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October, 21st , 2014
Des. Ind. Plant, 2
Compressible Flow along pipelines
a.
Consider a pipe (diameter D, lenght L) used to transport gas (molar mass M , ratio of specific heat evaluated at
constant pressure and at constant volume given by γ = cp /cv ) from point 1 to point 2.
1. Derive the relation between specific mass flow rate G and pressure at the upstream/downstream section of the
pipeline. Assume isothermal transformation for the gas (p/ρ = C) along the pipeline.
Since the pipeline works in steady state conditions, m
˙ = cost and if the pipe diameter is constant along the
length of pipe, also G = const. The flow along the line is described by Benrnoulli equation:
d
v2
dp
dlv
+
+ gdh = dws −
2
ρ
ρ
(1)
where gdh ≃ 0, dws = 0 (no compressor along the line) and dlv /ρ = 2f dxv 2 /D as for incompressible flow.
Bernoulli should be integrated between point 1 and 2, but it can not be integrated in the form of Equation 1
because we do not know the law of variation of velocity along the pipe, v(x), which is required to quantify the
viscous losses. To integrate that term, we need to rewrite Bernoulli as:
ρ2 d
v2
dp
+
2
ρ
=−
2f (v 2 ρ2 )dx
D
(2)
which, considering G = ρv, becomes
2f G2 dx
ρ2 G2 1
d 2 + ρdp = −
2
ρ
D
(3)
Integration of the first term gives
2
1
ρ2 G2 1
d 2 =
2
ρ
2
1
dρ
p2
p1
1
ρ2
ρ2 G2
= −G2 ln
= G2 ln
(−2) 3 dρ = −G2
= −G2 ln
2
ρ
ρ
ρ1
p1
p2
(4)
where we considered the relationship between p and ρ in isothermal flow. Integration of the second term gives
2
2
ρdp =
1
1
M
pM
dp =
(p2 − p21 )
RT
2RT 2
(5)
Integration of the right hand side gives
2
1
L
2f G2 dx
= 2f G2
D
D
(6)
The integral version of Bernoulli equation for isothermal flow along the pipeline is:
G2 ln
p1
M
L
+
(p2 − p21 ) + 2f G2 = 0 = F (G, p1 , p2 )
p2
2RT 2
D
(7)
from which we calculate
G=
M
2
2
2RT (p1 − p2 )
p1
L
ln p2 + 2f D
= f (p1 , p2 )
(8)
the relationship we were looking for. If we assume that the upstream pressure p1 is fixed, the specific flow rate
is a function of the downstream pressure p2 only. p2 can varies in the range [0 : p1 ]. G is positive definite, and
is null for both p2 = p1 and p2 = 0. For the Weierstass theorem it has maximum in the range [0 : p1 ]. To
find the maximu we should calculate dG(p2 )/dp2 = 0. Since the function G(p2 ) is rather complex, we can use
Equation 7 to derive the derivative of G in a simpler way. Considering the differential of F (G, p1 , p2 ) we get
1
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⌣
October, 21st , 2014
Des. Ind. Plant, 2
dF (G, p2 ) =
dG
∂F/∂p2
∂F
∂F
dp2 = 0 →
=−
dG +
∂G
∂p2
dp2
∂F/∂G
(9)
from which dG = 0 only if ∂F/∂p2 = 0. Therefore:
dp2
∂F
G2
M
=−
+
p2 = 0 → G =
∂p2
p2
RT
M 2 √
p = ρ2 p 2
RT 2
(10)
and the velocity of gas at the outlet section is v2 = G/ρ2 = p2 /ρ2 = vsound . When G is maximum, the velocity
of gas at the outlet section is the sound speed (evaluated in isothermal conditions). Substituting the value of
Gmax in Equation 7 we obtain
M 2 p1
M p22
1−
+
p2 ln
RT
p2
2RT
2
p1
p2
+ 2f
L M 2
p =0
D RT 2
(11)
which can be simplified as
ln
1
p1
1−
+
p2
2
p1
p2
2
+ 2f
L
=0
D
(12)
This is a funtion of (p1 /p2 ) only and gives the value of the critical pressure ratio corresponding to G = Gmax .
As already discussed for the efflux from a tank, when the gas velocity at the outlet section equals the speed of
sound, if the outer pressure is lowered there is no way for this information to propagate upstream along the pipe
toward the inlet section. Therefore, once the critical flow is established, no further variation of G is expected
even if p2 is reduced and G = Gmax .
2. Derive the formula for the specific mass flow rate at critical conditions when the gas undergoes adiabatic (irreversible, i.e. non isoentropic) transformation moving along the pipe.
In this case, the integral form of Bernoulli will be given by:
G2 ln
p1
+
p2
2
ρdp + 2f
1
L 2
G =0
D
(13)
where we need a relationship between ρ and p along the pipeline which is valid for irreversible adiabatic transformation. We cannot use p/ργ = cost which is only valid for revesible adiabatic. Nevertheless, we can derive
an alternative relationship from the total energy conservation equation:
d(e +
p 1 2
+ v + gh) = dq + dws
ρ 2
(14)
e is the internal energy, q is the heat flux added to the control volume of gas and ws is the mechanical work
made on the control volume. If we define the enthalpy as H = e + p/ρ, the total energy equation becomes
1
d(H + v 2 + gh) = dq + dws
2
(15)
where we can neglect gh, dws = 0 (there is no compressor on the pipeline) and dq = 0 (for adiabatic flow). We
obtain
1
dv 2
d(H + v 2 ) = 0 → cp dT +
=0
2
2
(16)
where, according to Mayer equation, cp is given by
cp =
γ R
γ −1M
2
(17)
October, 21st , 2014
Des. Ind. Plant, 2
and according to ideal gas law dT = M/Rd(p/ρ) and
γ
d
γ−1
p
ρ
1
+ dv 2 = 0
2
(18)
p
ρ
+
G2
= C = cost
ρ2
(19)
2γ
γ −1
Equation 19 gives a relationship between p and ρ which is valid for each point along the pipe and can be used
to integrate the term in Equation 13. From 19 we caan write
p=
C−
G2
ρ2
γ−1
ρ
2γ
(20)
and
dp = −(−2)
G2
G2
ρ+ C − 2
3
ρ
ρ
γ−1
G2 γ − 1
γ−1
2G2
G2
+C − 2
+C
dρ =
dρ =
dρ
2
2γ
ρ
ρ
2γ
ρ2
2γ
(21)
The integral in Bernoulli becomes
2
2
ρdp =
ρ
1
1
ρ2
C
γ−1
γ−1
G2
G2 ln
+C
+ (ρ22 − ρ21 )
dρ =
ρ2
2γ
2γ
ρ1
2
(22)
Further simplifications led to
2
ρdp =
1
ρ2
γ−1
G2 ln
+
2γ
ρ1
γ
G2
p 1 ρ1 +
γ−1
2
ρ2
ρ1
2
−1
(23)
and the final form for integral Bernoulli equation for adiabatic irreversible flow:
G2 ln
γ−1 2
ρ1 γ + 1
+ p 1 ρ1 +
G
ρ2 γ
γ
ρ2
ρ1
2
− 1 + 4f
L 2
G =0
D
(24)
which is again an implicit function F (G, ρ1 , ρ2 ) = 0. If we consider the upstream condition fixed, F (G, ρ2 ) = 0
describes the variation of G as the downstream condition change. Following the same procedure as before, we
can find the condition for which G = Gmax from
dG
dG
∂F/dρ2
∂F
=0→
=−
=0
→
dρ2
dρ2
∂F/∂G
dρ2
(25)
which gives
∂F
G2 γ + 1
γ−1 2
=−
+ ρ1 p 1 +
G
dρ2
ρ2 γ
2γ
2
ρ2 = 0
ρ21
(26)
and using Equation 19 to rewrite the term in parentesis, we get
2γ p2
γ − 1 G2
G2 γ + 1
2ρ2
+
=
ρ2 γ
2γ
ρ22
γ − 1 ρ2
(27)
G2 γ + 1 γ − 1
G2
√
= 2p2 → 2
−
= 2ρ2 p2 → Gmax = γp2 ρ2
ρ2
γ
γ
γ
(28)
and
3
October, 21st , 2014
Des. Ind. Plant, 2
The last equation indicates that when the specific flow rate is maximum, G is a function of pressure and density
at the outlet section and the formula is the same as that calculated for the flow exiting from a tank under
adiabatic conditions. The velocity of gas at the outlet section turns out to be v2 = G/ρ2 = γp2 /ρ2 which is
the speed of sound. Substituting the value of Gmax into Equation 24 we obtain the equation which gives the
link between densities at the upstream and downstream section of the pipe at critical flow conditions:
ln
1
ρ1
1−
+
ρ2
2
2
ρ1
ρ2
+ 4f
L γ
=0
Dγ+1
(29)
which together with
p1
ρ1 γ + 1 γ − 1 ρ2
=
−
p2
ρ2 2
2 ρ1
(30)
allows to link upstream and downstream conditions of the pipe. Equation 29 is the analogous of the critical
pressure ratio calculated for isothermal transport. Equation 30 is derived from Equation 19 using the result of
Equation 29.
4

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