To measure fluid velocity, the flow meter incorporates a bluff body shedder bar in the flow stream and measures the frequency of vortices created by the shedder bar. A platinum resistance temperature detector PRTD measures temperature. Pressure measurement is achieved using a solid-state pressure transducer.
Born in Germany Velocity measurement is based on the phenomenon of vortex shedding. A vortex is an eddy, or swirl, of fluid. Around the turn of the century, German scientist Theodore Von Karman investigated the vortexshedding phenomenon, which was later exploited to form the basis for vortex- shedding flow meters.
Von Karman demonstrated that when a fluid flows past a non-streamlined body called a bluff body or shedder bar an alternating series of vortices is shed from each side of the body. This alternating series of vortices is termed a Von Karman Street see Figure 3. Velocity sensor Vortex shedder bar Vortices. Various flow meter manufacturers have different shapes of shedder bars.
Their common trait is sharp corners, which enhance the strength, or energy, of the vortices and ensure boundary-layer separation at two defined points: the two sharp corners.
This feature is responsible for the extra-ordinary linearity of the frequency of vortex shedding over a wide velocity range. Multivariable mass vortex flow meters are available in two configurations: In-line flow meter replaces a section of the pipeline Insertion flow meter requires a cold tap or a hot tap into an existing pipeline Sierra Instruments is the only manufacturer of this configuration.
Both the in-line and insertion configurations are similar in that they both use identical electronics and have similar sensor heads. Besides installation differences, the main difference between an in-line flow meter and an insertion flow meter is their method of measurement. For an in-line meter, the shedder bar is located across the entire diameter of the flow body. Thus, the entire pipeline flow is included in the vortex formation and measurement. The sensing head, which directly measures velocity is located just downstream of the shedder bar.
An insertion flow meter has its sensing head at the end of a 0. The stem is inserted into the pipe until the sensing head is properly located in the pipes cross section. The sensing head fits through any entry port with a 1.
The vortex velocity sensor is located just downstream of the shedder bar. Von Karman vortices form downstream of the shedder bar into two distinct wakes. The vortices of one wake rotate clockwise, while those of the other wake rotate counterclockwise.
Vortices form one at a time, alternating from the left side to the right side of the shedder bar. They interact with their surrounding space and overpower every other nearby swirl on the verge of development. Thus, the volume encompassed by each vortex remains constant.
By sensing the number of vortices per unit time passing the vortex shedding frequency f by the velocity sensor, the flow meter is able to compute the fluids velocity V. Thus f is proportional to V.
This is the fundamental principal by which a vortex flow meter operates. 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. Key Words. Love, romance, differential equations, chaos. In his book, Strogatz has a short section on love affairs and several related mathematical exercises.
Essentially the same model was described earlier by Rapoport , and it has also been studied by Radzicki Although Strogatzs model was originally intended more to motivate students than as a serious description of love affairs, it makes several interesting and plausible predictions and suggests extensions that produce an even wider range of behavior.
This paper is written in the same spirit and extends the ideas to love triangles including nonlinearities, which are shown to produce chaos. There are many types of love, including intimacy, passion, and commitment Sternberg, , and each type consists of a Correspondence address: Department of Physics, University of Wisconsin, Madison, Wisconsin ; e-mail: sprott physics. In addition to love for another person, there is love of oneself, love of life, love of humanity, and so forth.
Furthermore, the opposite of love may not be hate, since the two feelings can coexist, and one can love some things about ones partner and hate others at the same time.
While there is no limit to the ways in which the models can be made more realistic by adding additional phenomena and parameters, these embellishments almost certainly only increase the likelihood of chaos, which is the main new observation reported here. The simplest model is linear with. The parameter a describes the extent to which Romeo is encouraged by his own feelings, and b is the extent to which he is encouraged by Juliets feelings.
Gottman et al. The resulting dynamics are two-dimensional, governed by the initial conditions and the four parameters, which may be positive or negative. A similar linear model has been proposed by Rinaldi a in which a constant term is added to each of the derivatives in Eq. Such a model is more realistic since it allows feelings to grow from a state of indifference and provides an equilibrium not characterized by complete apathy.
However, it does so at the expense of introducing two additional parameters. While the existence of a non-apathetic equilibrium may be very important to the individuals involved, it does not alter the dynamics other than to move the origin of the RJ state space. Gragnani, Rinaldi, and Feichtinger use the terms secure and synergic to refer to individuals with negative a and positive b, respectively, and such people probably represent the majority of the population.
Since Juliet can also exhibit four styles, there are 16 possible pairings, each with its own dynamics, although half of those correspond to an interchange of R and J. The corresponding dynamics in the RJ plane are summarized in Fig. The real solutions are of two types, a node if the eigenvalues are of the same sign and a saddle otherwise. The node may either be stable an attractor if both eigenvalues are negative or unstable a repellor if both are positive.
The saddle has a stable direction along which trajectories approach the origin the inset or stable manifold and an unstable direction along which they are repelled the outset or unstable manifold. Strogatz asks his students to consider a number of special pairings of individuals as described in the following sections. The outcome for Cases 1 and 2 depend on the initial conditions first impressions count as does the size of the oscillation in Case 3. The former case leads to a saddle in which the eager beaver and hermit are at odds and the narcissistic nerd and cautious lover are in love or at war, and the latter leads to a center.
Thus they can end up in any quadrant all four combinations of love and hate or in a never-ending cycle, but never apathetic. Oscillations are not possible. Suppose Romeo has a mistress, Guinevere, although the third person could be a child or other relative. The state space is then sixdimensional rather than two-dimensional because each person has feelings for two others and there are twelve parameters if each can adopt different styles toward the others, even when the natural appeal considered by Rinaldi a is ignored.
In the simplest case, Juliet and Guinevere would not know about one another and Romeo would adopt the same romantic style toward them both he is an equal opportunity lover. The resulting fourdimensional system becomes two decoupled two-dimensional systems unless Romeos feelings for Juliet are somehow affected by Guineveres feelings for him, and similarly for Guinevere.
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The role of perceived team effectiveness in improving chronic illness care. Care, , SING, D. Improving care for people with long term conditions: a review of UK and international frameworks.
TSAI, A. A meta-analysis of interventions to improve care for chronic illnesses. Chronic disease management : what will it take to improve care for chronic illness? O problema do conhecimento verdadeiro na epidemiologia.
O desafio do conhecimento. TORO A. La Communication Publique. PUF, Col. Que sais-je? Paris, FUNG, Archon. Belo Horizonte: Del Rey, ENA, , p. Deve ser feita em 2 a 5 minutos. Neoplasias tumores III. The largest Lyapunov exponents base-e are 0. The Kaplan-Yorke dimension is 2.
The regions of parameter space that admit chaos are relatively small, sandwiched between cases that produce limit cycles and unbounded solutions.
Strange attractor from the nonlinear love triangle in 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.
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.
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