Page 132 - Air and Gas Drilling Manual

4-18 Air and Gas Drilling Manual
T
Pv = R

1
1
1
Using the definitions for a polytropic process given above, and Equation 4-17, then
Equation 4-10 can be reduced to (4-17)
 k −1 
P
T
− ∫ 1 2 Pdv = R −1 1      P 2  k −1   (4-18)


k
  1  
Again, using the definitions for a polytropic process and Equation 4-16, then a
general relationship between P and T can be obtained. This is
−
k 1
 P  k T 2
2
  = (4-19)
 P  T 1
1
Substituting Equation 4-19 into Equation 4-18 yields


− ∫ 2 Pdv = R T 1 T 2 −  = R ( T − ) (4-20)
T
1
−  T
1 k 1  1   k −1 2 1

Substituting Equation 4-20 into Equation 4-8 gives
V 2 V 2
W + 1 − 2 + Pv − P v = R T − ) (4-21)
T
s 2 g 2 g 1 1 2 2 k −1 ( 2 1

Equation 4-17 was the definition of Pv at state 1. Similarly, the definition of Pv at
state 2 is
Pv = R T (4-22)
2 2 2
Substituting Equations 4-17 and 4-22 into Equation 4-21 and rearranging gives
T − )
V 2 2 V 1 2 R
T − ) +
W = − + ( 2 T 1 ( 2 T 1 (4-23)
R
s
2 g 2 g k −1
The above can be rearranged to give
 T  V 2 − V 2
W = k R T  2 −1  + 2 1 (4-24)
s 1
k −1  T 1  2 g