɟɧɢɟɦ
2
0
2
dP u
gdz d
U
§·
¨¸
©¹
.
ȼɫɥɟɞɫɬɜɢɟ ɦɚɥɨɣ ɩɥɨɬɧɨɫɬɢ ɝɚɡɚ ɞɨɩɭɫɬɢɦɨ ɩɪɟɧɟɛɪɟɱɶ ɩɟɪɜɵɦ
ɱɥɟɧɨɦ ɭɪɚɜɧɟɧɢɹ, ɬɚɤ ɤɚɤ ɞɥɹ ɱɚɫɬɢɰɵ ɝɚɡɚ ɫɢɥɵ ɜɟɫɚ ɩɪɟɧɟɛɪɟɠɢɦɨ
ɦɚɥɵ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɫɢɥɚɦɢ ɢɧɟɪɰɢɢ ɞɚɜɥɟɧɢɹ. Ɍɨɝɞɚ ɩɪɟɞɵɞɭɳɟɟ
ɭɪɚɜɧɟɧɢɟ ɩɪɢɨɛɪɟɬɚɟɬ ɜɢɞ:
2
0
2
dP u
d
U
§·
¨¸
©¹
.
ɗɬɨ ɜɵɪɚɠɟɧɢɟ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɭɪɚɜɧɟɧɢɟ ɷɧɟɪɝɢɢ ɞɥɹ ɝɚɡɚ ɜ
ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɣ ɮɨɪɦɟ.
17
ɋɱɢɬɚɹ ɬɟɱɟɧɢɟ ɝɚɡɚ ɚɞɢɚɛɚɬɧɵɦ, ɜɵɪɚɡɢɦ ɞɢɮɮɟɪɟɧɰɢɚɥ ɞɚɜɥɟɧɢɹ
ɫ ɩɨɦɨɳɶɸ ɭɪɚɜɧɟɧɢɹ ɚɞɢɚɛɚɬɵ:
1k
dP k c d
U
U
,
2k
dP
kc d
U
U
U
.
ɉɨɞɫɬɚɜɥɹɹ ɡɧɚɱɟɧɢɟ
dP
U
ɜ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɹ ɢ ɢɧɬɟɝ-
ɪɢɪɭɹ ɟɝɨ ɜɞɨɥɶ ɫɬɪɭɣɤɢ, ɩɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɟ ɷɧɟɪɝɢɢ ɜ ɢɧɬɟɝɪɚɥɶɧɨɣ
ɮɨɪɦɟ, ɢɥɢ ɭɪɚɜɧɟɧɢɟ Ȼɟɪɧɭɥɥɢ-ɋɟɧ-ȼɟɧɚɧɚ:
2
12
kPu
const
k
U
.
ɋɪɚɜɧɢɜɚɹ ɩɨɥɭɱɟɧɧɵɟ ɭɪɚɜɧɟɧɢɹ ɫ ɭɪɚɜɧɟɧɢɟɦ Ȼɟɪɧɭɥɥɢ ɞɥɹ ɢɞɟ-
ɚɥɶɧɨɣ ɧɟɫɠɢɦɚɟɦɨɣ ɠɢɞɤɨɫɬɢ, ɜɢɞɢɦ, ɱɬɨ ɨɬɥɢɱɢɟ ɫɨɫɬɚɜɢɬ ɜ ɦɧɨɠɢ-
ɬɟɥɟ
1
k
k
. ɉɨɹɜɥɟɧɢɟ ɷɬɨɝɨ ɦɧɨɠɢɬɟɥɹ ɫɜɹɡɚɧɨ ɫ ɬɟɦ, ɱɬɨ ɜ ɩɨɬɟɧɰɢɚɥɶ-
ɧɭɸ ɷɧɟɪɝɢɸ ɝɚɡɚ ɜɯɨɞɢɬ ɟɳɟ ɢ ɟɝɨ ɜɧɭɬɪɟɧɧɹɹ ɷɧɟɪɝɢɹ. ȼ ɫɥɭɱɚɟ ɦɟɬ-
ɪɢɱɟɫɤɨɣ ɡɚɩɢɫɢ ɷɬɢɯ ɭɪɚɜɧɟɧɢɣ ɝɨɜɨɪɹɬ, ɱɬɨ ɞɥɹ ɝɚɡɚ ɤ ɩɶɟɡɨɦɟɬɪɢɱɟ-
ɫɤɨɦɭ ɧɚɩɨɪɭ ɞɨɛɚɜɥɹɟɬɫɹ «ɬɟɦɩɟɪɚɬɭɪɧɵɣ ɧɚɩɨɪ».
ȼɵɪɚɠɚɹ ɜ ɭɪɚɜɧɟɧɢɢ ɷɧɟɪɝɢɢ ɨɬɧɨɲɟɧɢɟ
P
U
ɱɟɪɟɡ ɭɪɚɜɧɟɧɢɟ ɫɨ-
ɫɬɨɹɧɢɹ, ɩɨɥɭɱɢɦ
2
21
ukRT
const
k
.
ɂɡ ɩɨɥɭɱɟɧɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɫɥɟɞɭɟɬ, ɱɬɨ ɩɪɢ ɨɬɫɭɬɫɬɜɢɢ ɬɟɩɥɨɨɛɦɟ-
ɧɚ ɫ ɜɧɟɲɧɟɣ ɫɪɟɞɨɣ ɭɜɟɥɢɱɟɧɢɟ ɫɤɨɪɨɫɬɢ ɜɞɨɥɶ ɫɬɪɭɣɤɢ ɩɪɢɜɨɞɢɬ ɤ
ɩɚɞɟɧɢɸ ɬɟɦɩɟɪɚɬɭɪɵ ɝɚɡɚ ɢ ɧɚɨɛɨɪɨɬ.
ɂɫɩɨɥɶɡɭɹ ɮɨɪɦɭɥɭ ɞɥɹ ɫɤɨɪɨɫɬɢ ɡɜɭɤɚ, ɭɪɚɜɧɟɧɢɟ ɷɧɟɪɝɢɢ ɦɨɠɧɨ
ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ:
22
21
ua
const
k
,
ɨɬɤɭɞɚ ɜɢɞɧɨ, ɱɬɨ ɫɤɨɪɨɫɬɶ ɞɜɢɠɟɧɢɹ ɝɚɡɚ ɢ ɫɤɨɪɨɫɬɶ ɡɜɭɤɚ ɜɡɚɢɦɨɫɜɹ-
ɡɚɧɵ, ɚ ɢɦɟɧɧɨ: ɭɜɟɥɢɱɟɧɢɟ ɫɤɨɪɨɫɬɢ ɬɟɱɟɧɢɹ ɩɪɢɜɨɞɢɬ ɤ ɭɦɟɧɶɲɟɧɢɸ
ɫɤɨɪɨɫɬɢ ɡɜɭɤɚ. ɉɨɥɭɱɟɧɧɨɟ ɭɪɚɜɧɟɧɢɟ ɩɨɡɜɨɥɹɟɬ ɬɚɤɠɟ ɜɵɹɫɧɢɬɶ ɫɦɵɫɥ
ɩɨɫɬɨɹɧɧɨɣ ɜ ɩɪɚɜɨɣ ɱɚɫɬɢ. Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɜ ɩɨɤɨɹɳɟɦɫɹ ɝɚɡɟ 0u ɢ
ɫɤɨɪɨɫɬɶ ɡɜɭɤɚ ɞɨɫɬɢɝɚɟɬ ɫɜɨɟɣ ɧɚɢɛɨɥɶɲɟɣ ɜɟɥɢɱɢɧɵ
0
a
. ɋɥɟɞɨɜɚɬɟɥɶ-
ɧɨ,
2
0
1
a
const
k
ɢ ɭɪɚɜɧɟɧɢɟ ɷɧɟɪɝɢɢ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ
22 2
0
211
ua a
kk
.
18
ɇɚɤɨɧɟɰ, ɟɫɥɢ ɢɫɩɨɥɶɡɨɜɚɬɶ ɩɨɧɹɬɢɟ ɷɧɬɚɥɶɩɢɢ i , ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɟ
ɜ ɬɟɪɦɨɞɢɧɚɦɢɤɟ
1
p
k
icT RT
k
,
ɬɨ ɭɪɚɜɧɟɧɢɟ ɷɧɟɪɝɢɢ ɩɪɢɨɛɪɟɬɚɟɬ ɜɢɞ
2
2
u
i const
.
1.4.6. ɋɠɡɣɧɶ ɭɠɲɠɨɣɺ ɡɣɟɥɩɬɭɣ (ɩɪɶɭɶ ɋɠɤɨɩɦɷɟɬɛ)
Ɋɟɠɢɦɵ ɬɟɱɟɧɢɹ, ɥɚɦɢɧɚɪɧɵɣ ɢɥɢ ɬɭɪɛɭɥɟɧɬɧɵɣ, ɤɚɤ ɭɫɬɚɧɨɜɢɥ
Ɋɟɣɧɨɥɶɞɫ, ɡɚɜɢɫɢɬ ɨɬ ɛɟɡɪɚɡɦɟɪɧɨɝɨ ɤɨɦɩɥɟɤɫɚ, ɤɨɬɨɪɵɣ ɧɚɡɵɜɚɸɬ
ɱɢɫɥɨɦ Ɋɟɣɧɨɥɶɞɫɚ Re ɢ ɨɩɪɟɞɟɥɹɟɬɫɹ, ɤɚɤ Re
Vd
Q
. ɂɦ ɠɟ ɛɵɥɨ ɧɚɣ-
ɞɟɧɨ ɤɪɢɬɢɱɟɫɤɨɟ ɡɧɚɱɟɧɢɟ ɷɬɨɝɨ ɤɨɦɩɥɟɤɫɚ Re 2300
ɤɪ
| , ɡɧɚɹ ɤɨɬɨɪɨɟ,
ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɪɟɠɢɦ ɬɟɱɟɧɢɹ, ɬ. ɟ., ɟɫɥɢ: Re 2300
– ɥɚɦɢɧɚɪɧɨɟ
ɬɟɱɟɧɢɟ; Re 2300
! – ɬɭɪɛɭɥɟɧɬɧɨɟ (ɯɚɨɬɢɱɧɨɟ) ɬɟɱɟɧɢɟ. Ɉɛɳɢɣ ɜɢɞ
ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɨɣ ɭɫɬɚɧɨɜɤɢ Ɋɟɣɧɨɥɶɞɫɚ ɩɨɤɚɡɚɧ ɧɚ ɪɢɫ 1.4.6.1.
Ɋɢɫ. 1.4.6.1
1.4.7. Ɇɛɧɣɨɛɫɨɩɠ ɭɠɲɠɨɣɠ ɡɣɟɥɩɬɭɣ ɝ ɭɫɮɜɛɰ
Ɉɬɵɫɤɚɧɢɟ ɬɨɱɧɵɯ ɪɟɲɟɧɢɣ ɭɪɚɜɧɟɧɢɣ ɇɚɜɶɟ-ɋɬɨɤɫɚ ɧɚɬɚɥɤɢɜɚɟɬ-
ɫɹ ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ ɧɚ ɧɟɩɪɟɨɞɨɥɢɦɵɟ ɦɚɬɟɦɚɬɢɱɟɫɤɢɟ ɬɪɭɞɧɨɫɬɢ, ɜɵ-
ɡɜɚɧɧɵɟ ɧɟɥɢɧɟɣɧɨɣ (ɩɪɢɫɭɬɫɬɜɢɟ ɤɜɚɞɪɚɬɢɱɧɵɯ ɱɥɟɧɨɜ) ɩɪɚɜɨɣ ɱɚ-
ɫɬɶɸ. ȼ ɫɥɭɱɚɟ ɥɚɦɢɧɚɪɧɨɝɨ ɬɟɱɟɧɢɹ ɠɢɞɤɨɫɬɢ ɷɬɢ ɤɜɚɞɪɚɬɢɱɧɵɟ ɫɥɚ-
19
ɝɚɟɦɵɟ ɫɚɦɢ ɫɨɛɨɣ ɢɫɱɟɡɚɸɬ, ɬ.ɤ. ɫɨɫɬɚɜɥɹɸɳɢɟ ɫɤɨɪɨɫɬɢ 0, 0
yz
uu ɢ
ɢɡ ɭɪɚɜɧɟɧɢɹ ɧɟɪɚɡɪɵɜɧɨɫɬɢ ɫɪɚɡɭ ɜɵɬɟɤɚɟɬ, ɱɬɨ
0
x
u
x
w
w
, ɬ. ɟ. ɫɨɫɬɚɜ-
ɥɹɸɳɚɹ
x
u ɧɟ ɦɨɠɟɬ ɡɚɜɢɫɟɬɶ ɨɬ ɤɨɨɪɞɢɧɚɬɵ
x
, ɚ ɫɥɟɞɨɜɚɬɟɥɶɧɨ ɞɚɜɥɟ-
ɧɢɟ ɬɚɤɠɟ ɡɚɜɢɫɢɬ ɬɨɥɶɤɨ ɨɬ
x
.
ȼ ɫɥɭɱɚɟ ɭɫɬɚɧɨɜɢɜɲɟɝɨɫɹ ɬɟɱɟɧɢɹ, ɤɨɝɞɚ
0
x
u
t
w
w
, ɭɪɚɜɧɟɧɢɟ ɇɚ-
ɜɶɟ-ɋɬɨɤɫɚ ɩɪɢɧɢɦɚɟɬ ɛɨɥɟɟ ɩɪɨɫɬɨɣ ɜɢɞ:
22
22
xx
uuP
yz x
P
§·
ww w
¨¸
ww w
©¹
.
Ɋɚɫɫɦɨɬɪɢɦ ɭɫɬɚɧɨɜɢɜɲɟɟɫɹ ɥɚɦɢɧɚɪɧɨɟ ɬɟɱɟɧɢɟ ɜ ɤɪɭɝɥɨɣ ɬɪɭɛɟ
ɩɨɫɬɨɹɧɧɨɝɨ ɫɟɱɟɧɢɹ ɫ ɜɧɭɬɪɟɧɧɢɦ ɪɚɞɢɭɫɨɦ
0
r
(ɪɢɫ. 1.4.7.1).
Ɋɢɫ. 1.4.7.1
ɋɨɫɬɚɜɥɹɸɳɢɟ ɫɤɨɪɨɫɬɢ ɜ ɪɚɞɢɚɥɶɧɨɦ ɧɚɩɪɚɜɥɟɧɢɢ 0
r
u ɢ ɜ ɧɚ-
ɩɪɚɜɥɟɧɢɢ ɤɚɫɚɬɟɥɶɧɨɣ ɤ ɨɤɪɭɠɧɨɫɬɢ ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ 0
r
u . Ɍɨɝɞɚ
ɢɡ ɭɪɚɜɧɟɧɢɹ ɧɟɪɚɡɪɵɜɧɨɫɬɢ ɫɥɟɞɭɟɬ, ɱɬɨ
0
x
u
x
w
w
.
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɢɡ ɬɪɟɯ ɭɪɚɜɧɟɧɢɣ ɇɚɜɶɟ-ɋɬɨɤɫɚ ɜ ɰɢɥɢɧɞɪɢɱɟɫɤɢɯ
ɤɨɨɪɞɢɧɚɬɚɯ ɨɫɬɚɟɬɫɹ ɬɨɥɶɤɨ ɨɞɧɨ ɞɥɹ ɨɫɟɜɨɝɨ ɧɚɩɪɚɜɥɟɧɢɹ, ɤɨɬɨɪɨɟ
ɩɪɢɧɢɦɚɟɬ ɜɢɞ:
22
2
1
xx
du du dP
dr r dr dx
P
§·
¨¸
©¹
.
Ƚɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɛɭɞɭɬ 0
x
ddf ɩɪɢ:
0
;0;0; 0
x
x
du
rru r
dr
.
Ⱦɥɹ ɪɟɲɟɧɢɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɩɪɟɞɫɬɚɜɢɦ ɟɝɨ ɜ ɜɢɞɟ
1
x
ddudP
r
rdr dr dx
P
§·
¨¸
©¹
.
20
ɂɡ ɭɫɥɨɜɢɹ 0
x
u
x
w
w
ɫɥɟɞɭɟɬ, ɱɬɨ ɨɧɨ ɜɨɡɦɨɠɧɨ ɬɨɥɶɤɨ ɬɨɝɞɚ, ɤɨɝɞɚ
12
dP P P
const
dx
A
.
Ɂɧɚɤ ɦɢɧɭɫ ɝɨɜɨɪɢɬ ɨ ɬɨɦ, ɱɬɨ ɞɚɜɥɟɧɢɟ ɩɨ ɞɥɢɧɟ ɩɨɬɨɤɚ ɭɦɟɧɶɲɚ-
ɟɬɫɹ.
ɂɧɬɟɝɪɢɪɭɹ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɢ, ɨɩɪɟɞɟɥɹɹ ɩɨɫɬɨɹɧɧɵɟ ɢɧɬɟɝ-
ɪɢɪɨɜɚɧɢɹ ɢɡ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ, ɩɨɥɭɱɢɦ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɫɤɨɪɨɫɬɟɣ ɩɨ
ɩɨɩɟɪɟɱɧɨɦɭ ɫɟɱɟɧɢɸ ɬɪɭɛɵ:
22
0
4
x
P
urr
P
+
A
.
Ɍɚɤɨɟ ɬɟɱɟɧɢɟ ɧɚɡɵɜɚɸɬ ɬɟɱɟɧɢɟɦ ɏɚɝɟɧɚ-ɉɭɚɡɟɣɥɹ ɢ ɯɚɪɚɤɬɟɪɢɡɭ-
ɟɬɫɹ ɩɨ ɩɚɪɚɛɨɥɨɢɞɚɦ ɜɪɚɳɟɧɢɹ. Ɇɚɤɫɢɦɚɥɶɧɚɹ ɫɤɨɪɨɫɬɶ ɬɟɱɟɧɢɹ ɢɦɟ-
ɟɬ ɦɟɫɬɨ ɧɚ ɨɫɢ ɬɪɭɛɵ ɩɪɢ 0r
ɢ ɪɚɜɧɚ
2
0
4
ɦɚɯ
P
ur
P
+
A
.
ɋɪɟɞɧɹɹ ɫɤɨɪɨɫɬɶ ɩɨ ɫɟɱɟɧɢɸ ɬɪɭɛɵ ɛɭɞɟɬ ɪɚɜɧɚ
0
22 2
00
2
0
0
1
2
48
r
ɫɪ
QP P
Vrrrdrr
Fr
S
SP P
³
++
AA
.
ɋɪɚɜɧɢɜɚɹ ɜɵɪɚɠɟɧɢɹ ɞɥɹ ɦɚɤɫɢɦɚɥɶɧɨɣ ɢ ɫɪɟɞɧɟɣ ɫɤɨɪɨɫɬɟɣ,
ɢɦɟɟɦ:
max
1
2
ɫɪ
Vu
.
ɉɟɪɟɩɚɞ ɞɚɜɥɟɧɢɹ ɩɨ ɞɥɢɧɟ ɬɪɭɛɵ ɨɩɪɟɞɟɥɹɟɬɫɹ ɫɨɩɪɨɬɢɜɥɟɧɢɟɦ,
ɜɨɡɧɢɤɚɸɳɢɦ ɜ ɪɟɡɭɥɶɬɚɬɟ ɬɪɟɧɢɹ. ȼ ɬɟɯɧɢɱɟɫɤɢɯ ɪɚɫɱɟɬɚɯ ɩɪɢɧɹɬɨ
ɷɬɨɬ ɩɟɪɟɩɚɞ ɞɚɜɥɟɧɢɹ ɫɜɹɡɵɜɚɬɶ ɫɨ ɫɪɟɞɧɟɣ ɫɤɨɪɨɫɬɶɸ ɬɟɱɟɧɢɹ ɩɨɫɪɟɞ-
ɫɬɜɨɦ ɜɜɟɞɟɧɢɹ ɛɟɡɪɚɡɦɟɪɧɨɝɨ ɤɨɷɮɮɢɰɢɟɧɬɚ, ɧɚɡɵɜɚɟɦɨɝɨ ɤɨɷɮɮɢɰɢ-
ɟɧɬɨɦ ɫɨɩɪɨɬɢɜɥɟɧɢɹ
O
. Ⱦɥɹ ɷɬɨɣ ɰɟɥɢ ɩɪɢɧɢɦɚɟɬɫɹ, ɱɬɨ
2
2
ɬɪ
V
PP
dg
O
A
+
(ɮɨɪɦɭɥɚ Ⱦɚɪɫɢ).
ɉɨɞɫɬɚɜɥɹɹ ɜ ɩɪɢɜɟɞɟɧɧɭɸ ɮɨɪɦɭɥɭ ɡɧɚɱɟɧɢɟ ɫɪɟɞɧɟɣ ɫɤɨɪɨɫɬɢ,
ɩɨɥɭɱɢɦ
32 64 64
ReVr Vd
P
Q
O
U
.
ɉɨɫɥɟɞɧɟɟ ɪɚɜɟɧɫɬɜɨ ɜɵɪɚɠɚɟɬ ɫɨɛɨɣ ɡɚɤɨɧ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɞɥɹ
ɤɪɭɝɥɨɣ ɬɪɭɛɵ ɩɪɢ ɥɚɦɢɧɚɪɧɨɦ ɬɟɱɟɧɢɢ.