modelling of hydraulic elements for design and …

MODELLING OF HYDRAULIC ELEMENTS FOR DESIGN AND SUPERVISION OF HYDRAULIC PROCESSES

M. Golob, B. Bratina, B. Tovornik, University of Maribor, FERI, Slovenia

Corresponding Author: M. Golob

University of Maribor, Faculty of Electrical engineering and Computer Science Smetanova ulica 17, 2000 Maribor, Slovenia

Phone: +386 2 220 7161, Fax: +386 2 251 1178 email: [email protected]

Abstract. In this paper a different modelling approach for modelling of hydraulic systems is presented. The modelling approach deals with how to obtain adequate mathematical models of most common hydraulic elements as are control valves, tanks, pumps and pipelines, which are often used in process industries. Partial models should be suitable for behaviour simulation of the complete hydraulic system, for simulation of hydraulic process control applications, for supervision of hydraulic processes, and for educating purposes. All partial models (model of pump, model of pipeline, model of tank, and model of control valve) are logically and properly connected together in Matlab/Simulink environment and form the mathematical model of desired system. The difference between classical and presented approach is explained where a non-linear and linearized models of all hydraulic elements tanks are derived. The hydraulic process is simulated and results show the potential of the presented modelling approach. 1. Introduction Computer modelling in education process plays an important role for student’s understanding of the process structure, behaviour, function or control. By use of simulation environment the abstractive view of mathematical equations can be reduced, hence the behaviour of the process is more transparent. The process model is based on mathematical analysis and is usually derived from the first principles (mass, momentum and/or energy balance equations) which can be according to the process complexity sometimes very difficult to obtain. More simple processes like parts of hydraulic systems, heat exchangers, shock absorbers, inverted pendulums, can be derived with minimal effort, and lots of such mathematical models can already be found in modelling literature [1], [2], [3] and [4]. Of course, due to the un-modelled dynamics of the process, estimated values of parameters, non-linearities in the process, the mathematical model of the process is actually only approximate function of the real process behaviour and has to be properly validated to assure, that the model behaviour is adequate to the real process behaviour. By using the adequate process model, the behaviour of the process can be studied in many simulation environments such as Matlab/Simulink, Omola, Modelica, Dymola, for purposes of control design, process parameter identification, process supervision and fault detection, etc. Process units in hydraulic systems are usually connected by pipelines where pressure or flow of the media is controlled by control valves or pumps. In most cases, when modelling hydraulic process, elements along a pipeline (valve, pump) are deliberately neglected, although mathematical models of these elements can be obtained from literature. These models of hydraulic processes are based upon mass balance equations and the non-linear model of the process is due to the short pipelines adequate to the real process. In case of long distances between tanks, the affect of elements along pipeline should be taken into consideration. In our case we built partial models of hydraulic elements and then formed mathematical model of a real processes. In the beginning a simple mathematical model of two connected tanks is presented, from which partial models of hydraulic process (pipeline, pump, valve) were derived. The model of the pipeline together with valves and pump is derived by using the momentum balance equations, and the model of the tank is derived by the mass balance equations. The emphasis is how to properly describe behaviour of the partial hydraulic elements, by defining their main parameters which contributes to the system behaviour. Simulation results are presented and evaluated in the end. 2. Modelling of hydraulic processes Process units in hydraulic systems are usually connected by pipelines where pressure or flow of the media is controlled by control valves or pumps. Information and knowledge about their behaviour and function in the process are very important for determining the main parameters which affects the behaviour of the whole

Proceedings 5th MATHMOD Vienna, February 2006 (I.Troch, F.Breitenecker, eds.)

Modelling and Simulation for Control System Design, Coordination and Supervision 2 - 1

process. By considering these main parameters of the elements and characteristics of the medium, many less influential parameters can be deliberately neglected (friction factor in a short pipeline, viscosity of the fluid, etc.) during modelling. In this paper only ideal fluids or “non-compressible” media are considered, meaning that the possibility of mass conservation in the pipelines is neglected and hydraulic shocks in the system are intentionally avoided. Modelling of the pipeline, valve and pump First let’s focus to the modelling of hydraulic elements which can be found along pipelines. Consider a hydraulic process (Fig. 1.) with a horizontally placed pipeline of length L and a cross-section A that connects two tanks. The mass flow in the pipeline can be changed by a centrifugal pump, which is controlled by n (a number of revolutions in a minute), or by a valve with corresponding state Av. To describe dynamic changes of mass flow in a pipeline, an adequate mathematical model can be obtained using the momentum balance equations.

Fig. 1. Pressure differences along a pipeline between two tanks

Global momentum equation is defined by the mass m and the mass velocity v:

ms

kgG m v ⎡ ⎤= ⋅ ⎢ ⎥⎣ ⎦ (1)

And equation of momentum preservation is:

∑=

=n

ii dt

dGF1

(2)

where Fi stands for the sum of all forces that affect to the total mass of the fluid in the pipeline. The time variant difference of hydrostatic pressures between the tank 1 (P1) and in tank 2 (P2) is equal to the pressure differences caused by the pump (δPpump), pipeline (δPpipe) and the valve (δPvalve). This is described by the next equation:

( )1 2pump valve pipedvP P P P P A L Adt

δ δ δ ρ+ − − − = ⋅ ⋅ ⋅ (3)

Mass flow in a pipeline is defined by:

kgs

mq A v ρ ⎡ ⎤= ⋅ ⋅ ⎢ ⎥⎣ ⎦, (4)

where A denotes the cross-section of the pipe, ρ denotes the specific density and v denotes the velocity of the medium. If velocity v is expressed and inserted into (3):

1 2

mpump valve pipe

dqLP P P P PA dt

δ δ δ+ − − − = ⋅ (5)



The obtained equation describes the mass flow changes in the pipeline as a consequence of pressure changes (P1, P2, δPpump, δPpipe, δPvalve). The problem lies in the determination of partial pressures along the pipeline, valves and pumps. A direct relationship between partial pressure changes δP and corresponding mass flow changes qm is obvious for a pipeline, but when it comes to valves and pumps these pressure changes depend also on the state of the valve Av and on the number of revolutions of the pump, n. In order to obtain complete dynamic mass flow behaviour, all relationships between mass flow and pressure differences in the system must be known. Pressure difference in front of and behind of the pump can be written as a function of mass flow and a number of revolutions in a minute, n:

2

N

m( , ) pump pump mP P q nδ δ ⎡ ⎤= ⎢ ⎥⎣ ⎦

(6)

Pressure difference in front of and behind of the valve can be written as a function of mass flow and a state of the valve Av:

2

N

m( , ) valve valve m VP P q Aδ δ ⎡ ⎤= ⎢ ⎥⎣ ⎦

(7)

Pressure difference along the pipeline can be written as a function of mass flow:

2

N

m( ) pipe pipe mP P qδ δ ⎡ ⎤= ⎢ ⎥⎣ ⎦

(8)

Pressure difference along particular hydraulic elements can be determined by the element’s characteristic function (Fig. 2. and Fig. 3.) and resistance to the mass flow can be determined. The mass flow along a pipeline is usually turbulent due to the many valves and pipes, therefore a relationship between pressure and outlet mass flow must be known. Pressure of the pipe δPpipe for a turbulent mass flow can be written as:

2

222

2

N

m

2 2m

pipe pipe mqL v LP K

d d Aδ λ ρ λ

ρq ⎡ ⎤= = = ⋅ ⎢ ⎥⋅ ⋅ ⎣ ⎦

(9)

where λ denotes friction coefficient, L length, d diameter and A cross-section of the pipeline, v velocity and ρ specific density of the fluid. For easier treatment of equation a coefficient Kpipe is obtained:

2

1

kg m

2pipeLK

d Aλ

ρ⋅ ⎡ ⎤= ⎢ ⎥⋅ ⋅ ⋅ ⎣ ⎦

(10)

As mentioned before, the parameters of the valve and the pump can be determined from the corresponding characteristics. Pump curves, presented on Fig. 2, relate flow rate and pressure (head) developed by the pump at different rotational speeds n. The centrifugal pump operation should conform to the pump curves supplied by the manufacturer.

Fig. 2. Characteristics of the pump obtained experimentally or from datasheets.



Fig. 3. Characteristics of the valve obtained experimentally or from datasheets.

By considering the determination of pressure difference written in (9), it is obvious that (3) has a non-linear behaviour. In order to analyze the system at stationary conditions with only small changes around the working point, a linearization is proposed. Eq. (3) is expressed in a differential form:

1 2

( )mpump valve pipe

d dqLdP d P d P d P dPA dt

δ δ δ+ − − − = (11)

and a partial linearized model of the pump can be derived:

pump pumppump m pump m pump

m

P Pd P dq dn R dq b dnq n

δ δδ ∂ ∂= + = ⋅ +

∂ ∂⋅ (12)

where main pump parameters are:

pumppump

m

PR

qδ∂

=∂

, (13)

pump

pump

Pb

nδ∂

=∂

. (14)

Rpump denotes the resistance of the pump (gradient of the pressure when mass flow is changed) and bpump denotes the relationship between mass flow and number of revolutions of the pump. Since they are both defined in a linearized form from pump characteristic presented on Fig. 2, they are accurate only in the close vicinity of the working point of the pump. From valve characteristic presented on Fig. 3 also linearized model of the valve can be derived:

valve valvevalve m V valve m valve V

m V

d P d Pd P dq dA R dq b dAq A

δ ∂ ∂= + = +

∂ ∂ (15)

Main parameters are:

valvevalve

m

d PRq

∂=

∂ (16)

valve

valveV

d PbA

∂=

∂ (17)



where Rvalve denotes the resistance of the valve (gradient of the pressure when mass flow is changed) and bvalve denotes the gradient of the pressure when the state of the valve is changed. Since they are both in a linearized model they are accurate only in the close vicinity of the working point. And a linearized model of the pipeline can be derived:

pipepipe m

m

Pd P dqq

δδ ∂=

∂ (18)

Main parameter of the pipeline is:

pipepipe

m

PRq

δ∂=

∂ (19)

where Rpipe denotes the resistance along the pipeline and is accurate only in the close vicinity of the working point. From (9) the Rpipe is

2pipe pipe mR K q= . (20) Still, some knowledge and experience must be considered while obtaining these parameters, especially for the pump, since the hydraulic power of the pump is reduced when the mass flow is increased, which gives the resistance Rpump a negative sign. Similar goes for the parameter bvalve of the valve. By considering small differences ∆ near the set point the determination of pressure differences are

pump pump m pumpP R q bδ = − + n (21)

valve valve m valve VP R q b Aδ = − (22)

pipe pipe mP R qδ = (23) If (21), (22) and (23) are inserted into (11) we get:

1( )m

valve pipe pump m pump valve Vd qL R R R q P b n b A 2P

A dt+ + + = + + − (24)

The obtained equation consists of all main parameters along the pipeline. Expression L/A is named as the inertia coefficient I of the pipeline system. If the same model is used, where mass flow of the fluid is running along a pipeline of length L and cross-section A, then the total mass m of the fluid with specific density ρ in the pipeline is described by:

[ ]kg m A L ρ= ⋅ ⋅ (25) Using (25) and (4), inserted into (2) gives:

( ) m mdq dqd m v dv mP A m Ldt dt A dt dt

δρ

⋅⋅ = = = = (26)

And by rearranging:

mdqLPA dtI

δ = (27)

where I denotes the inertia coefficient of the pipeline. To obtain a dimension of time (time constant of the system) the inertia coefficient I is divided by resistance R:

m

m

L qI L LA v m vT GR AR AR q A P A P F

ρδ δ

= = = = = = (28)



And for our case:

1 21 1 pump valvem

m V

b bd qT q P P ndt R R R R

+ = − + + A (29)

T stands for the time constant and R stands for the sum of all partial resistances in the system. From (24) it is obvious that the small mass flow changes of the non-compressible media in the pipeline will result as a first order difference equation response. In the end all the partial pressures and resistances are derived, and by introducing a Laplace transform an adequate model of pipeline system is:

1 2

1

( )m

pump valve Vvalve pipe pump

qLP b n b A P s R R RA

=+ + − + + +

(30)

Modelling of the tank Still, the obtained mathematical model describes only the behaviour of the pipeline system without any fluid storage in the system (capacity). To derive the model of tanks in the system, the mass balance equation is used. The ideal tank is depicted in Fig. 4.

Fig. 4. The parameters of the tank

The mass balance in the tank is defined as:

inkg

s

outin mmdh dVq q Sdt dt

ρ ρ ⎡ ⎤− = ⋅ ⋅ = ⋅ ⎢ ⎥⎣ ⎦ (31)

where S denotes the cross-section of the tank, h denotes the level and ρ specific density of the fluid in the tank. The mass balance equation could be replaced by the volume balance equation, since the specific density ρ and the total volume V of the fluid are constant. Still one should consider the viscosity coefficient of the fluid since it affects the dynamic properties of the system. The outflow velocity v can be defined by Bernoulli equation, which must meet the Torricelli’s condition (outflow cross-section must be very small in comparison to the cross-section of the tank):

ms

2 v g h ⎡ ⎤= ⋅ ⋅ ⎢ ⎥⎣ ⎦ (32)

g denotes the gravity constant and h denotes the level of the fluid in the tank. The outlet mass flow is then:

kg

s2

out v imq K A ghρ ⎡ ⎤= ⋅ ⋅ ⋅ ⎢ ⎥⎣ ⎦ (33)



where Kv stands for the valve coefficient and is usually given by the manufacturer (datasheets). The mass flow which runs thru the valve also depends on the cross-section of the valve Ai and on the level of the fluid in the tank h. By combining equation (31) and (33) a non-linear first order equation is obtained:

inkgs

2 v i mdhS K A g h qdt

ρ ρ ⎡ ⎤⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ = ⎢ ⎥⎣ ⎦ (34)

Linearization of the non-linear outflow function (33) is proposed, where main mass flow changes depends on the change of the valve cross-section A and on the change of the level h in the tank:

out outiout

i

m mm

q qd q dA dh

A h∂ ∂

= +∂ ∂

(35)

By linearization and knowing the valve coefficient Kv, the next expressions can be written:

out outA

ii

m mq qK

A A∂

= =∂

(36)

1

2out outm mq q

h Rh∂

= =∂ ⋅

(37)

2out out

iouti

m mm

q qq A

A hh= ⋅ + ⋅ (38)

Final equation gives a linearized model of the tank:

21

out outiin

i

m mm

q qd hS h qdt h AC

R

ρ⋅ ⋅ + = − ⋅ A (39)

where R stands for resistance and is defined as a relation between level (pressure) and outlet mass flow of the tank, and C stands for capacity or mass of the fluid in the tank. Introducing resistance R and capacity C in to (39) the next equation is obtained:

in A i (40) m

d hT h R q R K Adt

⋅ + = ⋅ − ⋅ ⋅

Further analysis brings us to the interesting conclusion regarding the time constant of the system:

[ ]02 2 2 out outm m

sS h mT C R Tq q

ρ⋅ ⋅= ⋅ = = = (41)

where the time constant T0 gives the relationship between the total fluid mass and the mass flow in the tank, which also can be written as:

[ ]0out

s m

m total massTq mass flow

= = (42)

By rearranging and using Laplace transform on (40):

in1( ) ( ) ( ) ( )A

imK RRh s h s q s A s

T s T s T s⋅

= − ⋅ + ⋅ − ⋅⋅ ⋅ ⋅

(43)



3. Simulation of the two-tank system The goal was to be able to decouple a hydraulic process in a way that each element has its own mathematical model. For the proposed model of two tanks connected by a pump, control valve and pipelines, depicted in Fig. 1, decoupling can be done, where the mass flow and the pressure differences along the system have the main role. Obtained partial models of the tank, pump, valve and pipeline are properly connected where an output pressure of the first element is calculated on the basis of the elements input pressure and mass flow in the system. The calculated output pressure presents an input pressure to the next element (valve, pump, pipeline with different cross-section or length, etc.), where a corresponding output pressure is calculated, etc… In each partial model a pressure drop on the element can be extracted by the model’s resistance (Rvalve, Rpump, Rpipe) and pressure differences in front of and behind of the element. Instead of the output pressure also the output mass flow can be expressed and the complete model could also be decoupled by mass flow inputs and outputs. According to the process knowledge and needs the complete model of the process can consist of both decoupling approaches (by mass flow output or by pressure output) which contributes to better connectivity of the elements. The mass flow in the system is used to determine the resistance (pressure drop) of the hydraulic elements, and has always a positive value. To derive the partial models of the process an equation (30) is used, where appropriate decoupling method and basic resistances and pressures of the hydraulic element are chosen. For the above case, the model of the pipeline is decoupled by pressure differences in the tank, where the hydraulic resistance of the element is defined according to the mass flow. The resistance has a negative influence on the mass flow, the same as if the outlet of the pipeline would be placed higher then the inlet (Ph).

( ) ( ) ( ) ( )2 1 2 hm c m mLP s P s s q s K q q s PA

= − ⋅ − ⋅ ⋅ − (44)

Pressure P1 and P2 are the hydrostatic pressures at the bottom of the tanks, respectively. If instead of pressure input and output difference of the model the mass flow is required, then the next equation can be used:

( )( ) ( )1 2

1m

hpipe

q sLP s P s P s RA

=− − +

(45)

where model of pipeline calculates outlet mass flow on the basis of pressure difference between last partial element output and the second tank output. Of course, the outlet mass flow depends also on resistance of the pipeline Rpipe, its length L and cross-section of the pipe A. Ph stands for the pipeline’s vertical offset from the bottom level of the tank and affects the pressure difference. Model of the pump was obtained by (21), where non-linear behaviour of parameters Rpump and bpump must be known. By increasing the number of the pump’s revolutions in a minute, the mass flow is increasing and the output pressure is rising. Unfortunately it also has a negative affect because by increasing mass flow the hydraulic power of the pump is decreasing.

( ) ( ) ( ) ( )2 11

1pump pump m

pump

P s P s b n s R q ss T

= + ⋅ ⋅ − ⋅+ ⋅

(46)

According to the identification process we can describe the dynamic behaviour of the pump composition: frequency drive - asynchronous motor - centrifugal pump, can be approximated by a first-order transfer function with Tpump = 0.8 s. Model of the valve is obtained by knowing the behaviour of non-linear parameters Rvalve and bvalve, where state of the valve affects to the resistance and the output mass flow. The model of the valve is presented by next equation:

( ) ( ) ( ) ( )2 11

1valve V valve mvalve

P s P s b A s R q ss T

= + ⋅ ⋅ − ⋅+ ⋅

(47)

The first-order dynamic behavior of the control valve is expected and approximated by a first-order transfer function with Tvalve = 2 s. As mentioned before, non-linear parameters of the pump bpump, Rpump or valve bvalve, Rvalve can be obtained experimentally by measurement or by characteristics provided by manufacturer.



In Fig. 5 the complete model of the two-tank process in Matlab/Simulink environment is presented, where output pressures of the hydraulic elements are also monitored. By subtraction of input and output pressure of the model a pressure drop or resistance of the element on the particular element could be obtained. By simulating the complete model in Matlab/Simulink, the model was tested and compared to classical modelling approach, where it was shown that the model behaved as it should. The mass flow from one tank to another was generated by the input and output pressure differences caused by the fluid level in the tank, respectively. An initial water level h1 in the tank 1 was set up to 8 meters and the level h2 in the tank 2 was set up to 5 meters. The pump was not running at the start of the simulation but it was used only as hydraulic resistance to the mass flow. The state of the valve was set up to the initail value of 20% open. Non-linear parameters bpump, Rpump, bvalve, Rvalve of the valve and the pump were selected with two-dimensional lookup tables, which were defined experimentally from the measurements on the real hydraulic process.

Fig. 5. The two-tank model formed by partial models in Matlab/Simulink (partial pressures [Pa] at time t=450 s) Fig. 6. shows the results of the simulation. Due to the level difference between tanks (hydrostatic pressure difference), a mass flow from first to the second tank was generated. The mass flow depended mostly from the state of the valve (openness of the valve, Av = 20%). At time t = 50 seconds, the valve was fully opened (Av = 100 %), which consequently increased the mass flow qm to 0.5 kg/s and hence to a faster equalization of both tank levels. As the pressure difference between tanks was equalized, the mass flow fell to 0. At time t = 400 seconds a pump was started with a nominal revolutions per minute n=2820 rpm, reached by the asynchronous motor at frequency f = 50 Hz of the frequency drive. The mass flow qm increased to the value of 1.35 kg/s and the level in the second tank was adequately rising. At time t = 500 seconds, a valve was partially closed from Av=100 % to Av=60 %, which means that the hydraulic dumping in the system increased and consequently the mass flow qm fell to 1.4 kg/s. Therefore the rising level speed in the second tank was reduced until the level h2 reached final value h2 = 13 m. 4. Conclusion The derivation of partial hydraulic models for modelling of hydraulic systems on the basis of mass and momentum balance equations was presented. By considering the main hydraulic parameters that affect the mass flow along the pipeline or the particular hydraulic element of the system, partial models of the process were obtained (model of pipeline, valve, pump). The obtained partial hydraulic models have to be properly logically connected together so various hydraulic process models can be formed. The behaviour of the two-tank process



was simulated and tested in Matlab/Simulink environment, where dynamic properties and the behaviour of the system in comparison to the classic modelling approach is better defined. Still, some limitations of the presented approach exists, for instance the mass flow along the system can be mathematically positive or negative and the model (input and output pressure differences) should be properly adapted, though the partial resistances in the system are always positive regardless to the mass flow direction. Other mechanical characteristics of the materials and system structure should also be considered (viscosity, pipeline angles) in order to improve the mathematical models of the partial elements and therefore the complete model of the process ass well. By modular structure of hydraulic processes one can form various process models and study their behaviour or the behaviour of its’ partial elements (the process of pipeline, valve, pump and/or tank). The presented modelling approach is also appropriate for modelling and supervision of big scale water plant systems such as water distribution systems and remote heating systems.

0 100 200 300 400 500 600 700 8000

20

40

60

80

100

120Reference values of the frequency controlled pump and control valve openness

time [s]

f[Hz]

, Av

[%]

0 100 200 300 400 500 600 700 8000

2

4

6

8

10

12

14Tank 1 level and tank 2 level

time [s]

H1,

H2

[m]

0 100 200 300 400 500 600 700 800

0

0.5

1

1.5

Mass flow

time [s]

Q [

kg/s

]

Fig. 6. Time responses (tank levels and mass flow between two tanks) of the hydraulic system to the pump and

valve reference values. References [1] J.L. Meriam, L.G. Kraige, Engineering Mechanics –Volume 2: Dynamics, Fourth edition, John Willey & Sons, New York, 1998. [2] Y. A. Cengel, R. H. Turner, Fundamentals of Thermal-Fluid Sciences, McGraw-Hill International Edition, Boston, 2001. [3] D. E. Seborg, T. F. Edgar, P. A. Mellichamp, Process Dynamics and Control, John Willey & Sons, New York, 1989. [4] V. Kecman, Dinamika procesa (Processes Dynamics), Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Zagreb, 1990.



modelling of hydraulic elements for design and …

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