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ɍɪɚɜɧɟɧɢɹ ɱɟɬɵɪɟɯ ɨɫɧɨɜɧɵɯ ɩɪɨɰɟɫɫɨɜ, ɢɝɪɚɸɳɢɯ ɜɚɠɧɭɸ ɪɨɥɶ
X
ɜ ɬɟɪɦɨɞɢɧɚɦɢɤɟ, ɩɨɥɭɱɚɸɬɫɹ ɩɪɢ ɫɥɟɞɭɸɳɢɯ ɡɧɚɱɟɧɢɹɯ ɩɨɤɚɡɚɬɟɥɹ ɩɨɥɢɬɪɨɩɵ: ɢɡɨɯɨɪɧɵɣ ɩɪɨɰɟɫɫ
n = 0; ɢɡɨɬɟɪɦɢɱɟɫɤɢɣ ɩɪɨɰɟɫɫ Ɍ = const (ɢɥɢ ɪ
ɧɵɣ ɩɪɨɰɟɫɫ p
k
X
= const, n = k.
X
= const, n = f; ɢɡɨɛɚɪɧɵɣ ɩɪɨɰɟɫɫ ɪ = const,
X
= const), n = 1; ɚɞɢɚɛɚɬ-
ȼ ɬɚɛɥ. 1.1 ɞɚɧɚ ɫɜɨɞɤɚ ɩɨɤɚɡɚɬɟɥɟɣ ɷɬɢɯ ɩɪɨɰɟɫɫɨɜ. ɇɚ ɪɢɫ. 1.2 ɩɪɟɞ-
X
n = - k
ɢ Ɍs.
n = - f
n = 0
ɫɬɚɜɥɟɧɵ ɩɨɥɢɬɪɨɩɧɵɟ ɩɪɨɰɟɫɫɵ ɜ ɤɨɨɪɞɢɧɚɬɚɯ ɪ
p
n = - k
n = - 1
n = - f
Ɋɚɫɲɢɪɟɧɢɟ
T
ɋɠɚɬɢɟ
n = 0
ɋɠɚɬɢɟ
n = +f
Ɋɢɫ. 1.2. ɉɨɥɢɬɪɨɩɧɵɟ ɩɪɨɰɟɫɫɵ ɜ ɤɨɨɪɞɢɧɚɬɚɯ ɪX ɢ Ɍ s
n = 1
n = k
n = - 1
n = 0
n = +f
Ɋɚɫɲɢɪɟɧɢɟ
n = k
n = 1
s
11
Ɍɚɛɥɢɰɚ 1.1
ɚ
ɚ
X
X
X
X
q
n
c
u
12
TT
1
kn
u
n
X
c
1
n
nk
X
c
21p
cTT
p
c
,ln
2
X
21
RT
X
cTT
p
f
c
1
q = w
0 0
21p
cTT
'u 'h
w
ɋɜɹɡɶ ɦɟɠɞɭ
ɋɜɨɞɤɚ ɩɨɤɚɡɚɬɟɥɟɣ ɬɟɪɦɨɞɢɧɚɦɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ
ɍɪɚɜɧɟɧɢɟ
Ɂɧɚɱɟɧɢɟ n
12
X
cTT
/1
XX
22 11
pp n
;//
;//
n
n
1221
1221
XX
XX
pp
TT
ɩɚɪɚɦɟɬɪɚɦɢ
=const
n
X
ɩɪɨɰɟɫɫ
+ v… v ɪ
21p
21p
cTT
cTT
12
12
X
X
cTT
cTT
1
12
p
0
21
RT T n
1
n
2121
ppTT
//
2121
// TT
2121
// TTpp
= const
X
f
21p
cTT
0 0
,ln
2
1
X
RT
1221
//
pp
=const
X
T = const
ɪ
21
X
cTT
1;
/1
w = q
21
w = 'u
RT T k
XX
22 11
pp k
1
;//
k
k
k
1221
1221
XX
//
pp
2121
ppTT
XX
//
TT
=const
k
X
ɪ
p
X
c
c
k
3
ɩɪɨɰɟɫɫ
ɇɚɢɦɟɧɨɜɚɧɢɟ
ɂɡɨɛɚɪɧɵɣ 0 ɪ = const
ɉɨɥɢɬɪɨɩɧɵɣ
ɂɡɨɯɨɪɧɵɣ
ɂɡɨɬɟɪɦɢɱɟɫɤɢɣ 1
Ⱥɞɢɚɛɚɬɧɵɣ
1.3. Ɍɟɪɦɨɞɢɧɚɦɢɱɟɫɤɢɟ ɫɜɨɣɫɬɜɚ ɜɨɞɵ ɢ ɜɨɞɹɧɨɝɨ ɩɚɪɚ
X
Ʌɸɛɨɟ ɱɢɫɬɨɟ ɜɟɳɟɫɬɜɨ ɦɨɠɟɬ ɧɚɯɨɞɢɬɶɫɹ ɜ ɬɜɟɪɞɨɣ, ɠɢɞɤɨɣ ɢɥɢ ɝɚɡɨɨɛɪɚɡɧɨɣ ɮɚɡɟ. ɉɟɪɟɯɨɞ ɜɟɳɟɫɬɜɚ ɢɡ ɨɞɧɨɣ ɮɚɡɵ ɜ ɞɪɭɝɭɸ ɫɜɹɡɚɧ ɫ ɢɡɦɟɧɟɧɢɟɦ ɩɚɪɚɦɟɬɪɨɜ ɩɪɢ ɩɟɪɟɞɚɱɟ ɬɟɩɥɨɬɵ. ɉɪɨɦɟɠɭɬɨɱɧɨɟ ɫɨɫɬɨɹɧɢɟ
ɜɟɳɟɫɬɜɚ ɦɟɠɞɭ ɝɚɡɨɦ ɢ ɠɢɞɤɨɫɬɶɸ ɧɚɡɵɜɚɟɬɫɹ ɩɚɪɨɦ. ɉɪɢɛɥɢɠɟɧɧɨ ɫɨɨɬɧɨɲɟɧɢɟ ɦɟɠɞɭ ɩɚɪɚɦɟɬɪɚɦɢ ɩɚɪɚ ɦɨɠɧɨ ɯɚɪɚɤɬɟɪɢɡɨɜɚɬɶ ɭɪɚɜɧɟɧɢɟɦ
Ʉɥɚɩɟɣɪɨɧɚ Ɇɟɧɞɟɥɟɟɜɚ ɢɥɢ ɭɪɚɜɧɟɧɢɟɦ ȼɚɧ
ȿɫɥɢ ɫɠɢɦɚɬɶ ɝɚɡ ɩɪɢ ɩɨɫɬɨɹɧɧɨɣ ɬɟɦɩɟɪɚɬɭɪɟ, ɬɨ ɦɨɠɧɨ ɞɨɫɬɢɱɶ ɫɨɫɬɨɹɧɢɹ ɧɚɫɵɳɟɧɢɹ (ɫɠɢɠɟɧɢɹ ɝɚɡɚ), ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɝɨ ɷɬɨɣ ɬɟɦɩɟɪɚɬɭɪɟ
ɢ ɧɟɤɨɬɨɪɨɦɭ ɨɩɪɟɞɟɥɟɧɧɨɦɭ ɞɚɜɥɟɧɢɸ. ɉɪɢ ɞɚɥɶɧɟɣɲɟɦ ɫɠɚɬɢɢ ɩɚɪ ɛɭɞɟɬ ɤɨɧɞɟɧɫɢɪɨɜɚɬɶɫɹ ɢ ɜ ɨɩɪɟɞɟɥɟɧɧɵɣ ɦɨɦɟɧɬ ɩɨɥɧɨɫɬɶɸ ɩɪɟɜɪɚɬɢɬɫɹ
ɜ ɠɢɞɤɨɫɬɶ. ɉɪɨɰɟɫɫ ɩɟɪɟɯɨɞɚ ɩɚɪɚ ɜ ɠɢɞɤɨɫɬɶ ɩɪɨɢɫɯɨɞɢɬ ɩɪɢ ɩɨɫɬɨɹɧɧɵɯ ɬɟɦɩɟɪɚɬɭɪɟ ɢ ɞɚɜɥɟɧɢɢ, ɬɚɤ ɤɚɤ ɞɚɜɥɟɧɢɟ ɧɚɫɵɳɟɧɧɨɝɨ ɩɚɪɚ ɨɞɧɨɡɧɚɱɧɨ ɨɩɪɟɞɟɥɹɟɬɫɹ ɬɟɦɩɟɪɚɬɭɪɨɣ. ɇɚ ɪɞɜɭɯɮɚɡɧɵɯ ɫɨɫɬɨɹɧɢɣ (ɩɚɪ ɢ ɠɢɞɤɨɫɬɶ) ɥɟɠɢɬ ɦɟɠɞɭ ɤɪɢɜɵɦɢ ɤɢɩɹɳɟɣ
ɠɢɞɤɨɫɬɢ ɢ ɫɭɯɨɝɨ ɧɚɫɵɳɟɧɧɨɝɨ ɩɚɪɚ. ɉɪɢ ɭɜɟɥɢɱɟɧɢɢ ɞɚɜɥɟɧɢɹ ɷɬɢ ɤɪɢɜɵɟ ɫɛɥɢɠɚɸɬɫɹ. ɋɛɥɢɠɟɧɢɟ ɩɪɨɢɫɯɨɞɢɬ ɩɨɬɨɦɭ, ɱɬɨ ɨɛɴɟɦ ɩɚɪɚ ɭɦɟɧɶɲɚɟɬɫɹ, ɚ ɨɛɴɟɦ ɠɢɞɤɨɫɬɢ ɭɜɟɥɢɱɢɜɚɟɬɫɹ. ɉɪɢ ɧɟɤɨɬɨɪɨɦ ɨɩɪɟɞɟɥɟɧɧɨɦ
ɞɥɹ ɞɚɧɧɨɣ ɠɢɞɤɨɫɬɢ (ɩɚɪɚ) ɞɚɜɥɟɧɢɢ ɤɪɢɜɵɟ ɤɢɩɹɳɟɣ ɠɢɞɤɨɫɬɢ ɢ ɩɚɪɚ
ɜɫɬɪɟɱɚɸɬɫɹ ɜ ɤɪɢɬɢɱɟɫɤɨɣ ɬɨɱɤɟ Ʉ, ɤɨɬɨɪɨɣ ɫɨɨɬɜɟɬɫɬɜɭɸɬ ɤɪɢɬɢɱɟɫɤɢɟ
ɩɚɪɚɦɟɬɪɵ: ɞɚɜɥɟɧɢɟ ɪ
ɪɢɡɭɸɳɢɟ ɤɪɢɬɢɱɟɫɤɨɟ ɫɨɫɬɨɹɧɢɟ ɜɟɳɟɫɬɜɚ. ɉɪɢ ɤɪɢɬɢɱɟɫɤɨɦ ɫɨɫɬɨɹɧɢɢ
ɢɫɱɟɡɚɸɬ ɪɚɡɥɢɱɢɹ ɦɟɠɞɭ ɠɢɞɤɨɫɬɶɸ ɢ ɩɚɪɨɦ. ɉɪɢ ɬɟɦɩɟɪɚɬɭɪɟ ɛɨɥɟɟ ɜɵɫɨɤɨɣ, ɱɟɦ ɤɪɢɬɢɱɟɫɤɚɹ, ɝɚɡ ɧɢ ɩɪɢ ɤɚɤɨɦ ɞɚɜɥɟɧɢɢ ɧɟ ɦɨɠɟɬ ɫɤɨɧɞɟɧɫɢɪɨɜɚɬɶɫɹ, ɬ. ɟ. ɩɪɟɜɪɚɬɢɬɶɫɹ ɜ ɠɢɞɤɨɫɬɶ.
1.3.1. Ɍɟɪɦɨɞɢɧɚɦɢɱɟɫɤɢɟ ɫɜɨɣɫɬɜɚ ɩɚɪɨɜ
-ɞɟɪ-ȼɚɚɥɶɫɚ.
p
p
ɄP
Ɉɛɥɚɫɬɶ
ɠɢɞɤɨɫɬɢ
Ʉɪɢɜɚɹ
ɤɢɩɹɳɟɣ
ɠɢɞɤɨɫɬɢ
Ɋɢɫ. 1.3. Ⱦɢɚɝɪɚɦɦɚ ɪɟɚɥɶɧɨɝɨ ɝɚɡɚ
, ɬɟɦɩɟɪɚɬɭɪɚ ɌɄɊ, ɭɞɟɥɶɧɵɣ ɨɛɴɟɦ
ɄɊ
Ʉ Ɉɛɥɚɫɬɶ
ɩɚɪɚ
Ɉɛɥɚɫɬɶ
ɞɜɭɯɮɚɡɧɵɯ
ɫɨɫɬɨɹɧɢɣ
(ɩɚɪ ɢ ɠɢɞɤɨɫɬɶ)
X
-ɞɢɚɝɪɚɦɦɟ (ɪɢɫ. 1.3) ɨɛɥɚɫɬɶ
13
Ʉɪɢɜɚɹ
ɫɭɯɨɝɨ
ɧɚɫɵɳɟɧɧɨɝɨ
ɩɚɪɚ
X
, ɯɚɪɚɤɬɟ-
ɄɊ
1.3.2. ȼɨɞɹɧɨɣ ɩɚɪ.
ɉɚɪɨɨɛɪɚɡɨɜɚɧɢɟ ɩɪɢ ɩɨɫɬɨɹɧɧɨɦ ɞɚɜɥɟɧɢɢ
ȼɨɞɹɧɨɣ ɩɚɪ ɩɨɥɭɱɢɥ ɲɢɪɨɤɨɟ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɟ ɤɚɤ ɪɚɛɨɱɟɟ ɜɟɳɟɫɬɜɨ
ɜ ɬɟɩɥɨɜɵɯ ɞɜɢɝɚɬɟɥɹɯ ɢ ɤɚɤ ɞɜɢɠɭɳɚɹ ɫɪɟɞɚ, ɢɫɩɨɥɶɡɭɟɦɚɹ ɞɥɹ ɨɫɭɳɟɫɬɜɥɟɧɢɹ ɩɪɨɰɟɫɫɚ ɬɟɩɥɨɨɛɦɟɧɚ ɜ ɬɟɩɥɨɨɛɦɟɧɧɵɯ ɚɩɩɚɪɚɬɚɯ.
ȼɨɞɹɧɨɣ ɩɚɪ ɟɫɬɶ ɜɨɞɚ ɜ ɝɚɡɨɨɛɪɚɡɧɨɦ ɫɨɫɬɨɹɧɢɢ.
ɉɪɨɰɟɫɫ ɩɪɟɜɪɚɳɟɧɢɹ ɜɨɞɵ ɜ ɩɚɪ ɧɚɡɵɜɚɟɬɫɹ ɩɚɪɨɨɛɪɚɡɨɜɚɧɢɟɦ ɢ
ɦɨɠɟɬ ɨɫɭɳɟɫɬɜɥɹɬɶɫɹ ɞɜɭɦɹ ɪɚɡɥɢɱɧɵɦɢ ɩɨ ɢɧɬɟɧɫɢɜɧɨɫɬɢ ɢ ɯɚɪɚɤɬɟɪɭ
ɩɪɨɰɟɫɫɚɦɢ: ɢɫɩɚɪɟɧɢɟɦ ɢ ɤɢɩɟɧɢɟɦ
.
ɉɨɞ ɢɫɩɚɪɟɧɢɟɦ ɩɨɧɢɦɚɸɬ ɩɚɪɨɨɛɪɚɡɨɜɚɧɢɟ, ɩɪɨɢɫɯɨɞɹɳɟɟ ɧɚ ɫɜɨɛɨɞɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɜɨɞɵ ɩɪɢ ɬɟɦɩɟɪɚɬɭɪɟ ɧɢɠɟ ɬɨɱɤɢ ɤɢɩɟɧɢɹ ɩɪɢ ɞɚɧɧɨɦ ɞɚɜɥɟɧɢɢ.
Ʉɢɩɟɧɢɟ – ɩɪɨɰɟɫɫ ɢɧɬɟɧɫɢɜɧɨɝɨ ɢɫɩɚɪɟɧɢɹ ɧɟ ɬɨɥɶɤɨ ɫɨ ɫɜɨɛɨɞɧɨɣ
ɩɨɜɟɪɯɧɨɫɬɢ ɜɨɞɵ, ɧɨ ɢ ɫɨ ɜɫɟɝɨ ɨɛɴɟɦɚ ɨɛɪɚɡɭɸɳɢɯɫɹ ɜɧɭɬɪɢ ɩɭɡɵɪɶɤɨɜ
ɩɚɪɚ.
Ɋɚɫɫɦɨɬɪɢɦ ɩɪɨɰɟɫɫ ɩɚɪɨɨɛɪɚɡɨɜɚɧɢɹ (ɪɢɫ. 1.4) 1 ɤɝ ɜɨɞɵ, ɡɚɤɥɸɱɟɧɧɨɝɨ ɜ ɰɢɥɢɧɞɪ 1 ɫ ɩɨɞɜɢɠɧɵɦ ɩɨɪɲɧɟɦ 2, ɞɚɜɥɟɧɢɟ ɩɨɞ ɤɨɬɨɪɵɦ ɜ ɬɟɱɟɧɢɟ ɜɫɟɝɨ ɩɪɨɰɟɫɫɚ ɨɫɬɚɟɬɫɹ ɩɨɫɬɨɹɧɧɵɦ. ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɜ ɧɚɱɚɥɶɧɨɦ
ɫɨɫɬɨɹɧɢɢ (ɩɨɥɨɠɟɧɢɟ ɩɨɪɲɧɹ 0) ɜɨɞɚ ɧɚɯɨɞɢɬɫɹ ɩɪɢ t0 = 0 °ɋ ɢ ɡɚɧɢɦɚɟɬ
ɨɛɴɟɦ
X
= 0,001 ɦ3/ɤɝ.
0
p = const
3
t > t"
2
t = t" = t
ɉɪɢ ɢɡɨɛɚɪɧɨɦ ɩɪɨɰɟɫɫɟ ɩɨɞɜɨɞɚ ɬɟɩɥɨɬɵ
ɤ ɜɨɞɟ ɬɟɦɩɟɪɚɬɭɪɚ ɢ ɭɞɟɥɶɧɵɣ ɨɛɴɟɦ ɜɨɞɵ ɛɭɞɭɬ ɭɜɟɥɢɱɢɜɚɬɶɫɹ, ɢ ɩɪɢ ɞɨɫɬɢɠɟɧɢɢ ɧɟɤɨɬɨɪɨɣ
ɬɟɦɩɟɪɚɬɭɪɵ t
ɜɨɞɚ ɡɚɤɢɩɢɬ. ɇɚ ɪɢɫ. 1.4 ɫɨɫɬɨɹ-
S
ɧɢɸ ɜɨɞɵ ɧɚ ɝɪɚɧɢɰɟ ɤɢɩɟɧɢɹ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɩɨ-
S
ɥɨɠɟɧɢɟ ɩɨɪɲɧɹ 1.
1
ɉɪɢ ɞɚɥɶɧɟɣɲɟɦ ɩɨɞɜɨɞɟ ɬɟɩɥɨɬɵ ɧɚɱɢɧɚɟɬɫɹ
1
0
t
S
t
1 ɤɝ
0
Ɋɢɫ. 1.4.
ɉɪɨɰɟɫɫ
ɩɚɪɨɨɛɪɚɡɨɜɚɧɢɹ
2
ɩɚɪɨɨɛɪɚɡɨɜɚɧɢɟ. ɑɚɫɬɶ ɦɨɥɟɤɭɥ ɩɚɪɚ, ɞɜɢɠɭɳɢɯɫɹ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɩɨɞ ɩɨɪɲɧɟɦ ɪɚɜɧɨɦɟɪɧɨ ɩɨ
ɜɫɟɦ ɧɚɩɪɚɜɥɟɧɢɹɦ, ɫɨɩɪɢɤɚɫɚɟɬɫɹ ɫ ɩɨɜɟɪɯɧɨɫɬɶɸ ɜɨɞɵ ɢ ɜɨɡɜɪɚɳɚɟɬɫɹ ɨɛɪɚɬɧɨ ɜ ɧɟɟ. ɉɪɨɢɫɯɨɞɢɬ ɩɪɨɰɟɫɫ ɩɪɟɜɪɚɳɟɧɢɹ ɩɚɪɚ ɜ ɠɢɞɤɨɫɬɶ (ɤɨɧ-
ɞɟɧɫɚɰɢɹ). ȼ ɧɟɤɨɬɨɪɵɣ ɦɨɦɟɧɬ, ɤɨɝɞɚ ɫɤɨɪɨɫɬɢ
ɤɨɧɞɟɧɫɚɰɢɢ ɢ ɩɚɪɨɨɛɪɚɡɨɜɚɧɢɹ ɫɪɚɜɧɢɜɚɸɬɫɹ, ɜ
ɫɢɫɬɟɦɟ ɧɚɫɬɭɩɚɟɬ ɞɢɧɚɦɢɱɟɫɤɨɟ ɪɚɜɧɨɜɟɫɢɟ.
ɉɚɪ, ɧɚɯɨɞɹɳɢɣɫɹ ɜ ɬɟɪɦɨɞɢɧɚɦɢɱɟɫɤɨɦ ɪɚɜɧɨɜɟɫɢɢ ɫ ɜɨɞɨɣ, ɢɡ ɤɨ-
ɬɨɪɨɣ ɨɧ ɨɛɪɚɡɭɟɬɫɹ, ɧɚɡɵɜɚɟɬɫɹ ɧɚɫɵɳɟɧɧɵɦ.
ȼɥɚɠɧɵɣ ɧɚɫɵɳɟɧɧɵɣ ɩɚɪ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɫɦɟɫɶ ɩɚɪɚ ɫ ɠɢɞɤɨɫɬɶɸ, ɩɪɢɱɟɦ ɠɢɞɤɨɫɬɶ ɦɨɠɟɬ ɛɵɬɶ ɫɨɫɪɟɞɨɬɨɱɟɧɚ ɜ ɧɢɠɧɟɣ ɱɚɫɬɢ ɰɢɥɢɧɞɪɚ ɢɥɢ ɪɚɜɧɨɦɟɪɧɨ ɪɚɫɩɪɟɞɟɥɟɧɚ ɜ ɜɢɞɟ ɦɟɥɶɱɚɣɲɢɯ ɤɚɩɟɥɶ ɩɨ ɜɫɟɦɭ
ɨɛɴɟɦɭ.
14
ɉɪɨɰɟɫɫ ɩɚɪɨɨɛɪɚɡɨɜɚɧɢɹ ɢɞɟɬ ɩɪɢ ɩɨɫɬɨɹɧɧɵɯ ɞɚɜɥɟɧɢɢ ɢ ɬɟɦɩɟɪɚɬɭɪɟ (ɢɡɨɛɚɪɧɨ-ɢɡɨɬɟɪɦɢɱɟɫɤɢɣ ɩɪɨɰɟɫɫ). ȼɫɥɟɞɫɬɜɢɟ ɷɬɨɝɨ ɫɜɨɣɫɬɜɚ
ɧɚɫɵɳɟɧɧɨɝɨ ɩɚɪɚ ɨɩɪɟɞɟɥɹɸɬɫɹ ɬɟɦɩɟɪɚɬɭɪɨɣ, ɹɜɥɹɸɳɟɣɫɹ ɮɭɧɤɰɢɟɣ
ɞɚɜɥɟɧɢɹ ɫɪɟɞɵ, ɜ ɤɨɬɨɪɨɣ ɩɪɨɢɫɯɨɞɢɬ ɩɪɨɰɟɫɫ ɩɚɪɨɨɛɪɚɡɨɜɚɧɢɹ. ɉɪɢ
ɩɨɞɜɨɞɟ ɬɟɩɥɨɬɵ ɜ ɩɪɨɰɟɫɫɟ ɩɚɪɨɨɛɪɚɡɨɜɚɧɢɹ ɤɨɥɢɱɟɫɬɜɨ ɩɚɪɚ ɛɭɞɟɬ ɜɨɡɪɚɫɬɚɬɶ ɫ ɨɞɧɨɜɪɟɦɟɧɧɵɦ ɭɦɟɧɶɲɟɧɢɟɦ ɤɨɥɢɱɟɫɬɜɚ ɤɢɩɹɳɟɣ ɜɨɞɵ. ɉɨɥɨɠɟɧɢɟ ɩɨɪɲɧɹ 2 ɧɚ ɪɢɫ
. 1.4 ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɦɨɦɟɧɬɭ ɨɤɨɧɱɚɧɢɹ ɩɪɨɰɟɫɫɚ
ɩɚɪɨɨɛɪɚɡɨɜɚɧɢɹ.
Ʉɨɥɢɱɟɫɬɜɨ ɬɟɩɥɨɬɵ, ɧɟɨɛɯɨɞɢɦɨɟ ɞɥɹ ɩɪɟɜɪɚɳɟɧɢɹ 1 ɤɝ ɤɢɩɹɳɟɣ ɜɨɞɵ ɜ ɫɭɯɨɣ ɧɚɫɵɳɟɧɧɵɣ ɩɚɪ, ɧɚɡɵɜɚɟɬɫɹ ɬɟɩɥɨɬɨɣ ɩɚɪɨɨɛɪɚɡɨɜɚɧɢɹ.
ȼɨ ɜɫɟɯ ɩɪɨɦɟɠɭɬɨɱɧɵɯ ɫɨɫɬɨɹɧɢɹɯ ɦɟɠɞɭ ɩɟɪɜɵɦ ɢ ɜɬɨɪɵɦ ɩɨɥɨɠɟɧɢɹɦɢ ɩɨɪɲɧɹ (ɫɦ. ɪɢɫ. 1.4) ɩɨɞ ɧɢɦ ɧɚɯɨɞɢɬɫɹ ɜɥɚɠɧɵɣ ɧɚɫɵɳɟɧɧɵɣ
ɩɚɪ, ɩɪɟɞɫɬɚɜɥɹɸɳɢɣ ɫɨɛɨɣ ɫɦɟɫɶ m' ɤɝ ɤɢɩɹɳɟɣ ɠɢɞɤɨɫɬɢ ɢ m" ɤɝ ɫɭɯɨɝɨ
ɧɚɫɵɳɟɧɧɨɝɨ ɩɚɪɚ.
m
Ɉɬɧɨɲɟɧɢɟ
x
ɳɟɧɧɨɝɨ ɩɚɪɚ, ɚ ɜɟɥɢɱɢɧɭ
ɧɚɡɵɜɚɸɬ ɫɬɟɩɟɧɶɸ ɫɭɯɨɫɬɢ ɜɥɚɠɧɨɝɨ ɧɚɫɵ-
"'"mm
'
1
m
x
ɫɬɟɩɟɧɶɸ ɜɥɚɠɧɨɫɬɢ. ɋɬɟɩɟɧɶ
"'
mm
ɫɭɯɨɫɬɢ ɢɡɦɟɧɹɟɬɫɹ ɨɬ ɯ = 0 (ɤɢɩɹɳɚɹ ɜɨɞɚ) ɞɨ ɯ = 1 (ɫɭɯɨɣ ɧɚɫɵɳɟɧɧɵɣ
ɩɚɪ).
ɉɪɢ ɩɨɞɜɨɞɟ ɬɟɩɥɨɬɵ ɫɭɯɨɣ ɧɚɫɵɳɟɧɧɵɣ ɩɚɪ ɩɟɪɟɯɨɞɢɬ ɜ ɫɨɫɬɨɹɧɢɟ
ɩɟɪɟɝɪɟɬɨɝɨ ɩɚɪɚ (ɩɨɥɨɠɟɧɢɟ ɩɨɪɲɧɹ 3 ɧɚ ɪɢɫ. 1.4). ɉɨɞ ɩɟɪɟɝɪɟɬɵɦ ɩɨɧɢɦɚɸɬ ɩɚɪ, ɬɟɦɩɟɪɚɬɭɪɚ ɤɨɬɨɪɨɝɨ ɜɵɲɟ ɬɟɦɩɟɪɚɬɭɪɵ ɧɚɫɵɳɟɧɧɨɝɨ ɩɚɪɚ
ɬɨɝɨ ɠɟ ɞɚɜɥɟɧɢɹ.
1.3.3. Ɍɚɛɥɢɰɵ ɢ ɞɢɚɝɪɚɦɦɵ ɞɥɹ ɜɨɞɵ ɢ ɜɨɞɹɧɨɝɨ ɩɚɪɚ
Ʉɨɥɢɱɟɫɬɜɟɧɧɵɟ ɫɨɨɬɧɨɲɟɧɢɹ ɦɟɠɞɭ ɪɚɡɥɢɱɧɵɦɢ ɩɚɪɚɦɟɬɪɚɦɢ ɢ
ɮɭɧɤɰɢɹɦɢ ɫɨɫɬɨɹɧɢɹ ɜɨɞɵ, ɫɭɯɨɝɨ ɧɚɫɵɳɟɧɧɨɝɨ ɢ ɩɟɪɟɝɪɟɬɨɝɨ ɩɚɪɚ
ɭɫɬɚɧɚɜɥɢɜɚɸɬɫɹ ɩɨ ɮɨɪɦɭɥɚɦ ɢɥɢ ɫɩɟɰɢɚɥɶɧɵɦ ɬɚɛɥɢɰɚɦ, ɫɨɫɬɚɜɥɟɧɧɵɦ
ɧɚ ɨɫɧɨɜɚɧɢɢ ɬɟɨɪɟɬɢɱɟɫɤɢɯ ɢ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɯ ɢɫɫɥɟɞɨɜɚɧɢɣ.
Ɍɚɛɥɢɰɵ ɬɟɪɦɨɞɢɧɚɦɢɱɟɫɤɢɯ ɫɜɨɣɫɬɜ ɜɨɞɵ ɢ ɜɨɞɹɧɨɝɨ ɩɚɪɚ ɫɨɞɟɪɠɚɬ ɬɪɢ ɬɚɛɥɢɰɵ [6]: ɬɟɪɦɨɞɢɧɚɦɢɱɟɫɤɢɟ ɫɜɨɣɫɬɜɚ ɜɨɞɵ ɢ ɜɨɞɹɧɨɝɨ
ɩɚɪɚ (Ɍɋȼȼɉ) ɜ ɫɨɫɬɨɹɧɢɢ ɧɚɫɵɳɟɧɢɹ ɩɨ ɬɟɦɩɟɪɚɬɭɪɚɦ ɢ ɩɨ ɞɚɜɥɟɧɢɹɦ ɢ
ɬɟɪɦɨɞɢɧɚɦɢɱɟɫɤɢɟ ɫɜɨɣɫɬɜɚ ɜɨɞɵ ɢ ɩɟɪɟɝɪɟɬɨɝɨ ɩɚɪɚ.
ȼ ɩɪɢɥɨɠɟɧɢɢ ɩɪɢɜɟɞɟɧɵ ɬɚɛɥɢɰɵ ɬɟɪɦɨɞɢɧɚɦɢɱɟɫɤɢɯ ɫɜɨɣɫɬɜ ɜɨɞɵ
ɢ ɜɨɞɹɧɨɝɨ ɩɚɪɚ.
ȼ ɬɚɛɥ. ɉ.1 ɢ ɉ.2 ɩɪɢɜɨɞɹɬɫɹ ɡɧɚɱɟɧɢɹ ɭɞɟɥɶɧɨɝɨ ɨɛɴɟɦɚ, ɷɧɬɚɥɶɩɢɢ ɢ
ɷɧɬɪɨɩɢɢ ɜɨɞɵ, ɧɚɝɪɟɬɨɣ ɞɨ ɫɨɫɬɨɹɧɢɹ ɤɢɩɟɧɢɹ, ɢ ɫɭɯɨɝɨ ɧɚɫɵɳɟɧɧɨɝɨ
ɩɚɪɚ, ɬɟɦɩɟɪɚɬɭɪɵ ɤɢɩɟɧɢɹ, ɞɚɜɥɟɧɢɹ, ɩɪɢ ɤɨɬɨɪɨɦ ɩɪɨɢɫɯɨɞɢɬ ɩɚɪɨɨɛɪɚɡɨɜɚɧɢɟ, ɬɟɩɥɨɬɵ ɩɚɪɨɨɛɪɚɡɨɜɚɧɢɹ,
ɩɚɪɚɦɟɬɪɵ ɤɪɢɬɢɱɟɫɤɨɝɨ ɫɨɫɬɨɹɧɢɹ.
15
ȼ [6] ɩɪɢɜɟɞɟɧɵ ɭɞɟɥɶɧɵɣ ɨɛɴɟɦ, ɷɧɬɚɥɶɩɢɹ ɢ ɷɧɬɪɨɩɢɹ ɜɨɞɵ ɢ ɩɟɪɟ-
X
X
ɝɪɟɬɨɝɨ ɩɚɪɚ ɩɪɢ ɪɚɡɥɢɱɧɵɯ ɞɚɜɥɟɧɢɹɯ ɢ ɬɟɦɩɟɪɚɬɭɪɚɯ.
Ɍɚɛɥɢɰɵ ɞɚɸɬ ɥɢɲɶ ɞɢɫɤɪɟɬɧɵɟ ɡɧɚɱɟɧɢɹ ɢɫɤɨɦɵɯ ɜɟɥɢɱɢɧ. Ⱦɥɹ
ɢɡɨɛɪɚɠɟɧɢɹ ɧɟɩɪɟɪɵɜɧɵɯ ɢɡɦɟɧɟɧɢɣ ɩɚɪɚɦɟɬɪɨɜ (ɩɪɨɰɟɫɫɨɜ) ɧɚ ɩɪɚɤɬɢɤɟ ɱɚɫɬɨ ɢɫɩɨɥɶɡɭɟɬɫɹ ɩɥɨɫɤɚɹ ɫɢɫɬɟɦɚ ɤɨɨɪɞɢɧɚɬ (ɞɢɚɝɪɚɦɦɚ ɜɨɞɹɧɨɝɨ
ɩɚɪɚ).
Ⱦɢɚɝɪɚɦɦɚ ɪ,
ɜɨɞɹɧɨɝɨ ɩɚɪɚ ɩɪɢɜɟɞɟɧɚ ɧɚ ɪɢɫ. 1.5. ɇɚ ɞɢɚɝɪɚɦɦɟ
ɧɚɧɟɫɟɧɵ ɬɪɢ ɥɢɧɢɢ:
Ⱥ
Ⱥ' – ɥɢɧɢɹ ɯɨɥɨɞɧɨɣ ɜɨɞɵ. ɇɚ ɞɢɚɝɪɚɦɦɟ ɨɧɚ ɢɡɨɛɪɚɠɚɟɬɫɹ ɜ ɜɢɞɟ
Ɉ
ɩɪɹɦɨɣ ɥɢɧɢɢ, ɩɚɪɚɥɥɟɥɶɧɨɣ ɨɫɢ ɨɪɞɢɧɚɬ, ɬɚɤ ɤɚɤ ɜɨɞɚ ɩɪɚɤɬɢɱɟɫɤɢ ɧɟɫɠɢɦɚɟɦɚ. ɗɬɚ ɥɢɧɢɹ ɛɭɞɟɬ ɢ ɢɡɨɬɟɪɦɨɣ 0 °ɋ;
Ⱥ'Ʉ – ɥɢɧɢɹ ɤɢɩɹɳɟɣ ɜɨɞɵ (ɥɟɜɚɹ ɩɨɝɪɚɧɢɱɧɚɹ ɤɪɢɜɚɹ);
ɄȺ" – ɥɢɧɢɹ ɫɭɯɨɝɨ ɧɚɫɵɳɟɧɧɨɝɨ ɩɚɪɚ (ɩɪɚɜɚɹ ɩɨɝɪɚɧɢɱɧɚɹ ɤɪɢɜɚɹ).
p
Ʌɢɧɢɹ ɤɢɩɟɧɢɹ
A
0
I ɫɨɫɬɨɹɧɢɟ
ɧɟɞɨɝɪɟɬɨɣ
ɜɨɞɵ
a
0
A'
a'
II – ɫɨɫɬɨɹɧɢɟ
ɜɥɚɠɧɨɝɨ ɩɚɪɚ
Ʉ
III ɩɟɪɟɝɪɟɬɵɣ
a"
a
ɩɚɪ
Ʌɢɧɢɹ ɫɭɯɨɝɨ
ɧɚɫɵɳɟɧɧɨɝɨ ɩɚɪɚ
A"
Ɋɢɫ. 1.5. Ⱦɢɚɝɪɚɦɦɚ ɪ, X ɜɨɞɹɧɨɝɨ ɩɚɪɚ
Ƚɪɚɮɢɱɟɫɤɢɦ ɢɡɨɛɪɚɠɟɧɢɟɦ ɩɪɨɰɟɫɫɚ ɧɚɝɪɟɜɚ ɜɨɞɵ, ɩɚɪɨɨɛɪɚɡɨɜɚɧɢɹ
ɢ ɩɟɪɟɝɪɟɜɚ ɩɚɪɚ ɩɪɢ ɩɨɫɬɨɹɧɧɨɦ ɞɚɜɥɟɧɢɢ ɛɭɞɟɬ ɥɢɧɢɹ ɚɈɚ'ɚ"ɚ (ɪɢɫ. 1.5)
ɫ ɱɟɬɵɪɶɦɹ ɬɨɱɤɚɦɢ: ɚ
– ɯɨɥɨɞɧɚɹ ɜɨɞɚ ɩɪɢ 0 °ɋ; ɬɨɱɤɚ ɚ' – ɤɢɩɹɳɚɹ ɜɨɞɚ
Ɉ
(ɧɚɱɚɥɨ ɩɪɨɰɟɫɫɚ ɩɚɪɨɨɛɪɚɡɨɜɚɧɢɹ); ɬɨɱɤɚ ɚ" – ɫɭɯɨɣ ɧɚɫɵɳɟɧɧɵɣ ɩɚɪ
(ɨɤɨɧɱɚɧɢɟ ɩɪɨɰɟɫɫɚ ɩɚɪɨɨɛɪɚɡɨɜɚɧɢɹ); ɬɨɱɤɚ ɚ – ɩɟɪɟɝɪɟɬɵɣ ɩɚɪ.
Ɉɬɪɟɡɨɤ ɚ
ɚ' ɧɚ ɩɪɹɦɨɣ ɚɈɚ'ɚ"ɚ ɢɡɨɛɪɚɠɚɟɬ ɜ ɪX-ɞɢɚɝɪɚɦɦɟ ɩɪɨɰɟɫɫ
Ɉ
ɧɚɝɪɟɜɚ ɜɨɞɵ ɞɨ ɤɢɩɟɧɢɹ, ɨɬɪɟɡɨɤ ɚ'ɚ" – ɩɚɪɨɨɛɪɚɡɨɜɚɧɢɟ; ɨɬɪɟɡɨɤ ɚ"ɚ –
ɩɟɪɟɝɪɟɜ ɩɚɪɚ.
ɉɨ ɦɟɪɟ ɭɜɟɥɢɱɟɧɢɹ ɞɚɜɥɟɧɢɹ ɪɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ ɩɨɝɪɚɧɢɱɧɵɦɢ ɤɪɢɜɵɦɢ ɭɦɟɧɶɲɚɟɬɫɹ ɢ, ɧɚɤɨɧɟɰ, ɤɪɢɜɵɟ ɫɯɨɞɹɬɫɹ ɜ ɤɪɢɬɢɱɟɫɤɨɣ ɬɨɱɤɟ Ʉ.
ȼ ɷɬɨɦ ɫɨɫɬɨɹɧɢɢ ɢɫɱɟɡɚɟɬ ɪɚɡɥɢɱɢɟ ɜ ɫɜɨɣɫɬɜɚɯ ɩɚɪɚ ɢ ɜɨɞɵ. Ⱦɥɹ ɜɨɞɵ
ɩɚɪɚɦɟɬɪɵ ɤɪɢɬɢɱɟɫɤɨɝɨ ɫɨɫɬɨɹɧɢɹ ɫɥɟɞɭɸɳɢɟ: ɪ
tɄɊ = 374,12 °ɋ;
X
= 0,003147 ɦ3/ɤɝ; ɭɞɟɥɶɧɚɹ ɷɧɬɚɥɶɩɢɹ hɄɊ = 2095,2 ɤȾɠ/ɤɝ;
ɄɊ
16
= 22,115 Ɇɉɚ;
ɄɊ
ɭɞɟɥɶɧɚɹ ɷɧɬɪɨɩɢɹ sɄɊ = 4,4237 ɤȾɠ/(ɤɝɄ). ɉɪɢ ɬɟɦɩɟɪɚɬɭɪɚɯ ɜɵɲɟ ɤɪɢ-
XXX
X
X
ɬɢɱɟɫɤɨɣ t
ɜɨɡɦɨɠɧɨ ɫɨɫɭɳɟɫɬɜɨɜɚɧɢɟ ɬɨɥɶɤɨ ɩɟɪɟɝɪɟɬɨɝɨ ɩɚɪɚ.
ɄɊ
Ɍɨɱɤɚ Ⱥ' ɩɟɪɟɫɟɱɟɧɢɹ ɥɢɧɢɢ ɤɢɩɹɳɟɣ ɠɢɞɤɨɫɬɢ ɢ ɥɢɧɢɢ ɯɨɥɨɞɧɨɣ ɜɨɞɵ
ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɫɨɫɬɨɹɧɢɸ ɤɢɩɹɳɟɣ ɠɢɞɤɨɫɬɢ ɜ ɬɪɨɣɧɨɣ ɬɨɱɤɟ. ȼ ɷɬɨɣ ɬɨɱɤɟ
ɦɨɝɭɬ ɧɚɯɨɞɢɬɶɫɹ ɜ ɪɚɜɧɨɜɟɫɢɢ ɜɫɟ ɬɪɢ ɮɚɡɵ ɜɨɞɵ: ɥɟɞ, ɜɨɞɚ ɢ ɩɚɪ. ɉɚɪɚɦɟɬɪɵ ɬɪɨɣɧɨɣ ɬɨɱɤɢ ɞɥɹ ɜɨɞɵ: ɪ
= 0,61 ɤɉɚ; tɈ = 0,01 °ɋ;
Ɉ
X
= 0,001 ɦ3/ɤɝ.
Ɉ
Ʌɢɧɢɢ ɯɨɥɨɞɧɨɣ ɢ ɤɢɩɹɳɟɣ ɜɨɞɵ ɢ ɫɭɯɨɝɨ ɧɚɫɵɳɟɧɧɨɝɨ ɩɚɪɚ ɪɚɡɛɢɜɚɸɬ ɩɨɥɟ ɞɢɚɝɪɚɦɦɵ ɧɚ ɬɪɢ ɨɛɥɚɫɬɢ:
ɨɛɥɚɫɬɶ I, ɪɚɫɩɨɥɨɠɟɧɧɚɹ ɦɟɠɞɭ ɥɢɧɢɹɦɢ Ⱥ
Ⱥ' ɢ Ⱥ'Ʉ, ɯɚɪɚɤɬɟɪɢɡɭɟɬ
Ɉ
ɫɨɫɬɨɹɧɢɟ ɧɟɞɨɝɪɟɬɨɣ ɜɨɞɵ;
ɨɛɥɚɫɬɶ II, ɨɝɪɚɧɢɱɟɧɧɚɹ ɥɢɧɢɹɦɢ Ⱥ'Ʉ ɢ ɄȺ", ɪɚɜɧɨɜɟɫɧɨɟ ɫɨɫɬɨɹɧɢɟ ɜɨɞɵ ɢ ɜɥɚɠɧɨɝɨ ɧɚɫɵɳɟɧɧɨɝɨ ɩɚɪɚ ɫ ɪɚɡɥɢɱɧɵɦɢ ɫɬɟɩɟɧɹɦɢ ɫɭɯɨɫɬɢ
(ɞɜɭɯɮɚɡɧɨɟ ɫɨɫɬɨɹɧɢɟ);
ɨɛɥɚɫɬɶ III, ɧɚɯɨɞɢɬɫɹ ɩɪɚɜɟɟ ɤɪɢɜɨɣ ɄȺ", ɩɟɪɟɝɪɟɬɵɣ ɩɚɪ.
Ʉɨɥɢɱɟɫɬɜɨ ɬɟɩɥɨɬɵ, ɤɨɬɨɪɨɟ ɧɭɠɧɨ ɫɨɨɛɳɢɬɶ ɜɨɞɟ, ɱɬɨɛɵ ɧɚɝɪɟɬɶ ɟɟ
t0 = 0 °ɋ ɞɨ ɬɟɦɩɟɪɚɬɭɪɵ ɤɢɩɟɧɢɹ ɜ ɩɪɨɰɟɫɫɟ ɪ = const, ɧɚɡɵɜɚɟɬɫɹ ɬɟɩ-
ɨɬ
ɥɨɬɨɣ ɠɢɞɤɨɫɬɢ ɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ:
c
ɝɞɟ
ɞɨ t
ɫɪɟɞɧɹɹ ɬɟɩɥɨɟɦɤɨɫɬɶ ɜɨɞɵ ɜ ɢɧɬɟɪɜɚɥɟ ɬɟɦɩɟɪɚɬɭɪ ɨɬ 0 °ɋ
Pȼ
°ɋ.
ɇ
ɉɪɢ ɧɢɡɤɢɯ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ Ɍ
ɄɊ
,'
ttcq
0
HPȼ
ɬɟɦɩɟɪɚɬɭɪɚɯ ɦɨɠɧɨ ɫɱɢɬɚɬɶ
c
=
Pȼ
= 4,1865 ɤȾɠ/(ɤɝāɄ).
ɉɚɪɚɦɟɬɪɵ ɜɥɚɠɧɨɝɨ ɧɚɫɵɳɟɧɧɨɝɨ ɩɚɪɚ ɩɪɢ ɡɚɞɚɧɧɨɣ ɜɟɥɢɱɢɧɟ ɫɭɯɨɫɬɢ ɦɨɝɭɬ ɛɵɬɶ ɨɩɪɟɞɟɥɟɧɵ ɢɡ ɫɥɟɞɭɸɳɢɯ ɫɨɨɬɧɨɲɟɧɢɣ.
ɍɞɟɥɶɧɵɣ ɨɛɴɟɦ ɜɥɚɠɧɨɝɨ ɧɚɫɵɳɟɧɧɨɝɨ ɩɚɪɚ:
".'1
xx
Ɍɚɤ ɤɚɤ ɨɛɴɟɦ ɜɨɞɵ (1 ɯ)X' ɦɚɥ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɨɛɴɟɦɨɦ ɩɚɪɚ, ɬɨ
ɩɪɢ ɧɟɜɵɫɨɤɢɯ ɞɚɜɥɟɧɢɹɯ:
".
x
ɗɧɬɚɥɶɩɢɸ ɜɥɚɠɧɨɝɨ ɧɚɫɵɳɟɧɧɨɝɨ ɩɚɪɚ ɫ ɭɱɟɬɨɦ ɬɨɝɨ, ɱɬɨ ɧɚ ɩɪɟɜɪɚɳɟɧɢɟ ɜ ɩɚɪ ɯ ɤɝ ɠɢɞɤɨɫɬɢ ɧɟɨɛɯɨɞɢɦɨ ɡɚɬɪɚɬɢɬɶ xr ɤȾɠ/ɤɝ ɬɟɩɥɨɬɵ,
ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɩɨ ɮɨɪɦɭɥɚɦ:
,' xrhh
ɋɤɪɵɬɚɹ ɬɟɩɥɨɬɚ ɩɚɪɨɨɛɪɚɡɨɜɚɧɢɹ, ɤȾɠ/ɤɝ:
.1'" xhxhh
r = h" – h'.
17
ɍɞɟɥɶɧɚɹ ɷɧɬɪɨɩɢɹ ɜɥɚɠɧɨɝɨ ɧɚɫɵɳɟɧɧɨɝɨ ɩɚɪɚ, ɤȾɠ/(ɤɝɄ),
s = s" x + s' (1 – x).
ɉɟɪɟɝɪɟɬɵɣ ɩɚɪ ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɬɟɦ, ɱɬɨ ɟɝɨ ɬɟɦɩɟɪɚɬɭɪɚ ɜɵɲɟ ɬɟɦɩɟɪɚɬɭɪɵ ɩɚɪɨɨɛɪɚɡɨɜɚɧɢɹ Ɍ
ɩɪɢ ɬɨɦ ɠɟ ɞɚɜɥɟɧɢɢ, ɢ ɭɞɟɥɶɧɵɣ ɨɛɴɟɦ ɟɝɨ
ɇ
ɛɨɥɶɲɟ, ɱɟɦ ɫɭɯɨɝɨ ɧɚɫɵɳɟɧɧɨɝɨ ɩɚɪɚ ɩɪɢ ɬɨɦ ɠɟ ɞɚɜɥɟɧɢɢ.
Ʉɨɥɢɱɟɫɬɜɨ ɬɟɩɥɨɬɵ, ɧɟɨɛɯɨɞɢɦɨɟ ɞɥɹ ɩɟɪɟɜɨɞɚ 1 ɤɝ ɫɭɯɨɝɨ ɧɚɫɵɳɟɧɧɨɝɨ ɩɚɪɚ ɩɪɢ ɪ = const ɜ ɩɟɪɟɝɪɟɬɵɣ ɫ ɬɟɦɩɟɪɚɬɭɪɨɣ t, ɧɚɡɵɜɚɸɬ ɬɟɩ-
ɥɨɬɨɣ ɩɟɪɟɝɪɟɜɚ q
ɢ ɨɩɪɟɞɟɥɹɸɬ ɩɨ ɮɨɪɦɭɥɟ:
ɉ
T
dTɫq .
Ɋɉ
³
T
H
ȿɫɥɢ ɫɊm – ɫɪɟɞɧɹɹ ɦɚɫɫɨɜɚɹ ɬɟɩɥɨɟɦɤɨɫɬɶ ɩɟɪɟɝɪɟɬɨɝɨ ɩɚɪɚ ɩɪɢ ɩɨɫɬɨ-
ɹɧɧɨɦ ɞɚɜɥɟɧɢɢ, ɬɨ:
ɗɧɬɚɥɶɩɢɹ ɩɟɪɟɝɪɟɬɨɝɨ ɩɚɪɚ:
.
TTcq
HPmɉ
TTcrhqhh '"
HPmɉɉ
ɧɚɡɵɜɚɟɬɫɹ ɩɨɥɧɨɣ ɬɟɩɥɨɬɨɣ ɩɟɪɟɝɪɟɬɨɝɨ ɩɚɪɚ.
Ⱦɢɚɝɪɚɦɦɚ Ɍ, s ɜɨɞɹɧɨɝɨ ɩɚɪɚ. Ⱦɥɹ ɝɪɚɮɢɱɟɫɤɨɝɨ ɢɡɨɛɪɚɠɟɧɢɹ ɩɪɨɰɟɫɫɨɜ, ɩɪɨɢɫɯɨɞɹɳɢɯ ɜ ɩɚɪɟ, ɭɞɨɛɧɨ ɩɨɥɶɡɨɜɚɬɶɫɹ Ɍs-ɞɢɚɝɪɚɦɦɨɣ (ɪɢɫ. 1.6).
ɉɥɨɳɚɞɶ ɩɨɞ ɤɪɢɜɨɣ ɨɛɪɚɬɢɦɨɝɨ ɩɪɨɰɟɫɫɚ ɨɩɪɟɞɟɥɹɟɬ ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɨɬɵ, ɫɨɨɛɳɚɟɦɨɟ ɪɚɛɨɱɟɦɭ ɜɟɳɟɫɬɜɭ ɢɥɢ ɨɬɧɢɦɚɟɦɨɟ ɨɬ ɧɟɝɨ.
T
ɀɢɞɤɨɫɬɶ
b
p=const
ɯ = 0
Ʉ
e
c
ɉɟɪɟɝɪɟɬɵɣ
ɩɚɪ
a
ȼɥɚɠɧɵɣ
ɩɚɪ
Ɉ
A B C
Ɋɢɫ. 1.6. Ɍs-ɞɢɚɝɪɚɦɦɚ ɜɨɞɹɧɨɝɨ ɩɚɪɚ
18
ɯ = 1
ɯ = const
s
Ɉɛɥɚɫɬɶ, ɥɟɠɚɳɚɹ ɦɟɠɞɭ ɥɢɧɢɟɣ ɤɢɩɹɳɟɣ ɠɢɞɤɨɫɬɢ (ɚɄ) ɢ ɥɢɧɢɟɣ
ɫɭɯɨɝɨ ɧɚɫɵɳɟɧɧɨɝɨ ɩɚɪɚ (ɫɄ), ɷɬɨ ɨɛɥɚɫɬɶ ɜɥɚɠɧɨɝɨ ɧɚɫɵɳɟɧɧɨɝɨ ɩɚɪɚ.
Ɉɛɥɚɫɬɶ, ɥɟɠɚɳɚɹ ɩɪɚɜɟɟ ɥɢɧɢɢ ɫɄ, ɨɛɥɚɫɬɶ ɩɟɪɟɝɪɟɬɨɝɨ ɩɚɪɚ.
Ɍɚɤ ɤɚɤ ɩɪɨɰɟɫɫ ɩɚɪɨɨɛɪɚɡɨɜɚɧɢɹ ɢɞɟɬ ɩɪɢ Ɍɇ = const ɢ ɪ = const, ɢɡɨɬɟɪɦɚ b-c ɹɜɥɹɟɬɫɹ ɨɞɧɨɜɪɟɦɟɧɧɨ ɢ ɢɡɨɛɚɪɨɣ. Ⱦɚɥɶɧɟɣɲɢɣ ɩɨɞɜɨɞ ɬɟɩɥɨɬɵ ɫɧɨɜɚ ɫɨɩɪɨɜɨɠɞɚɟɬɫɹ ɭɜɟɥɢɱɟɧɢɟɦ ɬɟɦɩɟɪɚɬɭɪɵ ɢ ɷɧɬɪɨɩɢɢ ɩɪɨɰɟɫɫ
ɩɟɪɟɝɪɟɜɚ ɩɚɪɚ (ɤɪɢɜɚɹ ɫe).
Ɍɟɩɥɨɬɚ, ɩɨɞɜɟɞɟɧɧɚɹ ɤ ɠɢɞɤɨɫɬɢ ɜ ɩɪɨɰɟɫɫɟ ɧɚɝɪɟɜɚ ɞɨ ɫɨɫɬɨɹɧɢɹ
ɤɢɩɟɧɢɹ, ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɚ ɩɥɨɳɚɞɢ abȺɈ. ɉɥɨɳɚɞɶ bcȼȺ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɚ ɬɟɩɥɨɬɟ, ɩɨɞɜɨɞɢɦɨɣ ɤ ɜɨɞɟ ɜ ɩɪɨɰɟɫɫɟ ɩɚɪɨɨɛɪɚɡɨɜɚɧɢɹ
; ɩɥɨɳɚɞɶ
ɫɟɋȼ – ɬɟɩɥɨɬɟ, ɡɚɬɪɚɱɟɧɧɨɣ ɧɚ ɩɟɪɟɝɪɟɜ ɩɚɪɚ.
Ⱦɢɚɝɪɚɦɦɚ h, s ɜɨɞɹɧɨɝɨ ɩɚɪɚ. Ⱦɥɹ ɢɡɭɱɟɧɢɹ ɢ ɪɚɫɱɟɬɨɜ ɪɚɡɥɢɱɧɵɯ
ɬɟɪɦɨɞɢɧɚɦɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ, ɜ ɤɨɬɨɪɨɦ ɪɚɛɨɱɢɦ ɜɟɳɟɫɬɜɨɦ ɹɜɥɹɟɬɫɹ
ɧɚɫɵɳɟɧɧɵɣ ɢ ɩɟɪɟɝɪɟɬɵɣ ɩɚɪ, ɭɞɨɛɧɨ ɩɨɥɶɡɨɜɚɬɶɫɹ hs-ɞɢɚɝɪɚɦɦɨɣ
(ɪɢɫ. 1.7). ɇɚ ɞɢɚɝɪɚɦɦɟ ɥɢɧɢɢ ɤɢɩɹɳɟɣ ɠɢɞɤɨɫɬɢ ɢ ɫɭɯɨɝɨ ɧɚɫɵɳɟɧɧɨɝɨ
ɩɚɪɚ ɫɥɢɜɚɸɬɫɹ ɜ ɤɪɢɬɢɱɟɫɤɨɣ ɬɨɱɤɟ Ʉ. ȼ ɷɬɨɣ
ɞɢɚɝɪɚɦɦɟ ɬɟɩɥɨɬɚ ɠɢɞɤɨɫɬɟɣ, ɩɚɪɨɨɛɪɚɡɨɜɚɧɢɹ ɢ ɩɟɪɟɝɪɟɜɚ ɢɡɨɛɪɚɠɚɟɬɫɹ ɥɢɧɟɣɧɵɦɢ ɨɬɪɟɡɤɚɦɢ, ɚ
ɧɟ ɩɥɨɳɚɞɹɦɢ. Ɍɟɩɥɨɬɚ ɩɚɪɨɨɛɪɚɡɨɜɚɧɢɹ ɩɨ ɞɚɧɧɨɣ ɢɡɨɛɚɪɟ ɪɚɜɧɚ ɪɚɡɧɨɫɬɢ ɨɪɞɢɧɚɬ ɬɨɱɟɤ ɩɟɪɟɫɟɱɟɧɢɹ ɢɡɨɛɚɪɵ ɫ ɩɨɝɪɚɧɢɱɧɵɦɢ ɤɪɢɜɵɦɢ:
ȼ ɨɛɥɚɫɬɢ ɜɥɚɠɧɨɝɨ ɩɚɪɚ ɢɡɨɛɚɪɵ,
ɹɜɥɹɹɫɶ ɨɞɧɨɜɪɟɦɟɧɧɨ ɢ ɢɡɨɬɟɪɦɚɦɢ,
r = h" – h'.
h
p = const
ɩɪɟɞɫɬɚɜɥɹɸɬ ɫɨɛɨɣ ɩɪɹɦɵɟ ɥɢɧɢɢ.
ɂɡɨɛɚɪɵ ɩɟɪɟɫɟɤɚɸɬ ɩɨɝɪɚɧɢɱɧɵɟ ɤɪɢɜɵɟ ɛɟɡ ɢɡɥɨɦɚ.
ɂɡɨɛɚɪɵ ɜ ɨɛɥɚɫɬɢ ɩɟɪɟɝɪɟɬɨɝɨ ɩɚɪɚ
ɫɥɚɛɨɜɨɝɧɭɬɵɟ ɥɨɝɚɪɢɮɦɢɱɟɫɤɢɟ ɤɪɢɜɵɟ, ɢɡɨɬɟɪɦɵ – ɜɵɩɭɤɥɵɟ ɤɪɢɜɵɟ,
ɩɨɞɧɢɦɚɸɳɢɟɫɹ ɫɥɟɜɚ ɜɜɟɪɯ ɧɚɩɪɚɜɨ.
ɑɟɦ ɛɨɥɶɲɟ ɬɟɦɩɟɪɚɬɭɪɚ, ɬɟɦ ɜɵɲɟ
h"
h'
r
ɯ = 0
Ⱥ
Ʉ
p = const
ɯ = 0,5
ȼ
ɋ
t = const
ɯ = 1
ɯ = 0,95
ɪɚɫɩɨɥɚɝɚɟɬɫɹ ɢɡɨɬɟɪɦɚ. ɑɟɦ ɞɚɥɶɲɟ ɨɬ
ɩɨɝɪɚɧɢɱɧɨɣ ɤɪɢɜɨɣ (ɯ = 1) ɩɪɨɯɨɞɢɬ
ɢɡɨɬɟɪɦɚ, ɬɟɦ ɛɨɥɶɲɟ ɨɧɚ ɩɪɢɛɥɢɠɚɟɬ-
s'
ss
ɫɹ ɤ ɝɨɪɢɡɨɧɬɚɥɢ (h = const), ɬɚɤ ɤɚɤ ɜ
ɨɛɥɚɫɬɢ ɢɞɟɚɥɶɧɨɝɨ ɝɚɡɚ ɷɧɬɚɥɶɩɢɹ ɨɞɧɨɡɧɚɱɧɨ ɨɩɪɟɞɟɥɹɟɬɫɹ ɬɟɦɩɟɪɚɬɭɪɨɣ.
Ɋɢɫ. 1.7. hs-ɞɢɚɝɪɚɦɦɚ
ɜɨɞɹɧɨɝɨ ɩɚɪɚ
ɇɚ ɪɢɫ. 1.7 ɬɨɱɤɢ Ⱥ, ȼ, ɋ ɢɡɨɛɪɚɠɚɸɬ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɫɨɫɬɨɹɧɢɟ ɜɥɚɠ-
ɧɨɝɨ, ɫɭɯɨɝɨ ɢ ɩɟɪɟɝɪɟɬɨɝɨ ɩɚɪɚ.
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1.3.4. Ɉɫɧɨɜɧɵɟ ɬɟɪɦɨɞɢɧɚɦɢɱɟɫɤɢɟ ɩɪɨɰɟɫɫɵ ɜɨɞɹɧɨɝɨ ɩɚɪɚ
X
X
Ɉɫɧɨɜɧɵɦɢ ɡɚɞɚɱɚɦɢ ɚɧɚɥɢɡɚ ɬɟɪɦɨɞɢɧɚɦɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ ɜɨɞɹɧɨɝɨ ɩɚɪɚ ɹɜɥɹɸɬɫɹ ɧɚɯɨɠɞɟɧɢɟ ɧɚɱɚɥɶɧɵɯ, ɤɨɧɟɱɧɵɯ ɩɚɪɚɦɟɬɪɨɜ ɢ ɮɭɧɤɰɢɣ ɫɨɫɬɨɹɧɢɹ; ɨɩɪɟɞɟɥɟɧɢɟ ɜɟɥɢɱɢɧ, ɜɯɨɞɹɳɢɯ ɜ ɭɪɚɜɧɟɧɢɟ ɩɟɪɜɨɝɨ ɡɚɤɨɧɚ ɬɟɪɦɨɞɢɧɚɦɢɤɢ; ɩɨɫɬɪɨɟɧɢɟ ɝɪɚɮɢɱɟɫɤɨɝɨ ɢɡɨɛɪɚɠɟɧɢɹ ɩɪɨɰɟɫɫɨɜ ɜ
ɞɢɚɝɪɚɦɦɚɯ.
ɂɡɨɯɨɪɧɵɣ ɩɪɨɰɟɫɫ (
X
= const) 12, ɩɪɨɬɟɤɚɸɳɢɣ ɰɟɥɢɤɨɦ ɜ ɨɛɥɚɫ-
ɬɢ ɧɚɫɵɳɟɧɢɹ, ɢ 34, ɡɚɤɚɧɱɢɜɚɸɳɢɣɫɹ ɜ ɡɨɧɟ ɩɟɪɟɝɪɟɜɚ, ɩɪɟɞɫɬɚɜɥɟɧ
ɧɚ ɪɢɫ. 1.8.
ɉɪɨɰɟɫɫɵ 12 ɢ 34 ɩɪɨɬɟɤɚɸɬ ɜ ɨɞɢɧɚɤɨɜɨɦ ɢɧɬɟɪɜɚɥɟ ɞɚɜɥɟɧɢɣ ɪ1 ɢ ɪ2.
Ʉɨɧɮɢɝɭɪɚɰɢɹ ɢɡɨɯɨɪ ɜ Ɍs- ɢ hs-ɞɢɚɝɪɚɦɦɚɯ ɨɩɪɟɞɟɥɹɟɬɫɹ ɡɧɚɱɟɧɢɹɦɢ
ɫɬɟɩɟɧɢ ɫɭɯɨɫɬɢ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɞɚɜɥɟɧɢɹ ɜ ɨɛɥɚɫɬɢ ɧɚɫɵɳɟɧɢɹ ɢ ɡɧɚɱɟɧɢɹɦɢ ɞɚɜɥɟɧɢɹ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɨɛɥɚɫɬɢ ɩɟɪɟɝɪɟɜɚ.
p
ɪ
2
ɪ
1
Ʉ
2
4
ɯ
2
3 1
ɯ = 0 ɯ = 1
ɚ
T
T
4
t
4
t
2
T
2
T
1
Ʉ
4
ɪ
2
4
2
ɪ
1
1
3
ɯ = 1 ɯ = 0
s
h
h
4
h
2
h
3
h
1
ɛ
p
2
4
t
4
t
2
p
1
t
1
2
3
ɯ = 1
1
s
ɜ
Ɋɢɫ. 1.8. ɂɡɨɯɨɪɧɵɟ ɩɪɨɰɟɫɫɵ ɞɥɹ ɜɨɞɹɧɨɝɨ ɩɚɪɚ:
X
-ɞɢɚɝɪɚɦɦɚ; ɛ – Ɍ-s-ɞɢɚɝɪɚɦɦɚ; ɜ – h-s-ɞɢɚɝɪɚɦɦɚ
ɚ – ɪ-
ɂɡɨɯɨɪɧɵɟ ɩɪɨɰɟɫɫɵ ɧɚɛɥɸɞɚɸɬɫɹ ɜ ɛɚɪɚɛɚɧɚɯ ɩɚɪɨɜɵɯ ɤɨɬɥɨɜ ɢ
ɬɟɩɥɨɨɛɦɟɧɧɢɤɚɯ ɩɪɢ ɧɚɝɪɟɜɚɧɢɢ ɢɥɢ ɨɯɥɚɠɞɟɧɢɢ ɡɚɩɨɥɧɹɸɳɟɝɨ ɢɯ ɬɟɩɥɨɧɨɫɢɬɟɥɹ, ɟɫɥɢ ɨɛɨɪɭɞɨɜɚɧɢɟ ɨɬɤɥɸɱɟɧɨ ɨɬ ɜɧɟɲɧɢɯ ɤɨɦɦɭɧɢɤɚɰɢɣ.
Ⱦɥɹ ɬɟɪɦɨɞɢɧɚɦɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ, ɩɪɨɬɟɤɚɸɳɢɯ ɜ ɨɛɥɚɫɬɢ ɧɚɫɵɳɟɧɧɨɝɨ ɩɚɪɚ, ɡɧɚɱɟɧɢɹ ɷɧɬɚɥɶɩɢɣ ɩɚɪɚ ɦɨɝɭɬ ɛɵɬɶ ɧɚɣɞɟɧɵ ɧɚ h-s-ɞɢɚɝɪɚɦɦɟ
ɢɥɢ ɩɨ ɮɨɪɦɭɥɟ:
h
= h' + rx, (1.2)
X
ɝɞɟ h' – ɷɧɬɚɥɶɩɢɹ ɜɨɞɵ;
r – ɫɤɪɵɬɚɹ ɬɟɩɥɨɬɚ ɩɚɪɨɨɛɪɚɡɨɜɚɧɢɹ;
ɯ – ɫɬɟɩɟɧɶ ɫɭɯɨɫɬɢ ɩɚɪɚ.
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