Page 155 -

Gas Compressors 123

When D H (feet) is replaced by d H (inches), this equation can be simpli-
fied to

dP = aP + abðT s + GHÞ dH (6.12)
T s + GH P

where
2
b
a = S g Q g + 0:074d S s R p + 76:3ðS x Q x + S l Q f Þ (6.13)
53:3Q g
and

fQ 2 g
b =± 572:7 (6.14)
2
gA d H
The solution to Eq. (6.12) is
P 2 2ða−GÞ
G (6.15)
2
ða − GÞ + ab = ½CðT s + GHފ
ðT s + GHÞ
where C is an integration constant. It can be determined, using the bound-
ary condition at the top of the hole section (i.e., at depth H = 0), that
(6.16)
P = P s
and
(6.17)
T = T s
When these conditions are applied, Eq. (6.15) results in

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
v
2a
u

u
ab T s + GH G ab
P = t P + 2 − 2 (6.18)
2
s T s ðT s + GHÞ
a −G T s a −G
which is a general equation for pressure at the point of interest in the
hole at depth H.
Angel (1957) applied Weymouth’s (1912) friction factor, which was
derived before the friction factor was fully understood, to vertical flow
when he developed his annular pressure equation. The friction factor was
found to be a function of pipe wall roughness in the 1930s and 1940s
when Nikuradse’s (1933) correlation and Moody’s (1944) chart were
developed. Unfortunately, when Angel developed his model in 1957, he