Nonlinear Modeling of Butterfly Valves and Flow Rate Control Using the Circle Criterion Bode Plot

Transcription

1 American Control Conference Marriott Waterfront, Baltimore, MD, USA June 3-July, WeC.6 Nonlinear Modeling of Butterfly Valves and Flow Rate Control Using the Circle Criterion Bode Plot J.D. Taylor, Bruno Sinopoli, and William Messner Abstract Butterfly valves are widely used control components in piping systems. This paper derives the nonlinear mathematical relationship between flow rate, pressure difference, and valve angle in a butterfly valve. We use the Circle Criterion Bode plot and loop shaping to design a flow rate controller in a motor-gearbox-butterfly valve system. The design employs a complex proportional-integral-lead (CPIL) compensator, and we show the benefit of using a damping ratio less than unity. Finally, we demonstrate the effectiveness of the design through simulation. I. INTRODUCTION The distribution and control of water has been a significant engineering challenge throughout history. Notable examples include the irrigation techniques developed in ancient Egypt, the aqueducts constructed by the Roman Empire, and the water-wheel and steam powered mechanisms of the industrial revolution. Water is still used extensively in modern engineering systems. For instance, chilled water networks are often used to cool critical electronics in large-scale data centers and aboard naval ships. The robustness of these systems is of particular importance since loss of cooling water can lead to device failure in a matter of minutes. Rapid rerouting of water flow in piping systems subject to damage must consider the coupled hydro-mechanical dynamics of the system components. In this paper we consider the modeling and control of butterfly valves, which are widely used in chilled water systems. This paper presents a new derivation of the model of flow past a butterfly valve as a function of valve angle and pressure difference. It combines the mathematically formidable free streamline theory [] with the empirical relations for viscous pressure losses in expanding flows (Borda-Carnot) and within pipes (Darcy-Weisbach). The paper also presents an application of the Circle Criterion Bode plot (CBode plot) [] for controlling a motor-gearbox-butterfly valve system with flow rate feedback, and it shows the utility of the complexproportional-integral-lead (CPIL) compensator for this type of loop shaping design. II. VALVE MODELING A schematic of the butterfly valve used to derive the nonlinear relationship between volumetric flow rate, pressure difference, and valve plate angle of attack, α, is presented in Fig.. This derivation is carried out in -D; however, it has been verified experimentally that this approach can The authors are with the Departments of Mechanical, Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, PA 53 jdtaylor@andrew.cmu.edu,brunos@ece.cmu.edu, bmessner@andrew.cmu.edu D Stagnation Point V Fig.. d dc α Vj θ L Wake Region d Vj dc Schematic of Butterfly Valve accurately predict the fluid resistance of 3-D valves by integrating over cross-sections [3]. The Borda-Carnot equation, Eq., is an empirical relation describing viscous pressure losses in a fluid undergoing an abrupt expansion in flow area. These losses are due primarily to turbulence in the wake regions. V D P = ξ ρ( V ) () where P is the pressure lost due to the flow expansion and ρ is the fluid density. V = V j V is the reduction in average flow velocity between the vena contracta, V j, and valve outlet, V. ξ is an empirical, dimensionless loss coefficient with a value ranging from ξ. For an abrupt and wide expansion (assumed for the butterfly valve) the loss coefficient is equal to. Note that upstream of the vena contracta, the flow accelerates and pressure losses are negligible in this region. Bernoulli s equation can be modified to account for viscous losses in incompressible flows, Eq.. P + ρv = P + ρv + P () It is convenient to define the pressure loss coefficient, ζ. ζ = P ρv (3) In this analysis, the inlet pressure and flow velocity, P and V, are assumed to be uniform and known. The flow is incompressible, and the velocities at all the cross-sections are related by continuity, Eq. 4. D V =d V =(d c + d c )V j = D V (4) where D,D and V,V are the diameters and average velocities of the inlet and outlet respectively, d and V are the width and average velocity of the orifices between the //$6. AACC 967

2 valve plate and wall. d is maximum when the valve is fully open (α = o ) and minimum when the valve is fully closed (α =9 o ). d c and d c are the widths of the upper and lower jets at their minimum points (i.e. vena contracta) and V j is the corresponding jet velocity. Following [4] we take the jet velocities, V j, to be equal. However, the values of d c and d c are distinct due to the different angles at which the fluid impinges on the surface of the valve plate. We assume in this case that the inlet and outlet diameters are equal D = D D implying V = V V. Assuming that the valve plate length is equal to the diameter of the valve, the flow cross-section as a function of the angle of attack, α, is given by d = D ( sinα) (5) We now define the coefficients of contraction, C c and C c, to be the ratios of the minimum width of the jets to the width of the orifices. C c d c d C c d c d (6) Now from Eqs. 4-6 we have ( ) Cc + C c V = ( sinα)v j (7) The pressure loss coefficient due to the flow expansion, ζ e, from the vena contracta to the outlet of the butterfly valve is given by Eq. 8. [( ζ e = (C c + C c )( sinα) ) ] (8) This coefficient characterizes the pressure loss due to flow expansion. However, at small angles when the valve is nearly open and the wake region is very small, C c and C c approach unity, and the predicted pressure loss is zero. This result is unphysical, and an additional term due to pipe friction must be included. This pressure drop is described by the well known Darcy-Weisbach equation, Eq. 9. P = f L D ρv (9) here f is a dimensionless laminar or turbulent friction factor, L and D are the valve length and diameter respectively, and ρ and V are the density and velocity of the fluid. The friction factor may in general be determined from the Moody chart; however, for laminar flows it is given by the simple formula f =64/Re in which the Reynolds number is Re = ρv D/µ. Therefore, the pressure loss coefficient due to laminar pipe friction, ζ f, is readily determined to be ζ f = 64µL ρv D () The fluid resistance, R, is defined as the ratio of the overall static pressure difference, P P across the valve to the mass flow rate through the valve, ṁ. Since the inlet and outlet diameters are equal, there is no convective acceleration, and the static pressure difference is equal to the total pressure lost due to both flow expansion and pipe friction, P P = P. Note that the fluid resistance is a nonlinear function of both the valve attack angle and the fluid velocity. R = P ṁ =(ζ e + ζ f ) V D () In order to evaluate Eq. 9, the coefficients of contraction for the particular system must be specified. These parameters usually are determined empirically since analytic solutions are generally not available. Fortunately, however, the coefficients of contraction for butterfly valves have both been derived analytically and verified experimentally, [], [4], [5], [6]. These analytical derivations use Helmholtz free streamline theory to calculate the velocity field using successive conformal transformations. Though free streamline theory applies only to inviscid incompressible flow, it is well suited to this problem due to the significance of pressure and inertia in establishing the flow field, while shear and gravitational effects are secondary [5]. In this analysis, the upper and lower coefficients of contraction derived by Sarpkaya in [4] will be used. The interpolated coefficients are shown graphically in Fig.. Coefficients of Contraction C c C c Angle of Attack, α [deg] Fig.. Theoretical upper and lower contraction coefficients versus angle of attack for butterfly valves [4]. The fluid resistance is shown versus angle of attack in Fig. 3. The resistance is small but nonzero for small α due to the pipe friction, it grows exponentially in the approximate range 5 α 75, and it diverges at α =9 at which point the flow ceases. III. CIRCLE CRITERION BODE PLOT The well known Circle Criterion provides a sufficient condition for the stability of a nonlinear feedback system of the form given in Fig. 4, where L(s) is the transfer function of a linear time invariant (LTI) system and Ψ( ) is a memoryless and possibly time-varying sector nonlinearity [7]. The nonlinearity is said to belong to the sector [α, β], if for any input u to Ψ( ), αu uψ(u) βu. 968

3 Fluid Resistance, R [Pa (kg/s) ] Angle of Attack, α [deg] Fig. 3. Fluid resistance as a function of the angle of attack at constant velocity for butterfly valve. + - L( s ) Ψ( ) βu Ψ(u) Fig. 4. Block diagram of system with linear part L(s) and memory-less nonlinearity, Ψ( ). An example of a nonlinear function belonging to the sector [α, β]. Circle Criterion: For a SISO transfer function L(s) and a memory-less function Ψ( ) belonging to the sector [α, β], sufficient conditions for determining the stability of the closed-loop system are the following []. ) β>α, a) The Nyquist plot of L(jω) remains outside the disc with center on the real axis and intersecting the real axis at β and α for α>. b) If L(s) has N u unstable poles, the Nyquist plot of L(jω) encircles the disc anti-clockwise N u times. c) If α =, the Nyquist plot of L(jω) remains to the right of the vertical line x = β. ) β> >α, The Nyquist plot of L(jω) remains inside the disc with center on the real axis and intersecting the real axis at β and α. 3) α<β, Replace L(jω) with L(jω) and Ψ( ) with Ψ( ) which belongs to sector [ β, α]. Return to Case. Fig. shows the Nyquist plot and disc of a system satisfying Case of the Circle Criterion. The Circle Criterion, however, is difficult to use for loopshaping controller design because frequency is a hidden variable on the Nyquist plot. To address this difficulty, Messner and Xia developed the Circle Criterion Bode (CBode) plot αu u []. The CBode plot maps the disc from the Nyquist plot onto the Bode magnitude and phase plots by transforming the Circle Criterion into inequality relationships between the open-loop gain and phase that involve α and β. For a first quadrant nonlinearity (Case ), the inequality is ( Re [L(jω)] ( )) + () > ( α + β ( α β )) () That is, the distance of L(jω) from the disc center should always be greater than the disc radius. These relations are used to define forbidden regions on the magnitude plot based on the open-loop phase, and forbidden regions on the phase plot based on open-loop magnitude. In the CBode plots shown in Figs. 8 and 9, there are intersections between the open-loop frequency response and the forbidden regions depicted in gray, and therefore the Circle Criterion is not satisfied. In Fig. there are no intersections, so the Circle Criterion is satisfied, and the closed-loop system is stable. The objective of loop-shaping controller design using the CBode plot is to apply compensators to shape the open-loop response to eliminate intersections between the open-loop response and the forbidden regions. A useful property of the CBode magnitude plots is that the boundary functions depend only on the open-loop phase. Thus, multiplying the open-loop by a constant gain will move open-loop magnitude up or down while leaving the forbidden regions on the magnitude plot unchanged []. IV. CONTROLLER DESIGN In this section we illustrate the use of the CBode plot for controller design for an output nonlinearity. The design specifications for this system based on speed of response are Stability in the presence of the valve nonlinearity Zero steady-state error for a step disturbance Open-loop db crossover of rad/s Phase margin of 6 TABLE I Motor and Gearbox Parameters Motor Torque Constant k τ = N-m/A Motor Coil Resistance R = Ω Motor Coil Inductance L =.3 H Motor-Gearbox Inertia J =.4 kg-m Motor-Gearbox Damping b =.7 N-m/s Gearbox Gear Ratio N = 5 The transfer function for this system for a voltage input and valve angle output is P(s)= Nkt s(n (Js+ b)(ls + R)+kt ) where the parameters are given in Table I. (3) 969

4 A. Choosing the Optimum Output Variable Offset One of the important aspects of Circle Criterion design is choosing the output variable offset, which affects the slopes of the sector boundaries. Proper choice of this offset can reduce the ratio of slopes of the two bounding lines thus reducing the size of the Circle Criterion disc. The closer the ratio β n = β/α is to one, the closer the system is to being linear, and the closer the disc is to a point. Fig. 5 shows the flow rate as a function of the valve opening angle, θ =9 α, and the sector lines for the optimum offset of 6. Since the nonlinearity is part of the plant rather than the sensor, after selecting the offset, it is important to normalize the sector by multiplying the plant gain by α and multiplying the sector slopes by /α. This ensures that the linear portion of the plant accounts for the minimum gain. The slope of the lower boundary will be α n =and the slope of upper boundary is β n = β/α. The point +j is the critical point for stability, such that when the compensated openloop system has positive gain margin and has no intersection with the forbidden regions, the system satisfies the Circle Criterion..5 θ = θ = 6.5 θ = θ =. θ = 6. θ = Fig. 7. Circle criterion discs for output offsets θ =, θ =6, and θ = Flow Rate, Q [m 3 /s] Ψ( ) α 6 β CBode plot for offset θ = Fig. 5. The nonlinear function Ψ( ) belongs to the sector [α θ,β θ ] depending on the choice of operating point (6 shown). Fig. 6 shows the value of /β n = α/β as a function of the offset angle. The optimal offset θ maximizes this value. It has the effect of moving the right endpoint of the disc on the real axis at /β n = α/β away from the imaginary axis, as illustrated in Fig CBode plot for offset θ = α / β Fig. 6. α/β as a function of offset angle θ (c) CBode plot for offset θ =9 Fig. 8. CBode plots for different output variable offsets. θ =. θ =6. (c) θ =9. 97

5 Figs. 8-8(c) show the CBode plots for valve angle offsets of, 6, and 9 respectively with the gain normalized to at frequency rad/s. The gain of the plant has been multiplied by the corresponding α and the boundary slopes have been multiplied by /α. Fig. 7 shows the discs of the corresponding offsets. The offset of θ =6 gives the smallest value of β n = β/α. B. Loop-Shaping Controller Design The optimal output variable offset is 6, and so we begin the design by considering the CBode plot of Fig. 8. The idea is to design a compensator that provides integral action for zero steady-state error while lifting the phase plot over the phase forbidden region. The compensator must also provide sufficient phase to achieve a phase margin of 6 at ω db = rad/s. The recently invented complex proportional-integral-lead compensator (CPIL) is well suited for this task [8]. Eq. 4 shows the structure of the CPIL transfer function, C cpil (s)= s +ζω z s + ω z s(s + p) (4) where ω z = ω m ( ζtan(φ m )+ ) ζ tan (φ m )+, (5) p = ω m +sin (φ m ) sin (φ m ), (6) The phase contribution at ω m is 3φm 45. The phase angle must satisfy <φ m < 9. The zeros of this compensator are complex when ζ<. Lower damping ratios provide higher gains at low frequencies and a steeper phase peak. The highest point of the phase forbidden region in Fig. 8 is about 8 above the open-loop phase at frequency 4 rad/s. Therefore, as a first attempt, we choose a CPIL compensator with ζ =.7 providing φ m =8 at frequency ω m =4rad/s. The resulting CBode plot is shown in Fig. 9. The intersections with the forbidden regions are still present but have been reduced compared with the uncompensated system, Fig. 8. The intersection between the phase forbidden region and the phase response occur at the upper frequencies of the forbidden region. The amount of phase added seems to be enough, but the phase should be added at a slightly higher frequency. As a second attempt we keep the parameters of the CPIL the same, but change ω m to 64 rad/s. Fig. shows that with this compensator, all intersections with the forbidden regions are eliminated : : 44.4 : : 4-3 Fig.. CBode plot of open-loop after compensation with CPIL compensator with ζ =.7 providing φ m =8 at frequency ω m =64rad/s. The compensator transfer function in this case is given by C CPIL (s)= s +7.43s +8. s(s + ) (7) Fig. shows the corresponding Nyquist plot and the Circle Criterion disc, confirming that the Circle Criterion is, in fact, satisfied. Checking the phase at ω =rad/s confirms that the phase margin requirement is also satisfied; therefore, the closed-loop nonlinear feedback system is stable Fig. 9. CBode plot of open-loop after compensation with CPIL compensator with ζ =.7 providing φ m =8 at frequency ω m =4rad/s. Fig.. Nyquist plot of the compensated system along with the Circle Criterion. The system response does not intersect the circle with endpoints - and α/β, guaranteeing the system stability and verifying the CBode approach. is the overall Nyquist plot and is a zoomed in view near the origin. To show the benefit of using smaller damping ratios, Fig. shows the CBode plot for a CPIL compenator with ζ =. providing φ m =8 at frequency ω m =7rad/s, which also satisfies the Circle Criterion and the phase margin requirement; however, the gain at ω =rad/s is about 5.5 db higher than for the ζ =.7 compensator. 97

6 : : 39.5 : : 4-3 Fig.. CBode plot of open-loop after compensation with CPIL compensator with ζ =. providing φ m =8 at frequency ω m =7. V. CONCLUSION This paper derived the nonlinear relationship between flow rate, pressure difference, and valve angle in a butterfly valve from first principles and data in the literature. This derivation is easier to understand and more compact than previous approaches. This paper also applied the Circle Criterion Bode plot to the problem of flow rate control using voltage input to to motor-gear-valve feedback system. The controller design method employed iterative loop-shaping, and a complex proportional-integral-lead compensator was chosen with a damping ratio less than one, which provides larger gain at low frequencies compared to proportional-integral-lead compensators with real zeros. The Nyquist plot confirmed that the Circle Criterion was satisfied, and a simulation indicated an improvement in dynamic response. In order to verify the stability and performance of the compensated system, a simulation of the dynamic system response to a step input was created using MATLAB Simulink. The time-domain response of the uncompensated and CPIL compensated unity-gain feedback systems are presented in Fig. 3. Note that both systems are stable, which is consistent with fact that the Circle Criterion is a conservative indicator of stability (i.e. sufficient but not necessary). However, the compensated system does exhibit a marked improvement in dynamic response. Flow Rate [m 3 /s].5 x Reference Flow Rate REFERENCES [] W. Hassenpelug, Free-streamlines, Computers and Mathematics with Applications, vol. 36, no., pp. 69 9, 998. [] L. Xia and W. Messner, Loop shaping with the circle criterion-bode plot with application to active tape steering, in American Control Conference, 7. ACC 7, 7, pp [3] T. Kimura, T. Tanaka, K. Fujimoto, and K. Ogawa, Hydrodynamic characteristics of a butterfly valve: Prediction of pressure loss characteristics, ISA Transactions, vol. 34, no. 4, pp , 995. [4] T. Sarpkaya, Oblique impact of a bounded stream on a plane lamina, Journal of the Franklin Institute, vol. 67, pp. 9 4, 959. [5], Torque and cavitation characteristics of butterfly valves, ASME Journal of Applied Mechanics, vol. 8, pp. 5 58, 96. [6] J. Park and M. Chung, Study on hydrodynamic torque of a butterfly valve, Journal of Fluids Engineering, vol. 8, p. 9, 6. [7] H. Khalil, Nonlinear systems. Prentice-Hall,. [8] W. Messner, Classical control revisited: variations on a theme, AMC 8. th IEEE International Workshop, pp. 5, March Time [s].5 x 3 Flow Rate [m 3 /s].5 Reference Flow Rate Time [s] Fig. 3. Simulated time-domain responses to a step input of the uncompensated and CPIL compensated unity-gain feedback systems. 97