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Chapter 3: Background Theory For Modelling Pipeline Outflow

1780 words - 7 pages

3.5.3 Fanning Friction Factor (fw) Determination
The fanning friction factor, fw is a function of the flow Reynolds’ number. It is required for calculating the contribution of frictional force to the momentum equation (equation (3.28)).
In the developed UCL model, the Chen (Chen, 1979) correlation is employed for the calculation of the fanning friction factor for transition and turbulent flows in rough pipes. It is given by:
1/√(f_w )=3.48-1.7372 ln⁡[ε/r_in -16.2446/Re ln⁡A ] (3.76)
Where
A= [ε⁄r_in ]^1.0198/6.0983+ (7.149/Re)^0.8981 (3.77)
ε, is the pipe roughness and rin represents the pipe inner radius.

3.6. The Steady State Isothermal Flow Model (Existing UCL Model)
In this section, the steady state isothermal flow model developed by Atti (2006) and Garfield (2009) is presented. The model is for one-dimensional flow based on the continuity and momentum equations presented in section 3.2.1 and 3.2.2 respectively.
From equation (3.6), the steady state expression (i.e. when all fluid properties are time invariant) for continuity in one-dimension can be written as:
u ∂ρ/∂x+ ρ ∂u/∂x=0 (3.78)
The author derived the single-phase and two-phase isothermal steady state flow equations based on isothermal flow assumption.

3.6.1 Single-phase isothermal steady state flow equations
Due to the isothermal flow assumption, only the continuity together with the momentum or energy equation is required for the complete solution of the steady state problem. For single-phase flow, the continuity and momentum equations are employed as they contain terms that are easily solvable (Zhou and Adewumi, 1995a).
Rearranging equation (3.78) and separating variables, equation (3.78) becomes:

Integrating equation (3.79) gives:

Equation (3.80) can be rewritten as:

Equation (3.81) is the governing equation for mass conservation at steady state for flow in a uni-diameter pipeline.
From equation (3.27), the steady state momentum equation in one-dimension can be expressed as:

Where the steady state frictional force term (x) is given by (Atti 2006):

f is the fanning friction factor, which is a function of the flow Reynolds’ number, and can be calculated from a number of empirical correlations.

Rearranging (3.82) and (3.83) gives equations:

Multiplying equation (3.84) by dx/ and noting that for positive flow |u| = u results in:

Multiplying equation (3.85) with 2 gives:

Now equation (3.78) can be rewritten as:

Also, from equation (3.81) the fluid velocity (u) can be expressed as:

Substituting equations (3.82), (3.87), and (3.88) into equation (3.86) gives:

Dividing equation (3.89) by the coefficient of dx on the right hand side of the equation and setting the limits of...

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