lib_hydraulic_resistance

Hydraulic Resistance Library Rev 10 – November 2010 Copyright © LMS IMAGINE S.A. 1995-2010 AMESim® is the registered trademark of LMS IMAGINE S.A. AMESet® is the registered trademark of LMS IMAGINE S.A. AMERun® is the registered trademark of LMS IMAGINE S.A. AMECustom® is the registered trademark of LMS IMAGINE S.A. LMS Imagine.Lab is a registered trademark of LMS International N.V. LMS Virtual.Lab Motion is a registered trademark of LMS International N.V. ADAMS® is a registered United States trademark of MSC.Software Corporation. MATLAB and SIMULINK are registered trademarks of the Math Works, Inc. Modelica is a registered trademark of the Modelica Association. UNIX is a registered trademark in the United States and other countries exclusively licensed by X / Open Company Ltd. Python is a registered trademark of the Python Software Foundation. Windows is the registered trademark of the Microsoft Corporation. All other product names are trademarks or registered trademarks of their respective companies. TABLE OF CONTENTS 1. Introduction ............................................................................................................................1  2. Getting started with the Hydraulic Resistance Library .......................................................2  3. Pressures used in AMESim.................................................................................................11  3.1. AMESim® standard components 11  3.2. Hydraulic Resistance Library components 11  4. Modeling a network with Hydraulic Resistance components: important rules ..............13  5. Tutorial example...................................................................................................................14  6. Formulation of equations and underlying assumptions...................................................20  6.1. Basic equations 20  6.2. Further assumptions 21  7. Reynolds number.................................................................................................................22  8. Hydraulic Resistance submodels: classification ..............................................................24  8.1. Frictional drag category 24  8.2. Local resistance category 26  8.3. Frictional and local resistance category 27  8.4. Plain journal bearing category 27  8.4.1. Eccentricity calculation 28  8.4.2. Bearing submodels: Assumptions 29  8.4.3. Examples with bearings 30  9. Causality in the hydraulic resistance library .....................................................................35  10. The connectors: steady state and quasi-steady state problems ...................................36  References................................................................................................................................37  APPENDIX A .............................................................................................................................38  APPENDIX B .............................................................................................................................45  NNoovveemmbbeerr 22001100 TTaabbllee ooff CCoonntteennttss 1/1 Using the Hydraulic Resistance Library 1. Introduction Flow resistance has a strong influence on the design of fluid power circuits in which pressures are relatively low but flow rates are high. This is the motivation for creating the AMESim® Hydraulic Resistance Library. This library comprises a set of components from which it is easy to model large hydraulic networks, evaluate the pressure drops through the elements and, if required, modify the design of the system. Fluids are moved by a difference of pressure. Resistance to flow can be characterized by friction (regular pressure drops) and changes in stream direction or velocity (singular pressure drops). Pressure drops in each hydraulic resistance library component are evaluated based on the Idel'cik [1] formulae and experiment data. Before summarizing the particular characteristics of this library, we present a small tutorial example. Next rules and advice are given on how to take advantage of the possibilities offered by the hydraulic resistance library. An additional tutorial example is presented. This will introduce some concepts which will lead to a brief description of the background theory. Finally a classification of the submodels is given followed by some more general information. It is assumed that the reader is familiar with the use of AMESim®. If this is not the case, we suggest that you do the exercises in chapter 2 of the AMESim® manual before attempting the examples below. NNoovveemmbbeerr 22001100 UUssiinngg tthhee HHyyddrraauulliicc RReessiissttaannccee LLiibbrraarryy 1/45 2. Getting started with the Hydraulic Resistance Library Evaluating pressure drops in a network appears to be an easy task but studying the problem more deeply highlights several subtle points which need to be clarified. By modeling the system shown in Figure 1, we invite the user to discover several questions that will be answered in this manual. Hence we recommend you do this example yourself as a first contact with the hydraulic resistance library and to motivate the brief theoretical section that follows. The system below represents a "Venturi" tube. It consists of a convergent pipe, a neck and a diffuser and can be used to measure flow velocities and flow rates. convergent neck area1 area2 area3 =area1 diffuser flow Figure 1 To model this system, select the hydraulic resistance library category icon shown in Figure 2. Figure 2: Hydraulic resistance library category icon This will produce the popup shown in Figure 3. NNoovveemmbbeerr 22001100 UUssiinngg tthhee HHyyddrraauulliicc RReessiissttaannccee LLiibbrraarryy 2/45 Figure 3: components of the Hydraulic Resistance Library First look at all the components available in this library. Display the titles of each component by moving the pointer over the icons. When this is done, build the model of the "Venturi tube" as shown in Figure 4. Figure 4: model of a Venturi tube This model comprises nine elements from the hydraulic resistance library and various other AMESim® categories and each is referenced by a number. Fill in the parameters of these components as described in the table below leaving other parameters at their default values. Use the oil properties shown in Figure 5. Finally run a simulation with the default run parameters. NNoovveemmbbeerr 22001100 UUssiinngg tthhee HHyyddrraauulliicc RReessiissttaannccee LLiibbrraarryy 3/45 Submodel name and type Belongs to category Principal simulation parameters 1 QS00 flow source Hydraulic supplies a flow from 0 to 150 L/min in 10 sec 2 HRL00 hydraulic pipe (no friction) Hydraulic resistance diam = 30 mm length = 0.1 m 3 HR236 convergent pipe Hydraulic resistance large diam = 30 mm narrow diam = 10 mm length = 100 mm 4 HRPT3 hydraulic static pressure sensor Hydraulic resistance diam = 10 mm 5 HRL00 (neck) hydraulic pipe Hydraulic resistance diam = 10 mm length = 0.1 m 6 HR236 diffuser Hydraulic resistance large diam = 40 mm narrow diam = 10 mm length = 100 mm 7 HRPT3 hydraulic static pressure sensor Hydraulic resistance diam = 40 mm 8 JUN3M comparison junction Signals, controls and observers no parameters 9 TK00 hydraulic tank Hydraulic tank pressure = 3 bar Figure 5: fluid properties used NNoovveemmbbeerr 22001100 UUssiinngg tthhee HHyyddrraauulliicc RReessiissttaannccee LLiibbrraarryy 4/45 Most of the hydraulic resistance elements do not have state variables. We could describe them as steady state submodels or more precisely as instantaneous submodels. This means that they are assumed to react instantaneously to the pressures applied to them so that they are always in an equilibrium state. The exceptions are submodels such as HRL00. These are very commonly used in hydraulic resistance library systems and they will be described as connection elements. If you look at the variables available for plotting, you will discover that some pressures are described as static pressures and others as total pressures. These terms will be defined precisely in the theoretical section. At this stage we will simply say that the static pressure can be measured by a manometer and the total pressure by a Pitot tube inserted in the flow stream. If the fluid is in motion, total pressure is always greater than static pressure. The difference is due to the motion (kinetic energy) of the moving fluid and is called the dynamic pressure. total pressure = static pressure + dynamic pressure (1) The main point of interest in this example is to observe what happens between the neck and the outlet of the diffuser. If we compare the static pressure in the neck pstat_neck with the static pressure in the diffuser outlet pstat_diffuser we have diffuserstatneckstat pp __ < (2) In contrast, if we compare the corresponding total pressures ptot_neck and ptot_diffuser we have (3) diffusertotnecktot pp __ > What is happening is that the dynamic pressure (kinetic energy) in the neck is transformed into static pressure (potential energy) in the diffuser. Note that the comparison junction enables us to plot the difference between static pressures directly. The result of the simulation is given in Figure 6. Note that all the graphs are plotted according to the volumetric flow rate coming from the flow source submodel (QS00). The differential static pressure calculated between the diffuser and the neck is always negative. We say that there is a static pressure recovery between the neck and the diffuser. Every component which has an enlargement of cross-sectional area is subject to static pressure recovery under most conditions. The only case when this rule does not apply is when there is a large frictional component. Figure 6: comparison of static pressure in the neck / and static pressure at diffuser output NNoovveemmbbeerr 22001100 UUssiinngg tthhee HHyyddrraauulliicc RReessiissttaannccee LLiibbrraarryy 5/45 In each hydraulic resistance component, total pressures are calculated. If a static pressure is not available, it may be calculated from the total pressure by the hydraulic static pressure sensor in the hydraulic resistance library. In this tutorial example, we will compare the three types of pressure: total pressures, static pressures and dynamic pressures. First we will consider the evolution of total pressures with the flow rate. Bernoulli's formula, which will be studied in detail later in this document, implies that if the flow goes from a point A to a point B in a network: The total pressure at point A is always greater than the total pressure at point B. The total pressure drop between A and B is due to the loss of energy through the elements encountered from point A to point B, and each is characterized by a friction factor. Ptot conv > Ptot neck > Ptot Figure 7: evolution of total pressures against the volumetric flow rate In Figure 7, we see the way the three total pressures vary with the flow rate. At the end of the simulation, when the flow rate has reached 150 L/min, the total pressures at convergent pipe inlet, in the neck and at diffuser outlet are respectively, ptot_conv = 4.72 bar, ptot_neck = 4.03 bar and ptot_diff = 3 bar. The last pressure, of course, corresponds to the tank pressure. NNoovveemmbbeerr 22001100 UUssiinngg tthhee HHyyddrraauulliicc RReessiissttaannccee LLiibbrraarryy 6/45 Pstat conv > Pstat diff > Pstat Figure 8: evolution of static pressures against volumetric flow rate We will consider now the evolution of static pressures. If we have a flow going from point A to point B, it is not possible to conclude that the static pressure at point A is greater than the static pressure at point B. In Figure 8, we see how the three static pressures vary with the flow rate. At the end of the simulation, when the flow rate has reached 150 L/min, the static pressures at the convergent pipe inlet, in the neck and at diffuser outlet are respectively pstat_conv = 4.67 bar, pstat_neck = -0.02 bar and pstat_diff = 2.98 bar. Next we consider the evolution of the dynamic pressures in the Venturi tube. In Figure 9, we see how the three dynamic pressures vary with the flow. Remember that the dynamic pressure is obtained by subtracting the value of the static pressure from the value of the total pressure. In order to maintain mass conservation the fluid velocity is very much higher in the neck. It follows that the dynamic pressure is much higher at this point. At the end of the simulation, when the flow rate has reached 150 L/min, the dynamic pressures are respectively at convergent pipe inlet, in the neck and at diffuser outlet are respectively pdyn_conv = 0.05 bar, pdyn_neck = 4.05 bar and pdyn_diff = 0.02 bar. Pdyn neck > Pdyn diff > Pdyn Figure 9: evolution of dynamic pressures against volumetric flow rate NNoovveemmbbeerr 22001100 UUssiinngg tthhee HHyyddrraauulliicc RReessiissttaannccee LLiibbrraarryy 7/45 Now we can ask why the flow rates and pressures vary during the simulation? Clearly the flow rate source, which ramps from 0 to 150 L/min in 10 seconds, is a major cause of the variations. There is, however, another factor. You may have noticed that there are two state variables. These are the pressures in the two HRL00 submodels. If you look very carefully at figures 6, 7, 8 and 9 you will find there is some very fast transient behavior at the start of the simulation. This is because the starting value of the pressures was 0 bar whereas the pressure in the tank was 3 bar. Figure 10: transient behavior due to two pressure state variables If we run the simulation again over seconds with a suitable communication interval, we can see this behavior in detail as shown in Figure 10. Alternatively we can get rid of this behavior by using one of the two following methods: 310 − Either by setting the initial value of pressure in both HRL00 submodels to 3 bar. Or by running with the option Stabilizing run + Dynamic run + Minimum discontinuity handling. NNoovveemmbbeerr 22001100 UUssiinngg tthhee HHyyddrraauulliicc RReessiissttaannccee LLiibbrraarryy 8/45 Figure 11: system eigenvalues We can get some insight into the nature of the transient behavior resulting from the state variables by doing an eigenvalue analysis. The problem is non-linear and hence the eigenvalues can be expected to vary with time. However, we are only interested in representative values. Hence perform a single linear analysis at a time of 5 seconds and examine the eigenvalues. Figure 11 shows the results. We can say that there are two time constants involved each being approximately seconds. These are very fast transients. 510 − Quite clearly the dominant source of variations in the pressures and flow rates is the flow rate source term. Naturally if we had an extremely long length in HRL00 (e.g. 30 m) or very fast variations in the source term, the state variables would exert a strong influence. We find that it is very common in hydraulic resistance library applications that the eigenvalues of the system (which are due to the state variables) give rise to very small time constants and variations in pressure and flow rate are totally dominated by the source terms. NNoovveemmbbeerr 22001100 UUssiinngg tthhee HHyyddrraauulliicc RReessiissttaannccee LLiibbrraarryy 9/45 From this very basic example, the following questions are raised: - Which pressure should we work with? It turns out that there are many good reasons for working with total pressure in hydraulic resistance components. As was seen in this first example, if the flow goes from a point A to a point B we know that we always have a loss of total pressure whereas it is not possible to assert the same with static or dynamic pressures. (Remember the pressure recovery phenomenon.) However, it is often necessary to know the static pressure. Without the static pressure we would not know if cavitation was occurring. - Why are there special dynamic connection elements (HRL00 in this example) inserted between hydraulic resistance elements? - How do we know if the problem is quasi-steady state? These three questions are extremely important. They are answered in the following sections and the answers constitute the basis for the correct use of the hydraulic resistance library. Hence, the user is strongly advised to read the next section carefully. NNoovveemmbbeerr 22001100 UUssiinngg tthhee HHyyddrraauulliicc RReessiissttaannccee LLiibbrraarryy 10/45 3. Pressures used in AMESim 3.1. AMESim® standard components At this stage it is important to define formally the terms total pressure and dynamic pressure. Specialized books (such as Idel'cik [1] and Miller [2]) give a definition of the total pressure in terms of static pressure , density totp statp ρ and fluid velocity w as: dynstat 2 stattot ppw2 pp +=+= ρ (4) Total pressure is the sum of a static pressure (pstat) which derives from potential energy, and a dynamic pressure (pdyn) which derives from kinetic energy. The AMESim® standard component library used for the study of "Fluid Power Control" systems offers a set of submodels governed by classical assumptions adopted by the scientific community in this particular domain. One important assumption is that static pressures and total pressures involved are considered as being equal. This is perfectly acceptable because flow velocities are generally low in pipe systems. These velocities rarely exceed 4 m/s which with a density of 860 kg/m3 give a dynamic pressure less than 0.1 bar. The dynamic pressure is then neglected compared with the working pressures of the system which typically reaches tens and sometimes hundreds of bar. (In a few specialized applications, pressures of over 1000 bar are common). In other applications, for instance low pressure - high flow rate systems with complex geometry (comprising T-junctions, bends, sudden expansions, contractions, etc.), it becomes necessary to evaluate pressure drops in situations in which dynamic pressure is very significant. Some relevant examples include: - Design of the suction circuit of a pump in order to avoid cavitation; - Flow rate distribution in an engine lubrication circuit. In these particular cases it is necessary to maintain a clear distinction between static pressures and total pressures. 3.2. Hydraulic Resistance Library components In hydraulic resistance applications flow velocities can be much greater than 4 m/s. Velocities of 15 m/s are not uncommon which with a density of 860 kg/m3 gives a dynamic pressure of approximately 1 bar that cannot be neglected. This is particularly true when the system pressure is very low, the influence of the kinetic energy variations then dominates the calculation of the flow resistance. Consider the following example: NNoovveemmbbeerr 22001100 UUssiinngg tthhee HHyyddrraauulliicc RReessiissttaannccee LLiibbrraarryy 11/45 V1 V2 = 0Area1 We have a flow coming from a conduit of finite cross-sectional area entering a large (effectively infinite) tank volume. In the pipe the fluid has a non-zero velocity whereas in the tank the velocity is so small that it can be assumed to be zero. Between the pipe and the tank, all the kinetic energy (dynamic pressure) of the fluid is transformed into potential energy (static pressure). In the pipe, the dyna