Page 190 - Air and gas Drilling Field Guide 3rd Edition

7.5 Air and Gas Drilling Model 181

where
8 9
32
2 3 2
P g T av
> >
> >
Q g
> >
< =
6 _ w g 7 f 6 P T g 7
6 7 :
B a PðÞ ¼ 1 6 7
P g T av > 2gD h D p D D >
4 5 4 p 2 2 5
> h p >
> 4 >
P T g Q g : ;
Using Equations (7-6), (7-7), and (7-14), Equation (7-56) can be rearranged to give
ð ð H
PdP a a
P bh
¼ dh; (7-57)

2
P b a T 2
P in av T av 0
where
S
a a ¼ R e (7-58)
f 2 _ w 2 g
b a ¼ R e : (7-59)
2
p 2
2gD h D p S D D 2
2
4 h p
In this form, both sides of Equation (7-57) can be integrated. Using Equations
(7-58) and (7-59), the solution to Equation (7-57) is
P
bh
1 2 a a H
ln P b a T h jj : (7-60)
2 ¼
2 T av
av 0
P in
Evaluating Equation (7-60) at the limits and rearranging the results give
2 2
P b a T 2a a
ln bh av ¼ H: (7-61)
2
P b a T 2 T av
in av
Raising both sides of Equation (7-63) to the natural exponential exponent gives
2a a H
2
P 2 bh b a T av T av :
2
P b a T 2 av ¼ e (7-62)
in
Equation (7-62) can be rearranged and a solution obtained for P in . This is
13 0:5
0
2a i H
2
2
6 2 B e C7
av
ai
P þ b i T @ T av 1A7
6
6 7
6 7
P in ¼ 6 7 : (7-63)
6 2a i H 7
6 7
e T av
6 7
4 5
The von Karman empirical correlation can be used to determine the friction
factor given in Equation (7-59)[1]. This empirical expression is
2 3 2
1
f ¼ 4 5 ; (7-64)
2 log D h D p
e