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April 2012<br />
<strong>gwf</strong>-<strong>Gas</strong> <strong>Erdgas</strong> 265<br />
Rohrnetz<br />
Fachberichte<br />
Anhand <strong>die</strong>ses Beispiels lässt sich gut nachvollziehen,<br />
auf welche Weise verschiedene Gleichungen <strong>für</strong><br />
hydraulisch glattes und hydraulisch raues Verhalten auf<br />
der Basis der Colebrook-Whiteschen Interpolationsregel<br />
miteinander kombiniert werden können. Formale Voraussetzung<br />
hier<strong>für</strong> war <strong>die</strong> Darstellung der zu kombinierenden<br />
Gleichungssätze in der Form von Gl. (48).<br />
4.5 Rohrreibungszahl – Nebenbetrachtung III<br />
Nunmehr soll <strong>die</strong>se Art der „Rekombination“ von verfügbaren<br />
Teilgleichungen <strong>für</strong> das hydraulisch glatte und<br />
raue Verhalten mit dem Ansatz von Techo, Tickner und<br />
James (siehe Gl. (33)) <strong>für</strong> <strong>die</strong> Glattrohrströmung und<br />
dem Ansatz von Nikuradse <strong>für</strong> hydraulisch raues Verhalten<br />
(Gl. (21)) nochmals ausprobiert werden. Zunächst<br />
<strong>sind</strong> <strong>für</strong> den Term der Glattrohrströmung einige Umformungen<br />
erforderlich, um eine Gleichungsstruktur<br />
gemäß dem Ansatz (Gl. (48)) zu erreichen. Diese arithmetischen<br />
Manipulationen sollen nachfolgend kurz und<br />
weitestgehend unkommentiert erfolgen:<br />
Gl. (33) in der „klassischen“ Struktur lautet:<br />
1<br />
0 86859<br />
1 964 38215<br />
1<br />
086<br />
λ<br />
λ<br />
glatt<br />
glatt<br />
= ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=−<br />
, ln<br />
Re<br />
, lnRe ,<br />
, 859<br />
1 964 38215<br />
0 4343<br />
1<br />
⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
( ) =<br />
( )<br />
ln , lnRe ,<br />
Re<br />
ln<br />
lg<br />
,<br />
x<br />
x<br />
λ glatt<br />
= ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
= ⋅<br />
0 8686<br />
0 4343 1 964 38215<br />
1<br />
2<br />
,<br />
,<br />
lg<br />
Re<br />
, lnRe ,<br />
lg<br />
Re<br />
λ glatt 1 964 0 4343 3 8215<br />
2<br />
1964<br />
04343<br />
,<br />
lg Re<br />
,<br />
,<br />
lg<br />
Re<br />
,<br />
,<br />
lg R<br />
⋅<br />
( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎜<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎟<br />
⎟<br />
= ⋅<br />
⋅ e ,<br />
lg<br />
Re<br />
, lgRe ,<br />
( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎜<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎟<br />
⎟<br />
= ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
38215<br />
1<br />
2<br />
4 5222 3 8215<br />
λ glatt<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
1<br />
2<br />
4 5222 3 8215<br />
1<br />
2<br />
λ<br />
λ<br />
glatt<br />
glatt<br />
lg<br />
, lgRe ,<br />
Re<br />
lg 4 5222 3 8215<br />
2<br />
45222 3 821<br />
, lgRe<br />
Re<br />
,<br />
Re<br />
lg<br />
,<br />
Re<br />
lg Re<br />
,<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟ =− ⋅ ⋅ ( ) − 5<br />
1<br />
2<br />
4 5222 3 8215<br />
4 5222<br />
Re<br />
lg<br />
,<br />
Re<br />
lg Re<br />
,<br />
,<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅ ⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎛<br />
λ glatt ⎝ ⎜ ⎞<br />
⎠<br />
⎟<br />
=− ⋅ ⋅ ( ) −<br />
( )<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
( ) =<br />
1<br />
2<br />
4 5222<br />
0 845<br />
7 0 84<br />
λ glatt<br />
lg<br />
,<br />
Re<br />
lg Re ,<br />
lg , 5<br />
1 4 5222<br />
⎛<br />
⎞<br />
,<br />
(51)<br />
oder<br />
1<br />
0 86859<br />
1 964 38215<br />
1<br />
086<br />
λ<br />
λ<br />
glatt<br />
glatt<br />
= ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=−<br />
, ln<br />
Re<br />
, lnRe ,<br />
, 859<br />
1 964 38215<br />
0 4343<br />
1<br />
⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
( ) =<br />
( )<br />
ln , lnRe ,<br />
Re<br />
ln<br />
lg<br />
,<br />
x<br />
x<br />
λ glatt<br />
= ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
= ⋅<br />
0 8686<br />
0 4343 1 964 38215<br />
1<br />
2<br />
,<br />
,<br />
lg<br />
Re<br />
, lnRe ,<br />
lg<br />
Re<br />
λ glatt 1 964 0 4343 3 8215<br />
2<br />
1964<br />
04343<br />
,<br />
lg Re<br />
,<br />
,<br />
lg<br />
Re<br />
,<br />
,<br />
lg R<br />
⋅<br />
( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎜<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎟<br />
⎟<br />
= ⋅<br />
⋅ e ,<br />
lg<br />
Re<br />
, lgRe ,<br />
( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎜<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎟<br />
⎟<br />
= ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
38215<br />
1<br />
2<br />
4 5222 3 8215<br />
λ glatt<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
1<br />
2<br />
4 5222 3 8215<br />
1<br />
2<br />
λ<br />
λ<br />
glatt<br />
glatt<br />
lg<br />
, lgRe ,<br />
Re<br />
lg 4 5222 3 8215<br />
2<br />
45222 3 821<br />
, lgRe<br />
Re<br />
,<br />
Re<br />
lg<br />
,<br />
Re<br />
lg Re<br />
,<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟ =− ⋅ ⋅ ( ) − 5<br />
1<br />
2<br />
4 5222 3 8215<br />
4 5222<br />
Re<br />
lg<br />
,<br />
Re<br />
lg Re<br />
,<br />
,<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅ ⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎛<br />
λ glatt ⎝ ⎜ ⎞<br />
⎠<br />
⎟<br />
=− ⋅ ⋅ ( ) −<br />
( )<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
( ) =<br />
1<br />
2<br />
4 5222<br />
0 845<br />
7 0 84<br />
λ glatt<br />
lg<br />
,<br />
Re<br />
lg Re ,<br />
lg , 5<br />
(52)<br />
Es gilt:<br />
1<br />
0 86859<br />
1 964 38215<br />
1<br />
086<br />
λ<br />
λ<br />
glatt<br />
glatt<br />
= ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=−<br />
, ln<br />
Re<br />
, lnRe ,<br />
, 859<br />
1 964 38215<br />
0 4343<br />
1<br />
⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
( ) =<br />
( )<br />
ln , lnRe ,<br />
Re<br />
ln<br />
lg<br />
,<br />
x<br />
x<br />
λ glatt<br />
= ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
= ⋅<br />
0 8686<br />
0 4343 1 964 38215<br />
1<br />
2<br />
,<br />
,<br />
lg<br />
Re<br />
, lnRe ,<br />
lg<br />
Re<br />
λ glatt 1 964 0 4343 3 8215<br />
2<br />
1964<br />
04343<br />
,<br />
lg Re<br />
,<br />
,<br />
lg<br />
Re<br />
,<br />
,<br />
lg R<br />
⋅<br />
( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎜<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎟<br />
⎟<br />
= ⋅<br />
⋅ e ,<br />
lg<br />
Re<br />
, lgRe ,<br />
( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎜<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎟<br />
⎟<br />
= ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
38215<br />
1<br />
2<br />
4 5222 3 8215<br />
λ glatt<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
1<br />
2<br />
4 5222 3 8215<br />
1<br />
2<br />
λ<br />
λ<br />
glatt<br />
glatt<br />
lg<br />
, lgRe ,<br />
Re<br />
lg 4 5222 3 8215<br />
2<br />
45222 3 821<br />
, lgRe<br />
Re<br />
,<br />
Re<br />
lg<br />
,<br />
Re<br />
lg Re<br />
,<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟ =− ⋅ ⋅ ( ) − 5<br />
1<br />
2<br />
4 5222 3 8215<br />
4 5222<br />
Re<br />
lg<br />
,<br />
Re<br />
lg Re<br />
,<br />
,<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅ ⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎛<br />
λ glatt ⎝ ⎜ ⎞<br />
⎠<br />
⎟<br />
=− ⋅ ⋅ ( ) −<br />
( )<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
1<br />
2<br />
4 5222<br />
0 845<br />
λ glatt<br />
lg<br />
,<br />
Re<br />
lg Re ,<br />
Einsetzen in Gl. (51), Runden (0,86859 ≈ 0,8686) und<br />
Umformen:<br />
1<br />
0 86859<br />
1 964 38215<br />
1<br />
086<br />
λ<br />
λ<br />
glatt<br />
glatt<br />
= ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=−<br />
, ln<br />
Re<br />
, lnRe ,<br />
, 859<br />
1 964 38215<br />
0 4343<br />
1<br />
⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
( ) =<br />
( )<br />
ln , lnRe ,<br />
Re<br />
ln<br />
lg<br />
,<br />
x<br />
x<br />
λ glatt<br />
= ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
= ⋅<br />
0 8686<br />
0 4343 1 964 38215<br />
1<br />
2<br />
,<br />
,<br />
lg<br />
Re<br />
, lnRe ,<br />
lg<br />
Re<br />
λ glatt 1 964 0 4343 3 8215<br />
2<br />
1964<br />
04343<br />
,<br />
lg Re<br />
,<br />
,<br />
lg<br />
Re<br />
,<br />
,<br />
lg R<br />
⋅<br />
( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎜<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎟<br />
⎟<br />
= ⋅<br />
⋅ e ,<br />
lg<br />
Re<br />
, lgRe ,<br />
( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎜<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎟<br />
⎟<br />
= ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
38215<br />
1<br />
2<br />
4 5222 3 8215<br />
λ glatt<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
1<br />
2<br />
4 5222 3 8215<br />
1<br />
2<br />
λ<br />
λ<br />
glatt<br />
glatt<br />
lg<br />
, lgRe ,<br />
Re<br />
lg 4 5222 3 8215<br />
2<br />
45222 3 821<br />
, lgRe<br />
Re<br />
,<br />
Re<br />
lg<br />
,<br />
Re<br />
lg Re<br />
,<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟ =− ⋅ ⋅ ( ) − 5<br />
1<br />
2<br />
4 5222 3 8215<br />
4 5222<br />
Re<br />
lg<br />
,<br />
Re<br />
lg Re<br />
,<br />
,<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅ ⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎛<br />
λ glatt ⎝ ⎜ ⎞<br />
⎠<br />
⎟<br />
=− ⋅ ⋅ ( ) −<br />
⎛<br />
⎞<br />
1<br />
2<br />
4 5222<br />
0 845<br />
lg<br />
,<br />
lg Re ,<br />
1<br />
0 86859<br />
1 964 38215<br />
1<br />
086<br />
λ<br />
λ<br />
glatt<br />
glatt<br />
= ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=−<br />
, ln<br />
Re<br />
, lnRe ,<br />
, 859<br />
1 964 38215<br />
0 4343<br />
1<br />
⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
( ) =<br />
( )<br />
ln , lnRe ,<br />
Re<br />
ln<br />
lg<br />
,<br />
x<br />
x<br />
λ glatt<br />
= ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
= ⋅<br />
0 8686<br />
0 4343 1 964 38215<br />
1<br />
2<br />
,<br />
,<br />
lg<br />
Re<br />
, lnRe ,<br />
lg<br />
Re<br />
λ glatt 1 964 0 4343 3 8215<br />
2<br />
1964<br />
04343<br />
,<br />
lg Re<br />
,<br />
,<br />
lg<br />
Re<br />
,<br />
,<br />
lg R<br />
⋅<br />
( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎜<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎟<br />
⎟<br />
= ⋅<br />
⋅ e ,<br />
lg<br />
Re<br />
, lgRe ,<br />
( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎜<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎟<br />
⎟<br />
= ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
38215<br />
1<br />
2<br />
4 5222 3 8215<br />
λ glatt<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
1<br />
2<br />
4 5222 3 8215<br />
1<br />
2<br />
λ<br />
λ<br />
glatt<br />
glatt<br />
lg<br />
, lgRe ,<br />
Re<br />
lg 4 5222 3 8215<br />
2<br />
45222 3 821<br />
, lgRe<br />
Re<br />
,<br />
Re<br />
lg<br />
,<br />
Re<br />
lg Re<br />
,<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟ =− ⋅ ⋅ ( ) − 5<br />
1<br />
2<br />
4 5222 3 8215<br />
4 5222<br />
Re<br />
lg<br />
,<br />
Re<br />
lg Re<br />
,<br />
,<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅ ⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎛<br />
λ glatt ⎝ ⎜ ⎞<br />
⎠<br />
⎟<br />
=− ⋅ ⋅ ( ) −<br />
( )<br />
⎛<br />
⎜<br />
⎞<br />
⎟<br />
1<br />
2<br />
4 5222<br />
0 845<br />
lg<br />
,<br />
lg Re ,<br />
1 964 38215<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
ln<br />
Re<br />
, lnRe ,<br />
1 964 38215<br />
⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
ln , lnRe ,<br />
Re<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
1 964 38215<br />
Re<br />
, lnRe ,<br />
Re<br />
4 0 4343 3 8215<br />
2<br />
1964<br />
04343<br />
lg Re<br />
,<br />
,<br />
lg<br />
Re<br />
,<br />
,<br />
lg R<br />
⋅<br />
( ) − ⎞<br />
⎠<br />
⎟<br />
⎟<br />
⎟<br />
= ⋅<br />
⋅ e ,<br />
Re<br />
lgRe ,<br />
( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎜<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎟<br />
⎟<br />
⋅ ( ) −<br />
38215<br />
22 3 8215<br />
⎞<br />
⎠<br />
⎟<br />
⋅ ( ) − ⎞<br />
⎠<br />
⎟<br />
222 3 8215<br />
lgRe ,<br />
Re<br />
222 3 8215<br />
2<br />
45222 3 821<br />
lgRe<br />
Re<br />
,<br />
Re<br />
lg<br />
,<br />
Re<br />
lg Re<br />
,<br />
⋅ ( ) − ⎞<br />
⎠<br />
⎟ =− ⋅ ⋅ ( ) − 5<br />
Re<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
1<br />
0 86859<br />
1 964 38215<br />
1<br />
086<br />
λ<br />
λ<br />
glatt<br />
glatt<br />
= ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=−<br />
, ln<br />
Re<br />
, lnRe ,<br />
, 859<br />
1 964 38215<br />
0 4343<br />
1<br />
⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
( ) =<br />
( )<br />
ln , lnRe ,<br />
Re<br />
ln<br />
lg<br />
,<br />
x<br />
x<br />
λ glatt<br />
= ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
= ⋅<br />
0 8686<br />
0 4343 1 964 38215<br />
1<br />
2<br />
,<br />
,<br />
lg<br />
Re<br />
, lnRe ,<br />
lg<br />
Re<br />
λ glatt 1 964 0 4343 3 8215<br />
2<br />
1964<br />
04343<br />
,<br />
lg Re<br />
,<br />
,<br />
lg<br />
Re<br />
,<br />
,<br />
lg R<br />
⋅<br />
( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎜<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎟<br />
⎟<br />
= ⋅<br />
⋅ e ,<br />
lg<br />
Re<br />
, lgRe ,<br />
( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎜<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎟<br />
⎟<br />
= ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
38215<br />
1<br />
2<br />
4 5222 3 8215<br />
λ glatt<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
1<br />
2<br />
4 5222 3 8215<br />
1<br />
2<br />
λ<br />
λ<br />
glatt<br />
glatt<br />
lg<br />
, lgRe ,<br />
Re<br />
lg 4 5222 3 8215<br />
2<br />
45222 3 821<br />
, lgRe<br />
Re<br />
,<br />
Re<br />
lg<br />
,<br />
Re<br />
lg Re<br />
,<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟ =− ⋅ ⋅ ( ) − 5<br />
Re<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
1<br />
0 86859<br />
1 964 38215<br />
1<br />
086<br />
λ<br />
λ<br />
glatt<br />
glatt<br />
= ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=−<br />
, ln<br />
Re<br />
, lnRe ,<br />
, 859<br />
1 964 38215<br />
0 4343<br />
1<br />
⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
( ) =<br />
( )<br />
ln , lnRe ,<br />
Re<br />
ln<br />
lg<br />
,<br />
x<br />
x<br />
λ glatt<br />
= ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
= ⋅<br />
0 8686<br />
0 4343 1 964 38215<br />
1<br />
2<br />
,<br />
,<br />
lg<br />
Re<br />
, lnRe ,<br />
lg<br />
Re<br />
λ glatt 1 964 0 4343 3 8215<br />
2<br />
1964<br />
04343<br />
,<br />
lg Re<br />
,<br />
,<br />
lg<br />
Re<br />
,<br />
,<br />
lg R<br />
⋅<br />
( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎜<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎟<br />
⎟<br />
= ⋅<br />
⋅ e ,<br />
lg<br />
Re<br />
, lgRe ,<br />
( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎜<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎟<br />
⎟<br />
= ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
38215<br />
1<br />
2<br />
4 5222 3 8215<br />
λ glatt<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
1<br />
2<br />
4 5222 3 8215<br />
1<br />
2<br />
λ<br />
λ<br />
glatt<br />
glatt<br />
lg<br />
, lgRe ,<br />
Re<br />
lg 4 5222 3 8215<br />
2<br />
45222 3 821<br />
, lgRe<br />
Re<br />
,<br />
Re<br />
lg<br />
,<br />
Re<br />
lg Re<br />
,<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟ =− ⋅ ⋅ ( ) − 5<br />
Re<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
lg<br />
Re<br />
, lgRe ,<br />
= ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
1<br />
2<br />
4 5222 3 8215<br />
λ glatt<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
1<br />
2<br />
4 5222 3 8215<br />
1<br />
2<br />
λ<br />
λ<br />
glatt<br />
glatt<br />
lg<br />
, lgRe ,<br />
Re<br />
lg 4 5222 3 8215<br />
2<br />
45222 3 821<br />
, lgRe<br />
Re<br />
,<br />
Re<br />
lg<br />
,<br />
Re<br />
lg Re<br />
,<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟ =− ⋅ ⋅ ( ) − 5<br />
1<br />
2<br />
4 5222 3 8215<br />
4 5222<br />
Re<br />
lg<br />
,<br />
Re<br />
lg Re<br />
,<br />
,<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅ ⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎛<br />
λ glatt ⎝ ⎜ ⎞<br />
⎠<br />
⎟<br />
=− ⋅ ⋅ ( ) −<br />
( )<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
( ) =<br />
1<br />
2<br />
4 5222<br />
0 845<br />
7 0 84<br />
λ glatt<br />
lg<br />
,<br />
Re<br />
lg Re ,<br />
lg , 5<br />
1<br />
2<br />
4 5222<br />
7<br />
1<br />
2<br />
4<br />
λ<br />
λ<br />
glatt<br />
glatt<br />
=− ⋅ ⋅ ( ) − ( )<br />
( )<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
lg<br />
,<br />
Re<br />
lg Re<br />
lg<br />
lg<br />
,<br />
Re<br />
lg Re , ln<br />
Re<br />
, lnRe ,<br />
5222<br />
7<br />
08686<br />
1964 38215<br />
⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟ ≡ ⋅ ⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅ ⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟ +<br />
⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
1<br />
2<br />
4 5222<br />
7 3 71<br />
1<br />
λ<br />
λ<br />
turb<br />
turb<br />
k<br />
D<br />
lg<br />
,<br />
Re<br />
lg Re ,<br />
=− ⋅<br />
⋅ ( ) −<br />
+<br />
⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
0 8686<br />
1 964 38215<br />
371<br />
1<br />
2<br />
, ln , lnRe ,<br />
Re ,<br />
k<br />
D<br />
λ glatt<br />
lg<br />
,<br />
Re<br />
lg Re<br />
4 518 7<br />
⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
= ⋅<br />
2 lg<br />
λ glatt 1 964 0 4343 3 8215<br />
2<br />
1964<br />
04343<br />
,<br />
lg Re<br />
,<br />
,<br />
lg<br />
,<br />
,<br />
lg R<br />
⋅<br />
( ) −<br />
⎝<br />
⎜<br />
⎜<br />
⎠<br />
⎟<br />
⎟<br />
= ⋅<br />
⋅ e ,<br />
lg<br />
Re<br />
, lgRe ,<br />
( ) −<br />
⎝<br />
⎜<br />
⎜<br />
⎠<br />
⎟<br />
⎟<br />
= ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
38215<br />
1<br />
2<br />
4 5222 3 8215<br />
λ glatt<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
1<br />
2<br />
4 5222 3 8215<br />
1<br />
2<br />
λ<br />
λ<br />
glatt<br />
glatt<br />
lg<br />
, lgRe ,<br />
Re<br />
lg 4 5222 3 8215<br />
2<br />
45222 3 821<br />
, lgRe<br />
Re<br />
,<br />
Re<br />
lg<br />
,<br />
Re<br />
lg Re<br />
,<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟ =− ⋅ ⋅ ( ) − 5<br />
1<br />
2<br />
4 5222 3 8215<br />
4 5222<br />
Re<br />
lg<br />
,<br />
Re<br />
lg Re<br />
,<br />
,<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅ ⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎛<br />
λ glatt ⎝ ⎜ ⎞<br />
⎠<br />
⎟<br />
=− ⋅ ⋅ ( ) −<br />
( )<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
( ) =<br />
1<br />
2<br />
4 5222<br />
0 845<br />
7 0 84<br />
λ glatt<br />
lg<br />
,<br />
Re<br />
lg Re ,<br />
lg , 5<br />
1<br />
2<br />
4 5222<br />
7<br />
1<br />
2<br />
4<br />
λ<br />
λ<br />
glatt<br />
glatt<br />
=− ⋅ ⋅ ( ) − ( )<br />
( )<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
lg<br />
,<br />
Re<br />
lg Re<br />
lg<br />
lg<br />
,<br />
Re<br />
lg Re , ln<br />
Re<br />
, lnRe ,<br />
5222<br />
7<br />
08686<br />
1964 38215<br />
⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟ ≡ ⋅ ⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅ ⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟ +<br />
⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
1<br />
2<br />
4 5222<br />
7 3 71<br />
1<br />
λ<br />
λ<br />
turb<br />
turb<br />
k<br />
D<br />
lg<br />
,<br />
Re<br />
lg Re ,<br />
=− ⋅<br />
⋅ ( ) −<br />
+<br />
⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
0 8686<br />
1 964 38215<br />
371<br />
1<br />
2<br />
, ln , lnRe ,<br />
Re ,<br />
k<br />
D<br />
λ glatt<br />
lg<br />
,<br />
Re<br />
lg Re<br />
4 518 7<br />
⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
= ⋅<br />
1<br />
2 lg<br />
Re<br />
λ glatt 1 964 0 4343 3 8215<br />
2<br />
1964<br />
04343<br />
,<br />
lg Re<br />
,<br />
,<br />
lg<br />
Re<br />
,<br />
,<br />
lg R<br />
⋅<br />
( ) −<br />
⎝<br />
⎜<br />
⎜<br />
⎜<br />
⎠<br />
⎟<br />
⎟<br />
⎟<br />
= ⋅<br />
⋅ e ,<br />
lg<br />
Re<br />
, lgRe ,<br />
( ) −<br />
⎝<br />
⎜<br />
⎜<br />
⎜<br />
⎠<br />
⎟<br />
⎟<br />
⎟<br />
= ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
38215<br />
1<br />
2<br />
4 5222 3 8215<br />
λ glatt<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
1<br />
2<br />
4 5222 3 8215<br />
1<br />
2<br />
λ<br />
λ<br />
glatt<br />
glatt<br />
lg<br />
, lgRe ,<br />
Re<br />
lg 4 5222 3 8215<br />
2<br />
45222 3 821<br />
, lgRe<br />
Re<br />
,<br />
Re<br />
lg<br />
,<br />
Re<br />
lg Re<br />
,<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟ =− ⋅ ⋅ ( ) − 5<br />
1<br />
2<br />
4 5222 3 8215<br />
4 5222<br />
Re<br />
lg<br />
,<br />
Re<br />
lg Re<br />
,<br />
,<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅ ⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎛<br />
λ glatt ⎝ ⎜ ⎞<br />
⎠<br />
⎟<br />
=− ⋅ ⋅ ( ) −<br />
( )<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
( ) =<br />
1<br />
2<br />
4 5222<br />
0 845<br />
7 0 84<br />
λ glatt<br />
lg<br />
,<br />
Re<br />
lg Re ,<br />
lg , 5<br />
1<br />
2<br />
4 5222<br />
7<br />
1<br />
2<br />
4<br />
λ<br />
λ<br />
glatt<br />
glatt<br />
=− ⋅ ⋅ ( ) − ( )<br />
( )<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
lg<br />
,<br />
Re<br />
lg Re<br />
lg<br />
lg<br />
,<br />
Re<br />
lg Re , ln<br />
Re<br />
, lnRe ,<br />
5222<br />
7<br />
08686<br />
1964 38215<br />
⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟ ≡ ⋅ ⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅ ⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟ +<br />
⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
1<br />
2<br />
4 5222<br />
7 3 71<br />
1<br />
λ<br />
λ<br />
turb<br />
turb<br />
k<br />
D<br />
lg<br />
,<br />
Re<br />
lg Re ,<br />
=− ⋅<br />
⋅ ( ) −<br />
+<br />
⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
0 8686<br />
1 964 38215<br />
371<br />
1<br />
2<br />
, ln , lnRe ,<br />
Re ,<br />
k<br />
D<br />
λ glatt<br />
lg<br />
,<br />
Re<br />
lg Re<br />
4 518 7<br />
⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
= ⋅<br />
1<br />
2 lg<br />
Re<br />
λ glatt 1 964 0 4343 3 8215<br />
2<br />
1964<br />
04343<br />
,<br />
lg Re<br />
,<br />
,<br />
lg<br />
Re<br />
,<br />
,<br />
lg R<br />
⋅<br />
( ) −<br />
⎝<br />
⎜<br />
⎜<br />
⎜<br />
⎠<br />
⎟<br />
⎟<br />
⎟<br />
= ⋅<br />
⋅ e ,<br />
lg<br />
Re<br />
, lgRe ,<br />
( ) −<br />
⎝<br />
⎜<br />
⎜<br />
⎜<br />
⎠<br />
⎟<br />
⎟<br />
⎟<br />
= ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
38215<br />
1<br />
2<br />
4 5222 3 8215<br />
λ glatt<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
1<br />
2<br />
4 5222 3 8215<br />
1<br />
2<br />
λ<br />
λ<br />
glatt<br />
glatt<br />
lg<br />
, lgRe ,<br />
Re<br />
lg 4 5222 3 8215<br />
2<br />
45222 3 821<br />
, lgRe<br />
Re<br />
,<br />
Re<br />
lg<br />
,<br />
Re<br />
lg Re<br />
,<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟ =− ⋅ ⋅ ( ) − 5<br />
1<br />
2<br />
4 5222 3 8215<br />
4 5222<br />
Re<br />
lg<br />
,<br />
Re<br />
lg Re<br />
,<br />
,<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅ ⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎛<br />
λ glatt ⎝ ⎜ ⎞<br />
⎠<br />
⎟<br />
=− ⋅ ⋅ ( ) −<br />
( )<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
( ) =<br />
1<br />
2<br />
4 5222<br />
0 845<br />
7 0 84<br />
λ glatt<br />
lg<br />
,<br />
Re<br />
lg Re ,<br />
lg , 5<br />
1<br />
2<br />
4 5222<br />
7<br />
1<br />
2<br />
4<br />
λ<br />
λ<br />
glatt<br />
glatt<br />
=− ⋅ ⋅ ( ) − ( )<br />
( )<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
lg<br />
,<br />
Re<br />
lg Re<br />
lg<br />
lg<br />
,<br />
Re<br />
lg Re , ln<br />
Re<br />
, lnRe ,<br />
5222<br />
7<br />
08686<br />
1964 38215<br />
⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟ ≡ ⋅ ⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅ ⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟ +<br />
⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
1<br />
2<br />
4 5222<br />
7 3 71<br />
1<br />
λ<br />
λ<br />
turb<br />
turb<br />
k<br />
D<br />
lg<br />
,<br />
Re<br />
lg Re ,<br />
=− ⋅<br />
⋅ ( ) −<br />
+<br />
⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
0 8686<br />
1 964 38215<br />
371<br />
1<br />
2<br />
, ln , lnRe ,<br />
Re ,<br />
k<br />
D<br />
λ glatt<br />
lg<br />
,<br />
Re<br />
lg Re<br />
4 518 7<br />
⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
lg (7) = 0,845<br />
= ⋅<br />
1<br />
2 lg<br />
Re<br />
λ glatt 1 964 0 4343 3 8215<br />
2<br />
1964<br />
04343<br />
,<br />
lg Re<br />
,<br />
,<br />
lg<br />
Re<br />
,<br />
,<br />
lg R<br />
⋅<br />
( ) −<br />
⎝<br />
⎜<br />
⎜<br />
⎜<br />
⎠<br />
⎟<br />
⎟<br />
⎟<br />
= ⋅<br />
⋅ e ,<br />
lg<br />
Re<br />
, lgRe ,<br />
( ) −<br />
⎝<br />
⎜<br />
⎜<br />
⎜<br />
⎠<br />
⎟<br />
⎟<br />
⎟<br />
= ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
38215<br />
1<br />
2<br />
4 5222 3 8215<br />
λ glatt<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
1<br />
2<br />
4 5222 3 8215<br />
1<br />
2<br />
λ<br />
λ<br />
glatt<br />
glatt<br />
lg<br />
, lgRe ,<br />
Re<br />
lg 4 5222 3 8215<br />
2<br />
45222 3 821<br />
, lgRe<br />
Re<br />
,<br />
Re<br />
lg<br />
,<br />
Re<br />
lg Re<br />
,<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟ =− ⋅ ⋅ ( ) − 5<br />
1<br />
2<br />
4 5222 3 8215<br />
4 5222<br />
Re<br />
lg<br />
,<br />
Re<br />
lg Re<br />
,<br />
,<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅ ⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎛<br />
λ glatt ⎝ ⎜ ⎞<br />
⎠<br />
⎟<br />
=− ⋅ ⋅ ( ) −<br />
( )<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
( ) =<br />
1<br />
2<br />
4 5222<br />
0 845<br />
7 0 84<br />
λ glatt<br />
lg<br />
,<br />
Re<br />
lg Re ,<br />
lg , 5<br />
1<br />
2<br />
4 5222<br />
7<br />
1<br />
2<br />
4<br />
λ<br />
λ<br />
glatt<br />
glatt<br />
=− ⋅ ⋅ ( ) − ( )<br />
( )<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
lg<br />
,<br />
Re<br />
lg Re<br />
lg<br />
lg<br />
,<br />
Re<br />
lg Re , ln<br />
Re<br />
, lnRe ,<br />
5222<br />
7<br />
08686<br />
1964 38215<br />
⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟ ≡ ⋅ ⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅ ⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟ +<br />
⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
1<br />
2<br />
4 5222<br />
7 3 71<br />
1<br />
λ<br />
λ<br />
turb<br />
turb<br />
k<br />
D<br />
lg<br />
,<br />
Re<br />
lg Re ,<br />
=− ⋅<br />
⋅ ( ) −<br />
+<br />
⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
0 8686<br />
1 964 38215<br />
371<br />
1<br />
2<br />
, ln , lnRe ,<br />
Re ,<br />
k<br />
D<br />
λ glatt<br />
lg<br />
,<br />
Re<br />
lg Re<br />
4 518 7<br />
⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
Abschließend erhält man auf <strong>die</strong>sem Wege <strong>die</strong> Gleichung<br />
nach Techo, Tickner und James in modifizierter<br />
Schreibweise:<br />
λ glatt<br />
= ⋅<br />
⋅ ( ) −<br />
⎝<br />
⎜<br />
⎠<br />
⎟<br />
= ⋅<br />
0 4343 1 964 38215<br />
1<br />
2<br />
,<br />
lg , lnRe ,<br />
lg<br />
Re<br />
λ glatt 1 964 0 4343 3 8215<br />
2<br />
1964<br />
04343<br />
,<br />
lg Re<br />
,<br />
,<br />
lg<br />
Re<br />
,<br />
,<br />
lg R<br />
⋅<br />
( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎜<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎟<br />
⎟<br />
= ⋅<br />
⋅ e ,<br />
lg<br />
Re<br />
, lgRe ,<br />
( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎜<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎟<br />
⎟<br />
= ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
38215<br />
1<br />
2<br />
4 5222 3 8215<br />
λ glatt<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
1<br />
2<br />
4 5222 3 8215<br />
1<br />
2<br />
λ<br />
λ<br />
glatt<br />
glatt<br />
lg<br />
, lgRe ,<br />
Re<br />
lg 4 5222 3 8215<br />
2<br />
45222 3 821<br />
, lgRe<br />
Re<br />
,<br />
Re<br />
lg<br />
,<br />
Re<br />
lg Re<br />
,<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟ =− ⋅ ⋅ ( ) − 5<br />
1<br />
2<br />
4 5222 3 8215<br />
4 5222<br />
Re<br />
lg<br />
,<br />
Re<br />
lg Re<br />
,<br />
,<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅ ⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎛<br />
λ glatt ⎝ ⎜ ⎞<br />
⎠<br />
⎟<br />
=− ⋅ ⋅ ( ) −<br />
( )<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
( ) =<br />
1<br />
2<br />
4 5222<br />
0 845<br />
7 0 84<br />
λ glatt<br />
lg<br />
,<br />
Re<br />
lg Re ,<br />
lg , 5<br />
1<br />
2<br />
4 5222<br />
7<br />
1<br />
2<br />
4<br />
λ<br />
λ<br />
glatt<br />
glatt<br />
=− ⋅ ⋅ ( ) − ( )<br />
( )<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
lg<br />
,<br />
Re<br />
lg Re<br />
lg<br />
lg<br />
,<br />
Re<br />
lg Re , ln<br />
Re<br />
, lnRe ,<br />
5222<br />
7<br />
08686<br />
1964 38215<br />
⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟ ≡ ⋅ ⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅ ⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟ +<br />
⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
1<br />
2<br />
4 5222<br />
7 3 71<br />
1<br />
λ<br />
λ<br />
turb<br />
turb<br />
k<br />
D<br />
lg<br />
,<br />
Re<br />
lg Re ,<br />
=− ⋅<br />
⋅ ( ) −<br />
+<br />
⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
0 8686<br />
1 964 38215<br />
371<br />
1<br />
2<br />
, ln , lnRe ,<br />
Re ,<br />
k<br />
D<br />
λ glatt<br />
lg<br />
,<br />
Re<br />
lg Re<br />
4 518 7<br />
⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
0 4343<br />
1<br />
ln( ) =<br />
,<br />
x<br />
λ glatt<br />
= ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
= ⋅<br />
0 8686<br />
0 4343 1 964 38215<br />
1<br />
2<br />
,<br />
,<br />
lg<br />
Re<br />
, lnRe ,<br />
lg<br />
Re<br />
λ glatt 1 964 0 4343 3 8215<br />
2<br />
1964<br />
04343<br />
,<br />
lg Re<br />
,<br />
,<br />
lg<br />
Re<br />
,<br />
,<br />
lg R<br />
⋅<br />
( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎜<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎟<br />
⎟<br />
= ⋅<br />
⋅ e ,<br />
lg<br />
Re<br />
, lgRe ,<br />
( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎜<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎟<br />
⎟<br />
= ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
38215<br />
1<br />
2<br />
4 5222 3 8215<br />
λ glatt<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
1<br />
2<br />
4 5222 3 8215<br />
1<br />
2<br />
λ<br />
λ<br />
glatt<br />
glatt<br />
lg<br />
, lgRe ,<br />
Re<br />
lg 4 5222 3 8215<br />
2<br />
45222 3 821<br />
, lgRe<br />
Re<br />
,<br />
Re<br />
lg<br />
,<br />
Re<br />
lg Re<br />
,<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟ =− ⋅ ⋅ ( ) − 5<br />
1<br />
2<br />
4 5222 3 8215<br />
4 5222<br />
Re<br />
lg<br />
,<br />
Re<br />
lg Re<br />
,<br />
,<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅ ⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎛<br />
λ glatt ⎝ ⎜ ⎞<br />
⎠<br />
⎟<br />
=− ⋅ ⋅ ( ) −<br />
( )<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
( ) =<br />
1<br />
2<br />
4 5222<br />
0 845<br />
7 0 84<br />
λ glatt<br />
lg<br />
,<br />
Re<br />
lg Re ,<br />
lg , 5<br />
1<br />
2<br />
4 5222<br />
7<br />
1<br />
2<br />
4<br />
λ<br />
λ<br />
glatt<br />
glatt<br />
=− ⋅ ⋅ ( ) − ( )<br />
( )<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
lg<br />
,<br />
Re<br />
lg Re<br />
lg<br />
lg<br />
,<br />
Re<br />
lg Re , ln<br />
Re<br />
, lnRe ,<br />
5222<br />
7<br />
08686<br />
1964 38215<br />
⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟ ≡ ⋅ ⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅ ⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟ +<br />
⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
1<br />
2<br />
4 5222<br />
7 3 71<br />
1<br />
λ<br />
λ<br />
turb<br />
turb<br />
k<br />
D<br />
lg<br />
,<br />
Re<br />
lg Re ,<br />
=− ⋅<br />
⋅ ( ) −<br />
+<br />
⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
0 8686<br />
1 964 38215<br />
371<br />
1<br />
2<br />
, ln , lnRe ,<br />
Re ,<br />
k<br />
D<br />
λ glatt<br />
lg<br />
,<br />
Re<br />
lg Re<br />
4 518 7<br />
⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
(53)<br />
Kombiniert man <strong>die</strong>sen Ansatz <strong>für</strong> <strong>die</strong> Glattrohrströmung<br />
wieder in bewährter Weise mit dem Nikuradse-<br />
Term <strong>für</strong> hydraulisch raue Strömung, dann ergibt sich<br />
<strong>die</strong> Komplettgleichung zu:<br />
1<br />
086<br />
λ glatt =− , 859<br />
1 964 38215<br />
0 4343<br />
1<br />
⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
( ) =<br />
( )<br />
ln , lnRe ,<br />
Re<br />
ln<br />
lg<br />
,<br />
x<br />
x<br />
λ glatt<br />
= ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
= ⋅<br />
0 8686<br />
0 4343 1 964 38215<br />
1<br />
2<br />
,<br />
,<br />
lg<br />
Re<br />
, lnRe ,<br />
lg<br />
Re<br />
λ glatt 1 964 0 4343 3 8215<br />
2<br />
1964<br />
04343<br />
,<br />
lg Re<br />
,<br />
,<br />
lg<br />
Re<br />
,<br />
,<br />
lg R<br />
⋅<br />
( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎜<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎟<br />
⎟<br />
= ⋅<br />
⋅ e ,<br />
lg<br />
Re<br />
, lgRe ,<br />
( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎜<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎟<br />
⎟<br />
= ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
38215<br />
1<br />
2<br />
4 5222 3 8215<br />
λ glatt<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
1<br />
2<br />
4 5222 3 8215<br />
1<br />
2<br />
λ<br />
λ<br />
glatt<br />
glatt<br />
lg<br />
, lgRe ,<br />
Re<br />
lg 4 5222 3 8215<br />
2<br />
45222 3 821<br />
, lgRe<br />
Re<br />
,<br />
Re<br />
lg<br />
,<br />
Re<br />
lg Re<br />
,<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟ =− ⋅ ⋅ ( ) − 5<br />
1<br />
2<br />
4 5222 3 8215<br />
4 5222<br />
Re<br />
lg<br />
,<br />
Re<br />
lg Re<br />
,<br />
,<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅ ⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎛<br />
λ glatt ⎝ ⎜ ⎞<br />
⎠<br />
⎟<br />
=− ⋅ ⋅ ( ) −<br />
( )<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
( ) =<br />
1<br />
2<br />
4 5222<br />
0 845<br />
7 0 84<br />
λ glatt<br />
lg<br />
,<br />
Re<br />
lg Re ,<br />
lg , 5<br />
1<br />
2<br />
4 5222<br />
7<br />
1<br />
2<br />
4<br />
λ<br />
λ<br />
glatt<br />
glatt<br />
=− ⋅ ⋅ ( ) − ( )<br />
( )<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
lg<br />
,<br />
Re<br />
lg Re<br />
lg<br />
lg<br />
,<br />
Re<br />
lg Re , ln<br />
Re<br />
, lnRe ,<br />
5222<br />
7<br />
08686<br />
1964 38215<br />
⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟ ≡ ⋅ ⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅ ⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟ +<br />
⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
1<br />
2<br />
4 5222<br />
7 3 71<br />
1<br />
λ<br />
λ<br />
turb<br />
turb<br />
k<br />
D<br />
lg<br />
,<br />
Re<br />
lg Re ,<br />
=− ⋅<br />
⋅ ( ) −<br />
+<br />
⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
0 8686<br />
1 964 38215<br />
371<br />
1<br />
2<br />
, ln , lnRe ,<br />
Re ,<br />
k<br />
D<br />
λ glatt<br />
lg<br />
,<br />
Re<br />
lg Re<br />
4 518 7<br />
⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
(54)<br />
Dem entspricht vollkommen <strong>die</strong> in [70] (1988) verwendete<br />
Formulierung <strong>die</strong>ses Sachverhalts:<br />
1 964 38215<br />
1<br />
086<br />
λ<br />
λ<br />
glatt<br />
glatt<br />
⋅ ( ) −<br />
⎝<br />
⎠<br />
=−<br />
, lnRe ,<br />
, 859<br />
1 964 38215<br />
0 4343<br />
1<br />
⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
( ) =<br />
( )<br />
ln , lnRe ,<br />
Re<br />
ln<br />
lg<br />
,<br />
x<br />
x<br />
λ glatt<br />
= ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
= ⋅<br />
0 8686<br />
0 4343 1 964 38215<br />
1<br />
2<br />
,<br />
,<br />
lg<br />
Re<br />
, lnRe ,<br />
lg<br />
Re<br />
λ glatt 1 964 0 4343 3 8215<br />
2<br />
1964<br />
04343<br />
,<br />
lg Re<br />
,<br />
,<br />
lg<br />
Re<br />
,<br />
,<br />
lg R<br />
⋅<br />
( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎜<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎟<br />
⎟<br />
= ⋅<br />
⋅ e ,<br />
lg<br />
Re<br />
, lgRe ,<br />
( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎜<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎟<br />
⎟<br />
= ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
38215<br />
1<br />
2<br />
4 5222 3 8215<br />
λ glatt<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
1<br />
2<br />
4 5222 3 8215<br />
1<br />
2<br />
λ<br />
λ<br />
glatt<br />
glatt<br />
lg<br />
, lgRe ,<br />
Re<br />
lg 4 5222 3 8215<br />
2<br />
45222 3 821<br />
, lgRe<br />
Re<br />
,<br />
Re<br />
lg<br />
,<br />
Re<br />
lg Re<br />
,<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟ =− ⋅ ⋅ ( ) − 5<br />
1<br />
2<br />
4 5222 3 8215<br />
4 5222<br />
Re<br />
lg<br />
,<br />
Re<br />
lg Re<br />
,<br />
,<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅ ⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎛<br />
λ glatt ⎝ ⎜ ⎞<br />
⎠<br />
⎟<br />
=− ⋅ ⋅ ( ) −<br />
( )<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
( ) =<br />
1<br />
2<br />
4 5222<br />
0 845<br />
7 0 84<br />
λ glatt<br />
lg<br />
,<br />
Re<br />
lg Re ,<br />
lg , 5<br />
1<br />
2<br />
4 5222<br />
7<br />
1<br />
2<br />
4<br />
λ<br />
λ<br />
glatt<br />
glatt<br />
=− ⋅ ⋅ ( ) − ( )<br />
( )<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
lg<br />
,<br />
Re<br />
lg Re<br />
lg<br />
lg<br />
,<br />
Re<br />
lg Re , ln<br />
Re<br />
, lnRe ,<br />
5222<br />
7<br />
08686<br />
1964 38215<br />
⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟ ≡ ⋅ ⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅ ⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟ +<br />
⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
1<br />
2<br />
4 5222<br />
7 3 71<br />
1<br />
λ<br />
λ<br />
turb<br />
turb<br />
k<br />
D<br />
lg<br />
,<br />
Re<br />
lg Re ,<br />
=− ⋅<br />
⋅ ( ) −<br />
+<br />
⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
0 8686<br />
1 964 38215<br />
371<br />
1<br />
2<br />
, ln , lnRe ,<br />
Re ,<br />
k<br />
D<br />
λ glatt<br />
lg<br />
,<br />
Re<br />
lg Re<br />
4 518 7<br />
⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
1<br />
0 86859<br />
1 964 38215<br />
1<br />
086<br />
λ<br />
λ<br />
glatt<br />
glatt<br />
= ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=−<br />
, ln<br />
Re<br />
, lnRe ,<br />
, 859<br />
1 964 38215<br />
0 4343<br />
1<br />
⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
( ) =<br />
( )<br />
ln , lnRe ,<br />
Re<br />
ln<br />
lg<br />
,<br />
x<br />
x<br />
λ glatt<br />
= ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
= ⋅<br />
0 8686<br />
0 4343 1 964 38215<br />
1<br />
2<br />
,<br />
,<br />
lg<br />
Re<br />
, lnRe ,<br />
lg<br />
Re<br />
λ glatt 1 964 0 4343 3 8215<br />
2<br />
1964<br />
04343<br />
,<br />
lg Re<br />
,<br />
,<br />
lg<br />
Re<br />
,<br />
,<br />
lg R<br />
⋅<br />
( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎜<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎟<br />
⎟<br />
= ⋅<br />
⋅ e ,<br />
lg<br />
Re<br />
, lgRe ,<br />
( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎜<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎟<br />
⎟<br />
= ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
38215<br />
1<br />
2<br />
4 5222 3 8215<br />
λ glatt<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
1<br />
2<br />
4 5222 3 8215<br />
1<br />
2<br />
λ<br />
λ<br />
glatt<br />
glatt<br />
lg<br />
, lgRe ,<br />
Re<br />
lg 4 5222 3 8215<br />
2<br />
45222 3 821<br />
, lgRe<br />
Re<br />
,<br />
Re<br />
lg<br />
,<br />
Re<br />
lg Re<br />
,<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟ =− ⋅ ⋅ ( ) − 5<br />
1<br />
2<br />
4 5222 3 8215<br />
4 5222<br />
Re<br />
lg<br />
,<br />
Re<br />
lg Re<br />
,<br />
,<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅ ⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎛<br />
λ glatt ⎝ ⎜ ⎞<br />
⎠<br />
⎟<br />
=− ⋅ ⋅ ( ) −<br />
( )<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
( ) =<br />
1<br />
2<br />
4 5222<br />
0 845<br />
7 0 84<br />
λ glatt<br />
lg<br />
,<br />
Re<br />
lg Re ,<br />
lg , 5<br />
1<br />
2<br />
4 5222<br />
7<br />
1<br />
2<br />
4<br />
λ<br />
λ<br />
glatt<br />
glatt<br />
=− ⋅ ⋅ ( ) − ( )<br />
( )<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
lg<br />
,<br />
Re<br />
lg Re<br />
lg<br />
lg<br />
,<br />
Re<br />
lg Re , ln<br />
Re<br />
, lnRe ,<br />
5222<br />
7<br />
08686<br />
1964 38215<br />
⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟ ≡ ⋅ ⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅ ⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟ +<br />
⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
1<br />
2<br />
4 5222<br />
7 3 71<br />
1<br />
λ<br />
λ<br />
turb<br />
turb<br />
k<br />
D<br />
lg<br />
,<br />
Re<br />
lg Re ,<br />
=− ⋅<br />
⋅ ( ) −<br />
+<br />
⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
0 8686<br />
1 964 38215<br />
371<br />
1<br />
2<br />
, ln , lnRe ,<br />
Re ,<br />
k<br />
D<br />
λ glatt<br />
lg<br />
,<br />
Re<br />
lg Re<br />
4 518 7<br />
⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
(55)<br />
Gl. (54) stimmt bis auf eine geringe numerische Differenz<br />
(wahrscheinlich Rundungsfehler) mit Gl. (24)<br />
überein. Das heißt, mit Veröffentlichung der Gleichung<br />
von Techo, Tickner und James (1965) lagen alle „Zutaten“<br />
vor, um unter Verwendung der Interpolationsregel von<br />
Colebrook-White <strong>die</strong> Gleichung abzuleiten, <strong>die</strong> Hofer<br />
1973 in [51] angegeben hat. Barr [57] weist 1981 ausdrücklich<br />
darauf hin, dass <strong>die</strong> Gleichung von Techo,<br />
Tickner und James in der Form von Gl. (53) als<br />
1<br />
0 86859<br />
1 964 38215<br />
1<br />
086<br />
λ<br />
λ<br />
glatt<br />
glatt<br />
= ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=−<br />
, ln<br />
Re<br />
, lnRe ,<br />
, 859<br />
1 964 38215<br />
0 4343<br />
1<br />
⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
( ) =<br />
( )<br />
ln , lnRe ,<br />
Re<br />
ln<br />
lg<br />
,<br />
x<br />
x<br />
λ glatt<br />
= ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
= ⋅<br />
0 8686<br />
0 4343 1 964 38215<br />
1<br />
2<br />
,<br />
,<br />
lg<br />
Re<br />
, lnRe ,<br />
lg<br />
Re<br />
λ glatt 1 964 0 4343 3 8215<br />
2<br />
1964<br />
04343<br />
,<br />
lg Re<br />
,<br />
,<br />
lg<br />
Re<br />
,<br />
,<br />
lg R<br />
⋅<br />
( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎜<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎟<br />
⎟<br />
= ⋅<br />
⋅ e ,<br />
lg<br />
Re<br />
, lgRe ,<br />
( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎜<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎟<br />
⎟<br />
= ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
38215<br />
1<br />
2<br />
4 5222 3 8215<br />
λ glatt<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
1<br />
2<br />
4 5222 3 8215<br />
1<br />
2<br />
λ<br />
λ<br />
glatt<br />
glatt<br />
lg<br />
, lgRe ,<br />
Re<br />
lg 4 5222 3 8215<br />
2<br />
45222 3 821<br />
, lgRe<br />
Re<br />
,<br />
Re<br />
lg<br />
,<br />
Re<br />
lg Re<br />
,<br />
⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟ =− ⋅ ⋅ ( ) − 5<br />
1<br />
2<br />
4 5222 3 8215<br />
4 5222<br />
Re<br />
lg<br />
,<br />
Re<br />
lg Re<br />
,<br />
,<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅ ⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎛<br />
λ glatt ⎝ ⎜ ⎞<br />
⎠<br />
⎟<br />
=− ⋅ ⋅ ( ) −<br />
( )<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
( ) =<br />
1<br />
2<br />
4 5222<br />
0 845<br />
7 0 84<br />
λ glatt<br />
lg<br />
,<br />
Re<br />
lg Re ,<br />
lg , 5<br />
1<br />
2<br />
4 5222<br />
7<br />
1<br />
2<br />
4<br />
λ<br />
λ<br />
glatt<br />
glatt<br />
=− ⋅ ⋅ ( ) − ( )<br />
( )<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
lg<br />
,<br />
Re<br />
lg Re<br />
lg<br />
lg<br />
,<br />
Re<br />
lg Re , ln<br />
Re<br />
, lnRe ,<br />
5222<br />
7<br />
08686<br />
1964 38215<br />
⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟ ≡ ⋅ ⋅ ( ) −<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅ ⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟ +<br />
⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
1<br />
2<br />
4 5222<br />
7 3 71<br />
1<br />
λ<br />
λ<br />
turb<br />
turb<br />
k<br />
D<br />
lg<br />
,<br />
Re<br />
lg Re ,<br />
=− ⋅<br />
⋅ ( ) −<br />
+<br />
⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
=− ⋅<br />
0 8686<br />
1 964 38215<br />
371<br />
1<br />
2<br />
, ln , lnRe ,<br />
Re ,<br />
k<br />
D<br />
λ glatt<br />
lg<br />
,<br />
Re<br />
lg Re<br />
4 518 7<br />
⋅<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟<br />
⎛<br />
⎝<br />
⎜<br />
⎞<br />
⎠<br />
⎟ (56)
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