computational fluid dynamics authors: a. ghavrish, ass. prof. of nmu, m. packer, president of ldi...

24
Computational fluid dynamics Authors: A. Ghavrish, Ass. Prof. of NMU, M. Packer, President of LDI inc.

Upload: liliana-johnson

Post on 05-Jan-2016

218 views

Category:

Documents


3 download

TRANSCRIPT

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

Computational fluid dynamics

Authors: A. Ghavrish, Ass. Prof. of NMU, M. Packer, President of LDI inc.

Page 2: Computational fluid dynamics Authors: A. Ghavrish, Ass. Prof. of NMU, M. Packer, President of LDI inc

Analytical model

2

Page 3: Computational fluid dynamics Authors: A. Ghavrish, Ass. Prof. of NMU, M. Packer, President of LDI inc

Method of calculatingThe 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

Page 4: Computational fluid dynamics Authors: A. Ghavrish, Ass. Prof. of NMU, M. Packer, President of LDI inc

Method of characteristicsThe 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

Page 5: Computational fluid dynamics Authors: A. Ghavrish, Ass. Prof. of NMU, M. Packer, President of LDI inc

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

Page 6: Computational fluid dynamics Authors: A. Ghavrish, Ass. Prof. of NMU, M. Packer, President of LDI inc

Equations of motion

If

Then equation (3) becomes an ordinary differential equation:

6

Page 7: Computational fluid dynamics Authors: A. Ghavrish, Ass. Prof. of NMU, M. Packer, President of LDI inc

Equations of motionSolving (5), we obtain:

Substituting equation (7) (6), we obtain a system of total differential equations:

7

Page 8: Computational fluid dynamics Authors: A. Ghavrish, Ass. Prof. of NMU, M. Packer, President of LDI inc

Finite-difference scheme

x

Pt+t

t

SCR B x

For solving the nonlinear equations (8) - (11) a finite difference method was used.

Fig. 1.A

8

Page 9: Computational fluid dynamics Authors: A. Ghavrish, Ass. Prof. of NMU, M. Packer, President of LDI inc

Finite-difference schemeSpatial - 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

Page 10: Computational fluid dynamics Authors: A. Ghavrish, Ass. Prof. of NMU, M. Packer, President of LDI inc

10

Finite-difference schemeThe equations (8) - (11) as finite differences

We have used a constant time step - a special time interval

Page 11: Computational fluid dynamics Authors: A. Ghavrish, Ass. Prof. of NMU, M. Packer, President of LDI inc

11

Finite-difference schemeLet us to rewrite the equations (12) and (14)

as:

where

Page 12: Computational fluid dynamics Authors: A. Ghavrish, Ass. Prof. of NMU, M. Packer, President of LDI inc

Finite-difference schemeFrom Fig. 1 and (13):

WhereFrom Fig. 1 and (15):

Combine all these expressions:

,

12

Page 13: Computational fluid dynamics Authors: A. Ghavrish, Ass. Prof. of NMU, M. Packer, President of LDI inc

Finite-difference schemeSolving the equation (16) and (17) with the aR=aS, we get the pressure at P:

To calculate the rate VP can be with any of the equations (16) and (17). This completely determines the state at all interior points of the pipeline.

13

Page 14: Computational fluid dynamics Authors: A. Ghavrish, Ass. Prof. of NMU, M. Packer, President of LDI inc

Special time intervalTo 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

Page 15: Computational fluid dynamics Authors: A. Ghavrish, Ass. Prof. of NMU, M. Packer, President of LDI inc

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

Page 16: Computational fluid dynamics Authors: A. Ghavrish, Ass. Prof. of NMU, M. Packer, President of LDI inc

Block diagram for the calculation of hydrodynamic processes in the working fluid discharge pipe

Read DateRead Date

Go

InitializeInitialize

Increment Time

Increment Time

Completed?

Stop

Apply Initial Boundary Condition

Apply Initial Boundary Condition

Yes

No

Calculate Internal Section States

Calculate Internal Section States

Apply Terminal Boundary Condition

Apply Terminal Boundary Condition

Next Section or Device?

No

Yes

Output results

16

Page 17: Computational fluid dynamics Authors: A. Ghavrish, Ass. Prof. of NMU, M. Packer, President of LDI inc

Graphical user interface (GUI)

17

Page 18: Computational fluid dynamics Authors: A. Ghavrish, Ass. Prof. of NMU, M. Packer, President of LDI inc

18

Page 19: Computational fluid dynamics Authors: A. Ghavrish, Ass. Prof. of NMU, M. Packer, President of LDI inc

19

Page 20: Computational fluid dynamics Authors: A. Ghavrish, Ass. Prof. of NMU, M. Packer, President of LDI inc

20

Page 21: Computational fluid dynamics Authors: A. Ghavrish, Ass. Prof. of NMU, M. Packer, President of LDI inc

Results of calculationsFor 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

Page 22: Computational fluid dynamics Authors: A. Ghavrish, Ass. Prof. of NMU, M. Packer, President of LDI inc

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

Page 23: Computational fluid dynamics Authors: A. Ghavrish, Ass. Prof. of NMU, M. Packer, President of LDI inc

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

Page 24: Computational fluid dynamics Authors: A. Ghavrish, Ass. Prof. of NMU, M. Packer, President of LDI inc

ConclusionsThe 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