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Direct calculations of unsteady-state Weymouth equations for gas volumetric flow rate occur more frequently in the design and operation analysis of natural gas systems. Most of the existing gas pipelines design procedures are based on a particular friction factor and steady-state flow analysis. This paper examined the behavior of different friction factors and the need to develop model analysis capable of calculating unsteady-state gas flow rate in horizontal and inclined pipes. The results show different variation in flow rate with Panhandle A and Panhandle B attaining stability in accurate time with initial unsteadiness at the instance of flow. Chen and Jain friction factors have opposition to flow with high flow rate: The prediction also reveals that Colebrook-White degenerated to Nikuradse friction factor at high Reynolds number. The horizontal and inclined flow equations are considerably enhanced on the usage of different friction factors with the aid of Matlab to handle these calculations.

[

[7,8] compared a variety of transient models. Numerical solution of the partial differential equations, which characterize a dynamic model of the network, requires significant computational resources. The problem is to find, for a given mathematical model of a pipeline, a numerical method that meets the criteria of accuracy and relatively small computation time. The main goal of this paper is to characterize different transient models and existing numerical techniques to solve the transient equations. Reference [

In this work, the developed model equations in [

This paper considered different friction factor equations for use in Equations (1) and (2) and these are substituted into the unsteady state equations for solutions. These solutions can be used to calculate the instantaneous volumetric gas flow rate in both horizontal and inclined pipes if p_{1} and p_{2} are known. The developed equations for both the horizontal and incline pipes are stated as follows:

for horizontal pipeline and

For vertical pipeline. For a uniform slope:

where

and if the Moody friction factor for Weymouth equation is

All these friction factors below will be substituted into Equations (1) and (2) respectively to calculate the instantaneous volumetric gas flow rate in horizontal and inclined pipes.

The Panhandle A pipeline flow equation assumes that f varies as follows:

This is probably the most widely used equation for long lines (transmission and delivery). The modified Panhandle equation assumes that f varies as

This correlation is still the best one available for fully developed turbulent flow in rough pipes:

This is used for laminar flow.

This is applicable to smooth pipes and to flow in transition and fully rough zones of turbulent flow.

Equations (1) and (2) are for horizontal and inclined pipes and these provide a fundamental relationship between gas flow rate, inlet and outlet pressures, and the usual pipeline parameters. These two equations are based on fundamental fluid flow equations that governs compressible gas flow in pipes [9-14] considered a single phase real gas flow in pipe with uniform cross-sectional area using the mass conservation principle to developed a conservation equation for a control system that includes energy equation that so obtained the unsteady-state equations for gas flow in pipes and these can be applied to a variety of problems.

In deriving Equations (1) and (2), it was assumed that temperature and compressibility factor are constant, for a very short piece of pipeline, this assumption claimed to be valid and thus the equations should be accurate [

The volumetric gas flow rate for horizontal and inclined pipes is calculated for a period of 1000 hours using Equations (1) and (2) for the different friction factors. These are compared with steady-state gas flow for both horizontal and inclined pipes and are hereby discussed.

The steady-state gas flow rate is achievable as 56.5 MMScf/hour for horizontal pipe and 43.7 MMScf/hour for inclined flow.

The steady-state gas flow rate is achievable as 79.0 MMScf/hour for horizontal pipe and 61.1 MMScf/hour for inclined pipe.

The steady-state gas flow rate is achievable as 39.5 MMScf/hour for horizontal pipe and 30.5 MMScf/hour for inclined pipe.

The analysis of results and discussion shows that all the flow variations in the graphs are evident that there exists an initial transience at the instance of flow which later stabilizes with time. It is observed that the flow rate for inclined flow is smaller when compared to horizontal flow. It is also observed that it took a longer time to achieve steadiness in inclined pipes as compared to horizontal pipes and these observations can be attributed to gravity effect due to change in elevation. The gas volumetric

flow rate obtained when using Panhandle B is higher than the volumetric flow rate when using Panhandle A.

This can be attributed to the fact that Panhandle B equation is correlated for higher flow rates. We also observed that gas volumetric flow rate increased with decreasing friction factor.

Chen Equation which has a friction factor of 0.0045 has the highest flow rate while Jain Equation which has a friction factor of 0.01739 has the lowest flow rate. This can be attributed to friction factor that shows the degree

to which flow is opposed in a pipe. Hence, a small friction factor means that opposition to flow is low which implies high flow rate and vice versa.

The Nikuradse’s friction factor obtained is approximately equal to the friction factor of the ColebrookWhite Equation for a Reynolds number of 8 × 10^{5}. This is because Colebrook-White degenerates to Nikuradse correlation at high Reynolds number.

Direct calculation of unsteady-state Weymouth equations has been examined on different friction factors without

neglecting any of the terms in the fundamental gas flow equations. These friction factors show a functional relationship between flow rate, inlet and outlet pressure and are very useful in gas pipeline calculations where any of these variables needs to be estimated if the others are given. The friction factors show different variation in flow rate with more realistic results, however, we can predict that panhandle A and panhandle B are more accurate in attaining stability with initial unsteadiness and flow rate at any given time. It is observed that Chen and Jain friction factors have opposition to flow which implies high flow rate and vice versa. The examination also observed that Colebrook-White degenerate to Nikuradse friction factor at high Reynolds number. In conclusion, we observed that all unsteady-state processes tend towards steady-state with time. The initial unsteadiness at the instance of flow is enhanced at high Reynolds number. The usage have tremendous application when examined through different friction factors and is able to predict unsteady-state flow in pipeline compare to the steady-state process assumed in industry.

The authors acknowledged the support of the Chemical Engineering lab at the University of Lagos and the assistance and suggestions. Authors would like to thank the Nigerian Gas Company for making it possible to write this paper.

Inlet pressure,

Outlet pressure,

Acceleration due to gravity, ft/sec^{2}

Average flowing temperature,

Base of natural logarithm = 2.718

Base temperature,

Base pressure,

Constant cross-sectional area of the pipeline,

Effective length of the pipeline,.

Gas density

Gas deviation factor at average flowing temperature and average pressure

Volumetric gas flow rate, at and.

Gas specific gravity (air = 1)

Gas velocity, ft/sec

Gravitational conversion factor = 32.17 lbm-ft/lbf-sec^{2}

Inside diameter of pipe,

Length of pipe,

Moody friction factor

Outlet elevation – inlet elevation

Specific volume,

Angle of Inclination, degree.

Gas Constant,

Gas Compressibility Factor

Change in time

Psia = psi × 6.894757

Psi = ib/in^{2}

1k = 1.8^{0} R

Units based on US empirical units.

P1 = 933.45

P2 = 899.35

Tb = 520

Pb = 14.7

z = 0.9

T = 512.2

Gas gravity = 0.62

Diameter = 1

L = 101711

e = 0.0006

Time = 20 - 1000 hrs

Reynolds number:

1. Panhandle A: 800000

2. Panhandle B: 800000

3. Hagen-Poiseuille: 8000

4. Chen: 800000

5. Jain: 800000

6. Colebrook: 800000