Von Karman discovered that the distance between vortices, or the wavelength, is constant for higher Reynolds numbers. The Reynolds number is a parameter used to describe fluid flow. It encompasses the fluids velocity, density, and viscosity and the pipes diameter.
The Reynolds number is the ratio of the inertial forces to the viscous forces in a flowing fluid and is defined by Equation 1 as follows:. Strouhal, another German scientist, expanded on Von Karmans findings. He discovered that the frequency of the vortices times the width of the shedder bar divided by the velocity of the vortex street was constant for higher Reynolds numbers.
As shown in Figure 5, Innova-Mass flow meters exhibit a constant Strouhal number across a large range of Reynolds numbers, indicating a consistent linear output over a wide range of flows and fluid types.
Below this linear range, the intelligent electronics in the Innova-Mass automatically corrects for the variation in the Strouhal number.
The smart electronics correct for this non-linearity by calculating the www. Reynolds number based on constant values of the fluids density and viscosity stored in the instruments memory. Innova-Mass flow meters automatically correct down to a Reynolds number of 5, The Seven Calculations Performed 1. The sensing head of the flow meter directly measures the fluids velocity V , temperature T , and pressure P.
Then, in real time, the built-in flow computer calculates the following: 2. The fluid density, , using the P and T measurements and the fluids equation of state, which is stored in memory.
For incompressible fluids liquids , only T is needed for this calculation. The fluid viscosity, , using T and an onboard equation.
The Reynolds number from , , and the pipes inside diameter, D, which is stored in memory. A Reynolds number correction factor for low-velocity fluids. The volumetric flow rate, Q, from V and the pipes cross-sectional area and the mass flow rate using and Q. Three mA output signals for the users choice of three of five variables: , Q, T, P, or.
The following section discussed these measurements in more detail. Velocity The velocity sensor in the multivariable mass vortex flow meter consists of a fin immersed in the flow behind the shedder bar. The alternating lift forces created by the vortices cause the fin to deflect back and forth at the exact frequency of the vortices. The fin in the vortex mass flow meter is mechanically connected to piezoelectric elements.
Many techniques have been applied to sense the passage of vortices, including pressure sensors, ultrasonic sensors, capacitance-based sensors, heated thermistors, and strain gauges. Strain-gauge velocity sensors based on the piezoelectric effect have the highest sensitivity and range-ability. When piezoelectric elements are strained, they produce an electric charge, which is converted to a current.
Thus, the output of the velocity sensor is a sinusoidal current with a frequency equal to that of the vortices. Since frequency is the basic output of the velocity sensor, vortex flow meters have zero drift, in contrast to other flow meters with analog sensors. The insertion Innova-Mass vortex flow meters have the velocity sensor embedded in the shedder bar itself, while the in-line Innova-Mass has its lightweight fin located just behind the shedder bar where the strength of the vortices is highest.
Most real-world industrial pipelines experience vibration caused by pumps, machinery, and flow- induced oscillations. The velocity sensors in traditional vortex flow meters are designed for maximum sensitivity and often pick up the spurious signals generated by pipeline vibrations. The resulting noise creates erroneous vortex frequency signals, which degrade the meters accuracy. Sierras Innova-Mass eliminates this problem using a patented low mass sensor and sophisticated digital signal processing DSP.
The low mass sensor makes it very sensitive to the vortices formed and less sensitive to heavy vibration while the DSP conditions and filters the signal. The overall result is greater accuracy and reliability, particularly at lower flow rates. Temperature An error in measuring the temperature of flowing fluids is another annoying problem in real- world industrial mass flow monitoring systems. Conventional vortex and turbine meter systems and orificebased systems of either the traditional or multivariable-transmitter variety measure fluid temperature with.
Such temperature sensors have high stem conduction and therefore measure a value of flowing fluid temperature that lies somewhere between the actual fluid temperature and the temperature of the pipes wall. This problem is not so severe in applications where the convective heat-transfer coefficient is high, such as liquid flows and flows at high velocity.
On the other hand, the problem is exacerbated in gas and steam flows, especially at lower velocities and at higher temperatures. The design of the temperature sensor in the Innova-Mass solves this problem.
The temperature- sensing element is a 1,ohm PRTDa highly accurate temperature measurement device. This temperature sensor has low intrinsic stem conduction. The result is fast time response and superb accuracy within 1C. It is ideal for steam flow and other low- velocity, high-temperature fluid measurements. Pressure The Innova-Mass incorporates a solid-state pressure transducer isolated by a stainless steel diaphragm.
The transducer itself is micro- machined silicon, fabricated using integratedcircuit-processing technology. Digital compensation allows these transducers to operate within a 0. Why Multivariable? Some conventional vortex flow meters accept inputs from external temperature and pressure transmitters thereby providing an inferred mass flow rate output. In these conventional inferential mass flow devices, temperature and pressure sensors are located somewhere in the pipeline either upstream or downstream of the vortex flow meter, but typically not at the same location in the pipeline.
This causes errors in calculating fluid density from the temperature and pressure measurements resulting in mass flow rate errors.
Extensive testing has revealed that the only acceptable location for accurate temperature and pressure monitoring is just downstream of the shedder bar adjacent to the velocity sensor. In the Sierra Innova-Mass for example, all three sensors velocity, temperature, and pressureare located adjacent to eachother in a single sensing head.
This integrated multivariable design concept ensures accurate direct mass flow monitoring. Big Advantages Over Orifice Plates Every multivariable mass vortex simplifies process measurement because it provides output signals for five parameters: mass flow rate, volumetric flow rate, temperature, pressure, and density.
It does this with only one break in the pipeline. Multivariable mass vortex flow meters combine a differential pressure transducer, absolute pressure transducer, temperature-sensor electronics, and flow computer in one package.
Traditional mass flow systems using vortex provide a standard flow measurement, but they require a separate temperature sensor, four invasions of the pipeline, and the installation of tubing, valves, and manifolds. When factoring in the cost of installing electrical conduit and wiring and the associated engineering and equipment costs, the Innova-Mass clearly exhibits the lowest total cost of ownership see Table 1.
Accuracy figures are included in Table 1 also. Notes: 1Based on average list prices in U. Primary care: concept, evaluation, and policy. New York, Oxford University Press, The world health report health systems, improving performance. Geneva, WHO, Belo Horizonte, mimeo, Esse modelo tem sido exaustivamente avaliado.
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Lei Estadual Artigo 85 da lei Conversely, if Juliet is sufficiently hostile, Romeo might decide to be nice to her, in what Gottman et al. Thus we could replace the bJ in Eq. Qualitatively similar results follow from the function bJ 1 J2 , which is the case considered by Rinaldi b in his model of the love felt by the 14th century Italian poet Francis Petrarch toward his beautiful married platonic mistress Laura Jones There are now four equilibria, including the one at the origin.
Equations 4 apparently do not admit limit cycles, and there is no chaos since the system is only two-dimensional. The same nonlinearity can be applied to the love triangle in Eq. This system can exhibit chaos with strange attractors, an example of which is in Fig. The largest Lyapunov exponents base-e are 0. The Kaplan-Yorke dimension is 2. Figure 5 illustrates the effect of the positive Lyapunov exponent on the time evolution of Romeos love for Juliet for the same case as Fig.
The regions of parameter space that admit chaos are relatively small, sandwiched between cases that produce limit cycles and unbounded solutions.
Chaotic evolution of Romeos love for Juliet from Eq. Even simple nonlinearities can produce chaos when there are three or more variables. An interesting extension of the model would consider a group of interacting individuals a large family or love commune. The models are gross simplifications since they assume that love is a simple scalar variable and that individuals respond in a consistent and mechanical way to their own love and to the love of others toward them without external influences.
The mathematics of marriage. Gragnani, A. Cyclic dynamics in romantic relationships. International Journal of Bifurcation and Chaos, 7, Jones, F. The structure of Petrarchs Canzoniere. Cambridge: Brewer. Radzicki, M. Dyadic processes, tempestuous relationships, and system dynamics.
System Dynamics Review 9, Rapoport, A. Fights, games and debates. Ann Arbor: University of Michigan Press. Rinaldi, S. Love dynamics: the case of linear couples. Applied Mathematics and Computation, 95, Laura and Petrarch: An intriguing case of cyclical love dynamics. Love dynamics between secure individuals: A modeling approach. Nonlinear Dynamics, Psychology, and Life Sciences, 2, Scharfe, E.
Reliability and stability of adult attachment patterns. Personal Relationships, 1, Sprott, J. Chaos and time-series analysis. Oxford: Oxford University Press. Sternberg, R. The triangular theory of love. Psychological Review, 93, The psychology of love. Strogatz, S. Nonlinear dynamics and chaos: With applications to physics, biology, chemistry, and engineering. Reading, MA: AddisonWesley. Open navigation menu. Close suggestions Search Search. User Settings. Skip carousel. Carousel Previous.
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