Partial Differential Equations - Example: solution of the hyperbolic equation. Dynamics of pressures in the pipeline with a liquid
Example: solution of the hyperbolic equation
Dynamics of pressures in the pipeline with a liquid
Let's consider dynamic processes in the pipeline with a liquid with an input variable flow Q (t) and an output throttle regulation
f (t) (Fig.5).
Fig. 5. The simplified diagram of a pipeline with a liquid
After transition in the equations (19) to finite differences we’ll set in a point х0 the following boundary conditions for
a speed uok :
at linear variation of flow
at pulsation of flow (23)
Boundary conditions for a speed unk in a point хn look like:
where μ – flow coefficient; f ( t ) – the variable area of through passage section of a throttle (Fig.5),
similar dependence (22); р* – constant pressure behind a throttle.
Let's make the numerical analysis of a liquid dynamics in the pipeline at external influences (22) – (24).
Fig. 6. Dynamics of pressure in input in pipeline at volumetric regulation under the linear law
On Fig.6 the graph of change in time of pressure in input to pipeline in length L = 3 m at change of liquid flow in input under the linear
law (22), characteristic for volumetric regulation (for example, pump flow) is presented.
At flow variation from maximal U up to zero in time τ = 0.1 s fluctuations of pressure on the first resonant frequency
c/4L (in this case ~85 Hz) take place. At a stopping delivery of a liquid during the moment of time t = 0.1 s the
amplitude of fluctuations of pressure sharply increases because of jump of an inertial pressure duok /dt.
Fig. 7. Dynamics of pressures in input of pipelines at pulsing input flow
On Fig.7 dynamic processes of pressure in input of pipelines in length L = 1, 5 and 15 m are presented. As external influence a pulsing
input flow of a liquid Q (t) was given (23), simulating non-uniformity of pump flow with frequency ω ≈ 300 Hz and relative
amplitude а = 0.03, close to corresponding parameters of non-uniformity of flow of seven-cylinder axial-piston hydraulic machines. The
chosen frequency of influence exceeded the first resonant frequencies of pipelines c/4L , equal accordingly to 255, 51 and 17 Hz.
For pipelines in length of 5 and 15 m amplitudes of pressures were equal to 0.2 – 0.4 MPa, and for the pipeline in length of 1 m – approximately
in 6 times above as frequency of influence is close to its resonant frequency. In long pipelines in time moments of reflection of direct and
return waves occurring with time interval Т = 2L/c, failures of stationary fluctuations of pressure, accompanied by change of
amplitude take place.
At calculations of dynamics of the pressure arising at throttle regulation (24) the following values varied: the volumetric module of a liquid
elasticity, the pipeline length, the module of a pipe material elasticity, the throttle overlapping time.
It is known, that maximal pressure peaks in the pipeline take place in the event that time of a throttle overlapping does not more than time
of double run of a wave
τ < 2L/c. (25)
The condition (25) was considered as criterion at definition of a range of variation of parameters which numerical values for various versions of
calculation are presented in the table.
Here L – the pipeline length; Е1 – the volumetric module of an operating liquid elasticity; Е2
– the module of a pipe wall material elasticity; f – the minimal area of the throttle through passage section; τ – time of
a throttle overlapping.
| Version || L, m || Е1, MPa || Е2, MPa || f, cm2 || τ, s
| 1 || 1 || 980 || 1.96•105 || 1.0 || 0.1
|| 2 || 5 || 980 || 1.96•105 || 1.0 || 0.1
| 3 || 15 || 980 || 1.96•105 || 1.0 || 0.1
| 4 || 15 || 490 || 1.96•105 || 1.5 || 0.03
|| 5 || 15 || 245 || 1.96•105 || 1.5 || 0.1
| 6 || 15 || 245 || 1.96•105 || 1.5 || 0.03
|| 7 || 15 || 980 || 9.8•103 || 1.5 || 0.03
The fixed values are: the pipe internal diameter d = 30 mm; the liquid density ρ = 901.6 kg/m3; the liquid
kinematic viscosity ν = 30 mm2/s; a flow coefficient μ = 0.62; the pipe wall thickness δ = 3 mm;
F = 5 cm2 – the maximal area of the throttle through passage section; an integration step Δх = 0.1 m; the given
accuracy of an iterative process solution ε = 0.01.
In first three versions of calculation of the flow dynamics connected with wave processes, practically it was not observed, as τ
>> 2L/c (for example, for a version 3 we have 2L/c = 0.03 s < τ = 0.1 s).
Fig. 8. Dynamics of pressures on an input of pipelines at throttle regulation of an output flow by variation of the throttle
passage section area under the linear dependence
On Fig.8 graphs of variation of pressure on an input of pipelines for the specified three versions of mathematical model (18) are presented at
throttle regulation of an output flow (24) by variation of the throttle through passage section area f ( t ) according to the linear
Let's consider distribution of pressures on length of the pipeline for versions 4–7 (see tab.). As the pipeline length equal to 15 m, can be
considered as limiting one for some hydraulic systems, its further increase is inexpedient, therefore for reception of dynamics of a flow due to
wave processes, that is for fulfillment of the condition (25), versions 4–6 with the lowered values of the volumetric module of an operating liquid
elasticity Е1 (490 and 245 MPa) have been considered at overlapping of area of the throttle through passage section from
F = 5 cm2 up to f =1.5 cm2 in time τ = 0.03 s (but τ = 0.1 s in a version 5 ). Also the
version 7 with former value of the volumetric module of an operating liquid elasticity (Е1 = 980 MPa) and the lowered value
of the module of a pipe wall material elasticity (Е2 = 9.8•103 MPa), characteristic for sleeves of high pressure
has been considered.
Fig. 9. Dynamics of wave processes in pipelines with a liquid
For versions 4, 6 and 7 the condition (25) is carried out, so wave processes therefore should be shown, as has been received as a result of
calculation (Fig.9а,b). Values Е1 and Е2 for versions 4 and 7 give identical values of speed of
the wave propagation, therefore the received processes are completely identical (Fig.9b). The version 5 is characteristic slow overlapping
of a throttle at which the condition (25) is not carried out; therefore wave processes practically are not shown (Fig.9c). It's necessary
to note, that even at occurrence of wave processes the last have slightly the expressed character and quickly damp. It has allowed to come to the
conclusion in due time, that at the dynamic analysis of the hydraulic systems characterized by rather short lengths of pipelines and rather small
speed of the directional control valves (τ ≥ 0.1s), for the description of dynamics of pipelines with a liquid at throttle regulation the
model with the concentrated parameters without taking into account inertial component of an operating liquid flow is comprehensible.
Varying various boundary conditions (i.e. a kind of external influences in input and output), geometry and physical parameters of pipelines and
working liquid, it is possible to spend analysis of dynamic processes arising in pipelines of real hydraulic systems for a design stage.