Jamming and Flow in 2D Hoppers
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We consider jamming that occurs for the flow of dry granular materials from a hopper as the outlet size, D, is reduced relative to the particle size, d. We propose and test a model based on the simple idea that the probability of jamming on a short time, dt, is a constant, τ. This leads to the prediction that the probability of surviving until time t without jamming is an exponentially decaying function, Ps(t) = exp(−t/τ). The important physical question is what determines τ? Experiments to date have confirmed the exponential character of the survival probability. Work in progress characterizes τ(D, d), fluctuations and their possible relation to the jamming process.Keywords:
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Exponential decay
Intensity
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Matrix (chemical analysis)
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This chapter contains sections titled: Introduction Origin of e Distinction Between Exponential and Power Functions The Value of e The Exponential Series Properties of e and Those of Related Functions Comparison of Properties of Logarithm(s) to the Bases 10 and e A Little More About e Graphs of Exponential Function(s) General Logarithmic Function Derivatives of Exponential and Logarithmic Functions Exponential Rate of Growth Higher Exponential Rates of Growth An Important Standard Limit Applications of the Function ex: Exponential Growth and Decay
Exponential formula
Logarithmic growth
Exponential integral
Power function
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Exponential distribution is widely used in reliability and maintainability studies although it is well known that the constant failure rate assumption may not be valid. The purpose of this paper is to investigate the use of exponential distribution as an approximation. In fact, for components undergoing regular maintenance or replacement, the exponential assumption can be acceptable. In this paper, the exponential approximation for regularly maintained Weibull component is studied. The approximated exponential distribution using the average failure rate is compared with the exact reliability. The asymptotic relative error is derived, which can be used to adjust the exponential approximation when needed. Based on the framework of exponential approximation for Weibull distributed components, the problems of decision‐making regarding the optimal maintenance time and spare allocation are also addressed.
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Component (thermodynamics)
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The response of some biological processes is dependent on the frequency of stimulation. With first-order processes, the response is driven exponentially to an equilibrium determined by the value of the driving function. When the stimulus or driving function is viewed as switching between constant values the resulting response is piecewise exponential. With periodic excitation, the time course of a point fixed in time relative to the initiation time of each stimulus is shown to be exponential with a rate and steady state that are linearly dependent on the rates and equilibria associated with each component exponential. This linearity can be exploited and leads to a simple estimation procedure for the apparent state-dependent rates.
Exponential decay
Stimulus (psychology)
Time constant
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Abstract In conventional energy harvesting systems, energy can be extracted from a fixed-level source at a constant rate at best. The resulting growth of harvested energy is bound by a linear function. Here we show that exponential energy harvesting can be achieved in a system of reconfigurable energy storage elements. The exponential extraction results from the positive feedback of the system potential energy due to repetitive system reconfigurations. The concept is studied theoretically and validated with results from systems of droplet capacitors. A device with three 300 μL mercury drops can generate an exponentially growing voltage that reaches 168 V within a few cycles of a low-level and low-frequency mechanical excitation. The same device with water drops can generate a similarly growing voltage that reaches 56 V. This concept holds potential in DC power generation and may be applied in other energy domains.
Exponential decay
Electric potential energy
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A common Gamma-Ray Burst-light curve shape is the “FRED” or “fast-rise exponential-decay.” But how exponential is the tail? Are they merely decaying with some smoothly decreasing decline rate, or is the functional form an exponential to within the uncertainties? If the shape really is an exponential, then it would be reasonable to assign some physically significant time scale to the burst. That is, there would have to be some specific mechanism that produces the characteristic decay profile. So if an exponential is found, then we will know that the decay light curve profile is governed by one mechanism (at least for simple FREDs) instead of by complex/multiple mechanisms. As such, a specific number amenable to theory can be derived for each FRED. We report on the fitting of exponentials (and two other shapes) to the tails of ten bright BATSE bursts. The BATSE trigger numbers are 105, 257, 451, 907, 1406, 1578, 1883, 1885, 1989, and 2193. Our technique was to perform a least square fit to the tail from some time after peak until the light curve approaches background. We find that most FREDs are not exponentials, although a few come close. But since the other candidate shapes come close just as often, we conclude that the FREDs are misnamed.
Exponential decay
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The exponential function as a mathematical concept plays an important role in the corpus of mathematical knowledge, but unfortunately students have problems grasping it. Paper exposes example of exponential example of exponential function application in real world. One of the most prevalent application of exponential function involves growth and decay models. Exponential growth and decay show up in a host of natural application. From population Growth and continuously compounded interest to radioactive decay and Newton’s law of cooling, exponential function are ubiquitous in nature. In this section, we examine exponential growth and decay in the context of some of the application. In the preceding section, we examined a population growth problem in which the population grew at a fixed percentage each year. In that case, we found that the population can be described by exponential function. A similar analysis will show that any process in which a quantity grows by a fixed percentage each year can be modeled by an exponential function. Compound interest is good example of such a process. This work is motivated by the works of [1-15, 22]. Other example of exponential function are bacterial growth, bacterial decay, population decline, are obtained in this project.
Exponential decay
Exponential integral
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