Time Constant of the System

Material property and dynamic property are interfaced by the concept of time constant. Referring to the scale analysis of MEMS[185], Time constant T of the system is defined as required time of tip of the gel becomes vertical to the original angle. Then, the time constant is described with the parameters and variables as  

Additional description is required for length of the gel L. The orientation of the tip of the gel is a spatial integration of the curvature of the gel. Therefore, if the length of the gel becomes twice, the time to reach the same orientation of the tip becomes one half. 

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Liquid flows into a tank at the rate of Q m3/s. The tank has three vertical walls and one sloping outwards at an angle fi to the vertical. The base of the tank is a square with sides of length x m and the average operating level of liquid in the tank is Z0 m. If the relationship between liquid level and flow out of the tank at any instant is linear, develop an expression for the time constant of the system. 

For capacity measurements, several techniques are applicable. Impedance spectroscopy, lock-in technique or pulse measurements can be used, and the advantages and disadvantages of the various techniques are the same as for room temperature measurements. An important factor is the temperature dependent time constant of the system which shifts e.g. the capacitive branch in an impedance-frequency diagram with decreasing temperature to lower frequencies. Comparable changes with temperature are also observed in the potential transients due to galvanostatic pulses. 

If a proportional feedback temperature controller is used, calculate the con> troller gain that yields a closedloop damping coefficient of 0.707 and calculate the closedloop time constant of the system when (u) Jacket water only is used. 

The principle of this method, proposed originally by Davison (1966), is to neglect eigenvalues of the original system that are farthest from the origin (the nondominant modes) and retain only dominant eigenvalues and hence the dominant time constants of the system. If we consider the solution of the linearized model... 

Mathematical models that contain ordinary differential equations face an inherent computational difficulty associated with the stiffness of the equations. Stiffness of ordinary differential equations depends on the relative magnitudes of the response modes or the characteristic time constants of the system being modeled. In solid fuel conversion problems where particles of varying sizes are considered the differential equations for the thermal transients of the particles are usually stiff. Estab-... 

In a transient problem we are interested in the solution for tQ < t < tmax where the solution time (tmax-t0) is a few times larger than the maximum time constant of the system, i.e.,... 

It is clear from Fig. 8.15 that it is not possible to measure the IMPS response at sufficiently high frequencies to observe the limit where the quantum efficiency tends towards unity (i.e., where to )) kini). The limitations arise in this case from the dynamic response of the potentiostat. In other cases, attenuation due to the RC time constant of the system may obscure the injection semicircle. The upper limit to the majority carrier injection rate constants that can be obtained by IMPS is around 105 s-1. 

In the search for a more effective control strategy, the consequences of controlling the addition of air (rather than the addition of biomass) in order to maintain a constant bed height, even where operating conditions are changed markedly is investigated. This is done in order to reduce the largest time constant of the system. 

The overall heat transfer coefficient, U, does not remain the same for a long period of operation. Corrosion, dirt, or various other solids deposited on the internal or external surfaces of the heating coil result in a gradual decrease of the heat transfer coefficient. This, in turn, will cause the time constant of the system to vary. This example is characteristic of what can happen to even simple first-order systems. 

Where t is the time constant of the system. The use of single or multiple thermocouples in calorimetry is important. If the difference between the temperatures of the sample and reference thermocouple junctions is AT, then an emf [E] is produced which depends on the thermoelectric constant e and the number of thermocouples N). Therefore... 

A plot of current versus time is given in Fig. 2.20. At time f = 0 the exponential term is equal to 1 and the current is zero. This corresponds to the properties of a coil that resists fast changes in a current. Then the current increases to the limiting value Eq/R as the coil does not oppose the passage of the constant current [Ld (f)/df = 0 in Eq. (2.20)]. The characteristic time constant of the system is t = L/R. 

The dead time and the time constant of the system are large. Stable operation, more particularly in a kiln plant with planetary cooler, therefore requires that disturbances are compensated already before the burning zone. 

The sensitivity of the facility to potential faults should be minimised. Any failure, process perturbation or maloperation in a facility should produce either a change in plant state towards a safer condition or no significant response. If the change should be to a less safe condition, then the time constants of the systems involved should be long, so that key parameters deviate only slowly from their desired values. ... 

Further, it is easy to show that the maximal value of —%m corresponds to oimav = 1 /(RC), that is, to the inverse time constant of the system. Knowing R and o>max, C... 

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