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FORCE CALCULATION APPLICATIONS IN PIPENET 1. Introduction:
This document discusses two typical applications of force calculation in PIPENET: (1) pipe stress analysis and (2) reaction force on pipe support. In the stress analysis, we need to know the hydraulic force to strain pipe wall. In the reaction force calculation, the net hydraulic force passing along the t he pipeline and loading on the pipe support (anchor) is analysed and discussed. Finally several examples are presented to illuminate the use of elastic and rigid joints. 2. Modelling Equations:
Simple force: n
m
i 1
j 1
F tag s F s i tag f F f
j
(1)
Complex force n
F ( X ,Y , Z ) tag s ( X ,Y , Z ) F s i 1
Where F F b Fs Ff n m tags tagf
m
tag i
f ( X ,Y , Z )
j 1
F f
j
F b X ,Y , Z
(2)
total hydraulic force body force, such as pipe and fluid weight. hydraulic force created at control surface by fluid pressure and flow momentum. hydraulic force created in control volume by flow resistance. number of control surfaces. number of control volumes, which can contain several components in PIPENET. tag of boundary condition, 0 for elastic joint and 1 for rigid joint. tag of flow direction, tags= +/-1.
At a middle point of a straight pipe, there is no direction change of flow momentum and the projective area of pressure on the pipe wall is zero. The hydraulic force created at the control surface is zero. The The boundary condition should be set as “Elastic”, “Elastic” , i.e. tags = 0. On the contrary, at an elbow, the flow momentum changes direction and the projective area is equal to the pipe cross section area. The elbow should be set as “Rigid” to consider the hydraulic force due to fluid pressure and flow momentum, i.e. tag s = 1. 3. Pipe Stress Analysis
This section focuses on the stress in pipe wall due to hydraulic static and dynamic force. Figure 1 depicts a simple network with a straight pipe and two elbows. Some assumptions are made to simplify the problem, including (1) The length and volume of the elbows are negligible so that the inlet and outlet have equal flow parameters. (2) The study focuses on the stress caused by hydraulic force, does not consider body force, pretightening and thermal distortion.
1
Page 2 of 10 2016-03-02 PIPENET Technical Document Force In order to illuminate the problem clearly, the pipe is divided into two parts to study the stress at the middle point of the pipe, i.e. control surface 1 in Figure 1.
Figure 1: Force and Moment in the Control Volume and Control Surfaces
F s1 F s 2 pA mu
(3)
F i1 F j 2 F f 1 F s1 Ma1
(4a)
F i 2 F j1 F s 2 Ma2
(4b)
T 1 T 2 I F i
(5)
Where a A A’ Ff Fi F j Fs I p m M T u W y
(6)
A' F j
(7)
A' T
(8)
W linear acceleration cross section area of pipe cross section area of pipe wall flow friction on pipe wall tensile force at control surface shear force at control surface hydraulic force at elbow total inertial moment of fluid and pipe body fluid pressure at elbow mass flow rate at elbow total mass of fluid and pipe body moment at control surface flow velocity at elbow section modulus, W = Ix/y perpendicular distance to the neutral axis angular acceleration with the rotation axis of elbow cross point tensile stress at control surface shear stress at control surface bending stress at control surface
2
Page 3 of 10 PIPENET Technical Document Subscript 1 at control surface 1 or along its normal vector 2 at control surface 2 or along its normal vector
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At steady sate, the linear acceleration and angular acceleration are zero. Here we assume the shear force F j and moment T are negligible. The stresses at the control surface 1 can be given by Equation 4a, 6, 7 and 8, i.e.
F i1
A' F j
A' T
W
F f 1 F s1 A'
0
0
The tensile force Fi can be calculated by PIPENET. The boundary conditions at the straight pipe and elbow are elastic and rigid respectively. The figure below presents such a network.
Figure 2: A Simple Example If we would like to calculate the tensile stress at the middle point of pipe 3, the force Fi can be defined as:
Figure 3: Force Definition in PIPENET 3
Page 4 of 10 2016-03-02 PIPENET Technical Document Force In this example, the force Fi at initial steady state is 5804.7N, see the figure below. The pipe (Dn 200, AnsiB3610_40) has 202.692 mm internal diameter and 219.202 mm external diameter. The cross section area of the pipe wall is 0.00547 m 2. The strain stress in the pipe wall due to hydraulic force can be calculated as:
5804.7 0.00547
5
10.62 10 Pa
Figure 4: Hydraulic Force in the Downstream Section of Pipe 3 The force calculation at transient state becomes so complex that it is impossible to give accurate or even conservative results by hand calculation. For example, (1) (2) (3) (4)
The linear acceleration a is unknown. The angular acceleration is unknown. The shear force F j is unknown. The moment T is unknown.
The above variables (a, , F j and T) strongly depend on the pipeline elasticity and anchor position. It is difficult to calculate them by hand but it is not a problem for stress analysis software. The hydraulic force calculated by PIPENET can be included in the stress analysis. The hydraulic model must coordinate with the stress analysis model. The following settings should be cross-checked to insure calculation accuracy. (1) The stress analysis program may consider body weight, which requires zero body weight in the PIPENET model. (2) The hydraulic model has considered the hydraulic force produced by fluid pressure. If the stress model also calculates the tensile stress caused by fluid pressure, the input pressure in the stress model should be 0 barg to avoid duplicate calculation. (3) The boundary condition should be “Elastic” in PIPENET to obtain the strain force at the pipe section.
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4. Selection of Elastic Joint and Rigid Joint
Pipe stress analysis If we would like to calculate strain stress in a straight pipe, caused by hydraulic force F. The boundary condition should be set as “Elastic”, see Figure 8.
Figure 8. Stress Analysis in a Straight Pipe The force Fi presented in Figure 9 is used to c alculate strain stress at the middle point of pipe 3, which is set as an elastic joint. The pipe outlet is an elbow so it is a rigid joint.
Figure 9: Force Definition in PIPENET
Hydraulic force at elbow An elbow changes flow direction and the projective area of pressure on pipe wall is equal to the pipe area. Therefore, it is a rigid joint in force calculation, see Figure 8 and 9. Similarly an angle valve is a rigid joint as well.
Reaction force on anchor We discuss the force calculation in Section 4. In theory, an anchor is a break point for force transfer, which requires defining two forces for the upstream pipe and the downstream pipe respectively. The anchor is set as “Elastic” in both the forces, see Figure 10 and 11.
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Figure 10. Force Analysis on a Straight Pipe with an Anchor
Figure 11. Force Definition when the Anchor is an End Point The forces F3a and F3b can combine into a force F3 if other anchors are far away, see Figure 12 and 13. This simplification unlikely produces calculation error for the reaction force on the anchor.
Figure 12. Force Analysis with Several Anchors
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Figure 13. Force Definition when the Anchor is a Middle Point
Hydraulic force at elastic joint An elastic joint is a break point for force transfer. We must define two forces to calculate the hydraulic forces in the upstream pipe and the downstream pipe respectively because they can’t cancel each other at any situations. An elastic joint can be considered as a piece of straight pipe in hydraulic analysis which does not change flow direction and the projective area is zero. Therefore, it is an elastic joint in force calculation. A hose is a kind of elastic joint so it must be set as “Elastic” as well.
Figure 14. Stress Analysis on a Straight Pipe with an Elas tic Joint
Figure 15. Force Definition on a Straight Pipe with an Elastic Joint 7
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Hydraulic force at jet exit
A jet exit does not change flow direction and the local pressure is 0 barg. Therefore, the fluid does not produce hydraulic force at the outlet section. The exit should be set as “Elastic”. In PIPENET, the nozzle and valve models are set as elastic joints by default. You can use them to model a jet exit to calculate reaction force, see Figure 16.
2
1
1
Figure 16. Modelling of Jet Exit by a Nozzle Alternatively, you can use a valve together with a pipe to model the jet exit. In this case, the pipe outlet should be set as “Elastic”, see Figure 17.
2
1
3
1
Figure 17. Modelling of Jet Exit by a Valve and a Pipe 8
Page 9 of 10 PIPENET Technical Document Hydraulic force at tank entrance/exit
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A tank/reservoir generally changes flow direction, e.g. from horizontal to vertical, and the projective area of pressure on the tank wall is equal to the pipe area. Therefore, the tank/reservoir should be set as “Rigid”. In PIPENET, the tank models (accumulator, surge tank and receiving vessel) are “Rigid” by default in force calculation. You can use them to model a tank type entrance/exit directly, see Figure 18.
2
1
1
Figure 18. Modelling of a Tank Entrance/Exit by a Tank If the entrance/exit is simplified as a constant pressure specification, the local boundary condition should be set as “Rigid”, see Figure 19.
Reservoir 2
1
Figure 19. Modelling of a Tank Entrance/Exit by a Pressure Specification 9
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5. Conclusion:
(1) The tensile stress caused by hydraulic force can be estimated by hand calculation at steady state. However, it is recommended to carry out the analysis by programs at transient state. (2) PIPENET hydraulic model must coordinate with stress analysis model to avoid duplicate calculation on body force and fluid pressure. (3) An anchor is a breaker point for force transfer. It should be set as an elastic joint in PIPENET force calculation. If the anchor bears all loads in the control volume because other anchors are far away, the anchor can be included within the control volume or set as a rigid joint. This simplification unlikely produces calculation error on the rea ction force of the anchor. (4) The typical boundary conditions are summarized in the table below. Table 2: Typical End Points in Force Calculation Interruption Control surface for stress analysis Anchor (Axis restriction), Elastic joint (Hose) Jet Exit (nozzle, monitor, valve etc.) Elbow (Angle valve) Dead end, Tank Entrance/Exit (tank, reservoir, pressure vessel etc.)
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End Point Elastic Elastic Elastic Rigid Rigid
