Dynamic-weighing liquid flow calibration system - Realization of a model-based concept

Transcription

1 FLOMEKO 007, Johannesburg, South Africa, September 8 -, 007 Dynamic-weighing liquid flow calibration system Jesús Aguilera, Rainer Engel, Gudrun Wendt jesus.aguilera@ptb.de, rainer.engel@ptb.de, gudrun.wendt@ptb.de PTB Physikalisch-Technische Bundesanstalt Braunschweig, Germany Abstract The aim of this paper is to present the first experimental results of a new principle to measure liquid flow by applying a dynamic-weighing approach, which is being implemented in the PTB Hydrodynamic Test Field []. This new measurement principle relies on a physical analogous model comprising fluid, mechanical and electromechanical elements. They are basic elements representing the model-based approach [3], where appropriate real-time processing and synchronized data acquisition play particularly an important role to reproduce the instantaneous liquid flow rate. Concerning fluid-structure interaction, an exploratory research was conducted to identify the type of forces present in the process, their occurrence and effect upon the system elements to determine the instantaneous flow rate. Some of these system state variables turn out to be measurable and some others, due to their nature, are just assumed as disturbances. For modeling purposes such forces and elements are only possible to be analyzed in terms of their basic form as: sources of excitation, solid and fluid oscillators. First steps are also made to calibrate the output response of the system model in reference to the real system output signal, with the objective to validate the theoretical measurement principle, and to find out the optimum parameter estimation that meets the real process measurand within an acceptable degree of accuracy. Introduction - Needs for dynamic-weighing liquid flow measurements At present, the majority of primary standards for liquid flow measurements are based upon the static-weighing principle with flying or standing start-and-finish method []. Two types of dynamic weighing calibrators are in use: the mechanical (lever) balance with electronic force-metering devices, and the weighing tank with an immersed inlet pipe [3,4]. These systems enjoy acceptance and application, but reveal some disadvantages. For instance, the timing error due to inertial effects and lever deflection for the mechanical balance. For the case of the immersed inlet pipe design, fluid-induced vibration is generated by the immersed pipe, disturbing the collected water and therefore, biasing the balance readout. On the other hand, there is a rapid growth of demands from the industry for highly time-resolved, accurate process controlling. Especially chemistry, pharmacy and food production as well as energy and water supply that require real-time flow rate measurements to ensure efficient production processes and a high quality of their products. Therefore, the current activities of the liquid flow group at PTB are concentrated on investigations to realize and disseminate the instantaneous flow rate as a differential variable quantity in real time. Fluid-structural phenomena acting upon the system Before dealing with dynamic-weighing measurements of liquid flow, it is necessary to know what occurs in the process. One of the major difficulties in order to find an acceptable procedure for a dynamic characterization of a meter under test (MUT) by means of using the static weighing system is the

2 treatment of fluid-structure interaction. Once, these concepts are reasonably understood and included in the process, it is possible to implement a technique that determines the dynamic forces acting on the weighing system, and thus determining the liquid flowrate. The fluid-structure interaction fundamentals presented in this document describe the physics prior and during the collection time in order to see how the measuring system does response to dynamic and hydrodynamic forces at different water level magnitudes.. Free water jet As for the free water jet, the variables directly involved in its development are the acceleration of gravity and the instantaneous velocity component u r that is a function of the height h between i the nozzle outlet and the tank base (or water surface during the filling) as seen in Fig.. Fig. a) Free water jet and moment of momentum, b) Water jet impacting the tank base. Moment of the water jet momentum During the first stage of filling, the water jet flows at an angle α to the normal where a few moments later, it impacts the tank base, generating a stagnation force (Eq. ) that divides the jet into streams. These streams cause in addition, radial forces that act upon the tank walls. Fig. a shows the free water jet of thickness b jet impacting on O, where radial streams with thicknesses b and b.flow due to the stagnation. r r F = ρ u A cos α () y w i jet ( ) r Here F y is the magnitude of the normal force exerted on the tank base by the water stream and A jet is the crossed section of the water jet. Furthermore, a moment about O (see Fig. and Eq. ) is caused by the vertical component of the impacting force due to the fact that C is really the center of pressure that concentrates the normal force F r (refer to [5]). y r r r M = s ρ u A cos α () z oc w i jet ( ) What these statements prove is that not only a vertical component of force is involved in the process but also radial forces (Eq. 6) when the water jet is stagnated, in addition to a mechanical moment due to the eccentricity of the normal fluid force acting upon the tank base. /4

3 .3 Submerged water jet and penetration depth Perhaps the most important characteristic of any submerged water jet is that the fluid momentum (Eq. 3) carried through the collected water downstream is constant. This is true as long as we neglect a very small portion of flow entrainment from the surrounding water [6]. r r r P=ρ u h = ρ u h (3) w w, w w, The general behavior of the submerged turbulent water jet propagation inside the weighing tank is depicted in Fig.. Fig. Submerged water jet spreading inside a partially-filled tank The figure shows half of the longitudinal section of the tank with a diameter, wherein the jet is being discharged with an initial thickness of l at the water surface, and an initial velocity u r. As 0 the jet front goes deeper from the water surface, the axial velocity u r m drops, and at the depth MM, the jet begins to turn due to opposite buoyancy and viscous forces from the finite water volume, in addition to rigid bounds. These conditions create the backflow and limit the penetration depth of the stream [7]. The goal of this explanation is to show the importance about the maximum water-jet penetration depth in order to quantify the quasi-static portion of water volume below the point where the jet turns (Fig. 3b and c) The presence of a nearly steady fluid is a clear condition of effective energy dissipation, whereby it helps the weighing system to increase its critical damping coefficient, C c, and also gives the possibility to analyze the fluid as a mass element during system modeling (Fig. 3c)..4 Impact energy dissipation in weighing tanks The turbulent energy exchange and dissipation for the case of collection tanks occurs mainly in the region where the jet axis and water surface intersect. This is called the shear region.(see Fig. 3a). In this region, the jet is concentrated in a small water volume, so it is unable to loose momentum quickly. After the water jet impacts, the remaining impact energy is then dissipated through the impact region as well as the mixing region [8]. This statement is a useful design parameter for system modeling because it shows that most of the water jet force, represented by impact energy, strikes at the height of the water surface and it does change constantly upwards during the filling process. d T 3/4

4 Fig. 3 a) Regions of energy dissipation in a weighing tank b) High water jet penetration with non-steady conditions in the water volume c) Low water jet penetration with quasi-steady conditions in the water volume, to be analyzed as a damped fluid element (Photos taken from reference [9]) Fig. 4 Schematic of hydrodynamic forces exerted on the weighing system by the water jet and collected water.5 Fluid-structure interaction In order to understand more clearly the interaction that fluid and solid elements have on the system dynamic response as well as the coupling effects, the forces have to be classified in the following way: Turbulent Force (TF) and Flow-Induced Force (FIF). The aim of this approach is to identify and to split the total hydraulic force, which acts on the system, into its hydrodynamic and its hydrostatic part. The latter is representing the best approximation of the true water mass to calculate the liquid flow rate in small intervals of time (see Fig. 4) [0]. 4/4

5 For this application, the hydrostatic force will be determined by means of signal analysis applied to the weighing system and the tank-structure oscillation in dependence to the impinging fluid, collected water volume and its internal flow. 3 Force measurement experimentation and related process variables 3. Dynamic forces acting upon the dynamic weighing system The previous chapters described the way the water jet exerts forces upon the system, the type of forces and their occurrence. Now, the next step is to recognize the variables of influence on the system via experimentation, identifying the type of forces, and interpretating their magnitudes by applying some of the concepts previously described. In addition to it, the rate of change of momentum described by Eq. 4 and the mass flowrate definition (Eq. 5) are also applied. r r r ρ r r w(a u - A u ) dt F= =m & (u -u ) dt r r r & dm r m= = ρw A u =ρw A u dt In this connection, the force vector (Eq. 4) involved in the real process does not follow a normal trajectory but an angular path α. Whereby at least two force components must be taken into account, by the fact that radial forces ( F r ) have also an effect due to lateral sensitivity of the x weighing system when measurements are taken (see Fig. ). Therefore, the radial force component acting on the tank and the normal hydrodynamic force respectively are: r r r F=m(u-u)sin(α) & x (6) r r r F = m & (u - u ) cos(α) (7) y (4) (5) Fig. 5 Rate of change of fluid momentum at different nominal mass flowrates 5/4

6 Fig. 6 Pulse counting variation from a turbine flowmeter, mostly due to changes of fluid momentum in the MUT section, in addition to mechanical rotatory components. It also demonstrates that general assumptions of constant flowrate in primary liquid flow standards do not apply during dynamic-weighing measurements. Fig. 5 and 6 are good examples to show how the rate of fluid momentum, described by Eq. 4, varies through the time by observing two different signals: the balance based on the electromagnetic force-compensation (EFC) load cell and a turbine flowmeter installed in the MUT section. 3. SGT and EFC weighing system The strain-gauge transducer (SGT) is the most common type of force transducer used in liquid flow primary standards, where its features can also be used in the field of dynamic force measurements due to its fast signal response as demonstrated in Fig. 7. The Figure describes a dynamic force measurement at a nominal flowrate of 50 m³/h performed by SGT and EFC. The EFC weighing system is regulated by an optical positioning sensor and a PID controller. These are responsible to zero position the lever mechanism of weighing at all times. This is an excellent feature that brings good stability and reproducibility to the system. The only disadvantage from this measurement principle as already seen in Fig. 7, is the response delay []. 6/4

7 Fig. 7 Dynamic response of the two-weighing systems (EFC and SGT) to a nominal flow rate of 50 m³/h. 3.3 Measuring force by using the weighing system It should be stated that the EFC and SGT do not measure mass directly but the force caused by the pouring water and the collected mass. An explanation of this statement is given by Fig. 8, where the reading of the EFC at the time displays the final force value (or stable value), however the reading keeps increasing due to hydrodynamic acting forces, until it reaches the overshoot peak at a time. After t, the falling t slope of the overshoot is predominantly a result of the mechanical reacting forces that are responsible to dampen the system until the final force value is reached for a second time at. 3.4 Splitting the total hydraulic force into hydrostatic and hydrodynamic parts The aim to split the total hydraulic force of the process into its hydrodynamic and its hydrostatic part is in order to calculate the liquid flowrate during dynamic-weighing measurements by using the hydrostatic fluid force, which represents the best approximation of the true water mass contained in the system within a time interval dt (Eq. 4 and 5). The total hydraulic force for this case can be effectively divided from the process by defining a new force variable. In order to do this, the fluid is assumed to be standing still where an atmospheric pressure force F r acts upon the water surface. This condition can be stated into atm the momentum equation assuming zero fluid velocity [6]. r r r r F =F +ρ g dv + τ da T atm w Vw w (8) t 3 t 7/4

8 Fig. 8 EFC response to fluid-mechanical dynamic forces at the moment the fluid is diverted back to the bypass Fig. 9 Water volume element subjected to hydrodynamic forces in addition to an atmospheric pressure force. The forces highlighted in red refer to the sum of forces describing F r. dyn The solution of Eq.8 shows that the total hydraulic force F r is equal to the hydrostatic force, F r for T s r the fact that viscous stress τ disappears when there is no fluid motion ( u = 0 ). This analogy serves as a basis to understand that two different types of forces are involved: the hydrostatic force F r, caused by the collected water in the tank, and the hydrodynamic force F r, due to the s fact that the fluid is flowing (see Fig. 0 and ). Hence, the total hydraulic force is comprised by parts as described in Eq. 9. r r r F =F + F n (9) T s dy After these statements, the total hydraulic force is given as the sum of fluid forces in the following form: dyn 8/4

9 r r { } r r r r r F T =F atm +ρw g dv w + Fjet - τ Vw ρ V jet,sub w gdv+ w da+ ρ Vw air gdvw (0) The first and second components represent the hydrostatic part influenced by the atmospheric pressure force. The second component is the most important to determine the instantaneous mass flowrate. The third component represents the water jet impact force against the water surface. The fourth and fifth components comprise the buoyancy force of the collected water against the volume of the submerged water jet and the viscous force, respectively (see Fig. 9). Due to the complexity to measure the fourth and the fifth components, such forces are included into the water jet impact force to be part of a single hydrodynamic force F r. dyn The last component (air buoyancy) is included in Eq. 0 with the purpose to enclose all fluid forces acting upon the weighing tank, however the air buoyancy is already included in the measurement when the force is measured by the calibrated weighing system. The magnitude of these summed forces is clearly seen in Fig. 0 and where an experiment quantifies the fluid-mechanical dynamic forces by measuring the magnitude of the signal response overshoot when the water jet is suddenly diverted back to the bypass at different stages of the filling. It is important to clarify that the term fluid-mechanical dynamic forces is used in the last paragraph to emphasize that the force magnitude of the overshoot is not merely the sum of acting hydrodynamic forces but a product of fluid-structure interaction between acting hydrodynamic forces and reacting mechanical forces coming from the elastic elements of the weighing system. Fig. 0 System response to fluid-mechanical forces at different stages of the filling process (Nominal flowrate of 50 m³/h) 9/4

10 Fig. The total hydraulic force and quasi-hydrostatic force change as a function of the water volume collected in the tank. The magnitude of the summed fluid-mechanical dynamic forces is inversely proportional to the water depth in the collection tank. 4 First steps towards weighing system modeling The aim of the system modeling is to prove that the simulated fluid and mechanical designated variables, are in a reasonable accordance with the real process output signals to validate the measurement principle and to determine the instantaneous mass flow rate within a certain degree of accuracy. This analysis starts by identifying the individual mechanical variables that compound the dynamic system. Just to recall the inertial mass and translational elastic elements that store (k) and dissipate energy (c) in the system as illustrated in Fig. 3. These components correspond to the EFC and SGT weighing systems supporting the collection tank. The input signal acting on the system is the fluid force, and the output response of these elements are the deflection ( y ), the relative velocity of its connecting points ( y& ), and the acceleration due to all forces acting on the mass ( y&& ). Concerning to the formulation of the water force input signal, such a signal is analyzed by the momentum equation, however during the first steps made, the test input signals selected have been the impulse, step and ramp in the time domain []. This approach is quite useful, because there is a reasonable correlation between these standard input signals and the response of the real process during the filling process, the instantaneous impact water force and the impulse lateral force (or radial force) as illustrated in Fig.. The motion of the weighing system as it has been previously described, involves the alternating transfer of energy (or forces) between the acting fluid momentum and the reacting mechanical components that respond in opposite direction as stated by Eq.. 0/4

11 Fig. Acting fluid forces and their equivalent representation as test input signals Fig. 3 Analogous representation of the weighing system [3] r r F = F () Fluid Mech 4. Analogous model representation of the weighing system In order to determine the true instantaneous water mass magnitude, it is necessary to understand how the weighing system responds in accordance to the fluid-induced force. For this purpose a mathematical model has to be developed, to describe such a fluid-structure interaction. The system behavior as explained before does respond as a mass-spring-damper system model represented in its general form by Eq. and illustrated in Fig. 3. The matricial form of this equation indicates that more than one degree of freedom (DOF) is implied. r My+ && Cy+ & K y=f T () /4

12 4. 3 Degree-of-Freedom weighing system model The experiments and simulations performed in this section were made with the objective to observe how the 3-DOF model agrees (see Fig. 3) in relation to the real 3-DOF system by analyzing its step response (instantaneous water jet impact force) and its ramp response (water collection process). The results presented show a general concordance with the real process. For instance, Fig. 4 describes the system step response by applying a load of 500 N upon the weighing system. The objective of this test is to reproduce an ideal condition of quasi-static water collection by placing a load, and to discard any fluid motion that can disturb the measurement. The simulated output signals according to Eq. 3, and the real system output signals turn out to be nearly the same in magnitude and response. That is a good sign, which says that the 3-DOF equation of motion applied approaches to the real process. However, a better approximation of the SGT elastic coefficients is required in order to underdamp the simulated signal as displayed by the real SGT response. Fig. 4 System step response exerted by a reference load of 500 N upon the collection tank As for the EFC, refinements are in progress to incorporate the response delay caused by the inertial effects of the lever mechanism and the PID positional control loop. Parameter optimization will be also applied in future work to approximate the stiffness and damping coefficients to the real process. m && & 0 0 y c -c 0 y 0 ( m ) && & + m 0 y + -c (c + c ) -c3 y ( m + ) && & + m m 3 y 3 0 -c (c + c 3 ) y 3 k -k 0 y Fy Fy t...+ -k (k + k ) -k 3 y = 0 or 0 0 -k (k + k 3 ) y Step input signal Ram p input signal (3) The differential equations describing the dynamic performance of the system process are represented in form of a matrix (Eq. 3) to make clear how the variables and coefficients are correlated. /4

13 Fig.5 Simulated and real water collection process of 500 L, at a nominal flow rate of 50 m³/h 4.3 Degree-of-Freedom and 4-DOF multi-axis weighing system model Further considerations are in progress to analyze the two and four degree-of-freedom systems. The aim of modeling the weighing system as a -DOF system is to compare the inertial coupling effects between the SGT, EFC, and the foundation when the vibration isolation system (VIS) is activated and deactivated [3]. At present, a 4-DOF multi-axis model is being derived with the intention to understand how the weighing system not only reacts to normal forces acting upon the vertical tank but also to lateral forces due to internal radial flow and, the moment of fluid momentum (see Eq. 6 and 7). Fig. 6 4-DOF multi-axis weighing system representation with orthogonal elastic elements in the xyz-plane to reproduce the lateral sensitivity in the weighing system 3/4

14 This 4-DOF system model is basically a vertical -DOF system in relation to the y-axis and a - DOF orthogonal system that represents the lateral sensitivity effect. In other words, the displacement of one element will coincidentally have an effect in the orthogonal element attached. The free body diagram of this model is depicted in Fig Conclusions and future work A new proposal for dynamic-weighing liquid flow measurements has been presented by PTB to be applied in the Hydrodynamic test field. First results of system modeling prove feasibility about reproducing the system reaction in relation to fluid forces applied. Parameter estimation is the next step to determine more accurately the system variables and coefficients that describe the dynamic system response. Further theoretical analysis is also needed to combine the system dynamics theory and the fluid momentum equations into a real process input signal. References [] W. Pöschel, R. Engel, The concept of a new primary standard for liquid flow measurement at PTB Braunschweig, FLOMEKO 98, Lund, Sweden,998 [] M. Shaffer, F. Ruegg, Liquid-flowmeter calibration techniques, Transactions of ASME, October, Washington, DC, 953 [3] R. Engel, Dynamic weighing Improvements in gravimetric liquid flowmeter calibration, The 5 th International Symposium on Fluid Flow Measurement, Arlington, 00 [4] R. Engel, H. Baade, New design dual-balance gravimetric reference system with PTB s new 'hydrodynamic test field, FLOMEKO 003, Groningen, The Netherlands, 003 [5] G.K. Batchelor, An introduction to Fluid Dynamics, Cambridge university press, Great Britain,967 [6] R. L. Panton, Incompressible Flow, John Wiley and sons,984 [7] G.N. Abramovich, The theory of turbulent jets, Massachusetts institute of technology, Massachusetts,963 [8] X. Weilin, Turbulent flow and energy dissipation in plunge pool of high arch dam, Journal of hydraulic research, No 4,Vol. 40, China, 00 [9] R. Aaserude, J. Orsborn, "New concepts in fish ladder design, Volume II of IV; Results of laboratory and field research on new concepts in weir and pool fishways", final report, Project No , BPA Report DOE/BP , 984 [0] F. Axisa, José Antunes, Modeling of mechanical systems: Fluid-structure interaction, Volume 3, Elsevier, First edition, Great Britain, 007 [] D. Reber, Electromagnetic force compensation devices in mass comparators, Mettler Toledo AG Greifensee, Switzerland, 000 [] R. Dorf, R. Bishop, Modern control systems, Prentice hall, Tenth international edition, USA, 005 [3] W. Pöschel, R. Engel, D. Dopheide, A unique diverter design for water flow calibration facilities, 0 th International conference on flow measurement FLOMEKO 000, Salvador, Brazil, June 5-8, [ /4