Page 286 - Air and Gas Drilling Manual




BP() = 

i
     P   w ˙ T g av +  w ˙ m Chapter 6: Direct Circulation Models 6-31

g
     Q + Q m 
g
   P   T   
g
   2 
    P g   T av Q + Q  
    P     g m  
  T

1 − f  π g   

 2gD i  D 2  
  4 i  
      

The Fanning friction factor f given in the above equation is determined by the
standard fluid mechanics empirical expressions relating the friction factor to the
Reynolds number, diameter, and absolute pipe roughness. In general, the values for
Reynolds number, diameter, and absolute pipe roughness are known. The classic
expression for the Reynolds number is
DV
i
N R = ν (6-103)
There are three flow conditions that can exist in the annulus. These are laminar,
transitional, and turbulent flow conditions.
The empirical expression for the Fanning friction factor for laminar flow
conditions is
f = 64 (6-104)
N R
This equation can be solved directly once the Reynolds number is known. In
general, Equation 6-104 is valid for values for Reynolds numbers from 0 to 2,000.
The empirical expression for the Fanning friction factor for transitional flow
conditions is known as the Colebrook equation. This equation cannot be solved
directly and must be solved by trial and error. This empirical expression is
   
  e  
1   D i  251 
.
=− 2 log  +  (6-105)
f  37 . N R f 
 
   