4.4 Flow regimen profiles calculation
Velocity profiles for laminar, partially turbulent and fully turbulent flow are shown infiJ It may be noted that the velocity profile depends upon the surface condition of the pipewi smoother wall results in a more uniform velocity profile than a rough pipe wall. 4.3.9.1 General Flow Equations
Since the 1800s, a great deal of work has been performed to develop equations that I rately predict flow behavior of gas and liquids. Major contributions to the developmJ the flow equations are due to [15]:
Darcy
Euler
Reynolds
Fanning
Stanton
Pannel
Prandtl
Nikurasdse
Colebrook and Moody
Bernoulli
Weymouth
White
Hazen-Williams
The USBM and the American Gas Association (AGA) have had a part in the further development of gas flow equations. With the exception of two and three phase flows, the development of flow equations have evolved to a stage where, in most applications, the error inherent in the flow equation is insignificant in comparison to the error caused by the inaccuracy of the actual inputs into these equations (e.g., roughness, fluid properties, etc.). For example, research into the steady state hydraulic behavior of gas pipelines has shown that the AGA gas equation can predict pressure drop to within 0.3-1% of actual pressure that is measured. In comparison, the parameters used as input to the flow equations, such as flow, operating temperature, operating pressure and internal roughness generally involve a great deal of estimation and have a far greater impact on the overall error.
|
Figure 4-5. Laminar and turbulent flow velocity profiles |
The USBM and the American Gas Association (AGA) have had a part in the further development of gas flow equations. With the exception of two and three phase flows, the development of flow equations have evolved to a stage where, in most applications, the error inherent in the flow equation is insignificant in comparison to the error caused by the inaccuracy of the actual inputs into these equations (e.g., roughness, fluid properties, etc.). For example, research into the steady state hydraulic behavior of gas pipelines has shown that the AGA gas equation can predict pressure drop to within 0.3-1% of actual pressure that is measured. In comparison, the parameters used as input to the flow equations, such as flow, operating temperature, operating pressure and internal roughness generally involve a great deal of estimation and have a far greater impact on the overall error.
Equation Factors. A valid, usable flow equation is vitally important in both the design and operation of a gas transmission system. To be valid, such an equation must reasonably approximate the actual behavior of gas as it flows in a pipeline. It must incorporate the most significant properties of the gas and the conduit and express them in a form which properly reflects the manner in which the flow behavior is affected.
To be useful, the equation must permit the relatively simple determination of any one of the three prime design variables (throughput, diameter, and pressure drop) as a function of the other two, and the known properties of the gas and the pipe. It should express these relationships in practical industry terminology.
Of the several dozen flow equations which have been proposed to describe the behavior of gas flowing in transmission pipelines, very few completely satisfy all these requirements. Most of the equations reflect the limitations of the restricted conditions under which they were developed. Some are empirical fits of data taken over short ranges of flow conditions. Many of the early equations were developed from information on lines of smaller diameter and lower throughput than those in general use today. Some equations are purely analytical; others are combinations of analytical derivations, modified by field data. Almost all can be considered to accurately reflect pipeline flow behavior for only specific, limited situations.
The general flow equation was derived from a general mass momentum and energy balance taken around an element of fluid flowing through a differential length of pipe under conditions of steady state flow and thermodynamic equilibrium. However, the flow equations that mostly used in the pipeline transmission industry are: Darcy, Colebrook-White, Panhandle (A&B) AGANB 13 Equation, USBM
The most frequently used equations for large transmission lines are Colebrook-White and AGA correlations.
Darcy's Formula. Flow in a pipe, Fig. 4-6, is accompanied by friction of fluid particles rubbing against one another and, consequently, by loss of energy. This implies that in a horizontal pipe there must be a pressure drop in the direction of flow. If pressure gauges were connected to a horizontal pipe containing a flowing fluid, the pressure PI would be higher than P2.
|
Figure 4-6. Generalized fluid flaw through a circular conduit |
With the exception of completely laminar flow, the energy losses of actual systems cannot be predicted theoretically but must be determined by actual experiment and then correlated as some function of the flow variables. Such a relationship is the well known Darcy equation:
,
(4-27)
where:
h - fluid head
fD = Darcy friction factor
= 4 x fanning friction factor
This
equation may be written to express pressure drop, 
P,
as follows:
,
(4.28)
Darcy's formula ignores the effects of inertia and elevation and is, in fact, a simplified form of Euler' s equation. The Darcy equation is valid for laminar or turbulent flow and be used with natural gas when suitable restrictions apply.
Limitations of Darcy's Formula. For compressible flow when the pressure drop is less than 10% of the inlet pressure, use p based on either inlet or outlet conditions. When pressure drop is greater than 10% but less than 40% of the inlet pressure, use the average of p based on inlet and outlet conditions. When the pressure drop is greater than 40% I inlet pressure, use other estimating methods for p.
The generalized flow equation for constant temperature, steady-state flow in a pipeline elevation change, and for an average value of the compressibility factor is (Fig. 4-6):
,
(4-32)
where:
=
elevation factor
Tb= base temperature
Pb = base pressure
For situation involving no change in elevation, the general flaw equation reduces to:
,
(4-30)
Other relationships have often been used to determine the friction factor. The Panhandle, Weymouth, and Colebrook-White equations were developed by assuming a relationship for friction factor. These equations are satisfactory for limited flow ranges and may have other restrictions.
The Colebrook-White correlation defines the transition from partially turbulent flow to fully turbulent flow with one function. This is not entirely accurate because there is a discontinuity in the relationship which cannot be expressed in a single function.
However, Panhandle and AGA equations provide different correlations for partially and fully turbulent flows.
Only two examples of the industry use flow equations are provided below but details and use of other flow equations are fully described by reference [7].
Colebrook-White. The Colebrook-White formula has one relation for the transmission factor (1/f)05, which attempts to approximate both the fully turbulent and partially turbulent flow ranges. The equation shown below has been slightly modified from the equation published by Colebrook to reflect the data obtained by the USBM.
,
(4.31)
Panhandle "A ". The Panhandle "A" formula was developed by the Panhandle and Eastern Gas Co. for simulation of natural gas pipelines from 168.3 to 610 mm with Re's ranging from 5 x 106 to 14 x 106.
,
(4.32)
where:
Alternatively, the factor (1/f)0,5 can be substituted into the general equation as follows:
(1/f)0,5=6,892E0,92695Re0,07305 (4.33)
Where
E=flow efficiency
Panhandle "A ". The Panhandle "A" formula was developed by the Panhandle and Eastern Gas Co. for simulation of natural gas pipelines from 168.3 to 610 mm with Re's ranging from 5 x 106 to 14 x 106.
,
(4.34)
where:
Alternatively, the factor (1/f)0,5 can be substituted into the general equation as follows:
(1/f)0,5=7,490E0,93215Re0,6788 , (4.35)
The Panhandle equations are used extensively because the formulation for input into the General Flow Equation uses a pipe efficiency. This can be used in operating pipeline systems to "balance" the simulations in a pipe segment to actual measured data. Although this formulation is not as accurate for planning purposes as the general flow equation (using the effective roughness), it is easy to conceive of a pipeline efficiency.
Worked examples
Example
1: What is the maximum Re for which a flow regime remains partially
turbulent given a transmission factor or 
?
Using the following equation:
If the calculated Re for an actual pipeline with transmission factor 18 exceeds that value, the flow regime is fully turbulent.
Example 2: A gas transmission line is to be constructed to transport 1,500,000 m3/hr of natural gas from a gas refinery to the first compressor station located 100 km away. The route is almost horizontal with no considerable elevation changes. Determine the size of the pipeline required to transport the gas if the pipeline inlet pressure is 1140 psia and 300-psia pressure drop is allowable. Use Weymouth, Panhandle B, and AGA fully turbulent equations to compare the diameters that can be predicted by each flow equation. Assume an effective roughness value of Kc = 700 μ in. for the line. Solution for Pandle is provided below. Other equations can be obtained from reference.
Additional data:
Taye = 522.6 °R
G= 0.64
Tb = 520°R
Pb = 14.7psia
Zave = 1.0
Solution: using equations in imperial units:
L = 100 km = 62.1504 miles
P1 = 1140 psia
P2 = 1140-300 = 840 psia
D - ? (inches, inside diameter)
q = 1,500,000 x 35.31 = 52,965,000 SCF/HR = 1,271,160,000 SCF/D
Use Panhandle B equation with 100% efficiency.
E = 0 = Elevation Change, etc. Using Panhandle B equation:
Using Panhandle B equation:
And upon substitution of date
ID=35.380 in., OD(NPS 36)