A polytropic approximation of c - W. M. Kirkland.pdf

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A Polytropic Approximation of Compressible Flow in Pipes with Friction
1
William M. Kirkland
Research Reactors Division
Oak Ridge National Laboratory, Oak Ridge TN (USA)
kirklandwm@ornl.gov
ABSTRACT
This paper demonstrates the usefulness of treating subsonic Fanno flow (adiabatic flow, with
friction, of a perfect gas in a constant-area pipe) as a polytropic process. It is shown that the
polytropic model allows an explicit equation for mass flow rate to be developed. The concept of
the energy transfer ratio is used to develop a close approximation to the polytropic index. Explicit
equations for mass flow rate and net expansion factor in terms of upstream properties and
pressure ratio are developed for Fanno and isothermal flows. An approximation for choked flow is
also presented. The deviation of the results of this polytropic approximation from the values
obtained from a traditional gas dynamics analysis of subsonic Fanno flow is quantified and
discussed, and a typical design engineering problem is analyzed using the new method.
1.
INTRODUCTION
In the design and analysis of industrial systems and equipment, the engineer often encounters the
subsonic, compressible flow of gases in pipes, ducts, and other configurations in which friction
Notice: This manuscript has been authored by UT-Battelle, LLC, under contract DE-AC05-00OR22725 with
the US Department of Energy (DOE). The US government retains and the publisher, by accepting the article for
publication, acknowledges that the US government retains a nonexclusive, paid-up, irrevocable, worldwide license to
publish or reproduce the published form of this manuscript, or allow others to do so, for US government purposes.
DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public
Access Plan (http://energy.gov/downloads/doe-public-access-plan).
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Kirkland
plays a key role. The layout and analysis of process gas distribution systems, the sizing and
qualification of safety devices such as pressure relief valves and rupture discs, and the modeling
of accident consequences and release rates in the chemical and nuclear industries all require the
working engineer to calculate one-dimensional compressible flows with significant friction losses.
Subsonic compressible flow is a complex phenomenon, and in order to simplify the required
analysis and to allow broadly applicable techniques to be developed, this flow is often modeled as
the adiabatic flow of a calorically perfect gas in a constant-area pipe, with friction modeled as a
shear stress acting on the fluid from the pipe walls [1]. This model is known as Fanno flow. The
theoretical analysis of Fanno flow is well-developed, and practical engineering Fanno flow
problems may be solved with the methods of gas dynamics, using implicit equations or tables,
typically expressed in terms of critical ratios and Mach numbers [1–6]. Parameters of interest to
piping and process engineers, such as pressures and mass flow rates, can then be determined from
perfect gas relations.
Since the implicit solution of relatively complex equations is somewhat inconvenient, several
explicit, approximate methods are also in widespread use. Since an explicit equation for mass
flow rate exists for isothermal flow [4, 7], the flow is sometimes treated as isothermal even when
an adiabatic model would be more appropriate. Alternatively, the Darcy-Weisbach formula for
incompressible pipe flows, solved in terms of mass flow rate, can be multiplied by a net expansion
factor,
Y,
to match the results of the exact implicit equations. Graphical representations of the net
expansion factor are given by Crane [7], and more detailed charts are also available [3], but no
explicit equation is given in these references.
By assuming a polytropic relationship between pressure and volume, then applying the concept of
the energy transfer ratio [8] to estimate the polytropic index, this paper produces approximate
equations for the mass flow rate and net expansion factor for subsonic Fanno flow in terms of the
pressure drop
∆P/P
1
, friction losses
K,
and ratio of specific heats
γ.
These equations are explicit
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Kirkland
and fairly simple in form. An approximation for the pressure drop ratio under choked flow
conditions is also presented. These approximate explicit equations are compared to those from
classical Fanno flow theory, and the results are found to agree to within 1% over the range of
non-choked
∆P/P
1
,
K,
and
γ
typically encountered in engineering practice.
2.
TRADITIONAL APPROACH TO FANNO FLOW
The frictional losses of a piping system are represented by the Darcy friction factor
f
, which
represents the number of velocity heads lost along each diameter of pipe length. To account for
the frictional and viscous losses in fittings other than straight pipe, a resistance coefficient or
K
factor is determined for each fitting, and the system
K
factor is obtained by summing the
frictional losses for straight pipe and all fittings:
K
=
f L/D
+
i
K
i
. The
K
factor therefore
represents the number of velocity heads lost by the fluid during its flow through the system [7].
Shapiro [1] presented a method for finding the mass flow rate and other parameters of interest
once the system frictional losses have been determined. Subsequent references [2, 3, 9] present
the same procedure. In brief, the process consists of determining the upstream and downstream
Mach numbers of the flow by solving the following simultaneous equations [2]:
2
P
2
M
1
2
+
(γ
−
1)
M
1
=
2
P
1
M
2
2
+
(γ
−
1)
M
2
1
2
(1)
K
=
K
∗
(M
1
)
−
K
∗
(M
2
)
where
(γ
+
1)
M
2
1
−
M
2
γ
+
1
+
K
(M)
=
ln
γM
2
2γ
2
+
(γ
−
1)
M
2
∗
(2)
(3)
For subsonic flow in constant-area pipes, the requirement that entropy must increase through the
process produces the additional constraint that
M
1
<
M
2
<
1 [2]. Once the upstream and
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Kirkland
downstream Mach numbers have been determined, the densities, velocities, and mass flow rate
can then be calculated.
Given the complex form of
K
∗
(M), it is not possible in general to solve Eqs. (1) and (2) explicitly.
Instead, tabular [2], iterative [5], or graphical [9] techniques must be used. The method of
Hullender, Woods, and Huang [10] merits mentioning in that, though it is strictly speaking an
iterative method, it produces accurate values for mass flow rate on the second iteration, making it
effectively an explicit formula. For routine use in engineering design calculations, however, it is
fairly complex to implement in comparison to the common tabular and graphical methods.
3.
CALCULATION OF MASS FLOW RATE
An explicit equation for the mass flow rate of Fanno flow can be derived by following a method
similar to Shapiro’s derivation for the usual Fanno flow equations in terms of Mach number [1],
except that the continuity equation is used to express velocity in terms of mass flow rate.
Upstream and downstream control surfaces are placed along the flow path, and mechanical
conservation laws applied to the resulting control volume.
For Fanno flow, friction is modeled as a shearing force
τ
w
, which is proportional to the square of
fluid velocity and acts upon the fluid at the pipe wall. Taking this shearing force into account, the
momentum balance equation on the control volume is:
−AdP −
τ
w
dA
w
=
m du
˙
(4)
Defining the Darcy friction factor,
f
≡
8τ
w
/ρu
2
, and hydraulic diameter,
D
≡
4A
dL/dA
w
, allows
equation (4) to be expressed as
ρu
2
dL
=
ρu
du
−dP −
f
2
D
(5)
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Kirkland
Dividing by the velocity head,
ρu
2
/2,
and rearranging terms produces
f
2vdP 2du
+
+
dL
=
0
u
2
u
D
(6)
The continuity equation on the control volume is
ρuA
=
uA
=
m
=
const.
˙
v
(7)
allowing the fluid velocity to be expressed in terms of the mass flow rate. Since mass flow rate
and pipe area are constant,
du
=
m
˙
dv
A
(8)
Substituting Eq. (7) and Eq. (8) into Eq. (6) produces a combined momentum/continuity
equation:
2A
2
dP
2dv
f
+
+
dL
=
0
m
2
v
˙
v
D
Since friction is present, the flow is non-isentropic and the relation
Pv
γ
=
const.
does not hold.
However, for calculational convenience, it is useful here to assume that some polytropic relation
Pv
c
=
const.
may be used, to allow the specific volume to be expressed in terms of the pressure
ratio.
Pv
c
=
P
1
v
c
=
const.
1
P
1
v
=
P
1
c
(9)
(10)
(11)
v
1
The polytropic index
c
will be determined below through the use of energy considerations. It is
worth noting that, as will be shown,
c
is not a specific constant, but will be permitted to vary with
a system’s fluid properties, pressure difference, and friction losses.
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Kirkland

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