1. Computational fluid dynamics Authors: A. Ghavrish, Ass. Prof. of NMU, M. Packer, President of LDI inc.

2. Analytical model 2

3. Method of calculating The transient and frequency characteristics of the pipeline that contains the pump and the dampener, is based on nonlinear mathematical model. The basis of calculation is the method of characteristics applied to the simplified Navier-Stokes equations. The resulting nonlinear differential equations are solved using the finite difference method of first order. 3

4. Method of characteristics The method of characteristics converts partial differential equations, for which the solution can't be written in general terms (as, for example, the equations describing the fluid flow in a pipe) into the equations in total derivatives. The resulting nonlinear equations can then be integrated by the methods of the equations in finite differences. 4

5. Equations of motion Hydraulics equations that embody the principles of conservation of angular momentum and continuity in the one-line pipe, respectively, are as follows: These equations can be combined with the unknown factor and obtain the equation: 5

6. Equations of motion If Then equation (3) becomes an ordinary differential equation: 6

7. Equations of motion Solving (5), we obtain: Substituting equation (7) (6), we obtain a system of total differential equations: 7

8. Finite-difference scheme xx P t+  t t SCRB x For solving the nonlinear equations (8) - (11) a finite difference method was used. Fig. 1. A 8

9. Finite-difference scheme Spatial - temporal grid (Fig. 1) describes the state of the liquid at various points in the pipeline at time t and t +  t. Pressure and velocity at points A, C and D, which correspond to time t, are known either from the previous step, or from data on the steady flow. States at R and S correspond to the time t and should be calculated from the values at points A, C and B. State at the point P corresponds to the time t +  t is determined from equations (8) - (11). 9

10. 10 Finite-difference scheme The equations (8) - (11) as finite differences We have used a constant time step - a special time interval

11. 11 Finite-difference scheme Let us to rewrite the equations (12) and (14) as: where

12. Finite-difference scheme From Fig. 1 and (13): Where From Fig. 1 and (15): Combine all these expressions:, 12

13. Finite-difference scheme Solving the equation (16) and (17) with the a R =a S, we get the pressure at P: To calculate the rate V P can be with any of the equations (16) and (17). This completely determines the state at all interior points of the pipeline. 13

14. Special time interval To save the convergence of these equations imply the satisfaction with the Courant conditions: These conditions imply that in Fig. 1 points R and S are located between points A and B. Notice: Note the use of linear interpolation of pressure and velocity of the liquid in the pipeline. To maintain accuracy in the calculation of nonlinear systems, values  and  must satisfy the Courant inequalities that involve interpolation only a small step of the grid. 14

15. Boundary conditions (samples) A known pressure at the inlet to the pipeline : A known pressure at the outlet of the pipeline : Dampener at the cut-off valve inlet of a single pipe line: Instantaneous pressure change (pump  P): 15

16. Block diagram for the calculation of hydrodynamic processes in the working fluid discharge pipe Read Date Go Initialize Increment Time Completed? Stop Apply Initial Boundary Condition Yes No Calculate Internal Section States Apply Terminal Boundary Condition Next Section or Device? No Yes Output results 16

17. Graphical user interface (GUI) 17

18. Graphical user interface (GUI) 18

19. Graphical user interface (GUI) 19

20. Graphical user interface (GUI) 20

21. Results of calculations For watered pipe with the length of 3000 m, and a carbon steel wall thickness of 9.525mm, and ID of 205 mm. Nominal pump head is 15 bars with a nominal input pressure of 1 bar and nominal flow velocity at the outlet of the pump is 1.5 m / sec (steady state flow). Mass flow rate was 27.8kg/s. At the end of the pipe right before the valve a dampener was mounted. The pressure and volume of the gas bladder were 20 bar and 1 litre, respectively. The cut-off valve starts to close at 5 seconds. The time of the cut-off valve completely closing was 1.3 s. 21

22. For watered pipe with the length of 3000 m, and a carbon steel wall thickness of 9.525mm, and ID of 205 mm. Nominal pump head is 15 bars with a nominal input pressure of 1 bar and nominal flow velocity at the outlet of the pump is 1.5 m / sec (steady state flow). Mass flow rate was 27.8kg/s. At the end of the pipe right before the valve a dampener was mounted. The pressure and volume of the gas bladder were 20 bar and 50 litre, respectively. The cut-off valve starts to close at 5 seconds. The time of the cut-off valve completely closing was 1.3 s. 22 Results of calculations

23. For watered pipe with the length of 3000 m, and a carbon steel wall thickness of 9.525mm, and 205 mm ID. Nominal pump head is 15 bars with a nominal input pressure of 1 bar and nominal flow velocity at the outlet of the pump is 1.5 m / sec (steady state flow). Mass flow rate was 27.8kg/s. At the end of the pipe right before the valve a dampener was mounted. The pressure and volume of the gas bladder were 20 bar and 250 litre, respectively. The cut-off valve starts to close at 5 seconds. The time of the cut-off valve completely closing was 1.3 s. 23 Results of calculations

24. Conclusions The developed program with the theoretical basis on the method showed above can help to model the water hammer fluctuations in the pipes and make it clear the dampener parameters which satisfy to the fluid flow smoothness in the pumped piping system requested by customer 24