Time characteristic dimensions

Microelectrodes with several geometries are reported in the literature, from spherical to disc to line electrodes each geometry has its own critical characteristic dimension and diffusion field in the steady state. The difhisional flux to a spherical microelectrode surface may be regarded as planar at short times, therefore displaying a transient behaviour, but spherical at long times, displaying a steady-state behaviour [28, 34] - If a... 

To determine the risk value A , look along the horizontal axis until the characteristic dimension is found, and at the same time, locate the adjusted tolerance on the vertical axis. Read off the A value in the zone at which these lines intersect on the map by interpolating as required between the zone bands, A = 1 to A = 9. 

It will be noted that the relevant characteristic dimension in the Biot number is defined as the ratio of the volume to the external surface area of the particle (V/Ae), and the higher the value of V/Ae, then the slower will be the response time. With the characteristic dimension defined in this way, this analysis is valid for particles of any shape at values of the Biot number less than 0.1... 

One of the most interesting theorems of Worz et al. is that they see a serious potential for micro reactors to permit small-scale production of some different sort [110-112]. Micro channels serve as an ultra-precise measuring tool, whereas production is done in channels about 10 to 100 times larger, i.e. miUimeter-sized channels. The limit of tube diameter of industrial production reactors is reported to be 2 cm hence any new reactor of smaller characteristic dimensions bears some potential for improvement. Worz et al. conclude with the remark that the above strategy could be the most important result of their studies [110-112]. 

Here the Ac is the channel cross-section flow area, and the factor, fRe, is a numerical constant computed and tabulated for various channel geometries. The characteristic dimension has been replaced by a hydraulic diameter defined as four times the flow area divided by the channel perimeter. 

Zukoski and Marble [70, 71] held that the wake of a flame holder establishes a critical ignition time. Their experiments, as indicated earlier, established that the length of the recirculating zone was determined by the characteristic dimension of the stabilizer. At the blowoff condition, they assumed that the free-stream combustible mixture flowing past the stabilizer had a contact time equal to the ignition time associated with the mixture that is, rw = ri( where rw is the flow contact time with the wake and r, is the ignition time. Since the flow contact time is given by... 

Dodson, 1973). Both of these equations are developed assuming a linear decrease of temperature T with time t (see the analogy of treatment in section 5.9.1). T is a time constant corresponding to the time taken for D to diminish by a factor of exp( —1), a is the characteristic dimension of the phase, and the asterisk denotes the radiogenic isotope. 

Experiments have indicated that Reynold s Number for the flow regime cannot exceed 400, based on the hydraulic radius as the characteristic dimension. Based on this correlation, the minimum H times W for a given ( , can be determined from ... 

We have established that the volume change kinetics of responsive gels are usually diffusion-controlled processes. Even when the diffusion analysis failed, the rates were comparable to or slower than a classical diffusive process. The implications of this for practical applications are quite negative, since diffusive processes are quite slow. A gel slab 1 mm thick with a diffusion coefficient of 10-7 cm2/s will take over an hour to reach 50% of equilibrium and more than six hours to reach 90% of equilibrium in response to a stimulus. This is far too slow for almost all potential applications of these materials. Since diffusion times scale with the square of dimension, decreasing the characteristic dimension of a sample will increase the rates dramatically. Thus if an application can make use of submillimeter size gels, millisecond response times become possible. Unfortunately, it may not always practical to use gels of such small dimension. 

The criterion of homochronity tk/(P, where d is the characteristic dimension of the system, k is the thermal diffusivity and r is the characteristic time of the chemical reaction. Due to the strong dependence of the chemical reaction rate on temperature (see the next two criteria), it is necessary to define the temperature to which the quantity r relates. We will relate it to the theoretical temperature of combustion. 

Strictly speaking, except for the explicitly appearing reaction time r or rate mn, the form of the function depends on the concrete kinetics of the chemical reaction, i.e., on whether we are dealing with a first- or second-order reaction or with an autocatalytic reaction, just as, except for the characteristic dimension d, the form of the functions depends on the concrete geometric properties of the system and will be different for a round capillary and a plane slit. 

Often it is useful to combine variables that affect physical phenomenon into dimensionless parameters. For example, the transition from laminar to turbulent flow in a pipe depends on the Reynolds number, Re = pLv/p, where p is the fluid density, I is a characteristic dimension of the pipe, v is the velocity of flow, and // is the viscosity of the fluid. Experiments show that the transition from laminar to turbulent flow occurs at the same value of Re for different fluids, flow velocities, and pipe sizes. Analyzing dimensions is made easier if we designate mass as M, length as L, time as t, and force as F. With this notation, the dimensions of the variables in Re are ML 3 for p, (L) for L, (L/t) for v, and (FL 2t) for //. Combining these it is apparent that Re = pLu/p, is dimensionless. 

When the electrochemical properties of some materials are analyzed, the timescale of the phenomena involved requires the use of ultrafast voltammetry. Microelectrodes play an essential role for recording voltammograms at scan rates of megavolts-per-seconds, reaching nanoseconds timescales for which the perturbation is short enough, so it propagates only over a very small zone close to the electrode and the diffusion field can be considered almost planar. In these conditions, the current and the interfacial capacitance are proportional to the electrode area, whereas the ohmic drop and the cell time constant decrease linearly with the electrode characteristic dimension. For Cyclic Voltammetry, these can be written in terms of the dimensionless parameters yu and 6 given by... 

This expression compares the characteristic time of runaway (TMRad) with the characteristic cooling time. Thus, knowing the mass, specific heat capacity, heat transfer coefficient, and heat exchange area allows the assessment. It is worth noting that, since the thermal time constant contains the ratio V/A, heat losses are proportional to the characteristic dimension of the container. 

This problem was addressed and solved by Frank-Kamenetskii [6], who established the heat balance of a solid with a characteristic dimension r, an initial temperature T0 equal to the surrounding temperature, and containing a uniform heat source with a heat release rate q expressed in W m The object is to determine under which conditions a steady state, that is, a constant temperature profile with time, can be established. We further assume that there is no resistance to heat transfer at the wall, that is, there is no temperature gradient at the wall. The second Fourier Law can be written as (Figure 13.2)... 

There is no doubt that the ultimate development of process intensification leads to the novel field of microreaction technology (Figure 1) (7-9). Because of the small characteristic dimensions of microreaction devices, mass and heat transfer processes can be strongly enhanced, and, consequently, initial and boundary conditions as well as residence times can be precisely adjusted for optimizing yield and selectivity. Microreaction devices are evidently superior, due to their short response time, which simplifies the control of operation. In connection with the extremely small material holdup, nearly inherently safe plant concepts can be realized. Moreover, microreaction technology offers access to advanced approaches in plant design, like the concept of numbering-up instead of scale-up and, in particular, the possibility to utilize novel process routes not accessible with macroscopic devices. 

The averaging time duration should be chosen in such a way that thf T <characteristic time of the high-frequency component rHF may be estimated from the reciprocal of the characteristic spectral frequency of the fluctuation, while the characteristic time of the low-frequency component Tlf may be determined from the time required to travel the characteristic dimension of the physical system at the local characteristic low-frequency speed. Thus, a time averaging after volume averaging can be defined as... 

For the design of micro heat exchangers, it has to be considered that both heat and mass transport time-scales are strongly correlated with the characteristic dimensions of the exchanger according to diffusion theory [8,9] ... 

Measurement of Electrode Area. Because of surface toughness, the real or true surface area of a solid electrode is greater than the projected or geometric area. However, if the electrode is polished to a smooth surface finish, this will be of no consequence in most voltammetric work. The depth of the depleted region around the electrode surface (the diffusion-layer thickness) is substantially larger than the characteristic dimensions of surface toughness for electrolysis times that are greater than 1 s. [The diffusion-layer thickness may be crudely approximated by the term (Dt)m, where D is the diffusion coefficient (cm2 s"1) and t is the time.]... 

The area of a polished electrode (taken to be the projected or geometric area in most voltammetric experiments at times > 1 s) usually is measured directly or electrochemically. If the electrode is of regular geometry, such as a disk, sphere, or wire of uniform diameter, its characteristic dimensions can be measured by use of a micrometer, optical comparator, or traveling microscope and the area calculated. 

In order to design and dimension stirrers for the homogenization of liquid mixtures - and this is by far the most common task when it comes to stirring - it is vital to know the power characteristic and the mixing time characteristic of the type of stirrer in question. If this information is available for various types of stirrers, it is possible to determine both the best type of stirrer for the given mixing task and the optimum operating conditions for this particular type. 

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