Page 170 - Air and gas Drilling Field Guide 3rd Edition

6.5 Air and Gas Drilling Model 161

where
8 9
32
2 3 2
P g T av
> >
> >
Q g
> >
< =
6 _ w t 7 f 6 P T g 7
6 7 6 7 :
B a ðPÞ¼ 1 þ
4 5 4 2 2 5
P g T av > 2gðD h D p Þ p >
p
h
> ðD D Þ >
> 4 >
P T g Q g : ;
Using Equations (6-6), (6-7), and (6-13), Equation (6-53) can be rearranged to
give
ð ð H
P bh
PdP a a
¼ dh; (6-54)
2 2
ðP þ b a T Þ T av
P at av 0
where

S g _ w s
a a ¼ 1 þ (6-55)
_ w g
R e
2 2
f _ w g
b a ¼ R e : (6-56)
p 2

2g ðD h D p Þ S g 2 2 2
ðD D Þ
h
p
4
In the form just given, both sides of Equation (6-54) can be integrated. Using the
constants in Equations (6-55) and (6-56), the solution to Equation (6-54) is
P
bh
1 2 a a H
lnðP þ b a T Þ ¼ jhj : (6-57)
2
2 av T av 0
P at
Evaluating Equation (6-57) at the limits and rearranging the results give
2 2
P þ b a T 2a a
ln bh av ¼ H: (6-58)
2
P þ b a T 2 T av
at av
Raising both sides of Equation (6-58) to the natural exponent e gives
2a a H
2
P 2 bh þ b a T av ¼ e T av : (6-59)
2
P þ b a T 2
at av
Equation (6-59) can be rearranged and a solution obtained for P bh . This is
2 3 0:5
2a a H
2
2
P bh ¼ðP þ b a T Þe T av b a T 5 : (6-60)
2 7
6
av
at
av
4
The von Karman empirical correlation for wholly turbulent flow conditions can
be used to determine the friction factor in Equation (6-56)[1]. This empirical
expression is
2 3 2
1
6 7
f ¼ 6 7 : (6-61)
D h D p
4 5
2 log þ 1:14
e