Page 310 - Air and Gas Drilling Manual

dP
P bh
(7-58)
P
a
where ∫ P in B () = ∫ 0 H dh Chapter 7: Reverse Circulation Models 7-19
     P   T   2 
g
       av  Q  
 w ˙      P   T  g  

g
B () =  g − f   

P
a   1 gD − π
  P   T av    2 ( h D )  ( D 2 − D )  
2
g
p
     Q g    4 h p  
   P   T          
g

Using Equations 7-5, 7-6 or 7-10, and 7-7, Equation 7-58 can be rearranged to give
∫ P bh P ( PdP = a a ∫ H dh (7-59)
2
P in 2 − b T av) T av 0
a
where
a = S (7-60)
a
R
b = f  R  2 w ˙ 2 g (7-61)
a gD −  S  2
π
2 ( D )  
h p 2 − 2 2
  ( D h D )
p
4
In the above form, both sides of Equation 7-59 can be integrated. Using Equations
7-60 and 7-61 the solution to Equation 7-59 is
ln P ( 2 − b T av) = a h (7-62)
1 P bh a H
2
2 a T av 0
P in
Evaluating Equation 7-62 at the limits and rearranging the results gives
 P 2 − b T  2 a
2
ln  bh a av  = a H (7-63)
  P in 2 − b T   T av
2
a
av
Raising both sides of Equation 7-63 to the natural exponential exponent gives