4.13. Fast filling pipe

4.13.1. PIPE FFP (class)

Fig. 4.13.1 Wanda component for the fast filling pipe.

Fall type

type label	description	active
Pipe	Pipe model which can be used to model the fast filling of an empty pipeline. The water front moves vertically through the pipeline (perpendicular to the axis).	No

Fast Filling Pipe

The fast filling pipe can be used to simulate the fast filling of a single pipeline. Filling of branched systems is not possible with this component. In case of fast filling, the water front moves vertically through the pipe (perpendicular to the pipe axis). Filling can be modelled from either left or right side but filling from both sides simultaneously is not possible. The user does not have to specify the filling direction.

4.13.1.1. Mathematical model

The mathematical model for the moving water front is based on ref [1]. The behaviour of the fast filling pipe is divided into four phases:

1. Initialisation phase

2. Waiting to start with filling phase

3. Filling phase

4. Completely filled phase

The criterion for “fast filling” is derived from ref. [3]. The user needs to check the flow number. The flow number should be above 0.9 to make sure that all air is transported and does not remain trapped at local high points.

Initialisation phase

This phase is used in the steady state to initialize an empty or partly filled pipeline at the beginning of the transient simulations. The head in the pipeline is initialized at the elevation initial air pressure specified by the user of the pipeline.

Waiting to start with filling phase

The pipe is waiting to be filled from one of the connection points. As soon as the head at one of the connection points increases above the elevation of the pipe at the connection point, the pipe goes to the filling phase, in which the pipe is filled from the connection point where the head is raised. If the head is raised on both sides simultaneously, a warning is given and the calculations are continued with filling from connection point 1.

Filling phase

During the filling phase, the pipeline is filled from the side at which the head is increased first. The filling flow rate is determined from (see ref [2]):

(4.13.1)\[\frac{1}{A} \frac{d Q}{d t}=\frac{g}{L_{f}}\left(H_{1,2}-H_{a}-f \frac{L_{f}}{D} \frac{Q|Q|}{2 g A^{2}}-\left(1+\xi_{\text {entr }}\right) \frac{Q|Q|}{2 g A^{2}}\right)\]

In which:

Variable	Description	Units
\(A\)	Area of the pipeline	m2
\(Q\)	Flow rate	m3/s
\(dt\)	Time step	s
\(g\)	Gravitational acceleration	m/s2
\(L_f\)	Length of the fluid column	m
\(H_{1,2}\)	Head at side from which the pipe is filling	m
\(H_a\)	Air pressure head at the end of the liquid column	m
\(f\)	Friction factor	-
\(D\)	Diameter pipe	m
\(\xi_{entry}\)	Entry loss	-

The air pressure head is calculated from:

(4.13.2)\[\frac{d H_{a}}{d t}=\lambda \frac{H_{a}}{L_{a}}\left(Q-\left(\frac{P_{a t m}}{P_{a i r}}\right)^{\frac{1}{\lambda}} Q_{a i r}\right)\]

with:

Variable	Description	Units
\(\lambda\)	Laplace coefficient	-
\(L_a\)	Length of air column	m
\(Q_{air}\)	Air flow rate at atmospheric pressure	Nm3/s

The air discharge is calculated based on the air pressure, in the same way as for air valves.

Equation 1 is based on the rigid column model. During the calculations, the length of the fluid column is followed. The head is set to the height of the pipe for locations which are not filled yet. For the remainder (elements that are filled) the head decreases linearly to the initialized head. When the length of the liquid column becomes identical with the length of the pipeline, the pipeline is completely filled and phase 4 is started. This transition results in strong transients, which may depend upon the time step used. It is therefore recommended to do a sensitivity analysis on the time step to ensure physical correct results are obtained.

Completely filled phase

When the pipeline is completely filled, the calculation method is changed to water hammer. Based on the boundary condition, e.g. closed valve or reservoir, the resulting head and flow rate are calculated.

4.13.2. Fast filling pipe properties

4.13.2.1. Hydraulic specifications

description	input	Units	default	remarks
Inner diameter	real	m		if Cross section=Circle
Wave speed mode	Physical, specified		Physical	if Calculation mode = Waterhammer
Wall thickness	real	m		if Wave speed mode = physical
Young’s modulus	real	N/m2		if Wave speed mode = physical
Specified wave speed	real	m/s		if Wave speed mode = specified
Wall roughness	real	mm		if Friction model = D-W k
Dynamic friction	Quasi-steady, none			Only in transient mode and if Friction model = D-W k
Geometry input	Length , l-h, xyz, xyz diff
Length	real	m		if Geometry input = Length
Profile	table			if Geometry input = L-h, xyz or xyz diff
Upper limit pressure	real	N/m2		Only visible if checked in “Mode&options” window
Lower limit pressure	real	N/m2		Only visible if checked in “Mode&options” window
Location	real	m		If empty, chart button creates a location serie. If filled in, chart button creates a time serie for nearest internal node
Discharge coefficient air	Real			Discharge coefficient for air outflow
Discharge area air	Real	m2		Area of the air outlet
Entrance loss coefficient flow	Real			Additional energy loss coefficient for the liquid inflow
Laplace coefficient	Real			Laplace coefficient for ideal-gas law
Initial air pressure	Real	N/m2.a	1.014	Initial air pressure of air volume
Initial length liquid column	Real	m	0	Initial length of liquid column in pipe

Remarks

Please note that the fast filling pipe is a dedicated component and can only be used to simulate the filling of a single pipeline. Furthermore, it is recommended for stability to use a small time step, to ensure the transition from filling to water hammer phase is taken place smoothly.

4.13.2.2. Component specific output

Output	Description
Pipe length [m]	Total pipe length based on entered profile
Wave speed [m/s]	Wave propagation speed based on fluid properties, pipe properties and time step
Pipe element count [-]	Amount of elements of same length in which pipe is divided.
Adapted wave speed [m/s]	Wave speed is adapted in such way that an integer number of elements (minimal 1) fits in the total length
Deviation adapted [%]	Ratio between wave speed and adapted wave speed. Must be less than 0.25 for a valid deviation
Cavitation fraction [-]	Ratio between cavitation volume and element volume Must be less than 0.20 for valid range of cavitation algorithm
H-node height check	Input validation between H-node height with corresponding pipe node elevation. OK means the difference is less than 0.5 D, otherwise “Warning” is displayed.
Wall roughness	The wall roughness based on the friction factor and including local losses
dH total t= 0	Total head losses across the pipe at steady state
dH local losses t=0	Head loss due to local losses at steady state
Flow number	Flow number, can be used to determine if it is a fast or slow filling.
Air pressure	Pressure of the air
Length liquid column	The length of the liquid column
Air flow rate	Flow rate of air out (+) or in (-)

Message	Type	Explanation
Filling starts from connection point 1	info
Filling starts from connection point 2	info
Head increases on both sides,	warning
filling starts from connection point 1
Pipe completely filled	info

References

[1] Zhou, F., Hicks, F.E., Steffler, P.M., 2002, Transient Flow in a Rapidly Filling Horizontal Pipe Containing Trapped Air, J. Hydraul. Eng., 128 (6), 625-634. [2] van ’t Westende, J.M.C. ,Pothof, I.W.M., Heinsbroek A.G.T.J., Tukker M.J., Guijt W. Incident analysis of a fuel loading line, 14th International Conference on Multiphase Production Technology, Cannes, France: 17th – 19th June 2009 [3] Pothof, I.W.M., Co-currunt air-water flow in downward sloping pipes, 2011, ISBN: 978-90-89577-018-5